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abstract: 'In this paper, we give a characterization of the class of all circular-arc graphs whose schemes are association. Moreover, all association schemes which are the scheme of a circular-arc graph are characterized, specially it is proved that they are Schurian.'
address:
- 'Department of Mathematics, K. N. Toosi University of Technology, P. O. Box: 16315-1618, Tehran, Iran. '
- ' School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6\'
author:
- Fatemeh Raei Barandagh
- Amir Rahnamai Barghi
title: 'On circular-arc graphs with association schemes'
---
Introduction {#sect1}
============
In [@Wax], B. Weisfeiler and A. Leman have shown that a special matrix algebra is assigned to a given graph which contains the adjacency matrix of the graph. In fact, this algebra is the adjacency algebra of a scheme. The [*scheme of a graph*]{} is the smallest scheme on the vertex set of the graph such that the edge set of which is the union of some basic relations of the scheme. The scheme of forests, interval graphs and some special classes of graphs have been studied in [@EPT]. In this paper, we study the scheme of a circular-arc graph which is the intersection graph of a family of arcs of a circle.
Circular-arc graphs received considerable attention since a series of papers by Tucker in [@T1; @T2; @T3], and by Dur$\acute{a}$n, Lin and McConnel in [@D; @LS; @M]. Various subclasses of circular-arc graphs have been also studied. Among these are the proper circular-arc graphs, unit circular-arc graphs, Helly circular-arc graphs and co-bipartite circular-arc graphs. Several characterizations and recognition algorithms have been formulated for circular-arc graphs and its subclasses. But in this paper, we correspond a finite or algebraic description for circular arc graphs instead of description according arcs of a circle.
Let $n$ be a positive integer and let $S$ be a subset of $\mathbb{Z}_n$ such that $S=\{\pm 1,\ldots, \pm k\}$ for $0\leq
2k+1< n$. Then the Cayley graph $\operatorname{Cay}(\mathbb{Z}_n,S)$ is a circular-arc graph (see Sec. \[cmn graph\]), and we call it [*elementary circular-arc graph*]{}. For $k=0$ it is empty graph and for $k=1$ it is an undirected cycle.
We say that $\Gamma$ is a [*graph with association scheme*]{} if the scheme of $\Gamma$ is association. Our main results give a characterization of circular-arc graphs with association schemes in the terms of lexicographic product of graphs.
In graph theory, what we have called the [*lexicographic product*]{} or [*composition*]{} of graphs is also often called the [*wreath product*]{}. The term wreath product comes from group theory, and it is also defined in scheme theory.
In fact, analysis duplicating the vertices of an elementary circular-arc graph shows that the lexicographic product of an elementary circular-arc graph and a complete graph is a circular-arc graph.
The rest of this section is to state our results. The following theorem provides a necessary and sufficient condition for a circular-arc graph whose scheme is association.
\[graphtheorem\] A graph is a circular-arc graph with association scheme if and only if it is isomorphic to the lexicographic product of an elementary circular-arc graph and a complete graph.
One can associate to any finite permutation group $G$ a scheme, denoted by $\operatorname{Inv}(G)$. The scheme associated to the dihedral group $D_{2n}$ is called [*dihedral*]{} scheme.
The class of [*forestal schemes*]{} have been defined inductively by means of direct sums and wreath products in [@EPT]. The scheme of cographs, trees and interval graphs are forestal (see [@EPT]).
A scheme is said to be [*circular-arc scheme*]{} if it is the scheme of a circular-arc graph. In the following theorem we give a characterization of association circular-arc schemes.
\[schemetheorem\] A circular-arc scheme is association if and only if it is isomorphic to the wreath product of a rank $2$ scheme and a scheme which is either forestal or dihedral.
\[remark\] Forestal scheme in Theorem \[schemetheorem\] occurs as a wreath product of the scheme of at most on 2 points and a rank 2 scheme. For example: the scheme of a disjoint union of copies of a complete graph, namely $mK_n$, or the scheme of the lexicographic product of a complete graph and a complete graph without perfect matching.
A scheme $\mathcal{X}$ is said to be [*Schurian*]{} if $\mathcal{X}=\operatorname{Inv}(G)$ for some permutation group $G$, see [@Zi1]. In fact, any rank 2 scheme and any dihedral scheme are Schurian. Moreover, the wreath product of two Schurian schemes is Schurian, see [@Zi1]. Thus we have the following corollary:
Any association circular-arc scheme is Schurian.
It is known that the automorphism group of each graph is equal to automorphism group of its scheme, see [@Wax]. Moreover, it is well-known that the automorphism group of wreath product of two schemes is equal to the wreath product of their automorphism groups. Denote by $\operatorname{Sym}(n)$ the symmetric group on $n$ points, and denote by $G\wr H$ the wreath product of two groups $G$ and $H$. The following corollary is an immediate consequence of Theorems \[graphtheorem\] and \[schemetheorem\].
Let $\Gamma$ be a circular-arc graph with association scheme on $n$ vertices. Then there is an even integer $k$, $k | n$, such that $\operatorname{Aut}({\Gamma})$ is isomorphic to $\operatorname{Sym}(\frac{n}{k}) \wr G$, where $G$ is $\operatorname{Sym}(k)$ or $\operatorname{Sym}(2)\wr \operatorname{Sym}(\frac{k}{2})$ or $D_{2k}$.
This paper is organized as follows. In Section 2, we present some preliminaries on graph theory and scheme theory. In Section 3, we first remind the concept of circular-arc graphs and then we introduce arc-function and reduced arc-function of a circular-arc graph. Moreover, we characterize non-empty regular circular-arc graphs without twins. Section 4 contains relationship between lexicographic product of graphs and wreath product of their schemes. In Section 5, we define elementary circular-arc graphs. Then, we characterize the scheme of graphs which belong to this class. Finally, in Section 6 we give the proof of Theorem \[graphtheorem\] and Theorem \[schemetheorem\].
Throughout the paper, $V$ denotes a finite set. The diagonal of the Cartesian product $V^2$ is denoted by $1_V$.
For $r,s\subset V^2$ and $X,Y \subset V$ we have the following notations:
$$r^*=\{(u,v)\in V^2:\ (v,u)\in r\},$$ $$r_{X,Y}=r\cap(X\times Y) ~,
~r_X=r_{X,X},$$ $$r\cdot s=\{(v,u)\in V^2:\ (v,w)\in r,\
(w,u)\in s ~{\text{ for some}}~ w\in V\},$$ $$r\otimes s=\{((v_1,v_2),(u_1,u_2))\in V^2\times V^2:\
(v_1,u_1)\in r ~{\text{ and}} ~ (v_2,u_2)\in s\},$$
Also for any $v \in V$, set $vr=\{u\in V :\
(v,u)\in r \}$ and $n_r(v)=|vr|$.
For $S\in 2^{V^2}$ denote by $S^\cup$ the set of all unions of the elements of $S$, and set $S^*=\{s^*:\ s\in S\}$ and $v
S=\cup_{s\in S}v s$. For $T\in 2^{V^2}$ set $$S\cdot T=\{s\cdot
t:\ s\in S,\, t\in T\}.$$
For an integer $n$, let $\mathbb{Z}_n$ be the ring of integer numbers modulo $n$. Set $$A\mathbb{Z}_n:=\{\{i, i+1,\ldots, i+k\}: i,k \in \mathbb{Z}_n
~{\text {and}}~ k \neq n-1 \}.$$ For each set $\{i, i+1,\ldots, i+k\}$, the points $i$ and $i+k$ are called the end-points of the set.
Preliminaries
==============
Graphs. {#graph}
-------
All terminologies and definitions about graph theory have been adapted from [@bondy]. In this paper, we consider finite and undirected graphs which contains no loops and multiple edges. We denote complete graph on $n$ vertices by $K_n$, and an undirected cycle on $n$ vertices by $C_n$.
Let $\Gamma=(V,R)$ be a graph with vertex set $V$ and edge set $R$. Let $E$ be an equivalence relation on $V$, then $\Gamma_{V/E}$ is a graph with vertex set $V/E$ in which distinct vertices $X$ and $Y$ are adjacent if and only if at least one vertex in $X $ is adjacent in $\Gamma$ with some vertex in $Y$. For every subset $X$ of $V$, the graph $\Gamma_X$ is the subgraph of $\Gamma$ induced by $X$.
Let $\Gamma_i$ be a graph on $V_i$, for $i=1,2$. The graphs $\Gamma_1$ and $\Gamma_2$ are [*isomorphic* ]{} if there is a bijection $f: V_1 \to V_2$, such that two vertices $u$ and $v$ in $V_1$ are adjacent in $\Gamma_1$ if and only if their images $f(u)$ and $f(v)$ are adjacent in $\Gamma_2$. Such a bijection is called an [*isomorphism*]{} between $\Gamma_1$ and $\Gamma_2$. The set of all isomorphism between $\Gamma_1$ and $\Gamma_2$ is denoted by $\operatorname{Iso}(\Gamma_1,\Gamma_2)$. An isomorphism from a graph to itself is called an [*automorphism*]{}. The set of all automorphisms of a graph $\Gamma$ is the [*automorphism group*]{} of $\Gamma$, and denoted by $\operatorname{Aut}(\Gamma)$.
The [*lexicographic product*]{} or [*composition*]{} of graphs $\Gamma_1$ and $\Gamma_2$ is the graph $\Gamma_1[\Gamma_2]$ with vertex set $V_1\times V_2$ in which $(u_1,u_2)$ is adjacent to $(v_1,v_2)$ if and only if either $u_1$ and $v_1$ are adjacent in $\Gamma_1$ or $u_1=v_1$ and also $u_2$ and $v_2$ are adjacent in $\Gamma_2$.
Let $\Gamma=(V,R)$ be a graph. Two vertices $u,v \in V$ are [*twins*]{} if $u$ and $v$ are adjacent in $\Gamma$ and $vR \backslash\{u\}=uR
\backslash\{v \}$, where the set of neighbors of a vertex $v$ in the graph $\Gamma$ is denoted by $vR$.
Schemes. {#scheme}
---------
In this part all terminologies and notations are based on [@EP].
A pair $\mathcal{X}=(V,S)$, where $V$ is a finite set and $S$ a partition of $V^2$, is called a [*scheme*]{} on $V$ if $1_V\in S^\cup$, $S^*=S$, and for any $r,s,t\in S$ the number $$c_{rs}^t:=|v r\cap us^*|$$ does not depend on the choice of $(v,u)\in t$. The scheme $\mathcal{X}$ is called [*association*]{} if $1_V\in S$.
The elements of $V$, $S$, $S^\cup$ and the numbers $c_{rs}^t$ are called the [*points*]{}, the [*basic relations*]{}, the [*relations*]{} and the [*intersection numbers*]{} of the scheme $\mathcal{X}$, respectively. The numbers $|V|$ and $|S|$ are called the [*degree*]{} and the [*rank*]{} of $\mathcal{X}$. A unique basic relation containing a pair $(v,u)\in V^2$ is denoted by $r_{\mathcal{X}}(v,u)$ or $r(v,u)$.
An equivalence relation on a subset of $V$ belonging to $S^\cup$ is called an equivalence relation of the scheme $\mathcal{X}$. Any scheme has [*trivial*]{} equivalence relations: $1_V$ and $V^2$. Let $e\in S^\cup$ be an equivalence relation. For a given $X\in V/e$ the [*restriction*]{} of the scheme $\mathcal{X}$ to $X$ is the scheme $$\mathcal{X}_X=(X,S_X),$$ where $S_X$ is the set of all non-empty relations $r_X$ with $r\in S$. The [*quotient*]{} of the scheme $\mathcal{X}$ modulo $e$ is defined to be the scheme $$\mathcal{X}_{V/e}=(V/e,S_{V/e}),$$ where $S_{V/e}$ is the set of all non-empty relations of the form $\{(X,Y):\ s_{X,Y}\neq \emptyset\}$, with $s\in S$.
Two schemes $\mathcal{X}_1$ and $\mathcal{X}_2$ are called [*isomorphic*]{} if there exists a bijection between their point sets in such a way that induces a bijection between their sets of basic relations. Such a bijection is called an [*isomorphism*]{} between $\mathcal{X}_1$ and $\mathcal{X}_2$. The set of all isomorphism between $\mathcal{X}_1$ and $\mathcal{X}_2$ is denoted by $\operatorname{Iso}(\mathcal{X}_1,\mathcal{X}_2)$. The group of all isomorphisms of a scheme $\mathcal{X}$ to itself contains a normal subgroup $$\operatorname{Aut}(\mathcal{X})=\{f\in \operatorname{Sym}(V): s^f=s, s\in S\}$$ called the [*automorphism group*]{} of $\mathcal{X}$ where $s^f=\{(u^f,v^f):(u,v)\in s\}$.
The [*wreath product*]{} $\mathcal{X}_1\wr \mathcal{X}_2$ of two schemes $\mathcal{X}_1=(V_1,S_1)$ and $\mathcal{X}_2=(V_2,S_2)$ is a scheme on $V_1\otimes V_2$ with the following basic relations $$V^2_1\otimes r,\ ~{\text{for}}~ r\in S_2\backslash 1_{V_2}
~~{\text{and}}~~ s\otimes 1_{V_2} ,\ ~{\text{for}}~ s\in S_1.$$
The scheme of a graph. {#s of g}
-----------------------
There is a natural partial order $"\leq"
$ on the set of all schemes on $V$. Namely, given two schemes $\mathcal{X}=(V,S)$ and $\mathcal{X}'=(V,S')$ we set $$\mathcal{X}\leq \mathcal{X}' \ \Leftrightarrow\ S^\cup\subseteq
(S')^\cup.$$ In this case $\mathcal{X}$ is called a [*fusion*]{} (subscheme) of $\mathcal{X}'$ and $\mathcal{X}'$ is called a [*fission*]{} (extension) of $\mathcal{X}$. The minimal and maximal elements with respect to $"\leq"$ are the [*trivial*]{} and the [*complete*]{} schemes on $V$ respectively: the basic relations of the former one are the reflexive relation $1_V$ and (if $|V|>1$) its complement in $V^2$, whereas the relations of the later one are all binary relations on $V$.
Let $R$ be an arbitrary relation on the set $V$. Denote by $\operatorname{Fis}(R)$ the smallest scheme with respect to $"\leq"$ such that $R$ is a union of its basic relations. Let $\Gamma=(V,R)$ be a graph with vertex set $V$ and edge set $R$. By the scheme of $\Gamma$ we mean $\operatorname{Fis}(\Gamma)=\operatorname{Fis}(R)$. For example, if $\Gamma$ is a complete graph or empty graph with at least 2 vertices, then its scheme is the trivial scheme of rank 2. One can check that if $\operatorname{Fis}(\Gamma)$ is an association scheme, then $\Gamma$ is a regular graph.
In general, it is quite difficult to find the scheme of an arbitrary graph. In [@EPT], the scheme of a graph has been studied for some classes of graphs.
Circular-arc graphs
===================
From [@bondy], for a given family $ \mathcal{F}$ of subsets of $V$, one may associate an [*intersection graph*]{}. This is the graph whose vertex set is $\mathcal{F}$, two different sets in $\mathcal{F}$ being adjacent if their intersection is non-empty. Circular-arc graph is the intersection graph of a family of arcs of a circle.
\[arcfun\] Let $\Gamma$ be a graph on $V$ with $n$ vertices. Then $\Gamma$ is a circular-arc graph if and only if there exists a function $f:V \to A\mathbb{Z}_{m}$, for some $m$, such that $\Gamma$ is the intersection graph of the family $\operatorname{Im}(f)=\{f(v):
v\in V\}$. Moreover, this function can be chosen such that
1. any element of $\mathbb{Z}_{m}$ is the end-point of at least one set in $\operatorname{Im}(f)$,
2. each set in $\operatorname{Im}(f)$ contains at least two elements.
To prove sufficient part let $\Gamma$ be a circular-arc graph. Then by the definition it is the intersection graph of some arcs of a circle $C$. Without loss of generality, we may assume that the end-points of any of these arcs are distinct. Let $m$ be the number of these end-points and let $A=\{a_0, a_1,\ldots,
a_{m-1}\}$ be the set of all of them. It is clear that $m\leq
2n$. Here the indices of the points $a_i$ are the elements of $\mathbb{Z}_m$; they are chosen in such a way that the point $a_i$ precedes the point $a_{i+1}$ in the clockwise order of the circle $C$. Then for any vertex $v\in V$ there exist uniquely determined elements $i_v, j_v \in \mathbb{Z}_m$ such that $$A_v:=C_v \cap A= \{a_{i_v},a_{i_v+1}\ldots, a_{j_v}\},$$ where $C_v$ is a subset of $C$ which is the arc corresponding to $v$. Moreover, it is easily seen that $C_u \cap C_v$ is not empty if and only if $i_v\in A_u$ or $j_v\in A_u$ or $i_u,j_u\in
A_v$. Therefore, the vertices $u$ and $v$ are adjacent if and only if the set $A_u \cap A_v$ is not empty. Now define a function $f:V \to A\mathbb{Z}_{m}$ by $$f(v)=\{i_v, i_v+1, \ldots, j_v\}.$$ Then $\Gamma$ is the intersection graph of the family $\operatorname{Im}(f)$. Moreover, statements (1) and (2) immediately follows from the definition of $f$.
Conversely, let $m\leq 2n$ and $f:V \rightarrow
A\mathbb{Z}_{m}$ be a function such that $\Gamma$ is the intersection graph of $\operatorname{Im}(f)$. Consider a circle $C$ and choose $m$ distinct points on it. We may label these points by the elements of $\mathbb{Z}_m$ such that these points appears in $C$ in clockwise order. Since $\operatorname{Im}(f)\subset A\mathbb{Z}_m$ for each vertex $v\in V$ there exist $i_v,j_v\in \mathbb{Z}_m$ such that $f(v)= \{i_v, i_v+1, \ldots, j_v\}$. We correspond an arc $C_v
\subset C$, from $i_v$ to $j_v$ in clockwise order to the vertex $v$. It is clear that the set $f(u)\cap f(v)$ is not empty if and only if $C_u \cap C_v$ is not empty. It follows that $\Gamma$ is the intersection graph of the set $\{C_v : v\in V\}$. So it is a circular-arc graph. This completes the proof of the lemma. $\Box$
The function $f:V\to A\mathbb{Z}_m$ satisfying statements (1) and (2) and conditions of Lemma \[arcfun\] is called the [*arc-function*]{} of the graph $\Gamma$ and the number $m$ is called the [*length*]{} of $f$.\
\[reduced\] Let $\Gamma=(V,R)$ be a non-empty circular-arc graph on $n$ vertices. Suppose that for any two vertices $u$ and $v$ in $V$ we have: $$\label{subset}
v\in uR ~~~ \Rightarrow~~~ uR \not\subset \{v\}\cup vR.$$ Then there exists an arc-function $f$ of $\Gamma$ such that the following statements hold:
1. no set of $\operatorname{Im}(f)$ is a subset of another set of $\operatorname{Im}(f)$,
2. the length of $f$ is equal to $n$,
3. any element $i \in \mathbb{Z}_{n}$ is the end-point of exactly two sets in $\operatorname{Im}(f)$.
\[remark1\] The graph $\Gamma$ satisfying condition is a connected graph. Indeed, otherwise, it is easily seen that $\Gamma$ is an interval graph. On the other hand, each interval graph is chordal, and so it has a vertex whose neighborhood is a complete graph (see [@bondy]) which contradicts the condition .
By Lemma \[arcfun\], there exists an arc-function $f'$ of $\Gamma$ of length $m'\leq 2n$. Denote by $\sim$ the binary relation on $\mathbb{Z}_{m'}$ defined by $i\sim j$ if and only if for any $ v\in V$ $$i,j\in f'(v) {\text{~or~}} i,j\notin f'(v).$$ One can check that $\sim$ is an equivalence relation, and its equivalence classes belong to $A\mathbb{Z}_{m'}$. By the definition of $"\sim"$ any element of $\operatorname{Im}(f')$ is a disjoint union of some classes. Let us define a function $f$ such that for each $v\in V$, $f(v)$ is the set of $\sim$-classes contained in $f'(v)$. The equivalence classes of $\sim$ can be identified by $\mathbb{Z}_{m}$. By this identification we have $f(v)\in
A\mathbb{Z}_{m}$.
We claim that $f$ is an arc-function of $\Gamma$. Indeed, from the definition of $f$ it follows that for each two vertices $u$ and $v$, the set $f(u)\cap f(v)$ is not empty if and only if the set $f'(u)\cap
f'(v)$ is not empty. Moreover, statement (1) of Lemma \[arcfun\] is obvious. Thus, it is sufficient to verify that statement (2) of Lemma \[arcfun\] occurs. Suppose on the contrary that there is a vertex $v\in V$ such that $f(v)$ contains exactly one element. Then $f'(v)$ is a class of the equivalence $\sim$. By condition this implies that $v $ is an isolated vertex in $\Gamma$, which is impossible by Remark \[remark1\]. Thus $f$ is an arc-function of $\Gamma$.
By Lemma \[arcfun\], the graph $\Gamma$ is isomorphic to the intersection graph of the family $\operatorname{Im}(f')$. Thus for two adjacent vertices $u$ and $v$ in $V$, if $f'(u)\subseteq f'(v)$ then any vertex in $V\backslash \{v \}$ which is adjacent to $u$ in $\Gamma$ is also adjacent to $v$. On the other hand, it is easy to see that $f'(u)\subseteq f'(v)$ is equivalent to $f(u)\subseteq f(v)$. Therefore, we have $$\label{vr}
f(u)\subseteq f(v) ~~~ \Rightarrow~~~ uR \subset \{v\}\cup vR.$$ Hence, statement ([*i*]{}) follows from condition .
Statement ([*ii*]{}) is a consequence of statement ([*iii*]{}). First we will show that any element $i\in \mathbb{Z}_{m}$ is the end-point of exactly two sets in $\operatorname{Im}(f)$. Suppose on the contrary that there is an element $i\in \mathbb{Z}_{m}$ which is an end-point of at least three sets of $\operatorname{Im}(f)$. By statement (2) of Lemma \[arcfun\], there are at least two sets $f(u)$ and $f(v)$ such that $$i+1\in f(v)\cap f(u)~~{\text or }~~i-1\in f(v)\cap f(u).$$ It follows that in any case $f(v)\subseteq f(u)$ or $f(u)\subseteq f(v)$. From , this contradicts condition . Thus any element $i\in \mathbb{Z}_{m}$ is the end-point of at most two sets in $\operatorname{Im}(f)$.
To complete the proof, suppose that there exists $i\in
\mathbb{Z}_{m} $ which is an end-point of exactly one set, say $f(v)$, in $\operatorname{Im}(f)$. By statement (2) of Lemma \[arcfun\], we have $i+1\in f(v)$ or $i-1\in f(v)$. Suppose that the former inclusion holds. Then by we have $$\label{uvr}
u\in vR ~ \Rightarrow ~ i \not\in f(v)\cap f(u).$$ If $i+1$ is an end-point of $f(v)$ then, since $\Gamma$ does not contain any isolated vertex, so there is a vertex $u\in V $ such that $f(u)\cap f(v)$ is not empty. From it follows that $i+1$ is an end-point of $f(u)$. Let $w\in vR$, then the set $f(v)\cap f(w)$ is not empty. From we have $i+1\in f(w)$ and so $f(w)\cap
f(u)$ is not empty. So, $w\in uR$. Therefore, in this case $vR
\subseteq \{u\} \cup uR$, that contradicts the condition . Thus we suppose $i+1$ is not an end-point of $f(v)$. In this case, statement (1) of Lemma \[arcfun\] implies that $i+1$ is an end-point of a set in $\operatorname{Im}(f)$, say $f(u)$. From Lemma \[arcfun\], we have $f(u)\nsubseteq f(v)$ and it follows that $f(v)\backslash \{i\} \subseteq f(u)$. Now from , we have $vR\subseteq \{u\} \cup uR$, which contradicts the condition . If $i-1\in f(v)$, by the same argument we have a contradiction again. Thus, any element $i\in \mathbb{Z}_{m}$ is the end-point of exactly two sets in $\operatorname{Im}(f)$. This completes the proof of the theorem. $\Box$
Any arc-function $f:V\to A\mathbb{Z}_n$, satisfying conditions ([*i*]{}), ([*ii*]{}) and ([*iii*]{}) of Theorem \[reduced\] is called the [*reduced arc-function*]{} of the graph $\Gamma$.
\[n(v)\] Let $\Gamma=(V,R)$ be a graph which satisfies the conditions of Theorem \[reduced\]. Then $n_R(v)=2|f(v)|-2$ for each vertex $v\in V$, where $f$ is the reduced arc-function of $\Gamma$.
Let $v\in V$. From statement ([*iii*]{}) of Theorem \[reduced\] any $i\in f(v)$ is the end-point of exactly two elements in $\operatorname{Im}(f)$. Therefore, we get $$\label{nrf}
n_R(v) \leq 2|f(v)|-2.$$ In fact, by statement ([*i*]{}) of Theorem \[reduced\] for each $u\in vR$, exactly one of the end-points of $f(u)$ belongs to $f(v)$. Thus we have equality in , and we are done. $\Box$
\[regular\] Let $\Gamma=(V,R)$ be a non-empty circular-arc graph without twins. Then $\Gamma$ is regular if and only if for any two vertices $u$ and $v$ in $V$ we have: $$\label{subset1}
v\in uR ~~~ \Rightarrow~~~ uR \not\subset \{v\}\cup vR.$$
Suppose that $\Gamma$ is regular and $u$ and $v$ are two adjacent vertices of the graph $\Gamma$. Then $| \{v\} \cup vR|=|\{u\}\cup uR|$. However, $ \{v\} \cup vR \neq \{u\}\cup uR$ because $u$ and $v$ are not twins. It follows that there exists a vertex in $uR$, different from $v$, which is not in $vR$; and there exists a vertex in $vR$, different from $u$, which is not in $uR$. Therefore, the condition holds.
Conversely, suppose that $\Gamma$ satisfies condition . By the same argument as Remark \[remark1\] the graph $\Gamma$ is connected. Thus, it is sufficient to show that any two adjacent vertices $u$ and $v$ have the same degree. On the other hand, by Theorem \[reduced\] there is a reduced arc-function $f:V\to A\mathbb{Z}_n$, where $n=|V|$. So by Corollary \[n(v)\] it is sufficient to show that $|f(v)|=|f(u)|$, or equivalently $$\label{feq}
|f(u)\backslash f(v)|=|f(v)\backslash f(u)|.$$ Note that the set $f(u)\cap f(v)$ is not empty because $u$ and $v$ are adjacent. Moreover, by hypothesis, $u$ and $v$ are not twins so $f(v)\neq f(u)$. We may assume that $$\label{cap}
f(u)\cup f(v) \neq \mathbb{Z}_n.$$ Indeed, otherwise, we would have $f(u)\cup f(v)=\mathbb{Z}_n$ and then from statement ([*i*]{}) of Theorem \[reduced\], any set in $\operatorname{Im}(f)$ different from both $f(u)$ and $ f(v) $ has one end-point in $f(v)$ and one end-point in $f(u)$. This implies that any vertex in $V\backslash \{u,v\}$ is adjacent to both $u$ and $v$, which is impossible because $u$ and $v$ are not twins.
Let $i\in f(u)\backslash f(v)$. Then by statement ([*iii*]{}) of Theorem \[reduced\], there are exactly two vertices $u_i, v_i\in V$, for which $i$ is the end-point of both $f(u_i)$ and $f(v_i)$, or equivalently due to we have $$\{i\}=f(u_i) \cap f(v_i).$$ Moreover, by statement ([*i*]{}) of Theorem \[reduced\], neither $f(u_i)$ nor $f(v_i)$ is a subset of $f(u)$. Now, from it follows that the end-points of $f(u_i)$ and $f(v_i)$ different from $i$ is not in the set $f(u)$. Thus, exactly one of $f(u_i)$ or $f(v_i)$ contains the set $f(v)\cap f(u)$. Assume that $f(u)\cap f(v)\subset
f(u_i)$. Since, by statement ([*i*]{}) of Theorem \[reduced\], $f(v)$ is not a subset of $f(u_i)$, it follows that the end-point of $f(u_i)$, different from $i$, is in the set $f(v)\backslash f(u)$, denote this end-point by $j_{i}$, (see Fig. 1).
(6,4)
(5,0.5)(8,2)(11,0.9)
(5,0.5) (7.15,1.22) (11,0.9) (7,0.6)[$i$]{} (10.9,0.4)[$j_i$]{} (8.2,0.6 )[$f(u_i)$]{} (5.5,0.3 )[$f(v_i)$]{}
(6,1.2)(8,2)(10,1.55) (6,1.2) (10,1.55) (6.3,1.9)[$f(u)$]{}
(8,2)(10.5,2.2)(12.5,1) (8,2) (12.5,1) (11,1.9)[$f(v)$]{}
(3.5,-0.8)[Fig. 1: Some arcs of reduced arc-function $f$ of $\Gamma$ ]{}
So far we could define the mapping, $i\to j_i$, from $f(u)\backslash f(v)$ to $f(v)\backslash f(u)$. Now we claim that this mapping is bijection. To do so, we first prove that it is injective. Suppose on the contrary that there are $i,i'\in f(u)\backslash f(v)$ such that $j_{i} = j_{i'}$. Then $f(u_i)\subset f(u_{i'})$ or $f(u_{i'})\subset f(u_{i})$. However, in both cases this contradicts statement ([*i*]{}) of Theorem \[reduced\]. Now, let $j\in f(v)\backslash f(u)$ then in a similar way there is a corresponding element of $f(u)\backslash f(v)$, say $i$. By statement ([*i*]{}) of Theorem \[reduced\], it is clear that $j_i=j$. This shows that the mapping is surjective.
The same argument can be done for the case $f(u)\cap f(v)\subset f(v_i)$. Thus follows and this proves the proposition. $\Box$
\[correduced\] Let $\Gamma$ be an $m$-regular circular-arc graph on $n$ vertices. Suppose that $m\geq 1$ and the graph has no twins. Then for each reduced arc-function $f$ of $\Gamma$ and each $v\in
V$, we have $|f(v)|=\frac{m+2}{2}$.
By the hypothesis the graph $\Gamma$ is non-empty and without twins. Therefore, the hypothesis of Corollary \[n(v)\] holds and we are done. $\Box$
Graphs and schemes
==================
In this section we prove some results on the scheme of a graph. In particular we will study a relationship between the lexicographic product of two graphs and the wreath product of their schemes.
\[rk\] Let $\Gamma=(V,R)$ be a graph. For each integer $k$, let $$R_k=\{(u,v) \in R : ~~ |uR \cap vR|=k \}.$$ Then $R_k$ is a union of some basic relations of the scheme $\operatorname{Fis}(\Gamma)$.
Let $S$ be the set of basic relations of $\operatorname{Fis}(\Gamma)$. Then $R=\bigcup_{s\in S'} s$ where $S' \subset S$. It is sufficient to show that $R_k$ contains any relation from $S'$ whose intersection with $R_k$ is not empty. To do this, let $s$ be such a relation. Then there exists a pair $(u,v)\in s$ such that $|uR
\cap vR|=k$. On the other hand, by the definition of intersection numbers we have $$|uR \cap vR|= \sum_{r,t\in S'}c^s_{rt}.$$ Thus the number $|uR \cap vR|$ does not depend on the choice of $(u,v)\in s$. By definition of $R_k$ this implies that $s \subset R_k$ as required. $\Box$
\[equ\] Let $\Gamma$ be a graph on the vertex set $V$ and let $$E=\{(u,v)\in V\times V~ :~ u ~{\rm{and}}~ v ~{\rm{are ~twins
~or}}~ u=v\}.$$ Then $E$ is an equivalence relation of the scheme $\operatorname{Fis}(\Gamma)$. Moreover, if $\Gamma$ is a graph with association scheme then, it is isomorphic to lexicographic product of the graph $\Gamma_{V/E}$ and a complete graph.
Let $S$ be the set of basic relations of the scheme $\operatorname{Fis}(\Gamma)$ and let $R$ be the edge set of $\Gamma$. Then there exists a subset $S'$ of $S$ such that $$\label{r}
R=\bigcup_{s\in S'} s.$$ To prove the first statement, we need to check that any non-reflexive basic relation $r\in S$ such that $r\cap E\neq
\emptyset$ is contained in $E$. To do this, let $(u,v)\in r$. We claim that $(u,v)\in E$, or equivalently $u$ and $v$ are twins.
First we show that $$\label{sub}
uR\backslash \{v\} \subseteq vR\backslash \{u\}.$$ If the set $uR\backslash \{v\}$ is empty, then is clear. Now, we may assume that there exists an element $w$ in $V$ such that $w\in uR\backslash \{v\}$. It is enough to show that $v$ is adjacent to $w$ in $\Gamma$. By there exists a basic relation $s\in S'$ so that $(u,w)\in s$. Denote by $t$ the basic relation which contains $(v,w)$. It is sufficient to show that $t\in S'$.
We have $w\in us\cap vt^*$, thus $|us\cap vt^*|=c^r_{st^*}\neq
0$. Since intersection number does not depend on the choice of $(u,v)\in r$, for $(u',v')\in r$ we have $$\label{crst}
|u's\cap v't^*|=c^r_{st^*}\neq 0.$$ On the other hand, by the choice of $r$ there exists $(u',v')\in r\cap E$. So by , there exists a vertex $w'\in V$ such that $$\label{prim}
w'\in u's\cap v't^*.$$ Moreover, since $s\in S'$, we have $w'\in u'R\backslash \{v'\}$. On the other hand, $u'R\backslash \{v'\}
=v'R\backslash \{u'\}$, since $u'$ and $v'$ are twins. It follows that $w'\in v'R\backslash \{u'\}$, so from we conclude that $w'$ is adjacent to $v'$ in $\Gamma$ and so from we have $t\in S'$. The converse inclusion of can be proved in a similar way. Thus $u$ and $v$ are twins and the first statement follows.
To prove the second statement, suppose that $\Gamma$ is a graph with association scheme. It is well-known fact that all classes of an equivalence relation of an association scheme have the same size, say $m$, where $m$ divides $n=|V|$. Moreover, for each $X\in V/E$ we have $u, v \in X $ if and only if $u$ and $v$ are twins. Thus for each $X\in V/E$ we have $$\label{twin}
\Gamma_X \simeq K_m.$$ Fix a class $X_0 \in V/E$. For each $X\in V/E$ choose an isomorphism $f_X
\in \operatorname{Iso}(\Gamma_{X_0},\Gamma_{X})$. Then the mapping $$\label{iso1}
f:V \to V/E \times X_0 \\$$ $$~~~~~~~v \mapsto (X, f^{-1}_{X}(v)),$$ is a bijection, where $X$ is a class of $E$ containing $v$. In order to complete the proof, we show that the above bijection is a required isomorphism: $$\label{iso}
f\in \operatorname{Iso}(\Gamma, \Gamma_{V/E}[\Gamma_{X_0}]).$$ Take two different vertices $u$ and $v$ in $V$, then $f(u)=(X,u_0)$ and $f(v)=(Y, v_0)$, where $X,Y\in V/E$, $u\in X$, $v\in Y$ and $u_0, v_0 \in X_0$. It is enough to show that $u$ and $v$ are adjacent in $\Gamma$ if and only if $f(u)$ and $f(v)$ are adjacent in $\Gamma_{V/E}[\Gamma_{X_0}]$.
First, we assume that $u$ and $v$ are not twins. Then $X \neq Y$. In this case, by definition of $E$, if $u$ and $v$ are adjacent in $\Gamma$ then any vertices in $X$ and any vertices in $Y$ are adjacent to each other. Also if $X$ and $Y$ be adjacent in $\Gamma_{V/E}$, by definition of $\Gamma_{V/E}$, there is a vertex in $X$ and a vertex in $Y$ which are adjacent. However, since all of the vertices in each class of $V/E$ are twins, then $u$ and $v$ are adjacent in $\Gamma$. Thus $u$ and $v$ are adjacent in $\Gamma$ if and only if $X$ and $Y$ are adjacent in $\Gamma_{V/E}$. Therefore, $u$ and $v$ are adjacent in $\Gamma$ if and only if $f(u)$ and $f(v)$ are adjacent in $\Gamma_{V/E}[\Gamma_{X_0}]$.
Now, we may assume that $u$ and $v$ are twins. Then $X=Y$. However, $u$ and $v$ are twins, so they are adjacent in $\Gamma$. Since, $f_X$ is an isomorphism thus $f^{-1}_{X}(u)\neq f^{-1}_{X}(v)$, so $u_0\neq v_0$. From , it follows that $u_0$ and $v_0$ are adjacent in $\Gamma_X$. Therefore, $f(u)$ and $f(v)$ are adjacent in $\Gamma_{V/E}[\Gamma_{X_0}]$. Thus $f$ is an isomorphism and follows, as desired. $\Box$
In the next theorem we show that the scheme of the lexicographic product of two graphs is smaller than the wreath product of their schemes. In general, we do not have equality here. For example, the scheme of the lexicographic product of two complete graphs is a scheme of rank 2, but the wreath product of their schemes has rank 3.
\[wreath\] Let $\Gamma_1$ and $\Gamma_2$ be two graphs. Then the scheme $\mathcal{X}=\operatorname{Fis}(\Gamma_2[\Gamma_1])$ is isomorphic to a fusion of the scheme $\mathcal{Y}= \operatorname{Fis}(\Gamma_1) \wr \operatorname{Fis}(\Gamma_2)$. Moreover, if $\Gamma_1$ is a complete graph and $\Gamma_2$ is a graph without twins such that its scheme is association, then $\mathcal{X}$ and $\mathcal{Y}$ are isomorphic.
Let $\Gamma_i=(V_i,R_i)$, $\mathcal{X}_i=\operatorname{Fis}(\Gamma_i)$ and $S_i$ be the set of basic relations of $\mathcal{X}_i$ for $i=1,2$. Then there exists $S'_i\subset S_i$ such that $$\label{ri}
R_i=\bigcup_{s\in S'_i}s.$$ Let $\Gamma$ be the lexicographic product of $\Gamma_2$ and $\Gamma_1$, and let $R$ be the edge set of $\Gamma$. Then we have $$R=\{((i,k),(j,l)) \in (V_2\times V_1)^2 :~ (i,j)\in R_2 ~~{\text {or}}~~~ (k,l)\in R_1 ~{\text {with}}~ i=j \}.$$ Let $\Gamma'$ be a graph on vertex set $V_1\times V_2$ such that $(k,i)$ and $(l,j)$ are adjacent in $\Gamma'$ if and only if $(i,k)$ and $(j,l)$ are adjacent in $\Gamma$. Define $\sigma:V_2\times V_1 \to V_1\times V_2 $ such that $(i,k)^{\sigma}=(k,i)$. Then $\sigma \in \operatorname{Iso}(\Gamma,\Gamma')$. Thus $\Gamma$ and $\Gamma'$ are isomorphic and it follows that $$\label{fis}
\operatorname{Fis}(\Gamma)^{\sigma}= \operatorname{Fis}(\Gamma').$$ Let $R'$ be the edge set of $\Gamma'$, then $$R'=\{((k,i),(l,j)) \in (V_1\times V_2)^2 : (i,j)\in R_2 ~~{\text {or}}~~~ (k,l)\in
R_1 ~{\text {with}}~ i=j \}$$ $$=\{((k,i),(l,j)) \in (V_1\times V_2)^2 : (i,j)\in R_2 \}\cup~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$\{((k,i),(l,j)) \in (V_1\times V_2)^2 : (k,l)\in R_1 ~{\text {with}}~ i=j \}.~~~~~~~~~~~~~$$ So, by we have $$\label{r'}
R'=\{ (V_1)^2 \otimes s : s\in S'_2\}~\cup ~\{ s \otimes 1_{V_2}
: s \in S'_1\}.$$ Therefore, $R'$ is a union of some basic relations of the scheme $\mathcal{Y}=\mathcal{X}_1\wr \mathcal{X}_2$. Thus we conclude that $$\label{fis3}
\operatorname{Fis}(\Gamma') \leq \mathcal{Y},$$ and from the first statement follows. To prove the second statement, let $\Gamma_1$ be a complete graph on $n$ vertices and let $\Gamma_2$ be a graph without twins such that $\mathcal{X}_2$ is an association scheme. Then, $\mathcal{X}_1\wr \mathcal{X}_2$ is association and from the first statement it follows that $\operatorname{Fis}(\Gamma')$ is association. Thus $1_{V_1}\otimes 1_{V_2}$ is a basic relation of $\operatorname{Fis}(\Gamma')$.
If $\Gamma_2$ be an empty graph, then it is easy to see that we have equality in . So we may suppose that $\Gamma_2$ is a non-empty graph. Since it is a graph with association scheme, there exists a positive integer $t$ such that $\Gamma_2$ is a $t$-regular graph. Let $t_0(i,j)$ be the number of common neighbors of two adjacent vertices $i$ and $j$ in $\Gamma_2$. Since $\Gamma_2$ is without twins, we have $$\label{d0}
t_0(i,j) < t-1.$$ Let $u=(k,i)$ and $v=(l,j)$ be two adjacent vertices in $\Gamma'$. Since, for each $i\in V_2$ the graph $\Gamma_{V_1\times i}$ is isomorphic to $\Gamma_1$, and for each two adjacent vertices $i$ and $j$ in $\Gamma_2$ the set $(V_1\times i)\times (V_1\times j)$ is a subset of $R'$, thus we have $$\label{delta}
|uR'\cap vR'|=\left\{\begin{array}{lll}
(n-2) +tn, ~~~~~~~~~~~~~~~~~~~~i=j \\[0.2cm]
2(n-1)+t_0(i,j)n, ~~~~~~~~~~~ i\neq j. \\[0.1cm]
\end{array}\right.$$ Using , for each $i\neq j$ we have $$\label{delta1}
2(n-1)+t_0(i,j)n< (n-2) +tn.$$ Define $$E:=\{(u,v)\in R': |uR'\cap vR'|= (n-2) +tn\}.$$ From and it follows that $$E=\cup_{i\in V_2}(V_1\times i)^2\backslash ( 1_{V_1}\otimes
1_{V_2}).$$ Now, from Lemma \[rk\], the set $E$ is a union of some of the basic relations of $\operatorname{Fis}(\Gamma')$. On the other hand, $E=s
\otimes 1_{V_2}$, where $s$ is the non-reflexive basic relation of the scheme $\mathcal{X}_1$ of rank 2. However, $s \otimes
1_{V_2}$ is a basic relation of the scheme $\mathcal{X}_1\wr
\mathcal{X}_2$. Therefore, from it is obvious that $E$ is a basic relation of $\operatorname{Fis}(\Gamma')$. Hence, $$\label{f}
F=E\cup( 1_{V_1}\otimes 1_{V_2})$$ is an equivalence relation of the scheme $\operatorname{Fis}(\Gamma')$.
The scheme $\mathcal{X}_1 \wr \mathcal{X}_2$ is the minimal scheme which contains an equivalence $F$ such that for each class $X\in V/F$, the scheme $(\mathcal{X}_1 \wr \mathcal{X}_2)_X$ is isomorphic to $\mathcal{X}_1$ and $(\mathcal{X}_1 \wr \mathcal{X}_2)_{V/F}$ is isomorphic to $\mathcal{X}_2$. In order to prove equality in , it is sufficient to show that $\operatorname{Fis}(\Gamma')$ have the above property.
Let $X\in V/F$. Then by , the scheme $\operatorname{Fis}(\Gamma')_X$ is isomorphic to the scheme $\mathcal{X}_1$. Moreover, from it follows that $$\label{ve}
\operatorname{Fis}(\Gamma')_{V/F}\leq \mathcal{X}_2.$$ However, the edge set of $\Gamma_2$ is a union of some of the basic relations of $\operatorname{Fis}(\Gamma')_{V/F}$. Thus we have equality in , and we are done. $\Box$
\[subg\] Let $\Gamma$ and $\Gamma'$ be two graphs with the same vertex set. Suppose that the edge set of $\Gamma'$ is a union of some basic relations of the scheme of $\Gamma$. Then $\operatorname{Fis}(\Gamma')
\leq \operatorname{Fis}(\Gamma)$.
Elementary circular-arc graphs {#cmn graph}
==============================
Given integers $n$ and $k$ such that $0\leq
2k+1< n$, set $C_{n,k}=\operatorname{Cay}(\mathbb{Z}_n,S)$ where $S=\{\pm
1,\ldots, \pm k\}$. It immediately follows that $C_{n,k}$ is a $2k$-regular graph without twins. Note that $C_{n,0}$ is an empty graph and $C_{n,1}$ is an undirected cycle on $n$ vertices.
From definition, one can verify that $C_{n,k}$ is the graph with vertex set $V=\mathbb{Z}_{n} $ in which two vertices $i$ and $j$ are adjacent if and only if $$\{i,\ldots, k+i\}\cap \{j,\ldots, k+j\} \neq \emptyset.$$ Suppose that $f:V\to A\mathbb{Z}_{n}$, such that $f(i)=\{i,\ldots,
k+i\}$. Then the graph $C_{n,k}$ is the intersection graph of the family $\operatorname{Im}(f)$. Thus, by Lemma \[arcfun\] we conclude that $C_{n,k}$ is a circular-arc graph, and we call it [*elementary circular-arc graph*]{}.
\[ex4\] For $n=2k+2$, two different vertices $i$ and $j$ are adjacent in $C_{n,k}$ if and only if $j\neq i+k+1$. Thus $C_{n,k}$ is isomorphic to a graph on $n$ vertices which is obtained from a complete graph by removing the edges of a perfect matching. It is easy to check that the scheme of this graph is isomorphic to $\mathcal{X}_1\wr \mathcal{X}_2$, where $\mathcal{X}_1$ is a rank 2 scheme on 2 points and $\mathcal{X}_2$ is a rank 2 scheme on $k+1$ points.
\[regularcag\] A regular circular-arc graph without twins is elementary.
Let $\Gamma$ be a circular-arc graph with the vertex set $V$ where $n=|V|$. Suppose that it has no twins. Then by Proposition \[regular\] and Theorem \[reduced\], there exists a reduced arc-function $f$ of $\Gamma$, such that for each vertex $v\in V$ we have $f(v)=\{i_v,\ldots, j_v\} $. Define a bijection from $ V$ to $\mathbb{Z}_n$, the vertex set of $C_{n,k}$, such that $v\to i_v$. From Corollary \[n(v)\] we conclude that $\Gamma$ is a $2k$-regular graph. Then $k=|f(v)|-1$ for any vertex $v$ by Corollary \[correduced\]. Hence $j_v= i_v+ k$. By Lemma \[arcfun\], two vertices $u$ and $v$ in $V$ are adjacent if and only if $f(u)\cap f(v)\neq \emptyset$. It follows that $i_u$ and $i_v$ are adjacent if and only if $\{i_u,\ldots,
k+i_u\}\cap \{i_v,\ldots, k+i_v\}\neq \emptyset$. Therefore, the bijection defined above gives the required isomorphism. $\Box$
\[dihedral\] Let $n$ and $k$ be two positive integers such that $2k+2 < n$. Then the scheme of the graph $C_{n,k}$ is isomorphic to a dihedral scheme.
Let $R$ be the edge set of $C_{n,k}$. For two vertices $i,j\in\mathbb{Z}_n$, we define $d(i,j)$ be the distance of $i$ and $j$ in the graph $C_{n,1}$. Suppose that $i$ and $j$ be two adjacent vertices in $C_{n,k}$. Then by definition of $C_{n,k}$, we have $d(i,j)\leq
k$. Without loss of generality we may assume that the vertices $\{i+1,i+2,\ldots, j-1\}$ are between $i$ and $j$. Thus they are adjacent to both of $i$ and $j$ in the graph $C_{n,k}$. Moreover, the vertices $\{j-k,j-k+1\ldots, i-1\}$ and $ \{j+1,j+2,\ldots,
i+k\}$ are adjacent to both of $i$ and $j$ too. Thus the latter three sets are subsets of $iR\cap jR$, which are of size $d(i,j)-1$, $k-d(i,j)$ and $k-d(i,j)$ respectively. Moreover, since $n>2k+2$, they are disjoint. On the other hand, $B_i:=\{i-k,i-k+1, \ldots , j-k-1 \}$ is the set of all other vertices which are adjacent to $i$, and $B_j:=\{i+k+1,i+k+2,
\ldots , j+k \}$ is the set of all other vertices which are adjacent to $j$, (see Fig. 2).
(6,4)
(10.9,2.0)(10.9,2.787)(10.3435,3.3435) (10.3435,3.3435)(9.787,3.9)(9.0,3.9) (9.0,3.9)(8.213,3.9)(7.6565,3.3435) (7.6565,3.3435)(7.1,2.787)(7.1,2.0) (7.1,2.0)(7.1,1.213)(7.6565,0.6565) (7.6565,0.6565)(8.213,0.1)(9.0,0.1) (9.0,0.1)(9.787,0.1)(10.3435,0.6565) (10.3435,0.6565)(10.9,1.213)(10.9,2.0)
(7.7,3.4) (7.3,3.6)[$i$]{}
(10.4,3.3) (10.6,3.6)[$j$]{}
(7.2,1.4) (5.8,1.3)[$j-k$]{}
(7.42,0.9) (5,0.6)[$j-k-1$]{}
(8.4,0.2) (7.5,-0.3)[$j+k$]{} (9.7,0.2) (9.8,-0.3)[$i-k$]{}
(10.8,1.4) (11.2,1.3)[$i+k$]{}
(10.58,0.9) (11,0.6)[$i+k+1$]{}
(7.6,1)(8.5,0)(9.6,0.4) (7.9,0.85)[$B_i$]{}
(8.4,0.6)(9.5,0.2)(10.3,1.1) (9.6,1.1)[$B_j$]{}
(1.5,-1.5)[Fig. 2: Some vertices of the graph $C_{n,1}$ and the sets $B_i$ and $B_j$]{}
It is clear that $B_i$ and $B_j$ are disjoint from the above three subsets. In addition, we have $|B_i|=|B_j|=d(i,j)$. Moreover, since $n>2k+2$, the vertex $j-k-1$ is not in $B_j$ and the vertex $i+k+1$ is not in $B_i$. It follows that $B_i\neq
B_j$, and $$\label{bij}
|B_i\cap B_j|<d(i,j).$$ On the other hand, $B_i\cap B_j\subset iR\cap jR$ and thus $$\label{ir}
~~|iR\cap jR|= (d(i,j)-1)+2(k-d(i,j))+|B_i\cap B_j|$$ $$= 2k-d(i,j)-1+|B_i\cap B_j|.$$ Now, set $$R_{2k-2}:=\{(i,j) \in R : ~~ |iR \cap jR|=2k-2 \}.$$ Then $R_{2k-2}$ is a symmetric relation. Moreover, from we see that $$\label{mod1}
(i,j)\in R_{2k-2} ~\Leftrightarrow~ |B_i\cap B_j|=d(i,j)-1.$$ If $d(i,j)=1$, then by we have $|B_i\cap B_j|=0$. Thus, from we see that $$\label{dij1}
d(i,j)=1 ~\Rightarrow ~(i,j)\in R_{2k-2}.$$ If $1<d(i,j)\leq k$, then $j-k-2\in B_i\backslash B_j$ and $i+k+2\in B_j\backslash B_i$. It follows that $|B_i\cap
B_j|<d(i,j)-1$. Thus, from we have $(i,j)\notin
R_{2k-2}$. Then using we have $(i,j)\in R_{2k-2}$ if and only if $d(i,j)=1$.
It follows that the graph $(\mathbb{Z}_n, R_{2k-2})$ is isomorphic to an undirected cycle on $n$ points, say $C_n$. By Remark \[rk\], we conclude that $R_{2k-2}$ is union of some basic relations of $\operatorname{Fis}(C_{n,k})$. Thus by Lemma \[subg\], we have $$\operatorname{Fis}(C_n) \leq \operatorname{Fis}(C_{n,k}).$$ It is well-known that $\operatorname{Fis}(C_n)=\operatorname{Inv}(D_{2n})$, where $D_{2n}$ is a dihedral group on $n$ elements. So, in order to complete the proof of the theorem it is enough to show that $$\operatorname{Fis}(C_{n,k})\leq \operatorname{Fis}(C_n).$$ Equivalently, it is suffices to verify that $$\operatorname{Aut}(\operatorname{Fis}(C_n)) \leq \operatorname{Aut}(\operatorname{Fis}(C_{n,k})).$$ Since the automorphism group of a graph is equal to the automorphism group of its scheme, it is sufficient to show that $\operatorname{Aut}(C_n) \leq
\operatorname{Aut}(C_{n,k})$.
Since $\operatorname{Aut}(C_n)$ is the dihedral group $D_{2n}$, and $D_{2n}$ is generated by automorphisms $\sigma$ and $ \delta$ where $i^{\sigma}=i+1$ and $i^{\delta}=n-i$ for each $i\in
\mathbb{Z}_n$. It is enough to show that $$\sigma, \delta\in
\operatorname{Aut}(C_{n,k}).$$ Let $i,j \in \mathbb{Z}_n$. Two vertices $i$ and $j$ are adjacent in $C_{n,k}$ if and only if $$\{i,\ldots, k+i\}\cap \{j,\ldots, k+j\} \neq \emptyset
\Leftrightarrow$$ $$\{i+1,\ldots, k+i+1\}\cap \{j+1,\ldots, k+j+1\} \neq \emptyset
\Leftrightarrow$$ $$\label{sigma}
\{i^{\sigma},\ldots, k+i^{\sigma}\}\cap \{j^{\sigma},\ldots,
k+j^{\sigma}\} \neq \emptyset.$$ Moreover, we have if and only if $i^{\sigma}$ and $j^{\sigma}$ are adjacent in $C_{n,k}$. Thus $\sigma \in
\operatorname{Aut}(C_{n,k})$. In a similar way we can show that $\delta \in
\operatorname{Aut}(C_{n,k})$, this completes the proof. $\Box$
Proof of the main theorems
==========================
Proof of Theorem \[graphtheorem\]. {#proof-of-theorem-graphtheorem. .unnumbered}
----------------------------------
We first prove the necessity condition of the theorem. Let $\Gamma=(V,R)$ be a circular-arc graph with association scheme and $|V|=n$. Denote by $E$ the equivalence relation on $V$ defined in Theorem \[equ\]. Then by this theorem the graph $\Gamma$ is isomorphic to lexicographic product of the graph $\Gamma_{V/E}$ and a complete graph. In particular, it is easy to see that $\Gamma_{V/E}$ is also a circular-arc graph and has no twins. So, to complete the proof it is enough to show that $\Gamma_{V/E}$ is an elementary circular-arc graph.
If $\Gamma_{V/E}$ is empty, then it is isomorphic to $C_{m,0}$ with $m=|V/E|$, and we are done. We suppose that $\Gamma_{V/E}$ is non-empty. The graph $\Gamma$ is regular, because it is a graph with association scheme. By the definition of $E$ this implies that the graph $\Gamma_{V/E}$ is regular too. From Corollary \[correduced\], the graph $\Gamma_{V/E}$ is $2k$-regular for some integer $k> 0$. Therefore, from Theorem \[regularcag\] the latter graph is isomorphic to the elementary circular-arc graph $C_{m,k}$. Thus $\Gamma$ is isomorphic to lexicographic product of an elementary circular-arc graph and a complete graph.
Conversely, let $\Gamma_1$ be a complete graph and let $\Gamma_2$ be an elementary circular-arc graph. If $\Gamma_2$ be an empty graph then it is a graph with association scheme. If it is a non-empty elementary circular-arc graph then from Example \[ex4\] and Theorem \[dihedral\] we conclude that $\Gamma_2$ is a graph with association scheme. On the other hand, the wreath product of two association scheme is association. Thus, from Theorem \[wreath\] the scheme of $\Gamma_2[\Gamma_1]$ is association. This completes the proof of the theorem. $\Box$
Proof of Theorem \[schemetheorem\]. {#proof-of-theorem-schemetheorem. .unnumbered}
-----------------------------------
We first assume that $\mathcal{X}$ is an association circular-arc scheme. Then there is a circular-arc graph $\Gamma$ such that $\operatorname{Fis}(\Gamma)=\mathcal{X}$. From Theorem \[graphtheorem\], the graph $\Gamma$ is isomorphic to the lexicographic product of an elementary circular-arc graph and a complete graph. On the other hand, if the elementary circular-arc graph is empty then its scheme is of rank 2. Otherwise, from Example \[ex4\] and Theorem \[dihedral\] the scheme of an elementary circular-arc graph is isomorphic to the wreath product of a rank 2 scheme on 2 points and a rank 2 scheme, or it is isomorphic to a dihedral scheme. Therefore, in any case the scheme of an elementary circular-arc graph is association. Note that in the first two cases the scheme of an elementary circular-arc graph is forestal. Moreover, any elementary circular-arc graph is without twins. Hence, from Theorem \[wreath\] it follows that $\operatorname{Fis}(\Gamma)$ is isomorphic to the wreath product of a rank 2 scheme and the scheme of an elementary circular-arc graph which is either forestal or dihedral.
Conversely, assume that $\mathcal{X}$ is a circular-arc scheme such that it is isomorphic to the wreath product of a rank 2 scheme and a scheme which is either forestal or dihedral. Since any rank 2 scheme, any forestal scheme and any dihedral scheme are association, it is enough to note that the wreath product of two association schemes is an association scheme. Thus $\mathcal{X}$ is an association scheme and the proof is complete. $\Box$
The authors are extremely grateful to Professor Ilia Ponomarenko for his useful discussions and valuable helps. The first author was visiting the Euler Institute of Mathematics, St. Petersburg, Russia during the time this paper was written and she thanks the Euler Institute for its hospitality.
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---
abstract: 'Physics beyond the Standard Model could be measured indirectly, through its effects on Standard Model observables. One place to look for such effects is in the semileptonic decays of $B$ mesons. In order to constrain the possible role of new physics on semileptonic $B$ decays, we present the most general low energy effective Lagrangian constructed from all dimension six four-fermion operators that contribute to the process $b\rightarrow c\ell{\bar\nu}_\ell$. We then use it to compute the corrections due to new physics to the differential decay rates for the exclusive processes ${\bar B}\rightarrow D^{(*)}\ell{\bar\nu}_\ell$, as well as the inclusive decays ${\bar B}\rightarrow X_c\ell{\bar\nu}_\ell$. Both inclusive and exclusive rates are expressed in terms of a set of parameters that characterize the types of four-fermion interactions that can be induced by physics beyond the Standard Model. Although it is not particularly useful to carry out a full analysis until data from the next generation of $B$ factories becomes available, here we illustrate how the existing experimental results can be used to constrain some of these parameters.'
address: 'California Institute of Technology, Pasadena, CA 91125'
author:
- 'Walter D. Goldberger[^1]\'
title: Semileptonic $B$ Decays as a Probe of New Physics
---
Introduction
============
Despite the fact that the Standard Model successfully accounts for most of the observed experimental results (see [@hewett] for a review), it is incomplete, and must be replaced by a more fundamental theory at some high energy scale. Whatever form physics at a higher scale might take, it will generate low energy effective couplings that contribute to Standard Model processes. These couplings provide the opportunity to search for signatures of new physics indirectly, by measurements of the deviation of experimental observables from their Standard Model predictions.
Rather than compute the effect of physics beyond the Standard Model within the framework of a specific model, our ignorance dictates that we include all effective interactions consistent with the symmetries of the Standard Model. The resulting observables are expressed in terms of the parameters of the generalized interaction, which can be constrained by the experimental data. Since these parameters characterize the most general interaction, their experimental bounds can be used to rule out or guide in the construction of specific models.
Anticipating the precision data from the upcoming generation of $B$ factories, one place to look for the effects of new physics is in observables related to $B$ meson physics. Therefore, in this paper we set up a parametrization for a set of observables associated with semileptonic $B$ decays into charmed mesons. Specifically, in Section \[sec:def\] we construct the low energy effective Lagrangian for the process $b\rightarrow c\ell{\bar {\nu}}_\ell$ induced by new physics. We do so by including all the dimension six, four-fermion Lorentz scalars that contribute to the process, but excluding higher dimension operators. We also introduce a set of real parameters bilinear in the coupling constants of this effective Lagrangian. These parameters can be used to characterize the low energy effects of the new physics.
Using the generalized interaction of Section \[sec:def\], we then calculate at leading order in HQET [@isgur; @hqet; @neubert] the contribution from new physics to the differential decay rates for ${\bar B}\rightarrow D\ell{\bar {\nu}}_\ell$ and ${\bar B}\rightarrow D^*\ell{\bar {\nu}}_\ell$. The results for these observables, expressed in the parametrization of Section \[sec:def\] appear in Section \[sec:D\] and Section \[sec:D\*\] respectively. The parametrization of Section \[sec:def\] also appears when we consider the corrections induced by physics beyond the Standard Model on the inclusive semileptonic decay of $B$ mesons into final states containing a $c$ quark, ${\bar B}\rightarrow X_c \ell{\bar {\nu}}_\ell.$ In Section \[sec:Xc\], these deviations are calculated at leading order in perturbation theory, which is justified for $m_b\gg \Lambda_{\mbox{\tiny QCD}}$ [@ope]. In Section \[sec:disc\] we use the results of the previous sections to discuss briefly the constraints which the existing experimental data imposes on the parametrization of Section \[sec:def\].
Finally, we note that new physics effects on observables related to $b\rightarrow c\ell{\bar\nu}_\ell$ have been studied extensively for restricted types of non-Standard model interactions. The decays $B\rightarrow X\ell{\nu}_\ell$ (specially for decays into $\tau$-leptons) have been used to constrain classes of models with scalar interactions [@kraw; @kal; @grzad1; @hou; @isidori; @gross; @coarasa], such as those which arise in models with extended Higgs sectors. Other constraints on such types of models have also come from observables related to the exclusive decays ${\bar B}\rightarrow D\ell{\nu}_\ell$ [@garisto; @tanaka; @kiers] as well as ${\bar B}\rightarrow D^*\ell{\nu}_\ell$ [@garisto; @tanaka; @grzad2]. The process $b\rightarrow c\ell{\bar\nu}_\ell$ has also been used to analyze possible vector and axial vector couplings beyond the Standard Model $V-A$. In particular, model-independent analyses of right-handed $b\rightarrow c$ quark current contributions to $B\rightarrow X\ell{\nu}_\ell$ and ${\bar B}\rightarrow D^*\ell{\nu}_\ell$ were performed in [@voloshin; @rizzo]. These were motivated by an extension of the Standard Model, the Left-Right Symmetric model [@gronau1], which contains gauge bosons that couple to right-handed fermions. Also inspired by this class of models were [@garisto; @gronau2], who constructed observables based on the exclusive decays ${\bar B}\rightarrow D^{(*)}\ell{\nu}_\ell$ in order to constrain right-handed vector couplings. The combined effects of non-Standard Model scalar, pseudoscalar, vector, and axial vector interactions were studied in [@wu] within the context of $CP$ violating polarization observables related to ${\bar B}\rightarrow D\tau{\nu}_\tau$ and ${\bar B}\rightarrow D^*\ell{\nu}_\ell.$
Definitions {#sec:def}
===========
Our starting point for the calculation of semileptonic $B$ decays to charmed mesons is the following interaction Hamiltonian: $$\label{eq:int}
{\cal H}_{\mbox{\tiny int}}={\cal H}_{\mbox{\tiny{SM}}} + \left[{4 G_F \over\sqrt{2}}V_{cb}\sum_{\gamma,\mu,\epsilon} g^\gamma_{\mu\epsilon} [{\bar c}\Gamma^\gamma b_\mu] [{\bar \ell}_\epsilon\Gamma^\gamma{\nu_\ell}] + \mbox{h.c.}\right].$$ ${\cal H}_{\mbox{\tiny{SM}}}$ is the effective $V-A$ Hamiltonian which mediates the Standard Model Weak interaction process $b\rightarrow c \ell{\bar \nu}_\ell$ for energy scales much less than $m_W$. The second term represents the low-energy effective interaction generated by new physics at some energy higher than $m_W$. Rather than adopt a particular model for this new physics, we include all the lowest dimension operators which contribute to $b\rightarrow c \ell{\bar \nu}_\ell$ at tree level and which also respect the Lorentz and gauge invariance of the Standard Model. These are the dimension six four-fermi operators which appear in Eq. (\[eq:int\]). One way to organize their contribution [@tau] is to sum the index $\gamma$ over $S,V,T,$ with $$\begin{aligned}
\Gamma^S=1; & (\Gamma^V)^\mu =\gamma^\mu; & (\Gamma^T)^{\mu\nu} =\sigma^{\mu\nu}\equiv\frac{i}{2}[\gamma^\mu,\gamma^\nu],\end{aligned}$$ while projecting the $b$-quark and lepton $\ell$ into fields of definite chirality (respectively denoted by $b_\mu$ and $\ell_\epsilon$, with $\mu,\epsilon\in\{L,R\}$) and summing over all values of $\mu,$ $\epsilon$. Given $\gamma,$ the chirality of the $c$ quark is fixed by $\mu$, and the chirality of the neutrino field $\nu_\ell$ is fixed by the value of $\epsilon.$ Note in particular that we are including operators which contain a right-handed neutrino field. These occcur for $\gamma=S,T$ and $\epsilon=L,$ or $\gamma=V$ and $\epsilon=R$.
The complex coefficients $g^\gamma_{\mu\epsilon}$ are a set of twelve dimensionless coupling constants (scaled by the Fermi constant $G_F$), one for each choice of $\gamma,$ $\mu,$ and $\epsilon.$ (In the abscence of right-handed neutrinos, $g^S_{\mu L}=g^V_{\mu R}=g^T_{\mu L}=0,$ and the number of coefficients would be reduced to six.) If written in terms of the fields that transform as definite representations of the Standard Model $SU(3)\times SU(2)\times U(1)$ gauge group, no insertions of the Standard Model Higgs doublet are needed to make the dimension six operators of Eq. (\[eq:int\]) gauge invariant (we take the right-handed neutrino to be a Standard Model gauge singlet). Dimensional analysis then implies that $g^\gamma_{\mu\epsilon}\sim{\mathcal O}(m_W^2/M^2),$ where $M$ is the energy scale of the new physics.
Only certain real combinations of these coupling constants will appear in the expressions for the calculated observables. To first order in $g^\gamma_{\mu\epsilon},$ we have $$\begin{aligned}
\label{eq:1st}
\nonumber \beta_0 &=& 2\hbox{Re}\left\{g^V_{LL}\right\},\\
\nonumber \beta''_0 &=& 2 \hbox{Re}\left\{g^V_{RL}\right\},\\
\nonumber \epsilon_0 &=& 2 \hbox{Re}\left\{g^S_{LR}\right\},\\
\nonumber \epsilon'_0 &=& 2 \hbox{Re}\left\{g^S_{RR}\right\},\\
\rho_0 &=& 2 \hbox{Re}\left\{g^T_{LR}\right\},\end{aligned}$$ which in principle could be present even in the abscence of a right-handed neutrino. However, the last three of these can only appear for non-zero lepton mass. We expect these five parameters, which arise by interference with ${\cal H}_{\mbox{\tiny{SM}}}$, to give the most significant contribution to signals of new physics. Eq. (\[eq:1st\]) includes only the interference with the leading order part of the Standard Model term. In reality, it should be modified by higher order Standard Model effects (such as radiative corrections), which we ignore here.
The remaining parameters are second order in the $g^\gamma_{\mu\epsilon}$ and are suppressed in comparison to the previous set. We classify them by the following types:
- Scalar-scalar: $$\begin{aligned}
\nonumber \alpha &=& \left|g^S_{LL}\right|^2 + \left|g^S_{LR}\right|^2 + \left|g^S_{RL}\right|^2 + \left|g^S_{RR}\right|^2, \\
\alpha'&=& 2 \hbox{Re}\left\{g^S_{RL} {g^S_{LL}}^\ast + g^S_{LR} {g^S_{RR}}^\ast\right\}.\end{aligned}$$
- Vector-vector: $$\begin{aligned}
\nonumber \beta &=& \left|g^V_{LL}\right|^2 + \left|g^V_{RR}\right|^2, \\
\nonumber \beta'&=& \left|g^V_{LR}\right|^2 + \left|g^V_{RL}\right|^2, \\
\beta''&=& 2 \hbox{Re}\left\{g^V_{RL} {g^V_{LL}}^\ast + g^V_{LR} {g^V_{RR}}^\ast\right\}.\end{aligned}$$
- Tensor-tensor: $$\gamma = \left|g^T_{LR}\right|^2 + \left|g^T_{RL}\right|^2.$$
- Scalar-tensor $$\begin{aligned}
\nonumber \delta &=& 2 \hbox{Re}\left\{g^T_{RL} {g^S_{RL}}^\ast + g^T_{LR} {g^S_{LR}}^\ast\right\},\\
\delta' &=& 2 \hbox{Re}\left\{g^T_{RL} {g^S_{LL}}^\ast + g^T_{LR} {g^S_{RR}}^\ast\right\}.\end{aligned}$$
- Scalar-vector (only appear for non-zero lepton mass): $$\begin{aligned}
\nonumber \epsilon &=& 2 \hbox{Re}\left\{g^V_{LL} {g^S_{LR}}^\ast + g^V_{RL} {g^S_{RR}}^\ast+ g^V_{LR} {g^S_{LL}}^\ast + g^V_{RR} {g^S_{RL}}^\ast\right\},\\
\epsilon' &=& 2 \hbox{Re}\left\{g^V_{RL} {g^S_{LR}}^\ast + g^V_{LL} {g^S_{RR}}^\ast+ g^V_{RR} {g^S_{LL}}^\ast + g^V_{LR} {g^S_{RL}}^\ast\right\}.\end{aligned}$$
- Tensor-vector (only appear for non-zero lepton mass): $$\begin{aligned}
\nonumber \rho &=& 2 \hbox{Re}\left\{g^V_{LL} {g^T_{LR}}^\ast + g^V_{RR} {g^T_{RL}}^\ast\right\},\\
\rho'&=& 2\hbox{Re}\left\{g^V_{RL} {g^T_{LR}}^\ast + g^V_{LR} {g^T_{RL}}^\ast\right\}.\end{aligned}$$
${\bar B}\rightarrow D \ell\bar{\nu}_\ell$ {#sec:D}
==========================================
As in [@neubert], we write, in the rest frame of the $\bar B$-meson: $$\label{eq:D}
{d^2 \Gamma\over dq^2 d(\cos\theta)} = {G_F^2 \left|V_{cb}\right|^2 \over 768\pi^3}\left|{\mathbf p}_D\right|{q^2-m_\ell^2 \over m_B^2}\left[(1+\cos\theta)^2\left|H_+\right|^2 + (1-\cos\theta)^2\left|H_-\right|^2 + 2 \sin^2\theta\left|H_0\right|^2\right],$$ where ${\bf p}_D$ is the $D$ three-momentum in the $\bar B$ rest frame, $\theta$ is the angle between the lepton and the $D$ meson in the rest frame of the ‘virtual $W$-boson,’ (i.e, the frame in which ${\mathbf p}_B = {\mathbf p}_D$) and $q^2=(p_B-p_D)^2$. $\left|H_\pm\right|^2$ and $\left|H_0\right|^2$ are helicity amplitudes, which we decompose as: $$\begin{aligned}
\nonumber \left|H_\pm\right|^2 &=& \left|H^{\mbox{\tiny (SM)}}_\pm\right|^2+\Delta^{(1)}_\pm + \Delta^{(2)}_\pm,\\
\left|H_0\right|^2 &=&\left|H^{\mbox{\tiny (SM)}}_0\right|^2 +\Delta^{(1)}_0+ \Delta^{(2)}_0.\end{aligned}$$ In these expressions, $\left|H^{\mbox{\tiny (SM)}}_\pm\right|^2$ and $\left|H^{\mbox{\tiny (SM)}}_0\right|^2$ are the helicity amplitudes computed assuming only the Standard Model $V-A$ coupling. The terms $\Delta^{(1)}_\pm$, $\Delta^{(1)}_0$ are contributions from the interference of the Standard Model part of the amplitude (taken only to leading order) and other terms in Eq. (\[eq:int\]). Finally, the terms $\Delta^{(2)}_\pm$ and $\Delta^{(2)}_0$ are second order in the new interactions of Eq. (\[eq:int\]). Here we calculate the interference and the second order terms by matching hadronic matrix elements of the quark-quark operators in Eq. (\[eq:int\]) to HQET matrix elements, neglecting both perturbative ${\mathcal O}(\alpha_s)$ and non-perturbative ${\mathcal O}(1/m_Q)$ corrections. Higher order corrections to the Standard Model contribution to the decays ${\bar B}\rightarrow D^{(*)}\ell{\bar \nu}_\ell$ can be found in [@ligeti].
Writing $w=(m_B^2 + m_D^2 -q^2)/2 m_B m_D,$ and denoting the Isgur-Wise function [@isgur] by $\xi(w)$, we find for the interference tems: $$\begin{aligned}
\nonumber \Delta^{(1)}_\pm&=& \frac{3 m_B m_D}{q^2} (1+w)\xi(w)^2\biggl[{m_\ell^2\over q^2}(\beta_0+\beta''_0)\left\{\left(w\pm\sqrt{w^2-1}\right)m_B^2\right.\\
\nonumber & &\hspace{0.5cm}\left. \mbox{}- 2 m_B m_D + \left(w\mp\sqrt{w^2-1}\right)m_D^2\right\}\\
\nonumber & & \mbox{}+ \frac{m_\ell}{2}\epsilon_0\left\{(m_B - m_D) (1+w) \pm (m_B+m_D)\sqrt{w^2-1}\right\}\\
& &\mbox{}+ 2m_\ell\rho_0\left\{(m_B+m_D) (w-1) \pm (m_B-m_D)\sqrt{w^2-1}\right\}\biggr],\end{aligned}$$ and $$\begin{aligned}
\nonumber \Delta^{(1)}_0 &=& \frac{3 m_B m_D}{q^2}(1+w)\xi(w)^2\biggl[ (\beta_0+\beta''_0)\Bigl\{m_\ell^2+(m_B+m_D)^2(w-1)\Bigr\}\\
& & \mbox{}+ {m_\ell\over 2} \epsilon_0 (1+w)(m_B-m_D) + 2 m_\ell \rho_0 (w-1) (m_B + m_D) \biggr].\end{aligned}$$ Likewise, the second order terms are $$\begin{aligned}
\nonumber \Delta^{(2)}_\pm&=& {3\over 2} m_B m_D (1+w) \xi(w)^2\Bigl[(\alpha+\alpha')(1+w) + 16\gamma(w-1)\\
& & \mbox{}\pm 4(\delta+\delta')\sqrt{w^2-1}\Bigr] + \Delta^m_\pm,\\
\nonumber \Delta^{(2)}_0&=& {3 m_B m_D\over 2 q^2}(1+w)\xi(w)^2\Bigl[(\alpha+\alpha')(1+w)q^2\\
& & \mbox{}+2\big\{(\beta+\beta'+\beta'')(m_B + m_D)^2 - 8q^2\gamma\big\}(w-1)\Bigl] + \Delta^m_0,\end{aligned}$$ where $\Delta^m_\pm$ and $\Delta^m_0$ represent corrections due to non-zero lepton mass (which are only non-negligible for decays into $\tau$-leptons): $$\begin{aligned}
\nonumber \Delta^m_\pm &=& \frac{3 m_B m_D}{q^2}(1+w)\xi(w)^2 \biggl[\frac{m_\ell^2}{q^2}(\beta + \beta'+ \beta'')\left\{\left(w\pm\sqrt{w^2-1}\right)m_B^2\right.\\
\nonumber& &\left. \hspace{0.5cm} \mbox{} - 2 m_B m_D + \left(w\mp\sqrt{w^2-1}\right)m_D^2\right\}\\
\nonumber& & \mbox{} +\frac{m_\ell}{2}\epsilon\left\{(m_B-m_D)(1+w) \pm (m_B + m_D) \sqrt{w^2-1}\right\}\\
& & \mbox{} + 2m_\ell\rho\left\{(m_B+m_D) (w-1) \pm (m_B-m_D)\sqrt{w^2-1}\right\}\biggr],\\
\nonumber \Delta^m_0 &=& \frac{3 m_B m_D}{q^2}(1+w)\xi(w)^2 \Bigl[m_\ell^2\{(\beta + \beta'+ \beta'') + 16\gamma (w-1) \}\\
& & \mbox{} + \frac{m_\ell}{2}\epsilon(1+w)(m_B-m_D) + 2m_\ell\rho (w-1)(m_B + m_D)\Bigr].\end{aligned}$$
${\bar B}\rightarrow D^* \ell\bar{\nu}_\ell$ {#sec:D*}
============================================
As in the previous case, we write in the $\bar B$ rest frame $$\label{eq:D*}
{d^2 \Gamma\over dq^2 d(\cos\theta)} = {G_F^2 \left|V_{cb}\right|^2\over768\pi^3}\left|{\bf p}_{D^*}\right|{{q^2-m_\ell^2}\over m_B^2}\left[(1+\cos\theta)^2\left|H_+\right|^2 + (1-\cos\theta)^2\left|H_-\right|^2 + 2 \sin^2\theta\left|H_0\right|^2\right].$$ The kinematic variables ${\bf p}_{D^*}$, $\theta$ and $q^2$ are defined as before (just replace $D$ with $D^*$ in the definitions of the previous section). Also as before, we separate the helicity amplitudes as $$\begin{aligned}
\nonumber \left|H_\pm\right|^2 &=& \left|H^{\mbox{\tiny (SM)}}_\pm\right|^2+\Delta^{(1)}_\pm + \Delta^{(2)}_\pm,\\
\left|H_0\right|^2 &=&\left|H^{\mbox{\tiny (SM)}}_0\right|^2 +\Delta^{(1)}_0+ \Delta^{(2)}_0.\end{aligned}$$ Once again, the interference terms only take into account the leading order part of the Standard Model amplitude. Calculating these as well as the second order terms by matching hadronic matrix elements onto HQET ones (neglecting ${\mathcal O}(\alpha_s)$ or ${\mathcal O}(1/m_Q)$ corrections), we find: $$\begin{aligned}
\nonumber \Delta^{(1)}_\pm &=& \frac{3 m_B m_{D^*}}{q^2}(1+w)\xi(w)^2 \biggl[2 (\beta_0 w -\beta''_0)q^2 + \frac{m_\ell^2}{q^2}(\beta_0-\beta''_0)\left\{\left(w\pm\sqrt{w^2-1}\right) m_B^2\right.\\
\nonumber& & \left.\hspace{0.5cm}\mbox{} - 2 m_B m_{D^*} + \left(w\mp\sqrt{w^2-1}\right)m_{D^*}^2\right\}\\
\nonumber& & \mbox{} + \frac{m_\ell}{2}(\epsilon_0'-\epsilon_0)\left\{(m_B + m_{D^*})(w-1)\pm(m_B-m_{D^*})\sqrt{w^2-1}\right\}\\
& & \mbox{} - 2m_\ell\rho_0\left\{m_B(5+w)-m_{D^*}(1+5w)\pm (m_B+5 m_{D^*})\sqrt{w^2-1}\right\}\biggr],\\
\nonumber \Delta^{(1)}_0 &=& {3 m_B m_{D^*}\over q^2}(1+w)\xi(w)^2\Bigl[ (1+w)(m_B-m_{D^*})^2 (\beta_0-\beta''_0) \\
\nonumber & & \mbox{} + m_\ell^2\{\beta_0(2w-1)-\beta''_0\} + \frac{m_\ell}{2}(\epsilon'_0-\epsilon_0)(m_B+m_{D^*})(w-1)\\
& & \mbox{} - 2m_\ell \rho_0\{m_B(5+w) - m_{D^*}(1+5w)\}\Bigr].\end{aligned}$$ The second order terms are $$\begin{aligned}
\nonumber \Delta^{(2)}_\pm &=& {3\over 2} m_B m_{D^*} (1+w) \xi(w)^2\Bigl[(\alpha-\alpha')(w-1) + 4\left\{(\beta+\beta')w-\beta''\right\}\\
& & \mbox{}+ 16\gamma(1+w) \pm4\left(\beta'-\beta + \delta-\delta'\right)\sqrt{w^2-1}\Bigr] + \Delta^m_\pm,\\
\nonumber \Delta^{(2)}_0 &=& {3 m_B m_{D^*}\over 2 q^2} (1+w) \xi(w)^2\Bigl[(\alpha-\alpha')(w-1)q^2\\
\nonumber& & \mbox{} + 2(\beta+\beta'-\beta'') (m_B-m_{D^*})^2 (1+w)\\
& & \mbox{} + 16\gamma\left\{(m_B^2+m_{D^*}^2)(3w-1) + 2 m_B m_{D^*} (w^2 + w -4)\right\}\Bigr] +\Delta^m_0,\end{aligned}$$ with the following corrections due to the effects of a non-zero lepton mass: $$\begin{aligned}
\nonumber\Delta^m_\pm &=& \frac{3 m_B m_{D^*}}{q^2}(1+w)\xi(w)^2 \biggl[\frac{m_\ell^2}{q^2} (\beta+\beta'+32\gamma -\beta'')\left\{\left(w\pm\sqrt{w^2-1}\right) m_B^2\right. \\
\nonumber& & \hspace{0,5cm}\left. \mbox{} - 2 m_B m_{D^*}+\left(w\mp\sqrt{w^2-1}\right)m_{D^*}^2\right\}\\
\nonumber & & \mbox{} +\frac{m_\ell}{2}(\epsilon'-\epsilon)\left\{(m_B + m_{D^*})(w-1)\pm(m_B-m_{D^*})\sqrt{w^2-1}\right\}\\
\nonumber & & \mbox{}+ 2m_\ell\left\{(\rho' m_B +\rho m_{D^*})(1+5w)\pm 5(\rho' m_B - \rho m_{D^*})\sqrt{w^2-1}\right.\\
& &\left. \hspace{0,5cm}\mbox{} - (\rho m_B +\rho' m_{D^*}) (5+w) \mp (\rho m_B - \rho' m_{D^*})\sqrt{w^2-1}\right\}\biggr],\\
\nonumber \Delta^m_0 &=& \frac{3 m_B m_{D^*}}{q^2}(1+w)\xi(w)^2 \Bigl[m_\ell^2\{(\beta+\beta')(2w-1) - \beta'' + 16\gamma(1+w)\}\\
\nonumber & & \mbox{} + \frac{m_\ell}{2}(\epsilon'-\epsilon)(m_B+m_{D^*})(w-1) + 2 m_\ell m_B\{\rho'(1+5 w) -\rho(5+w)\}\\
& & \mbox{}+ 2 m_\ell m_{D^*}\{\rho(1+5w) - \rho'(5+w)\}\Bigl].\end{aligned}$$
${\bar B}\rightarrow X \ell\bar{\nu}_\ell$ {#sec:Xc}
==========================================
Starting from the operator product expansion (OPE), it can be shown that in the limit $m_b\gg \Lambda_{\mbox{\tiny QCD}}$ the inclusive semileptonic $B$ decay rate is equivalent to the perturbative quark level $b$ decay rate [@ope]. This makes it possible to calculate the effects of new physics on the inclusive decay ${\bar B}\rightarrow X_c \ell\bar{\nu}_\ell$ directly from Eq. (\[eq:int\]), by calculating the rate for the quark level process $b\rightarrow c\ell{\bar \nu}_\ell$. We write the differential decay rate as $$\label{eq:diff}
{d^3\Gamma\over {dq^2 dE_\ell dE_\nu}} = \left({d^3\Gamma\over {dq^2 dE_\ell dE_\nu}}\right)_{\mbox{\tiny (SM)}} + \left({d^3\Gamma\over {dq^2 dE_\ell dE_\nu}}\right)_{(1)} + \left({d^3\Gamma\over {dq^2 dE_\ell dE_\nu}}\right)_{(2)},$$ where $q^2=(p_b-p_c)^2$ and $E_\ell$, $E_\nu$ are the $b$ quark rest frame energies of the lepton and neutrino respectively. Eq. (\[eq:diff\]) is not meant to imply that the differential rate is a contribution from three distinct processes. Instead, we are merely decomposing the observable into a piece arising solely from Standard Model physics, an interference term, and a term that is second order in the new physics. At present, nonperturbative corrections to the Standard Model term are known to order $(\Lambda_{\mbox{\tiny QCD}}/m_b)^3$ [@gremm0], while only the part proportional to $\alpha_s^2\beta_0$ of the NLO perturbative corrections is known [@gremm1] ($\beta_0$ being the coefficient of the one-loop QCD beta function). Here, we calculate the last two terms in Eq. (\[eq:diff\]), neglecting ${\mathcal O}(\alpha_s)$ or ${\mathcal O}(\Lambda_{\mbox{\tiny QCD}}/m_b)$ corrections. Introducing the kinematic variables $$\begin{aligned}
\nonumber E_c &=&(m_b^2+m_c^2 - q^2)/2m_b,\\
\nonumber {\hat q}^2 &=& q^2/m_b^2,\\
{\hat E}_\nu &=& 2 E_\nu/m_b,\end{aligned}$$ and $y=E_\ell/{E_\ell}_{\mbox{\tiny{max}}}$ with ${E_\ell}_{\mbox{\tiny{max}}}=(m_b^2-m_c^2+m_\ell^2)/2 m_b$, as well as $x=m_\ell/m_b$ and $z=m_c/m_b$, we find, in the rest frame of the $b$ quark $$\begin{aligned}
\label{eq:diff1}
\nonumber \left({d^3\Gamma\over {dq^2 dE_\ell dE_\nu}}\right)_{(1)} &=& {G_F^2\left|V_{cb}\right|^2 m_b^2\over 32\pi^3}\delta\left(m_b-E_c-E_e-E_\nu\right)\biggl[-4z\beta''_0\left({\hat q}^2-x^2\right)\\
\nonumber & & \mbox{} + 4\beta_0 {\hat E}_\nu\Bigl\{\left(x^2(1+y)+y(1-z^2)\right)-{\hat q}^2\Bigr\}\\
\nonumber & & \mbox{} + 2x\left((1-z^2)(1-y)-(1+y)x^2\right)\epsilon_0'\\
& & +2{\hat E}_\nu xz(\epsilon_0-12\rho_0)\biggr],\end{aligned}$$ which, as in the exclusive decays, includes only the interference with the leading order Standard Model contribution. The remaining contributions are of second order in the new physics: $$\begin{aligned}
\label{eq:diff2}
\nonumber \left({d^3\Gamma\over {dq^2 dE_\ell dE_\nu}}\right)_{(2)} &=& {G_F^2\left|V_{cb}\right|^2 m_b^2\over 32\pi^3}\delta\left(m_b-E_c-E_e-E_\nu\right)\biggl[\left({\hat q}^2-x^2\right)\Bigl\{\alpha(1+z^2-{\hat q}^2)\\
\nonumber & &\hspace{0.5cm} \mbox{} + 2z(\alpha'-2\beta'')\Bigr\}\\
\nonumber & & \mbox{} + 4\beta {\hat E}_\nu\Bigl\{\left(x^2(1+y)+y(1-z^2)\right)-{\hat q}^2\Bigr\}\\
\nonumber & & \mbox{} + 2\left((1-z^2)(1-y)-(1+y)x^2\right)\Bigl\{2y(1+x^2-z^2)\beta'\\
\nonumber & & \hspace{0.5cm} \mbox{} + x(\epsilon'+12\rho')\Bigr\} \\
\nonumber & & \mbox{} -4\delta\Bigl\{{\hat E}_\nu\left(x^2(1+y)+y(1-z^2)\right)\\
\nonumber & & \hspace{0.5cm} \mbox{} + y(1+x^2-z^2)\left(x^2(1+y)-(1-z^2)(1-y)\right) -{\hat E}_\nu {\hat q}^2\Bigr\}\\
\nonumber & & \mbox{} + 16\gamma \Bigl\{2{\hat E}_\nu\left(x^2(1+y)+y(1-z^2)\right) + \left(2y(1-y)(1-z^2)^2\right. \\
\nonumber & & \hspace{0.5cm} \left. \mbox{} + \left(1+z^2 -4(1-z^2)y^2\right)x^2-2y(1+y)x^4\right)\\
\nonumber & & \hspace{0.5cm}\mbox{} -{\hat q}^2(1+x^2+z^2+ 2{\hat E}_\nu)-{\hat q}^4\Bigr\}\\
& & \mbox{}+2{\hat E}_\nu xz(\epsilon-12\rho)\biggr].\end{aligned}$$ Integration of these expressions over $E_\nu$ is trivial. Performing the $q^2$ integral over its physically allowed region gives the lepton spectrum $${d\Gamma\over dE_\ell} = \left({d\Gamma\over dE_\ell}\right)_{\mbox{\tiny (SM)}} + \left({d\Gamma\over dE_\ell}\right)_{(1)} + \left({d\Gamma\over dE_\ell}\right)_{(2)},$$ where $$\begin{aligned}
\label{eq:lep1}
\nonumber \left({d\Gamma\over dE_\ell}\right)_{(1)} &=& {{G_F^2 \left|V_{cb}\right|^2 m_b^4}\over 48\pi^3} F(x,y,z)\biggl[G_\beta (x,y,z)\beta_0\\
\nonumber & & \mbox{} - 3z\left(1-(1-z^2)y + (1-y)x^2\right) \left((1-z^2)y-(2-y)x^2\right)\beta''_0 \\
\nonumber & & \mbox{} + \frac{3}{2}xz\left(2-y(1-z^2)-yx^2\right)\left(1-(1-z^2)y + (1-y)x^2\right)(\epsilon_0 - 12\rho_0)\\
& & \mbox{} + 3x\left(1-(1-z^2)y + (1-y)x^2\right)^2\epsilon'_0 \biggr]\end{aligned}$$ is the $q^2$ integral of Eq. (\[eq:diff1\]) and $$\begin{aligned}
\label{eq:lep2}
\nonumber \left({d\Gamma\over dE_\ell}\right)_{(2)} &=& {{G_F^2 \left|V_{cb}\right|^2 m_b^4}\over 48\pi^3} F(x,y,z)\biggl[G_\beta (x,y,z)(\beta+\frac{1}{4}\alpha)\\
\nonumber & & \mbox{} + \frac{3}{2}z\left(1-(1-z^2)y + (1-y)x^2\right)\left((1-z^2)y-(2-y)x^2\right)(\alpha'-2\beta'')\\
\nonumber & & \mbox{} + 3\left(1-(1-z^2)y + (1-y)x^2\right)^2(2y(1-z^2 + x^2)\beta'+x\epsilon'+12x\rho')\\
\nonumber & & \mbox{} +\frac{3}{2}xz\left(2-y(1-z^2) - yx^2\right)\left(1-(1-z^2)y + (1-y)x^2\right)(\epsilon - 12\rho)\\
& & \mbox{} + G_\delta(x,y,z)\delta + 4G_\gamma(x,y,z)\gamma\biggr] \end{aligned}$$ is that of Eq. (\[eq:diff2\]). In these last two expressions, the coefficient functions are defined by: $$\begin{aligned}
F(x,y,z) &=& (1-y)^2 (1+ x^2 - z^2)^2 \sqrt{y^2(1-z^2+x^2)^2 - 4x^2}\over [1-(1-z^2)y + (1-y)x^2]^3,\\
\nonumber G_\beta (x,y,z) &=& y(1-z^2)\left\{3(1+z^2) -(5-4z^2-z^4)y + 2y^2(1-z^2)^2\right\} \\
\nonumber & & - \left\{4(1+2z^2)-(13-4z^2-3z^4)y + 3(5-6z^2+z^4)y^2\right.\\
\nonumber & & \hspace{0.5cm}\left. -6(1-z^2)^2 y^3\right\} x^2\\
\nonumber & & - \left\{4-13y +3(5-3z^2)-6(1-z^2)y^3\right\}x^4\\
& & +y\left\{3-5y+2y^2\right\}x^6,\\
\nonumber G_{\delta}(x,y,z) &=& y (1-z^2)^2 \left\{3-(7-z^2)y+4(1-z^2)y^2\right\} \\
\nonumber & & + \left\{4+8z^2+(5-8z^2+3z^4)y-3(7-10z^2+3z^4)y^2\right.\\
\nonumber & & \hspace{0.5cm}\left.+ 12(1-z^2)^2y^3\right\}x^2\\
\nonumber & & + \left\{4+(5-6z^2)y-3(7-5z^2)y^2+12(1-z^2)y^3\right\}x^4\\
& & +y\left\{3-7y+4y^2\right\}x^6,\\
\nonumber G_\gamma(x,y,z) &=& y(1-z^2) \left\{3(5+z^2)-(29-28z^2-z^4)y+14(1-z^2)^2 y^2\right\}\\
\nonumber & & - \left\{4(1+2z^2)-(49-28z^2-3z^4)y+3(29-38z^2+9z^4)y^2\right.\\
\nonumber & & \left.\hspace{0.5cm} -42(1-z^2)^2 y^3\right\}x^2\\
\nonumber & & - \left\{4-(49-12z^2)y + 3(29-19z^2)y^2-42(1-z^2)y^3\right\}x^4\\
& & +y\left\{15-29y+14y^2\right\}x^6.\end{aligned}$$
Discussion {#sec:disc}
==========
Any effects of physics beyond the Standard Model should become more apparent with the improved data from the next generation $B$ factories, some of which are scheduled to go on line in the near future. Therefore, a detailed extraction of our parametrization of new physics should await these results. For completeness, however, we will use the formulas derived in the previous sections together with the existing data to obtain bounds on some of the parameters of Section \[sec:def\]. Our analysis is only meant to be illustrative and therefore should not be taken too seriously.
One place to look for constraints on the parametrization of Section \[sec:def\] is in the extraction of $|V_{cb}|$ from the exclusive decays ${\bar B}\rightarrow D^{(*)}e{\bar \nu}_e$. Because in the zero recoil limit ($w\rightarrow 1$), the ${\mathcal O}(1/m_Q)$ HQET corrections to ${\bar B}\rightarrow D^{(*)}\ell{\bar \nu}_\ell$ vanish [@luke], and because $\xi(1)=1$ [@isgur], these observables provide a theoretically clean way of extracting the CKM matrix element $|V_{cb}|$ from experiment [@athanas; @barish]. In the abscence of new physics, we expect the value of $|V_{cb}|$ obtained from ${\bar B}\rightarrow D e{\bar \nu}_e$ to agree with that from the decay ${\bar B}\rightarrow D^* e{\bar \nu}_e$. Therefore, a discrepancy between the two measurements of $|V_{cb}|$ could be used to put constraints on the parametrization of Section \[sec:def\].
Integrating Eq. (\[eq:D\]) and Eq. (\[eq:D\*\]) over $\cos\theta$, and making a change of variables from $q^2$ to $w$, the exclusive decay rates can be written in the form [@neubert] $$\begin{aligned}
\nonumber {d\Gamma\over dw}({\bar B}\rightarrow D\ell{\bar \nu}_\ell) = {G_F^2 |V_{cb}|^2 m_B^5\over 48\pi^3} (w^2-1)^{3/2} r^3 (1+r)^2 {\mathcal F}_D(w)^2,\\
\nonumber {d\Gamma\over dw}({\bar B}\rightarrow D^*\ell{\bar \nu}_\ell) = {G_F^2 |V_{cb}|^2 m_B^5\over 48\pi^3} (w^2-1)^{1/2}(w+1){r^*}^3 (1-r^*)^2\\
\times \left[1+{4w\over 1+w}{1-2r^*+{r^*}^2 \over (1-r^*)^2}\right] {\mathcal F}_{D^*}(w)^2,\end{aligned}$$ where $r=m_D/m_B$ and $r^*=m_{D^*}/m_B$. From the results of Section \[sec:D\] and Section \[sec:D\*\], we find $$\begin{aligned}
\nonumber {\mathcal F}_D (w)^2 &=& {{\mathcal F}^{\mbox{\tiny (SM)}}_D (w)}^2 + (\beta_0 +\beta_0'')\xi(w)^2 + \biggl[\beta+\beta'+\beta''\\
\label{eq:FD}& & \mbox{} + {3(\alpha+\alpha')\over 2(1+r)^2}{w+1\over w-1}(1-2rw+r^2) +{8\gamma\over (1+r)^2}(1-2rw+r^2)\biggr]\xi(w)^2,\\
\nonumber {\mathcal F}_{D^*} (w)^2 &=& {{\mathcal F}^{\mbox{\tiny (SM)}}_{D^*} (w)}^2 + \left(\beta_0-{{(5+w){r^*}^2 -2 (1+5w)r^* + 5 +w}\over (1+5w){r^*}^2 -2(1+w+4w^2)r^* + 1 + 5w}\beta_0''\right)\xi(w)^2\\
\nonumber & & \mbox{} + \biggl[\beta + \beta' + \biggl\{{3\over 2}(w-1)(1-2wr^* +{r^*}^2)(\alpha-\alpha')\\
\nonumber & & \hspace{0.5cm}\mbox{}-\left((5+w){r^*}^2 -2 (1+5w)r^* + 5 +w\right)\beta''\\
\nonumber& & \hspace{0.5cm}\mbox{}+8\left((1+5w){r^*}^2-2(4+w+w^2)r^* + 1+5w\right)\gamma\biggr\}\\
\label{eq:FD*}& & \times\left\{(1+5w){r^*}^2 -2(1+w+4w^2)r^* + 1 + 5w\right\}^{-1}\biggr]\xi(w)^2.\end{aligned}$$ Note that although the decay rate for ${\bar B}\rightarrow D e{\bar \nu}_e$ is well-behaved at $w=1$, the quantity ${\mathcal F}_D (w)$ actually has a pole there. This is only because the scalar contribution to the decay rate (the terms involving $\alpha$ and $\alpha'$) vanishes more slowly than $(w^2-1)^{3/2}$ as $w\rightarrow 1$. In fact, the observation of this behavior as $w\rightarrow 1$ in the experimental data for ${\mathcal F}_D (w)$ could be seen as evidence for the possibility of scalar contributions to the process $b\rightarrow c\ell{\bar \nu}_\ell.$
We expect the second order parameters in Eq. (\[eq:FD\]) and Eq. (\[eq:FD\*\]) to be suppressed by a factor of $(m_W/M)^2$ in comparison to the interference terms. Ignoring their contribution, we find in the zero recoil limit: $$\label{eq:ratio}
{{\mathcal F}_D (1)^2 \over {\mathcal F}_{D^*}(1)^2} \simeq\left({{{\mathcal F}^{\mbox{\tiny (SM)}}_D (1)}^2 + \beta_0 +\beta_0''\over {{\mathcal F}^{\mbox{\tiny (SM)}}_{D^*} (1)}^2 + \beta_0 - \beta_0''}\right).$$ The CLEO Collaboration has used data on the exclusive decay ${\bar B}\rightarrow De{\bar \nu}_e$ at zero recoil to extract $|V_{cb}|{\mathcal F}_D (1)=0.0337\pm 0.044\pm 0.048^{+0.0053}_{-0.0012}$ [@athanas]. From the data on ${\bar B}\rightarrow D^*e{\bar \nu}_e$, it has also found $|V_{cb}|{\mathcal F}_{D^*} (1)=0.0351\pm 0.0019\pm 0.0018\pm 0.0008$ [@barish]. If we neglect ${\mathcal O}(1/m_Q^2)$ corrections, ${{\mathcal F}_D^{\mbox{\tiny (SM)}}}(1)={{\mathcal F}_{D^*}^{\mbox{\tiny (SM)}}}(1)=1$ (the $1/m_Q$ corrections vanish automatically by Luke’s theorem [@luke]). Eq. (\[eq:ratio\]) then reads $$0.96\simeq {1+\beta_0+\beta_0''\over 1+\beta_0-\beta_0''},$$ which to lowest order in the parameters, gives $\beta_0''\simeq-0.02.$
We can also use the data on the inclusive decays to put constraints on the parameters of Section \[sec:def\]. The OPE for the lepton spectrum does not agree locally with the physical spectrum near the maximum lepton energy. Therefore, the result of Section \[sec:Xc\] can only be compared with experiment by constructing an observable which integrates the OPE result over a sufficiently large region. A suitable observable (introduced in [@gremm2]) for comparison of the OPE with the inclusive data is given by $$R_1 = {{\int_{1.5\mbox{\scriptsize GeV}} E_\ell {d\Gamma\over dE_\ell}dE_\ell}\over \int_{1.5\mbox{\scriptsize GeV}} {d\Gamma\over dE_\ell}dE_\ell},$$ which is independent of the value of $|V_{cb}|$. Using the inclusive $B\rightarrow Xe\bar{\nu}_e$ data from CLEO, the authors of [@gremm2] extracted a central value of $R_1=1.7831\mbox{ GeV}.$ (This includes contributions from $b\rightarrow u e\bar{\nu}_e,$ which introduce an error of only a few percent.) If we split $R_1$ into a piece coming from Standard Model physics alone, and a correction $\delta R_1$ due to new physics, we find $$\label{eq:dR1}
\delta R_1 \simeq {{\int_{1.5\mbox{\scriptsize GeV}} E_\ell {d\Gamma\over dE_\ell}dE_\ell}\over \int_{1.5\mbox{\scriptsize GeV}} \left({d\Gamma\over dE_\ell}\right)_{\mbox{\tiny SM}}dE_\ell}-{{R^{\mbox{\tiny (SM)}}_1\int_{1.5\mbox{\scriptsize GeV}}{d\Gamma\over dE_\ell} dE_\ell}\over \int_{1.5\mbox{\scriptsize GeV}} \left({d\Gamma\over dE_\ell}\right)_{\mbox{\tiny SM}}dE_\ell},$$ where $R^{\mbox{\tiny (SM)}}_1$ is the Standard Model contribution to $R_1.$ For the purposes of this crude analysis, we can replace it by the full $R_1$ with negligible error. Similar reasoning allows us to approximate the Standard Model part of the lepton spectrum by its leading order contribution. Using Eq. (\[eq:lep1\]) and Eq. (\[eq:lep2\]) to do the integrals of Eq. (\[eq:dR1\]), $$\begin{aligned}
\label{eq:dR1n}
\nonumber \delta R_1 &\simeq& -0.001\alpha - 0.002\alpha' -0.005(\beta_0+\beta) -0.036\beta'\\
& & \mbox{} + 0.004(\beta_0''+\beta'')-0.031\delta-0.309\gamma,\end{aligned}$$ where we have used the values $m_b=4.8\mbox{ GeV}$ and $m_c=1.4\mbox{ GeV}$ [@gremm2]. Note, however, that the magnitude of the coefficients in Eq. (\[eq:dR1n\]) is rather sensitive to the particular values of $m_b$ and $z=m_c/m_b$. Therefore, the bounds on new physics that are obtained from this equation will be highly dependent on the numerical values of the $b$ and $c$ quark masses that are used to evaluate them.
In [@gremm2], the Standard Model ${\mathcal O}(1/m_b^3)$ corrections to the theoretical value of $R_1$ are estimated to be $$\begin{aligned}
\label{eq:dR1t}
\delta R_1 &=& -(0.4{\bar\Lambda}^3 + 5.5{\bar\Lambda}\lambda_1 + 6.8{\bar\Lambda}\lambda_2 + 7.7\rho_1)/{\bar m}_B^3,\end{aligned}$$ where ${\bar m}_B=(m_B+3 m_{B^*})/4$ is the spin-averaged $B$ meson mass, which we take to be ${\bar m}_B=5.31\mbox{ GeV}$ [@pdb]. Assume that the first three terms of this equation are fixed by the values ${\bar \Lambda}=0.39\mbox{ GeV},$ $\lambda_1=-0.19\mbox{ GeV}^2,$ $\lambda_2=0.12\mbox{ GeV}^2$ of [@gremm2]. The term involving $\rho_1$ has larger uncertainty. Varying it between $\rho_1=0$ and the estimated value $\rho_1\simeq (300\mbox{ MeV})^3$ of [@gremm2] gives $$\label{eq:range}
-8.5\times 10^{-4}\mbox{ GeV}\leq \delta R_1 \leq 5.3\times 10^{-4}\mbox{ GeV}.$$
Without a better knowledge of $\rho_1$, a contribution to $R_1$ from new physics in the range of Eq. (\[eq:range\]) cannot be excluded, even if the other HQET parameters were known to arbitrary precision. Therefore, assuming that the corrections to the theoretical value of $R_1$ from new physics also lie in this range, we can derive constraints on our parameters by comparing Eq. (\[eq:range\]) with Eq. (\[eq:dR1\]). We will do so by choosing only one parameter from Eq. (\[eq:dR1\]) to be non-zero at a time. Then the bounds are derived by taking the ratio of Eq. (\[eq:range\]) to the coefficient of that parameter in Eq. (\[eq:dR1\]). Of course, this is not entirely satisfactory, since some of these parameters are not independent of each other. For instance, setting $\alpha=0$ forces $\alpha'$ to vanish as well. However, this procedure should be sufficient if all we are interested in is a rough numerical estimate. Thus, given the uncertainty on $\rho_1$, we cannot rule out the following range for the scalar terms: $$\begin{aligned}
0 \leq \alpha \leq 0.72; & -0.26 \leq\alpha' \leq 0.41,\end{aligned}$$ Likewise, for the vector parameters $$\begin{aligned}
\nonumber -0.11\leq\beta_0 \leq0.18; & |\beta|<|\beta_0|,\\
\nonumber \begin{array}{c}0\leq \beta' \leq 0.023,\end{array}\\
-0.20\leq\beta_0'' \leq 0.13 ;& |\beta''|<|\beta_0''|,\end{aligned}$$ and finally $-0.017\leq\delta \leq 0.027$ and $0\leq\gamma\leq 0.003$. Note in particular that the bounds on $\beta_0''$ are consistent with the value derived from the exclusive data.
I would like to thank Mark Wise for guidance throughout the completion of this work. Also, I would like to thank Zoltan Ligeti for helpful comments on the manuscript. This work was supported in part by the Department of Energy under grant number DE-FG03-92-ER 40701.
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[^1]: walter@theory.caltech.edu
|
---
author:
- Steffen Mieske
- Michael Hilker
- Leopoldo Infante
date: 'Received 30 October 2001 / Accepted 19 December 2001'
title: 'Ultra Compact Objects in the Fornax Cluster of Galaxies: Globular clusters or dwarf galaxies?'
---
Introduction
============
Magnitude - surface brightness relation of early type dwarf galaxies
--------------------------------------------------------------------
The faint end of the galaxy luminosity function is mainly populated by dwarf elliptical galaxies (dEs) and dwarf spheroidals (dSphs, the faintest dEs in the Local Group). These galaxies are the most numerous type of galaxies in the nearby universe, having absolute magnitudes fainter than $M_{V}$ $\simeq$ $-$17 mag. They follow a tight magnitude-surface brightness relation in the sense that central surface brightness increases with increasing luminosity (Ferguson & Sandage [@FergSan88], [@FergSan89]). The validity of this relation has been a subject of lively debate over the last decade. A number of authors have argued against the existence of a magnitude-surface brightness relation for dEs (Davies et al. [@Davies88], Phillipps et al. [@Philli88], Irwin et al. [@Irwin90]) and questioned the cluster membership assignement to dEs based on morphology.\
Recently, spectroscopic membership confirmation has shed light into this matter. Drinkwater et al. ([@Drinkw01a]) do confirm the brightness - surface brightness relation for Fornax dwarfs based on the data of their Fornax Cluster Spectroscopic Survey (FCSS). They obtain spectra for all objects in the central 2 degrees of the Fornax cluster down to a limiting magnitude of $M_V\sim-12.5$ mag and find that the magnitude surface brightness relation for dEs is well defined. Under this scheme, galaxies whose total luminosity and/or surface brightness lie significantly outside this relation are hard to classify. Two of the few examples for such peculiar objects that have been known for a long time are the high surface brightness compact dwarf elliptical (cdE) M32 and the very extended low surface brightness spiral Malin 1 (Bothun et al. [@Bothun87], Impey et al. [@Impey88], Bothun et al. [@Bothun91]).\
New Ultra Compact Objects
-------------------------
Most recently, in the course of the FCSS, Drinkwater et al. ([@Drinkw00a]) detected five ultra compact objects (UCOs, UCDs in their papers) within 30$'$ projected distance from the Fornax cluster’s central galaxy, NGC 1399[^1]. Although as bright as average size dEs, they are by far more compact. Four of the five UCOs have absolute magnitudes of about $M_V = -12$ mag, one is significantly brighter with $M_V = -13.3$ mag. On HST-STIS images, the four fainter UCOs have King profile effective radii between 10 and 17 pc, while the brightest one has about 50 pc (Drinkwater et al. [@Drinkw01b]). In Table \[ucocor\], the properties of the UCOs are summarized. All of them are significantly brighter than the brightest galactic globular cluster ($\omega$ Centauri has $M_V=-10.2$ mag) but significantly fainter than M 32 ($M_V=-16$ mag). Neither Hilker et al. ([@Hilker99]), Drinkwater et al. ([@Drinkw00a], [@Drinkw01b]) nor Phillipps et al. ([@Philli01]) could draw definite conclusions about the nature of the UCOs.\
Name $\alpha$ (2000) $\delta$ (2000) $V$ \[mag\] $M_V$ \[mag\] $(V-I)$ \[mag\] $r_{\rm eff}$ \[pc\] $d$(NGC 1399) \[$'$\]
----------- ----------------- ----------------- ------------- --------------- ----------------- ---------------------- -----------------------
UCO 1 03:37:03.30 -35:38:04.6 19.31 -11.99 1.17 12 20.74
UCO 2 03:38:06.33 -35:28:58.8 19.23 -12.07 1.14 15 5.10
UCO 3$^*$ 03:38:54.10 -35:33:33.6 18.06 -13.24 1.20 50 8.26
UCO 4$^*$ 03:39:35.95 -35:28:24.5 19.12 -12.18 1.14 17 13.64
UCO 5 03:39:52.58 -35:04:24.1 19.50 -11.80 1.04 10 28.26
One possibility is that UCOs are bright globular clusters. NGC 1399 has a very rich globular cluster system with about 6000 GCs within 10$'$ (about 40 kpc at Fornax distance) from its center (Kohle et al. [@Kohle96], Forbes et al. [@Forbes98]) which may contain GCs as bright as the UCOs. Another possibility is that UCOs are the nuclei of stripped dwarf galaxies. Threshing dE,Ns in the cluster potential has been shown to work (Bekki et al. [@Bekki01]). Lotz et al. ([@Lotz01]) find that the luminosity function of 27 Fornax and Virgo dwarf nuclei peaks at $V=21.7$ mag ($M_{V}=-9.6$ mag) with a dispersion of $\sigma=1.2$ mag, whereas the Globular Cluster Luminosity Function (GCLF) peaks at about $M_V=-7.4$ mag. So, nuclei are on average more than 2 mag brighter than GCs and might mix up with the bright tail of the GCLF. Or do the UCOs represent a new group of ultra compact dwarf galaxies, extreme cases of M 32? No discriminating statement could be made until now.\
It would be very interesting to know whether the UCOs had an origin different from the population of globular clusters of NGC 1399.\
The UCOs have magnitudes roughly equal to the completeness magnitude limit of the FCSS ($V=19$ mag or $M_V=-11.7$ mag). This is about 4.5 magnitudes brighter than the turnover of the globular cluster luminosity function (Kohle et al. [@Kohle96]). In Fig. \[introlf\], the three relevant luminosity functions are shown in one plot: the LF of all observed sources in Drinkwater et al.’s FCSS (to indicate its completeness limit); the LF of the UCOs; and the bright end of NGC 1399’s GCLF, respresented by a Gaussian with $V_{\rm to}=23.9$ mag and $\sigma=1.2$ mag, as taken from Kohle et al.’s GCLF. A total number of 8100 GCs was adopted, which is the number contained within 20$'$ from NGC 1399 (see Sect. \[totalnum\]).\
\
Aim of this paper
-----------------
As one can see in Fig. \[introlf\], the magnitude range between the UCOs and the bright globular clusters must be probed more thoroughly, both to know whether UCOs extend to fainter magnitudes and to determine the bright end of the GCLF. A gap in magnitude space between both populations would imply that the UCOs are very compact dEs or nuclei of stripped dwarfs rather than globular clusters.\
In this paper, we describe and analyse a survey of compact objects in the central region of the Fornax cluster of galaxies that closes the magnitude gap between the FCSS and the globular cluster regime.\
In Sect. \[Selcand\] we describe the selection of candidates for the survey. In Sect. \[obsdatared\] the observations and data reduction are described. Sect. \[measureradvel\] shows the radial velocity measurement. In Sect. \[analysis\] the results are analyzed, the question whether or not the UCOs are a distinct population is discussed and metallicities of a number of Fornax dE,Ns are measured. These results are discussed in Sect. \[discussion\]. Finally, in Sect. \[summary\] a summary and conclusions are presented.\
Selection of candidates {#Selcand}
=======================
To choose candidates for our spectroscopic survey, we analyzed existing wide field images of the Fornax cluster. Those had been obtained over four nights in December of 1999 with the 2.5m Du Pont Telescope at Las Campanas Observatory, Chile. Their field of view was 25$'$ $\times$ 25$'$; 14 fields were observed. These fields map a circular region of 2 degrees diameter around the central giant elliptical galaxy, NGC 1399. The seeing ranged from 1.5$''$ to 2$''$, which corresponds to a spatial resolution of $\approx$ 120 - 160 pc at the distance of the Fornax cluster, assumed to be 18 Mpc (Kohle et al. 1996). The pixel scale was 0.774$''$/pixel. Images in $V$ and $I$ bandpasses were obtained for all fields. One of the central fields was also observed in $B$. Details of the photometry and of the calibrations will be presented in an accompanying paper (Hilker et al. [@Hilker02], in prep). All object magnitudes mentioned in this work are taken from that paper.\
Unresolved objects
------------------
In turn, we describe the [*compact*]{} objects selection criteria for spectroscopy. These criteria are based on object brightness, colour and morphology.\
The faint magnitude limit was set by analyzing the GCLF of NGC 1399, as determined by Kohle et al. ([@Kohle96]). They adopt a Gaussian distribution for the GCLF and estimate a turn over magnitude of $V_{\rm to}$=23.9 mag ($M_V=-7.4$ mag) and a dispersion of $\sigma$=1.2 mag. Since one of our goals was to include bright globular clusters (GCs), we defined $V_{\rm faint}$=21 mag as the faint magnitude limit for our observations, which is 2.4 $\sigma$ brighter than the turn over. This means that about 0.8% of all globular clusters around NGC 1399 would be accessible to our survey. Adopting 6000 as the total number (Kohle et al. [@Kohle96]), then about 50 GCs would enter in our survey. The bright magnitude limit is given by the FCSS, which covers basically all objects down to $V$ $\approx$ 19 mag.\
Apart from pure brightness limits, other restrictions had to be made. Four of the five UCOs are unresolved on our CCD images due to their small diameter of about 0.2$''$ or 15 pc at Fornax distance (Drinkwater et al. [@Drinkw01b]). We did therefore not expect compact Fornax members in the magnitude regime fainter than the UCOs to be resolved on our images. Thus, unresolved sources were interesting in the first place. A source on our images was defined as unresolved, when SExtractors star-classifier value (0 for a “perfect” galaxy, 1 for a “perfect” star, see Bertin & Arnouts [@Bertin96]) was larger than 0.45 in both $V$ and $I$. It was defined as resolved when this was not the case.\
Of the unresolved objects in the mentioned magnitude regime, only those which had $(V-I)<1.5$ mag were accepted as candidates. The value 1.5 comes from the following consideration: we wanted to include all possible kinds of galactic nuclei or bright globular cluster type objects, but exclude background galaxies at high redshift or very red giants in the foreground. According to Worthey’s ([@Worthe94]) stellar population models, for a very old population of 15 Gyrs with $[\frac{Fe}{H}]=0$ one gets $(V-I)$=1.33 mag. Taking into account model uncertainties, photometric errors - which were smaller than 0.05 mag - and reddening by dust (Schlegel et al. ([@Schleg98]) found E(B-V) = 0.013 towards Fornax), $(V-I)$=1.5 mag was chosen as the red colour limit. To avoid missing very young star forming nuclei, no blue limit was applied.\
Fig. \[cmdseldw2\] shows a colour magnitude diagram for one of the selected fields, with the selected candidates for the survey and the objects marked.\
The only UCO included in our survey is UCO 2. The other ones lie outside the fields we covered (see Sect. \[analysis\]).\
Resolved objects
----------------
The only UCO resolved on our images (UCO 3) is about 1 mag brighter than the magnitude completeness limit of the FCSS. Yet, it cannot be ruled out a priori that there are compact, but resolved Fornax members fainter than the completeness limit of the FCSS. Therefore we selected in addition to the unresolved objects the smallest resolved sources in the same magnitude colour range like the unresolved objects. They were given lower priorities in the mask creation process than the unresolved ones.\
Apart from compact objects, we included the dE,Ns cataloged in Fergusons’s Fornax Cluster Catalog (FCC) (Ferguson [@Fergus89]) that were bright enough and in the field we wanted to observe. This made it possible to compare the spectral properties of these nucleus-dominated spectra with those of the UCOs.\
Observations and data reduction {#obsdatared}
===============================
The observations were performed in the three nights of 2000/12/30 to 2001/01/01 at the 2.5m Du Pont at Las Campanas. The instrument was the Wide Field CCD (WFCCD) camera, which reimages a 25$'$ field onto a Tek$\#$5-Detector of 2048x2048 pixel with a pixel scale of 0.774$''$/ pixel. Multi-slit masks and the H & K grism, which has a good transmission between 3500 and 6300 [Å]{} and a resolution of about 1.3 [Å]{} per pixel, were used. As the slit width in our observation was $\approx$ 1.5$''$ (= 2 pixel), the effective resolution was in the order of 2.5 - 3 Å, corresponding to 200 km/s at 4000 Å. Four fields were observed, with integration times of roughly 3 hrs each. In total we obtained spectra of 160 objects (40 per field), of which about 100 were unresolved on our images.\
All the data reduction was performed with the IRAF-packages IMRED and TWODSPEC. After bias subtraction, the cosmic rays were removed. The combined dome-flats were normalized to unity and the sky-flats were divided by the normalized dome-flats. The remaining illumination change along the slit was measured on these sky-flats. The single object exposures were then flat-field divided and corrected for the remaining illumination change. The object exposures were corrected for the tilting caused by the optics of the WFCCD camera and then combined. After these reduction steps, the spectra were extracted.\
Radial velocity measurement {#measureradvel}
===========================
Radial velocities were determined by performing Fourier cross correlation between object and template spectra with the IRAF-task FXCOR.\
Templates
---------
As templates for the radial velocity measurement we used two of our standard star spectra (HD 54810 and HD 22879) and one synthetic spectrum taken from Quintana et al. ([@Quinta96]). These three templates showed the highest correlation peaks when cross correlating them with the four brightest Fornax dE,Ns included in our survey. When measuring the radial velocities with these templates, we accept the result as correct if the confidence level $R$ (the $r$-ratio of Tonry & Davis [@Tonry79], defined as the relative height of the cross-correlation peak with respect to the neighbouring peaks) of the cross correlation peak between the template and the object spectrum is larger than 3.5 for all three templates. This corresponds to a S/N of about 4 between 4500 and 5000 Å.\
Results
-------
In total we obtained 164 spectra in four fields. For 40 of those, the S/N was too low to reliably determine radial velocities. We successfully determined radial velocities for 66 foreground stars, 18 Fornax members and 40 background galaxies. Besides five dE,N candidates and UCO 2, 12 unresolved objects turned out to have a radial velocity between 600 and 2500 km/s, so they are probable Fornax members. We regard the unresolved objects as GC candidates (GCCs) from now on, since they are fainter than the UCOs. They are all within 20$'$ projected distance from NGC 1399. Eight of them have not been measured spectroscopically before, they are newly discovered members of the Fornax cluster. Two of the remaining four objects were known to be cluster members from Hilker (NTT 1998, private communication); the other two from Kissler-Patig et al. ([@Kissle99]). The five dE,Ns and UCO 2 can be confirmed as cluster members. The parameters of all Fornax members included in our survey are given in Table \[fornmem\]. The names of the dE,Ns are from the Fornax Cluster Catalogue (FCC) of Ferguson ([@Fergus89]). The names of our GC candidates are “FCOS Field Number$-$Object number”. “FCOS” stands for “Fornax Compact Object Survey”. The field number is: 1 for the south-east field, 2 for the south-west field and 4 for the north-west field, as shown in Fig. \[cmdcvd\]c). The object number is taken from the mask-creation file. The radial velocities, their errors and the confidence level $R$ were computed by averaging the values given from FXCOR for each of the three templates. A list of all foreground and background sources is given in the Appendix. They are available electronically at http://www.XXX. As the highest object number is 105 (FCOS 2-105, a foreground star), one leading zero is used for originally two-digit numbers and two leading zeroes for originally one-digit numbers, such that all object numbers consist of three digits.\
Name $\alpha$ (2000.0) $\delta$ (2000.0) $v_{\rm rad} [km/s]$ $R$ $V$ $(V-I)$ Type Comment
------------ ------------------- ------------------- ---------------------- ------ ------- --------- ------ -----------------------------------
FCC 222 3:39:13.23 $-$35:22:17.6 825 $\pm$ 25 10.7 14.91 1.11 dE,N 850 $\pm$ 50$^{c}$
FCC 207 3:38:19.42 $-$35:07:44.3 1420 $\pm$ 20 20.3 15.34 1.03 dE,N 1425 $\pm$ 35$^{d}$
FCC 211 3:38:21.65 $-$35:15:35.0 2325 $\pm$ 15 31.0 15.65 1.04 dE,N 2190 $\pm$ 85$^{e}$
FCC B1241 3:38:16.79 $-$35:30:27.0 2115 $\pm$ 25 11.9 16.84 0.87 dE,N 1997 $\pm$ 78$^{b}$
FCC 208 3:38:18.88 $-$35:31:50.8 1720 $\pm$ 50 7.0 17.18 1.06 dE,N 1694 $\pm$ 84$^{c}$
UCO 2 3:38:06.41 $-$35:28:58.2 1245 $\pm$ 15 18.8 19.15 1.13 UCO 1312 $\pm$ 57$^a$
FCOS 1-021 3:38:41.96 $-$35:33:12.9 2010 $\pm$ 40 7.7 19.70 1.18 GCC 1993 $\pm$ 55$^{f}$
FCOS 1-060 3:39:17.66 $-$35:25:30.0 980 $\pm$ 45 8.2 20.19 1.27 GCC
FCOS 1-063 3:38:56.14 $-$35:24:49.1 645 $\pm$ 45 8.1 20.29 1.05 GCC
FCOS 2-073 3:38:11.98 $-$35:39:56.9 1300 $\pm$ 45 4.6 20.40 1.20 GCC $v_{rad}$=1300 [**or**]{} 280
FCOS 1-019 3:38:54.59 $-$35:29:45.8 1680 $\pm$ 35 5.6 20.62 1.01 GCC 1730 $\pm$ 80$^f$
FCOS 1-058 3:38:39.30 $-$35:27:06.4 1610 $\pm$ 40 5.8 20.67 1.04 GCC 1540 $\pm$ 150$^g$
FCOS 2-078 3:37:41.83 $-$35:41:22.2 1025 $\pm$ 60 4.6 20.69 1.21 GCC
FCOS 2-086 3:37:46.77 $-$35:34:41.7 1400 $\pm$ 50 5.74 20.81 0.92 GCC
FCOS 4-049 3:37:43.09 $-$35:22:12.9 1330 $\pm$ 50 5.0 20.85 0.98 GCC
FCOS 2-089 3:38:14.02 $-$35:29:43.0 1235 $\pm$ 45 8.21 20.87 1.08 GCC
FCOS 2-095 3:37:46.55 $-$35:28:04.8 1495 $\pm$ 45 3.9 20.96 1.14 GCC $R$ $<$ limit, 1186 $\pm$ 150$^g$
FCOS 1-064 3:38:49.77 $-$35:23:35.6 900 $\pm$ 85 3.8 20.96 1.21 GCC $R$ $<$ limit
For three of the 12 GC candidates, namely FCOS 1-064, 2-095 and 2-073, the membership assignment is not definite. For objects 1-064 and 2-095, the confidence level $R$ is lower than 3.5 for one of the templates, but larger for the other two. Cross correlation with all three templates yields the same (Fornax-) velocity within the error range. 2-073 has two cross correlation peaks at 1300 km/s and 280 km/s, both with $R$ larger than 3.5 for all templates. The peak at 1300 km/s is 20% more pronounced than the peak at 280 km/s. Object 2-095 is one of the four objects that were known as Fornax members from before (Kissler-Patig et al. [@Kissle99]). This indicates that our limits for $R$ may be too strict. We therefore include all 12 GC candidates in our analysis.\
Analysis
========
In this section a detailed analysis of the objects discovered is presented. From the total number of GCs associated with NGC 1399 (Dirsch et al. [@Dirsch01], Forbes et al. [@Forbes98], Kohle et al. [@Kohle96]), the form of its GCLF (Kohle et al. [@Kohle96]) and the completeness of our survey we will calculate how many GCs should be included in our survey and compare that with the number we found. This will enable us to restrict the form of NGC 1399’s GCLF at the bright end. We also determine whether there is a statistically signifcant gap in magnitude between the UCOs and our GC candidates. Lick line-indices are calculated for UCO 2 (see Table \[fornmem\]) and the dE,Ns included in our survey.\
In Fig. \[cmdcvd\]a–d a colour magnitude diagram, a map of the loci of all successfully observed point sources, a colour velocity and a magnitude velocity diagram are shown. Note that in field 6, which is more than 1 degree away from the central galaxy NGC 1399, no GC candidates were found. In Fig. \[lfpso\] the luminosity function for all our detections and for the GC candidates is given, including the 3 UCOs within 20$'$ from NGC 1399. The 2 UCOs outside 20$'$ that were included in Fig. \[introlf\] have been omitted in Fig. \[lfpso\], because our area coverage outside 20$'$ is basically zero.\
The mean radial velocity of the GC candidates is 1300 $\pm$ 109 km/s with a standard deviation of $\sigma$=377 km/s. Richtler et al. (2001, private communication), who obtained spectroscopy of about 350 GCs around NGC 1399, get 1447 $\pm$ 16 km/s as the mean radial velocity. This is more than 1 $\sigma$ away from our result. However, we can rule out a systematic shift in our velocity values with respect to the measurements of other authors (see Table \[fornmem\]). For the ten objects with known radial velocities, the median of the difference between literature value and our value is -20 km/s $\pm$ 35 km/s. With smaller samples biased to brighter GCs also other authors get smaller mean values: Minniti et al. ([@Minnit98]) found for a sample of 18 GCs around NGC 1399 a mean of 1353 $\pm$ 79 km/s with $\sigma$=338 km/s. Kissler-Patig et al. ([@Kissle99]) suggest with a larger sample of 74 GCs, that the velocity distribution of GCs around 1399 may even be bimodal with two peaks at 1200 and 1900 km/s, respectively.\
The colour magnitude diagram in Fig. \[cmdcvd\]a shows that the GC candidates have colours typical for GCs. Having a mean colour of $(V-I)=1.11$ with a standard deviation of 0.11, they are only slightly redder than the average $(V-I)\simeq 1$ for GCs around NGC 1399.\
Comparing the luminosity function of all unresolved objects in our survey with the GC candidates (see Fig. \[lfpso\]) shows that the latter ones are concentrated towards the faint end of our magnitude regime. Only one of the 12 GC candidates is brighter than $V=20$ mag, although we cover the whole magnitude range between $19<V<21$ mag ($-10.3<M_V<-12.3$ mag). This drop in frequency towards brighter magnitudes should be expected if all of the candidates are GCs. We are probing the regime 3 - 5 magnitudes brighter than the GCLF turnover magnitude ($V_{\rm
to}\simeq -7.4$ mag) in which GC number counts should decrease towards brighter magnitudes. This is of course only a qualitative statement. To draw more quantitative conclusions, two questions need to be answered: can the number of our GC candidates be explained only by bright GCs belonging to NGC 1399’s GCS? and, is there a significant magnitude gap between both populations? The next two subsections deal with these questions.\
The expected number of globular clusters {#numgc}
----------------------------------------
How does the number of our GC candidates compare with the number expected from Kohle et al.’s ([@Kohle96]) GCLF? To find that out, three steps are necessary: First, the total number of GCs associated with NGC 1399 is calculated (Sect. \[totalnum\]). Second, the completeness of our survey is determined (Sections \[photometricincompleteness\] and \[geometricincompleteness\]). Then the total number is multiplied with the completeness and the fraction of GCs brighter than our survey limit (Sect. \[resexpnum\]). As the GC surface density and the completeness vary with radius, the calculations are performed separately for the three rings as shown in Fig. \[cmdcvd\]b.\
### Total number of globular clusters associated with NGC 1399
Our detections reach out as far as almost 20$'$ projected distance. Previous photometric studies of 1399’s GCS, like those of Forbes et al. ([@Forbes98]) or Kohle et al. ([@Kohle96]), reached only 10$'$. To cover the zone between 10-20$'$, the latest results of Dirsch et al. ([@Dirsch01]) were used. Their data was obtained with the CTIO’s MOSAIC camera and map the GCS of NGC 1399 out to a radius of 20$'$. At 20$'$ they still find a GC surface density 2-3 times higher than in a comparison field 3.5 degrees away from the cluster center. Dirsch et al. subdivide the region around NGC 1399 into three rings with limits 0$'$–2.5$'$, 2.5$'$–8$'$ and 8$'$–20$'$. In Fig. \[cmdcvd\][**b**]{} we adopted the ring limits equal to theirs (except for the innermost ring). There is a different slope in the radial density profile in all three rings. It holds in units of arcmin$^{-2}$:\
$$\begin{aligned}
r \epsilon [0;2.5]' : \rho(r) = 10^{1.28} \\
r \epsilon [2.5;8]' : \rho(r) = 10^{0.98}\cdot r^{-0.77} + 10^{1.42}\cdot r^{-1.46}\\
r \epsilon [8;20]' : \rho(r) = 10^{1.46}\cdot r^{-1.41} + 10^{1.27}\cdot r^{-1.50}\end{aligned}$$ The numbers of Dirsch et al. ([@Dirsch01]) are not corrected for incompleteness. In order to estimate the fraction of objects missing, we integrated their number counts within 10$'$, yielding 1545 $\pm$ 150 GCs, and divided this by the total number of GCs within that region derived from previous surveys. We adopted the mean of the two values of Kohle et al. ([@Kohle96]) and Forbes et al. ([@Forbes98]), who get 5940 $\pm$ 570 and 5700 $\pm$ 500, respectively. The ratio is then 0.27 $\pm$ 0.03.\
With that information at hand, the number of GCs within each of the three rings, as plotted in Fig. \[cmdcvd\][**b**]{}, is calculated by integrating the surface density law within the rings and dividing the numbers by 0.27. The GC candidate closest to NGC 1399 is only 2$'$ away, so we integrated Dirsch et al.’s values from 2$'$ to 8$'$ instead of 2.5$'$ to 8$'$. The results are: 3800 $\pm$ 460 for ring 1 (2$'$–8$'$), 2340 $\pm$ 280 for ring 2 (8$'$–14$'$) and 1960 $\pm$ 230 for ring 3 (14$'$–20$'$).\[totalnum\]\
### Photometric completeness {#photometricincompleteness}
To determine the incompleteness involved in the source extraction on the photometric images, artificial star experiments were performed. With the IRAF task MKOBJECTS in the ARTDATA package 5000 artificial stars with Gaussian profiles were put into the $V$ and $I$ images of the three fields where the new members were found. Their magnitude ranged between $21.5>V>19$ mag and their colour indices between $2>(V-I)>0$ mag. Poisson noise was negligible compared to the stars’ signal and was therefore not included. Magnitudes, colours and positions in the frame were randomly created using a C code.\
To gain a statistically significant result without altering the crowding properties in our images, we added 50 times 100 artificial stars to our observed images. Then SExtractor was run on each of them, using the same parameters as on the original extraction. The photometric selection criteria for point sources as defined in Sect. \[Selcand\] were then applied to the SExtractor output catalog.\
The ratio between the number of artificial stars that match the selection criteria and the number of input artificial stars in that magnitude-colour-range defines the completeness in the photometric detection. For the objects in the magnitude-colour range of the selected candidates ($19.5<V<21$ mag and $0.4<(V-I)<1.5$ mag, see Fig. \[cmdseldw2\]) the result is 0.79 $\pm$ 0.07. There is no significant magnitude dependence of the photometric completeness in the given magnitude regime. The differences between input and output magnitudes were on average smaller than 0.07 mag, with a standard deviation of $\simeq$ 0.1 mag. For all object magnitudes mentioned in this paper, the internal error is therefore on the order of 0.05 mag and for the colours it is about 0.07 mag.\
### Geometric completeness {#geometricincompleteness}
The geometric completeness is obtained by dividing the surface density $n_{\rm obs}$ of successfully[^2] observed point objects by the surface density $n_{\rm sel}$ of point objects that satisfy our selection criteria and were not included in the FCSS:\
Geometric completeness = $n_{\rm obs}/n_{\rm sel}$.\
$n_{\rm sel}$ is distance independent. It was determined by counting the number $N_{\rm sel}$ of objects satisfying our selection criteria and not included in the FCSS in the central 20$'$ of the CCD frames of field 1, 2 and 4. The area covered is $\pi * 10^2=314.2$ arcmin$^2$, and therefore the resulting surface density is $n_{\rm sel}=\frac{N_{\rm
sel}}{314.2'^2}$. The numbers for both $N_{\rm sel}$ and $n_{\rm sel}$ are given in Table \[nselect\]. The mean of $n_{\rm sel}$ is 0.197 $\pm$ 0.022 arcmin$^{-2}$.\
Field-Nr. $N_{\rm sel}$ $n_{\rm sel}$ \[arcmin$^{-2}$\]
----------- --------------- ---------------------------------
1 52 0.166
2 74 0.235
4 59 0.188
62 $\pm$ 7 0.197 $\pm$ 0.022
: \[nselect\]Total number $N_{\rm sel}$ and surface density $n_{\rm sel}$ of objects that satisfy our selection criteria and are in the central 20$'$ of the three fields where new members were found.
[ccccc]{} Ring \[$'$\]& $N_{\rm obs}$&$n_{\rm obs}$ \[arcmin$^{-2}$\] & $C_{\rm 0}=\frac{n_{\rm obs}}{n_{\rm sel}}\cdot0.79$\
2–8 & 13&0.069 $\pm$ 0.023 & 0.28 $\pm$ 0.08\
8–14 & 34&0.082 $\pm$ 0.014 & 0.33 $\pm$ 0.07\
14–20 & 19&0.030 $\pm$ 0.007 & 0.12 $\pm$ 0.03\
2–20 & 66&0.053 $\pm$ 0.007 & 0.21 $\pm$ 0.04\
To get $n_{\rm obs}$ for the three rings, we counted the number $N_{\rm obs}$ of point sources within each one and divided this value by the respective area: $n_{\rm obs}=\frac{N_{\rm obs}}{Area}$. These numbers, together with the magnitude independent geometric completeness $\frac{n_{\rm obs}}{n_{\rm sel}}$, are given in Table \[nobserv\]. Multiplying photometric and magnitude independent geometric completeness yields the total magnitude independent completeness $C_{\rm 0}$ of our survey, ranging between 12 and 33%. The exact values for the three rings are summarized in Table \[nobserv\].\
Up to now, the magnitude dependence of the geometric completeness has not been considered; bright selected objects get higher priorities in the mask creation than faint ones, and the faint sources more likely have too low S/N. To take this into account, we subdivide the magnitude regime $19<V<21$ in two bins of 1 mag width. For $19<V<20$ mag the geometric completeness is 1.18$\cdot
C_{\rm 0}$, for $20<V<21$ mag it is 0.92$\cdot C_{\rm 0}$.\
### Observed number of Globular Clusters {#resexpnum}
As result of our radial velocity measurements we found 6 GC candidates inside ring 1, 5 in ring 2, and 1 in ring 3. If they are all GCs, the expected number of observed GCs should be close to these values for each ring.\
In the last three subsections, the [*existing*]{} number of GCs in each of the three rings around NGC 1399, the photometric and the geometric completeness were calculated. The number of [*expected*]{} GCs is the number of existing ones multiplied by 0.8$\%$ (the bright end of the LF) and the total completeness. As only one of the 12 GC candidates is brighter than $V=20$ mag, 0.92$\cdot C_{\rm 0}$ is adopted as the completeness (see Sect. \[geometricincompleteness\]). The results of these calculations are given in Table \[nexpected\].\
------------------------------------------------------------------------------------------------
Ring \[$'$\] $N_{\rm exist}\cdot 0.008$ $C_{\rm $N_{\rm exp}$ $N_{\rm obs}$
0}$$\cdot0.92$
-------------- ---------------------------- ------------------- ---------------- ---------------
2–8 30 $\pm$ 6 0.253 $\pm$ 0.082 7.6 $\pm$ 2.8 6
8–14 19 $\pm$ 4 0.305 $\pm$ 0.074 5.8 $\pm$ 2.4 5
14–20 16 $\pm$ 4 0.112 $\pm$ 0.040 1.8 $\pm$ 1.4 1
15.2 $\pm$ 3.9 12
------------------------------------------------------------------------------------------------
As one can see, the expected number of GCs matches well the number of GC candidates in all three rings. $N_{\rm obs}$ is always lower than $N_{\rm exp}$, but the error ranges overlap. In other words, the radial distribution of our GC candidates agrees within the errors with the distribution of the whole GCS. Summing the values up, the total number of expected globulars is between 11 and 19 assuming a Gaussian LF with $V_{\rm to}$=23.9 mag and $\sigma$=1.2, when we observe 12. This is a good agreement, too. It implies that the bright tail of the GCLF is well described by a Gaussian and that the great majority of GC candidates really are GCs.\
There are of course other representation for the GCLF. Kohle et al. ([@Kohle96]) fit a $t_5$-function to their data and find that the best value for $\sigma$ is 1.1 $\pm$ 0.1 mag, with the same turnover than for the Gaussian. Hilker et al. ([@Hilker99]) made estimates of the number of existing GCs in the bright manitude regime where they find 2 of the new UCOs ($V$=18.0 and $V$=19.1 mag, respectively). They adopt both a Gaussian and a $t_5$-function and find that GCs as bright as the brightest UCO can statistically only exist if the bright LF wings are described by a $t_5$-function. The fraction of GCs brighter than $V$=21 mag, when adopting the $t_5$-function as in Kohle et al. becomes 2.3% (1.7% for $\sigma$=1.0 mag). This is almost three times higher than for the Gaussian. The expected number of observed GCs in our survey would become 44 $\pm$ 8, by far higher than our result, even when including the UCOs as possible GCs.\
In Fig. \[lfgausst5\]a, the Gaussian ($\sigma$=1.2 mag) and the $t_5$ functions ($\sigma$=1.1 and $\sigma$=1.0 mag) are plotted together with the two luminosity functions. For the total number of GCs we adopt 8100, which is the sum of the values calculated for the three rings around NGC 1399 (see Sect. \[totalnum\]). The LF of our GC candidates is multiplied by the total completeness as determined in Sect. \[geometricincompleteness\]. Apparently, the LF is fit better by the adopted Gaussian than by the $t_5$ functions, which significantly overestimate the number of bright GCs. The existence of GCs as bright as $V=18$ mag is therefore very unlikely.\
A gap between the UCOs and our GC candidates? {#numcomp}
---------------------------------------------
From Fig. \[lfgausst5\] one can see that the binning independent representation of the joint LF of GCs and UCOs is very well fit by the Gaussian. In the magnitude regime between $19.2$ and $20.2$ mag there is no evidence for a pronounced difference between the Gaussian and the data. We can therefore not confirm a significant gap in magnitude space between our newly found GC candidates and the 2 UCOs. There seems to be a smooth transition between both populations. This is consistent with the faint UCOs and the GC candidates belonging to the same group of objects, namely the brightest GCs of NGC 1399. However, the smooth transition observed by us is only a necessary and not a sufficient condition to link UCOs and GCs. Our data are consistent with a slight enhancement of number counts between $V=19$ and $V=20$ mag due to the presence of a small number of stripped nuclei, too (for further discussion, cf. Sect. \[discussion\]).\
Line indices and metallicities
------------------------------
### Line indices
One of the UCOs (UCO 2) was included in our survey. In this section its line indices are compared with the observed Fornax dE,Ns and the brightest GC candidate (object FCOS 1-021). The S/N of the other GC candidates is too low (4 - 6) to reliably measure line indices. In order to estimate metallicities and ages, we measured 8 line indices as defined by Faber and Burstein ([@Faber73]) and Brodie and Hanes ([@Brodie86]).\
Brodie & Huchra ([@Brodie90]) determine a linear metallicity calibration for 7 of the 8 indices we used (not for Fe53), based on the Zinn & West ([@Zinn84]) scale for Galactic Globular Cluster metallicities: \[$\frac{Fe}{H}$\]=$a\cdot I+b$. Here, $I$ is the line index, $a$ and $b$ are the coefficients of the linear calibration.\
To measure the line indices and their error for each of the confirmed Fornax members we followed closely the reduction procedure of Brodie & Huchra (1990). The object spectra were flux calibrated and all bandpasses where shifted according to the objects’ radial velocity. The statistical error was determined from the original, not flux calibrated spectra.\
In Fig. \[Mg2-Hb-Fe\], the equivalent widths of H$\beta$ (lower panel) and $<Fe>$ (=(Fe52+Fe53)/2) (upper panel) are plotted vs. Mg2. The values for UCO 3 as determined by Hilker et al. ([@Hilker99]) are plotted as well. The red cutoff of object FCOS 1-021’s spectrum is at about 5200 Å, therefore the measurement of Mg2 and $<Fe>$ is not possible for this object. For comparison, evolutionary tracks for different ages, taken from Worthey ([@Worthe94]) are overplotted. The metallicity range is $-$2 $< [\frac{Fe}{H}] <$ 0.5 dex for 17 and 8 Gyr and $-$1.7 $<
[\frac{Fe}{H}] <$ 0.5 dex for 3 and 1.5 Gyr.\
Of the 6 plotted objects, only FCC 207 and FCC 211 (both dE,Ns), have sufficiently high S/N to make a reliable age estimation. They appear to be quite old (age $\geq$ 12 Gyr) with metallicities between $-$1.0 and $-$1.5 dex according to Worthey’s values. One can see that UCO 2 shows a high metallicity in comparison to the other objects (about $-$0.5 in dex), but appears to be old as well, although error bars are large. The metallicity is calculated more accurately in the next section. UCO 3 (the brightest UCO, value from Hilker et al. [@Hilker99]) appears very old, too. It is even more metal rich than UCO 2.\
In the $<Fe>$ vs. Mg2 plot, five of our six objects and UCO 3 fall nicely into Worthey’s evolutionary tracks, only FCC 222 apparently has a very high Fe abundance compared to the other objects.\
### Metallicities derived from line indices
We calculated the metallicity for each index (except Fe53) using the coefficients defined by Brodie and Huchra ([@Brodie90]) and then determined the weighted average $[\frac{Fe}{H}]_{\rm W}$ of the six different values.\
The results for the seven Fornax members are given in Table \[resmet\]. Only for UCO 2, FCC 207 and FCC 211 and FCC B1241 the weighted means have errors smaller than 0.5 dex. The metallicities of FCC 207 and FCC 211 lie well in the range of metallicities derived spectroscopically for bright Fornax dEs. Examples are Held & Mould (1994) ($-$0.75 to $-$1.45), Brodie & Huchra ([@Brodie91]) ($-$1.11 $\pm$ 0.22) or the most recent study from Rakos et al. ([@Rakos01]) ($-$0.4 to $-$1.6), the latter one based on Strömgren photometry. Held & Mould get $[\frac{Fe}{H}]$=$-$1.19 $\pm$ 0.05 for FCC 207, Rakos et al. get $-$1.50 for both FCC 207 and FCC 211 and obtain $[\frac{Fe}{H}]$=$-$0.9 for FCC 222. Thus, our values are consistent with the findings of other groups.\
Name $[\frac{Fe}{H}]_{\rm W}$ \[dex\]
------------ ----------------------------------
UCO 2 $-$0.57 $\pm$ 0.28
FCC B1241 $-$1.41 $\pm$ 0.37
FCC 208 $-$1.85 $\pm$ 0.74
FCC 207 $-$1.34 $\pm$ 0.41
FCC 211 $-$1.39 $\pm$ 0.24
FCOS 1-021 $-$0.17 $\pm$ 0.74
FCC 222 $-$0.74 $\pm$ 0.68
: \[resmet\]Results for the weighted mean $[\frac{Fe}{H}]_{\rm W}$ derived from 6 different line indices (see text).
UCO 2 is metal rich compared to average dE,Ns. For NGC 1399, the average GC metallicity is about $-$1.0 $\pm$ 0.2 dex (Brodie & Huchra [@Brodie91], Kissler-Patig et al. [@Kissle98]), but the metallicity distribution is bimodal with peaks at about $-$1.5 and $-$0.5 dex (Kissler-Patig et al. [@Kissle98]). Thus, UCO 2 is more metal rich than the mean, but could belong to the metal rich GC population. Hilker et al. ([@Hilker99]) find that UCO 3 is with $[\frac{Fe}{H}]$ $\simeq$ 0 significantly metal richer than the dE,Ns, while UCO 4 shows similar line index values like the dE,Ns, however with very large errors.\
The most compact known dE M 32 is known to have a metallicity close to the solar one (e.g. del Burgo et al. [@delBur01]). Thus, the relatively high metallicities of UCO 2 and 3 may not be too surprising, if it is an intrinsic property of these very compact objects. The nucleus dominated spectrum of FCC 222 shows a high metallicity (with large error) as well, so UCO 2 and 3 could be nuclei of entirely stripped dE,Ns like FCC 222, too.\
Discussion
==========
What can we learn about the nature of the UCOs from our data?\
The statistical arguments presented in Sect. \[numgc\] suggest that all our GC candidates can be accounted for by GCs and that there is no significant gap between our GC candidates and the fainter UCOs: the expected number of GCs detected in our survey (15 $\pm$ 4) is very close to the number of our GC candidates (12) when assuming a Gaussian LF for the GCS around NGC 1399. Adopting a LF with more extended wings like a $t_5$-function overestimates the number of bright globulars by far (44 $\pm$ 8). We therefore conclude that the luminosity function at the bright end is significantly better fit by a Gaussian than a $t_5$. This makes the existence of bright GCs with magnitudes like the [*brightest*]{} UCO ($M_{V}=-13.2$ mag) very unlikely.\
The [*fainter*]{} UCOs ($M_{V}\simeq-12$ mag) do not appear as special as they were thought to be, considering the shape of the LF as shown in Fig. \[lfgausst5\]c. It is remarkable in this context that Minniti et al. ([@Minnit98]) already found two GC candidates with $V=19.6$ and $V=19.8$ mag ($M_V$=$-$11.7 and $-$11.5 mag) about 5$'$ away from NGC 1399 in a region not covered by us in this survey. This supports our conclusions that there exists no significant gap between GCs and faint UCOs and that our incompleteness corrections are in the right order. Including the 2 UCOs outside 20$'$ we have confirmed Fornax members at $V$ magnitudes of 19.1, 19.2, 19.3, 19.5, 19.6, 19.7, 19.8, 20.2 and fainter. The distinction between both groups of objects based only on their different brightnesses might not be appropiate. The FCSS (Drinkwater [@Drinkw00a]) was not deep enough to detect the extention of the UCOs to fainter magnitudes. Therefore the detected UCOs seemed to be a separate group of objects. Now we can state that the GCS of NGC 1399 is so rich that it probably contains GCs as bright as the four fainter UCOs. However, this does not exclude that some of them are isolated nuclei of dissolved dE,Ns.\
To link the UCOs to GCs, they not only have to share the same magnitude space, but need to have a mean radial velocity and spatial distribution consistent with the GCS of NGC 1399. Both properties can only be determined very unprecisely for a sample of five objects. Nevertheless we remark that the UCOs’ mean radial velocity of 1530 $\pm$ 110 km/s (Drinkwater et al. [@Drinkw00a]) is consistent with that of the whole GCS (1447 $\pm$ 16 km/s, Richtler et al. 2001 private communication). Drinkwater et al. ([@Drinkw00a]) find that the UCOs are significantly more concentrated towards the cluster center than the Fornax dE,Ns, but have a shallower distribution than the GCS as derived from previous surveys (Grillmair et al. [@Grillm94], Forbes et al. [@Forbes98]). However, the latest results by Dirsch et al. ([@Dirsch01]) show that the GCS of NGC 1399 does extend to well beyond 20$'$ projected distance from the galaxy’s center. The distribution of our 12 GC candidates is consistent with that (cf. Section \[resexpnum\]). Therefore even the furthest UCO at 28.3$'$ projected distance can still be a member of the GCS.\
In the FCSS, Drinkwater et al. ([@Drinkw00a]) found that for all the UCOs, cross correlation yielded higher confidence levels when using K-type stars as templates than younger F-type stars. We can confirm this result in so far as that cross correlation with HD 54810, a K star, gives higher $R$ values than for HD 22879, an F star. For the nucleated dEs included in our survey it is vice versa. Thus, the integrated stellar type of UCO 2 appears to be more similar to those of GCs than to those of dE,Ns.\
If nuclei of entirely stripped dwarf galaxies exist they would be expected to mix up with the bright GCs, because on average they are more than 2 mag brighter than GCs (cf. Introduction). The results obtained by us do not speak against the existence of these “naked” dwarf elliptical nuclei. Depending on the frequency of stripping, a certain fraction of the UCOs and the GC candidates observed by us, could be accounted for by stripped nuclei. We could be observing the sum of the bright tail of the GCLF and the nucleus-LF. Without thorough theoretical treatment of the frequency of entire stripping, this is very hard to quantify, although the nuclei are certainly less frequent than GCs. The number of dE,Ns in the central Fornax region is only in the order of a few dozen (as well in Virgo, see Lotz et al. [@Lotz01]). The timescale for a total stripping is in the order of several Gyr (Bekki et al. [@Bekki01]). So the number of stripped nuclei should be much smaller than the number of GCs, but due to their higher average brightness might become significant at bright magnitudes. A possible effect of the “naked nuclei” would be that the number counts at the very bright end ($V\simeq 19.5$ mag) are enhanced with respect to the extrapolation of the GCLF from the magnitude range between $V=20$ and $V=21$ mag to fainter magnitudes. With our limited statistical sample this cannot be traced. We have only probed about 25% of the area within 20$'$ from NGC 1399. A much better coverage of the inner 20$'$ and extension to a distance of 30$'$ (where the remaining two UCOs at $V\simeq 19.4$ mag are found) is needed to make more definite statements. Although our results together with Minniti et al.’s ([@Minnit98]) detections are indicative for an extension of NGC 1399’s GCS to about $M_{V}=-12$ mag, they have to be confirmed on a more profound statistical basis. Observing time to substantially increase the area coverage of our survey has been approved.\
The brightest UCO (UCO 3) clearly stands out from the others. It is more than 1 mag brighter than the rest and based on our results it cannot be explained by the GCS. UCO 3 thus probably is galaxian, either the compact remnant of a once extended dE,N (“naked nucleus”) or a cdE which was already “born” as compact as it is now, or a non nucleated dE changed by some tidal process to a more compact shape.\
From the metallicity and morphology of the UCOs no conclusive statement can be made: our metallicity value for UCO 2 is quite high (about $-$0.55 in dex), implying a metallicity higher than the majority of Fornax dE,Ns. Still, dE,Ns with similar metallicities have been found (Rakos et al. [@Rakos01]). Of special interest is that M 32 – the most similar dwarf galaxy to the UCOs – has a very high metallicity, about solar (Burgos et al. 2001). As well, UCO 2 could belong to the metal rich GC population of NGC 1399 (Kissler-Patig et al. [@Kissle98]). No discriminating statement can be made from metallicity, and due to the still too low S/N of the spectrum, a reliable age determination is not possible.\
In our images, UCO 3 is the only one which is not classified as a stellar-like object according to the selection criteria given in Sect. \[Selcand\]. Its FWHM is 2.3$''$, when the seeing was 1.85$''$. This corresponds to a diameter of about 170 pc, by far larger than any GC found so far. But, the other UCOs are all unresolved, they have effective radii of about 15 pc (HST-STIS data published in Drinkwater et al. [@Drinkw01b]). None of the UCOs shows low surface brightness features in the outer part away from the central peak, although we can detect features down to $\simeq$ 26.5 mag/arcsec$^2$ in $V$ (Hilker et al. [@Hilker02], in prep.). If the UCOs are remnants of dE,Ns, the stripping has been extremely efficient.\
Summary and conclusions {#summary}
=======================
In this paper, we presented a spectroscopic survey on compact objects in the central region of the Fornax cluster. The aim was to survey the magnitude regime framed by the newly discovered ultra compact objects (UCOs) in Fornax (Drinkwater et al. [@Drinkw00a], Hilker et al. [@Hilker99]) and the brightest GCs around NGC 1399. In the FCSS, performed by Drinkwater et al., the UCOs are at the faint magnitude limit of their survey. We wanted to know whether these detections constitute the bright tail of a continous luminosity distribution with a smooth transition into the GC regime or whether they are a separate population.\
The velocity measurements (cf. Sect. \[measureradvel\]) resulted in the discovery of 12 GC candidates in the magnitude range between $19.70<V<20.95$ mag ($-11.6<M_{V}<-10.35$), located all within 20$'$ from NGC 1399. For four of them, cluster membership was known before. Their mean colour is $<V-I>$ = 1.11 $\pm$ 0.11 mag and their mean radial velocity 1300 $\pm$ 109 km/s with $\sigma$=377 km/s.\
In the subsequent analysis of the discoveries (cf. Sect. \[analysis\]), the following results were obtained:\
1. The expected number of observed GCs originating from NGC 1399’s GCS is 15 $\pm$ 4, in good agreement with the 12 objects found. To calculate the expected number we assumed a Gaussian LF with $V_{\rm
to}$ = 23.9 mag and $\sigma$ = 1.2 mag taken from Kohle et al. ([@Kohle96]), used the GC surface density from Dirsch et al. ([@Dirsch01]) and took into account photometric and geometric incompleteness.\
2. Assuming a more extended $t_5$ distribution for the LF as adopted by Kohle et al. ([@Kohle96]), the expected number of observed GCs rises to 44 $\pm$ 8, which is more than three $\sigma$ higher than the number of observed objects. This implies that the LF has no extended bright wings. Hilker et al. ([@Hilker99]) found that only by assuming such an extended LF the brightest UCO can be explained as a GC.\
3. There is no significant gap in magnitude space between our GC candidates and the four fainter UCOs within 20$'$ of NGC 1399. The GCS of NGC 1399 appears to extent to $M_{V}\simeq -12$ mag.\
4. The only UCO included in our survey is slightly better fit by early stellar types, in contrast to the dE,Ns.\
5. The only UCO included in our survey has a relatively high metallicity compared to the dE,Ns, but is in the range of metal rich GCs or very compact dwarfs with almost solar metallicity like M 32.\
Considering only point 5, we can neither rule out nor confirm that the UCOs are bright GCs. From points 1 to 4 we conclude: the four UCOs fainter than $V=19.1$ mag ($M_{V}=-12.2$ mag) can be well explained by the bright tail of the GCLF of NGC 1399. However, the apparent overlap of the two LFs is not sufficient to exclude the existence of stripped nuclei of formerly extended dE,Ns mixing up with the bright GCs. The faint UCOs are probably no extremely faint examples of cdEs, but are the brightest members of their object class (GCs or stripped nuclei). UCO 3, however, is so bright and large, that it probably is not a GC. It remains the most puzzling object.\
\
We thank our referee Michael Drinkwater for his useful comments which helped to improve the paper. SM acknowledges support by the Heinrich-Hertz-Stiftung of the Ministerium für Bildung und Wissenschaft des Landes Nordrhein-Westfalen. MH and LI acknowlegde support through “Proyecto FONDECYT 3980032” and “8970009”, respectively. LI thanks [*Proyecto Puente PUC*]{}, CONICYT and Pontificia Universidad Católica de Chile for partial funding.\
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Tables of all foreground stars and background galaxies
======================================================
Table \[foreground\] contains the parameters of all foreground stars and Table \[background\] of all background galaxies for which a radial velocity was determined.\
The name given in the first column consists of the acronym FCOS (Fornax Cluster Compact Survey) followed by a sequence number of the field and the object reference number used in the mask creation process. An asterix $^*$ means that the object’s radial velocity could be determined only by identification of emission lines, because cross correlation yielded unreliable results. These were objects that show $one$ strong emission line on top of a faint continuum. In all cases with stronger continuum and clearly identifiable absorption lines present (H&K), this emission line proved to be the redshifted OII line at restframe 3727.3 Å. Therefore, whenever the continuum was faint, we looked for absorption features close to the emission line and calculated their restframe wavelength assuming that the emission line was OII. If the wavelenght were equal to important absorption lines around 3727 [Å]{} such as H8, H9 and H10, we accepted the assumption that the emission line was OII and determined the corresponding redshift. When no reliable identification was possible, no redshift determination was done.\
The second and third column are the right ascension and declination for epoch 2000.\
In the fourth column, the radial velocity with its error is given. Both values were computed by averaging the values given from FXCOR for each of the three templates used for cross correlation. In case of emission line objects with very faint continuum, the error was derived from the uncertainty of the emission line position. For the four quasars discovered, redshift instead of radial velocity is given.\
In the fifth and sixth column, apparent magnitude $V$ and colour $(V-I)$ are given. Both values are from Hilker et al. ([@Hilker02], in prep.).\
Table \[background\] contains an additional comments-column. Here, ELO stands for “Emission Line Object”. Quasars were identified by their extremely broad emission line features.\
Name $\alpha$ (2000.0) $\delta$ (2000.0) $v_{\rm rad} [km/s]$ $V$ $(V-I)$
------------ ------------------- ------------------- ---------------------- ------- ---------
FCOS 6-052 3:34:55.50 $-$35:03:59.6 230 $\pm$ 60 20.70 0.52
FCOS 6-042 3:34:57.65 $-$35:03:23.0 195 $\pm$ 45 20.35 0.62
FCOS 6-050 3:35:01.50 $-$35:07:54.0 170 $\pm$ 25 20.66 1.26
FCOS 6-043 3:35:03.10 $-$35:17:31.9 325 $\pm$ 75 20.36 $-$0.13
FCOS 6-047 3:35:03.87 $-$35:14:01.7 180 $\pm$ 25 20.59 1.07
FCOS 6-041 3:35:05.55 $-$35:10:10.6 120 $\pm$ 25 20.31 1.04
FCOS 6-029 3:35:07.03 $-$35:06:16.8 45 $\pm$ 25 19.46 0.70
FCOS 6-048 3:35:07.30 $-$34:58:13.1 130 $\pm$ 25 20.61 0.87
FCOS 6-038 3:35:09.32 $-$35:18:47.1 175 $\pm$ 35 20.27 0.55
FCOS 6-030 3:35:13.04 $-$35:04:46.3 135 $\pm$ 25 19.49 1.38
FCOS 6-034 3:35:13.99 $-$35:05:02.6 80 $\pm$ 15 19.80 1.11
FCOS 6-027 3:35:17.25 $-$35:20:20.8 240 $\pm$ 35 19.06 0.61
FCOS 6-051 3:35:18.31 $-$34:58:21.7 160 $\pm$ 35 20.70 0.55
FCOS 6-037 3:35:19.47 $-$34:59:37.2 115 $\pm$ 15 20.17 1.44
FCOS 6-039 3:35:22.54 $-$35:07:03.6 75 $\pm$ 15 20.27 0.69
FCOS 6-033 3:35:22.66 $-$35:06:34.3 95 $\pm$ 15 19.76 1.46
FCOS 6-056 3:35:24.22 $-$35:16:50.8 135 $\pm$ 25 20.88 1.12
FCOS 6-066 3:35:25.08 $-$35:21:46.2 $-$35 $\pm$ 15 16.45 1.14
FCOS 6-028 3:35:26.75 $-$35:16:35.8 85 $\pm$ 15 19.2 1.38
FCOS 6-036 3:35:32.04 $-$35:16:01.0 110 $\pm$ 25 19.96 0.53
FCOS 2-050 3:37:42.65 $-$35:25:41.4 190 $\pm$ 25 19.34 0.69
FCOS 2-080 3:37:42.98 $-$35:42:14.5 140 $\pm$ 45 20.70 0.59
FCOS 2-105 3:37:43.04 $-$35:38:28.6 135 $\pm$ 35 19.19 0.64
FCOS 2-059 3:37:43.49 $-$35:25:57.7 190 $\pm$ 35 19.71 1.47
FCOS 4-021 3:37:44.01 $-$35:15:09.3 95 $\pm$ 25 19.81 1.04
FCOS 2-051 3:37:45.96 $-$35:27:23.4 315 $\pm$ 35 19.42 0.86
FCOS 4-037 3:37:46.43 $-$35:13:01.4 20 $\pm$ 35 20.40 1.23
FCOS 4-033 3:37:47.42 $-$35:17:18.1 80 $\pm$ 25 20.30 1.26
FCOS 2-054 3:37:49.39 $-$35:44:07.5 145 $\pm$ 25 19.55 1.21
FCOS 4-043 3:37:49.66 $-$35:19:43.0 165 $\pm$ 70 20.7 0.59
FCOS 4-036 3:37:51.18 $-$35:12:42.3 $-$35 $\pm$ 60 20.37 0.60
FCOS 4-026 3:37:51.66 $-$35:05:19.3 95 $\pm$ 25 20.02 0.68
FCOS 2-057 3:37:52.02 $-$35:33:07.7 125 $\pm$ 15 19.65 0.99
FCOS 4-029 3:37:52.49 $-$35:07:34.3 70 $\pm$ 45 20.06 0.64
FCOS 4-031 3:37:53.43 $-$35:22:00.7 270 $\pm$ 50 20.27 0.72
FCOS 4-027 3:37:54.57 $-$35:18:25.4 140 $\pm$ 35 20.02 1.18
FCOS 4-020 3:37:55.11 $-$35:16:57.7 145 $\pm$ 85 19.27 0.07
FCOS 4-056 3:38:00.85 $-$35:00:10.1 100 $\pm$ 45 16.48 0.93
FCOS 4-022 3:38:01.58 $-$35:02:23.3 20 $\pm$ 25 19.82 1.45
FCOS 2-070 3:38:05.23 $-$35:35:31.8 20 $\pm$ 110 20.21 1.45
FCOS 2-058 3:38:08.05 $-$35:33:28.9 $-$5 $\pm$ 15 19.66 1.11
FCOS 4-041 3:38:08.40 $-$35:20:46.2 70 $\pm$ 15 20.54 0.60
FCOS 4-028 3:38:09.46 $-$35:08:35.5 20 $\pm$ 25 20.06 1.23
FCOS 4-035 3:38:10.01 $-$35:04:33.5 80 $\pm$ 45 20.35 0.59
FCOS 2-079 3:38:12.93 $-$35:41:44.2 205 $\pm$ 35 20.69 0.63
FCOS 2-060 3:38:15.50 $-$35:43:28.2 125 $\pm$ 25 19.75 0.66
FCOS 1-035 3:38:38.45 $-$35:34:51.4 30 $\pm$ 60 19.65 0.78
FCOS 1-028 3:38:40.67 $-$35:36:52.3 285 $\pm$ 30 19.86 1.17
FCOS 1-023 3:38:42.69 $-$35:43:37.7 160 $\pm$ 45 20.37 0.79
FCOS 1-033 3:38:43.67 $-$35:35:05.0 200 $\pm$ 35 19.76 0.87
FCOS 1-061 3:38:45.81 $-$35:25:12.6 170 $\pm$ 45 20.89 0.83
FCOS 1-047 3:38:45.91 $-$35:32:40.6 60 $\pm$ 45 20.31 0.75
FCOS 1-024 3:38:46.11 $-$35:42:11.4 70 $\pm$ 35 19.75 0.85
FCOS 1-073 3:38:48.85 $-$35:40:16.4 115 $\pm$ 15 19.09 1.33
FCOS 1-049 3:38:49.43 $-$35:30:01.8 10 $\pm$ 50 20.84 0.73
FCOS 1-057 3:38:51.15 $-$35:27:41.2 175 $\pm$ 50 20.95 0.72
FCOS 1-025 3:38:59.26 $-$35:41:14.4 140 $\pm$ 50 20.69 1.31
FCOS 1-043 3:39:02.39 $-$35:33:53.4 175 $\pm$ 25 20.31 1.06
FCOS 1-041 3:39:06.29 $-$35:34:11.4 100 $\pm$ 15 19.76 1.17
FCOS 1-030 3:39:07.31 $-$35:36:02.2 255 $\pm$ 45 20.24 0.72
FCOS 1-053 3:39:10.81 $-$35:29:09.7 120 $\pm$ 35 20.48 1.44
FCOS 1-066 3:39:12.55 $-$35:21:22.7 60 $\pm$ 35 19.63 1.19
Name $\alpha$ (2000.0) $\delta$ (2000.0) $v_{\rm rad} [km/s]$ $V$ $(V-I)$ Comment
---------------- ------------------- ------------------- ---------------------- ------- --------- ---------
FCOS 6-023$^*$ 3:34:57.34 $-$35:12:24.0 16830 $\pm$ 85 18.97 0.72 ELO
FCOS 6-032$^*$ 3:34:59.92 $-$35:11:41.6 57820 $\pm$ 130 19.71 1.17 ELO
FCOS 6-057$^*$ 3:35:00.09 $-$35:09:25.3 z=2.17 20.95 0.70 Quasar
FCOS 6-005$^*$ 3:35:03.08 $-$35:20:53.4 49220 $\pm$ 105 19.57 0.85
FCOS 6-006$^*$ 3:35:03.29 $-$35:00:33.8 94500 $\pm$ 175 19.69 0.76
FCOS 6-001 3:35:06.61 $-$35:10:53.5 61840 $\pm$ 50 19.24 1.41
FCOS 6-045$^*$ 3:35:26.93 $-$34:59:51.0 100900 $\pm$ 85 20.52 1.30 ELO
FCOS 6-031 3:35:31.62 $-$35:02:49.9 56950 $\pm$ 50 19.66 1.34
FCOS 4-040$^*$ 3:37:39.46 $-$35:01:34.8 75515 $\pm$ 85 20.45 1.48
FCOS 2-055$^*$ 3:37:42.69 $-$35:32:29.2 71045 $\pm$ 140 19.58 1.13 ELO
FCOS 4-007$^*$ 3:37:43.02 $-$35:13:56.1 82650 $\pm$ 155 20.27 1.44
FCOS 4-023$^*$ 3:37:43.30 $-$35:11:02.1 z=2.29 19.87 0.53 Quasar
FCOS 2-052 3:37:44.46 $-$35:36:18.0 48195 $\pm$ 45 19.43 1.33
FCOS 2-084$^*$ 3:37:44.59 $-$35:34:50.9 41635 $\pm$ 85 20.79 1.08 ELO
FCOS 4-001$^*$ 3:37:44.62 $-$35:18:32.6 84230 $\pm$ 175 19.63 1.09 ELO
FCOS 4-006$^*$ 3:37:45.37 $-$35:00:22.2 49950 $\pm$ 140 20.12 0.91 ELO
FCOS 4-016$^*$ 3:37:45.63 $-$35:03:35.4 76910 $\pm$ 210 21.04 0.83 ELO
FCOS 2-001$^*$ 3:37:46.47 $-$35:24:11.4 64460 $\pm$ 105 19.17 1.40 ELO
FCOS 2-065$^*$ 3:37:46.65 $-$35:27:50.0 33960 $\pm$ 120 20.02 1.42
FCOS 2-071$^*$ 3:37:48.12 $-$35:33:57.8 41755 $\pm$ 120 20.24 0.86 ELO
FCOS 2-031$^*$ 3:37:48.21 $-$35:29:12.1 64800 $\pm$ 140 20.50 0.91 ELO
FCOS 4-025$^*$ 3:37:48.69 $-$35:04:23.1 z=2.57 19.97 0.92 Quasar
FCOS 2-085$^*$ 3:37:50.94 $-$35:39:47.2 95380 $\pm$ 175 20.80 1.05
FCOS 4-044$^*$ 3:37:52.48 $-$35:13:13.6 87600 $\pm$ 140 20.72 1.00
FCOS 4-024 3:37:55.60 $-$35:09:36.1 50165 $\pm$ 50 19.93 1.35
FCOS 4-008$^*$ 3:37:55.70 $-$35:06:53.0 92200 $\pm$ 120 20.31 1.24
FCOS 2-008 3:37:59.86 $-$35:39:04.4 61900 $\pm$ 80 19.68 1.40
FCOS 4-010$^*$ 3:38:02.74 $-$35:01:19.2 33350 $\pm$ 140 20.45 0.58
FCOS 4-032$^*$ 3:38:02.79 $-$35:08:55.4 73095 $\pm$ 160 20.30 1.37 ELO
FCOS 2-066$^*$ 3:38:03.76 $-$35:36:07.9 48325 $\pm$ 85 20.07 0.92 ELO
FCOS 4-030$^*$ 3:38:12.67 $-$35:16:20.3 z=2.53 20.13 0.62 Quasar
FCOS 4-003$^*$ 3:38:21.76 $-$35:09:22.5 50260 $\pm$ 140 19.77 1.03
FCOS 1-011 3:38:31.12 $-$35:39:13.4 33455 $\pm$ 85 18.08 1.14 ELO
FCOS 1-007 3:38:42.45 $-$35:30:50.2 33215 $\pm$ 50 20.10 1.24
FCOS 1-003$^*$ 3:38:49.09 $-$35:24:12.0 87600 $\pm$ 210 19.79 1.11 ELO
FCOS 1-006$^*$ 3:38:49.14 $-$35:29:20.2 82330 $\pm$ 225 19.63 0.93 ELO
FCOS 1-008$^*$ 3:38:56.91 $-$35:31:10.9 128801 $\pm$ 300 21.02 0.98 ELO
FCOS 1-013$^*$ 3:39:02.64 $-$35:41:49.5 88492 $\pm$ 190 20.68 0.85 ELO
FCOS 1-029$^*$ 3:39:03.63 $-$35:36:14.6 55305 $\pm$ 155 19.74 0.92 ELO
FCOS 1-075 3:39:08.18 $-$35:38:41.3 48075 $\pm$ 85 18.68 0.99 ELO
[^1]: Two of them had already been detected by Hilker et al. ([@Hilker99])
[^2]: Here we implicitly treat the objects observed with too low S/N like the ones rejected for geometric reasons
|
---
abstract: 'A family of composite black brane solutions in the model with scalar fields and fields of forms is presented. The metric of any solution is defined on a manifold which contains a product of several Ricci-flat “internal” spaces. The solutions are governed by moduli functions $H_s$ ($s = 1, ..., m$) obeying non-linear differential equations with certain boundary conditions imposed. These master equations are equivalent to Toda-like equations and depend upon the non-degenerate ($m \times m$) matrix $A$. It was conjectured earlier that the functions $H_s$ should be polynomials if $A$ is a Cartan matrix for some semisimple finite-dimensional Lie algebra (of rank $m$). It is shown that the solutions to master equations may be found by using so-called fluxbrane polynomials which can be calculated (in principle) for any semisimple finite-dimensional Lie algebra. Examples of dilatonic charged black hole ($0$-brane) solutions related to Lie algebras $A_1$, $A_2$, $C_2$ and $G_2$ are considered.'
---
**Black brane solutions governed by fluxbrane polynomials**
**V. D. Ivashchuk\[1\]\[1\][e-mail: ivashchuk@mail.ru]{}**
*Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya ul., Moscow 119361, Russia*
*Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia, 6 Miklukho-Maklaya ul., Moscow 117198, Russia*
Introduction
============
In this paper we deal with spherically-symmetric solutions with horizon defined on product manifolds containing several Ricci-flat factor-spaces (with diverse signatures and dimensions). Solutions of such type appear either in models with antisymmetric forms and scalar fields [@BIM]-[@IMtop] or in models with multi-component anisotropic fluid (MCAF) [@IMS1]-[@I-10]. For black brane solutions with $1$-dimensional factor-spaces (of Euclidean signatures) see [@CT; @AIV; @Oh] and references therein.
These and more general brane cosmological and spherically symmetric solutions were obtained by reduction of the field equations to the Lagrange equations corresponding to Toda-like systems [@IMJ; @IK].
Here we consider black brane solutions in the model with scalar field and fields of forms, when certain relations on parameters are imposed. The solutions are governed by a set of functions $H_s$, $s = 1, ..., m$, obeying non-linear differential equations with certain boundary conditions imposed. These equations depend upon the non-degenerate $m \times m$ matrix $A$. It was conjectured in [@IMp1] that the moduli functions $H_s$ should be polynomials when $A$ is a Cartan matrix for some semisimple finite-dimensional Lie algebra ${\cal G}$ of rank $m$. In this case we deal with special solutions to open Toda chain equations related to the Lie algebra ${\cal G}$ [@T; @B; @OP; @K] which are integrable in quadratures. The conjecture from [@IMp1] was verified for the Lie algebras $A_m$, $C_{m+1}$, $m \geq 1$ in [@IMp2; @IMp3] by using the solutions to Toda chain equations corresponding to the Lie algebras $A_m$ from [@And].
Here we show that the black brane solutions under consideration may be also found by using so-called fluxbrane polynomials [@Iflux], which may be calculated (in principle) for any (semi)simple finite-dimensional Lie algebra using MATHEMATICA [@Iv-02] or MAPLE [@GoIv-08; @GoIv-09]. For any solution we find the Hawking temperature as a function of $p_i$-parameters of fluxbrane polynomials. Recently, in [@LY] a similar approach appeared in a context of special Toda charged black hole solutions corresponding to Lie algebras $A_m$, where a formal relation for $A_m$ fluxbrane polynomials was obtained in a way similar to our earlier consideration [@IMp2; @IMp3] based on the Anderson solution [@And]. Here we illustrate the general approach by applying it to dilatonic charged black hole solutions related to Lie algebras $A_1$, $A_2$, $C_2$ and $G_2$.
Black brane solutions
=====================
We start with a model governed by the action
S=d\^Dx {R\[g\]-h\_g\^[MN]{}\_M\^ \_N\^-\_[a]{} (F\^a)\^2}, where $g=g_{MN}(x)dx^M\otimes dx^N$ is a metric, $\varphi=(\varphi^\alpha)\in{ {\mathbb R} }^l$ is a vector of scalar fields, $(h_{\alpha\beta})$ is a constant symmetric non-degenerate $l\times l$ matrix $(l\in { {\mathbb N} })$, $\theta_a=\pm1$,
F\^a = dA\^a = F\^a\_[M\_1 …M\_[n\_a]{}]{} dz\^[M\_1]{} …dz\^[M\_[n\_a]{}]{} is a $n_a$-form ($n_a\ge1$), $\lambda_a$ is a 1-form on ${ {\mathbb R} }^l$: $\lambda_a(\varphi)=\lambda_{a \alpha }\varphi^\alpha$, $a\in{\triangle}$, $\alpha=1,\dots,l$. In (\[1a.1\]) we denote $|g| =
|\det (g_{MN})|$, $(F^a)^2_g = F^a_{M_1 \ldots M_{n_a}} F^a_{N_1
\ldots N_{n_a}} g^{M_1 N_1} \ldots g^{M_{n_a} N_{n_a}}$, $a \in
{\triangle}$. Here ${\triangle}$ is some finite set. In the models with one time all $\theta_a = 1$ when the signature of the metric is $(-1,+1,
\ldots, +1)$.
In [@IMp1; @IMp2; @IMp3] we have obtained a family of black brane solutions to the field equations corresponding to the action (\[1a.1\]). These solutions are defined on the manifold
M = (R\_[0]{}, + ) (M\_0 = S\^[d\_0]{}) (M\_1 = [ [R]{} ]{}) …M\_n, and have the following form
g= (\_[s S]{} H\_s\^[2 h\_s d(I\_s)/(D-2)]{} ) { f\^[-1]{} dR dR + R\^2 g\^0\
- (\_[s S]{} H\_s\^[-2 h\_s]{} ) f dt dt + \_[i = 2]{}\^[n]{} (\_[sS]{} H\_s\^[-2 h\_s \_[iI\_s]{}]{} ) g\^i },\
\[2a.31\] (\^)= \_[sS]{} H\_s\^[h\_s \_s \_[a\_s]{}\^]{},\
\[2.32a\] F\^a= \_[s S]{} \^a\_[a\_s]{} [F]{}\^[s]{}, where $f =1 - 2\mu/R^{d}$,
\^s= - ( \_[s’ S]{} H\_[s’]{}\^[- A\_[s s’]{}]{} ) dR (I\_s), s S\_e,
\^s= Q\_s (|I\_s), sS\_m.
Here $Q_s \neq 0$, $s\in S$, are charge densities, $R >R_0$, $R_0
= (2 \mu)^{1/d} > 0$ and $ d = d_0 -1$.
In (\[2a.30\]) $g^0$ is the canonical metric on the unit sphere $M_0 =S^{d_0}$ and $g^i$ is a Ricci-flat metric on $M_{i}$, $i= 2,\ldots,n$ and $\delta_{iI}= \sum_{j\in
I} \delta_{ij}$ is the indicator of $i$ belonging to $I$: $\delta_{iI}= 1$ for $i\in I$ and $\delta_{iI}= 0$ otherwise. We also denote $g^1 = -dt \otimes dt$.
The brane set $S$ is by definition S= S\_e S\_m, S\_v= \_[a]{}{a}{v}\_[a,v]{},
$v= e,m$ and $\Omega_{a,e}, \Omega_{a,m} \subset \Omega$, where $\Omega = \Omega(n)$ is the set of all non-empty subsets of $\{
1, \ldots, n \}$, i.e. all branes do not “live” in $M_0$.
Any brane index $s \in S$ has the form $s = (a_s,v_s, I_s)$, where $a_s \in {\triangle}$, $v_s = e,m$ and $I_s \in \Omega_{a_s,v_s}$. The sets $S_e$ and $S_m$ define electric and magnetic branes correspondingly. In (\[2a.31\]) $\chi_s = +1, -1$ for $s \in
S_e, S_m$, respectively. All branes contain the time manifold $M_1
= { {\mathbb R} }$, i.e.
1 I\_s, s S.
All manifolds $M_{i}$, $i > 0$, are assumed to be oriented and connected and the volume $d_i$-forms
\_i dy\_i\^[1]{} …dy\_i\^[d\_i]{},
and signature parameters
\(i) ( (g\^i\_[m\_i n\_i]{})) = 1
are well-defined for all $i= 1,\ldots,n$. Here $d_{i} = {\rm
dim} M_{i}$, $i = 0, \ldots, n$, $d_0 > 1$, $d_1 = 1$ and for any $I = \{ i_1, \ldots, i_k \} \in \Omega$, $i_1 < \ldots < i_k$, we denote
\(I) \_[i\_1]{} … \_[i\_k]{}, d(I) = \_[i I]{} d\_i, (I) (i\_1) …(i\_k).
Forms ${\cal F}^s$ correspond to electric and magnetic branes for $s\in S_e, S_m$, respectively. In (\[2a.33\]) $\bar I = \{0,\ldots,n\}\setminus I$ and $\tau (\bar I) =
\tau_0 \wedge \tau(\{1,\ldots,n\}\setminus I)$, where $\tau_0$ is the volume form on $M_0 =S^{d_0}$.
The parameters $h_s$ appearing in the solution satisfy the relations: $h_s = (B_{s s})^{-1}$, where
B\_[ss’]{} = d(I\_sI\_[s’]{})+ + \_s\_[s’]{}\_[a\_s ]{}\_[ a\_[s’]{} ]{} h\^,
$s, s' \in S$, with $(h^{\alpha\beta})=(h_{\alpha\beta})^{-1}$ and $D = 1 + \sum_{i = 1}^{n} d_{i}$. In (\[2a.31\]) $\lambda_{a_s}^{\alpha } = h^{\alpha \beta} \lambda_{a_s \beta }$.
The parameters $B_{ss'}$ are scalar products of certain $U^s$-vectors belonging to ${ {\mathbb R} }^{n+l+1}$: $B_{ss'} = (U^s,U^{s'})$ [@IMJ; @IMtop].
Here we assume that ([**i**]{}) K\_s = B\_[ss]{} 0, s S,\
\[1a.B2\] ([**ii**]{}) [det]{}(B\_[s s’]{}) 0, i.e. the matrix $(B_{ss'})$ is a non-degenerate one.
We consider the matrix (A\_[ss’]{}) = ( 2 B\_[s s’]{}/B\_[s’ s’]{} ), $s, s' \in S$. Here some ordering in $S$ is assumed.
We denote \_s= (I\_s) \_[a\_s]{} [for]{} v\_s = e,\
\[1.eps2\] \_s= -\[g\] (I\_s) \_[a\_s]{}, [for]{} v\_s = m, $s\in S$, ${\varepsilon}[g] = \sign\det(g_{MN})$.
Functions $H_s > 0$ obey the equations
R\^[d\_0]{} = B\_s \_[s’ S]{} H\_[s’]{}\^[- A\_[s s’]{}]{}, with $B_s = {\varepsilon}_s B_{s s} Q_s^2$ and the boundary conditions imposed: H\_s |\_[R =R\_0 + 0]{} = H\_[s0]{} (0, +), and H\_s|\_[R = +]{} = 1, $s \in S$.
Here we also impose the following condition H\_s [is smooth in]{} (R\_, + ), $s \in S$, where $R_{\epsilon}= (2 \mu)^{1/d} e^{- \epsilon}$, $\epsilon > 0$. Then the metric has a regular horizon at $R^{d} = 2 \mu$ and has an asymptotically flat $(2 + d_0)$-dimensional section.
Due to (\[2a.32\]) and (\[2a.33\]), the dimension of brane worldvolume is defined by relations $d(I_s)= n_{a_s}-1$ for $s \in S_e$ and $d(I_s)= D- n_{a_s} -1$ for $S_m$. For a $p$-brane: $p = p_s = d(I_s)-1$.
[**Restrictions on brane intersections.**]{}
The composite black brane solutions under consideration take place if two restrictions on the sets of branes are obeyed. These restrictions guarantee the block-diagonal form of the energy-momentum tensor.
[**Restriction 1.**]{}
*For any colour index $a\in{\triangle}$ and electromagnetic index $v= e,m$*
d(I J) d(I) - 2, for any $I,J \in \Omega_{a,v}$, $I \neq J$ (here $d(I) = d(J)$).
[**Restriction 2**]{} [*For any colour index $a \in{\triangle}$, $I \in
\Omega_{a,e}$ (electric brane set) and $J \in \Omega_{a,m}$ (magnetic brane set) d(I J) 0.* ]{}
[**The Hawking temperature.**]{} The Hawking temperature corresponding to the solution reads T\_H= \_[s S]{} H\_[s0]{}\^[-h\_s]{}, where $H_{s0}$ are defined in (\[2.2.1a\]).
This solution describes a set of charged (by forms) overlapping branes “living” on submanifolds of $M_1 \times \dots \times
M_n$.
Polynomial structure of $H_s$ for semisimple Lie algebras
=========================================================
Black brane polynomials
-----------------------
Now we deal with solutions to second order non-linear differential equations (\[2.2.1\]) which may be rewritten as follows
( H\_s ) = |B\_s \_[l =1]{}\^[m]{} H\_[l]{}\^[- A\_[s l]{}]{},
where $H_s(z) > 0$, $z = R^{-d} \in (0, (2\mu)^{-1})$ and $\bar B_s =
B_s/ d^2 \neq 0$.
Eqs. (\[2.2.1a\]) and (\[2.2.1b\]) read
H\_[s]{}((2)\^[-1]{} -0) = H\_[s0]{} (0, + ),\
\[3.3.2b\] H\_[s]{}(+ 0) = 1, $s = 1,..., m$. Here we identify for simplicity $S$ with the set of $m = |S|$ numbers: $S = \{1,..., m \}$.
The condition (\[2.2.1c\]) reads as follows H\_s(z) > 0 [is smooth in]{} (0, z\_), $s = 1,..., m$, where $z_{\epsilon}= (2 \mu)^{-1} e^{ \epsilon d}$, $\epsilon > 0$.
Equations (\[3.3.1\]) are equivalent to Toda-type equations [@IMp2; @IMp3].
It was conjectured in [@IMp1] that equations (\[3.3.1\])-(\[3.3.2b\]) have polynomial solutions when $(A_{s l})$ is a Cartan matrix for some semisimple finite-dimensional Lie algebra $\cal G$ of rank $m$. In this case we get
H\_[s]{}(z) = 1 + \_[k = 1]{}\^[n\_s]{} P\_s\^[(k)]{} z\^k,
where $P_s^{(k)}$ are constants, $k = 1,\ldots, n_s$; $P_s^{(n_s)} \neq 0$, and
n\_s = 2 \_[l = 1]{}\^m A\^[s l]{} $s = 1,..., m$, are the components of twice the dual Weyl vector in the basis of simple co-roots [@FS]. Here $(A^{sl}) = (A_{sl})^{-1}$.
This conjecture was verified for ${\bf A_m}$ and ${\bf C_{m+1}}$ series of Lie algebras in [@IMp2; @IMp3]. In the extremal case ($\mu
= + 0$) an analogue of this conjecture was suggested (implicitly) in [@LMMP].
[**${\bf A_1} \oplus \ldots \oplus {\bf A_1}$ -case.**]{} The simplest example occurs in the orthogonal case : $B_{sl} =
(U^s,U^{l})= 0$, for $s \neq l$ [@BIM; @IMJ] (see also [@CT; @AIV; @Oh] and refs. therein). In this case $(A_{s l}) =
{\rm diag}(2,\ldots,2)$ is a Cartan matrix for the semisimple Lie algebra ${\bf A_1} \oplus \ldots \oplus {\bf A_1}$ and
H\_[s]{}(z) = 1 + P\_s z, with $P_s \neq 0$, satisfying
P\_s(P\_s + 2) = -|B\_s = - \_s K\_s Q\_s\^2/d\^2, $s = 1,..., m$. When all ${\varepsilon}_s K_s < 0$ there exists a unique set of numbers $P_s > 0$ obeying (\[3.3.5a\]).
[**$A_2$-case.**]{} For the Lie algebra $\cal G$ coinciding with ${\bf A_2} = sl(3)$ we get $n_1 = n_2 =2$ and
H\_[s]{} = 1 + P\_s z + P\_s\^[(2)]{} z\^[2]{},
where $P_s= P_s^{(1)}$ and $P_s^{(2)} \neq 0$ are constants, $s = 1,2$.
It was found in [@IMp1] that for $P_1 +P_2 + 4\mu \neq 0$ (e.g. when all $P_s >0 $) the following relations take place
P\_s\^[(2)]{} = , |B\_s = - , $s = 1,2$.
Here we denote $s+ 1 = 2, 1$ for $s = 1,2$, respectively.
[**Other solutions.**]{} The “master” equations were integrated (using Maple) in [@GrIvKim1; @GrIvMel2] for Lie algebras ${\bf C_2}$ and ${\bf A_3}$, respectively. Recently, in [@LY] the solutions to master equations were found for $A_m$ Lie algebras using the general solutions to Toda chain equations from [@And].
Special solutions $H_{s}(z) = (1 + P_s z)^{b_s}$ appeared earlier in [@Br1] and later in [@IMJ2; @CIM] in a context of so-called block-orthogonal configurations.
Fluxbrane polynomials
=====================
Here we deal with the so-called fluxbrane polynomials which will be used in the next section for solving the black brane master equations (\[3.3.1\]) with the boundary conditions (\[3.3.2a\]) and (\[3.3.2b\]) imposed.
The conjecture on fluxbrane polynomials
---------------------------------------
Now, we deal with the polynomials which obey the following set of equations ( \_s ) = [P]{}\_s \_[l = 1]{}\^[m]{} [H]{}\_[l]{}\^[- A\_[s l]{}]{}, with the boundary conditions imposed \_[s]{}(+ 0) = 1, $s = 1,...,m$. Here functions ${\cal H}_s(z) > 0$ are defined on the interval $(0, +\infty)$ if ${\cal P}_s > 0$ for all $s$ and $(A_{s l})$ is the Cartan matrix for some finite-dimensional semisimple Lie algebra $\cal G$ of rank $m$.
The functions ${\cal H}_s(z) > 0$ appeared in generalized fluxbrane solutions which were obtained in [@Iflux]. Parameter ${\cal P}_s$ is proportional to brane charge density squared $q_s^2$, $s = 1,...,m$ and $z = \rho^2$, where $\rho$ is a radial coordinate. The boundary condition (\[4.2\]) guarantees the absence of singularity (in the metric) for $\rho = +0$. For fluxbrane solutions in supergravities (or stringy-inspired models) see [@Iflux; @CGS; @GutSt; @CGSaf] and references therein.
The fluxbrane solutions from [@Iflux] and similar $S$-brane solutions from [@GonIM] are special classes of more general solutions from [@IK]. The simplest “fluxbrane” solution is a well-known Melvin solution [@Melv] corresponding to the Lie algebra $A_1 = sl(2)$.
It was conjectured in [@Iflux] that eqs. (\[4.1\]), (\[4.2\]) have polynomial solutions \_[s]{}(z) = 1 + \_[k = 1]{}\^[n\_s]{} [P]{}\_s\^[(k)]{} z\^k, where ${\cal P}_s^{(k)}$ are constants, $k = 1,\ldots, n_s$. Here ${\cal P}_s^{(n_s)} \neq 0$ and $n_s$ are defined in (\[3.2.20\])
It was pointed in [@Iflux] that the conjecture on polynomial structure of ${\cal H}_{s}$ may be verified for $A_n$ and $C_n$ Lie algebras along a line as it was done for black brane polynomials in [@IMp2; @IMp3].
The substitution of (\[4.3\]) into (\[4.1\]) gives a chain of relations on parameters ${\cal P}_s^{(k)}$ and ${\cal P}_s $. For $A-D-E$ (simply laced) Lie algebras these relations were used for calculations of polynomials (by using MATHEMATICA) in [@Iv-02]. The first relation in this chain is \_s\^[(1)]{} = [P]{}\_s , $s = 1, ..., m$.
We note that for a special choice of parameters: ${\cal P}_s = n_s P$, $P > 0$, the polynomials have the following simple form [@Iflux] \_[s]{}(z) = (1 + P z)\^[n\_s]{}, $s = 1, ..., m$. This ansatz is a nice tool for verification of general solutions obtained by either analytical or computer calculations.
In [@GoIv-08; @GoIv-09] a computational program for calculations of polynomials (using MAPLE) corresponding to classical series of simple Lie algebras was suggested.
The calculations of ${\cal P}_s^{(k)}$ (which are polynomials of $k$-th power in ${\cal P}_s$) give huge denominators for big $k$. This may be avoided by using new parameters $p_s$ instead of ${\cal P}_s$ p\_s = [P]{}\_s/n\_s, $s = 1, ..., m$.
Examples of polynomials
------------------------
Here we present certain examples of polynomials corresponding to the Lie algebras $A_2$, $C_2$ and $G_2$.
### $A_r$-polynomials, $r = 1,2,3$.
[**$A_1$-case.**]{} The simplest example occurs in the case of the Lie algebra $A_1 = sl(2)$. Here $n_1 = 1$. We get
\_[1]{}(z) = 1 + p\_1 z.
[**$A_2$-case.**]{} For the Lie algebra $A_2 = sl(3)$ with the Cartan matrix
(A\_[ss’]{})= (
[\*[6]{}[c]{}]{} 2 & -1\
-1& 2\
) we have [@Iflux] $n_1 = n_2 =2$ and \_[1]{} = 1 + 2 p\_1 z + p\_1 p\_2 z\^[2]{},\
\[A.7\] [H]{}\_[2]{} = 1 + 2 p\_2 z + p\_1 p\_2 z\^[2]{}.
[**$A_3$-case.**]{} The polynomials for the $A_3$-case read as follows
\_[1]{} = 1 + 3 p\_1 z + 3 p\_1 p\_2 z\^[2]{} + p\_1 p\_2 p\_3 z\^[3]{},\
\[A.9\] [H]{}\_[2]{} = 1 + 4 p\_2 z + 3 ( p\_1 p\_2 + p\_2 p\_3 ) z\^[2]{} + 4 p\_1 p\_2 p\_3 z\^[3]{} + p\_1 p\_2\^[2]{} p\_3 z\^[4]{},\
\[A.10\] [H]{}\_[3]{} = 1 + 3 p\_3 z + 3 p\_2 p\_3 z\^[2]{} + p\_1 p\_2 p\_3 z\^[3]{}.
### $C_2$-polynomials.
For the Lie algebra $C_2 = so(5)$ with the Cartan matrix
(A\_[ss’]{})= (
[\*[6]{}[c]{}]{} 2 & -1\
-2& 2\
) we get from (\[3.2.20\]) $n_1 = 3$ and $n_2 = 4$.
For fluxbrane polynomials we obtain from [@GonIM] \_1= 1+ 3 p\_1 z+ 3 p\_1 p\_2 z\^2 + p\_1\^2 p\_2 z\^3,\
\[C.3\] [H]{}\_2 = 1+ 4 p\_2 z+ 6 p\_1 p\_2 z\^2 + 4 p\_1\^2 p\_2 z\^3 + p\_1\^2 p\_2\^2 z\^4.
### $G_2$-polynomials.
For the Lie algebra $G_2$ with the Cartan matrix
(A\_[ss’]{})= (
[\*[6]{}[c]{}]{} 2 & -1\
-3& 2\
) we get from (\[3.2.20\]) $n_1 = 6$ and $n_2 = 10$. The fluxbrane polynomials read [@GonIM] \_[1]{} = 1+ 6 p\_1 z+ 15 p\_1 p\_2 z\^2 + 20 p\_1\^2 p\_2 z\^3 +\
15 p\_1\^3 p\_2 z\^4 + 6 p\_1\^3 p\_2\^2 z\^5 + p\_1\^4 p\_2\^2 z\^6 ,\
\[G.3\] [H]{}\_2 = 1+ 10 p\_2 z + 45 p\_1 p\_2 z\^2 + 120 p\_1\^2 p\_2 z\^3 + p\_1\^2 p\_2 (135 p\_1 + 75 p\_2) z\^4\
+ 252 p\_1\^3 p\_2\^2 z\^5 + p\_1\^3 p\_2\^2 (75 p\_1 + 135 p\_2 )z\^6 + 120 p\_1\^4 p\_2\^3 z\^7\
+ 45 p\_1\^5 p\_2\^3 z\^8 + 10 p\_1\^6 p\_2\^3 z\^9 + p\_1\^[6]{} p\_2\^[4]{} z\^[10]{}.
In all examples presented above the substitution $p_1 = \dots =
p_n = P$ into polynomials gives us relations (\[4.s\]).
In what follows we denote \_[s]{}(z) = [H]{}\_[s]{}(z; p), $s = 1, ..., m$, where $p = (p_1,...,p_m)$.
It should be noted that the set of fluxbrane polynomials for any semisimple Lie algebra ${\cal G} = {\cal G}_1 \oplus \dots \oplus {\cal G}_k$, where all ${\cal G}_i$ are simple Lie algebras, is just a union of sets of fluxbrane polynomials, corresponding to ${\cal G}_1$, ... , ${\cal
G}_k$, respectively.
Toda chain solutions
--------------------
Fluxbrane polynomials ${\cal H}_s$ define special solutions to open Toda chain equations [@T; @B; @K; @OP] = - 4 n\_s p\_s ( \_[l = 1]{}\^[m]{} A\_[s l]{} q\^[l]{} ), which correspond to a semisimple finite dimensional Lie algebra $\cal G$ with the Cartan matrix $(A_{s l})$.
These solution read q\^s(u) = - \_s(e\^[-2u]{};p) - n\_s u, $s = 1, ..., m$ (i.e. we put here $z = e^{-2u})$.
Relations (\[4.h\]) imply the following asymptotic formulae
q\^s(u) = - C\_s + n\_s u + o(1), u - ,\
\[4.s2\] q\^s(u) = - n\_s p\_s e\^[-2u]{} (1 + o(1) ) - n\_s u, u + , $s = 1, ..., m$. Here the scattering data $C_s > 0$ appear in the leading terms of polynomials \_s = 1 + n\_s p\_s z + …+ C\_s z\^[n\_s]{}, $s = 1, ..., m$. The calculations (see examples above) give the following relations
C\_s = \_[i=1]{}\^[m]{} p\_[i]{}\^[(s,i)]{},
where ${\nu(s,i)}$ are non-negative integer numbers which obey the relations $\sum_{i=1}^m {\ \nu(s,i)} = n_s$, $s = 1, ..., m$. (They are natural ones for simple finite-dimensional Lie algebras.)
It follows from (\[4.s1\]) and (\[4.s2\]) that
v\^s () = n\_s, u , $s = 1, ..., m$, i.e. the asymptotical values of velocities for “in” and “out” asymptotics are opposite in sign and coinciding by absolute value with the vector $(n_1, \dots, n_m)$. This is the main specific feature of the special solutions (\[4.h\]) to open Toda chain equations which are described by fluxbrane polynomials.
Black brane solutions governed by fluxbrane polynomials
=======================================================
In this section we show that the “black brane” master equations (\[3.3.1\])-(\[3.3.2b\]) may be solved (at least for small enough values of charge parameters $Q_s$) in terms of fluxbrane polynomials and give several examples of solutions.
Reduction to fluxbrane polynomials
----------------------------------
Let us denote $f =f(z)= 1 - 2\mu z$. Then the relations (\[3.3.1\]) may be rewritten as ( H\_s ) = B\_s (2 d )\^[-2]{} \_[l =1]{}\^[m]{} H\_[l]{}\^[- A\_[s l]{}]{}. $s = 1, ..., m$. These relations could be solved by using fluxbrane polynomials ${\cal H}_{s}(f) = {\cal H}_{s}(f; p)$, corresponding to $m \times m$ Cartan matrix $(A_{s l})$, where $p =
(p_1,...,p_m)$ is the set of reduced parameters (\[4.g\]). Here we impose the restrictions $p_i \neq 0$ for all $i$ instead of $p_i > 0$ from the previous section.
We put H\_s(z) = [H]{}\_[s]{}(f(z);p)/[H]{}\_[s]{}(1;p) for all $s = 1, ..., m$. \[2\]\[2\][This trick was known to the author for many years as well as the schemes of calculations of ${\cal H}_{s}$ [@Iv-02].]{}
Then the relations (\[5.1\]), or, equivalently, (\[3.3.1\]) are satisfied identically if
n\_s p\_s \_[l =1]{}\^[m]{} ([H]{}\_[l]{}(1;p))\^[- A\_[s l]{}]{} = B\_s /(2 d)\^2 = \_s K\_[s]{} Q\_s\^2/(2 d)\^2, $s = 1, ..., m$.
We call the set of parameters $p = (p_1,...,p_m)$ ($p_i \neq 0$) as proper one if \_[s]{}(f;p) > 0 for all $f \in [0,1]$ and $s = 1,..., m$.
We denote the set of all proper $p$ by ${\cal D}$. In what follows we consider only proper $p$.
Relations (\[5.3\]) imply p\_s = [sign]{} B\_s = \_s, $s = 1, ..., m$.
The boundary conditions (\[3.3.2a\]) are valid since H\_[s]{}((2)\^[-1]{} -0) = 1/[H]{}\_[s]{}(1;p) > 0, $s = 1,..., m$, and conditions (\[3.3.2b\]) are satisfied just due to definition (\[5.2\]) ($p \in {\cal D})$.
Relations (\[5.6\]) imply the following formula for the Hawking temperature (\[2a.36\]) T\_H= \_[s = 1]{}\^[m]{} ([H]{}\_[s]{}(1;p))\^[h\_s]{},
It should be noted that usual black brane solutions deal with negative $B_s$ (since ${\varepsilon}_s < 0$ and $B_{ss} >0$) and hence $p_s$ should be negative. Fluxbrane polynomials with negative $p_s$ were considered earlier in cosmological ($S$-brane) solutions which describe an accelerated expansion of 3-dimensional factor-space [@IKM; @Gol]. The original fluxbrane polynomials [@Iflux] responsible for fluxbrane solutions have positive parameters $p_s$. For black brane applications positive $p_s$ correspond to positive $B_s$. This takes place for a certain family of phantom black brane solutions which may be a subject of a separate publication.
[**Remark.**]{} For all examples of fluxbrane polynomials we know, all $p$ = $(p_i)$ with positive entries $p_i >0$ are proper since in this case ${\cal H}_{s}(f;p) >0$ for all $s$ and $f \geq 0$.
For fixed signs in (\[5.5\]) we denote \_ = {p | p = (p\_1, ...,p\_m) , [sign]{} p\_i = \_i, i = 1,..., m }. Here ${\epsilon} = (\epsilon_i)$.
Then relations (\[5.3\]) define the map f\_: [D]{}\_ [ [R]{} ]{}\_[+]{}\^[m]{}, with $f_{\epsilon}(p) = (Q_1^2, \dots, Q_m^2)$. Locally, for small enough $p_i$ the function $f_{\epsilon}$ defines one-to-one correspondence between the sets of parameters $(p_1, ...,p_m)$ and $(Q_1^2, \dots, Q_m^2)$. An open question here is to verify whether this is correct globally for certain choices of $\epsilon = (\epsilon_i)$ and semisimple Lie algebras ${\cal G}$ (i.e. whether the function $f_{\epsilon}$ is bijective or not for certain cases).
Examples
--------
Now we illustrate the general approach by considering several examples of charged black hole ($0$-brane) solutions corresponding to Lie algebras of rank $m = 1, 2$.
### Black hole for $A_1$.
First we consider the gravitational model with one scalar field and one 2-form: S=d\^Dx {R\[g\]- g\^[MN]{} \_M \_N - (2 )F\^2 }.
Here $g$ is a D-dimensional metric, $F = dA$ is $2$-form; $\varphi$ is scalar field and $\lambda \in { {\mathbb R} }$ is dilatonic coupling. We deal with a charged black hole solution defined on the manifold
M = (0, + ) M\_0 M\_1 M\_2, where $M_0 = S^{d_0}$, $M_1 = { {\mathbb R} }$ is a one-dimensional (time) manifold of signature $(-)$ and $M_2$ is $d_2$-dimensional Ricci-flat manifold.
We put the brane multi-index corresponding to 1-forms $A$ as $I_1 = \{ 1 \}$. We get from (\[1a.B1\]) K\_1 = + \^2 > 0.
In this case relation (\[1a.18\]) is valid for the Cartan matrix $A = (2)$, corresponding to the Lie algebra $A_1$.
The charged black hole solution has the following form (see (\[2a.30\])-(\[2a.32\]))
g = H\^[2 h/(D-2)]{} { f\^[-1]{} dR dR + R\^2 g\^0 - H\^[-2 h]{} f dt dt + g\^2 },\
\[6.5b\] ()= H\^[h ]{}\
\[6.6b\] F = H\^[- 2]{} dt dR .
Here $f =1 - 2\mu/R^{d}$, $\mu > 0$, $d = d_0 -1$, $h = K_1^{-1} >0$, $Q = Q_1 \neq 0$ is (electric) charge, $R > R_0 = (2\mu)^{1/d}$, and $g^2$ is a Ricci-flat metric on $M_{2}$.
The moduli function $H$ is given by (\[A.1\]) and general prescriptions (\[5.2\]) and (\[5.3\]):
H(z) = ,
where $f(z) = 1 - 2 \mu z$, and the parameter $ p_1$ is negative due to relation
p\_1 / (1 + p\_1)\^2 = - K\_1 Q\^2 /(2 d)\^2, following from (\[5.3\]).
The parameter $p_1$ should be proper, i.e. ${\cal H}_{1}(f; p_1) = 1 + p_1 f
> 0$ for all $f \in [0,1]$, hence we get $-1 < p_1 < 0$. Function (\[5.8\]) in this case is given by (\[6.8b\]) $f_{\epsilon}: {\cal D}_{\epsilon} = (-1,0) \to { {\mathbb R} }_{+}$, $\epsilon = (-1)$. It is a bijection (moreover, $f_{\epsilon}$ is a diffeomorphism).
Relation (\[6.7b\]) may be rewritten as follows H(z) = 1 + P z, P = > 0.
It follows from (\[6.8b\]) and (\[6.9bb\]) that
P(P + 2) = K\_1 Q\^2/d\^2 = , A() = D-3 + \^2 (D -2), in agreement with (\[3.3.5a\]) and hence ($P >0$) P = - + .
Thus, relations (\[6.4b\])-(\[6.6b\]) with the moduli function H = 1 + P R\^[-d]{} and $P > 0$ from (\[6.11b\]) give us a charged dilatonic black hole solution which coincides up to notations with the solutions from [@BBFM; @BI]. In [@BI] another radial variable $r =
(R^d + P)^{1/d}$ and parameters $B_{-} = P$ and $B_{+} = P + 2
\mu$ were used. For special cases of this solution see [@GM] ($d_0 = 2$), [@MP] ($\lambda = 0$, $\varphi$ is absent).
### Black holes for $A_2$, $C_2$ and $G_2$.
Let us consider the gravitational model with two scalar fields and two forms of rank 2: S=d\^Dx {R\[g\]- g\^[MN]{} \_M \_N - \_[s =1]{}\^[2]{}(F\^s)\^2 }.
Here $g$ is a D-dimensional metric, $F^1 = dA^1$ and $F^2 = dA^2$ are $2$-forms; $\vec{\varphi}=(\varphi^1,\varphi^2) \in { {\mathbb R} }^2$ is a vector of two scalar fields, $\vec{\lambda_1} = (\lambda_{1 \alpha}), \vec{\lambda_2}
= (\lambda_{2 \alpha}) \in { {\mathbb R} }^2$ are dilatonic coupling vectors.
In what follows we consider electrically charged black hole solutions defined on the manifold (\[6.2\]). We put the brane multi-indices corresponding to 1-forms $A^1$ and $A^2$ as $I_1 =
I_2 = \{ 1 \}$, respectively.
We get from (\[1a.17\]) B\_[ss’]{} = + , $s, s' = 1,2$, and K\_s = B\_[ss]{} = + \^2 > 0, $s = 1,2$.
We impose intersection rules (\[1a.18\]) corresponding to the Lie algebras $A_2$, $C_2$, $G_2$ with the Cartan matrices
(A\_[ss’]{})= (
[\*[6]{}[c]{}]{} 2 & -1\
-k& 2\
) for $k = 1, 2, 3$, respectively. For our case these rules are equivalent to the set of two relations: (i) $2 B_{12} = A_{12} K_2
= -K_2$, (ii) $2 B_{21} = A_{21} K_1 = - k K_1$, which imply (due to $B_{12} = B_{21}$) $K_2 = k K_1$, or \^2 = k \^2 + (k-1) , while relation (ii) reads 2 = - k \^2 - (k+2) , $k = 1, 2, 3$.
The set of relations (\[6.5\]), (\[6.6\]) is equivalent to intersection rules (\[1a.18\]).
[**Remark.**]{} [*It may be readily verified that the vectors $\vec{\lambda_{1}}, \vec{\lambda_{2}} \in { {\mathbb R} }^2$ obeying (\[6.5\]), (\[6.6\]) do exist. This may be done by fixing $\vec{\lambda_{1}}^2 = N >0$ and writing the Gramian matrix $(\vec{\lambda_{i}} \vec{\lambda_{j}})$ by using (\[6.5\]) and (\[6.6\]) in terms of $N$, $\frac{D-3}{D-2}$ and $k$. For big enough value of $N$ ($N > N_0$) the Gramian matrix is positive definite and hence there exist dilatonic coupling vectors which obey (\[6.5\]) and (\[6.6\]). One may choose: $\vec{\lambda_{i}} = \sqrt{N} (\vec{u_{i}} + o(1))$, for $N \to +
\infty$, where $\vec{u_{i}}$, $i = 1,2$, are simple roots of the Lie algebra ${\cal G} = A_2, C_2, G_2$ for $k = 1, 2, 3$, respectively.*]{}
The charged (by forms $F^1$ and $F^2$) electric black hole solutions have the following form (see (\[2a.30\])-(\[2a.32\]))
g= ( H\_1\^[2 h\_1]{}H\_2\^[2 h\_2]{} )\^[1/(D-2)]{} { f\^[-1]{} dR dR + R\^2 g\^0\
- H\_1\^[-2 h\_1]{}H\_2\^[-2 h\_2]{} f dt dt + g\^2 },\
\[6.8\] (\^)= H\_1\^[h\_1 \_[1 ]{}]{} H\_2\^[h\_2 \_[2 ]{}]{},\
\[6.9a\] F\^1= H\_[1]{}\^[- 2]{} H\_[2]{} dt dR\
\[6.9b\] F\^2= H\_[2]{}\^[- 2]{} H\_[1]{}\^[k]{} dt dR where $\alpha = 1, 2$ and $k = 1, 2, 3$ for Lie algebras $A_2$, $C_2$, $G_2$, respectively.
Here $f =1 - 2\mu/R^{d}$, $\mu > 0$, $d = d_0 -1$, $h_s = K_s^{-1} >0$, $Q_s \neq 0$ ($s =1,2$) are charges, $R > R_0 = (2\mu)^{1/d}$, and $g^2$ is a Ricci-flat metric on $M_{2}$.
The moduli functions $H_s$ are given by general prescriptions (\[5.2\]) and (\[5.3\]):
H\_s(z) = \_s\^[-1]{} [H]{}\_[s]{}(f(z); p), \_s = [H]{}\_[s]{}(1; p),
$s = 1,2$, where $f(z) = 1 - 2 \mu z$ and the vector $p =
(p_1, p_2)$ has negative components: $p_1 < 0$ and $p_2 < 0$, which are related to charges $Q_1,
Q_2$ by formulae
n\_1 p\_1 \_1\^[-2]{} \_2 = - K\_1 Q\_1\^2 /(2 d)\^2,\
\[6.7a\] n\_2 p\_2 \_2\^[-2]{} \_1\^[k]{} = - K\_2 Q\_2\^2 /(2 d)\^2, $k =1,2,3.$
The fluxbrane polynomials ${\cal H}_{1},
{\cal H}_{2}$ are given by relations: (i) (\[A.6\]), (\[A.7\]) for $k = 1$; (ii) (\[C.2\]), (\[C.3\]) for $k = 2$; and (iii) (\[G.2\]), (\[G.3\]) for $k = 3$. The powers of polynomials read: $(n_1,n_2) = (2,2), (3,4),
(6,10)$ for $k = 1, 2, 3$, respectively.
For the Hawking temperatures of black holes ($k = 1,2,3$) we get from (\[5.7\]) T\_H= \_[s = 1]{}\^[2]{} \_[s]{}\^[h\_s]{}.
We remind that here as in general case the set of parameters $p =
(p_1, p_2)$ should be proper.
The $A_2$-solution without “internal space” $(M_2,g^2)$ and with one scalar field was obtained recently in [@LY]. In [@LY] $A_n$ black hole solutions with $n$ vectors fields and $(n-1)$ scalar fields were found.
Conclusions and discussions
===========================
Here we have described a family of composite black brane solutions corresponding to semisimple Lie algebras in the models with scalar field and fields of forms. Intersection rules for branes are given by Cartan matrices for these Lie algebras. The metric of any solution contains $(n -1)$ Ricci-flat “internal” metrics and certain restrictions on brane intersections are imposed.
The moduli functions of solutions are given by fluxbrane polynomials which define special solutions to open Toda chain equations corresponding to semisimple Lie algebras. These polynomials may be calculated (in principle) for any simple or semisimple Lie algebra, e.g. by using MAPLE or MATHEMATICA. One can use also formal relations for $A_m$- polynomials which were obtained recently in [@LY]. $C_m$-polynomials may be obtained from $A_{2m+1}$-polynomials by using the identifications of parameters: $p_1 = p_{2m+1}$, $p_2 =
p_{2m}$ etc [@IMp3; @Iflux].
Here we have applied our formalism to few examples of dilatonic charged black hole ($0$-brane) solutions related to Lie algebras $A_1$, $A_2$, $C_2$ and $G_2$. To our knowledge the last solution is a new one while others are covered by earlier publications, e.g. [@BBFM; @BI] ($A_1$), [@IMp3] ($A_2$) and [@GrIvKim1] ($C_2$). (In the $C_2$-case the parametrization of polynomials in [@GrIvKim1] differs from the one considered in this paper).
We have also obtained formulae for Hawking temperatures corresponding to black brane solutions (under consideration) in terms of polynomial parameters. The calculation of other thermodynamic quantities and studying the global structures of certain black brane/hole solutions will be a subject of a separate publication.
[**Acknowledgments**]{}
This work was supported in part by Templan grant of PFUR (No 200312-1-174) in 2014.
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0.8truecm
**Hyperfine Structure and Zeeman Splitting**
0.3truecm
**in Two-Fermion Bound-State Systems**
0.6truecm
Andrei G. Terekidi$^{a }$, Jurij W. Darewych$^{b }$, Marko Horbatsch$^{c}$
0.5truecm
*Department of Physics and Astronomy, York University, Toronto, Ontario, M3J 1P3, Canada*
*$^{a }$terekidi@yorku.ca, $^{b}$darewych@yorku.ca,$^{c}$marko@yorku.ca*
1.6truecm
**Abstract**
A relativistic wave equation for bound states of two fermions with arbitrary masses which are exposed to a magnetic field is derived from quantum electrodynamics. The interaction kernels are based upon the generalized invariant $\widetilde{\mathcal{M}}\mathcal{\,}$-matrices for inter-fermion and fermion-field interactions. As an application we calculate the energy corrections in a weak homogeneous $\mathbf{B}$ field to obtain the Zeeman splitting of the hyperfine structure (HFS) and $g$-factors in the lowest order (*i.e.* to $O\left( \alpha^{4}\right) )$. Landé $g$-factors are presented for several of the first excited states of hydrogen, muonium, and muonic-hydrogen.
0.6truecm
**[1. Introduction]{}**
0.4truecm
The relativistic treatment of energy levels of two-fermion atomic systems (including atomic hydrogen, hydrogen-like ions, helium-3 ion, muonium, muonic-hydrogen), as well as their fine structure (FS) and hyperfine structure (HFS) in an external uniform magnetic field (Zeeman effect), is an important problem. The theoretical knowledge of energy spectra and transition frequencies provides a test of two-body bound-state QED \[1\]. One can then obtain information about the character of the coupling in the system, the gyromagnetic ratios of the bound particles, the magnetic moments \[2\], the mass ratio \[2-5\], and fundamental physical constants such as the Rydberg constant $R_{\infty}$, and the fine structure constant $\alpha$ \[6\]. The Zeeman effect in the HFS can be used as a diagnostic tool for solar photospheric magnetic fields \[7\], fusion research and plasma physics, where the magnetic field is applied to control the shape and position of the plasma \[8\].
In the lowest-order approximation the linearly dependent part of the energy splitting for a two-fermion system placed in a weak static magnetic field $\mathbf{B}$ can be written as \[1,9-11\]$$\Delta E_{F,m_{J},j_{1},\ell,s_{1},I}^{ext}=\left( \mu_{B1}g_{1}+\mu
_{B2}g_{2}\right) Bm_{F},$$ where $F$, $m_{J}$, $j_{1}$, $\ell$, $s_{1}$, $I$ are quantum numbers, which characterize the system: $s_{1}$ and $I$ are the spins of the first and second particle respectively, $\ell$ and $j_{1}$ represent the orbital and total angular momentum quantum numbers of the first particle. The total angular momentum of the system is denoted by the quantum number $F=j_{1}+I,$ $j_{1}+I-1,...,\left\vert j_{1}-I\right\vert $. The projection of the total angular momentum on the $\mathbf{B}$ direction is $m_{F}=-F,-F+1,...F-1,F$. The Bohr magnetonsfor the two particles are defined as $\displaystyle\mu_{B1}=Q_{1}\hbar/2m_{1}c$, and $\mu
_{B2}=-Q_{2}\hbar/2m_{2}c$, where $Q_{1}$, $Q_{2}>0$). Usually, in our notation $m_{1}$ and $m_{2}$ correspond to the light and heavy particle respectively. Assuming that the energy-level splitting (1) is much smaller then the HFS splitting, $\Delta E^{ext}<<\Delta E^{HFS}$, the Landé ($g$-) factors $g_{1}$ and $g_{2}$ take the form \[9-11\]$$g_{1}=g_{j_{1}}\frac{F\left( F+1\right) +j_{1}\left( j_{1}+1\right)
-I\left( I+1\right) }{2F\left( F+1\right) },$$ where$$g_{j_{1}}=1+\left( g_{s_{1}}-1\right) \frac{j_{1}\left( j_{1}+1\right)
+s_{1}\left( s_{1}+1\right) -\ell\left( \ell+1\right) }{2j_{1}\left(
j_{1}+1\right) },$$ and$$g_{2}=g_{s_{2}}\frac{F\left( F+1\right) -j_{1}\left( j_{1}+1\right)
+I\left( I+1\right) }{2F\left( F+1\right) }.$$ Here $g_{s_{1}}$ and $g_{s_{2}}$ are the intrinsic spin magnetic moments of the constituent particles. According to the Dirac theory a free particle at rest has $g_{s}=2$. In QED $g_{s}$ is corrected by the anomaly, which to lowest order is given by the Schwinger correction. For bound particles the intrinsic moment can be expressed as $$g_{s_{1,2}}=2+\bigtriangleup g_{s_{1,2}}^{REL}+\bigtriangleup g_{s_{12}}^{QED},$$ where the terms $\bigtriangleup g^{REL}$, $\bigtriangleup g^{QED}$ represent the relativistic \[9,12,13\], and QED corrections respectively (cf. the review \[14\]). There is also an additional higher-order contribution to (1), $\bigtriangleup g_{1,2}^{HFS}\mu_{B1,2}Bm_{F}$, which is caused by the hyperfine structure (HFS) \[15\].
The $g$-factors (2) and (4) are not symmetrical, because they were obtained under the assumption that the orbital motion of the heavy particle can be neglected. In hydrogen the nucleus contributes a fraction of $m_{1}/\left(
m_{1}+m_{2}\right) \approx5\times10^{-4}$ to the orbital angular momentum, while for muonic hydrogen this fraction is $\approx0.1$. The relativistic and QED corrections in (5) can be comparable with the orbital angular momentum effects of the heavy particle. Recent high-precision measurements of the $g$-factor in hydrogen-like systems have reached an accuracy of about $5\times10^{-9}$ \[16,17\]. Thus, it is desirable to obtain a more general result for the $g$-factor in order to overcome the shortcomings of Eqs. (2,4). It will be shown that this is particularly important for excited states.
In this work we present an analysis of the HFS of a two-fermion system in an external magnetic field based upon a reformulation of QED and the variational Hamiltonian formalism developed earlier \[18-20\]. A relativistic two-fermion wave equation for arbitrary fermion masses is, thus, derived from first principles. A solution of this equation permits, in principle, to obtain all QED energy corrections to any order of the coupling constant \[18\]. In the present paper we extend the method to derive the integral wave equation in momentum space for the case where a uniform weak magnetic field is present. We calculate the Zeeman splitting of the HFS energy levels to $O\left(
\alpha^{4}\right) $ for all quantum states and unrestricted values for the fermion masses. We obtain a novel result for the $g$-factor, Eqs. (38-41), and demonstrate that it coincides with Eqs. (2-4) in the case of $m_{2}>>m_{1}$, as long as the intrinsic moment of $m_{1}$ is restricted to the Dirac value $g_{s}=2$.
The modification of the wave equations due to the external magnetic field is presented in Section 2. In Section 3 we provide the classification of the quantum states, and a partial-wave decomposition of the momentum-space equations. Section 4 contains expressions for the Zeeman energy splittings of the HFS levels, and the $g$-factor results. Numerical values for the Landé factors are compared with data from Eqs. (2,4) for various excited states of hydrogen, muonium and muonic hydrogen. In most expressions we use natural units $\hbar= c = 1$. 1.6truecm
**[2. Bound-State Variational Wave Equation]{}**
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For two-fermion systems without external fields wave equations were derived in \[18-19\] on the basis of a modified QED Lagrangian \[21-22\]. With this Lagrangian a simple Fock-space trial state $$\left\vert \psi_{trial}\right\rangle =\underset{s_{1}s_{2}}{\sum}\int
d^{3}\mathbf{p}_{1}d^{3}\mathbf{p}_{2}F_{s_{1}s_{2}}(\mathbf{p}_{1},\mathbf{p}_{2})b_{\mathbf{p}_{1}s_{1}}^{\dagger}D_{\mathbf{p}_{2}s_{2}}^{\dagger}\left\vert 0\right\rangle ,$$ sufficed to obtain HFS levels correct to fourth order in the fine-structure constant. Here $b_{\mathbf{q}_{1}s_{1}}^{\dagger}$ and $D_{\mathbf{q}_{2}s_{2}}^{\dagger}$ are creation operators for a free fermion of mass $m_{1}$ and an (anti-)fermion of mass $m_{2}$ respectively, and $\left\vert
0\right\rangle $ is the trial vacuum state such that $b_{\mathbf{q}_{1}s_{1}}\left\vert 0\right\rangle =D_{\mathbf{q}_{2}s_{2}}\left\vert 0\right\rangle
=0$.
As discussed in section 3 below, the four adjustable functions $F_{s_{1}s_{2}}$ must be chosen so that the trial state (17) is an eigenstate of the relativistic total angular momentum operator, its projection, and parity (as well as charge conjugation for the case $m_{1}=m_{2}$ such as positronium).
A variational principle is invoked to obtain a momentum-space wave equation for the amplitudes \[18\]: $$\begin{aligned}
0 & =\sum_{s_{1}s_{2}}\int d^{3}\mathbf{p}_{1}d^{3}\mathbf{p}_{2}\left(
\omega_{p_{1}}+\Omega_{p_{2}}-E\right) F_{s_{1}s_{2}}(\mathbf{p}_{1},\mathbf{p}_{2})\delta F_{s_{1}s_{2}}^{\ast}(\mathbf{p}_{1},\mathbf{p}_{2})\\
& -\frac{m_{1}m_{2}}{\left( 2\pi\right) ^{3}}\underset{\sigma_{1}\sigma
_{2}s_{1}s_{2}}{\sum}\int\frac{d^{3}\mathbf{p}_{1}d^{3}\mathbf{p}_{2}d^{3}\mathbf{q}_{1}d^{3}\mathbf{q}_{2}}{\sqrt{\omega_{p_{1}}\omega_{q_{1}}\Omega_{p_{2}}\Omega_{q_{2}}}}\nonumber\\
& \times F_{\sigma_{1}\sigma_{2}}(\mathbf{q}_{1},\mathbf{q}_{2})\left(
-i\right) \widetilde{\mathcal{M}}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}\left(
\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{q}_{1}\mathbf{,q}_{2}\right) \delta
F_{s_{1}s_{2}}^{\ast}(\mathbf{p}_{1},\mathbf{p}_{2}),\nonumber\end{aligned}$$ where $\omega_{p_{1}}^{2}=\mathbf{p}_{1}^{2}+m_{1}^{2}$ and $\Omega_{p_{1}}^{2}=\mathbf{p}_{1}^{2}+m_{2}^{2}$. The interaction is governed by the generalized invariant $\mathcal{M}$-matrix $\widetilde{\mathcal{M}}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}\left( \mathbf{p}_{1},\mathbf{p}_{2},\mathbf{q}_{1}\mathbf{,q}_{2}\right) $. It has the form $$\mathcal{M}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}^{\left( 1\right) }\left(
\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{q}_{1}\mathbf{,q}_{2}\right)
\equiv\mathcal{M}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}^{ope}\left( \mathbf{p}_{1},\mathbf{p}_{2},\mathbf{q}_{1}\mathbf{,q}_{2}\right) +\mathcal{M}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}^{ext}\left( \mathbf{p}_{1},\mathbf{p}_{2},\mathbf{q}_{1}\mathbf{,q}_{2}\right) ,$$ where $\mathcal{M}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}^{ope}\left(
\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{q}_{1}\mathbf{,q}_{2}\right) $ is the usual invariant matrix element, corresponding to the one-photon exchange Feynman diagram \[19-20\].
The element $\mathcal{M}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}^{ext}$ represents the interaction with a given external classical field $A_{\mu}^{ext}$$$\begin{aligned}
& \mathcal{M}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}^{ext}\left( \mathbf{p}_{1},\mathbf{p}_{2},\mathbf{q}_{1}\mathbf{,q}_{2}\right) \\
& =i\left( 2\pi\right) ^{3/2}\left(
\begin{array}
[c]{c}\frac{\sqrt{\Omega_{\mathbf{p}_{2}}\Omega_{q_{2}}}}{m_{2}}A_{\mu}^{ext}(\mathbf{p}_{1}-\mathbf{q}_{1})\overline{u}\left( \mathbf{p}_{1},s_{1}\right) \left( -iQ_{1}\right) \gamma^{\mu}u\left( \mathbf{q}_{1},\sigma_{1}\right) \delta_{s_{2}\sigma_{2}}\\
+\frac{\sqrt{\omega_{\mathbf{p}_{1}}\omega_{\mathbf{q}_{1}}}}{m_{1}}A_{\mu
}^{ext}(\mathbf{q}_{2}-\mathbf{p}_{2})\overline{V}\left( \mathbf{p}_{2},\sigma_{2}\right) \left( -iQ_{2}\right) \gamma^{\mu}V\left(
\mathbf{q}_{2},s_{2}\right) \delta_{s_{1}\sigma_{1}}\end{array}
\right) .\nonumber\end{aligned}$$
The Ansatz (6) can not accommodate processes that include the emission or absorption of real, physical (as opposed to virtual) photons. Such radiative processes could be included by generalizing the trial state. Here we limit ourselves to the form (6), *i.e.*, the effects of radiative decay or absorption of radiation are ignored in the present work.
In order to obtain the Lande factors we evaluate the $\mathcal{M}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}^{ext}$ matrix (9) in a stationary uniform magnetic field $\mathbf{B}=B\mathbf{\hat{z}}$. The vector potential can be chosen as$$A_{1}^{ext}\left( \mathbf{x}\right) =-\frac{1}{2}yB,\;\;\ \ \;A_{2}^{ext}\left( \mathbf{x}\right) =\frac{1}{2}xB,\ \ \ \ \ \ \ \ A_{0}^{ext}\left( \mathbf{x}\right) =A_{3}^{ext}\left( \mathbf{x}\right) =0.$$ The inverse Fourier transform of the non-zero components yields$$A_{1}^{ext}(\mathbf{k})=\frac{\left( 2\pi\right) ^{3/2}iB}{2}\delta\left(
k_{x}\right) \frac{d\delta\left( k_{y}\right) }{dk_{y}}\delta\left(
k_{z}\right) ,\ \ \ \ A_{2}^{ext}(\mathbf{k})=\mathbf{-}\frac{\left(
2\pi\right) ^{3/2}iB}{2}\frac{d\delta\left( k_{x}\right) }{dk_{x}}\delta\left( k_{y}\right) \delta\left( k_{z}\right) .$$ Using the semi-relativistic expansion$$\begin{aligned}
\overline{u}\left( \mathbf{p}_{1},s_{1}\right) \gamma^{1}u\left(
\mathbf{q}_{1},\sigma_{1}\right) & =\frac{1}{2m_{1}c}\varphi_{s_{1}}^{\dagger}\left( i\left[ \overrightarrow{\mathbf{\sigma}}_{1}\times\left(
\mathbf{p}_{1}-\mathbf{q}_{1}\right) \right] +\mathbf{q}_{1}+\mathbf{p}_{1}\right) _{1}\varphi_{\sigma_{1}},\\
\overline{u}\left( \mathbf{p}_{1},s_{1}\right) \gamma^{2}u\left(
\mathbf{q}_{1},\sigma_{1}\right) & =\frac{1}{2m_{1}c}\varphi_{s_{1}}^{\dagger}\left( i\left[ \overrightarrow{\mathbf{\sigma}}_{1}\times\left(
\mathbf{p}_{1}-\mathbf{q}_{1}\right) \right] +\mathbf{q}_{1}+\mathbf{p}_{1}\right) _{2}\varphi_{\sigma_{1}},\nonumber\end{aligned}$$ where $\varphi_{1}^{\dagger}=[1\ 0]$, $\varphi_{2}^{\dagger}=[0\ 1]$,and $\left( \omega_{p_{1}}\omega_{q_{1}}\right) ^{1/2}\simeq m_{1}$, and a similar expansion for anti-particle spinors we obtain$$\begin{aligned}
& \mathcal{M}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}^{ext}\left( \mathbf{p}_{1},\mathbf{p}_{2},\mathbf{q}_{1}\mathbf{,q}_{2}\right) \\
& =\frac{\left( 2\pi\right) ^{3/2}}{2c}\left(
\begin{array}
[c]{c}\frac{Q_{1}}{m_{1}}A_{j}^{ext}(\mathbf{p}_{1}-\mathbf{q}_{1})\varphi_{s_{1}}^{\dagger}\left( i\left[ \overrightarrow{\mathbf{\sigma}}_{1}\times\left(
\mathbf{p}_{1}-\mathbf{q}_{1}\right) \right] +\mathbf{q}_{1}+\mathbf{p}_{1}\right) _{j}\varphi_{\sigma_{1}}\delta_{s_{2}\sigma_{2}}\\
+\frac{Q_{2}}{m_{2}}A_{j}^{ext}(\mathbf{q}_{2}-\mathbf{p}_{2})\chi_{\sigma
_{2}}^{\dagger}\left( i\left[ \overrightarrow{\mathbf{\sigma}}_{2}\times\left( \mathbf{p}_{2}-\mathbf{q}_{2}\right) \right] +\mathbf{q}_{2}+\mathbf{p}_{2}\right) _{j}\chi_{s_{2}}\delta_{s_{1}\sigma_{1}}\end{array}
\right) ,\nonumber\end{aligned}$$ where $\chi_{1}^{\dagger}=[0\ 1]$, $\chi_{2}^{\dagger}=-[1\ 0]$, and $j=1,2$. It is straightforward to show that$$\left( \mathbf{q}_{1}\right) _{j}\ A_{j}^{ext}(\mathbf{p}_{1}-\mathbf{q}_{1})=-\frac{\left( 2\pi\right) ^{3/2}B}{2}\widehat{L}_{1z}\left(
\mathbf{q}_{1}\right) \delta\left( \mathbf{p}_{1}-\mathbf{q}_{1}\right) ,$$ and$$\begin{aligned}
& A_{j}^{ext}(\mathbf{p}_{1}-\mathbf{q}_{1})\varphi_{s_{1}}^{\dagger}\left(
i\left[ \overrightarrow{\mathbf{\sigma}}_{1}\times\left( \mathbf{p}_{1}-\mathbf{q}_{1}\right) \right] +\mathbf{q}_{1}+\mathbf{p}_{1}\right)
_{j}\varphi_{\sigma_{1}}\\
& =-\left( 2\pi\right) ^{3/2}B\left( \varphi_{s_{1}}^{\dagger}\sigma
_{1z}\varphi_{\sigma_{1}}+\delta_{s_{1}\sigma_{1}}\widehat{L}_{1z}\left(
\mathbf{q}_{1}\right) \right) \delta^{3}\left( \mathbf{p}_{1}-\mathbf{q}_{1}\right) ,\nonumber\end{aligned}$$ where $\widehat{L}_{1z}\left( \mathbf{q}_{1}\right) $ is the $z$-component of the angular momentum operator of the particle with mass $m_{1}$$$\widehat{L}_{1z}\left( \mathbf{q}_{1}\right) =-i\left( q_{1x}\frac
{\partial}{\partial q_{1y}}-q_{1y}\frac{\partial}{\partial q_{1x}}\right) .$$ Taking $\varphi_{s_{1}}$ to be the eigenstates of the spin operator $\widehat{S}_{1z}=\frac{1}{2}\widehat{\sigma}_{1z}$, and using a similar procedure for the second particle, we obtain$$\begin{aligned}
& \mathcal{M}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}^{ext}\left( \mathbf{p}_{1},\mathbf{p}_{2},\mathbf{q}_{1}\mathbf{,q}_{2}\right) \\
& =-\frac{\left( 2\pi\right) ^{3}B}{2c}\left(
\begin{array}
[c]{c}\frac{Q_{1}}{m_{1}}\left( 2\varphi_{s_{1}}^{\dagger}\widehat{S}_{1z}\varphi_{\sigma_{1}}+\delta_{s_{1}\sigma_{1}}\widehat{L}_{1z}\left(
\mathbf{q}_{1}\right) \right) \delta_{s_{2}\sigma_{2}}\delta^{3}\left(
\mathbf{p}_{1}-\mathbf{q}_{1}\right) \\
-\frac{Q_{2}}{m_{2}}\left( 2\chi_{\sigma_{2}}^{\dagger}\widehat{S}_{2z}\chi_{s_{2}}+\delta_{\sigma_{2}s_{2}}\widehat{L}_{2z}\left( \mathbf{q}_{2}\right) \right) \delta_{s_{1}\sigma_{1}}\delta^{3}\left( \mathbf{p}_{2}-\mathbf{q}_{2}\right)
\end{array}
\right) ,\nonumber\end{aligned}$$ or$$\begin{aligned}
& \mathcal{M}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}^{ext}\left( \mathbf{p}_{1},\mathbf{p}_{2},\mathbf{q}_{1}\mathbf{,q}_{2}\right) \\
& =-\left( 2\pi\right) ^{3}B\left(
\begin{array}
[c]{c}\mu_{B1}\left( 2\tilde{m}_{\sigma_{1}}+\widehat{L}_{1z}\left( \mathbf{q}_{1}\right) \right) \delta^{3}\left( \mathbf{p}_{1}-\mathbf{q}_{1}\right)
\\
-\mu_{B2}\left( 2\tilde{m}_{\sigma_{2}}+\widehat{L}_{2z}\left(
\mathbf{q}_{2}\right) \right) \delta^{3}\left( \mathbf{p}_{2}-\mathbf{q}_{2}\right)
\end{array}
\right) \delta_{s_{2}\sigma_{2}}\delta_{s_{1}\sigma_{1}},\nonumber\end{aligned}$$ where the spin projection quantum numbers $\tilde{m}_{\sigma}$ can take the values $\pm1/2$. The quantities $\mu_{B1}$ and $\mu_{B2}$ are the Bohr magnetons" defined in the previous section. As expected, a unit of spin interacts with a magnetic field twice as strongly as a unit of orbital angular momentum.
By going to the next order in the expansion of the invariant $\mathcal{M}$ matrix one can obtain self-energy corrections, which lead to divergent loop integrals that have to be cured by charge renormalization. The vertex term modifies the Dirac value of the magnetic moment by a factor $\left(
1+k\right) $, where $k$ is the anomaly (Schwinger correction). This factor can be included in our calculation by a replacement $2 \tilde m_{\sigma_{1}}$ and $2 \tilde m_{\sigma_{2}}$ in Eq. (18) by $g_{s_{1}}\tilde m_{\sigma
_{1}}$ and $g_{s_{2}}\tilde m_{\sigma_{2}}$ respectively, where $g_{s_{1,2}}/2=1+k_{1,2}$. The anomaly is the lowest-order QED correction to the $g$ factor $\bigtriangleup g_{s_{12}}^{QED}=2k_{1,2}$ in Eq. (5).
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**[3. Partial-wave decomposition and radial wave equations]{}**
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The present work is an extension of Ref. \[18\], in which the partial-wave decomposition of the wave equation has been provided. The external magnetic field is treated as a first-order perturbation which implies that the quantum labels for the eigenstates do not change. The restrictions on the magnetic field strength to justify a perturbative treatment of Eq.(18) are$$B\lesssim\min\left[ \frac{\alpha^{4}m_{r}c^{2}}{\mu_{B1}},\frac{\alpha
^{4}m_{r}c^{2}}{\mu_{B2}}\right] ,$$ where $\alpha=Q_{1}Q_{2}/4\pi$, and $m_{r}=m_{1}m_{2}/\left( m_{1}+m_{2}\right) $ is the reduced mass. A more explicit restriction on $B$ will be presented in Section 4.
As outlined in Ref. \[18\] the trial state (6) is taken to be an eigenstate of total linear momentum $\widehat{\mathbf{P}}$, total angular momentum squared $\widehat{\mathbf{J}}^{2}$, its projection $\widehat{J}_{3}$, and parity $\widehat{\mathcal{P}}$. It is natural to work in the rest frame, where the total linear momentum vanishes. In this frame the adjustable functions take the form $F_{s_{1}s_{2}}(\mathbf{p}_{1},\mathbf{p}_{2})=\delta\left(
\mathbf{p}_{1}+\mathbf{p}_{2}\right) F_{s_{1}s_{2}}(\mathbf{p}_{1})$, where $F_{s_{1}s_{2}}(\mathbf{p}_{1})$ (using $\mathbf{p}_{1} \equiv\mathbf{p}$) can be written as $$F_{s_{1}s_{2}}(\mathbf{p})=\sum_{\ell_{s_{1}s_{2}}}\sum_{m_{s_{1}s_{2}}}f_{s_{1}s_{2}}^{\ell_{s_{1}s_{2}}m_{s_{1}s_{2}}}\left( p\right)
Y_{\ell_{s_{1}s_{2}}}^{m_{s_{1}s_{2}}}(\widehat{\mathbf{p}}),$$ and $Y_{\ell_{s_{1}s_{2}}}^{m_{s_{1}s_{2}}}(\widehat{\mathbf{p}})$ are the usual spherical harmonics. Here and henceforth we will use the notation $p=\left\vert \mathbf{p}\right\vert $ etc., while four-vectors will be written as $p^{\mu}$. The orbital indices $\ell_{s_{1}s_{2}}$and $m_{s_{1}s_{2}}$ and the radial functions $f_{s_{1}s_{2}}^{\ell_{s_{1}s_{2}}m_{s_{1}s_{2}}}\left(
p\right) $ depend on the spin variables $s_{1}$ and $s_{2}$. In the rest frame, the operators $\widehat{L}_{1z}\left( \mathbf{q}\right) $ and $\widehat{L}_{2z}\left( \mathbf{q}\right) $ can be expressed in terms of the orbital angular momentum operator, $\widehat{L}_{z}\left( \mathbf{q}\right) $, of the relative motion: $$\widehat{L}_{1z}\left( \mathbf{q}\right) =\frac{m_{2}}{m_{1}+m_{2}}\widehat{L}_{z}\left( \mathbf{q}\right) ,\;\;\;\;\;\;\;\;\widehat{L}_{2z}\left( -\mathbf{q}\right) =\frac{m_{1}}{m_{1}+m_{2}}\widehat{L}_{z}\left( \mathbf{q}\right) .$$
The substitution of the partial-wave expansion (20) into the rest-frame form of Ansatz (6) leads to two categories of relations among the adjustable functions $F_{s_{1}s_{2}}(\mathbf{p})$:
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*(i) The spin-mixed (quasi-singlet and quasi-triplet) states*
In this case we have $\ell_{s_{1}s_{2}}\equiv\ell=J$, and the general solution under the condition of well-defined $\widehat{\mathbf{P}}$, $\widehat{\mathbf{J}}^{2}$, $\widehat{J}_{3}$, and $\widehat{\mathcal{P}}$ can be expressed with the help of Dirac $\Gamma$ matrices as \[18\] $$F_{s_{1}s_{2}}\left( \mathbf{p}\right) =\overline{u}_{\mathbf{p}s_{1}}\Gamma_{m_{s_{1}s_{2}}}^{J\left( sg\right) }\left( \widehat{\mathbf{p}}\right) V_{-\mathbf{p}s_{2}}f_{J}^{\left( sg\right) }(p)+\overline
{u}_{\mathbf{p}s_{1}}\Gamma_{m_{s_{1}s_{2}}}^{J\left( tr\right) }\left(
\widehat{\mathbf{p}}\right) V_{-\mathbf{p}s_{2}}f_{J}^{\left( tr\right)
}(p).$$ Here $f_{J}^{\left( sg\right) }(p)$ and $f_{J}^{\left( tr\right) }(p)$ are radial functions to be determined. They represent the contributions of spin-singlet and spin-triplet states, *i.e.*, the total spin is not conserved in general.
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*(ii) The* $\ell$-*mixed triplet states*
These states occur for $\ell_{s_{1}s_{2}}\equiv\ell=J\mp1$. Their radial decomposition can be written as$$F_{s_{1}s_{2}}\left( \mathbf{p}\right) =\overline{u}_{\mathbf{p}s_{1}}\Gamma_{m_{s_{1}s_{2}}}^{J-1}\left( \widehat{\mathbf{p}}\right)
V_{-\mathbf{p}s_{2}}f_{J-1}(p)+\overline{u}_{\mathbf{p}s_{1}}\Gamma
_{m_{s_{1}s_{2}}}^{J+1}\left( \widehat{\mathbf{p}}\right) V_{-\mathbf{p}s_{2}}f_{J+1}(p).$$ The system in these states is characterized by $J,$ $m_{J},$ and $\mathcal{P}=(-1)^{J}$, and $\ell$ is not a good quantum number. The two radial functions $f_{J-1}(p)$ and $f_{J+1}(p)$ correspond to the cases $\ell=J-1$ and $\ell=J+1$. Mixing of this type occurs only for principal quantum number $n \ge3$.
From the variational method we obtain a system of coupled radial equations expressed in matrix form as $$\left( \omega_{p}+\Omega_{p}-E\right) \mathbb{F}\left( p\right)
=\frac{m_{1}m_{2}}{\left( 2\pi\right) ^{3}}\int\frac{q^{2}dq}{\sqrt
{\omega_{p}\omega_{q}\Omega_{p}\Omega_{q}}}\mathbb{K}\left( p,q\right)
\mathbb{F}\left( q\right) ,$$ where $\omega_{p}^{2}=\mathbf{p}^{2}+m_{1}^{2}$ and $\Omega_{p}^{2}=\mathbf{p}^{2}+m_{2}^{2}$, and $q=\left\vert \mathbf{q}\right\vert $ as already mentioned. Here $\mathbb{F}\left( p\right) $ and $\mathbb{K}\left(
p,q\right) $ are matrices composed of radial functions and kernels respectively. The kernel matrix $\mathbb{K}=\mathbb{K}^{ope}+\mathbb{K}^{ext}$ is made up of one-photon-exchange and external-field parts. Explicit expressions for $\mathbb{K}^{ope}$ can be found in Ref.\[18\], while the external-field contributions are calculated in this work.
For the spin-mixed states the two-component Fock-space amplitude is given as $$\mathbb{F}\left( p\right) =\left[
\begin{array}
[c]{c}f_{J}^{\left( sg\right) }(p)\\
f_{J}^{\left( tr\right) }(p)
\end{array}
\right] .$$ The equations imply a mixing of spin and radial variables, and the radial equations are usually coupled. We apply a unitary transformation with rotation angle $\beta$ to the spin part of function (22) to diagonalize the kernel-matrix. The diagonalization can be carried out for arbitrary $p$ and $q$ (cf. Eq. (55) in the Appendix), and defines a new quasi-spin basis $$\left\vert s_{1},s_{2},\ell,\widetilde{s},J,m_{J}\right\rangle =C_{1}\left\vert s_{1},s_{2},\ell,S=0,J,m_{J}\right\rangle +C_{2}\left\vert
s_{1},s_{2},\ell,S=1,J,m_{J}\right\rangle ,$$ where $\ell=J$, $S$ is the total spin of the system, and $\widetilde{s}=0$ for quasi-singlet and $\widetilde{s}=1$ for quasi-triplet states. The coefficients used to express the new basis states in terms of the previously defined singlet and triplet states are found to be $C_{1}=\sqrt{\left( 1+\xi\right)
/2}$, $C_{2}=-\sqrt{\left( 1-\xi\right) /2}$, for the quasi-singlet states, and $C_{1}=\sqrt{\left( 1-\xi\right) /2}$, $C_{2}=\sqrt{\left(
1+\xi\right) /2}$ for the quasi-triplet states. Here the rotation angle $\beta$ has been replaced for convenience according to $\tan{2\beta}=\sqrt{1-\xi^{2}}/\xi$.
The quasi-singlet and quasi-triplet states are both characterized by the same quantum numbers $J$, $m_{J}$ and $\mathcal{P}=(-1)^{J+1}$, and they mix the states given in the $LS$ coupling representation. The states are labeled for convenience not by the quasi-spin $z$-projection $t_{3}=\mp1/2$, but rather by $\widetilde{s}=t_{3}+1/2$, which takes on the values of $0,1$. In the Appendix the kernels for spin-mixed states are given explicitly in order to solve for the angle $\beta$, *i.e.*, to determine the $\xi$-values.
In the limit $m_{2}>>m_{1}$ the total angular momenta of the first and the second particles are $j_{1}=\ell_{1}\pm1/2$, $j_{2}=s_{2}=1/2$, where $\ell_{1}=\ell$. In this case $j_{1}$ can be used as a good quantum number, and the role of the indices $\tilde s_{s}$, $\tilde s_{t}$ are played by $j_{1}=\ell_{1}+1/2$ and $j_{1}=\ell_{1}-1/2$ respectively. In this case the coefficients $C_{1}$ and $C_{2}$ reduce to C-G coefficients $$C_{1,2}=\left( -1\right) ^{1/2+1/2+\ell_{1}+j_{1}}\sqrt{\left( 2S+1\right)
\left( 2j_{1}+1\right) }\left\{
\begin{array}
[c]{ccc}1/2 & 1/2 & S\\
\ell_{1} & \ell_{1} & j_{1}\end{array}
\right\} .$$ Note that the one-body limit corresponds to the $j_{1}j_{2}$ coupling representation, which can not be used in the general case of arbitrary masses since $j_{1}$ and $j_{2}$ are not independent (they are related through the common angular momentum $\ell$). For positronium the quasi-states become true singlet ($C_{2}=0$) and triplet ($C_{1}=0$) states with different charge conjugation quantum numbers.
We now proceed to calculate the kernels $\mathcal{K}_{mn}^{ext}\left(
p,q\right) $ associated with the classical external field $A_{\mu}^{ext}$. Using Eq. (9) for $\mathcal{M}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}^{ext}$ taken in the rest frame, we obtain$$\begin{aligned}
& \mathcal{K}_{mn}^{ext}\left( p,q\right) =-\frac{\left( \pi/2\right)
^{3/2}}{N\left( m_{1}m_{2}\right) ^{2}}\int d^{3}\widehat{\mathbf{p}}d^{3}\widehat{\mathbf{q}}\\
& \times Tr\left(
\begin{array}
[c]{c}Q_{1}\sqrt{\Omega_{q}\Omega_{q}}A_{\mu}^{ext}(\mathbf{p}-\mathbf{q})\left(
\gamma^{\lambda}q_{\lambda}+m_{1}\right) \gamma^{\mu}\left( \gamma^{\lambda
}q_{\lambda}+m_{1}\right) \Gamma^{n}\left( \widehat{\mathbf{q}}\right)
\left( \gamma^{\lambda}\widetilde{q}_{\lambda}-m_{2}\right) \Gamma^{\prime
m}\left( \widehat{\mathbf{p}}\right) \\
-Q_{2}\sqrt{\omega_{p}\omega_{p}}A_{\mu}^{ext}(\mathbf{q}-\mathbf{p})\left(
\gamma^{\lambda}q_{\lambda}+m_{1}\right) \Gamma^{n}\left( \widehat
{\mathbf{q}}\right) \left( \gamma^{\lambda}\widetilde{q}_{\lambda}-m_{2}\right) \gamma^{\mu}\left( \gamma^{\lambda}\widetilde{q}_{\lambda
}-m_{2}\right) \Gamma^{\prime m}\left( \widehat{\mathbf{p}}\right)
\end{array}
\right) ,\nonumber\end{aligned}$$ where $q=\left( \omega_{p},\mathbf{q}\right) $, and $\widetilde{q}=\left(
\Omega_{q},-\mathbf{q}\right) $. The $\Gamma$ -matrices correspond to the various $J^{\mathcal{P}}$ states. The evaluation of these kernels would allow one to obtain all relativistic corrections to the $g$-factor (5), however this is a formidable task. To determine the lowest-order effect it is sufficient to use the nonrelativistic limit ($q^{2}/m^{2}<<1$). In this case the kernels (28) take the form$$\begin{aligned}
& \mathcal{K}_{mn}^{ext}\left( p,q\right) =-\frac{\left( \pi/2\right)
^{3/2}}{N}\int d^{3}\widehat{\mathbf{p}}d^{3}\widehat{\mathbf{q}}\\
& \times Tr\left(
\begin{array}
[c]{c}Q_{1}A_{\mu}^{ext}(\mathbf{p}-\mathbf{q})\left( \gamma^{0}+I\right)
\gamma^{\mu}\left( \gamma^{0}+I\right) \Gamma^{n}\left( \widehat
{\mathbf{q}}\right) \left( \gamma^{0}-I\right) \Gamma^{\prime m}\left(
\widehat{\mathbf{p}}\right) \\
-Q_{2}A_{\mu}^{ext}(\mathbf{q}-\mathbf{p})\left( \gamma^{0}+I\right)
\Gamma^{n}\left( \widehat{\mathbf{q}}\right) \left( \gamma^{0}-I\right)
\gamma^{\mu}\left( \gamma^{0}-I\right) \Gamma^{\prime m}\left(
\widehat{\mathbf{p}}\right)
\end{array}
\right) .\nonumber\end{aligned}$$ These are evaluated for a stationary uniform magnetic field (10). The results are given separately for the two types of states:
*(i) The spin-mixed states* ($\ell
=J,\;\ J\geq1,\;\mathcal{P}=(-1)^{J+1}$)
In contrast to $\mathbb{K}^{\left( ope\right) }\left( p,q\right)
$ the kernel matrix $\mathbb{K}^{\left( ext\right) }\left( p,q\right) $ is not diagonal in the basis of the quasi-singlet $\left\vert sg_{q}\right\rangle
$ and quasi-triplet $\left\vert tr_{q}\right\rangle $ states, and can be written as $$\begin{aligned}
& \mathcal{K}_{11}^{\left( ext\right) }\left( p,q\right) \\
& =-\frac{\left( 2\pi\right) ^{3}}{2c}\left(
\begin{array}
[c]{c}\frac{Q_{1}}{m_{1}}\left( \left( 1-\frac{1-\xi}{2J\left( J+1\right)
}\right) \frac{m_{2}}{m_{1}+m_{2}}+\frac{g_{s_{1}}}{2}\left( \frac{1-\xi
}{2J\left( J+1\right) }-2\frac{\left\vert m_{1}-m_{2}\right\vert }{m_{1}+m_{2}}\xi\right) \right) \\
-\frac{Q_{2}}{m_{2}}\left( \left( 1-\frac{1-\xi}{2J\left( J+1\right)
}\right) \frac{m_{1}}{m_{1}+m_{2}}+\frac{g_{s_{2}}}{2}\left( \frac{1-\xi
}{2J\left( J+1\right) }-2\frac{\left\vert m_{1}-m_{2}\right\vert }{m_{1}+m_{2}}\xi\right) \right)
\end{array}
\right) Bm_{J},\nonumber\end{aligned}$$$$\begin{aligned}
& \mathcal{K}_{22}^{\left( ext\right) }\left( p,q\right) \\
& =-\frac{\left( 2\pi\right) ^{3}}{2c}\left(
\begin{array}
[c]{c}\frac{Q_{1}}{2m_{1}c}\left( \left( 1-\frac{1+\xi}{2J\left( J+1\right)
}\right) \frac{m_{2}}{m_{1}+m_{2}}+\frac{g_{s_{1}}}{2}\left( \frac{1+\xi
}{2J\left( J+1\right) }+2\frac{\left\vert m_{1}-m_{2}\right\vert }{m_{1}+m_{2}}\xi\right) \right) \\
-\frac{Q_{2}}{2m_{2}c}\left( \left( 1-\frac{1+\xi}{2J\left( J+1\right)
}\right) \frac{m_{1}}{m_{1}+m_{2}}+\frac{g_{s_{2}}}{2}\left( \frac{1+\xi
}{2J\left( J+1\right) }+2\frac{\left\vert m_{1}-m_{2}\right\vert }{m_{1}+m_{2}}\xi\right) \right)
\end{array}
\right) Bm_{J},\nonumber\end{aligned}$$$$\begin{aligned}
\mathcal{K}_{12}^{\left( ext\right) }\left( p,q\right) & =\mathcal{K}_{21}^{\left( ext\right) }\left( p,q\right) \\
& =-\frac{\left( 2\pi\right) ^{3}}{2c}\left(
\begin{array}
[c]{c}\frac{Q_{1}}{m_{1}}\left( \frac{\xi}{\sqrt{J\left( J+1\right) }}\frac{g_{s_{1}}}{2}+2\left( \frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right) ^{2}\xi^{2}\left( 1-\frac{g_{s_{1}}}{2}\right) \right) \\
-\frac{Q_{2}}{m_{2}}\left( \frac{\xi}{\sqrt{J\left( J+1\right) }}\frac{g_{s_{2}}}{2}+2\left( \frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right) ^{2}\xi^{2}\left( 1-\frac{g_{s_{2}}}{2}\right) \right)
\end{array}
\right) Bm_{J}.\nonumber\end{aligned}$$ Thus, it couples the system (24).
0.2truecm
*(ii) The pure triplet and* $\ell$*-mixed states* ($\ell=J\mp1,\;\ J\geq1,\;\mathcal{P}=(-1)^{J}$)
The system (24) can not be decoupled for these states, and the matrix $\mathbb{K}^{\left( ope\right) }\left( p,q\right) $ is not diagonal \[19\]. The magnetic part of the kernel is, however, diagonal$$\mathbb{K}^{\left( ext\right) }\left( p,q\right) =-\frac{\left(
2\pi\right) ^{3}}{2c}\left( \frac{Q_{1}}{m_{1}}-\frac{Q_{2}}{m_{2}}\right)
\left[
\begin{array}
[c]{cc}1 & 0\\
0 & 1
\end{array}
\right] Bm_{J}.$$ All kernels $\mathbb{K}^{\left( ext\right) }$ vanish in the case of equal masses and opposite charges ($Q_{1}=Q_{2}$), as occurs in the positronium case, where magnetic effects appear only in $O\left( B^{2}\right) $ \[23\].
0.8truecm
**[4. HFS to ]{}**$O\left( \alpha^{4}\right) $ **[order in a magnetic field ]{}**
0.4truecm
To obtain results for energy levels to $O\left( \alpha^{4}\right) $ we solve the radial equations (24) perturbatively using hydrogen-like radial functions (non-relativistic Schrödinger form $f_{n,J,m_{J}}^{Sch}\left( p\right)
$) in momentum space \[9\]. The energy eigenvalues can be calculated from the matrix equation, which follows from (24)$$\begin{aligned}
E\int p^{2}dp\mathbb{F}^{\dagger}\left( p\right) \mathbb{F}\left( p\right)
& =\int p^{2}dp\left( \omega_{p}+\Omega_{p}\right) \mathbb{F}^{\dagger
}\left( p\right) \mathbb{F}\left( p\right) \\
& -\frac{m_{1}m_{2}}{\left( 2\pi\right) ^{3}}\int_{0}^{\infty}\frac
{p^{2}dp}{\sqrt{\omega_{p}\Omega_{p}}}\int_{0}^{\infty}\frac{q^{2}dq}{\sqrt{\omega_{q}\Omega_{q}}}\mathbb{F}^{\dagger}\left( p\right)
\mathbb{K}\left( p,q\right) \mathbb{F}\left( q\right) ,\nonumber\end{aligned}$$ If the system (24) has been decoupled, or the contribution of nondiagonal elements of the $\mathbb{K}\left( p,q\right) $ matrix with given radial functions in (34) is zero, Eq. (34) immediately gives the perturbative solution for the energy levels. As shown in Ref. \[19\], the contribution of the nondiagonal elements $\mathcal{K}_{12}^{ope}$ and $\mathcal{K}_{21}^{ope}$ in Eq. (34) to order $O\left( \alpha^{4}\right) $ is zero for the $\ell$-mixing states. Thus, in the present scheme the energy corrections for $\ell$-mixing states can be calculated independently for $\ell=J-1$ and $\ell=J+1$ states. As a result, all triplet states with $\ell=J\mp1$ can be treated as pure states. In the case of spin-mixed states the kernel matrix $\mathbb{K}^{ope}$ has been diagonalized in the basis of quasi-states (26), however the magnetic part of the interaction gives rise to the non-diagonal terms (32). Since we are solving the system (34) perturbatively, we can use a new basis $\left\vert ext\right\rangle =C_{1}^{\prime}\left\vert sg_{q}\right\rangle
+C_{2}^{\prime}\left\vert tr_{q}\right\rangle $ which mixes the quasi-states with arbitrary constants $C_{1}^{\prime}$ and $C_{2}^{\prime}$. This leads to a two-level problem with the solution $E_{n,J,m_{J}}=\left( H_{11}+H_{22}\right) /2\pm\left( \left( \left( H_{11}-H_{22}\right) /2\right)
^{2}+H_{12}H_{21}\right) ^{1/2}$, where $H_{11}=H_{11}^{ope}+H_{11}^{ext}$, $H_{22}=H_{22}^{ope}+H_{22}^{ext}$, $H_{12}=H_{21}=H_{12}^{ext}=H_{21}^{ext}$. In our case $\left\vert H_{11}-H_{22}\right\vert >>H_{12}H_{21}$, because the difference $\left\vert H_{11}-H_{22}\right\vert $ is of the order of the fine structure which dominates over the hyperfine splitting and the magnetic perturbation $H_{12}$. Therefore, we can approximate $E_{n,J,m_{J}}\approx
H_{11},\ H_{22}$.
The results are presented in the form$$\Delta E_{n,J,m_{J}}=E_{n,J,m_{J}}-\left( m_{1}+m_{2}\right) +\frac{\left(
Z\alpha\right) ^{2}m_{r}}{2n^{2}}=\Delta E_{n,J}\left( \alpha^{4}\right)
+\Delta E_{J,m_{J}}^{ext},$$ where $Q_{2}=Z Q_{1}$. The energy corrections $\Delta E_{n,J}\left(
\alpha^{4}\right) $ due to the kernels $\mathbb{K}^{\left( ope\right)
}\left( p,q\right) $ were obtained previously \[19\]. The corrections $\Delta
E_{n,J}\left( \alpha^{4}\right) $ contain spin-spin interactions that lead to the HFS which is illustrated in Fig. 1 for the low-lying excited states. A detailed analysis of the HFS to $O\left( \alpha^{4}\right) $ is provided in \[19\]. We note that the HFS of the $1S_{1/2}$ and $2S_{1/2}$ states is obtained in agreement with the known Fermi splittings \[9\], *i.e.*, $\Delta E_{HFS}\left( 1S_{1/2}\right) =\left( Z\alpha
\right) ^{4}m_{r}\left( 8m_{r}/3M\right) $, and $\Delta E_{HFS}\left(
2S_{1/2}\right) = \left( Z\alpha\right) ^{4}m_{r}\left( m_{r}/3M\right)
$, where $M=m_{1}+m_{2}$. The HFS of states with $\ell>0$, however, is more complicated \[19\]. In standard spectroscopic notation it has the form $$\begin{aligned}
\Delta E_{HFS}\left( n,\ell,s_{s}\right) & \equiv\Delta E_{n,J=\ell
+1}-\Delta E_{n,J=\ell,s_{s}}\\
& =\frac{\left( Z\alpha\right) ^{4}m_{r}}{n^{3}}\frac{1}{2\ell+1}\left(
\frac{2\ell+1-\xi^{-1}}{4\ell\left( \ell+1\right) }+\frac{2m_{r}}{M}\frac
{1}{2\ell+3}\right) ,\nonumber\end{aligned}$$$$\begin{aligned}
\Delta E_{HFS}\left( n,\ell,s_{t}\right) & \equiv\Delta E_{n,J=\ell,s_{t}}-\Delta E_{n,J=\ell-1}\\
& =\frac{\left( Z\alpha\right) ^{4}m_{r}}{n^{3}}\frac{1}{2\ell+1}\left(
\frac{2\ell+1-\xi^{-1}}{4\ell\left( \ell+1\right) }+\frac{2m_{r}}{M}\frac
{1}{2\ell-1}\right) ,\nonumber\end{aligned}$$ where the quantity $\xi$ is defined by Eq. (56), but with the quantum number $J$ replaced by $\ell$. The formulae (36) and (37) are valid for all quantum numbers $n$, $\ell$ and for any mass values $m_{1}, m_{2}$. The weak external field further splits the energy levels. Eqs. (36) and (37) give excellent agreement with experiment for the HFS \[19\].
The energy corrections $\Delta E_{J,m_{J}}^{ext}$ remove the degeneracy with respect to the $m_{J}$ quantum number. The solution of Eq. (34) in the above-made approximation can be written in the form of Eq. (1) for all states.
For all *pure states* ($\ell=J\mp1$) we obtain the following results:
for $\ell=J-1$:$$g_{1,2}=1-\frac{m_{1,2}}{m_{1}+m_{2}}\frac{J-1}{J}+\left( \frac{g_{s_{1,2}}}{2}-1\right) \frac{1}{J},$$ for $\ell=J+1$:$$g_{1,2}=1-\frac{m_{1,2}}{m_{1}+m_{2}}\frac{J+2}{J+1}-\left( \frac{g_{s_{1,2}}}{2}-1\right) \frac{1}{J+1} .$$
For *spin–mixed* states $\ell=J\neq0$ the solution of Eq. (34), as mentioned, reduces to a standard two-energy level problem. The diagonal elements of the kernel matrix give the first-order Zeeman splitting (in $O\left( B\right) $) in the quasi-spin representation (26), which was used to derive the HFS energies (36) and (37). Note that the non-diagonal elements give a contribution to higher-order Zeeman splitting corrections.
To first order in the magnetic field strength we obtain the Landé factors to be$$g_{1}=\frac{m_{2}}{m_{1}+m_{2}}\left( 1-\frac{1\pm\xi}{2J\left( J+1\right)
}\right) +\frac{g_{s_{1}}}{2}\left( \frac{1\pm\xi}{2J\left( J+1\right)
}\pm2\frac{\left\vert m_{1}-m_{2}\right\vert }{m_{1}+m_{2}}\xi\right) ,$$$$g_{2}=\frac{m_{1}}{m_{1}+m_{2}}\left( 1-\frac{1\mp\xi}{2J\left( J+1\right)
}\right) +\frac{g_{s_{2}}}{2}\left( \frac{1\mp\xi}{2J\left( J+1\right)
}\mp2\frac{\left\vert m_{1}-m_{2}\right\vert }{m_{1}+m_{2}}\xi\right) ,$$ where the upper sign is taken for $sg_{q}$ and lower sign for $tr_{q}$ states respectively. Our expressions (40-41) are symmetrical with respect to the masses of the two particles. Obviously all these first-order Zeeman corrections, $\Delta E_{J,m_{J}}^{ext}$, vanish for the positronium case ($m_{1}=m_{2}=m_{e},$ $Z=1$), as expected. The intrinsic factors $g_{s_{1,2}}$ associated with the spins of the individual particles can include QED corrections.
In the case when $m_{2}>>m_{1}$ our general results agree with the result from Eqs. (2,4) in which the orbital motion of the heavy particle is ignored. It is only in this limit (as discussed below Eq. (27)), that the total angular momenta of the individual particles are not related through the common angular momentum $\ell$, and can be written as $j_{1}=\ell\pm1/2$, and $j_{2}=1/2$. In $j_{1}$-$j_{2}$ coupling, the eigenstates are taken to be the eigenstates of the operators $\widehat{\mathbf{j}}_{1}^{2}=\left( \widehat{\mathbf{L}}+\widehat{\mathbf{s}}_{1}\right) ^{2}$, $\widehat{\mathbf{j}}_{2}^{2}=\widehat{\mathbf{s}}_{2}^{2}$, $\widehat{\mathbf{J}}^{2}$, and $\widehat{J}_{z}$, and are designated as $\left\vert j_{1}j_{2}Jm_{J}\right\rangle $ in contrast to the spin-mixed $\left\vert LsJm_{J}\right\rangle $ and pure states $\left\vert LSJm_{J}\right\rangle $ which diagonalize the expectation value of the Hamiltonian to order $O\left(
\alpha^{4}\right) $. To facilitate the comparison we make the following replacement of quantum numbers: $F\rightarrow J$, $J\rightarrow j_{1}$, $L\rightarrow\ell_{1}=\ell$, $S\rightarrow s_{1}$, $I\rightarrow s_{2}$. It follows that for all pure states $\ell=J\mp1$, formulae (38-39) and (2-4) give the same result, namely, $$g_{1}=1+\left( \frac{g_{s_{1}}}{2}-1\right) \frac{1}{J},\ \ \ \ \ \ \ \ g_{2}=\frac{g_{s_{2}}}{2}\frac{1}{J},$$ for $\ell=j_{1}-1/2\overset{or}{=}J-1$ and $$g_{1}=1-\left( \frac{g_{s_{1}}}{2}-1\right) \frac{1}{J+1},\ \ \ \ \ \ \ \ \ g_{2}=-\frac{g_{s_{2}}}{2}\frac{1}{J+1}$$ for $\ell=j_{1}+1/2\overset{or}{=}J+1$.
In the limit $m_{2}>>m_{1}$ the energy levels of spin-mixed states $\Delta
E_{J,m_{J}}^{ext\left( sg_{q}\right) }$ and $\Delta E_{J,m_{J}}^{ext\left(
tr_{q}\right) }$ reduce to $\Delta E_{j_{1}=\ell+1/2,J,m_{J}}^{ext}$ and $\Delta E_{j_{1}=\ell-1/2,J,m_{J}}^{ext}$ respectively, and the Landé factors given by (40-41) take the form $$g_{1}=\frac{2J+3}{2J+1}+\left( \frac{g_{s_{1}}}{2}-1\right) \frac{1}{J},\ \ \ \ g_{2}=-\frac{1}{J+1}-\left( \frac{g_{s_{2}}}{2}-1\right)
\frac{1}{J+1},$$ and for $\ell=j_{1}+1/2\overset{or}{=}J\ $($tr_{q}$)$$g_{1}=\frac{2J-1}{2J+1}-\left( \frac{g_{s_{1}}}{2}-1\right) \frac{1}{J+1},\ \ \ \ \ \ g_{2}=\frac{1}{J}+\left( \frac{g_{s_{2}}}{2}-1\right)
\frac{1}{J}$$ for $\ell=j_{1}-1/2\overset{or}{=}J\ $($sg_{q}$). Here the decoupling angle $\beta$ is given by $\xi\approx1/\left( 2\ell+1\right) $ in the $m_{2}>>m_{1}$ case.
Formula (4) gives a similar result for the second particle, but for the lighter particle Eq. (2) yields$$g_{1}=\frac{2J+3}{2J+1}+\left( \frac{g_{s_{1}}}{2}-1\right) \frac
{2J+3}{\left( 2J+1\right) \left( J+1\right) }$$ for ($sg_{q}$) states, and$$g_{1}=\frac{2J-1}{2J+1}-\left( \frac{g_{s_{1}}}{2}-1\right) \frac
{2J-1}{J\left( 2J+1\right) }$$ for ($tr_{q}$) states. This result agrees with (44-45) only in the particular case of $g_{s_{1}}=2$. Note that most theoretical and experimental results are concerned with $nS_{1/2}\left( J=1\right) $ states for which the mass ratiocorrection in (38) disappears. Thus our results will be most useful for $\ell>0$ states.
In Tables 1-3 we present results of our calculations of the $g$-factors for the first excited states in hydrogen, muonium, and muonic hydrogen respectively. Only states with non-zero total angular momentum are included. Eqs. (40-41) are used for the spin-mixed states $P_{1/2\left( J=1\right) }$, $P_{3/2\left( J=1\right) }$, $D_{3/2\left( J=2\right) }$, $D_{5/2\left(
J=2\right) }$, Eq. (38) is used for the pure state $P_{3/2\left( J=2\right)
}$.
0.4truecm
**Table 1.** $g$-factors for the electron ($g_{1}$) and proton ($g_{2}$) respectively in excited atomic hydrogen states. Results from the present calculation, Eqs. (38,40) for electrons, are compared with Eq. (2) in the top half of the table. For protons the bottom half displays the present results from Eqs.(38,41) in comparison with Eq. (4). Each row contains in the upper part the Landé factor where the intrinsic $g_{s}$-value is corrected for the anomaly (see text), while the numbers below are based upon the Dirac value $g_{s}=2$.
0.2truecm
$\begin{tabular}
[c]{||c|r|r|r|r|r||}\hline
$pe\^[-]{}$ & $P\_[1/2( J=1) ]{}$ & $P\_[3/2( J=1) ]{}$ &
$P\_[3/2( J=2) ]{}$ & $D\_[3/2( J=2) ]{}$ & $D\_[5/2( J=2) ]{}$\\\hline
$g\_[1]{}$ using Eqs.~(38), (40) & $
\[c\][c]{}0.33237\
0.33296
$ & $
\[c\][c]{}1.66740\
1.66622
$ & $
\[c\][c]{}1.00032\
0.99973
$ & $
\[c\][c]{}0.59912\
0.59951
$ & $
\[c\][c]{}1.40008\
1.39949
$\\\hline
$g\_[1]{}$ using Eq.~(2) & $
\[c\][c]{}0.33294\
1/3
$ & $
\[c\][c]{}1.66765\
5/3
$ & $
\[c\][c]{}1.00059\
1
$ & $
\[c\][c]{}0.59965\
3/5
$ & $
\[c\][c]{}1.39945\
7/5
$\\\hline
$g\_[2]{}$ using Eqs.~(38), (41) & $
\[c\][c]{}1.79321\
1.00036
$ & $
\[c\][c]{}-0.89597\
-0.49955
$ & $
\[c\][c]{}0.89670\
0.50027
$ & $
\[c\][c]{}0.89691\
0.50049
$ & $
\[c\][c]{}-0.59711\
-0.33283
$\\\hline
$g\_[2]{}$ using Eq.~(4) & $
\[c\][c]{}1.79285\
1
$ & $
\[c\][c]{}-0.89642\
-1/2
$ & $
\[c\][c]{}0.89642\
1/2
$ & $
\[c\][c]{}0.89642\
1/2
$ & $
\[c\][c]{}-0.59762\
-1/3
$\\\hline
\end{tabular}
\ $
0.4truecm
**Table 2.** Same as in Table 1, but for muonium. The Landé factor for the electron is $g_{1}$, and for the muon it is $g_{2}$.
0.2truecm
$\begin{tabular}
[c]{||c|r|r|r|r|r||}\hline
$\^[+]{}e\^[-]{}$ & $P\_[1/2( J=1) ]{}$ & $P\_[3/2( J=1) ]{}$
& $P\_[3/2( J=2) ]{}$ & $D\_[3/2( J=2) ]{}$ &
$D\_[5/2( J=2) ]{}$\\\hline
$g\_[1]{}$ using Eqs.~(38), (40) & $
\[c\][c]{}0.329451\
0.33004
$ & $
\[c\][c]{}1.66392\
1.66274
$ & $
\[c\][c]{}0.99818\
0.99759
$ & $
\[c\][c]{}0.59527\
0.59566
$ & $
\[c\][c]{}1.39610\
1.39551
$\\\hline
$g\_[1]{}$ using Eq.~(2) & $
\[c\][c]{}0.33294\
1/3
$ & $
\[c\][c]{}1.66765\
5/3
$ & $
\[c\][c]{}1.0006\
1
$ & $
\[c\][c]{}0.59965\
3/5
$ & $
\[c\][c]{}1.39945\
7/5
$\\\hline
$g\_[2]{}$ using Eqs.~(38), (41) & $
\[c\][c]{}1.00434\
1.00320
$ & $
\[c\][c]{}-0.49657\
-0.49598
$ & $
\[c\][c]{}0.50299\
0.50241
$ & $
\[c\][c]{}0.50491\
0.50433
$ & $
\[c\][c]{}-0.32923\
-0.32884
$\\\hline
$g\_[2]{}$ using Eq.~(4) & $
\[c\][c]{}1.00117\
1
$ & $
\[c\][c]{}-0.50058\
-1/2
$ & $
\[c\][c]{}0.50058\
1/2
$ & $
\[c\][c]{}0.50058\
1/2
$ & $
\[c\][c]{}-0.33372\
-1/3
$\\\hline
\end{tabular}
$
0.4truecm
**Table 3.** Same as in Table 1, but for muonic hydrogen. The Landé factor for the muon is $g_{1}$, and for the proton it is $g_{2}$.
0.2truecm
$\begin{tabular}
[c]{||c|r|r|r|r|r||}\hline
$p\^[+]{}\^[-]{}$ & $P\_[1/2( J=1) ]{}$ & $P\_[3/2( J=1) ]{}$
& $P\_[3/2( J=2) ]{}$ & $D\_[3/2( J=2) ]{}$ &
$D\_[5/2( J=2) ]{}$\\\hline
$g\_[1]{}$ using Eqs.~(38), (40) & $
\[c\][c]{}0.26707\
0.26765
$ & $
\[c\][c]{}1.58232\
1.58116
$ & $
\[c\][c]{}0.95019\
0.94960
$ & $
\[c\][c]{}0.50961\
0.51000
$ & $
\[c\][c]{}1.30580\
1.30521
$\\\hline
$g\_[1]{}$ using Eq.~(2) & $
\[c\][c]{}0.33295\
1/3
$ & $
\[c\][c]{}1.66764\
5/3
$ & $
\[c\][c]{}1.00058\
1
$ & $
\[c\][c]{}0.59965\
3/5
$ & $
\[c\][c]{}1.39946\
7/5
$\\\hline
$g\_[2]{}$ using Eqs.~(38), (41) & $
\[c\][c]{}1.85425\
1.06317
$ & $
\[c\][c]{}-0.80664\
-0.41198
$ & $
\[c\][c]{}0.94682\
0.55040
$ & $
\[c\][c]{}0.98584\
0.58982
$ & $
\[c\][c]{}-0.50225\
-0.23836
$\\\hline
$g\_[2]{}$ using Eq.~(4) & $
\[c\][c]{}1.79285\
1
$ & $
\[c\][c]{}-0.89642\
-1/2
$ & $
\[c\][c]{}0.89642\
1/2
$ & $
\[c\][c]{}0.89642\
1/2
$ & $
\[c\][c]{}-0.59762\
-1/3
$\\\hline
\end{tabular}
$
0.4truecm
Our calculations (given to five digits after the decimal point) are to be compared with the ($m_{2}\rightarrow\infty$) results (2-4). Upper values for each $g$-factor have taken into account the following anomalous magnetic moment values: $g_{e}/2=1.00118$, $g_{p}/2=1.792847$ $g_{\mu
}/2=1.001166$ \[1,9,14\]. The intrinsic proton anomaly reflects the fact that it is not a fundamental particle, while in the case of electrons and muons the lowest-order radiative correction was included. The lower values in each row were calculated with $g_{s_{1,2}}=2$. We used the following values for the mass ratios: $m_{p}/m_{e}\approx1836.15267$ and $m_{\mu}/m_{e}\approx
206.76828$ \[1,9,14\].
For the case of muonium we find that the deviations between the present results and those obtained from the one-body limit are in the few-percent range. The muon as the heavier of the two particles acquires a systematically increased Landé factor, while the values are always lowered for the electron.
For muonic hydrogen the effects are more pronounced, and range from 3 to 25 % for the states shown in Table 3. Only those results which take the anomalous magnetic moment of the proton into account should be considered as physically relevant. The systematics are similar to those shown in Table 2 for muonium, with the largest decrease in the Landé factor observed for the muon in the $P_{1/2(J=1)}$ state (-25 %), while the largest increase (19 %) for the proton $g$-value occurs in the $D_{5/2(J=2)}$ state.
For atomic hydrogen the effect is smallest due to the small $e/p$ mass ratio. Given that atomic spectroscopy is far more advanced in hydrogen than in muonic atoms one should not neglect these corrections. For the two above-mentioned states which are most affected we observe about 0.1 % deviations in the electron and proton Landé factors respectively.
As mentioned above, our results are applicable only in low magnetic fields, such that the hyperfine energy splitting exceeds the Zeeman splitting, namely$$B<<\frac{\Delta E_{HFS}\left( n,\ell\right) }{\mu_{B}^{\ast}g}.$$ Thus, formula (48), for $2P_{3/2}$ states, requires that $B<<300\ $gauss for muonium and $B<<100\ $gauss for hydrogen.
0.8truecm
**[5. Conclusion]{}**
0.4truecm
We have used the Hamiltonian variational method in reformulated QED to derive relativistic stationary-state equations for two-fermion systems in an external magnetic field. These equations can include interactions to any order of the coupling constant, at least in principle. The classification of the states follows naturally from the conserved quantum numbers which appear in the trial state (6). For given total angular momentum $J$ there are, in general, coupled equations, both for mixed-spin states, and for triplet mixed-$\ell$ states (cf. Eq. (24)). We present explicit forms for the kernels (momentum-space potentials) for the case of a constant, weak external magnetic field.
We solved the radial equations perturbatively to obtain the Zeeman splitting of the HFS to order $O\left( \alpha^{4}\right) $, and calculated the $g$-factors for the system of two bound fermions. Our results are applicable to all states (*i.e.* for all quantum numbers) and any fermion masses. In the limit $m_{2}>>m_{1}$ our formulae reproduce the well-known $g$-factor result. For the spin-mixed states, however, Eq. (2) is found to be not exact if the intrinsic magnetic moment is different from the Dirac value $g_{s_{1}}=
2$.
0.8truecm
[[**[Acknowledgment]{}**]{} ]{}
[0.4truecm ]{}
The financial support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
0.8truecm
**[Appendix. One-photon exchange kernels for the spin-mixed states to order ]{}**${\Large \alpha}^{4}$
0.4truecm
We use the notation $z=\left( p^{2}+q^{2}\right) /2pq$, and $\mathrm{Q}_{J}(z)$ is the Legendre function of the second kind \[24\]. The contributions of the various terms to the kernel are as follows ($\ell=J\ (J\geq
1),\;\mathcal{P}=(-1)^{J+1}$):
*(i)* orbital term$$\begin{aligned}
\mathcal{K}_{11}^{\left( orb\right) }\left( p,q\right) & =\mathcal{K}_{22}^{\left( orb\right) }\left( p,q\right) =\frac{2\pi Q_{1}Q_{2}}{pq}\mathrm{Q}_{J}(z)\\
& +\frac{\pi Q_{1}Q_{2}}{2m_{1}m_{2}}\left( \left( \frac{m_{1}}{m_{2}}+\frac{m_{2}}{m_{1}}-\left( J-1\right) \right) \left( \frac{p}{q}+\frac{q}{p}\right) \mathrm{Q}_{J}(z)+2\left( J+1\right) \mathrm{Q}_{J+1}(z)\right) ,\nonumber\end{aligned}$$ *(ii)* spin-orbit interaction$$\mathcal{K}_{11}^{\left( s-o\right) }\left( p,q\right) =0,$$$$\mathcal{K}_{12}^{\left( s-o\right) }(p,q)=-\frac{\pi Q_{1}Q_{2}}{2m_{1}m_{2}}\left\vert \frac{m_{1}}{m_{2}}-\frac{m_{2}}{m_{1}}\right\vert
\frac{2\sqrt{J\left( J+1\right) }}{2J+1}\left( \mathrm{Q}_{J+1}\left(
z\right) -\mathrm{Q}_{J-1}\left( z\right) \right) ,$$$$\mathcal{K}_{22}^{\left( s-o\right) }(p,q)=-\frac{\pi Q_{1}Q_{2}}{2m_{1}m_{2}}\left( \frac{m_{1}}{m_{2}}+\frac{m_{2}}{m_{1}}+4\right)
\frac{1}{2J+1}\left( \mathrm{Q}_{J+1}\left( z\right) -\mathrm{Q}_{J-1}\left( z\right) \right) ,$$ *(iii)* spin-spin interaction$$\mathcal{K}_{11}^{\left( s-s\right) }\left( p,q\right) =0,$$$$\mathcal{K}_{22}^{\left( s-s\right) }(p,q)=\frac{\pi Q_{1}Q_{2}}{m_{1}m_{2}}\frac{1}{2J+1}\left( \mathrm{Q}_{J+1}\left( z\right) -\mathrm{Q}_{J-1}\left( z\right) \right) .$$ The diagonalization condition $$\tan2\beta\left( \mathcal{K}_{22}\left( p,q\right) -\mathcal{K}_{11}\left(
p,q\right) \right) =2\mathcal{K}_{12}\left( p,q\right) .$$ determines the parameters $\beta$ and $\xi$.:$$\tan2\beta=2\left\vert \frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right\vert
\sqrt{J\left( J+J\right) },$$ and$$\xi=\left( 4\left( \frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right) ^{2}J\left(
J+1\right) +1\right) ^{-1/2}.$$ Therefore, we obtain the diagonalized kernels for the quasi-states$$\begin{aligned}
& \mathcal{K}^{\left( sg_{q}\right) },\mathcal{K}^{\left( tr_{q}\right)
}\\
& =\mathcal{K}_{11}^{\left( orb\right) }+\frac{\xi\pm1}{\sqrt{1-\xi^{2}}}\mathcal{K}_{12}^{\left( s-o\right) }\nonumber\\
& =\frac{2\pi Q_{1}Q_{2}}{pq}\mathrm{Q}_{J}(z)\nonumber\\
& +\frac{\pi Q_{1}Q_{2}}{2m_{1}m_{2}}\left( \left( \frac{m_{1}}{m_{2}}+\frac{m_{2}}{m_{1}}-\left( J-1\right) \right) \left( \frac{p}{q}+\frac{q}{p}\right) \mathrm{Q}_{J}(z)+2\left( J+1\right) \mathrm{Q}_{J+1}(z)\right) \nonumber\\
& -\frac{\pi Q_{1}Q_{2}}{2m_{1}m_{2}}\frac{\xi\pm1}{\xi\left( 2J+1\right)
}\left( \mathrm{Q}_{J+1}\left( z\right) -\mathrm{Q}_{J-1}\left( z\right)
\right) .\nonumber\end{aligned}$$ 0.2truecm
[0.8truecm ]{}
[[**[References]{}**]{} ]{}
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---
abstract: 'In this letter we consider the superradiant phase transition of a two-component Fermi gas in a cavity across a Feshbach resonance. It is known that quantum statistics plays a crucial role for the superradiant phase transition in atomic gases; in contrast to bosons, in a Fermi gas this transition exhibits strong density dependence. We show that across a Feshbach resonance, while the two-component Fermi gas passes through the BEC-BCS crossover, the superradiant phase transition undergoes a corresponding crossover from a fermionic behavior on the weakly interacting BCS side, to a bosonic behavior on the molecular BEC side. This intricate statistics crossover makes the superradiance maximally enhanced either in the unitary regime for low densities, in the BCS regime for moderate densities close to Fermi surface nesting, or in the BEC regime for high densities.'
author:
- Yu Chen
- Hui Zhai
- Zhenhua Yu
title: Superradiant Phase Transition of Fermi Gases in a Cavity across a Feshbach Resonance
---
Recent experiment has combined atomic Bose-Einstein condensates and cavity quantum electrodynamics together where atom-light interactions are strongly enhanced [@Kimble; @Esslinger0; @Reichel]. A superradiant phase transition driven by external pumping field has been observed, across which atoms form a density-wave order [@Black; @Esslinger1; @Esslinger2], and roton mode softening has been found in the vicinity of this superradiant phase transition [@Esslinger_Roton]. Theoretical studies have extended to investigate noninteracting Fermi gases inside a cavity [@Larson; @Meystre; @Subir; @Chen; @Simons; @Piazza]. It is shown that the Fermi statistics plays a crucial role in the superrandiant phase transition at moderate and high densities [@Chen; @Simons; @Piazza]. At moderate densities, Fermi surface displays a nesting structure and strongly enhances superradiance, when the momentum of the cavity light field matches the nesting momentum. While at high densities, Pauli blocking effect forbids a large part of atom-light scattering processes, and consequently, strongly suppresses superradiance. The strong density dependence marks the major difference between superradiances in ideal Fermi gases and Bose gases.
During the past decade, another important development in cold atom physics is the study of strongly interacting two-component Fermi gases and the BEC-BCS crossover utilizing Feshbach resonance [@Giorgini; @Chin]. The inter-atomic $s$-wave scattering length $a_\text{s}$ can be continuously changed by a Feshbach resonance, and the dimensionless parameter $-1/k_\text{F}a_\text{s}$ ($k_\text{F}$ is the Fermi momentum in the noninteracting limit) controls the BEC-BCS crossover. In the BCS limit of the crossover $-1/k_\text{F}a_\text{s}\rightarrow +\infty$, fermions form loosely bound Cooper pairs and the low-energy response is dominated by fermionic quasi-particles; the system recovers a noninteracting Fermi gas. In the BEC limit $-1/k_\text{F}a_\text{s}\rightarrow -\infty$, Cooper pairs transform into tightly bound bosonic molecules, and the system responses to external fields mainly as bosons. In between, when $a_\text{s}$ is so large that $-1/k_\text{F}a_\text{s}\approx 0$, the system is in a strongly interacting regime and its response shall exhibit both bosonic and fermionic characters.
So far, Fermi gases with inter-atomic interactions in a cavity have been barely studied. In this work we consider a two-component Fermi gas in a cavity across a Feshbach resonance. Given that the gas can be continuously tuned between the fermion limit and the boson limit, and that atoms with different statistics have been shown to behave differently in the superradiant phase transition [@Chen; @Simons; @Piazza], the motivation of our study is to address how the statistics crossover manifests itself in the superradiant phase transition across a Feshbach resonance, and the physical consequence of this crossover. In experiments, the superradiant phase transition is usually driven by increasing the strength of pumping field. In this work we will reveal nontrivial dependence of the critical pumping strength on the density of fermions $n$ and the inter-atomic interaction strength characterized by $-1/k_\text{F}a_\text{s}$. Our results represent a manifestation of the interplay between strong interactions from Feshbach resonance and strong atom-light coupling in a cavity, and will provide insight for future experiments.
*Model.* Our system is schematically shown in Fig. (\[setup\]). Applied on the Fermi gas is a pumping field that consists of two laser beams counter-propagating along the $\hat{y}$ direction, with frequency $\omega_p$ and polarization in the $\hat z$ direction. The single-mode cavity field of interest varies in the $\hat{x}$ direction, with frequency $\omega_c$ close to $\omega_p$. The system is described by the Hamiltonian $\hat{\mathcal{H}}=\hat{\mathcal{H}}_\text{at}-\delta_c \hat{a}^\dag\hat{a}$, where $\hat a$ is the field operator for the cavity mode and $\delta_c=\omega_c-\omega_p$ is the cavity field detuning.
![ Experimental setup scheme. The pumping field propagates along the $\hat y$ direction shown by the red arrows. The cavity field is represented by the wiggled lines in the $\hat x$ direction. Fermions of different spins are shown in different colors. []{data-label="setup"}](Setup){width="6.0cm"}
The Hamiltonian experienced by the fermions has two parts $\hat{\mathcal{H}}_\text{at}=\hat{\mathcal{H}}_0+\hat{\mathcal{H}}_\text{int}$. The free fermion part is [@Chen] $$\begin{aligned}
{\label{eq:Hamil_BECBCS}}
& \hat{\mathcal{H}}_0=\sum_{\sigma=\uparrow,\downarrow}\int d^3{\bf r}\hat{\psi}_\sigma^\dag({\bf r})H_0\hat{\psi}_\sigma({\bf r}),\\
& H_0= \frac{{\bf p}^2}{2m}-\mu+V({\bf r})+ \eta({\bf r})(\hat{a}^\dag+\hat{a})+U({\bf r})\hat{a}^\dag \hat{a},\label{fh}\end{aligned}$$ where $\hat{\psi}_\sigma({\bf r})$ are the fermion field operators with (peudo) spin index $\sigma\in\{\uparrow, \downarrow\}$. The pumping field and the cavity field generate respectively the optical potentials $V({\bf r})=V_0\cos^2(k_0y)$ and $U({\bf r})=U_0\cos^2(k_0x)$, and the coupling between the pumping field and the cavity field comes from an interference term $$\eta({\bf r})=\eta_0\cos k_0x\cos k_0y$$ with $\eta_0=\sqrt{U_0 V_0}$, $k_0$ is the wavevector magnitude of both the pumping field and the cavity mode [@supple]. The recoil energy $E_{\text{r}}= \hbar^2 k_0^2/2m$ is defined for latter use.
The inter-atomic interaction nearby a Feshbach resonance is described by the Hamiltonian $$\begin{aligned}
{\label{eq:Hamil_BECBCS1}}
\hat{\mathcal{H}}_{\rm int}=g\int d^3{\bf r}\hat{\psi}^\dag_\uparrow({\bf r})\hat{\psi}^\dag_\downarrow({\bf r})\hat{\psi}_\downarrow({\bf r})\hat{\psi}_\downarrow({\bf r}).\end{aligned}$$ The bare attractive inter-fermion interaction coupling $g$ is renormalized to the $s$-wave scattering length $a_s$ via $m/4\pi a_s=1/g+m\Lambda/2\pi^2$ with $\Lambda$ the momentum cutoff. This attractive interaction between fermions lead to fermion pairing and a Fermi superfluid ground state.
*Ground State in Non-superradiant Phase.* Before entering the suprradiance phase, $\langle \hat{a}^\dag\rangle=\langle \hat{a}\rangle=\langle \hat{a}^\dag\hat{a}\rangle=0$, fermions only experience a one-dimensional lattice $V({\bf r})$ along the direction of the pumping field, and the single-particle eigenstates are the Bloch states $|{\bf k}\rangle$ satisfying $H_0|{\bf k}\rangle=\xi_{\bf k}|{\bf k}\rangle$. By expanding $\hat{\psi}_\sigma({\bf r})=\sqrt{1/V}\sum_{\bf k}\phi_{\bf k}({\bf r})\hat{c}_{\bf k\sigma}$ with $\langle {\bf r}|{\bf k}\rangle=\phi_{\bf k}({\bf r})$ and $V$ the gas volume, we introduce fermion pairing order parameter $\Delta_0={(g/V)}\sum_{\bf k}\langle c_{\bf k\uparrow}c_{-\bf k\downarrow}\rangle$$(\neq0)$. Here we assume the lattice $V({\bf r})$ is weak and we have ignored pairing at non-zero crystal momentum. With this assumption, the order parameter $\Delta_0$ is determined by the gap equation and the number equation [@supple]. In this Fermi superfluid state, the single-particle Green’s functions are given by $$\begin{aligned}
G_0^{\uparrow\uparrow(\downarrow\downarrow)}({\bf k},i\omega_n)=-\frac{i\omega_n+(-)\xi_{\bf k}}{\omega_n^2+E_{\bf k}^2},\end{aligned}$$ $$\begin{aligned}
G_0^{\uparrow\downarrow}({\bf k},i\omega_n)=G_0^{\downarrow\uparrow}({\bf k},i\omega_n)=\frac{\Delta_0}{\omega_n^2+E_{\bf k}^2},\end{aligned}$$ with $E_{\mathbf k}=\sqrt{\xi_{\mathbf k}^2+\Delta_0^2}$ and the fermionic Matsubara frequencies $\omega_n=(2n+1)\pi/\beta$ for $n=0,\pm1,\pm2,\dots$, and $\beta$ the inverse of temperature. Their diagrams are shown in Fig. (\[diagram\])(a1). The components $G_0^{\uparrow\uparrow}(k)$ and $G_0^{\downarrow\downarrow}(k) (k\equiv({\bf k},i\omega_n))$ describe the propagation of particles and holes, respectively, while $G_0^{\uparrow\downarrow}(k)$ and $G_0^{\downarrow\uparrow}(k)$ are the anomalous Green’s functions proportional to the pairing gap $\Delta_0$ which we take to be real.
![(a1): the Feynman diagrams for propagators; the first line for the particle and hole propagators and the second line for the anomalous Green’s functions. (a2): the interaction vertex between the cavity field and the fermions. (b,c), the Feynman diagrams corresponding to fermonic and bosonic contribution to density-wave susceptibility. The last line is the propagate of cooper pairs. \[diagram\]](SusceptibilityC "fig:"){width="5.5cm"}\
*Condition for Superradiant Phase Transition.* The superradiant phase transition is determined by the instability of non-superradiance toward developing non-zero $\langle \hat{a}^\dag \hat{a}\rangle$. As shown in Ref. [@Esslinger1], the superradiant phase transition occurs simultaneously with the formation of density-wave order of atoms with momentum ${\bf Q}_{\pm\pm}$, where ${\bf Q}_{\pm\pm}=(\pm k_0,\pm k_0,0)$ is the momentum transfer between the cavity field and the pumping field. That is to say, $\langle\hat{a}\rangle$ is proportional to the density-wave order parameter $\Theta=\int d^3{\bf r}\langle \hat n({\bf r})\rangle\eta({\bf r})/\eta_0$ with $\hat n({\bf r})=\sum_\sigma\psi^\dagger_\sigma({\bf r})\psi_\sigma({\bf r})$. By integrating out the fermion fields, one can obtain the free-energy of the system in the form $\mathcal{F}=\mathcal{C}\Theta^2$ [@supple], where $\mathcal{C}$ changing sign from positive to negative gives the critical pumping field strength $\eta_0^{\rm cr}$ for the superradiance transition [@Esslinger1; @Chen; @supple] $$\begin{aligned}
\eta_0^{\rm cr}=\frac{1}{2}\sqrt{\frac{\tilde{\delta}_c^2+\kappa^2}{-\tilde{\delta}_c\chi}}\label{cri}.\end{aligned}$$ Here $\kappa$ is the cavity mode decay rate, and $\tilde{\delta}_c$ is the shifted cavity mode detuning $\tilde{\delta}_c=\delta_c-\int d^3\mathbf r \langle \hat n({\bf r})\rangle U(\mathbf r)$, which is assumed to be red-detuned ($\tilde{\delta}_c<0$). The most essential quality determining this transition is the density-wave order susceptibility of the Fermi superfluid defined as $$\begin{aligned}
\chi=-\frac1{2\beta\eta_0^2}{\rm Tr}[
\langle T\hat n(\mathbf r_1,t_1)\hat n(\mathbf r_2,t_2) \rangle \eta(\mathbf r_1)\eta(\mathbf r_2)].\end{aligned}$$ Here $\rm Tr$ includes the integration of the spatial coordinates and the imaginary times, $T$ is the time ordered operator. The expectation value of the fermion operators $\langle \dots\rangle$ is taken in the non-superradiant Fermi superfluid phase. A larger $\chi$ means that the Fermi gas has stronger tendency toward forming a density-wave order at a momentum ${\bf Q}_{\pm\pm}$, and it is easier for the Fermi gas to enter the superradiant phase; in another word, the critical pumping strength shall be smaller.
*Density-Wave Order Susceptibility.* In order to capture both fermionic and bosonic responses of a Fermi superfluid, the density-wave order susceptibility $\chi$ should be calculated by the random phase approximation. This approximation maintains conservation laws [@baym; @yu] and guarantees that one can recover the free fermion and the free boson results in the limits $-1/(k_\text{F}a_\text{s})\rightarrow \pm\infty$, respectively. Within this approximation, we have $\chi=\chi_\text{F}+\chi_\text{B}$ and
$$\begin{aligned}
\!\!\chi_F\!&=&\!\!-\frac{1}{2\beta\eta_0^2}\!\sum_{{\bf k},{\bf k}',n}\!\!\!\!\left(G_0^{\uparrow\uparrow}({\bf k}',i\omega_n)G_0^{\uparrow\uparrow}({\bf k},i\omega_n)+G_0^{\downarrow\downarrow}({\bf k}',i\omega_n)G_0^{\downarrow\downarrow}({\bf k},i\omega_n)\right)|\langle{\bf k}|\eta({\bf \hat{r}})|{\bf k}'\rangle|^2, \label{chiF}\\
\!\!\chi_B\!&=&\frac1V\!\!\sum_{\mathbf q={\bf Q}_{\pm\pm}}A_{\mathbf q}^*\Pi_{\mathbf q}A_{\mathbf q}, \label{chiB1}\\
A_{\bf q}&=&\!\!-\frac{1}{2\beta\eta_0}\sum_{{\bf k},{\bf k}',n}\left(G_0^{\uparrow\uparrow}({\bf k}',i\omega_n)
G_0^{\downarrow\uparrow}({\bf k},i\omega_n)-G_0^{\downarrow\downarrow}({\bf k}',i\omega_n)
G_0^{\uparrow\downarrow}({\bf k},i\omega_n)\right)\langle{\bf k}'|\eta({\bf \hat{r}})|{\bf k}\rangle\langle{\bf k}|\gamma_{\bf q}({\bf \hat{r}})|{\bf k}'\rangle, \label{chiB2}\\
\!\!\Pi^{-1}_{\mathbf q}\!&=&\!\!-\frac{1}{{g}}+\frac{1}{V\beta}\sum_{{\bf k},{\bf k}',n}\sum_{\bf q'={\bf Q}_{\pm\pm}}\left(
G_0^{\uparrow\uparrow}({\bf k}',i\omega_n)G_0^{\downarrow\downarrow}({\bf k},i\omega_n)+
G_0^{\uparrow\downarrow}({\bf k}',i\omega_n)G_0^{\downarrow\uparrow}({\bf k},i\omega_n)\right)
\!\!\langle{\bf k}'|\gamma_{\bf q}({\bf \hat{r}})|{\bf k}\rangle\!\langle{\bf k}|\gamma_{\mathbf q'}({\bf \hat{r}})|{\bf k}'\rangle, \label{chiB3}\end{aligned}$$
where $\gamma_{\bf q}({\bf r})\!=\!\cos({\bf q}\cdot{\bf r})$ is the mode factor for Cooper pair fluctuations.
The fermionic response $\chi_\text{F}$ is due to that the cavity field couples to the fermonic excitations of the Fermi superfluid by breaking up Cooper pairs. The Feynman diagrams corresponding to $\chi_\text{F}$ are shown in Fig. (\[diagram\])(b). The diagrams describe the process that a fermion with momentum ${\bf k}$ is scattered to momentum ${\bf k}^\prime$, where the momentum transfer comes from the photon momentum change from the pumping field to the cavity field, as denoted by the vertex in Fig. (\[diagram\])(a2). Since all fermions are paired in the Fermi superfluid phase, this process must be accompanied by pair breaking. In the BCS limit where the pairing gap $\Delta_0$ is small and pairs are easy to break, $\chi_\text{F}$ is dominant in $\chi$ and could recover the transition for free fermions in the limit of vanishing pairing gap [@supple]. While in the BEC limit this process is strongly suppressed because of large pairing gap.
The bosonic response $\chi_\text{B}$ originates from the process that the cavity field excites nonzero momentum Cooper pairs and corresponds to the diagram shown in Fig. (\[diagram\])(c). In this process, one of the two fermions in the Cooper pair, say, the one with momentum ${\bf k}$, is scattered to momentum ${\bf k+q}$ by a photon. Thus, the Cooper pair acquires a finite momentum ${\bf q}$ and propagates with this fixed momentum ${\bf q}$ (up to a reciprocal lattice vector along $\hat{y}$). After another scattering with a photon, the Cooper pair returns to zero-momentum. Because of weak lattice $V(\mathbf r)$ we only take into account the contributions from the scattered Cooper pairs of momentum $|q_y|\le q_0$. The Cooper pair propagator $\Pi_{{\bf q}}^{-1}$ is given in Eq. (\[chiB3\]) and its diagram in Fig. (\[diagram\])(c) which is a summation of ladder diagrams. There are two ways for a Cooper pair to propagate, through multiple scattering and through vacuum fluctuations, respectively. Both are included in Eq. (\[chiB3\]) and in the bottom of Fig. (\[diagram\])(c). In the BEC limit $\chi_\text{B}$ is dominant in $\chi$ and $\chi_\text{B}\sim a_s\Delta_0^2\sim n$ recovering the free boson result. While in the BCS limit, $\chi_\text{B}\sim \Delta_0^2/k_F$ is exponentially suppressed [@supple].
![Dimensionless susceptibilities $f_F$, $f_B$ and $f$ vs $-1/k_{\text F}a_{\text s}$ are plotted in (a), (b), (c) respectively with the pumping strength $V_0/E_r$ fixed at $0.1$ and $\nu$ taking $0.2$, $0.5$ and $4.0$. The bottom row is a pictorial representation of the BEC-BCS crossover. \[fffbf\]](StatisticalCrossover){width="7.0cm"}
We plot in Fig. (\[fffbf\]) the dimensionless susceptibility $f\equiv E_\text{r}\chi/N$, as well as its fermionic and bosonic constituent $f_F\equiv E_\text{r}\chi_F/N$ and $f_B\equiv E_\text{r}\chi_B/N$, as functions of the BEC-BCS crossover controlling parameter $-1/k_\text{F}a_\text{s}$, for different filling fractions $\nu=(k_F/k_0)^3$.
First, Fig. (\[fffbf\])(a) shows that $f_\text{F}$ exhibits strong density dependence on the BCS side. Around a moderate density of $(k_\text{F}/k_0)^3=0.5$, the Fermi surface nesting is optimal, and $f_\text{F}$ becomes much larger than the low-density limit value $f_\text{F}=1/2$ [@Chen]. This is the regime where Fermi surface nesting strongly enhances superradiance, as discussed in noninteracting Fermi systems [@Chen; @Simons; @Piazza]. On the other hand, for high densities, say, $(k_\text{F}/k_0)^3=4$ in Fig. (\[fffbf\])(a), $f_\text{F}$ is much smaller than $1/2$ on the BCS side. This is the regime where the Pauli exclusion principle strongly suppresses superradiance. As approaching the BEC side, $f_\text{F}$ is strongly suppressed for all densities. Second, as shown in Fig. (\[fffbf\])(b), $f_\text{B}$ approaches the value of noninteracting bosons (also $=1/2$) in the BEC limit, independent of densities. While on the BCS side, for all densities $f_\text{B}$ is strongly suppressed.
Figure (\[fffbf\])(c) shows the central result of this work. The total $f$ exhibits different features for different densities as $-1/k_{\text F}a_{\text s}$ varies. The most intriguing case is at relatively low-densities, say, $(k_\text{F}/k_0)^3=0.2$, where $f$ displays a maximum in the unitary regime ($1/a_\text{s}\approx0$). This maximum is because in this regime, the bosonic contribution already takes off while the fermionic contribution has not damped out. While for moderate densities of Fermi surface nesting regime, $f$ monotonically increases as $-1/k_\text{F}a_\text{s}$ increases from the BEC limit to the BCS limit, due to the Fermi-surface nesting enhancement of $\chi$ on the BCS side. In contrast, for high densities, $f$ monotonically decreases, due to the Pauli blocking suppression of $\chi$ on the BCS side. The total $f$ has strong density dependence on the BCS side where it is dominated by the fermionic behavior, and becomes less and less sensitive to density in the BEC limit where it is dominated by the bosonic behavior. This change of $f$ with $-1/k_\text{F}a_\text{s}$ between the two limits is the manifestation of statistics crossover in superradiance.
*Phase Diagram.* The boundary separating the normal and the superradiant phases can be obtained by solving Eq. (\[cri\]) [@supple]. In Fig. (\[pd\]), we plot the phase diagram in term of $V_0$ and $\tilde{\delta}$ for different densities and interaction strengths. In the BCS region, Fig. (\[pd\]) (a) shows that the moderate density $\nu=0.5$ is the easiest to be superradiant. In the unitary region as shown in Fig. (\[pd\]) (b) the low density $\nu=0.2$ is the easiest to be superradiant primarily due to the maximum of $f$ mentioned above in this part of the parameter space. On the BEC side, Fig. (\[pd\]) (c) shows that the density dependence diminishes since it shall be washed out completely in the BEC limit.
![Phase diagram for superradiance for different interaction parameter $1/k_{\text F}a_{\text s}=-2.0$ (a), $0.8$(b) and $5.6$ (c), and with different densities $\nu=0.2$, $0.5$ and $4.0$. For all cases, we take the typical experimental parameters $\kappa/E_r=250$ and $U_0N_{\rm at}/E_r=10^3$.[]{data-label="pd"}](PhaseDiagramC){width="8.0cm"}
*Conclusion.* We have presented basic features of the superradiant phase transition of two-component Fermi gases across a Feshbach resonance. The main results are: i) On the BCS side of resonance the superradiant phase transition shows strong density dependence, similar as noninteracting Fermi gas; While on the BEC side it gradually becomes density independent, similar as noninteracting bosons. ii) Superradiance is mostly enhanced in the unitary regime for low density, in the BCS regime for moderate density, and in the BEC regime for high density. In this work, we have only focused on the superradiant phase transition itself. Inside the superradiant phase, the additional lattice due to the cavity field will further modify the single-particle dispersion, which will feedback to the Fermi superfluid. Furthermore, the quantum fluctuation of the cavity field will also generate additional effect on the Fermi superfluid. The properties of Fermi superfluids in the superradiant phase would be a subject for future studies.
*Acknowledgements.* This work is supported by Tsinghua University Initiative Scientific Research Program, NSFC under Grant No. 11004118, No. 11174176, No. 11104157, No. 11474179 and No. 11204152, and NKBRSFC under Grant No. 2011CB921500.
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Supplementary Material
======================
**Model Parameters.** The single fermion Hamiltonian Eq. (\[fh\]) is obtained by adiabatically integrating out all the electronic excitation states of the atoms in the rotating wave frame. The parameters in Eq. (\[fh\]) are related to the experimental tunable parameters as $V_0=\Omega_p^2/\delta_a$, $U_0=g_0^2/\delta_a$ and $\eta_0=\sqrt{U_0 V_0}=\Omega_p g_0/\delta_a$. Here $\Omega_p$ is the pumping field strength, $\delta_a$ is the pumping laser frequency detuning with respect to electronic transitions of atoms, and $g_0$ is the coupling strength between the cavity mode and the fermions.
**Mean Field Equation for Fermi Superfluids.** When the lattice induced by the pumping field is not strong, we can approximate $\Delta({\bf r})={g}\langle\hat{\psi}({\bf r})\hat{\psi}({\bf r})\rangle=\Delta_0$ as a constant. The mean field gap equation becomes $$\begin{aligned}
\Delta_0=\frac{{g}}{\beta V}\sum_{{\bf k},i\omega_n}G^{\uparrow\downarrow}({\bf k},i\omega_n)=-\frac{1}{V}\sum_{\bf k}\frac{{g}\Delta_0}{2E_{\bf k}}.\end{aligned}$$ Together with the number equation $n=\frac{1}{V}\sum_{\bf k\sigma}\langle c_{\bf k\sigma}^\dag c_{\bf k\sigma}\rangle=\frac{1}{\beta V}\sum_{\bf{k},i\omega_n}(G^{\uparrow\uparrow}({\bf k},i\omega_n)-G^{\downarrow\downarrow}({\bf k},i\omega_n))$, or more explicitly, $$\begin{aligned}
n=\frac{1}{V}\sum_{\bf k}\left(1-\frac{\xi_{\bf k}}{E_{\bf k}}\right),\end{aligned}$$ we can determine $\Delta_0$ and $\mu$ self-consistently for a given pumping strength $V_0/E_r$ and given density $n$.
**Instability Condition for Superradiant Phase Transition.** The mean field value of the cavity field $\alpha=\langle \hat a\rangle$ satisfies [@Esslinger1] $$\begin{aligned}
i\frac{\partial\alpha}{\partial t}=(-\tilde{\delta}_c-i\kappa)\alpha+\eta_0\Theta,\end{aligned}$$ where $\Theta=\int d^3{\bf r}\langle \hat n({\bf r})\rangle\eta({\bf r})/\eta_0$ is the fermion density order parameter. The introduced decay rate $\kappa$ is to model the weak leakage of electromagnetic field from the high-*Q* cavity. In a steady state, $\partial_t\alpha=0$; we have $$\begin{aligned}
\label{eq:PhaseLock}
\alpha=\frac{\eta_0\Theta}{\tilde{\delta}_c+i\kappa}, \label{mean_alpha}\end{aligned}$$ which locks the cavity field to the fermion density order parameter. Both $\alpha$ and $\Theta$ are zero in the normal phase and become nonzero in the superradiant phase.
To the second order of $\alpha$, the effective free energy can be obtained as $$\begin{aligned}
F_\alpha=-\frac{1}{\beta}\ln {\cal Z}_\alpha=-\tilde{\delta}_c\alpha^*\alpha-\chi(\alpha^*+\alpha)^2,\label{fa}\end{aligned}$$ where ${\cal Z}_\alpha={\rm Tr}e^{-\beta H}$ with a specified $\alpha$. By substituting (\[mean\_alpha\]) into Eq. (\[fa\]), we have $$\begin{aligned}
F=-\left[\frac{\tilde{\delta}_c}{\tilde{\delta}_c^2+\kappa^2}+\chi
\frac{4\tilde{\delta}_c^2\eta_0^2}{(\tilde{\delta}_c^2+\kappa^2)^2}\right]\eta_0^2\Theta^2,\label{freeenergy}\end{aligned}$$ which determines the superradiant transition when the quadratic coefficient of $\Theta$ changes its sign.
**Explicit Expression for Density-Wave Order Susceptibility.** The explicit expressions for the density-wave order susceptibility within the BCS theory are $$\begin{aligned}
\chi_F=&\sum_{{\bf k},{\bf k}'}\frac{\left|\langle{\bf k}'|\eta({\bf \hat{r}})|{\bf k}\rangle\right|^2}{2\eta_0^2(E_{\bf k}+E_{{\bf k}'})}\left(1-\frac{\xi_{\bf k}\xi_{{\bf k}'}-\Delta_0^2}{E_{\bf k}E_{{\bf k}'}}\right),\label{chi33e}\\
A_{{\bf q}}=&\sum_{{\bf k},{\bf k}'}\frac{\langle {\bf k}'|\eta({\bf \hat{r}})|{\bf k}\rangle\langle {\bf k}|\gamma_{\bf q}(\bf \hat{r})
|{\bf k}'\rangle
}{2\eta_0(E_{\bf k}+E_{{\bf k}'})}
\frac{\Delta_0(\xi_{\bf k}+\xi_{{\bf k}'})}{E_{\bf k}E_{{\bf k}'}},\\
\Pi_{{\bf q}}^{-1}=&-\frac{V}{g}+\sum_{{\bf k},{\bf k}'}\sum_{\bf q={\bf Q}_{\pm\pm}}
\frac{2\langle {\bf k}'|\gamma_{\bf q}(\bf \hat{r})|{\bf k}\rangle \langle {\bf k}|\gamma_{\bf q'}(\bf \hat{r})|{\bf k}'\rangle
}{2(E_{\bf k}+E_{{\bf k}'})}\nonumber\\
&\times\left(1+\frac{\xi_{\bf k}\xi_{{\bf k}'}-\Delta_0^2}{E_{\bf k}E_{{\bf k}'}}\right).\end{aligned}$$ In the BSC limit, the factor $1-\xi_{\bf k}\xi_{{\bf k}'}/E_{\bf k}E_{\bf k'}\approx n_F(\xi_{\bf k})-n_F(\xi_{\bf k'})$ with $n_F$ the Fermi-Dirac distribution; $\chi_F$ becomes the same as it is for free fermions [@Chen]. In the BEC limit, $A_{{\bf q}}\approx m^2a_s \Delta_0V\delta_{\mathbf q, \bf Q_{\pm,\pm}}/16\pi$ and $\Pi_{\bf q}\approx-16\pi/k_0^2ma_s$, $f\approx1/2$ which is the same as it is for condensed noninteracting bosons [@Esslinger1; @Chen].
**Determination of Phase Boundary.** The boundary between the non-superradiant and superradiant phases is determined by Eq. (\[cri\]). Since $\eta_0=\sqrt{U_0V_0}$, $\chi=Nf/E_r$, and $f$ is a dimensionless function of dimensionless parameters $V_0/E_r$ and $\nu=(k_F/k_0)^3$, we could recast Eq. (\[cri\]) in the form $$\begin{aligned}
\frac{V_0}{E_r}\frac{NU_0}{E_r}f\left(\nu,\frac{V_0}{E_r}\right)=\frac{x^2+(\kappa/E_r)^2}{-x}\end{aligned}$$ by introducing $x=\tilde{\delta}_c/E_r$. We take typical experimental values $NU_0/E_r=10^3$ and $\kappa/E_r=250$. Thus at each given pumping strength $V_0/E_r$ we can obtain the critical strengths of the cavity detuning $\tilde{\delta}_c$.
|
---
abstract: 'Motivated by recent bounds for charge diffusion in critical matter, we investigate the question: What sets the scale for charge diffusion in a scale-invariant system? To make our statements precise, we analyze the diffusion pole in an exactly solvable model for a Mott transition in the presence of a interaction term. To achieve scale invariance, we limit our discussion to the flat-band regime. We find in this limit that the diffusion pole which would normally obtain at finite is pushed to zero resulting in a vanishing of the diffusion constant. Consequently, scale-invariance precludes any reasonable definition of the diffusion constant. Nonetheless, we do find that a scale can be defined, all be it, irrelevant to diffusion, which is the product of the band velocity and the density of states.'
author:
- 'Philip W. Phillips'
- Chandan Setty
- Shuyi Zhang
bibliography:
- 'FlatBand.bib'
title: 'Absence of a Charge Diffusion Pole in an Exactly Solvable Interacting Flat Band Model in d-dimensions'
---
Scale invariance is both a simplifying and problematic organizing principle for strongly correlated systems. On the one hand, it dictates that the correlation functions must obey power-law decay with a universal length scale but on the other, it precludes the presence of a natural energy scale from entering the transport properties. For electronic systems, this implies that the Fermi energy cannot enter any transport property if scale invariance is present. This is particularly problematic in describing the strange metal in the cuprates as both scale-invariance and a breakdown of the particle concept have been advocated[@homes2004; @qcrit2; @Marel2003; @Valla2110; @Anderson1997; @abrahams; @pll2015] to be operative. In fact, the key characteristic of quantum critical systems, namely the presence of a dissipation rate that scales linearly with temperature, the Planckian limit of dissipation, has been shown[@zaanen] to undergird the experimental observation of Homes’ law[@homes2004] in the cuprates. Since Homes’ law is about the dc conductivity just above the normal state, some natural scale should govern charge diffusion in critical matter. Hence the question as to what sets the scale for transport in quantum critical systems and scale-invariant systems emerges more generally. Since the Planckian[@zaanen] rate, $\hbar/k_BT$ is the natural scale for dissipation, a secondary question is does this set an upper bound for dissipation and hence serve to bound any subsequent charge diffusion? Neither of these questions has been answered definitively since violations[@pakhira2014; @kuang2015; @kovtun2008; @kovtun2003; @myers2007; @pang2009] to charge diffusion bounds[@hartnoll; @blake], based on the Planckian upper bound coupled with a diffusion constant parameterized by some phenomenological velocity, abound.
In this paper, we limit our discussion to the former question, namely what sets the scale in scale-invariant systems for charge diffusion? To make our conclusions precise, we focus on an exactly solvable model for strongly correlated electrons that exhibits a metal-insulator transition regardless of the dimension. To impose scale invariance, we focus on the flat-band limit. In the flat band limit, the energy-spacing between energy levels, $\Delta E$, is the smallest scale in the problem and as a consequence, the band velocity vanishes. Hence, the resultant diffusion constant defined by $D\sim v^2\tau$ must vanish if $\tau$ is finite. The scattering time in our problem is governed by scalar impurity interactions and hence is finite. We show explicitly that even for an interacting system, , the diffusion pole which would normally occur at finite is pushed to zero- and continues to be dictated by scale invariance. Hence, , and strictly , there is no energy scale that emerges which permits a reasonable definition of the charge diffusion constant in scale-invariant system. Nonetheless, we do find that a flat-band scale can be defined, , if one were to consider the product of the band velocity and the density of states. Since the density of states diverges, the product can be finite. We analyze this constant here and show that
The model[@hk1992; @hk1996] we analyze has long-range non-local interactions with standard tight-binding hoppings, $$\begin{aligned}
\nonumber
H &=& -t \sum_{\langle j,l\rangle,\sigma} \pqty{ c^\dagger_{j\sigma} c^{}_{l\sigma} + h.c. } - \mu \sum_{j\sigma} c^\dagger_{j\sigma} c^{}_{j\sigma}\\
&& + \frac{U}{N} \sum_{j_1..j_4}\delta_{j_1+ j_3, j_2+ j_4} c^\dagger_{j_1\uparrow} c^{}_{j_2\uparrow} c^\dagger_{j_3\downarrow} c^{}_{j_4\downarrow},\\ \nonumber
$$ where the first and second terms denote the local hopping and chemical potential and are set by the scale $\gamma$. The last term is the infinite-range Hubbard-like interaction $U$; this term is non-zero for electrons that scatter in such a way that their position vectors satisfy the constraint of center of mass conservation given by $j_1+j_3 = j_2 + j_4$. This model predates the SYK[@sy1993; @k2015] model by 2 years, though it is considerably less studied. Although both models contain non-local interactions, the current model is exactly solvable as a result of the conservation of the center of mass in the interaction term. Similar models with long range correlations were studied in Refs [@baskaran1991exactly; @muthukumar1994toy; @continentino1994scaling]. The integrability of this model, without resorting to a $1/N$ expansion as in the SYK model[@sy1993; @k2015], is best seen in momentum space $$H = \sum_{\vec k}H_{\vec k} = \sum_{\vec k} \left (\xi(\vec k)(\hat{n}_{\vec k\uparrow} + \hat{n}_{\vec k\downarrow} ) + U \hat{n}_{\vec k\uparrow} \, \hat{n}_{\vec k\downarrow}\right),
\label{eq:kSpaceHK}$$ from which it is clear that the kinetic and potential energy terms commute. Here $\xi(\vec k) \equiv \epsilon(\vec k ) - \mu$ and $\hat{n}_{\vec k \sigma} \equiv c_{\vec k \sigma}^{\dagger} c_{\vec k \sigma}$. We see that in this model, different momentum states are decoupled, and the Hamiltonian can be diagonalized by states in the number representation for each $\vec k$. The basis of states which spans Eq. \[eq:kSpaceHK\] are given by $\left(\mid0\rangle_{\vec k}, \mid \uparrow\rangle_{\vec k}, \mid\downarrow \rangle_{\vec k}, \mid\uparrow\downarrow \rangle_{\vec k}\right)$ with eigenvalues $\left(0,\xi(\vec k),\xi(\vec k), 2 \xi(\vec k) + U\right)$, for each momentum point $\vec k$. Regardless of the simplicity of this model, a non-trivial Mott transition exists as can be seen from the single-particle Green function, $$G_0(\vec k,i\omega_n)_U = \left(\frac{g(\vec k, U)}{i \omega_n - \xi(\vec k)+\frac{U}{2}} + \frac{1- g(\vec k, U)}{i\omega_n - \left(\xi(\vec k) - \frac{U}{2}\right)}\right),
\label{eq:GreensFunction}$$ where the function $g(\vec k, U)$ is defined by $$g(\vec k, U) = \frac{1+ e^{-\beta \xi(\vec k)}}{1+ 2 e^{-\beta \xi(\vec k)} + e^{-\beta\left(2\xi(\vec k) + U\right)}},$$ with $\beta$ being the inverse temperature, and $i\omega_n$, the fermionic Matsubara frequency. At half-filling, the Green function is a sum of poles at $E^\pm_k=\xi(\vec k)\pm U/2$. A gap exists between the two bands when $U> 4td$, that is, when $U$ exceeds the non-interacting bandwidth. The transition to the gapped state is of the Mott type because ${\cal \Re }G(\omega,\vec k)=0$ identically when $\xi_{\vec k}=0$. That is, in the gapped state, the Fermi surface of the non-interacting system is converted into a surface of zeros, the fingerprint[@stanescu2007] of Mottness. This coincidence obtains entirely because of particle-hole symmetry[@stanescu2007] of the underlying Hamiltonian. We have as our starting point then an exactly solvable model which exhibits a Mott transition regardless of the spatial dimension.
Before we use this model to analyze the existence or lack there of a diffusion pole, we review the standard formulation for a free-electron gas in which scalar impurities act as the source of momentum relaxation. One natural way to obtain such a response is to extend the corresponding density response function of the electron liquid by replacing the infinitesimally small adiabatic continuation parameter $\eta$ with the inverse impurity scattering life time $\frac{1}{\tau}$, i.e., by making the substitution $\chi^{imp}(\vec q, \omega + i\eta) \rightarrow \chi(\vec q, \omega + \frac{i}{\tau})$, where $\chi(\vec q, \omega)$ ($\chi^{imp}(\vec q, \omega)$) is the density response function of the electron liquid without (with) impurities. However, it was pointed out [@mermin1970lindhard] that such a naive substitution does not respect the continuity equation. To remedy this defect, Mermin instead proposed an alternate approach where, provided one can define a local chemical potential $\mu(\vec q,\omega)$, it is possible to use the continuity equation to relate the impurity response to the analytically continued density response function as $$\chi^{imp}(\vec q, \omega) = \frac{\chi(\vec q, \omega + \frac{i}{\tau})}{1+ (1- i \omega \tau)^{-1}\left( \frac{\chi(\vec q, \omega + \frac{i}{\tau})}{\chi(\vec q, 0)}-1 \right)}.
\label{Eq:ImpurityResponse}$$ It can be clearly seen that, in general, $\chi^{imp}(\vec q, \omega ) \neq \chi(\vec q, \omega + \frac{i}{\tau})$. The two of them become the same only when the density response without impurities is energy independent.\
*Electron gas:* To lead our discussion toward a flat band response, let us quickly recall the linear response behavior of a $d$-dimensional electron gas in the presence of scalar impurities. By substituting the $d-$dimensional Lindhard function, $\chi_0(\vec q, \omega)$, in place of the density response into Eq. \[Eq:ImpurityResponse\] and making the replacement $\chi_0(\vec q, \omega + i\eta) \rightarrow \chi_0(\vec q, \omega + \frac{i}{\tau})$, we obtain in the diffusive limit ( $\omega \tau\ll 1$ and $qv_F \tau \ll 1$ where $v_F$ is the Fermi velocity and $q$ is the magnitude of $\vec q$) [@rammer2004quantum] $$\chi^{imp}_0(\vec q, \omega) \simeq \frac{-N_0 D q^2}{ - i \omega + D q^2}.
\label{eq:DiffusionResponse}$$ Here $D$ is the diffusion constant given by $D = \frac{v_F^2 \tau}{d}$, and $N_0$ is the density of states at the Fermi level. An important feature of the form of the impurity response appearing in Eq. \[eq:DiffusionResponse\] is the presence of a diffusion pole at $\omega = -i D q^2$. The diffusion pole results in a strong enhancement of the density fluctuation spectrum at low energies and, depending on the spatial dimension, plays a crucial role in drastically modifying the quasiparticle scattering rates. Importantly, the form of the diffusion pole appearing the denominator of Eq. \[eq:DiffusionResponse\] indicates that the electron density, $n(\vec r, t)$, relaxes according to the diffusion equation given by $$\frac{\partial n(\vec r, t)}{\partial t} = D \nabla^2 n(\vec r, t),
\label{eq:DiffusionEquation}$$ and the average mean square electron displacement scales as $\langle \vec r~^2(t) \rangle \sim 2 d D t$.\
*Flat band with no interactions:* To begin our analysis for the impurity diffusion in a free flat band, it is useful to provide a precise definition of what we mean by a flat band at the very outset. At an operational level, we define a flat band as a system satisfying two conditions:\
(i) the bandwidth/dispersion is the smallest energy scale in the problem, i.e., if $\epsilon(\vec k)$ is the energy dispersion, $\epsilon(\vec k) - \epsilon(\vec k + \vec q) \equiv \gamma \ll \omega, \tau^{-1}, T$ and,\
(ii) $\gamma\rightarrow 0$ and $\Lambda^d \equiv a^{-d} \rightarrow \infty$ such that the product $\gamma \Lambda^d $ goes to a constant ($a$ is the lattice spacing).\
From now on, we will choose $\gamma$ to be a small constant that tends to zero. It is even possible, in principle, to supplement this constant with momentum dependence. Our results will, however, remain unchanged as long as its width goes to zero by satisfying the two above conditions. Note that as a consequence of the conditions stated above, the zero temperature dependence appearing in the sections below are obtained by *first* taking $\frac{\epsilon(\vec k)}{T}\rightarrow0$ *and then* $\frac{T}{E}\rightarrow 0$, where $E$ could be any other remaining energy scale in the problem. Interchanging this order of limits will, in general, yield results not relevant to a flat band. With these points in mind and noting that the system is set at half filling by choosing the chemical potential $\mu$ to zero, it is easy to write the non-interacting flat band Lindhard function as (in the limit $\gamma\rightarrow0$) $$\chi_0\left(\omega+ \frac{i}{\tau}\right)_{FB} = -\frac{R}{t_0^2}.
\label{eq:FreeFlatBand}$$ Here we have defined the *flat band constant* $R = 2 \gamma \Lambda^d \tau^2$ which takes a finite non-zero value in the $\gamma \rightarrow 0$ limit and characterizes the flat band, and $t_0 = t + i$ where $t\equiv \omega \tau$. It is worth noticing that, unlike the case of the $d-$dimensional electron gas, both the flat band Lindhard function as well as its resulting impurity response function (that appears below) is independent of the momentum transfer $\vec q$. This is not unexpected given that, by definition, a flat band has no spatial dynamics. To obtain the impurity response for the flat band, we also need the flat band Lindhard function at zero frequency. This quantity can be seen to diverge as $\chi_0(0)_{FB} = \frac{R}{\gamma^2}$ as $\gamma\rightarrow 0$ due to the chemical potential being set to zero. Substituting for $\chi_0\left(\omega+ \frac{i}{\tau}\right)_{FB}$ and $\chi_0(0)_{FB}$ into Eq \[Eq:ImpurityResponse\], we obtain in the diffusion limit ($\omega\tau\ll 1$) $$\chi_0^{imp}(\omega)_{FB} \simeq \frac{R}{-i t} = \frac{R'}{-i \omega},
\label{eq:FreeFlatBandImpurity}$$ where $R' \equiv R/\tau$. The real part of $\chi_0^{imp}(\omega)_{FB}$ (of order $R$) has not been included in Eq \[eq:FreeFlatBandImpurity\] as it is of order $\omega\tau$ smaller than the imaginary part. Through a comparison of the structure of the poles in Eqs \[eq:FreeFlatBandImpurity\] and \[eq:DiffusionResponse\], one can conclude that the corresponding electron density, $n(\vec r, t)$, does not relax according to the diffusion equation Eq \[eq:DiffusionEquation\], entirely due to the lack of spatial dynamics. Therefore, in the strictest sense, there is no electron diffusion in a flat band. However, Eq. \[eq:FreeFlatBandImpurity\] is still a meaningful quantity as the weight factor $R'$ (or $R$) plays the analogous role of the diffusion constant $D$ times the density of states at the Fermi level, and can therefore be extracted experimentally. $R'$ also plays a central role in defining the characteristics of the electron gas and its properties in the presence of long range interactions will be further explored in the next section. A significant feature of the flat band impurity response (in Eq. \[eq:FreeFlatBandImpurity\]) is that the ‘diffusion’ pole shifts to zero energy as opposed to a non-zero momentum dependent value in a $d-$dimensional electron gas. This shift to zero energy is expected to result in a serious toll on the quasiparticle lifetime near the Fermi surface, which is already shortened considerably in a $d-$dimensional electron gas in the presence of impurities. A more quantitative analysis of the quasiparticle lifetime in a dirty flat band is beyond the scope of this paper and will be the focus of future work.\
The susceptibility bubble for a flat band in the presence of long range interactions is given by (FBLR denotes flat band, long range) $$\chi_0(\vec q, iq_n)_{FBLR} = \frac{1}{\beta V} \sum_{ik_n \vec k \sigma} G_0(\vec k,ik_n)_U G_0(\vec k + \vec q,ik_n + i q_n)_U.$$ This expression can be evaluated in the flat band limit and found to be momentum independent (just as in the case of the free flat band) and will be denoted as $\chi_0(\omega)_{FBLR}$ after analytic continuation. The expression for $\chi_0(\omega)_{FBLR}$ can be substituted into Eq. \[Eq:ImpurityResponse\] to yield an impurity response (see Appendix A) of the form $$\chi_0^{imp}(\omega)_{FBLR} \simeq -\frac{R t_0}{t} \left[ \frac{\kappa_0(g)}{t_0^2} + \frac{\kappa_+(g)}{t_+^2} + \frac{\kappa_-(g)}{t_-^2} \right].
\label{eq:ImpurityFBLR}$$ Here we have defined the dimensionless variables $t_{\pm} \equiv t \pm u + i$, where $u\equiv U \tau$ and $\kappa_0,\kappa_{\pm}$ are functions of $g \equiv g(U) = \frac{2}{3+ e^{-\beta U}}$, and $t$ and $R$ have been previously defined. The functions $\kappa_0,\kappa_{\pm}$ are given by $$\begin{aligned}
\kappa_0(g) &=& \left(g^2 +\frac{2(1-g)^2}{e^{\beta U} + 1}\right)\\
\kappa_+(g) &=& \frac{2g(1-g)}{e^{\beta U} + 1}\\
\kappa_-(g) &=& g(1-g).\end{aligned}$$ There are several aspects of $\chi_0^{imp}(\omega)_{FBLR}$ appearing in Eq. \[eq:ImpurityFBLR\] that are worthy of mention. Firstly, the general form of Eq. \[eq:ImpurityFBLR\] bears some similarities to Eq. \[eq:FreeFlatBandImpurity\]. While the first term is just the free flat band result times a $g$ (and hence temperature) dependent ’weight’ factor, $\kappa_0(g)$, the rest of the two terms are modified by the presence of a non-zero interaction strength $U$. These two terms also appear with their own weight factors $\kappa_{\pm}(g)$. Thus, the impurity response of a flat band system, in the presence of a constant long range interactions, is distributed between pure flat band and interaction-modified flat band terms with their respective weights. For $U=0$ we have $\kappa_0 = \frac{1}{2}$ and $\kappa_{\pm} = \frac{1}{4}$ and, as must be expected, $\chi_0^{imp}(\omega)_{FBLR}$ reduces to $\chi_0^{imp}(\omega)_{FB}$. Secondly, it is useful to study the expression for $\chi_0^{imp}(\omega)_{FBLR}$ in different limits. For a repulsive, finite long range interaction ($U>0$) and $T\rightarrow0$, we have $\kappa_0\rightarrow \frac{4}{9}$, $\kappa_+ \rightarrow 0$ and $\kappa_- \rightarrow \frac{2}{9}$. Thus, the maximum fraction of the response comes from the free flat band while there is no contribution from the $\kappa_+$ term. If we extend the repulsive interaction to $U\rightarrow \infty$, then the contribution from the $\kappa_-$ term is negligible, and we are left with $\chi_0^{imp}(\omega)_{FBLR}\simeq \frac{4}{9}\chi_0^{imp}(\omega)_{FB} < \chi_0^{imp}(\omega)_{FB} $. Thus, one can define an effective flat band constant $\tilde{R}$ which is reduced by a factor of $\frac{4}{9}$ compared to the free flat band value. Similarly when $T\rightarrow0$ for a repulsive interaction with $U\ll\omega$, we have $\chi_0^{imp}(\omega)_{FBLR}\simeq \frac{2}{3}\chi_0^{imp}(\omega)_{FB} < \chi_0^{imp}(\omega)_{FB} $. It is also possible to obtain an enhanced effective flat band constant by choosing an attractive interaction $U$. For example, in the limit when $U<0$ and $T\rightarrow0$, we have $\kappa_0 \rightarrow 2$ and $\kappa_{\pm}\rightarrow0$. This results in $\chi_0^{imp}(\omega)_{FBLR}\simeq 2\chi_0^{imp}(\omega)_{FB} > \chi_0^{imp}(\omega)_{FB}$, where the effective flat band constant is twice the free flat band value. This state of affairs obtains because choosing $U<0$ amounts to each $\vec k$ point being pairwise occupied and hence doubling the effective flat band constant and the impurity response. In contrast, when $U>0$ and large, single occupancy of each momentum point is energetically more favorable, and hence the response is lower than the free flat band value.\
*Momentum dependent interaction $U(\vec k)$:* In the previous section, we assumed the interaction $U$ to be a constant. It would be more meaningful to explore the effect of a momentum-dependent interaction as in the case of a Coulomb interaction. To this end, we modify our interaction to be of the more generalized Yukawa form $U(\vec k) = U + \frac{\alpha}{\lambda^2 + k^2}$ (this amounts to a Coulomb interaction when $U=\lambda =0$), so that we recover Eq \[eq:ImpurityFBLR\] for $\chi_0^{imp}(\omega)_{FBLR}$ when $\alpha=0$. Formally, such a generalization from a constant $U$ to $U(\vec k)$ is fairly straightforward$-$ the variable $U$ appearing in the momentum space representation of the Hamiltonian in Eq. \[eq:kSpaceHK\] and Green function in Eq. \[eq:GreensFunction\] has to be replaced by $U(\vec k)$. However, the ensuing momentum integrals are fairly convoluted and can only be solved in certain limits and simplifying assumptions. Once we understand how the integrals behave in these limits, we can gain insight into what they look like for other parallel cases. For analytical tractability, we will assume that $U$ is repulsive so that $U(\vec k)$ is always positive. In this limit, the momentum integrals can be solved exactly as $T\rightarrow0$ in all three dimensions (see Appendix B) and the response (which we donote as $\chi_0(\omega)_{FBY}$, where $FBY$ stands for Flat Band Yukawa) in the presence of the interaction $U(\vec k)$ becomes
$$\chi_0(\omega)_{FBY}\simeq
\begin{cases}
\hphantom{-}\chi_0(\omega)_{FBLR} + lim_{\gamma\rightarrow0}\left[ \frac{\pi \alpha}{z'^2} \sqrt{\lambda^2 + \frac{\alpha}{z'}} - \frac{\pi \alpha}{z^2} \sqrt{\lambda^2 - \frac{\alpha}{z}} \right]+ \mathscr{O}(\frac{1}{\Lambda})&\text{$d=3$},\\[2ex]
\hphantom{-}\chi_0(\omega)_{FBLR} + lim_{\gamma\rightarrow0}\left[ \alpha~Log\left(\Lambda\right)\left[\frac{1}{z^2} - \frac{1}{z'^2}\right]\right] +\mathscr{O}(\frac{1}{\Lambda}) &\text{$d=2$},\\[2ex]
\hphantom{-}\chi_0(\omega)_{FBLR} + lim_{\gamma\rightarrow0}\left[\frac{\pi \alpha}{z^2 \sqrt{\lambda^2 - \frac{\alpha}{z}}} - \frac{\pi \alpha}{z'^2 \sqrt{\lambda^2 + \frac{\alpha}{z'}}}\right] + \mathscr{O}(\frac{1}{\Lambda}) &\text{$d=1$}.
\end{cases}$$
Here we have defined $z = -\omega - \frac{i}{\tau} + \gamma + U$, $z' = \omega + \frac{i}{\tau} +\gamma - U$, and $\Lambda$ is the large momentum cutoff. To derive the above expressions, we have only kept leading order contributions in $\Lambda$. For each dimension $d$ except $d=2$, the largest contributing terms are of $\mathscr{O}(\Lambda^d)$ and the next highest order is $\mathscr{O}(1)$. In the case of $d=2$, the next highest order is of $\mathscr{O}(Log(\Lambda))$. However, in each dimension, the $\mathscr{O}(\Lambda^d)$ terms recombine to give $\chi_0(\omega)_{FBLR}$, which is $\mathscr{O}(1)$ because we have assumed that $\Lambda^d \gamma$ is a constant of order unity. We must therefore keep the next highest order terms in each case. However, in the limit $\gamma \rightarrow 0$, we have $z' = -z$, and according to our definition of a flat band, all the terms proportional to $\alpha$ vanish (see the formula for $\chi_0(\omega)_{FBY}$ above). *Thus we can conclude that a momentum-dependent interaction of the form $\frac{\alpha}{\lambda^2 + k^2}$ has no effect on the impurity response in the flat band limit*. Although our conclusion is derived for the case when $U$ is repulsive and $T\rightarrow0$, it is easy to see why it holds for other cases as well. The key reason why an interaction of the Yukawa or Coulomb form does not affect the flat band response function (with or without impurities) is that only electrons near the Brillouin zone edges contribute to the response. This can be seen from from our assumption that $\Lambda^d \gamma$ is a constant; hence, electrons not at the boundary do not contribute because $\gamma$ goes to zero faster than the enclosed zone volume at finite $\vec k$. Thus, given that only the edge electrons contribute in a flat band, in the limit that $\Lambda^d\rightarrow \infty$, $\frac{\alpha}{\lambda^2 + k^2}$ has no effect as it goes to zero at the Brillouin zone edges. Therefore a potential of the form of $\frac{\alpha}{\lambda^2 + k^2}$ does not change a flat band response with or without impurities for either attractive or repulsive interactions, even at non-zero temperatures. However, when $U(\vec k)$ is a constant (as in the previous section), the potential has a finite value even at the zone edges and, as a consequence, has a non-trivial effect on the response properties. The irrelevance of a Yukawa or Coulomb-like term is exclusively a property of the flat band and does not hold in the case of a dispersive band due to the fact that the $\mathscr{O}(1)$ $\alpha$ term contributions are, in general, finite and non-zero.\
![Plot of $\chi_0^{imp}(\omega)_{FBLR}$ normalized with the free flat band impurity response (and hence independent of $\omega$) as function of the dimensionless parameter $r= \beta U$. Positive (negative) $r$ corresponds to repulsive (attractive) long range interaction. []{data-label="Temperature"}](Temperature){width="3.5in" height="2"}
*Temperature dependence:* For $d-$dimensional electrons, in the absence of any interactions, the band width sets the only energy scale in the problem. It is with respect to this scale that electrons can be thermally excited into higher energy states away from the Fermi level leading to temperature-dependent response functions. However, in the case of a flat band, where all the states have an equal probability of occupation, there is no notion of ordering of states energetically. Hence, the free flat band response with or without impurities is independent of temperature as reflected in Eqs. \[eq:FreeFlatBand\] and \[eq:FreeFlatBandImpurity\]. An equivalent statement is that in the extremely high temperature limit ($T$ much larger than the band width), the temperature is already large enough so that all momentum states are energetically accessible and, therefore, any smaller changes in temperatures will not have any effect on occupation number dependent observable properties. In the presence of long range interactions of the type appearing in Eq. \[eq:kSpaceHK\], however, $U$ sets the only scale in the problem. It is, therefore, reasonable to expect that observable quantities depend on temperature through the dimensionless parameter $\beta U \equiv r$; Eq. \[eq:ImpurityFBLR\] reflects this expectation. A plot of the $\chi_0^{imp}(\omega)_{FBLR}$ (normalized with the free flat band impurity response, and hence independent of $\omega$) as a function of $r$, for $U$ much smaller than $\omega$, appears in Fig \[Temperature\]. There is a significant temperature dependence of $\chi_0^{imp}(\omega)_{FBLR}$ ($\propto \beta$) only for small $r$ and is featureless asymptotically. This behavior can be understood from the form of the function $g(U)$ which determines the fraction of the response that is split between the free flat band and the part dominated by long-range interactions.\
*Conclusions:* To conclude, we studied the impurity response of a flat band in the diffusion limit with and without long range interactions of the type proposed in Ref. [@hk1992]. Starting from the non-interacting case, we argued that the system does not ’diffuse’ in the traditional sense of a $d-$dimensional electron gas, but found it useful to define the notion of a *flat band constant* that takes a non-zero value inspite of the fact that the Fermi velocity of a dispersionless band is zero. . In the presence of long range interactions, we saw that the flat band constant could be effectively enhanced, suppressed or unaffected depending on whether the interaction is a constant and attractive, a constant and repulsive, or momentum dependent of the Yukawa/Coulomb form respectively. . Finally, we argued the temperature independence of the impurity response for the case of the dirty non-interacting flat band. In the interacting case, the calculated response is $\propto \beta$ for small $\beta U$ and saturates to a constant for larger values. Looking ahead, it would be of interest to further explore the effect of the shift of the diffusion pole to zero energy in the exact scaling form of the quasiparticle life time in dirty flat band systems.\
*Acknowledgments:* We acknowledge support from Center for Emergent Superconductivity, a DOE Energy Frontier Research Center, Grant No. DE-AC0298CH1088. We also thank the NSF DMR-1461952 for partial funding of this project.
APPENDIX A
==========
The form of the Green function for the model proposed in ref. [@hk1992] has been derived in ref. [@hk1996] which we simply re-state here for completeness. Given a quasiparticle dispersion $\xi(\vec k)$, the Green function is given as $$G_0(\vec k,i\omega_n)_U = \left(\frac{g(\vec k, U)}{i \omega_n - \xi(\vec k)} + \frac{1- g(\vec k, U)}{i\omega_n - \left(\xi(\vec k) + U\right)}\right),$$ where the function $g(\vec k, U)$ is defined by $$g(\vec k, U) = \frac{1+ e^{-\beta \xi(\vec k)}}{1+ 2 e^{-\beta \xi(\vec k)} + e^{-\beta\left(2\xi(\vec k) + U\right)}},$$ with $\beta$ being the inverse temperature, $i\omega_n$ is the fermionic Matsubara frequency, and $U$ is the constant long range interaction. The density-density correlation function for such a Green function is given (see, for example, Bruus-Flensberg(2004)) by the generalized pair susceptibility bubble $$\begin{aligned}
\chi_0(\vec q, i q_n) &=& \frac{1}{\beta V} \sum_{i k_n}\sum_{\vec k \sigma} G_0(\vec k,i\omega_n)_U G_0(\vec k + \vec q,i\omega_n + iq_n)_U \\ \nonumber
&=& \frac{1}{\beta V} \sum_{i k_n}\sum_{\vec k \sigma} \Biggl[ \frac{g(\vec k, U) g(\vec k + \vec q, U)}{\left(i k_n - \xi(\vec k)\right)\left( i k_n + i q_n - \xi(\vec k+ \vec q ) \right)} + \frac{\left(1-g(\vec k, U)\right)\left(1- g(\vec k + \vec q, U)\right)}{\left(i k_n - \xi(\vec k) -U\right)\left( i k_n + i q_n - \xi(\vec k+ \vec q) - U \right)} \\
&+& \frac{g(\vec k, U) \left( 1- g(\vec k + \vec q, U)\right)}{\left(i k_n - \xi(\vec k)\right)\left( i k_n + i q_n - \xi(\vec k+ \vec q ) - U\right)} + \frac{\left(1- g(\vec k, U)\right) g(\vec k + \vec q, U)}{\left(i k_n - \xi(\vec k) - U\right)\left( i k_n + i q_n - \xi(\vec k+ \vec q ) \right)} \Biggr].\end{aligned}$$ Performing the Matsubara sums and shifting the momentum from $\vec k \rightarrow -\vec k - \vec q$ in the second of each resulting Fermi distribution, we obtain $$\begin{aligned}
\chi_0(\vec q, \omega + \frac{i}{\tau}) &=& \frac{1}{V} \sum_{\vec k \sigma} \Biggl[g(\vec k, U) g(\vec k + \vec q, U) \Biggl(\frac{n_{\vec k}}{\omega + \frac{i}{\tau} + \xi(\vec k) - \xi(\vec k + \vec q)} + \frac{n_{\vec k}}{-\omega - \frac{i}{\tau} + \xi(\vec k) - \xi(\vec k + \vec q)} \Biggr) \\ \nonumber
&& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! + \left(1- g(\vec k, U)\right) \left(1- g(\vec k + \vec q, U) \right) \Biggl(\frac{n_{\vec k U}}{\omega + \frac{i}{\tau} + \xi(\vec k)_U - \xi(\vec k + \vec q)_U} + \frac{n_{\vec k U}}{-\omega - \frac{i}{\tau} + \xi(\vec k)_U - \xi(\vec k + \vec q)_U} \Biggr) \\ \nonumber
&& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! + g(\vec k, U) \left(1- g(\vec k + \vec q, U)\right) \Biggl(\frac{n_{\vec k}}{ \omega + \frac{i}{\tau} + \xi(\vec k) - \xi(\vec k + \vec q)_U} \Biggr) + g(\vec k + \vec q, U) \left(1- g(\vec k, U) \right) \Biggl(\frac{n_{\vec k U}}{ -\omega - \frac{i}{\tau} + \xi(\vec k) - \xi(\vec k + \vec q)_U} \Biggr) \\ \nonumber
&& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! + g(\vec k + \vec q, U) \left(1- g(\vec k, U)\right) \Biggl(\frac{n_{\vec k U}}{ \omega + \frac{i}{\tau} + \xi(\vec k)_U - \xi(\vec k + \vec q)} \Biggr) + g(\vec k, U) \left(1- g(\vec k + \vec q, U) \right) \Biggl(\frac{n_{\vec k }}{ -\omega - \frac{i}{\tau} + \xi(\vec k)_U - \xi(\vec k + \vec q)} \Biggr) \Biggr],\end{aligned}$$ where we have defined $n_{\vec k} \equiv n_F\left(\xi(\vec k)\right)$, $n_{\vec k U} = n_F\left(\xi_{\vec k} + U\right)$, $\xi(\vec k )_U = \xi(\vec k) + U$, and $n_F(x)$ is the Fermi function. Substituting for $\xi(\vec k) - \xi(\vec k + \vec q) = \gamma$, using the definition of the flat band, and with the help of Eq \[Eq:ImpurityResponse\] appearing in the main text, we obtain the flat band response in the presence of impurties and in the absence of interactions as $$\chi_0^{imp}(\omega)_{FBLR} \simeq -\frac{R t_0}{t} \left[ \frac{\kappa_0(g)}{t_0^2} + \frac{\kappa_+(g)}{t_+^2} + \frac{\kappa_-(g)}{t_-^2} \right].
\label{eq:ImpurityFBLR}$$ Here we have defined the dimensionless variables $t_0 = t + i$, $t_{\pm} \equiv t \pm u + i$, where $t \equiv \omega \tau$, $u\equiv U \tau$ and $\kappa_0,\kappa_{\pm}$ are functions of $g \equiv g(U) = \frac{2}{3+ e^{-\beta U}}$. The flat band constant $R \equiv2 \gamma \Lambda^d \tau^2$. In deriving the above expression for $\chi_0^{imp}(\omega)_{FBLR}$, we have used the fact that $\xi(\vec k)$ is equal to zero in the Fermi function ($n_{\vec k}$) and hence every $\vec k$ point has a probability of being occupied as $\frac{1}{2}$. Thus the occupied density of electrons translates to a momentum space volume $\Lambda^d$. Similarly, it is easily seen that $\frac{1}{V} \sum_{\vec k \sigma} n_{\vec k U} = \frac{2 \Lambda^d}{e^{\beta U} + 1}$.
APPENDIX B
==========
As briefly mentioned in the main text, for analytical tractability, we will assume repulsive interactions and work in the limit of zero temperature. In this limit, we can ignore the terms proportional to $n_{\vec k U}$ to the lowest order as they are exponentially smaller than the terms proportional to $n_{\vec k}$; additionally, the function $g(\vec k, U)$ goes to a constant equal to $\frac{2}{3}\equiv g$. With these assumptions, we can write the response function as $$\chi_0\left(\vec q, \omega + \frac{i}{\tau}\right) = \frac{1}{V} \sum_{\vec k} \Biggl[ g^2 \left( \frac{1}{\omega + \frac{i}{\tau} + \gamma} + \frac{1}{-\omega - \frac{i}{\tau} + \gamma}\right) + g(1-g) \left( \frac{1}{\omega + \frac{i}{\tau} + \gamma - U(\vec k + \vec q)} + \frac{1}{-\omega - \frac{i}{\tau} + \gamma + U(\vec k) }\right)\Biggr],$$ where $U(\vec k) = U + \frac{\alpha}{\lambda^2 + k^2}$. The first term proportional to $g^2$ is independent of $\vec k$ and the momentum sum simply gives the non-interacting flat band response modified by a factor of $g^2= \frac{4}{9}$. The effect of the interactions appears in the second term proportional to $g(1-g)$ and has a momentum dependence due to $U(\vec k)$. Hence the resulting momentum integrals need to be performed in all three dimensions.\
Consider the integral $$I_1(3D) = \frac{g(1-g)}{\left(2 \pi \right)^3} \int_0^{\Lambda} \int_{\Omega_{d=3}} \left[ \frac{dk k^2 d\Omega_{d=3}}{\omega + \frac{i}{\tau} + \gamma - \left(U + \frac{\alpha}{ \left(\vec k + \vec q\right)^2 + \lambda^2}\right)}\right].$$ Defining $z\equiv \left( \omega + \frac{i}{\tau} + \gamma -U \right)$, performing the azimuthal $\phi$ integral, and changing variables $t = cos\theta$, we obtain $$I_1(3D) = \frac{g(1-g)}{4 \pi^2} \int_0^{\Lambda} \int_{-1}^{1} \frac{dk k^2 dt \left(X+ Yt \right)}{z\left( X + Y t\right) - \alpha},$$ where we have defined $X\equiv k^2 + q^2 + \lambda^2$, $Y = 2 kq$. The $t$ integral can easily be performed to give $$I_1(3D) = \frac{g(1-g)}{4 \pi^2} \int_0^{\Lambda} k^2 dk \left[ \frac{2}{z} + \frac{\alpha}{2 k q z^2} Log\left(\frac{z(k+q)^2 + \lambda^2 z - \alpha}{z(k-q)^2 + \lambda^2 z - \alpha} \right) \right].$$ The radial momentum integral of the first term is simply $\frac{2 \Lambda^3}{3 z}$. The radial momentum integrals of the second term are given as $$\int_0^{\Lambda} k~dk Log\left( \mu k^2 \pm \nu k + \delta \right) = \biggl[ I_+(k) - I_-(k) \biggr]_0^{\Lambda},$$ where, $$I_{\pm}(k) = \frac{1}{4 \mu^2} \left[2 k \mu (- k \mu \pm \nu) \mp 2 \nu \sqrt{4 \delta \mu - \nu^2}~Arctan\left( \frac{2 k \mu \pm \nu}{\sqrt{4 \delta \mu - \nu^2}}\right) + \left( 2 \mu (\delta + k^2 \mu) - \nu^2\right) Log\left( \delta + k^2 \mu \pm k \nu\right)\right]$$ and $\mu\equiv z$, $\nu\equiv 2 q z$ and $\delta \equiv z\left(q^2 + \lambda^2 \right) - \alpha$. Substituting for $I_{\pm}(k)$ into the integral, taking the limit of $\Lambda\rightarrow\infty$ and keeping only the most divergent terms in $\Lambda$ we obtain $$I_1(3D) \simeq \frac{g(1-g)}{4 \pi^2} \left( \frac{2 \Lambda^3}{3 z} + \frac{2 \Lambda \alpha}{z^2} \right) + \mathscr{O}(\Lambda^0).$$ Similarly, the integral $$I_2(3D) = \frac{g(1-g)}{\left(2 \pi \right)^3} \int_0^{\Lambda} \int_{\Omega_{d=3}} \left[ \frac{dk k^2 d\Omega_{d=3}}{-\omega - \frac{i}{\tau} + \gamma + \left(U + \frac{\alpha}{ \vec k^2 + \lambda^2}\right)}\right]\simeq \frac{g(1-g)}{4 \pi^2} \left( \frac{2 \Lambda^3}{3 z'} - \frac{2 \Lambda \alpha}{z'^2} \right) + \mathscr{O}(\Lambda^0),$$ where we have defined $z' \equiv -\omega -\frac{i}{\tau} + \gamma + U$. Note that $I_1(3D)$ and $I_2(3D)$ differ by a sign in the second term proportional to $\alpha$ and $z$ is replaced with $z'$. Thus, combining $I_1(3D)$ and $I_2(3D)$, we see that the terms proportional to $\Lambda$ cancel out, whereas terms proportional to $\Lambda^3$ combine with $\gamma$ to give a constant of $\mathscr{O}(1)$. Therefore, we need to include other $\mathscr{O}(1)$ contributions which were left out in $I_1(3D)$ and $I_2(3D)$. These $\mathscr{O}(1)$ contributions can be evaluated to be $\frac{\pi \alpha}{z'^2}\sqrt{\lambda^2 + \frac{\alpha}{z'}} - \frac{\pi \alpha}{z^2}\sqrt{\lambda^2 - \frac{\alpha}{z}}$, which is the expression that appears in the main text. Note that there is an additional prefactor of $\frac{1}{6\pi^2}$ that appears in these expressions compared to free flat band case. This is only because of the difference in the normalization used while working in spherical co-ordinates, and can be trivially absorbed into the defintion of the flat band constant $R$.\
As the remaining two cases can be derived similar to the $d=3$ case, we will only give a sketch of what the integrals look like. In the $d=2$ case we consider the integral $$I_1(2D) = \frac{g(1-g)}{4 \pi^2}\int_0^{\Lambda} \int_0^{2\pi}\frac{k~dk~d\phi}{z - \frac{\alpha}{\left(\vec k + \vec q\right)^2 + \lambda^2}} = \frac{g(1-g)}{4 \pi^2} \int_0^{\Lambda}\int_0^{2\pi} \frac{k~dk~d\phi ~(X + Y cos\phi)}{z(X+ Y cos\phi) - \alpha},$$ where again $z\equiv \left( \omega + \frac{i}{\tau} + \gamma -U \right)$, and $X\equiv k^2 + q^2 +\lambda^2$, $Y \equiv 2 k q$. The azimuthal $\phi$ integral can be performed to give $$I_1(2D) = \frac{g(1-g)}{4\pi^2} \int_0^{\Lambda} k~dk\left[ \frac{2\pi}{z} + \frac{\alpha}{z \sqrt{z(k+q)^2 + z \lambda^2 -\alpha} \sqrt{z(k-q)^2 + z \lambda^2 - \alpha}}\right].$$ In the large $\Lambda$ limit, the remaining radial $k$ integral yields $$I_1(2D)\simeq \frac{g(1-g)}{4 \pi^2} \left[\frac{\pi \Lambda^2}{z} + \frac{\alpha}{z^2}Log(\Lambda)\right]$$ where we have collected the two largest contributions in $\Lambda$. Similarly, we have $$I_2(2D)= \frac{g(1-g)}{4 \pi^2}\int_0^{\Lambda} \int_0^{2\pi}\frac{k~dk~d\phi}{z' +\frac{\alpha}{\vec k^2 + \lambda^2}}\simeq \frac{g(1-g)}{4 \pi^2} \left[\frac{\pi \Lambda^2}{z'} - \frac{\alpha}{z'^2}Log(\Lambda)\right].$$ Combining $I_1(2D)$ and $I_2(2D)$, we see that the terms proportional to $\Lambda$ are multiplied by $\gamma$ and are just of order unity. And the remaining terms proportional to $\alpha Log(\Lambda)$ just obtain $\alpha Log(\Lambda) \left[\frac{1}{z^2} - \frac{1}{z'^2}\right]$ which is the expression that appears in the main text.\
The one dimensional case follows along similar lines. Consider the integral $$I_1(1D) = \int_{-\frac{\Lambda}{2}}^{\frac{\Lambda}{2}} \frac{dk}{z - \frac{\alpha}{(k+q)^2 + \lambda^2}} = \int_{\frac{-\Lambda}{2}}^{\frac{\Lambda}{2}} \frac{d k \left(k^2 + a k +b \right)}{z\left(k^2 + a k + b \right) - \alpha},$$ where we have defined $a=2q$ and $b=q^2 + \lambda^2$. This integral can be performed exactly and is written as $$I_1(1D) = \left[\frac{k}{z} + \frac{2 \alpha~Arctan\left(\frac{(a+ 2 k)\sqrt{z}}{\sqrt{-a^2 z + 4 b z - 4 \alpha}}\right)}{z^{3/2}\sqrt{-a^2 z + 4 b z - 4 \alpha}}\right]_{-\frac{\Lambda}{2}}^{\frac{\Lambda}{2}}.$$ In the large $\Lambda$ limit, the leading terms are given as $$I_1(1D) \simeq \left( \frac{\Lambda}{z} + \frac{2 \pi \alpha}{z^{3/2} \sqrt{-a^2 z + 4 b z - 4 \alpha}}\right)+ \mathscr{O}(\frac{1}{\Lambda}).$$ Note that we have also included the $\mathscr{O}(1)$ contributions since we learned from the previous two cases that the $\mathscr{O}(\Lambda^d)$ terms combine to give $\mathscr{O}(1)$ terms. On similar lines we can write $$I_2(1D) = \int_{\frac{-\Lambda}{2}}^{\frac{\Lambda}{2}} \frac{dk}{z' + \frac{\alpha}{k^2 + \lambda^2}} \simeq \left( \frac{\Lambda}{z'} - \frac{2 \pi \alpha}{z'^{3/2} \sqrt{-a^2 z' + 4 b z' + 4 \alpha}}\right)+ \mathscr{O}(\frac{1}{\Lambda}).$$ As before, adding the $I_1(1D)$ and $I_2(1D)$ terms, we see that the terms proportional to $\Lambda$ combine with $\gamma$ gives us a term of $\mathscr{O}(1)$. The remainder of the terms yield $2 \pi \alpha \left[\frac{1}{z^2 \sqrt{4 \lambda^2 - \frac{4\alpha}{z}}} - \frac{1}{z'^2 \sqrt{4 \lambda^2 + \frac{4\alpha}{z'}}} \right]$ which is the expression appearing in the main text.
|
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abstract: |
A projective moving average $\{ X_t, t \in \Z \}$ is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel $Q$ and a linear combination of projections of $X_t$ on “intermediate” lagged innovation subspaces with given coefficients $\alpha_i, \beta_{i,j}$. The class of such equations include usual moving-average processes and the Volterra series of the LARCH model. Solvability of projective equations is studied, including a nested Volterra series representation of the solution $X_t$. We show that under natural conditions on $Q, \alpha_i, \beta_{i,j}$, this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.
[Keywords:]{} [projective stochastic equations; long memory; nested Volterra series; LARCH model; Bernoulli shift; invariance principle]{}
author:
- Ieva Grublytė and Donatas Surgailis
date: |
\
Vilnius University
title:
- Projective stochastic equations and nonlinear long memory
- Projective stochastic equations and nonlinear long memory
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Introduction
============
A discrete-time second-order stationary process $\{X_t, t \in \Z\}$ is called [*long memory*]{} if its covariance $\gamma(k) = {\rm cov}(X_0,X_k)$ decays slowly with the lag in such a way that its absolute series diverges: $\sum_{k=1}^\infty |\gamma(k)| = \infty $. In the converse case when $\sum_{k=1}^\infty |\gamma(k)| < \infty $ and $\sum_{k=1}^\infty \gamma(k) \neq 0 $ the process $\{X_t\}$ is said [*short memory*]{}. Long memory processes have different properties from short memory (in particular, i.i.d.) processes. Long memory processes have been found to arise in a variety of physical and social sciences. See, e.g., the monographs Beran [@ref2], Doukhan et al. [@ref6], Giraitis et al. [@ref11] and the references therein.
Probably, the most important model of long memory processes is the linear, or moving average process $$\begin{aligned}
\label{linear}
X_t&=&\sum_{s\le t} b_{t-s} \zeta_s, \qquad t \in \Z,\end{aligned}$$ where $\{\zeta_s, s\in \Z\}$ is a standardized i.i.d. sequence, and the moving average coefficients $b_j$’s decay slowly so that $\sum_{j=0}^\infty |b_j| = \infty,\,
\sum_{j=0}^\infty b^2_j < \infty $. The last condition guarantees that the series in (\[linear\]) converges in mean square and satisfies $\E X_t = 0,\, \E X^2_t = \sum_{j=0}^\infty b^2_j < \infty$. In the literature it is often assumed that the coefficients regularly decay as $$\label{bj}
b_j \sim \kappa j^{d-1}, \qquad j \to \infty, \qquad \exists \ \kappa >0, \ d \in (0,1/2).$$ Condition (\[bj\]) guarantees that $$\label{covk}
\gamma(k)\ = \ \sum_{j=0}^\infty b_j b_{k+j} \ \sim \ \kappa^2 B(d,1-2d) k^{1-2d}, \qquad k \to \infty$$ and hence $\sum_{k=1}^\infty |\gamma(k)| = \infty. $ The parameter $d$ in (\[bj\]) is called the [*long memory parameter*]{} of $\{X_t\}$. A particular case of linear processes (\[linear\])-(\[bj\]) is the parametric class ARFIMA$(p,d,q)$, in which case $d \in (0, 1/2) $ is the order of fractional integration. An important property of the linear process in (\[linear\])-(\[bj\]) is the fact that its (normalized) partial sums process $S_n(\tau) := \sum_{j=1}^{[nt]} X_j, \, \tau \ge 0$ tends to a fractional Brownian motion ([@ref4]), viz., $$\label{dalX}
n^{-d - 1/2} S_n(\tau) \ \to_{D[0,1]} \ \sigma(d) B_H(t),$$ where $H = d + \frac{1}{2} $ is the Hurst parameter, $\sigma(d)^2 := \kappa^2 B(d,1-2d)/d(1+2d) >0 $ and $\to_{D[0,1]}$ denotes the weak convergence of random processes in the Skorohod space $D[0,1]$.
On the other hand, the linear model (\[linear\]) has its drawbacks and sometimes is not capable of incorporating empirical features (“stylized facts”) of some observed time series. The “stylized facts” may include typical asymmetries, clusterings, and other nonlinearities which are often observed in financial data, together with long memory.
The present paper introduces a new class of nonlinear processes which generalize the linear model in (\[linear\])-(\[bj\]) and enjoy similar long memory properties to (\[covk\]) and (\[dalX\]). These processes are defined through solutions of the so-called [*projective stochastic equations*]{}. Here, the term “projective” refers to the fact that these equations contain linear combinations of projections, or conditional expectations, of $X_t$’s on lagged innovation subspaces which enter the equation in a nonlinear way.
Let us explain the main idea of our construction. We call a [*projective moving average*]{} a random process $\{X_t\}$ of the form $$\begin{aligned}
\label{proma0}
X_t&=&\sum_{s\le t} g_{s,t} \zeta_s, \qquad t \in \Z,\end{aligned}$$ where $\{\zeta_s\} $ is a sequence of standardized i.i.d. r.v.’s as in (\[linear\]), $g_{t,t} \equiv g_0$ is a deterministic constant and $g_{s,t}, \, s< t$ are r.v.’s depending only on $\zeta_{s+1}, \dots, \zeta_t$ such that $$\label{gst0}
g_{s,t} = g_{t-s}(\zeta_{s+1}, \dots, \zeta_t), \qquad s < t,$$ where $g_j: \R^j \to \R, \, j=1,2, \dots$ are [*nonrandom*]{} functions satisfying $$\label{gsum}
\sum_{s \le t} \E g^2_{s,t}\ = \ \sum_{s \le 0} \E g^2_{-s}(\zeta_{s+1}, \dots, \zeta_0) \ < \ \infty.$$ It follows easily that under condition (\[gsum\]) the series in (\[proma0\]) converges in mean square and define a stationary process with zero mean and finite variance $\E X^2_t = \sum_{s \le t} \E g^2_{s,t}$. The next question - how to choose the “coefficients” $g_{s,t}$ (\[gst0\]) so that they depend on $X_t$ and behave like (\[bj\]) when $j= t-s \to \infty $?
A particularly simple choice of the $g_{s,t}$’s to achieve the above goals is $$\label{Qst}
g_{s,t} \ =\ b_{t-s} Q( \E_{[s+1,t]} X_t), \qquad s \le t$$ where $b_j$ are as in (\[bj\]), $Q: \R \to \R$ is a given deterministic kernel, and $\E_{[s+1,t]} X_t := \E [X_t | \zeta_v, s+1 \le v \le t]$ is the projection of $X_t$ onto the subspace of $L^2 $ generated by the innovations $\zeta_v, s+1\le v\le t $ (the conditional expectation). The corresponding projective stochastic equation has the form $$\begin{aligned}
\label{proma1}
X_t&=&\sum_{s\le t} b_{t-s} Q( \E_{[s+1,t]} X_t) \zeta_s.\end{aligned}$$ Notice that when $s \to -\infty $ then $ \E_{[s+1,t]} X_t \to X_t$ by a general property of a conditional expectation and then $g_{s,t} \sim b_{t-s} Q(X_t)$ if $Q$ is continuous. This means that the $g_{s,t}$’s in (\[Qst\]) feature both the long memory in (\[bj\]) and the dependence on the “current” value $X_t$ through $Q(X_t)$. In particular, for $Q(x) = \max(0,x)$, the behavior of $g_{s,t}$ in (\[Qst\]) strongly depends on the sign of $X_t$ and the trajectory of (\[proma1\]) appears very asymmetric (see Fig. \[lmQint2\], top).
Let us briefly describe the remaining sections. Sec. 2 contains basic definitions and properties of projective processes. Sec. 3 introduces the notion of nested Volterra series which plays an important role for solving of projective equations. Sec. 4 introduces a general class of projective stochastic equations, (\[proma1\]) being a particular case. We obtain sufficient conditions of solvability of these equations, and a recurrent formula for computation of “coefficients” $g_{s,t}$ (Theorem \[thm1\]). Sec. 5 and 6 present some examples and simulated trajectories and histograms of projective equations. It turns out that the LARCH model studied in [@ref7] and elsewhere is a particular case of projective equations corresponding to linear kernel $Q(x)$ (Sec. 5). Some modifications of projective equations are discussed in Sec.7. Sec. 8 deals with long memory properties of stationary solutions of stochastic projective equations. We show that under some additional conditions these solutions have long memory properties similar to (\[covk\]) and (\[dalX\]).
Finally, we remark that “nonlinear long memory” is a general term and that other time series models different from ours for such behavior were proposed in the literature. Among them, probably the most studied class are subordinated processes of the form $\{Q(X_t)\}$, where $\{X_t\}$ is a Gaussian or linear long memory process and $Q: \R \to \R$ is a nonlinear function. See [@ref19], [@ref14] and [@ref11] for a detailed discussion. A related class of Gaussian subordinated stochastic volatility models is studied in [@ref16]. [@ref6a] discuss a class of long memory Bernoulli shifts. [@ref1] consider fractionally integrated process with nonlinear autoregressive innovations. A general invariance principle for fractionally integrated models with weakly dependent innovations satisfying a projective dependence condition of [@ref4a] is established in [@ref17]. See also [@ref21] and Remark \[Wu0\] below.
We expect that the results of this paper can be extended in several directions, e.g., projective equations with initial condition, continuous time processes, random field set-up, infinite variance processes. For applications, a major challenge is estimation of “parameters” of projective equations. We plan to study some of these questions in the future.
Projective processes and their properties
=========================================
Let $\{\zeta_t, t \in {\Z}\} $ be a sequence of i.i.d. r.v.’s with $\E \zeta_0 = 0, \, \E \zeta^2_0 = 1$. For any integers $s\le t $ we denote ${\mathcal F}_{[s,t]} := \sigma\{\zeta_u: u \in [s,t]\} $ the sigma-algebra generated by $\zeta_u, u \in [s,t]$, ${\mathcal F}_{(-\infty,t]} := \sigma\{\zeta_u: u \le t\}, \,
{\mathcal F} := \sigma\{\zeta_u: u \in \Z\}. $ For $s > t$, we define ${\mathcal F}_{[s,t]} := \{\emptyset, \Omega \} $ as the trivial sigma-algebra. Let $L^2_{[s,t]}, \, L^2_{(-\infty,t]}, \,
L^2$ be the spaces of all square integrable r.v.’s $\xi $ measurable w.r.t. ${\mathcal F}_{[s,t]}, \,
{\mathcal F}_{(-\infty,t]}, \,
{\mathcal F}$, respectively. For any $s, t \in \Z$ let $$\E_{[s,t]} [\xi ] := \E\left[ \xi \big| {\mathcal F}_{[s,t]} \right], \qquad \xi \in L^2$$ be the conditional expectation. Then $\xi \mapsto \E_{[s,t]} [\xi ] $ is a bounded linear operator in $L^2$; moreover, $\E_{[s,t]}, \, s,t \in \Z $ is a projection family satisfying $\E_{[s_2, t_2]} \E_{[s_1,t_1]} = \E_{[s_2, t_2]\cap [s_1,t_1]} $ for any intervals $[s_1,t_1], [s_2,t_2] \subset \Z$. From the definition of conditional expectation it follows that if $g_u: \R \to \R, u \in \Z $ are arbitrary measurable functions with $\E g^2_u(\zeta_u) < \infty, $ $[s_2,t_2] \subset \Z$ is a given interval and $\xi = \prod_{u\in [s_2,t_2]} g_u(\zeta_u)$ is a product of independent r.v.’s, then for any interval $[s_1, t_1] \subset \Z$ $$\begin{aligned}
\E_{[s_1,t_1]} \prod_{u\in [s_2,t_2]} g_u(\zeta_u)&=&\prod_{u\in [s_1,t_1] \cap [s_2,t_2]} g_u(\zeta_u)
\prod_{v \in [s_2,t_2]\setminus [s_1,t_1]} \E [g_v(\zeta_v)].\end{aligned}$$ In particular, if $\E g_u(\zeta_u) = 0, \, u \in \Z $ then $$\begin{aligned}
\label{projP}
\E_{[s_1,t_1]} \prod_{u\in [s_2,t_2]} g_u(\zeta_u)&=&\begin{cases}
\prod_{u\in [s_1,t_1]} g_u(\zeta_u), &[s_2,t_2] \subset [s_1,t_1], \\
0, &[s_2,t_2] \not\subset [s_1,t_1].
\end{cases}\end{aligned}$$ Any r.v. $Y_t \in L^2_{(-\infty,t]} $ can be expanded into orthogonal series $Y_t = \E Y_t + \sum_{s \le t} P_{s,t} Y_t, $ where $P_{s,t} Y_t := (\E_{[s,t]} - \E_{[s+1,t]})Y_t.$ Note that $\{ P_{s,t} Y_t, {\cal F}_{s,t}, s\le t\}$ is a backward martingale difference sequence and $\E Y^2_t = (\E Y_t)^2 + \sum_{s \le t} \E (P_{s,t} Y_t)^2. $
\[propro\] A [*projective process*]{} is a random sequence $\{Y_t \in L^2_{(-\infty,t]}, \, t \in \Z\}$ of the form $$Y_t\ = \ \E Y_t + \sum_{s\le t} g_{s,t} \zeta_s, \label{XMA}$$ where $g_{s,t} $ are r.v.’s satisfying the following conditions (i) and (ii):
\(i) $g_{s,t}$ is ${\mathcal F}_{[s+1,t]}$-measurable, $ \forall s, t \in \Z, \, s < t; \, g_{t,t} $ is a deterministic number;
\(ii) $\sum_{s \le t} \E g^2_{s,t} < \infty, \ \forall \, t \in \Z. $
In other words, a projective process has the property that the projections $\E_{[s,t]} Y_t = \E Y_t + \sum_{i=s}^{t} P_{i,t} Y_t =
\E Y_t + \sum_{i=s}^t \zeta_i g_{i,t}, \, s \le t $ form a backward martingale transform w.r.t. the nondecreasing family ${\{\cal F}_{[s,t]}, s \le t\} $ of sigma-algebras, for each $t \in \Z$ fixed. A consequence of the last fact is the following moment inequality which is an easy consequence of Rosenthal’s inequality ([@ref12], p.24). See also [@ref11], Lemma 2.5.3.
\[propmom\] Let $\{Y_t \} $ be a projective process in (\[XMA\]). Assume that $\mu_p := \E |\zeta_0|^p < \infty $ and $\sum_{s \le t} (\E |g_{s,t}|^p )^{2/p} < \infty $ for some $p \ge 2$. Then $\E |Y_t|^p < \infty $. Moreover, there exists a constant $C_p < \infty $ depending on $p$ alone and such that $$\E |Y_t|^p \ \le \ C_p \Big( |\E Y_t|^p + \mu_p \big(\sum_{s \le t} (\E |g_{s,t}|^p )^{2/p} \big)^{p/2}\Big).$$
\[proma\] A [*projective moving average*]{} is a projective process of (\[XMA\]) such that the mean $\E Y_t = \mu $ is constant and there exist a number $g_0 \in \R$ and nonrandom measurable functions $g_j: \R^j \to \R, j=1,2, \dots $ such that $$g_{s,t} = g_{t-s}(\zeta_{s+1}, \dots, \zeta_t) \qquad \text{a.s., \ for any} \quad s \le t, \, s,t \in \Z.$$
By definition, a projective moving average is a stationary Bernoulli shift ([@ref5], p.21): $$\label{bernoulli}
Y_t \ = \ \mu + \sum_{s \le t} \zeta_s g_{t-s} (\zeta_{s+1}, \dots, \zeta_t)$$ with mean $\mu$ and covariance $$\begin{aligned}
\hskip-.3cm \cov (Y_s, Y_t)&=&\sum_{u\le s} \E [g_{s-u } (\zeta_{u+1}, \dots, \zeta_s) g_{t-u}(\zeta_{u+1}, \dots, \zeta_t) ] {\nonumber}\\
&=& \sum_{u\le 0} \E[ g_{-u} (\zeta_{u+1}, \dots, \zeta_0) g_{t-s -u}(\zeta_{u+1}, \dots, \zeta_{t-s-u}) ], \quad s\le t.\label{XMA1}\end{aligned}$$ These facts together with the ergodicity of Bernoulli shifts (implied by a general result in [@ref18], Thm.3.5.8) are summarized in the following corollary.
\[rem1\] A projective moving average is a strictly stationary and ergodic stationary process with finite variance and covariance given in (\[XMA1\]).
\[rem2\] [If the coefficients $g_{s,t}$ are nonrandom, a projective moving average is a linear process $Y_t = \mu + \sum_{s\le t} g_{t-s} \zeta_s, \, t \in \Z.$]{}
\[propfilter\] Let $\{Y_t\}$ be a projective process of (\[XMA\]) and $\{a_j, j \ge 0\}$ a deterministic sequence, $\sum_{j=0}^\infty |a_j| < \infty, \sum_{j=0}^\infty |a_j| |\E Y_{t-j}| < \infty $. Then $\{u_t := \sum_{j=0}^\infty a_j Y_{t-j}, \, t \in \Z\} $ is a projective process $u_t = \E u_t + \sum_{s \le t} \zeta_s G_{s,t} $ with $\E u_t = \sum_{j=0}^\infty a_j \E Y_{t-j}$ and coefficients $G_{s,t} := \sum_{j=0}^{t-s} a_j g_{s,t-j}$.
[*Proof*]{} follows easily by the Cauchy-Schwarz inequality and is omitted. $\Box$
\[propunique\] If $\{Y_t\}$ is a projective process of (\[XMA\]), then for any $s\le t$ $$\begin{aligned}
\label{Eg}
\hskip-.3cm \E_{[s,t]} Y_t&=&\E Y_t + \sum_{s \le u \le t} \zeta_u g_{u,t}, \quad
P_{s,t} Y_t \ = \ (\E_{[s,t]} - \E_{[s+1,t]}) Y_t\ =\ \zeta_s g_{s,t}.\end{aligned}$$ The representation (\[XMA\]) is unique: if (\[XMA\]) and $Y_t = \sum_{s\le t} g'_{s,t} \zeta_s $ are two representations, with $g'_{s,t}$ satisfying conditions (i) and (ii) of Definition \[propro\], then $g'_{s,t} = g_{s,t}\ \forall \ s\le t. $
(\[Eg\]) is immediate by definition of projective process. From (\[Eg\]) it follows that $\zeta_s
g''_{s,t} = 0,$ where $g''_{s,t} := g_{s,t} - g'_{s,t}$ is independent of $\zeta_s$. Relation $\E \zeta^2_s = 1$ implies $\P (|\zeta_s|^2 > \epsilon) >0$ for all $\epsilon >0 $ small enough. Hence, $0= \P (|\zeta_s g''_{s,t}|> \epsilon)
\ge \P (|\zeta_s| > \sqrt{\epsilon}, |g''_{s,t}|> \sqrt{\epsilon}) =
\P (|\zeta_s| > \sqrt{\epsilon}) \P( |g''_{s,t}|> \sqrt{\epsilon}),$ implying $ \P( |g''_{s,t}|> \sqrt{\epsilon}) = 0$ for any $\epsilon >0$. $\Box$
The following invariance principle is due to Dedecker and Merlevède ([@ref4a], Cor. 3), see also ([@ref20], Thm. 3 (i)).
Let $\{Y_t\}$ be a projective moving average of (\[XMA\]) such that $\mu = 0$ and $$\label{DedMer}
\Omega(2) \ := \ \sum_{t=0}^\infty \| g_{0,t} \| \ < \infty,$$ where $\| \xi \| = \E^{1/2} [\xi^2], \, \xi \in L^2.$ Then $$\label{DedMer1}
n^{-1/2} \sum_{t=1}^{[n\tau]}Y_t\, \longrightarrow_{D[0,1]}\, c_Y B(\tau),$$ where $B$ is a standard Brownian motion and $c^2_Y := \| \sum_{t=0}^\infty g_{0,t} \|^2 = \sum_{t\in \Z} \E [Y_0 Y_t]. $
Nested Volterra series
=======================
First we introduce some notation. Let $T \subset \Z $ be a set of integers which is bounded from above, i.e., $\sup \{s: s \in T\} < \infty $. Let ${\cal S}_T $ be a class of finite nonempty subsets $S = \{s_1, \dots, s_n \} \subset T, \, s_1 < \dots < s_n, \, n\ge 1$. Write $|S |$ for the cardinality of $S \subset Z$. For any $S = \{s_1, \dots, s_n\} \in {\cal S}_T, \, S' = \{s'_1, \dots, s'_m \} \in {\cal S}_T,
$ the notation $S \prec S' $ means that $m= n+1 $ and $s_1= s'_1, \dots, s_n =s'_n < s'_{n+1} = s'_m $. In particular, $S \prec S' $ implies $S \subset S'$ and $|S'\setminus S| = 1$. Note that $\prec$ is not a partial order in ${\cal S}_T $ since $S \prec S', \, S'\prec S'' $ do not imply $S \prec S'' $. A set $ S \in {\cal S}_T $ is said [*maximal*]{} if there is no $S' \in {\cal S}_T $ such that $S \prec S'$. Let ${\cal S}^{\rm max}_T $ denote the class of all maximal elements of ${\cal S}_T. $
\[nested\] Let $T$ and ${\cal S}_T $ be as above. Let ${\cal G}_T := \{G_S, S \in {\cal S}_T \} $ be a family of measurable functions $G_S = G_{s_1, \dots, s_m}: \R \to \R$ indexed by sets $ S = \{s_1, \dots, s_m \} \in {\cal S}_T $ and such that $G_S =: a_S$ is a constant function for any maximal set $S \in {\cal S}^{\rm max}_T $. A [*nested Volterra series*]{} is a sum $$\begin{aligned}
\hskip-1cmV({\cal G}_T)&=&\sum_{S_1 \in {\cal S}_T: |S_1|=1} \zeta_{S_1} G_{S_1} \Big( \sum_{S_1\prec S_2} \zeta_{S_2\setminus S_1} G_{S_2}
\Big( \dots \zeta_{S_{p-1}\setminus S_{p-2}}{\nonumber}\\
&&\hskip3cm \times G_{S_{p-1}} \Big(\sum_{S_{p-1}\prec S_p }
\zeta_{S_p\setminus S_{p-1}} G_{S_p}
\Big)
\Big) \Big), \label{nestvolt0}\end{aligned}$$ where the nested summation is taken over all sequences $S_1 \prec S_2 \prec \dots \prec S_p \in {\cal S}^{\rm max}_T, p=1,2,\dots$, with the convention $G_S = a_S, \, S \in {\cal S}^{\rm max}_T, $ and $\zeta_S := \zeta_s $ for $S = \{s\}, |S|=1.$
In particular, when ${\cal S}_T = \{ S: S \subset T\} $ is the class of all subsets of $T$, (\[nestvolt0\]) can be rewritten as $$\begin{aligned}
\hskip-.5cm V({\cal G}_T)&=&\sum_{s_1 \in T} \zeta_{s_1} G_{s_1} \Big( \sum_{s_1< s_2 \in T} \zeta_{s_2} G_{s_1,s_2}
\Big( \dots \zeta_{s_{p-1}} {\nonumber}\\
&&\hskip3cm \times G_{s_1,\dots,s_{p-1}}\Big(\sum_{s_{p-1} < s_p \in T} \zeta_{s_p} G_{s_1, \dots, s_p}
\Big) \Big) \Big), \label{nestvolt}\end{aligned}$$ where the last sum is taken over all $s_p > s_{p-1}$ such that $\{s_1, \dots, s_p\} \in {\cal S}^{\rm max}_T.$
The following example clarifies the above definition and its relation to the usual Volterra series ([@ref5], p.22).
\[exnest\] Let $ T(t) = (-\infty, t] \cap \Z, \, t \in \Z$ and ${\cal S}_{T(t)} $ be the class of all subsets $S = \{s_1,\dots, s_k \} \subset T(t) $ having $k$ points. Let ${\cal G}_{T(t)} = \{G_S, S \in {\cal S}_{T(t)} \} $ be a family of linear functions $$G_{S}(x):=
\begin{cases}
x,&S \in {\cal S}_T, S \not \in {\cal S}^{\rm max}_{T(t)}, \\
a_S = a_{s_1,\dots, s_k}, &S =\{s_1,\dots, s_k\} \in {\cal S}^{\rm max}_{T(t)}.
\cr
\end{cases}$$ Then $$\begin{aligned}
\label{volt}
V({\cal G}_{T(t)})&=&\sum_{s_1 < \dots < s_k \le t} a_{s_1, \dots, s_k} \zeta_{s_1} \zeta_{s_2} \cdots \zeta_{s_k} \
= \ \sum_{S \subset T, |S| = k} a_S \zeta^S,\end{aligned}$$ $\zeta^S := \zeta_{s_1} \zeta_{s_2} \cdots \zeta_{s_k},$ is the (usual) Volterra series of order $k$. The series in (\[volt\]) converges in mean square if and only if $$\label{volt1}
A_{T(t)}\ :=\ \sum_{s_1 < \dots < s_k \le t} a^2_{s_1, \dots, s_k} < \infty,$$ in which case $\E V^2({\cal G}_{T(t)})= A_{T(t)}, \, \E V({\cal G}_{T(t)}) = 0$.
\[propvolt\] Let $ T(t) := (-\infty, t] \cap \Z, \, t \in \Z$ as in Example \[exnest\]. Assume that the system ${\cal G}_{T(t)} = \{G_S, S \in {\cal S}_{T(t)}\} $ in Definition \[nested\] satisfies the following condition $$\label{GSbdd}
|G_S(x)|^2\ \le \ \begin{cases}
\alpha^2_S + \beta^2_S x^2, &S \in {\cal S}_{T(t)}, \, S \not\in {\cal S}^{\rm max}_{T(t)}, \\
\alpha^2_S(= a^2_S), &S \in {\cal S}^{\rm max}_{T(t)},
\end{cases}$$ where $\alpha_S, \beta_S$ are real numbers satisfying $$\begin{aligned}
{\cal A}_{T(t)}&:=&
\sum_{p \ge 1} \sum_{S_1 \prec S_2 \prec \dots \prec S_p}
\beta^2_{S_1} \beta^2_{S_2} \cdots \beta^2_{S_{p-1}} \alpha^2_{S_p} \ < \ \infty,
\label{calAt}\end{aligned}$$ where the inner sums are taken over all sequences $S_1 \prec S_2 \prec \dots \prec S_p, \, S_i \in {\cal S}_{T(t)}, \, 1\le i \le p$ with $|S_1|=1 $ and $S_p \in {\cal S}^{\rm max}_{T(t)}$.
Then, the nested Volterra series $V({\cal G}_{T(t)})$ in (\[nestvolt\]) converges in mean square and satisfies $\E V({\cal G}_{T(t)})^2 \le {\cal A}_{T(t)}, \, \E V({\cal G}_{T(t)}) = 0.$ Moreover, $ X_t := V({\cal G}_{T(t)})$ is a projective process with zero mean and coefficients $$\begin{aligned}
\label{gst}
g_{s,t}&:=&
G_{S_1} \big( \sum_{S_1\prec S_2} \zeta_{S_2\setminus S_1} G_{S_2}
\big(\cdots \zeta_{S_{p-1}\setminus S_{p-2}} {\nonumber}\\
&&\hskip3cm \times G_{S_{p-1}} \big(\sum_{S_{p-1}\prec S_p }
\zeta_{S_p\setminus S_{p-1}} G_{S_p}
\big) \big) \big)\end{aligned}$$ if $S_1= \{s\} \in {\cal S}_{T(t)}$, $g_{s,t}:=0$ otherwise, where the nested summation is defined as in (\[nestvolt0\]).
Clearly, the coefficients $g_{s,t}$ in (\[gst\]) satisfy the measurability condition (i) of Definition \[propro\]. Condition (ii) for these coefficients follows by recurrent application of (\[GSbdd\]): $$\begin{aligned}
\sum_{s\le t} \E g^2_{s,t}
&=&\sum_{S_1 \in {\cal S}_{T(t)}: |S_1|=1} \E G^2_{S_1} \big( \sum_{S_1\prec S_2} \zeta_{S_2\setminus S_1} G_{S_2}
(\dots ) \big)\\
&\le& \sum_{S_1 \in {\cal S}_{T(t)}: |S_1|=1} \big( \alpha_{S_1}^2 + \beta_{S_1}^2\E \big(\sum_{S_1\prec S_2} \zeta_{S_2\setminus S_1} G_{S_2}( \dots ) \big)^2\big) {\nonumber}\\
&\le& \sum_{S_1 \in {\cal S}_{T(t)}: |S_1|=1} \big( \alpha_{S_1}^2 + \beta_{S_1}^2 \sum_{S_1\prec S_2} \big( \alpha_{S_2}^2+\beta_{S_2}^2 \E \big(\sum_{S_2\prec S_3 } \zeta_{S_3\setminus S_2} G_{S_3}(\dots )
\big)^2 \big) \big){\nonumber}\\
&\le&\sum_{S_1 \in {\cal S}_{T(t)}: |S_1|=1} \Big(\alpha_{S_1}^2+ \beta_{S_1}^2 \sum_{S_1\prec S_2} \alpha_{S_2}^2+
\beta_{S_1}^2 \sum_{S_1\prec S_2\prec S_3} \beta_{S_2}^2 \alpha_{S_3}^2 +\dots \Big) {\nonumber}\\
&=& \sum_{p\ge 1} \sum_{S_1\prec S_2\prec \dots \prec S_p}\beta_{S_1}^2 \beta_{S_2}^2 \cdots \beta_{S_{p-1}}^2 \alpha_{S_p}^2 = {\cal A}_{T(t)} < \infty.\end{aligned}$$ Thus, $ X_t = \sum_{s\le t} g_{s,t} \zeta_s$ is a well-defined projective process and $X_t = V({\cal G}_{T(t)})$. $\Box$
\[rem4\] [In the case of a usual Volterra series in (\[volt\]), condition (\[GSbdd\]) is satisfied with $\alpha_S =0, \beta_S =1 $ for $S \in {\cal S}_{T(t)}, \, S \not\in {\cal S}^{\rm max}_{T(t)}, $ and the sums ${\cal A}_{T(t)}$ of (\[calAt\]) and $A_{T(t)}$ of (\[volt1\]) coincide: ${\cal A}_{T(t)} = A_{T(t)}. $ This fact confirms that condition (\[calAt\]) for the convergence of nested Volterra series cannot be generally improved.]{}
Projective stochastic equations
===============================
Let $Q_{s,t} = Q_{s,t} (x_{u,v}, s < u \le v \le t), \, s, t \in \Z, s < t$ be some given measurable deterministic functions depending on $(t-s)(t-s+1)/2 $ real variables $x_{u,v},\, s <t,$ and $\mu_t, \, Q_{t,t}, \, t \in \Z$ be some given constants. A [*projective stochastic equation*]{} has the form $$\begin{aligned}
\label{proSE}
X_t&=&\mu_t + \sum_{s\le t} \zeta_s Q_{s,t}(\E_{[u,v]} X_v, s < u \le v \le t).
$$
By [*solution of (\[proSE\])*]{} we mean a projective process $\{ X_t, t \in \Z \}$ satisfying $$\sum_{s\le t}\E [ Q^2_{s,t}(\E_{[u,v]}X_v, s< u \le v \le t)] \ < \ \infty$$ and (\[proSE\]) for any $t \in \Z$.
\[proseGen\] Assume that that $\mu_t = \mu$ does not depend on $t \in \R,$ the functions $Q_{s,t} = Q_{t-s}, \, s\le t$ in (\[proSE\]) depend only on $t-s,$ and that $\{ X_t \}$ is a solution of (\[proSE\]). Then $\{ X_t \}$ is a projective moving average of (\[bernoulli\]) with $\E X_t = \mu$ and $g_n: \R^n \to \R, \, n = 0,1, \dots$ defined recursively by $$\label{grecurs}
g_{n}(x_{-n+1}, \dots, x_0) := Q_n\Big(\mu+ \sum_{k=u}^v x_k g_{v-k}(x_{u+1}, \dots, x_v), -n< u \le v \le 0\Big), \quad n \ge 1.$$ $g_0 := Q_0.$ Moreover, such a solution is unique.
From (\[proSE\]) and the uniqueness of (\[XMA\]) (Proposition \[propunique\]) we have $X_t = \mu+ \sum_{s\le t} g_{s,t} \zeta_s, $ where $g_{s,t} = Q_{t-s}(\E_{[u,v]} X_v, s < u \le v \le t).$ For $s = t$ this yields $g_{t,t} = Q_0 = g_0 \, \forall t \in \Z$ as in (\[grecurs\]). Similarly, $g_{t-1,t} = Q_{1}(\E_{[t,t]} X_t) = Q_{1}(\mu+ g_0 \zeta_t) = g_1(\zeta_t)$, where $g_1$ is defined in (\[grecurs\]). Assume by induction that $$\label{ginduct}
g_{t-m, t} = g_m(\zeta_{t-m+1}, \dots, \zeta_t), \qquad \forall \ t \in \Z$$ with $g_m$ defined in (\[grecurs\]), hold for any $0\le m < n$ and some $n \ge 1$; we need to show that (\[ginduct\]) holds for $m=n$, too. Using (\[ginduct\]), (\[Eg\]) and (\[grecurs\]) we obtain $$\begin{aligned}
g_{t-n,t}&=&Q_{n}(\E_{[u,v]} X_v, t-n < u \le v \le t) \\
&=&Q_n\Big(\mu+\sum_{k=u}^v \zeta_k g_{v-k}(\zeta_{u+1}, \dots, \zeta_v), t-n< u \le v \le t\Big) \\
&=&g_n(\zeta_{t-n+1}, \dots, \zeta_t).\end{aligned}$$ This proves the induction step $n-1 \to n$ and hence the proposition, too, since the uniqueness follows trivially. $\Box$
Clearly, the choice of possible kernels $Q_{s,t}$ in (\[proSE\]) is very large. In this paper we focus on the following class of projective stochastic equations: $$\begin{aligned}
\label{proSEI}
X_t &=&\mu+ \sum_{s\le t} \zeta_s Q\Big(\alpha_{t-s}+ \sum_{s<u\le t} \beta_{t-u, u-s} \left(\E_{[u,t]} X_{t} - \E_{[u+1,t]} X_{t}\right) \Big),\end{aligned}$$ where $\{\alpha_i, i \ge 0\}, \ \{\beta_{i,j}, i \ge 0, j\ge 1 \} $ are given arrays of real numbers, $\mu \in \R$ is a constant, and $Q = Q(x)$ is a measurable function of a single variable $x \in \R$. Two modifications of (\[proSEI\]) are briefly discussed below, see (\[proSEII\]) and (\[proSE0\]). Particular cases of (\[proSEI\]) are $$\begin{aligned}
\label{proSE1}
X_t &=&\sum_{s\le t} \zeta_s Q\Big(\alpha_{t-s}+ \beta_{t-s} \E_{[s+1,t]} X_t \Big),\end{aligned}$$ and $$\begin{aligned}
\label{proSE2}
X_t &=&\mu + \sum_{s\le t} \zeta_s Q\Big(\alpha_{t-s}+ \sum_{s<u\le t} \beta_{u-s}
\left(\E_{[u,t]} X_t - \E_{[u+1,t]} X_t\right) \Big),\end{aligned}$$ corresponding to $\beta_{i,j} = \beta_{i+j}$ and $\beta_{i,j} = \beta_{j},$ respectively.
Next, we study the solvability of projective equation (\[proSEI\]). We assume that $Q$ satisfies the following dominating bound: there exists a constant $c_Q >0$ such that $$\begin{aligned}
|Q(x)|&\le&c_Q|x|, \qquad \forall \ x \in {\R}. \label{Qdom}\end{aligned}$$ Denote $$\begin{aligned}
K_Q
&:=&\sum_{i=0}^\infty \alpha_i^2 \sum_{k=0}^\infty c^{2k+2}_Q \sum_{j_1=1}^\infty \beta^2_{i,j_1} \cdots
\sum_{j_k=1}^\infty \beta^2_{i+ j_1 + \dots + j_{k-1}, j_k}. \label{KQ}\end{aligned}$$ The main result of this section is the following theorem.
\[thm1\] (i) Assume condition (\[Qdom\]) and $$\label{Kcond}
K_{Q} \ < \ \infty.$$ Then there exists a unique solution $\{ X_t \} $ of (\[proSEI\]), which is written as a projective moving average in (\[XMA\]) with coefficients $g_{t-k,t}$ recursively defined as $$\begin{aligned}
\label{giter}
g_{t-k,t}&:=&\begin{cases}
Q\big(\alpha_k+ \sum_{i=0}^{k-1} \beta_{i,k-i} \zeta_{t-i} g_{t-i,t} \big), &k =1,2, \dots, \\
Q(\alpha_k), &k=0.
\end{cases}\end{aligned}$$ The above solution is represented by the following nested Volterra series $$\begin{aligned}
\hskip-.5cm X_t&=&\mu + \sum_{s_1 \le t} \zeta_{s_1} G_{s_1} \Big( \sum_{s_1< s_2 \le t} \zeta_{s_2} G_{s_1,s_2}
\Big( \dots
\zeta_{s_{p-1}} G_{s_1,\dots,s_{p-1}}\Big(\sum_{s_{p-1} < s_p \le t} \zeta_{s_p} G_{s_1, \dots, s_p}
\Big) \Big) \Big),\end{aligned}$$ where $$\begin{aligned}
G_S(x)&=&
\begin{cases}
Q(\alpha_0), &S = \{t \}, \\
Q(\alpha_{t-s} + x), &S = \{s \}, \ s <t, \\
\beta_{t-s_{k}, s_{k}-s_{k-1}}Q (\alpha_{t-s_k}), &S = \{s_1, \dots ,s_k \}, \ s_1 < \dots < s_k = t,\, k>1, \\
\beta_{t-s_{k}, s_{k}-s_{k-1}}Q (\alpha_{t-s_k}+x), &S = \{s_1, \dots ,s_k \}, \ s_1 < \dots < s_k < t, \, k>1.
\end{cases}\end{aligned}$$ More explicitly, $$\begin{aligned}
X_t&=&\mu + Q(\alpha_0) \zeta_t + Q\big(\alpha_1 + \beta_{0,1} \zeta_t Q(\alpha_{0})\big)\zeta_{t-1} \\
&+&Q\Big(\alpha_2 + \beta_{0,2}\zeta_t Q(\alpha_{0}) + \beta_{1,1}\zeta_{t-1} Q\big(\alpha_1 +\beta_{0,1} \zeta_t Q(\alpha_{0}) \big) \Big)\zeta_{t-2} + \dots.\end{aligned}$$
\(ii) In the case of linear function $Q(x) = c_Q x$, condition (\[Kcond\]) is also necessary for the existence of a solution of (\[proSEI\]).
\(i) Let us show that the $g_{k-t,t}$’s as defined in (\[giter\]) satisfy $\sum_{k=0}^\infty \E g^2_{t-k,t} < \infty$. From (\[Qdom\]) and (\[giter\]) we have the recurrent inequality: $$\begin{aligned}
\hskip-.3cm \E g^2_{t-k,t}
&\le&c^2_Q \E \Big(\alpha_k+ \sum_{i=0}^{k-1} \beta_{i,k-i} \zeta_{t-i} g_{t-i,t} \Big)^2 \
=\ c^2_Q \Big(\alpha^2_k+ \sum_{i=0}^{k-1} \beta^2_{i,k-i} \E g^2_{t-i,t} \Big).
\label{gineq}\end{aligned}$$ Iterating (\[gineq\]) we obtain $$\begin{aligned}
\hskip-.4cm \E g^2_{t-k,t}
&\le&c^2_Q \Big(\alpha^2_k+ c^2_Q\sum_{i=0}^{k-1} \beta^2_{i,k-i} \Big(\alpha^2_i + \sum_{j=0}^{i-1}
\beta^2_{j, i-j} \E g^2_{t-j,t} \Big) \Big) {\nonumber}\\
&=&c^2_Q \alpha^2_k + c^4_Q \sum_{i=0}^{k-1} \alpha^2_i \beta^2_{i,k-i} + c^6_Q \sum_{i=0}^{k-1} \alpha^2_i
\sum_{j_1=1}^{k-1-i} \beta^2_{i,j_1} \beta^2_{i+j_1, k-i-j_1} + \dots \label{gkiter}\end{aligned}$$ and hence $$\begin{aligned}
\sum_{k=0}^\infty \E g^2_{t-k,t}
&\le&c^2_Q \sum_{i=0}^\infty \alpha^2_i + c^4_Q \sum_{i=0}^\infty \alpha^2_i \sum_{j_1=1}^\infty \beta^2_{i,j_1}
+ c^6_Q \sum_{i=0}^\infty \alpha^2_i \sum_{j_1=1}^\infty \beta^2_{i,j_1} \sum_{j_2=1}^\infty \beta^2_{i+j_1, j_2} + \dots {\nonumber}\\
&=&K_{Q} \ < \ \infty \label{Kineq}\end{aligned}$$ according to (\[Kcond\]). Therefore, $X_t = \mu + \sum_{s\le t} g_{s,t} \zeta_s$ is a well-defined projective moving-average. The remaining statements about $X_t$ follow from Proposition \[proseGen\].
\(ii) Similarly to (\[gineq\]), (\[Kineq\]) in the case $Q(x) = c_Q x$ we obtain $$\begin{aligned}
\E g^2_{t-k,t}
&=&c^2_Q \E \Big(\alpha_k+ \sum_{i=0}^{k-1} \beta_{i,k-i} \zeta_{t-i} g_{t-i,t} \Big)^2 \
=\ c^2_Q \Big(\alpha^2_k+ \sum_{i=0}^{k-1} \beta^2_{i,k-i} \E g^2_{t-i,t} \Big)\end{aligned}$$ and hence ${\rm Var}(X_t) = \sum_{k=0}^\infty \E g^2_{t-k,t} = K_Q $. This proves (ii) and Theorem \[thm1\], too. $\Box$
In the case of equations (\[proSE1\]) and (\[proSE2\]), condition (\[Kcond\]) can be simplified, see below. Note that for $ A^2 := \sum_{i=0}^\infty \alpha^2_i = 0$, equations (\[giter\]) admit a trivial solution $g_{t-k,t} = 0 $ since $Q(0) = 0$ by (\[Qdom\]), leading to the constant process $X = \mu $ in (\[proSEI\]).
\[propBeta\] (i) Let $A^2 >0, $ $\beta_{i,j} = \beta_{i+j}, \, i\ge 0, \ j \ge 1, $ and $B^2 := \sum_{i=0}^\infty \beta^2_i$. Then $K_Q<\infty $ is equivalent to $A^2 < \infty $ and $ B^2 < \infty $.
\(ii) Let $A^2 >0$, $\beta_{i,j} = \beta_{j}, \, i\ge 0, \ j \ge 1$ and $B^2 := \sum_{i=1}^\infty \beta^2_i$. Then $K_Q <\infty $ is equivalent to $A^2 < \infty $ and $c^2_Q B^2 < 1 $. Moreover, $K_Q = c^2_Q A^2/(1 - c^2_Q B^2).$
\(i) By definition, $$\begin{aligned}
K_{Q}
&=&\sum_{k=0}^\infty c^{2k+2}_Q \sum_{i=0}^\infty \alpha_i^2 \sum_{j_1=1}^\infty \beta^2_{i+j_1} \cdots
\sum_{j_k=1}^\infty \beta^2_{i+ j_1 + \dots + j_{k-1}+ j_k} \\
&=&\sum_{k=0}^\infty c^{2k+2}_Q \sum_{0\le i< j_1 < \dots < j_k< \infty} \alpha_i^2 \beta^2_{j_1} \cdots
\beta^2_{j_k} {\nonumber}\\
&\le&\sum_{k=0}^\infty c^{2k+2}_Q A^2 B^2_1 \cdots B^2_k, {\nonumber}\end{aligned}$$ where $B^2_k := \sum_{j=k}^\infty \beta^2_j.$ Since $B^2 < \infty $ entails $\lim_{k\to \infty} B^2_k = 0$, $\forall \epsilon>0 \, \exists \, K \ge 1$ such that $B^2_k < \epsilon/ c^2_Q $ $ \forall \, k > K$. Hence, $$\begin{aligned}
K_{Q}&\le&c^2_Q A^2\Big(\sum_{k=0}^K (c^{2}_Q B^2)^k + \sum_{k=K}^\infty \epsilon^k \Big) \ < \infty.\end{aligned}$$ Therefore, $A^2 < \infty $ and $B^2 < \infty $ imply $K_{Q} < \infty$. The converse implication is obvious.
\(ii) Follows by $$\begin{aligned}
K_{Q}
&=&\sum_{k=0}^\infty c^{2k+2}_Q \sum_{i=0}^\infty \alpha_i^2 \sum_{j_1=1}^\infty \beta^2_{j_1} \cdots
\sum_{j_k=1}^\infty \beta^2_{j_k} \
=\ \sum_{k=0}^\infty c^{2k+2}_Q A^2 (B^2)^k \ = \ \frac{c^2_Q A^2}{1 - c^2_Q B^2}.\end{aligned}$$
\[rem5\] [It is not difficult to show that conditions on the $\beta_{i,j}$’s in Proposition \[propBeta\] (i) and (ii) are part of the following more general condition: $\limsup_{i\to \infty} \sum_{j=1}^\infty c_Q^2 \beta^2_{i,j} < 1, $ which also guarantees that $K_Q < \infty $.]{}
The following Proposition \[propmom1\] obtains a sufficient condition for the existence of higher moments $\E |X_t|^p < \infty, p > 2$ of the solution of projective equation (\[proSEI\]). The proof of Proposition \[propmom1\] is based on a recurrent use of Rosenthal-type inequality of Proposition \[propmom\], which contains an absolute constant $C_p$ depending only on $p$. For $p\ge 2$, denote $$\begin{aligned}
K_{Q, p}
&:=&C_p^{2/p} \sum_{i=0}^\infty \alpha_i^2 \sum_{k=0}^\infty (c_Q C^{1/p}_p \mu^{1/p}_p)^{2k+2} \sum_{j_1=1}^\infty \beta^2_{i,j_1} \cdots
\sum_{j_k=1}^\infty \beta^2_{i+ j_1 + \dots + j_{k-1}, j_k}. \label{KQp}\end{aligned}$$ where (recall) $\mu_p = \E |\zeta_0|^p$. Note $C_2 = \mu_2 = 1 $, hence $K_{Q,2} = K_Q $ coincides with (\[KQ\]).
\[propmom1\] Assume conditions of Theorem \[thm1\] and $K_{Q, p}<\infty, $ for some $p\ge 2$. Then $\E |X_t|^p < \infty $.
The proof is similar to that of Theorem \[thm1\] (i). By Proposition \[propmom\], $$\begin{aligned}
\big(\E \big|X_t\big|^p\big)^{2/p} & \le & C_p^{2/p} \Big( \big|\E X_t\big|^p + \mu_p \big(\sum_{s \le t} \big(\E |g_{s,t}|^p \big)^{2/p} \big)^{p/2}\Big)^{2/p}\ =\ C_p^{2/p} \mu_p^{2/p}\sum_{s \le t} (\E |g_{s,t}|^p )^{2/p}.\end{aligned}$$ Using condition , Proposition \[propmom\] and inequality $(a+b)^q \le a^q+b^q, \, 0<q\le1$ we obtain the following recurrent inequality: $$\begin{aligned}
\big(\E |g_{s,t}|^p\big)^{2/p} & \leq & \big(c_Q^p\E \big|\alpha_{t-s}+ \sum\nolimits_{s<u\le t} \beta_{t-u, u-s} \zeta_u g_{u,t} \big|^p \big)^{2/p} \\
& \leq & c_Q^2 C_p^{2/p} \Big(|\alpha_{t-s}|^p+ \mu_p \big( \sum\nolimits_{s<u\le t} ( |\beta_{t-u, u-s}|^p\,\E |g_{u,t}|^p)^{2/p} \big)^{p/2} \Big)^{2/p}. \\
& \leq & c_Q^2 C_p^{2/p} \Big(|\alpha_{t-s}|^2+ \mu_p^{2/p} \sum_{s<u\le t} \beta_{t-u, u-s}^2 (\E |g_{u,t}|^p)^{2/p}
\Big).\end{aligned}$$ Iterating the last inequality as in the proof of Theorem \[thm1\] we obtain $(\E |X_t|^p)^{2/p} \le {K}_{Q,p}<\infty$, with $K_{Q,p} $ given in (\[KQp\]). $\Box$
Finally, let us discuss the question when $X_t$ of (\[proSEI\]) satisfies the weak dependence condition in (\[DedMer\]) for the invariance principle.
Let $\{X_t \}$ satisfy the conditions of Theorem \[thm1\] and $\Omega(2)$ be defined in (\[DedMer\]). Then $$\begin{aligned}
\Omega(2)
&\le&\sum_{i=0}^\infty |\alpha_i| \sum_{k=0}^\infty c^{k+1}_Q \sum_{j_1=1}^\infty |\beta_{i,j_1}| \cdots
\sum_{j_k=1}^\infty |\beta_{i+ j_1 + \dots + j_{k-1}, j_k}|. \label{KKQ}\end{aligned}$$ In particular, if the quantity on the r.h.s. of (\[KKQ\]) is finite, $\{X_t \}$ satisfies the functional central limit theorem in (\[DedMer1\]).
Follows from (\[gkiter\]) and the inequality $|\sum x_i |^{1/2} \le \sum |x_i|^{1/2}$. $\Box$
Examples
========
[*1. Finitely dependent projective equations.*]{} Consider equation , where $\alpha_i = \beta_{i,j} = 0 $ for all $i > m $ and some $m \ge 0$. Since $Q(0) = 0$, the corresponding equation writes as $$\begin{aligned}
\label{proSEI0}
X_t &=&\mu+ \sum_{t-m< s\le t} \zeta_s Q\Big(\alpha_{t-s}+ \sum_{s<u\le t} \beta_{t-u, u-s} \left(\E_{[u,t]} X_{t} - \E_{[u+1,t]} X_{t}\right) \Big),\end{aligned}$$ where the r.h.s. is ${\cal F}_{[t-m+1,t]}$-measurable. In particular, $\{X_t\}$ of is an $m$-dependent process. We may ask if the above process can be represented as a moving-average of length $m$ w.r.t. to some i.i.d. innovations? In other words, if there exists an i.i.d. standardized sequence $\{\eta_s, s\in \Z\}$ and coefficients $c_j, 0\le j < m$ such that $$\label{Xma}
X_t = \sum_{t-m < s \le t} c_{t-s} \eta_s, \qquad t \in \Z.$$
To construct a negative counter-example to the above question, consider the simple case of with $m=2$, $\mu = 0$, $\alpha_1 = 0, \beta_{0,1} = 1, Q(\alpha_0) = 1$: $$\begin{aligned}
\label{proSEI00}
X_t&=&\zeta_t Q(\alpha_0) + \zeta_{t-1} Q(\alpha_1 + \beta_{0,1} \E_{[t,t]} X_t) = \zeta_t + \zeta_{t-1} Q( \zeta_t).\end{aligned}$$ Assume that $\E Q(\zeta_t)= 0$. Then $\E X_t X_{t-1} = 0, \E X^2_t = 1 + \E Q^2(\zeta_0)$. On the other hand, from with $m=2 $ we obtain $0= \E X_t X_{t-1} = c_0 c_1, $ implying that $\{X_t \}$ is an i.i.d. sequence.
Let us show that the last conclusion contradicts the form of $X_t$ in under general assumptions on $Q$ and the distribution of $\zeta = \zeta_0$. Assume that $\zeta$ is symmetric, $ \infty > \E \zeta^4 > (\E \zeta^2)^2 = 1 $ and $Q$ is antisymmetric. Then $$\begin{aligned}
{\rm cov}(X^2_t,X^2_{t-1})
&=&\E Q^2(\zeta)\big\{(\E \zeta^4 - 1) + (\E \zeta^2 Q^2(\zeta) - \E Q^2(\zeta))\big\}.\end{aligned}$$ Assume, in addition, that $Q$ is monotone nondecreasing on $[0, \infty)$. Then $\E \zeta^2 Q^2(\zeta) \ge \E \zeta^2 \E Q^2(\zeta) = \E Q^2(\zeta)$, implying ${\rm cov}(X^2_t,X^2_{t-1}) >0$. As a consequence, is not a moving average of length 2 in some standardized i.i.d. sequence.
[*2. Linear kernel $Q$.*]{} For linear kernel $Q(x) = c_Q x $, the solution of (\[proSEI\]) of Theorem \[thm1\] can be written explicitly as $X_t = \mu+ \sum_{k=1}^\infty X^{(k)}_t$, where $X^{(1)}_t = c_Q\sum_{i=0}^\infty \alpha_i \zeta_{t-i} $ is a linear process and $$\begin{aligned}
X^{(k+1)}_t&=&c_Q^{k+1}\sum_{i=0}^\infty \alpha_i \sum_{j_1, \dots, j_k =1}^\infty
\beta_{i,j_1} \cdots \beta_{i+j_1+ \dots + j_{k-1}, j_k} \zeta_{t-i} \zeta_{t-i-j_1} \cdots \zeta_{t-j_1 - \dots - j_k}\end{aligned}$$ for $k\ge 1$ is a Volterra series of order $k+1$, see ([@ref5], p.22), which are orthogonal in sense that $\E X^{(k)}_t X^{(\ell)}_s =0, \, t,s \in \Z, \, k,\ell \ge 1,\, k \ne \ell$.
Let $H^2_{(-\infty,t]} \subset L^2_{(-\infty,t]} $ be the subspace spanned by products $1, \zeta_{s_1} \cdots \zeta_{s_k}, \, s_1 < \dots < s_k \le t, k\ge 1.$ Clearly, the above Volterra series $X_t, X^{(k)}_t \in H^2_{(-\infty, t]}, \, \forall t\in \Z $ (corresponding to linear $Q$) constitute a very special class of projective processes. For example, the process in cannot be expanded in such series unless $Q$ is linear. To show the last fact, decompose as $X_t = Y_t + Z_t$, where $Y_t := \zeta_t + \alpha \zeta_{t-1} \zeta_t \in H^2_{(-\infty,t]}, \, \alpha := \E \zeta Q(\alpha) $ and $Z_t := \zeta_{t-1} (Q(\zeta_t) - \alpha \zeta_t)$ is orthogonal to $H^2_{(-\infty,t]}, \, Z_t \neq 0,$ hence $X_t \not\in H^2_{(-\infty,t]}$.
[*3. The LARCH model.*]{} The Linear ARCH (LARCH) model, introduced by Robinson [@ref15], is defined by the equations $$\label{larch} r_t = \sigma_t \zeta_t, \ \ \ \sigma_t= \alpha + \sum_{j= 1}^\infty \beta_j r_{t-j},$$ where $\{\zeta_t\}$ is a standardized i.i.d. sequence, and the coefficients $\beta_j$ satisfy $B := \Big\{\sum_{j=1}^\infty
\beta^2_j \Big\}^{1/2} < \infty $. The LARCH model was studied in [@ref7], [@ref8], [@ref10]), [@ref9], [@ref3] and other papers. In financial modeling, $r_t$ are interpreted as (asset) returns and $\sigma_t$ as volatilities. Of particular interest is the case when the $\beta_j$’s in (\[larch\]) are proportional to ARFIMA coefficients, in which case it is possible to rigorously prove long memory of the volatility and the (squared) returns. It is well-known ([@ref9]) that a second order strictly stationary solution $\{r_t\}$ to (\[larch\]) exists if and only if $$\label{B2}
B \ < \ 1,$$ in which case it can be represented by the convergent orthogonal Volterra series $$\label{orthog}
r_t = \sigma_t\zeta_t, \ \ \ \sigma_t= \alpha\Big(1+\sum_{k= 1}^\infty \sum_{j_1, \dots, j_k= 1}^\infty \beta_{j_1}\cdots\beta_{j_k}
\zeta_{t-j_1}\cdots\zeta_{t-j_1-\dots-j_k}\Big).$$ Clearly, the last series is a particular case of the Volterra series of the previous example. We conclude that under the condition (\[B2\]), the volatility process $\{X_t = \sigma_t\}$ of the LARCH model satisfies the projective equation (\[proSE2\]) with linear function $Q(x) = x $ and $\alpha_j = \alpha \beta_j$. Note that (\[B2\]) coincides with the condition $c^2_Q B^2 < 1 $ of Proposition \[propBeta\] (ii) for the existence of solution of (\[proSE2\]).
From Proposition \[propmom1\] the following new result about the existence of higher order moments of the LARCH model is derived.
Assume that $$\label{larchp}
C_p^{1/p} \mu^{1/p}_p B < 1,$$ where $\mu_p = \E |\zeta_0|^p $ and $C_p$ is the absolute constant from Proposition \[propmom\], $p \ge 2$. Then $\E |r_t|^p = \mu_p \E |\sigma_t|^p < \infty $. Moreover, $$\label{larchmom}
\E |\sigma_t|^p \ \le \ \frac{\alpha^2 C_p^{4/p} \mu_p^{2/p} B^2}{1- C_p^{2/p} \mu_p^{2/p} B^2}.$$
Follows from Proposition \[propmom1\] and the easy fact that for the LARCH model, $K_{Q,p}$ of (\[KQp\]) coincides with the r.h.s. of (\[larchmom\]). $\Box$
Condition (\[larchp\]) can be compared with the sufficient condition for $\E |r_t|^p < \infty, p = 2,4,\dots $ in ([@ref7], Lemma 3.1): $$\label{larchp1}
(2^p - p -1)^{1/2} \mu^{1/p}_p B < 1.$$ Although the best constant $C_p$ in the Rosenthal’s inequality is not known, (\[larchp\]) seems much weaker than (\[larchp1\]), especially when $p$ is large. See, e.g. [@ref13], where it is shown that $C^{1/p}_p = O(p/\log p),\, p \to \infty. $
[*4. Projective “threshold” equations.* ]{} Consider projective equation $$\begin{aligned}
\label{proSET}
X_t &=&\zeta_t + \sum_{j=1}^p \zeta_{t-j} Q\big(\E_{[t-j+1,t]} X_{t} \big),\end{aligned}$$ where $1\le p < \infty$ and $Q $ is a bounded measurable function with $Q(0) = 1$. If $Q$ is a step function: $Q(x) = \sum_{k=1}^q c_k \1(x \in I_k), $ where $\cup_{k=1}^q I_k = \R$ is a partition of $\R$ into disjoint intervals $I_k, 1\le k \le q$, the process in follows different “moving-average regimes” in the regions $\E_{[t-j+1,t]} X_{t} \in I_k, 1\le j \le p $ exhibiting a “projective threshold effect”. See Fig. \[QTR\], where the left graph shows a trajectory having a single threshold at $x=0$ and the right graph a trajectory with two threshold points at $x=0$ and $x=2$.
Simulations
===========
Solutions of projective equations can be easily simulated using a truncated expansion $X^{(M)}_t = \sum_{t-M \le s \le t} g_{s,t} \zeta_s $ instead of infinite series in (\[proma0\]). We chose the truncation level $M$ equal to the sample size $M= n = 3000 $ in the subsequent simulations. The coefficients $g_{s,t} $ of projective equations are computed very fast from recurrent formula (\[giter\]) and simulated values $\zeta_s, -M \le s \le n$. The innovations were taken standard normal. For better comparisons, we used the same sequence $\zeta_s, -M \le s \le n$ in all simulations.
Stationary solution of equation (\[proSE2\]) was simulated for three different choices of $Q$ and two choices of coefficients $\alpha_j, \beta_j$. The first choice of coefficients is $\alpha_j = 0.5^j, \beta_j= c\, 0.9^j $ and corresponds to a short memory process $\{X_t\}$. The second choice is $\alpha_j = \Gamma(d+ j)/\Gamma(d) \Gamma (j+1), \, \beta_j = c \alpha_j$ with $d = 0.4$ corresponds to a long memory process $\{X_t\}$ with coefficients as in ARFIMA$(0,d,0)$. The value of $c>0$ was chosen so that $ c^2_Q B^2 = 0.9 < 1 $. The latter condition guarantees the existence of a stationary solution of (\[proSE2\]), see Proposition \[propBeta\].
The simulated trajectories and (smoothed) histograms of marginal densities strongly depend on the kernel $Q$. We used the following functions: $$\label{Qsimul}
Q_1(x) = x, \qquad Q_2(x)=\max(0,x), \qquad Q_3(x) = \begin{cases}
x,&x\in [0, 1],\\
2-x,&x \in [1, 2],\\
0,&\text{otherwise}.
\end{cases}$$ Clearly, $Q_i, i=1,2,3$ in (\[Qsimul\]) satisfy (\[Qdom\]) with $c_Q =1 $ and the Lipschitz condition (\[QLip\]). Note that $Q_3$ is bounded and supported in the compact interval $[0,2]$ while $Q_1, Q_2 $ are unbounded, the latter being bounded from below. Also note that for $\beta_j \equiv 0$ and the choice of $\alpha_j$ as above, the projective process $\{X_t \}$ of (\[proSE2\]) agrees with AR(.5) for $\alpha_j = 0.5^j$ and with ARFIMA$(0,0.4,0)$ for $\alpha_j = \Gamma(d+ j)/\Gamma(d) \Gamma (j+1)$ in all three cases in (\[Qsimul\])
\
\
\
A general impression from our simulations is that in all cases of $Q$ in (\[Qsimul\]), the coefficients $\alpha_j$ account for the persistence and $\beta_{j}$ for the clustering of the process. We observe that as $\beta_j$’s increase, the process becomes more asymmetric and its empirical density diverges from the normal density (plotted in red in Figures \[lmQint1\]-\[lmQint3\] with parameters equal to the empirical mean and variance of the simulated series). In the case of unbounded $Q = Q_1, Q_2$ and long memory ARFIMA coefficients, the marginal distribution seems strongly skewed to the left and having a very light left tail and a much heavier right tail. On the other hand, in the case of geometric coefficients, the density for $Q = Q_1, Q_2$ seems rather symmetric although heavy tailed. Case of $Q=Q_3$ corresponding to bounded $Q$ seems to result in asymmetric distribution with light tails.
Modifications
=============
Equation (\[proSEI\]) can be modified in several ways. The first modification is obtained by taking the $\alpha_{t-s}$’s “outside of $Q$”: $$\begin{aligned}
\label{proSEII}
X_t&=&\mu + \sum_{s\le t} \zeta_s \alpha_{t-s} Q\Big(\sum_{s<u\le t} \beta_{t-u, u-s} \left(\E_{[u,t]} X_t - \E_{[u+1,t]} X_t\right) \Big),\end{aligned}$$ where $\alpha_i, \beta_{i,j}, Q$ satisfy similar conditions as in (\[proSEI\]). However, note that (\[Qdom\]) implies $Q(0) = 0$ in which case (\[proSEII\]) has a trivial solution $X_t \equiv \mu $. To avoid the last eventuality, condition (\[Qdom\]) must be changed. Instead, we shall assume that $Q$ is a measurable function satisfying $$\begin{aligned}
Q(x)^2&\le&c^2_0 + c^2_1x^2, \quad x \in {\R} \label{QdomII}\end{aligned}$$ for some $c_0, c_1 \ge 0$. Denote $$\begin{aligned}
\tilde K_{Q}&:=&c^2_0
\sum_{k=0}^\infty c_1^{2k} \sum_{i=0}^\infty \alpha_i^2 \sum_{j_1=1}^\infty \alpha^2_{i+j_1}\beta^2_{i,j_1} \cdots
\sum_{j_k=1}^\infty \alpha^2_{i+ j_1 + \dots + j_k} \beta^2_{i+ j_1 + \dots + j_{k-1}, j_k}.\end{aligned}$$ Proposition \[thm2\] can be proved similarly to Theorem \[thm1\] and its proof is omitted.
\[thm2\] (i) Assume condition (\[QdomII\]) and $$\label{Kcond2}
\tilde K_{Q} \ < \ \infty.$$ Then there exists a unique solution $\{ X_t \} $ of (\[proSEII\]), which is written as a projective moving average of (\[XMA\]) with coefficients $g_{t-k,t}$ recursively defined as $$\begin{aligned}
g_{t-k,t}&:=&\alpha_k
Q\Big(\sum_{i=0}^{k-1} \beta_{i,k-i} \zeta_{t-i} g_{t-i,t} \Big), \qquad k =1,2, \dots, \quad g_{t,t}:= \alpha_0 Q(0).
\label{giterII}\end{aligned}$$
\(ii) In the case of linear function $Q(x) = c_0 + c_1 x$, condition (\[Kcond2\]) is also necessary for the existence of a solution of (\[proSEII\]).
\[remMod\] [Let $A^2_k := \sum_{i=k}^\infty \alpha^2_i$ and $|\beta_{i,j}| \le \bar \beta $. Then $$\begin{aligned}
\tilde K_{Q}&\le&c^2_0
\sum_{k=0}^\infty (c_1 \bar \beta)^{2k} \sum_{i=0}^\infty \alpha_i^2 \sum_{j_1=1}^\infty \alpha^2_{i+j_1} \cdots
\sum_{j_k=1}^\infty \alpha^2_{i+ j_1 + \dots + j_k} \
\le\ c^2_0
\sum_{k=0}^\infty (c_1 \bar \beta)^{2k} A^2_0 A^2_1 \cdots A^2_k.\end{aligned}$$ Hence, $A^2 = A^2_0 < \infty $ and $\bar \beta < \infty $ imply $\tilde K_{Q}< \infty, $ for any $c_0, c_1, \bar \beta $; see the proof of Proposition \[propBeta\]. ]{}
Projective stochastic equations (\[proSEI\]) and (\[proSEII\]) can be further modified by including projections of lagged variables. Consider the following extension of (\[proSEI\]): $$\begin{aligned}
\label{proSE0}
\hskip-.6cm &X_t= \mu
+ \sum\limits_{s\le t} \zeta_s Q\Big(\alpha_{t-s}+ \sum\limits_{u=s+1}^{t-1} \beta_{t-1-u, u-s} \left(\E_{[u,t-1]} X_{t-1} - \E_{[u+1,t-1]}\right) X_{t-1} \Big),\end{aligned}$$ where $\alpha_i, \beta_{i,j}, \, Q $ are the same as in (\[proSEI\]) and the only new feature is that $t$ is replaced by $t-1$ in the inner sum on the r.h.s. of the equation. This fact allows to study [*nonstationary* ]{} solutions of (\[proSE0\]) with a given projective initial condition $X_t = X^0_t, t \le 0$ and the convergence of $X_t$ to the equilibrium as $t \to \infty$; however, we shall not pursue this topic in the present paper. The following proposition is a simple extension of Theorem \[thm1\] and its proof is omitted.
\[prop11\] Let $\alpha_i, \beta_{i,j}, Q$ satisfy the conditions of Theorem \[thm1\], including (\[Qdom\]) and (\[Kcond\]). Then there exists a unique solution $\{ X_t \} $ of (\[proSE0\]), which is written as a projective moving average of (\[XMA\]) with coefficients $g_{t-k,t}$ recursively defined as $g_{t-k,t} := Q(\alpha_k), k=0,1 $ and $$\begin{aligned}
g_{t-k,t}&:=&
Q\big(\alpha_k+ \sum_{i=0}^{k-2} \beta_{i,k-1-i} \zeta_{t-1-i} g_{t-1-i,t-1} \big), \qquad k \ge 2.
\label{giter11}\end{aligned}$$
Finally, consider a projective equation (\[proSE\]) with $\mu_t \equiv 0$ and kernels $Q_{s,t} $ $= $ $Q_{t-s} (x_{s+1,t-1}, \dots,$ $ x_{s+1,s})$ depending on $t-s$ real variables, where $Q_0 = 1$ and $$\begin{aligned}
\label{Qj}
Q_j (x_1, \dots, x_j)&=&\frac{d(x_1)}{1} \cdot \frac{d(x_2) + 1}{2} \cdot \frac{d(x_3) + 2}{3}
\cdots
\frac{d(x_j) +j - 1}{j},\end{aligned}$$ $j\ge 1$, where $d(x), x \in \R$ is a measurable function taking values in the interval $(-1/2, 1/2)$. More explicitly, $$\begin{aligned}
\label{proseARFIMA}
\hskip-.8cm X_t&=&\sum_{j=0}^\infty Q_j \big(\E_{[t-j+1,t-1]} X_{t-1}, \E_{[t-j+1,t-2]} X_{t-2}, \dots, \E_{[t -j+1,t-j]} X_{t-j} \big) \zeta_{t-j},\end{aligned}$$ where $ \E_{[t -j+1,t-j]} X_{t-j} = \E X_t = 0$. Note that when $d(x) = d$ is constant, $\{X_t \} $ (\[proseARFIMA\]) is a stationary ARFIMA$(0,d,0)$ process. Time-varying fractionally integrated processes with deterministic coefficients of the form (\[Qj\]) were studied in [@ref16a], [@ref16b]. We expect that (\[proseARFIMA\]) feature a “random” memory intensity depending on the values of the process. A rigorous study of long memory properties of this model does not seem easy. On the other hand, solvability of (\[proseARFIMA\]) can be established similarly to the previous cases (see below).
\[prop1d\] Let $d(x)$ be a measurable function taking values in $(-1/2,1/2) $ and such that $\sup_{x \in \R} d(x) \le \bar d$, where $\bar d \in (0, 1/2)$. Then there exists a unique stationary solution $\{ X_t \} $ of (\[proseARFIMA\]), which is written as a projective moving average of (\[XMA\]) with coefficients $g_{s,t}$ recursively defined as $g_{t,t} := 1 $ and $$\begin{aligned}
g_{s,t}&:=&
Q_{t-s}\big(\sum_{s< u \le t-1} \zeta_u g_{u, t-1}, \sum_{s< u \le t-2} \zeta_u g_{u, t-2}, \dots, 0\big), \qquad s<t,
\label{gARFIMA}\end{aligned}$$ with $Q_{t-s} $ defined at (\[Qj\]).
Note that $\sup_{x_1, \dots, x_j \in \R} |Q_j(x_1,\dots, x_j)| \le \Gamma (\bar d +j)/\Gamma (\bar d) \Gamma(j) =:
\psi_j$ and $\sum_{j=0}^\infty \psi_j^2 < \infty$. Therefore the $g_{s,t}$’s in (\[gARFIMA\]) satisfy $\sum_{s\le t} \E g^2_{s,t} < \infty $ for any $t \in \Z$. The rest of the proof is analogous as the case of Theorem \[thm1\]. $\Box$
Long memory
===========
In this section we study long memory properties (the decay of covariance and partial sums’ limits) of projective equations (\[proSEI\]) and (\[proSEII\]) in the case when the coefficients $\alpha_j$’s decay slowly as $j^{d-1}, 0< d < 1/2.$
\[LM1\] Let $\{ X_t \} $ be the solution of projective equation (\[proSEI\]) satisfying the conditions of Theorem \[thm1\] and $\mu = \E X_t = 0$. Assume, in addition, that $Q$ is a Lipschitz function, viz., there exists a constant $c_L >0$ such that $$\label{QLip}
|Q(x)- Q(y)| \ \ < \ c_L|x-y|, \qquad x, y \in \R$$ and that there exist $\kappa >0 $ and $0< d < 1/2$ such that $$\begin{aligned}
\label{mainterm}
b_j&:=&Q(\alpha_j) \ \sim \ \kappa \, j^{d-1}, \qquad j \to \infty\end{aligned}$$ and $$\begin{aligned}
\label{barbeta1}
\bar \beta_j&:=&\max_{0\le i <j} |\beta_{i,j-i}| \ =\ o(b_j), \qquad j \to \infty.\end{aligned}$$ Then, as $t \to \infty $ $$\begin{aligned}
\label{CovLM}
\E X_0 X_t&\sim&\sum_{k= 0}^\infty b_k b_{t+k}
\ \sim \ \kappa^2_d t^{2d-1},\end{aligned}$$ where $\kappa^2_d := \kappa^2 B(d,1-d)$ and $B(d,1-d)$ is beta function. Moreover, as $n \to \infty$ $$\begin{aligned}
\label{FCLT}
n^{-1/2 -d}\sum_{t=1}^{[n\tau]}X_t&\longrightarrow_{D[0,1]}&c_{\kappa,d} B_H(\tau),\end{aligned}$$ where $B_H$ is a fractional Brownian motion with parameter $H = d+ (1/2)$ and variance $\E B^2_H(t) = t^{2H}$ and $c^2_{\kappa,d} := \frac{\kappa^2 B(d,1-d)}{d(1+2d)}$.
Let us note that the statements (\[CovLM\]) and (\[FCLT\]) are well-known when $\beta_{i,j} \equiv 0$, in which case $X_t$ coincides with the linear process $Y_t := \sum_{s\le t} b_{t-s} \zeta_s$. See, e.g., [@ref11], Proposition 3.2.1 and Corollary 4.4.1.
The natural idea of the proof is to approximate $\{X_t\}$ by the linear process $\{Y_t\}$. For $t \ge 0, k \ge 0$, denote $$\begin{aligned}
r^X_t&:=&\E X_0 X_t \ = \
\sum_{s\le 0} \E [g_{s,0}\, g_{s,t}], \qquad r^Y_t \ := \ \E Y_0 Y_t \ = \ \sum_{s \le 0} b_{-s} b_{t-s}, \\
\phi_{t-k,t}&:=&g_{t-k,t} - b_k \ = \ Q\Big(\alpha_k+ \sum_{i=0}^{k-1} \beta_{i,k-i} \zeta_{t-i} g_{t-i,t} \Big)- Q(\alpha_k).\end{aligned}$$ Then $$\begin{aligned}
r^X_t- r^Y_t&=&
\sum_{s\le 0} \E [(b_{-s} + \phi_{s,0}) (b_{t-s} + \phi_{s,t}) - b_{-s} b_{t-s}]\\
&=&\sum_{s\le 0} b_{-s} \E [\phi_{s,t} ] + \sum_{s\le 0} b_{t-s} \E [\phi_{s,0} ] + \sum_{s\le 0} \E [\phi_{s,0} \, \phi_{s,t} ]
\ =: \ \sum_{i=1}^3 \rho_{i,t}.\end{aligned}$$ Using (\[QLip\]) we obtain $$\begin{aligned}
|\E \phi_{t-k,t}|^2\ \le \ \E \phi^2_{t-k,t}
&\le&c^2_L \E \Big(\sum_{i=0}^{k-1} \beta_{i,k-i} \zeta_{t-i} g_{t-i,t} \Big)^2 \\
&=&c^2_L \Big(\sum_{i=0}^{k-1} \beta^2_{i,k-i} \E g^2_{t-i,t} \Big) \\
&\le&\bar \beta^2_k c^2_L \Big(\sum_{i=0}^{\infty} \E g^2_{t-i,t} \Big) \\
&\le&\bar \beta^2_k c^2_L K_{Q}.\end{aligned}$$ This and condition (\[barbeta1\]) imply that $$\begin{aligned}
|\E \phi_{t-k,t}| + \E^{1/2} \phi^2_{t-k,t}&\le&\delta_k k^{d-1}, \qquad \forall \ t, k \ge 0
$$ where $\delta_k \to 0 \, (k \to \infty)$. Therefore for any $t \ge 1$ $$\begin{aligned}
|\rho_{1,t}|&\le&C\sum_{k=1}^\infty k^{d-1} (t+ k)^{d-1} \delta_{t+k} \ \le \ C \delta'_t t^{2d-1}, \\
|\rho_{2,t}|&\le&C\sum_{k=1}^\infty k^{d-1} \delta_k (t+ k)^{d-1} \ \le \ C \delta'_t t^{2d-1}, \\
|\rho_{3,t}|&\le&\sum_{s\le t} \E^{1/2} [\phi^2_{s,0}] \, \E^{1/2}[ \phi^2_{s,t} ] \ \le \
C\sum_{k=1}^\infty k^{d-1} (t+ k)^{d-1} \delta_k \delta_{t+k}\ \le \ C \delta'_t t^{2d-1},
\end{aligned}$$ where $\delta'_k \to 0 \ (k \to \infty)$. This proves (\[CovLM\]).
To show (\[FCLT\]), consider $Z_t := X_t - Y_t = \sum_{u\le t} \phi_{u,t} \zeta_u, t \in \Z. $ By stationarity of $\{Z_t \}$, for any $s\le t$ we have $\cov (Z_t, Z_s)=
\sum_{u\le 0}\E [\phi_{u,0} \, \phi_{u,t-s} ] \le \sum_{u\le 0}\E^{1/2} [\phi^2_{u,0}] \, \E^{1/2} [\phi^2_{u,t-s} ]$ $= o((t-s)^{2d-1}),$ see above, and therefore $ \E \big(\sum_{t=1}^n Z_t\big)^2 = o(n^{2d+1})$, implying $$n^{-d -(1/2)} \sum_{t=1}^{[n\tau]} X_t\ = \ n^{-d -(1/2)} \sum_{t=1}^{[n\tau]} Y_t + o_p(1).$$ Therefore partial sums of $\{X_t\}$ and $\{Y_t\}$ tend to the same limit $c_{\kappa,d} B_H(\tau)$, in the sense of weak convergence of finite dimensional distributions. The tightness in $D[0,1]$ follows from (\[CovLM\]) and the Kolmogorov criterion. Theorem \[LM1\] is proved. $\Box$
A similar but somewhat different approximation by a linear process applies in the case of projective equations of (\[proSEII\]). Let us discuss a special case of $\beta_{i,j}$: $$\label{betaK}
\beta_{i,j} = 1, \qquad \text{for all} \ i=0,1, \dots, \, j=1,2,\dots.$$ Note that for such $\beta_{i,j}$, $\sum_{s<u\le t} \beta_{t-u,u-s} (\E_{[u,t]} - \E_{[u+1,t]}) X_t =
\E_{[s+1,t]} X_t, \, s < t$ and the corresponding projective equation (\[proSEII\]) with $\mu = 0, \alpha_i = b_i$ coincides with (\[proma1\]). Recall that for bounded $\beta_{i,j}$’s as in (\[betaK\]), condition (\[QdomII\]) on $Q$ together with $\sum_{i=0}^\infty \alpha^2_i < \infty $ guarantee the existence of the stationary solution $\{X_t\} $ (see Remark \[remMod\]). We shall also need the following additional condition: $$\label{Qextra}
\E \big(Q(\E_{[s,0]} X_0) - Q(X_0)\big)^2 \ \to \ 0, \qquad \text{as} \quad s \to - \infty.$$ Since $\E \big(\E_{[s,0]} X_0 - X_0\big)^2 \to 0, \, s \to - \infty$, so (\[Qextra\]) is satisfied if $Q$ is Lipschitz, but otherwise conditions (\[Qextra\]) and (\[QdomII\]) allow $Q$ to be even discontinuous. Denote $$c^2_{Q,d} := \big(\E [Q (X_0)] \big)^2 B(d,1-d).$$
\[LM2\] Let $\{ X_t \} $ be the solution of projective equation (\[proSEII\]) with $\mu = 0, \, \beta_{i,j}$ as in (\[betaK\]), $Q$ satisfying (\[QdomII\]) and $$\label{alfaLM}
\alpha_k\ \sim\ \, k^{d-1}, \qquad k \to \infty, \qquad \exists \ \ 0< d < 1/2.$$ In addition, let (\[Qextra\]) hold. Then $$\label{CovLMII}
\E X_0 X_t\ \sim\ c^2_{Q,d} t^{2d-1}, \qquad t \to \infty$$ and $$\begin{aligned}
\label{FCLTII}
n^{-1/2 -d}\sum_{t=1}^{[n\tau]}X_t&\longrightarrow_{D[0,1]}&c'_{Q,d} B_H(\tau), \qquad
c'_{Q,d} := c_{Q,d}/(d(1+2d)^{1/2}.\end{aligned}$$
Similarly as in the proof of the previous theorem, let $Y_t := \sum_{s\le t} b_{t-s} \zeta_s$, $b_k := \ \alpha_k \E [Q ( X_0)]$, $r^X_t:=\E X_0 X_t, \,
r^Y_t:=\E Y_0 Y_t, \, t\ge 0$. Relation (\[CovLMII\]) follows from $$\label{rXY}
r^X_t - r^Y_t \ = \ o(t^{2d-1}), \qquad t \to \infty.$$ We have $X_t = \sum_{s\le t} g_{s,t} \zeta_s, \, g_{s,t} = \alpha_{t-s} Q(\E_{[s+1,t]} X_t), \, \E X^2_t = \sum_{s\le t} \E g^2_{s,t} < \infty $ and $\E [Q(\E_{[s+1,t]} X_t)^2] \le c_0^2 + c_1^2 \E (\E_{[s+1,t]} X_t)^2 \le c_0^2 + c_1^2 \E X^2_t < C.
$ Decompose $r^X_t = r^X_{1,t} + r^X_{2,t}$, where $$\begin{aligned}
r^X_{1,t}
&:=&\sum_{s\le 0} \alpha_s \alpha_{t+s} \E [Q(\E_{[s+1,0]} X_0)] \,\E[ Q(\E_{[1,t]} X_t) ], \qquad
r^X_{2,t} \ := \ \sum_{s\le 0} \alpha_s \alpha_{t+s} \gamma_{s,t},$$ and where $$\begin{aligned}
|\gamma_{s,t}|&:=&\big|\E \big[Q(\E_{[s+1,0]} X_0) \big\{ Q(\E_{[s+1,t]} X_t) - Q(\E_{[1,t]} X_t)\big\} \big]\big|
\ \le \ \tilde \gamma^{1/2}_{1,s} \tilde \gamma^{1/2}_{2,s,t},\end{aligned}$$ Here, $\tilde \gamma_{1,s} := \E[Q^2(\E_{[s+1,0]} X_0)] \le C, $ see above, while $$\begin{aligned}
|\tilde \gamma_{2,s,t}|&:=&\E \big[\big(Q(\E_{[s+1,t]} X_t) - Q(\E_{[1,t]} X_t)\big)^2 \big]{\nonumber}\\
&=&\E \big[\big(Q(\E_{[s+1-t,0]} X_0) - Q(\E_{[1-t,0]} X_0)\big)^2 \big]\ \to \ 0, \qquad t \to \infty
\label{Qgamma}\end{aligned}$$ uniformly in $s \le 0$, according to (\[Qextra\]). Hence and from (\[alfaLM\]) it follows that $$\label{rliek}
|r^X_{2,t}|
\ = \ o(t^{2d-1}), \qquad t \to \infty.$$ Accordingly, it suffices to prove (\[rXY\]) with $r^X_t$ replaced by $r^X_{1,t}$. We have $$\begin{aligned}
r^X_{1,t}
&=&r^Y_t + \sum_{s\le 0} \alpha_s \alpha_{t+s}
\phi_{1, s,t}
+ \sum_{s\le 0} \alpha_s \alpha_{t+s}
\phi_{2, s,t} + \sum_{s\le 0} \alpha_s \alpha_{t+s}
\phi_{3, s,t},\end{aligned}$$ where the “remainders” $\phi_{1, s,t} := \E [Q(X_0)] \big\{\E [Q(\E_{[s+1,0]} X_0)] - \E [Q(X_0)]\big\}$, $\
\phi_{2, s,t} :=$ $\E [Q(X_0)]$ $\times \big\{\E [Q(\E_{[1-t,0]} X_0)] - \E [Q(X_0)]\big\} $ and $ \phi_{3, s,t} := \big(\E [Q(\E_{[s+1,0]} X_0)] - \E [Q(X_0)]\big)\big(\E [Q(\E_{[1-t,0]} X_0)] - \E [Q(X_0)]\big) $ can be estimated similarly to (\[Qgamma\]), leading to the asymptotics of (\[rliek\]) for each of the three sums in the above decomposition of $r^X_{1,t}$. This proves (\[CovLMII\]).
Let us prove (\[FCLTII\]). Consider the convergence of one-dimensional distributions for $\tau = 1$, viz., $$\label{clt2}
n^{-d-1/2} S^X_n \ \to \ {\cal N}(0, \sigma^2), \qquad \sigma = c'_{Q,d}$$ where $S^X_n := \sum_{t=1}^n X_t$. Then (\[clt2\]) follows from $$\label{clt2Z}
\E (S^X_n - S^Y_n)^2 \ = \ o(n^{1+ 2d}),$$ where $S^Y_n := \sum_{t=1}^n Y_t$ and $Y_t$ is as above. We have $$\begin{aligned}
\label{SXY}
\E (S^X_n - S^Y_n)^2&=&\E \Big( \sum_{s\le n} \zeta_s \sum_{t=1\vee s}^n \alpha_{t-s} \tilde Q_{s,t} \Big)^2 {\nonumber}\\
&= &
\sum_{s \le n} \sum_{t_1, t_2=1\vee s}^n \alpha_{t_1-s} \alpha_{t_2-s} \E [ \tilde Q_{s,t_1} \tilde Q_{s,t_2}],\end{aligned}$$ where $\tilde Q_{s,t} := Q\big(\E_{[s+1,t]} X_t \big)-
\E [Q (X_0)]$. Let us prove that uniformly in $s \le t_1 $ $$\begin{aligned}
\label{Q?}
\E [\tilde Q_{s,t_1} \tilde Q_{s,t_2}]&=&o(1), \qquad \text{as} \qquad t_2-t_1 \to \infty.\end{aligned}$$ We have for $s \le t_1 \le t_2$ that $$\begin{aligned}
\E [\tilde Q_{s,t_1} \tilde Q_{s,t_2}]&=&\E \Big[\tilde Q_{s,t_1} \big\{Q\big(\E_{[s+1,t_2]} X_{t_2} \big)-
\E [Q (X_0)]\big\} \Big] \\
&=&\E [\tilde Q_{s,t_1}] \big\{ \E \big[Q\big(\E_{[t_1+1,t_2]} X_{t_2} \big)\big] -
\E [Q (X_0)] \big\}\\
&+&\E \Big[\tilde Q_{s,t_1} \big\{ Q\big(\E_{[s+1,t_2]} X_{t_2} \big)- Q\big(\E_{[t_1+1,t_2]} X_{t_2} \big)\big\} \Big] \
=: \ \psi'_{s,t_1,t_2} + \psi''_{s,t_1,t_2},\end{aligned}$$ where we used the fact that $\tilde Q_{s,t_1}$ and $Q\big(\E_{[t_1+1,t_2]} X_{t_2} \big)$ are independent. Here, thanks to (\[Qextra\]), we see that $|\psi'_{s,t_1,t_2}| \le \E^{1/2} [\tilde Q^2_{s,t_1}] \,\E^{1/2} \big[\big\{Q\big(\E_{[t_1-t_2+1,0]} X_{0}\big) - Q(X_0)\big\}^2\big]
\le C \E^{1/2} \big[\big\{Q\big(\E_{[t_1-t_2+1,0]} X_{0}\big) - Q(X_0)\big\}^2\big]$ $ \to 0$ uniformly in $s \le t_1 \le t_2$ as $t_2 - t_1 \to \infty $. The same is true for $|\psi''_{s,t_1,t_2}|$ since it is completely analogous to (\[Qgamma\]). This proves (\[Q?\]). Next, with (\[SXY\]) in mind, split $\E (S^X_n - S^Y_n)^2 =: T_n = T_{1,n} + T_{2,n} $, where $$T_{1,n} := \sum_{s \le n} \sum_{t_1, t_2=1\vee s}^n \1(|t_1-t_2| >K) \dots,
\qquad T_{2,n} :=
\sum_{s \le n} \sum_{t_1, t_2=1\vee s}^n \1 (|t_1-t_2|\le K) \dots,
$$ where $K$ is a large number. By (\[Q?\]), for any $\epsilon>0 $ we can find $K>0$ such that $\sup_{s \le t_1 < t_2: t_2 -t_1 > K} $ $|\E [\tilde Q_{s,t_1} \tilde Q_{s,t_2}]| < \epsilon $ and therefore $$|T_{1,n}|\ <\ \epsilon \sum_{s \le n} \sum_{t_1, t_2=1\vee s}^n
|\alpha_{t_1-s} \alpha_{t_2-s}|\ \le \ C\epsilon \sum_{t_1, t_2=1}^n |\bar r_{t_1-t_2}| \
\le \ C \epsilon n^{1+2d}$$ holds for all $n>1$ large enough, where $\bar r_t := \sum_{i=0}^\infty |\alpha_i \alpha_{t+i}| = O(t^{2d-1})$ in view of (\[alfaLM\]). On the other hand, $|T_{2,n}| \le C K n = o(n^{1+2d})$ for any $ K < \infty $ fixed. Then (\[clt2Z\]) follows, implying the finite-dimensional convergence in (\[FCLTII\]). The tightness in (\[FCLTII\]) follows from (\[CovLMII\]) and the Kolmogorov criterion, similarly as in the proof of Theorem \[LM1\]. Theorem \[LM2\] is proved. $\Box$
\[Wu0\]
Shao and Wu [@ref17] discussed partial sums limits of fractionally integrated nonlinear processes $Y_t = (1- L)^{-d} u_t, \, t \in \Z, $ where $L X_t = X_{t-1}$ is the backward shift, $(1- L)^d = \sum_{j=0}^\infty \psi_j (d) L^j, \,
d \in (-1,1)$ is the fractional differentiation operator, and $\{u_t \} $ is a causal Bernoulli shift: $$\label{ut}
u_t = F(\dots, \zeta_{t-1}, \zeta_t), \qquad t \in \Z$$ in i.i.d. r.v.’s $\{\zeta_t, t \in \Z \} $. The weak dependence condition on $\{u_t\} $ (\[ut\]), analogous to (\[DedMer\]) and guaranteeing the weak convergence of normalized partial sums of $\{Y_t \}$ towards a fractional Brownian motion, is written in terms of projections $P_0 u_t = (\E_{[0,t]} - \E_{[1,t]})u_t $: $$\begin{aligned}
\label{Wu}
\Omega(q) \ := \ \sum_{t=1}^{\infty} \|P_0 u_t\|_q \ < \ \infty,\end{aligned}$$ where $\| \xi \|_q := \E^{1/q} |\xi|^q $ and $q= 2$ for $0< d < 1/2$; see ([@ref17], Thm. 2.1), also [@ref21], [@ref20]. The above mentioned papers verify (\[Wu\]) for several classes of Bernoulli shifts. It is of interest to verify (\[Wu\]) for projective moving averages. For $X_t $ of (\[proma0\]) and $0< d < 1/2$, $u_t := (1-L)^d X_t
= \sum_{s \le t} \zeta_s G_{s,t}$ is a well-defined projective moving average with coefficients $$G_{s,t} := \sum_{s\le v \le t} \psi_{t-v}(d) g_{s, v}, \qquad s \le t,$$ see Proposition \[propfilter\]. For concreteness, let $g_{s,t} = \psi_{t-s}(-d) Q(\E_{[s+1,t]} X_t) $ as in Theorem \[LM2\] with $\alpha_j = \psi_j(-d)$. We have $\Omega(2) = \sum_{t=1}^{\infty} \|G_{0,t}\|_2 $, where $$\begin{aligned}
\hskip-1.5cm &&\|G_{0,t}\|^2_2
= \E\Big[ \sum_{v=0}^t
\psi_{t-v}(d) \psi_{v}(-d) Q(\E_{[1,v]} X_v) \Big]^2\
= \E \Big[ \sum_{v=0}^{t-1}
\psi_{t-v}(d) \psi_{v}(-d) Q_{v,t}\Big]^2, \label{Wu1}\end{aligned}$$ where $Q_{v,t} := Q(\E_{[1,v]} X_v) - Q(\E_{[1,t]} X_t)$ and we used $ \sum_{v=0}^t
\psi_{t-v}(d) \psi_{v}(-d) = 0, \, t \ge 1$ in the last equality. Note that $ \psi_{t-v}(d) \psi_{v}(-d) < 0$ have the same sign and $Q_{v,t} \approx Q(X_{v}) - Q(X_t)$ are not negligible in (\[Wu1\]). Therefore we conjecture that $\|G_{0,t}\|^2_2 = O\big(\sum_{v=0}^{t-1}
|\psi_{t-v}(d) \psi_{v}(-d)|\big)^2$ $ = O(t^{-2(1-d)})$ and hence $\Omega(2) = \infty $ for $0< d < 1/2$. The above argument suggests that projective moving averages posses a different “memory mechanism” from fractionally integrated processes in [@ref17].
Acknowledgements
================
This work was supported by a grant (No. MIP-063/2013) from the Research Council of Lithuania. The authors also thank an anonymous referee for useful remarks.
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|
---
author:
- 'H. Andrews, [^1] D. Fenech, R. K. Prinja, J. S. Clark,'
- 'L. Hindson'
bibliography:
- 'ref.bib'
date: 'Received 8th July, 2019; accepted 4th September, 2019'
title: A Radio Census of the Massive Stellar Cluster Westerlund 1
---
=1
[Massive stars and their stellar winds are important for a number of feedback processes. The mass lost in the stellar wind can help determine the end-point of the star as a NS or a BH. However, the impact of mass-loss on the post-Main Sequence evolutionary stage of massive stars is not well understood. Westerlund 1 is an ideal astrophysical laboratory in which to study massive stars and their winds in great detail over a large range of different evolutionary phases.]{} [We aim to study the radio emission from Westerlund 1, in order to measure radio fluxes from the population of massive stars, and determine mass-loss rates and spectral indices where possible.]{} [Observations were carried out in 2015 and 2016 with the Australia telescope compact array (ATCA) at 5.5 and 9$\,$GHz using multiple configurations, with maximum baselines ranging from 750$\,$m to 6$\,$km.]{} [30 stars were detected in the radio from the fully concatenated dataset, 10 of which were WRs (predominantly late type WN stars), 5 YHGs, 4 RSGs, 1 LBV star, the sgB\[e\] star W9, and several O and B supergiants. New source detections in the radio were found for 5 WR stars, and 5 OB supergiants. These detections have led to evidence for 3 new OB supergiant binary candidates, inferred from derived spectral index limits.]{} [Spectral indices and index limits were determined for massive stars in Westerlund 1. For cluster members found to have partially optically thick emission, mass-loss rates were calculated. Under the approximation of a thermally emitting stellar wind and a steady mass-loss rate, clumping ratios were then estimated for 8 WRs. Diffuse radio emission was detected throughout the cluster. Detections of knots of radio emission with no known stellar counterparts indicate the highly clumped structure of this intra-cluster medium, likely shaped by a dense cluster wind.]{}
Introduction
============
Massive stars and the outflows formed from their stellar winds are responsible for a number of important feedback processes, of chemical and mechanical origin, both to their immediate surroundings as well as further afield. Massive stars are predominantly found to form in massive stellar clusters or associations, and so understanding not only the stars themselves and their individual evolution, but also the impact on and from their environment is important for quantifying all the physics involved. This includes the chemical and mechanical feedback from individual stars to their circumstellar environment, as well as larger scale feedback processes, such as the production of cosmic rays, or galactic superwinds. Understanding the cluster environments can also help us to understand how outflows from these massive clusters may trigger or inhibit star formation in nearby regions.
These massive stars are also progenitors of the some of the most exotic endpoints in astrophysics, ending their lives as either a neutron star (NS), or directly collapsing to a black hole (BH), with many massive stars also experiencing a corresponding supernovae (SNe) explosion. This is of particular importance when considering the recent discovery of gravitational waves, which occur due to the mergers of these important astrophysical objects, with progenitors of these mergers believed to be of very large masses themselves [@Abbott2016]. These final endpoints are characterised primarily by the initial mass of the star, and the mass lost through its stellar lifetime. The characterisation of the SNe itself can also be found to be influenced by the surrounding environment. By understanding the cluster environments that many of these massive stars live and die in, this can help us to understand the origins of the geometries of SNe remnants observed today, and any associated asymmetries.
Despite the clear motivations behind understanding this field of astrophysics, the evolutionary pathways of these massive stars are not well understood. Even the consideration of the relative phases of stars beyond the main-sequence, where the most massive stars are able to become hydrogen stripped Wolf-Rayet (WR) stars, are not well understood. The range of evolutionary stages experienced by massive stars have direct consequences on their final endpoints.
Massive stars are impacted by factors such as mass-loss, rotation speeds, and the possible effect of magnetism, as well as binarity. Quantifying the mass loss through stellar winds can help determine the possible evolutionary pathways for different initial stellar masses, helping to constrain and determine possible progenitors for each type of final stellar endpoint.
Mass-loss rates are not always consistent between those assumed in evolutionary codes and from observations, especially for the case of early-type O and B stars, where discrepancies of up to a factor of 10 have recorded$\,$[@Puls2006; @Fullerton2006]. One of the ways in which this is impacted is the presence of clumping in the wind, which has a direct impact on the final value of mass-loss determined in the wind, with the necessary inclusion of a clumping factor in calculations quenching the observed mass-loss rates$\,$[@Prinja2010; @Prinja2013; @Sundqvist2011; @Surlan2012].
Another factor that significantly impacts stellar evolution is binarity. Many WR stars have been determined to be located in binary systems. Almost 70% of massive stars are believed to reside in binary systems$\,$ [@Sana2012; @Sana2013], and interactions between the primary and secondary stars, especially for those with close separations, could cause significant levels of mass transfer, affecting the evolution of both stellar components. There are several diagnostic factors by which you can determine the presence of binarity for a stellar system, from radial velocity variations to the presence of hard X-rays. Another way of discovering a binary candidate is by measuring a negative spectral index for the star in the radio regime, indicating the presence of non-thermal emission which may be attributed to colliding stellar winds$\,$[@Blomme2010].
Binarity is also of importance in terms of providing a possible source, via the colliding winds of binaries, as a source for cosmic rays. Single massive stars, especially stars with dense winds, such as WRs, are also a potential source for generating cosmic rays, via shocks between the stellar winds and their circumstellar environments$\,$[@Cesarsky1983]. The generation of cosmic rays in the stellar winds of massive stars are believed to be possible in the locations of open clusters in particular, where the regions of interstellar material and a possible strong radiation field can help to sustain cosmic rays produced by stellar sources, either via their winds or from the resultant SNe$\,$[@Bednarek2014].
Westerlund 1$\,$(Wd1) provides a unique astrophysical laboratory in order to investigate the stellar evolution of massive stars at a multitude of evolutionary stages, with hundreds of O and B stars as well as the largest coeval population of yellow hypergiants (YHGs) in our galaxy. Westerlund 1 is the most massive stellar cluster in the Milky Way, discovered in 1961 by Bengt Westerlund$\,$[@Westerlund1961]. It has a total mass estimate of $\sim$ 10$^{5}$ M$_{\odot}\,$[@Clark2005]. It has a current age estimate of 5$\,$Myr and a distance estimate of 5$\,$kpc. Recent estimates from Gaia DR2 have suggested a smaller distance of $\sim$ 3$\,$kpc may be more appropriate$\,$[@Aghakhanloo2019], but other discussions of the cluster distance in light of Gaia data have suggested significant limitations in the use of Gaia to determine Wd1’s distance, due to the extended structure of some of the cluster’s brighter members$\,$[@Clark2018]. For this paper, the estimate of 5$\,$kpc is used.
Wd1 has been previously observed across most of the electromagnetic spectrum, with observations carried out in the IR, optical, millimetre, and X-ray [@Clark2005; @Negueruela2010; @Crowther2006; @Bonanos2007; @Clark2008; @Damineli2016; @Muno2006]. It has been determined to be a source of highly energetic cosmic rays from the observations of the products of CRs, TeV $\gamma$-rays, observed from the cluster$\,$[@Abramowski2012; @Aharonian2019]. It has also previously been observed in the radio with the use of the Australia telescope compact array (ATCA) [@Clark1998; @Dougher2010 henceforth ], and most recently, in the millimetre [@Fenech2018 henceforth ]. The observations discussed in this paper provide a direct follow-on from the prior radio observations, with improved sensitivity and resolution.
Section 2 details the observations, as well as the data reduction. Section 3 describes the data analysis then carried out in order to determine fluxes and spatial extents. Section 4 introduces the result of the diffuse radio emission detected throughout the cluster. In Section 5 we introduce the new radio source detections found, separated into stellar sources, uncatalogued sources found in the previous mm-observations, and further previously uncatalogued radio sources determined (with no optical or mm counterpart). Section 6 goes into detail about the results of the stellar sources, considering the WR population, the cool supergiant and hypergiant population, and other stellar sources, including Wd1-9, the luminous blue variable (LBV) Wd1-243 and the O and B supergiants. The extended emission of the cluster and its possible origin is discussed in section 7. A summary of our conclusions and possible future avenues for investigation is then provided in section 8.
{width="\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
Observations and data reduction
===============================
------------- ------------------- --------------- ----------------------
Array Observing Obs. Duration Beam
Config. Dates (hours) Size
6A 27th$\,$-$\,$29th 31.77 1.71“ $\times$ 0.99”
(6$\,$km) October 2015 (0.790$^{\circ}$)
1.5A 25th$\,$-$\,$27th 16.45 3.59“ $\times$ 1.99”
(1.5$\,$km) November 2015 (-7.23$^{\circ}$)
1.5B 3rd June 2016 15.76 4.40“ $\times$ 2.41”
(1.5$\,$km) (-2.73$^{\circ}$)
750C 14th$\,$-$\,$15th 28.37 6.98“ $\times$ 4.44”
(750$\,$m) December 2015 (6.28$^{\circ}$)
------------- ------------------- --------------- ----------------------
: Summary of observations taken by ATCA.[]{data-label="table:obs_details"}
Observations were made with the use of ATCA over 4 different observing periods from October 2015 to June 2016. ATCA contains 6 radio dishes with a diameter of 22$\,$m each. Data were taken in four different configurations, with maximum baselines set at 750$\,$m (750C), 1.5$\,$km (1.5A, 1.5B) and 6$\,$km (6A), with details listed in Table \[table:obs\_details\]. Two of these configurations were set at a largest baseline of 1.5$\,$km, due to a repetition of observations carried out to account for bad weather on the first set of 1.5$\,$km observations. The data were collected over two spectral windows, with central frequencies 5.5$\,$GHz, and 9$\,$GHz, and a bandwidth of 2$\,$GHz for each band. Each spectral window contained 2048 channels, with bandwidths of 1$\,$MHz. Integration time on source per pointing was $\sim$16$\,$hours. J1636-4101 was used as a phase calibrator for the 6$\,$km configuration and 1600-48 was used as a phase calibrator for all other configurations. 1934-638 was observed as a bandpass calibrator for all configurations and was also used for flux amplitude calibration. A secondary calibrator, 0823-500, was also observed.
Data reduction and calibration was carried out with the use of <span style="font-variant:small-caps;">miriad</span>$\,$[@Miriad], with additional flagging of data carried out in astronomical image processing software (<span style="font-variant:small-caps;">aips</span>) with the use of scripted e-MERLIN RFI mitigation pipeline for interferometry (SERPent$\,$[^2]) on the 9$\,$GHz data [@Peck2013]. This involved the flagging of radio frequency interference (RFI) and any erroneous data, as well as calibrating for the bandpass, phase, flux density and any polarisation leakage, following standard procedures as laid out in the Miriad User Guide [@MiriadGuide]. After calibration, data were concatenated in several forms, discussed in detail below, with a run-through of both phase and amplitude self-calibration then additionally carried out on these final datasets. This involved the use of <span style="font-variant:small-caps;">casa</span>$\,$[@Casa], where the functions <span style="font-variant:small-caps;">gaincal</span> and <span style="font-variant:small-caps;">applycal</span> were carried out alongside the use of the deconvolution imaging tool <span style="font-variant:small-caps;">tclean</span>, cleaned in the mode ‘multi-scale multi-frequency synthesis’ with Briggs weighting applied (with the robust parameter set to 0). The psf mode was set to ‘clark’. The resolution was set to be equivalent to the smallest primary beam size present in each dataset. For the fully concatenated dataset, this beam size was taken from the 6$\,$km observations, 1.71“ $\times$ 0.99” (position angle 0.790$^{\circ}$), as calculated automatically in <span style="font-variant:small-caps;">CASA</span>, and the cell size was set to 0.3$\times$0.3$\,$arcseconds. The multiple deconvolution scales applied were at the size of the beam, as well as additional scales set at 5, 10, 12, 15, 18, 25, 47 pixels, in order to consider the varying resolutions when concatenating datasets from different configurations and spectral windows.
This led to a final set of fits files containing fully calibrated and cleaned images. For data analysis, <span style="font-variant:small-caps;">pbcor</span> (primary beam correction) was applied to mitigate the effects of attenuation from the primary beam for sources further from the pointing centre.
A note on different datasets
----------------------------
Several datasets were considered for the analysis of the radio observations. The definitive fluxes of radio sources from the cluster were taken from the fully integrated dataset containing all configurations and all spectral windows observed by ATCA, hereby referred to as the dataset <span style="font-variant:small-caps;">FullConcat</span>. Fluxes were also considered for comparative use from datasets split into the 5.5$\,$GHz and 9$\,$GHz, combined over all array configurations, <span style="font-variant:small-caps;">Full5</span> and <span style="font-variant:small-caps;">Full9</span>. These datasets are used for considering the mass-loss rates from each spectral window for thermally emitting sources, and the <span style="font-variant:small-caps;">Full9</span> is used to compare flux values to the prior radio observations taken of Wd1 by .
Additional datasets generated from the ATCA observations are tapered datasets of both the 5.5 and 9$\,$GHz observations, <span style="font-variant:small-caps;">Taper5</span> and <span style="font-variant:small-caps;">Taper9</span>. The datasets of tapered visibilities were created in order to compare datasets at different wavelength regimes more accurately, by comparing emission detected only from common ranges of u-v visibilities for both the radio and the millimetre ALMA observations . These datasets were used for the calculation of spectral indices, as well as mass-loss rates, and is discussed in more detail in Section 3.5 and Section 6.1.1. The corresponding mass-loss rates were then used to calculate clumping gradients, as discussed in Section 6.1.2. The tapered millimetre observations are referred to as <span style="font-variant:small-caps;">TaperALMA</span>, in order to distinguish from the results presented in , which contains all the full u-v range from the ALMA observations.
Analysis
========
After data reduction was completed, analysis was carried out on the datasets using SEAC[^3], a source extraction software tool [@Peck2014; @Morford2019]. This tool involves the use of <span style="font-variant:small-caps;">parseltongue</span>, a way to use a python interface with the software <span style="font-variant:small-caps;">aips</span>. The data was converted for use from fits files into the AIPS environment. A consideration of the possible offset of sources to previous datasets was determined following the procedure of previous radio and millimetre wavelength measurements . The ATCA datasets were run through the <span style="font-variant:small-caps;">aips</span> task <span style="font-variant:small-caps;">hgeom</span>, where the images were aligned to the same geometry as the FORS and ALMA dataset. The data was then run through <span style="font-variant:small-caps;">jmfit</span> to check any possible offset for the most radio luminous source W9, and only a small offset of the peak position was found. Figure $\ref{fig:fig2}$ shows the aligned dataset (without primary-beam correction) overlaid as contours on a FORS camera R-band image (655$\,$nm)$\,$, where contours are scaled by the overall rms uncertainty, $\sigma$, taken to be 0.015$\,$mJy beam$^{-1}$.
The SEAC software utilises a floodfill algorithm. This algorithm selects initial pixels as possible sources which have values above a specified ‘seed’ threshold. An ‘island’ is then generated by appending adjacent pixels to an array containing just the ’seed’ pixel with a peak value initially, until the flux values of these surrounding pixels no longer reach above a specified ‘flood’ threshold. These were set at 5$\sigma$ for the seed threshold and 3$\sigma$ for the flood threshold. The value of $\sigma$ is the rms value of the local spatial region. Local rms values were determined from a generated noise map, created from dividing the full image into a grid with a user-specified number of cells. The rms level is then calculated within each individual cell, and the corresponding cell gives the value of the local noise level, $\sigma$, for each source. This allows for changes in the background radio emission to be considered, especially with regards to possible effects from extended emission of the particularly radio luminous objects W9 and W26, which otherwise could have affected the rms of the total image.
Selection of source fluxes
--------------------------
With the use of SEAC, fluxes were determined from each dataset, for each source. The definitive source fluxes were determined from the detections made in the <span style="font-variant:small-caps;">FullConcat</span> dataset. Many of the sources detected are surrounded by significant levels of radio emission, as discussed further in section 4. Due to this, a measurement of the core components of the sources was carried out using an additional segmentation tool within SEAC. This involved the implementation of the Watershed algorithm$\,$[@Watershed]. SEAC allows the use of two different options in applying this algorithm, either involving a Gaussian filter approach, or using a noise elevation map. Both versions of the segmentation tool were applied to the datasets when trying to pick out accurate core and extended components of the sources, leading to 3 source output lists for each dataset.
[0.4]{}
[0.4]{}
[0.4]{}
[0.4]{}
The two possible segmentation algorithms led to different representations of source emission. Figure \[fig:fig4\] demonstrates how applying the two segmentation approaches have clearly different results, shown here for the source WR B, with a non-segmented image taken from the <span style="font-variant:small-caps;">Taper9</span> dataset, shown in Figure \[fig:fig4b\]. The <span style="font-variant:small-caps;">Noise</span> segmentation uses two user-defined thresholds to select the strength of the flux that defines a core component. The ‘bottom’ threshold defines the background noise level to dismiss, and the ‘top’ threshold defines the level of emission above which peaks can be detected. This results in the selection of core components of emission embedded within a larger general area of extended emission, as shown by the brighter shading of the two core components detected in Figure \[fig:fig4c\].
The <span style="font-variant:small-caps;">Gauss</span> segmentation picks out local maxima of peaks of radio emission, and then applies a Gaussian filter. These Gaussians are then smoothed by a user-defined level. This leads to the segmentation of large areas of diffuse emission into smaller groups, helping to segregate out sources which may be effected by crowding, as is seen clearly in Figure \[fig:fig4d\]. This method leads to the inclusion of a larger proportion of extended emission than when determining the core flux for a source from the <span style="font-variant:small-caps;">Noise</span> segmentation. A higher level of smoothing will lead to a smaller number of final segments considered, and will attribute larger levels of surrounding extended emission to the source.
Segmentation was carried out when considering the definitive fluxes of sources taken from <span style="font-variant:small-caps;">FullConcat</span>, as well as when measuring the fluxes to compare the radio observations to the millimetre observations, taken from <span style="font-variant:small-caps;">Taper5</span> and <span style="font-variant:small-caps;">Taper9</span>. It allowed for the core components of sources to be detected, and allowed for further detection of sources embedded in larger regions of continuous diffuse emission surrounding several of the known stellar sources. Segmentation was not carried out for the <span style="font-variant:small-caps;">TaperALMA</span>, as source structure in ALMA was considered sufficiently compact for all sources, as previously seen in the analysis of the full dataset$\,$.
Confirmation of ATCA detected Sources
-------------------------------------
The definitive flux density values were determined from the <span style="font-variant:small-caps;">FullConcat</span> dataset. This was the dataset with the lowest resultant noise level, and the largest integrated flux surrounding each of the sources, so gave the best chance of determining the most complete number of source detections. The decision of final flux density values took into consideration the size of the emission area associated with the source, picked out by SEAC - this required a thorough visual inspection of the SEAC results, for all segmentation options, on each dataset. After source identification was carried out and flux density values were confirmed, further analysis could be applied to the results to calculate physical quantities associated with the sources, including spatial sizes, spectral indices and mass-loss values.
Errors on the flux densities are given from the combination of the error found by the calculation of integrated flux density in SEAC and the error given for the flux amplitude calibrator 1934-638 [@ATCAcaldatabase online calibrator database]. Calibrator errors were listed as 0.1$\,$% at 5.5$\,$GHz, and 0.2$\,$% at 9$\,$GHz. These two values were then combined in quadrature to give the error from amplitude calibration for the <span style="font-variant:small-caps;">FullConcat</span> dataset. The combined amplitude calibrator error was then combined in quadrature with the SEAC flux error to give final errors for the flux densities. Errors for the fluxes given from the <span style="font-variant:small-caps;">TaperALMA</span> measurements used a combination in quadrature of the errors found from the flux determination in SEAC and a 5% error for the absolute amplitude calibration error, following the prescription in . However, due the presence of the diffuse emission, it is unclear whether the boundary measured is the true boundary between emission truly related to nearby stellar sources, and emission from the diffuse background. Errors quoted in this paper are therefore potentially underestimates of the true uncertainty on these values, and should be considered as conservative.
Spatial extent
--------------
The determination of spatial sizes follows a similar prescription to the method carried out in . Source sizes were measured from the <span style="font-variant:small-caps;">FullConcat</span> dataset. Measurements of the spatial extents were made using <span style="font-variant:small-caps;">jmfit</span> in <span style="font-variant:small-caps;">aips</span> to determine a Gaussian fit for the source. Peak flux values and pixel locations for sources as calculated from SEAC were used as central positions from which to measure the source structure and provide a Gaussian model fit to the data. A threshold of surrounding pixels above 3$\sigma$ of the local rms was applied. Convolved and deconvolved source sizes were then determined from this. For sources where there were large levels of surrounding extended emission, the size of the core component was given an initial estimate manually, instead of using the 3$\sigma$ threshold. There were still difficulties experienced in applying a Gaussian fit due to the extended levels of diffuse radio background throughout the cluster. For sources that were found to be clearly non-Gaussian, a largest angular size (LAS) was determined instead, via visual inspection of the goodness of fit.
Both convolved and deconvolved sizes are given in Table \[table:atca\_known\_Sizes\], when applicable. For non-Gaussian sources, only the convolved size is given. For sizes determined from a Gaussian fit, errors are taken directly for the convolved sizes, with errors from the associated minimum and maximum ranges of the deconvolved size used to calculate the deconvolved spatial dimension errors. Sources that are fully resolved can all be separated into point-like sources that fit well to a Gaussian, or sources with significant asymmetric extended emission.
Most of the sources were found to have resolved, extended emission. The large de-convolved sizes that were measured can also be considered in comparison to the prior ALMA observations in where many of the stellar detections were unexpectedly found to be resolved, especially many of the WRs in the cluster. The diffuse radio emission detected throughout the cluster may have impacted the measurements. The impact of the extended emission is discussed further in Section 4.
Comparison to Do10 measurements
-------------------------------
The preliminary results revealed the presence of diffuse emission throughout the cluster. The possible astrophysical origin of this and its impact on our conclusions about sources present in the cluster is discussed further in section 4 and 7. Motivated by this diffuse emission, and as a sanity check on the possible effect of this onto the source fluxes, a consideration of the source size and flux was carried out on the <span style="font-variant:small-caps;">Full9</span> dataset in comparison to the previous 8.6$\,$GHz radio observations$\,$. This consideration involved the use of altering the user-parameters applied in the segmentation tool in SEAC to adjust the final size of the emission measured for core components of each source. The number of pixels with which flux was measured was adjusted indirectly with the alteration of the user-parameters, so that the relative sizes were found to be as similar in the output as possible, using the images as a reference. This involved the consideration of the relative pixel sizes of each image, and was carried out on a source by source basis.
In general, this comparison showed that we could make conservative measurements of sources by selecting core components of the source from SEAC, with increased fluxes still typically found for sources in the new radio observations. A small number of sources that were exceptions to this were found to be so due to their location within the cluster, where they were in close proximity to much brighter radio sources and surrounded by large extended regions of emission. Other than the effect of larger components being picked up for the total emission regions, including extended components, the increase in flux seen for the <span style="font-variant:small-caps;">Full9</span> dataset is likely due to increased u-v coverage, resulting from the use of a wider bandwidth in these more recent ATCA observations. The increased bandwidth will allow for a larger number of u-v visibilities to be sampled, allowing for the <span style="font-variant:small-caps;">Full9</span> dataset to be sensitive to a larger number of spatial scales within the same emission region. We can therefore conclude that the Do10 and Full9 datasets provide source fluxes that are broadly consistent with each other.
The comparative fluxes determined from this analysis are listed in Table \[table:atca\_do10comp\], with a comparative plot also presented in Figure \[fig:figB2\]. A note was also made of whether sources were particularly isolated with respect to other sources and the diffuse emission, or were located in more crowded regions within the cluster. Where possible, both fluxes from core components and total source fluxes, including an extended component, were measured.
Spectral indices
----------------
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The spectral indices of all prior known sources were calculated by comparing source fluxes at three frequencies; 5.5$\,$GHz, 9$\,$GHz and 100$\,$GHz. In order to compare the observations for calculating spectral indices, tapering was applied to datasets of ATCA and ALMA observations, so that only equivalent u-v ranges were considered. However, this led to a reduction in the absolute fluxes found for the sources due to the removal of several short spacings in u-v space from the data considered for the image. The reduction in absolute flux density values found from these images was not considered critical to the spectral index derived, as the final value calculated was reliant on the relative flux values at each frequency, rather than the absolute values.
Using tapered visibilities allowed for similar source sizes to be considered from the ALMA and ATCA observations, as seen in Figure \[fig:fig5\]. The W237 images from <span style="font-variant:small-caps;">Taper5</span> and <span style="font-variant:small-caps;">Taper9</span>, as shown in Figures \[fig:fig5b\] and \[fig:fig5c\] show only the compact bright emission detected for W237, with no inclusion of dimmer more extended emission from the longer spatial scales, that can be seen around W237 in the <span style="font-variant:small-caps;">FullConcat</span> dataset, shown in Figure \[fig:fig5d\]. However, an initial run-through of SEAC on these datasets still led to inconsistencies in spectral variance when considering the relationship found between the two radio spectral windows, and the overall relationship calculated in the radio-mm spectral index.
Upon further investigation on a source-by-source basis, the initial inconsistencies in spectral variance were found to be due to differing source sizes determined for the core component of the emission. In order to give the closest comparison possible of the flux when considering the spectral index, a variety of segmentation thresholds were applied to the data, in order to measure core components of the sources. This involved editing the user-defined noise levels in the <span style="font-variant:small-caps;">Noise</span> segmentation tool, and adjusting the smoothing parameter in the <span style="font-variant:small-caps;">Gauss</span> segmentation, until the source size determined for known stellar objects with previous detections were as similar in size as possible, across the different datasets (separated by spectral window). As the pixel sizes of the ALMA and ATCA datasets differed, this was taken into consideration when finding smaller source sizes representative of core emission. The revision of comparative fluxes and spectral indices was not found to significantly affect the overall spectral index determined for the sources from the comparison across all 3 spectral windows at 5.5$\,$GHz, 9$\,$GHz, and 100$\,$GHz. However, it did reduce the level of conflict found when comparing the spectral variance across the two radio wavelengths, 5.5$\,$GHz and 9$\,$GHz, to the index measured across all three frequency bands.
Extended emission
=================
One of the main results to become immediately apparent when viewing the images generated by these observations was the large level of diffuse radio emission present throughout the cluster, as seen clearly in Figures \[fig:fig1\] - \[fig:fig3\]. The presence of this pervasive diffuse emission calls into question the origin of the material (and therefore the resulting flux measured) surrounding the majority of sources, where the radio emission is found to be resolved and extended. It is unclear whether this material can be truly attributed to these stellar sources due to their stellar wind, or whether the material is actually part of the diffuse background that has become captured by, or is simply interacting with, far more compact nebulae surrounding the sources. If these extended nebulae are truly part of the diffuse background, it is unclear why the majority of this emission is still seen to be directly surrounding the stellar objects, with the most significantly extended structures found around cluster members that are expected to have dense winds and prior periods of extreme mass-loss, such as the cool supergiants, hypergiants and the WR stars. The possible origin of this diffuse emission is discussed further in Section 7.
Due to the high levels of diffuse radio emission observed, we had to acknowledge a caveat in our analysis of these observations that radio fluxes measured may include radio emission that has been misattributed to stellar sources. The definitive radio flux densities of the sources, given in Table \[table:atca\_known\], were calculated from the image generated from the <span style="font-variant:small-caps;">FullConcat</span> dataset, as shown in Figure \[fig:fig3\], where the inclusion of the full set of data allowed for the highest sensitivity.
This diffuse emission may have affected the fluxes measured, the spatial extents found for sources, and all parameters derived from these measurements, including the spectral indices and the mass-loss rates. We were able to mitigate for the impact of this with the use of conservative flux measurements, and by considering only the core components of emission from sources wherever possible. This is discussed in more detail in Section 3.
The spectral indices and mass-loss rates used radio and ALMA measurements in comparison. When calculating the spectral indices, the effect from the extended emission could lead to over-estimates for the radio measurements. As this background flux is believed to be optically-thin in nature, it would be expected to have a flat spectra, and so have a greater impact on the ATCA flux values than on the higher frequency ALMA values. It would also preferentially effect the ATCA radio observations, due to the lower relative fluxes of the stellar sources, if they are expected to emit thermal emission, and be impacted by the lower resolution of the radio observations. Accounting for all possible impacts led to the consideration of source fluxes from comparative source sizes across the ALMA and the ATCA tapered datasets when calculating spectral indices and mass-loss values, as was described in Section 3.5.
We can also consider the possible impact of external ionisation on the resultant spectral indices. Previous observations of externally ionised winds of hot stars, including the wind of the red supergiant (RSG) IRS 7, and the YHG HR 8752, have shown spectral indices that are consistent with values found for internally ionised stellar winds. This indicates that although the stellar winds of these cool evolved stars are ionised by surrounding hot stars (in the case of hot stars in the nearby nuclear cluster for the RSG, and the case of an early B companion for the YHG [@Stickland1978]), the external ionisation doesn’t have a significant effect on the spectral indices [@Higgs1978; @YusefZadeh1991].
As the extended emission can be seen to be asymmetric in structure over the cluster, with the strongest illumination orientated towards the cluster centre, it is also of importance to consider whether this would have an impact on the resulting stellar winds. The Pistol star, an LBV located in the Quintuplet cluster, has one hemisphere ionised by its host cluster, though is surrounded by a shell of cold dust, as has been seen in the infra-red [@Lau2014]. The spectral index of this star has been found to be consistent with free-free emission from optically thin ionised plasma, which is evidence that even with only part of the shell externally ionised, the end result can be seen to be thermal emission [@YusefZadeh1989].
The result of over-estimated fluxes would lead to flattened spectral indices. This means any impact on the spectral indices would be to make thermal sources appear less thermal. For the majority of sources, especially in the case of the WRs, these objects are already thermal, as shown in Section 6.1, and so any removal of this extended emission would only lead to spectral indices that could still be considered to be in line with the canonical value for a thermally-emitting stellar wind. It would also lead to flattened spectral indices for several of the cool hypergiants and supergiants, which are already significantly deviated from the canonical wind value, in line with expectations of composite emission, containing optically thick and thin components. The extended emission would also have led to effects causing over-estimates of the spatial extent of sources, and over-estimates of mass-loss rates. In order to counteract any possible impact of this, the mass-loss rates are calculated using only core components of flux from the stellar sources as a conservative estimate, and the resulting mass-loss rates are compared throughout the discussion to rates calculated for analogous sources of each spectral type.
In order to try to quantify the impact of the extended emission on our results, a comparison was carried out between the prior 8.6$\,$GHz observations and the 9$\,$GHz results from the new radio census, with the outcome shown in Table \[table:atca\_do10comp\] and Figure \[fig:figB2\]. This was discussed previously in Section 3.4, which concluded that the fluxes were broadly consistent between the observations, with slight increases generally found for the new radio results due to the larger u-v coverage resulting in higher sensitivity to more spatial scales. Any decrease in fluxes was found to be due to the location of a source embedded within a diffuse region, and so only a core component could be selected from within the background, resulting in conservative source fluxes measured.
One example of source consideration is the ATCA flux value attributed to one of the most radio luminous sources in the cluster, W9, found to be significantly higher at 9$\,$GHz in the more recent results, with a flux of 30.47$\,\pm\,$0.09$\,$mJy, in comparison to the core flux component measured by , 24.9$\,\pm\,$2.5$\,$mJy. For the measurement including the extended component, vastly different fluxes were found, with our flux measurement of 80.8$\,\pm\,$0.5$\,$mJy, versus the detection of 30.5$\,\pm\,$3.0$\,$mJy. This is due in part to a larger amount of surrounding radio emission detected around W9. However, it was found that due to the increased sensitivity to a range of spatial scales of emission, despite core components for the <span style="font-variant:small-caps;">Full9</span> dataset picked out by SEAC typically resulting in comparatively *smaller* sizes than the core components selected in , the core region measured in the <span style="font-variant:small-caps;">Full9</span> still contained a much larger level of flux.
Results of the radio census
===========================
Stellar sources
---------------
Category Source Number
---------------- ---------------
ATCA+optical 30
ATCA+ALMA 30
ATCA-only 53
Total 113
WRs 10
YHGs 5
RSGs 4
BSGs 2
LBV 1
sgB\[e\] 1
OB supergiants 7
Total 30
: Summary of the radio detections of different types of cluster members from the <span style="font-variant:small-caps;">FullConcat</span> dataset.[]{data-label="table:detection_summary"}
These radio observations led to 30 radio detections of known stellar sources in Westerlund 1, summarised in Table \[table:detection\_summary\]. Out of the 30 stellar sources with confirmed radio fluxes found in <span style="font-variant:small-caps;">FullConcat</span>, 5 of the 10 detections of WRs and 5 of the 7 detections of OB supergiants were new radio detections. The WRs newly detected in the radio were WR D and WR G, WN7o stars, WR I, a WN8o star, and WR E and M, both WC9 stars. OB supergiants with new detections were W10, W18 and W19, all early B supergiants, and W1031 and W1056, both late O-type giants. Flux densities for stellar sources with optical counterparts are given in Table$\,$\[table:atca\_known\], with flux densities taken from the integrated island flux found by SEAC. Sources with core and total flux components as detected by SEAC are listed with both flux values, specified by the superscript *c* and *t* respectively.
By investigating the <span style="font-variant:small-caps;">Full9</span> dataset, we could compare our radio detections directly to the previous 8.6$\,$GHz radio observations of Westerlund 1 cluster members$\,$. Of the 21 sources detected by at 8.6$\,$GHz, all but 1 source was also detected in the <span style="font-variant:small-caps;">Full9</span> dataset. The lack of detection for this source, WR V, can be linked to its location embedded within an area of diffuse emission in the cluster (discussed in more detail in section 6.1).
Spectral indices and spectral index limits were determined for all sources where at least one flux density from the tapered datasets <span style="font-variant:small-caps;">TaperALMA</span> , <span style="font-variant:small-caps;">Taper5</span> or <span style="font-variant:small-caps;">Taper9</span> could be found. These are presented in Table \[table:spec\_index\]. This table also includes spectral index limits found for the new detections of OB supergiants with no corresponding millimetre detections, with the radio flux density taken from <span style="font-variant:small-caps;">FullConcat</span> due to the faintness of the sources, and using the flux density limit from the full ALMA observations .
The spectral index limits derived from the new radio detections of OB supergiants led to the conclusion that these stars were binary candidates. The spectral index limits were found to constrain the spectral indices to negative values, implying the presence of non-thermal emission. These spectral index limits included the cluster members currently categorised as OB supergiants, W15, W18, and W1031, as well as OB supergiants with new radio detections, W10, W19, and W1056. This is discussed in more detail in section 6.3.1.
Non-Stellar sources
-------------------
### ALMA sources
Previous observations made in the millimetre , detected a large number of sources that were not associated with any catalogued stellar sources. 30 of these objects have now been found to have associated detections in the radio. Flux densities were determined, with the positions and fluxes listed in Table \[table:alma\_uncat\], with sources presented in a corresponding plot of the cluster, in Figure \[fig:figB3\]. The number assigned to these sources is the FCP18 source number, as given in .
One of the sources can clearly be attributed to a knot of emission associated with the extended nebulae around the YHG W4a. 4 of the sources are found to be aligned with a significant area of extended emission to the west of Wd1 (to the RHS side of the figure). A large number of the rest of the sources can be found clustered within other regions of diffuse radio background or nearby radio luminous known stellar sources, such as W9, WR B and D09-R1. At least 7 of the sources detected were found to be in isolated positions and not linked to any known stellar sources in the cluster.
15 of these sources were strong enough in the radio for spectral indices to be calculated from the tapered datasets, <span style="font-variant:small-caps;">Taper5</span>, <span style="font-variant:small-caps;">Taper9</span> and <span style="font-variant:small-caps;">TaperALMA</span>, as shown in Table \[table:spec\_index\_almaonly\]. Overwhelmingly, all of these sources were found to have clearly non-thermal spectral indices. Some of the sources that are seen to have a negative spectral index are clearly embedded within a larger diffuse radio background. This indicates that shocks within the cluster wind may be responsible for these knots of emission [@Bell1978]. Some of the uncategorised sources are in the near vicinity of stellar sources and the impact of extended emission from these nearby stars may influence the spectral behaviour observed [@YusefZadeh2003].
However, some sources cannot be linked to either of these external influences, and so must experience a non-thermal spectral index for another reason, most notably in the case of sources FCP18-16, -22, -27, and -95. Possible explanations for this spectral behaviour are the presence of synchrotron emission in the radio resulting from an as of yet undetected binary stellar system, where there is an interacting shock between two stellar winds. Another possible explanation is that the knots of radio and millimetre emission could be of extra-galactic origin rather than from Wd1 itself, though this likelihood was discussed and largely dismissed previously in .
### ATCA-only sources
Alongside the detection of previously observed sources in other wave-bands, 53 additional sources of radio emission were detected through the use of SEAC, listed in Table \[table:atca\_uncat\], with the source number assigned from the full SEAC output list, ordered by RA. A conservative limit in determining the number of these radio-only sources meant that only knots of radio emission detected without the use of the segmentation tools within SEAC were considered. Some of these sources may be related to extended radio emission from nearby radio luminous sources, including HA19-23 in the nearby vicinity of W12a, and HA19-43 in the close vicinity of the large extended tail of W20. A large number can also be seen (as displayed in Figure \[fig:figB4\]) to be in isolated regions away from any of the previously determined sources in Wd1. Additionally, many of these isolated radio sources display a geometry that indicate the emission cannot be considered an extension to other known sources, due to the compact, point-source like morphology of the emission, including examples such as HA19-65 to the north of the cluster, HA19-3, and HA19-4 to the west of the cluster, and HA19-85, -88 and -89 to the east.
Stellar sources
===============
[p[1.2cm]{}p[2.5cm]{}p[2.5cm]{}\*2[p[2.8cm]{}]{}p[2.2cm]{}]{} Source & Spectral Type & RA & DEC & Flux$_{\textsc{FullConcat}}$ (mJy)\
WR A & WN7b+OB? & 16 47 8.36412 & -45 50 45.5948 & 0.89 $\pm$0.06\
&&&& 1.38 $\pm$ 0.09\
WR D & WN7o & 16 47 6.29738 & -45 51 26.6985 & 0.71 $\pm$ 0.05\
WR B & WN7o+OB? & 16 47 5.34956 & -45 51 5.3995 & 3.38 $\pm$ 0.03$^{c}$\
&&&& 16.53 $\pm$ 0.15$^{t}$\
WR G & WN7o & 16 47 4.05743 & -45 51 23.7000 & 0.34 $\pm$ 0.04\
WR P & WN7o & 16 47 1.61624 & -45 51 45.5984 & 0.05$\pm$0.02$^{m}$\
WR I & WN8o & 16 47 0.89863 & -45 51 20.6974 & 0.42 $\pm$ 0.02\
WR L & WN9h+OB? & 16 47 4.20100 & -45 51 7.5000 & 0.55 $\pm$ 0.04\
WR S & WN10-11h/BHG & 16 47 2.96652 & -45 50 19.7997 & 0.16 $\pm$ 0.04\
WR E & WC9 & 16 47 6.06807 & -45 52 8.6988 & 0.13 $\pm$ 0.02\
WR F & WC9d+OB? & 16 47 5.20647 & -45 52 25.1996 & 0.29 $\pm$ 0.02\
WR M & WC9d & 16 47 3.94256 & -45 51 38.1000 & 0.24 $\pm$ 0.04\
W16a & A5Ia$^{+}$& 16 47 6.69880 & -45 50 41.0980 & 2.70 $\pm$ 0.11\
W12a &F1Ia$^{+}$& 16 47 2.19106 & -45 50 59.3991 & 1.76 $\pm$ 0.03$^{c}$\
&&&& 3.77 $\pm$ 0.07$^{t}$\
W4a &F3Ia$^{+}$ & 16 47 1.44479 & -45 50 37.4982 & 3.18 $\pm$ 0.04$^{c}$\
&&&& 4.72 $\pm$ 0.07$^{t}$\
W32 &F5Ia$^{+}$ & 16 47 3.71289 & -45 50 43.8000 & 0.18 $\pm$ 0.04\
W265 &F5Ia$^{+}$ & 16 47 6.29597 & -45 49 24.2985 & 1.02 $\pm$ 0.04$^{c}$\
&&&& 3.34 $\pm$ 0.11$^{t}$\
W237 & M3Ia & 16 47 3.10953 & -45 52 19.1998 & 1.31 $\pm$ 0.02$^{c}$\
&&&& 9.57 $\pm$ 0.19$^{t}$\
W75 &M4Ia & 16 47 8.93719 & -45 49 58.7933 & 0.34 $\pm$ 0.03\
W20 &M5Ia& 16 47 4.68920 & -45 51 24.2999 & 2.54 $\pm$ 0.03$^{c}$\
&&&& 20.65 $\pm$ 0.23$^{t}$\
W26 &M5-6Ia& 16 47 5.40679 & -45 50 36.5994 & 153.11 $\pm$ 0.17\
W17 & O9Iab & 16 47 6.18210 & -45 50 49.4987 & 0.98 $\pm$ 0.04$^{c}$\
&&&& 2.04$\pm$0.08$^{t}$\
W243 & A2Ia (LBV) & 16 47 7.50460 & -45 52 29.3966 & 1.38 $\pm$ 0.04\
W9 &sgB\[e\]& 16 47 4.14355 & -45 50 31.5000 & 27.38 $\pm$ 0.06$^{c}$\
&&&& 83.14 $\pm$ 0.41$^{t}$\
D09-R1 & BSG & 16 47 9.08253 & -45 51 10.1929 & 1.29 $\pm$ 0.03$^{c}$\
&&&&5.44 $\pm$ 0.10 $^{t}$\
D09-R2 &BSG& 16 47 6.89972 & -45 50 37.1977 & 0.96 $\pm$ 0.06\
W15 & O9Ib & 16 47 6.72735 & -45 50 28.7980 & 1.65 $\pm$ 0.09\
W10 & B0.5I + OB & 16 47 3.42580 & -45 50 35.3999 & 0.16 $\pm$ 0.05\
W18 & B0.5Ia & 16 47 5.60787 & -45 50 50.3993 & 0.36 $\pm$ 0.06\
W19 &B1Ia& 16 47 4.86139 & -45 50 57.8998 & 0.08 $\pm$ 0.03\
W1031 & O9III & 16 47 1.96137 & -45 50 57.8988 & 0.90 $\pm$ 0.05\
W1056 & O9.5II & 16 47 8.70904 & -45 51 2.0939 & 0.09 $\pm$ 0.02\
[p[1.2cm]{}\*2[p[3.2cm]{}]{}p[3cm]{}p[2.8cm]{}]{} Source & Flux$_{\textsc{Taper5}}$ (mJy) & Flux$_{\textsc{Taper9}}$ (mJy) & Flux$_{\textsc{TaperALMA}}$ (mJy) & Spectral Index ($\alpha$)\
WR O & $<$ 0.10 & $<$ 0.10& 0.28 $\pm$ 0.04 & $>$ 0.37\
WR U & $<$ 0.19 & $<$ 0.15 & 0.17 $\pm$ 0.05 & $>$ -0.01\
WR Q &$<$ 0.12 & $<$ 0.10 & 0.18 $\pm$ 0.05 & $>$ 0.17\
WR A & 0.62 $\pm$ 0.08 & 0.94 $\pm$ 0.05 & 3.89 $\pm$ 0.22 & 0.61 $\pm$ 0.04\
WR D & $<$0.17 & 0.17 $\pm$ 0.05 & 0.58 $\pm$ 0.07 & 0.50 $\pm$ 0.16\
WR B & 3.51 $\pm$ 0.12 & 2.96 $\pm$ 0.09 & 2.62 $\pm$ 0.23 & -0.10 $\pm$ 0.03\
WR G & $<$0.16 & 0.13 $\pm$ 0.04 & 0.72 $\pm$ 0.08 & 0.71 $\pm$ 0.14\
WR P &$<$ 0.12 & $<$ 0.12 & 0.50 $\pm$ 0.06 & $>$ 0.52\
WR I & 0.16 $\pm$ 0.04 & 0.47 $\pm$ 0.04 & 2.02 $\pm$ 0.11 & 0.79 $\pm$ 0.08\
WR V & $<$ 0.24 & $<$ 0.28 & 0.90 $\pm$ 0.09 & $>$ 0.47\
WR L & 0.24 $\pm$ 0.06 & 0.48 $\pm$ 0.04 & 3.37 $\pm$ 0.18 & 0.88 $\pm$ 0.07\
WR S & $<$ 0.13 & $<$ 0.09 & 0.36 $\pm$ 0.07 & $>$ 0.42\
W13 & $<$ 0.20 & $<$ 0.21 & 0.18 $\pm$ 0.05 & $>$ -0.05\
WR K &$<$ 0.16 & $<$ 0.14 & 0.16 $\pm$ 0.05 & $>$ 0.02\
WR E & $<$ 0.16 &$<$ 0.10 & 0.81 $\pm$ 0.06 & $>$ 0.66\
WR F & $<$0.12 & 0.37 $\pm$ 0.04 & 1.26 $\pm$ 0.09 & 0.51 $\pm$ 0.07\
WR C &$<$ 0.17 & $<$ 0.11 & 0.22 $\pm$ 0.04 & $>$ 0.15\
WR H &$<$ 0.12 & $<$ 0.09 & 0.28 $\pm$ 0.06 & $>$ 0.35\
WR M & $<$0.16 & 0.11 $\pm$ 0.03 & 0.39 $\pm$ 0.05 & 0.53 $\pm$ 0.12\
W16a & 1.59 $\pm$0.11 & 1.67 $\pm$ 0.11 & 1.23 $\pm$ 0.15 & -0.10 $\pm$ 0.05\
W12a & 1.77 $\pm$ 0.05& 1.70 $\pm$ 0.08 & 0.78 $\pm$ 0.13 & -0.30 $\pm$ 0.07\
W4 & 1.85 $\pm$ 0.04 & 1.84 $\pm$ 0.04 & 1.78 $\pm$ 0.16 & -0.01 $\pm$ 0.04\
W265 & 1.00 $\pm$ 0.04 & 0.94 $\pm$ 0.08 & - & -0.13 $\pm$ 0.23\
W237 & 1.07 $\pm$ 0.06 & 1.03 $\pm$ 0.05 & 1.05 $\pm$ 0.08 & 0.00$\pm$ 0.04\
W75 & $<$0.13 & 0.23 $\pm$ 0.05 & 0.22 $\pm$ 0.05 & -0.01 $\pm$ 0.12\
W20 & 2.23 $\pm$ 0.07 & 2.10 $\pm$ 0.06 & 1.99 $\pm$ 0.16 & -0.03 $\pm$ 0.03\
W26 & 106.3 $\pm$ 0.2 & 117.7 $\pm$ 0.4 & 103.3 $\pm$ 5.2 & -0.02 $\pm$ 0.02\
W17 & 0.81 $\pm$ 0.09 & 0.86 $\pm$ 0.09 & 0.76 $\pm$ 0.12 & -0.03 $\pm$ 0.07\
W46a &$<$ 0.11 &$<$ 0.09 & 0.18 $\pm$ 0.04 & $>$ 0.21\
W243 & 0.82 $\pm$ 0.05 & 1.54 $\pm$ 0.07 & 9.69 $\pm$ 0.49 & 0.82 $\pm$ 0.03\
W7 & $<$ 0.09 & $<$ 0.11 & 0.20 $\pm$ 0.04& $>$ 0.27\
W9 & 22.10 $\pm$ 0.07 & 30.34 $\pm$ 0.11 & 158.6 $\pm$ 7.9 & 0.68 $\pm$ 0.02\
D09-R1 & 0.82 $\pm$ 0.07 & 0.96 $\pm$ 0.07 & 0.85 $\pm$ 0.14 & -0.01 $\pm$ 0.07\
D09-R2 & 0.54 $\pm$ 0.06 & 0.59 $\pm$ 0.07 & 0.51 $\pm$ 0.07 & -0.03 $\pm$ 0.07\
Source && Flux$_{\textsc{FullConcat}}$ (mJy) & Flux$_{\textsc{ALMA}}$ (mJy) & Spectral Index ($\alpha$)\
W15 && 1.65$\pm$0.09 & $<$ 0.10 & $<$ -1.15\
W10 && 0.16$\pm$0.05 & $<$ 0.15 & $<$ -0.04\
W18 && 0.36$\pm$0.06 &$<$ 0.127 & $<$ -0.43\
W19 && 0.08$\pm$0.03 &$<$ 0.104& $<$ 0.10\
W1031 && 0.90$\pm$ 0.05& $<$ 0.09 & $<$ -0.95\
W1056 && 0.09$\pm$0.02 & $<$ 0.100 & $<$ 0.02\
[p[1.2cm]{}p[1.6cm]{}p[1.8cm]{}p[0.7cm]{}\*4[p[2.2cm]{}]{}]{} Source & RA (16 47)& DEC (-45) & Offset & &\
&&&&Major axis (“) &Minor axis (”) &Major axis (“) &Minor axis (”)\
WR A & 8.36412 & 50 45.5948 & 0.47 & 2.49 $\pm$ 0.07 & 1.34 $\pm$ 0.04 & 1.82 $\pm$ 0.11 & 0.89 $\pm$ 0.08\
WR D & 6.29738 & 51 26.6985 & 0.65 & 4.96 & LAS & - & -\
WR B & 5.34956 & 51 5.3995 & 0.40 &3.23 $\pm$ 0.03 & 2.32 $\pm$ 0.02 & 2.77 $\pm$ 0.04 & 2.06 $\pm$ 0.03\
&&&& 6.98 & LAS & - &-\
WR G & 4.05743 & 51 23.7000 & 1.57 & 3.05 $\pm$ 0.19 & 1.19 $\pm$ 0.08 & 2.53 $\pm$ 0.24 & 0.63 $\pm$ 0.18\
WR I & 0.89863 & 51 20.6974 & 0.22 & 1.86 $\pm$ 0.07 & 1.05 $\pm$ 0.04 & 0.80 $\pm$ 0.21 & -\
WR L & 4.20100 & 51 7.5000 & 0.14 & 1.76 $\pm$ 0.05 & 1.13 $\pm$ 0.03 & 0.55 $\pm$ 0.34 & 0.39 $\pm$ 0.17\
WR S & 2.96652 & 50 19.7997 & 0.25 & 2.25 $\pm$ 0.26 & 1.96 $\pm$ 0.22 & 1.74 $\pm$ 0.51 & 1.42 $\pm$ 0.65\
WR E & 6.06807 & 52 8.6988 & 0.54 & 1.78 $\pm$ 0.19 & 1.10 $\pm$ 0.12 & 0.51 $\pm$ 0.53 & 0.47 $\pm$ 0.41\
WR F & 5.20647 & 52 25.1996 & 0.25 & 1.89 $\pm$ 0.10 & 1.06 $\pm$ 0.06 & 0.81 $\pm$ 0.30 & 0.37 $\pm$ 0.28\
WR M & 3.94256 & 51 38.1000 & 0.36 & 2.18 $\pm$ 0.17 & 1.61 $\pm$ 0.12 & 1.49 $\pm$ 0.33 & 1.10 $\pm$ 0.4\
W16a & 6.69880 & 50 41.0980 & 1.37 & 3.66 $\pm$ 0.09 & 2.41 $\pm$ 0.06 & 3.25 $\pm$ 0.10 & 2.18 $\pm$ 0.07\
W12a & 2.19106 & 50 59.3991 & 0.61 & 2.39 $\pm$ 0.03 & 2.05 $\pm$ 0.03 & 1.80 $\pm$ 0.05 & 1.67 $\pm$ 0.06\
&&& & 4.81 & LAS & - & -\
W4a & 1.44479 & 50 37.4982 & 0.47 & 2.35 $\pm$ 0.02 & 1.55 $\pm$ 0.01 & 1.62 $\pm$ 0.03 & 1.19 $\pm$ 0.02\
&&&& 6.91 &LAS & - & -\
W32 & 3.71289 & 50 43.8000 & 0.50 & 1.78 $\pm$ 0.09 & 1.04 $\pm$ 0.05 & 0.50 $\pm$ 0.4 & 0.31 $\pm$ 0.28\
W265 & 6.29597 & 49 24.2985 & 0.72 & 2.46 $\pm$ 0.05 & 1.92 $\pm$ 0.04 & 1.94 $\pm$ 0.09 & 1.44 $\pm$ 0.10\
&&&& 12.59 &LAS & - & -\
W237 & 3.10953 & 52 19.1998 & 0.43 & 2.60 $\pm$ 0.04 & 1.94 $\pm$ 0.03 & 1.97 $\pm$ 0.06 & 1.64 $\pm$ 0.04\
&&&&15.36 &LAS & - & -\
W75 & 8.93719 & 49 58.7933 & 0.22 & 2.74 $\pm$ 0.15 & 1.46 $\pm$ 0.08 & 2.16 $\pm$ 0.19 & 1.05 $\pm$ 0.13\
W20 & 4.68920 & 51 24.2999 & 0.50 & 2.89 $\pm$ 0.03 & 2.21 $\pm$ 0.02 & 2.35 $\pm$ 0.04 & 1.96 $\pm$ 0.03\
&&& & 21.05 & LAS &- & -\
W26 & 5.40679 & 50 36.5994 & 0.18 & 17.04 & LAS & - & -\
W17 & 6.18210 & 50 49.4987 & 0.76 & 2.52 $\pm$ 0.05 & 2.04 $\pm$ 0.04 & 2.18 $\pm$ 0.13 & 1.38 $\pm$ 0.11\
W243 & 7.50460 & 52 29.3966 & 0.25 & 1.75 $\pm$ 0.02 & 1.00 $\pm$ 0.01 & 0.40 $\pm$ 0.11 & -\
W9 & 4.14355 & 50 31.5000 & 0.58 & 2.212 $\pm$ 0.001 & 1.235 $\pm$ 0.001 & 1.405 $\pm$ 0.002 & 0.735 $\pm$ 0.001\
&&& & 9.39 & LAS & - & -\
D09-R1 & 9.08253 & 51 10.1929 & 0.07 & 2.37 $\pm$ 0.03 & 1.88 $\pm$ 0.02 & 1.92 $\pm$ 0.05 & 1.24 $\pm$ 0.07\
&&&& 12.85 &LAS & - & -\
D09-R2 & 6.89972 & 50 37.1977 & 0.07 & 2.13 $\pm$ 0.05 & 1.49 $\pm$ 0.04 & 1.45 $\pm$ 0.10 & 0.86 $\pm$ 0.13\
W15 & 6.72735 & 50 28.7980 & 1.40 & 3.95 $\pm$ 0.10 & 1.61 $\pm$ 0.04 & 3.55 $\pm$ 0.12 & 1.27 $\pm$ 0.06\
W10 & 3.42580 & 50 35.3999 & 1.15 & 2.55 $\pm$ 0.22 & 1.41$\pm$ 0.12 & 1.90 $\pm$ 0.23 & 0.98 $\pm$ 0.22\
W18 & 5.60787 & 50 50.3993 & 1.08 &3.86 $\pm$ 0.67 & 1.79 $\pm$ 0.31 & 3.47 $\pm$ 0.8 & 1.49 $\pm$ 0.4\
W19 & 4.86139 & 50 57.8998 & 1.19 & 3.75 $\pm$ 0.16 & 2.39 $\pm$ 0.10 & 3.35 $\pm$ 0.19 & 2.15 $\pm$ 0.13\
W1031 & 1.96137 & 50 57.8988 & 1.55 & 2.90 $\pm$ 0.13 & 2.53 $\pm$ 0.11 & 2.73 $\pm$ 0.15 & 1.87 $\pm$ 0.17\
W1056 & 8.70904 & 51 2.0939 & 1.01 &2.90 $\pm$ 0.28 & 1.28 $\pm$ 0.12 &2.36 $\pm$ 0.36 & 0.74 $\pm$ 0.29\
Wolf-Rayet stars
----------------
[0.36]{}
[0.28]{}
[0.32]{}
\[fig:WR\_postage\]
Westerlund 1 contains 24 known WR stars, comprising almost 4% of the galactic population$\,$[online catalogue, @Crowther2019]. 10 WR stars were detected, with 5 WRs not previously detected in the radio. WR stars were detected across a variety of subtypes, with many late-type WN stars detected (WN7, WN8, WN9), as well as half of the WC WRs in the cluster. 4 of the WN stars detected have prior evidence suggesting they may be binary systems, and all of the detected WC stars detected have some evidence of binarity from other observations. Of the 24 WR stars present in Westerlund 1, 3 were outside of the field of view, meaning that almost half of the available WR stars were detected with associated radio emission (10 detections and 1 marginal detection of 21 available WR stars).
WR P was detected in the ATCA observations, but this result is not presented in Table \[table:atca\_known\] as it was a marginal detection, with a flux of 0.05$\,\pm\,$0.02$\,$mJy. Due to a low SNR value of $\sim$ 2.8, it cannot be listed as a confirmed source, though it’s worth noting the caveat that with a slightly different defined background noise level (for example, by considering a sub-image created of this source from the full field of view, which leads to a different absolute cell-size measured in generating the background noise level), the flux reaches above the SNR limit of $\sim$ 3.
Of the WR stars not detected but within the central field of view, many of these were located in a nearby proximity to a stellar source with high levels of surrounding extended radio emission. Obscuration from this surrounding emission is highly likely so that any radio flux emitted by the fainter stars was not separable from the extended emission. This is a likely cause of the non-detection of WR V, located close to W9, one of the most luminous radio sources present in the cluster. At the location of WR V, the peak flux was found to be 0.3$\,$mJy, with a mean value over the immediate surrounding area of the source of $\sim$0.1$\,$mJy, on the same order as the level of 3$\sigma$ in that region, with $\sigma$ the rms value of the local spatial region. This is especially of note considering this source was detected in previous radio observations$\,$.
Other sources affected by this include WR U, located near to W16a, WR R, located near to W26, and WR J, located near W12a. The other WRs not detected by ATCA, WR W, O, Q, K, C and H, may emit radio fluxes, but any emission was too faint to be detected above the noise level of the observations. This is especially likely if we assume a thermal spectral index for these sources.
Fluxes of the WRs from the <span style="font-variant:small-caps;">Full9</span> dataset were compared to , as shown in Table \[table:atca\_do10comp\]. Of sources detected in both datasets, consistent fluxes were found for all sources that were not surrounded by extended emission, or in the proximity of nearby radio luminous sources. For sources surrounded by extended emission, the conservative choice of flux determination on these sources (as discussed in section 3), lead to lower flux values, as seen for the source WR S. In other cases, the fluxes were found to be typically slightly higher, due to the increased sensitivity of the more recent observations.
Spectral indices and limits were determined from the tapered datasets, <span style="font-variant:small-caps;">Taper5</span>, <span style="font-variant:small-caps;">Taper9</span> and <span style="font-variant:small-caps;">TaperALMA</span>, with spectral index values found for 8 WRs and spectral index limits found for 11 WRs. The majority of the WR stars were found to have a spectral index in line with the canonical value for a thermal stellar wind. This indicates that even though a majority of the WRs detected have evidence of binarity from other observations [@Clark2019b], thermal emission is still the dominant mechanism from the outer wind regions. Any colliding winds present must be located at close separations, likely inside of the radio photospheres of the stars. Spectral index limits determined all indicated flattened or positive spectral indices, consistent with expectations of single stars with thermal wind emission.
Out of the WRs, only WR B was found to have a negative spectral index, found to be -0.10$\pm$0.03 as shown in \[table:spec\_index\]. This indicates the presence of non-thermal emission. WR B also shows the most complex morphology of the WRs, and is surrounded by a large region of extended emission, as seen in Figure \[fig:WRB\_postage\]. The core component of the source was used to determine the spectral index. The negative spectral index determined, $\alpha$ = -0.10 is in line with that found in previous millimetre and radio comparisons$\,$.
WR A, as shown in Figure \[fig:WRA\_postage\], was found to have a positive spectral index of 0.61$\,\pm\,$0.04. This revised spectral index is flatter than the previous spectral index found in , more in line with the canonical value for partially optically thick free-free emission. WR D, as shown in Figure \[fig:WRD\_postage\], was found to have a slightly flattened positive spectral index of $\sim$ 0.50, also indicating partially optically thick and thin emission. A revised spectral index limit on WR V , $\alpha$ > 0.47, is consistent with leading to the canonical value for a stellar wind, containing partially optically thick and thin emission. This is in rough agreement with the radio-millimetre spectral index found in .
WR G was found to have a highly thermal spectral index $\alpha \sim$ 0.71, and the WN8 and WN9+OB? subtypes WR I and WR L were found to have spectral indices $\sim$ 0.79 and $\sim$ 0.88 respectively. These values are suggestive of a partially optically thick stellar wind with the increased optically thick component possibly due to colliding wind regions. This is because for close in binaries with short period, the material from the wind collision zone (WCZ) can become cooled and cause optically thick thermal emission to dominate the spectrum, causing a steepening of the index over the mm-radio continuum [@Pittard2010; @Stevens1992]. The suggestion is supported by previous evidence of binarity for sources WR G and L, although WR I has no current evidence indicative of binarity.
WR S (WN10-11h/BHG) is detected, but it is not bright enough to be seen in the <span style="font-variant:small-caps;">Taper9</span> or <span style="font-variant:small-caps;">Taper5</span> datasets, so can only be given a spectral index limit. The spectral index limit, $\alpha$ > 0.42, is consistent with the canonical stellar wind value, suggestive of partially optically thick emission, and is higher than the previous radio-millimetre spectral index value determined in . The flux value for WR S, much fainter than in the observations, was likely affected by its proximity to the radio luminous object W9.
WR F (WC9d+OB) and WR M (WC9d) both have spectral indices slightly flattened with comparison to the canonical stellar wind value, of $\sim$ 0.51 and $\sim$ 0.53 respectively. This could be due to a higher presence of optically thin or non-thermal components, and is likely related to the binarity of these sources. Spectral index limits are also derived for a further 11 WR stars, all in line with either a slightly flattened spectral index or the canonical wind spectral index value. WR E, C, and H are all binary candidates, but no evidence is found of a significant non-thermal component for these sources. For 9 of the WRs with definitive flux detections in the radio, only excepting WR B, the spectral indices and spectral index limits are in line with the value expected for stellar wind emission, making it a reasonable assumption to use their fluxes to determine mass-loss rates.
### Mass-loss rates
[p[1.5cm]{}p[2.5cm]{}\*3[p[2.5cm]{}]{}]{} Source & Spectral Type & (M$_{\odot}$$\,$yr$^{-1}$)\
& & All & 5.5GHz & 9 GHz\
WR A & WN7b + OB? & 7.02$\pm$0.35e-05 & - & 4.01$\pm$2.50e-05\
WR D & WN7o & 5.92$\pm$0.31e-05 & 2.97$\pm$0.74e-05 & 3.09$\pm$0.41e-05\
WR G & WN7o & 3.41$\pm$0.30e-05 & - & -\
WR I & WN8o & 3.10$\pm$0.11e-05 & 1.62$\pm$0.32e-05 & 2.96$\pm$0.15e-05\
WR L & WN9h+OB? & 2.72$\pm$0.15e-05 & 1.55$\pm$0.32e-05 & 2.35$\pm$0.16e-05\
WR S & WN10-11h/BHG &0.63$\pm$0.12e-05 & - & 0.34$\pm$0.10e-05\
WR E & WC9 & 3.62$\pm$0.42e-05 & - & 2.90$\pm$0.40e-05\
WR F & WC9d+OB? & 6.62$\pm$0.34e-05 & 3.62$\pm$0.74e-05 & 6.76$\pm$0.44e-05\
WR M & WC9d & 5.74$\pm$0.71e-05 &- & 3.84$\pm$0.72e-05\
For all sources with spectral indices found to be consistent with optically thick or thin free-free emission ($\alpha$ > 0), mass-loss rates were determined from their associated fluxes. The Wright and Barlow equation was applied to determine radio mass-loss rates$\,$[@WrightBarlow1975]. This equation includes the assumptions that the stellar wind is spherically symmetric and only emits thermal free-free emission. The mass-loss rate, $\dot{M}$ (combined with the clumping factor, $f_{cl}$) can be determined from the source flux as follows,
$$\dot{M}f_{cl}^{\frac{1}{2}}= 0.095 \times \frac{\mu v_{\infty} S_{\nu}^{\frac{3}{4}}D^{\frac{3}{2}}}{Z \gamma^{\frac{1}{2}} g_{ff}^{\frac{1}{2}} \nu^{\frac{1}{2}} } \quad \textrm{M$_{\odot}$$\,$yr$^{-1}$}
\label{eq:wb}$$
where $\dot{M}$ is the mass-loss rate in M$_{\odot}\,$$\,$yr$^{-1}$, $\mu$ is the mean molecular weight per ion, $v_{\infty}$ is the terminal velocity of the wind (in km$\,$s$^{-1}$), $S_{\nu}$ is the observed flux (in mJy), measured at the frequency $\nu$ (in Hz), $D$ is the distance (in kpc), $Z$ is the ratio of electron to ion density, $\gamma$ is the mean number of electrons per ion, and $g_{ff}$ is the gaunt factor, defined by,
$$g_{ff} \sim 9.77 \left( 1 + 0.13 \log\left(T_{e}^{\frac{3}{2}}/\nu \sqrt{(\bar{Z^{2}})} \right) \right)
\label{eq:gaunt}$$
where $T_{e}$ is the electron temperature$\,$[@LeithererRobert1991]. $f_{cl}$ is the clumping factor, defined by $f_{cl} = < \rho^{2} > / < \rho >^{2}$, where $\rho$ is the density of the wind. This demonstrates the underlying relationship between the mass-loss rate derived from the wind and the intrinsic density. If $f_{cl}$ is taken to be 1, it represents a smooth-wind, with larger values indicating the presence of structure, suggesting a clumped wind.
For the different stellar types considered, different assumptions are made about the various parameters, $T_{e}, Z, \mu$ and $\gamma$. Many of these follow the assumptions made in . For OB stars, $\mu$ was taken to be 1.4, and $Z$ and $\gamma$ were both taken to be 1.0. This relates to a chemistry where H is fully ionised, the dominant form of He is singly ionised, and the He abundance is n$_{He}$ / n$_{H}$ = 0.1. $Z$ and $\gamma$ were also taken to be 1.0 for WRs, and $\mu$ was taken as 4 for WN6 and earlier spectral types, with 2.0 used for spectral types later than WN6 and 4.7 used for WC8 and WC9 stars. For RSGs, LBVs and YHGs, $\mu$ was assumed to be 1.4, $Z$ was taken to be 0.9, and $\gamma$ was taken as 0.8. For all early-type hot stars, $T_{e}$ was taken to be $0.5 \times T_{eff}$, where $T_{eff}$ is the effective temperature of the star. For cool stars, including the YHGs and RSGs, $T_{e}$ was taken to be 10,000 K. The distance, $D$, was assumed to be 5$\,$kpc, as justified in the Introduction.
$\dot{M}f_{cl}^{\frac{1}{2}}$ values for 9 of the 11 WRs detected, where spectral indices or index limits indicated optically thick or thin emission, in line with thermal stellar winds, are presented in Table \[table:mass\_loss\]. The absolute mass-loss rates are calculated for the <span style="font-variant:small-caps;">FullConcat</span>, <span style="font-variant:small-caps;">Full9</span>, and <span style="font-variant:small-caps;">Full5</span> datasets. The majority of these sources show consistency with the expected value for WRs, of $\sim$ 10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$.
A slightly lower value is determined for WR S, though this could be due to the possibility that WR S is believed to be a post-interaction product of a previously close binary, and so has already experienced extra stripping of its outer layers during that phase. Our value for WR S, $\dot{M}f_{cl}^{\frac{1}{2}}$ 6.3$\,\pm\,$1.2$\,\times\,$10$^{-6}$ $M_{\odot}$$\,$yr$^{-1}$, is larger than determined by previous modelling of this object, where $\dot{M}f_{cl}^{\frac{1}{2}}$ was found to be 2.16$\,\times\,$10$^{-6}$ $M_{\odot}$$\,$yr$^{-1}$$\,$[@Clark2014], but is consistent with the measurement from millimetre observations of 5.18$\,\times\,$10$^{-6}$ $M_{\odot}$$\,$yr$^{-1}$$\,$.
As WR B was found to have a negative spectral index, this indicates significant non-thermal emission associated with the source, and so the approximation for a thermally emitting stellar wind is not valid for this star. An estimate of $\dot{M}f_{cl}^{\frac{1}{2}}$ for the WR B flux in the <span style="font-variant:small-caps;">FullConcat</span> dataset was still made, found to be $\sim$ 1.90$\,\times\,$10$^{-4}$ M$_{\odot}$$\,$yr$^{-1}$. This is much higher than the rest of the cohort, supporting the conclusion that this source cannot be realistically approximated with a purely thermally emitting stellar wind.
The radio flux of WR A gives an $\dot{M}f_{cl}^{\frac{1}{2}}$ value, 7.02$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$, consistent within uncertainties to the value found in the millimetre of 7.19$\,\times\,$10$^{-5}$ M$_{\odot}$$yr^{-1}$. As WR A was found to have extended emission, only the flux from its core component was used to calculate the mass-loss rate. As a core component could not be resolved for this source at 5.5$\,$GHz, $\dot{M}f_{cl}^{\frac{1}{2}}$ values were only determined for the <span style="font-variant:small-caps;">FullConcat</span> and <span style="font-variant:small-caps;">Full9</span> datasets.
WR D was found to have a $\dot{M}f_{cl}^{\frac{1}{2}}$ value of 5.92$\,\pm\,$0.31$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$, larger than the mm value of 2.07$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$$\,$. WR G was also found to have a larger mass-loss rate calculated from the radio fluxes, with a value of 3.41$\,\pm\,$0.30$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$ from the radio observations, in contrast to the mm value of 2.22$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$$\,$. This was also the case for WR M, where the radio mass-loss rate was found to be 5.74$\,\pm\,$0.71$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$, with the mm mass-loss rate previously measured as 3.57$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$.
WR I is found to give a lower value in the radio than in the millimetre. The mass-loss rate determined in the radio was 3.10$\,\times\,$10$^{-5}$ M$_{\odot}$$yr^{-1}$, in comparison to the $\dot{M}f_{cl}^{\frac{1}{2}}$ value calculated from the mm of 3.55$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$ $\,$, however due to the uncertainties on the radio $\dot{M}f_{cl}$, the two values could still potentially be consistent. Higher values are also found in the millimetre for $\dot{M}f_{cl}^{\frac{1}{2}}$ for WR E. The $\dot{M}f_{cl}^{\frac{1}{2}}$ value from the radio observations was calculated to be 3.62$\,\pm\,$0.42$\times$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$ for WR E, which can be compared to the mm mass-loss rate from the millimetre, of 5.24$\,\times\,$10$^{-5}$.
We can also compare our mass-loss rates to the previous radio results in .$\,$ find mass-loss values for two of the WR stars they detect - WR L and WR F. For WR L, they find a mass-loss rate of 2$\,\times\,$10$^{-5}$(v$_{\infty}$/1000km$\,$s$^{-1}$)M$_{\odot}$$\,$yr$^{-1}$. By taking the terminal velocity of this source to be v$_{\infty}$ $\sim$ 700$\,$km$\,$s$^{-1}$, this leads to a value of 1.4$\times$10$^{-5}$M$_{\odot}$$\,$yr$^{-1}$. In our observations, we determine a higher mass-loss rate of 2.72$\,\times\,$10$^{-5}$$\,$ M$_{\odot}$$\,$yr$^{-1}$.
found a mass-loss rate of 3.3$\,\times\,$10$^{-5}$(v$_{\infty}$/1000km$\,$s$^{-1}$) M$_{\odot}$$\,$yr$^{-1}$ for WR F. With an expected v$_{\infty}$ value of 1200$\,$km$\,$s$^{-1}$, this would lead to a mass-loss rate of 3.96$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$. Our mass-loss calculation of WR F results in a higher value, 6.62$\pm\,$0.34$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$. We can also consider the predicted mass-loss rate derived for WR F of $\sim$ 3.16$\,\times\,$ 10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$ from [@Clark2011]. These values can be reconciled if there are higher levels of clumping in the inner parts of the winds, as the [@Clark2011] value is derived from optical and near-IR data.
Mean mass-loss rates could be determined for the different classes of WR stars, with a mean mass-loss rate over the total ensemble calculated to be 4.61 $\times$ 10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$. The average mass-loss rate for the WN classes was determined as 4.26 $\times$ 10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$, and for WC subclasses, it was found to be 5.33 $\times$ 10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$. In general, the WC subtype can be seen to have higher values of mass-loss rates *on average*, but there were only measurements from three WC stars, and one of these is a likely binary (WR F).
These values can be considered to be fairly consistent with previous mass-loss determinations of WR stars, with a value from [@Leitherer1997], of $\dot{M} \sim$ 4 $\times$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$ over all spectral sub-types. Measurements from [@Cappa2004] gave $\dot{M} \sim$ 4$\,\pm\,$3$\times$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$ for WN, so our WN mean value can be seen to be consistent, but their measurement for the WC subtype was $\sim$2$\,\pm\,$1$\times$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$, lower than our average value found for this spectral subtype. The overall spread of the mass-loss rates is found to be similar to that seen in , from 1.52 - 9.75 $\times$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$ (ignoring the WN10-11h star/BHG, WR S).
### Clumping ratios
[p[1.5cm]{}p[2.3cm]{}\*2[p[2.3cm]{}]{}p[3.5cm]{}\*2[p[2cm]{}]{}]{} Source & Spectral Type & (M$_{\odot}$$\,$yr$^{-1}$) & ALMA $\dot{M}f_{cl}^{\frac{1}{2}}$ (M$_{\odot}$$\,$yr$^{-1}$) & $\frac{f_{cl, 5.5}}{f_{cl, ALMA}}$ & $\frac{f_{cl,9}}{f_{cl, ALMA}}$\
& & 5.5GHz & 9 GHz &&&\
WR A & WN7b + OB? & 6.06$\pm$0.59e-05 & 6.63$\pm$ 0.26e-05 & 6.63$\pm$0.14e-05 & 0.84 $\pm$0.21 & 1.00 $\pm$ 0.09\
WR D & WN7o & - & 1.84$\pm$0.41e-05 & 1.59$\pm$0.12e-05 & - & 1.34 $\pm$ 0.63\
WR G & WN7o & - & 1.50$\pm$0.35e-05 &1.87$\pm$0.14e-05 & - & 0.64 $\pm$ 0.31\
WR I & WN8o & 1.70$\pm$0.32e-05 & 3.06$\pm$0.20e-05 & 3.15$\pm$0.05e-05 & 0.29$\pm$0.11 & 1.47 $\pm$ 0.19\
WR L & WN9h+OB? & 1.66$\pm$0.31e-05 & 2.67$\pm$0.16e-05 & 3.36$\pm$0.05e-05 & 0.24$\pm$0.09 & 0.63 $\pm$ 0.08\
WR F & WC9d+OB? & - & 3.07$\pm$0.25e-05 & 6.24$\pm$0.22e-05 & - & 0.24 $\pm$ 0.04\
WR M & WC9d & - & 1.23$\pm$0.25e-05 & 2.59$\pm$0.25e-05 & - & 0.23 $\pm$ 0.10\
Due to the different wavelength regimes utilised for the calculating $\dot{M}f_{cl}^{\frac{1}{2}}$, then different clumping factors may play a role in the value of $\dot{M}f_{cl}^{\frac{1}{2}}$ calculated, assuming a constant mass-loss rate throughout the wind. Unfortunately, the intrinsic mass-loss rate for these winds is unknown, so we can only consider the relative degrees of clumping for different wavelength regimes. This still gives an indication of the different levels of clumping present at varying radial geometries within the stellar winds.
Clumping ratios were determined by comparing stellar mass-loss rates at different frequencies. The datasets which contained tapered u-v visibilities were used, to allow for the comparison between the millimetre and radio observations as previously discussed. Clumping ratios were derived for 7 WR stars from the tapered datasets, shown in Table \[table:clumping\_ratio\]. These ratios were calculated assuming a normalisation of the mass-loss rate (using $\dot{M}f_{cl}^{\frac{1}{2}}$ values from <span style="font-variant:small-caps;">TaperALMA</span>) and clumping ratios were derived separately for the <span style="font-variant:small-caps;">Taper5</span> and the <span style="font-variant:small-caps;">Taper9</span> mass-loss rates against this normalised value.
A large spread of clumping ratios were found over the different Wolf-Rayet stars. The majority of the WR stars (WR A, WR G, WR L, WR F and WR M) demonstrate the expected behaviour of the clumping factor, decreasing as the wavelength increases. This is not seen for WR D, which has a clumping ratio of 1.34, significantly above 1. This could indicate that the wind is highly clumped even out at radio wavelengths, in the most extended part of the wind. The adverse clumping ratio (above the expected fraction for an outer part of the wind) could be related to the fact that WR D is a binary candidate. As the calculated mass-loss rate is reliant on assuming fluxes are dominated by thermal emission from stellar winds of a single star, non-thermal radio components due to binarity could have led to these unexpected results.
These sources don’t give a clear indication of a uniform trend for clumping ratios between the mm and the radio-photosphere, but in general support the belief that in stellar winds, clumping should decrease within the wind as a function of radial distance from the star.
Cool supergiants and hypergiants
--------------------------------
Source Spectral Type ATCA $\dot{M}f_{cl}^{\frac{1}{2}}$ (M$_{\odot}$$\,$yr$^{-1}$)
-------- --------------- ---------------------------------------------------------------
W12a F1Ia$^{+}$ 1.66$\pm$0.02$\times$10$^{-5}$
W4a F3Ia$^{+}$ 2.59$\pm$0.02$\,$$\times$10$^{-5}$
W32 F5Ia$^{+}$ 3.01$\pm$0.50$\times$ 10$^{-6}$
W265 F5Ia$^{+}$ 1.10$\pm$0.03$\times$ 10$^{-5}$
W237 M3Ia 2.00$\pm$0.02 $\times$$\,$10$^{-6}$
W75 M4Ia 7.27$\pm$0.48$\times$$\,$$10^{-7}$
W20 M5Ia 3.28$\pm$0.03 $\times$$\,$10$^{-6}$
: Mass-loss estimates for the YHGs and RSGs.[]{data-label="table:mass_loss_sgs"}
Nine of the ten RSGs and YHGs present in Westerlund 1 were detected, with the singular exception of the YHG W8a. 8 of these sources are resolved, with extended nebulae detected around all of these sources. As these are all cool supergiants and hypergiants, the typical photospheric temperatures ($\sim$ 4000K for RSGs, 4000 - 8000K for YHGs) would not be enough to cause the ionisation of the nebulae material and lead to radio emission. Instead, a diffuse radiation field caused by the hundreds of O and B stars present in the cluster was postulated as the source of the ionisation, allowing for the stellar nebulae to be observed.
The structures of the extended emission around these sources were seen to be asymmetric and complex in nature. A mixture of morphologies were seen around several of the sources, including the bow shock structure as seen around W237 in Figure \[fig:W237\_postage\] and the cometary morphology around W265, as shown in Figure \[fig:W265\_postage\].
Spectral indices were determined for all detected RSGs and YHGs except the source W32. A marginal detection could be found from the <span style="font-variant:small-caps;">Taper9</span> dataset at $\sim$ 0.13$\,$mJy, (with a SNR of only $\sim$ 2.69), alongside a marginal detection of $\sim$0.08$\,$mJy in the <span style="font-variant:small-caps;">Taper5</span> dataset (at a SNR of $\sim$ 1.26). The ALMA flux of W32 from , 0.38$\,\pm\,$0.07$\,$mJy, in combination with marginal detections suggest that the emission is of a thermal nature.
Core components of the source emission were used to calculate spectral indices. For most of the cool supergiants and hypergiants, spectral indices were found to be extremely flat, such as 0.00$\,\pm\,$0.04 for W237, -0.01$\,\pm\,$0.12 for W75, and -0.01$\,\pm\,$0.04 for W4a, consistent with composite emission of both optically thin and thick components, as seen previously in the millimetre . Amongst the YHGs, some sources were found to have more negative spectral indices, including W265 and W12a, which had spectral indices of -0.13$\,\pm\,$0.23 and -0.30$\,\pm\,$0.07 respectively. Due to the relatively high uncertainty on these values, this is consistent with the presence of non-thermal emission, but also consistent with optically thin emission surrounding these sources.
W16a, shown in Figure \[fig:W16a\_postage\], was found to have a spectral index -0.10$\,\pm\,$0.05, which would suggest the presence of non-thermal or optically thin emission components. This object may actually have a much flatter spectral index, as there is possible source confusion due to extremely close presence of WR U, which is not readily distinguished from the emission associated with W16a.
Mass-loss rates were calculated for sources for which the emission present could be attributed to optically thin and thick components in the stellar wind, excluding sources with potential non-thermal emission. The core component of flux detected for these stellar sources were used in the mass-loss calculation, with the exception of W75, which was found to be far fainter than the rest of the sources. W26 and W16a were not considered as no core component could be isolated within from the emission associated with these sources. The mass-loss estimates are presented in Table \[table:mass\_loss\_sgs\].
We can compare the range of mass-loss rates found for RSGs, $\sim$ 10$^{-7}$ - 10$^{-6}$ M$_{\odot}$$\,$yr$^{-1}$, to that found for field RSGs, which vary from 10$^{-6}$ - 10$^{-4}$ M$_{\odot}$$\,$yr$^{-1}$ [@Jura1990; @Sylvester1998]. An RSG with similar morphology, VY CMa, has been found to have mass-loss values of $\dot{M}f_{cl}^{\frac{1}{2}} \sim$ 10$^{-3}$ M$_{\odot}$$\,$yr$^{-1}$ or greater$\,$[@Shenoy2016]. For W237, a time-averaged mass-loss rate was previously calculated in , $\sim$ 10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$, which is significantly higher than the value determined here, but uses the measure of the ionised mass of the extended nebula surrounding the source.
There is a clear caveat in comparing the mass-loss estimates determined here to those calculated in the literature, as the methods used to determine the $\dot{M}f_{cl}^{\frac{1}{2}}$ values here are derived from the free-free emission in the wind, whereas $\dot{M}f_{cl}^{\frac{1}{2}}$ values in the literature are typically calculated from fitting models to the metal lines of stars, as seen for YHGs such as $\rho$ Cas, [@Lobel1998; @Lobel2003], or from the modelling of low excitation CO lines and using models to extrapolate the mass-loss rates, as has been carried out for the YHG IRC+10420 [@Oudmaijer1996; @CastroCarrizo2007], and the RSG VY CMa [@Decin2006]. These models often require many assumptions to be made about the velocity and temperature structure of the stellar wind, as well as the heating and cooling processes occurring within the wind, and the relative gas and dust densities. These methods also use information from the emission of different wavelength regimes, and so the effect of clumping and structure within the wind could lead to inconsistent results between our estimates and the literature.
The range of $\dot{M}f_{cl}^{\frac{1}{2}}$ $\sim$ 10$^{-6}$ - 10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$ is found for the YHGs present in Westerlund 1. Some direct comparisons can be made to previous observations, including W4a, where a mass-loss rate estimate was inferred from millimetre observations, $\sim$ 10$^{-5}$M$_{\odot}$$\,$yr$^{-1}$$\,$. This is found to be consistent with our estimate. suggest an estimate of $\sim$ 10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$ for W265. Our mass-loss rate for this source, 1.10$\,\pm\,$0.03$\times$ 10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$, is therefore also consistent with this earlier estimate, even though the mass-loss rate was derived from more the extended region of emission.
OB stars, supergiants, hypergiants, LBVs, and sgB\[e\] stars
------------------------------------------------------------
Of the radio detections made from the survey of Westerlund 1, 11 of these are OBA stars. This includes the LBV A2 Ia star W243, the sgB\[e\] star Wd1-9, 7 confirmed OB supergiants and 2 radio sources previously discovered in Do10, D09-R1 and D09-R2, which have the assigned spectral type of blue supergiants (BSGs). The detections cover a range of spectral types, from late type O supergiants (O9Ib) to early type B supergiants (B1Ia). No stars from a later spectral type were detected. This indicates that the majority of stars within the cluster exhibit typical thermal spectral behaviour, and so their expected radio fluxes fall well below the noise level for these observations .
### OB super and hypergiants
In total, 9 OB stars were detected in the cluster with 5 new radio detections. W10 is a known double-lined spectroscopic binary (SB2) binary in the cluster, with stellar components B0.5I + OB, and was found to have an associated flux density of 0.16$\,\pm\,$0.05$\,$mJy. W18 and W19, both early type B supergiants, with spectral types B0.5Ia and B1Ia, were found to have fluxes 0.36$\,\pm\,$0.06$\,$mJy and 0.08$\,\pm\,$0.03$\,$mJy respectively. Additionally, two late type O supergiants were newly detected, W1031 of spectral type 09III with a flux density of 0.90$\,\pm\,$0.05$\,$mJy, and W1056, a 09.5II star, with a flux density measured of 0.09$\,\pm\,$0.02$\,$mJy.
Cluster members D09-R1 and D09-R2, which have the current spectral type designation of BSG, were previously discovered in the radio . These sources currently lack an accurate spectral classification. Fluxes were found for D09-R1 for both a core component and total value of 1.29$\,\pm\,$0.03$\,$mJy and 5.44$\,\pm\,$0.10$\,$mJy. D09-R1 is embedded in an area of diffuse radio emission, as was previously indicated in . A flux was also determined for the source D09-R2, and found to be 0.96$\,\pm\,$0.06$\,$mJy.
As these sources were also detected in the millimetre, spectral indices could be determined. They were found to have fairly flat, slightly negative spectral indices of $\sim$ -0.01 and -0.03 for D09-R1 and D09-R2 respectively. This suggests that these objects experience a composite behaviour. This could include a combination of partially optically thick, optically thin and non-thermal emission components. The spectral indices are roughly consistent with previous radio-mm spectral indices, with the spectral index determined for D09-R1 slightly flatter in comparison to a previous radio-mm measurement of $\sim$ 0.17$\,$.
W17 and W15 are both OB stellar sources with previous radio detections . W15 is currently assigned the spectral classification O9Ib, and was found to have a flux density of 1.65$\,\pm\,$0.09$\,$mJy. W17, a late O supergiant, was found to have a radio flux of 0.98$\,\pm\,$0.04$\,$mJy for the core component of the source, surrounded by an extended component with a total flux density of 2.04$\,\pm\,$0.08$\,$mJy. W17 was also detected in the millimetre, and so a spectral index was derived for this source of $\sim$ -0.03, suggesting a composite spectra, which may be made up of optically thin and thick, as well as non-thermal components. This revised spectral index was found to be significantly flattened in comparison to the previous radio-mm spectral index found of $\sim$ 0.33$\,$.
As the emission from W17 could be attributed to optically thick and thin components, an approximation could be made to a thermal stellar wind in order to determine the mass-loss rate. A mass-loss rate was determined for this source of 1.38$\,\pm\,$0.03$\,\times\,$10$^{-4}$ M$_{\odot}$$\,$yr$^{-1}$. This is higher than would be expected for a late O supergiant. This suggests that the approximation of a thermally emitting stellar wind is not suitable, and may indicate the presence of binarity, supported by the fact that W17 has already been found to exhibit variability [@Bonanos2007].
6 of the OB supergiants did not have counterpart millimetre detections. This included a range of spectral types, over B1Ia - 09Ib. W15 was the only one of these sources previously detected in the radio in .
By using the flux limits of the ALMA dataset presented in , spectral index limits could be determined. These limits involved the use of the <span style="font-variant:small-caps;">FullConcat</span> dataset, rather than the tapered datasets, due to the intrinsic faintness of the sources. All of the stars were found to have spectral index limits consistent with non-thermal emission, with especially strongly negative spectral indices for W15 and W1031. We considered the approximation of the radio fluxes to thermal stellar wind emission, to determine mass-loss rate estimates. The range of mass-loss rates found for the OB supergiants varied from 4.8$\,\times\,$10$^{-6}$ - 1.31$\,\times\,$10$^{-4}$ M$_{\odot}$$\,$yr$^{-1}$.
Although the lower end of this range is in line with expectations, in general these values are much higher than would be expected for late O/early B type supergiants, $\sim$ 10$^{-6}$ - 10$^{-7}$ M$_{\odot}$$\,$yr$^{-1}$. The high mass-loss rates, along with the negative spectral index limits derived, indicate that attributing the radio emission of these stars to stellar winds from *single stars* is not valid. It is then worth considering that these stars may be binaries.
W15 has no prior evidence from hard X-ray detections or RV variability, but is most clearly constrained by its spectral index limit of $<$ -1.15, strongly suggesting non-thermal emission. If W15 is a binary system, then this system must be either in an eccentric or a very wide orbit in order to be consistent with other observational results for this star. A wide system would allow for the WCZ to occur outside of the radio photosphere, so non-thermal emission would not be obscured by surrounding thermal emission from a stellar wind, and the proximity between the WCZ and the star would be insufficient for shocks to produce hard X-rays, as would typically be expected for binaries [@Pittard2018]. This is also supported by recent observational evidence that binary stars have been seen to present X-ray signatures that are consistent with single stars of a similar spectral type [@Clark2019 and references within].
This system can be compared to the WR star, WR 140. WR 140 is considered to be the archetypal wind colliding binary (WCB), consisting of a WC7 WR and an O5 companion. W140 has been found to have variable emission due to the high eccentricity of the orbit [$\sim$ 0.90, @Fahed2011], with different variabilities measured at various radio wavelengths$\,$[@Dougherty2011]. WR 140 has been detected to have X-ray variability, including in the hardness of its X-ray spectra over the orbital phase$\,$[@Corcoran2011]. If W15 is a similar system, then this scenario could explain the lack of a hard X-ray detection.
Another piece of evidence supporting this hypothesis is the variability seen in the radio flux between the 8.6$\,$GHz and ATCA 9$\,$GHz observations - where the increase in flux is almost of a factor 2. This variance could be due to a change in orbital phase between the two measurements. W15 is an object that would benefit from a further time-domain study, to map out possible changes of orientation over orbital phase, in order to provide constraints on the properties of the system.
W10 is the only previously confirmed binary of the detected OB supergiants, first presented as an SB2 in [@Clark2008]. W10 has also been previously found to be an RV variable. W10 was recently found to experience softer and dimmer X-ray emission than other SB2s in Westerlund 1, suggesting an eccentric binary where the emission from a WCZ was less present in observations taken in 2005 [@Clark2019]. This may be supported by the weak constraint on the nature of emission for W10, with a spectral index limit of $<$-0.04, consistent with a composite spectra.
W18 was found to have a negative spectral index limit, $<$ -0.43, strongly indicative of non-thermal emission. This stellar source has no current associated evidence for binarity from X-ray observations and no RV detections. This doesn’t rule out binarity for this source, as bias in the observational limits of these methods could mean that a binary system is present that was not easily detected by RV variations. The surrounding dense cluster wind and material from other stars in the cluster could also obscure X-ray emission, as there is diffuse X-ray emission known to be present throughout the cluster [@Muno2006]. W19 is also a cluster member with detected radio emission but no prior evidence of binarity from RV or X-ray measurements, although it has previously been found to be H$\alpha$ variable. It has the weakest constraint on its spectral index limit, allowing for a flat composite spectrum of emission. It is also of note that W18 and W19, alongside W10 and W15, have previously been found to be associated with long period/aperiodic variability, supportive of the conclusion of binarity [@Bonanos2007].
The other two stellar cluster members detected, W1031 and W1056, are both confirmed RV variables. The period for W1031 is not currently determined. W1056 with a candidate period of less than 10 days. It’s of note that W1056 has a much looser constraint on the spectral index, $<$ 0.02, in comparison to the other stars with late O spectral classification, which may be consistent with a flat composite spectra, which would be expected if stellar winds are obscuring some of the radio emission from the colliding wind region.
We suggest that we would not expect to detect any emission from purely thermal stellar wind emitters in the radio. This is supported by the lack of millimetre detections in for the majority of O/B supergiants in Westerlund 1. With this in mind, along with the previously observed binarity characteristics for many of these objects in the UV and X-ray, we can infer that *all* of the OB supergiants we detect are either binaries or binary candidates. We believe most of these objects are highly likely to be long period binaries, where the separation between the stellar components is sufficiently high for non-thermal emission from colliding wind regions to occur outside the region of the stellar wind radio photospheres. Therefore, we have managed to detect 3 new binary candidates, that are likely long period binaries where RV variations could not be detected.
### W243
W243, a LBV with assigned spectral type A2 Ia, is detected in the ATCA observations. The spectral index was found to be $\sim$ 0.82, consistent with previous measurements. This is indicative of a thermal spectrum, with a slight steepening in comparison to the canonical stellar wind value. Our mass-loss determination for this object was found to be 3.12$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$.
determined a mass-loss estimate for W243 on the order 1.1$\times$10$^{-5}$(v$_{\infty}$/200km$\,$s$^{-1}$)M$_{\odot}$$\,$yr$^{-1}$. From an assumed v$_{\infty}$ $\sim$ 550$\,$km$\,$s$^{-1}$ for W243, the mass-loss rates therefore can be considered to be consistent. We can also compare this mass-loss rate to the previous millimetre observations, which gave a $\dot{M}$ of $\sim$ 2.5 $\times$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$ $\,$. The radio and mm values are both higher than the expected mass-loss rate, $\dot{M} \sim$ 8 $\times$ 10$^{-7}$ M$_{\odot}$$\,$yr$^{-1}$, given from modelling in$\,$@Ritchie2009. Clear limitations for this modelling included the issue of fitting to all H and He emission features prominent in the spectra, suggesting that the spectral type approximation used was not suitable in order to generate an accurate fit.
This LBV $\dot{M}f_{cl}^{\frac{1}{2}}$ value is also larger than a mass-loss rate of $\sim$ 1.35$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$, found from radio observations of a post-RSG LBV HD160529 $\,$[@LeithererRobert1995]. It can also be compared to other LBV measurements from sources including AFGL2338, AG Car, FMM362 and the Pistol Star$\,$[@ClarkLBV; @Najarro2009; @Groh2009], which have mass-loss rates over the range of 3 - 6$\,\times\,$10$^{-5}$M$_{\odot}$$\,$yr$^{-1}$, all larger than the mass-loss rate calculated for W243.
An estimate of the clumping ratios are also found for this object, between the millimetre and radio data. The clumping ratios are found to be $\sim$ 0.027 and $\sim$ 0.07 when comparing 5.5$\,$GHz and 9$\,$GHz respectively to the millimetre observations, suggesting a much higher level of structure and clumping within the inner regions of the wind sampled by the mm observations, that then decreases with increasing distance from the star.
### W9
W9 can be seen in Fig \[fig:Wd19\_postage\], where the two filaments of emission to the south-east and south-west of the source as observed previously in millimetre observations$\,$ can be seen at much lower resolution in the radio, as well as an additional third filament to the north-west. The flux of the core compact source region of W9 was found to be 27.38$\pm$0.06$\,$mJy, and the extended region around it gives a total flux value of 83.14$\,\pm\,$0.41$\,$mJy. The spectral index of W9 indicates thermal emission, in line with the canonical value for a stellar wind, with a value of $\alpha$ = 0.68$\,\pm\,$0.02. This value is consistent with the previous measurements.
As W9 was found to have a thermal nature derived from the spectral index, we could then make an estimate of the approximate mass-loss rate for this source, despite the evidence that this source is known to be a binary from its X-ray emission [@Clark2013]. The mass-loss rate was calculated to be $\dot{M} f_{cl}^{\frac{1}{2}} \sim$ 13.03$\,\pm\,$0.01$\,\times\,$10$^{-5}$ M$_{\odot}$$\,$yr$^{-1}$, similar to the value found from Model 2 of Wd1-9 in .
Origin of the extended emission within the cluster
==================================================
As was previously introduced in Section 4, there is a pervasive level of diffuse radio emission that can be seen throughout the cluster. This can be seen most clearly by looking at Figure \[fig:fig1\], which shows the ATCA observations separated by configuration. The spatial scales measured by the shortest configuration with a maximum baseline of 750m demonstrate a significant amount of diffuse radio emission present throughout the cluster. Much of this extended radio emission appears to be centred on the radio bright sources, such as W9, W20, or even the WR star WR B, but some of the extended emission cannot be attributed to any of the known stellar sources. These areas still contain compact sources with no stellar counterpart, as discussed in section 5.2. These areas of concentrated emission that appear throughout the cluster indicate the high level of varying structure present, and that the overall structure of the diffuse emission is clumped and non-spherical.
Stellar and cluster wind interaction
------------------------------------
From the morphology of this diffuse background, as seen clearly in all the figures displaying the whole cluster region (Figures \[fig:fig1\] - \[fig:fig3\], \[fig:figB1\]), we can see this diffuse emission is more concentrated towards the centre of the cluster. At extended distances from stellar sources, a large level of radio emission is detected, indicating interactions between the stellar winds of the sources and a cluster wind. The interaction of stellar winds with the surrounding environment has previously been discussed in @Mackey2015 as an origin of the sculpted nebulae surrounding Wd1-26. Typically, this diffuse emission would be expected to arise as a result of interaction between young massive stars and remnants of their parent giant molecular cloud (GMC), but at an expected age of 5$\,$Myr, this is not believed to be the case for Wd1.
This is also supported by the general lack of evidence to support any ongoing star formation within the cluster. This is currently supported by results from many mid- and near-infrared surveys (2MASS, GLIMPSE, MIPSGAL) that have included the cluster within their field of view [@Clark2015], as well as a lack of any detections of massive young stellar objects (YSOs) in the vicinity of the cluster, or the presence of any methanol masers, a typical signature of star forming regions (SFRs). We can also compare the general geometry of the diffuse emission to stellar clusters that are known to be located in star forming regions, such as the clusters Danks 1 and 2, located in G305. The structure of the diffuse emission around these clusters are seen to follow a much simpler classical wind blown shell geometry [@ClarkPorter2004; @Davies2012].
Some future observations that could help us to constrain the origin of the gas within this cluster would be to search more explicitly for spectral signatures that constrain the origin to natal emission (with linked evidence of ongoing SFRs present), or from the outflows of the stellar population. A signature of recent star formation (and hence, diffuse emission from the natal remnants) already mentioned is that of methanol masers. Future studies of transition frequencies, such as 12.2$\,$GHz for methanol, or 1.7$\,$GHz for OH masers, would provide evidence of star formation and hence constrain the origin of the gas; or a lack of detection would help suggest that the current conclusion of stellar emission is more valid.
A more explicit method to constrain the nature of the gas due to stellar emission would be to search for He and N emission lines, using optical/near-IR spectroscopy. N \[II\] emission lines have previously been detected around some cluster members, including the RSG population [@Clark2010]. N \[II\] emission lines are known to occur in the ejecta of other cool evolved stars with nearby stellar companions, where the external ionisation of the ejecta is due to material from the stellar companion’s wind, as in the case of the YHG HR 8752 [@Stickland1978]. Observational evidence of these emission lines within the diffuse regions of the gas would therefore provide further evidence that the extended emission can be related to stellar winds of massive stars in the cluster.
Previous evidence of a cluster wind in Wd1 has been found in the X-ray$\,$[@Muno2006], radio$\,$ and millimetre$\,$. This cluster wind is also indicated by the common orientation of several of the resolved extended nebulae around the cool super and hypergiants in the cluster, where bow shocks and cometary tails are indicative of a general outward direction, originating from the centre of the cluster$\,$[@Andrews2018]. The origins of cluster winds are believed to be related to the feedback of stellar winds that become thermalized and then drive the cluster winds. The properties of the mechanisms of cluster wind propagation are not fully determined, due to uncertainties in the conversion of stellar kinetic energy to thermal energy$\,$[@Chevalier1985; @Canto2000; @StevensHartwell2003]. The origin of the possible cluster wind in Westerlund 1 is not confirmed, but is believed to be due to a diffuse radiation field arising in the cluster from the radiation pressure of the hundreds of OB stars present.
The overall flux of the cluster wind can be considered. A value of the total flux in the cluster was determined for the central 1.5’ of the cluster (as carried out in ). This led to a value of the total flux present of 635$\,$mJy at 9$\,$GHz and 640$\,$mJy at 5.5$\,$GHz. This was found to be significantly larger than the flux determined by at 8.6$\,$GHz of 422$\,$mJy and 461$\,$mJy at 4.8$\,$GHz, as well as the single dish measurements of the flux of the cluster from @Kothes2007, of 450$\,$mJy and 499$\,$mJy at 4.8$\,$ and 8.6$\,$GHz respectively. From a total flux in the <span style="font-variant:small-caps;">FullConcat</span> dataset of 647$\,$mJy, this leads a to flux for the extended emission (with the stellar components subtracted by considering flux from known detections), of 408$\,$mJy over the <span style="font-variant:small-caps;">FullConcat</span> dataset. A general estimate of ionised mass present in the cluster can be considered from $\sim$ 408$\,$mJy flux found for the extended emission at 9$\,$GHz over a assumed spherical region of diameter 1.5$\,$arcminutes), gave a value of 29$\,$M$_{\odot}$, (assuming a general plasma temperature of 10kK), almost twice the ionised mass previously determined for the extended emission in the cluster$\,$.
By characterising the cluster wind, we can hope to learn more about the stellar ejecta, and how it is impacted not only by the physics of stellar outflows, but also by the differing surrounding environments of these stars, and how it may sculpt and shape the ejecta as is clearly seen here. The presence of this cluster wind indicates the important role that an ionisation field may play on stellar populations within large clusters and associations.
Understanding the diffuse emission and its origin is important not only for determining accurately the level of material that is associated with each stellar object, but also in starting to consider how the stellar ejecta are affected by a cluster wind. Characterising the cluster wind and the interaction between an intra-cluster medium and stellar winds is important for many areas of astrophysics. The processes involved have an impact on the complete life-cycles of stars, from start to end. Cluster winds may play a role in both inhibiting and triggering the process of star formation, and may also lead to the disruption of natal giant molecular clouds. It also helps in formalising the feedback from stars into their surrounding environment, understanding resultant galactic superwinds as well as the production of cosmic rays, for which the production mechanisms require a significant level of interstellar material, or a strong radiation field, evidence for both of which can be seen in Westerlund 1.
A dense background may also affect the resulting SNe that will occur from many stars when they reach their stellar endpoint. Interactions between stars and their surrounding circumstellar nebulae can impact the resultant SNe light curve, and so affect the relative ratios observed of each SNe type. This may in turn impact the expected ratios of observed SNe in the early universe. This shows a clear wider implication of this environment for a wide range of fields of study within astrophysics.
Conclusion
==========
This paper has presented the full results of a radio census of the massive stellar cluster Westerlund 1. Observations were made using ATCA, over 4 different pointings of varying maximum baselines from 750$\,$m to 6$\,$km, and over two spectral windows centred at 5.5$\,$GHz and 9$\,$GHz. The census aimed to present a direct follow up to previous observations of the rich co-eval population of Westerlund 1, especially the previous radio ATCA observations$\,$. 30 stellar sources were detected in the radio, over a range of spectral types. A strong and diffuse radio background was also clearly seen to be present in the cluster. 5 new detections were made in the radio for WR stars in the cluster, and 5 new radio detections were made of OB supergiants in the cluster. For sources that had previously been detected in the radio, new details of surrounding extended structures of emission have come to light, due to the better resolution and sensitivity now possible with ATCA. From the 5 new OB radio detections, we were able to determine the presence of 3 new binary candidates.
Many of the sources have been found to exhibit emission that has a spectral index fully consistent with the canonical value for thermal emission from a single stellar wind. The use of the Wright and Barlow formalism, [@WrightBarlow1975], allowed for mass-loss rates to be determined for a number of the sources. This was carried out for sources which were found to have spectral indices indicative of thermal emission. Clumping ratios could then be determined for many of the WR stars, with comparison between the radio observations and previously published millimetre observations$\,$.
The majority of sources detected were found to be resolved. The origin of extended emission around most of these sources is not clear. One possible explanation is due to binarity, but many of sources detected with extended emission are not binary candidates. Many of the sources are also not expected to have experienced large levels of variability or pulsations, which could have been another solution to explain the presence of the extended nebulae. The majority of the sources are post-Main Sequence objects, RSGs, YHGs or WRs. There is a possibility that all these stars may have experienced previous epochs of extreme mass-loss, but the degree of structure surrounding sources is not seen to vary uniformly by spectral type, which indicates it may not be due to the evolutionary status of the star.
Understanding the origin of the diffuse background and the extent of its interaction with the stellar wind material is of importance. The boundary between the stellar wind material and the surrounding diffuse emission is not clear, and shows a clear need for more observations of similar environments as well as more developed models, taking into account the interaction between stellar winds of massive stars and their parent clusters. Future observations, especially of the neutral material present in the cluster, would allow for additional consideration of the intra-cluster material present as well as the extended nebulae of each source, allowing for updated considerations to be made of the nebulae masses and geometries around many of the stars, especially the population of the YHGs and RSGs present in Wd1.
Higher resolution, longer baseline, radio observations of these objects would also be of great use. The use of higher resolution observations would allow for constraints to be applied on how to distinguish between the diffuse cluster material and the outflows from the stellar sources. Such observations may allow for any possible CWB regions to finally be resolved for many of the binary candidates present in the cluster. This includes the non-thermal emitters Wd1-15 and the other OB supergiants, W10, W18, and W1031.
More tailored modelling of mass-loss rates and consideration of clumping at different radial geometries of the wind that quantifies the radial stratification taking place would also help to provide better constraints on the observations of the stellar winds across a large variety of different stellar spectral types, as seen to populate Westerlund 1.
This paper makes use of the following ALMA data: ADS/JAO.ALMA/2013.1.00897.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. H. Andrews wishes to acknowledge STFC for the funding of a PhD Studentship. D. Fenech wishes to acknowledge funding from a STFC consolidated grant (ST/M001334/1).
Tables
======
Source Flux$_{Full9}$ Flux$_{Do10}$ Notes
-------- ------------------------ ---------------------- ----------
WR A 0.48 $\pm$ 0.04 0.5 $\pm$ 0.06
1.34 $\pm$ 0.09 -
WR D 0.34 $\pm$ 0.06 -
WR B 3.24 $\pm$ 0.06$^{c}$ - Crowded
9.50$\pm$ 0.18$^{t}$ 4.3 $\pm$ 0.4$^{r}$
WR I 0.45 $\pm$ 0.03 - Isolated
WR V - 0.4 $\pm$ 0.06 Crowded
WR L 0.40 $\pm$ 0.04 0.4 $\pm$ 0.06 Isolated
WR S 0.08 $\pm$ 0.03 0.3 $\pm$ 0.06
WR E 0.11 $\pm$ 0.02 - Isolated
WR F 0.34 $\pm$ 0.03 0.3 $\pm$ 0.06 Isolated
WR M 0.16 $\pm$ 0.04 -
W16a 1.79 $\pm$ 0.09 1.6 $\pm$ 0.3 Crowded
W12a 1.66 $\pm$ 0.04$^{c}$ -
3.28 $\pm$ 0.08 2.9 $\pm$ 0.3$^{r}$
W4a 1.79 $\pm$ 0.04$^{c}$ 0.8 $\pm$ 0.08
4.19 $\pm$ 0.11$^{t}$ 2.2 $\pm$ 0.2$^{r}$
W32 0.16 $\pm$ 0.04 0.4 $\pm$ 0.06 Crowded
W265 0.94 $\pm$ 0.06$^{c}$ - Isolated
2.72 $\pm$ 0.16$^{t}$ 2.3 $\pm$ 0.3$^{r}$
W237 1.26 $\pm$ 0.04$^{c}$ 1.8 $\pm$ 0.2 Isolated
7.01 $\pm$ 0.22$^{t}$ 5.6$\pm$ 2.2$^{r}$
W75 0.26 $\pm$ 0.04 0.3 $\pm$ 0.06 Isolated
W20 2.39 $\pm$ 0.05$^{c}$ -
16.62 $\pm$ 0.27$^{t}$ 3.8 $\pm$ 0.4$^{r}$
W26 152.15 $\pm$ 0.33 20.1 $\pm$ 2.0$^{r}$
W17 0.98$\pm$ 0.05 - Crowded
1.27 $\pm$ 0.07 1.7$\pm$0.2
W243 1.65 $\pm$ 0.05 1.5 $\pm$ 0.2 Isolated
W9 30.47$\pm$0.09$^{c}$ 24.9 $\pm$ 2.5 Crowded
80.8 $\pm$ 0.5$^{t}$ 30.5 $\pm$ 3.0$^{r}$
D09-R1 1.20 $\pm$ 0.04$^{c}$ 0.7 $\pm$ 0.07 Crowded
4.66 $\pm$ 0.14$^{t}$ 6.5 $\pm$ 1.2$^{r}$
D09-R2 0.88 $\pm$ 0.08 0.7 $\pm$ 0.06 Crowded
W15 0.84 $\pm$ 0.10 0.6 $\pm$ 0.06
[p[3cm]{}p[2.5cm]{}\*2[p[2.8cm]{}]{}p[3.5cm]{}]{} FCP18 Source & RA & DEC & Flux$_{\textsc{FullConcat}}$ (mJy)\
2 & 16 46 58.60265 & -45 50 31.4920 & 1.09 $\pm$ 0.02\
3 & 16 46 58.88981 & -45 50 28.7929 & 1.88$\pm$0.02$^{c}$\
&&& 3.92$\pm$0.03$^{t}$\
6 & 16 46 59.40638 & -45 50 37.1942 & 2.11$\pm$0.02$^{c}$\
&&& 4.88$\pm$0.04$^{t}$\
8 & 16 46 59.57887 & -45 50 26.6947 & 0.31 $\pm$ 0.04\
12 & 16 47 1.01416 & -45 50 36.2975 & 0.75 $\pm$ 0.04\
16 & 16 47 1.81753 & -45 51 23.0987 & 2.11 $\pm$ 0.04\
22 & 16 47 2.33444 & -45 51 21.5992 & 0.27 $\pm$ 0.02$^{c}$\
&& & 0.66 $\pm$ 0.04$^{t}$\
27 & 16 47 2.85135 & -45 51 19.1996 & 1.12 $\pm$ 0.05\
35 & 16 47 3.68413 & -45 51 16.5000 & 0.31 $\pm$ 0.04\
37 & 16 47 3.74162 & -45 50 24.0000 & 0.38 $\pm$ 0.06\
39 & 16 47 3.88514 & -45 51 6.3000 & 1.80$\pm$0.03$^{c}$\
&&& 4.63 $\pm$0.09$^{t}$\
49 & 16 47 4.22972 & -45 51 16.2000 & 0.35 $\pm$ 0.05\
60 & 16 47 5.06241 & -45 51 2.3997 & 1.59 $\pm$ 0.02\
64 & 16 47 5.75158 & -45 51 7.1991 & 2.81 $\pm$ 0.03\
72 & 16 47 6.29690 & -45 50 44.9985 & 0.60 $\pm$ 0.03\
73 & 16 47 6.38337 & -45 51 13.7984 & 1.56 $\pm$ 0.02\
77 & 16 47 6.46965 & -45 51 23.9983 & 0.33 $\pm$ 0.03\
80 & 16 47 6.81406 & -45 51 10.4978 & 1.24 $\pm$ 0.05\
82 & 16 47 7.30222 & -45 51 11.0970 & 0.45 $\pm$ 0.04\
85 & 16 47 7.87660 & -45 51 15.2959 & 0.09 $\pm$ 0.02\
86 & 16 47 8.30665 & -45 50 43.4949 & 0.32 $\pm$ 0.03\
87 & 16 47 8.33615 & -45 51 19.7949 & 0.40 $\pm$ 0.03\
90 & 16 47 8.50825 & -45 51 11.0944 & 2.13 $\pm$ 0.05\
91 & 16 47 8.53623 & -45 50 38.6944 & 0.30 $\pm$ 0.05\
93 & 16 47 8.70934 & -45 51 14.6939 & 2.28 $\pm$ 0.04\
95 & 16 47 8.87997 & -45 50 6.2935 & 0.48 $\pm$ 0.02\
97 & 16 47 8.93797 & -45 50 30.2933 & 0.28 $\pm$ 0.03\
99 & 16 47 9.16849 & -45 51 3.2927 & 0.28 $\pm$ 0.03\
100 & 16 47 9.36973 & -45 51 12.2921 & 0.76 $\pm$ 0.04\
101 & 16 47 10.77535 & -45 50 30.2875 & 0.35 $\pm$ 0.04\
[p[3cm]{}\*3[p[3.2cm]{}]{}p[3cm]{}]{} FCP18 Source & Flux$_{\textsc{Taper5}}$ (mJy) & Flux$_{\textsc{Taper9}}$ (mJy) & Flux$_{\textsc{TaperALMA}}$ (mJy) & Spectral Index ($\alpha$)\
3 & 2.69 $\pm$ 0.04 & 1.91 $\pm$ 0.06 & 1.18 $\pm$ 0.14 & -0.26 $\pm$ 0.05\
6 & 3.98$\pm$0.05 & 3.75 $\pm$ 0.08 & 1.04 $\pm$ 0.12 & -0.49 $\pm$ 0.05\
16 & 1.66 $\pm$ 0.07 & 1.91 $\pm$ 0.07 & 0.96 $\pm$ 0.06 & -0.23 $\pm$ 0.03\
22 & - & 0.30 $\pm$ 0.05 & 0.10 $\pm$ 0.03 & -0.81 $\pm$ 0.12\
27 & 0.54 $\pm$ 0.09 & 0.85 $\pm$ 0.06 & 0.34 $\pm$ 0.05 & -0.27 $\pm$ 0.07\
39 & 2.04 $\pm$ 0.08 & 2.28 $\pm$ 0.07 & 1.57 $\pm$ 0.15 & -0.10 $\pm$ 0.04\
60 & 1.42 $\pm$ 0.07 & 1.05 $\pm$ 0.06 & 0.68 $\pm$ 0.12 & -0.23 $\pm$ 0.07\
64 & 2.33 $\pm$ 0.08 & 1.89 $\pm$ 0.07 & 1.67 $\pm$ 0.17 & -0.09 $\pm$ 0.04\
72 & 0.40 $\pm$ 0.08 & 0.49 $\pm$ 0.09 & 0.26 $\pm$ 0.07 & -0.19 $\pm$ 0.13\
73 & 1.20 $\pm$ 0.05 & 1.26 $\pm$ 0.06 & 0.72 $\pm$ 0.12 & -0.19 $\pm$ 0.07\
82 & - & 0.34 $\pm$ 0.05 & 0.16 $\pm$ 0.05 & -0.82 $\pm$ 0.13\
90 & 2.00 $\pm$ 0.15 & 1.81 $\pm$ 0.09 & 0.74 $\pm$ 0.12 & -0.35 $\pm$ 0.07\
93 & 2.00 $\pm$ 0.14 & 2.28 $\pm$ 0.10 & 0.82 $\pm$ 0.13 & -0.34 $\pm$ 0.07\
95 & 0.57 $\pm$ 0.06 & 0.62 $\pm$ 0.06 & 0.33 $\pm$ 0.06 & -0.21 $\pm$ 0.08\
99 &- & 0.25 $\pm$ 0.07 & 0.11 $\pm$ 0.04 & -0.50 $\pm$ 0.16\
[\*4[p[3cm]{}]{}]{} HA19 Source ID & RA & DEC & Flux$_{ATCA}$ (mJy)\
3 & 16 46 55.96080 & -45 50 46.1824 & 0.12 $\pm$ 0.03\
4 & 16 46 56.10430 & -45 50 47.6830 & 0.05 $\pm$ 0.01\
5 & 16 46 56.22091 & -45 50 2.3835 & 0.27 $\pm$ 0.04\
6 & 16 46 56.47741 & -45 50 51.5846 & 0.31 $\pm$ 0.03\
8 & 16 46 56.88056 & -45 50 18.2862 & 0.08 $\pm$ 0.02\
10 & 16 46 57.16625 & -45 50 58.7872 & 0.06 $\pm$ 0.02\
11 & 16 46 57.59801 & -45 50 25.7888 & 0.25 $\pm$ 0.04\
12 & 16 46 59.20651 & -45 49 50.9937 & 0.21 $\pm$ 0.03\
14 & 16 46 59.52162 & -45 50 19.1945 & 0.09 $\pm$ 0.03\
16 & 16 47 0.38207 & -45 51 2.3964 & 0.25 $\pm$ 0.04\
17 & 16 47 0.66916 & -45 51 5.3970 & 0.06 $\pm$ 0.02\
23 & 16 47 1.61678 & -45 51 0.5984 & 0.10 $\pm$ 0.03\
31 & 16 47 3.10981 & -45 51 17.3998 & 0.12 $\pm$ 0.03\
32 & 16 47 3.25367 & -45 49 59.9998 & 1.21 $\pm$ 0.07\
33 & 16 47 3.42563 & -45 51 36.5999 &0.13 $\pm$ 0.02\
43 & 16 47 5.52210 & -45 51 37.7993 & 0.38 $\pm$ 0.04\
44 & 16 47 5.69449 & -45 51 47.0992 & 0.07 $\pm$ 0.02\
48 & 16 47 6.64016 & -45 49 8.6981 & 0.11 $\pm$ 0.03\
50 & 16 47 7.04369 & -45 51 4.7975 & 0.32 $\pm$ 0.04\
51 & 16 47 7.27263 & -45 50 17.6971 & 0.12 $\pm$ 0.02\
54 & 16 47 7.67579 & -45 51 25.7963 & 0.32 $\pm$ 0.03\
55 & 16 47 7.76142 & -45 50 58.4961 & 0.10 $\pm$ 0.03\
56 & 16 47 7.87526 & -45 50 5.9959 & 0.49 $\pm$ 0.04\
57 & 16 47 7.90541 & -45 51 19.7958 & 0.67 $\pm$ 0.05\
58 & 16 47 7.93390 & -45 51 8.3958 & 0.29 $\pm$ 0.04\
59 & 16 47 7.99108 & -45 50 56.0956 & 0.35 $\pm$ 0.04\
60 & 16 47 8.07573 & -45 49 42.8954 & 0.10 $\pm$ 0.02\
61 & 16 47 8.07736 & -45 51 2.9954 & 0.55 $\pm$ 0.04\
62 & 16 47 8.33501 & -45 50 27.2949 & 0.61 $\pm$ 0.08\
65 & 16 47 8.70722 & -45 49 44.3939 & 0.09 $\pm$ 0.02\
67 & 16 47 8.90891 & -45 50 16.1934 & 0.07 $\pm$ 0.02\
69 & 16 47 9.05354 & -45 50 59.3930 & 0.07 $\pm$ 0.02\
70 & 16 47 9.08041 & -45 49 46.7929 & 0.19 $\pm$ 0.03\
72 & 16 47 9.34005 & -45 50 35.9922 & 0.94 $\pm$ 0.05\
73 & 16 47 9.56953 & -45 50 28.7915 & 0.63 $\pm$ 0.04\
74 & 16 47 9.68496 & -45 50 49.7912 & 0.39 $\pm$ 0.03\
76 & 16 47 10.17335 & -45 50 59.0896 & 0.21 $\pm$ 0.03\
77 & 16 47 10.28717 & -45 50 26.0892 & 0.47 $\pm$ 0.04\
78 & 16 47 10.43165 & -45 50 55.4887 & 1.63 $\pm$ 0.06\
79 & 16 47 10.46117 & -45 51 20.3886 & 0.19 $\pm$ 0.03\
81 & 16 47 10.83417 & -45 51 11.3872 & 0.08 $\pm$ 0.02\
82 & 16 47 10.94981 & -45 51 33.8868 & 1.20 $\pm$ 0.06\
83 & 16 47 11.17729 & -45 50 30.5859 & 0.55 $\pm$ 0.05\
85 & 16 47 11.34994 & -45 50 41.3853 & 0.05 $\pm$ 0.01\
86 & 16 47 11.52155 & -45 50 23.9846 & 0.32 $\pm$ 0.04\
87 & 16 47 11.55173 & -45 51 2.9844 & 0.14 $\pm$ 0.03\
88 & 16 47 11.75329 & -45 51 17.3836 &0.14 $\pm$ 0.02\
89 & 16 47 11.78273 & -45 51 35.9835 & 0.14 $\pm$ 0.02\
90 & 16 47 11.86639 & -45 50 32.6831 & 0.11 $\pm$ 0.02\
91 & 16 47 12.03810 & -45 50 18.8824 & 0.36 $\pm$ 0.04\
92 & 16 47 12.09525 & -45 50 12.2821 & 0.84 $\pm$ 0.07\
93 & 16 47 12.38490 & -45 51 14.0808 & 0.14 $\pm$ 0.03\
94 & 16 47 14.33810 & -45 51 24.8708 & 0.24 $\pm$ 0.04\
Figures
=======
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[^1]: e-mail: holly.andrews.16@ucl.ac.uk (UCL)
[^2]: https://github.com/daniellefenech/SERPent
[^3]: https://github.com/daniellefenech/SEAC.
|
---
abstract: '[An inexpensive protection device capable of disconnecting a load upon detecting a voltage sag or surge is presented. Said device is aimed towards the residential voltage of Venezuela’s electrical system. The device is based on the Arduino platform, and uses an ATmega 328P. It detects the presence of a voltage sag or surge by sampling the residential voltage used to feed the load, and continuously measuring the RMS $\frac{1}{2}$ voltage. It disconnects the load after 3 semi-cycles where the residential voltage is outside the boundaries established by Venezuela’s national electrical code, and therefore there’s the presence of a voltage sag or surge. The device is capable of transmitting the data of the sampled signal, for making further analysis.]{}'
address: 'College of Electrical Engineering, Engineering Faculty, Universidad Rafael Urdaneta, Maracaibo, Zulia, Venezuela'
author:
- Daniel Pérez
title: Design of a Sag and Surge Detector For Residential Voltage
---
Power quality; sag; surge; Venezuela; Arduino.
Introduction
============
In present day Venezuela various power disturbances such as voltage sags and surges are very prominent. Even though there isn’t any official data disclosed on the topic, it is agreed among many Venezuelans that one of the main sources of damage of electric appliances are volt-age sags and surges. These disturbances arise because Venezuela’s electrical system is in a bad state, which is due to mismanagement. The only permanent solution to this issue would be to make the necessary improvements to the electrical system in order to restore its proper functioning.\
For this reason it can be asserted that any permanent solution to the main power quality issues of Venezuela is mid-term, at least. For as long as the improvements aren’t done or even started to do, the voltage sags and surges are going to keep reappearing. This problem gene-rates the necessity of creating a short-term solution, one that doesn’t have to be permanent but that definitely has to have swift means of implementation. The solution has to be very low cost as well, due to the precarious si-tuation of the Venezuelan economy.\
The design of a voltage sag and surge detector based on the Arduino platform is presented as a possible solution. Said device disconnects any load attached to it, whenever the RMS $\frac{1}{2}$ value of the residential voltage signal is outside the boundaries established by Venezuela’s national electrical code (NEC) [@CEN], for a period of time equal or higher than 3 semi-cycles. The boundaries set by Venezuela’s NEC coincide with the ones established by IEEE [@1159]. Nonetheless, being that in Venezuela residential systems can either use 110 or 120 VRMS, it was decided to use 110 VRMS to set the lower boundary and 120 VRMS to set the upper boundary.\
To do the detection, the device samples the residential voltage signal at a rate of 3600 Hz, using the code of a power quality monitor developed to address Venezuela’s power quality issues as well [@dape]. After establishing the presence of either a sag or surge, the device disconnects the load attached to it by triggering a relay connected in series with the load. The device doesn’t reconnect the load until the residential voltage signal is in compliance with the set boundaries. The device is capable of transmitting the sampled data it retrieves, which can be used for either a deep analysis after the fact, or real-time mo-nitoring.\
The device distinguishes itself from other protection devices using the term “detector”, because usually protectors that are marketed as voltage sag or surge protectors offer some sort of relief upon the disturbance, such as peak suppression, instead of just disconnecting the load.
Hardware
========
The hardware used for this device is comprised by a small and simple circuit used for sampling the residential voltage signal, a circuit for managing the load attached to the device and the ATmega328P that controls the entire system.\
In figure \[pcbatmega\] can be seen the PCB design of the entire system. It is shown directly instead of presenting the schematic first to maintain simplicity in the presentation, and because it lets showcase the positioning of the elements of the system.
![Preview of PCB design that encompasses entire design. Doesn’t include DC sources[]{data-label="pcbatmega"}](pcbatmega328p.png){width="85mm"}
\
For this design it was assumed that separate modules are going to be used to provide the DC voltages required by the system. Being that this design doesn’t include a model for the case, this gives total freedom in terms of making a design for it. Also, it is worth noting that each pin used for the serial port interface (SPI) is connected to a pad, in such way that is easy to solder any module that can be used for transmitting the sampled data, which opens up the possibility of assembling an Internet of Things infrastructure [@ARD] with this design. If a component to send the data isn’t to be added, then the pads of the pins of the SPI should be shorted to ground to avoid leaving them floating.\
To lower the amount of noise in the system as much as possible, a series of measures taken in the power qua-lity monitor [@dape] were employed here as well. Those measures were taken because floating pins generate noise and can trigger $\frac{di}{dt}$ transients in the system [@CDC], and are based on various concepts of transmission lines[@MOT], grounding[@GND], power electronics[@VR] and electromagnetic compatibility[@EMC],[@EMG]. It should be pointed out that in order to reduce the overall inductance of the system tracks were kept as short as possible, and to increase the overall capacitance the gap between power lines was kept as short as possible.\
The circuit utilized for sampling is the same used for a power quality monitor targeted for Venezuela’s residential voltage [@dape]. Said circuit transforms a 120 VRMS at 60 Hz sine wave into one of 0.6 VRMS at 60 Hz with an offset of 3.3 V. This measure was taken to stay in compliance with the limits of the analog to digital converter (ADC) of the ATmega 328P [@AVR]. This design was preferred because using a transformer would unnecessarily add noise to the system, and it would incorporate losses as well. The use of resistors in this scenario is valid because the measurement of an analog signal is done with a shunt connection, and the value of the internal resistance of the ADC of the ATmega 328P is 100M$\Omega$, which is 20000 times higher than the value of the resistor that perceives the input analog signal.\
To manage the load, a relay with a normally open switch is connected in series with it. The ATmega 328P only closes the switch when the code determines that proper conditions for powering the load are met, according to the set boundaries. As can be seen in figure \[pcbatmega\], there’s a diode connected to the coil of the relay, and it has the function to prevent that a negative voltage is seen by the pins of the ATmega 328p, scenario that can happen due to a $\frac{di}{dt}$ transient.\
A couple of LEDs are used to indicate if the residential voltage signal is currently meeting the requirements. The red LED indicates that the voltage is beyond the 10% above the 120 VRMS or 10% under 110 VRMS, while the green LED points that the system is providing adequate energy. Adding a small display to indicate the current stats of the residential voltage signal is planned to be added in the future. An alternative design using an Arduino NANO V3 is shown in figure \[pcbnanov3\]:
![Preview of an alternative PCB design that uses an Arduino NANO V3 board[]{data-label="pcbnanov3"}](pcbnanov3.png){width="85mm"}
\
This alternative version works exactly the same than the original one. The main difference lies in the fact that updating the software is a simple and straightforward process in this alternative design, because it only requires using a PC with the Arduino IDE and a mini USB cable. Also, it is capable of transmitting data only using the mini USB cable if desired.
Software
========
Although this device can be connected to others in order to send data, its only function beyond providing output is to manage the connection of the load attached to it, based on the current status of the residential voltage signal it is sampling. Therefore, all the code that it requires to do these tasks has to be run by the ATmega 328P. For this version, all the code used is based on the Arduino platform’s set of C/C++ instructions. This requires flashing to the ATmega 328P the bootloader of Arduino, and the program to be used. There’s two core mechanics present in the software of this device: the data sampling and data processing.\
Data Sampling
-------------
It is based on the code used for sampling in a power quality monitor targeted for Venezuela’s residential voltage [@dape]. The code ensures that sampling occurs at 3600 Hz, which is well beyond the Nyquist frequency for this scenario [@DSP], which would be 120 Hz. It is worth noting that although a much higher sampling rate isn’t required, the more values taken from the residential voltage, more precise the RMS $\frac{1}{2}$ value will be. The code used for this device is planned to be updated in the future, being that the code it’s based on is subject to future updates as well. This is because it is desired to ensure compatibility between this device and the power quality monitor.\
Data Processing
---------------
The data processing in this device is fairly simple, it only requires to recognize when the semi-cycles begin, and to adjust the data accordingly to calculate the RMS $\frac{1}{2}$ value of each semi-cycle. In addition to this, the code that corresponds to the data processing also keeps track of the overall behavior of the RMS $\frac{1}{2}$ values.\
The expression used to calculate the RMS $\frac{1}{2}$ value for each semi-cycle, is the following: $$VRMS\frac{1}{2} = \frac{\sqrt{\sum_{n=1}^{60}V_{n}^{2}}}{60}$$ Where $V_{n}$ corresponds to each sampled voltage, and 60 represents the amount of samples taken per each semi-cycle.\
The code determines the presence of either a voltage sag or surge after the RMS $\frac{1}{2}$ value meets the criteria for either of these disturbances, for 3 consecutive semi-cycles. Once the presence is established the code orders the disconnection of the load, and it doesn’t get reconnected until the RMS $\frac{1}{2}$ values are in compliance with the set boundaries, for 360 consecutive semi-cycles. This way of operation was chosen based on the fact that all types of loads don’t suffer any consequences from sags with a duration under 5 cycles [@PSERC], and because most voltage surges that appear in Venezuela’s electrical system come after voltage sags, as a consequence of the FIDVR phenomenon [@FIDVR]. This ensures that the impact of voltage sags and surges to the load is kept at a minimum, but it also implies that this device is merely a palliative. The abrupt disconnection of an appliance isn’t a harm-free operation, it is just the most desirable option upon a voltage sag or surge under the current scenario that Venezuela is in.\
A graphical representation of the algorithm that governs the sag and surge detection and load management can be seen in figure \[alg\]:
![Algorithm used for detecting voltage sags and surges, and sensing proper conditions for reconnecting the load[]{data-label="alg"}](algorithm.png){width="85mm"}
Conclusions
===========
The design requires the addition of a case in order to be ready for commercial use, although the core mechanics are ensured to work with the current version. To this design can be added any component that can be used for data transmission, which opens up the possibility of building an Internet of Things infrastructure based on it [@ARD]. It can be used even as a sensor for a micro-grid system, due to the scope it can reach. The design is very simple and uses low cost parts.
[00]{}
——, [*National Electrical Code FONDONORMA 200:2004.*]{} CODELECTRA, 2004. ——, [*IEEE Standard 1159-2019.*]{} IEEE, 2019. V. Nundloll, B. Porter, G. Blair, J. Cosby, B. Emmett, B. Winterboum, G. Dean, P. Beattie, R. Shaw, D. Jones, D. Chadwick, M. Brown, W. Shelley and I. Ullah, [*“The Design and Deployment of an End-to-end IoT Infrastructure for the Natural Environment’.*]{} arXiv preprint, arXiv:1901.06270, 2019. D. Pérez, [*“Power Quality Monitor for Residential Voltage”.*]{} arXiv preprint, arXiv:2005.06045, 2020. P. Wilson, [*The Circuit Designer’s Companion.*]{} Newnes, 2013. ——, [*Motorola application note AN1501: Transmission line effects in PCB applications,*]{} Motorola, 1990. R. Morrison, [*Grounding and Shielding Techniques in Instrumentation.*]{} Wiley-Interscience, 1986. N. Ross, [*The Essence of Power Electronics.*]{} Prentice Hall, 1997. J. Goedbloed, [*Electromagnetic Compatibility.*]{} Prentice Hall, 1992. J. Kraus, [*Electromagnetics.*]{} McGraw Hill, 1991. ——, [*megaAVR Data Sheet Rev. A.*]{} Microchip Technology Inc., 2018. S. Smith, [*The Scientist and Engineer’s Guide to Digital Signal Processing.*]{} California Technical Publishing, 1999. G. Karady, S. Saksena, B. Shi and N. Senroy, [*“Effects of Voltage Sags on Loads in a Distribution System”.*]{} Power Systems Engineering Research Center, Cornell University, 2005. A. Gómez and G. Puche, [*“Voltage surges in the Venezuelan occidental electrical system taking into account the fault-induced delayed voltage recovery (FIDVR) phenomenon: A Case Study”.*]{} Universidad Rafael Urdaneta, 2014.
|
---
author:
- Jiyang Chen
- Zhiwei Feng
- 'Jen-Yang Wen'
- 'Bo Liu^\*^ [^1]'
- Lui Sha
bibliography:
- 'ref.bib'
title: 'A Container-based DoS Attack-Resilient Control Framework for Real-Time UAV Systems'
---
Acknowledgment {#acknowledgment .unnumbered}
==============
This project is sponsored in part by NSF 1739732 and by N00014-17-1-2783, and China Scholarship Council under Grant No.: 201706080092. The work was carried out at the Intelligent Robotics Laboratory, Coordinated Science Laboratory, University of Illinois at Urbana-Champaign.
[^1]: ^\*^Bo Liu is now with [NVIDIA]{} Corporation, USA
|
---
abstract: 'Extensive calculations of properties of supernova matter are presented, using the extended Nuclear Statistical Equilibrium model of Ref. [@eNSE_PRC2015] based on a statistical distribution of Wigner-Seitz cells modeled using realistic nuclear mass and level density tables, complemented with a non-relativistic Skyrme functional for unbound particles and beyond drip-line nuclei. Both thermodynamic quantities and matter composition are examined as a function of baryonic density, temperature, and proton fraction, within a large domain adapted for applications in supernova simulations. The results are also provided in the form of a table, with grid mesh and format compatible with the CompOSE platform [@compose] for direct use in supernova simulations. Detailed comparisons are also presented with other existing databases, all based on relativistic mean-field functionals, and the differences between the different models are outlined. We show that the strongest impact on the predictions is due to the different hypotheses used to define the cluster functional and its modifications due to the presence of a nuclear medium.'
address:
- 'National Institute for Physics and Nuclear Engineering (IFIN-HH), RO-077125, Bucharest-Magurele, Romania'
- 'Université de Caen Normandie, ENSICAEN, LPC, UMR6534, F-14050 Caen, France'
author:
- 'Ad. R. Raduta'
- 'F. Gulminelli'
title: Nuclear Statistical Equilibrium Equation of State for Core Collapse
---
equation of state at sub-saturation densities, nuclear statistical equilibrium, core-collapse supernova
Introduction
============
During the in-fall and post bounce stages of the core collapse evolution of massive stars huge domains of density, temperature and charge fraction are explored. Matter consists of baryons, leptons (electrons, positrons, neutrinos and antineutrinos) and photons, and it has a homogeneous/unhomogeneous structure at supra-/sub-saturation densities. Leptons and photons interact weakly and are customarily treated as ideal Fermi and, respectively, Bose gases [@Lattimer_NPA_1985]. Composition and thermodynamics of baryonic matter, generically called equation of state (EoS), is still a matter of research. The reasons rely on both uncertainties related to the effective interactions and difficulties in the modelling. The core-collapse supernova dynamics does not only depend on the EoS, but also, and more importantly, on the progenitor models, on the weak interaction rates (electron capture, $\beta$-decay and neutrino absorption and scattering), and on the modelling of neutrino transport; however all these different aspects are closely inter-correlated, and a reliable modelling of matter composition is very important to limit the propagation of uncertainties.
Because of the complexity of core-collapse supernova dynamics [@Janka_2012], which requires full relativistic hydrodynamics in three dimensions to be coupled with a complete solution of the Boltzmann equation for neutrino transport, energetics and composition of nuclear matter is implemented via external tables covering within a small mesh the huge ranges of thermodynamic conditions explored during the astrophysical evolution. In this way the coupling of very different length scales is avoided and the sensitivity studies on the different ingredients is much simplified.
The first and so far most intensively used equation of state (EoS) models employed simplifying hypothesis on matter composition at densities below the saturation density of symmetric nuclear matter. Indeed, the Single Nucleus Approximation (SNA) on which Refs. [@Lattimer_NPA_1985; @LS_NPA_1991; @HShen_NPA_1998; @HShen_ApJSS_2011] rely, assumes that nuclear matter consists of a homogeneous gas of self-interacting neutrons and protons, a free gas of alpha particles, and a unique cluster of nucleons, all of which in thermal, strong, but not necessarily weak equilibrium. It is analogous to the one-component plasma (OCP) description of the catalyzed crust of neutron stars [@BPS_ApJ_1971], that relies on the energy minimization condition within a solid lattice structure. The nuclear cluster treatment, realized either within the Compressible Liquid Drop Model (CLDM) [@Lattimer_NPA_1985; @LS_NPA_1991] or the Thomas Fermi (TF) approximation [@HShen_NPA_1998; @HShen_ApJSS_2011], allowed to account for in-medium modifications. Coherent use of the same energy density functional for the unbound nucleon gas and the cluster warranted a correct treatment of the transition between unhomogeneous and homogeneous matter, taking place at densities slightly lower than the saturation density, and made possible the first studies on the interplay between star matter EoS and EoS of nuclear matter. It also allows, at least in principle, to encode in this modelling any recent development of effective interactions coming from experimental measurements or ab-initio modelling.
The limit of this approach comes from the fact that most of the time, according to collapse simulations, temperatures are above the crystallization temperature and the minimization of the thermodynamic potential naturally leads to a whole distribution of different nuclear species. The increasing importance of the distribution of nuclear clusters as the temperature increases was acknowledged for the first time in Ref. [@Lattimer_NPA_1985], where approximate formulas have been proposed. The limitations related to SNA have been recently addressed in Refs. [@eNSE_PRC2015; @NSE_SNA]. Ref. [@NSE_SNA] showed that the cluster distributions can be approximated by the most probable nucleus only when they are close to a Gaussian, as it comes out to be the case when one adopts, for the cluster energy functional, a smooth and continuous function [@Botvina_NPA_2010]. However, due to the nuclear shell effects, the nuclear free energy is strongly discontinuous up to temperatures of the order of a few MeV. Under a wide range of thermodynamic conditions, competition between neutron and proton magic numbers leads to multi-modal cluster distributions and, thus, a composition that is very different from the one predicted by the SNA [@NSE_SNA]. While this was argued to only marginally affect the EoS [@Burrows_ApJ_1984], important effects on the electron capture (EC) rates were reported [@Juodagalvis_NPA_2010; @Fischer_EPJA_2014; @Magic_PRC_2016; @Furusawa_PRC_2017]. They are obviously due to the nuclear structure dependence of the weak interaction rates. Considering that EC rates determine the deleptonization dynamics and the global neutrino production [@Lattimer_PhysRep_2000; @Langanke_PRL_2003; @Hix_PRL_2003; @Janka_PhysRep_2007; @Sullivan_ApJ_2016; @EC_PRC_2017], it is easy to understand that it is of key importance to account for realistic nuclear distributions.
SNA-related drawbacks can be overcome within the Nuclear Statistical Equilibrium (NSE) approach [@HNW_AA_1984], valid at $T \gtrsim 0.5$ MeV. In the recent years a number of improved EoS models including the full nuclear distribution has started to become available [@Heckel_PRC_2009; @RG_PRC_2010; @Botvina_NPA_2010; @Hempel_NPA_2010; @Blinnikov_AA_2011; @Furusawa_ApJ_2011; @Hempel_ApJ_2012; @SHF_ApJ_2013; @Furusawa_ApJ_2013; @eNSE_PRC2015; @Furusawa_NPA_2017; @Furusawa_JPG_2017]. Hybrid EoS models that combine, over complementary density-temperature domains, SNA and NSE have been proposed as well [@GShen_PRC_2011; @Schneider_PRC_2017] and their EoS tables are publicly available. At variance with the original NSE [@HNW_AA_1984] model based of the solution of the Saha equations, the recently proposed extended NSE models effectively account for nuclear interactions among the unbound nucleons as well as interactions between the unbound nucleons and nuclei. The interaction among the unbound nucleons is accounted for by the employed energy density functional. The interaction between the unbound nucleons and the nuclei is mimicked via the classical excluded volume approximation, which prevents different species to occupy the same volume. This excluded volume formalism extends to a nuclear distribution the treatment which is employed within the SNA treatment. The effective nuclear interaction being still largely unknown in dense and strongly isospin asymmetric matter, the proposed models span different behaviors in both isoscalar and isovector channel. In addition to this, differences in cluster modelling lead to some model dependence [@Buyukcizmeci_NPA_2013], especially in what regards the chemical composition and the transition to uniform matter. The impact the nuclear EoS has on collapse evolution can only be assessed by performing simulations [@Hempel_ApJ_2012; @Fischer_EPJA_2014; @Furusawa_JPG_2017; @Schneider_PRC_2017]. To this aim a sufficiently large number of EoS tables have to be available.
To contribute to this collective effort of the nuclear astrophysics community, in this paper we present complete EoS tables from the extended NSE model of ref. [@eNSE_PRC2015]. This table, where the energy functional is taken from Skyrme interactions, can be considered as complementary to the existing NSE models which employ relativistic mean field (RMF) parametrizations. Indeed it is well known that relativistic (RMF) and non-relativistic (Skyrme) functionals do not cover the same range of empirical EoS parameters [@Dutra_PRC_2008]. Apart from the choice of the energy functional, other more technical differences exist among the different models, essentially concerning the treatment of clusters which, even if the functional was perfectly known, remains a complex many-body problem only solved within strong approximations [@Buyukcizmeci_NPA_2013]. In the next sections, the different ingredients of the model that can lead to model dependencies are explained in detail, and a throughout comparison is presented with other EoS tables available in the literature.
As a general result, we observe a good agreement on the different thermodynamic quantities despite considerable differences in the composition. We conclude that the EoS uncertainties in supernova modelling essentially concern the approximations of the many-body treatment of nuclear clusters embedded in a nuclear medium, much more than the present uncertainties in the nuclear energy functional.
The model
=========
The extended NSE model was initially proposed in Ref. [@RG_PRC_2010] and subsequently developed in Refs. [@Raduta_EPJA_2014; @eNSE_PRC2015]. Applications on core collapse have been presented in Refs. [@Magic_PRC_2016; @EC_PRC_2017]. The basic idea of the grand-canonical version of the model [@eNSE_PRC2015], used for the generation of the present tables, is to replace the NSE distribution of non-interacting nuclei with a distribution of non-interacting Wigner-Seitz (WS) cells, with appropriate boundary conditions. The distribution of clusters is then obtained by factorizing out of the total partition sum the free nucleon contribution, which is treated as a self-interacting homogeneous gas in the mean-field approximation.
This main simplifying hypothesis of non-interacting WS cells, which is shared with all EoS models we are aware of, means that nuclear and, more important, Coulomb interactions among clusters are completely disregarded. The use of Wigner-Seitz cells as degrees of freedom guarantees that the correct zero temperature limit is properly recovered [@eNSE_PRC2015], that is that the predictions of the extended NSE coincide with the standard minimization of the energy density in the limit of vanishing temperature. Also, this formalism guarantees that the most probable cluster of the distribution exactly coincides with the unique cluster of the SNA approximation. Thus, under the thermodynamic conditions where the SNA approximation is justified, our approach naturally converges to the SNA thus avoiding possible discontinuities when different regimes are matched [@Schneider_PRC_2017]. Within this statistical treatment, the possible model dependence is uniquely due to the expression employed for the free energy of a WS cell, the other equations resulting by general statistical mechanics expressions. The expression of the free energy of a WS cell requires a choice for the energy functional and contains different approximations, which sometimes involve a certain degree of arbitrariness. These approximations are explained in details in this section.
The general derivation of the formalism and the main equations can be found in Refs. [@Raduta_EPJA_2014; @eNSE_PRC2015]. Since modifications and improvements were added during the years, in this section we briefly recall all the main ingredients which are used in the present version of the code.
From distribution of WS cells to distribution of clusters
---------------------------------------------------------
The main hypothesis of all extended NSE models including ours is the absence of nuclear and Coulomb interactions among the different clusters. In this hypothesis, the system in a given thermodynamic condition $(n_B,T,Y_p)$ can be viewed as a collection of non-interacting Wigner-Seitz cells, defined as electrically neutral volumes centered on each cluster. Each Wigner-Seitz cell $(i)$ contains a single cluster and is characterized by a baryon number $A_i$ and atomic number $Z_i$. Because of the boundary conditions, the densities of free electrons ($n_e$) and unbound protons ($n_{gp}$) and neutrons ($n_{gn}$) in the different cells are the same. The cell volume is given by the neutrality condition, $V_i=Z_i/n_e$. The total Helmholtz free-energy of the system in a given configuration $k$ is given by: $$F_{tot}(n_B,T,Y_p)=\sum_i n_i^k F_i,$$ where $n_i^k$ is the number of occurrences of the cell $i$ in the total volume for the configuration $k$, and $F_i$ is the free energy of the cell. The grand-canonical partition sum is computed as: $$Z_{\beta,\mu_n,\mu_p}=\sum_k \exp \left ( -\beta\sum_i n_i^k \left (F_i-\mu_n N_i - \mu_p Z_i \right)\right ),$$ where the sum extends to all possible configurations (or microstates, in statistical mechanics language) and the usual notation $\beta=1/k_B T$ is used. The cell free energy $F_i$ depends on the variables of the cell $A_i,Z_i,V_i$ but also on the densities of the unbound particles $n_e$, $n_{gn}$,$n_{gp}$ which are common to all cells. These quantities are implicitly dependent on the occupations $n_i^k$ through the conservation laws valid for each configuration $k$: $$\begin{aligned}
n_B&=&\frac{1}{V_{tot}}\sum_i n_i^k A_i ,\\
Y_p n_B&=&\frac{1}{V_{tot}}\sum_i n_i^k Z_i=n_e.\end{aligned}$$ For this reason, additional rearrangement terms arise, and the free energy appearing in the probability distributions is given by $F_i + \partial F_{i}/\partial n_i|_{n_j}$. These rearrangement terms amount to shift the chemical potentials with respect to the free system value [@eNSE_PRC2015; @NSE_SNA]. We account for the shift which is trivially due to mass conservation, as it is shown below. This same shift appears in the Hempel and Schaffner-Bielich formalism [@Hempel_NPA_2010] through the excluded volume correction implemented by those authors. In principle, extra rearrangement terms should arise from the explicit density dependence of the cluster functional due to electron polarization effects. To our knowledge, these effects are neglected in all NSE models, including ours.
The cell free energy is rearranged such as to sort out the contribution of a uniform gas of unbound particles as: $$F_i-\mu_n N_i - \mu_p Z_i=F_i^{(e)}-\mu^{(e)}_i + V_i \left ( f_{HM}-\mu_n n_{gn}-\mu_p n_{gp} + f_e-\mu_e n_e\right ).
\label{gi}$$ Here, $f_e(n_e)$ is the free energy density of an ideal gas of electrons, $\mu_e=\partial f_e/ \partial n_e$ and $f_{HM}(n_{gn},n_{gp})$ is the free energy density of homogeneous nuclear matter. Their respective expressions are given by Refs. [@Copperstein_NPA_1985; @LS_NPA_1991] and the standard density functional theory [@Vautherin_NPA1996]. For all the numerical applications presented in this paper and in the associated tables, the Skyrme SLy4 functional [@SLY4] is used, which provides a good description of binding energies and radii of atomic nuclei as well as pure neutron matter as calculated by ab-initio models. Equation (\[gi\]) defines the in-medium modified free energies $F_i^{(e)}$ and chemical potentials $\mu^{(e)}_i$ of the clusters, whose expressions are given in the next section. The introduction of $G_i^{(e)}=F_i^{(e)}-\mu^{(e)}_i$ allows factorizing the partition sum as: $$Z_{\beta,\mu_n,\mu_p}= \left( z_e z_{HM}\right )^{V_{tot}} Z^{cl}_{\beta,\mu_n,\mu_p}, \label{fact}$$ with standard notations for the homogeneous components partition sums, $-k_BT\ln z_{HM}=f_{HM}-\mu_n n_{gn}-\mu_p n_{gp}$, $-k_BT\ln z_{e}=f_{e}-\mu_e n_e$. The cluster partition sum comes out to be identical to the one of the Fisher cluster model [@SMM], where however the vacuum expression for the cluster Gibbs energy is replaced by the in-medium modified one (here noted by the superscript $(e)$) [@eNSE_PRC2015], $$Z^{cl}_{\beta,\mu_n,\mu_p}=\prod_i \sum_{m=0}^{\infty} \frac{ \left[ \exp \left[-\beta G_i^{(e)} \right]
\right] ^m}{m!}=
\prod_i \exp \omega_{\beta,\mu_n,\mu_p} (i),$$ with $ \omega_{\beta,\mu_n,\mu_p} (i)= \exp \left[-\beta G_i^{(e)} \right]$. The statistical average prediction for the number of occurrences of the cell $i$ is then given by: $$\langle n_i\rangle = \frac{\partial Z_{\beta, \mu_n, \mu_p}^{cl}}{\partial \left(\beta\mu_i \right)}
=\omega_{\beta,\mu_n,\mu_p} (i).$$ A given thermodynamic condition in the grand-canonical ensemble $(\beta,\mu_n,\mu_p)$ is associated to well defined values of the unbound components densities $n_{q}$ ($q=gn,gp,e$) as $n_q=k_BT\partial \ln z_q/\ln \mu_q$. Therefore at a given thermodynamic condition there is a one-to-one correspondence between a WS cell $i$ and the cluster species $(A,Z)$ which is present in that cell, implicitly defined by eq.(\[gi\]). This finally gives the cluster distribution as: $$p_i=\frac{\exp \left(-\beta G_i^{(e)}\right)}{\sum_i \exp \left(-\beta G_i^{(e)} \right)}.\label{pnse}$$ The extended NSE numerical code then consists in the self-consistent solution, for a given input set $(T,n_B,Y_p)$, of the coupled equations $\langle V\rangle n_B=\sum_i p_i A_i$, $\langle V\rangle Y_p n_B=\sum_i p_i Z_i$, where $p_i$ is given by eq. (\[pnse\]).
The WS free energy and the definition of e-clusters {#ssec:WScellandcl}
---------------------------------------------------
Let us consider a WS cell of volume $V$ composed of a dense component (or “cluster”) of atomic and mass number $A_r,Z_r$, and a uniform density of unbound protons $n_{gp}$, neutrons $n_{gn}$ and electrons $n_e$. The free energy of the cell is given by
$$\begin{aligned}
F&=& E^{vac}(A_r,Z_r)-TS^{vac}(A_r,Z_r) \\ \label{fcell}
&+&(V-V_0)f_{HM}(n_{gp},n_{gn}) +\delta E_{surf} +\delta E_{Coul} + V f_e(n_e), \nonumber\end{aligned}$$
where $E^{vac}(S^{vac})$ is the vacuum energy (entropy) of the cluster, $V_0$ is the cluster volume related to the average cluster density $n_0$ by $V_0=A_r/n_0$, $\delta E_{Coul}$ is the electron-electron and electron-cluster Coulomb interaction energy and $\delta E_{surf}$ is the modification of the cluster surface energy due to the interaction with the external gas. The reduced volume $\left( V-V_0 \right)$ available to the unbound component accounts for the excluded volume effect [@Hempel_NPA_2010] and is sometimes referred to as “coexisting phase approximation” [@Avancini_PRC_2008; @Avancini_PRC_2009].
To achieve the decomposition of eq.(\[gi\]) we write: $$\begin{aligned}
F&=&E^{(e)}- T S^{(e)}+ V \left ( f_{HM} + f_e \right ), \label{fi} \\
A&=&A_e +(n_{gp}+n_{gn}) V \\
Z&=&Z_e +n_{gp} V\end{aligned}$$ where $A_e (Z_e)$ represent the number of bound nucleons (protons), and $F^{(e)}=E^{(e)}- T S^{(e)}$ gives the cluster free energy, modified by the interactions with the unbound nucleons and electrons.
The cluster chemical potential appearing in eq.(\[gi\]) is thus given by $$\mu_i=\mu_p Z_{i,e} +\mu_n (A_{i,e}-Z_{i,e})$$
We can see from eq.(\[pnse\]) that the equilibrium abundances do not depend on the baryonic and atomic number of the dense component $A_r,Z_r$, but only on its bound part, or “e-cluster” [@Panagiota_PRC2013], given by the left over part of the cluster after subtracting the contribution of the nucleons of the gas they are embedded in [@Panagiota_PRC2013], $$\begin{aligned}
A_e=A_r \left(1-\frac{n_g}{n_0} \right), Z_e=Z_r \left(1-\frac{n_{gp}}{n_{0p}} \right),
\label{eq:AeZe}\end{aligned}$$ where $n_0$ ($n_{0p}$) is the bulk total (proton) density of the cluster. Comparing eq.(\[gi\]) and eq.(\[fi\]) we get for the in-medium modified energy: $$\begin{aligned}
E^e=E^{vac}+\delta E_{bulk}+\delta E_{Coul}+\delta E_{surf},
\label{eq:Ee}\end{aligned}$$ where $\delta E_{bulk}=-\epsilon_{HM} \frac{A_r}{n_0}$ and $\epsilon_{HM}(n_{gn},n_{gp})$ is the energy density of the nucleons in the gas.
We can see that a bulk nuclear energy shift naturally arises from the factorization condition of the partition sum eq. (\[fact\]). This shift is due to the excluded volume appearing in eq. (\[fcell\]), but it can also be physically interpreted as a Pauli blocking shift in the Thomas-Fermi approximation [@Roepke_PRC_2015; @Pais_PRC_2018]: if we consider that the continuum single particle states of the cluster can be approximated by plane waves, such states are occupied by the unbound component and must therefore be excluded in the energy evaluation of the cluster. This approximation to the Pauli-blocking energy shift is only justified for heavy clusters, while the shifts should be calculated microscopically in the case of light particles [@Roepke_PRC_2015]. Such microscopic shifts are included for deuterons, tritons, helions and $\alpha$ particles in the NSE model by Furusawa et al. [@Furusawa_ApJ_2011; @Furusawa_ApJ_2013; @Furusawa_NPA_2017; @Furusawa_JPG_2017]. A phenomenological expression inspired by the microscopic shifts [@Typel_PRC_2010] is used in the gRDF model of Ref. [@Pais-Typel] for all clusters, instead than the excluded volume effect. In this work, we do not attempt to make the distinction between light and heavy clusters, and use the simple excluded-volume shift for all clusters, similar to Ref. [@Hempel_NPA_2010]. For a comparison between the excluded volume mechanism and the gRDF prescription, see Ref. [@Pais-Typel].
The extra in medium surface correction $\delta E_{surf}$ reflects a possible modification of the cluster surface tension due to the presence of an external nucleon gas. Several authors effectively include this effect in the isospin dependence of the surface tension [@LS_NPA_1991; @Steiner_2005; @Newton_ApJSS_2013]. However more complex dependencies on both cluster size and composition and gas are observed in self-consistent Thomas-Fermi calculations in beta-equilibrium matter [@Centelles_NPA_1998; @Douchin_NPA_2000; @Sil_PRC2002]. Inspired by these self-consistent calculations, simple analytic expressions are proposed of an in-medium modification of the surface tension [@Pais_PRC_2016; @Grams_PRC_2017]. In the context of NSE models, an explicit correction depending both on the gas density and on the temperature is introduced in the NSE model by Furusawa et al. [@Furusawa_ApJ_2011; @Furusawa_ApJ_2013; @Furusawa_NPA_2017; @Furusawa_JPG_2017]. Thorough investigation of this aspect is under work and will be published elsewhere. As such, the approximation $\delta E_{surf} \approx 0$ will be done through this paper. The Coulomb energy shift due to electron screening is treated in our model, as frequently done in the literature, within the Wigner-Seitz approximation, $$\delta E_{Coulomb}=a_c f_{WS} Z_r^2/A_r^{1/3},
\label{eq:ECoulomb}$$ with $$f_{WS}=\frac32 \left[\frac{n_{e}}{n_{0p}} \right]^{1/3}
-\frac12 \left[ \frac{n_{e}}{n_{0p}}\right],$$ and the Coulomb parameter $a_c=0.69$.
For the energy functional of the clusters in vacuum, $E^{vac}$, the following choices are made. For nuclei for which experimental masses are known, the AME2012 mass tables of Audi [*el al.*]{} [@Audi_2013] are used. Then, up to the drip lines, evaluated masses of the 10-parameter model by Duflo and Zuker [@DZ10], here after referred to as DZ10, are employed. Beyond drip lines, nuclear binding energies are described according to the Liquid Drop Model (LDM)- like parametrization of Ref. [@Danielewicz], which accounts for isospin effects in the surface tension including the effect of extra neutrons in the neutron skin [@Steiner_2005]. This expression is modified in two respects. First, a phenomenological pairing term, $\Delta(A)=\pm 12/\sqrt{A}$, where $+(-)$ corresponds to even-even (odd-odd) nuclei, is added. Then, two correction terms are included such as to smoothly match, for each isotopic chain, the liquid-drop predictions with the limiting values of DZ10. Based on Hartree-Fock calculations liquid-drop parameters are proposed in Ref. [@Danielewicz] for several dozens of Skyrme effective interactions. For the sake of consistency of the energy functional, we use the parametrization corresponding to the same effective interaction as the one employed for describing the homogeneous gas, namely SLy4 [@SLY4]. Inclusion of nuclei beyond drip-lines is motivated by the fact that, in medium, other species than those existing in vacuum may exist [@Sil_PRC2002] and accounting for them might, in principle, modify the sharing of matter between clusterized and homogeneous components as well as isotopic abundances. The allowed mass range of clusters is $2 \leq A \leq 300$. Note that, at low temperatures and densities close to the transition to homogeneous matter, larger structures can be formed if the allowed mass range is extended accordingly [@Furusawa_ApJ_2011]. Their existence is nevertheless much dependent on the in-medium surface modification of the energy functional and additional shape degrees of freedom, all of which poorly known.
The bulk total $n_0$ (proton $n_{0p}$) cluster density is taken to be the total (proton) number density of saturated nuclear matter of isospin asymmetry $\delta=\left(1-2 n_{0p}/n_0 \right)$ . It can be expressed [@Panagiota_PRC2013] as a function of the saturation density of symmetric matter $n_0(0)$ by the equation, $$n_0(\delta)=n_0(0) \left(1-\frac{3 L_{\rm{sym}} \delta^2}{K_{\rm{sat}}+K_{\rm{sym}} \delta^2} \right),
\label{eq:rho0_delta}$$ where $L_{\rm{sym}}$, $K_{\rm{sym}}$ and $K_{\rm{sat}}$ represent the slope and curvature of the symmetry energy $J_{\rm{sym}}$ and, respectively, the incompressibility of nuclear matter, all calculated for symmetric saturated matter.
Due to skin effects, the bulk isospin asymmetry $\delta$ obviously differs from the total isopin asymmetry, $(1-2Z_r/A_r)$. For the case of a nucleus in the vacuum (where $Z_r=Z_e$, $A_r=A_e$), the following analytic expression has been derived within the liquid drop model [@LD_NPA1980], $$\delta=\delta_0=\frac{1-\frac{2Z_e}{A_e}+\frac{3a_c}{8Q} \frac{Z_e^2}{A_e^{5/3}}}{1+\frac{9 J_{\rm{sym}}}{4Q} \frac{1}{A_e^{1/3}}},
\label{eq:deltabulk_vacuum}$$ where contributions of the symmetry energy, surface stiffness and Coulomb are readily identified in addition to the size dependence. $Q$ represents the surface stiffness coefficient. For the general case of a nucleus in a dilute medium, of interest here, we employ the expression proposed in Ref. [@Panagiota_PRC2013], $$\delta(n_g,n_{gp})=\left[1-\frac{n_g}{n_0(\delta)} \right]\delta_0(Z_e,A_e)+
\frac{n_g}{n_0(\delta)}\left( 1-\frac{2n_{gp}}{n_g}\right).
\label{eq:deltabulk_medium}$$
The entropy term includes both translational and internal degrees of freedom, $$S^{(e)} \left(A_r,Z_r,n_g, n_{gp} \right)= \ln V + \ln c_{T}(A_r,Z_r)
+\frac32 \ln A_e,$$ where $c_{T}=g_T(A_r,Z_r)(mT/(2\pi\hbar^2))^{3/2}$, $m$ denotes the mass of a nucleon and the internal state partition sum is: $$g_T(A_r,Z_r)=\sum_{i, discrete} \left(2 J_i+1 \right) \exp \left(-E^*/T \right)+
\int_{(cont)} dE^* \rho_{A_r,Z_r}(E^*) \exp \left ( -E^* /T \right ).
\label{eq:degen}$$
Different corrections must be applied to the cluster entropy in order to avoid double counting with the gas states [@few; @tubbs; @rauscher; @Bonche_NPA_1984; @Bonche_NPA_1985]. First, the translation term must only be computed for the bound part $A_e$ of the cluster. Moreover, the gas states should be subtracted from the internal state partition sum. The simplest approach [@few] consists in cutting the partition sum at the nucleon separation energy, such as to exclude all continuum states. This is a crude treatment, while a thermodynamically consistent subtraction of the gas partition sum was proposed in Ref. [@Bonche_NPA_1984; @Bonche_NPA_1985; @tubbs; @rauscher]. If the interaction in the gas is neglected, the subtraction can be done analytically, and it was shown in Refs. [@tubbs; @rauscher] that the simple approach of Ref. [@few] underestimates the partition sum, which can be easily understood because the resonant states are also cut in that procedure. However, an exact state counting in the self-consistent mean-field approximation in the case of nuclei beyond the dripline at zero temperature was presented in Ref. [@Panagiota_PRC2013], showing that the correct number of particles is obtained for the cluster if all the continuum states are excluded, and the contribution of resonant states is relatively small. Because of the complexity of the issue, in this work we stick to the simplest procedure of cutting the internal energy partition sum at the minimum between the average neutron and proton separation energies, $\langle S \rangle =\min \left( \langle S_n \rangle, \langle S_p \rangle\right)$.
The full list of low-lying resonances for nuclei with $4 \leq A \leq 10 $ has been considered. For the level density $\rho(A_r,Z_r)(E^*)$ we used the realistic expression of Ref. [@Bucurescu2005], fitted on experimental data. For more details see Ref. [@eNSE_PRC2015].
Thermodynamic quantities
------------------------
The total and proton number densities are given by: $$\begin{aligned}
n_B&=&n_{g}+\frac1V \sum_{A_r,Z_r} A_e n(A_r,Z_r)\\
Y_p n_B&=&n_{gp}+\frac1V \sum_{A_r,Z_r} Z_e n(A_r,Z_r).
\label{eq:conserv}\end{aligned}$$
Clusterized phase pressure, entropy density and internal energy density may be readily calculated from their thermodynamic definitions. The following expressions are obtained for the pressure, $$p_{cl}=T \frac{\partial \ln Z_{\beta,\mu_n,\mu_p}^{cl}}{\partial V}=\frac TV \ln Z_{\beta,\mu_n,\mu_p}^{cl}=
\frac TV \sum_{A_r,Z_r} n(A_r,Z_r),
\label{eq:pcl}$$ entropy density, $$\begin{aligned}
s_{cl}&=&\frac 1V \frac{\partial(T \ln Z_{\beta,\mu_n,\mu_p}^{cl})}{\partial T} \nonumber \\
&=&
\frac 1V \sum_{A_r,Z_r} n(A_r,Z_r) \left( \frac52 +T \frac{\partial \ln g_T(A_r,Z_r)}{\partial T}
+\frac{E^e(A_r,Z_r)-\mu_n (A_e-Z_e)- \mu_p Z_e}{T}
\right),
\label{eq:scl}\end{aligned}$$ and internal energy density, $$e_{cl}=\frac 1V \sum_{A_r,Z_r} n(A_r,Z_r) \left(\frac32 T+\langle E^*(A_r,Z_r)\rangle+ E^e (A_r,Z_r)
\right),
\label{eq:ecl}$$ where the average excitation energy of the cluster $(A,Z)$ is, $$\langle E^*(A_r,Z_r)\rangle=\frac{\int_{0}^{S(A_r,Z_r)} dE^* E^* \rho_{A_r,Z_r}(E^*) \exp(-E^*/T)}{\int_{0}^{S(A_r,Z_r)} dE^* \rho_{A_r,Z_r}(E^*) \exp(-E^*/T)}
=T^2 \frac{\partial \ln g_T \left(A_r,Z_r \right)}{\partial T}.$$
Baryonic pressure, entropy and energy densities are obtained by summing up clusterized and homogeneous phases contributions, $$\begin{aligned}
p_B&=&p_{cl}+p_g \nonumber \\
s_B&=&s_{cl}+s_g \nonumber \\
e_B&=&s_{cl}+e_g.\end{aligned}$$ Note that free volume corrections do not appear explicitly, as in Ref. [@Hempel_NPA_2010], as they have been already taken into account in the definition of e-clusters.
Total pressure, entropy and energy densities are obtained by adding to the baryonic quantities the contributions of the electron (including contribution of positrons) and photon gases. Analytic expressions for the electron gas with temperature larger than 1 MeV and the photon gas are given in Ref. [@LS_NPA_1991]. Expressions for the electron gas at $T \lesssim 1$ MeV have been proposed in Ref. [@Copperstein_NPA_1985]. The total values of the fundamental thermodynamic quantities thus write, $$\begin{aligned}
p_T&=&p_B+p_{el}+p_{\gamma}-\sum_{A,Z} n(A,Z) a_c \frac{Z^2}{2 A^{1/3}}
\left[ \frac{n_{el} A}{n_0(\delta) Z}- \left( \frac{n_{el} A}{n_0(\delta) Z}\right)^3\right], \nonumber \\
s_T&=&s_B+s_{el}+s_{\gamma}, \nonumber \\
e_T&=&e_B+e_{el}+e_{\gamma}.\end{aligned}$$ We note the extra negative pressure term coming from the Coulomb lattice and remind that lattice contribution to energy and entropy enters in eqs. (\[eq:ecl\]) and (\[eq:scl\]) via the Coulomb energy shift, eq. (\[eq:ECoulomb\]).
Transition to homogeneous matter
--------------------------------
At densities of the order of $n_0/2-2 n_0/3$ the nonuniform nuclear matter phase is replaced by a uniform phase which persists up to several times the value of symmetric saturated nuclear matter density. Physically this transition occurs in order to minimize the system free energy. The exact value of the transition density and the way in which it takes place, [*i.e.*]{} via phase coexistence or not, depend on a number of issues as effective interactions, shape degrees of freedom and in-medium surface modification of the energy functional, which make the phenomenology strongly model dependent.
For the purpose of building an EoS database suitable for astrophysics use, the details of the transition are less important than the thermodynamic stability and consistency. As such, for fixed values of $Y_p$ and $T$ the clusterized phase is computed, as described in the above sections, up to maximum density where the NSE procedure still converges, typically $4 \cdot 10^{-2} - 9 \cdot 10^{-2}$ fm$^{-3}$. Homogeneous matter is supposed to onset, independently on temperature and proton fraction, at $n_t=10^{-1}$ fm$^{-3}$. For intermediate values of density, chemical composition and thermodynamic observables are computed by linear interpolation between the boundary values.
Results {#section:results}
=======
EoS databases are usually delivered as three dimensional tables of baryon number density, charge fraction and temperature. In order to emphasize the transition from clusterized matter to homogeneous matter or in medium cluster dissolution matter, composition and thermodynamic quantities are most frequently plotted and discussed as a function of baryon number density at fixed values of the charge fraction and temperature [@Hempel_NPA_2010; @Furusawa_ApJ_2011; @GShen_PRC_2011; @Furusawa_ApJ_2013; @Furusawa_NPA_2017; @Furusawa_JPG_2017; @Schneider_PRC_2017]. Evolution as a function of temperature when the values of baryon number density and charge fraction are fixed is considered mainly when the focus is put on the effects of in-medium interactions, as is the case of cluster mass fractions from Ref. [@Buyukcizmeci_NPA_2013; @Pais-Typel]. As the whole body of literature shows, in the sub-saturation domain and for moderate values of the charge fraction, thermodynamic quantities bear little sensitivity to the employed effective interactions, nuclear cluster definition, approximations (NSE vs SNA) and degree of sophistication of the approaches. This is due to the fact that the thermodynamic variables of a clusterised medium are largely determined by the properties of nuclei in vacuum, which are relatively well known and upon which all effective interactions have been fitted. The situation is different for matter composition. Effects of the cluster definition in terms of maximum size, isospin asymmetry and excitation energy, density dependence of the symmetry energy, in-medium interactions of light nuclei [@Pais-Typel], evolution of shell effects far from stability [@Magic_PRC_2016] or temperature [@Furusawa_NPA_2017] etc. have been identified. Though systematic analyses are not available yet, these effects are expected to influence the core collapse evolution via the neutrino opacity, electron capture rates and energy dissipation of the shock wave.
Two representations are used in this work for investigating the results of our NSE model. For general survey of global chemical and energetic behaviors (subsections \[ssec:compo\] and \[ssec:thermo\]), we prefer plots as a function of charge fraction for constant values of the two remaining grid parameters, $T$ and $n_B$. This view offers technical advantages. First, it allows to straightforwardly see the effect of $Y_p$. Then, for sufficiently different values of $T$ and $n_B$, the curves are outdistanced, which enhances the plots readability. When confronting our results with those of other models in the literature (subsection \[ssec:comparison\]) we prefer the investigation as a function of baryonic number density when $T$ and $Y_p$ are fixed. Different thermodynamic conditions are considered: in subsections \[ssec:compo\] and \[ssec:thermo\] we focus on $T$ and $n_B$-values under which the employed microphysics hypothesis and approximations are well justified. In subsection \[ssec:comparison\] we focus on states populated in proto-neutron stars and late stage evolution of core-collapse, where the differences in microphysics treatments can be maximized.
Composition {#ssec:compo}
-----------
![Average neutron (1st row panels) and proton (2nd row panels) numbers of nuclei with $A \geq 20$ as a function of proton fraction for fixed values of the baryon number density and temperature, see legend. The 3rd and 4th row panels give, under the same conditions, the standard deviation of the proton number, $\sigma_{Z_{heavy}}$, and, respectively, the fraction of mass bound in these nuclei. $N$ and $Z$ magic numbers are marked on the 1st and, respectively, 2nd row panels by horizontal dotted lines. The values are given at the r.h.s. of the right Y-axis. []{data-label="fig:NhZh"}](AhZh_vs_Yp.eps){width="99.00000%"}
Figs. \[fig:NhZh\] and \[fig:Xnuc\] illustrate matter composition as a function of $Y_p$ for three representative values of the baryon number density, $n_B=10^{-6}$, $10^{-4}$ and $2 \cdot 10^{-2}$ fm$^{-3}$ and four values of the temperature, $T=0.7$, 1, 4 and 9 MeV.
Fig. \[fig:NhZh\] depicts the average neutron and proton numbers of clusters of mass number $A \geq 20$ (dubbed as “heavy clusters” in the following) together with the standard deviation of the proton number, $\sigma_{Z_{heavy}}$, and fraction of mass bound in these nuclei, $X_{heavy} = \sum_{A \geq20,Z} A n(A,Z)/n_B$. The standard deviation of the neutron number, $\sigma_{N_{heavy}}$, (not plotted) shows features similar to those of $\sigma_{Z_{heavy}}$.
As one may notice, for all thermodynamic conditions under which they are produced, heavy cluster population is determined by the competition between neutron and proton magic numbers. At low densities and temperatures (e.g. $n_B=10^{-6}, 10^{-4}$ fm$^{-3}$ and $T$=0.7, 1 MeV) most abundant nuclei have neutron and proton numbers close to $N=8, 20, 28, 50$ and, respectively, $Z=8, 20, 28$. The relatively small values of $\sigma_{Z_{heavy}} \lesssim 5$ are explained by the small difference between the two most frequently competing $Z$-magic numbers, 20 and 28, and the dominance of one of the peaks. Given the increased variety of competing $N$-magic numbers, slightly larger values characterize $\sigma_{N_{heavy}}$. As the density increases, much more numerous and massive clusters are populated up to higher temperatures. Depending on thermodynamic conditions, including the proton fraction, the abundance peaks are the result of competition between a broad range of $N$ and $Z$ magic numbers: $N$ =8, 20, 28, 50, 82, 126, 184 and $Z$=8, 20, 28, 50, 82, 114 (the last superheavy shell closure $N=184,Z=114$ obviously depends on the mass model employed, here DZ10 [@DZ10]). This feature explains the large values of both $\sigma_{Z_{heavy}}$ and $\sigma_{N_{heavy}}$ as well as their steep evolution with $Y_p$. Nuclide abundances dominated by magic numbers and, consequently, large values of the standard deviation of heavy cluster mass distributions have been already signaled in Ref. [@Furusawa_ApJ_2011].
Whether heavy clusters bind a significant amount of matter or not depends on thermodynamic conditions (bottom panel of Fig. \[fig:NhZh\]): around isospin symmetry and for the lowest values of temperature $X_{heavy}$ exhausts a large fraction of matter even at densities as low as $10^{-6}$ fm$^{-3}$. On the contrary, at high temperatures or in very neutron-rich matter practically no heavy cluster exists.
![Upper (lower) panels: neutron (deuteron) (thin lines) and proton ($\alpha$-particle) (thick lines) mass fraction as a function of proton fraction for fixed values of the baryon number density and temperature, see legend.[]{data-label="fig:Xnuc"}](Xnucdalpha_vs_Yp.eps){width="99.00000%"}
The mass fractions of neutrons, protons and the two most abundant light clusters, $d$ and $^4$He, are plotted in Fig. \[fig:Xnuc\]. The top panels of this figure give mass fractions of unbound neutrons and protons. Both $X_p$ and $X_n$ show strong sensitivities to $T$ and $n_B$. At high temperatures, unbound nucleons exist over large domains of density and proton fraction and their abundances monotonically increase (decrease) with $Y_p$. At variance, at low $T$-values unbound neutrons (protons) exist only in neutron (proton)-rich matter. At the highest density, the change of slope of the neutron fraction is due to the abrupt appearance of the heavy clusters (see Fig.\[fig:NhZh\] above), which are close to isospin symmetry. $d$ and $^4$He abundances, represented in the bottom panels of Fig. \[fig:Xnuc\], present a complex and non-trivial evolution with temperature, baryonic number density and proton fraction. For the lowest considered density, $n_B=10^{-6}$ fm$^{-3}$, $^4$He is produced only at the lowest temperatures. For $T= 0.7$ MeV, $^4$He exists only in isospin symmetric matter and the associated mass fraction barely exceeds 13%. The $T= 1$ MeV results show that, quite interesting, $^4$He exists even in very asymmetric matter. Indeed, at $Y_p \approx 0.1$, $X_{\alpha} \approx 10\%$ while values as high as $X_{\alpha} \approx 80\%$ are attained for $Y_p \approx 0.5$. For the other two values of baryonic density, significant $\alpha$-production occurs at higher temperatures, e.g $T$=9 MeV for $n_B=2 \cdot 10^{-2}$ fm$^{-3}$, extends over limited domains of $Y_p$ and gets maximized in symmetric matter. $d$-production shows features qualitatively similar to those of $\alpha$-production. Quantitatively, higher densities and temperatures are required to produce $d$ than $\alpha$. One may notice that, under specific thermodynamic conditions, the loosely bound $d$ may dominate over the strongly bound $^4$He, as already observed in previous works [@Pais_PRC_2018; @Typel_PRC_2010; @Avancini2012; @Sedrakian_2017]. This is the case of $T=4$ and 9 MeV and $n_B=10^{-4}$ fm$^{-3}$ and $T=9$ MeV and $n_B=2 \cdot 10^{-2}$ fm$^{-3}$, irrespective the value of $Y_p$. As for $\alpha$, for all considered $n_B$ and $T$-values, the mass fraction of the isospin symmetric deuteron gets maximized in symmetric matter.
![Domains in $(n_B,Y_p)$ (top left), $(n_B,T)$ (top right) and $(S_B,Y_p)$ (bottom left) where the mass fraction of $^2$H, $^3$H, $^4$H, $^5$H, $^6$H and $^4$He and $^5$He exceeds certain values, see legend.[]{data-label="fig:HeandHe"}](HandHe_v3.eps){width="99.00000%"}
Population of $d$ and $^4$He is further considered in Fig. \[fig:HeandHe\] together with that of other isotopes of H and He in terms of mass fraction as a function of $T$, $n_B$, $Y_p$ and baryonic entropy per baryon $S_B$. The different contours correspond to the mass fraction thresholds shown in the legend. We note that by far the most abundant light cluster is $^4$He. Its mass fraction exceeds 70% over considerable domains of baryonic density ($10^{-12} \lesssim n_B \lesssim 10^{-4}$ fm$^{-3}$) and proton fraction ($0.35 \lesssim Y_p \lesssim 0.6$) for temperatures up to a couple MeV ($0.3 \lesssim T \lesssim 2$ MeV), which correspond to entropies per baryon ranging from 3 to 10. Mass fractions larger than 10% are obtained for $^5$He, $^4$H, $^5$H, $^6$H for $T \gtrsim 2$ MeV, $n_B \gtrsim 10^{-4}$ fm$^{-3}$ over various domains of $Y_p$, which correspond to few $S_B$. As easy to anticipate, extreme neutron-rich nuclei ([*e.g.*]{} $^5H$, $^6H$) are preferentially produced in neutron-rich environments ($Y_p \lesssim 0.3$). Indeed, as already observed in Ref. [@eNSE_PRC2015; @Burrello], in very asymmetric matter and beta-equilibrium matter at high density and temperature, heavy hydrogen and helium isotopes strongly dominate over the isospin symmetric $d$ and $^4$He.
Thermodynamic quantities {#ssec:thermo}
------------------------
Baryonic and total pressure, entropy and energy per nucleon are shown as a function of $Y_p$ in Figs. \[fig:baryon\_esp\] and \[fig:total\_esp\]. The same thermodynamic conditions as in Figs. \[fig:Xnuc\] and \[fig:NhZh\] are considered. Note that the baryonic pressure already includes the contribution of the Coulomb lattice. For the sake of convenience the internal energy per nucleon is scaled and shifted by the nucleon mass. The presently considered EoS model shows features similar to those of other SNA and NSE models in the literature. Roughly speaking, two domains may be identified based on the relative dominance of homogeneous or clusterized matter. Whenever matter is mainly composed of free protons and neutrons, e.g. low densities and/or high temperatures and/or extreme values of the proton fraction, the baryonic pressure scales with density and temperature. When, at the contrary, the thermodynamic conditions are such that an important amount of matter is bound in nuclei, the pressure decreases. Even negative values may be reached, due to the Coulomb lattice. This behavior is easy to understand considering the proportionality between the baryonic pressure and the total multiplicity per unit volume, see eq. (\[eq:pcl\]).
The baryonic entropy per nucleon shows a similar behavior. As expected, it increases with increasing temperature and decreasing $n_B$ as more nucleons and light clusters are populated, that is the effective number of degrees of freedom increases. The most significant decrease of $s_B/n_B$ arises in symmetric matter at low temperatures, where matter almost entirely consists of almost isospin symmetric, and thus strongly bound, massive nuclei. For low $T$-values, the high sensitivity of isotope population to the global proton fraction, shown in Fig. \[fig:NhZh\], leads to a wobbling behavior of $s_B(Y_p)$. At variance with this, dominance of nucleons and light clusters at high temperatures makes $s_B$ independent on $Y_p$.
The energy per baryon roughly replicates the $T$- and $n_B$-dependencies of $s_B/n_B$. This fact is easy to understand considering the similarity of Eqs. (\[eq:ecl\]) and (\[eq:scl\]) and can be summarized as follows. Dominance of heavy clusters, that occurs at low temperature and/or $Y_p \approx 0.5$ and/or high densities, is signaled by small or negative values of $e_B/(n_B m_n)-1$ while dominance of nucleons and light clusters gives high values of the considered quantity.
Accounting for electron and photon contributions leads to a global increase of the pressure, energy and entropy. This is shown in Fig. \[fig:total\_esp\]. We can see that the highest effects concern the pressure and the internal energy per baryon at high densities. With the addition of the electron and, to a less extent, photon contributions, the total pressure is always positive meaning that a baryonic matter which would be unstable if no leptons were present, is stabilized by the presence of the electrons. This is the physical reason behind the well-known quenching of the nuclear liquid-gas phase transition in stellar matter [@RG_PRC_2010]. The effect on $e_B/(n_B m_n)-1$ is less spectacular and trivially due to the increase of electron energy with density.
![Baryonic+lattice pressure (top), entropy per baryon (middle) and shifted energy per baryon (bottom) as a function of proton fraction for fixed values of the baryonic number density and temperature, see legend.[]{data-label="fig:baryon_esp"}](baryon_esp_vs_Yp.eps){width="99.00000%"}
![Total pressure (top), entropy per baryon (middle) and shifted energy per baryon (bottom) as a function of proton fraction for fixed values of the baryonic number density and temperature, see legend.[]{data-label="fig:total_esp"}](total_esp_vs_Yp.eps){width="99.00000%"}
Baryon and charge chemical potentials, equal by definition to the neutron chemical potential ($\mu_B=\mu_n$) and, respectively, the difference between proton and neutron chemical potentials ($\mu_Q=\mu_p-\mu_n$), are plotted in Fig. \[fig:mu\] after subtracting the neutron mass and, in the case of $\mu_B$, applying a constant shift for better visibility. The same thermodynamic conditions as in Figs. \[fig:Xnuc\]-\[fig:total\_esp\] are considered. The monotonic decrease (increase) of $\mu_B$ ($\mu_Q$) with $Y_p$ at constant $n_B$-values and the evolution of $\mu_B$ with respect to $n_B$ are trivially due to the proportionality between one species chemical potential and its density, in stable nuclear matter. The decrease of $\mu_B$ as a function of $T$ recalls the usual behavior of an ideal gas. On top of these expected behaviors, the heavy clusters component is responsible for non trivial small scale variations of $\mu_B$ and $\mu_Q$.
![Scaled and shifted baryon chemical potential and scaled charge chemical potential as a function of proton fraction for fixed values of the baryonic number density and temperature. Legend as in Fig. \[fig:baryon\_esp\].[]{data-label="fig:mu"}](muBmuQ_vs_Yp.eps){width="99.00000%"}
Comparison with other NSE models for the EoS {#ssec:comparison}
--------------------------------------------
### General survey {#sssec:general}
As already mentioned, detailed microphysics input in NSE modelling, e.g. the nuclear matter energy functional and the cluster definition in terms of maximum baryonic number, isospin asymmetry and excitation energy, were proven to have little impact on thermodynamic quantities such as pressure, energy and entropy densities and chemical potentials. At variance, a certain sensitivity was identified on the matter composition, namely the sharing between homogeneous and clusterized components, as well as the elemental and isotopic abundances. Two examples in this sense are offered by the inner crust of proto-neutron stars and the central element of the core during the late stages of the collapse, when $Y_p$-values as low as 0.1 are attained. In the first case the temperatures are of the order of few MeV and the baryonic number densities are around $n_0/2-2 n_0/3$. In the later case, $n_B \lesssim n_0$ and $ T \approx 20$ MeV. Under these circumstances, other nuclei than those existing in the vacuum are expected to be produced, notably extremely neutron rich nuclei, and even nuclei beyond drip-lines. Because of the lack of experimental information on these exotic species, the predictions become strongly model dependent.
In order to quantify the model dependence we compare the predictions of the present model, which, in addition to AME2012 [@Audi_2013], employs the DZ10 [@DZ10] mass model prolonged, for larger isospin asymmetries, with a SLy4-based LDM parametrization [@Danielewicz], and introduces a cut-off on the internal state density to avoid double counting with the nuclear gas (see \[ssec:WScellandcl\]), with those of other two NSE models in literature. The first considered EoS model is HS DD2 [@Hempel_NPA_2010] which, in addition to a number of light species (d, t, $^3$He, $^4$He), accounts for 8979 nuclei ranging from $^{16}$O to $^{339}$136 and extending from the proton drip line to the neutron drip line. Binding energies are implemented according to the Finite Range Droplet Model (FRDM) mass model [@FRDM_1997]. The second EoS model is FYSS [@Furusawa_NPA_2017]. It allows for nuclei far beyond drip lines and nuclei whose proton number may reach values as high as 1000 units. Binding energies are calculated via a LD-parametrization supplemented by phenomenological (temperature-dependent) shell corrections and in-medium modification of the surface energy. Concerning the unbound nucleon interactions, these models employ RMF functionals, DD2 [@Typel_PRC_2010] (HS DD2) and TM1 [@TM1] (FYSS). Both HS DD2 and FYSS are available on the CompOSE [@compose] database.
The mass fractions of unbound neutrons and protons, scaled and shifted baryon chemical potential, scaled charge chemical potential and total pressure, energy and entropy per nucleon are displayed in Figs. \[fig:comparison\_1\] and \[fig:comparison\_2\] as a function of the baryonic number density for $T=2, 5, 10, 20$ MeV and $Y_p=0.1$. The mass sharing between nuclear clusters is not represented since the information is not available for HS DD2 and FYSS. The neutron mass fraction shows a strong sensitivity to the details of the models, especially at low temperatures. For instance at $T=2$ MeV and $5 \cdot 10^{-3} ~ {\rm fm^{-3}} \lesssim n_B \lesssim n_t$ HS DD2 predicts between 44% and 86% and more neutrons than the present model. For $T=5$ MeV, the difference amounts to 66% around $n_B=3 \cdot 10^{-2} ~ {\rm fm}^{-3}$. The explanation relies in the much limited isospin asymmetry allowed for clusters in HS DD2. The maximum dispersion among the different NSE models concerning $X_p$ is obtained at $T=10$ MeV and $1 \cdot 10^{-2} \lesssim n_B \lesssim n_t$, with a maximum ratio $X_p^{HS DD2} /X_p^{present} \approx 4$. Concerning the neutrons, the difference would be strongly reduced if we would count the extra neutrons beyond the dripline in vacuum as free neutrons. Given their small values, the differences in what concerns $X_p$ are not expected to significantly influence the electron capture rates and, thus, affect the collapse. The baryon and charge chemical potential show different features: $\mu_b$ shows no sensitivity to the details of the models while $\mu_q$ shows a certain sensitivity, especially at the highest considered densities, where differences up to 50% are obtained. Fig. \[fig:comparison\_2\] shows that the other three thermodynamic observables, $p/n_B$, $s/n_B$ and $e/n_B m_n -1$, are mostly sensitive to microphysics input at $2 \lesssim T \lesssim 10$ MeV and $10^{-3} ~ {\rm fm^{-3}} \leq n_B \leq n_t$. The much more significant differences that occur at densities higher than the transition density to homogeneous matter are due to the different EoS and are not relevant for the NSE treatment.
![(1st and 2nd row panels) Unbound neutrons and protons mass fractions and (3rd and 4th row panels) scaled baryon and charge chemical potentials as a function of $n_B$ for $T=2, 5, 10, 20$ MeV and $Y_p=0.1$. The predictions of the present model are confronted with those of HS DD2 [@Hempel_NPA_2010] and FYSS [@Furusawa_NPA_2017].[]{data-label="fig:comparison_1"}](comparison_vs_nB_1.eps){width="99.00000%"}
![The same as in Fig. \[fig:comparison\_1\] for total pressure per nucleon, entropy per nucleon and energy per nucleon. The last quantity is additionally shifted and scaled with the neutron mass.[]{data-label="fig:comparison_2"}](comparison_vs_nB_2.eps){width="99.00000%"}
### Detailed chemical composition
Before the first extended NSE models were developed, less than ten years ago, astrophysical simulations typically modeled the chemical composition using the Saha equation, which neglects interactions among unbound nucleons and between nucleons and nuclear clusters. At low densities, this is certainly a very good approximation. At high densities excluded volume effects become important but, along a core collapse trajectory, their effect tends to be somehow limited by the increasing temperature, which favors the production of lighter nuclei.
To quantify the effect of extended NSE, we compare in Fig. \[fig:nse\_saha\] the predictions of the present EoS model with those of the Saha equation, $$n(A,Z)=g_T(A,Z) \left( \frac{M_{A,Z} T}{2 \pi \hbar^2}\right)^{3/2}
\exp\left[\frac{\left( A-Z\right) \mu_n + Z \mu_p - M(A,Z)+\delta E_{Coulomb}}T \right],$$ where $M(A,Z)$ stands for the nuclear mass and the same clusters definition has been employed. The thermodynamical conditions, $n_B=6.1 \cdot 10^{-3}$ fm$^{-3}$, $Y_p=0.26$ and $T$=3.25 MeV, correspond to the most compressed state of the central element of mass $0.01M_{\odot}$ of a 25$M_{\odot}$-progenitor, whose in-fall evolution was followed in Ref. [@Juodagalvis_NPA_2010]. The entropy per baryon equals $\approx 1.7$. The mass distribution predicted by the Saha-equation presents the same features as the NSE-distribution, including the multi-peak structure induced by shell effects. Quantitative differences are nevertheless apparent: the most abundant masses have multiplicities that might differ by one order of magnitude when one switches from one model to the other and the “valley” between to peaks gets correspondingly reduced/enhanced, because of conservation laws.
![Atomic mass number distribution for the last point in the in-fall evolution of a 25$M_{\odot}$-progenitor considered in Ref. [@Juodagalvis_NPA_2010]. The thermodynamic conditions, mentioned on the figure, correspond to the central element of mass $0.01M_{\odot}$. Predictions of the present EoS model care confronted with those of the Saha equation.[]{data-label="fig:nse_saha"}](NSE_vs_Saha_T=3.27_Yp=0.256.eps){width="70.00000%"}
As already discussed in Ref. [@Buyukcizmeci_NPA_2013], the differences among NSE model predictions essentially stem from the different fragment definition. To get extra insight on the issue we consider here the effect of different assumptions made on the maximum allowed excitation energy $E^*_{max}$ ([*i.e.*]{} the upper limit of the sum/integral in Eq. (\[eq:degen\])), and on the level density. Thermodynamic conditions similar to those in Fig. \[fig:nse\_saha\] are considered. For nuclear binding energies we use experimental data [@Audi_2013] and FRDM [@FRDM_1995] predictions for $2 \leq A \leq 15$ and, respectively, $A \geq 16$. To keep the modelling as simple as possible, here we restrict ourselves to the predictions of the Saha equation. The following cases are investigated: (1) $E^*_{max}=\min(S_n, S_p)$, as assumed in Ref. [@eNSE_PRC2015] and in the present EoS model, and $\rho_{A,Z}(E^*)$ of Ref. [@Bucurescu2005], (2) $E^*_{max}=B(A,Z)$, where $B(A,Z)$ is the binding energy as in ref.[@Hempel_ApJ_2012], and $\rho_{A,Z}(E^*)$ of Ref. [@Bucurescu2005], (3) $E^*_{max}=B(A,Z)$ and $\rho_{A,Z}(E^*)$ as in Eq. (3) of Ref. [@Hempel_NPA_2010]. The results are plotted in Fig. \[fig:saha\_ingredients\], together with the predictions (4) of the extended NSE (DD2-FRDM) model of Ref. [@Hempel_NPA_2010], as available at [@Hempel_Basel]. Several observations are in order. Allowing for high excitation energies washes out to a large extent the staggering due to odd-even and shell structure effects, and also favors heavy nuclei with higher state density. Different level density formulas lead to different abundancies. Finally, the curves (3) and (4) compare Saha and extended-NSE. In all cases, magic nuclei with $Z=28$, $N=50$ and $N=82$ lead to abundance peaks, though their heights prove very sensitive to the working hypotheses.
![Proton (left) and neutron (right) number distributions of nuclei produced under thermodynamic conditions similar to those of Fig. \[fig:nse\_saha\]. Predictions of Saha equation (1-3) corresponding to different nuclear cluster definition (see text) are confronted with those (4) of Ref. [@Hempel_NPA_2010]. In all cases experimental data and predictions of FRDM are used for nuclear masses.[]{data-label="fig:saha_ingredients"}](Saha_ingredients_T=3.27_Yp=0.256.eps){width="99.00000%"}
Conclusions
===========
In this paper we have presented extensive calculations of the composition of supernova matter and its thermodynamic properties, in the framework of the extended NSE model of Ref. [@eNSE_PRC2015], whose main formalism is recalled. The corresponding EoS database covers large intervals of temperature $0.3 \leq T \leq 50$ MeV, baryonic density $10^{-12} \leq n_B \leq 1.5~{\rm fm}^{-3}$, and proton fraction $0.01 \leq Y_p \leq 0.6$, with a discretization adapted for direct applications in supernova simulation. This database is provided in the form of tables following the standards of the CompOSE database [@compose] (see the Appendix). The present NSE-EoS database is complementary to the other, already available, NSE-EoS databases on the CompOSE platform. Indeed it is the first complete NSE table based on a non-relativistic Skyrme energy functional, and as such it can be used to quantify the effect of the EoS model dependence on the supernova dynamics. A detailed comparison with other available models shows that the most important differences among the present modellings concern the treatment of the cluster functionals, and this model dependence is more important than the one due to the value of the EoS empirical parameters explored for instance in Ref. [@Fischer_EPJA_2014]. Some of the different treatments can be easily justified. This is notably the case of the nuclear pool considered in the probability distribution. The same is true for the cluster functional, which must be sophisticated enough to include realistic shell effects, which are the essential ingredient that determine the nuclear distributions, and therefore the electron capture rates. Other differences between the existing models are more subtle and concern the many-body treatment inside the WS cell. In particular, the in-medium energy shifts of the light clusters and the in-medium modification of the surface tension are important and physically sound ingredients, which however are difficult to implement consistently in a NSE model. As a consequence, different models use phenomenological parametrizations which contain a certain degree of arbitrariness. This aspect deserves further progress for a model-independent treatment of the microphysics of supernova dynamics.
Appendix: The CompOSE database
==============================
The CompOSE database, available in the public domain at http://compose.obspm.fr, is a repository of EoSs for astrophysics purposes. Detailed thermodynamic and composition data are provided in standardized formats as three dimensional arrays as a function of $T$, $n_B$ and $Y_p$. Mash points on these quantities are specified in the eos.t, eos.nb, eos.yp files. The domains covered by the presently discussed EoS model and the number of points are given in Table \[tabel:griddetails\].
$T$ $n_{B}$ $Y_p$
------------------ ---------- ---------------------- -------
number of points 120 140 59
minimum 0.3 MeV 10$^{-12}$ fm$^{-3}$ 0.01
maximum 50.0 MeV 1.5 fm$^{-3}$ 0.60
: Domains of temperature, baryonic number density and charge fraction covered by the present EoS database and the corresponding numbers of mash points. []{data-label="tabel:griddetails"}
The thermodynamic quantities, stored in eos.thermo, are: pressure divided by baryon number density $p/n_B$ \[MeV\], entropy per baryon $s/n_B$, scaled and shifted baryon chemical potential $\mu_B /m_n-1$, scaled charge chemical potential $\mu_Q /m_n$, scaled electron chemical potential $\mu_{el} /m_n$, scaled and shifted free energy per baryon $f /(n_B m_n )-1$ and scaled and shifted energy per baryon $e/(n_B m_n )-1$. $m_n$ is the nucleon mass, specified in eos.thermo.
The composition data, stored in eos.compo, consists in the particle fractions of neutrons ($n_n/n_B$) and protons ($n_p/n_B$) together with those, $n(A,Z)/n_B$, of the at maximum 500 most probable nuclides whose multiplicity per unit volume is not less than $\left(f_{lim} Y_{max} \right)$, where $Y_{max}$ is the multiplicity per unit volume of the most abundant nucleus with $A \geq 2$. For $f_{lim}$, over complementary domains, two values are used: $10^{-5}$ and $10^{-8}$. Note that, because of excluded volume effects, mass and charge conservation equation are expressed in terms of e-clusters, as already specified in eqs. (\[eq:conserv\]).
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been partially funded by Pharos, COST Action CA16214. Ad. R. R. acknowledges useful discussions with Micaela Oertel and kind hospitality from LUTH-Meudon.
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https://astro.physik.unibas.ch/people/matthias-hempel/equations-of-state.html
|
---
abstract: 'Generative models for source code are an interesting structured prediction problem, requiring to reason about both hard syntactic and semantic constraints as well as about natural, likely programs. We present a novel model for this problem that uses a graph to represent the intermediate state of the generated output. Our model generates code by interleaving grammar-driven expansion steps with graph augmentation and neural message passing steps. An experimental evaluation shows that our new model can generate semantically meaningful expressions, outperforming a range of strong baselines.'
author:
- |
Marc Brockschmidt, Miltiadis Allamanis, Alexander Gaunt\
Microsoft Research\
Cambridge, UK\
`{mabrocks,miallama,algaunt}@microsoft.com`\
Oleksandr Polozov\
Microsoft Research\
Redmond, WA, USA\
`polozov@microsoft.com`\
bibliography:
- 'bibliography.bib'
title: '[Generative Code Modeling with Graphs]{}'
---
Introduction {#sec:introduction}
============
Background & Task {#sec:task}
=================
Graph Decoding for Source Code {#sec:models}
==============================
Related Work {#sec:relatedWork}
============
Evaluation {#sec:evaluation}
==========
Discussion & Conclusions {#sec:conclusion}
========================
|
=23.8cm
Introduction
============
The $S=1$ bilinear-biquadratic chain $$H =\sum_{i} \left[\cos\theta\,
{\rm\bf S}_i {\rm\bf S}_{i+1}
+ \sin\theta\, ({\rm\bf S}_i {\rm\bf S}_{i+1})^2\right]
\label{spin-1}$$ is one of the prototype models for the physics of Haldane gap[@Hal] antiferromagnets. Its zero temperature phase diagram has been the subject of intensive studies in recent years.[@Aff]$^-$[@Sch-Mut] By now it is well-established that the energy gap (Haldane gap) persists in a wide range $-\pi/4<\theta<\pi/4$ around the conventional Heisenberg point $\theta=0$. The model is gapless at the special points[@TakBabSuth] $\theta=\pm\pi/4$ beyond which other phases with qualitatively different physical properties appear. Although the gap and the hidden (string) order[@denNijs] characterizing the Haldane-gap systems persist for the whole Haldane phase $-\pi/4<\theta<\pi/4$, one can divide this interval into (at least) two, somewhat different subphases.[@Bur-Xia-Ger; @Sch-Jol-Gar] These subphases are separated by the so called valence-bond-solid (VBS) point[@AKLT] $\theta_{\rm vbs}=\tan^{-1}{1/3}\approx 0.1024\pi$ where the ground state properties can be obtained exactly. The two subphases differ in the form of the long distance asymptotics of the two-point correlation function $\langle S^z_i S^z_{i+n} \rangle$. In the “commensurate” Haldane phase (C-phase) for $-\pi/4<\theta<\theta_{\rm vbs}$ the leading behavior is expected to be $$\langle S^z_i S^z_{i+n} \rangle \sim (-1)^n\frac{e^{-n/\xi}}{n^{1/2}}
\qquad \mbox{for $n\to\infty$},
\label{G_c}$$ while in the “incommensurate” Haldane phase (IC-phase) for $\theta_{\rm vbs}<\theta<\pi/4$ this was predicted to take the form $$\langle S^z_i S^z_{i+n} \rangle
\sim \frac{e^{-n/\xi}}{n^{1/2}} \cos (q n +\phi)
\qquad \mbox{for $n\to\infty$}
\label{G_ic}$$ where $q=q(\theta)\in (\pi,2\pi/3)$ is a $\theta$-dependent incommensurate wavenumber, $\phi$ is a phase shift, and $\xi=\xi(\theta)$ is the correlation length. At the “C-IC transition point” (also called the “disorder point”) $\theta_{\rm cic}=
\theta_{\rm vbs}$ the correlation function is known rigorously and it is purely exponential without any algebraic prefactor $$\langle S^z_i S^z_{i+n} \rangle =
\frac{4}{3} (-)^n e^{-n/\xi}- \frac{2}{3}\delta_{n0},
\label{G_VBS}$$ with $\xi(\theta_{\rm vbs})=1/\ln 3\approx 0.9102$. Correlation functions of some other operators can also be studied rigorously at the VBS point.[@Sch-Mut] In particular, $\langle (S^z)^2_i (S^z)^2_{i+n} \rangle$ has a similar purely exponential decay with $\xi=1/\ln 3$ at $\theta=\theta_{\rm vbs}$.
The commensurate-incommensurate (C-IC) transition of the spin-1 chain bears much similarity to C-IC transitions found in other models, e.g., in the anisotropic 2D Ising model on the triangular lattice at finite temperature, where it can be analyzed rigorously using the exact solution.[@Stephenson] Other, higher dimensional, not exactly solvable models with disorder points were investigated using an RPA approach to the susceptibility in Ref. \[Gar-Mai\]. In general, C-IC transitions can be divided into two categories (two kinds). In this classification scheme the transition at the VBS point of the $S=1$ chain is a C-IC transition of the [*first*]{} kind, with the property that the incommensurate wavenumber $q$ in the IC regime is parameter dependent. For C-IC transitions of the first kind the correlation length is predicted to behave on the C and IC sides of the disorder point $\theta_{\rm cic}$ as $$\left. \frac{d\xi}{d\theta} \right|_{\rm C}=-\infty, \qquad
\left. \frac{d\xi}{d\theta} \right|_{\rm IC}={\rm finite}
\label{xi_limit}$$ with $\xi(\theta_{\rm cic})\ne 0$ at the transition point. The characteristic wavenumber is expected to vary on the IC side as $$q(\theta) - q(\theta_{\rm cic}) \sim |\theta-\theta_{\rm cic}|^{1/2}.
\label{q_limit}$$ Moreover, the RPA theory[@Gar-Mai] also predicts the change of the asymptotic form of the correlation functions at the disorder point: the algebraic prefactor is $n^{-(D-1)/2}$ in $D$ dimensions except at the disorder point where $D\to D^\prime$, $D^\prime<D$ should be taken, reflecting a “dimensional reduction”. In our case $D=1+1=2$, $D^\prime=1$ as is seen in Eqs. (\[G\_c\]-\[G\_VBS\]). These general features of the C-IC transitions of the first kind have been tested numerically for the spin-1 chain, and except for some observed deviation from the predicted form in Eq. (\[G\_ic\]) slightly above $\theta_{\rm vbs}$, all were justified.[@Sch-Jol-Gar] Note that around the VBS point finite size corrections are very small, thus the numerical results (exact diagonalization and DMRG) are extremely precise.
Within the IC subphase some other special points can be defined. The first one is $\theta_{\rm disp}\approx 0.1210\pi$, where the second derivative of the magnon dispersion at $k=\pi$ vanishes, and the dispersion becomes quartic. When the chain is subject to a uniform magnetic field the magnetization-vs-field curve has an anomalous, non-square-root-like singularity at this point.[@Oku-Hie-Aku] For $\theta>\theta_{\rm disp}$, the second derivative at $k=\pi$ is negative and the gap takes its minimum value for a momentum different from $k=\pi$.[@Gol-Jol-Sor] In this region a strong enough field causes the Haldane gap to collapse into a [*two*]{}-component Luttinger liquid (LL) phase instead of a conventional one-component LL phase.[@Fat-Lit] The next special point is $\theta_{\rm max}\approx 0.123\pi$ where the Haldane gap takes its maximum value. Note that this point has less physical relevance, since it strongly depends on the actual definition of the Hamiltonian; a $\theta$-dependent rescaling of the model can easily change its location. Finally one can define the Lifsitz point $\theta_{\rm Lifs}\approx 0.1314\pi$, where the incommensurability manifests itself in the structure factor. This is different from the disordered point in massive models due to the finite linewidth of the peaks.[@Sch-Jol-Gar] Figure \[fig:specpoints\] shows these special points.
=
In both the C and IC Haldane phases the energy spectrum consists of a discrete triplet branch of “magnons” separated by finite gaps from the singlet ground state, and also from a continuum of higher lying “multi-magnon” excitations in a wide momentum range. This one-magnon branch is clearly discernible around the edges of the Brillouin zone $k=\pm\pi$. However, it merges into the continuum and vanishes due to magnon scattering processes in an extended range around $k=0$. The energy spectrum, especially in the Heisenberg point and the VBS point was studied numerically by many authors.[@Sch-Mut; @Whi-Hus; @Fat-Sol-CM; @Aro-Aue-Hal] All concluded that the magnon-magnon interactions are rather weak, and bound states do not play a role at low energies. Many properties of the system, from ground-state correlation functions[@Whi-Hus; @Sor-Aff] to the onset of magnetization in uniform fields[@Oku-Hie-Aku; @Sak-Tak; @Aff-Sor] can be extremely well approximated using a massive relativistic free boson theory. Such an approximate theory can be derived directly from the non-linear $\sigma$-model (NL$\sigma$M) description of the spin chain at the Heisenberg point,[@NLSM] or from the Majorana fermion representation of the integrable Takhtajan-Babujian model in the vicinity of $\theta=-\pi/4$.[@Tsvelik] However, all these microscopic theories utilize the [*a priori*]{} assumption that the important low-energy fluctuations are at momenta $k=0$ and $\pi$, and thus they cannot account for the C-IC transition, nor can they give any reliable description of the IC regime.
Although we do not have a rigorous microscopic justification all results we possess are consistent with the assumption that the elementary excitations are essentially free bosons in the IC regime, too. Thus the aim of the present paper is to extend the free boson description to the whole Haldane phase, and give a general theory which is capable of accounting for the C-IC transition in simple terms. In the lack of a detailed microscopic formulation, however, this theory, at present, is only phenomenological.
Effective theory - continuum version
====================================
A continuum field theory to describe Heisenberg antiferromagnetic chains ($\theta=0$) with integer $S$ was developed by Haldane.[@Hal] This is the nonlinear $\sigma$-model (NL$\sigma$M) – without topological terms – which can be derived in the large $S$ limit, but whose implications are believed to hold for $S=1$ too. The NL$\sigma$M is defined by the Lagrangian[@NLSM] $${\cal L}=\frac{1}{2g}\left[{1\over v}
(\partial_t{\vec\phi})^2 -v(\partial_x{\vec\phi})^2\right],
\label{Lagr-NLsigmaM}$$ where ${\vec\phi}(x,t)$ is a vector field with unit length ${\vec\phi}^2=1$, $v$ is a nonuniversal constant (velocity) setting the energy scale, and $g=2/S$ is the coupling constant. The associated Hamiltonian is $${\cal H}=\frac{v}{2}\left[
g {\vec{l}}^2 +{1\over g}(\partial_x{\vec\phi})^2\right],
\label{Ham-NLsigmaM}$$ where the momentum canonically conjugate to ${\vec\phi}$ turns out to be $${\vec{l}}(x,t) \equiv {1\over gv} \left[{\vec\phi}\times\frac{\partial{\vec\phi}}{\partial t}
\right](x,t).
\label{l-term}$$ The spin operator ${\vec{S}}_n$ can be expressed in terms of ${\vec\phi}$ and ${\vec{l}}$ as $${\vec{S}}_n(t) = (-1)^n S {\vec\phi}(x_n,t) + \delta x\, {\vec{l}}(x_n,t)
\label{S-phi-l}$$ where $x_n=n\delta x$, $\delta x$ being the lattice constant (usually set to unity), and $S=1$ in the present case. At the mean field level the NL$\sigma$M can be well approximated by a massive, essentially free vector-boson theory\cite{} $${\cal L}={1\over v}
(\partial_t{\vec\phi})^2 -v(\partial_x{\vec\phi})^2 - m^2 {\vec\phi}^2,
\label{relat-Lagr}$$ now without any constraint on the ${\vec\phi}$ field. The boson field ${\vec\phi}$ varies smoothly on the scale of the lattice constant in the commensurate regime (the pure Heisenberg model, which the theory applies for, is in the C phase) and thus higher derivatives, neglected in Eq. (\[relat-Lagr\]), are indeed small. The Lagrangian gives rise to a relativistic dispersion $\omega(k)=
\sqrt{m^2+v^2 k^2/2m}$ which takes its minimum at $k=0$. \[Note the factor $(-1)^n$ introduced in Eq. (\[S-phi-l\]) shifting all momenta by $\pi$.\]
The NL$\sigma$M summarized above gives a good description of the low-energy behavior of the $S=1$ Heisenberg chain, but is unable to describe the C-IC transition in the bilinear-biquadratic chain. The key feature missing is that in the vicinity of our special points the shape of the magnon dispersion changes drastically, and the minimum at $k=\pi$ ($k=0$ in the momentum-shifted boson language) splits. Obviously, such an effect can never be obtained from a relativistic field theory such as Eq. (\[relat-Lagr\]) and we need to consider a non-relativistic model. The simplest continuum Lagrangian with this property is $$\begin{aligned}
{\cal L} &=&
(\partial_t{\vec\phi})^2 -a\, (\partial_x{\vec\phi})^2
-b\,(\partial_x^2{\vec\phi})^2 - m^2 {\vec\phi}^2.
\label{Lagr}\end{aligned}$$ At this level $a$, $b$, and $m$ are $\theta$-dependent phenomenological parameters. For stability $b$ is supposed to be positive, and $a$ is assumed to change sign at the special point $\theta_{\rm disp}$. \[Note that in the relativistic case $b=0$, and $a$ is in fact $v^2$, where $v$ is the “speed of light”. Formally taking $a<0$, which we will consider in the sequel, implies an imaginary $v$. This only means, however, that one should redefine the physical (quasi)particles to live around the two new minima of the dispersion. In fact, one should introduce two new particles, now with $v=$real, for the two minima. This immediately leads to a two-particle (two-band) description which was analyzed in detail in Ref. \[Fat-Lit-98\] for the massless case (above the critical magnetic field) and was shown to result a two-component (two-band) Luttinger liquid there. Here we shortcut these complications by simply assuming that the two particles are two chiral components ($k<0$ and $k>0$) of the same boson with $a<0$.\] Since our phenomenological model is not derived directly from the microscopic Hamiltonian in Eq. (\[spin-1\]) the connection between the boson field and the original spin variables is not known rigorously. In principle ${\vec{S}}_n$ can be expanded in powers of the field ${\vec\phi}$ and its derivatives, and any term permitted by the symmetries of the model can appear. Being guided by the NL$\sigma$M description at the Heisenberg point, in the following we will assume that the first two terms (linear and quadratic in ${\vec\phi}$) are $${\vec{S}}_n(t) = g_\phi (-1)^n{\vec\phi}(x_n,t) +
g_l {\vec{l}}(x_n,t)
+{\cal O}(|{\vec\phi}|^3),
\label{Szphi}$$ where $g_\phi$ and $g_l$ are unknown constants. Note that the ${\vec{l}}$ term, defined by Eq. (\[l-term\]), is the most relevant 2-boson term which is even under parity and odd under time reversal as it should be. Although, we cannot exclude the possibility that up to ${\cal O}({\vec\phi}^2)$ some other terms with higher derivatives also appear, such extra derivatives, due to the finite mass gap, would not modify the long distance asymptotics of $\langle {\vec{S}}_0 {\vec{S}}_n \rangle$. Equation (\[Szphi\]) shows that in the C phase where spin-spin correlations are antiferromagnetic $\phi$ varies smoothly, and higher order derivatives in the Langrangian have minor role. Around the C-CI transition point, and in the IC-phase, however, this is no longer the case: as ${\vec{S}}_n$ picks up incommensuration, ${\vec\phi}(x)$ must do so, as well. It no longer varies smoothly, and higher derivatives in the Lagrangian cannot be neglected. This is the effect which we try to take into consideration by the $b$ term in Eq.(\[Lagr\]).
The asymptotic behavior of the correlation function $\langle S^z_i S^z_{i+n}
\rangle$ is determined by the one-boson term; the two-boson term and in general any multi-boson terms only constitute minute corrections which decay at least twice as rapidly. Thus, up to the two-boson term in Eq. (\[Szphi\]) the spin-spin correlation function has the behavior $$\langle S_n^z(t) S_0^z(0)\rangle = g_\phi^2 (-)^{n} G_\phi(x_n,t) +
g_l^2 G_l(x_n,t),
\label{corr}$$ with $$\begin{aligned}
G_\phi(x,t) &\equiv& \langle \phi^z(x,t)\phi^z(0,0) \rangle , \\
G_l(x,t)&\equiv& \langle l^z(x,t) l^z(0,0) \rangle .\end{aligned}$$ These are the quantities we will now calculate.
The Euler-Lagrange equation associated with our Lagrangian gives a generalized Klein-Gordon equation for each component $\alpha=x,y,z$ of ${\vec\phi}$ $$\partial_t^2\phi^\alpha - a\, \partial_x^2\phi^\alpha
+b\, \partial_x^4\phi^\alpha+ m^2\,\phi^\alpha =0,$$ whose Green’s function is $$G_\phi(\omega,k) = \frac{1}{\omega^2-a k^2-b k^4-m^2 +i\varepsilon}.$$ This defines the non-relativistic dispersion $$\omega(k)=\sqrt{m^2+a k^2+b k^4},
\label{disp}$$ which reduces to the relativistic dispersion of Ref. \[Sor-Aff\] when $b=0$.
The generalized theory can be quantized identically to the relativistic Klein-Gordon theory.[@Kak] The field $\phi^\alpha$, $\alpha=x,y,z$, has the following mode expansion $$\phi^\alpha(x,t)=\int \frac{dk}{\sqrt{4\pi\omega(k)}}
\left[d^\alpha_k e^{i{\bf K\cdot X}}+
d^{\alpha\dag}_k e^{-i{\bf K\cdot X}}\right]
\label{mode}$$ where ${\bf K\cdot X}=\omega t-k x$, and the normalization is $[d^\beta_k,d^{\alpha\dag}_{k^\prime} ]=\delta_{\beta\alpha}\delta(k-k')$. Using the mode expansion the equal time expressions $G_\phi(x)$ and $G_l(x)$ can be easily reduced to Fourier transforms[@Sor-Aff] $$\begin{aligned}
G_\phi(x) &=&
\int \frac{dk}{4\pi} \frac{e^{ikx}}{\omega(k)}, \nonumber\\
\label{phiphi} \\
G_l(x) &=& {1\over 2} \int \frac{dk'}{4\pi} \omega(k')e^{ik'x}
\int \frac{dk}{4\pi} \frac{e^{ikx}}{\omega(k)} -{1\over 2}\delta^2(x).
\nonumber\end{aligned}$$ Note that the integral determining $G_\phi$ also appears as a multiplicative factor in $G_l$.
Since the asymptotic behavior is determined by $G_\phi$, we start our analysis with this. The evaluation of the Fourier transform starts with locating the zeros of $\omega(k)$ in the complex plane. The four zeros are given by $$\label{zeros}
k=\pm
\left[{1\over 2b}\left(-a\pm\sqrt{a^2-4m^2b}\right)\right]^{1\over 2}$$ and depend on the single parameter $\theta$, via $a$, $b$ and $m$. Since our phenomenological model is not derived directly from the microscopic one, we have to make several assumptions about the $\theta$-dependence of $a$, $b$ and $m$. We will assume that this dependence is smooth (analytical), $m(\theta)$ and $b(\theta)$ are nonnegative whereas $a$ decreases with increasing $\theta$ and changes sign at $\theta_{\rm disp}$. We introduce the discriminant $$D(\theta)=a^2-4m^2b,
\label{discriminant}$$ which is hence an analytic function of $\theta$, and we suppose that it is positive for $\theta<\theta_{\rm vbs}$, vanishes at the VBS point, and it is negative from $\theta_{\rm vbs}$ to $\theta=\pi/4$ where it vanishes again. As we show below, under the above hypotheses the phenomenological model provides the expected asymptotic behavior of the correlation function in all parts of the Haldane phase.
=
At this point $a>0$ and $D=0$. The expression under the square-root in $\omega(k)$ is a complete square, thus $\omega$ becomes quadratic in $k$ and has two purely imaginary zeros, $\pm i\sqrt{2m^2/a}=\pm i\sqrt{a/2b}$ \[see Fig. \[fig:cuts\](b)\]. The contour of integration can be closed in the upper half plane and the result is $$\label{VBS}
G_\phi(x)={1\over 2\sqrt{2a}}e^{-\sqrt{a\over 2b}|x|}\ .$$ Because of the purely exponential decay, this indeed corresponds to $\theta=\theta_{\rm vbs}$.
In this region $a>0$ and $D>0$. We may suppose $b>0$; the case when $b=0$ can be obtained by continuity. We get four purely imaginary zeros $\pm iv_\pm$ where $$\label{vpm}
v_\pm=
\left[{1\over 2b}\left(a\pm\sqrt{D}\right)\right]^{1\over 2}\ .$$ Now $$\omega(k)=\sqrt{b}(k^2+v_+^2)^{1\over 2}(k^2+v_-^2)^{1\over 2}$$ is single-valued in the complex plane with two cuts, one between $-iv_+$ and $-iv_-$ and another one between $iv_-$ and $iv_+$ \[see Fig. \[fig:cuts\](a)\]. (We use the convention that $z^{1/2}$ has a cut along the negative real axis.) The contour of integration can be closed in the upper half-plane and drawn onto the upper cut, giving $$\label{AF}
G_\phi(x)={1\over 2\pi\sqrt{b}}\int_{v_-}^{v_+}{e^{-t|x|}\over
\left[(t^2-v_-^2)(v_+^2-t^2)\right]^{1\over 2}} dt \ .$$ This function is positive for all $x$, so with the factor $(-1)^n$ introduced in Eq. (\[Szphi\]) we obtain the expected antiferromagnetic modulation. As $|x|$ goes to infinity, the main contribution to the integral is coming from the vicinity of the end point $v_-$, and it is legitimate[@Watsonlemma] to expand the integrand around this point. For $|x|>|D|^{-1/2}$ this yields $$\label{AFasy}
G_\phi(x)={e^{-v_-|x|}\over 2[2\pi v_-\sqrt{D}]^{1\over 2}}
\left[|x|^{-{1\over 2}}+O(|x|^{-{3\over 2}})\right]\ .$$ Together with Eq. (\[Szphi\]), we find the expected asymptotic form Eq. (\[G\_c\]) of the spin-spin correlation function with a correlation length $\xi=1/v_-$ which is continuous at $\theta=\theta_{\rm vbs}$.
Here D becomes negative and $\omega$ has four complex zeros $k_j=\pm u\pm iv$ ($j=1,\ldots,4$) with $$\label{uv}
u=\left[{m\over 2\sqrt{b}}-{a\over 4b}\right]^{1\over 2}
\qquad
v=\left[{m\over 2\sqrt{b}}+{a\over 4b}\right]^{1\over 2} \ .$$ If we write $\omega$ in the form $$\omega(k)=\sqrt{b}\prod_{j=1}^4(k-k_j)^{1\over 2}$$ we see that it is single-valued on the complex plane with a cut between $-u-iv$ and $u-iv$ and another cut between $-u+iv$ and $u+iv$ \[see Fig. \[fig:cuts\](c)\]. For $k$ real we get back the original positive function. The integration can be carried out along a contour which starts at $-u+i\infty$, goes vertically down to $-u+iv$, passes below the upper cut and goes vertically to $u+i\infty$. Next, we replace the integral below the cut by an integral going above the cut and in the opposite sense, and this latter by the sum of the two integrals along the vertical half-lines. These are complex conjugate to each other, so finally we obtain $$\begin{aligned}
\label{IC}
G_\phi &&(x)= \\
&& {1\over\pi\sqrt{b}}\int_v^\infty{e^{-|x|t}\cos[u|x|-\varphi(t)]\; dt\over
\sqrt{t^2-v^2}[4u^2+(t-v)^2]^{1\over 4}[4u^2+(t+v)^2]^{1\over 4}},
\nonumber\end{aligned}$$ where $$\varphi(t)={1\over 2}\left(\arctan{t-v\over 2u}+\arctan{t+v\over 2u}\right).$$ Now $G_\phi(x)$ changes sign periodically, and we can identify $\pi-u$ with the wave number $q$ of the incommensurate oscillation. At the VBS point $u=0$, and as it is shown by Eq. (\[uv\]), the assumed analyticity of $m$, $a$ and $b$ assures that it has a square-root-type singularity above the VBS point in accordance with Eq. (\[q\_limit\]). The large-$|x|$ asymptotics of $G_\phi(x)$ can be obtained by Watson’s lemma[@Watsonlemma] or by a direct expansion, $$\begin{aligned}
\label{ICasy}
G_\phi &&(x)= \\
&& {e^{-v|x|}\cos(u|x|-{1\over 2}\arctan{v\over u}) \over
(2\pi)^{1\over 2} \left(\frac{-mD}{\sqrt{b}}\right)^{1\over 4}}
\left[|x|^{-{1\over 2}}+O(|x|^{-{3\over 2}})\right].\nonumber
$$
Formulas (\[IC\]) and (\[ICasy\]) apply for any $\theta$ between $\theta_{\rm vbs}$ and $\pi/4$, including $\theta_{\rm disp}$ which is a symmetry point of the domain of incommensurate oscillations ($u=v$). As $\theta$ approaches $\pi/4$, $D$ goes to zero and the zeros of $\omega(k)$ tend to the real axis. Thus, $v$ goes to zero and the correlation length diverges. This is what we expect at the boundary of the Haldane phase $\theta=\pi/4$ where the gap disappears. At this special point the ground state has a tripled periodicity,[@FS123; @TakBabSuth] implying $u(\theta=\pi/4)=2\pi/3$.
For [*any fixed*]{} $x$, $G_\phi(x)$ depends analytically on $\theta$ inside the whole Haldane phase. This can be seen from the original form Eq. (\[phiphi\]) of $G_\phi(x)$ by inserting the original, non-factorized expression Eq. (\[disp\]) for $\omega(k)$. A proof can be found in \[\[Tit\]\]. The argument makes use of the continuity of the integrand in $k$ real, its (supposed) analyticity for any fixed $k$ as a function of $\theta$ in suitable complex domains, and the uniform convergence of the integral for $\theta$ in any of these domains. It is interesting to examine another kind of asymptotics, valid in a close neighborhood of $\theta_{\rm vbs}$, when $|D|/a^2\ll 1$. In this case Eq.(\[AF\]) reduces to the form $$\begin{aligned}
G_\phi(x) &\approx& {e^{-\sqrt{a\over 2b}|x|}\over \sqrt{8a}}{1\over \pi}\int_0^\pi
\cosh\left[\sqrt{D\over 8ab}|x|\sin\alpha\right] d\alpha \nonumber\\
&\equiv&
{e^{-\sqrt{a\over 2b}|x|}\over \sqrt{8a}}
I_0\left(\sqrt{D\over 8ab} |x|\right) \label{approx}\end{aligned}$$ where $I_0$ is the zeroth order Bessel function. We can arrive at the same equation from Eq. (\[IC\]), by changing the contour of integration (integrating around the upper cut). Now analyticity at the VBS point is manifest, because in the expansion of the hyperbolic cosine about zero only the even powers of $\sqrt{D}$ appear. Equation (\[approx\]) shows that the crossover to the decay with the $|x|^{-{1\over2}}$ prefactor sets in at the characteristic distance $|x|\sim D^{-{1\over 2}}$, which diverges at the VBS point. This explains the numerical difficulties[@Sch-Jol-Gar] verifying the expected asymptotic behavior very close to the VBS point.
At the VBS point there is an infinite jump in the derivative of the correlation length, as predicted by Eq. (\[xi\_limit\]). Indeed, Eq. (\[vpm\]) yields $$\left.{d\xi\over d\theta}\right|_{\theta_{\rm vbs}-0}
\left.\approx {D'\over4\sqrt{2a}m}\right|_
{\theta_{\rm vbs}}D^{-{1\over 2}}
\sim -(\theta_{\rm vbs}-\theta)^{-{1\over2}}$$ because $D'(\theta_{\rm vbs})<0$. On the other hand, from Eq. (\[uv\]) $$\left.{d\xi\over d\theta}\right|_{\theta_{\rm vbs}+0}
=\left.-{1\over2\sqrt{a}}\left({m'\over 2m}+{a'\over 2a}+{m^2\over
a^2}\right)\right|_{\theta_{\rm vbs}},$$ which is finite. The singularity of the correlation length at $\theta_{\rm vbs}$ is in no contradiction with the analyticity of $G(x)$ at a [*fixed*]{} $x$. Indeed, the divergence of the derivative of $\xi$ was extracted from the single-exponential asymptotic form Eq. (\[AFasy\]) which, again, is valid only for $|x|>(v_+-v_-)^{-1}\sim D^{-{1\over 2}}$.
To see the role of the two-boson term $G_l$ in the correlation functions, we can use the identity (after proper regularization) $$\int \frac{dk}{4\pi} \omega(k) e^{-ikx}=
\omega^2\!\left(i\frac{\partial}{\partial x}\right)
\int \frac{dk}{4\pi} \frac{e^{-ikx}}{\omega(k)}.
$$ where $\omega^2(i\partial/\partial x)$ is a shorthand for $m^2-a\,{\partial^2}
/{\partial x^2}+b\,{\partial^4}/{\partial x^4}$ in the present case. With this $$G_l(x)=
G_\phi(x)\; \omega^2\!\left(i\frac{\partial}{\partial x}\right) G_\phi(x),$$ where we have neglected the singular, delta-function term of Eq. (\[phiphi\]). This term can be evaluated directly for large $x$, knowing the asymptotic form of $G_\phi$ in the different regimes. Using Eqs. (\[VBS\]), (\[AFasy\]) and (\[ICasy\]) we obtain $$\begin{aligned}
G_l(x)\sim \left\{
\begin{array}{ll}
\displaystyle{\frac{e^{-2v_-x}}{x^2}} &
\mbox{if $\theta<\theta_{\rm vbs}$} \\
0^{} & \mbox{if $\theta=\theta_{\rm vbs}$} \\
\displaystyle\frac{e^{-2vx}}{x^2} [c_1+c_2 \cos(2ux+\alpha)]
& \mbox{if $\theta>\theta_{\rm vbs}$},
\end{array}
\right. \end{aligned}$$ where the constants $c_1,c_2$ and $\alpha$ can be expressed straightforwardly with $a,b$ and $m$. It is interesting to remark that $G_l$ is exactly zero for any $x>0$ at the disorder point, i.e., the 2-boson processes do not contribute to the equal time correlation function there. This can be easily verified by calculating $\omega^2 (i\partial/\partial x)
G_\phi(x)$ using Eq. (\[VBS\]) and the fact that $D=0$ at the VBS point. Here $\langle S^z_n S^z_0 \rangle$ only contains the $G_\phi$ term in full accordance with the exact solution. For other values of $\theta$, $G_l$ decays twice as rapidly as $G_\phi$.
It is also of interest to see what predictions our simple field theory gives for the analytic properties of the ground state energy density and the gap. Using the mode expansion in Eq. (\[mode\]), the Hamiltonian can be written as $$H = \int_{-\Lambda}^{\Lambda} dk\; \omega(k) \left[
d^{\dag}_k d^{}_k + {1\over 2} \right],$$ where $\Lambda$ is an appropriate UV momentum cutoff, proportional to the inverse of the lattice constant. From this the ground state energy is $$E = {1\over 2}\int_{-\Lambda}^{\Lambda} dk\; \omega(k).$$ The ground state energy depends analytically on $a$, $b$ and $m$ whenever the zeros of $\omega(k)$ are not on the real axis. Together with the supposed analyticity of $a(\theta)$, etc., this means that $E(\theta)$ is also analytic inside the whole Haldane phase.[@Tit] A straightforward expansion around $\theta_{\rm vbs}$ yields $$\begin{aligned}
\lim_{\Lambda\to\infty} &&\left[ E(\theta)-E(\theta_{\rm vbs})\right] =\\
&& -\frac{8\pi a^{3/2}}{\sqrt{2} b}\sum_{n=1}^\infty
\frac{2^{-6n}(4n-4)!}{(2n-2)!n!(n-1)!} \left( \frac{D}{a^2} \right)^n,
\nonumber\end{aligned}$$ which is convergent if $|D|/a^2<1$. We notice that in general $E(\theta)$ can be expressed in a closed form in terms of elliptic integrals of the first and second kind.
The energy gap of the model is by definition $$\Delta = {\rm min}_k\; \omega(k) .$$ This is obviously analytic at $\theta_{\rm cic}=\theta_{\rm vbs}$ but has a singularity at $\theta_{\rm disp}$, where the minima of $\omega(k)$ move away from $k=0$ as the parameter $a=0$ changes sign. While for $\theta\le \theta_{\rm disp}$ the minimum is taken at $k=0$, for $\theta>\theta_{\rm disp}$ it is taken at $k=\pm\sqrt{-a/2b}$. The gap $\Delta$ and its derivatives with respect to $\theta$ on the two sides of $\theta_{\rm disp}$ turn out ot be, resp., $$\begin{aligned}
\Delta(\theta_{\rm disp}-0)&=&m, \quad\;\;\;\,
\Delta(\theta_{\rm disp}+0)=m, \nonumber\\ \\
\Delta' (\theta_{\rm disp}-0)&=&m', \quad\;\,
\Delta' (\theta_{\rm disp}+0)=m',\nonumber\\
\Delta'' (\theta_{\rm disp}-0)&=&m'', \quad
\Delta'' (\theta_{\rm disp}+0)=
m''-\left.\frac{\displaystyle{a'^2}}{\displaystyle{4bm}}
\right|_{\theta_{\rm disp}}. \nonumber\end{aligned}$$ We see that there is a discontinuity in the second derivative. This behavior of the effective theory seems consistent with the numerical results shown in Fig. 9 of Ref. \[\[Sch-Jol-Gar\]\].
Effective theory - lattice version
==================================
The effective theory presented above is capable of providing a complete qualitative description of the C-IC transition of the spin-1 bilinear-biquadratic model in accordance with the available numerical data. However, in order to give quantitative predictions, too, the theory needs some refinement. We have seen earlier that our simple theory identifies the disorder point $\theta_{\rm cic}$ with the point where the discriminant $D$ defined by Eq. (\[discriminant\]) vanishes. It is easy to verify that the second and fourth derivatives of $\omega(k)$ at $k=0$ are, resp., $$\frac{\partial^2\omega}{\partial k^2}(k=0)={a\over m},\qquad
\frac{\partial^4\omega}{\partial k^4}(k=0)=-{3D\over m^3},
\label{derivs}$$ thus in the above theory the fourth derivative vanishes at the disorder point. In contrast with this, Golinelli et al.[@Gol-Jol-Sor] measured numerically the second and fourth derivatives at the known disorder point $\theta_{\rm vbs}$ and found $$\frac{\partial^2\omega}{\partial k^2}(k=0)=0.9778(1),\quad
\frac{\partial^4\omega}{\partial k^4}(k=0)=-1.202(1).
\label{numVBS}$$ Note that the fourth derivative is only zero far inside the IC regime, which seems inconsistent with the above theory.
One step to improve the theory is to realize that the model is defined on a lattice, and thus the dispersion $\omega(k)$ must be a $2\pi$-periodic function of $k$ (from now on the lattice constant is set $\delta x=1$). This can be incorporated into the Lagrangian in Eq. (\[Lagr\]) by the standard replacement $\partial_x\phi \to
[\phi(n+\delta x)-\phi(n)]/\delta x$, leading to the substitution $k^2\to 2[1-\cos(k)]$ in the dispersion $$\begin{aligned}
\omega (k)&& = \\
&& \sqrt{m^2+2a+6b - (2a+8b)\cos(k) + 2b\cos(2k)}. \nonumber \end{aligned}$$ The Green’s function now reads $$G_\phi(n) =
\int_{-\pi}^\pi \frac{dk}{4\pi} \frac{1}{\omega(k)}e^{ikn}.
\label{G_discrete}$$ The condition that the expression under the square root in $\omega(k)$ is a complete square is again $D(\theta)=0$ with $D$ defined in Eq.(\[discriminant\]). When this is satisfied the dispersion simplifies to $$\omega_{\rm vbs}(k)=
m_{\rm vbs}+\frac{a_{\rm vbs}}{m_{\rm vbs}}[1-\cos(k)],
\label{dispCIC2}$$ and $1/\omega_{\rm vbs}(k)$ has poles instead of branch cuts. \[One pole within the Brillouin zone $-\pi<{\rm Re}(k)\le \pi$ with $0<{\rm Im}(k)$.\] Now the second and fourth derivatives at $k=0$ are \[cf. Eq. (\[derivs\])\] $$\frac{\partial^2\omega}{\partial k^2}(k=0)={a\over m},\qquad
\frac{\partial^4\omega}{\partial k^4}(k=0)=-{3D\over m^3}-{a\over m}.$$ At the disorder point we find $$\frac{\partial^4\omega}{\partial k^4}(k=0)=-{a_{\rm vbs}\over
m_{\rm vbs}},
\label{om4}$$ which is nonzero. The correlation length $\xi=1/{\rm Im}(k)$ at the disordered point is determined by the position of the pole, i.e., by the solution of the transcendental equation $$1+\frac{a_{\rm vbs}}{m^2_{\rm vbs}}[1-\cos(k)] = 0.
\label{tranc}$$ Working the other way around, knowing that at the VBS (C-IC) point $\xi_{\rm vbs}=1/\ln 3$, Eq. (\[tranc\]) gives $a_{\rm vbs}=3m^2_{\rm vbs}/2$ and thus the dispersion $$\omega_{\rm vbs} = m_{\rm vbs}\left[{5\over 2}-{3\over 2}\cos(k)
\right].
\label{singlemode}$$ With this expression the lattice Green’s function defined in Eq. (\[G\_discrete\]) reads $$G_\phi(n) = {1\over 4m_{\rm vbs}} e^{-|n|\ln 3},
\label{lG}$$ which should be conferred to Eq. (\[G\_VBS\]). The functional form of the dispersion relation in Eq. (\[singlemode\]) is exactly the same as the one appearing in the [*single mode approximation*]{} of the VBS model.[@Fat-Sol-CM; @Aro-Aue-Hal] There, one derives an upper bound for the gap $\Delta_{\rm vbs}=m_{\rm vbs}=\sqrt{40}/9$, whereas here we should use the phenomenological (numerical) value $\Delta_{\rm vbs}=0.664314$ in Eq. (\[singlemode\]). This, together with Eq. (\[om4\]) yields the value $\partial^4\omega/\partial k^4=-3m_{\rm vbs}/2 \approx -1.0$, which is rather close to the numerical estimate in Eq. (\[numVBS\]).
The split of the double root in the vicinity of the disorder point can be analyzed similarly to the continuum theory. We do not go into details here, but emphasize that the critical exponents characterizing the behavior of $\xi(\theta)$ and $q(\theta)$ at $\theta_{\rm vbs}$, and the type of the singularity of the gap at $\theta_{\rm disp}$ remain the same. Similarly, we find that the ground state energy is analytic everywhere.
One can wonder about the possible consequences of keeping higher order terms in the continuum Lagrangian Eq. (\[Lagr\]) or in the improved lattice version. If the theory remains free the only effect is to bring about additional branch cuts or poles in $1/\omega(k)$. If the higher order terms are small the additional branching points are far in $k$ space, and the C-IC transition remains intact. The long distance asymptotics of the correlation functions do not change. Since at the VBS point the exact correlation function in Eq. (\[G\_VBS\]) only contains a single exponential term, the free boson approximation does not allow any higher order spatial derivatives in the effective Lagrangian there.
Although the dispersion $\omega_{\rm vbs}$ in Eq. (\[singlemode\]) gives rise to the exponential term in Eq. (\[G\_VBS\]), it misses the $\delta_{n,0}$ contribution. This is, however, another artifact of the continuum approach, which necessarily neglects some important short distance details. In fact, we should recall that in a quantum spin liquid with short-range valence bond ground state the elementary excitations (bosons) are physically triplet bonds living [*between*]{} lattice sites, rather than on the sites themselves. As was argued in Ref. \[\[Fat-Sol-CM\]\], $S^z_i$ acting on the VBS ground state produces the linear combination of two states, one of which containing a boson at site $i-1/2$, the other a boson at site $i+1/2$. Hence the one-boson term of $\langle S_i S_{i+n} \rangle$ is in fact $$\langle S_n^z S_0^z\rangle = g_\phi^2 (-)^{n} [G_\phi(n-1)+
2G_\phi(n)+G_\phi(n+1)].
\label{GGGG}$$ Using Eq. (\[lG\]) this leads directly to Eq. (\[G\_VBS\]), including the $\delta$-function piece, if $g_\phi^2=m_{\rm vbs}$.
Summary and discussion
======================
In summary, we proposed a simple effective field theory to describe the commensurate-incommensurate transition in the Haldane phase of the spin-1 bilinear-biquadratic chain. The theory is capable of reproducing many features of this transition previously seen in the numerical studies. Moreover it also has some new predictions. The effective theory predicts that [*the C-IC transition at $\theta_{\rm vbs}$ is not a phase transition in the conventional sense*]{}, since the ground state energy remains an analytic function of the control parameter $\theta$. The only singularity occurring is in the correlation length. We should emphasize, however, that the correlation function itself remains analytic as a function of $\theta$ for any fixed distance, unlike in conventional phase transitions.
There is another point $\theta_{\rm disp}$ close to the disorder point where another quantity becomes singular. This is the energy gap (Haldane gap) whose second derivative produces a jump. This singularity in the singlet-triplet gap becomes important when a high enough magnetic field is applied, producing a crossing between these levels, and thus leading to the collapse of the gap. At the critical field, as $\theta$ is varied a real phase transition takes place at $\theta_{\rm disp}$, thus this point is the endpoint of a phase transition line on the magnetic field vs $\theta$ plain separating two Luttinger liquid type phases.
In a technical sense the C-IC transition is a consequence of an accidental degeneracy of roots of the dispersion relation. We have shown in the free boson approach that this degeneracy causes the “dimensional reduction” of the correlation function, and makes it to be a pure exponential at the VBS point. We also found that the two-magnon contributions, and presumably any higher order, multi-magnon contributions too, vanish exactly at this point. We derived a formula valid in the vicinity of the VBS point showing how the pure exponential decay emerges from the standard form with algebraic prefactors. In particular we found that there is a crossover between a pure exponential decay and a decay containing algebraic prefactors. The characteristic distance of this crossover tends to infinity as the VBS point is approached.
The spin-1 bilinear-biquadratic model studied in this paper is not the only model which produces a C-IC transition. Another example is the spin-1/2 chain with next-nearest-neighbor interaction (this can also be visualized as a two-leg zig-zag ladder).[@watanabe] By now it is well established that the Majumdar-Ghosh point of this model is a disorder point where a C-IC transition of the first kind occurs. On the same footing as described here, it seems possible to develop an effective theory which supposes that the elementary excitations (spin-1/2 solitons in that case) are essentially free particles with a non-relativistic dispersion. Care should, however, be taken on the facts that solitons are always created in pairs and that they are spin-1/2 particles. Beside the $S=1/2$ case, the appearence of disorder points has been demonstrated in other $S\ge 1$ frustrated Heisenberg chains too.[@KoleRoth] Another interesting quasi-one-dimensional system where a commensurate-incommensurate transition have been reported in a numerical investigation is the SU(2)$\times$SU(2) symmetric coupled spin-orbit model.[@Patietal] The elaboration and testing of effective theories, similar to the one described in this paper, for these models could be a possible direction of future research.
We thank L. Balents and J. Sólyom for valuable discussions. This work was financially supported by the Hungarian Scientific Research Found (OTKA) under grant Nos. 30173, 30543, and F31949.
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author:
- 'M. Brusa [^1]'
- 'C. Feruglio'
- 'G. Cresci'
- 'V. Mainieri'
- 'M. T. Sargent'
- 'M. Perna'
- 'P. Santini'
- 'F. Vito'
- 'A. Marconi'
- 'A. Merloni'
- 'D. Lutz'
- 'E. Piconcelli'
- 'G. Lanzuisi'
- 'R. Maiolino'
- 'D. Rosario'
- 'E. Daddi'
- 'A. Bongiorno'
- 'F. Fiore'
- 'E. Lusso'
date: 'Received 9 December 2014; accepted 3 March 2015'
title: 'Evidence for feedback in action from the molecular gas content in the $z\sim1.6$ outflowing QSO XID2028[^2]'
---
[Gas outflows are believed to play a pivotal role in shaping galaxies, as they regulate both star formation and black hole growth. Despite their ubiquitous presence, the origin and the acceleration mechanism of such powerful and extended winds is not yet understood. Direct observations of the cold gas component in objects with detected outflows at other wavelengths are needed to assess the impact of the outflow on the host galaxy interstellar medium (ISM). ]{} [We observed with the Plateau de Bure Interferometer an obscured quasar at z$\sim$1.5, XID2028, for which the presence of an ionised outflow has been unambiguously signalled by NIR spectroscopy. The detection of $^{12}$CO(3–2) emission in this source allows us to infer the molecular gas content and compare it to the ISM mass derived from the dust emission. We then analyze the results in the context of recent insights on scaling relations, which describe the gas content of the overall population of star-forming galaxies at a similar redshifts. ]{} [ The Star formation efficiency ($\sim100$) and gas mass (M$_{\rm gas}=2.1-9.5\times10^{10}$ M$_\odot$) inferred from the CO(3-2) line depend on the underlying assumptions on the excitation of the transition and the CO-to-H2 conversion factor. However, the combination of this information and the ISM mass estimated from the dust mass suggests that the ISM/gas content of XID2028 is significantly lower than expected for its observed M$_\star$, sSFR and redshift, based on the most up-to-date calibrations (with gas fraction $<$20% and depletion time scale $<$340 Myr).]{} [Overall, the constraints we obtain from the far infrared and millimeter data suggest that we are observing QSO feedback able to remove the gas from the host.]{}
Introduction
============
There are both theoretical (e.g. @Hopkins2008) and observational (e.g. @Sanders1988 [@Yan2010]) arguments that support the notion that luminous star-forming galaxies (hereafter: ‘Starbursts’) and luminous, unobscured Active Galactic Nuclei (AGN; hereafter luminous AGN or ‘QSO’) are basically the same systems caught in different stages of the co-eval growth of (massive) galaxies and the Super Massive Black Holes (SMBH) sitting in their centres. In particular, Starbursts should trace objects caught in the rapid SMBH growth phase characterized by efficient Star Formation (SF), in a dust-enshrouded, dense environment, while the unobscured QSOs are systems radiating at the Eddington limit, where the SMBH is almost fully assembled. Given that both SF and AGN activity are thought to be sustained by the availability of cold gas in galaxies (see e.g @Menci2008 [@Vito2014_gas]), millimeter observations of molecular transitions are needed to directly probe the presence and state of this gas. In the past decade, observations of cold molecular gas reservoirs at high redshift (see @CW2013 for a comprehensive review) turned out to be crucial in studying the gas content and consumption rate in both normal and peculiar systems. For example, the gas properties of “normal" galaxies are being investigated in increasing details up to high-z [@Tacconi2013; @Genzel2014_gas; @Sargent2014], and as a function of many of the structural and physical properties of the systems (e.g. Star Formation Rate, SFR; stellar mass; colors; see e.g. @Genzel2014_gas [@Sargent2014]). This has become possible thanks to the large investment of time at millimeter arrays, mainly the Plateau de Bure Interferometer (PdBI). In particular, it has been reported that, among massive systems, (M$_{\star}>10^{10}$ M$_\odot$), the gas fraction increases across the main sequence (MS; defined between the SFR and the stellar mass of galaxies) at fixed redshift (see @Magdis2012a [@Magdis2012b; @Saintonge2012; @Tacconi2013; @Sargent2014]) and is hence closely related to the Specific Star Formation rate (sSFR). This fits in a scenario where the redshift evolution of the sSFR is consistent with being driven by the gas fraction (see also @Lilly2013). Similar conclusions are reached in works involving dust fitting methods to derive the gas mass (see e.g. @Santini2014).
The first molecular studies on local Ultra Luminous Infrared Galaxies (ULIRGs) and Submillimeter Galaxies (SMG) at higher redshifts, i.e. targetting objects in the ‘Starburst’ phase, showed that these systems typically have a low molecular gas content with respect to their current SFR, or alternatively higher star-formation efficiencies. Indeed, defining empirically the Star Formation Efficiency (SFE) as the ratio of the IR luminosity to the CO luminosity (in units of L$_{\rm \odot}$/(K km s$^{-1}$ pc$^2$)), ‘Starbursts’ have SFE$>$200 (see e.g. @Daddi2010 [@Genzel2010]) larger than those observed in normal star forming galaxies with the same molecular gas content (@Tacconi2010, SFE$\sim50-200$). In other words, their consumption time scale is shorter with respect to normal galaxies and they will exhaust their gas reservoirs in a short timescale ($\lsimeq100$ Myr). This is consistent with the hypothesis that ‘Starbursts’ in general (and ULIRGs/SMGs in particular) are objects at the peak of their SF activity in the heavily obscured phase.
On the other hand, high values of the L$_{\rm IR}$/L’(CO) ratio have been also observed in high-z unobscured QSO host galaxies (SFE$>200$; e.g. @Solomon2005 [@Riechers2011; @Riechers2011_QSOz3]), although, being a subsequent phase of ‘Starbursts’ in the evolutionary sequence, their SFR is expected to be already substantially suppressed. In this case a significant fraction of the gas could have been previously removed during the ‘blow-out’ phase, and the observed high SFE in unobscured QSOs can be ascribed to region of residual, on-going SF, pointing towards a possible effect of ’positive feedback’ on the galaxy from the AGN [@Silk2013; @Zubovas2014].
{width="8.7cm"} {width="6.8cm"}
What is still missing for a full understanding of the results of the aforementioned studies, in terms of the role of the physical processes which govern the co-eval BH-galaxy growth, is a full characterization of the gas properties of objects caught in the short-lived “transition” phase between the Starburst and QSO stages. This phase is expected to be characterized by gas reservoirs not yet depleted and by complex kinematics, including strong winds and outflows. @Brusa2010 proposed that sources in the ‘blow-out’ phase at z$\sim1.5$ can be isolated on the basis of their observed X-ray-to-optical-to-NIR colors and presented the source XID2028 (z=1.5927), detected in the XMM-COSMOS survey, as the prototype of this class. XID2028 is a luminous (L$_{\rm bol}\sim2\times10^{46}$ erg s$^{-1}$), mildly obscured QSO hosted in a massive galaxy, with M$_{*}\sim4.5\times10^{11}$ M$_\odot$ and a SFR$\sim270$ M$_\odot$ yr$^{-1}$ as measured by [*Herschel*]{} from PEP and SPIRE data [@Lutz2011; @Bethermin2012]. At its center, XID2028 has a supermassive black hole with mass M$_{\rm BH}\sim3\times10^9$ M$_\odot$ [@Bongiorno2014], which is accreting at $\sim5$% of its Eddington luminosity.
The presence of a massive outflow in the ionized gas component of XID2028, traced by the \[O III\]$\lambda$5007 emission, has been unambiguosly and independently confirmed by X-shooter slit spectroscopy [@Brusa2015; @Perna2015] and SINFONI J-band IFU observations: in fact, XID2028 hosts one of the most massive ($\dot{M}_{ion}>250$ M$_\odot$ yr$^{-1}$, with v$>1500$ km s$^{-1}$) and most extended (out to scales of $\sim13$ kpc) outflows detected in a high-z QSO [@Cresci2015]. Most importantly, the outflow lies exactly in the center of a cavity in star forming regions in the host galaxy (as traced by narrow H$\alpha$ emission line map and rest frame U band imaging; see @Cresci2015) thus suggesting that the wind is removing the gas from the host galaxy (‘negative feedback’), and at the same time is also triggering star formation by outflow induced pressure at the edges (‘positive feedback’; e.g. @Zubovas2014). XID2028 therefore represents a test case to study QSO ‘feedback in action’. However, the evidence of feedback in this source mostly comes from measurements of the on-going star formation in the source traced by the narrow H$\alpha$ emission line that in principle may be affected by e.g. differential extinction effects in the host galaxy.
Direct observations of the cold gas component in this galaxy are needed to assess whether the ionized outflow has an impact on the cold gas reservoir. With this aim, here we present observations of the [CO(3-2) ]{}transition of XID2028, redshifted to 2mm, obtained with the PdBI Interferometer. We compare the gas masses derived from CO with that inferred from the dust mass and based on Far Infrared (FIR) data. These two methods allow us to investigate whether AGN feedback has already been effective in diminishing the cold gas mass in the host, or whether the feedback phase is still associated with cold-gas-rich galaxies similarly to MS star-forming galaxies, with important consequences for galaxy-AGN coevolutionary models.
The paper is organised as follows: Section 2 presents the PdBI observations and data analysis, Section 3 discusses the results, while Section 4 summarizes our conclusions. Throughout the paper, we adopt the cosmological parameters $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_m$=0.3 and $\Omega_{\Lambda}$=0.7 (Spergel 2003). In quoting magnitudes, the AB system will be used, unless otherwise stated. We adopt a Chabrier Initial Mass Function to derive stellar masses and SFRs for the target and comparison samples. The physical scale is 1"$\sim8.5$ kpc at the redshift of the source.
Millimeter observations
=======================
Data reduction
--------------
XID2028 was observed with receivers tuned to a frequency of 133.37 GHz, corresponding to the expected frequency of the CO(3-2) emission line, with the PdBI array in the most compact (D) configuration. The observations were split in 3 tracks (31-May, 1, 6, June 2014). The system temperature (T$_{sys}$) was between 100 and 300 K, and water vapor 4-6 mm. The quasar 1005+058 (0.3 Jy at 133.7 GHz) was used as a phase and amplitude calibrator. MCW349 (with a flux of 1.8 Jy) was used for absolute flux calibration, which yields an absolute flux accuracy of about 5% at the observed frequency. Calibration and mapping were done in the GILDAS environment. The flagging of the phase visibilities was fixed at $<45\degree$ rms.
The total observing time was 5.6 hrs (3.06 hrs on source), for a total of 3673 visibilities available, before applying any flag. We then removed one scan (3994 in 01-June track) due to problems with the tracking. Data from antenna 1 from the 06-June track were not used in the final dataset due to the presence of a tuning parasite that produced a spurious signal at a frequency (133.21 GHz) close to the observed frame frequency of the CO(3-2) transition. After flagging of bad visibilities, the total on source time is 2.54 hours (six-antenna equivalent), and the 1$\sigma$ sensitivity is 1.36 mJy/beam in 20 MHz channels, for a total of 3052 visibilities. The clean beam of the observations is 4.5“x3.4”, with an angle of 38 degrees. The phase center of the data set was set to the HST position of the QSO nucleus (RA=10:02:11.29, DEC=+01:37:06.79).
Analysis
--------
We estimated the 2 mm continuum by collapsing the line-free channels of the data set and fitting the visibilities. The continuum is not detected with a 3$\sigma$ upper limit on its flux of 0.3 mJy. The redshift of the host galaxy (z=1.5927) was adopted to convert the frequency to velocity space. Figure \[coline\] shows the line spectrum integrated over the beam. The line displays two peaks: one centered around the systemic redshift (FWHM$\sim$$550\pm200$ km s$^{-1}$ from a Gaussian fit), and another centered at $\sim1000$ km s$^{-1}$ (henceforth referred to as “red feature"). The peak at the systemic position is significant at $5\sigma$, while the “red feature" is at a lower significance ($\sim$3$\sigma$). Moreover, the “red feature" peaks at $\sim133.0$ GHz, close to a known parasite signal at 132.9 coming from antenna 4 and identified in all tracks. We created a new table flagging data on Antenna 4. The total exposure time decreased to 1.8hr and the total number of visibilities to be used for the scientific analysis also considerably decreased. The red feature is not significant anymore (S/N$<3$). However, the significance of the detection over the systemic line also decreased at S/N$\sim$4. For this reason we decided to keep the full datasets in the analysis and, in the absence of deeper, more highly resolving observations which could confirm the presence of a second dynamically distinct component, we will consider the red feature as spurious.
The zero spacing flux estimated by fitting the averaged visibilities in the velocities range from -340 to +440 km s$^{-1}$ with a point source function is [*S’(CO)*]{}=1.6$\pm$0.3 mJy (5.3$\sigma$), and returns a centroid at (RA,DEC=10:02:11.24 01:37:05.48). The integrated flux over the full velocity range of the systemic line (with Full Width Zero Intensity, FWZI$\sim770$ km s$^{-1}$) is therefore $\int S’(CO)dv$= 1.23$\pm$0.23 Jy km s$^{-1}$. This measure depends only on the data calibration (including flagging of the Antennas) and does not depend on any other assumption, like e.g. masking, extraction region, ad-hoc centroid. The quoted errors take into account the statistical errors of the [*uv*]{} plane fit and the errors on the absolute flux calibration (5%). The right panel of Figure \[coline\] shows the integrated map over the systemic line emission. We verified that the flux extracted from the integrated map (S=1.55$\pm0.3$ mJy) on a region slightly larger than the beam, is in agreement with the one estimated by fitting the visibilities.
Figure \[FigHST\] shows the HST/ACS image (background) with superimposed the contours from the K-band image (blue), which should trace the extension of the host galaxy. The black contours are from the map obtained on the line detected at the systemic position (e.g. from Figure 1 right, in steps of S/N, starting from 1$\sigma$) and the black cross marks the line centroid.
From both Figures 1 and 2 it is clear that the line peak is offset by $\sim1\arcsec$ from the QSO nucleus position. From previous observations with the same phase calibrator (1005+058), we can exclude errors in the absolute astrometry. The error associated with the beam and the S/N of the source translates into a positional uncertainty of 0.46“ x 0.36”. We note, however, that the displacement may be due to the limited [*uv*]{} coverage of the data, and that a CO-offset is typical of low S/N data (see e.g. @Casey2011). Better signal-to-noise ratio and [*uv*]{} coverage are needed to refine the location of the gas reservoir.
A dynamical mass can be estimated from the CO line width assuming a size ($\rm R$) and an inclination ($i$) of a rotating molecular gas disc. The size can be inferred using the spectroastrometric technique [@Gnerucci2011; @Carniani2013 and references therein], applied to the CO data cube. By integrating the CO data in the red (0,+400 km s$^{-1}$) and blue (-400,0 km s$^{-1}$) line channels, we measure a difference in the line centroids of $\sim1.5\pm0.2\arcsec$ (with an error of 0.3 pixels for each detection). The centroids of these detections are also shown in Figure 1 (right panel) as blue and red crosses to mark the blue and red line channels, respectively. The measured shift corresponds to $\sim$13 kpc at the source redshift and translates to R$\sim$$6.5\pm0.8$ kpc, in agreement with the extension seen in the K-band data (see also @Cresci2015). Applying Equation 5 of @Gnerucci2011, we infer a M$_{\rm dyn}$(sin i)$^2$=4.5$\times10^{11}$ M$_\odot$ and, assuming an inclination of 60 deg, a M$_{\rm dyn}$$\sim$6.0$\pm2.3\times10^{11}$ M$_\odot$ once all the uncertainties in the quantities are taken in account. We will discuss in the following how this compares with the total mass derived from M$_{\star}$+M$_{\rm gas}$.
![HST/ACS image (F814W filter) with superimposed K-band contours from CFHT (blue, arbitrary levels chosen to trace the whole K-band emission). Black contours represent CO(3-2) emission from the integrated map in the channels corresponding to the “systemic” peak of the line (same levels as in right panel of Figure 1; starting from 1 sigma). The black cross (with associated ellipse) marks the line centroid. The image is about 10“ across. The beam size is 4.5”x3.4", with an angle of 38 degrees. []{data-label="FigHST"}](hst_cont.png){width="8cm"}
Results and discussion
======================
Deriving the luminosity L’CO\[1-0\] of the ground-state transition (which is generally regarded as the best indicator of the total gas reservoir) requires an assumption on r$_{31}$, the luminosity ratio between the [CO(3-2) ]{}and the CO(1-0) transitions, which depends on the nature of the systems: a ratio of $\sim0.7-1$ is typically reported for SMG galaxies and QSOs (see @CW2013 and references therein), while an average ratio of $\sim0.42$ has been determined for MS star-forming galaxies at a similar redshift as XID2028 (e.g. @Daddi2015).
The [CO(3-2) ]{}luminosity of XID028 is L’(CO\[3-2\])=1.9$\pm0.4\times10^{10}$ K km s$^{-1}$ pc$^2$ ($\sim9.3\pm2\times10^5$ L$_\odot$), following @Solomon2005. This value is in between the average values observed in this molecular transition for U/LIRGs (L\[CO(3-2)\]=$2.6\pm0.5$$\times10^{9}$ K km s$^{-1}$ pc$^2$) and SMGs (L\[CO(3-2)\]$4.4\pm1.1$$\times10^{10}$ K km s$^{-1}$ pc$^2$), as reported in the work of @Iono2009. On the other hand, the SFR ($\sim270$ M$_\odot$ yr$^{-1}$) and M$_\star$ ($\sim4.5\times10^{11}$ M$_\odot$) of XID2028 are consistent with those observed in a MS galaxy at z$\sim1.5$ (see @Mainieri2011 [@Bongiorno2014; @Brusa2015]).
The CO(2-1) transition in XID2028 is not detected down to a sensitivity of 0.23 mJy/beam over the full 770 km/s line width (corresponding to a 3$\sigma$ upper limit on the line integrated flux of 0.53 Jy km/s), from a separate, 3mm-band PdBI observation of XID2028 in October 2014 (M. Sargent, private communication). This suggests a near thermal CO-excitation state[^3], and therefore r$_{31}$ around unity, i.e. larger than the standard value usually adopted for MS galaxies, more consistent with the QSO/Starburst scenario. Given the complex nature of the system, we derive the CO(1-0) luminosity under the [*conservative*]{} assumption that r$_{31}$=0.7 (consistent with the constraints we have from millimeter data alone), and we will apply different $\alpha_{\rm CO}$ factors to derive molecular gas masses under the QSO/ULIRG and MS assumptions, discussing the implications of the findings in the different cases. The inferred L’(CO\[1-0\]) luminosity for XID2028 (abbreviated as L’(CO) in the following) is therefore L’(CO)=2.6$\times10^{10}$ K km s$^{-1}$ pc$^2$ ($\sim1.2\times10^{6}$ L$_\odot$).
Star Formation Efficiency
-------------------------
Figure \[SFE\] (left panel) shows L’(CO) against the total Infrared Luminosity (L$_{\rm IR}$, computed between 8-1000 $\mu$m) for XID2028 (red circle). The IR luminosity of XID2028 is very well constrained by Herschel/PACS+SPIRE data (logL$_{\rm IR}$=12.47; see @Brusa2015 [@Perna2015]) and has been estimated from fitting all bands with photometry with rest-frame wavelength $>50\mu$m with @Dale2002 Starbursts templates, using the same technique as in @Santini2009. Although recent works on FIR emission of AGN show that even the flux observed at rest frame wavelengths longer than 60$\mu$m can be AGN-dominated (e.g. @Mullaney2011), in XID2028 the QSO contribution is expected to be negligible, as shown in Figure 2 of @Perna2015, where the most recent SED fitting decomposition for this object is presented. The observed IR luminosity corresponds to a SFR of $\sim270^{+50}_{-100}$ M$_\odot$ yr$^{-1}$, using the SFR-IR luminosity relation [@Kennicutt1998], and taking into account the uncertainties on the flux normalization of the starburst component related to the AGN-host SED decomposition.
We compare this measurement with the compilation of low- and high-redshift normal star-forming galaxies with measured $\alpha_{\rm CO}$ presented in @Sargent2014 and the SMG-sample from @Bothwell2013, involving both outliers and galaxies consistent with the locus of the main sequence at their redshift. We also plot the unobscured QSOs at 1$<z<$4 presented in @Riechers2011[^4]. For these sources, the IR luminosities are extracted from the @CW2013 compilation. Finally, in Figure 3 we show SW022550 and SW022513 at z$\sim3.4$ [@Polletta2011], ULASJ1539 at z$\sim2.5$ [@Feruglio2014], and the MIPS-selected sources at z$\sim$2 from @Yan2010. All these systems have been proposed to be in the “transition phase" between an heavily obscured Starburst phase and the unobscured QSO phase.
The SFE of XID2028 (SFE$\sim110$) is on the lower side of the SFEs measured for high-z SMG and unobscured QSOs (SFE$\sim100-1000$). Instead, the SFE is consistent with those reported (albeit with much larger uncertanties due to the lack of a complete multiwavelength coverage and reliable measurements of L$_{\rm IR}$) in the obscured QSOs systems proposed to be in the “transition phase" mentioned above.
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Molecular gas mass from CO data
-------------------------------
Estimating the molecular gas mass based on the CO luminosity critically hinges on the CO-to-H2 conversion factor $\alpha_{\rm CO}$, defined as the ratio between the mass of molecular gas (M$_{\rm mol}$) to the integrated CO(1-0) luminosity ($\alpha_{\rm CO}$=M$_{\rm mol}$/L’(CO)\[1-0\] in units of M$_\odot$/(K km s$^{-1}$ pc$^{2}$)). This value depends on the ISM conditions, and two distinct assumptions are often adopted: $\alpha_{\rm CO}\sim4$ for extended SF disks/MS galaxies of solar metallicity, and $\alpha_{\rm CO}\sim0.8$ for compact luminous systems (@Downes1998; see @CW2013 and @Bolatto2013 for in-depth discussions).
From a morphological point of view we do not have a clear classification on the properties of the host galaxy. Given that the HST image suffers from substantial extinction (A$_V\sim3$; see discussion in @Perna2015) and, in any case, is dominated by the central active nucleus, it cannot be used for a reliable morphological analysis. However, no clear signatures of merging structures are visible in the rest-frame U-band. The low-resolution (with respect of HST) K-band image is instead consistent with both an elliptical galaxy and a spiral galaxy, possibly interacting with a north-east system (see Figure 2). Even if the MS is mainly populated by “normal” spiral and disk galaxies (see e.g. @Wuyts2011a), we note that XID2028 would lie among the population which occupies the upper envelope of the MS at z$\sim1.5$. These galaxies may have also cushier light profiles, intermediate between disky galaxies and red and dead systems (see @Wuyts2011b, their Figure 1, right panel). In any case, if after point-source subtraction this galaxy were to be shown to have an early-type or disturbed host galaxy morphology, it would actually be highly consistent with the statistical findings of @Wuyts2011b.
In the vast majority of studies targeting SMGs, QSOs and ULIRGs systems (see e.g. @Aravena2008 [@Riechers2011_QSOz3; @Polletta2011; @Feruglio2014], among others), $\alpha_{\rm CO}=0.8$ has been adopted even in absence of better information on the physical properties of the system (e.g. compactness of the source). Under the assumption of starbursts/QSO scenario, we obtain for XID2028 a gas mass M$_{\rm mol}\sim$2.1$\pm0.4\times10^{10}$ M$_\odot$.
To infer the molecular gas mass under the MS hypothesis, we consider a metallicity dependent conversion factor $\alpha_{\rm CO}$ (see e.g.@Genzel2012 [@Bolatto2013]). In the following we will assume for XID2028 a value of $12+log(O/H)=9.07$, the metallicity inferred from the so-called Fundamental Metallicity Relation (FMR, @Mannucci2010), that relates the metal content with the stellar mass and the SF of the galaxy independently of redshift [@Cresci2012]. Applying the relations describing redshift-dependent variations of $\alpha_{\rm CO}$ in the SFR-M$_\star$ plane of @Sargent2014 to XID2028 one would expect[^5] $\alpha_{\rm CO}\sim3.6$, and the corresponding molecular gas mass would then be M$_{\rm mol}\sim9.5\pm1.9\times10^{10}$ M$_\odot$.
The two values for M$_{\rm gas}$ inferred under the two different assumptions are plotted in Figure 3b (with the statistical errors associated to the line detection), where the molecular gas mass is shown as a function of the SFR (for the same samples presented in Fig. 3a).
Molecular gas mass from FIR emission
------------------------------------
We also adopt an independent method to compute the total gas mass in this source, using the dust mass derived from FIR photometry. For this purpose, we assume a metallicity-dependent gas to dust ratio, following the calibration presented by @Santini2014 and recently extended to AGN samples in the work by @Vito2014_gas. This estimate is independent from $\alpha_{\rm CO}$, although it depends on the metallicity ($Z$) of the system and on the assumptions that the dust-to-gas ratio scales linearly with $Z$ through a constant factor [@Draine2007].
The dust mass is obtained via SED decomposition of the AGN and host galaxy contributions, using a combination of the @Silva2004 AGN templates and the @DraineLi2007 dust templates to fit the 100-500$\mu$m range. For objects at z$>$1, submillimeter data are in principle required to properly sample the dust emission free from AGN contamination. However, we note that for XID2028 the best fit SED decomposition performed following @Vito2014_gas is consistent with the upper limit of the continuum at 2mm (see Section 2.2).
We obtain a total dust mass of M$_{\rm dust}=7.7\pm4.2\times10^{8}$ M$_\odot$. Assuming the FMR metallicity (see above) and the @Santini2014 calibration, this translates into a M$_{\rm gas}\sim4.5\pm2.4\times10^{10}$ M$_\odot$ (without considering the uncertainty on the dust-to-gas-ratio calibrations, e.g. a factor of $\sim2$, see @Sandstrom2013). We note that using the @Leroy2011 metallicity dependence of the dust-to-gas ratio would yield consistent results, within the errors.
The value inferred from the dust fit approach is plotted as a green star in Fig. 3b. If we use this estimate for M$_{\rm gas}$, and the observed L’(CO), we can derive an [*effective*]{} $\alpha_{\rm CO}$ for this source, $\alpha_{\rm CO(dust)}\sim2.4\times r_{31}$ ($\alpha_{\rm CO(dust)}\sim$1.7 given our adopted excitation correction r$_{31}$=0.7).
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Gas fraction and depletion timescale
------------------------------------
The uncertainty in the derived gas mass from the CO data is dominated by the assumption in $\alpha_{\rm CO}$ (a factor of 4.5) with respect to the statistical uncertainties (20%). Given that the value derived from the dust fit is in between those for the two different assumptions in $\alpha_{\rm CO}$, in the following we will refer to this value as our best estimate for the molecular gas mass, and those from the CO data under the MS and QSO/Starburst assumptions as upper and lower limit, respectively, i.e. M$_{\rm gas}=4.5 (1.7-11.4)\times10^{10}$ M$_\odot$, where the lower and upper limits in parenthesis take also into account the statistical uncertainties of the detection, and overall also the uncertainty in the assumed dust-to-gas ratio. We note that the total gas mass inferred from the dust continuum fit includes both the molecular and atomic components. However, the atomic mass usually constitutes a negligible fraction of the total gas mass.
The stellar mass of XID2028 is M$_{\star}\sim4.5\times10^{11}$ M$_\odot$ from the most recent SED fitting decomposition [@Perna2015]. This value is a result of the inclusion in the multicomponent SED fitting of a mildly obscured QSO component, given that we observe the Broad Line Region (BLR) emission in the H$\alpha$ line complex [@Bongiorno2014]. In Section 2.2 we reported a dynamical mass M$_{\rm dyn}$$\sim$6$\pm2\times10^{11}$ M$_{\odot}$. Although the estimate of the dynamical mass suffer from large uncertainties, it is quite reassuring that it is consistent with the value we obtain from the sum of the stellar and molecular mass components (M$_{\rm tot}$$\sim4.6-5.6\times10^{11}$ M$_\odot$, taking into account the range of M$_{gas}$).
We can then calculate the molecular gas fraction, $\mu_{\rm mol}$, defined as the ratio of the molecular gas mass and the stellar mass ($\mu_{\rm mol}$=M$_{\rm mol}$/M$_{\star}$; see e.g. @Sargent2014 [@Genzel2014_gas]). Given the molecular gas masses inferred in the previous Section, the gas fraction translates into $\sim5$% for the QSO/Starburst and $\sim21$% for the MS scenarios. The value from the dust mass measurement is in between these two estimates ($\sim10\%$). Similarly, we can estimate the depletion time scale (defined as M$_{\rm gas}$/SFR; e.g. the rate at which the gas is converted into stars) and we infer t$_{\rm depl}$=75, 340 and 160 Myr using the QSO/Starburst, MS and dust-fit derived gas masses, respectively.
Evidence for QSO feedback
-------------------------
Figure \[fgas\] (left panel) shows the gas fraction in XID2028 for the three assumptions described above, plotted against the sSFR-excess with respect to the main sequence, e.g. sSFR/sSFR$_{\rm MS}$=$0.86$, where the mass- and redshift-dependence of the characteristic sSFR of MS galaxies follows the calibration in [@Sargent2014] which is based on a large compilation of literature data. In this plot we show the same samples used in Figure 3 (with the exception of unobscured QSOs and the MIPS selected sources for which no stellar mass estimates are available) and we plot as a solid line the median trend with normalized sSFR, expected for a galaxy of the same mass and redshift of XID2028 (taken from the 2-Star Formation Mode description of normal and starbursting (off-MS) galaxies in @Sargent2014).
Taking the best M$_{\rm gas}$ estimate for our target, and even taking into account the uncertainty on $\alpha_{\rm CO}$ assumption, XID2028 is among the objects with the lowest gas fraction for its sSFR detected so far in the high-z Universe and associated to normal star-forming galaxies (green star in Figure 4), especially when compared to systems with similar masses (solid line). The $\mu_{\rm mol}$ is instead more similar to that expected for ‘Starburst’ galaxies of a similar mass and redshift (see value of black trend line at sSFR/<sSFR>$_{\rm MS}\gsimeq4$), but XID2028 does not share with these sources the same burst of star formation. Instead, the gas fraction of XID2028 is similar to normal galaxies in the local universe (open triangles), despite its higher redshift.
An alternative way of visualizing the gas content and consumption is illustrated in the right panel of Figure \[fgas\], where the depletion time scale is plotted against the MS-normalised sSFR of the host galaxy. Assuming our best M$_{\rm gas}$ estimate, XID2028 lies at shorter depletion time scales with respect to MS galaxies (at any redshift), i.e. it is consuming its residual gas more rapidly than normal star-forming galaxies. This qualifies XID2028 as a clear outlier with respect to the average population, and a rare object, consistent with the hypothesis that it is caught in the very short transition phase in which the QSO feedback is released.
Similar conclusions can be reached examining the position of our source with respect to the Kennicutt-Schmidt relation [@Kennicutt1998]: assuming the physical scales inferred in Section 2.2, and that the molecular gas and the SF episodes are distributed uniformly over this region, XID2028 would lie slightly above (a factor $\sim2.5$) the correlation observed for normal and starburst galaxies. However, the SFR density measured in this way is to be considered a lower limit, given that the SF regions seem to be patchy (see Cresci et al. 2015). Therefore, XID2028 would further deviate above the K-S relation, towards regions of short depletion timescales.
It is important to note that, even when using the MS assumption, the molecular gas fraction and depletion timescale would be considerably lower than those expected for systems of the same host galaxies properties of XID2028 (blue point in Figure 4). In particular, the depletion timescale observed in XID2028 for the MS scenario is a factor of $\sim2$ lower than the expectations of @Sargent2014 and a factor $\sim3$ lower than that obtained by the parameterization of MS and off-MS galaxies presented in @Genzel2014_gas [using their [*global fit*]{} we expect for XID2028 $\rm{t_{dep}(G15)_{global}}\sim970$ Myr]. The discrepancy with the calibrations is more extreme if the values obtained in the QSO scenario are adopted (red circles in Figure 4). We also note that our chain of assumptions in deriving M$_{\rm gas}$ has been very conservative. For example, we used r$_{31}$=0.7 instead of $r_{31} \gsimeq 0.9$ as suggested by the non detection of CO\[2-1\] emission, which would have instead provided a 20% smaller CO\[1-0\] flux. This conservative assumption also compensates a possible overestimate of the value of the CO\[3-2\] flux, which could result from measuring the line flux at the phase centre rather than at the slightly offset centroid. The result of the lack of molecular gas in XID2028, with M$_{\rm gas}\lsimeq10^{11}$ M$_\odot$, is therefore quite robust.
A short depletion time scale with respect to MS galaxies has also been found for SMGs in the @Bothwell2013 sample, and other AGN/Starburst systems plotted in Figure 4 [@Aravena2008; @Polletta2011; @Feruglio2014]. @Yan2010 also reported a short depletion timescale of $\sim$40 Myr for the sample of MIPS-selected ULIRGs. The short depletion time scale in SMGs has been interpreted as higher star formation efficiency in the galaxy (e.g. @Genzel2010 [@Daddi2010]), probably due to higher density of the ISM in these compact systems. These may also be the case for ULASJ1534 and the COSBO11, which have sSFRs comparable to SMGs, and for which we expect compact gas reservoirs.
Instead, in XID2028 a significant fraction of the gas is expected to be already expelled from the galaxy. The SF is then probably maintained only in the denser environments, less affected by the negative feedback, and possibly enhanced by positive feedback due to the outflow induced pressure (e.g. @Silk2013). The fact that XID2028 has a smaller gas reservoir and shorter depletion time than that measured for MS galaxies of similar sSFR therefore constitutes a new probe, in addition to the analysis presented in @Cresci2015 based on NIR data, that QSO feedback in the form of powerful outflows is able to affect star formation in the host and expel a significant fraction of gas from the host galaxy.
Summary
=======
We presented the first molecular line luminosity measurement, via [CO(3-2) ]{}observations obtained at the PdBI interferometer, in a luminous obscured QSO at z$\sim$1.5. The target is thought to be in the ‘blow-out’ phase, and the presence of a powerful outflow with significant impact on the host galaxy has been unveiled through previous NIR observations [@Perna2015; @Cresci2015]. We complemented the PdBI data with FIR dust fitting, and report the following results:
- We measure a SFE ($\simeq$110) at the lower end of those reported in the literature for a large number of QSOs and Starburts/SMG galaxies (see @Iono2009 [@CW2013]), and consistent with that inferred for obscured QSOs at higher redshift;
- We infer a molecular gas mass (M$_{\rm mol}$) in the range $2.1\pm0.4\div9.5\pm1.9\times10^{10}$ M$_\odot$ applying the QSO/Starburst or MS conversion factors to the measured L’CO line luminosity, respectively, and a total gas mass M$_{\rm gas}\sim4.5\times10^{10}$ M$_\odot$ from dust continuum fitting;
- A value for the molecular gas mass $<10^{11}$ M$_\odot$ is also remarkably consistent with our estimates of the dynamical mass through spectroastrometric methods (see Section 2.2), given the high stellar mass of XID2028;
- We also infer a molecular gas fraction $\mu_{\rm mol}\sim5-20$%. This translates into a gas depletion time scale t$_{\rm depl}\sim$70-340 Myr, depending on the assumptions on $\alpha_{\rm CO}$ (see Figure 4).
- The value of t$_{\rm depl}$ is considerably lower ($\lsimeq30$%) than those observed in systems hosted in similar massive (M$_{\star}>10^{11}$ M$_\odot$) MS galaxies (MS-normalised sSFR$\sim1$), and consistent with those observed for SMGs and for the other few systems proposed to be in the transition phase.
We propose that in XID2028 the QSO wind, detected in the ionised gas component out to 10-kpc scales, has already removed most of the molecular gas from the host galaxy. All the observational constraints (low molecular gas content, lowest $\mu_{\rm mol}$ at a fixed sSFR when compared to M$_\star>10^{11}$ M$_\odot$ systems, and lowest sSFR at a fixed $\mu_{\rm mol}$) are consistent with such a scenario, where the gas in the host galaxy of XID2028 is indeed already depleted/dispersed by the effects of the strong QSO feedback (see also @Coppin2008 and @Yan2010 for similar interpretation). In dense regions (e.g. clumpy M$_{gas}$ reservoirs), possibly located at the edge of the outflow cavity [@Cresci2015], the residual gas is converted into stars at a high rate similar to that observed in SMGs, where the low depletion time scale is indeed ascribed to the efficient SF triggered in dense and compact gas reservoirs.
The measure of the intensity of the [CO(3-2) ]{}emission in XID2028 represents a first step towards a mapping experiment using high spatial resolution to study the morphology and the kinematics of the molecular gas reservoir and of the clumpy structures in the distribution of SF regions seen in HST and SINFONI maps. Sensitive ALMA and/or NOEMA observations of XID2028 will finally give the spatial resolution to locate molecular clouds (see, e.g., @Aravena2014) and reveal any possible molecular outflow component.
Based on observations carried out under project number X–8 with the IRAM PdBI Interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). We gratefully acknowledge the allocation of IRAM DDT time, and we thank the staff of the IRAM Observatory for their support of this program. MB, MP and GL acknowledge support from the FP7 Career Integration Grant “eEASy” (“SMBH evolution through cosmic time: from current surveys to eROSITA-Euclid AGN Synergies", CIG 321913). MB gratefully acknowledges fundings from the DFG cluster of excellence ‘Origin and Structure of the Universe’ (www.universe-cluster.de). We acknowledge financial support from INAF under the contracts PRIN-INAF-2011 (“Black Hole growth and AGN feedback through cosmic time"), PRIN-INAF-2012 (“The Lifecycle of early Black Holes”) and PRIN MIUR 2010-2011 (“The dark Universe and the cosmic evolution of baryons"). We thank Dennis Downes and Andrea Comastri for enlightening discussion. We thank the anonymous referee for his/her interest towards the results of our work, a very careful reading of the paper and useful suggestions which improved the presentation of the results.
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[^1]: email:marcella.brusa3@unibo.it
[^2]: Based on observations with the Plateau de Bure millimetre interferometer, operated by the Institute for Radio Astronomy in the Millimetre Range (IRAM), which is funded by a partnership of INSU/CNRS (France), MPG (Germany) and IGN (Spain).
[^3]: r$_{31}\gsimeq1.0$ assuming CO\[2-1\] is thermalized, and r$_{31}\gsimeq0.9$ assuming CO\[2-1\] is sub-thermally excited $r_{21}=0.84$, standard value for MS objects
[^4]: In the case of lensed quasars, the values are corrected for the amplification, as reported in Riechers 2011.
[^5]: A virtually identical conversion factor would be inferred using the relation between metallicity and $\alpha_{\rm CO}$ calibrated in @Genzel2012 once the offsets between different metallicity calibrations are taken into account.
|
---
author:
- Haoran Man
- Jiangang Guo
- Rui Zhang
- 'Rico U. Schönemann'
- Zhiping Yin
- Mingxuan Fu
- 'M. B. Stone'
- Qingzhen Huang
- Yu Song
- Weiyi Wang
- David Singh
- Felix Lochner
- Tillman Hickel
- Ilya Eremin
- Leland Harriger
- 'Jeffrey W. Lynn'
- Collin Broholm
- Luis Balicas
- Qimiao Si
- Pengcheng Dai
title: Spin excitations and the Fermi surface of superconducting FeS
---
*Correspondence and requests for materials should be addressed to Z.P.Y (yinzhiping@bnu.edu.cn), L.B. (balicas@magnet.fsu.edu), or P.D. (e-mail: pdai@rice.edu)*
Abstract
========
High-temperature superconductivity occurs near antiferromagnetic instabilities and nematic state. Debate remains on the origin of nematic order in FeSe and its relation with superconductivity. Here, we use transport, neutron scattering and Fermi surface measurements to demonstrate that hydro-thermo grown superconducting FeS, an isostructure of FeSe, is a tetragonal paramagnet without nematic order and with a quasiparticle mass significantly reduced from that of FeSe. Only stripe-type spin excitation is observed up to 100 meV. No direct coupling between spin excitation and superconductivity in FeS is found, suggesting that FeS is less correlated and the nematic order in FeSe is due to competing checkerboard and stripe spin fluctuations.
Introduction
============
High-transition temperature (high-$T_c$) superconductivity in copper oxides and iron-based materials occurs near checkerboard and stripe antiferromagnetic (AF) instabilities, respectively [@scalapino; @dai; @si2016]. Although there is also ample evidence for the existence of a nematic order, where a translationally invariant metallic phase spontaneously breaks rotational symmetry [@fradkin; @fisher; @Fernandes2011; @anna; @xylu14], and nematic quantum critical point (QCP) near optimal superconductivity in iron-based superconductors [@Dai2009; @Kuo2016], much is unclear concerning its microscopic origin and relationship to superconductivity [@dai; @si2016]. In particular, recent debates focus on whether the nematic order in superconducting FeSe below the tetragonal-to-orthorhombic transition temperature $T_s=91$ K without static AF order [@McQueen; @baek15; @anna15] is due to competing magnetic instabilities or to orbital ordering [@Rahn; @WangQ; @WangQa; @YuR; @WangF; @Glasbrenner; @CaoHY; @Chubukov15; @Yamakawa]. Here, we use transport, neutron scattering and Fermi surface measurements to demonstrate that superconducting FeS, an isostructure of FeSe [@XLai; @borg], is a tetragonal paramagnet without nematic order and with a quasiparticle mass significantly reduced from that of FeSe. Our neutron scattering experiments in the energy regime below 100 meV reveal only stripe-type spin fluctuations in FeS that are not directly coupled to superconductivity. These properties suggest that FeS is a weakly correlated analog of FeSe and, moreover, that the nematic order in FeSe is due to the frustrated magnetic interactions underlying the competing checkerboard and stripe spin fluctuations [@WangQa; @YuR; @WangF].
A key to understanding the physics of the iron-based superconductors is to determine the role of magnetism and electronic nematic phase to superconductivity [@scalapino; @dai; @si2016; @fisher; @Fernandes2011; @anna]. In a typical AF ordered iron-pnictide, a tetragonal-to-orthorhombic lattice distortion $T_s$ occurs at temperatures above or at the AF ordering temperature $T_N$ [@dai], and the nematic phase is observed in the paramagnetic orthorhombic phase between $T_s$ and $T_N$ [@fisher; @Fernandes2011; @anna]. Although iron chalcogenide FeSe single crystals \[Fig. 1(a) and 1(b)\] also undergo a nematic transition at $T_s$ and become superconducting at $T_c=9.3$ K [@McQueen], the low-temperature static AF ordered phase is absent [@baek15; @anna15]. This has fueled debates concerning the role of AF order and spin fluctuations to the nematic phase and superconductivity [@baek15; @anna15; @Rahn; @WangQ; @WangQa; @YuR; @WangF; @Glasbrenner; @CaoHY; @Chubukov15; @Yamakawa]. Initially, nuclear magnetic resonance (NMR) experiments on FeSe suggested that magnetism plays no role in its nematic transition [@baek15; @anna15]. However, subsequent neutron scattering measurements reveal strong low-energy spin fluctuations at the stripe AF ordering wave vector and a resonance coupled to superconductivity [@Rahn; @WangQ], similar to spin fluctuations in the iron pnictides [@dai]. In addition, recent spin excitation measurements suggest that the nematic transition in FeSe is due to a competition between the checkerboard and the stripe spin fluctuations at AF wave vectors $(1,1)$ and $(1,0)$, respectively \[Fig. 1(c) and 1(d)\] [@WangQa], consistent with the frustrating magnetic interactions [@YuR; @WangF]. In this picture, one would expect that S-substituted FeSe$_{1-x}$S$_x$, which reduces $T_s$ and lattice orthorhombicity [@Watson15; @LRWang], should have reduced spin fluctuations associated with the checkerboard order. As FeS single crystals are isostructural to FeSe but with a reduced $T_c=4.3$ K, it should allow a direct comparison with FeSe [@Rahn; @WangQ; @WangQa], and thus elucidate the role of spin fluctuations to the nematic phase and to superconductivity.
Results
=======
Here, we use transport (Fig. 1), neutron scattering (Figs. 2 and 3), quantum oscillation experiments (Fig. 4), as well as density function theory (DFT) [@Subedi08] and DFT combined with dynamical mean field theory (DMFT) calculations [@Yin; @Yin14] to study single crystals of FeS [@SI]. To search for the presence of a nematic phase in FeS, we performed elastoresistance measurements on single crystals of FeS and BaFe$_{1.97}$Ni$_{0.03}$As$_2$ [@xylu16] using a piezo electric device \[Fig. 1(g)\] [@Kuo2016].
Figure 1(h) compares thestrain dependence of the elastoresistance at different temperatures for FeS, FeSe, and BaFe$_{1.97}$Ni$_{0.03}$As$_2$, respectively. While there is a clear resistivity anisotropy for FeSe and BaFe$_{1.97}$Ni$_{0.03}$As$_2$, indicative of a nematic phase, FeS reveals no anisotropy in measurements of the elastoresistance from 5 K to 105 K. We therefore conclude that FeS has no nematic order, which is consistent with the previous reports on FeS [@XLai; @borg] and with the notion that the nematic phase vanishes for FeSe$_{1-x}$S$_x$ for $x\geq 0.17$ [@Hosoi; @XX1; @XX2]. The results from the transport measurements are complemented by those from elastic neutron scattering measurements, which reveal that the system is paramagnetic at all temperatures [@SI], suggesting that the previous observation of magnetic order in FeS is likely due to impurity phases [@Holenstein; @Kirschner].
Having established the absence of any nematic order in FeS, we turn to probing the spin excitation spectrum by inelastic neutron scattering experiments. Figure 2 summarizes our neutron time-of-flight measurements on FeS to determine the overall wave vector and energy dependence of the spin fluctuations [@SI]. For these measurements, we use orthorhombic unit cell notation and define momentum transfer ${\bf Q}$ in three-dimensional (3D) reciprocal space in Å$^{-1}$ as $\textbf{Q}=H\textbf{a}^\ast+K\textbf{b}^\ast+L\textbf{c}^\ast$, where $H$, $K$, and $L$ are Miller indices and ${\bf a}^\ast=\hat{{\bf a}}2\pi/a$, ${\bf b}^\ast=\hat{{\bf b}}2\pi/b$, ${\bf c}^\ast=\hat{{\bf c}}2\pi/c$ \[Fig. 1(c) and 1(d)\]. Our single crystals are aligned with the $c$-axis along the incident beam and with the $a$-axis in the horizontal plane. In this geometry, we expect that the checkerboard and stripe AF correlations occurs at $(\pm 1,\pm1)$ and $(\pm1,0)$ in-plane wave vectors, respectively. Figure 2(a)-2(d) shows the spin excitations of FeS at energy transfers of $E=20\pm 4, 40\pm 5, 50\pm 7,$ and $59\pm 7$ meV, respectively. In all cases, we see transversely elongated spin excitations centered around the stripe wave vector $(1,0)$ with no obvious magnetic signal at the checkerboard wave vector $(1,1)$. Since magnetic scattering is normalized to absolute units using a vanadium standard [@dai], we can quantitatively compare the results with those of FeSe [@WangQ; @WangQa]. Figures 2(e)-2(h) show the transverse cuts for FeS (solid circles) and FeSe (solid lines) corresponding to energies in Figs. 2(a)-2(d) along the $[1,k]$ direction \[see red dashed lines in Fig. 2(a) for scan direction\]. Integrating the scattering over the same energy interval, we see that the FeS scattering is much weaker, and we do not observe magnetic scattering associated with the checkerboard correlations for energies below 100 meV, in contrast with the clear magnetic scattering of FeSe at $(1,1)$ as marked by vertical arrows in Fig. 2(e)-2(h). Figure 2(i) compares the energy dependence of the local dynamic susceptibility $\chi^{\prime\prime}(E)$, defined as the dynamic susceptibility integrated over the dashed white box in Fig. 2(a) [@dai], for both FeS and FeSe [@WangQa]. Within the energy region probed, $\chi^{\prime\prime}(E)$ increases with increasing energy but has about a quarter of the intensity of FeSe \[Fig. 2(i)\].
To determine if spin excitations in FeS couple to superconductivity, we carried out temperature dependence measurements of the low-energy spin fluctuations near the stripe ordering wave vector $(1,0)$. For this purpose, single crystals of FeS were aligned in the $[H,0,L]$ scattering plane, and maps of scattering intensity at different energies above and below $T_c$ were measured using a cold neutron spectrometer. Figures 3(a)-3(d) show background subtracted scattering maps at $E=0.75$, 2, 4 and 6 meV, respectively, well below $T_c$ at $T=1.5$ K. In all cases, we see rod-like scattering centered at $(1,0,L)$ with extended scattering along the $L$ direction, consistent with short-range $c$-axis spin correlations. In the case of FeSe, a neutron spin resonance coupled to superconductivity was found near $E_r=4$ meV, which correspond to approximately $5.3k_BT_c$ where $k_B$ is the Boltzmann constant, at $(1,0)$ [@Rahn; @WangQ]. Since the $T_c$ of FeS is about half of that of FeSe, the resonance in FeS should be present around $E_r\approx 2$ meV. To accurately determine the temperature dependence of the dynamic susceptibility near $(1,0)$, we integrate the scattering around $(1,0)$ along the $L$ direction, and then fit the profile to a Gaussian on a linear background \[see inset in Fig. 3(e)\]. After correcting for the Bose factor, we show in Fig. 3(e) the temperature dependence of the dynamic susceptibility $\chi^{\prime\prime}(E)$ near the wave vector $(1,0)$. The energy dependence of $\chi^{\prime\prime}(E)$ is weakly temperature dependent below about 10 meV and shows no evidence for a neutron spin resonance expected around $E_r\approx 2$ meV.
The contrast in the spin dynamics between FeS and FeSe is striking and provides the clue to the physics of both systems. We start from the observation that, as in the case of P-for-As substitution [@Dai2009], the reduction of Fe-pnictogen distance on moving from FeSe to FeS facilitates electron hopping, and thus reduces the electron correlations \[Fig. 1(b)\], as seen in spin excitations of BaFe$_2$(As$_{0.7}$P$_{0.3}$)$_2$ [@xx3]. The notion that FeS is a less correlated analogue of FeSe is qualitatively consistent with our conclusion that the spin spectral weight at low energies is much reduced in FeS compared to FeSe \[Fig. 2(i)\].
The stoichiometric nature of FeS facilitates both quantum oscillation measurements and electronic structure calculations, thereby providing the opportunity to address the correlation physics in a more quantitative way. We therefore turn to the understanding of both the Fermi surface and the effective quasiparticle mass. Figure 1(e) shows the calculated Fermi surfaces of FeS using combined DFT and DMFT [@SI]. Comparing with schematics of the measured Fermi surfaces of FeSe in Fig. 1(f) [@Watson15], substituting S for Se in FeSe induces the $d_{xy}$ orbital hole pocket near $(1,1)$ and changes the properties of the hole pockets near the $\Gamma$ point $(0,0)$ \[Fig. 1(e)\]. To quantitatively determine the differences in the Fermi surfaces of FeS and FeSe, we performed torque magnetometry and resistivity measurements under high magnetic fields. Figure 4 summarizes the quantum oscillatory phenomena observed on FeS investigated through torque magnetometry and resistivity measurements under fields as high as $\mu_0H = 35\,\mathrm{T}$ in resistive Bitter magnets equipped with either a $^{3}\mathrm{He}$ refrigerator or $^{4}\mathrm{He}$ cryostat. Resistivity measurements were performed on a sample characterized by a residual resistivity ratio ($RRR = R_{300\,\mathrm{K}}/R_{6\,\mathrm{K}}$) of 41, using a standard four wire technique, while torque was measured through a cantilever beam set-up whose deflection was determined capacitively [@SI]. We were able to observe well pronounced Shubnikov-de Haas (SdH) and de Haas-van Alphen (dHvA) oscillations in the resistance and in torque measurements, respectively. Typical dHvA and SdH oscillations and their respective Fast Fourier Transformations (FFT’s) for $H\parallel c-$axis are shown in Figs. 4(a) and 4(b), respectively. Although their amplitudes differ, most of the SdH frequencies observed below $1\,\mathrm{kT}$, which are indicated by the peaks labeled as $\alpha$, $\beta$, $\kappa$, $\delta$ and $\epsilon$, are reproduced in the dHvA spectrum. Only $\nu$ and $\gamma$ are not visible in the dHvA data. Furthermore, the prominent dHvA peaks at $F = 370\,\mathrm{T}$ and $400\,\mathrm{T}$ seem to be suppressed in the SdH data, which is attributable to the lower temperature for the torque measurements. Here, it is important to emphasize that the SdH-effect is superimposed onto an electrical transport quantity (resistivity) which is driven by scattering processes, while the dHvA one is superimposed onto a thermodynamic variable (magnetic susceptibility) which, in a metal is dependent upon the density of states at the Fermi level. Therefore, it is not surprising that the relative amplitude between peaks observed in the FFT spectra is technique dependent. In addition, different crystals from a given synthesis batch are likely to display variations in mobility. This should affect the detection of some of the orbits and hence also produce comparative differences in the FFT spectra collected from the different crystals, as seen in our experiments.
The effective mass $\mu$ of the different orbits can be extracted from the temperature dependence of the FFT amplitude as depicted in Fig. 4(c). The decrease of the FFT amplitude with increasing temperature is described by the Lifshitz-Kosevich damping factor $R_{T} = \pi\lambda/\sinh(\pi\lambda)$. Considering only the first harmonic, one gets $\lambda = 2\pi k_{\mathrm{B}}T/\beta H$, where $\beta \propto 1/\mu$. This analysis yields effective masses of $1.1(1)m_{0}$ for the $\alpha$, $\beta$ and $\kappa$ orbits as well as $1.7(1)$, $1.8(2)$, $1.9(2)$ and $1.8(2)m_{0}$ for the $\delta$, $\epsilon$, $\nu$ and $\gamma$ orbits. Thus charge carriers in FeS have lower effective masses than those of FeSe whose masses range from 1.9 to $7.2m_{0}$ [@terashima_anomalous_2014].Notice that we obtain somewhat heavier masses for the $\alpha$ and $\beta$ orbits than the values reported in Ref. . We re-analyzed our data by, for instance, extracting the effective masses from different field windows. However, we found that this does not explain the difference between the effective mass values extracted from both studies.
This is consistent with our DFT+DMFT calculations with mass enhancement $m^\ast/m_{band}$ of 1.9/1.6 for $t_{2g}$/$e_g$ orbitals in FeS, which is much smaller than that in FeSe [@Yin]. The whole angular dependence of the SdH and dHvA frequencies as a function of $\theta$ is shown in Fig. 4(d), where $\theta$ denotes the angle between $H$ and the crystallographic $c$-axis. Based on the dHvA measurements, we observe a multitude of frequencies especially in the region between $0.3$ and $0.6\,\mathrm{kT}$ as well as at least three additional Fermi surface pockets with $F\geq 1\,\mathrm{kT}$. While tracking the individual frequencies that belong to certain Fermi surface sheets is a difficult task in the dHvA data, the picture seems to become clearer for the SdH oscillations. Nevertheless, we were not able to observe SdH oscillations for $\theta > 30^{\circ}$. The lines depicted in Fig. 4(d) are intended to provide a hint on the evolution of the frequencies as a function of $\theta$. However, a precise comparison with band structure calculations is required to associate the observed frequencies with specific Fermi surface sheets [@SI]. Band structure calculations find that the Fermi surface consists of two-dimensional (2D) cylindrical Fermi surface sheets at the center and at the corners of the Brillouin zone, respectively [@Subedi08]. 2D orbits would lead to a $F \propto (\cos(\theta))^{-1}$ dependence which are not clearly observed here. Although the angular dependence of some of the frequencies (e. g. $\alpha$ and $\beta$) could match a cylindrical Fermi surface, the bulk of the observed frequencies are clearly 3D in character and cannot be described by the currently available band structure calculations. A recent report on the SdH on FeS crystals detected only the two main peaks observed in our FFT spectra, probably because the measurements were performed at much lower fields [@Terashima_FeS]. However, the authors conclude that the Fermi surface of FeS has a 2D character in contrast to our observations. Nevertheless, in their study the SdH oscillations were observed in a quite narrow angular range, i.e. $\Delta \theta \sim \pm 30^{\circ}$ with respect to the *c*-axis, which is not a wide enough range to reach a definitive conclusion on the dimensionality of its Fermi surface. On the other hand, the observation of two of the same frequencies, or cross sectional areas, in samples grown by different groups further confirms that we are detecting the intrinsic Fermi surface of FeS.
Discussion
==========
In an attempt to further understand the observed quantum oscillations in Figs. 4(a)-4(d), we carried out first-principles DFT plus single-site DMFT calculations in the paramagnetic phase of FeS, using the experimentally determined FeS crystal structure [@SI] and Hubbard $U=5.0$ eV and Hund’s $J=0.8$ eV. When computing the 3D Fermi surface and the dHvA frequencies, we further incorporated the corrections from the long-range exchange interaction by shifting the hole (electron) Fermi surface down (up) by 50 meV. The calculated 3D Fermi surfaces are shown in Fig. 4(e). In particular, the middle hole Fermi surface and the two electron Fermi surfaces are quite 3D like, with large variation of the pocket size (cross section along the $[0,0,1]$ direction) along the $k_z$ direction. As shown in Fig. 4(f), the DFT+DMFT calculated dHvA frequencies agree well with experimental values. We further assign each dHvA frequency to its corresponding position on the 3D Fermi surface [@SI].
The reduced strength of the electrons correlations in FeS compared to FeSe also provides the understanding of the contrast in the spin dynamics of FeS to those of FeSe. Figures 4(g) and 4(h) show the energy dependence of the ground state magnetic scattering $S({\bf Q},E)$ for FeS and FeSe, respectively, calculated through a combination of DFT and DMFT methods [@Yin; @Yin14]. The main conclusion from these calculations is that the spin excitations are much more energetic for FeS than for FeSe, with the strongest scattering for FeSe occurring below 170 meV, while for FeS they extend to well beyond 400 meV similar to the case of iron phosphites [@Yin14].
It is also instructive to compare the spin dynamics of the superconducting state in FeS with the results on FeSe and iron pnictide superconductors. For most iron-based superconductors, the appearance of superconductivity is coupled with changes in the spin excitations with the opening of a spin gap, and inducing a neutron spin resonance near the stripe AF wave vector [@dai]. The presence of a resonance has mostly been interpreted as due to quasiparticle excitations between the hole-Fermi surfaces near the $\Gamma-$point and the electron Fermi surfaces near $(1,0)$ as a consequence of Fermi surface nesting [@dai]. Given the hole and electron Fermi surfaces in FeS \[Fig. 1(e)\] and FeSe \[Fig. 1(f)\], one would expect the presence of spin fluctuations in both materials at the commensurate stripe AF wave vector $(1,0)$. Our finding that FeS is a weakly correlated analog of FeSe provides a natural understanding of the lack of a neutron resonance. More quantitatively, from magnetic and transport measurements, it was argued that FeSe is deep inside Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein-condensate (BCS-BES) cross-over regime, where the ratio of superconducting gap $\Delta$ to Fermi energy $\epsilon_F$ is of the order of unity [@Kasahara; @Kasahara16]. From the experimentally obtained values for the SdH frequencies $F$ and the effective masses $\mu$ in FeS, we can estimate the Fermi energy $\epsilon_F$ by using: $\epsilon_F = \hbar^{2}k_{\mathrm{F}}^{2}/2\mu$, $A = \pi k_{\mathrm{F}}{^2}$ and $F = \hbar A/2\pi e$. Assuming that the superconducting gap $\Delta$ can be estimated by using the BCS formula for a weakly coupled superconductor for FeS: $\Delta(T\rightarrow 0) = 1.764\,k_{\mathrm{B}}T_{\mathrm{c}} = 0.65\,\mathrm{meV}$ with $T_{\mathrm{c}}=4.3\,\mathrm{K}$, we can calculate the ratio of superconducting gap to Fermi energy as shown in the table below. It clearly shows that the electron pairing in FeS is much closer to a BCS superconductor, again in line with our finding of a correlation strength in FeS that is considerably reduced than that of FeSe.
To summarize, our inelastic neutron scattering experiments below 100 meV indicate that the spin excitations in FeS occur at the stripe AF wave vector $(1,0)$ with no observable signal at the checkerboard ordering wave vector $(1,1)$, and are much weaker than those of FeSe (Fig. 2). The weaker correlations in FeS, established by our observation via quantum oscillation measurements of minute enhancement in the effective mass over its non-interacting counterpart, both reduce the low-energy spin spectral weight and push up the energy scale for the $(1,1)$ excitations. The weaker correlations also imply that FeS is much closer to a BCS superconductor, which allow us to understand why the low-energy spin excitations do not directly respond to superconductivity (Fig. 3). These results for the isostructural and stoichiometric FeS highlight the strongly correlated nature of FeSe. Indeed, the electron spectral weight in FeSe mainly resides in the incoherent part, which induces quasi-local moments. The ensuing physics of frustrated magnetism not only yields the nematic order but also is manifested in the co-existing spin excitations at $(1,0)$ and $(1,1)$ wave vectors [@YuR; @WangF]. The strong correlations in FeSe also enhance the effective quasiparticle interactions in its superconducting state, giving rise to a resonance spin excitation in FeSe [@Rahn; @WangQ]. As such, our findings elucidate both the origin of the nematic order and the nature of the superconductivity in FeSe.
Methods
=======
Our quantum oscillation transport measurements on FeS were carried out at National High-magnetic Field laboratory in Tallahassee, Florida [@SI]. Our inelastic neutron scattering measurements were carried out at the Fine-Resolution Fermi Chopper Spectrometer (SEQUOIA) at the Spallation Neutron Source, Oak Ridge National Laboratory and at the Multi Axis Crystal Spectrometer (MACS) at NIST Center for Neutron Research (NCNR), National Institute of Standards and Technology. Sample alignment for MACS and initial charactization is done at Spin Polarized Inelastic Neutron Spectrometer (SPINS), National Institute of Standards and Technology. We have also performed neutron powder diffraction experiments on the BT-1, NCNR. Single crystals of FeS (6.0 g for SEQUOIA and 6.5g for MACS) were grown using hydro-thermo method and characterizations of our samples are discussed in [@SI]. Pieces with size larger than 3\*3 mm$^2$ were used in the neutron scattering experiment. The elasto-resistance measurements were carried out using PPMS with a strain gauge attached on the piezo stack to measure strain at different temperatures. Measurements were performed by changing voltage on piezo stack and results presented here were scaled to actual strain in the sample. To facilitate an easy comparison with the results on FeSe [@Rahn; @WangQ; @WangQa], we used the orthorhombic notation with $a= b\approx 5.19$ Å and $c=5.03$ Å for FeS. In this notation, the stripe AF spin excitations for FeS occur at $(\pm 1,0,L)$ positions in reciprocal space. Samples are co-aligned in the $[H,0,L]$ scattering plane with a mosaic of 8$^\circ$. In the SEQUOIA experiment, the incident beam with $E_i=80,150$ meV was along the $c$-axis of the crystals. In the MACS experiment, $E_f=5$ meV was used for excitations above 1.6 meV and $E_f=3.7$ meV was used for excitations below 1.25 meV. Details of DFT+DMFT calculations are described in [@SI].
Acknowledgments
===============
The single crystal growth and neutron scattering work at Rice is supported by the U.S. DOE, BES under contract no. DE-SC0012311 (P.D.). A part of the materials work at Rice is supported by the Robert A. Welch Foundation Grants No. C-1839 (P.D.). The theoretical work at Rice is supported by the NSF Grant No. DMR-1611392 and the Robert A. Welch Foundation Grant No. C-1411 (Q.S.). Z.P.Y acknowledges financial support by the National Natural Science Foundation of China, Grant No. 11674030, the National Key Research and Development Program of China under contract No. 2016YFA0302300. L. B. is supported by DOE-BES through award DE-SC0002613. The NHMFL is supported by NSF through NSF-DMR-1157490 and the State of Florida. The use of ORNL’s SNS was sponsored by the Scientific User Facilities Division, Office of BES, U.S. DOE.
Author contributions
====================
Single crystal growth and neutron scattering experiments were carried out by H.M., J.G., R.Z. with assistance from M.F., M.S., Q.H., Y.S.,J.W.L., C.B. and P.D.. W.Y.W. and Y.S. performed ICP measurement. Quantum oscillation measurements and analysis were performed by R.U.S. and L.B.. Theoretical understandings were performed by Z.P.Y. (DFT+DMFT), D.J.S, T.H., I.E. (DFT), and Q.S. (Electron correlations and magnetic frustration). P.D. oversees the entire project. The paper was written by P.D., H.M., Z.P.Y., L.B., I.E., Q.S., and all authors made comments.
Competing interest
==================
The authors declare no competing financial interests.
Additional information
======================
Correspondence and requests for materials should be addressed to Z.P.Y (yinzhiping@bnu.edu.cn), L.B. (balicas@magnet.fsu.edu), or P.D. (e-mail: pdai@rice.edu)
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Figure Legends
==============
![[**Crystal structures, real/reciprocal spaces, Fermi surfaces, and transport measurements of FeS and FeSe.**]{} (a) The crystal structures of FeS or FeSe in orthorhombic notation. The sulfur (S) can be fully substituted by selenium (Se) to form FeSe. (b) Schematic illustration of sulfur and selenium atoms in FeS and FeSe compounds. (c) Illustration of stripe (red) and checkerboard (green) static long range AF order in real space. The orthorhombic long-axis direction is along the $a$-axis for stripe AF order. (d) The corresponding positions for stripe and checkerboard orders and excitations in reciprocal space. The areas of the Brillouin zones are marked as pink and blue, respectively. Schematics of Fermi surfaces corresponding to FeS (e) and FeSe (f) with possible nesting wave vectors marked by arrows. The orbital components ([$d_{xz}$, $d_{yz}$, $d_{xy}$]{}) for different Fermi surfaces are shown in different colors. (g) Schematics of the setup used to measure elasto-resistance using a physical property measurement system [@Kuo2016]. (h) Strain dependence of the resistivity anisotropy $\Delta\rho/\rho= 2(\rho_a-\rho_b)/(\rho_a+\rho_b)$ for FeS, FeSe, and BaFe$_{1.97}$Ni$_{0.03}$As$_2$ at different temperatures. ](Figure1)
![ [**Spin excitations of FeS obtained by time-of-flight neutron spectroscopy.**]{} (a)-(d) Constant energy cuts measured at $T=4$ K at the energy transfers indicated on top of each panel. Red dashed lines in (a) indicate integrating area in reciprocal space for the 1D cuts in panels (e)-(h). The white dashed box indicates the area of integration to estimate the local dynamic susceptibility $\chi^{\prime\prime}(E)$ in panel (i). (e)-(h) Constant energy cuts through reciprocal space stripe AF wave vectors along the $[1,K,]$ direction at energies corresponding to panels (a)-(d). The gray solid lines indicate fits to the data extracted from excitations in FeSe at the same energy range [@WangQa]. Gray arrow indicates the checkerboard wave vector observed in FeSe, which is absent in FeS. (i) Comparison of the energy dependence of the local dynamic susceptibility $\chi^{\prime\prime}(E)$ for FeS and FeSe [@WangQa]. The open and filled circles are data taken at $L=0.5, 1.5, \cdots$, and $0,1,2,\cdots$, respectively. ](Figure2)
![ [**Temperature dependence of the low-energy spin excitations of FeS.**]{} 2D images of neutron scattering intensity in the $[H,0,L]$ scattering plane at energies of (a) $E = 1.25$, (b) 2, (c) 4, and (d) 6 meV [@SI]. The high scattering intensity near the Bragg peak positions of $(0,0,\pm 1)$ is due to acoustic phonon scattering. Spin excitations in FeS form a ridge of scattering centered at $(1,0,L)$ positions. (e) Temperature dependence of the stripe AF spin excitations at different energies below and above $T_c=4$ K, respectively. Spin excitations are obtained by integrating $L$ from $-0.7\leq L\leq 0.7$, and fitted with a linear background and a Gaussian peak as shown in the inset. The black line is a fit of the energy dependence of the spin excitations with a relaxation form $\chi^{\prime\prime}(E)=A\Gamma E/[(\Gamma/2)^2+E^2]$, where $\Gamma=8.2\pm 2.8$ meV. Inset: $H$-scans at $E=2$ meV and at 1.5 K and 6 K respectively. The solid lines are Gaussian fits on a linear background. ](Figure3)
![ [**Quantum oscillations, Fermi surfaces and spin fluctuation spectra for FeS.**]{} (a) and (b) de Haas-van Alphen and Shubnikov-de Haas oscillations after background subtraction (red lines) with their associated Fast Fourier transformations (black lines) for magnetic fields applied parallel to the crystallographic $c$-axis. The dHvA signal was obtained at $T = 0.35\,\mathrm{K}$ and the SdH signal at $T= 1.35\,\mathrm{K}$, respectively. Greek letters ($\alpha$, $\beta$, $\gamma$, ...) indicate the most prominent peaks in the FFT spectra which can be assigned to extremal cross sectional areas of the Fermi surface. (c) Fourier transform spectra of the SdH oscillations for $H\parallel c$ at selected temperatures ranging from $1.35\,\mathrm{K}$ to $6.1\,\mathrm{K}$. Insets: temperature dependence of the FFT amplitude for the $\beta$ and $\gamma$ orbits as well as their effective masses as obtained from the Lifshitz-Kosevich formalism (magenta lines). (d) Angular dependence of the dHvA (open diamonds) and SdH (filled symbols) frequencies. The dHvA and SdH measurements cover angles ranging from $H\parallel c$ ($\theta = 0^{\circ}$) to $H\perp c$ ($\theta = 90^{\circ}$). Solid lines represent a suggestion as to how the individual frequencies might evolve as a function of $\theta$. (e) DFT+DMFT calculated 3D Fermi surfaces [@SI]. The Fermi surface drawing is using the tetragonal structure and the corresponding orthorhombic directions $[1,0]$ and $[0,1]$ are marked by arrows. (f) Comparison of frequencies of quantum oscillations with DFT+DMFT calculations. Peak position (black square) is obtained by FFT the magnitude of resistance data shown in blue. The expected ground state spin excitations of FeS (g) and FeSe (h) as calculated by combined DFT+DMFT. ](Figure4)
|
---
abstract: |
This paper presents a factor analysis model for symbolic data, focusing on the particular case of interval-valued variables. The proposed method describes the correlation structure among the measured interval-valued variables in terms of a few underlying, but unobservable, uncorrelated interval-valued variables, called *common factors*. Uniform and Triangular distributions are considered within each observed interval. We obtain the corresponding sample mean, variance and covariance assuming a general Triangular distribution.
In our proposal, factors are extracted either by Principal Component or by Principal Axis Factoring, performed on the interval-valued variables correlation matrix. To estimate the values of the common factors, usually called *factor scores*, two approaches are considered, which are inspired in methods for real-valued data: the Bartlett and the Anderson-Rubin methods. In both cases, the estimated values are obtained solving an optimization problem that minimizes a function of the weighted squared Mallows distance between quantile functions. Explicit expressions for the quantile function and the squared Mallows distance are derived assuming a general Triangular distribution.
The applicability of the method is illustrated using two sets of data: temperature and precipitation in cities of the United States of America between the years 1971 and 2000 and measures of car characteristics of different makes and models. Moreover, the method is evaluated on synthetic data with predefined correlation structures.
author:
- Paula Cheira
- Paula Brito
- 'A. Pedro Duarte Silva'
date: 'Received: date / Accepted: date'
title: Factor Analysis of Interval Data
---
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Introduction {#intro}
============
When, in 1987, Diday introduced symbolic data, he added a new dimension to data analysis and made us think about data in a new manner. Until then, in multivariate data analysis, data were represented in a data table where for each statistical unit (individual/object) a single value, numerical or categorical, was observed for each variable. However, this structure is unable to represent more complete and complex data, where the information for a statistical unit on each variable cannot be reduced to one single value. Symbolic Data Analysis extends the classical data model with the introduction of new types of statistical units and new types of variables. In this new model the statistical units (entities of interest) may be individuals/objects or classes of individuals/objects, usually called symbolic objects, described by variables, which allow representing explicitly any inherent data variability. We distinguish three new types of variables: interval, multi-valued (numerical and categorical) and distributional (histogram and categorical modal) variables [@Billard2006; @Bock2000; @Brito2014].
In this paper, we are interested in the analysis of interval data, i.e., where the statistical units are characterized by variables whose realizations are intervals of $\mathbb{R}$. There are numerous situations that give rise to interval data. A natural source of interval data is the aggregation of individual observations described by real values, in groups/classes according to some question of interest, when the databases are too large for direct analysis. Another source of interval data are original symbolic data - examples are descriptions of biological species or technical specifications. *Native* interval data, which occur when describing ranges of variables values, constitute yet another source of interval data - daily stock prices and daily temperatures, are examples of this type of data. Imprecise data, from repeated measures or confidence interval estimation, can also be represented by interval-valued variables.
Since its introduction, the analysis of symbolic data has known a considerable development, becoming one of the new lines of research in Multivariate Statistics and Data Analysis. The present paper introduces a factor analysis model for symbolic data, focusing on the particular case of interval-valued variables. The essential purpose of factor analysis is to explain the covariance and/or correlation structure among the measured variables [@Johnson2002; @Johnson1998]. In fact, when a large number of variables is measured on each statistical unit, the study of its dependence structure may be of interest. The proposed factor analysis model assumes that there is a smaller set of uncorrelated interval-valued variables - factors - that explain the relations between the interval-valued variables that were actually measured. With the new variables it is expected to get a better understanding of the data being analyzed, moreover, they may be used in future analysis. Two cases are considered for the distribution assumed within each interval: an Uniform distribution and a Triangular distribution [@Bertrand2000; @Billard2008].
In our proposal, factors are extracted by Principal Components or by Principal Axis Factoring, performed on the correlation matrix of the interval-valued variables. First and second sample order moments for interval-valued variables have been derived for the Uniform distribution by [@Bertrand2000; @Billard2003; @Billard2008] and for the Symmetric Triangular distribution by [@Billard2008]. In this paper, we obtain the formula for the sample mean, variance and covariance assuming a general Triangular distribution.
To estimate the factor scores, two approaches will be considered, which are inspired in methods for real-valued data: the Bartlett and the Anderson-Rubin methods [@DiStefano2009]. In both cases, the estimated values are obtained by solving an optimization problem that uses as criterion to be minimized a function of the weighted squared Mallows distance [@Mallows1972] between quantile functions. In the first method the factor scores are highly correlated with their corresponding factor and weakly (or not at all) with other factors. However, the estimated factor scores of different factors may still be correlated. In the second proposed method, the function to minimize is adapted to ensure that the factor scores are themselves not correlated with each other.
In this work the Mallows distance will be the measure used to evaluate the dissimilarity between distributions. In the last century, several dissimilarity measures between probability distributions were proposed, that Gibbs and Su [@Gibbs2002] and Bock and Diday [@Bock2000] reviewed and summarized. They present some of the most important metrics on probability measures that are used by statisticians and probabilists, performing a rigorous analysis of their properties and of the relations between them. Among all, the Mallows distance[^1] is considered by many researchers in various areas of study, the appropriate measure to evaluate the dissimilarity between probability distributions [@Mallows1972; @Arroyo2008; @Verde2007; @Irpino2008]. In the search of the best measure to quantify the error of a forecast, Arroyo [@Arroyo2008] studied several divergence measures concluding that only the Mallows and Wasserstein distances are adequate to represent an error measure. According to Arroyo and Maté [@Arroyo2008; @Arroyo2009] these measures have a clear and intuitive interpretation and are the ones that better adjust to the concept of distance as assessed by the human eye. Naturally, the Mallows distance was chosen in the Arroyo and Maté works [@Arroyo2008; @Arroyo2009], on forecasting time series of histogram-valued variables, to measure the error between the observed and forecasted distributions and to calculate the forecasts. It is noteworthy that the Mallows distance has been successfully applied in several others works in the context of Symbolic Data Analysis. Irpino and Verde [@Irpino2006] derived the Mallows distance between intervals assuming an Uniform distribution within them. They then used this measure for an agglomerative hierarchical clustering of histogram data [@Irpino2006] and as criterion function in a Dynamic Clustering Algorithm (DCA) applied to interval data and histogram data [@Irpino2008; @Verde2007]. Irpino and Verde [@Irpino2015; @Verde2010] and Dias and Brito [@Dias2015] also used this distance in a linear regression context. Here we deduce the explicit expression of the Mallows distance assuming a general Triangular distribution within each interval.\
The structure of this paper is as follows: In Section 2 we start by introducing interval-valued variables, review existing interval representations and fix notation. Then we present sample moments, quantile function representations and Mallows distance, assuming an Uniform or a general Triangular distribution within intervals. Section 3 presents a factor analysis model for interval-valued variables, where factor extraction is done by Principal Components or by Principal Axis Factoring on the correlation matrix between the interval-valued variables. Two approaches are considered to estimate the factor scores which are inspired in the Bartlett and the Anderson-Rubin methods. Section 4 shows the soundness of our proposal with synthetic data. Section 5 presents applications to one data set of measures of car characteristics of different makes and models and another data set of meteorological data. Section 6 concludes de paper.
Interval-valued variables {#sec:2}
=========================
Consider a set of $n$ units $S=\{s_1,\ldots,s_n\}$ under study. An interval-valued variable is defined by an application $Y : S \rightarrow B$ such that for each $s_i \in S$, $Y(s_i)=[l_i, u_i]$, with $l_i \le u_i$, where $B$ is the set of intervals of an underlying set $O \subseteq \mathbb{R}$. Thus the value of an interval-valued variable $Y$ for each $s_i \in S$ is defined by the bounds $l_i$ and $u_i$. Alternatively, the interval $Y(s_i)$ can be represented by its center (midpoint of the interval) $c_i= \displaystyle\frac{l_i+u_i}{2}$ and half-range $r_i= \displaystyle\frac{u_i-l_i}{2}$, then $Y(s_i)=[c_i-r_i, c_i+r_i]$.\
Let $I$ be an $n \times p$ matrix representing the values of $p$ interval-valued variables on $S$. Each unit $s_i \in S$ is represented by a $p$-uple vector of intervals, $I_i=(I_{i1}, ... , I_{ip}), i=1, ... , n,$ with $I_{ij} = [l_{ij}, u_{ij}], j=1,\ldots, p$ (see Table \[table1\]).
[|c|ccccc|]{} & $Y_{1}$ & …& $Y_{j}$ & …& $Y_{p}$\
$s_{1}$ &
$\left[l_{11},u_{11}\right]$
& …&
$\left[l_{1j},u_{1j}\right]$
& …&
$\left[l_{1p},u_{1p}\right]$
\
& & & & &\
$s_{i}$ &
$\left[l_{i1},u_{i1}\right]$
& …&
$\left[l_{ij},u_{ij}\right]$
& …&
$\left[l_{ip},u_{ip}\right]$
\
& & & & &\
$s_{n}$ &
$\left[l_{n1},u_{n1}\right]$
& …&
$\left[l_{nj},u_{nj}\right]$
& …&
$\left[l_{np},u_{np}\right]$
\
\[table1\]
**Notation:** From now on, in order to simplify, we will denote by $ Y_i $ the value of the interval-valued variable $Y$ measured on unit $ s_i $ when we are working with only one variable, instead of $ Y(s_i) $. If we measure various variables on the same unit $ s_i $, and we want to mention the values of the variables $ Y_j $ and $ Y_{j'} $, we will write $ Y_{ij} $ and $ Y_{ij'} $, instead of $ Y_j(s_i) $ and $ Y_{j'}(s_i) $.
Interval data as quantile functions {#sec2:1}
-----------------------------------
In this work we will also resort to the representation of the interval $Y_i$ by the respective quantile function (the inverse of the distribution function $\Psi_{Y_i}(y)$), $\Psi_{Y_i}^{-1}(t)$ with $t \in \left[0,1\right]$, assuming a specific distribution within the interval: Uniform distribution or Triangular distribution.\
If we assume that values within the interval $Y_i=[l_i, u_i]$ follow an Uniform distribution, its representation by the associated quantile function is given by $$\label{QFU1}
\Psi_{Y_i}^{-1}(t)=l_{i}+(u_{i}-l_{i})t, \qquad 0\leq t \leq 1,$$
or
$$\label{QFU2}
\Psi_{Y_i}^{-1}(t)=c_{i}+r_{i}(2t-1), \qquad 0\leq t \leq 1,$$
as a function of the center $c_i$ and half-range $r_i$.\
Assuming a Triangular distribution within interval $Y_i=[l_i, u_i]$, with mode $m_i$, the associated quantile function is as follows : $$\label{QFT1}
\Psi_{Y_i}^{-1}(t)=
\begin{cases}
l_{i}+\sqrt{(u_{i}-l_{i})(m_{i}-l_{i})t} , & \quad 0\leq t \leq \dfrac{m_{i}-l_{i}}{u_{i}-l_{i}}\\
u_{i}-\sqrt{(u_{i}-l_{i})(u_{i}-m_{i})(1-t)} , & \quad \dfrac{m_{i}-l_{i}}{u_{i}-l_{i}} < t \leq 1
\end{cases}$$
or, using the center $c_i$ and half-range $r_i$,
$$\label{QFT2}
\Psi_{Y_i}^{-1}(t)=
\begin{cases}
c_{i}-r_{i}+\sqrt{2r_{i}(m_{i}-c_{i}+r_{i})t} , & \quad 0\leq t \leq \dfrac{m_{i}-c_{i}}{2r_{i}}+\dfrac{1}{2}\\
c_{i}+r_{i}-\sqrt{2r_{i}(c_{i}+r_{i}-m_{i})(1-t)} , & \quad \dfrac{m_{i}-c_{i}}{2r_{i}}+\dfrac{1}{2} < t \leq 1
\end{cases}$$
In the particular case of the Symmetric Triangular distribution, that is, when $m_i=\dfrac{l_{i}+u_{i}}{2}$, expressions and become, respectively, $$\label{QFT3}
\Psi_{Y_i}^{-1}(t)=
\begin{cases}
l_{i}+\dfrac{u_{i}-l_{i}}{\sqrt{2}}\sqrt{t} , & \quad 0\leq t \leq \dfrac{1}{2}\\
u_{i}-\dfrac{u_{i}-l_{i}}{\sqrt{2}}\sqrt{1-t} , & \quad \dfrac{1}{2} < t \leq 1
\end{cases}$$
and
$$\label{QFT4}
\Psi_{Y_i}^{-1}(t)=
\begin{cases}
c_{i}-r_{i}+r_{i}\sqrt{2t} , & \quad 0\leq t \leq \dfrac{1}{2}\\
c_{i}+r_{i}-r_{i}\sqrt{2(1-t)} , & \quad \dfrac{1}{2} < t \leq 1.
\end{cases}$$
\
It is important to note that if we multiply an interval $Y_i$ by a positive real number $ \lambda $, $ \lambda \Psi_{Y_i}^{-1}(t) $ is the quantile function that represents the resulting interval $\lambda Y_i=[\lambda l_i, \lambda u_i]$, but if $ \lambda $ is a negative real number, the interval $\lambda Y_i=[\lambda u_i, \lambda l_i]$ is represented by the quantile function $ \lambda \Psi_{Y_i}^{-1}(1-t) $. For more details about the behavior of quantile functions, see Dias [@Dias2014].
Descriptive statistics {#sec2:2}
----------------------
All factor analysis models rely on properly defined correlation matrices. In our proposed model, correlations between interval-valued variables are defined as the quotients between covariances and products of standard deviation for interval-valued variables, which depend on the distribution assumed within each interval. Bertrand and Goupil [@Bertrand2000] were the first to propose the univariate statistics for this type of numerical symbolic variables: the mean and variance of a interval variable $Y$, defined on the set of $n$ units $S=\{s_1,\ldots,s_n\}$, correspond to those of a finite mixture of $n$ probability density (or frequency) functions, in which it is assumed that each unit is equally likely to be observed with probability $ \dfrac{1}{n}$. It is well known (see, for instance, [@Fruhwirth-Schnatter2006]) that given $n$ variables $Y_i$ with probability density functions $f_i$, $i=1,\ldots,n$, with means $\mu_{i}$ and variances $\sigma_{i}^2$, the variable $Y$ with the finite mixture probability density function $f$: $$\label{Fdp_finmix}
f(y)= \sum_{i=1}^{n} \frac{1}{n} f_i(y) = \frac{1}{n} \sum_{i=1}^{n} f_i(y),$$
has mean and variance, respectively, given by,
$$\label{Mean_finmix}
\mu=E \left[Y \right] = \frac{1}{n} \sum_{i=1}^{n} \mu_{i}$$
$$\label{Var_finmix}
\sigma^2 = E \left[ \left( Y-\mu \right) ^2 \right]
= \frac{1}{n} \sum_{i=1}^{n} \left( \mu_{i}^2+ \sigma_{i}^2 \right) - \mu^2.$$\
In the next subsection we present expressions for the symbolic sample mean, the symbolic sample variance and the symbolic sample covariance assuming a Uniform distribution or a Triangular distribution within the intervals.\
### Uniform distribution {#sec2:2:1}
We now assume an Uniform distribution within each interval $Y(s_i) = I_{i}=[l_{i},u_{i}]$, $i=1,\ldots,n$. Under these conditions, Bertrand and Goupil [@Bertrand2000] obtained the symbolic sample mean and the symbolic sample variance of the interval-valued variable $Y$, respectively, as $$\label{MeanU_1}
\overline{Y} = \frac{1}{2n}\sum_{i=1}^{n}(l_{i}+u_{i})$$ $$\label{VarU1_1}
S_{Y}^2 = \displaystyle \frac{1}{3n}\sum_{i=1}^{n}(l_{i}^2+l_{i}u_{i}+u_{i}^2)-\overline{Y}^2$$
or, expressed in terms of the centers $c_i$ and half-ranges $r_i$ of the interval $I_{i}$,
$$\label{MeanU_2}
\overline{Y} = \frac{1}{n}\sum_{i=1}^{n}c_{i} = \frac{1}{n}\sum_{i=1}^{n}\mu_{i}$$
$$\label{VarU_2}
S_{Y}^2 = \displaystyle \frac{1}{n}\sum_{i=1}^{n} \Big(\frac{r_{i}^2}{3}+c_{i}^2\Big)-\overline{Y}^2
= \displaystyle \frac{1}{n}\sum_{i=1}^{n}\sigma_{i}^2 +\displaystyle \frac{1}{n}\sum_{i=1}^{n}\mu_{i}^2-\overline{Y}^2\\,$$
\
obtained from the empirical density function for an interval variable. As it can be seen above, the sample variance of the interval-valued variable $Y$ is the sum of the average of the (within) variances of observed intervals with the variance of the means (or centers) of the intervals.\
For the symbolic sample covariance three definitions were proposed.\
Let $Y_{j}$ and $Y_{j'}$ be two interval-valued variables such that for each $s_i \in S=\{s_1,\ldots,s_n\}$, the observed $Y_{ij}$ and $Y_{ij'}$ values, $i=1,\ldots,n$, are uniformly distributed within each interval $I_{ik}=\left[ l_{ik},u_{ik} \right] = \left[ c_{ik}-r_{ik},c_{ik}+r_{ik} \right]$, $k=j,j'$, respectively.\
The first expression for the sample covariance between two interval-valued variables $Y_{j}$ and $Y_{j'}$ was obtained in 2003, by Billard and Diday [@Billard2003], from the joint density function, it is denoted Covariance 1 ($Cov_{1}$) : $$\label{Cov1U_1}
Cov_{1}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{4n}\sum_{i=1}^{n}(l_{ij}+u_{ij})(l_{ij'}+u_{ij'})-\overline{Y}_{j} \overline{Y}_{j'}$$
or, equivalently,
$$\label{Cov1U_2}
Cov_{1}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{n}\sum_{i=1}^{n} c_{ij} c_{ij'}-\overline{Y}_{j} \overline{Y}_{j'}.$$
\
Expression is the classic definition of covariance applied to the centers of the intervals $Y_{ij}$ and $Y_{ij'}$. However, it is noted that when one considers the variables $Y_{j} = Y_{j'}$, the expression of Covariance 1 does not reduce to the variance in expression . Furthermore, the resulting Covariance 1 function would not reflect the internal cross-variations between $Y_{j}$ and $Y_{j'}$.\
In 2006, Billard and Diday [@Billard2006] proposed a new expression for the symbolic sample covariance, denoted by Covariance 2 ($Cov_{2}$), incorporating more accurately both between and within interval variations into the overall covariance, $$\label{Cov2U}
Cov_{2}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{3n}\sum_{i=1}^{n}G_{j}G_{j'}\left[Q_{j}Q_{j'}\right]^{1/2}$$
where, for $k=j,j'$,
$$Q_{k}=(l_{ik}-\overline{Y}_{k})^2+(l_{ik}-\overline{Y}_{k})(u_{ik}-\overline{Y}_{k})+(u_{ik}-\overline{Y}_{k})^2$$
$$G_{k}=
\begin{cases}
-1 & \text{if} \quad \overline{Y}_{ik} \leq \overline{Y}_{k}\\
\;1 & \text{if} \quad \overline{Y}_{ik} > \overline{Y}_{k}
\end{cases}$$
and $\overline{Y}_{ik} = \displaystyle \frac{l_{ik}+u_{ik}}{2}$. The $Q_{k}$ and $G_{k}$ expressions can be rewritten in terms of the center $c_{ik}$ and half-range $r_{ik}$ of the interval $I_{ik}$ as
$$Q_{k}=3(c_{ik}-\overline{Y}_{k})^2+r_{ik}^2 \qquad \text{and} \qquad
G_{k}=
\begin{cases}
-1 & \text{if} \quad c_{ik} \leq \overline{Y}_{k}\\
\;1 & \text{if} \quad c_{ik} > \overline{Y}_{k}
\end{cases},\quad \text{for} \; k=j,j'.$$
When the variables $Y_{j} = Y_{j'}$, the expression of Covariance 2 coincides with the expression (or ) of variance. In 2008 Billard [@Billard2008] presented a new formulation, considering a decomposition of the Total Sum of Products (TotalSP), between the variables $Y_{j}$ and $Y_{j'}$, into Within Observations Sum of Products (WithinSP) and Between Observations Sum of Products (BetweenSP), named Covariance 3 ($Cov_{3}$), $$\label{Cov3U_1_1}
Cov_{3}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{n}
\underbrace{\sum\limits_{i=1}^{n}\frac{(u_{ij}-l_{ij})(u_{ij'}-l_{ij'})}{12}}_\text{WithinSP}+\frac{1}{n}
\underbrace{\sum\limits_{i=1}^{n}\Big(\frac{l_{ij}+u_{ij}}{2}-\overline{Y}_{j}\Big)\Big(\frac{l_{ij'}+u_{ij'}}{2}-\overline{Y}_{j'}\Big)}_\text{BetweenSP}$$ $$\label{Cov3U_1_2}
\qquad\qquad\qquad\quad = \displaystyle \frac{1}{12n}
\sum\limits_{i=1}^{n}(u_{ij}-l_{ij})(u_{ij'}-l_{ij'})+\frac{1}{4n}
\sum\limits_{i=1}^{n}(l_{ij}+u_{ij})(l_{ij'}+u_{ij'})-\overline{Y}_{j}\overline{Y}_{j'}$$
or, equivalently,
$$\label{Cov3U_2_1}
Cov_{3}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{3n}
\sum\limits_{i=1}^{n}{r_{ij}r_{ij'}}+\frac{1}{n}
\sum\limits_{i=1}^{n}c_{ij}c_{ij'}-\overline{Y}_{j}\overline{Y}_{j'}$$
$$\label{Cov3U_2_2}
\qquad\qquad\qquad = \displaystyle \frac{1}{n}
\sum\limits_{i=1}^{n}{\sigma_{ij}\sigma_{ij'}}+\frac{1}{n}
\sum\limits_{i=1}^{n}\mu_{ij}\mu_{ij'}-\overline{Y}_{j}\overline{Y}_{j'}\\$$
\
Note that the sample Covariance 3 between two interval-valued variables $Y_{j}$ and $Y_{j'}$ is the sum of the average of the product of the standard deviations of the intervals $Y_{ij}$ and $Y_{ij'}$ with the covariance between the means (or centers) of the intervals. Also noteworthy that expression of covariance reduces to the expression of variance when one considers the variables $Y_{j} = Y_{j'}$ and is equivalent to $$\label{Cov3U_2_3}
Cov_{3}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{n}
\sum\limits_{i=1}^{n}{\sigma_{ij}\sigma_{ij'}} + Cov_{1}(Y_{j},Y_{j'})\\$$ which shows the relation between the expressions of the covariance 1 and 3.\
### Triangular distribution {#sec2:2:2}
If a random variable $Y_i$ follows a Triangular distribution within each interval $\left[ l_i, u_i \right]$ with mode $m_{i}$, i.e., $Y_i \sim \mathcal{T}(l_i, u_i, m_i)$ then, the probability density function of $Y_i$ is
$$\label{FdpTriangular}
f_i(x)=
\begin{cases}
\displaystyle \frac{2(x-l_i)}{(u_i-l_i)(m_i-l_i)} & \text{for} \quad l_i \leq x \leq m_i\\
\displaystyle \frac{2(u_i-x)}{(u_i-l_i)(u_i-m_i)} & \text{for} \quad m_i < x \leq u_i\\
\qquad \quad 0 & \text{for any other case}
\end{cases}$$
with mean and variance
$$\label{MeanVarT}
\mu_{i} = E(Y_{i}) = \frac{l_{i}+u_{i}+m_{i}}{3} = \frac{2c_{i}+m_{i}}{3}$$
$$\label{VarVarT}
\sigma_{i}^2 = Var(Y_{i}) = \frac{l_{i}^2+u_{i}^2+m_{i}^2-l_{i}u_{i}-l_{i}m_{i}-u_{i}m_{i}}{18} = \frac{\left(c_{i}-m_{i}\right)^2}{18} + \frac{r_{i}^2}{6},$$
\
where $c_{i}$ and $r_{i}$ are, respectively, the center and the half-range of the interval $\left[l_i, u_i\right]$.\
\
Under the assumption that the observed $Y_i$ values, for each $s_i \in S=\{s_1,\ldots,s_n\}$, follow a Triangular distribution within each interval $[l_i, u_i]$ with mode $m_{i}$, $i=1,\ldots,n$, the symbolic sample mean and the symbolic sample variance of the interval variable $Y$ are given, respectively, by, $$\label{MeanT_1}
\overline{Y}= \displaystyle \frac{1}{3n}\sum_{i=1}^{n} \left( l_{i}+u_{i}+m_{i} \right)$$ $$\label{VarT_1}
S_{Y}^2=\displaystyle \frac{1}{6n}\sum_{i=1}^{n} \left( l_{i}^2+u_{i}^2+m_{i}^2+l_{i}u_{i}+l_{i}m_{i}+u_{i}m_{i} \right) -\overline{Y}^2,$$
obtained from the empirical density function, following the same line of reasoning of Bertrand and Goupil [@Bertrand2000], in determining expressions and . Expressions and may be rewritten as
$$\label{MeanT_2}
\overline{Y}= \displaystyle \frac{1}{3n}\sum_{i=1}^{n} \left( 2c_{i}+m_{i}\right) = \frac{1}{n}\sum_{i=1}^{n}\mu_{i}$$
$$\label{VarT_2}
S_{Y}^2=\displaystyle \frac{1}{6n} \sum_{i=1}^{n} \left( 3c_{i}^2+2c_{i}m_{i}+m_{i}^2+r_{i}^2 \right) -\overline{Y}^2
= \displaystyle \frac{1}{n}\sum_{i=1}^{n}\sigma_{i}^2 +\displaystyle \frac{1}{n}\sum_{i=1}^{n}\mu_{i}^2-\overline{Y}^2,$$
\
when expressed in terms of the center $c_i$ and half-range $r_i$ of the interval $[l_i, u_i]$. Similarly to what occurs with the Uniform distribution, also the sample variance of the interval-valued variable $Y$ with Triangular distribution can be written as the sum of the average of the variances of the intervals with the variance of the centers of the intervals.\
\
Following the same reasoning presented for the Uniform distribution [@Billard2003; @Billard2006; @Billard2008] we can also get three expressions for the symbolic sample covariance between two interval variables.\
Suppose $Y_{j}$ and $Y_{j'}$ are two interval-valued variables, such that for each $s_i \in S=\{s_1,\ldots,s_n\}$, the observed $Y_{ij}$ and $Y_{ij'}$ values follow a Triangular distribution within each interval $I_{ik}=\left[ l_{ik},u_{ik} \right] = \left[ c_{ik}-r_{ik},c_{ik}+r_{ik} \right]$ with mode $m_{ik}$, $k=j,j'$, respectively, $i=1,\ldots,n$.\
From the joint density function an expression for the covariance was obtained, which we will denote by Covariance 1 ($Cov_{1}$), $$\label{Cov1T_1}
Cov_{1}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{9n}\sum_{i=1}^{n}
(l_{ij}+u_{ij}+m_{ij})(l_{ij'}+u_{ij'}+m_{ij'})-\overline{Y}_{j}.\overline{Y}_{j'}$$
or, equivalently,
$$\label{Cov1T_2}
Cov_{1}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{9n}\sum_{i=1}^{n} (2c_{ij}+m_{ij})(2c_{ij'}+m_{ij'})-\overline{Y}_{j}.\overline{Y}_{j'}.$$
Incorporating more accurately both between and within interval variations into the overall covariance, a new expression can be defined for symbolic sample covariance, denoted by Covariance 2 ($Cov_{2}$), as follows $$\label{Cov2T}
Cov_{2}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{6n}\sum_{i=1}^{n}G_{j}G_{j'}\left[Q_{j},Q_{j'}\right]^{1/2}$$
with,
$Q_{k}=(l_{ik}-\overline{Y}_{k})^2+(u_{ik}-\overline{Y}_{k})^2+(m_{ik}-\overline{Y}_{k})^2+(l_{ik}-\overline{Y}_{k})(u_{ik}-\overline{Y}_{k})
+$\
$(l_{ik}-\overline{Y}_{k})(m_{ik}-\overline{Y}_{k})
+(u_{ik}-\overline{Y}_{k})(m_{ik}-\overline{Y}_{k})$,\
$$G_{k}=
\begin{cases}
-1 & \text{if} \quad \overline{Y}_{ik} \leq \overline{Y}_{k}\\
\;1 & \text{if} \quad \overline{Y}_{ik} > \overline{Y}_{k}
\end{cases}$$
\
and $\overline{Y}_{ik}= \displaystyle \frac{l_{ik}+u_{ik}+m_{ik}}{3}$, for $k=j,j'$. $Q_{k}$ can be rewritten in terms of the center $c_{ik}$ and half-range $r_{ik}$ of the interval $I_{ik}$ as $$Q_{k}=3(c_{ik}-\overline{Y}_{k})^2+2(c_{ik}-\overline{Y}_{k})(m_{ik}-\overline{Y}_{k})+(m_{ik}-\overline{Y}_{k})^2+r_{ik}^2.$$
Considering a decomposition of the Total Sum of Products (TotalSP), between the variables $Y_{j}$ and $Y_{j'}$, into Within Observations Sum of Products (WithinSP) and Between Observations Sum of Products (BetweenSP), as suggested by Billard [@Billard2008], we obtain a new expression for the covariance named Covariance 3, ($Cov_{3}$): $$\begin{aligned}
\label{Cov3T_1}
Cov_{3}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{n}
\underbrace{\sum\limits_{i=1}^{n}\sqrt{\dfrac{W_{ij}}{18}\dfrac{W_{ij'}}{18}}}_\text{WithinSP}
+\frac{1}{n} \underbrace{\sum\limits_{i=1}^{n}\Big(\frac{l_{ij}+u_{ij}+m_{ij}}{3}-\overline{Y}_{j}\Big)\Big(\frac{l_{ij'}+u_{ij'}+m_{ij'}}{3}-\overline{Y}_{j'}\Big)}_\text{BetweenSP}\end{aligned}$$ $$\begin{aligned}
\label{Cov3T_2}
\qquad\qquad\;= \displaystyle \frac{1}{18n}\sum\limits_{i=1}^{n}\sqrt{W_{ij}W_{ij'}}
+\frac{1}{9n}\sum\limits_{i=1}^{n}(l_{ij}+u_{ij}+m_{ij})(l_{ij'}+u_{ij'}+m_{ij'})-\overline{Y}_{j}\overline{Y}_{j'}\end{aligned}$$
where,\
$$W_{ik}=l_{ik}^2+u_{ik}^2+m_{ik}^2-l_{ik}u_{ik}-l_{ik}m_{ik}-u_{ik}m_{ik},\quad \text{for} \; k=j,j' .$$
This last expression of covariance may be written in terms of the center $c_{ik}$ and half-range $r_{ik}$ of the interval $I_{ik}$ as: $$\begin{aligned}
\label{Cov3T_3_1}
Cov_{3}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{18n}\sum\limits_{i=1}^{n}\sqrt{\Big( \left( c_{ij}-m_{ij} \right)^2 + 3r_{ij}^2 \Big)\Big( \left( c_{ij'}-m_{ij'} \right)^2 + 3r_{ij'}^2 \Big)}\nonumber\\
+\frac{1}{9n}\sum\limits_{i=1}^{n}(2c_{ij}+m_{ij})(2c_{ij'}+m_{ij'})-\overline{Y}_{j}\overline{Y}_{j'}\end{aligned}$$ $$\label{Cov3T_3_2}
= \displaystyle \frac{1}{n}
\sum\limits_{i=1}^{n}{\sigma_{ij}\sigma_{ij'}}+\frac{1}{n}
\sum\limits_{i=1}^{n}\mu_{ij}\mu_{ij'}-\overline{Y}_{j}\overline{Y}_{j'} \quad \quad \quad \quad$$\
When the variables $Y_{j} = Y_{j'}$, the expressions of Covariance 2 and 3 coincide with the expression (or ) of variance, but this is not the case for the expression of Covariance 1. As previously, the sample Covariance 3 between two interval-valued variables $Y_{j}$ and $Y_{j'}$ is the sum of the average of the product of the standard deviations of the intervals $Y_{ij}$ and $Y_{ij'}$ with the covariance between the means of the intervals. In addition, it is related with Covariance 1 according to the following expression, $$\label{Cov3T_3_3}
Cov_{3}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{n}
\sum\limits_{i=1}^{n}{\sigma_{ij}\sigma_{ij'}} + Cov_{1}(Y_{j},Y_{j'})$$\
When the observed $Y_i$ values, for each $s_i \in S=\{s_1,\ldots,s_n\}$, follow a Triangular distribution within the interval $[l_i, u_i]$ and the mode $m_{i}$ coincides with the center of the interval, i.e. $Y_i \sim \mathcal{T}(l_i, u_i, c_i)$, it is said that it follows a Symmetric Triangular distribution. In this case, the expressions presented above become much simpler. The mean and variance of $Y_i$ are given by [@Billard2008]: $$\label{MeanVarTS}
\mu_{i} = E(Y_{i}) = \frac{l_{i}+u_{i}}{2} = c_{i}$$ $$\label{VarVarTS}
\sigma_{i}^2 = Var(Y_{i}) = \frac{(u_{i}-l_{i})^2}{24} = \frac{r_{i}^2}{6}.\\$$\
The sample mean and variance of the interval variable Y becomes [@Billard2008]: $$\overline{Y}= \displaystyle \frac{1}{2n}\sum_{i=1}^{n} (l_{i}+u_{i})$$ $$\label{MeanTS}
\qquad\quad = \displaystyle \frac{1}{n}\sum_{i=1}^{n} c_{i}=\displaystyle \frac{1}{n}\sum_{i=1}^{n} \mu_{i}$$ $$S_{Y}^2=\displaystyle \frac{1}{24n}\sum_{i=1}^{n} (7l_{i}^2+10l_{i}u_{i}+7u_{i}^2) -\overline{Y}^2$$ $$\label{VarTS}
\quad\qquad\qquad\qquad\qquad = \displaystyle \frac{1}{n}\sum_{i=1}^{n} \Big(\frac{r_{i}^2}{6}+c_{i}^2\Big) -\overline{Y}^2
= \displaystyle \frac{1}{n}\sum_{i=1}^{n}\sigma_{i}^2 +\displaystyle \frac{1}{n}\sum_{i=1}^{n}\mu_{i}^2-\overline{Y}^2,$$
and the expressions of covariance between $Y_j$ and $Y_{j'}$ interval-valued variables as follows,
$$\label{Cov1TS}
Cov_{1}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{4n}\sum_{i=1}^{n}
(l_{ij}+u_{ij})(l_{ij'}+u_{ij'})-\overline{Y}_{j}.\overline{Y}_{j'}
= \displaystyle \frac{1}{n}\sum_{i=1}^{n} c_{ij}c_{ij'}-\overline{Y}_{j}.\overline{Y}_{j'}.$$
$$\label{Cov2TS}
Cov_{2}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{6n}\sum_{i=1}^{n}G_{j}G_{j'}\left[Q_{j},Q_{j'}\right]^{1/2}$$
with,
$Q_{k}= \displaystyle \frac{7}{4}(l_{ik}-\overline{Y}_{k})^2+\frac{7}{4}(u_{ik}-\overline{Y}_{k})^2+
\frac{5}{2}(l_{ik}-\overline{Y}_{k})(u_{ik}-\overline{Y}_{k})=6(c_{ik}-\overline{Y}_{k})^2+r_{ik}^2$\
$$G_{k}=
\begin{cases}
-1 & \text{if} \quad \overline{Y}_{ik} \leq \overline{Y}_{k}\\
\;1 & \text{if} \quad \overline{Y}_{ik} > \overline{Y}_{k}
\end{cases}$$
\
and $\overline{Y}_{ik}= \displaystyle \frac{l_{ik}+u_{ik}}{2}=c_{ik}$, for $k=j,j'$.\
$$\label{Cov3TS_1}
Cov_{3}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{24n}
\sum\limits_{i=1}^{n}(u_{ij}-l_{ij})(u_{ij'}-l_{ij'})+\frac{1}{4n}
\sum\limits_{i=1}^{n}(l_{ij}+u_{ij})(l_{ij'}+u_{ij'})-\overline{Y}_{j}\overline{Y}_{j'}$$
or
$$\label{Cov3TS_2}
Cov_{3}(Y_{j},Y_{j'}) = \displaystyle \frac{1}{6n}
\sum\limits_{i=1}^{n}{r_{ij}r_{ij'}}+\frac{1}{n}
\sum\limits_{i=1}^{n}c_{ij}c_{ij'}-\overline{Y}_{j}\overline{Y}_{j'}.$$
$$\label{Cov3TS_3}
\qquad \qquad \qquad = \displaystyle \frac{1}{n}
\sum\limits_{i=1}^{n}{\sigma_{ij}\sigma_{ij'}}+\frac{1}{n}
\sum\limits_{i=1}^{n}\mu_{ij}\mu_{ij'}-\overline{Y}_{j}\overline{Y}_{j'}$$
\
It should be noted that expression is equivalent to that presented in 2008 by Billard [@Billard2008].
Measuring distances by quantile functions {#sec2:3}
-----------------------------------------
The goal of this work is to propose a factor model, in which the observed interval variables are written as linear combinations of a few unobservable variables, the *common factors*, also defined as ranges of values. Although in factor analysis, the interest is usually focused on the parameters of the factor model, the estimated values of the common factors, the *factor scores*, are also often required.\
In classical factor analysis, one of the methods of estimation of factor scores was suggested by Bartlett, and is known as the Weighted Least Squares method. Bartlett [@Bartlett1937] proposed to choose as estimates of factor scores those that minimize the sum of squared errors, weighted by the reciprocal of their variances. In this case, the error is calculated as the difference between two real numbers, the observed variable value and the linear combination of common factors, and will be the smaller the closer these numbers are.\
In the factor model that we propose, both the observed variables values and the result of the linear combination of the common factors are intervals and therefore it would be natural that the error evaluation was measured by the difference between those intervals. However, the difference between intervals is not the appropriate measure to calculate the similarity between them [@Moore2009]. In fact, the difference between two equal intervals is not equal to the null interval \[0,0\], but in a interval with center zero and symmetrical bounds. Moreover the difference between any two ranges of values, non-degenerate, never results in the null interval. This happens because in interval arithmetic the resulting interval, from any of the four basic arithmetic operations between intervals, includes all results that are possible to be obtained with all pairs of numbers, one from each of the two intervals, respectively (with the only restriction that zero can not belong to the second interval if the operation is the division). That is, if we consider two intervals $X$ and $Y$, $X \odot Y = \lbrace x \odot y : x\in X, y\in Y\rbrace$, where $\odot$ represents any of the four arithmetic operations. In the particular case of the arithmetic *difference*, we obtain $$X - Y = \lbrace x - y : x\in X, y\in Y\rbrace$$
or,
$$X - Y = \left[ l_{x}-u_{y},u_{x}-l_{y}\right],$$
for $X=\left[ l_{x},u_{x}\right] $ and $Y=\left[ l_{y},u_{y}\right]$.
Therefore, at this step, it is necessary to select an appropriate measure of similarity between intervals, since it is inappropriate to apply arithmetic operations. Two measures considered to be good choice to study the dissimilarity between data with variability, in particular data with interval-valued variables or histogram-valued variables, are the Mallows and the Wassertein distances [@Arroyo2008; @Verde2007; @Irpino2008]. As stated above, the Mallows and Wasserstein distances are appropriate to represent an error measure. Furthermore, as distances they present interesting properties: they are positive definite measures, symmetric, and satisfy the triangular inequality. Both the Mallows distance and the Wasserstein distance are defined in terms of quantile functions, and the further apart these functions are, the greater the distance between them.\
Following is the definition of Mallows and Wasserstein distances:
\[defMallows\] If $Y_{j}$ and $Y_{j'}$ are interval-valued variables represented, respectively, by their quantile functions $\Psi_{Y_{ij}}^{-1}$ and $\Psi_{Y_{ij'}}^{-1}$ for an observation $s_i$, then the **Wasserstein distance** between intervals $Y_{ij}$ and $Y_{ij'}$ is defined by: $$D_W(\Psi_{Y_{ij}}^{-1}(t),\Psi_{Y_{ij'}}^{-1}(t))=\int_{0}^{1} | \Psi_{Y_{ij}}^{-1}(t)-\Psi_{Y_{ij'}}^{-1}(t) | dt,$$
and the **squared Mallows distance**,
$$\label{SquareMallowsDist}
D^{2}_M(\Psi_{Y_{ij}}^{-1}(t),\Psi_{Y_{ij'}}^{-1}(t))=\int_{0}^{1}(\Psi_{Y_{ij}}^{-1}(t)-\Psi_{Y_{ij'}}^{-1}(t))^2dt.$$
According to Arroyo [@Arroyo2008], one can establish a parallelism between the Wasserstein and the Manhattan distances, and between the Mallows and the Euclidean distances. In fact, taking into account the general expression $$\label{DistGeral}
D(\Psi_{Y_{ij}}^{-1}(t),\Psi_{Y_{ij'}}^{-1}(t))= \Big(\int_{0}^{1}(\Psi_{Y_{ij}}^{-1}(t)-\Psi_{Y_{ij'}}^{-1}(t))^pdt \Big)^{\frac{1}{p}}$$
the Wasserstein and Mallows distances are obtained when *p* = 1 and *p* = 2, respectively.
Also notice that when $Y_{ij}$ and $Y_{ij'}$ are degenerate intervals, and considering that expression is similar to the Minkowski metric, we then obtain the Manhattan distance for the particular case of *p* = 1, and the Euclidean distance for *p* = 2.\
Irpino and Verde [@Irpino2006] have rewritten expression assuming the Uniform distribution within each interval, using the centres and half-ranges:
\[defMallowsUnif\] [@Irpino2006] If $Y_{j}$ and $Y_{j'}$ are interval-valued variables represented, respectively, by their quantile functions $\Psi_{Y_{ij}}^{-1}$ and $\Psi_{Y_{ij'}}^{-1}$ for an observation $s_i$, and the **Uniform distribution** is assumed within the intervals $ I_{ik} $, $ k=j,j' $, respectively, then the **squared Mallows distance** between intervals is given by: $$\label{DMUnif}
D^{2}_M(\Psi_{Y_{ij}}^{-1}(t),\Psi_{Y_{ij'}}^{-1}(t))=\displaystyle (c_{ij}-c_{ij'})^2+\dfrac{1}{3}(r_{ij}-r_{ij'})^2$$ where, $c_{ij}$, $c_{ij'}$ and $r_{ij}$, $r_{ij'}$ are, respectively, the centers and the half-ranges of the observed $Y_{ij}$ and $Y_{ij'}$ intervals.\
Next we deduce the expression of the (square) Mallows distance for the case where the Triangular distribution is assumed within the observed intervals.
\[defMallowsTriang\] If $Y_{j}$ and $Y_{j'}$ are interval-valued variables represented by their quantile functions $\Psi_{Y_{ij}}^{-1}$ and $\Psi_{Y_{ij'}}^{-1}$ for an observation $s_i$, and the **Triangular distribution** is assumed within the intervals $I_{ik} $, with mode $ m_{ik} $, $ k=j,j' $, then the **square of the Mallows distance** between intervals is given by, respectively:\
<span style="font-variant:small-caps;">(i)</span> **non-degenerated intervals**, i.e., intervals with non-zero half-ranges:
if $ \dfrac{m_j-c_j}{2r_j} \leq \dfrac{m_{ij'}-c_{ij'}}{2r_{ij'}} $,
$$\; \; \; \; \; \; D^{2}_M(\Psi_{Y_{ij}}^{-1}(t),\Psi_{Y_{ij'}}^{-1}(t))=
\displaystyle (c_{ij}-c_{ij'})^2+\dfrac{1}{6}(r_{ij}-r_{ij'})^2+\dfrac{1}{6}(m_{ij}-c_{ij})^2+\dfrac{1}{6}(m_{ij'}-c_{ij'})^2 \qquad\qquad\qquad\qquad\qquad$$
$$\quad-\dfrac{5}{3}r_{ij}r_{ij'}+\dfrac{2}{3}(m_{ij}-c_{ij})(c_{ij}-c_{ij'}+r_{ij'})-\dfrac{2}{3}(m_{ij'}-c_{ij'})(c_{ij}-c_{ij'}+r_{ij})\qquad\qquad\qquad$$
$$+\dfrac{1}{6}\sqrt{r_jr{ij}_{ij'}(m_{ij}-c_{ij}+r_{ij})(m_{ij'}-c_{ij'}+r_{ij'})}\Big(5-\dfrac{m_{ij}-c_{ij}}{r_{ij}}\Big)\qquad\qquad\qquad\qquad\quad\;$$
$$+\dfrac{1}{6}\sqrt{r_{ij}r_{ij'}(c_{ij}+r_{ij}-m_{ij})(c_{ij'}+r_{ij'}-m_{ij'})}\Big(5+\dfrac{m_{ij'}-c_{ij'}}{r_{ij'}}\Big) \qquad\qquad\qquad\qquad\quad$$
$$\label{DMTGeral-ramo1}
\quad\quad\;+\dfrac{1}{2}\sqrt{r_{ij}r_{ij'}(c_{ij}+r_{ij}-m_{ij})(m_{ij'}-c_{ij'}+r_{ij'})}\Big(arcsin\dfrac{m_{ij'}-c_{ij'}}{r_{ij'}}-arcsin\dfrac{m_{ij}-c_{ij}}{r_{ij}}\Big)$$
if $ \dfrac{m_{ij}-c_{ij}}{2r_{ij}} > \dfrac{m_{ij'}-c_{ij'}}{2r_{ij'}} $,
$$\; \; \; \; \; \; D^{2}_M(\Psi_{Y_{ij}}^{-1}(t),\Psi_{Y_{ij'}}^{-1}(t))=
\displaystyle (c_{ij}-c_{ij'})^2+\dfrac{1}{6}(r_{ij}-r_{ij'})^2+\dfrac{1}{6}(m_{ij}-c_{ij})^2+\dfrac{1}{6}(m_{ij'}-c_{ij'})^2\qquad\qquad\qquad\qquad\qquad$$
$$\quad-\dfrac{5}{3}r_{ij}r_{ij'}+\dfrac{2}{3}(m_{ij}-c_{ij})(c_{ij}-c_{ij'}-r_{ij'})-\dfrac{2}{3}(m_{ij'}-c_{ij'})(c_{ij}-c_{ij'}-r_{ij}) \qquad\qquad\qquad$$
$$+\dfrac{1}{6}\sqrt{r_{ij}r_{ij'}(m_{ij}-c_{ij}+r_{ij})(m_{ij'}-c_{ij'}+r_{ij'})}\Big(5-\dfrac{m_{ij'}-c_{ij'}}{r_{ij'}}\Big)\qquad\qquad\qquad\qquad\quad\;$$
$$+\dfrac{1}{6}\sqrt{r_{ij}r_{ij'}(c_{ij}+r_{ij}-m_{ij})(c_{ij'}+r_{ij'}-m_{ij'})}\Big(5+\dfrac{m_{ij}-c_{ij}}{r_{ij}}\Big) \qquad\qquad\qquad\qquad\quad\quad$$
$$\label{DMTGeral-ramo2}
\quad\quad\;+\dfrac{1}{2}\sqrt{r_{ij}r_{ij'}(c_{ij'}+r_{ij'}-m_{ij'})(m_{ij}-c_{ij}+r_{ij})}\Big(arcsin\dfrac{m_{ij}-c_{ij}}{r_{ij}}-arcsin\dfrac{m_{ij'}-c_{ij'}}{r_{ij'}}\Big)$$
where, $c_{ij}$,$c_{ij'}$ and $r_{ij}$, $r_{ij'}$ are, respectively, the centers and the half-ranges of the observed $Y_{ij}$ and $Y_{ij'}$ intervals;\
<span style="font-variant:small-caps;">(ii)</span> only one **non-degenerate interval**: for instance $ I_{ij'}=[c_{ij'},c_{ij'}] $ $$\; \; \; D^{2}_M(\Psi_{Y_{ij}}^{-1}(t),\Psi_{Y_{ij'}}^{-1}(t))=\displaystyle (c_{ij}-c_{ij'})^2-\dfrac{4}{3}(m_{ij}-c_{ij})^2-\dfrac{1}{3}r_{ij}^2+\dfrac{2}{3}(m_{ij}-c_{ij})(c_{ij}-c_{ij'}) \qquad$$ $$+\dfrac{(m_{ij}-c_{ij}+r_{ij})^3}{4r_{ij}}+\dfrac{(c_{ij}+r_{ij}-m_{ij})^3}{4r_{ij}}$$\
<span style="font-variant:small-caps;">(iii)</span> **degenerated intervals**: $ I_{ij}=[c_{ij},c_{ij}] $ and $ I_{ij'}=[c_{ij'},c_{ij'}] $ $$D^{2}_M(\Psi_{Y_{ij}}^{-1}(t),\Psi_{Y_{ij'}}^{-1}(t))=\displaystyle (c_{ij}-c_{ij'})^2$$ or equivalently $$\label{DMReais}
D_M(\Psi_{Y_{ij}}^{-1}(t),\Psi_{Y_{ij'}}^{-1}(t))=|c_{ij}-c_{ij'}|.$$
Expression is the Euclidean distance between two real numbers, $ c_{ij} $ and $ c_{ij'} $, as expected. Note that we obtain exactly the same result if we consider null $r_{ij} $ and $r_{ij'} $ in expression .\
\[cor\] In the particular case where a **Symmetric Triangular distribution** is assumed within the intervals $ I_{ik} $, $ k=j,j' $, for the observation $s_i$ of the variables $Y_{j}$ and $Y_{j'}$, the **square of the Mallows distance** between intervals simplifies to: $$\label{DMTSimétrica}
D^{2}_M(\Psi_{Y_{ij}}^{-1}(t),\Psi_{Y_{ij'}}^{-1}(t))=\displaystyle (c_{ij}-c_{ij'})^2+\dfrac{1}{6}(r_{ij}-r_{ij'})^2,$$ where $\Psi_{Y_{ij}}^{-1}$ and $\Psi_{Y_{ij'}}^{-1}$ are the quantile functions representing $Y_{ij}$ and $Y_{ij'}$ respectively.
Factor Analysis of Interval Data {#sec:3}
================================
Factor Model {#sec3:1}
------------
Let $Y_{1},Y_{2},\ldots,Y_{p}$ be the observed interval-valued variables measured on a set of n units $S=\{s_1,\ldots,s_n\}$, as exemplified in Table 1 of Section 2. To specify the factor model we will use the standardized interval-valued variable $Z_{1},Z_{2},\ldots,Z_{p}$ defined by $Z_{ij}= \Big[\frac{l_{ij}-\overline{Y}_{j}}{S_{j}}, \frac{u_{ij}-\overline{Y}_{j}}{S_{j}}\Big]$. All variables $Z_{j}$, $ j = 1,\ldots,p $, have null sample mean and unit sample variance.
The proposed factor model presumes that these variables are linearly dependent on few unobservable interval-valued variables $f_{1},f_{2},\ldots,f_{m}$ $(m<<p)$ called **common factors** and $p$ interval-valued sources of variation $\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{p}$ called **specific factors**, or **errors**, such that $$\label{ModeloFat_1}
{Z_{j}}= \ell_{j1}f_{1}+\ell_{j2}f_{2}+\ldots+\ell_{jm}f_{m}+
\varepsilon_{j}, \quad j = 1,\ldots,p.$$ where $\ell_{jk}$’s, $k=1,\ldots,m$, are the model coefficients, real values, usually termed as **factor loadings** [@Johnson2002; @Johnson1998]. The interval $\varepsilon_{j}$ describes the residual variation specific to the $j$th variable $Z_{j}$ and its variance, $S^2_{\varepsilon_{j}}$, is called the **specific variance** of the $j$th variable.\
\
If we replace in the previous model , each interval by the associated quantile function we obtain the model rewritten as follows, $$\label{ModeloFat_2}
\Psi_{Z_{j}}^{-1}(t)= \underbrace{\ell_{j1}\Psi_{f_{1}}^{-1}(*)+\ell_{j2}\Psi_{f_{2}}^{-1}(*)+\ldots+\ell_{jm}\Psi_{f_{m}}^{-1}(*)}_{\Psi_{CL_{j}}^{-1}}+\Psi_{\varepsilon_{j}}^{-1}(t), \: j = 1,\ldots,p, \: 0 \leq t \leq 1$$ with $ * = t $ if $\ell_{ji}>0 $ and $ * = 1-t $ if $\ell_{ji}<0 $, where\
$\Psi_{Z_{j}}^{-1}$ is the quantile function associated with the standardized interval-valued variable $Z_{j}$;\
$\Psi_{f_{k}}^{-1}$ is the quantile function associated with the interval-valued variable $f_{k}$;\
$\Psi_{\varepsilon_{j}}^{-1}$ is the quantile function associated with the interval-valued variable $\varepsilon_{j}$.\
\
In the previous model it is necessary to assume that:\
<span style="font-variant:small-caps;">(i)</span> the $Z_{j}$ variables, the common factors $f_{k}$’s and specific factors $\varepsilon_{j}$’s have null mean;\
<span style="font-variant:small-caps;">(ii)</span> the common factors $f_{k}$’s and specific factors $\varepsilon_{j}$’s are uncorrelated for all combinations of *k* and *j*;\
<span style="font-variant:small-caps;">(iii)</span> the common factors $f_{k}$’s are uncorrelated and have unit variance.\
These assumptions and the preceding model constitute the *orthogonal factor model*. If (III) is not verified we have the so-called *model of oblique factors*. We can thus say that the orthogonal model of factor analysis assumes that there is a smaller set of uncorrelated interval-valued variables that explain the relations between the observed interval-valued variables.\
\
As in the classic case, the factors may be extracted by different methods. Here we consider extraction by Principal Component and by Principal Axis Factoring on the interval-valued variables correlation matrix.
Principal Component is the perhaps the most commonly used extraction method, but it implicitly assumes that all communalities are equal to one, so that the variables’ variances could (theorethically) be completely explained by common factors. Principal Axis Factoring, on the other hand, assumes a model with common and unique factors, and therefore variance cannot be explained just by common factors; the method proceeds iteratively, by first estimating communalities and then trying to identify the common factors responsible for these communalities and the correlations between variables (see, e.g. [@Sharma1996]). Therefore, a factor model is implicitly assumed in the Principal Axis Factoring.\
In both cases, and taking into account the model assumptions,\
<span style="font-variant:small-caps;">(i)</span> the correlation between $Z_{j}$ and $f_{k}$, denoted by $ Corr(Z_{j},f_{k}) $, is $\ell_{jk}$, the loading of the $j$th variable on the $k$th factor. For this reason it is said that each loading $\ell_{jk}$ measures the contribution of the $k$th common factor to the $j$th variable;\
<span style="font-variant:small-caps;">(ii)</span> the variance of $Z_{j}$ can be partitioned as $ S^2_{Z_{j}} = \displaystyle \sum_{k=1}^{m} \ell^2_{jk} + S^2_{\varepsilon_{j}} = 1 $, and the proportion of the variance of $Z_{j}$ that is explained by the common factors, $ \displaystyle \sum_{k=1}^{m} \ell^2_{jk} $, is named the **communality** of the $j$th variable;\
<span style="font-variant:small-caps;">(iii)</span> the correlation between $Z_{j}$ and $Z_{j'}$ is $ Corr(Z_{j},Z_{j'}) = \displaystyle \sum_{k=1}^{m} \ell_{jk}\ell_{j'k} $.
Factor Scores {#sec3:2}
-------------
In this section we will present two approaches to interval-valued factor scores estimation, inspired in methods for real-valued data, namely, the Bartlett and the Anderson-Rubin methods [@DiStefano2009].
The method suggested by Bartlett, also known as the Weighted Least Squares method, chooses as estimates of factor scores those that minimize the sum of squared errors, weighted by the reciprocal of their variances. It can be shown [@Johnson1998] that for real-valued variables the resulting factor scores are nothing more than the values of the (scaled) principal components.
Our first proposal, inspired by this idea, is to consider the sum of the squared Mallows distances between $ \Psi_{Z_{j}}^{-1} $ and $ \Psi_{CL_{j}}^{-1} $, taking into account model and choose the interval-valued factor scores estimates that minimize that sum, weighted by the reciprocal of the interval variable $ \varepsilon_{j} $ variance, $ S^2_{\varepsilon_{j}} $, that is,
Minimize $ \quad \displaystyle \sum_{j=1}^{p}\dfrac{D^{2}_M(\Psi_{Z_{j}}^{-1},\Psi_{CL_{j}}^{-1})}{S^2_{\varepsilon_{j}}}$.\
It is important to underline that the factor scores are no longer the values of the (scaled) principal components and, to the best of our knowledge, cannot be obtained by a closed formula.\
The method proposed by Anderson and Rubin adapts the approach of Bartlett such that the factor scores are not only uncorrelated with other factors, but also uncorrelated with each other. Thus, our second proposal is to
Minimize $ \quad \displaystyle \sum_{j=1}^{p}\dfrac{D^{2}_M(\Psi_{Z_{j}}^{-1},\Psi_{CL_{j}}^{-1})}{S^2_{\varepsilon_{j}}}$
subject to the condition $Corr(\widehat{f}_{k},\widehat{f}_{k'})=0$, for $ k \ne k' $, $ \forall \; k,k'=1,...,m $.\
In both approaches the estimates are obtained by solving an optimization problem.
In order to find the factor scores, we relied in the optimization routines of the R system. In particular, in the ’Bartlett method’, for each unit we specified an error function, *SumDist*, for the weighted sum of Mallows distances, which takes as its arguments the relevant distribution parameters. Then we minimize *SumDist* by the *nlminb* routine of the R system, using as starting points the U(0,1) (Uniform distribution) and the Tr(0,1,2) (Triangular distribution) for the *nlminb* search. We have found that convergence was usually obtained whitin a few dozen iterations. To check for potential problems created by local optima, we conducted some experiments with different starting points, and concluded that our procedure was robust, with the search converging always to the same solutions, even after large perturbations of the search origin. For the ’Anderson-Rubin method’, we defined a global error function, *SumDistFactort*, that adds the sum of weighted Mallows distances for all entities with sum of squared correlations between factor scores, multiplied by a large penalty. We used again the *nlminb* routine using as starting points the parameters of distributions found by the ’Bartlett method’. However, in this case we found evidence of local optima, and in order to mitigate this dependence we repeted the local search for different starting points until the best solution found did not change after many different iterations of this procedure.\
Synthetic Data {#sec:4}
==============
In this section we analyse the behaviour of the proposed method on synthetic data with predefined correlation structures. We consider cases where all interval-valued variables are highly or only moderatly correlated and cases where there are differents blocks of highly and$/$or moderatly correlated variables. Is assumed high correlations if values are between 0.8 and 1 and moderate correlations between 0.5 and 0.8.\
The generation of the synthetic data was done in three main steps:
1. Generate two different matrices with similar correlation structures: the correlation matrix between the centers R$_{c}$ and the correlation matrix between the half-ranges R$_{r}$. These correlation matrices were generated by application of Algorithm 1 suggested by Hardin. For more details on correlation matrices simulation with or without a given structure, see Hardin [@Hardin2013].
2. Generate the matrix of the centers of the intervals as the product of two matrices: $C_{ini}\times L_{C}$ where,
- R$_{c}$ = $L_{C}^{t}$ $\times$ $L_{C}$ is the Choleski decomposition of R$_{c}$.
- elements of matrix $C_{ini}$ are randomly selected from a Uniform distribution in the interval $(a, b)$ such that $a \frown U (0,5)$ and $b \frown U (5,15)$.
3. Generate the matrix of the half-ranges of the intervals as the product of two matrices: $R_{ini}\times L_{R}$ where,
- R$_{r}$ = $L_{R}^{t}$ $\times$ $L_{R}$ is the Choleski decomposition of R$_{r}$.
- elements of matrix $R_{ini}$ are randomly selected from a Uniform distribution in the interval $[0.1, 1]$.
Below, we define 6 different correlation matrix structures and present the correlation matrices generated between the centers R$_{c}$ and between the half-ranges R$_{r}$ for each of the cases. Problems with 10 interval-valued variables are analysed. In each case a set of 100 values of centers and half-ranges are generated.\
Case 1: All variables highly correlated.
[$$R_{c}=\left[
\begin{array}{cccccccccc}
1 & 0.898 & 0.914 & 0.910 & 0.915 & 0.891 & 0.890 & 0.907 & 0.909 & 0.892 \\
& 1 & 0.893 & 0.920 & 0.907 & 0.871 & 0.907 & 0.915 & 0.902 & 0.920\\
& & 1 & 0.907 & 0.920 & 0.912 & 0.915 & 0.919 & 0.920 & 0.903 \\
& & & 1 & 0.912 & 0.899 & 0.905 & 0.872 & 0.888 & 0.931 \\
& & & & 1 & 0.906 & 0.917 & 0.920 & 0.938 & 0.928 \\
& & & & & 1 & 0.926 & 0.929 & 0.931 & 0.888 \\
& & & & & & 1 & 0.912 & 0.925 & 0.922 \\
& & & & & & & 1 & 0.943 & 0.915 \\
& & & & & & & & 1 & 0.903 \\
& & & & & & & & & 1
\end{array}
\right]$$ ]{} [$$R_{r}=\left[
\begin{array}{cccccccccc}
1 & 0.818 & 0.888 & 0.919 & 0.862 & 0.861 & 0.883 & 0.887 & 0.911 & 0.910 \\
& 1 & 0.852 & 0.825 & 0.837 & 0.779 & 0.856 & 0.782 & 0.830 & 0.797 \\
& & 1 & 0.910 & 0.913 & 0.824 & 0.929 & 0.901 & 0.921 & 0.884 \\
& & & 1 & 0.926 & 0.853 & 0.918 & 0.878 & 0.918 & 0.902 \\
& & & & 1 & 0.878 & 0.996 & 0.902 & 0.910 & 0.864 \\
& & & & & 1 & 0.887 & 0.894 & 0.881 & 0.895 \\
& & & & & & 1 & 0.872 & 0.927 & 0.887 \\
& & & & & & & 1 & 0.918 & 0.897 \\
& & & & & & & & 1 & 0.912 \\
& & & & & & & & & 1
\end{array}
\right]$$]{}
Case 2: All variables moderatly correlated.
[$$R_{c}=\left[
\begin{array}{cccccccccc}
1 & 0.661 & 0.700 & 0.611 & 0.623 & 0.693 & 0.706 & 0.773 & 0.705 & 0.709 \\
& 1 & 0.768 & 0.686 & 0.683 & 0.695 & 0.685 & 0.726 & 0.719 & 0.723\\
& & 1 & 0.662 & 0.667 & 0.735 & 0.697 & 0.781 & 0.667 & 0.748 \\
& & & 1 & 0.726 & 0.686 & 0.760 & 0.686 & 0.677 & 0.660 \\
& & & & 1 & 0.688 & 0.714 & 0.652 & 0.779 & 0.724 \\
& & & & & 1 & 0.612 & 0.763 & 0.671 & 0.715 \\
& & & & & & 1 & 0.649 & 0.703 & 0.707 \\
& & & & & & & 1 & 0.723 & 0.665 \\
& & & & & & & & 1 & 0.629 \\
& & & & & & & & & 1
\end{array}
\right]$$ ]{}
[$$R_{r}=\left[
\begin{array}{cccccccccc}
1 & 0.760 & 0.715 & 0.767 & 0.730 & 0.564 & 0.721 & 0.464 & 0.711 & 0.798 \\
& 1 & 0.634 & 0.691 & 0.762 & 0.644 & 0.710 & 0.546 & 0.716 & 0.708 \\
& & 1 & 0.759 & 0.695 & 0.534 & 0.694 & 0.536 & 0.768 & 0.696 \\
& & & 1 & 0.735 & 0.601 & 0.711 & 0.496 & 0.757 & 0.817 \\
& & & & 1 & 0.683 & 0.766 & 0.677 & 0.798 & 0.710 \\
& & & & & 1 & 0.573 & 0.458 & 0.619 & 0.617 \\
& & & & & & 1 & 0.548 & 0.686 & 0.751 \\
& & & & & & & 1 & 0.573 & 0.475 \\
& & & & & & & & 1 & 0.727 \\
& & & & & & & & & 1
\end{array}
\right]$$]{}
Case 3: Two blocks of highly correlated variables.
[$$R_{c}=\left[
\begin{array}{cccccccccc}
1 & 0.858 & 0.891 & 0.220 & 0.203 & 0.230 & 0.170 & 0.234 & 0.228 & 0.192 \\
& 1 & 0.893 & 0.183 & 0.216 & 0.206 & 0.210 & 0.173 & 0.213 & 0.173 \\
& & 1 & 0.134 & 0.196 & 0.193 & 0.245 & 0.223 & 0.235 & 0.159 \\
& & & 1 & 0.831 & 0.805 & 0.792 & 0.806 & 0.797 & 0.808 \\
& & & & 1 & 0.824 & 0.813 & 0.799 & 0.793 & 0.821 \\
& & & & & 1 & 0.796 & 0.865 & 0.792 & 0.770 \\
& & & & & & 1 & 0.818 & 0.786 & 0.812 \\
& & & & & & & 1 & 0.807 & 0.808 \\
& & & & & & & & 1 & 0.779 \\
& & & & & & & & & 1
\end{array}
\right]$$ ]{}
[$$R_{r}=\left[
\begin{array}{cccccccccc}
1 & 0.843 & 0.829 & 0.254 & 0.238 & 0.249 & 0.233 & 0.234 & 0.281 & 0.255 \\
& 1 & 0.877 & 0.238 & 0.212 & 0.229 & 0.248 & 0.264 & 0.272 & 0.256 \\
& & 1 & 0.267 & 0.276 & 0.260 & 0.221 & 0.260 & 0.259 & 0.260 \\
& & & 1 & 0.918 & 0.857 & 0.880 & 0.926 & 0.906 & 0.900 \\
& & & & 1 & 0.941 & 0.918 & 0.900 & 0.904 & 0.890 \\
& & & & & 1 & 0.896 & 0.866 & 0.919 & 0.890 \\
& & & & & & 1 & 0.866 & 0.873 & 0.885 \\
& & & & & & & 1 & 0.917 & 0.906 \\
& & & & & & & & 1 & 0.914 \\
& & & & & & & & & 1
\end{array}
\right]$$]{}
Case 4: Two blocks of moderatly correlated variables.
[$$R_{c}=\left[
\begin{array}{cccccccccc}
1 & 0.541 & 0.529 & 0.551 & 0.068 & 0.102 & 0.048 & 0.097 & 0.094 & 0.118 \\
& 1 & 0.567 & 0.570 & 0.095 & 0.093 & 0.095 & 0.127 & 0.117 & 0.129 \\
& & 1 & 0.573 & 0.142 & 0.059 & 0.099 & 0.103 & 0.145 & 0.095 \\
& & & 1 & 0.059 & 0.058 & 0.098 & 0.130 & 0.108 & 0.087 \\
& & & & 1 & 0.624 & 0.674 & 0.625 & 0.673 & 0.636 \\
& & & & & 1 & 0.645 & 0.649 & 0.642 & 0.634 \\
& & & & & & 1 & 0.653 & 0.606 & 0.635 \\
& & & & & & & 1 & 0.594 & 0.650 \\
& & & & & & & & 1 & 0.640 \\
& & & & & & & & & 1
\end{array}
\right]$$ ]{}
[$$R_{r}=\left[
\begin{array}{cccccccccc}
1 & 0.566 & 0.629 & 0.577 & 0.192 & 0.192 & 0.238 & 0.222 & 0.215 & 0.205 \\
& 1 & 0.580 & 0.594 & 0.205 & 0.224 & 0.185 & 0.232 & 0.194 & 0.217\\
& & 1 & 0.567 & 0.215 & 0.245 & 0.189 & 0.217 & 0.231 & 0.197 \\
& & & 1 & 0.199 & 0.224 & 0.158 & 0.202 & 0.215 & 0.169 \\
& & & & 1 & 0.631 & 0.608 & 0.648 & 0.626 & 0.648 \\
& & & & & 1 & 0.611 & 0.622 & 0.654 & 0.623 \\
& & & & & & 1 & 0.611 & 0.619 & 0.640 \\
& & & & & & & 1 & 0.618 & 0.660 \\
& & & & & & & & 1 & 0.662 \\
& & & & & & & & & 1
\end{array}
\right]$$]{}
Case 5: One block of highly correlated variables and one block of moderatly correlated variables.
[$$R_{c}=\left[
\begin{array}{cccccccccc}
1 & 0.909 & 0.956 & 0.104 & 0.122 & 0.096 & 0.121 & 0.091 & 0.141 & 0.092 \\
& 1 & 0.913 & 0.082 & 0.130 & 0.133 & 0.108 & 0.080 & 0.076 & 0.088 \\
& & 1 & 0.111 & 0.126 & 0.097 & 0.082 & 0.0780 & 0.102 & 0.134 \\
& & & 1 & 0.587 & 0.564 & 0.595 & 0.563 & 0.617 & 0.560 \\
& & & & 1 & 0.599 & 0.562 & 0.587 & 0.613 & 0.557 \\
& & & & & 1 & 0.552 & 0.581 & 0.564 & 0.599 \\
& & & & & & 1 & 0.618 & 0.586 & 0.549 \\
& & & & & & & 1 & 0.533 & 0.569 \\
& & & & & & & & 1 & 0.544 \\
& & & & & & & & & 1
\end{array}
\right]$$ ]{}
[$$R_{r}=\left[
\begin{array}{cccccccccc}
1 & 0.912 & 0.931 & 0.201 & 0.173 & 0.207 & 0.224 & 0.179 & 0.231 & 0.184 \\
& 1 & 0.938 & 0.204 & 0.195 & 0.204 & 0.181 & 0.221 & 0.230 & 0.180 \\
& & 1 & 0.218 & 0.183 & 0.194 & 0.230 & 0.189 & 0.202 & 0.236 \\
& & & 1 & 0.608 & 0.617 & 0.575 & 0.597 & 0.618 & 0.613 \\
& & & & 1 & 0.626 & 0.610 & 0.631 & 0.587 & 0.606 \\
& & & & & 1 & 0.602 & 0.620 & 0.601 & 0.576 \\
& & & & & & 1 & 0.607 & 0.609 & 0.641 \\
& & & & & & & 1 & 0.551 & 0.620 \\
& & & & & & & & 1 & 0.590 \\
& & & & & & & & & 1
\end{array}
\right]$$]{}
Case 6: Three blocks of highly correlated variables.
[$$R_{c}=\left[
\begin{array}{cccccccccc}
1 & 0.901 & 0.786 & 0.048 & 0.117 & 0.044 & 0.083 & 0.133 & 0.110 & 0.104 \\
& 1 & 0.849 & 0.089 & 0.116 & 0.086 & 0.130 & 0.149 & 0.107 & 0.117 \\
& & 1 & 0.120 & 0.080 & 0.128 & 0.126 & 0.143 & 0.077 & 0.121 \\
& & & 1 & 0.889 & 0.900 & 0.910 & 0.101 & 0.085 & 0.110 \\
& & & & 1 & 0.900 & 0.940 & 0.131 & 0.177 & 0.114 \\
& & & & & 1 & 0.867 & 0.067 & 0.116 & 0.066 \\
& & & & & & 1 & 0.166 & 0.076 & 0.109 \\
& & & & & & & 1 & 0.881 & 0.948 \\
& & & & & & & & 1 & 0.929 \\
& & & & & & & & & 1
\end{array}
\right]$$ ]{}
[$$R_{r}=\left[
\begin{array}{cccccccccc}
1 & 0.941 & 0.938 & 0.191 & 0.126 & 0.161 & 0.142 & 0.185 & 0.192 & 0.159 \\
& 1 & 0.945 & 0.174 & 0.135 & 0.163 & 0.130& 0.168 & 0.137 & 0.166 \\
& & 1 & 0.179 & 0.155 & 0.145 & 0.143 & 0.138 & 0.159 & 0.117 \\
& & & 1 & 0.846 & 0.867 & 0.828 & 0.175 & 0.135 & 0.134 \\
& & & & 1 & 0.845 & 0.856 & 0.177 & 0.158 & 0.108 \\
& & & & & 1 & 0.817 & 0.168 & 0.114 & 0.125 \\
& & & & & & 1 & 0.183 & 0.208 & 0.177 \\
& & & & & & & 1 & 0.799 & 0.866 \\
& & & & & & & & 1 & 0.785 \\
& & & & & & & & & 1
\end{array}
\right]$$]{}
For each case, a factor analysis according to the proposed model (\[ModeloFat\_1\]) was performed assuming three distinct distributions within each interval: Uniform, Triangular Symmetric and Triangular (with the mode randomly chosen within interval).\
The number of common factors retained was defined according to the rule of eigenvalues greater than one and in line with the cumulative proportion of total variation. Both Principal Component and Principal Axis Factoring leads to the extraction of same number of factors in each case.\
Table \[TabfinalSim\] presents the number of interval-valued factors extracted in each case, which was the same for all the assumed distributions.
\[h!\]
-------- ------------ ------------------------ --------------------------------- ------------------------
Number
of factors 1st factor 2nd factor 3rd factor
Case 1 1 All —- —-
Case 2 1 All —- —-
Case 3 2 Variables of 1st group Variables of 2nd group —-
Case 4 2 Variables of 1st group Variables of 2nd group —-
Case 5 2 Variables of 1st group Variables of 1st and 2nd groups —-
Case 6 3 Variables of 1st group Variables of 2nd group Variables of 3rd group
-------- ------------ ------------------------ --------------------------------- ------------------------
: - Number of factors extracted considering the Uniform, Triangular Symmetric and Triangular distribution.
\[TabfinalSim\]
We can see from Table 2 that the factor analysis of this data succeeds in recovering their original structure in cases 1, 2, 3, 4 and 6. Case 5, where there are both groups of higly and moderatly correlated variables, is somehow more difficult, and the variables of the strongly correlated group sometimes also appear in the definition of the second factor. Nevertheless, this is not much different from similar data conditions in the factor analysis of classic data, and the basic group correlation structure is still recognized by the analysis.
Application {#sec:5}
===========
In this section, we illustrate the methodology proposed above on a car data set and on meteorological data, for the different alternatives concerning **(a)** the distribution within the intervals: Uniform and Triangular distributions, **(b)** the technique of factor extraction: Principal Component and Principal Axis Factoring, **(c)** and the estimation of factor scores: the Bartlett and the Anderson-Rubin methods. The number of common factors retained was defined according to the rule of eigenvalues $ \hat{\lambda}_{j} $, $ j = 1,\ldots,p $, greater than one and in line with the cumulative proportion of total variation, $ \dfrac{\sum_{k=1}^{j}\hat{\lambda}_{k}}{p} $.
Cars Data {#sec5:1}
---------
A factor analysis was performed on a set of 33 car models described by 8 interval-valued variables: Price, Engine Capacity, Top Speed, Acceleration, Wheelbase, Lenght, Width and Height (see Table \[Cars\]).
\[h!\]
Price Engine Capacity … Height
-------------- ------------------------------ -------------------------- --- ------------------------
Alfa 145 $\left[27806,33596\right]$ $\left[1370,1910\right]$ … $\left[143,143\right]$
Alfa 156 $\left[41593,62291\right]$ $\left[1598,2492\right]$ … $\left[142,142\right]$
Aston Martin $\left[260500,460000\right]$ $\left[5935,5935\right]$ … $\left[124,132\right]$
Porsche $\left[147704,246412\right]$ $\left[3387,3600\right]$ … $\left[130,131\right]$
Rover 25 $\left[21492,33042\right]$ $\left[1119,1994\right]$ … $\left[142,142\right]$
Passat $\left[39676,63455\right]$ $\left[1595,2496\right]$ … $\left[146,146\right]$
: - Cars data set (partial view).
\[Cars\]
### Uniform Distribution {#sec5:1:1}
In this section we assume that the values within the observed intervals are distributed according to an Uniform distribution.\
The following is the sample correlation matrix **R** obtained from the third definition of covariance $ Cov_3 $, using formula (\[Cov3U\_1\_2\]):\
Price EngCap TopSpeed Acceler Wheelbase Lenght Width Height
$$\mathbf{R}=\left[
\begin{array}{cccccccc}
1 & +0.9580 & +0.8712 & -0.7559 & +0.3732 & +0.5159 & +0.8261 & -0.6776 \\
& 1 & +0.8659 & -0.7296 & +0.4834 & +0.6260 & +0.8502 & -0.6269 \\
& & 1 & -0.8768 & +0.3396 & +0.5747 & +0.8529 & -0.7281 \\
& & & 1 & -0.3973 & -0.5991 & -0.8138 & +0.6037 \\
& & & & 1 & +0.8657 & +0.5944 & +0.1581 \\
& & & & & 1 & +0.7635 & -0.0373 \\
& & & & & & 1 & -0.5431 \\
& & & & & & & 1
\end{array}
\right]$$
Both Principal Component and Principal Axis Factoring of this matrix lead to the extraction of two factors, which together represent 89.9 % and 86.7 % of the total variance, respectively. Table \[ResumeUnif\] summarizes the estimated factor loadings for each variable in the two interval-valued factors, its eigenvalues, the communality of each variable and the cumulative proportion of total sample variance explained, for both methods.
\[h!\]
----------------------- ------------- ------------------- -------- ------------------- ------------- --------
**Variable** **Communalities** **Communalities**
$f_1$ $f_2$ $f_1$ $f_2$
Price $ -0.9219 $ $ -0.2059 $ 0.8923 $ -0.9113 $ $ -0.2029 $ 0.8717
Eng Capacity $ -0.9388 $ $ -0.0724 $ 0.8865 $ -0.9273 $ $ -0.0748 $ 0.8655
Top Speed $ -0.9384 $ $-0.2149 $ 0.9268 $ -0.9363 $ $ -0.2174 $ 0.9239
Acceleration +0.8832 +0.0960 0.7893 +0.8509 +0.0877 0.7317
Wheelbase $ -0.5636 $ +0.7825 0.9300 $ -0.5573 $ +0.7410 0.8596
Lenght $ -0.7386 $ +0.6271 0.9387 $ -0.7391 $ +0.6203 0.9310
Width $ -0.9483 $ +0.0998 0.9092 $ -0.9418 $ +0.0925 0.8956
Height +0.6364 +0.7145 0.9155 +0.6263 +0.6803 0.8550
Eingenvalues 5.5594 1.6290 5.4271 1.5069
Cumulative proportion
of total sample 0.6949 0.8986 0.6784 0.8668
variance explained
----------------------- ------------- ------------------- -------- ------------------- ------------- --------
: - Summary of factor analysis, assuming the Uniform distribution within intervals.
\[ResumeUnif\]
Analyzing the values presented in Table \[ResumeUnif\] we can see that there are very little differences between the results obtained by Principal Component and Principal Axis Factoring. From the figures in Table \[ResumeUnif\], we may now write the factor model, which in the case of Principal Axis Factoring is as follows:\
$Price= -0.9113f_{1}-0.2029f_{2}+\varepsilon_{Price}$\
$Eng Capacity= -0.9273f_{1}-0.0748f_{2}+\varepsilon_{Eng Capacity}$\
$Top Speed= -0.9363f_{1}-0.2174f_{2}+\varepsilon_{Top Speed}$\
$Acceleration= +0.8509f_{1}+0.0877f_{2}+\varepsilon_{Acceleration}$\
$Wheelbase= -0.5573f_{1}+0.7410f_{2}+\varepsilon_{Wheelbase}$\
$Lenght= -0.7391f_{1}+0.6203f_{2}+\varepsilon_{Lenght}$\
$Width= -0.9418f_{1}+0.0925f_{2}+\varepsilon_{Width}$\
$Height= +0.6263f_{1}+0.6803f_{2}+\varepsilon_{Height}$
or, if we represent each interval-valued variable by the respective quantile function,
$\Psi_{Price}^{-1}(t)= -0.9113\Psi_{f_{1}}^{-1}(1-t)-0.2029\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{Price}}^{-1}(t)$\
$\Psi_{Eng Capacity}^{-1}(t)= -0.9273\Psi_{f_{1}}^{-1}(1-t)-0.0748\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{Eng Capacity}}^{-1}(t)$\
$\Psi_{Top Speed}^{-1}(t)= -0.9363\Psi_{f_{1}}^{-1}(1-t)-0.2174\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{Top Speed}}^{-1}(t)$\
$\Psi_{Acceleration}^{-1}(t)= +0.8509\Psi_{f_{1}}^{-1}(t)+0.0877\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{Acceleration}}^{-1}(t)$\
$\Psi_{Wheelbase}^{-1}(t)= -0.5573\Psi_{f_{1}}^{-1}(1-t)+0.7410\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{Wheelbase}}^{-1}(t)$\
$\Psi_{Lenght}^{-1}(t)= -0.7391\Psi_{f_{1}}^{-1}(1-t)+0.6203\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{Lenght}}^{-1}(t)$\
$\Psi_{Width}^{-1}(t)= -0.9418\Psi_{f_{1}}^{-1}(1-t)+0.0925\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{Width}}^{-1}(t)$\
$\Psi_{Height}^{-1}(t)= +0.6263\Psi_{f_{1}}^{-1}(t)+0.6803\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{Height}}^{-1}(t)$\
with $0 \leq t \leq 1 $.\
The 1st factor presents high factor loadings for Price, Engine Capacity, Top Speed, Acceleration and Width and explains 67.8% of the total variance. The 2nd factor, with high factor loading for Wheelbase, explains 18.8% of total variance. It is noted that the Length and Height have high factor loadings on both factors, reflecting the fact that these characteristics do not contribute to the distinction of car models. Additionally, all communalities are high indicating that the two retained factors are suitable for describing the latent relational structure between characteristics of the car models.\
We can thus say that the factor model distinguishes the car models with higher price, higher engine capacity, higher top speed, greater width and shorter acceleration time from those with opposite characteristics; furthermore it separates car models with larger wheelbase from the others.\
The model based on Principal Component extraction may be written in a similar way, using the correspondent values of Table \[ResumeUnif\].\
Figures \[fig\_Unif\_PCF\] and \[fig\_Unif\_PAF\] show the 33 car models in the plane defined by the two interval-valued factors, obtained by the two extracting methods and by the two factor scores estimation methods considerated. As it can be seen, whereas the first factor distinguishes the upscale car models from the low cost car models, the second factor differentiates essentially car models with greater wheelbase from the smaller ones.
\[h!\]
\[h!\]
From the observation of the Figures \[fig\_Unif\_PCF\] and \[fig\_Unif\_PAF\] we can conclude that factor scores obtained by the two methods are very similar. However, we note that less degenerate intervals were obtained in the 2nd factor on the factor scores obtained by the model based on Principal Axis Factoring, and this difference is more noticeable when the ‘Anderson-Rubin method’ is choosen.
### Triangular Distribution {#sec5:1:2}
We now assume a Triangular distribution within each observed interval with a randomly generated mode. The data may hence be represented by triplets (min, mode, max) as in Table \[CarsT\].
\[h!\]
Price Engine Capacity … Height
-------------- -------------------------- -------------------- --- -----------------
Alfa 145 $(27806,32566,33596)$ $(1370,1609,1910)$ … $(143,143,143)$
Alfa 156 $(41593,61491,62291)$ $(1598,2249,2492)$ … $(142,142,142)$
Aston Martin $(260500,386054,460000)$ $(5935,5935,5935)$ … $(124,131,132)$
Porsche $(147704,242211,246412)$ $(3387,3578,3600)$ … $(130,130,131)$
Rover 25 $(21492,29242,33042)$ $(1119,1532,1994)$ … $(142,142,142)$
Passat $(39676,45063,63455)$ $(1595,2360,2496)$ … $(146,146,146)$
: - Car data set (partial view) described by triplets (min, mode, max).
\[CarsT\]
Applying the covariance definition $ Cov_3 $ as in formula (\[Cov3T\_2\]) we obtain the following correlation matrix:\
Price EngCap TopSpeed Acceler Wheelbase Lenght Width Height
$$\mathbf{R}=\left[
\begin{array}{cccccccc}
1 & +0.9527 & +0.8942 & -0.7920 & +0.3475 & +0.5091 & +0.8415 & -0.7280 \\
& 1 & +0.8694 & -0.7659 & +0.4897 & +0.6352 & +0.8672 & -0.6275 \\
& & 1 & -0.9108 & +0.3354 & +0.5720 & +0.8596 & -0.7326 \\
& & & 1 & -0.4101 & -0.6035 & -0.8238 & +0.6099 \\
& & & & 1 & +0.8672 & +0.5912 & +0.1570 \\
& & & & & 1 & +0.7627 & -0.0346 \\
& & & & & & 1 & -0.5470 \\
& & & & & & & 1
\end{array}
\right]$$
Principal Component and Principal Axis Factoring of this matrix leads to the extraction of two factors, with values of estimated factor loadings in interval-valued factors, eigenvalues and communalities which are very similar. For this reason in the Table \[ResumeTriang\] we only indicate those values for the Principal Axis Factoring method.
\[h!\]
--------------------------------- ------------- ------------------- --------
**Variable** **Communalities**
$f_1$ $f_2$
Price $ -0.9222 $ $ -0.2266 $ 0.9018
Eng Capacity $ -0.9327 $ $ -0.0584 $ 0.8733
Top Speed $ -0.9438 $ $-0.2155 $ 0.9372
Acceleration +0.8737 +0.0836 0.7704
Wheelbase $ -0.5485 $ +0.7465 0.8581
Lenght $ -0.7327 $ +0.6312 0.9354
Width $ -0.9442 $ +0.0967 0.9009
Height +0.6310 +0.6710 0.8484
Eingenvalues 5.5019 1.5235
Cumulative proportion of
total sample variance explained 0.6877 0.8782
--------------------------------- ------------- ------------------- --------
: - Summary of factor analysis, obtained by Principal Axis Factoring, considering the Triangular distribution within intervals.
\[ResumeTriang\]
Based on the factor loadings of the model we can conclude that the variables Price, Engine Capacity, Top Speed, Acceleration and Width are strongly related to the 1st factor and weakly associated with the 2nd factor, whereas the variable Wheelbase is strongly associated with the 2nd factor and more weakly associated with the 1st factor. Length and Height have high factor loadings on both factors, so do not contribute to the distinction of car models. 68.8% of total variance is explained by the 1st factor and 19.0% by the 2nd factor, which together represent 87.8% of the total variance. Furthermore, all communalities are high indicating that the two factors retained are suitable for describing the latent relational structure between characteristics of the car models.\
The resulting factor model is,\
$Price= -0.9222f_{1}-0.2266f_{2}+\varepsilon_{Price}$\
$Eng Capacity= -0.9327f_{1}-0.0584f_{2}+\varepsilon_{Eng Capacity}$\
$Top Speed= -0.9438f_{1}-0.2155f_{2}+\varepsilon_{Top Speed}$\
$Acceleration= +0.8737f_{1}+0.0836f_{2}+\varepsilon_{Acceleration}$\
$Wheelbase= -0.5485f_{1}+0.7465f_{2}+\varepsilon_{Wheelbase}$\
$Lenght= -0.7327f_{1}+0.6312f_{2}+\varepsilon_{Lenght}$\
$Width= -0.9442f_{1}+0.0967f_{2}+\varepsilon_{Width}$\
$Height= +0.6410f_{1}+0.6710f_{2}+\varepsilon_{Height}$
or, if we represent each interval-valued variable by the respective quantile function,
$\Psi_{Price}^{-1}(t)= -0.9222\Psi_{f_{1}}^{-1}(1-t)-0.2266\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{Price}}^{-1}(t)$\
$\Psi_{Eng Capacity}^{-1}(t)= -0.9327\Psi_{f_{1}}^{-1}(1-t)-0.0584\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{Eng Capacity}}^{-1}(t)$\
$\Psi_{Top Speed}^{-1}(t)= -0.9438\Psi_{f_{1}}^{-1}(1-t)-0.2155\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{Top Speed}}^{-1}(t)$\
$\Psi_{Acceleration}^{-1}(t)= +0.8737\Psi_{f_{1}}^{-1}(t)+0.0836\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{Acceleration}}^{-1}(t)$\
$\Psi_{Wheelbase}^{-1}(t)= -0.5485\Psi_{f_{1}}^{-1}(1-t)+0.7465\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{Wheelbase}}^{-1}(t)$\
$\Psi_{Lenght}^{-1}(t)= -0.7327\Psi_{f_{1}}^{-1}(1-t)+0.6312\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{Lenght}}^{-1}(t)$\
$\Psi_{Width}^{-1}(t)= -0.9442\Psi_{f_{1}}^{-1}(1-t)+0.0967\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{Width}}^{-1}(t)$\
$\Psi_{Height}^{-1}(t)= +0.6310\Psi_{f_{1}}^{-1}(t)+0.6710\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{Height}}^{-1}(t)$
with $0 \leq t \leq 1 $.\
The factor scores obtained by the ‘Bartlett method’ and by the ‘Anderson-Rubin method’ are represented in Figure \[fig\_Tri\_PAF\] showing the 33 car models in the plane defined by the two interval-valued factors. From their observation we can conclude that factor scores obtained by the two methods are very similar in the 1st fator, while much less degenerate intervals were obtained in the 2nd factor by ‘Anderson-Rubin method’.
\[h!\]
As it can be observed, the position in the plan of car models assuming the Triangular distribution is very similar to that obtained when the Uniform distribution was assumed, and therefore the conclusions are analogous. Essentially, the 1st factor distinguishes upscale from low cost car models and the 2nd factor differentiates car models with greater wheelbase from smaller ones.\
Meteorological Data {#sec5:2}
-------------------
In this section a factor analysis is performed on a set of 283 cities of the United States of America described by 13 interval-valued variables: the temperatures (in Fahrenheit degrees) of the 12 months of the year and the annual precipitation (in mm) between the years 1971 e 2000 (see Table \[TempPrcUSA\]).
\[h!\]
January February … December Precipitation
------------------------- -------------------------- -------------------------- --- -------------------------- --------------------------
BIRMINGHAM AP, AL $\left[32.3,52.8\right]$ $\left[35.4,58.3\right]$ … $\left[35.2,56.0\right]$ $\left[41.8,90.6\right]$
HUNTSVILLE, AL $\left[30.7,48.9\right]$ $\left[34.0,54.6\right]$ … $\left[33.8,52.4\right]$ $\left[40.7,89.4\right]$
MOBILE, AL $\left[39.5,60.7\right]$ $\left[42.4,64.5\right]$ … $\left[41.6,62.9\right]$ $\left[47.8,91.2\right]$
MONTGOMERY, AL $\left[35.5,57.6\right]$ $\left[38.6,62.4\right]$ … $\left[37.6,60.3\right]$ $\left[43.5,92.7\right]$
ANCHORAGE, AK $\left[9.3,22.2\right]$ $\left[11.7,25.8\right]$ … $\left[11.4,23.7\right]$ $\left[15.9,65.3\right]$
WAKE ISLAND, PC $\left[73.1,82.4\right]$ $\left[72.4,82.1\right]$ … $\left[74.7,83.9\right]$ $\left[76.3,88.8\right]$
YAP, W CAROLINE IS., PC $\left[73.7,86.5\right]$ $\left[73.8,86.7\right]$ … $\left[74.2,87.0\right]$ $\left[73.7,87.7\right]$
SAN JUAN, PR $\left[70.8,82.4\right]$ $\left[70.9,82.8\right]$ … $\left[72.1,83.2\right]$ $\left[74.0,87.8\right]$
: - Meteorological data set (partial view).
\[TempPrcUSA\]
### Uniform Distribution {#sec5:2:1}
In this section it is assumed that the values within the observed intervals are distributed according to an Uniform distribution.\
The following is the sample correlation matrix **R** obtained from the third definition of covariance $ Cov_3 $, using formula (\[Cov3U\_1\_2\]):\
Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec Prec $$\mathbf{R}=\left[
\begin{array}{ccccccccccccc}
1 & 0.9942 & 0.9701 & 0.9136 & 0.8232 & 0.7100 & 0.6437 & 0.7029 & 0.8406 & 0.9274 & 0.9745 & 0.9954 & 0.7310 \\
& 1 & 0.9875 & 0.9414 & 0.8588 & 0.7549 & 0.6943 & 0.7501 & 0.8721 & 0.9470 & 0.9760 & 0.9886 & 0.7620\\
& & 1 & 0.9808 & 0.9217 & 0.8372 & 0.7827 & 0.8294 & 0.9254 & 0.9756 & 0.9794 & 0.9725 & 0.8150 \\
& & & 1 & 0.9772 & 0.9198 & 0.8731 & 0.9054 & 0.9683 & 0.9855 & 0.9556 & 0.9262 & 0.8720 \\
& & & & 1 & 0.9785 & 0.9438 & 0.9588 & 0.9841 & 0.9595 & 0.8945 & 0.8441 & 0.9003 \\
& & & & & 1 & 0.9873 & 0.9861 & 0.9688 & 0.9023 & 0.8034 & 0.7346 & 0.8950 \\
& & & & & & 1 & 0.9944 & 0.9483 & 0.8604 & 0.7436 & 0.6665 & 0.8911 \\
& & & & & & & 1 & 0.9709 & 0.8996 & 0.7949 & 0.7236 & 0.9052 \\
& & & & & & & & 1 & 0.9733 & 0.9091 & 0.8584 & 0.9092 \\
& & & & & & & & & 1 & 0.9757 & 0.9427 & 0.8749 \\
& & & & & & & & & & 1 & 0.9885 & 0.7958 \\
& & & & & & & & & & & 1 & 0.7462 \\
& & & & & & & & & & & & 1 \\
\end{array}
\right]$$
Both Principal Component and Principal Axis Factoring of this matrix leads to the extraction of two factors, with values of estimated factor loadings in interval-valued factors, eigenvalues and communalities nearly equal. For this reason we only indicate those values for the Principal Axis Factoring method in Table \[ResumeUnifMeteo\].
\[h!\]
--------------------------------- ------------- ------------------- --------
**Variable** **Communalities**
$f_1$ $f_2$
January $ -0.9131 $ $ -0.4000 $ 0.9937
February $ -0.9371 $ $ -0.3338 $ 0.9895
March $ -0.9735 $ $-0.2035 $ 0.9890
April $ -0.9925 $ +0.0281 0.9858
May $ -0.9777 $ +0.1641 0.9827
June $ -0.9317 $ +0.3476 0.9889
July $ -0.8954 $ +0.4407 0.9960
August $ -0.9271 $ +0.3627 0.9910
September $ -0.9852 $ +0.1496 0.9931
October $ -0.9932 $ $ -0.0594 $ 0.9900
November $ -0.9579 $ $ -0.2549 $ 0.9825
December $ -0.9262 $ $ -0.3703 $ 0.9950
Precipitation $ -0.8884 $ +0.2112 0.8338
Eingenvalues 11.6513 1.0598
Cumulative proportion of
total sample variance explained 0.8963 0.9778
--------------------------------- ------------- ------------------- --------
: - Summary of factor analysis, obtained by Principal Axis Factoring, considering the Triangular distribution within intervals.
\[ResumeUnifMeteo\]
Based on the factor loadings of the model we can conclude that all variables: the temperatures of the 12 months of the year and the annual precipitation are strongly related to the 1st factor. Moreover the temperature variables in the months of January and July are moderatly associated with the 2nd factor. 89.6% of total variance is explained by the 1st factor and 8.2% by the 2nd factor, which together represent 97.8% of total variance. All communalities are high indicating that the two factors retained are suitable for describing the latent relational structure between the temperatures of the 12 months of the year and the annual precipitation.\
The resulting factor model is,\
$January= -0.9131f_{1}-0.4000f_{2}+\varepsilon_{January}$\
$February = -0.9371f_{1}-0.3338f_{2}+\varepsilon_{February}$\
$March= -0.9735f_{1}-0.2035f_{2}+\varepsilon_{March}$\
$April= -0.9925f_{1}-0.0281f_{2}+\varepsilon_{April}$\
$May= -0.9777f_{1}+0.1641f_{2}+\varepsilon_{May}$\
$June= -0.9317f_{1}+0.3476f_{2}+\varepsilon_{June}$\
$July= -0.8954f_{1}+0.4407f_{2}+\varepsilon_{July}$\
$August= -0.9271f_{1}+0.3627f_{2}+\varepsilon_{August}$\
$September= -0.9852f_{1}+0.1496f_{2}+\varepsilon_{September}$\
$October = -0.9932f_{1}-0.0594f_{2}+\varepsilon_{October}$\
$November= -0.9579f_{1}-0.2549f_{2}+\varepsilon_{November}$\
$December= -0.9262f_{1}-0.3703f_{2}+\varepsilon_{December}$\
$Precipitation= -0.8884f_{1}+0.2112f_{2}+\varepsilon_{Precipitation}$
or,
$\Psi_{January}^{-1}(t)= -0.9131\Psi_{f_{1}}^{-1}(1-t)-0.4000\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{January}}^{-1}(t)$\
$\Psi_{February}^{-1}(t)= -0.9371\Psi_{f_{1}}^{-1}(1-t)-0.3338\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{February}}^{-1}(t)$\
$\Psi_{March}^{-1}(t)= -0.9735\Psi_{f_{1}}^{-1}(1-t)-0.2035\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{March}}^{-1}(t)$\
$\Psi_{April}^{-1}(t)= -0.9925\Psi_{f_{1}}^{-1}(1-t)-0.0281\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{April}}^{-1}(t)$\
$\Psi_{May}^{-1}(t)= -0.9777\Psi_{f_{1}}^{-1}(1-t)+0.1641\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{May}}^{-1}(t)$\
$\Psi_{June}^{-1}(t)= -0.9317\Psi_{f_{1}}^{-1}(1-t)+0.3476\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{June}}^{-1}(t)$\
$\Psi_{July}^{-1}(t)= -0.8954\Psi_{f_{1}}^{-1}(1-t)+0.4407\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{July}}^{-1}(t)$\
$\Psi_{August}^{-1}(t)= -0.9271\Psi_{f_{1}}^{-1}(1-t)+0.3627\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{August}}^{-1}(t)$\
$\Psi_{September}^{-1}(t)= -0.9852\Psi_{f_{1}}^{-1}(1-t)+0.1496\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{September}}^{-1}(t)$\
$\Psi_{October}^{-1}(t)= -0.9932\Psi_{f_{1}}^{-1}(1-t)-0.0594\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{October}}^{-1}(t)$\
$\Psi_{November}^{-1}(t)= -0.9579\Psi_{f_{1}}^{-1}(1-t)-0.2549\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{November}}^{-1}(t)$\
$\Psi_{December}^{-1}(t)= -0.9262\Psi_{f_{1}}^{-1}(1-t)-0.3703\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{December}}^{-1}(t)$\
$\Psi_{Precipitation}^{-1}(t)= -0.8884\Psi_{f_{1}}^{-1}(1-t)+0.2112\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{Precipitation}}^{-1}(t)$
with $0 \leq t \leq 1 $.
The factor scores obtained by the ‘Anderson-Rubin method’ for Meteorological data set are displayed in Table \[AndRubUnifMeteo\] and represented in Figure \[figARUMeteo\].
Factor 1 Factor 2
------------------------- -------------------------------- --------------------------------
BIRMINGHAM AP, AL $\left[-1.4742,+0.3535\right]$ $\left[+0.2627,+0.2763\right]$
HUNTSVILLE, AL $\left[-1.3915,+0.3445\right]$ $\left[+0.3439,+0.3532\right]$
MOBILE, AL $[-1.7866,-0.0858]$ $[+0.0167,+0.0497]$
MONTGOMERY, AL $\left[-1.8090,+0.1110\right]$ $\left[+0.2519,+0.2783\right]$
ANCHORAGE, AK $[+0.9322,+2.2412]$ $[-0.7550,-0.7504]$
WAKE ISLAND, PC $\left[-2.0926,-1.3101\right]$ $\left[-1.6814,-1.3420\right]$
YAP, W CAROLINE IS., PC $\left[-2.0968,-1.0362\right]$ $\left[-2.1523,-1.8845\right]$
SAN JUAN, PR $\left[-2.0600,-1.1508\right]$ $\left[-1.6548,-1.4880\right]$
: - Factor scores obtained for the metereological data by the ‘Anderson-Rubin method’ and considering the Uniform distribution within intervals.
\[AndRubUnifMeteo\]
Figure \[figARUMeteo\] shows the 283 cities of the United States of America in the plane defined by the two interval-valued factors. It can be observed that, while the 1st factor distinguishes warm cities with high humidity from cold and dry cities, the 2nd factor basically differentiates cities with larger thermal amplitude from those with short thermal amplitude.\
\[h!\] ![ - Factor scores obtained for the metereological data by the ‘Anderson-Rubin method’ and considering the Uniform distribution within intervals.[]{data-label="figARUMeteo"}](MeteoARUnifPAF "fig:"){width="9.5cm" height="9.5cm"}
### Triangular Distribution {#sec6:1:2}
In this section a Triangular distribution within each observed interval with a randomly generated mode is assumed. The data may hence be represented by triplets (min, mode, max) as illustrated in Table \[MeteoT\].
\[h!\]
January … December Precipitation
------------------------- ------------------------------- --- ------------------------------- -------------------------------
BIRMINGHAM AP, AL $\left(32.3,40.8,52.8\right)$ … $\left(35.2,54.2,56.0\right)$ $\left(41.8,68.8,90.6\right)$
HUNTSVILLE, AL $\left(30.7,46.1,48.9\right)$ … $\left(33.8,43.1,52.4\right)$ $\left(40.7,67.6,89.4\right)$
MOBILE, AL $\left(39.5,59.0,60.7\right)$ … $\left(41.6,61.0,62.9\right)$ $\left(47.8,48.0,91.2\right)$
MONTGOMERY, AL $\left(35.5,57.5,57.6\right)$ … $\left(37.6,50.4,60.3\right)$ $\left(43.5,48.2,92.7\right)$
ANCHORAGE, AK $\left(9.3,15.1,22.2\right)$ … $\left(11.4,13.9,23.7\right)$ $\left(15.9,49.4,65.3\right)$
WAKE ISLAND, PC $\left(73.1,77.8,82.4\right)$ … $\left(74.7,80.9,83.9\right)$ $\left(76.3,80.3,88.8\right)$
YAP, W CAROLINE IS., PC $\left(73.7,83.6,86.5\right)$ … $\left(74.2,86.0,87.0\right)$ $\left(73.7,83.3,87.7\right)$
SAN JUAN, PR $\left(70.8,73.4,82.4\right)$ … $\left(72.1,75.5,83.2\right)$ $\left(74.0,81.7,87.8\right)$
: - Meteorological data set (partial view) described by triplets (min, mode, max).
\[MeteoT\]
Applying the covariance definition $ Cov_3 $ as in formula (\[Cov3T\_2\]) we obtain the following correlation matrix:
Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec Prec $$\mathbf{R}=\left[
\begin{array}{ccccccccccccc}
1 & 0.9815 & 0.9531 & 0.8915 & 0.7978 & 0.6760 & 0.6115 & 0.6757 & 0.8195 & 0.9119 & 0.9643 & 0.9869 & 0.7284 \\
& 1 & 0.9720 & 0.9212 & 0.8338 & 0.7230 & 0.6676 & 0.7289 & 0.8558 & 0.9246 & 0.9591 & 0.9779 & 0.7506\\
& & 1 & 0.9582 & 0.8946 & 0.8063 & 0.7485 & 0.8057 & 0.9035 & 0.9526 & 0.9623 & 0.9595 & 0.7945 \\
& & & 1 & 0.9459 & 0.8860 & 0.8383 & 0.8727 & 0.9367 & 0.9563 & 0.9332 & 0.9087 & 0.8292 \\
& & & & 1 & 0.9400 & 0.8972 & 0.9114 & 0.9452 & 0.9234 & 0.8689 & 0.8225 & 0.8355 \\
& & & & & 1 & 0.9396 & 0.9443 & 0.9268 & 0.8641 & 0.7737 & 0.7079 & 0.8152 \\
& & & & & & 1 & 0.9427 & 0.8990 & 0.8160 & 0.7119 & 0.6425 & 0.8018 \\
& & & & & & & 1 & 0.9228 & 0.8595 & 0.7652 & 0.7023 & 0.8265 \\
& & & & & & & & 1 & 0.9402 & 0.8877 & 0.8430 & 0.8584 \\
& & & & & & & & & 1 & 0.9541 & 0.9268 & 0.8327 \\
& & & & & & & & & & 1 & 0.9768 & 0.7864 \\
& & & & & & & & & & & 1 & 0.7440 \\
& & & & & & & & & & & & 1 \\
\end{array}
\right]$$
Both Principal Component and Principal Axis Factoring of this matrix leads to the extraction of two factors, with values of estimated factor loadings in interval-valued factors, eigenvalues and communalities very similar. For this reason we only indicate those values for the Principal Axis Factoring method in Table \[ResumeTriangMeteo\].
\[h!\]
--------------------------------- ------------- ------------------- --------
**Variable** **Communalities**
$f_1$ $f_2$
January $ -0.9105 $ $ -0.3979 $ 0.9874
February $ -0.9337 $ $ -0.3185 $ 0.9732
March $ -0.9671 $ $-0.1871 $ 0.9703
April $ -0.9790 $ $-0.0104 $ 0.9586
May $ -0.9551 $ +0.1730 0.9421
June $ -0.9042 $ +0.3668 0.9521
July $ -0.8624 $ +0.4429 0.9399
August $ -0.8993 $ +0.3630 0.9404
September $ -0.9663 $ +0.1488 0.9558
October $ -0.9780 $ $ -0.0550 $ 0.9960
November $ -0.9533 $ $ -0.2430 $ 0.9679
December $ -0.9270 $ $ -0.3577 $ 0.9872
Precipitation $ -0.8556 $ +0.1389 0.7512
Eingenvalues 11.2671 1.0188
Cumulative proportion of
total sample variance explained 0.8667 0.9451
--------------------------------- ------------- ------------------- --------
: - Summary of factor analysis, obtained by Principal Axis Factoring, considering the Triangular distribution within intervals.
\[ResumeTriangMeteo\]
Based on the factor loadings of the model we can conclude that all variables the temperatures on the 12 months of the year and the annual precipitation are strongly related to the 1st factor. The temperature variables in the months of December and July are moderatly associated with the 2nd factor. 86.7% of total variance is explained by the 1st factor and only 7.8% by the 2nd factor, which together represent 94.5% of total variance. All communalities are high indicating that the two factors retained are suitable for describing the latent relational structure between the temperatures of the 12 months of the year and the annual precipitation.\
The resulting factor model written as,\
$January= -0.9105f_{1}-0.3979f_{2}+\varepsilon_{January}$\
$February = -0.9337f_{1}-0.3185f_{2}+\varepsilon_{February}$\
$March= -0.9671f_{1}-0.1871f_{2}+\varepsilon_{March}$\
$April= -0.9790f_{1}-0.0104f_{2}+\varepsilon_{April}$\
$May= -0.9551f_{1}+0.1730f_{2}+\varepsilon_{May}$\
$June= -0.9042f_{1}+0.3668f_{2}+\varepsilon_{June}$\
$July= -0.8624f_{1}+0.4429f_{2}+\varepsilon_{July}$\
$August= -0.8993f_{1}+0.3630f_{2}+\varepsilon_{August}$\
$September= -0.9663f_{1}+0.1488f_{2}+\varepsilon_{September}$\
$October = -0.9780f_{1}-0.0550f_{2}+\varepsilon_{October}$\
$November= -0.9533f_{1}-0.2430f_{2}+\varepsilon_{November}$\
$December= -0.9270f_{1}-0.3577f_{2}+\varepsilon_{December}$\
$Precipitation= -0.8556f_{1}+0.1389f_{2}+\varepsilon_{Precipitation}$
or, using quantile functions,
$\Psi_{January}^{-1}(t)= -0.9105\Psi_{f_{1}}^{-1}(1-t)-0.3979\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{January}}^{-1}(t)$\
$\Psi_{February}^{-1}(t)= -0.9337\Psi_{f_{1}}^{-1}(1-t)-0.3185\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{February}}^{-1}(t)$\
$\Psi_{March}^{-1}(t)= -0.9671\Psi_{f_{1}}^{-1}(1-t)-0.1871\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{March}}^{-1}(t)$\
$\Psi_{April}^{-1}(t)= -0.9790\Psi_{f_{1}}^{-1}(1-t)-0.0104\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{April}}^{-1}(t)$\
$\Psi_{May}^{-1}(t)= -0.9551\Psi_{f_{1}}^{-1}(1-t)+0.1730\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{May}}^{-1}(t)$\
$\Psi_{June}^{-1}(t)= -0.9042\Psi_{f_{1}}^{-1}(1-t)+0.3668\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{June}}^{-1}(t)$\
$\Psi_{July}^{-1}(t)= -0.8624\Psi_{f_{1}}^{-1}(1-t)+0.4429\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{July}}^{-1}(t)$\
$\Psi_{August}^{-1}(t)= -0.8993\Psi_{f_{1}}^{-1}(1-t)+0.3630\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{August}}^{-1}(t)$\
$\Psi_{September}^{-1}(t)= -0.9663\Psi_{f_{1}}^{-1}(1-t)+0.1488\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{September}}^{-1}(t)$\
$\Psi_{October}^{-1}(t)= -0.9780\Psi_{f_{1}}^{-1}(1-t)-0.0550\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{October}}^{-1}(t)$\
$\Psi_{November}^{-1}(t)= -0.9533\Psi_{f_{1}}^{-1}(1-t)-0.2430\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{November}}^{-1}(t)$\
$\Psi_{December}^{-1}(t)= -0.9270\Psi_{f_{1}}^{-1}(1-t)-0.3577\Psi_{f_{2}}^{-1}(1-t)+\Psi_{\varepsilon_{December}}^{-1}(t)$\
$\Psi_{Precipitation}^{-1}(t)= -0.8556\Psi_{f_{1}}^{-1}(1-t)+0.1389\Psi_{f_{2}}^{-1}(t)+\Psi_{\varepsilon_{Precipitation}}^{-1}(t)$
with $0 \leq t \leq 1 $.
The factor scores obtained by the ‘Anderson-Rubin method’ for Meteorological data set are displayed in Table \[AndRubTriMeteo\] and represented in Figure \[figARTriMeteo\].
Factor 1 Factor 2
------------------------- ---------------------------------------- -----------------------------------------
BIRMINGHAM AP, AL $\left(-1.5239,-0.5693,+0.3565\right)$ $\left( +0.2263,+0.2907,+0.2725\right)$
HUNTSVILLE, AL $\left(-1.3746,-0.6350,+0.3916\right)$ $\left(+0.4210,+0.4615,+0.4779\right)$
MOBILE, AL $(-1.7899,-0.9609,-0.0203)$ $(-0.1760,-0.1071,+0.0522)$
MONTGOMERY, AL $\left(-1.7872,-0.7832,+0.2274\right)$ $\left(-0.0588,-0.0061,+0.1055\right)$
ANCHORAGE, AK $(+0.8832,+1.5452,+2.1446)$ $(-0.7528,-0.7528,-0.7499)$
WAKE ISLAND, PC $\left(-2.0054,-1.9622,-1.4283\right)$ $\left(-1.9039,-0.9689,-0.9016\right)$
YAP, W CAROLINE IS., PC $\left(-2.2681,-1.8660,-1.1843\right)$ $\left(-2.0128,-1.6979,-1.6445\right)$
SAN JUAN, PR $\left(-2.2222,-1.6949,-1.2837\right)$ $\left(-1.3940,-1.2853,-1.2500\right)$
: - Factor scores obtained for the metereological data by the ‘Anderson-Rubin method’ and considering the Uniform distribution within intervals.
\[AndRubTriMeteo\]
Figure \[figARTriMeteo\] shows the 283 cities of the United States of America in the plane defined by the two interval-valued factors. The conclusions are exactly the same as those taken earlier, the 1st factor distinguishes warm cities with high humidity from cold and dry cities and the 2nd factor differentiates cities with larger thermal amplitude from those with short thermal amplitude. We notice however that in this analysis there are less degenerate intervals in the second factor, than in the analysis that assumed an Uniform distribution.
\[h!\] ![ - Factor scores obtained for the metereological data by the ‘Anderson-Rubin method’ and considering the Triangular distribution within intervals.[]{data-label="figARTriMeteo"}](MeteoARTriPAF "fig:"){width="9.5cm" height="9.5cm"}
Concluding remarks {#sec:6}
==================
Most of the methodologies developed for Symbolic Data Analysis rely on distribution free approaches. In this paper we addressed the analysis of the dependence structure of interval-valued variables, proposing a factor model for interval data based on quantile function representations. In our proposal, factor extraction is carried on by performing a Principal Component or a Principal Axis Factoring based on the correlation matrix between the observed interval-valued variables. For that purpose we rely on, and extend, appropriate definitions of variance, covariance and correlation, for interval variables under the assumptions of uniform or triangular distributions to model the within variability of each interval. Factor scores were derived by solving optimization problems, that adapt the Bartllet, and Anderson-Rubin methods for real-valued data. However, unlike the original problems, in the case of interval data, the resulting optimization problems do not have a closed-formal analytical solution, and need to be solved numerically. The research presented in this paper may be extended in several ways. In the first place, alternative methods of factor extraction can be devised. One important avenue of research is the study of parametric methods of factor extraction, based on existing models for interval data, such as those proposed in Brito and Duarte Silva [@Brito2012]. Secondly, factor rotation of interval-valued factors may be adressed. Finally, the basic approach proposed here can be extended to establish factor models for other types of symbolic data such as distributional or histogram data.\
**Acknowledgements**\
This work is financed by National Funds through the FCT - Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) as part of projects UID/EEA/50014/2013 and UID/GES/00731/2013.
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[^1]: In many works the Mallows distance is denominated Wasserstein distance. The reason is related to the fact that, historically, this metric has been introduced several times and from different perspectives, and therefore is known under different names [@Irpino2015]. However, it was Mallows who introduced this metric in a statistical context, so we will name it henceforth as Mallows distance.
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abstract: |
Consider the generalized Navier-Stokes equations on $ {\mathbb R}^n$: $$\partial_tu+u\cdot\nabla u + D^s u+\nabla P=0,\quad \mathrm{div}\;u=0.$$ For some appropriate number $s$, we prove that the Cauchy problem with initial data of the form $$\nonumber
u_0^\epsilon(x) = (v_0^h(x_\epsilon), \epsilon^{-1}v_0^n(x_\epsilon))^T,\quad x_\epsilon = (x_{h}, \epsilon x_n)^T,$$ is globally well-posed for all small $\epsilon > 0$, provided that the initial velocity profile $v_0$ is analytic in $x_n$ and certain norm of $v_0$ is sufficiently small but independent of $\epsilon$. In particular, for $n\geq4$, our result is applicable to the n-dimensional classical Navier-Stokes equations.
author:
- Yukang Chen
- Bin Han
- Zhen Lei
title: 'Global Regularity to the Navier-Stokes Equations for A Class of Large Initial Data'
---
Introduction
============
The Cauchy problem of the incompressible Navier-Stokes equations on ${\mathbb R}^n$ is described by the following system $$\label{e1.1}
\left\{
\begin{array}{rlll}
&\partial_tu+u\cdot\nabla u-\Delta u+\nabla P=0,
\ \ &\ x\in{\mathbb R}^n,\ t>0,\\
&\hbox{div}\, u=0, \ \ &\ x\in{\mathbb R}^n,\ t>0,\\
&u(0)=u_0, \ \ &\ x\in {\mathbb R}^n,
\end{array}
\right.$$ where $u$ represents the velocity field and $P$ is the scalar pressure.
First of all, let us recall some known results on the small-data global regularity of the Navier-Stokes equations on ${\mathbb R}^3$. In the seminal paper [@Le], Leray proved that the 3D incompressible Navier-Stokes equations are globally well-posed if the initial data $u_0$ is such that $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2}$ is small enough. This quantity is invariant under the natural scaling of the Navier-Stokes equations. Later on, many authors studied different scaling invariant spaces in which Navier-Stokes equations are well-posed at least for small initial data, which include but are not limited to $$\dot{H}^{\frac{1}{2}}({\mathbb R}^3)\hookrightarrow L^3({\mathbb R}^3)\hookrightarrow
\dot{B}^{-1+\frac{3}{p}}_{p,\infty}({\mathbb R}^3)\hookrightarrow BMO^{-1}({\mathbb R}^3),$$ where $3<p<\infty$. The space $BMO^{-1}({\mathbb R}^3)$ is known to be the largest scaling invariant space so that the Navier-Stokes equations (\[e1.1\]) are globally well-posed under small initial data. The readers are referred to [@FK; @Ka; @CMP; @KT] as references. We also mention that the work of Lei and Lin [@LL] was the first to quantify the smallness of the initial data to be 1 by introducing a new space $\mathcal{X}^{-1}$.
We remark that the norm in the above scaling invariant spaces are always greater than the norm in the Besov space $\dot{B}^{-1}_{\infty,\infty}$ defined by $$\|u\|_{\dot{B}^{-1}_{\infty,\infty}}\overset{\mathrm{def}}{=}\sup\limits_{t>0}t^{\frac{1}{2}}\|e^{t\Delta}u_0\|_{L^\infty}.$$ Bourgain and Pavlovic in [@Bour] showed that the cauchy problem of the 3D Navier-Stokes equations is ill-posed in the sense of norm inflation. Partially because of the result of Bourgain and Pavlovic, data with a large $\dot{B}^{-1}_{\infty, \infty}$ are usually called large data to the Navier-Stokes equations (for instance, see [@CG; @PZ]).
Towards this line of research, a well-oiled case is the family of initial data which is slowly varying in vertical variable. The initial velocity field $u_0^{\epsilon}$ is of the form $$\label{e1.2}
u_0^{\epsilon}(x) = ( v_0^{h}(x_{\epsilon}),{\epsilon}^{-1}v_0^{3}(x_{\epsilon}))^T,\quad x_{\epsilon}= (x_h, \epsilon x_3)^T,$$ which allows slowly varying in the vertical variable $x_3$ when ${\epsilon}> 0$ is a small parameter. This family of initial data are very interesting (as has been pointed out by , see the acknowledgement in [@CGP]) and considered by Chemin, Gallagher and Paicu in [@CGP]. They proved the global regularity of solutions to the Navier-Stokes equations when $v_0$ is analytic in $x_3$ and periodic in $x_h$, and certain norm of $v_0$ is sufficiently small but independent of ${\epsilon}> 0$. More precisely, they proved the following Theorem:
\[t1.1\] Let $\alpha$ be a positive number. There are two positive numbers ${\epsilon}_0$ and $\eta$ such that for any divergence free vector field $v_0$ satisfying $$\|e^{\alpha D_3 }v_0\|_{{B}^{\frac{7}{2}}_{2, 1}}\leq \eta,$$ then, for any positive ${\epsilon}$ smaller than ${\epsilon}_0$, the initial data generates a global smooth solution to (\[e1.1\]) on $\mathbb{T}^2\times {\mathbb R}.$
The notation ${B}^{\frac{7}{2}}_{2, 1}$ in the above Theorem denotes the usual inhomogeneous Besov space. The significance of the result lies in that the global regularity of the 3D incompressible Navier-Stokes equations in [@CGP] only requires *very little smallness* imposed on the initial data. It is clear that the $\dot{B}^{-1}_{\infty, \infty}$ norm of $u_0^{\epsilon}$ can tend to infinity as $\epsilon \to 0$. Let us first focus on the periodic constraint imposed on the initial data in Theorem \[t1.1\].
As has been pointed out by Chemin, Gallagher and Paicu, the reason why the horizontal variable of the initial data in [@CGP] is restricted to a torus is to be able to deal with very low horizontal frequencies. In the proof of Theorem \[t1.1\] in [@CGP], functions with zero horizontal average are treated differently to the others, and it is important that no small horizontal frequencies appear other than zero. Later on, many efforts are made towards removing the periodic constraint of $v_0$ on the horizontal variables. For instance, see [@CG; @Ha; @GHZ; @PZ] and so on. We will review those results a little bit later.
In this paper, we consider the Cauchy problem of the following generalized Navier-Stokes equations on ${\mathbb R}^n$: $$\label{e1.3}
\left\{
\begin{array}{rlll}
&\partial_tu+u\cdot\nabla u+D^su+\nabla P=0,
\ \ &\ x\in{\mathbb R}^n,\ t>0,\\
&\hbox{div}\, u=0, \ \ &\ x\in{\mathbb R}^n,\ t>0,\\
&u(0)=u^{\epsilon}_0, \ \ &\ x\in {\mathbb R}^n,
\end{array}
\right.$$ where $D=\sqrt{-\Delta}$. The initial velocity field $u_0^{\epsilon}$ is of the form $$\label{e1.4}
u_0^{\epsilon}(x) = ( v_0^{h}(x_{\epsilon}),{\epsilon}^{-1}v_0^{n}(x_{\epsilon}))^T,\quad x_{\epsilon}= (x_h, \epsilon x_n)^T.$$ The horizontal variable $x_h=(x_1,x_2,\cdots,x_{n-1})$.
Our main result is the following theorem which generalizes the theorem of Chemin, Gallagher and Paicu to the whole space for the generalized Navier-Stokes equations with some appropriate number $s$. Definition of notations will be given in Section 2.
\[t1.2\] Let $\alpha$, $\epsilon_0$, $p$ and $s$ be four positive constants and $(p,s)$ satisfy $1\leq p < n-1, 1 \leq s < \min(n-1,\frac{2(n-1)}{p})$. There exists a positive constant $\eta$ such that for any $0 < {\epsilon}< {\epsilon}_0$ and any divergence free vector field $v_0$ satisfying $$\label{e1.5}
\|e^{\alpha D_n }v_0\|_{\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1}\cap \dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1}}\leq \eta,$$ then the generalized Navier-Stokes equations (\[e1.3\]) with initial data generate a global smooth solution on ${\mathbb R}^n$.
1\) When $n\geq4$, one can find that the classical Navier-Stokes equations on ${\mathbb R}^n$ satisfy the assumption $1\leq s=2 < \min(n-1,\frac{2(n-1)}{p})$ for $1\leq p <n-1$. Then according to Theorem \[t1.2\], the n-dimensional incompressible Navier-Stokes equations with initial data (\[e1.4\]) have a global smooth solution in the whole space case.\
2) In the case $n=3$, we require that $1\leq s<2$. The main obstacle when $s=2$ is that we can not get the product law in $\dot{B}^{\frac{2}{p}-2,\frac{1}{2}}_{p,1}({\mathbb R}^3).$ This anisotropic Besov space is induced by the $a \ priori$ estimate of anisotropic pressure $(\nabla_hq, {\epsilon}^2\partial_3q)$ (see equation (\[e1.6\]) and Step 5 for details). From this point of view, the global well-posedness of 3D incompressible Navier-Stokes equations with initial data (\[e1.4\]) on $\mathbb{R}^3$ is still unclear, even though the higher dimensional cases are settled down.\
3) In the present paper, we establish the global solution in the $L^p$-type Besov space, in which the bilinear estimate of the solution can not be derived by the classical $L^2$ energy method. Particularly, one can obtain the $L^1$ estimate in time of the solution by introducing the new quantity $$\int_0^t \|v^n\|_{\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}}\|\partial_nv^h\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}d\tau$$ in the $a\ priori$ estimate. We also mention that in [@CGP; @PZ; @PZ2], authors did not get the $L^1$-time estimate of the solution.
Now we mention that many authors make efforts to remove the periodic restriction on horizontal variable. J. Chemin and I. Gallagher considered the well-prepared initial data in [@CG]. They proved the global well-posedness of (\[e1.1\]) when $u_0^{\epsilon}$ is of the form $$u_0^{\epsilon}{=}(v^h_0+{\epsilon}w^h_0, w_0^3)(x_h,{\epsilon}x_3).$$ Later, G. Gui, J. Huang, and P. Zhang in [@GHZ] generalized this result to the density dependent Navier-Stokes equations with the same initial velocity. Recently, B. Han in [@Ha] considered the global regularity of (\[e1.1\]) if $u_0^{\epsilon}$ satisfies the form of $${u}^{\epsilon}_0(x){=}{\epsilon}^\delta(v_0^{h}( x_h,{\epsilon}x_3),{\epsilon}^{-1}v_0^{3}( x_h,{\epsilon}x_3))$$ for any $0<\delta<1$, then $u^{\epsilon}_0$ generates a global solution of (\[e1.1\]) on ${\mathbb R}^3.$ The case $\delta=\frac{1}{2}$ and $\delta\in(0,\frac{1}{2})$ were considered by M. Paicu and Z. Zhang in [@PZ; @PZ2] if the initial data is allowed to be in Gevery class. All of the initial data is large in $\dot{B}^{-1}_{\infty,\infty}$, but still generates a global solution.
[**[Main ideas of the Proof.]{}**]{} We will prove our main result by constructing the bilinear estimate (independent of ${\epsilon}$). Our strategy can be stated as follows.
[**[Step 1.]{}**]{} Rescaled system and simplification.
As in [@CGP], we define $$u^{\epsilon}(t,x)=( v^h(t,x_{\epsilon}), {\epsilon}^{-1}v^n(t,x_{\epsilon}))^T,\quad P^{\epsilon}(t,x)=q(t,x_{\epsilon}).$$ Denote $$\Delta_{\epsilon}=\Delta_h+{\epsilon}^2\partial_n^2, \ \Delta_h=\partial_1^2+\cdots+\partial_{n-1}^2,\ D_{\epsilon}=\sqrt{-\Delta_{\epsilon}}.$$ Using the Navier-Stokes equations , it is easy to derive the equations governing the rescaled variables $v$ and $q$ (they are still depending on ${\epsilon}$): $$\label{e1.6}
\left\{
\begin{array}{rlll}
&\partial_tv^h+ v\cdot\nabla v^h+D_{\epsilon}^s v^h+\nabla_h q=0,\\
&\partial_tv^n+v\cdot\nabla v^n+D_{\epsilon}^s v^n+{\epsilon}^2\partial_n q=0,\\
& \hbox{div}\,v=0,\quad v(0)=v_0(x),
\end{array}
\right.$$ where $v^h=(v^1,\cdots,v^{n-1})$. The rescaled pressure $q$ can be recovered by the divergence free condition as $$-\Delta_{\epsilon}q= \sum\limits_{i,j}\partial_i\partial_j(v^iv^j).$$ The global regularity of solutions to system for small initial data $v_0$ will be presented in Section 3 and 4 for any positive ${\epsilon}$. But to best illustrate our ideas, let us here focus on the case of ${\epsilon}= 0$. Formally, by taking ${\epsilon}=0$ in system , we have the following limiting system: $$\label{elimit}
\left\{
\begin{array}{rlll}
&\partial_tv^h+ v\cdot\nabla v^h+D_h^s v^h+\nabla_h q=0,\\
&\partial_tv^n+v\cdot\nabla v^n+D_h^s v^n=0,\\
&\hbox{div}\,v=0,\quad v(0)=v_0(x),
\end{array}
\right.$$ where $D_h=\sqrt{-\Delta_h}$. The pressure $q$ in (\[elimit\]) is given by $$-\Delta_h q= \sum\limits_{i,j}\partial_i\partial_j(v^iv^j).$$
[**[Step 2.]{}**]{} Set-up of the $a\ priori$ estimate.
Observing that in the rescaled system (\[elimit\]), the viscosity is absent in the vertical direction. To make the full use of smoothing effect from operator $\partial_t +D_h^s$, particularly in low frequency parts, we will apply the tool of anisotropic homogeneous Besov spaces. The goal is to derive certain $a\ priori$ estimate of the form: $$\Psi(t)\lesssim \Psi(0)+\Psi(t)^2.$$ Note that pressure term doesn’t explicitly appear in the equation of $v^n$ of the limiting system (\[elimit\]), which makes the estimate for $v^n$ easier. So here let us just focus on the equation of $v^h$. Naturally, we define $$\Psi(t)= \|v^h\|_{\widetilde{L}^\infty_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}+\|v^h\|_{L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1,\frac{1}{2}})}+\cdots.$$ At this step, we assume that the initial data $v_0^h$ belongs to $\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}$. This ensures that $\Psi(t)$ is a critical quantity with respect to the natural scaling of the generalized Navier-Stokes equations. By Duhamel’s principle, we can write the integral equation of $v^h$ by $$v^h = e^{-tD_h^s}v_0^h - \int_0^te^{-(t - \tau)D_h^s}(v^h\cdot\nabla_h v^h + v^n\partial_nv^h + \nabla_hq)d\tau.$$ According to the estimates of heat equation, one can formally has $$\Psi(t)\lesssim \Psi(0)+\|v^h\cdot\nabla_h v^h\|_{L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})} +\| v^n\partial_nv^h\|_{L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})} + \|\nabla_hq\|_{L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}.$$ It will be shown that $$\aligned
\|v^h\cdot\nabla_h v^h\|_{L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}\lesssim \|v^h\|_{\widetilde{L}^\infty_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}\|\nabla_h v^h\|_{L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}})} \lesssim \Psi(t)^2.
\endaligned$$
[**[Step 3.]{}**]{} Derivative loss: input the estimate of $\partial_nv^h$.
For the quantity $\Psi(t)$, we need to prove the bilinear estimate in the following form: $$\Psi(t)\lesssim \Psi(0)+\Psi(t)^2+ \|v^n\partial_nv^h\|_{ L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}+\cdots.$$ Certainly, there is $\partial_n$-derivative loss! By the product law in anisotropic Besov spaces (Lemma \[p2.5\]), the strategy to bound the last term is $$\|v^n\partial_nv^h\|_{L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}\lesssim \int_0^t \|v^n\|_{\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}}\|\partial_nv^h\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}d\tau,$$ and then we should add the new quantity $$\int_0^t \|v^n\|_{\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}}\|\partial_nv^h\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}d\tau$$ in the definition of $\Psi(t).$ We find that $\|\partial_nv^h\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}$ is the hardest term to estimate. Since we note that by Duhamel’s principle, $$\partial_nv^h = e^{-tD_h^s}\partial_nv_0^h - \int_0^te^{-(t - \tau)D_h^s}\partial_n(v^n\partial_nv^h)d\tau+\cdots.$$ There is still $\partial_n$-derivative loss in the $a\ priori$ estimates!
[**[Step 4.]{}**]{} Recover the derivative loss: analyticity in $x_n$.
Motivated by Chemin-Gallagher-Paicu [@CGP], we add an exponential weight $e^{\Phi(t, D_n )}$ with $$\Phi(t, |\xi_n|)=(\alpha-\lambda\theta(t))|\xi_n|.$$ Here $\theta(t)$ is defined by $$\theta(t)=\int_0^t\|v_\Phi^n\|_{\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}}d\tau,$$ which will be shown to be small to ensure that $\Phi(t, |\xi_n|)$ satisfies the subadditivity. Denoting $f_\Phi=e^{\Phi(t, D_n )}f$, we then have $$\partial_nv_\Phi^h = e^{-tD_h^s}e^{\Phi(t, D_n )}\partial_nv_0^h - \int_0^te^{-(t - \tau)D_h^s}e^{-\lambda\int_\tau^t\dot{\theta}(t')dt' D_n }\partial_n(v^n\partial_nv^h)_\Phi d\tau+\cdots.$$ Hence, we can recover the derivative loss by $$\aligned
&\int_0^t \|v_\Phi^n\|_{\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}} \|\partial_nv_\Phi^h\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}d\tau\\
&\lesssim \sum\limits_{k,j}2^{k(\frac{n-1}{p}+1-s)}2^{\frac{1}{2}j}\int_0^t\dot{ \theta}(\tau)\int_0^{\tau}e^{-c\lambda\int_{t'}^\tau\dot{\theta}(t'')dt''2^j}2^j\|\Delta_{k,j}(v^n\partial_nv^h)_\Phi\|_{L^p_h(L^2_v)}
dt' d\tau+\cdots\\
&\lesssim \sum\limits_{k,j}2^{k(\frac{n-1}{p}+1-s)}2^{\frac{1}{2}j}\int_0^t\int_{t'}^{t}e^{-c\lambda\int_{t'}^\tau\dot{\theta}(t'')dt''2^j}2^j\dot{ \theta}(\tau)d\tau\|\Delta_{k,j}(v^n\partial_nv^h)_\Phi\|_{L^p_h(L^2_v)}
dt' +\cdots\\
&\lesssim \frac{1}{\lambda}\int_0^t\|v^n_\Phi\|_{\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}}\|\partial_nv^h_\Phi\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}
dt' +\cdots.
\endaligned$$ In this way, the losing derivative term can be absorbed by the left hand side of the above inequality by choosing $\lambda$ sufficient large.
[**[Step 5.]{}**]{} The estimate of the pressure term $\nabla_hq$.\
To estimate the pressure term, we write $$\nabla_hq = -2\nabla_h(-\Delta_h)^{-1}(v^n\partial_n\mathrm{div}_hv^h)+2\nabla_h(-\Delta_h)^{-1}(\mathrm{div}_hv^h\mathrm{div}_hv^h)+\cdots$$ The $ L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})$ norm of the first term in $\nabla_hq$ can be estimated by $$\|\nabla_h(-\Delta_h)^{-1}(v^n\partial_n\mathrm{div}_hv^h)\|_{L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}\lesssim
\|(v^n\partial_n\mathrm{div}_hv^h)\|_{L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}-s,\frac{1}{2}})}.$$ Here we should point out that when $s=2,n=3$, system (\[e1.3\]) is nothing but the 3D incompressible Navier-Stokes equations. In this case we have to deal with $\|fg\|_{\dot{B}_{p,1}^{\frac{2}{p}-2,\frac{1}{2}}}$ type estimate. Unfortunately, the product law in $\dot{B}_{p,1}^{\frac{2}{p}-2,\frac{1}{2}}$ is hard to obtain since we can not control the low horizontal frequency part.
[**[Step 6.]{}**]{} Estimate of $\theta(t).$
In this step, we want to prove that for any time $t$, $\theta(t)$ is a small quantity. This ensures that the phase function $\Phi$ satisfies the subadditivity property. We will go to derive a stronger estimate for
$$\aligned
Y(t)=&\|{v}^n_{\Phi}\|_{\widetilde{L}_t^\infty(\dot{B}_{p,1}^{\frac{n-1}{p}-s,\frac{1}{2}})}
+\|{v}^n_{\Phi}\|_{{L}_t^1(\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}})}.
\endaligned$$ However, when ${\epsilon}>0$, we can not get the closed estimate for $Y(t)$. Our strategy is to add an extra term $\epsilon v^h$ under the same norm which is hidden in the pressure term ${\epsilon}^2\partial_nq.$ See section 4 for details.
[**[Step 7.]{}**]{} Estimate of $v^n_\Phi.$
If this is done, we could get a closed *a priori* estimate (see Lemma \[l3.2\]) and finish the proof of Theorem \[t1.2\]. Observing that the nonlinear term $v^n\partial_nv^n$ can be rewritten as $-v^n\mathrm{div}_h{v}^h$ due to divergence free condition. Hence, in the limiting system (\[elimit\]), there is no loss of derivative in vertical direction on $v^n$. Thus the estimate on $v^n$ is much easier than $v^h.$
There are also some other type of large initial data so that the Navier-Stokes equations are globally well-posed. For instance, when the domain is thin in the vertical direction, G. Raugel and G. Sell [@RS] were able to establish global solutions for a family of large initial data by using anisotropic Sobolev imbedding theorems (see also the paper [@IRS] by D. Iftimie, G. Raugel and G. Sell). By choosing the initial data to transform the equation into a rotating fluid equations, A. Mahalov and B. Nicolaenko [@MN] obtained global solutions generated by a family of large initial data. A family of axi-symmetric large solutions were established in [@TLL] by Thomas Hou, Z. Lei and C. M. Li. Recently, Z. Lei, F. Lin and Y. Zhou in [@LLZ] proved the global well-posedness of 3D Navier-Stokes equations for a family of large initial data by making use of the structure of Helicity. The data in [@LLZ] are not small in $\dot{B}^{-1}_{\infty,\infty}$ even in the anisotropic sense. We also mention that for the general 3D incompressible Navier-Stokes equations which possess hyper-dissipation in horizontal direction, D. Fang and B. Han in [@DH] obtain the global existence result when the initial data belongs to the anisotropic Besov spaces.
The remaining part of the paper is organized as follows. In Section 2, we present the basic theories of anisotropic Littlewood-Paley decomposition and anisotropic Besov spaces. Section 3 is devoted to obtaining the $a\ priori$ estimates of solution. The $\theta(t)$ will be studied in Section 4. Finally, the proof of the main result will be given in Section 5.
Anisotropic Littlewood-Paley theories and preliminary lemmas
============================================================
In this section, we first recall the definition of the anisotropic Littlewood-Paley decomposition and some properties about anisotropic Besov spaces. It was introduced by D. Iftimie in [@If] for the study of incompressible Navier-Stokes equations in thin domains. Let us briefly explain how this may be built in ${\mathbb R}^n$. Let $(\chi,\varphi)$ be a couple of $C^\infty$ functions satisfying $$\hbox{Supp}\chi\subset\{r\leq\frac{4}{3}\},
\ \ \ \
\hbox{Supp}\varphi\subset\{\frac{3}{4}\leq r\leq\frac{8}{3}\},$$ and $$\chi(r)+\sum_{k\in \mathbb{N}}\varphi(2^{-k}r)=1\ \ \mathrm{for}\ \ r\in{\mathbb R},$$ $$\sum_{j\in \mathbb{Z}}\varphi(2^{-j}r)=1\ \ \mathrm{for}\ \ r\in{\mathbb R}\backslash \{0\}.$$ For $u\in\mathcal {S}'({\mathbb R}^n)/\mathcal{P}({\mathbb R}^n)$, we define the homogeneous dyadic decomposition on the horizontal variables by $${{\Delta}}^h_ku=\mathcal {F}^{-1}(\varphi(2^{-k}|\xi_h|)\widehat{u})\ \ \hbox{for}\ \ k\in\mathbb{Z}.$$ Similarly, on the vertical variable, we define the homogeneous dyadic decomposition by $${{{\Delta}}}^v_{j}u=\mathcal {F}^{-1}(\varphi(2^{-j}|\xi_n|)\widehat{u})\ \ \hbox{for}\ \ j\in\mathbb{Z}.$$ The anisotropic Littlewood-Paley decomposition satisfies the property of almost orthogonality: $${{{\Delta}}}_k^h{{{\Delta}}}_l^h u\equiv0\quad \mathrm{if}\quad |k-l|\geq2\quad\quad\mathrm{and}\quad\quad{{{\Delta}}}_k^h(S_{l-1}^hu{{{\Delta}}}_l^hu)\equiv0\quad\mathrm{if}\quad|k-l|\geq5,$$ where $S_{l}^h$ is defined by $$S_{l}^hu=\sum\limits_{l'\leq l-1}{{\Delta}}_{l'}^hu.$$ Similar properties hold for ${{{\Delta}}}_{j}^v$. In this paper, we shall use the following anisotropic version of Besov spaces [@If]. In what follows, we denote for abbreviation $$\Delta_{k,j}f\overset{\text{def}}={{\Delta}}_k^h{{\Delta}}_j^vf.$$
\[D2.1\] Let $(p,r)\in[1,\infty]^2$, $\sigma,s\in{\mathbb R}$ and $u\in\mathcal {S}'({\mathbb R}^n)/\mathcal{P}({\mathbb R}^n),$ we set
$$\|u\|_{\dot{B}^{\sigma,s}_{p,r}}\overset{\mathrm{def}}=\|2^{k\sigma}2^{js}\|{\Delta}_{k,j}u\|_{L^p_h(L_v^2)}\|_{l^r(\mathbb{Z}^2)}.$$ (1) For $\sigma<\frac{n-1}{p},s<\frac{1}{2}$( $\sigma=\frac{n-1}{p}$ or $s=\frac{1}{2}$ if $r=1$), we define $$\dot{B}^{\sigma,s}_{p,r}({\mathbb R}^n)\overset{\mathrm{def}}=\{u\in\mathcal {S}'({\mathbb R}^n)\mid \|u\|_{\dot{B}^{\sigma,s}_{p,r}}<\infty \}.$$ (2) If $k,l\in\mathbb{N}$ and $\frac{n-1}{p}+k < \sigma<\frac{n-1}{p}+k+1$, $\frac{1}{2}+l < s<\frac{1}{2}+l+1$ ( $\sigma=\frac{n-1}{p}+k+1$ or $s=\frac{1}{2}+l+1$ if $r=1$), then $\dot{B}^{\sigma,s}_{p,r}({\mathbb R}^n)$ is defined as the subset of $u\in\mathcal {S}'({\mathbb R}^n)$ such that $\partial_h^\beta \partial_3^\alpha u\in \dot{B}^{\sigma-k,s-l}_{p,r}({\mathbb R}^n)$ whenever $|\beta|=k, \alpha=l.$
The study of non-stationary equation requires spaces of the type $L^\rho_T(X)$ for appropriate Banach spaces $X$. In our case, we expect $X$ to be an anisotropic Besov space. So it is natural to localize the equations through anisotropic Littlewood-Paley decomposition. We then get estimates for each dyadic block and perform integration in time. As in [@Ch2], we define the so called Chemin-Lerner type spaces:
\[D2.3\] Let $(p,r)\in[1,\infty]^2$, $\sigma,s\in{\mathbb R}$ and $T\in(0,\infty],$ we set $$\|u\|_{\widetilde{L}^\rho_T(\dot{B}^{\sigma,s}_{p,r})}\overset{\mathrm{def}}=
\|2^{k\sigma}2^{js}\|{\Delta}_{k,j}u\|_{L_T^\rho(L^p_h(L_v^2))}\|_{l^r(\mathbb{Z}^2)}$$ and define the space $\widetilde{L}^\rho_T(\dot{B}^{\sigma,s}_{p,r})({\mathbb R}^n)$to be the subset of distributions in $u\in\mathcal {S}'(0,T)\times\mathcal{S}'({\mathbb R}^n)$ with finite $\widetilde{L}^\rho_T(\dot{B}^{\sigma,s}_{p,r})$ norm.
In order to investigate the continuity properties of the products of two temperate distributions $f$ and $g$ in anisotropic Besov spaces, we then recall the isotropic product decomposition which is a simple splitting device going back to the pioneering work by J.-M. Bony [@Bo]. Let $f,g\in\mathcal {S}'({\mathbb R}^n)$, $$fg=T(f,g)+\widetilde{T}(f,g)+R(f,g),$$ where the paraproducts $T(f,g)$ and $\widetilde{T}(f,g)$ are defined by $$T(f,g)=\sum\limits_{k\in\mathbb{Z}}S_{k-1}f{{\Delta}}_kg,\quad \widetilde{T}(f,g)=\sum\limits_{k\in\mathbb{Z}}{{\Delta}}_kf S_{k-1}g$$ and the remainder $$R(f,g)=\sum\limits_{k\in\mathbb{Z}}{{\Delta}}_{k}f\widetilde{{{\Delta}}}_kg\quad\mathrm{with}\quad\widetilde{{{\Delta}}}_kg
=\sum\limits_{k'=k-1}^{k+1}{{\Delta}}_{k'}g.$$ Similarly, we can define the decompositions for both horizontal variable $x_h$ and vertical variable $x_n$. Indeed, we have the following split in $x_h.$ $$fg=T^h(f,g)+\widetilde{T}^h(f,g)+R^h(f,g),$$ with $$T^h(f,g)=\sum\limits_{k\in\mathbb{Z}}S^h_{k-1}f{{\Delta}}^h_kg,\quad \widetilde{T}^h(f,g)=\sum\limits_{k\in\mathbb{Z}}{{\Delta}}^h_kf S^h_{k-1}g$$ and $$R^h(f,g)=\sum\limits_{k\in\mathbb{Z}}{{\Delta}}^h_{k}f\widetilde{{{\Delta}}}^h_kg\quad\mathrm{where}
\quad\widetilde{{{\Delta}}}^h_kg=\sum\limits_{k'=k-1}^{k+1}{{\Delta}}^h_{k'}g.$$ The decomposition in vertical variable $x_n$ can be defined by the same line. Thus, we can write $fg$ as $$\begin{aligned}
\label{e2.3}
\begin{split}
fg&=(T^h+\widetilde{T}^h+R^h)(T^v+\widetilde{T}^v+R^v)(f,g)\\
&=T^hT^v(f,g)+T^h\widetilde{T}^v(f,g)+T^hR^v(f,g)\\
&\quad+\widetilde{T}^hT^v(f,g)+\widetilde{T}^h\widetilde{T}^v(f,g)+\widetilde{T}^hR^v(f,g)\\
&\quad+R^hT^v(f,g)+R^h\widetilde{T}^v(f,g)+R^hR^v(f,g).
\end{split}\end{aligned}$$ Each term of (\[e2.3\]) has an explicit definition. Here $$T^hT^v(f,g)=\sum\limits_{(k,j)\in \mathbb{Z}^2}S_{k-1}^hS_{j-1}^vf{\Delta}_{k,j}g,\quad \widetilde{T}^hT^v(f,g)=\sum\limits_{(k,j)\in \mathbb{Z}^2}{{\Delta}}_{k}^hS_{j-1}^vfS_{k-1}^h{{\Delta}}_j^vg.$$ Similarly, $$R^hT^v(f,g)=\sum\limits_{(k,j)\in \mathbb{Z}^2}{{\Delta}}_{k}^hS_{j-1}^vf\widetilde{{{\Delta}}}_k^h{{\Delta}}_j^vg,\quad R^hR^v(f,g)=\sum\limits_{(k,j)\in \mathbb{Z}^2}\Delta_{k,j} f\widetilde{\Delta}_{k,j}g,$$ and so on.
At this moment, we can state an important product law in anisotropic Besov spaces. The case $p=2, n=3$ was proved in [@GHZ]. For completeness, here we prove a similar result in the $L^p$ framework.
\[p2.5\] Let $1\leq p<n-1$ and $(\sigma_1,\sigma_2)$ be in ${\mathbb R}^2$. If $\sigma_1,\sigma_2\leq \frac{n-1}{p}$ and $$\aligned
&\sigma_1+\sigma_2>(n-1)\max(0,\frac{2}{p}-1),
\endaligned$$ then we have for any $f\in \dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}(\mathbb{R}^n)$ and $g\in \dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}(\mathbb{R}^n)$, $$\begin{aligned}
\label{e2.4}
\|fg\|_{\dot{B}^{\sigma_1+\sigma_2-\frac{n-1}{p},\frac{1}{2}}_{p,1}}\lesssim \|f\|_{\dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}}\|g\|_{\dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}}.\end{aligned}$$
According to (\[e2.3\]), we first give the bound of $T^hT^v(f,g)$. Indeed, applying Hölder and Bernstein inequality, we get that $$\aligned
\|{\Delta}_{k,j}(T^hT^v(f,g))&\|_{L_h^p(L^2_v)}\lesssim \sum\limits_{\substack{|k-k'|\leq4\\|j-j'|\leq4}}
\|S^h_{k'-1}S^v_{j'-1}f\|_{L^\infty}\|\Delta_{k',j'}g\|_{L_h^p(L^2_v)}\\
&\lesssim \sum\limits_{\substack{|k-k'|\leq4\\|j-j'|\leq4}} \sum\limits_{\substack{k''\leq k'-2\\j''\leq j'-2}}2^{k''\sigma_1}2^{\frac{1}{2}j''}\|\Delta_{k'',j''}f\|_{L_h^p(L^2_v)}
\cdot2^{k'\sigma_2}2^{\frac{1}{2}j'}\|\Delta_{k',j'}g\|_{L_h^p(L^2_v)}\\
&\quad\times 2^{(k''-k')(\frac{n-1}{p}-\sigma_1)}2^{(k-k')(\sigma_1+\sigma_2-\frac{n-1}{p})}2^{\frac{1}{2}(j-j')}2^{-k(\sigma_1+\sigma_2-\frac{n-1}{p})}2^{-\frac{1}{2}j}.
\endaligned$$ Since $\sigma_1 \leq \frac{n-1}{p}$, we obtain that $$\aligned
\|{\Delta}_{k,j}(T^hT^v(f,g))\|_{L_h^p(L^2_v)}&\lesssim c_{k,j}2^{-k(\sigma_1+\sigma_2-\frac{n-1}{p})}2^{-\frac{1}{2}j}\|f\|_{\dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}}\|g\|_{\dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}},
\endaligned$$ where the sequence $\{c_{k,j}\}_{(k,j)\in\mathbb{Z}^2}$ satisfies $\|c_{k,j}\|_{l^1(\mathbb{Z}^2)}=1$. This gives the estimate of $T^hT^v(f,g)$.
Similarly, for $\widetilde{T}^hT^v(f,g)$, we have $$\aligned
\|{\Delta}_{k,j}(\widetilde{T}^hT^v(f,g))&\|_{L_h^p(L^2_v)}\lesssim \sum\limits_{\substack{|k-k'|\leq4\\|j-j'|\leq4}}
\|{{\Delta}}^h_{k'}S^v_{j'-1}f\|_{L^p_h(L_v^\infty)}\|S^h_{k'-1}{{\Delta}}^v_{j'}g\|_{L^\infty_h(L_v^2)}\\
&\lesssim \sum\limits_{\substack{|k-k'|\leq4\\|j-j'|\leq4}} \sum\limits_{\substack{k''\leq k'-2\\j''\leq j'-2}}2^{k'\sigma_1}2^{\frac{1}{2}j''}\|\Delta_{k',j''}f\|_{L^p_h(L^2_v)}
\cdot2^{k''\sigma_2}2^{\frac{1}{2}j'}\|\Delta_{k'',j'}g\|_{L^p_h(L^2_v)}\\
&\quad\times 2^{(k''-k')(\frac{n-1}{p}-\sigma_2)}2^{(k-k')(\sigma_1+\sigma_2-\frac{n-1}{p})}2^{\frac{1}{2}(j-j')}2^{-k(\sigma_1+\sigma_2-\frac{n-1}{p})}2^{-\frac{1}{2}j}.
\endaligned$$ Again, $\sigma_2\leq \frac{n-1}{p}$ implies that $$\aligned
\|{\Delta}_{k,j}(\widetilde{T}^hT^v(f,g))\|_{L_h^p(L^2_v)}&\lesssim c_{k,j}2^{-k(\sigma_1+\sigma_2-\frac{n-1}{p})}2^{-\frac{1}{2}j}\|f\|_{\dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}}\|g\|_{\dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}}.
\endaligned$$
The estimate on the remainder operator which concerns the horizontal variable $R^hT^v(f,g)$ may be more complicated. When $2\leq p$, the strategy is following: $$\aligned
\|{\Delta}_{k,j}(R^hT^v(f,g))&\|_{L_h^p(L^2_v)}\lesssim \sum\limits_{\substack{k'\geq k-2\\|j-j'|\leq4}}
\|{{\Delta}}^h_{k'}S^v_{j'-1}f\widetilde{{{\Delta}}}^h_{k'}{{\Delta}}^v_{j'}g\|_{L^{\frac{p}{2}}_h(L_v^2)}2^{\frac{n-1}{p}k}\\
&\lesssim \sum\limits_{\substack{k'\geq k-2\\|j-j'|\leq4}} \sum\limits_{j''\leq j'-2}2^{k'\sigma_1}2^{\frac{1}{2}j''}\|\Delta_{k',j''}f\|_{L^{p}_h(L_v^2)}
\cdot2^{k'\sigma_2}2^{\frac{1}{2}j'}\|\widetilde{{{\Delta}}}^h_{k'}{{\Delta}}^v_{j'}g\|_{L^p_h(L_v^2)}\\
&\quad\times2^{(k-k')(\sigma_1+\sigma_2)}2^{\frac{1}{2}(j-j')}2^{-k(\sigma_1+\sigma_2-\frac{n-1}{p})}2^{-\frac{1}{2}j}.
\endaligned$$ As $\sigma_1+\sigma_2>0$ if $2\leq p$, we have $$\aligned
\|{\Delta}_{k,j}(R^hT^v(f,g))\|_{L_h^p(L^2_v)}&\lesssim c_{k,j}2^{-k(\sigma_1+\sigma_2-\frac{n-1}{p})}2^{-\frac{1}{2}j}\|f\|_{\dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}}\|g\|_{\dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}}.
\endaligned$$ In the case $1\leq p <2$, we have $$\aligned
\|{\Delta}_{k,j}(R^hT^v(f,g))&\|_{L_h^p(L^2_v)}\lesssim \sum\limits_{\substack{k'\geq k-2\\|j-j'|\leq4}}
\|{{\Delta}}^h_{k'}S^v_{j'-1}f\widetilde{{{\Delta}}}^h_{k'}{{\Delta}}^v_{j'}g\|_{L^{1}_h(L_v^2)}2^{(n-1)(1-\frac{1}{p})k}\\
&\lesssim \sum\limits_{\substack{k'\geq k-2\\|j-j'|\leq4}} \sum\limits_{j''\leq j'-2}2^{\frac{1}{2}j''}\|\Delta_{k',j''}f\|_{L^{2}}
\|\widetilde{{{\Delta}}}^h_{k'}{{\Delta}}^v_{j'}g\|_{L^2}2^{(n-1)(1-\frac{1}{p})k}\\
&\lesssim \sum\limits_{\substack{k'\geq k-2\\|j-j'|\leq4}} \sum\limits_{j''\leq j'-2}2^{k'\sigma_1}2^{\frac{1}{2}j''}\|\Delta_{k',j''}f\|_{L_h^p(L^2_v)}
\cdot2^{k'\sigma_2}2^{\frac{1}{2}j'}\|\widetilde{{{\Delta}}}^h_{k'}{{\Delta}}^v_{j'}g\|_{L_h^p(L^2_v)}\\
&\quad\times2^{(k-k')(\sigma_1+\sigma_2-(n-1)(\frac{2}{p}-1))}2^{\frac{1}{2}(j-j')}2^{-k(\sigma_1+\sigma_2-\frac{n-1}{p})}2^{-\frac{1}{2}j}.
\endaligned$$ As $\sigma_1+\sigma_2>(n-1)(\frac{2}{p}-1)$ if $1\leq p<2$, we have $$\aligned
\|{\Delta}_{k,j}(R^hT^v(f,g))\|_{L_h^p(L^2_v)}&\lesssim c_{k,j}2^{-k(\sigma_1+\sigma_2-\frac{n-1}{p})}2^{-\frac{1}{2}j}\|f\|_{\dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}}\|g\|_{\dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}}.
\endaligned$$ The other terms can be followed exactly in the same way, here we omit the details. These complete the proof of this lemma.
Throughout this paper, $\Phi$ denotes a locally bounded function on ${\mathbb R}^+\times{\mathbb R}$ which satisfies the following subadditivity (see for the explicit expression of $\Phi$) $$\Phi(t,\xi_n)\leq \Phi(t,\xi_n-\eta_n)+\Phi(t,\eta_n).$$ For any function $f$ in $\mathcal {S}'(0,T)\times\mathcal{S}'({\mathbb R}^n)$, we define $$\begin{aligned}
\label{ephi}
f_\Phi(t,x_h,x_n)=\mathcal {F}^{-1}\left(e^{\Phi(t,\xi_n)}\hat{f}(t,x_h,\xi_n)\right)\end{aligned}$$ Let us keep the following fact in mind that the map $f\mapsto f^+$ preserves the norm of $L^p_h(L_v^2)$, where $f^+(t,x_h,x_n)$ represents the inverse Fourier transform of $|\hat{f}(t,x_h,\xi_n)|$ on vertical variable, defined as $$f^+(t,x_h,x_n)\overset{\mathrm{def}}=\mathcal {F}^{-1}|\hat{f}(t,x_h,\xi_n)|.$$ On the basis of these facts, we have the following weighted inequality as in Lemma \[p2.5\].
\[p2.8\] Let $1\leq p<n-1$ and $(\sigma_1,\sigma_2)$ be in ${\mathbb R}^2$. If $\sigma_1,\sigma_2\leq \frac{n-1}{p}$ and $$\aligned
&\sigma_1+\sigma_2>(n-1)\max(0,\frac{2}{p}-1),
\endaligned$$ then we have for any $f_\Phi \in \dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}(\mathbb{R}^n)$ and $g_\Phi \in \dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}(\mathbb{R}^n)$, $$\begin{aligned}
\nonumber
\|(fg)_\Phi\|_{\dot{B}^{\sigma_1+\sigma_2-\frac{n-1}{p},\frac{1}{2}}_{p,1}}\lesssim \|f_\Phi\|_{\dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}}\|g_\Phi\|_{\dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}}.\end{aligned}$$
We only prove the $\dot{B}^{\sigma_1+\sigma_2-\frac{n-1}{p},\frac{1}{2}}_{p,1}$ norm of $T^hT^v(f,g)_\Phi$. For fixed $k,j$, we have $$\begin{aligned}
\begin{split}\nonumber
&\|{\Delta}_{k,j}(T^hT^v(f,g)_\Phi)\|_{L_h^p(L^2_v)}\\
&\lesssim \sum\limits_{\substack{|k-k'|\leq 4\\|j-j'|\leq 4}}
\|e^{\Psi(t,\xi_n)}\mathcal {F}(S^h_{k'-1} S^v_{j'-1}f)(x_h,\cdot)\star\mathcal {F}(\Delta_{k',j'}g)(x_h,\cdot)\|_{L_h^p(L^2_v)}\\
&\lesssim \sum\limits_{\substack{|k-k'|\leq 4\\|j-j'|\leq 4}}
\||\mathcal {F}(S^h_{k'-1} S^v_{j'-1}f_\Phi)(x_h,\cdot)|\star|\mathcal {F}(\Delta_{k',j'}g_\Phi)(x_h,\cdot)|\|_{L_h^p(L^2_v)}\\
&\lesssim \sum\limits_{\substack{|k-k'|\leq 4\\|j-j'|\leq 4}}\sum\limits_{\substack{k''\leq k'-2\\j''\leq j '-2}}
\||\mathcal {F}(\Delta_{k'',j''} f_\Phi)(x_h,\cdot)|\star|\mathcal {F}(\Delta_{k',j'}g_\Phi)(x_h,\cdot)|\|_{L_h^p(L^2_v)}\\\
&\lesssim \sum\limits_{\substack{|k-k'|\leq 4\\|j-j'|\leq 4}}\sum\limits_{\substack{k''\leq k'-2\\j''\leq j '-2}}
\|(\Delta_{k'',j''}f_\Phi)^+\|_{L_h^p(L^2_v)}\|(\Delta_{k',j'}g_\Phi)^+\|_{L_h^p(L^2_v)}2^{\frac{n-1}{p}k''}2^{\frac{1}{2}j''}.
\end{split}\end{aligned}$$ Using the fact that $f\mapsto f^+$ preserves the norm of $L_h^p(L^2_v)$, we then get by the similar method as in Lemma \[p2.5\] that $$\begin{aligned}
\begin{split}\nonumber
\|(T^hT^v(f,g)_\Phi)\|_{\dot{B}^{\sigma_1+\sigma_2-\frac{n-1}{p},\frac{1}{2}}_{p,1}}
\lesssim \|f_\Phi\|_{\dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}}
\|g_\Phi\|_{\dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}}.
\end{split}\end{aligned}$$ The other terms in (\[e2.3\]) can be estimated by the same method and finally, we have $$\begin{aligned}
\begin{split}\nonumber
\|(fg)_\Phi\|_{\dot{B}^{\sigma_1+\sigma_2-\frac{n-1}{p},\frac{1}{2}}_{p,1}}
\lesssim \|f_\Phi\|_{\dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}}
\|g_\Phi\|_{\dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}}.
\end{split}\end{aligned}$$
The following lemma is a direct consequence of Lemma \[p2.8\].
\[p2.9\] Let $1\leq p<n-1$, $\rho\in[1,\infty]$, $(\rho_1,\rho_2)\in[1,\infty]^2$ and $(\sigma_1,\sigma_2)$ be in ${\mathbb R}^2$. Assume that $$\frac{1}{\rho}\overset{\mathrm{def}}=\frac{1}{\rho_1}+\frac{1}{\rho_2}.$$ If $\sigma_1,\sigma_2\leq \frac{n-1}{p}$ and $$\aligned
&\sigma_1+\sigma_2>(n-1)\max(0,\frac{2}{p}-1),
\endaligned$$ then we have for any $f_\Phi \in \widetilde{L}_T^{\rho_1}(\dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}(\mathbb{R}^n))$ and $g_\Phi \in \widetilde{L}_T^{\rho_2}(\dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}(\mathbb{R}^n))$, $$\begin{aligned}
\nonumber
\|(fg)_\Phi\|_{\widetilde{L}_T^{\rho}(\dot{B}^{\sigma_1+\sigma_2-\frac{n-1}{p},\frac{1}{2}}_{p,1})}\lesssim \|f_\Phi\|_{\widetilde{L}_T^{\rho_1}(\dot{B}^{\sigma_1,\frac{1}{2}}_{p,1})}\|g_\Phi\|_{\widetilde{L}_T^{\rho_2}(\dot{B}^{\sigma_2,\frac{1}{2}}_{p,1})}.\end{aligned}$$
Estimates for the re-scaled system
===================================
This section is devoted to obtaining the $a\ priori$ estimate for the following system $$\label{e3.1}
\left\{
\begin{array}{rlll}
\partial_tv^h+ v\cdot\nabla {v}^h+D_\epsilon^s v^h+\nabla_h{q}&=&0,\\
\partial_tv^n+{v}\cdot\nabla v^n+D_\epsilon^sv^n+{\epsilon}^2 \partial_nq&=&0,\\
\hbox{div}\,{v}&=&0,\\
{v}(0)&=&{v}_0(x).
\end{array}
\right.$$ The pressure ${q}$ can be computed by the formula $$-\Delta_{\epsilon}{q}=\sum\limits_{i,j}\partial_i\partial_j({v}^i{v}^j).$$ Due to the divergence free condition, the pressure can be split into the following three parts $$\label{e3.2}
\left\{
\begin{array}{llll}
q^1=(-\Delta_{\epsilon})^{-1}\sum\limits_{i,j=1}^{n-1}\partial_i\partial_j(v^iv^j),\\
q^2=2(-\Delta_{\epsilon})^{-1}\sum\limits_{i=1}^{n-1}\partial_i\partial_n(v^iv^n),\\
q^3=-2(-\Delta_{\epsilon})^{-1}\partial_n(v^n\mathrm{div_h}{v}^h).
\end{array}
\right.$$ It is worthwhile to note that there will lose one vertical derivative owing to the term $v^n\partial_n {v}^h$ and pressure terms ${q}^2,{q}^3$ which appear in the equation on ${v}^h$. Thus, we assume that the initial data is analytic in the vertical variable. This method was introduced in [@Ch] to compensate the losing derivative in $x_n$. Therefore, we introduce two key quantities which we want to control in order to obtain the global bound of ${v}$ in a certain space. We define the function $\theta(t)$ by $$\label{e3.3}
\begin{split}
\theta(t)=\int_0^t \|{v}^n_{\Phi}(\tau)\|_{\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1}}d\tau,
\end{split}$$ and denote $$\label{e3.4}
\begin{split}
\Psi(t)=&\|{v}_\Phi\|_{\widetilde{L}_t^\infty(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}
+\|{v}_\Phi\|_{L_t^1(\dot{B}_{p,1}^{\frac{n-1}{p}+1,\frac{1}{2}})}\\
&+\int_0^t \|v^n_\Phi\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1,\frac{1}{2}}}\|\partial_nv^h_\Phi\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}} d\tau,\\
\Psi(0)=&\|e^{\alpha D_n }v_0\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}.
\end{split}$$ The phase function $\Phi(t,D_n)$ is defined by $$\label{e3.5}
\begin{split}
\Phi(t,\xi_n)=(\alpha-\lambda\theta(t))|\xi_n|,
\end{split}$$ for some $\lambda $ that will be chosen later on, $\alpha$ is a positive number. Obviously, we need to ensure that $\theta(t)< \frac{\alpha}{\lambda}$ which implies the subadditivity of $\Phi$.
The following lemma provides the $a\ priori$ estimate of ${v}_\Phi$ in the anisotropic Besov spaces, which is the key bilinear estimate.
\[l3.2\] There exist two constants $\lambda_0$ and $C_1$ such that for any $\lambda>\lambda_0$ and $t$ satisfying $\theta(t)\leq \frac{\alpha}{2\lambda}$, we have $$\Psi(t)\leq C_1\Psi(0) + C_1\Psi(t)^2.$$
Estimates on the horizontal component ${v}^h$
---------------------------------------------
According to the definition of ${v}_\Phi^h$, we find that in each dyadic block, it verifies the following equation $$\begin{aligned}
\label{e3.6}
\begin{split}
{\Delta}_{k,j}v_\Phi^h(t,x)&=e^{-tD_{\epsilon}^s+\Phi(t,D_n)}{\Delta}_{k,j}v^h_0\\
&-\int_0^te^{-(t-\tau)D_{\epsilon}^s}e^{-\lambda D_n
\int_\tau^t\dot\theta(t')dt'}{\Delta}_{k,j}(v\cdot\nabla v^h)_\Phi(\tau)d\tau\\
&-\int_0^te^{-(t-\tau)D_{\epsilon}^s}e^{-\lambda D_n \int_\tau^t\dot\theta(t')dt'}\nabla_h{\Delta}_{k,j}q_\Phi(\tau)d\tau.
\end{split}\end{aligned}$$ Taking the $L_h^p(L_v^2)$ norm, we deduce that $$\begin{aligned}
\label{e3.8}
\begin{split}
\|{\Delta}_{k,j}v_\Phi^h\|_{L_h^p(L_v^2)}&\lesssim e^{-c(2^{ks}+{\epsilon}^s2^{js})t}\|{\Delta}_{k,j}e^{\alpha D_n }v_0^h\|_{L_h^p(L_v^2)}\\
&+\int_0^te^{-c(2^{ks}+{\epsilon}^s2^{js})(t-\tau)}e^{-c\lambda2^{j}\int_\tau^t\dot\theta(t')dt'}\|{\Delta}_{k,j}(v\cdot\nabla v^h)_\Phi\|_{L_h^p(L_v^2)}d\tau\\
&+\int_0^te^{-c(2^{ks}+{\epsilon}^s2^{js})(t-\tau)}e^{-c\lambda2^{j}\int_\tau^t\dot\theta(t')dt'}\|\nabla_h{\Delta}_{k,j}q_\Phi\|_{L_h^p(L_v^2)}d\tau\\
&\overset{\mathrm{def}}{=}I_1+I_2+I_3.
\end{split}\end{aligned}$$
We first estimate the linear term $I_1$. In fact, we have $$\begin{aligned}
\label{e3.9}
\begin{split}
\| I_1\|_{L^\infty_t}+2^{ks}\| I_1\|_{L^1_t}&\lesssim \|{\Delta}_{k,j}e^{\alpha D_n }v_0^h\|_{L_h^p(L_v^2)}\\
&\lesssim c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|e^{\alpha D_n }v_0^h\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}},
\end{split}\end{aligned}$$ where $\{c_{k,j}\}_{(k,j)\in\mathbb{Z}^2}$ is a two dimensional sequence satisfying $\|c_{k,j}\|_{l^1(\mathbb{Z}^2)}=1.$
The term $I_{2}$ can be rewritten as $$\aligned
I_{2}&\lesssim \int_0^te^{-c2^{ks}(t-\tau)}\|{\Delta}_{k,j}(v\cdot\nabla v^h)_\Phi\|_{L^p_h(L^2_v)}d\tau.
\endaligned$$ By Young’s inequality, we have $$\aligned
\| I_{2}\|_{L^\infty_t}+ 2^{ks}\|I_{2}\|_{L^1_t} &\lesssim \|\Delta_{k,j}(v\cdot\nabla v^h)_\Phi\|_{L^1_t(L_h^p(L_v^2))}\\
&\lesssim c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|(v\cdot\nabla v^h)_\Phi\|_{L_t^1(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}.
\endaligned$$ Thus, we can get by Lemma \[p2.8\] and \[p2.9\] that $$\begin{aligned}
\nonumber
\begin{split}
\| I_{2}\|_{L^\infty_t}+ 2^{ks}\|I_{2}\|_{L^1_t}&\lesssim c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|v^h_\Phi\|_{\widetilde{L}_t^\infty(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}
\|v^h_\Phi\|_{L_t^1(\dot{B}_{p,1}^{\frac{n-1}{p}+1,\frac{1}{2}})}\\
&+ c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\int_0^t\|v^n_\Phi\|_{ \dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}} \|\partial_nv^h_\Phi\|_{ \dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}d\tau.
\end{split}\end{aligned}$$
Now we are left with the study of the pressure term $I_3$. The pressure can be split into ${q}={q}^1+{q}^2+{q}^3$ with ${q}^1,{q}^2,{q}^3$ defined in (\[e3.2\]). For convenience, we denote that $$I_{31}=\int_0^te^{-c(2^{ks}+{\epsilon}^s2^{js})(t-\tau)}e^{-c\lambda2^{j}\int_\tau^t\dot\theta(t')dt'}\|\nabla_h{\Delta}_{k,j}q^1_\Phi\|_{L^p_h(L^2_v)}d\tau,$$ $$I_{32}=\int_0^te^{-c(2^{ks}+{\epsilon}^s2^{js})(t-\tau)}e^{-c\lambda2^{j}\int_\tau^t\dot\theta(t')dt'}\|\nabla_h{\Delta}_{k,j}q^2_\Phi\|_{L^p_h(L^2_v)}d\tau,$$ $$I_{33}=\int_0^te^{-c(2^{ks}+{\epsilon}^s2^{js})(t-\tau)}e^{-c\lambda2^{j}\int_\tau^t\dot\theta(t')dt'}\|\nabla_h{\Delta}_{k,j}q^3_\Phi\|_{L^p_h(L^2_v)}d\tau.$$ Hence, using the fact that $(-\Delta_{\epsilon})^{-1}\partial_i\partial_j$ is a bounded operator applied for frequency localized functions in $L^p_h(L^2_v)$ when $i,j=1,2,\cdots,n-1$, we get $$\|\nabla_h {\Delta}_{k,j}q^1_\Phi\|_{L^p_h(L^2_v)}\lesssim \|{\Delta}_{k,j}(v^h\cdot \nabla_hv^h)\|_{L^p_h(L^2_v)}.$$ By the same method as in the estimate of $I_{2}$, we have $$\begin{aligned}
\label{e3.14}
\begin{split}
&\|I_{31}\|_{L^\infty_t}+ 2^{ks}\|I_{31}\|_{L^1_t}\lesssim c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|v^h_\Phi\|_{\widetilde{L}_t^\infty(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}
\|v^h_\Phi\|_{L_t^1(\dot{B}_{p,1}^{\frac{n-1}{p}+1,\frac{1}{2}})}.
\end{split}\end{aligned}$$ Noting that $$\nabla_hq^2=2(-\Delta_{\epsilon})^{-1}\nabla_h\partial_i(v^n\partial_nv^h-v^h\mathrm{div}_hv^h),$$ and as in the estimate of $I_2$, it holds that $$\begin{aligned}
\begin{split}
\|I_{32}\|_{L_t^\infty}+2^{ks}\|I_{32}\|_{L_t^1}&\lesssim c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\int_0^t\|v^n_\Phi\|_{ \dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}} \|\partial_nv^h_\Phi\|_{ \dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}d\tau\\
&+c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}
\|v^h_\Phi\|_{L_t^1(\dot{B}_{p,1}^{\frac{n-1}{p}+1,\frac{1}{2}})}\|v^h_\Phi\|_{\widetilde{L}_t^\infty(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}.
\end{split}\end{aligned}$$
Using $$\nabla_hq^3=2(-\Delta_{\epsilon})^{-1}\nabla_h(\mathrm{div}_hv^h\mathrm{div}_hv^h-v^n\partial_n\mathrm{div}_hv^h),$$ we write $I_{33}$ as follows $$\aligned
I_{33}&\lesssim 2^{-k}\int_0^te^{-c2^{ks}(t-\tau)}\Big(\|\Delta_{k,j}(\mathrm{div}_hv^h \mathrm{div}_hv^h)_\Phi\|_{L^p_h(L^2_v)}+\|\Delta_{k,j}(v^n\partial_n\mathrm{div}_hv^h)_\Phi\|_{L^p_h(L^2_v)}\Big)d\tau.
\endaligned$$ By Young’s inequality and Lemma \[p2.9\], as $1\leq s < \min\{n-1,2\frac{n-1}{p}\}$, we have $$\begin{aligned}
\begin{split}\label{e3.15}
\|I_{33}\|_{L_t^\infty}+ 2^{ks}\|I_{33}\|_{L_t^1}&\lesssim c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|v^h_\Phi\|_{L_t^\infty(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}\|v^h_\Phi\|_{L_t^1(\dot{B}_{p,1}^{\frac{n-1}{p}+1,\frac{1}{2}})}\\
&+c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\int_0^t\|v^n_\Phi\|_{ \dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}} \|\partial_nv^h_\Phi\|_{ \dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}d\tau.
\end{split}\end{aligned}$$
Now we are going to estimate the key quantity $$\int_0^t\|v^n_\Phi\|_{ \dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}} \|\partial_nv^h_\Phi\|_{ \dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}d\tau.$$ According to (\[e3.6\]) , we find that in each dyadic block $\partial_n{v}_\Phi^h$ verifies $$\begin{aligned}
\label{1}
\begin{split}
{\Delta}_{k,j}\partial_nv_\Phi^h(\tau,x)&=e^{-\tau D_\epsilon^s+\Phi(\tau,D_n)}{\Delta}_{k,j}\partial_nv^h_0\\
&-\int_0^\tau e^{-(\tau-t')D_\epsilon^s}e^{-\lambda D_n
\int_{t'}^\tau\dot\theta(t'')dt''}{\Delta}_{k,j}\partial_n(v\cdot\nabla v^h)_\Phi(t')dt'\\
&-\int_0^\tau e^{-(\tau-t')D_\epsilon^s}e^{-\lambda D_n \int_{t'}^\tau\dot\theta(t'')dt''}\nabla_h{\Delta}_{k,j}\partial_nq_\Phi(t')dt'.
\end{split}\end{aligned}$$ Taking the $L^p_h(L^2_v)$ norm on both sides of (\[1\]), we have $$\begin{aligned}
\label{2}
\begin{split}
\|{\Delta}_{k,j}\partial_nv_\Phi^h\|_{L^p_h(L^2_v)}&\lesssim e^{-c(2^{ks}+{\epsilon}^s2^{js})\tau}e^{-c\lambda2^{j}\int_{0}^\tau \dot\theta(t'')d{t''}}\|{\Delta}_{k,j}e^{\alpha D_n }\partial_nv_0^h\|_{L^p_h(L^2_v)}\\
&+\int_0^\tau e^{-c(2^{ks}+{\epsilon}^s2^{js})(\tau-t')}e^{-c\lambda2^{j}\int_{t'}^\tau \dot\theta(t'')d{t''}}\|{\Delta}_{k,j}\partial_n(v\cdot\nabla v^h)_\Phi\|_{L^p_h(L^2_v)}dt'\\
&+\int_0^\tau e^{-c(2^{ks}+{\epsilon}^s2^{js})(\tau-t')}e^{-c\lambda2^{j}\int_{t'}^\tau\dot\theta(t'')d{t''}}\|\nabla_h{\Delta}_{k,j}\partial_nq_\Phi\|_{L^p_h(L^2_v)}dt'.
\end{split}\end{aligned}$$ For fixed $k,j$, multiplying the (\[2\]) by $\dot \theta(\tau)$ and integrating over $(0,t)$, one can have $$\begin{aligned}
\nonumber
\begin{split}
\int_0^t\dot\theta(\tau)\|{\Delta}_{k,j}\partial_nv_\Phi^h\|_{L^p_h(L^2_v)}d\tau
&\lesssim\int_0^te^{-c\lambda2^{j}\int_{0}^\tau \dot\theta(t'')d{t''}}\dot\theta(\tau)\|e^{\alpha D_n }\partial_n{\Delta}_{k,j}v_0^h\|_{L^p_h(L^2_v)}d\tau\\
&+\int_0^t\int_0^\tau e^{-c\lambda2^{j}\int_{t'}^\tau \dot\theta(t'')d{t''}}2^j\dot\theta(\tau)\|{\Delta}_{k,j}(v\cdot\nabla v^h)_\Phi\|_{L^p_h(L^2_v)}dt'd\tau\\
&+\int_0^t\int_0^\tau e^{-c\lambda2^{j}\int_{t'}^\tau\dot\theta(t'')d{t''}}2^j\dot\theta(\tau)\|\nabla_h{\Delta}_{k,j}q_\Phi\|_{L^p_h(L^2_v)}dt'd\tau\\
&\overset{\mathrm{def}}{=}I_4+I_5+I_6.
\end{split}\end{aligned}$$ The term $I_4$ containing initial data can be bounded by $$\begin{aligned}
\label{3}
\begin{split}
I_4&\lesssim \int_0^t e^{-c\lambda2^{j}\int_{0}^\tau \dot\theta(t'')d{t''}}2^j\dot\theta(\tau)d\tau\|{\Delta}_{k,j}e^{\alpha D_n }v_0^h\|_{L^p_h(L^2_v)}\\
&\lesssim \frac{1}{\lambda}c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|e^{\alpha D_n }v_0^h\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}.
\end{split}\end{aligned}$$
By Fubini’s theorem, the term $I_5$ can be rewritten as $$\aligned
I_{5}&\lesssim \int_0^t \int_{t'}^t e^{-c\lambda2^{j}\int_{t'}^\tau \dot\theta(t'')dt''}2^j\dot\theta(\tau)d\tau \|{\Delta}_{k,j}(v\cdot\nabla v^h)_\Phi\|_{L^p_h(L^2_v)}dt' \\
&\lesssim \frac{1}{\lambda} \int_0^t \|{\Delta}_{k,j}(v\cdot\nabla v^h)_\Phi\|_{L^p_h(L^2_v)}dt'\\
&\lesssim \frac{1}{\lambda}c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|(v\cdot\nabla v^h)_\Phi\|_{L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}.
\endaligned$$ Thus, we can get by Lemma \[p2.8\] and \[p2.9\] that $$\begin{aligned}
\nonumber
\begin{split}
I_{5}&\lesssim \frac{1}{\lambda}c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|v^h_\Phi\|_{\widetilde L^\infty_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}\|v^h_\Phi\|_{L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1,\frac{1}{2}})}\\
&+\frac{1}{\lambda} c_{k,j} 2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j} \int_0^t \|v^n_\Phi\|_{\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}}\|\partial_nv^h_\Phi\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}dt'.
\end{split}\end{aligned}$$
As for $I_6$, for convenience, we denote that $$I_{61}=\int_0^t\int_0^\tau e^{-c\lambda2^{j}\int_{t'}^\tau \dot\theta(t'')dt''}2^j\dot\theta(\tau)\|\nabla_h{\Delta}_{k,j}q^1_\Phi\|_{L^p_h(L^2_v)}(t')dt'd\tau,$$ $$I_{62}=\int_0^t \int_0^\tau e^{-c\lambda2^{j}\int_{t'}^\tau \dot\theta(t'')dt''}2^j\dot \theta(\tau)\|\nabla_h{\Delta}_{k,j}q^2_\Phi\|_{L^p_h(L^2_v)}(t')dt' d\tau,$$ $$I_{63}=\int_0^t \int_0^\tau e^{-c\lambda2^{j}\int_{t'}^\tau\dot\theta(t'')dt''}2^j\dot \theta(\tau)\|\nabla_h{\Delta}_{k,j}q^3_\Phi\|_{L^p_h(L^2_v)}(t')dt'd\tau.$$ By the same method as in the estimate of $I_{5}$, we have $$\label{5}
\begin{split}
I_{61}+I_{62}&\lesssim \frac{1}{\lambda}c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|v^h_\Phi\|_{\widetilde{L}_t^\infty(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}
\|v^h_\Phi\|_{{L}_t^1(\dot{B}_{p,1}^{\frac{n-1}{p}+1,\frac{1}{2}})}\\
&+\frac{1}{\lambda} c_{k,j} 2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j} \int_0^t \|v^n_\Phi\|_{\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}}\|\partial_nv^h_\Phi\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}dt'.
\end{split}$$ Finally, $I_{63}$ can be estimated as follows $$\aligned
I_{63}&\lesssim\int_0^t\int_0^\tau e^{-c\lambda2^{j}\int_{t'}^\tau \dot\theta(t'')dt''}2^j \dot \theta(\tau)
\|{\Delta}_{k,j}(-\Delta_{\epsilon})^{-1}\nabla_h\partial_n(v^n \mathrm{div}_hv^h)_\Phi\|_{L^p_h(L^2_v)}(t')dt'd\tau\\
&\lesssim2^{-k}\int_0^t\int_{t'}^\tau e^{-c\lambda2^{j}\int_{t'}^\tau \dot\theta(t'')dt''}2^j \dot \theta(\tau)
d\tau \|{\Delta}_{k,j}\partial_n(v^n\mathrm{div}_hv^h)_\Phi\|_{L^p_h(L^2_v)}(t')dt'.
\endaligned$$ Thus, we can obtain that $$\begin{aligned}
\begin{split}\label{6}
I_{63}&\lesssim \frac{1}{\lambda}c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|\partial_n(v^n \mathrm{div}_hv^h)_\Phi\|_{L^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}-s,\frac{1}{2}})}\\
&\lesssim \frac{1}{\lambda}c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|v^h_\Phi\|_{\widetilde{L}_t^\infty(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}
\|v^h_\Phi\|_{{L}_t^1(\dot{B}_{p,1}^{\frac{n-1}{p}+1,\frac{1}{2}})}\\
& \ +\frac{1}{\lambda} c_{k,j} 2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j} \int_0^t \|v^n_\Phi\|_{\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}}\|\partial_nv^h_\Phi\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}dt'.
\end{split}\end{aligned}$$ Together with the above estimates on $I_1-I_6$, we get that $$\begin{aligned}
\begin{split}\label{e3.17}
&\|v_\Phi^h\|_{\widetilde{L}^\infty_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}
+\|v_\Phi^h\|_{{L}^1_t(\dot{B}_{p,1}^{\frac{n-1}{p}+1,\frac{1}{2}})}+\int_0^t \|v^n_\Phi\|_{\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}}}\|\partial_nv^h_\Phi\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}} d\tau\\
&\lesssim \Psi(0)+\frac{1}{\lambda}\Psi(t)+\Psi(t)^2.
\end{split}\end{aligned}$$
Estimates on the vertical component $v^n$
-----------------------------------------
We begin this part by studying the equation of $v^n$, which is stated as follows $$\partial_tv^n+D_{\epsilon}^s v^n+{v}\cdot\nabla v^n+{\epsilon}^2\partial_nq=0.$$ Observing that in the above equation, one can expect that there is no loss of derivative in vertical direction. More precisely, due to divergence free condition, the nonlinear term $v^n\partial_nv^n$ can be rewritten as $-v^n\mathrm{div}_h{v}^h$. Thus the estimate on $v^n$ is different from $v^h.$
Applying the anisotropic dyadic decomposition operator ${\Delta}_{k,j}$ to the equation of $v^n$, then in each dyadic block, $v^n$ satisfies $$\label{e3.23}
\partial_t{\Delta}_{k,j}v^n+D_{\epsilon}^s{\Delta}_{k,j}v^n
=-{\Delta}_{k,j}(v^h\cdot\nabla_h v^n)+{\Delta}_{k,j}(v^n\mathrm{div}_hv^h)-{\epsilon}^2{\Delta}_{k,j}\partial_nq.$$ Let us define $G\overset{\mathrm{def}}=v^h\cdot\nabla_h v^n-v^n\mathrm{div}_hv^h$. We write the solution of (\[e3.23\]) as follows $$\begin{aligned}
\label{e3.24}
\begin{split}
{\Delta}_{k,j}{v}^n_{\Phi}&=e^{-tD_\epsilon^s+\Phi(t,D_n)}{\Delta}_{k,j}v^n_0+\int_0^te^{-(t-\tau)D_\epsilon^s}e^{-\lambda D_n \int_\tau^t\dot{\theta}(t')dt'}{\Delta}_{k,j}G_{\Phi}d\tau\\
&\quad+{\epsilon}^2\int_0^te^{-(t-\tau)D_\epsilon^s}e^{-\lambda D_n \int_\tau^t\dot{\theta}(t')dt'}{\Delta}_{k,j}\partial_nq_{\Phi}d\tau.
\end{split}\end{aligned}$$ Taking the $L^p_h(L^2_v)$ norm, we infer that $$\begin{aligned}
\label{e3.25}
\begin{split}
\|{\Delta}_{k,j}{v}^n_{\Phi}\|_{L^p_h(L^2_v)}&\lesssim e^{-c2^{ks}t}\|{\Delta}_{k,j} e^{\alpha D_n }v^n_0\|_{L^p_h(L^2_v)}\\
&\quad+\int_0^te^{-c2^{ks}(t-\tau)}\|{\Delta}_{k,j}G_{\Phi}\|_{L^p_h(L^2_v)}d\tau\\
&\quad+{\epsilon}^2\int_0^te^{-c2^{ks}(t-\tau)}\|{\Delta}_{k,j}\partial_nq_{\Phi}\|_{L^p_h(L^2_v)}d\tau.
\end{split}\end{aligned}$$ By the Young’s inequality, we deduce that $$\begin{aligned}
\label{e3.26}
\begin{split}
\|{\Delta}_{k,j}{v}^n_{\Phi}\|_{L_t^\infty(L^p_h(L^2_v))}&+2^{ks}
\|{\Delta}_{k,j}{v}^n_{\Phi}\|_{L_t^1(L^p_h(L^2_v))}\\
&\lesssim\|{\Delta}_{k,j} e^{\alpha D_n }v^n_0\|_{L^p_h(L^2_v)}
+\|{\Delta}_{k,j}G_{\Phi}\|_{L^1_t(L^p_h(L^2_v))}\\
&\quad+{\epsilon}^2\|{\Delta}_{k,j}\partial_nq_{\Phi}\|_{L^1_t(L^p_h(L^2_v))}.
\end{split}\end{aligned}$$ Multiplying both sides of (\[e3.26\]) by $2^{k(\frac{n-1}{p}+1-s)}2^{j\frac{1}{2}}$ and taking the sum over $k,j$, we have $$\begin{aligned}
\label{e3.27}
\begin{split}
&\|{v}^n_{\Phi}\|_{\widetilde{L}_t^\infty(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}+
\|{v}^n_{\Phi}\|_{L_t^1(\dot{B}^{\frac{n-1}{p}+1,\frac{1}{2}}_{p,1})}\\
&\lesssim\|e^{\alpha D_n }v^n_0\|_{\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1}}
+\|G_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}
+{\epsilon}^2\|\partial_nq_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}.
\end{split}\end{aligned}$$ According to Lemma \[p2.9\], we can obtain the estimates of nonlinear term by the following: $$\|(v^h\cdot\nabla_h v^n)_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}\lesssim \|{v}^h_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}
\|{v}^n_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1,\frac{1}{2}}_{p,1})},$$ $$\|(v^n\mathrm{div}_h v^h)_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}\lesssim \|{v}^n_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}
\|{v}^h_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1,\frac{1}{2}}_{p,1})}.$$ This implies that $$\|G_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}\lesssim \|{v}_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}
\|{v}_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1,\frac {1}{2}}_{p,1})}.$$
While for the pressure term, we use the decomposition $q=q^1+q^2+q^3$ in (\[e3.2\]). For $q^1$, since ${\epsilon}(-\Delta_\epsilon)^{-1}\partial_i\partial_n$ is a bounded operator applied for frequency localized functions in $L^p_h(L^2_v)$ if $i =1,2,\cdots,n-1$, we have $$\aligned
{\epsilon}^2\|\partial_nq^1_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}
&={\epsilon}^2\|(-\Delta_\epsilon)^{-1}\partial_i\partial_j\partial_n
(v^iv^j)_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}\\
&\lesssim {\epsilon}\|\nabla_h(v^hv^h)_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}.
\endaligned$$ Therefore, we get by using Lemma \[p2.9\] that $${\epsilon}^2\|\partial_nq^1_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}\lesssim {\epsilon}\|{v}^h_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}
\|{v}^h_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1,\frac {1}{2}}_{p,1})}.$$ Similarly, the fact that ${\epsilon}^2(-\Delta_\epsilon)^{-1}\partial^2_n$ is a bounded operator applied for frequency localized functions in $L^p_h(L^2_v)$ implies $${\epsilon}^2\|\partial_nq^2_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}\lesssim \|\nabla_h(v^nv^h)_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})},$$ $${\epsilon}^2\|\partial_nq^3_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}\lesssim \|(v^n\mathrm{div}_hv^h)_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}.$$ Thus, we have $$\aligned
{\epsilon}^2\|\partial_nq^2_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}
+{\epsilon}^2\|\partial_nq^3_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}\lesssim \|{v}_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}
\|{v}_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1,\frac {1}{2}}_{p,1})}.
\endaligned$$ Then we obtain that $$\aligned
{\epsilon}^2\|\partial_nq_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}\lesssim \|{v}_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}
\|{v}_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}+1,\frac {1}{2}}_{p,1})}.
\endaligned$$ Combining the above estimates, we can get the bound of ${v}^n_{\Phi}$ as follows: $$\begin{aligned}
\label{e3.28}
\begin{split}
\|{v}^n_{\Phi}\|_{\widetilde{L}_t^\infty(\dot{B}^{\frac{n-1}{p}+1-s,\frac {1}{2}}_{p,1})}+
\|{v}^n_{\Phi}\|_{L_t^1(\dot{B}^{\frac{n-1}{p}+1,\frac {1}{2}}_{p,1})}
\lesssim \Psi(0) + \Psi(t)^2.
\end{split}\end{aligned}$$
Together (\[e3.28\]) with (\[e3.17\]), we finally get that $$\begin{aligned}
\label{e3.35}
\begin{split}
\Psi(t) \lesssim \Psi(0)+\frac{1}{\lambda}\Psi(t)+\Psi(t)^2.
\end{split}\end{aligned}$$ This completes the proof of Lemma \[l3.2\] by choosing $\lambda$ large enough.
Estimates for $\theta(t)$
==========================
In the above section, we have used the fact that $\Phi(t)$ is a subadditivity function. This means we should ensure that $\theta(t)< \frac{\alpha}{\lambda}$. Thus, it is sufficient to prove that for any time $t$, $\theta(t)$ is a small quantity. By the definition of $\theta(t)$, naturally, we assume that $e^{\alpha D_n }v^n_0$ belongs to $\dot{B}_{p,1}^{\frac{n-1}{p}-s,\frac{1}{2}}$. According to the property of the operator $\partial_t + D_\epsilon ^s$, then we can get the bound for ${v}^n_{\Phi}$ in ${L}_t^1(\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}})$. However, we can not enclose the estimate for ${v}^n_{\Phi}$ in $\widetilde{L}_t^\infty(\dot{B}_{p,1}^{\frac{n-1}{p}-s,\frac{1}{2}})\cap{L}_t^1(\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}})$. Our observation is to add an extra term $\epsilon v^h$ under the same norm which is hidden in the pressure term ${\epsilon}^2\partial_nq^1.$ Hence, we first denote that $$X_0={\epsilon}\|e^{\alpha D_n }v^h_0\|_{\dot{B}_{p,1}^{\frac{n-1}{p}-s,\frac{1}{2}}},$$ $$Y_0=\|e^{\alpha D_n }v^n_0\|_{\dot{B}_{p,1}^{\frac{n-1}{p}-s,\frac{1}{2}}},$$ $$X(t)={\epsilon}\|{v}_\Phi^h\|_{\widetilde{L}_t^\infty(\dot{B}_{p,1}^{\frac{n-1}{p}-s,\frac{1}{2}})}+{\epsilon}\|{v}_\Phi^h\|_{{L}_t^1(\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}})},$$ $$Y(t)=\|{v}^n_{\Phi}\|_{\widetilde{L}_t^\infty(\dot{B}_{p,1}^{\frac{n-1}{p}-s,\frac{1}{2}})}
+\|{v}^n_{\Phi}\|_{{L}_t^1(\dot{B}_{p,1}^{\frac{n-1}{p},\frac{1}{2}})}.$$ In order to get the desired estimates, it suffices to prove the following lemma.
\[l3.1\] There exists a constant $C_2$ such that for any $\lambda>0$ and $t$ satisfying $\theta(t)\leq \frac{\alpha}{2\lambda}$, we have $$X(t)+Y(t)\leq C_2(X_0+Y_0)+C_2(X(t)+Y(t))\Psi(t).$$
We apply the same method as in the above section to prove $Y(t).$ Indeed, multiplying both sides of (\[e3.26\]) by $2^{(\frac{n-1}{p}-s)k}2^{\frac{1}{2}j}$ and taking the sum over $k,j$, we can get that $$\begin{aligned}
\label{e3.37}
\begin{split}
\|{v}^n_{\Phi}\|_{\widetilde{L}_t^\infty(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}+
\|{v}^n_{\Phi}\|_{{L}_t^1(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}&\leq\|e^{\alpha D_n }v^n_0\|_{\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1}}\\
&\quad+\|G_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}
+{\epsilon}^2\|\partial_nq_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}.
\end{split}\end{aligned}$$ According to Lemma \[p2.9\], we can obtain the estimates of nonlinear terms that $$\aligned\|\mathrm{div}_h(v^n{v}^h)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}
&+\|(v^n\mathrm{div}_h{v}^h)_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}\\
&\lesssim \|{v}^n_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}
\|{v}^h_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}.
\endaligned$$ This implies that $$\|G_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}\lesssim Y(t)\Psi(t).$$
While for the pressure term, we use the decomposition $q=q^1+q^2+q^3$ in (\[e3.2\]). For $q_1$, since ${\epsilon}(-\Delta_{\epsilon})^{-1}\partial_i\partial_n$ is a bounded operator applied for frequency localized functions in $L^p_h(L^2_v)$ if $i =1,2,\cdots,n-1$, we have $$\aligned
{\epsilon}^2\|\partial_nq^1_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}
&={\epsilon}^2\|(-\Delta_{\epsilon})^{-1}\partial_i\partial_j\partial_n
(v^iv^j)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}\\
&\lesssim {\epsilon}\|\nabla_h(v^hv^h)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}\\
&\lesssim {\epsilon}\|{v}^h_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}
\|{v}^h_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}.
\endaligned$$ Similarly, the fact that ${\epsilon}^2(-\Delta_{\epsilon})^{-1}\partial^2_n$ is a bounded operator applied for frequency localized functions in $L^p_h(L^2_v)$ implies $${\epsilon}^2\|\partial_nq^2_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}\lesssim \|\nabla_h(v^nv^h)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})},$$ $${\epsilon}^2\|\partial_nq^3_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}\lesssim \|(v^n\mathrm{div}_hv^h)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}.$$ Thus, applying Lemma \[p2.9\], we have $$\aligned
{\epsilon}^2\|\partial_nq^2_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}
+{\epsilon}^2\|\partial_nq^3_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}
\lesssim \|{v}^n_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}
\|{v}^h_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}.
\endaligned$$ Then we obtain that $$\aligned
{\epsilon}^2\|\partial_nq_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}\lesssim Y(t)\Psi(t) +X(t)\Psi(t).
\endaligned$$ Combining all the above estimates, we can get the bound of ${v}^n_{\Phi}$ in $\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})\cap L^1_t(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})$ by the following: $$\begin{aligned}
\label{e3.38}
\begin{split}
\|{v}^n_{\Phi}\|_{\widetilde{L}_t^\infty(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}+
\|{v}^n_{\Phi}\|_{L_t^1(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}
\lesssim Y_0
+(X(t)+Y(t))\Psi(t).
\end{split}\end{aligned}$$ This completes the proof of $Y(t)$ in Lemma \[l3.1\].
The following is devoted to getting the estimate of $X(t).$ The horizontal component ${v}^h$ in each dyadic block satisfies $$\label{e3.43}
\partial_t{\Delta}_{k,j}{v}^h+D_{\epsilon}^s{\Delta}_{k,j}{v}^h
=-{\Delta}_{k,j}\mathrm{div}_h({v}^h\otimes{v}^h)-{\Delta}_{k,j}\partial_n(v^n{v}^h)-\nabla_h{\Delta}_{k,j}q.$$ Denote $F\overset{\mathrm{def}}=-\mathrm{div}_h({v}^h\otimes{v}^h)-\partial_n(v^n{v}^h)$, then we infer that $$\begin{aligned}
\label{e3.44}
\begin{split}
\|{\Delta}_{k,j}{v}_\Phi^h\|_{L^p_h(L^2_v)}&\lesssim e^{-c(2^{ks}+{\epsilon}^s2^{js})t}\|{\Delta}_{k,j} e^{\alpha D_n }v^h_0\|_{L^p_h(L^2_v)}\\
&\quad+\int_0^te^{-c(2^{ks}+{\epsilon}^s2^{js})(t-\tau)}\|{\Delta}_{k,j}F_{\Phi}\|_{L^p_h(L^2_v)}d\tau\\
&\quad+\int_0^te^{-c(2^{ks}+{\epsilon}^s2^{js})(t-\tau)}\|{\Delta}_{k,j}\nabla_hq_{\Phi}\|_{L^p_h(L^2_v)}d\tau.
\end{split}\end{aligned}$$ By Young’s inequality, it holds that $$\aligned
\|\int_0^te^{-c(2^{ks}+{\epsilon}^s2^{js})(t-\tau)}\|{\Delta}_{k,j}
\partial_n(v^nv^h)_{\Phi}\|_{L^p_h(L^2_v)}d\tau\|_{L^\infty_t}\lesssim \frac{1}{{\epsilon}}\|{\Delta}_{k,j}(v^nv^h)_{\Phi}\|_{L^\frac{s}{s-1}_t(L^p_h(L^2_v))},
\endaligned$$ $$\aligned
2^{ks}\|\int_0^te^{-c(2^{ks}+{\epsilon}^s2^{js})(t-\tau)}\|{\Delta}_{k,j}
\partial_n(v^nv^h)_{\Phi}\|_{L^p_h(L^2_v)}d\tau\|_{L^1_t}\lesssim \frac{1}{{\epsilon}}2^k\|{\Delta}_{k,j}(v^nv^h)_{\Phi}\|_{L^1_t(L^p_h(L^2_v))}.
\endaligned$$ Here and in what follows, if $s=1$, the quantity $\|{\Delta}_{k,j}(v^nv^h)_{\Phi}\|_{L^\frac{s}{s-1}_t(L^p_h(L^2_v))}$ should be regarded as $\|{\Delta}_{k,j}(v^nv^h)_{\Phi}\|_{L^\infty_t(L^p_h(L^2_v))}.$ Therefore, taking $L^\infty$ norm and $L^1$ norm on $[0,t]$, we deduce that $$\begin{aligned}
\label{e3.45}
\begin{split}
\|{\Delta}_{k,j}&{v}_\Phi^h\|_{L_t^\infty(L^p_h(L^2_v))}+2^{ks}
\|{\Delta}_{k,j}{v}_\Phi^h\|_{L_t^1(L^p_h(L^2_v))}\\
&\lesssim \|e^{\alpha D_n }{\Delta}_{k,j}v^h_0\|_{L^p_h(L^2_v)}+2^k\|{\Delta}_{k,j}(v^h\otimes v^h)_{\Phi}\|_{L^1_t(L^p_h(L^2_v))}\\
&\quad+\frac{1}{{\epsilon}}2^k\|{\Delta}_{k,j}(v^n v^h)_{\Phi}\|_{L^1_t(L^p_h(L^2_v))}+\frac{1}{{\epsilon}}\|{\Delta}_{k,j}(v^n v^h)_{\Phi}\|_{L^\frac{s}{s-1}_t(L^p_h(L^2_v))}\\
&\quad+\|{\Delta}_{k,j}\nabla_hq_{\Phi}\|_{L^1_t(L^p_h(L^2_v))}.
\end{split}\end{aligned}$$ Multiplying both sides of (\[e3.45\]) by $2^{(\frac{n-1}{p}-s)k}2^{\frac{1}{2}j}$ and taking the sum over $k,j$, we finally get $$\begin{aligned}
\label{e3.46}
\begin{split}
{\epsilon}\|{v}_\Phi^h\|_{\widetilde{L}_t^\infty(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}&+
{\epsilon}\|{v}_\Phi^h\|_{{L}_t^1(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}\\
&\lesssim{\epsilon}\|e^{\alpha D_n }v^h_0\|_{\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1}}
+{\epsilon}\|(v^h\otimes v^h)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}\\
&\quad+\|(v^n v^h)_{\Phi}\|_{\widetilde{L}^\frac{s}{s-1}_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}+\|(v^n v^h)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}\\
&\quad+{\epsilon}\|\nabla_hq_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}.
\end{split}\end{aligned}$$
For the pressure term $q=q^1+q^2+q^3$, we find that $$\aligned
\|\nabla_hq^1_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}&=\|(-\Delta_{\epsilon})^{-1}\nabla_h\partial_i\partial_j
(v^iv^j)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}\lesssim \|\nabla_h(v^hv^h)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})},
\endaligned$$ where we have used that $(-\Delta_{\epsilon})^{-1}\partial_i\partial_j$ is a bounded operator for frequency localized functions in $L^p_h(L^2_v)$. Similarly, $$\begin{aligned}
\nonumber
\|\nabla_hq^2_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})} &=& 2\|(-\Delta_{\epsilon})^{-1}\nabla_h\partial_i\partial_n
(v^iv^n)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}\\\nonumber &\lesssim& \frac{1}{{\epsilon}}\|\nabla_h(v^nv^h)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})},\end{aligned}$$ $$\begin{aligned}
\nonumber
\|\nabla_hq^3_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})} &=& 2\|(-\Delta_{\epsilon})^{-1}\nabla_h\partial_n
(v^n\mathrm{div}_hv^h)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}\\\nonumber &\lesssim& \frac{1}{{\epsilon}}\|(v^n\mathrm{div}_hv^h)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}.\end{aligned}$$ According to Lemma \[p2.9\], the right hand side of (\[e3.46\]) can be bounded by following: $${\epsilon}\|({v}^h\otimes{v}^h)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}\lesssim {\epsilon}\|{v}^h_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}
\|{v}^h_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})},$$ $$\|(v^n{v}^h)_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}\lesssim \|{v}^h_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}
\|{v}^n_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})},$$ $$\aligned
{\epsilon}\|\nabla_hq_{\Phi}\|_{L^1_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}&\lesssim \Big({\epsilon}\|{v}^h_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}+\|{v}^n_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}\Big)
\|{v}^h_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})},\\
\endaligned$$ $$\aligned\|(v^n{v}^h)_{\Phi}\|_{\widetilde{L}^\frac{s}{s-1}_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}&\lesssim \|{v}^n_{\Phi}\|_{\widetilde{L}^\frac{s}{s-1}_t(\dot{B}^{\frac{n-1}{p}-1,\frac{1}{2}}_{p,1})}
\|{v}^h_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}\\
&\lesssim \Big(\|{v}^n_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}+\|{v}^n_{\Phi}\|_{{L}^1_t(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}\Big)
\|{v}^h_{\Phi}\|_{\widetilde{L}^\infty_t(\dot{B}^{\frac{n-1}{p}+1-s,\frac{1}{2}}_{p,1})}.
\endaligned$$ These imply that $$\begin{aligned}
\label{e3.47}
\begin{split}
{\epsilon}\|{v}_\Phi^h\|_{\widetilde{L}_t^\infty(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}&+{\epsilon}\|{v}_\Phi^h\|_{L_t^1(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}\lesssim X_0 +(X(t)+Y(t))\Psi(t).
\end{split}\end{aligned}$$
Combining (\[e3.47\]) with (\[e3.38\]), we finally obtain that there exists a constant $C_2$ such that $$\begin{aligned}
\label{e3.50}
\begin{split}
X(t)+Y(t)\leq C_2 (X_0+Y_0) +C_2(X(t)+Y(t))\Psi(t).
\end{split}\end{aligned}$$ This completes the proof of Lemma \[l3.1\].
Proof of the main result
========================
In this section, we will prove the Theorem \[t1.2\]. It relies on a continuation argument. For any $\lambda>\lambda_0$ and $\eta_1$, we define $\tau$ by $$\label{e3.51}
\tau=\max\{t\geq0|\ X(t)+Y(t)\leq \eta_1,\quad \ \Psi(t)\leq \eta_1\}.$$ In what follows, we shall prove that $\tau=\infty$ under the assumption (\[e1.3\]) for some small number $\eta_1$. Assume that this is not true. We choose $\eta_1$ small enough such that $$\theta(\tau)\leq C Y(\tau)\leq C \eta_1 \leq \frac{\alpha}{2\lambda},\quad (C_1 +C_2)\eta_1\leq\frac{1}{4}.$$ For such fixed $\eta_1$, we select the following norms of initial data sufficiently small enough such that $$C_1\Psi(0)+C_2(X(0)+Y(0))\leq C\eta \leq \frac{\eta_1}{4}.$$ Hence, we obtain from Lemma \[l3.2\] and \[l3.1\] that $$\begin{aligned}
\label{e3.52}
\begin{split}
&\Psi(\tau)\leq C_1\Psi(0)+C_1\eta_1^2,\quad X(\tau)+Y(\tau)\leq C_2(X(0)+Y(0))+C_2\eta_1^2.
\end{split}\end{aligned}$$ This implies that $$\begin{aligned}
\label{e3.53}
\begin{split}
X(\tau)+Y(\tau)\leq \frac{\eta_1}{2},\ \ \Psi(\tau)\leq \frac{\eta_1}{2}.
\end{split}\end{aligned}$$ However, this contradicts (\[e3.51\]) and hence completes the proof.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors were in part supported by NSFC (grants No. 11171072, 11421061 and 11222107), Shanghai Talent Development Fund and SGST 09DZ2272900.
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abstract: 'Nambu proposed an extension of dynamical system through the introduction of a new bracket (Nambu bracket) in 1973. This article is a short review of the developments after his paper. Some emphasis are put on a viewpoint that the Nambu bracket naturally describes extended objects which appear in M-theory and the fluid dynamics. The latter part of the paper is devoted to a review of the studies on the Nambu bracket (Lie 3-algebra) in Bagger-Lambert-Gustavsson theory of multiple M2-branes. This paper is a contribution to the proceedings of Nambu memorial symposium (Osaka City University, September 29, 2015).'
author:
- 'Pei-Ming Ho'
- Yutaka Matsuo
bibliography:
- 'HM\_0331.bib'
title: 'Nambu bracket and M-theory'
---
Introduction
============
Nambu’s contributions to Physics are profound and diverse. While creating great ideas such as spontaneous symmetry breaking which becomes standard in the contemporary Physics, he sometimes presented ideas which were mysterious in the beginning but became gradually recognized after years. Nambu bracket [@Nambu:1973qe] may be one of latter examples. The importance might not be so obvious even for himself. According to the paper, he kept the idea for more than twenty years before the publication. If we take it as was written, it started in early 50s when he moved from Osaka City University to Princeton. The reason why he needed so long period to decide the publication is understandable from his paper. Just after the definition of the bracket, he pointed out serious obstacles for his generalized dynamical system. During the long period that he kept his idea, he developed various new ideas which are useful and stimulating even from the current viewpoints.
As described in [@Nambu:1973qe], there are two major challenges in the subject. One is how to quantize the Nambu bracket and the other is multi-variable extensions. This turned out to be difficult or impossible (there appeared the no-go theorems). We have to relax “natural" requirements of the Nambu bracket which are the direct generalization of the Poisson bracket. The ways to relax the conditions are not unique and depend on the problem we are considering. It explains the existence of many proposals to define (quantum) Nambu bracket.
The purpose of this article is to give a brief review of the Nambu bracket and to illuminate some applications in M-theory. In section 2, we explain the basic material in the original paper [@Nambu:1973qe] where many ideas were already written. We also briefly quote some of the important results since then. It turned out that Nambu bracket fits with M-theory well and there appeared varieties of applications. We put some emphasis on the matrix model description of M-theory. In section 3, we review a proposal by Takhtajan [@Takhtajan:1993vr] that the Nambu bracket naturally describes the extended object. For the 3-bracket case, it corresponds to strings. In this respect, it fits non-canonical string such as the self-dual string on M5-brane and the vortex in the incompressible fluid. We explain the quantization of Takhtajan’s action which might be relevant to describe these non-canonical strings. Finally in section 4, we review the developments of the Nambu bracket and associated Fillipov Lie 3-algebras to describe the multiple M2-branes by Bagger, Lambert and Gustavsson (BLG model) [@Bagger:2006sk; @Bagger:2007jr; @Bagger:2007vi; @Gustavsson]. Special emphasis is put on our works where we introduced varieties of Lie 3-algebras with Lorentzian signature in BLG formalism to describe different types of extended objects appearing in M-theory and string theory.
Nambu bracket
=============
An introduction of Nambu bracket
--------------------------------
In 1973 [@Nambu:1973qe], Nambu proposed a generalization of Poisson bracket defined on a canonical pair $x,p$ $$\begin{aligned}
\left\{ f, g\right\}=\frac{\partial f}{\partial x}\frac{\partial g}{\partial p}-
\frac{\partial f}{\partial p}\frac{\partial g}{\partial x}\,,
$$ by the introduction of new dynamical system based on a canonical [*triple*]{} $x_1, x_2, x_3$: $$\begin{aligned}
\label{NB1}
\{f,g,h\}=\sum_{ijk}\epsilon_{ijk}\frac{\partial f}{\partial x_i}\frac{\partial g}{\partial x_j}
\frac{\partial h}{\partial x_k}=:
\frac{\partial(f,g, h)}{\partial(x_1, x_2, x_3)}
\,.\end{aligned}$$ This bracket was later referred to as the Nambu bracket. Instead of the canonical Hamiltonian equation, $$\begin{aligned}
\label{PD}
\dot f=\left\{f, H\right\}\,,\end{aligned}$$ the time evolution is defined by the new bracket with two Hamiltonians $H, G$, $$\begin{aligned}
\label{ND}
\dot f=\{f, H, G\}\,.\end{aligned}$$ As the Hamiltonian is a constant of the motion in (\[PD\]), two Hamiltonians $H, G$ are constant of the motion under the Nambu dynamics (\[ND\]), $$\begin{aligned}
\dot H=\{H, H, G\}=0,\quad
\dot G=\{G, H, G\}=0\,,\end{aligned}$$ due to the antisymmetry of the bracket. Just as the canonical Hamiltonian equation (\[PD\]) keeps the infinitesimal area of the phase space, $\Delta x \Delta p$, the generalized system (\[ND\]) keeps the volume of the triple $\Delta x_1\Delta x_2\Delta x_3$: $$\begin{aligned}
\vec \nabla \cdot \vec{v}=0,\quad
\vec{v}:=\dot{\vec{x}}=\{\vec x, H, G\}=\vec\nabla H\times \vec\nabla G\,.\end{aligned}$$ In this sense, it defines a dynamical system which has a generalized Liouville property (conservation of phase volume). This was one of the reasons why Nambu introduced such bracket.
As an example which is described by the new bracket, Nambu considered the rotational motion of a rigid body which is described by angular momentum $J_x, J_y, J_z$. In this case, we have two conserved quantities, the energy and the total momentum: $$\begin{aligned}
H=\frac{J_x^2}{2I_x}+\frac{J_y^2}{2I_y} +\frac{J_z^2}{2I_z}\,,\quad
G=\frac{J_x^2+J_y^2+J_z^2}{2}=\frac{\mathbf{J}^2}{2}\,,\end{aligned}$$ where $I_x, I_y, I_z$ are the moment of inertial along each axis. We introduce the Nambu bracket by $$\begin{aligned}
\{f, g, h\}=\frac{\partial(f,g,h)}{\partial(J_x, J_y, J_z)}\,.\end{aligned}$$ Some computation shows that the equation (\[ND\]) gives Euler’s equation for the rigid body: $$\begin{aligned}
\dot J_x=\left(\frac{1}{I_y}-\frac{1}{I_z}\right) J_y J_z,\quad
\dot J_y=\left(\frac{1}{I_z}-\frac{1}{I_x}\right) J_z J_x,\quad
\dot J_z=\left(\frac{1}{I_x}-\frac{1}{I_y}\right) J_x J_y\,.\end{aligned}$$
Generalizations of Nambu bracket
--------------------------------
### Mathematical definition
Nambu bracket is defined in more abstractly through the following requirements which generalize those for the Poisson bracket. It is defined on the ring of $C^\infty$ functions $\mathcal{A}$ with $M$ variables $x_1,\cdots, x_M$. The Nambu bracket in a generalized sense is defined by a map $\mathcal{A}^{\otimes N}\rightarrow \mathcal{A}$ $$\begin{aligned}
f_1,\cdots,f_N \in \mathcal{A}\Rightarrow \{f_1,\cdots,f_N\}\in \mathcal{A}\end{aligned}$$ which satisfies the following three conditions [@Takhtajan:1993vr]:
- Alternation law (skew symmetry): $$\begin{aligned}
\label{alternation}
\{f_{\sigma(1)},\cdots, f_{\sigma(N)}\}=(-1)^\sigma \{f_1,\cdots,f_N\}\,\quad\mbox{for arbitrary } \sigma\in\mathfrak{S}_N\,.
\end{aligned}$$
- Derivative law (Leibniz rule): $$\begin{aligned}
\label{derivative}
\{fg,f_2,\cdots,f_N\}=f\{g,f_2,\cdots,f_N\}+g\{f,f_2,\cdots, f_N\}\,.
\end{aligned}$$
- Generalized Jacobi law (fundamental identity): $$\begin{aligned}
\label{Fundamental}
\{\{f_1,\cdots, f_{N}\},g_1,\cdots, g_{N-1}\}=\sum_{i=1}^N
\{f_1,\cdots \{f_i, g_1,\cdots, g_{N-1}\},\cdots, f_N\}\,.
\end{aligned}$$
These rules are essential to define the time evolution of Nambu equation with $N-1$ Hamiltonians: $$\begin{aligned}
\label{higherNambu}
\frac{df}{dt}=\{f,H_1,\cdots, H_{N-1}\}\,.\end{aligned}$$ or a canonical transformation of variables defined by generating functions $S_1,\cdots, S_{N-1}$ (for $N=M$): $$\begin{aligned}
\label{hC}
\delta x_i=\{x_i,S_1,\cdots, S_{N-1}\}\,.\end{aligned}$$ They are natural in the sense to ensure the basic properties of the dynamics. Firstly, the alternation law I) ensures the Hamiltonians are constants of the motion[^1]: $$\begin{aligned}
\label{higher_conservation}
\frac{dH_i}{dt}=\{H_i,H_1,\cdots, H_{N-1}\}=0.\end{aligned}$$ The derivative law II) implies Leibniz rule for the time derivative: $$\begin{aligned}
\frac{d(fg)}{dt}&=&\{fg,H_1,\cdots, H_{N-1}\}\nonumber\\
&=& f\{g,H_1,\cdots,H_{N-1}\}+\{f,H_1,\cdots, H_{N-1}\}g=\frac{df}{dt}g+f\frac{dg}{dt}\,.\end{aligned}$$ Finally the fundamental identity III) (in the following we abbreviate it as FI) implies the distribution law of the time derivative in the bracket: $$\begin{aligned}
\frac{d}{dt}\{f_1,\cdots, f_N\}=\sum_{i=1}^N \{f_1,\cdots, \frac{df_i}{dt},\cdots, f_N\}\,.\end{aligned}$$
### Some properties of the generalized Nambu bracket
Here is a few comments on the generalized Nambu bracket and Liouville theorem:
- The Jacobian [@Nambu:1973qe] $$\begin{aligned}
\label{higher}
\{f_1,\cdots, f_n\}:=\frac{\partial(f_1,\cdots,f_n)}{\partial(
x_1,\cdots, x_n)}
\end{aligned}$$ satisfies all conditions I)–III) for $N=M=n$. The time evolution defined by this bracket keeps the $n$-dimensional phase volume $\Delta x_1\cdots \Delta x_n$, thus the dynamics satisfies the Liouville theorem.
- In [@Takhtajan:1993vr], possible solutions to the conditions I) II) III) are examined. The bracket which satisfies I) and II) may be written in the form: $$\begin{aligned}
\{f_1,\cdots, f_N\}=\sum_{i_1,\cdots, i_N} \eta_{i_1\cdots i_N}(x)
\partial_{i_1} f_1\cdots \partial_{i_N} f_N
\end{aligned}$$ where $\eta_{i_1\cdots i_N}$ is anti-symmetric for the indices. The fundamental identity is written as the bilinear identities among Nambu tensor $\eta_{i_1\cdots i_N}(x)$. It was proved that Nambu bracket should be decomposable $$\begin{aligned}
\eta:=\eta_{i_1\cdots i_N}\partial_{i_1}\wedge\cdots \wedge\partial_{i_N}
=V_1\wedge\cdots \wedge V_N,\quad V_a=\sum_i v_a^i\partial_{x_i}\end{aligned}$$ to satisfy the constraint [@gautheron1996some]. In particular, a natural multi-variable extension such as $\eta=\partial_1\wedge\partial_2\wedge \partial_3+\partial_4\wedge\partial_5\wedge \partial_6$ does not satisfy FI.
- In order to keep the phase volume, it is possible to generalize (\[higherNambu\]) to $$\begin{aligned}
\label{general_Nambu}
\frac{df}{dt}=\sum_{\alpha=1}^Q \{f, H_1^{(\alpha)},\cdots, H_{N-1}^{(\alpha)}\}\,,
\end{aligned}$$ with $Q(N-1)$ Hamiltonians $H_i^{(\alpha)}$. These generalized Hamiltonians, however, are not preserved by the equation of motion. In terms of the canonical variables, the equation of motion is written as $$\begin{aligned}
\dot{x}_i=\sum_{j=1}^N \partial_j f_{ij}(x),\quad
f_{ij}:=\sum_{k_1,\cdots, k_{N-2}} \epsilon_{ijk_1\cdots k_{N-1}}\sum_{\alpha}^{Q}
H_1^{(\alpha)}\frac{\partial(H_2^{(\alpha)},\cdots, H_{N-1}^{(\alpha)})}{\partial(x_{k_1},\cdots, x_{k_{N-2}})}\,.
\end{aligned}$$ The quantity $f_{ij}$ is antisymmetric $f_{ij}=-f_{ji}$. The first equation is the most general form to preserve phase volume.
- For $N=3$ case, the canonical equation is rewritten as $$\begin{aligned}
\label{eom}
\dot{\vec{x}}=\vec\nabla\times \vec{A}, \quad \vec{A}=\sum_{\alpha=1} H_\alpha
\vec\nabla G_\alpha\,.
\end{aligned}$$ It was noted [@Nambu:1973qe] that there are some arbitrariness in the choice of $H_\alpha, G_\alpha$ to give the same equation. Namely a different set $H'_\alpha, G'_\alpha$ of Hamiltonian gives the same equation of motion as long as it satisfies canonical transformation with $(H_\alpha, G_\alpha)$ as the canonical pair in the Poisson sense, $$\begin{aligned}
[H'_\alpha, G'_\beta]:=\sum_{\gamma=1}^N \frac{\partial(H'_\alpha,G'_\beta)}{\partial(H_\gamma,G_\gamma)}
=\delta_{\alpha\beta},\quad [H'_\alpha, H'_\beta]=[G'_\alpha, G'_\beta]=0\,.
\end{aligned}$$ One may check the statement for the infinitesimal variations. Let us use $\delta H_\alpha=H'_\alpha-H_\alpha=\epsilon\frac{\partial S(H,G)
}{\partial G_\alpha}$ and $\delta G_\alpha=G'_\alpha-G_\alpha=-\epsilon\frac{\partial S(H,G)
}{\partial H_\alpha}$. The variation of the equation (\[eom\]) is absorbed in the variation of $\vec A$ as, $
\delta \vec{A} = \epsilon\vec\nabla\left(S-\sum_\alpha H_\alpha\frac{\partial S}{
\partial H_\alpha}\right)
$ which may be interpreted as the infinitesimal gauge transformation. It is obvious that it leads to the same equation of motion.
- The other type of the hierarchy structure exists for general $n$ [@Takhtajan:1993vr]. Starting from arbitrary $n+1$ bracket $\{f_1,\cdots, f_{n+1}\}$ which satisfies I)-III), one may define the $n$ bracket by using arbitrary $K$, $$\begin{aligned}
\{f_1,\cdots, f_n\}_K:=\{f_1,\cdots, f_n, K\}\,.\end{aligned}$$ One may show easily that the new bracket satisfies the three conditions. By continuing the same procedure, one may obtain Nambu $m$ bracket from Nambu $n$ bracket for $m<n$.
As an example, let us take the Nambu bracket for the rigid rotor. The original Nambu bracket was defined as $$\begin{aligned}
\{f,g, h\}=\frac{\partial(f,g,h)}{\partial(J_x,J_y, J_z)}\,.\end{aligned}$$ If we take $K=\frac{1}{2}(J_x^2+J_y^2+J_z^2)$, the Poisson bracket $\{\bullet,\bullet\}_K:=\{\bullet,\bullet,K\}$ gives $$\begin{aligned}
\{J_x, J_y\}_K= J_z,\quad
\{J_y, J_z\}_K= J_x,\quad
\{J_z, J_x\}_K= J_y,\end{aligned}$$ which is the standard Poisson bracket for the angular momentum.
Difficulties in Nambu bracket
-----------------------------
In [@Nambu:1973qe], it was already mentioned some serious difficulties in the formulation. They are not the technical problems and there is no way to overcome them. All we can do is to relax some of the conditions I), II), III) as long as they do not produce serious troubles in the applications which we consider.
#### Multi-variable extension
In Poisson bracket, it is straightforward to extend the formalism to $2N$ canonical pairs, $x^i, p_i$ ($i,j=1,\cdots, N$) as $$\begin{aligned}
\{f,g\}=
\sum_{i=1}^N \left(\frac{\partial f}{\partial x_j}\frac{\partial g}{\partial p^j}-\frac{\partial f}{\partial p^j}\frac{\partial g}{\partial x_j}\right)\end{aligned}$$ It satisfies the consistency condition of the Poisson bracket (Jacobi identity), $$\begin{aligned}
\label{Jacobi}
\{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0\,,\end{aligned}$$ for any $N$. The existence of such identity is necessary for the compatibility of the time evolution (\[PD\]). In the Nambu bracket, the analog of (\[Jacobi\]) is played by the fundamental identity (FI). A difficulty of the Nambu bracket is that the FI is too strict that there is almost no room for the generalization. As already mentioned, a naive multi-variable extension of (\[NB1\]) $$\begin{aligned}
\label{mNambu}
\{f,g,h\}=\sum_{a=1}^N
\frac{\partial(f,g, h)}{\partial(x^a_1, x^a_2, x^a_3)}\end{aligned}$$ for $3N$ variables $x^a_i$ ($a=1,\cdots, N$, $i=1,2,3$) does not satisfy FI. In [@Nambu:1973qe], Nambu examined the canonical transformation defined by the bracket (\[mNambu\]) and the generating function $S_i$ in (\[hC\]) should be decomposed as $S_i=\sum_a S_i^a(x^a)$ from the consistency conditions. It implies that the variable set $(x_1^a,x_2^a,x_3^a)$ should transform within themselves. While the fundamental identity was not proposed explicitly but this analysis has already shown the difficulty in the multi-variable extension.
#### Quantization
In the Poisson bracket, the quantization procedure is to replace the bracket into the commutator $$\begin{aligned}
\{f,g\}=\frac{\partial(f,g)}{\partial(x,p)} \rightarrow
\left[\hat f, \hat g\right]=\hat{f}\hat{g}-\hat{g}\hat{h}\,.\end{aligned}$$ The commutator satisfies a noncommutative version of the three consistency conditions. For the Nambu bracket, the most straightforward generalization of the commutator is, $$\label{canonical triple}
[X,Y,Z]=XYZ+YZX+ZXY-YXZ-XZY-ZYX=X[Y,Z]+Y[Z,X]+Z[X, Y]\,.$$ While it satisfies I), the conditions II) and III) are not kept.
#### Solutions to canonical quantization condition
While it does not satisfy the conditions, it may be possible to use it relaxing some conditions. In [@Nambu:1973qe], Nambu tried to find a set of operators which satisfies an analog of canonical quantization condition: $$\begin{aligned}
\label{canN}
[X_a, Y_b, Z_c]=i\delta_{abc}\end{aligned}$$ while neglecting the constraints (2,3) for the moment. Here $\delta_{abc}=1$ when $a=b=c$ and $=0$ otherwise. Assuming the set $\{X_a, Y_a, Z_a\}$ ($a=1,\cdots, N$) are the basis of some Lie algebra $\mathfrak{g}$. By writing $$\begin{aligned}
[X_1, Y_1]=i Z', \quad [Y_1, Z_1]=i X',\quad [Z_1, X_1]=i Y'\end{aligned}$$ for the first three generators and $X', Y', Z'\in \mathfrak{g}$. Eq.(\[canN\]) implies that $$\begin{aligned}
X_1 X'+Y_1 Y' +Z_1 Z'=1\,.\end{aligned}$$ The right hand side is c-number and should commute with arbitrary generators in $\mathfrak{g}$. So it may be implemented by Casimir operator for the Lie algebra. From this observation, assuming $\mathfrak{g}$ is semisimple, one may classify the possible algebras. The result is: $$\begin{aligned}
SO(3), SO(2,1), SO(4), SO(3,1)\,.\end{aligned}$$ If the algebra is not semi-simple, there are futher choices after contractions: $$\begin{aligned}
E(3), E(2,1), E(2), E(1,1)\end{aligned}$$ Here $E(3)$ is the euclidean algebra generated by $\vec P, \vec L$ (momentum and angular momentum operators). The others are similar algebra with different dimensions and signature.
#### Use of nonassociative algebras
Nambu also considered a possibility to use nonassociative algebra to define the quantization. In this case, the associator $$\begin{aligned}
(a,b,c)=(ab)c-a(bc)\end{aligned}$$ does not in general vanish. If we require that the associator be skew symmetric with respect to all elements, the algebra is restricted to Cayley number. It nevertheless does not satisfy the derivative property.
He then modified the bracket to keep the derivative property: $$\begin{aligned}
D(a,b;x)=D(a,b)x:= a(bx)-b(ax)+(xb)a-(xa)b+(bx)a-b(xa)\end{aligned}$$ for the Cayley number. This time, we do not have total skewness but only the partial one: $D(a,b;x)=-D(b,a;x)$. The time evolution generated by $$\begin{aligned}
\frac{dx}{dt}=\sum_i D(H_i,G_i)x\end{aligned}$$ generates the $G_2$ automorphism.[^2]
He also examined to use a commutative and nonassociative algebra (Jordan algebra). In this case, the derivative operator is written in the form: $$\begin{aligned}
D(a,b)x=(a,b,x)-(b,a,x)\,.\end{aligned}$$ Jordan algebra, in general, is written in terms of noncommutative and associative algebra by modification of the multiplication $a\cdot b=(ab+ba)/2$. If we use this realization, the derivative operator is rewritten as $D(a,b)x=[x,[a,b]]$. So the equation of motion is reduced to the conventional Hamiltonian flow where Hamiltonian is written in the form $[H,G]$.
Some attempts to quantize Nambu bracket
---------------------------------------
A natural approach to quantize the Nambu bracket is through the deformation quantization. It is a generalization of Moyal bracket, $$\begin{aligned}
&& f\star g:=e^{\frac12\hbar \epsilon_{ij} \partial^{(1)}_i \partial^{(2)}_j}f(x^{(1)})
g(x^{(2)})|_{x^{(1)}=x^{(2)}=x}\nonumber\\
&&\quad \rightarrow
(f, g,h):=e^{\frac16 \hbar \epsilon_{ijk} \partial^{(1)}_i \partial^{(2)}_j\partial^{(3)}_k}f(x^{(1)})
g(x^{(2)})h(x^{(3)})|_{x^{(1)}=x^{(2)}=x^{(3)}=x}\,.\end{aligned}$$ The quantum Nambu bracket thus defined failed to satisfy FI [@Takhtajan:1993vr]. There are a few alternative approaches for the deformation quantization (see for example, [@gautheron1996some; @Curtright:2002sr]). Later, Dito et. al. [@Dito:1996xr] proposed a deformation quantization based on Zariski quantization which satisfies FI. It is very different from conventional quantization method but some efforts have been made to use it for the M-theory [@Minic:1999js].
Curtight and Zachos tried to formulate the quantum Nambu bracket in the line of (\[canonical triple\]). Instead of the modification of the bracket (\[canonical triple\]), they proposed an alternative to the fundamental identity [@Curtright:2002fd]. This reference contains a nice review on the Nambu bracket.
In the connection with the matrix model approach to M-theory [@Banks:1996vh], the Nambu dynamics is natural to realize the generalized uncertainty relation $\Delta p\Delta q\Delta r\geq \hbar$. Awata, Li, Minic and Yoneya [@Awata:1999dz] defined a quantization of Nambu bracket through the matrices as $$\begin{aligned}
[A,B,C]:=\mbox{Tr}(A)[B,C]+\mbox{Tr}(B)[C,A]+\mbox{Tr}(C)[A,B]\,,\end{aligned}$$ which satisfies the fundamental identity. Very recently, Yoneya suggested a similar bracket [@Yoneya:2016wqw] to describe the covariant M-theory matrix model.
In the context of M-theory, the degree of freedom is predicted to behave as $O(N^3)$ for $N$ five-branes from AdS/CFT correspondence. In this sense, it may be natural that the quantum degree of freedom is described by a tensor with three indices $A_{ijk}$ (cubic matrix). Such direction was pursued by Kawamura in [@Kawamura:2002yz; @Kawamura:2003cw]. The triple matrix for the cubic matrix was defined as $$\begin{aligned}
(ABC)_{lmn}=\sum_k A_{lmk}B_{lkn}C_{kmn}\,,\end{aligned}$$ and quantum Nambu bracket is defined by anti-symmetrization. While FI is not satisfied with this bracket, a consistent dynamical system can be constructed if the Hamiltonians are restricted to the normal form, $$\begin{aligned}
H_{lmn}=\delta_{lm} h_{mn}+\delta_{nm}h_{ln}+\delta_{ln}h_{ml}.\end{aligned}$$ Due to this restriction, the time evolution becomes essentially diagonal. We note that the choice of the product of the cubic matrix is not unique. For example, in [@Ho:2007vk], a different choice, $(ABC)_{lmn}=\sum_{ijk} A_{ij n}B_{jkl}C_{kim}$ was used. It is more natural to associate the cubic matrix with the triangle which covers the membrane: the index is assigned to the edges of a triangle and the triple product is interpreted as gluing edges of three triangles to produce three open edges. It is a natural framework to implement discretized quantum gravity [@Turaev:1992hq] but the analog of FI is difficult to be realized.
Nambu bracket and the extended objects
======================================
Takhtajan’s action
------------------
In [@Takhtajan:1993vr], Takhtajan introduced an action principle which describes the Nambu dynamics as the motion of the extended objects. Let new variables $X^i(\sigma, t)$ ($i=1,2,3$) describe a string-like object in $\mathbb{R}^3$ (three spacial dimensions). We assume that the Hamiltonians $H, K$ are the functions of $X^i(\sigma, t)$ at the same world-sheet point. $$\begin{aligned}
\label{Tk_action}
S= \frac{1}{3}\int dtd\sigma \epsilon_{ijk} X^i\partial_t X^j \partial_\sigma X^k+
\int dtd\sigma H\partial_\sigma K\,.\end{aligned}$$ Variation of the action gives $$\begin{aligned}
\label{varTA}
\delta S=\int dtd\sigma
\left(\frac{1}{3}\epsilon_{ijk}\partial_t X^i- \frac{\partial(H,K)}{\partial(X^j,X^k)}\right)
\frac{\partial X^j}{\partial \sigma} \delta X^k\end{aligned}$$ It implies the equation of motion for the string-like object, $$\begin{aligned}
\partial_t X^i-\frac12 \epsilon_{ijk}\frac{\partial(H,K)}{\partial(X^j,X^k)}\propto
\partial_\sigma X^i\,.\end{aligned}$$ The left hand side of the equation is Nambu’s equation and the right hand side is the arbitrariness due to the reprametrization invariance with respect to $\sigma$. When we need to consider more general Nambu action of the form (\[general\_Nambu\]), one may simply replace it by $$\begin{aligned}
S=\int dt d^{N-2}\sigma
\left( \frac{1}{N!}
\epsilon_{i_1,\cdots, i_N}
X^{i_1}\frac{\partial(X^{i_2}\cdots X^{i_{N}})}{\partial(t,
\sigma_1,\cdots,\sigma_{N-2})}
- H_1 \sum_\alpha \frac{\partial(H^\alpha_2,\cdots, H^\alpha_{N-1})}{\partial(\sigma_1,\cdots,\sigma_{N-2})} \right)\end{aligned}$$ In this case, the variable $X^i(\sigma, t)$ describes an $(N-2)$-brane.
Takhtajan’s action is relevant to the study of self-dual string on M5-brane [@Bergshoeff:2000jn; @Kawamoto:2000zt] and the fluid motion in 3 dimensions. The connection with the fluid motion is discussed in the next subsection. In the context of M-theory, the fundamental degree of freedom is described by M2-brane (and the dual M5-brane) whereas the effective description by supergravity is described by anti-symmetric 3-form field $C$ and its dual 6-form. In the low energy, the effective description of the membrane is given by Nambu-Goto type action and the coupling to three-form $C$, $$\begin{aligned}
S=\int d^3\sigma T \det\left(-G\right)+\int_V C,\end{aligned}$$ where $T$ is the membrane tension and $V$ is the world volume of the membrane. Suppose we are considering an extreme situation where $C$ is constant and large enough such that one may neglect the Nambu-Goto part, we are left with the coupling of the membrane world-volume to the constant 3-form field. In the simplest case where $C_{012}\neq 0$, the latter term coincides with the Takhtajan action when the world-volume has the boundary since $$\begin{aligned}
\frac{1}{3!}C_{012}\int_V \epsilon_{ijk} dX^i\wedge dX^j\wedge dX^k=\frac{1}{3!} C_{012} \int_{\partial V} \epsilon_{ijk}X^i dX^j\wedge dX^k.\end{aligned}$$ It is known that the the boundary of M2-brane is located on M5-brane. On M5-brane, the two-form gauge field should be self dual, namely $C=\star C$. In this sense, Takhtajan string describes the self-dual string on M5.
Connections with incompressible fluid dynamics
----------------------------------------------
As Nambu himself pursued for a long time, (due to a review in [@Saitou:2014vwa]), the Nambu dynamics is a natural framework to describe the incompressible fluid motion. The incompressibility implies that the volume element $\Delta v$ does not change in the time evolution. It implies that the coordinates $\vec x(\vec x_0,t)$ has to satisfy $\frac{\partial(\vec x)}{\partial(\vec x_0)}=1$ in the Lagrangian formulation where $\vec x(\vec x_0,t)$ is the location of fluid which was at $\vec x_0$ at $t=t_0$. It implies that the time evolution should be written in the form, $$\begin{aligned}
\partial_t \vec x(\vec x_0, t)=\sum_\alpha \left\{\vec x,H_\alpha(\vec x_0,t), K_\alpha(\vec x_0,t)\right\}.\end{aligned}$$ In this subsection, we collect some descriptions of fluid motion by the Nambu-bracket.
### Vortex string dynamics
Takhtajan’s action for the Nambu dynamics can be directly related with the vortex motion where there is no dissipation. In the following, we use the description in [@Lund:1976ze; @Matsuo:1993ie]. We consider the Euler equation, $$\begin{aligned}
\frac{\partial V^i}{\partial t} = V^j\partial^i V_j -V^j\partial_j V^i \end{aligned}$$ for the velocity $\vec V(z)$. In such a system, the fluid motion is governed by the center of vorticity, described by strings localized at $\vec x=\vec X_I(\sigma_I,\tau)$. As long as there is no dissipation, the delta-function shape vorticity retains its form and motion of the vortex string determines the flow. Here we assume there are $N$ vortex filaments and $I=1,\cdots, N$. The vorticity is described by $$\begin{aligned}
\vec \omega(x)&=& \vec{\nabla} \times \vec V= \sum_{I=1}^N\Gamma_I\int d\sigma_I
\frac{\partial \vec X_I(\sigma_i,t)}{\partial \sigma_I}\delta^{(3)}(\vec x-\vec X_I(\sigma_I, t))\,.\end{aligned}$$ From this expression, one obtains the velocity field by Biot-Savart law, $$\begin{aligned}
\vec V(x) &=& \sum_{I} \frac{\Gamma_I}{4\pi}\int d\sigma_I \frac{\partial \vec X_I}{\partial \sigma_I}
\times\frac{\vec x-\vec X_I}{|\vec x-\vec X_I|^3}
=\vec{\nabla}\times\sum_I \frac{\Gamma_I}{4\pi} \int d\sigma_I \frac{\partial \vec X_I}{\partial \sigma_I} \frac{1}{|x-\vec X_I|}\,.\end{aligned}$$ Plug it into the Euler equation for the vorticity, $$\begin{aligned}
\frac{\partial\vec \omega}{\partial t}= -\nabla\times (\vec{\omega}\times \vec V),\end{aligned}$$ one finds that the Euler equation is solved if $\vec X_I$ satisfies the equation, $$\begin{aligned}
\frac{\partial \vec X_I}{\partial \sigma_I}
\times \frac{\partial \vec X_I}{\partial t}=
\frac{\partial \vec X_I}{\partial \sigma_I}\times \vec V(X_I(\sigma_I, t))\,.\end{aligned}$$ It implies that $\frac{\partial \vec X_I}{\partial t}=\vec V(X_I(\sigma_I, t))
+\alpha \frac{\partial \vec X_I}{\partial \sigma_I}$, namely the velocity of the string is identical to the flow velocity up to reparametrization. The fact that the above equation takes the same form as (\[varTA\]) implies that the action can be written in the Takhtajan form: $$\begin{aligned}
S&=& \int dt (L_0-E)\,,\\
L_0&=& \sum_{I=1}^N \frac{\Gamma_I}{3!}\int d\sigma_I \vec X_I\cdot
\frac{\partial\vec X_I}{\partial \sigma_I}\times \frac{\partial \vec X_I}{\partial t}\,,\\
E&=& \frac12 \int d^3 x |\vec V(x)|^2=\frac{1}{8\pi}\sum_{IJ}
\int d\sigma_I d\sigma_J \Gamma_I\Gamma_J
\left(\frac{\partial \vec X_I}{\partial \sigma_I}\cdot
\frac{\partial \vec X_J}{\partial \sigma_J}\right) \frac{1}{|\vec X_I-\vec X_j|}\,.\end{aligned}$$ The second term may be rewritten as $$\begin{aligned}
\sum_I \int d\sigma \Gamma_I \vec{U}(\vec X_I)\cdot\frac{\partial\vec X_I}{\partial \sigma},\quad\mbox{where }
\vec U(x)=\sum_{J}\frac{\Gamma_J}{8\pi }
\int d\sigma_J \frac{\partial \vec X_J}{\partial \sigma}\frac{1}{|\vec x-\vec X_J(\sigma)|}\,.\end{aligned}$$ One may regard it as a generalization of Takhtajan action with the Hamiltonians replaced by $H^i_I=\Gamma_I X^i_I, K^i_I=U^i(\vec X_I)$ with $\alpha$ replaced by multiple indices $i,I$.
### Fluid dynamics in shallow water
More recently, a totally different way of rewriting fluid dynamics as Nambu equation was developed in [@salmon2007general; @nevir2009energy; @sommer2009conservative]. The shallow water equation, $$\begin{aligned}
\dot u=h\omega v-\Phi_x,\quad
\dot v=-h\omega u-\Phi_y,\quad
\dot h = (-hu)_x-(hv)_x\end{aligned}$$ where $(u,v)$ is the velocity for horizontal directions, $h$ is the fluid depth, $\omega=(v_x-u_y)/h$ is the vorticity, and $\Phi=\frac{u^2+v^2}{2}+gh$ is the energy density. It was realized that the equations can be written in the form of Nambu dynamics $\dot F=\{F, H, Z\}$ where $H=\int d^2 x h\Phi(x,y)$ and $Z=\int d^2x h G(q(x,y))$, where $G$ is an arbitrary function. The bracket is defined as the functional deferentiation by $u,v,h$ which is more involved. See for example, eq.(1.15) in [@salmon2007general].
Quantization of Takhtajan’s action
----------------------------------
One may apply the standard quantization method to Takhtajan action. We refer to [@Matsuo:1993ie; @Bergshoeff:2000jn; @Kawamoto:2000zt; @Pioline:2002ba] for 3-bracket cases and [@Matsuo:2000fh] for higher cases.
We note that in the action (\[Tk\_action\]), the time derivative is contained in the first term. The momentum variable is therefore given as, $
\Pi_i(\sigma,t)=\frac{1}{3} \epsilon_{ijk}X^j \frac{\partial X^k}{\partial \sigma}.
$ Since it is expressed in terms of the coordinate variables, we have a constrained system with three constraints: $$\begin{aligned}
\phi_i=\Pi_i -\frac{1}{3} \epsilon_{ijk}X^j \frac{\partial X^k}{\partial \sigma} \approx 0.\end{aligned}$$ The Poisson brackets among the constraints are given by $$\begin{aligned}
\{\phi_i(\sigma),\phi_j(\sigma')\}=-\epsilon_{ijk}\frac{\partial X^k}{\partial \sigma}\delta(\sigma-\sigma')\,.\end{aligned}$$ This $3\times 3$ matrix has rank two. It implies that a combination of the constraints $\phi_i$ is the first class. By inspection, one finds that $$\begin{aligned}
T(\sigma)=-\frac{\partial X^i}{\partial \sigma}\phi_i\end{aligned}$$ has vanishing bracket and becomes first class. It satisfies a classical version of the Virasoro algebra, $$\begin{aligned}
\left\{T(\sigma), T(\sigma')\right\}=2 T(\sigma')\partial_{\sigma'}\delta(\sigma-\sigma')
+\partial_{\sigma}T(\sigma)\delta(\sigma-\sigma')\,.\end{aligned}$$ The appearance of the Virasoro algebra is natural since we have the reparametrization invariance. One may turn the first class constraints into the second class by adding the gauge fixing condition. There are some choices. The simplest one is to use “static gauge”, $$\begin{aligned}
\label{gf1}
\chi=X^3-\sigma\approx 0.\end{aligned}$$ The Dirac bracket associated with it gives $$\begin{aligned}
\label{staticgauge}
\{X^1(\sigma), X^2(\sigma')\}_D=\delta(\sigma-\sigma')\,.\end{aligned}$$ The other possibility is to use $O(3)$ invariant gauge, $$\begin{aligned}
\label{gf2}
\chi=(\partial_\sigma \vec X)^2-1\approx 0.\end{aligned}$$ The Dirac bracket for this gauge choice gives $$\begin{aligned}
\{X^i(\sigma),X^j(\sigma')\}_D=\epsilon_{ijk}\frac{\partial X^k}{\partial \sigma}\delta(\sigma-\sigma')\,.\end{aligned}$$ In either case, the Nambu dynamics is described in the form of Dirac bracket as $$\begin{aligned}
\frac{\partial X^i}{\partial t}=\left\{X^i, \omega(H,K)\right\}_D+\cdots,
\quad
\omega(H,K)=\int d\sigma H(X)\partial_\sigma K(X)\,.\end{aligned}$$ where $\cdots$ terms are changes associated with the reparametrization of $\sigma$ to keep the consistency of gauge fixing conditions (\[gf1\],\[gf2\]).
This procedure seems to produce a simple 2D conformal field theory. For example, the commutator (\[staticgauge\]) is the same as the commutator of $\beta-\gamma$ ghosts. A subtlety is how to regularize the volume preserving diffeomorphism generator $\omega(H,K)$ which are nonlinear functions of coordinates $\vec X$. It is also nontrivial how to recover the rotational symmetry $O(3)$. These issues have not been fixed in our understanding.
Nambu bracket in M-theory
=========================
In string theory, the Lie algebra is needed when one promotes the low energy effective theory of a single D-brane [@Leigh:1989jq] to that of a stack of multiple D-branes [@Witten:1995im]. Similarly, in M theory, the Nambu bracket is needed to promote the theory of a single membrance [@Bergshoeff:1987cm] to multiple membranes [@Bagger:2006sk; @Bagger:2007jr; @Bagger:2007vi]. On the other hand, the commutator is needed for the noncommuative D-brane in the $B$-field background [@Chu:1998qz; @Schomerus:1999ug; @Seiberg:1999vs], and similarly the Nambu bracket is needed to formulate an M5-brane in the $C$-field background [@HM; @HIMS; @Chen:2010br]. [^3] In this section, we review these theories of M-branes and D-branes in which the Nambu bracket and its generalizations appear to characterize the effect of interactions among branes, or the interaction with a particular background.
As an extension of M(atrix) theories
------------------------------------
The low-energy effective theories of D$p$-branes are well known to be supersymmetric Yang-Mills theories [@Witten:1995im], in which transverse coordinates $X^a$ of the target space are represented by matrices. It was learned in the study of M(atrix) theories that higher dimensional branes can be constructed out of lower dimensional ones through certain matrix configurations [@Banks:1996nn]. For instance, solutions to the Nahm equation [@Nahm:1979yw] $$\frac{dX_a}{d\sigma} + \frac{1}{2} \epsilon_{abc} [X^b, X^c] = 0$$ for the multiple D1-brane theory describe a bound state of D1-branes ending on a D3-brane [@Diaconescu:1996rk]. (The parameter $\sigma$ is the spatial world-sheet coordinate of the D1-brane.) This was generalized to the Basu-Harvey equation [@Basu:2004ed] $$\frac{dX_a}{d\sigma} + \frac{1}{6} \epsilon_{abcd} [G, X^b, X^c, X^d] = 0,$$ to describe M2-branes ending on an M5-brane. Here $\sigma$ is the spatial coordinate of the M2-branes parametrizing their extension orthogonal to the M5-brane, and $X^a$’s are the matrices representing transverse coordinates. The 4-bracket is defined as a sum over permutations $P$ of 4 indices: $$[A_1, A_2, A_3, A_4] =
\sum_{P} sgn(P) A_{P(1)}A_{P(2)}A_{P(3)}A_{P(4)}.$$ As the matrix $G$ is fixed, effectively a three-bracket $[G, \,\cdot\;, \,\cdot\;, \,\cdot\; ]$ appears here. Note that a 3-bracket structure must appear as the M5-brane is 3-dimensional higher than an M2-brane. Although the 3-bracket defined this way does not enjoy enough nice algebraic properties to allow one to define a supersymmetric action for multiple M2-branes, this is one of the first hints that one should replace the Lie bracket by something like the Nambu bracket when one considers M theory. Another hint for the relevance of the 3-bracket to M theory was obtained through calculations of scattering amplitudes of membranes in the $C$-field background [@Ho:2007vk].
As an alternative to the use of the matrix algebra to realize the Nambu bracket, one can also define Lie 3-algebra abstractly as an analogue of the Lie algebra. The Lie 3-algebra is defined as a linear space equipped with a totally anti-symmetrized bracket of 3 slots $[\,\cdot\; , \,\cdot\; , \,\cdot\; ]$, which maps three elements to an element in the linear space. For a given basis $\{ T^A \}$ of the linear space, the Lie 3-bracket $$[T^A, T^B, T^C] = f^{ABC}{}_D T^D$$ is given in terms of the structure constants $f^{ABC}{}_D \in \mathbb{C}$. The Lie 3-bracket is required to satisfy the fundamental identity $$[F_1, F_2, [F_3, F_4, F_5]] =
[[F_1, F_2, F_3], F_4, F_5] + [F_3, [F_1, F_2, F_4], F_5] + [F_3, F_4, [F_1, F_2, F_5]]
\label{Fundamental-Identity}$$ for all elements $F_1, F_2, \cdots, F_5$ of the algebra. Lie 3-algebra is essentially the algebra of the Nambu bracket without demanding algebraic rules of multiplication among the elements. Hence we will refer to the Lie 3-algebra bracket also as the Nambu bracket.
A symmetric bilinear map $\langle \,\cdot\; | \,\cdot\; \rangle \in \mathbb{C}$ that maps two elements to a number is said to be an invariant metric if we have $$\begin{aligned}
&\langle F_1 | F_2 \rangle = \langle F_2 | F_1 \rangle,
\\
&\langle [F_1, F_2, F_3] | F_4 \rangle + \langle F_3 | [F_1, F_2, F_4] \rangle = 0,
\label{invariant-inner-product}\end{aligned}$$ for all elements $F_1, F_2, F_3, F_4$.
Unlike Lie algebra, it is not clear how to realize Lie 3-algebras in terms of matrices. Let ${\cal F}$ denote a Lie 3-algebra. Then the Lie 3-bracket defines a set of maps $G(F_1, F_2) \equiv [F_1, F_2, \,\cdot\;]$ as derivatives acting on ${\cal F}$ for every anti-symmetric pair of elements $F_1, F_2 \in {\cal F}$. Define ${\cal G}$ to be the set of such maps; it is obviously a Lie algebra, of which ${\cal F}$ is a representation. The fundamental identity (\[Fundamental-Identity\]) implies that the Lie bracket of ${\cal G}$ is given by [^4] $$[G(F_1, F_2), G(F_3, F_4)]
= G([F_1, F_2, F_3], F_4) + G(F_3, [F_1, F_2, F_4]).$$ Note that whenever there is a continuous symmetry, there is an associated Lie group and hence a Lie algebra. The appearance of ${\cal G}$ and its Lie bracket is always implied by the Lie 3-algebra. One can define gauge theories for a Lie 3-algebra ${\cal F}$ by identifying the Lie algebra ${\cal G}$ as the gauge symmetry. For a Lie 3-algebra ${\cal F}$ with generators $\{T^A\}$, the generators of the Lie algebra ${\cal G}$ are $\{ [ T^A, T^B, \,\cdot\; ] \}$. A matter field $\Phi = \Phi_A T^A$ taking values in ${\cal F}$ changes by $$\delta \Phi = \Lambda_{AB}[T^A, T^B, \Phi]$$ under a gauge transformation with the transformation parameters $\Lambda_{AB}$. Equivalently, $$(\delta \Phi)_A = \Lambda_{CD} f^{CDB}{}_A \Phi_B = \tilde{\Lambda}^B{}_A \Phi_B,$$ where $f^{CDB}{}_A$ is the Lie 3-algebra structure constant in the basis $\{T^A\}$, and $\tilde{\Lambda}$ is defined by $$\tilde{\Lambda}^B{}_A \equiv \Lambda_{CD} f^{CDB}{}_A.$$
The gauge potential $A_{\mu}$ takes its value in the Lie algebra ${\cal G}$: $$A_{\mu} = A_{\mu AB} [T^A, T^B, \,\cdot\; ].
\label{gauge-potential}$$ The covariant derivative $D_{\mu}$ on the base space with coordinates $\sigma^{\mu}$ is thus $$D_{\mu} \Phi = \frac{\partial}{\partial\sigma^{\mu}} \Phi + A_{\mu AB} [T^A, T^B, \Phi],$$ or equivalently $$(D_{\mu} \Phi)_A = \frac{\partial}{\partial\sigma^{\mu}} \Phi_A + \tilde{A}_{\mu}{}^{B}{}_{A} \Phi_B,$$ where $A_{\mu AB}$ is the gauge potential and $$\tilde{A}_{\mu}{}^B{}_A \equiv A_{\mu CD} f^{CDB}{}_A.$$ Notice that the structure constants may be such that a change in $A_{\mu AB}$ does not always lead to a change in $\tilde{A}_{\mu}{}^B{}_A$, but only the components $\tilde{A}_{\mu}{}^B{}_A$ are relevant in the covariant derivative.
We refer to Ref.[@Park:2008qe] for a related idea to use the Nambu bracket in matrix model and to Ref.[@Trzetrzelewski:2012re] where it was used to describe the matrix regularization of higher dimensional spheres.
BLG model
---------
The Lie 3-algebra turns out to be the appropriate symmetry structure for constructing a manifestly supersymmetric effective theory for multiple M2-branes – the Bagger-Lambert-Gustavsson (BLG) model [^5] [@Bagger:2006sk; @Bagger:2007jr; @Bagger:2007vi; @Gustavsson].
Let $x^{\mu}$ ($\mu = 0, 1, 2$) be the world-volume coordinates of M2-branes. In addition to the gauge potential $A_{\mu}$ (\[gauge-potential\]), the scalar fields $X^a(x) = X^a_A(x) T^A$ ($a = 3, \cdots, 10$) represent the transverse coordinates, and the 11D Majorana spinors $\Psi(x) = \Psi_A(x) T^A$ their super-partners, which should satisfy the chirality condition $\Gamma_{012}\Psi = - \Psi$. With $T_2 = 1/(2\pi \ell^3_p)$ denoting the M2-brane tension ($\ell_p$ is the M theory Planck length scale), the action for the BLG model is [@Bagger:2006sk; @Bagger:2007jr; @Bagger:2007vi] $$\begin{aligned}
S &=& T_2 \int d^3 x \left[
- \frac{1}{2} \langle D_{\mu}X^a | D^{\mu}X^a \rangle
- \frac{1}{12} \langle [X^a, X^b, X^c] | [X^a, X^b, X^c] \rangle
\right.
\nonumber \\
&&
+ \frac{i}{2} \langle \bar{\Psi} | \Gamma^{\mu}D_{\mu} \Psi \rangle
+ \frac{i}{4} \langle \bar{\Psi} | \Gamma_{ab} [X^a, X^b, \Psi] \rangle
\nonumber \\
&&
\left.
+ \epsilon^{\mu\nu\lambda} \left(
\frac{1}{2} f^{ABCD} A_{\mu AB} \partial_{\nu} A_{\lambda CD}
- \frac{1}{3} f^{ACD}{}_{G} g^{GH} f^{BEF}{}_{H} A_{\mu AB}A_{\nu CD}A_{\lambda EF}
\right)
\right],
\label{BLG-action}\end{aligned}$$ where the invariant metric $g^{AB}$ is needed to define the action.
In addition to the gauge symmetry characterized by a Lie 3-algebra, this action has the supersymmetry of 16 Grassmannian paramters. Its SUSY transformation laws are [@Bagger:2006sk; @Bagger:2007jr; @Bagger:2007vi] $$\begin{aligned}
\delta X^a_A &=& i\bar{\epsilon}\Gamma^a \Psi_A, \\
\delta \Psi_A &=& D_{\mu}X^a_A \Gamma^\mu\Gamma_a \epsilon
- \frac{1}{6} X^a_B X^b_C X^c_D f^{BCD}{}_A \Gamma^{abc}\epsilon, \\
\delta \tilde{A}_{\mu}{}^B{}_A &=&
i\bar{\epsilon}\Gamma_{\mu}\Gamma_a X^a_C \Psi_D f^{CDB}{}_A, \end{aligned}$$ where the SUSY transformation parameter $\epsilon$ is an 11D Majorana spinor satisfying the chirality condition $
\Gamma_{012}\epsilon = \epsilon
$.
A different choice of the Lie 3-algebra corresponds to a different background for the membranes. At the time of the proposal of the BLG model, there were few examples of the Lie 3-algebra. An example is the 4-generator algebra ${\cal A}_4$ [@Filippov1986; @Kasymov1987] defined by $$[T^A, T^B, T^C] = \epsilon_{ABCD} T^D,$$ where $A, B, C, D = 1, 2, 3, 4$, and the structure constant $\epsilon_{ABCD}$ is the totally anti-symmetric tensor. The invariant metric is positive-definite and can be normalized as $$\langle T^A | T^B \rangle = \delta_{AB}.$$ The algebra ${\cal A}_4$ is formally a natural generalization of the Lie algebra $su(2)$, and the corresponding BLG model describes two M2-branes on an M-fold [@Lambert:2008et; @Distler:2008mk]. More examples of Lie 3-algebras were discussed in [@HHM; @DeMedeiros:2008zm].
For a model to be physically interesting, we often demand that it is free of ghosts. Naively this seems to say that the Killing metric of the Lie 3-algebra should be positive definite, in order for the kinetic terms to have the correct sign for all fields. It turns out that, however, it is possible to define physically interesting theories for invariant metrics with the Lorentzian signature.
### BLG model for Lorentzian 3-algebra
[**D2-branes**]{}
It was found [@GMR; @Benvenuti:2008bt; @HIM] that there is a Lie 3-algebra associated with each Lie algebra, and the BLG model defined for this Lie 3-algebra is exactly the super Yang-Mills (SYM) action for D2-branes [@HIM]. The duality between M theory and type IIA superstring theory is respected by the BLG model in a novel way.
Let us describe the promotion of a Lie algebra to a Lie 3-algebra in terms of a basis of generators $\{ T^A \}_{A = 1}^{N}$ with the Lie bracket $$[T^A, T^B] = f^{AB}{}_C T^C,$$ and the Killing form $$\langle T^A | T^B \rangle = h^{AB}.$$ The associated Lie 3-algebra [@GMR; @Benvenuti:2008bt; @HIM] can be defined by the following Nambu brackets: $$\begin{aligned}
{}[T^A, T^B, T^C] &=& f^{ABC} v,
\label{3-alg-1}
\\
{}[u, T^A, T^B] &=& f^{AB}{}_C T^C,
\\
{}[v, T^A, T^B] &=& 0,
\\
{}[u, v, T^A] &=& 0,
\label{3-alg-4}\end{aligned}$$ where $f^{ABC} \equiv f^{AB}{}_D h^{DC}$, with two new generators $u$ and $v$. The generator $v$ is central, i.e., the Nambu bracket vanishes whenever it appears. The generator $u$ has the special feature that it never shows up on the right hand side of the Nambu bracket. A shift of $u$ by a constant times $v$ is hence an algebra homomorphism.
The Killing form $h^{AB}$ of the Lie algebra also induces an invariant metric for the Lie 3-algebra: $$\begin{aligned}
&\langle T^A | T^B \rangle = h^{AB},
\label{metric-TT}
\\
&\langle u | T^A \rangle = 0,
\qquad
\langle v | T^A \rangle = 0,
\\
&\langle u | u \rangle = 0,
\qquad
\langle v | v \rangle = 0,
\qquad
\langle u | v \rangle = 1.
\label{metric-uv}\end{aligned}$$ As a convention, we have normalized the metric so that $\langle u | v \rangle = 1$. This is not the unique invariant metric, as the requirement (\[invariant-inner-product\]) that the inner product be invariant allows $\langle u | u \rangle$ to be non-zero. However, the algebra homomorphism $$u \rightarrow u + \alpha v
\qquad (\alpha \in \mathbb{C})$$ allows us to set it to zero without loss of generality.
Due to eq.(\[metric-uv\]), the signature of the metric is Lorentzian even if the Killing form $h^{AB}$ is positive definite. As the kinetic terms of the BLG model are defined by the metric, one should worry about the presence of negative-norm states. The components $X_u^a, X_v^a, \Psi_u, \Psi_v$ of the matter fields $$X^a = X_A^a T^A + X_u^a u + X_v^a v
\qquad \mbox{and} \qquad
\Psi = \Psi_A T^A + \Psi_u u + \Psi_v v$$ are the degrees of freedom in danger of giving negative-norm states. Due to the special algebraic properties of the generators $u, v$ mentioned above, the components $X_v^a$ and $\Psi_v$ only appear as Lagrange multipliers. The constraints they impose are free field equations for $X_u^a$ and $\Psi_u$, although the latter also appear in the interaction terms. A different choice of the solution of the constraints leads to differences in the interactions, and one obtains a slightly different model from the BLG model. The idea of the “Higgs mechanism” of the BLG model [@Mukhi:2008ux], which was originally proposed for a different Lie 3-algebra ${\cal A}_4$, suggests one to consider the special cases when $X_u^a, \Psi_u$ as constants $$\begin{aligned}
\label{vev}
X_u^a = 2\pi R \delta^a_{10},
\qquad
\Psi_u = 0,\end{aligned}$$ which are solutions to the free field equations. We have labelled the direction of the constant vector $X_u^a$ as the tenth direction in space-time without loss of generality. It is remarkable that in this way the BLG model leads to exactly the super Yang-Mills theory for multiple D2-branes [@HIM] obtained from compactifying M2-branes on a circle in the tenth direction of radius $R$.
[**D$p$-branes**]{}
The Lie 3-algebra upon which the BLG model reduces to the effective action for D2-branes can be generalized such that the BLG Model becomes the super Yang-Mills action for D$p$-branes for any $p \geq 2$ [@Ho:2009nk].
In order to obtain the D$p$-brane action from the BLG model, we have to enlarge the base space from $2+1$ dimensions to $p+1$ dimensions. The additional $p-2$ coordinates $x_a$ ($a = 3, 4, \cdots, p$) can be introduced through $p-2$ indices $\vec{m} = (m_3, m_4, \cdots, m_p)$ on the generators $T^A$, now denoted as $T^{\vec{m}i}$, which can be viewed as the product of a Lie algebra generator $T^i$ with a function $e^{i\vec{m}\cdot\vec{x}}$ of the coordinates $\vec{x} = (x_3, x_4, \cdots, x_p)$, and $\vec{m}$ represents the wave vector. The Lie bracket for $T^{\vec{m}i}$ should therefore be defined by $$[T^{\vec{m}i}, T^{\vec{n}j}] = f^{ij}{}_k T^{(\vec{m} + \vec{n})k}.
\label{TT=fT}$$ In terms of this kind of Lie algebra, in which the base-space dependence of the gauge group is incorporated explicitly in the Lie algebra, one can express a $q'+1$ dimensional SYM theory as a $q+1$ dimensional SYM theory for any $q' > q$. If the base space is a noncommutative space due to a constant $B$-field background [@Chu:1998qz; @Schomerus:1999ug; @Seiberg:1999vs], the Lie algebra has to be a matrix algebra (e.g. $U(N)$), and the bracket above (\[TT=fT\]) should be changed to [^6] $$[T^{\vec{m}i}, T^{\vec{n}j}] = f^{ij}{}_k \cos\left(\frac{1}{2}\theta^{ab}m_a n_b\right)T^{(\vec{m} + \vec{n})k}
+ id^{ij}{}_k \sin\left(\frac{1}{2}\theta^{ab}m_a n_b\right)T^{(\vec{m} + \vec{n})k},$$ where $d_{ij}{}^k$ is defined by the anti-commutator of the Lie algebra generators $\{ T^i, T^j \} = d^{ij}{}_k T^k$.
The Lie algebra (\[TT=fT\]) can be further extended by introducing generators $u_a$ corresponding to the derivatives of the coordinates $x_a$. The Lie bracket is given by $$\begin{aligned}
{}[u_a, u_b] &=& C_{ab} T^{\vec{0}0}, \\
{}[u_a, T^{\vec{m}i}] &=& m_a T^{\vec{m}i} - C_{ab} \delta^{\vec{m}}_{\vec{0}} \delta^i_0 v^b,
\\
{}[T^{\vec{m}i}, T^{\vec{n}j}] &=& m_a h^{ij} \delta^{\vec{m}+\vec{n}}_{\vec{0}} v^a + f^{ij}{}_k T^{(\vec{m}+\vec{n})k},
\\
{}[v^a, T^{\vec{m}i}] &=& 0,
\\
{}[u_a, v^b] &=& 0,\end{aligned}$$ with constant parameters $C_{ab}$. In the above, we have used the label $0$ for the identity matrix $T^0 = I$. (For Lie algebras in which there is no corresponding element, one can set it to zero in the equations above.) The Killing form is defined as $$\begin{aligned}
\langle u_a | v^b \rangle &=& \delta_a^b, \\
\langle T^{\vec{m}i} | T^{\vec{n}j} \rangle &=& h^{ij} \delta^{\vec{m} + \vec{n}}_{\vec{0}},\end{aligned}$$ with all other inner products vanishing. This is a higher loop generalization of current algebra. As far as we know, it has never been examined in the literature and is worth to be studied in more detail in the future.[^7]
The Lie algebra with generators $\{T^{\vec{m}i}, u_a, v^a\}$ can be promoted to a Lie 3-algebra in the way described above in eqs.(\[3-alg-1\])–(\[3-alg-4\]) by adjoining two more generators $u, v$. The invariant metric can be given by (\[metric-TT\])–(\[metric-uv\]), too. The BLG model with this Lie 3-algebra is then equivalent to the super Yang-Mills theory in $p+1$ dimensions [@Ho:2009nk]. The constant parameters $C_{ab}$ specify constant gauge field backgrounds.
M5 from M2
----------
D$p$-branes in $B$-field background can be constructed out of infinitely many D$(p-2)$-branes [@Banks:1996nn] (which in turn can be constructed out of lower dimensional branes in the same fashion). This is achieved mathematically by setting the background values of two infinite-dimensional matrix coordinates $X_{p-1}, X_{p}$ of the D$(p-2)$-branes to satisfy the commutation relation $[X_{p-1}, X_p] = c I$, where $I$ is the identity matrix and $c$ is a constant corresponding to the gauge field background. Similarly, an M5-brane in $C$-field background can be decomposed into infinitely M2-branes [@HM; @HIMS]. This is achieved by using the Nambu algebra as the Lie 3-algebra in the BLG model [@HIM]. Although this correspondence between M2-branes and M5-brane is expected, mathematically it is remarkable that it can be realized explicitly for the BLG model.
In terms of a complete basis of functions $\{\chi^A(y)\}$ on a 3-manifold ${\cal M}_3$, the Nambu bracket is $$\{\chi^A, \chi^B, \chi^C\} = \frac{1}{\rho} \epsilon^{\dot{\mu}\dot{\nu}\dot{\lambda}}
\frac{\partial \chi^A}{\partial y^{\dot{\mu}}}\frac{\partial \chi^B}{\partial y^{\dot{\nu}}}\frac{\partial \chi^C}{\partial y^{\dot{\lambda}}},$$ where $\rho$ defines the volume form $\rho dy^{\dot{1}} dy^{\dot{1}} dy^{\dot{3}}$. We shall consider the BLG model with this algebra as the symmetry algebra, and use the coordinates $y^{\dot{\mu}}$ with dotted indices for the internal space ${\cal M}_3$, to be distinguished from the M2-brane world-volume coordinates $x^{\mu}$ ($\mu = 0, 1, 2$).
Since the space of functions on ${\cal M}_3$ is infinite dimensional, the BLG model represents infinitely many M2-branes. If a field $\Phi$ (e.g. $X^a(x)$ and $\Psi(x)$) in the BLG model takes values in the Nambu algebra $$\Phi(x, y) = \Phi_A(x)\chi^A(y),$$ it can be interpreted as a field living on the M5-brane world-volume, which is the product of the 3-manifold ${\cal M}_3$ and the M2-brane world-volume.
Transformations defined by the Nambu bracket $$\delta \Phi(x, y) = \Lambda_{AB}(x) \{\chi^A(y), \chi^B(y), \Phi(x, y) \}
\label{covariant-transform}$$ is the same as a coordinate transformation in $y$, $$\delta \Phi = \delta y^{\dot{\mu}}(x) \partial_{\dot{\mu}} \Phi,$$ that preserves the 3-form $\rho d^3 y$. This 3-form $\rho d^3 y$ shall be interpreted as the $C$-field background in M theory. Recall that a $B$-field background turns the world-volume of a D-brane into a non-commutative space [@Chu:1998qz; @Schomerus:1999ug; @Seiberg:1999vs], and in the Poisson limit the gauge symmetry on the D-brane can be identified with the diffeomorphisms preserving the 2-form $B$-field background. Similarly, M5-branes in $C$-field background develops the gauge symmetry of diffeomorphisms preserving the 3-form $C$-field background.
The invariant metric can be identified with the integral $$\langle \chi^A | \chi^B \rangle = \int d^3 y \; \rho(y) \chi^A(y) \chi^B(y).$$ The action of the BLG model (\[BLG-action\]) is thus an integral over the M5-brane world-volume.
We will focus on the special case that ${\cal M}_3 = \mathbb{T}^3$, and choose $y$ to be the Cartesian coordinates. Then $\rho$ is just a constant, which can be scaled to $1$ without loss of generality.
The set of functions on 3-torus $\mathbb{T}^3$ is spanned by $\chi_{\vec n}(y)=\exp(2\pi i \vec n\cdot \vec y)$ ($\vec n\in \mathbb{Z}^3$) assuming all the radius are set to $1$ for simplicity. In addition to them, the linear functions $u^{\dot \mu}=y^{\dot \mu}$ may enter the Nambu bracket since the derivative gives the periodic function. They do not show up on the right hand side of the algebra. In this sense, they play the role similar to $u$ generator in (\[3-alg-1\]). We have to add three $v_{\dot\mu}$ generators to form a Lorentzian triple. As a whole, the three algebra of Nambu-Poisson bracket is spanned by $\chi_{\vec n}$ ($\vec n\in \mathbb{Z}^3$), $(u^{\dot\mu},v_{\dot\mu})$ and the explicit form of 3-algebra can be found in [@Ho:2009nk]. We note that a similar infinite dimensional Lie 3-algebra based on Nambu bracket was also considered in [@Curtright:2008jj; @Curtright:2009qf].
When we try to rewrite the BLG model in the form of a 6-dimensional field theory for the M5-brane, it is less obvious how to replace the gauge potential 1-form $A_{\mu}$ on the M2-brane world-volume by a 2-form gauge potential on the M5-brane. First, the potential $A_{\mu}(x)$ takes values in the tensor product of the Lie 3-algebra, so superficially it is a non-local field on the M5-brane world-volume: $$A_{\mu}(x, y, y') = A_{\mu AB}(x)\chi^A(y)\chi^B(y').$$ However, since the gauge potential appears in the BLG model only through the form $\tilde{A}_{\mu}{}^B{}_A \equiv A_{\mu CD} f^{CDB}{}_A$, the BLG model only depends on $A_{\mu}$ through the local field $$b_{\mu\dot{\mu}}(x, y) \equiv \left[\frac{\partial}{\partial y^{\prime\dot{\mu}}}A_{\mu}(x, y, y')\right]_{y' = y}.
\label{bdot}$$ Hence we have some of the components of the 2-form potential derived from $A_{\mu}$.
Next we consider the scalars $X^3, X^4, X^5$ representing the coordinates transverse to the M2-branes but parallel to the M5-brane. In order for the M5-brane to extend in these directions, we choose the background values $X^3 = y^{\dot{1}}/g, X^4 = y^{\dot{2}}/g, X^5 = y^{\dot{3}}/g$ for these scalars, where $g$ is an arbitrary constant factor of normalization. This is parallel to (\[vev\]). Hence a field is defined for each of the 3 scalars as the fluctuation field: $$X^{3} = \frac{y^{\dot{1}}}{g} + b^{\dot{1}}(x, y),
\qquad
X^{4} = \frac{y^{\dot{2}}}{g} + b^{\dot{2}}(x, y),
\qquad
X^{5} = \frac{y^{\dot{3}}}{g} + b^{\dot{3}}(x, y).$$ Then we can define another set of components for the M5-brane 2-form gauge potential $$b_{\dot\mu\dot\nu} \equiv \epsilon_{\dot\mu\dot\nu\dot\lambda} b^{\dot\lambda}.
\label{bdotdot}$$
So far we have $b_{\mu\dot{\mu}}$ and $b_{\dot\mu\dot\nu}$ of the M5-brane potential, while $b_{\mu\nu}$ is still missing. It turns out that, as the 3-form field strength is self-dual in the M5-brane theory, one can formulate the gauge theory in terms of only part of the components of the gauge potential [@HM; @HIMS]. A generalization of this formulation of self-dual gauge theories is available for self-dual theories in arbitrary dimensions [@Chen:2010jgb] (whenever the self-duality condition can be defined).
The covariant derivatives for this gauge symmetry can be defined as $$\begin{aligned}
{\cal D}_{\mu}\Phi &=& \partial_{\mu}\Phi - g\{ b_{\mu\dot\mu}, y^{\dot\mu}, \Phi \},
\\
{\cal D}_{\dot\mu}\Phi &=& \frac{g^2}{2} \epsilon_{\dot\mu\dot\nu\dot\lambda}
\{ X^{\dot\nu}, X^{\dot\lambda}, \Phi \}.\end{aligned}$$ They transform covariantly under gauge transformations if $\Phi$ transforms covariantly as (\[covariant-transform\]). It is interesting to see how the 2-form gauge potential appears in the covariant derivatives.
The field strength can be defined from the components (\[bdot\]) and (\[bdotdot\]) of the 2-form potential. In the free field limit (or weak field limit), they are expected to be given by $$\begin{aligned}
{\cal H}_{\mu\dot\mu\dot\nu} &\simeq&
\partial_{\mu}b_{\dot\mu\dot\nu} - \partial_{\dot\mu} b_{\mu\dot\nu} + \partial_{\dot\nu}b_{\mu\dot\mu}
+ \cdots,
\\
{\cal H}_{\dot{1}\dot{2}\dot{3}} &\simeq&
\partial_{\dot{1}}b_{\dot{2}\dot{3}} + \partial_{\dot{2}}b_{\dot{3}\dot{1}} + \partial_{\dot{3}}b_{\dot{1}\dot{2}}
+ \cdots.\end{aligned}$$ Furthermore, they should be covariant under gauge transformations (i.e., they transform like $\Phi$ in (\[covariant-transform\])). One can check that the field strength can be defined as $$\begin{aligned}
{\cal H}_{\mu\dot\mu\dot\nu} &\equiv&
\epsilon_{\dot\mu\dot\nu\dot\lambda} {\cal D}_{\mu}X^{\dot\lambda},
\\
{\cal H}_{\dot{1}\dot{2}\dot{3}} &\equiv&
g^2\{ X^3, X^4, X^5 \} - \frac{1}{g}.\end{aligned}$$ For self-dual gauge theories, the rest of the components of the field strength are redundant.
The action of the M5-brane in large $C$-field background derived from the BLG model this way is [@HIMS] $$S = S_{B} + S_{F} + S_{CS},$$ where the bosonic part is $$\begin{aligned}
S_B &=&
\int d^3 x d^3 y\; \Big[
- \frac{1}{2}({\cal D}_{\mu} X^a)^2 - \frac{1}{2}({\cal D}_{\dot\lambda} X^a)^2
- \frac{1}{4}{\cal H}_{\lambda\dot\mu\dot\nu}^2 - \frac{1}{12}{\cal H}_{\dot\lambda\dot\mu\dot\nu}^2
\nonumber\\
&&- \frac{g^4}{4}\{X^{\dot\mu}, X^a, X^b\}^2 - \frac{g^4}{12}\{X^a, X^b, X^c\}^2
- \frac{1}{2g^2}
\Big],
\nonumber\end{aligned}$$ the fermionic part is $$S_F = \int d^3 x d^3 y \; \Big[
\frac{i}{2} \bar\Psi\Gamma^{\mu}{\cal D}_{\mu}\Psi
+ \frac{i}{2} \bar\Psi\Gamma^{\dot\mu}{\cal D}_{\dot\mu}\Psi
+ \frac{ig^2}{2} \bar\Psi\Gamma_{\dot\mu a} \{X^{\dot\mu}, X^a, \Psi\}
- \frac{ig^2}{4} \bar\Psi\Gamma_{ab}\Gamma_{\dot{1}\dot{2}\dot{3}} \{X^a, X^b, \Psi\}
\Big]
\nonumber$$ and the Chern-Simons part is $$S_{CS} = \int d^3 x d^3 y \; \epsilon^{\mu\nu\lambda} \epsilon^{\dot\mu\dot\nu\dot\lambda}\Big[
- \frac{1}{2} \partial_{\dot\mu} b_{\mu\dot\nu} \partial_{\nu} b_{\lambda\dot\lambda}
+ \frac{g}{6} \partial_{\dot\mu} b_{\nu\dot\nu} \epsilon^{\dot\rho\dot\sigma\dot\tau}
\partial_{\dot\sigma} b_{\lambda\dot\rho} (\partial_{\dot\lambda} b_{\mu\dot\tau} - \partial_{\dot\tau} b_{\mu\dot\lambda})
\Big].
\nonumber$$ The fermion satisfies the chirality condition $$\Gamma^{123}\Gamma^{\dot{1}\dot{2}\dot{3}}\Psi = - \Psi.$$ The components $b_{\mu\nu}$ that are hidden in this formulation can be defined when solving the field equations of this action [@Pasti:2009xc].
Note that the resulting gauge theory is the first of its kind: higher-form self-dual gauge theories with non-Abelian gauge symmetry. The action has the correct global symmetry, including supersymmetry, for an M5-brane in a large $C$-field background. If we compactify this action on a circile in one of the $y$ directions, we obtain the D4-brane theory in a large $B$-field background [@Chen:2010br] — in the Poisson limit of the noncommutative gauge theory. On the other hand, if we compactify one of the $x$ directions, we obtain the D4-brane theory in a large 3-form RR-field background. Through T-dualities [@Ho:2011yr], one can derive effective theories of D$p$-branes in NS-NS $B$-field or RR-field background from these D4-brane theories [@Ho:2013paa].
[**D$p$-brane in R-R $(p-1)$-form field background**]{}
While D$p$-branes in NS-NS $B$-field background are well known to be non-commutative gauge theories, the effective theories for D$p$-branes in R-R $(p-1)$-form potential backgrounds were not known before. What we learned from the theory of an M5-brane in the $C$-field background is that, in addition to the usual $U(1)$ gauge symmetry for a D$p$-brane, the R-R background turns on an additional gauge symmetry [@Ho:2013paa], which is the symmetry of diffeomorphisms preserving the $(p-1)$-form background. (Although the R-R $(p-1)$-form is not the volume form of the D$p$-brane, we often refer to this symmetry as the volume preserving diffeomorphism.)
Under a coordinate transformation $\delta y^{\dot\mu} = \kappa^{\dot\mu}$, a scalar field $\Phi$ transforms as $$\delta \Phi = \kappa^{\dot\mu}\partial_{\dot\mu}\Phi,$$ and this transformation preserves the $(p-1)$-form $d^{p-1}y$ if $\kappa^{\dot\mu}$ is divergenceless: $$\partial_{\dot\mu}\kappa^{\dot\mu} = 0.$$ Here the $y^{\dot\mu}$’s represent coordinates along the directions of the R-R $(p-1)$-form, and we shall use $x^{\mu}$ ($\mu = 0, 1$) to denote the rest of the world-volume coordinates on the D$p$-brane.
To parametrize the transformations through unconstrained functional parameters, one can use a generalized Nambu bracket that has $(p-1)$ slots $$\{f_1, f_2, \cdots, f_{p-1}\} = \frac{\partial (f_1, f_2, \cdots, f_{p-1})}{\partial (y^{\dot{1}}, y^{\dot{2}}, \cdots, y^{\dot{p}-\dot{1}})}.$$ A covariant quantity $\Phi$ transforms like $$\delta \Phi = \sum_{\alpha} \{f^{(\alpha)}_1, f^{(\alpha)}_2, \cdots, f^{(\alpha)}_{p-2}, \Phi\}$$ under a gauge transformation. Identifying the right hand side with $\kappa^{\dot\mu}\partial_{\dot\mu}\Phi$ to determine $\kappa^{\dot\mu}$, one sees that the divergenceless condition of $\kappa^{\dot\mu}$ is automatically satisfied.
In the following we shall focus on the bosonic sector of the D$p$-brane theory in the R-R $(p-1)$-form background (the fermionic sector has not been worked out yet). In the effective theory for a D$p$-brane, the bosonic sector includes the scalars $X^a$ and an 1-form potential $a_{\hat{\mu}} = (a_{\mu}, a_{\dot\mu})$. (We shall use the hatted indices $\hat{\mu}$ to refer to both the dotted ($y^{\dot\mu}$) and undotted ($x^{\mu}$) indices.) These fields are originated from the boundary states of open strings ending on D$p$-brane [@Dai:1989ua]. In the large R-R $(p-2)$-form background, the D$(p-2)$-branes also plays an important role, so that by analogy (or through a series of S- and T-dualities), there is a $(p-2)$-form potential $b_{\dot\mu_1\cdots\dot\mu_{p-2}}$ associated with the boundary states of open D$(p-2)$-branes. This tensor field is related to the 1-form gauge potential through a duality condition that generalizes the self-duality condition on M5-branes, so that there is no new physical degrees of freedom on the D$p$-brane world-volume. They play the role of the gauge potential for the gauge symmetry of volume-preserving diffeomorphisms.
It is convenient to define scalar fields $X^{\dot\mu}$ by $$X^{\dot\mu} \equiv \frac{y^{\dot\mu}}{g} + b^{\dot\mu}
\qquad
\left(
b^{\dot\mu} \equiv \frac{1}{(p-2)!}\epsilon^{\dot\mu\dot\mu_1\cdots\dot\mu_{p-2}}b_{\dot\mu_1\cdots\dot\mu_{p-2}}
\right),$$ so that the gauge transformation property of the gauge field $b_{\dot\mu_1\cdots\dot\mu_{p-2}}$ is equivalent to the condition that $X^{\dot\mu}$ transform covariantly. While both $X^{\dot\mu}$ and $X^a$ transform covariantly, the 1-form potential transforms by $$\delta a_{\hat{\mu}} = \partial_{\hat{\mu}} \lambda
+ g(\kappa^{\dot{\nu}}\partial_{\dot{\nu}}a_{\hat{\mu}} + a_{\dot\nu}\partial_{\hat{\mu}}\kappa^{\dot\nu}),$$ where the first term is the usual $U(1)$ gauge transformation.
In terms of the following definitions $$\begin{aligned}
F_{\hat{\mu}\hat{\nu}} &\equiv& \partial_{\hat{\mu}}a_{\hat{\nu}} - \partial_{\hat{\nu}}a_{\hat{\mu}},
\\
V_{\dot\nu}{}^{\dot\mu} &\equiv& \delta_{\dot\nu}^{\dot\mu} + g \partial_{\dot\nu}b^{\dot\mu},
\\
M_{\dot\mu\dot\nu}{}^{\mu\nu} &\equiv&
V_{\dot\mu}{}^{\dot\lambda}V_{\dot\nu\dot\lambda}\delta^{\mu\nu}
- g\epsilon^{\mu\nu}F_{\dot\mu\dot\nu},
\\
\hat{B}_{\mu}{}^{\dot\mu} &\equiv&
(M^{-1})_{\mu\nu}{}^{\dot\mu\dot\nu}
(V_{\dot\nu}{}^{\dot\lambda}\partial^{\nu}b_{\dot\lambda} + \epsilon^{\nu\lambda}F_{\lambda\dot\nu}
+ g\partial_{\dot\nu}X^a \partial^{\nu}X^a),\end{aligned}$$ where $M^{-1}$ is defined by $M_{\dot\mu\dot\nu}{}^{\mu\nu}M^{-1}{}_{\nu\lambda}{}^{\dot\nu\dot\lambda} = \delta_{\dot\mu}^{\dot\lambda}\delta_{\lambda}^{\mu}$, the covariant derivatives of a covariant field $\Phi$ are defined as $$\begin{aligned}
{\cal D}_{\mu} \Phi &=& \partial_{\mu}\Phi - g\hat{B}_{\mu}{}^{\dot\mu}\partial_{\dot\mu}\Phi,
\\
{\cal D}_{\dot\mu} \Phi &=& \frac{(-1)^p}{(p-2)!}g^{p-2}
\epsilon_{\dot\mu\dot\mu_1\cdots\dot\mu_{p-2}}\{X^{\dot\mu_1}, \cdots, X^{\dot\mu_{p-2}}, \Phi\}.\end{aligned}$$
The usual definition of the Abelian field strength $F_{\hat{\mu}\hat{\nu}}$ is no longer suitable as it is not covariant. Proper definitions of the field strength for $a_{\hat{\mu}}$ are $$\begin{aligned}
{\cal F}_{\dot\mu\dot\nu} &=& \frac{g^{p-3}}{(p-3)!}
\epsilon_{\dot\mu\dot\nu\dot\mu_1\cdots\dot\mu_{p-3}}
\{X^{\dot\mu_1}, \cdots, X^{\dot\mu_{p-3}}, a_{\dot\nu}, y^{\dot\nu}\},
\\
{\cal F}_{\mu\dot\mu} &=& V^{-1}_{\dot\mu}{}^{\dot\nu}
(F_{\mu\dot\nu} + gF_{\dot\nu\dot\lambda} \hat{B}_{\mu}{}^{\dot\lambda}),
\\
{\cal F}_{\mu\nu} &=& F_{\mu\nu} + g[-F_{\mu\dot\mu}\hat{B}_{\nu}{}^{\dot\mu}
+ F_{\nu\dot\mu}\hat{B}_{\mu}{}^{\dot\mu} + gF_{\dot\mu\dot\nu}\hat{B}_{\mu}{}^{\dot\mu}\hat{B}_{\nu}{}^{\dot\nu}].\end{aligned}$$ On the other hand, the field strength for $b_{\dot\mu_1\cdots\dot\mu_{p-2}}$, $$\begin{aligned}
{\cal H}^{\dot\mu_1\cdots\dot\mu_{p-1}} &\equiv&
g^{p-2} \{ X^{\dot\mu_1}, \cdots, X^{\dot\mu_{p-1}} \}
- \frac{1}{g} \epsilon^{\dot\mu_1\cdots\dot\mu_{p-1}}\end{aligned}$$ is (up to a constant) just one of a class of covariant quantities defined by $$\begin{aligned}
{\cal O}^{\dot\mu_1\cdots\dot\mu_{\ell} a_1\cdots a_{p-1-\ell-2m}}_{(\ell m)}
&=&
\{ X^{\dot\mu_1}, \cdots, X^{\dot\mu_{\ell}}, a_{\dot\nu_1}, \cdots, a_{\dot\nu_m},
\frac{y^{\dot\nu_1}}{g}, \cdots, \frac{y^{\dot\nu_m}}{g}, X^{a_1}, \cdots, X^{a_{p-1-\ell-2m}} \},\end{aligned}$$ where $\ell$, $m$ are arbitrary non-negative integers such that $\ell + 2m \leq p-1$.
The bosonic part of the action is found to be given by $$\begin{aligned}
S_{Dp} &=& \int d^{p+1} x\; \Big[
- \frac{1}{2} ({\cal D}_{\mu}X^a)^2 + \frac{1}{2g} \epsilon^{\mu\nu}{\cal F}_{\mu\nu}
+ \frac{1}{2} {\cal F}_{\mu\dot\mu}^2
\nonumber \\
&& - \frac{g^{2(p-2)}}{2} \sum_{(\ell, m) \in S} \frac{1}{\ell ! (m!)^2 (q-\ell-2m)!} {\cal O}_{\ell m}^2
\Big],\end{aligned}$$ where $${\cal O}^2_{\ell m} \equiv \{ X^{\dot\mu_1}, \cdots, X^{\dot\mu_{\ell}}, a_{\dot\nu_1}, \cdots, a_{\dot\nu_m},
\frac{y^{\dot\nu_1}}{g}, \cdots, \frac{y^{\dot\nu_m}}{g}, X^{a_1}, \cdots, X^{a_{p-1-\ell-2m}} \}^2$$ and $$S = \{ (\ell, m) | \ell, m \geq 0; \ell + 2m \leq p-1 \}.$$ This result allows one to check explicitly the S-duality for D3-branes in the NS-NS and R-R field background [@Ho:2013opa]. Unlike the case of trivial background, where the S-duality is a quantum theory that cannot be verified directly by field redefinitions, the D3-brane in large NS-NS and R-R 2-form backgrounds can be explicitly verified.
Conclusion
==========
The Nambu bracket was first proposed as a generalization of the Poisson bracket for the canonical formulation of physical systems. In particular, the Nambu bracket and its generalizations found its natural applications to systems involving extended objects.
One may wonder whether the use of Nambu bracket is unavoidable, or how much advantage it can bring to us. On this aspect, we recall that in the canonical formulation, the Poisson bracket cannot be fixed without a complete gauge fixing when there is gauge symmetry. The definition of the Poisson bracket depends on the choice of gauge. On the other hand, it was shown [@Chu:2010eb] that, in certain examples, a Nambu bracket can be defined without gauge fixing, such that when a gauge-fixing condition $f=0$ is chosen, the Poisson bracket $\{\cdot, \cdot\}_f$ for that gauge is simply given by $$\{A, B\}_f = \{A, B, f\},$$ for any choice of gauge $f$. It is therefore a generalization of the canonical formulation that is gauge-independent. This trick can be extended to a generic constrained system [@curtright2003quantizing; @Horikoshi:2013yxa; @HM-future]. In general, a constrained system with $N$ constraints can be formulated with a generalized Nambu bracket with $N+2$ slots.
Like the Poisson bracket, the Nambu bracket and its generalizations also found their use in describing symmetries and interactions for various systems, including vortices and branes. The Nambu bracket is used in the description of a system of multiple M2-branes and a single M5-brane in $C$-field background. A $(p-1)$-bracket is used in the theory of a single D$p$-brane in the R-R $(p-1)$-form background.
The quantization of the Nambu bracket remains elusive. People have tried using matrices and even nonassociative algebras to define Nambu brackets, but it seems hard to satisfy the fundamental identity, at least not in the same fashion that the Jacobi identity is satisfied by the commutator of an associative algebra. The Zariski algebra provides a quantization of the Nambu algebra, but it is unclear how it can be applied in a physical theory as a small deformation of the classical Nambu algebra. For instance, the theory of a single M5-brane in $C$-field background involves the Nambu bracket. [^8] Upon double dimension reduction, it reduces to the Poisson limit of the noncommutative D4-brane. One would like to deform the Nambu-Poisson algebra in the M5-brane theory such that the double dimension reduction leads to the full noncommutative D4-brane. But there is a no-go theorem [@Chen:2010ny] against this possibility.
In the case of D-branes, a single D-brane in $B$-field background and a multiple D-brane system share the same algebraic structure of non-Abelian gauge symmetry characterized by the definition of commutators. This leads us to suspect that if one can quantize the Nambu-Poisson bracket, it would perhaps lead us to the mysterious non-Abelian gauge symmetry of multiple M5-branes. Over 40 years after Nambu’s introduction, reviewing the fruitful results inspired by the idea of the Nambu bracket, we believe that there are still much more remarkable results to come related to the Nambu bracket.
Acknowledgment {#acknowledgment .unnumbered}
==============
YM would like to thank the organizers of Nambu memorial symposium to provide an opportunity to give a talk on the Nambu bracket. He is obliged to Prof. Nambu for his hospitality during his stay at the University of Chicago as a postdoc fellow in 1989. The discussions and the conversations with Prof. Nambu have been invaluable experience for him. He is partially supported by Grants-in-Aid for Scientific Research (Kakenhi \#25400246) from MEXT, Japan. The work of PMH is supported in part by the Ministry of Science and Technology, R.O.C., and by National Taiwan University.
[^1]: It implies that the Nambu dynamical system has higher conserved quantities $H_1,\cdots, H_{N-1}$. In this sense, it has some connection with the integrable models. See, for example [@Curtright:2002sr], for a study in this direction.
[^2]: It looks like Nahm equation with $G_2$ holonomy if $H$ and $G$ are properly chosen. It may provide another link with M-theory. See for example, [@Cherkis:2014xua].
[^3]: The theory of a single M5-brane [@Pasti:1997gx; @Bandos:1997ui; @Aganagic:1997zq] was promoted to that of multiple M5-branes in [@Ho:2011ni] when they are compactified on a finite circle, yet only the Lie bracket is used.
[^4]: This expression is not manifestly antisymmetric in the exchange of $(F_1, F_2)$ with $(F_3, F_4)$, but the skew-symmetry is guaranteed by the fundamental identify (\[Fundamental-Identity\]). One can thus think of Lie 3-algebras as a special class of Lie algebras with additional internal structures.
[^5]: The use of the algebra with a 3-bracket is crucial for the full supersymmetry to be manifest. An effective theory defined with the usual Lie algebra is possible [@Aharony:2008ug], but only part of the supersymmetry is manifest.
[^6]: Eqs.(32) and (34) in Ref.[@Ho:2009mi] are incorrect.
[^7]: In mathematical literature, there is two-loop symmetry which is known as elliptic Hall algebra $\mathfrak{gl}(1)[x^{\pm 1}, y^{\pm 1}]$ [@burban2005hall] which is known to be equivalent to the quantum deformation of $W_{1+\infty}$ algebra, see for example, [@miki2007q; @feigin2009heisenberg; @feigin2016finite]. Since it is a quantum symmetry and not Lie algebra, it is different from the multi-loop algebra considered here. It describes the instanton partition functions in 5D super Yang-Mills [@BMZ; @BFMZZ; @Kimura-Pestun; @Mironov:2016yue] and the role played by the algebra seems to be similar.
[^8]: To emphasize that this Nambu bracket is classical, i.e., before quantization, the classical Nambu bracket is often referred to as the Nambu-Poisson bracket, and hence this M5-brane theory is referred to as the NP M5-brane.
|
---
abstract: 'The properties of three-jet events with total transverse energy greater than 320 GeV and individual jet energy greater than 20 GeV have been analyzed and compared to absolute predictions from a next-to-leading order (NLO) perturbative QCD calculation. These data, of integrated luminosity 86 pb$^{-1}$, were recorded by the CDF Experiment for $\pbarp$ collisions at $\sqrt{s}=1.8$ TeV. This study tests a model of higher order QCD processes that result in gluon emission and can be used to estimate the magnitude of the contribution of processes higher than NLO. The total cross section is measured to be $466\pm 3({\rm stat.})^{+207}_{-70}({\rm syst.})$ pb. The differential cross section is furthermore measured for all kinematically accessible regions of the Dalitz plane, including those for which the theoretical prediction is unreliable. While the measured cross section is consistent with the theoretical prediction in magnitude, the two differ somewhat in shape in the Dalitz plane.'
author:
- |
Sally Seidel\
Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131 USA\
\
[*For the CDF Collaboration*]{}
title: 'Comparison of Three-jet Events in $\pbarp$ Collisions at $\sqrt{s}=1.8$ TeV to Predictions from a Next-to-leading Order QCD Calculation'
---
In perturbative QCD, hard scattering of the constituent partons in the proton and antiproton results in events with large total transverse energy, $\sum{E_{\rm T}}$. Outgoing scattered partons hadronize and may be detected as hadronic jets. Three-jet events can be produced when a hard gluon is radiated from any of the initial, intermediate, or final state partons in an event with two primary outgoing partons.
We analyze here some properties of the cross section for three-jet event production in proton-antiproton collisions at the Fermilab Tevatron Collider at center-of-mass energy 1.8 TeV. The data, which were recorded by the Collider Detector at Fermilab (CDF) [@kn:Det], are compared with predictions by the first complete next-to-leading order (NLO) QCD generator, Trirad [@kn:Kilgore], for hadronic three-jet production at hadron colliders. We compare the measured and predicted absolute cross sections to test our understanding of the higher order QCD processes that result in gluon emission and to estimate the magnitude of the contribution of processes higher than NLO.[^1] In some kinematical regions, we provide a measurement of the cross section where the theoretical prediction is not reliable; this measurement may be a useful guide for theoretical calculations. We also compared the shapes of the measured and predicted cross sections when normalized, to examine the sensitivity of the cross section to variations in the value of the strong coupling, $\alpha_{\rm s}$. The data sample corresponds to an integrated luminosity of 86 pb$^{-1}$ collected during the 1994-1995 run (Run 1b).
A previous paper [@kn:Abe] examined a smaller dataset and was limited to a comparison with leading order theoretical calculations [@kn:Kunszt]. A subsequent analysis [@kn:Geer1] compared a larger dataset to predictions from the HERWIG [@kn:March] parton shower Monte Carlo program and to the NJETS [@kn:Berends] leading order $2 \rightarrow N$ parton-level prediction. The NLO calculation used here has the benefit of reduced renormalization scale dependence (and consequently lower systematic uncertainty) as well as a more reliable description of multijet production throughout phase space. This study expands upon the previous investigations by comparing the data to absolute cross section predictions. The measurements presented here include differential cross sections that may be useful constraints upon parton distribution functions.
We use a coordinate system with the $z$ axis along the proton beam, transverse coordinate perpendicular to the beam, azimuthal angle $\phi$, polar angle $\theta$, and pseudorapidity $\eta=-\ln \tan(\theta/2)$. The analysis uses the CDF calorimeters [@calor], which cover the pseudorapidity range $|\eta|<4.2$. The calorimeters are constructed in a tower geometry and are segmented in depth into electromagnetic and hadronic components. The calorimeter towers are 0.1 unit wide in $\eta$. The tower widths in $\phi$ are $15^\circ$ in the central region and $5^\circ$ for $|\eta|$ greater than approximately 1.2.
We begin by considering events from the data sample selected by the trigger requirement $\sum{E_{\rm T}} >$ 175 GeV. We refer to this 175 GeV as $E_{\rm tot}^{\rm thr}$ below. Event reconstruction uses a cone algorithm [@kn:Abe] described in more detail below. The transverse energy is defined as ${E_{\rm T}} \equiv {E} \; {\rm sin} \, \theta$, where $E$ is the scalar sum of energy deposited in the calorimeter within a particular cone and $\theta$ is the angle between the beam direction in the laboratory frame and the cone axis. All calorimeter energy clusters [@kn:Abe] in the event with $E_{\rm T} >$ 10 GeV are summed. The three leading jets in the laboratory frame are used as the basis of transformation into the three-jet rest frame. In the three-jet rest frame, the incoming partons are, by convention [@kn:UA1], labeled partons 1 and 2, and their momenta are designated $\vec p_1$ and $\vec p_2$, respectively. The highest energy jets in this frame have energies labeled $E_{\rm 3}$, $E_{\rm 4}$, and $E_{\rm 5}$ and are ordered such that ${E_{3}} > {E_{4}} > {E_{5}}$. The outgoing partons associated with these jets are correspondingly labeled partons 3, 4, and 5.
A three-jet system in the massless parton approximation can be uniquely described by five independent variables (see Figure 5 in [@kn:Geer2]). We use the following:
1. the invariant mass of the three-jet system, $m_{\rm 3J}$
2. the cosine of the angle $\theta_3^*$ between the average beam direction ($\vec {p}_{\rm AV} \equiv \vec{p}_1 - \vec{p}_2$) and parton 3 in the three-jet rest frame: $${\rm cos} \, \theta_{3}^{*} \equiv \frac{\vec {p}_{\rm AV} \cdot
\vec {p}_3}{ \left| \vec {p}_{\rm AV} \right|
\left| \vec {p}_3 \right|}$$
3. the cosine of the angle $\psi^*$ between the plane containing the average beam direction and parton 3 and the plane containing partons 3, 4, and 5 in their center of mass frame: $${\rm cos} \, \psi^{*} \equiv \frac{\left
( \vec {p}_3 \times \vec {p}_{\rm AV} \right) \cdot
\left( \vec {p}_4 \times \vec {p}_5 \right)}
{\left| \vec {p}_3 \times \vec {p}_{\rm AV} \right|
\left| \vec {p}_4 \times \vec {p}_5 \right|}$$
4. the Dalitz variable $X_3$ (see below) for the leading jet, and
5. the Dalitz variable $X_4$ (see below) for the next-to-leading jet.
The invariant $m_{\rm 3J}$ is calculated by sorting jets by their energies in the laboratory frame, boosting to the rest frame of those with the three highest energies, re-sorting jets by energy in that frame, then computing $m_{\rm 3J}=\sum_{i=3}^5 E_{\rm i}$, where the $E_{\rm i}$ are the energies of jets 3, 4, and 5 in the rest frame. We have investigated the probability that a jet with energy less than the weakest of the three jets in the laboratory frame may have an energy greater than $E_{5}$ in the 3-jet rest frame from which it is excluded by this algorithm. The restriction imposed by the cut on full trigger efficiency (see below) makes this probability negligible.
The Dalitz variables, $X_{\rm i}$, are defined as ${X}_{\rm i} \equiv {2 \cdot {E}_{\rm i}}/{m_{\rm 3J}}, (i=3,4,5)$. Momentum conservation restricts the ranges of the Dalitz variables to $$\begin{aligned}
\frac{2}{3}&\leq \; {X}_{3} \; \leq&1, \nonumber \\
\frac{1}{2}&\leq \; {X}_{4} \; \leq&1, \; {\rm and} \nonumber \\
0&\leq \; {X}_{5} \; \leq&\frac{2}{3}. \nonumber\end{aligned}$$
A set of trigger and offline requirements [@kn:Abe2] rejects events associated with cosmic rays, beam halo, and calorimeter malfunctions. Events are required to have a reconstructed primary vertex, defined as the vertex with the largest $\sum_{i} {P}_{\rm i}$ (where $P_{\rm i}$ is the total momentum of particle [*i*]{} leaving the vertex in the event), within $|{z}| <$ 60 cm. Events are defined to have resolved multiple interactions if a second vertex with at least ten associated tracks is reconstructed in the vertex track detector, and if that vertex is separated from the primary one by at least 10 cm. Because multiple interactions can change the jet multiplicity in an event, for example, misidentifying two-jet events as three-jet events, events with resolved multiple interactions are removed. The number of events with unresolved multiple interactions in which an additional jet could be misidentified is estimated to be less than 2% so no correction for them is applied. The resulting effective total integrated luminosity of the data sample is 77$\pm$4 pb$^{-1}$, where the uncertainty reflects both the overall luminosity uncertainty (4.2%) and the uncertainty (0.5%) associated with the removal of resolved multiple interactions.
An iterative cone algorithm [@kn:Abe] with cone radius ${R} \equiv \sqrt{(\Delta \eta)^{2} + (\Delta \phi)^{2}}=0.7$ is used to identify jets. Here $\Delta \eta = \eta_{2} - \eta_{1}$ and $\Delta \phi = \phi_{2} - \phi_{1}$. The subscripts 1 and 2 correspond to the axes of the cone and calorimeter tower, respectively.
Jets that share towers are combined if the total $E_{\rm T}$ of the shared towers is greater than 75% of the $E_{\rm T}$ of either jet; otherwise the towers are assigned to the nearest jet. Jet energies are corrected [@kn:Abe] for errors in the absolute and relative energy scales and for additional energy associated with the underlying event. Since partons that are radiated out of the cone lead to the same losses in the theoretical calculation and in the data, out-of-cone corrections are not applied. The $E_{\rm T}$ of a jet is calculated from the reconstructed position of the primary event vertex. All three leading jets are required to have $E_{\rm T}$ $>$ 20 GeV and $|\eta|$ $<$ 2.0. Events with fewer than three jets are rejected. To avoid collinear soft gluon instability in the iterative jet clustering algorithm [@kn:Giele], a cone overlap cut is imposed: events are rejected if the distance $\Delta R$ in $\eta$-$\phi$ space between the axes of any two of the three leading jets is less than 1.0 (see Figure 5 of [@kn:Abe], which shows that this selection requirement reduces to approximately zero the probability of the two jets being merged by the clustering algorithm). To exclude regions in which the geometrical acceptance [@kn:Geer2] is less than about 95%, we require $|{\rm cos}\, \theta_{3}^{*}| <
\sqrt{1 - \left( E_{\rm tot}^{\rm thr}/{\rm c}^2/{m_{\rm 2J}}
\right)^{2}}$, where $m_{\rm 2J}$ is the mass of the two leading jets in the three-jet system and is defined analogously to $m_{\rm 3J}$.
We require full trigger efficiency, which occurs when $\sum_{\rm 3~jets}{E_{\rm T}} >$ 320 GeV, where the sum is over the three highest energy jets in the event with corrected $E_{\rm T}$ $>$ 20 GeV [@kn:Abe3]. The data are compared to the theoretical prediction by sorting events into bins of size $0.02 \times 0.02$ in $X_{\rm 3}$-$X_{\rm 4}$ space, the Dalitz plane. Figure \[usec4\] shows the Dalitz distribution of data that remain after all of the selection requirements have been applied.
Before the final binning is done, the data are corrected for the effects of the combination of detector resolution and energy mismeasurement. A correction factor is determined for each bin in the plane as follows. A sample of events is generated at the parton level with the HERWIG Monte Carlo. The final state partons are hadronized. The events are then binned in the Dalitz plane. The same events are next passed through the CDF detector simulation and rebinned. For each bin the ratio of the number of events after and before detector simulation is computed. This ratio (ranging from 0.85 to 1.5) is the factor subsequently used to correct the number of events in each data bin. The data are also corrected for the $z$-vertex cut efficiency and then normalized to the effective total luminosity.
The principal sources of systematic uncertainty [@kn:syst] on the cross section are those on the absolute and relative ($\eta$-dependent) jet energy scales. The uncertainty on absolute jet energy derives from the resolution on the calibration of the calorimeter (uncertainty 1.3%-1.8%, and $E_{\rm T}$-dependent), the uncertainty associated with choice of jet fragmentation model (decreasing from 1.7% to 1.2% with increasing $E_{\rm T}$), the uncertainty associated with calorimeter stability over time (1%), and the uncertainty on the correction for the contribution of the underlying event (1 GeV). The uncertainty on the relative jet energy scale ranges from 2% to 6%. Uncertainties are also associated with the measurement of the effective total integrated luminosity (4.2%) and with the $z$-vertex cut efficiency (2%). There is also an uncertainty of less than 5% associated with the implementation of simulated events in the correction procedure. The upper (lower) limits on these uncertainties are added (subtracted) from the four-momenta of the jets in the data sample to obtain the systematic uncertainties on the cross section associated with each contribution. The uncertainties are then combined to produce the total experimental systematic uncertainty.
The Trirad calculation consists of $2 \rightarrow 3$ parton processes at one loop and $2 \rightarrow 4$ parton processes at tree level. For gluons $g$, incoming quarks $q$, and outgoing quarks $Q$ or $Q^\prime$, the subprocesses involved are $gg \rightarrow ggg$, ${\overline q}q \rightarrow
ggg$, ${\overline q}q \rightarrow {\overline Q}Qg$, and those related by crossing symmetry, all computed to one loop; and $gg \rightarrow gggg$, ${\overline q}q \rightarrow gggg$, ${\overline q}q \rightarrow
{\overline Q}Qgg$, and ${\overline q}q \rightarrow
{\overline Q} Q {\overline Q^\prime} Q^\prime$ and the crossed processes computed at tree level. The program uses the “subtraction improved” [@kn:Giele] phase space slicing method to implement infrared cancellation.
The cross section is predicted with the CTEQ4M [@cteq4] parton distribution function (PDF) for each bin in the Dalitz plane. The result is multiplied by the effective total integrated luminosity of the data to predict a number of events in each bin. We restrict the prediction to bins for which $X_3<0.98$; this is necessary as the perturbative expansion is not reliable where the three-jet configuration approaches a two-jet configuration. The comparison between the data and the calculation is made for 215 bins.
Figure \[shape\] compares Dalitz distributions of the data and the absolute theoretical prediction. The theoretical distribution is more strongly peaked—a trend that persists in comparisons with all members of the CTEQ4 family[^2] of PDFs. This trend, in which the edges of the Dalitz plane are more populated by data than by the prediction, may give some indication of the size of the higher order contributions to the cross section.
The data and theory are compared in two different ways. In Figure \[trendc4\], we compare the shapes of their Dalitz distributions by normalizing the data and theory predictions to the same number of events. In Figure \[frac4c7\], we normalize theory to the experimental luminosity and compare the absolute values of the cross sections that are observed and predicted. In both figures, the prediction is made using the CTEQ4M parton distribution function, and the difference between observed and predicted number of events, scaled by the number of predicted events, is computed.
The theoretical prediction for the cross section, using CTEQ4M and all bins in the Dalitz plane but those with $X_3>0.98$, is $473\pm 2({\rm stat.})^{+38}_{-66}({\rm scale})^{+21}_{-28}({\rm PDF})$ pb. The theoretical uncertainty associated with choice of renormalization and factorization scales, $\mu_{\rm R}$ and $\mu_{\rm F}$ respectively, is estimated by varying the scales, whose default value is $E_{\rm T}$, to values of $E_{\rm T}/2$ and $2E_{\rm T}$ while keeping $\mu_{\rm R}=\mu_{\rm F}$. The theoretical uncertainty associated with choice of PDF is estimated from the spread in the predictions generated with all members of the CTEQ4 family. The measurement is not sensitive to the value of $\alpha_{\rm s}$ as is also shown in [@kn:CTEQ6]. The measured cross section, using all bins in the Dalitz plane but those with $X_3>0.98$, is 458$\pm$3(stat.)$^{+203}_{-68}$(syst.) pb. This is consistent with the theoretical prediction and with a previous CDF measurement [@kn:Abe3] after corrections are made for the efficiencies of additional cuts introduced in this analysis. The measured cross section, using all bins in the Dalitz plane, is 466$\pm$3(stat.)$^{+207}_{-70}$(syst.) pb.
The measurements at high $X_3$ may provide useful constraints on future theoretical models in that region. It appears that up to NLO the theory predicts more soft radiation than the data have in the region where the primary partons are approximately back-to-back. The data, especially in the region above $X_3=0.98$ where a perturbative expansion is not reliable, may be useful input to theoretical models of gluon-emission processes.
We thank William Kilgore and Walter Giele for providing us with the code that computes the next-to-leading order calculation, and for their guidance concerning its use. We acknowledge the Center for High Performance Computing at the University of New Mexico and the University of Wisconsin Condor Project for providing a combined 26,000 CPU-hours for the Trirad NLO computations. We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Science, Sports and Culture of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Foundation; the A. P. Sloan Foundation; the Bundesministerium fuer Bildung und Forschung, Germany; and the Korea Science and Engineering Foundation.
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[^1]: While no quantitative estimate of the contribution of next-to-next-to-leading order processes to the cross section is available at this time, considerable progress has recently been made in calculating two loop $2 \rightarrow 2$ parton processes [@kn:glover], important groundwork for the future.
[^2]: The CTEQ4 family includes CTEQ4A1, CTEQ4A2, CTEQ4M, CTEQ4A5, and CTEQ4A6, which differ in the value of $\alpha_{\rm s}$ input to their global fit, and CTEQ4HJ, for which a higher statistical emphasis was given to the high $E_{\rm T}$ data from CDF.
|
---
abstract: 'Until now, double mode Cepheids (or beat Cepheids) were known in the Galaxy, Magellanic Clouds and M33. Curiously, none of more than 2000 Cepheids in M31 was claimed to show two pulsation modes. We conducted a systematic search for double mode Cepheids in the archival data and discovered four such objects. We identify one of the stars as a first and second overtone pulsator even though its secondary period is subject to strong aliasing. Two stars turn out to pulsate in the fundamental mode and the first overtone. Their fundamental periods are $9.392~{\rm d}$ and $9.163~{\rm d}$. This makes them the first candidates for the fundamental mode and the first overtone Cepheids, which double mode pulsations are caused by the 1:2 resonance of the fundamental mode and the second overtone.'
author:
- Radosław Poleski
title: 'Double-mode Cepheids in M31'
---
Introduction
============
Among many different types of pulsating stars, probably the most important group are classical (or type I) Cepheid variables. They obey a power-law period-luminosity relation, which allows measurement of distances to external galaxies. However, the details of this relation are subject of an active debate. The slope of the relation changes around a period of $\approx 10~{\rm d}$ at some wavelengths [e.g. @ngeow09; @tammann11]. Also the zero point of the relation is affected by metallicity. Different star formation history in different galaxies result in different histograms of Cepheid periods, which impacts the distance estimates if the period-luminosity relation is not a pure power-law. The other effects that affect the observed brightness of Cepheids are: extinction, infrared emission of circumstellar dust and blending. The above problems can not be tackled without a deep understanding of the Cepheid internal structure. Only recently has the discrepancy of Cepheid masses derived from the stellar evolution and pulsation theories been solved [@pietrzynski10].
Among the Cepheid variables an important group constitute multimode radial pulsators. The identification of two modes with the expected period ratio clearly shows that the object is a classical Cepheid, not a different type of the variable star. It also yields an additional constraint on stellar parameters as the two modes probe different parts of the star. In this context even more important are triple mode pulsators [@moskalik05] of which we know a dozen or so.
The highest number of Cepheids is known in the Magellanic Clouds. These galaxies contain altogether more than 8000 Cepheids [@soszynski08; @marquette09; @soszynski10cepsmc; @soszynski12]. Among them $6\%$ show at least two radial modes in the Small Magellanic Cloud. For the more metal-rich Large Magellanic Cloud the corresponding value is $10\%$. Except these, the double mode Cepheids are known only in the Milky Way and M33 [@beaulieu06b]. There are also more than 2000 Cepheids known in Andromeda galaxy [M31; @fliri12; @kodric13] and until now none of them has been claimed as a double mode pulsator. If the same single mode to double mode number ratio of Cepheids is in the M31 as in Magellanic Clouds one would expect more than a hundred double mode Cepheids to be detected by a deep enough survey. The number should be smaller because of more severe blending, which lowers the observed amplitudes of Cepheids and affects most the smaller amplitude mode. Even accounting for this, one would expect at least a few longest period (i.e., brightest) double mode Cepheids to be known in M31. We note that three double mode RR Lyr variables were found in the Hubble Space Telescope photometry by @reyner10 and they are fainter than Cepheids.
Here we present a search for double mode Cepheids in archival photometry for known variable stars in M31. The next section briefly presents data used in the analysis. We present the search method and its results in Sections 3 and 4. Finally, we discuss our findings. During our research, we found missclassifications for a few Cepheids, which are presented in the Appendix.
Data and method description
===========================
Our analysis is based on two sets of publicly available lightcurves for the Cepheids in M31. In their search for Cepheids in M31 @vilardell07 used data collected between 1999 and 2003 for stars in NE part of this galaxy. One field of $0.3~{\rm deg^2}$ was observed with 2.5 m Isaac Newton Telescope. The pixel scale was $0.33''$ and median seeing was $1.3''$ in the B band and $1.2''$ in the V band. The analysis, based on 265 epochs in B band and 259 in V band led to a discovery of 416 Cepheids. For more details we refer to @vilardell06 [@vilardell07].
The second set of lightcurves was presented by @kodric13. They analyzed data collected by $1.8$ m Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) with $1.4\times10^9~{\rm pixels}$ camera. The pixel scale is $0.258''$ and a total field of view is $7~{\rm deg^2}$ i.e., it allows covering the whole of M31 in a single image. The 183 epochs in $r_{\rm P1}$ and $i_{\rm P1}$ bands [@tonry12] collected in 2010 and 2011 were used to search for Cepheids. The median seeing of the 30 best seeing images was $0.86''$. Altogether 2009 Cepheids were found including type II objects and variables that could not be definitely assigned to type I or type II. The more detail description of the observation and their analysis was presented by @lee12 and @kodric13. Both @vilardell07 and @kodric13 used difference image analysis to preform photometric measurements.
The lightcurves of Cepheids were prewhitened with the period given in the original papers as well as periods independently found by us. The two periods differed only in questionable cases. The prewhitened lightcurves were searched for periods using both the discrete Fourier transform and multiharmonic analysis of variance [@schwarzenbergczerny96]. The final period estimates were taken using the latter method results. All the secondary periods found in a wide range around known structures in the Petersen diagram (period ratio vs. longer period) were visually verified. During our analysis we found a few stars that were incorrectly classified. We present these in the Appendix in order to help purify the M31 Cepheid sample used in future studies.
Results
=======
Below we present each double mode pulsator separately. Comparison to the Cepheids found in other environments and discussion of pulsation properties will be presented in next section. We denote fundamental mode and first overtone pulsators by F/1O and first and second mode pulsators by 1O/2O. The star identifiers come from @vilardell07 and @kodric13. Periods and Fourier parameters [phase differences $\phi_{21}=\phi_2-2\phi_1$ and amplitude ratios $R_{21}=A_2/A_1$; @simon81] are presented in Table \[tab:basic\] for each mode separately.
[lrrrrrr]{} J00450019+4129313 & $1.694919(16)$ & $4.13$ & $0.17$ & $1.361279(27)$ & $2.74$ & $0.12$\
PSO\_J010.6063+40.8608 & $9.3918(85)$ & $4.99$ & $0.09$ & $6.5551(49)$ & $4.11$ & $0.18$\
PSO\_J010.9364+41.2504 & $9.1633(80)$ & $4.79$ & $0.17$ & $6.3618(61)$ & $5.32$ & $0.07$\
PSO\_J011.3583+42.0404 & $10.4672(74)$ & $3.89$ & $0.10$ & $6.1610(52)$ & $4.41$ & $0.49$\
J00450019+4129313 – 1O/2O type pulsator
---------------------------------------
The photometry presented by @vilardell07 reveals the primary period of this star to be $1.694919(16)~{\rm d}$. We found a strong signal for a secondary period but its value cannot be found unambiguously because of the strong aliases in the power spectrum. The prewhitening of the light curve is illustrated in Figure \[fig:cep257\]. The four possible values are (starting from the most probable): $1.361279(27)$, $1.356241(44)$, $1.366328(47)$, or $1.351260(51)$. They give period ratios of $0.8032$, $0.8002$, $0.8061$, and $0.7972$, respectively. All of these values fall in the range typical for 1O/2O pulsators.
PSO\_J010.6063+40.8608 – F/1O type pulsator
-------------------------------------------
@an04 found the period of this star to be $9.42~{\rm d}$. The photometry presented by [@kodric13] results in a primary period of $9.3918(85)~{\rm d}$ and a secondary period of $6.5551(49)~{\rm d}$. The prewhitening of the light curve is presented in Figure \[fig:cep1300\]. The period ratio of $0.698$ is slightly lower than the period ratio for known F/1O pulsators but all of them have shorter periods. It lies on the extension of F/1O sequence on the Petersen diagram. Thus we classify it as F/1O Cepheid.
PSO\_J010.9364+41.2504 – F/1O type pulsator
-------------------------------------------
This object was already announced by @kaluzny99 and @joshi03 with periods of $9.173~{\rm d}$ and $9.160~{\rm d}$. Using @kodric13 photometry we found periods of $9.1633(80)~{\rm d}$ and $6.3618(61)~{\rm d}$ (Figure \[fig:cep1285\]). This object turns out to have periods similar to PSO\_J010.6063+40.8608 discussed above, with a slightly smaller period ratio of $0.694$. We conclude that this is a F/1O pulsator.
PSO\_J011.3583+42.0404 – candidate non-radial pulsator
------------------------------------------------------
The analysis of photometry presented by @kodric13 revealed two periods in this object: $10.4672(74)~{\rm d}$ and $6.1610(52)~{\rm d}$. The prewhitening shown in Figure \[fig:cep1364\] reveals a sound detection of secondary period. The period ratio of $0.589$ is not typical for any known combination of radial modes. This object can be either a non-radial pulsator or a blend of two stars. Non-radial modes are observed in other Cepheids [@moskalik09]. We note that @dziembowski12 tried to reproduce $\approx0.6$ period ratio observed in some of the LMC and SMC first overtone Cepheids. Their findings are not applicable to PSO\_J011.3583+42.0404 because neither analyzed sample nor the models extended to primary periods long enough.
Discussion
==========
We show the Petersen diagram for known classical double mode F/1O and 1O/2O Cepheids in Figure \[fig:peter\]. Separate symbols are used to present objects from the Milky Way [@soszynski11; @smolec10 and reference therein], LMC [@soszynski08; @marquette09; @soszynski12], SMC [@soszynski10cepsmc], M33 [@beaulieu06b] and M31. There are four different positions shown for J00450019+4129313, which correspond to the different aliases of the secondary period. All are consistent with 1O/2O pulsator. One can see that positions of PSO\_J010.6063+40.8608 and PSO\_J010.9364+41.2504 are consistent with extrapolation of the relation seen for F/1O pulsators. The fact that fundamental mode periods of these objects are close to $10~{\rm d}$ makes them unusual. The lightcurves of the fundamental mode Cepheids with similar periods show two maxima in each period. The appearance of the second maximum is caused by a 1:2 resonance of the second overtone and the fundamental mode. We note that the calculations of @buchler09, which were presented only in conference proceedings, suggest that F/1O pulsations with fundamental mode periods of around $10~{\rm d}$ are caused by the resonance mentioned above. The range of luminosities and effective temperatures in which this mechanism operates is very small. This should allow very detail modeling of these stars. Also the preliminary models for these stars indicate $P_{\rm F}/P_{\rm 2O}$ close to 2 and masses from 6 to $7~{\rm M_{\odot}}$ (Dziembowski & Smolec, in preparation). Based on the Petersen diagram presented in Figure \[fig:peter\] we suggest that the longest period F/1O Cepheids in LMC (OGLE-LMC-CEP-1082, $P_{\rm F}=7.86434~{\rm d}$ and $P_{\rm 1O}=5.56518~{\rm d}$) and SMC (OGLE-SMC-CEP-1497, $P_{\rm F}=4.9780~{\rm d}$ and $P_{\rm 1O}=3.588329~{\rm d}$) may also display double mode pulsations because of the 1:2 resonance of 2O and F modes. Both these objects are separated from the rest of F/1O Cepheids in a given galaxy by at least $2.3~{\rm d}$ in $P_{\rm F}$.
The double-mode Cepheids with periods as long as presented here might have been overlooked in previous analyses of the existing time-series photometry for variable stars in other galaxies. The double mode pulsators presented here fall outside not only the previously published Petersen diagrams constructed for known pulsators but also outside the period range studied in many theoretical investigations.
Finally, we would like to comment on a few different data access policies that we have seen during the literature search for data on the analyzed stars. One paper discussed the results of the search for M31 Cepheids and scientific results found using them but did not provide any information about the stars themselves, not even their coordinates. The next one presented the coordinates and basic properties of the variable stars but did not presented any time-resolved photometry. In other example the authors presented the list of variables and claimed that the timeseries photometry is made public but in fact it was not accessible. In one more example we have found that the photometry is published but the Julian Dates are rounded to integer values, which makes such photometry useless. There were only two examples in which the authors made photometry public in a proper form to allow scientific research. We call on authors to present both the properties of the variable stars found (at least position, brightness, and period) as well as their photometric timeseries for the astronomical community. With growing observing capabilities, observers typically are not able to extract the whole scientific information contained in their data. Allowing others to conduct more detail investigation is a good practice.
The author is grateful Wojciech Dziembowski and Radosław Smolec for fruitful discussion as well as Andy Gould for reading the manuscript. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France.
Misclassified objects
=====================
#### J00451769+4136367
@vilardell07 gives period of $7.862587~{\rm d}$. The true period is two times shorter.
#### J00452829+4139547
Photometry presented by @vilardell07 indicates that this is an artifact produced by the difference image analysis. The variations of measured flux are caused by the nearby brighter Cepheid J00452353+4140040. The periods of both objects agree.
#### PSO\_J009.9994+40.5558
Our analysis of photometry presented by @kodric13 revealed that this object varies with periods of $13.084(14)~{\rm d}$ and $9.5589(80)~{\rm d}$. The period ratio of $0.7306$ is similar to F/1O pulsators. However, this object cannot be classified as a double mode pulsator. As noted by @baade65 there are two Cepheids of similar brightness and separated by around $1.8''$. @baade65 identified them by numbers of 252 and 253. Their periods agree with quoted above. We conclude that @kodric13 presented photometry of two blended stars.
#### PSO\_J010.5263+40.7724
Classification of this object by @kodric13 as a Cepheid seems questionable as its light curve does not resemble typical Cepheids. The $i_{\rm P1}$-band photometry does not show clear periodic variability.
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|
---
abstract: 'Isospin-mixing corrections for superallowed Fermi transitions in [*fp*]{}-shell nuclei are computed within the framework of the shell model. The study includes three nuclei that are part of the set of nine accurately measured transitions as well as five cases that are expected to be measured in the future at radioactive-beam facilities. We also include some new calculations for $^{10}$C. With the isospin-mixing corrections applied to the nine accurately measured $ft$ values, the conserved-vector-current hypothesis and the unitarity condition of the Cabbibo-Kobayashi-Maskawa (CKM) matrix are tested.'
address:
- |
Physics Department, 401 Nielsen Hall, University of Tennessee,\
Knoxville, TN 37996-1200\
and\
Physics Division, Oak Ridge National Laboratory, P.O. Box 2008,\
MS-6373 Building 6003, Oak Ridge, TN 37831-6373
- |
National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy\
Michigan State University, East Lansing, MI 48824-1321
author:
- 'W. E. Ormand'
- 'B. A. Brown'
title: 'Isospin-mixing corrections for [*fp*]{}-shell Fermi transitions'
---
Superallowed Fermi $\beta$ transitions in nuclei, $(J^\pi=0^+,T=1)\rightarrow (J^\pi=0^+,T=1)$, provide an excellent laboratory for precise tests of the properties of the electroweak interaction, and have been the subject of intense study for several decades (cf. Refs. [@ref1; @Tow77; @Orm89; @Har90; @Sir78]). According to the conserved-vector-current (CVC) hypothesis, the $ft$ values for pure Fermi transitions should be nucleus independent, and given by $$ft=\frac{K}{G_V^2|M_F|^2},$$ where $K/(\hbar c)^6=2\pi^3\ln 2 \hbar /(m_ec^2)^5=
8.120270(12)\times 10^{-7}~{\rm GeV}^{-4}{\rm s}$, $G_V$ is the vector coupling constant for nuclear $\beta$ decay, and $M_F$ is the Fermi matrix element, $M_F=\langle\psi_f\mid T_\pm \mid\psi_i\rangle$. By comparing the decay rates for muon and nuclear Fermi $\beta$ decay, the Cabbibo-Kobayashi-Maskawa (CKM) mixing matrix element [@CKM] between $u$ and $d$ quarks ($v_{ud}$) can be determined and a precise test of the unitarity condition of the CKM matrix under the assumption of the three-generation standard model is possible [@Sir78; @CKM].
For tests of the standard model, two nucleus-dependent corrections must be applied to experimental $ft$ values. The first is a series of radiative corrections to the statistical rate function $f$, embodied in the factors $\delta_R$ and $\Delta_R$, giving $f_R=f(1+\delta_R+\Delta_R)$ [@Bli73; @Sir86; @Jau90; @Bar92; @Tow92; @foota]. Where $\delta_R$ is due to standard, electromagnetic (“inner”) radiative corrections (cf. p. 45 in Ref. [@Bli73]), while $\Delta_R$ is what has been referred to as the “outer” radiative correction (cf. p. 47 of Ref. [@Bli73]) and includes axial-vector interference terms [@Jau90; @Bar92; @Tow92]. The second correction is applied to the Fermi matrix element $M_F$, and is due to the presence of isospin-nonconserving (INC) forces in nuclei, and is denoted by $\delta_C$ [@Tow77; @Orm89; @Tow89]; namely $\mid M_F\mid^2=\mid M_{F0}\mid^2(1-\delta_C)$, where $M_{F0}=[T(T+1)-T_{Z_i}T_{Z_f}]^{1/2}$.
With the “nucleus-independent” ${\cal F}t$ values defined by $${\cal F}t=ft(1+\delta_R+\Delta_R)(1-\delta_C),$$ the CKM matrix element $v_{ud}$ is given by [@Bar92] $$\mid v_{ud}\mid^2 = \frac{\pi^3\ln 2}{{\cal F}t}\frac{\hbar^7}{G_F^2m_e^5c^4}
=\frac{2984.38(6)~{\rm s}}{{\cal F}t},$$ where the Fermi coupling constant, $G_F$ is obtained from muon $\beta$-decay, and includes radiative corrections. Currently, $ft$ values for nine superallowed transitions have been measured with an experimental precision of better than 0.2% [@Har90; @Sav95]. Prior to the recent measurement for $^{10}$C, the experimental $ft$-values gave some hint of an additional $Z$ dependence not presently accounted for. In addition, the unitarity condition for the CKM matrix was not satisfied. This prompted studies to empirically determine the “missing” correction and to satisfy the CVC requirement [@Wil93]. Recent results for $^{10}$C [@Sav95], however, do not support the conclusion that there may be a “missing” correction, as together all nine ${\cal F}t$ values satisfy the constancy requirement of the CVC hypothesis. The unitarity condition of the CKM matrix, however, is still violated at the level of $\sim~3\sigma$ [@Bar92; @Tow92; @Sav95], and can only be restored by the application of an across the board correction of approximately 0.3-0.4%. In the future, a possible $Z$ dependence in the ${\cal F}t$ values can be further tested by a remeasurement of $^{10}$C and precise measurements of heavier [*fp*]{}-shell Fermi transitions using radioactive beams.
The necessary formalism for computing $\delta_C$ is given in Refs. [@Tow77; @Orm85], and conventionally, $\delta_C$ is factored into two components, i.e., $\delta_C=\delta_{IM}+\delta_{RO}$ [@Tow77]. The correction $\delta_{IM}$ is due to isospin mixing between different valence shell-model configuration states (eg., the $0\hbar\omega$ $1s0d$ shell). The essential ingredients for $\delta_{IM}$ are a base isoscalar shell-model Hamiltonian that reproduces the spectra of excited $J=0$ states and an INC interaction that reproduces experimental mass splittings [@Orm85]. The second correction, $\delta_{RO}$, is due to the deviation from unity of the radial overlap between the converted proton and the corresponding neutron. This effect corresponds to the influence of states that lie outside the valence shell-model configuration space (eg., $2\hbar\omega$, one particle-one hole configurations). Currently, there are two approaches for evaluating $\delta_{RO}$ that give roughly the same agreement with the CVC hypothesis, but are in overall disagreement in magnitude. In the first approach [@Tow77], the radial wave functions were obtained using a Woods-Saxon (WS) plus Coulomb potential, while in the second [@Orm89; @Orm85], self-consistent Hartree-Fock (HF) calculations using Skyrme-type interactions (including Coulomb) were performed. The principal feature of the HF procedure is that since the mean field is proportional to the nuclear densities, the Coulomb force induces a one-body isovector potential that tends to counter Coulomb repulsion, therefore reducing $\delta_{RO}$. Because of this, the HF values of $\delta_{RO}$ are consistently smaller than the WS values by approximately 0.1-0.2 (in %).
In this paper, we re-evaluate the isospin-mixing corrections for the [*fp*]{}-shell transitions $^{46}$V, $^{50}$Mn, and $^{54}$Co that are included in the set of nine accurately measured transitions using expanded shell-model spaces and improved effective interactions. Comparisons with experimental data on the isospin-forbidden transition to the first excited ($J=0,T=1)$ state, which places some constraints on $\delta_{IM}$ [@Hag94], will also be made. In addition, one application of future radioactive beam facilities is to extend the data set to the heavier [*fp*]{}-shell nuclei $^{58}$Zn, $^{62}$Ga, $^{66}$As, $^{70}$Br, and $^{74}$Rb [@foot0]. Such a study may shed light on any possible $Z$ dependence in the ${\cal F}t$ values. As such, we present calculations for the important isospin-mixing corrections for these nuclei. We find for these nuclei that both $\delta_{IM}$ and $\delta_{RO}$ are much larger than in the case of the previous nine transitions. In addition, the difference between the Woods-Saxon and Hartree-Fock calculations for $\delta_{RO}$ is more pronounced for these nuclei, and precise measurements of these cases may be able to make a selection between the two approaches.
A calculation of $\delta_C$ begins with defining the shell-model configuration space and the base isoscalar shell-model Hamiltonian. Naturally, these are not independent choices, as model-space truncations may require renormalizations of the effective interaction. For the nuclei under consideration here, the base configuration space is comprised of the $0f_{7/2}$, $1p_{3/2}$, $1p_{1/2}$, and $0f_{5/2}$ orbitals, or [*fp*]{} shell. Because of computational restrictions, some model space truncations must be imposed on all nuclei except $^{46}$V and $^{74}$Rb. The active model space used for each nucleus is listed in Table I. These model-space truncations were found to be adequate except for the cases of $A=54$ and 74 as discussed below. In recent years, progress has been made towards the determination of effective interactions for use in [*fp*]{}-shell calculations, in particular for the lower part of the shell [@Ric90]. In this work, the FPD6 interaction of Ref. [@Ric90] was used for $A \le 50$. For $A=54$ the interaction was taken to be comprised of the two-body matrix elements of FPD6, while the single-particle energies were renormalized to reproduce the experimental binding energies of $^{57}$Ni assuming a closed $f_{7/2}$ core (FPD6$^*$). In the upper part of the [*fp*]{} shell, the interaction is less well determined, and for $58\le A \le 74$, we compare the results obtained using FPD6$^*$ and the FPVH interaction of Ref. [@FPVH]. The calculations presented here were performed using a unix version of the shell-model code OXBASH [@oxbash] on Silicon Graphics computers at Oak Ridge National Laboratory.
Another popular interaction used recently, but not here for the reasons outlined below, is a modified version of the original Kuo-Brown interaction referred to as KB3 [@Pov81]. Although this interaction gives very nearly the same results as FPD6 and FPD6$^*$ in the lower [*fp*]{} shell, it begins to diverge drastically from either FPD6$^*$ or FPVH for $A\ge 60$. The reason for this is that in the upper part of the shell, monopole terms in KB3 tend to push the $0f_{5/2}$ orbit up, creating a large gap between the [*p*]{} orbitals and the $0f_{5/2}$ orbit. In fact, for the single-hole nucleus $A=79$, KB3 predicts the ground state to be $J^\pi=5/2^-$ with excitation energies for the $1/2^-$ and $3/2^-$ hole states of 3.753 MeV and 7.010 MeV, respectively. This is in strong disagreement with spherical Hartree-Fock calculations, where, for example, the Skyrme M$^*$ force [@Bar82] predicts the ground state to be $J^\pi=1/2^-$, with excitation energies for the $5/2^-$ and $3/2^-$ hole states to be 0.591 MeV and 1.460 MeV, respectively. Both FPD6$^*$ and FPVH are in excellent agreement with the HF results.
To evaluate the configuration-mixing contribution $\delta_{IM}$ we use an INC interaction derived in the same manner as in Ref. [@Orm89a]. An important ingredient of the INC interaction is the mass scaling of the Coulomb two-body strength and single-particle energies as governed by the oscillator parameter $\hbar\omega$ (cf. Eq. (3.5) of Ref. [@Orm89a]). Since there are important deviations from the usual smooth formulae for $\hbar\omega$ around $A\sim
53-59$, and we want a uniform parameterization across the [*fp*]{} shell, we have chosen $\hbar\omega$ so as to reproduce the rms point proton radii obtained from with a spherical Hartree-Fock calculation using the Skyrme M$^*$ force. The values of $\hbar\omega$ used here are listed in Table I. Using these values of $\hbar\omega$, the parameters of the INC interaction of Ref. [@Orm89a] were redetermined. In addition, the single-particle energies of the $0f_{5/2}$ and $1p_{1/2}$ orbits were not well determined by the data set in Ref. [@Orm89a], and were chosen to reproduce the Coulomb splittings for the $J^\pi=5/2^-$ and $1/2^-$ $A=57$, $T=1/2$ multiplets [@Bha92] assuming a closed $^{56}$Ni core. The parameters of the INC interaction used are $\epsilon(0f_{7/2})=7.487$ MeV, $\epsilon(1p_{3/2})=7.312$ MeV, $\epsilon(0f_{5/2})=7.582$ MeV, $\epsilon(1p_{1/2})=7.240$ MeV, $S_C=1.006$, $S_0^{(1)}=0.0$, and $S_0^{(2)}=-4.2\times 10^{-2}$.
Shown in Tables II (FPD6$^*$ for $A\ge 58$) and III (FPVH for $A\ge 58$) are the results of shell-model calculations for $\delta_{IM}$ for the [*fp*]{}-shell nuclei under consideration. In addition, the theoretical and experimental values for the excitation energy of the first excited $J^\pi=0^+,T=1$ state are shown. Generally, for $A<58$ one finds that $\delta_{IM}$ is of the order 0.02-0.10%, while for the heavier nuclei it can be as large as 0.4%. One reason for the increase in $\delta_{IM}$ for $A \ge 62$ is that the excitation energy of the lowest $J=0, T=0$ state is steadily decreasing in these nuclei, eventually becoming equal to or less than that for the $J=0,T=1$ state. The effect of $T=0$ mixing in the $T_z=0$ parent is to remove Fermi strength from the transition, therefore increasing $\delta_{IM}$. The second reason for the enhancement in $\delta_{IM}$ is that the excitation energy of the first excited $J=0,T=1$ state is lower in these nuclei than for $A\le 54$. The contribution to $\delta_{IM}$ due to mixing with this state is given by $$\delta_{IM}^1=[\alpha(0)-\alpha(-1)]^2,
\label{dim}$$ where $\alpha(T_z)$ is the amplitude for mixing the first excited state into the ground state for the nucleus with third component of isospin $T_z=(Z-N)/2$, ($Z$ and $N$ denoting the number of protons and neutrons, respectively). In perturbation theory, the mixing amplitude $\alpha$ is is determined by the ratio of the matrix element of the INC interaction and the energy difference between the states, i.e. $$\alpha = \langle \psi_1 | V_{INC} | \psi_0 \rangle/\Delta E_{01}.$$ Therefore, a dependence in $\delta_{IM}$ on the isoscalar interaction and shell-model configuration space is manifested in the reproduction of the energy spectrum of $J=0$ states. Improved values for $\delta_{IM}^1$ and $\delta_{IM}$ maybe obtained by scaling $\delta_{IM}^1$ by the square of the ratio of the theoretical and experimental excitation energies, $(\Delta E_{01}^{th}/\Delta E_{01}^{exp})^2$. The results are tabulated in Tables II and III with the additional subscript $s$. In addition, for $^{46}$V the contribution due to the second excited state, $\delta_{IM}^2=0.012\%$ was also scaled by the ratio $(5.84/3.57)^2$ to account for the difference between the experimental and theoretical excitation energies for this state as well. As is pointed out in Ref. [@Hag94], the experimentally measured Fermi matrix element for the isospin-forbidden transition from the ground state of the parent to the first excited $J=0,T=1$ state in the daughter can be related to $\delta_{IM}^1$ [@footb]. The experimental and theoretical values are compared in Table II, where overall good agreement is achieved except for $A=54$.
Two nuclei in this study deserve special mention in regards to model-space truncations. The first is $A=74$. Towards the upper end of the [*fp*]{} shell, it is apparent that deformation effects are beginning to become important as can be seen by the steady decrease with nucleon number $A$ in the excitation energy of the lowest $J^\pi=2^+$ states in even-even $N=Z$ nuclei [@Led78; @Lis88] as shown in Table IV. Also shown in Table IV is a comparison between the experimental excitation energies and those obtained from a shell-model calculation using the FPD6$^*$ and FPVH interactions. A clear change is observed between $A$=72 and 76, and for this reason, a proper calculation for $A=74$ should probably include the $0g_{9/2}$ orbit. At present such a calculation is not feasible, and we express caution regarding the results for $A=74$ and the hope that more thorough calculations can be performed in the near future. The second case is $A=54$, where, to first order, the ground-state wave function is comprised of two $f_{7/2}$ holes. Excited $J=0$ states, which are important for $\delta_{IM}$, have at least two particles excited outside of the $0f_{7/2}$ orbit (i.e., a two particle-four hole ($2p-4h$) configuration relative to the $^{56}$Ni closed shell). The effect of including these configurations, however, is to decrease the binding energy of the ground state relative to the $2p-4h$ states, leading to an artificially large excitation energy for the excited states. In principle, if computational limitations permitted, the inclusion of $4p-6h$ states would decrease this gap. A calculation utilizing no restrictions with the $0f_{7/2}$ and $1p_{3/2}$ orbits is feasible, and the gap between the ground state and excited states is reduced considerably. The effects of isospin mixing in this space, however, are quite small, and are in disagreement the experimental results obtained in Ref. [@Hag94]. In addition, when excitations involving two particles into the $0f_{5/2}$ orbit are included, the gap worsens, indicating that $4p-6h$ excitations to the $0f_{5/2}$ orbit are important for describing the energy of the first excited state. An alternative approach is that of Ref. [@Tow89] where the isoscalar interaction was renormalized in the $2p-4h$ space so that the excitation spectrum had the correct energies. In that work, $\delta_{IM}^1$ and $\delta_{IM}$ were found to be 0.037(8)% and 0.045(5)%, respectively, and are in good agreement with the experimental value for $\delta_{IM}^1$ of 0.035(5). Given the computational limitations and the experimental data, probably the best value of $\delta_{IM}$ for $^{54}$Co when testing of CVC and the unitarity of the CKM matrix is 0.04(1)%.
The radial overlap correction $\delta_{RO}$ was evaluated using the procedures outlined in Refs. [@Tow77; @Orm85]. Shown in Tables II (FPD6$^*$) and III (FPVH) are the results for $\delta_{RO}$ using Hartree-Fock (HF) and Woods-Saxon (WS) single-particle wave functions. The HF results were computed using the Skyrme M$^*$ force [@Bar82], which generally gives better overall agreement with many experimental observables than do other Skyrme forces, in particular some isovector quantities such as the centroid energies for giant-dipole and giant isovector-monopole resonances [@Gle90]. Therefore, we have chosen to present all the results with Skyrme M$^*$. However, we believe the dependence on the parameters of the Skyrme interaction should be further investigated [@footc]. The WS values for $A\ge 58$ were computed using the Woods-Saxon parameters given in Ref. [@Str82].
An interesting feature of $\delta_{RO}$ is that it is much larger for the $A\ge 58$ cases. This is primarily due to: (1) the larger difference between the proton and neutron separation energies $\sim 10$ MeV; (2) the last proton being rather weakly bound $\sim 2.5$ MeV, as opposed to 5-6 MeV for $A \le 54$; and (3) $\delta_{RO}$ being dominated by the $0p_{3/2}$ orbit, which has a lower centrifugal barrier than in the case for $A \le 54$, which is dominated by the $0f_{7/2}$ orbit. Finally, it is apparent from Tables II and III that the difference between the HF and WS evaluations of $\delta_{RO}$ is considerably larger for the heavier nuclei, ranging from 0.3-0.7%, as opposed to 0.02-0.2% for the $A\le 54$ cases (cf., Ref. [@Orm89]). As such, CVC tests including accurate measurements of the [*ft*]{} values for the heavier [*fp*]{}-shell cases may lead to a differentiation between the two approaches.
To complete the survey of isospin-mixing corrections for Fermi transitions, the values of $\delta_{IM}$ and $\delta_{RO}^{HF}$ (and the sum $\delta_C$) for the nine accurately measured nuclei are listed in Table V. The $\delta_{RO}^{HF}$ values were obtained using the Skyrme M$^*$ force. The values presented for $^{10}$C were evaluated using the full $0p_{3/2},0p_{1/2}$ shell-model space and the CKPOT isoscalar interaction [@Coh65] and the INC interaction of Ref. [@Orm89].
Aside from the systematic difference between the HF and WS estimates of $\delta_{RO}$ the theoretical uncertainty in $\delta_C$ for $A\le 54$ is of the order 0.09% in most cases [@Orm89]. This arises from the addition in quadrature of 0.05% for $\delta_{IM}$, 0.06% for $\delta_{RO}$, and 0.05% as a conservative estimate for the spectator mismatch, which as discussed in Refs. [@Orm89; @Wil95] is expected to be negligible. For $A\ge 58$ there are some differences between the results obtained with the FPD6$^*$ and FPVH interactions. For the most part, the $\delta_{IM}$ values are in overall agreement with differences of the order 0.05%. For $\delta_{RO}$ the mean difference between the two interactions is 0.124%, but can be as large 0.33%. These differences are primarily attributed to differences in the excitation energies of the $T=3/2$ states in the $A-1$ parent. For more precise studies in the future, it will be necessary to improve upon the base shell-model isoscalar interaction. Nonetheless, both interactions predict large differences between the HF and WS approaches to $\delta_{RO}$.
A test of the CVC hypothesis is performed by applying $\delta_C$ to the $f_Rt$ values, which are also listed in Table V. Here, $f_Rt$ was computed by applying the radiative corrections listed in column 1 of Table 3 in Ref. [@Bar92] and the average of the $(\alpha/\pi)C_{NS}$ corrections listed in in Refs. [@Bar92; @Tow92] to the $ft$ values of the new Chalk River compilation [@Tow95]. Applying $\delta_C$ to $f_Rt$ (note that the ${\cal F}t$ are also listed in Table V) and taking the error-weighted average, we find ${\cal F}_{avg}t=3150.8\pm 1.2\pm 2.5$ s with $\chi^2/\nu=0.66$. Using Eq. (3) and $v_{us}=0.2199(17)$ [@Bar92] and $v_{ub} < 0.0075$ (90% confidence level) [@Tho88], the unitarity condition of the CKM matrix is found to be $0.9956\pm(0.0008)_{stat}\pm(0.0007)_{sys}$. Thus, from the constancy of the ${\cal F}t$ values, we conclude that CVC hypothesis is satisfied, but that the unitarity condition of the CKM matrix is violated at the level of 3-4 $\sigma$, and can only be achieved with an additional negative correction of 0.3-0.4% applied uniformly to each nucleus. It is important to note that a correction of this magnitude lies well outside the range of acceptable uncertainties in the nuclear corrections.
In summary, the isospin-mixing corrections for Fermi transitions in [*fp*]{}-shell nuclei were evaluated. The evaluation also included transitions involving heavier nuclei that are expected to be measured in the future radioactive-beam facilities. It was found that the isospin-mixing corrections were considerably larger for the $A \ge 58$ cases. In addition, the difference between the Hartree-Fock and Woods-Saxon method of evaluating $\delta_{RO}$ was much larger for these nuclei. As such, accurate measurements of the $ft$-values for these nuclei might lead to a discrimination between the two methods. In regard to the accurately measured transitions, it was found that the newer evaluations give better agreement with experiment for the configuration-mixing term $\delta_{IM}$, with the noted exception of $^{54}$Co, which poses a significant computational challenge. Lastly, it is found that the corrected ${\cal F}t$ values are in excellent agreement with the CVC hypothesis, but that the unitarity condition of the CKM matrix is violated at the level of 3-4 $\sigma$.
[**Acknowledgments**]{}
We wish to thank I. S. Towner for providing us the Chalk River $ft$ value data set, and for comments regarding this manuscript. Oak Ridge National Laboratory is managed for the U.S. Department of Energy by Martin Marietta Energy Systems, Inc. under contract No. DE–AC05–84OR21400. Theoretical nuclear physics research at the University of Tennessee is supported by the U.S. Department of Energy through contract No. DE–FG05–93ER40770. BAB acknowledges support from the National Science Foundation through grant no. PHY94-03666.
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----------- -------------------------------------------------------------------------------------------------------- ---------------------
Nucleus Configuration $\hbar\omega$ (MeV)
$^{46}$V full [*fp*]{} 10.952
$^{50}$Mn $(f_{7/2},p_{3/2})^{10}$ + $f_{7/2}^{n_7},f_{5/2}^{n_5},p_{1/2}^{n_1}$ $(n_5+n_1=1)$ 10.550
$^{54}$Co $(f_{7/2},p_{3/2})^{14}$ + $f_{7/2}^{n_7},p_{3/2}^{n_3},f_{5/2}^{n_5},p_{1/2}^{n_1}$ $(n_3+n_5+n_1=2)$ 10.486
$^{58}$Zn $f_{7/2}^{n_7},p_{3/2}^{n_3} 10.298
f_{5/2}^{n_5},p_{1/2}^{n_1}$ ($14 \le n_7 \le 16$)
$^{62}$Ga $f_{7/2}^{16},(p_{1/2},f_{5/2},p_{1/2})^6$ 10.017
$^{66}$As $f_{7/2}^{16},(p_{1/2},f_{5/2},p_{1/2})^{10}$ 9.681
$^{70}$Br $f_{7/2}^{16},(p_{1/2},f_{5/2},p_{1/2})^{14}$ 9.424
$^{74}$Ga full [*fp*]{} 9.203
----------- -------------------------------------------------------------------------------------------------------- ---------------------
: List of shell-model configuration spaces and $\hbar\omega$ used for each nucleus
----------- -------------- --------------- ----------------- ------------------- --------------------- --------------- ----------------- -------------------------- --------------------------
A $E_{x,th}^1$ $E_{x,exp}^1$ $\delta_{IM}^1$ $\delta_{IM,s}^1$ $\delta_{IM,exp}^1$ $\delta_{IM}$ $\delta_{IM,s}$ $\delta_{RO}^{{\rm HF}}$ $\delta_{RO}^{{\rm WS}}$
$^{46}$V 4.295 2.611 0.020 0.054 0.053(5)$^a$ 0.040 0.094 0.286 0.36(6)$^b$
$^{50}$Mn 3.620 3.69 0.014 0.015 $<$0.016$^a$ 0.026 0.017 0.325 0.40(9)$^b$
$^{54}$Co 6.423 2.561 0.0004 0.003 0.035(5)$^a$ 0.003 0.006 0.397 0.56(6)$^b$
$^{58}$Zn 2.850 2.943 0.196 0.183 - 0.227 0.214 0.974 1.677
$^{62}$Ga 1.876 2.33 0.261 0.169 - 0.471 0.379 0.885 1.217
$^{66}$As 0.848 - 0.066 - - 0.499 - 0.911 1.236
$^{70}$Br 1.083 - 0.089 - - 0.313 - 0.801 1.377
$^{74}$Rb 2.258 - 0.069 - - 0.223 - 0.831 1.716
----------- -------------- --------------- ----------------- ------------------- --------------------- --------------- ----------------- -------------------------- --------------------------
: List of isospin-mixing corrections $\delta_{IM}$ and $\delta_{RO}$ (in %), theoretical and experimental excitation energies for the first $J=0,T=1$ excited state (in MeV), theoretical and experimental values of $\delta_{IM}^1$. Values of $\delta_{IM}$ obtained by setting the theoretical excitations equal to experiment are indicated by the additional subscript $s$. Values of $\delta_{RO}$ for Hartree-Fock and Woods-Saxon wave functions are denoted by the superscripts HF and WS, respectively. The results obtained for $A \ge 58$ are shown for the FPD6$^*$ interaction.
$^a$ From Ref. [@Hag94]
$^b$ From Ref. [@Tow77]
----------- -------------- --------------- ----------------- ------------------- --------------- ----------------- -------------------------- -------------------------- --
A $E_{x,th}^1$ $E_{x,exp}^1$ $\delta_{IM}^1$ $\delta_{IM,s}^1$ $\delta_{IM}$ $\delta_{IM,s}$ $\delta_{RO}^{{\rm HF}}$ $\delta_{RO}^{{\rm WS}}$
$^{58}$Zn 2.850 2.943 0.224 0.258 0.231 0.265 0.997 1.762
$^{62}$Ga 1.460 2.33 0.201 0.079 0.408 0.286 1.029 1.409
$^{66}$As 1.250 - 0.019 - 0.388 - 1.243 1.577
$^{70}$Br 1.545 - 0.017 - 0.330 - 1.082 1.596
$^{74}$Rb 2.988 - 0.090 - 0.237 - 0.670 1.409
----------- -------------- --------------- ----------------- ------------------- --------------- ----------------- -------------------------- -------------------------- --
: Same as Table II for $A\ge 58$ using the FPVH interaction.
----------- ------- ---------- -----------
A FPVH FPD6$^*$ Exp
$^{60}$Zn 1.134 0.825 1.004$^a$
$^{64}$Ge 0.914 0.700 0.902$^c$
$^{68}$Se 0.939 0.600 0.854$^c$
$^{72}$Kr 0.976 0.707 0.709$^c$
$^{76}$Sr 0.892 0.752 0.261$^c$
$^{80}$Zr - - 0.289$^c$
----------- ------- ---------- -----------
: Comparion between theoretical and experimental excitation energies (in MeV) of the first $J^\pi=2^+$ state in even-even $N=Z$ nuclei.
$^a$ from Ref. [@Led78].
$^c$ from Ref. [@Lis88].
------------ --------------- -------------------------- ------------- ------------------------ -------------
A $\delta_{IM}$ $\delta_{RO}^{{\rm HF}}$ $\delta_C$ $f_Rt^c$ ${\cal F}t$
$^{10}$C 0.04 0.11 0.15(9) 3154.4$\pm 5.1\pm 2.4$ 3148.5(64)
$^{14}$O 0.01$^a$ 0.14 0.15(9) 3151.1$\pm 1.8\pm 2.4$ 3144.0(51)
$^{26m}$Al 0.01$^a$ 0.29 0.30(9) 3157.8$\pm 1.7\pm 2.4$ 3147.2(45)
$^{34}$Cl 0.06$^a$ 0.51 0.57(9) 3167.0$\pm 1.9\pm 2.4$ 3148.8(45)
$^{38m}$K 0.11$^a$ 0.48 0.59(9) 3166.5$\pm 2.6\pm 2.4$ 3146.3(49)
$^{42}$Sc 0.11$^a$ 0.31 0.42(9) 3168.1$\pm 1.4\pm 2.4$ 3148.7(46)
$^{46}$V 0.09 0.29 0.38(9) 3165.5$\pm 1.8\pm 2.4$ 3151.6(46)
$^{50}$Mn 0.02 0.33 0.35(9) 3164.2$\pm 1.6\pm 2.4$ 3149.6(56)
$^{54}$Co 0.04 0.40 0.44(9)$^b$ 3166.4$\pm 1.1\pm 2.4$ 3152.8(46)
------------ --------------- -------------------------- ------------- ------------------------ -------------
: List of isospin-mixing corrections $\delta_{IM}$, $\delta_{RO}$, and $\delta_C$ (in %), $f_R t$ and ${\cal F}t$ (in seconds) for the accurately measured cases.
$^a$ from Ref. [@Orm89].
$^b$ using $\delta_{IM}=0.04(1)$ as discussed in the text.
$^c$ From the new Chalk River data set [@Tow95]. The systematic uncertainty of 2.4 s is due to the systematic uncertainty of 0.08% in $\Delta_R$ [@Bar92].
|
---
abstract: |
This paper proposes a learning-based model predictive control (MPC) approach for the thermal control of a four-zone smart building. The objectives are to minimize energy consumption and maintain the residents’ comfort. The proposed control scheme incorporates learning with the model-based control. The occupancy profile in the building zones are estimated in a long-term horizon through the artificial neural network (ANN), and this data is fed into the model-based predictor to get the indoor temperature predictions. The Energy Plus software is utilized as the actual dataset provider (weather data, indoor temperature, energy consumption). The optimization problem, including the actual and predicted data, is solved in each step of the simulation and the input setpoint temperature for the heating/cooling system, is generated. Comparing the results of the proposed approach with the conventional MPC results proved the significantly better performance of the proposed method in energy savings ($40.56 \%$ less cooling power consumption and $16.73 \%$ less heating power consumption), and residents’ comfort.\
Keywords: Learning-based model predictive control; Model-based control; Smart building management and control; Artificial neural network; Occupancy estimation; Heating/cooling system.
author:
- 'Roja Eini$^{1}$ and Sherif Abdelwahed$^{2}$ [^1] [^2]'
title: '**Learning-based Model Predictive Control for Smart Building Thermal Management** '
---
INTRODUCTION
============
The residential and commercial building sector is known to use around 40% of the total end-use energy and, hence, is considered to be the largest energy-consuming sector in the world \[1\]. Approximately half of this energy is used for heating, cooling, ventilation, and air conditioning (HVAC), and this usage is increasing $0.5-5$% per year in developed countries \[2\]. This trend is similar to the rest of the world. Therefore, finding solutions to reduce energy use and/or increase energy efficiency in the building sectors, particularly for smart buildings in the smart city environment, is of crucial importance.
The majority of building thermal controls are based on model-based approaches. In model-based control designs, the controller is designed based on the mathematical model of the plant, assuming that the model represents the actual plant. However, model uncertainties and modeling errors always exist in the modeling process. One of the efficient model-based techniques in building thermal control is model predictive control (MPC) \[3\]. Just like other model-based strategies, MPC requires an accurate model (mathematical model) to predict the process inputs/outputs and obtain the control signal \[4-6\]. The performance of MPC is directly relevant to the accuracy of the model, and it diminishes by the model inaccuracy. In the context of building management, it is difficult to accurately identify the building’s thermal models, due to vast differences in construction materials and architectures, time-varying thermal dynamics, huge load of complex data processing, high cost of accurate modeling, and difficulties in modeling the residents’ behavior and occupancy.
Learning-based modeling is known to be efficient in accurate modeling of multi-zone buildings with nonlinearities, uncertainties, time-varying characteristics, the vast number of variables and components, nonuniform zone temperatures, and zonal couplings \[7\]. A machine learning algorithm can utilize the building’s historical datasets to improve the modeling or control framework over time by learning the model uncertainties and real-life conditions \[8\]. Machine learning algorithms can address the complex data such as occupant behavior and varying operating costs in a building management system, without requiring a detailed model and explicit programming \[9\].
By integrating learning-based algorithms with model-based controls, one can utilize both advantages of learning- and modeling-based designs. On the one hand, model-based design assists the learning-based design in learning explorations with maximum learning rate. On the other hand, occupants’ behaviors, and various cyber physical interactions can be handled by the learners and fed into the model-based management structure. The use of learning-based algorithms in building modeling and control have been studied in the literature \[10\]-\[12\]. In \[10\], an ANN model is used to decrease the temperature overshoot and undershoots in the HVACs, which result in the reduction of energy consumption. Authors in \[11\] and \[12\] employed feed-forward ANN to build a predictive thermal model to determine the ON/OFF time for the HVAC. None of the mentioned works considered the occupancy profile in the learning process. Moreover, none of them incorporated a model-based control framework with the learning-based algorithm to get the most advantage of both designs.
In this paper, a learning-based modeling strategy is incorporated with a model-based predictive control algorithm to manage a multi-zone building’s thermal property. An ANN is utilized to predict the occupancy profile; then this data is fed into the model-based control (MPC). The datasets of ANN are generated by simulating an actual building in Energy Plus software, considering the indoor temperature, time of day, weather data, energy consumption data, and setpoint temperatures. Through the MPC algorithm, the optimum setpoints are generated as the control inputs at each step to conserve energy and improve the comfort level. In contrast to the previous works, this work utilized both model-based and learning-based modeling in the MPC algorithm to enhance robustness and stability of the model-based control framework as well as improving the controlled system performance through learning from historical datasets.\
The rest of the paper is organized as follows. Section II describes the building model and its components. Section III and IV explain the learning-based and model-based modeling techniques, respectively. The next section introduces the proposed learning-based MPC approach. The simulation results and simulation assumptions are presented in section VI. Finally, section VII provides conclusions and discusses future research.
SYSTEM DEFINITION
=================
A two-story office building with four zones and one HVAC system per zone is considered in this work. Each zone thermostat is dual setpoint. Fig. \[fig:building\] shows the four-zone building CAD model. The floor area is $1600 m^2$ with the orientation to the north. Windows include shadings, overhangs, and fins. Several materials are used in various layers of the walls (exterior and interior), window frames, door, roof, ceiling, and inter-zone walls. Table \[tab:systemparams\] contains the building materials’ specifications.
[| c | m[1.3cm]{} | m[1.3cm]{} | m[1.3cm]{} | m[1.3cm]{} | m[1.3cm]{} | m[1.3cm]{} | m[1.3cm]{} | m[1.3cm]{} | m[1.3cm]{} |]{}
& 4 inch dense face brick & [2 inch insulation]{} & [4 inch concrete block]{} & [3/4 inch plaster board]{} & [1/8 inch hardwood]{} & [8 inch concrete block]{} & [acoustic tile]{} & [1/2 inch stone]{} & [3/8 inch membrane]{}\
& [Rough]{} & [Very rough]{} & [Medium rough]{} & [Smooth]{} & [Medium smooth]{} & [Rough]{} & [Medium smooth]{} & [Rough]{} & [Rough]{}\
& $0.1014684$ & $0.050901$ & $0.1014984$ & $0.019050$ & $0.003169$ & $0.2033016$ & $0.019050$ & $0.012710$ & $0.009540$\
& $1.245296$ & $0.043239$ & $0.3805070$ & $0.7264224$ & $0.1591211$ & $0.5707605$ & $0.060535$ & $1.435549$ & $0.1902535$\
& $2082.400$ & $32.03693$ & $608.7016$ & $1601.846$ & $720.8308$ & $608.7016$ & $480.5539$ & $881.0155$ & $1121.292$\
& $920.4800$ & $836.8000$ & $836.8000$ & $836.8000$ & $1255.200$ & $836.8000$ & $836.8000$ & $1673.600$ & $1673.600$\
& $0.900000$ & $0.900000$ & $0.900000$ & $0.900000$ & $0.900000$ & $0.900000$ & $0.900000$ & $0.900000$ & $0.900000$\
& $0.930000$ & $0.500000$ & $0.650000$ & $0.920000$ & $0.780000$ & $0.650000$ & $0.320000$ & $0.550000$ & $0.750000$\
& $0.930000$ & $0.500000$ & $0.650000$ & $0.920000$ & $0.780000$ & $0.650000$ & $0.320000$ & $0.550000$ & $0.750000$\
\[tab:systemparams\]
![four-zone building CAD model[]{data-label="fig:building"}](model2.png){height="6cm" width="8.5cm"}
OCCUPANCY PREDICTIONS
=====================
This section explains the learning-based approach to predict the occupancy impact on the indoor temperature. Occupancy information is important in energy-efficient building climate control since it can impact the temperature, environmental conditions, energy usage, and comfort constraints \[2\]. The goal of this work is to predict the occupancy information in the long-term through the ANN and investigate its influence on the building energy consumption and residents’ comfort. The selected ANN model is a Nonlinear Autoregressive netwoRk with eXogenous inputs (NARX). In the NARX model, at least three layers of nodes (input, output, and hidden layer) are used to approximate the outputs in (\[eq:NARX\]).\
$$\begin{aligned}[b]
y(t)&=f(x(t-1),...,x(t-d_x),y(t-1),y(t-2),...,y(t-d_y))
\end{aligned}
\label{eq:NARX}$$
$x(t)$, $y(t)$, $d_x$, and $d_y$ denote the inputs, outputs, input delays, and output delays of the ANN, respectively. $f$ is the mapping function. There are generally two architectures of NARX neural networks; i.e., series-parallel architecture and parallel architecture. The series-parallel configuration provides better prediction performance in training the model than the parallel configuration. The better performance is because the input of the feed-forward network is more accurate, and the static backpropagation can be used for training. Since in our work the actual output is available during the training of the network (from Energy Plus data), we have chosen the series-parallel architecture to train the network. The series-parallel NARX network is represented in Fig. \[fig:ANN\_typical\].
![A typical NARX network with input and output delays[]{data-label="fig:ANN_typical"}](NN_typical.png){height="3.3cm" width="6.5cm"}
The reason we used the NARX neural network for the occupancy predictions is that the occupancy profile is a time series, and one of the primary applications of NARX is predicting the time series models \[13\]. Moreover, since the occupancy produces heating, it is highly nonlinear, and NARX model is very beneficial for nonlinear models of this type. After training the network, it is validated. To evaluate the stopping criterion and the expected performance of the predicted data, the test data is used. Therefore, three datasets are used; training, validation, and test. Mean square error (MSE) and the regression value, representing the square error and the correlation between the output and the target values are utilized to validate the training performance. Thus, the NARX neural network algorithm is as follows:
- Define the input and output datasets.
- Define three sets of training, validation, and testing data.
- Choose a network architecture and a training algorithm by trial and error method.
- Train the network, and evaluate its performance.
- If the network performance is satisfactory, the problem is solved, otherwise, change the network size, retrain, or use larger datasets.
The ANN specifications of this work are described in section VI.
INDOOR TEMPERATURE PREDICTIONS
==============================
This section explains the model-based approach to predict the indoor temperature. Considering the thermal convection and conduction equations, the mathematical model of the indoor temperature is represented as (\[eq:thermal\]) \[14\], \[15\].\
$$\begin{aligned}[b]
\hat T_{in}(t)&=a[\hat T_{in}(t-1)+\frac {\Delta t}{C}[P(t-1)-U(\hat T_{in}(t-1)- T_{out}(t-1))]] \\
&+\hat b(t)
\end{aligned}
\label{eq:thermal}$$
where $\hat T_{in}$ and $T_{out}$ are the estimated indoor temperature and the outdoor temperature, respectively. $\Delta t$ is the time step, and $P$ is the heating power. $a$ and $U$ are the parameters to be identified. $\hat b(t)$ is the estimated occupancy at time $t$. In the learning-based simulation, the estimated value of occupancy is fed into the model-based predictor. In the conventional MPC; i.e., without learning, the occupancy profile is chosen constant at its average value ($\bar b(t)$).\
The parameters of the thermal model (\[eq:thermal\]) are identified through the recursive least square (RLS) identification algorithm using the Energy Plus input/output data. To evaluate the identification algorithm performance, the root mean square (RMS) criterion is used. The RLS algorithm is represented in brief as follows.\
$$\begin{aligned}[b]
&\hat F(t+1)=\frac{1}{\lambda}[F(t)-\frac{F(t) {\phi}^T(t) F(t)}{\lambda +{\phi}^T(t)F(t)\phi (t)}] \\
&e(t+1)=y(t+1)-\hat \theta (t) \phi (t) \\
&\hat \theta (t+1)=\hat \theta (t) + F(t+1) \phi (t) e(t+1)
\end{aligned}
\label{eq:RLS}$$
where $F$, $\lambda$, $\phi$, and $\hat \theta$ are the gain, forgetting factor, observations and estimated parameter, respectively. $e$ represents the error between the measurements and identification outputs.
LEARNING-BASED MODEL PREDICTIVE CONTROL
=======================================
![Learning-based model predictive control (MPC)[]{data-label="fig:block"}](MB_Learning.png){height="6.5cm" width="9cm"}
Having the weather and occupancy forecasts, the model predictive control (MPC) comes into play. MPC is a modern control technique that has been applied in many areas due to its ability to handle constrained control problems \[3\]. At each time instant, an optimal control problem is solved to obtain the optimal control action over the time horizon. Using MPC in the building temperature control, a plan for the HVAC system control is generated based on the predicted weather conditions and occupancy profiles over the time horizon. The first control action that minimizes the energy consumption and satisfies the comfort is applied to the building’s HVACs, then the control algorithm is repeated with the feedback information of building states and outputs at the next time instant. Fig. \[fig:block\] represents the proposed learning-based model predictive control approach.\
MPC cost function is defined as (\[eq:cost\]), such that it penalizes the deviations from the comfort level and optimum energy consumption.\
$$\begin{aligned}[b]
&J(t)=\sum\limits_{k=0}^{N} {\lVert{\hat T_{in}(t+k) - T_d\lVert}_Q }^2 + \sum\limits_{k=0}^{N} {\lVert{\Delta P(t+k-1)\lVert}_R }^2
\end{aligned}
\label{eq:cost}$$
where $Q$ and $R$ are the weighting factors associated with the states and inputs, respectively. $N$ is the time horizon, and $T_d$ is the comfort setpoint temperature. Therefore, the MPC problem is to minimize (\[eq:cost\]) subject to the performance constraints (\[eq:constraints1\]), robustness constraints (\[eq:constraints2\]), and limit constraints ((\[eq:constraints3\])). It is worth mentioning that equation (\[eq:constraints1\]) incorporates the learning while (\[eq:constraints2\]) is solely based on model-based design.
$$\begin{aligned}[b]
\hat T_{in}(t)&=a[\hat T_{in}(t-1)+\frac {\Delta t}{C}[P(t-1)-U(\hat T_{in}(t-1)- T_{out}(t-1))]] \\
&+\hat b(t)
\end{aligned}
\label{eq:constraints1}$$
$$\begin{aligned}[b]
\bar T_{in}(t)&=a[\bar T_{in}(t-1)+\frac {\Delta t}{C}[P(t-1)-U(\bar T_{in}(t-1)- T_{out}(t-1))]] \\
&+\bar b(t)
\end{aligned}
\label{eq:constraints2}$$
$$\begin{aligned}[b]
& T_{in}^{min} \leq \bar T_{in}(t+k) \leq T_{in}^{max}, \\
& P^{min} \leq P(t+k-1) \leq P^{max}
\end{aligned}
\label{eq:constraints3}$$
\
MPC algorithm is as follows:
- Define the system states and inputs at the current time, and their estimations up to the time horizon.
- Solve the optimization problem (cost function) to get the optimum inputs at time $t$.
- At time $t$, solve the optimization to get the input signal over the horizon.
- Apply the first control input, $t=t+1$, and go to the second step.
SIMULATION RESULTS
==================
In this section, all the simulation assumptions and results from the proposed learning-based MPC and the conventional MPC (without learning) are illustrated. The simulations are performed for one year, with 6 time steps per hour. To provide the ANN dataset, Energy Plus simulations on the building model of Fig. \[fig:building\] were completed from the 1st of July to 31st of December. The simulation assumptions are as follows.\
![NARX neural network series model[]{data-label="fig:NN"}](NNnew.PNG){height="2.5cm" width="48.00000%"}
![Neural network output response versus targets[]{data-label="fig:timeseries"}](timeseries.png){height="5cm" width="48.00000%"}
- The desired temperature of all zones are between 20 $^\circ$C and 25 $^\circ$C.
- The control variables are the HVAC setpoints.
- The maximum and minimum supply air temperatures are 50 $^\circ$C and 13 $^\circ$C, respectively.
- The maximum dry-bulb temperature for winter and summer days in Chicago Ohare location are considered -16.6 $^\circ$C and 31.6 $^\circ$C, respectively.
- The weather data at Chicago Ohare location is used.
- The number of people per zonal area is 0.1.
- The ANN input layer includes the environmental measures; the time of day, date, weather data, and the historical occupancy data.
- The input and output delays of NARX model are both chosen 2.
- One output layer and 10 hidden layers are chosen.
- The Levenbegrg-Marquardt backpropagation training algorithm is chosen.
![Regression and performance trajectories of datasets[]{data-label="fig:performance"}](regression1.png){height="4.9cm" width="50.00000%"}
![The identified model outputs versus real outputs, and the identification error trajectory[]{data-label="fig:id"}](idenew.png){height="4.8" width="40.00000%"}
The NARX neural network implemented in MATLAB is presented in Fig. \[fig:NN\]. Fig. \[fig:timeseries\] compares the network’s response with the actual vacancy profile, and shows the error values between the occupancy predictions and its actual profile throughout one month (To get a clear image, these plots are presented for one-month period). The maximum error value at each time is $1$; i.e., the target occupancy profile is well-tracked. Fig. \[fig:performance\] presents the regression and performance plots of the training, validation, and testing datasets. The regression values are all close to $1$ and the MSE error is $0.003189$; i.e., the training performance is satisfactory.
![The heating/cooling power consumption rate and zone 1 temperature using the proposed learning-based MPC[]{data-label="fig:proposed"}](zone1.png){width="48.00000%"}
\[t\]
[| c | m[1.4cm]{} | m[1.4cm]{} | m[1.1cm]{} |]{} Parameters & Conventional MPC & Learning-based MPC & Change\
Average cooling power & 396.28 W & 235.55 W & $\downarrow$ 40.56%\
Average heating power & 2.43 KW & 2.02 KW & $\downarrow$ 16.73%\
\[tab:Simulation results\]
Fig. \[fig:id\] shows the results of indoor temperature model identification throughout one-month simulation. From Fig. \[fig:id\], the identification error does not exceed 0.05; i.e., the identified outputs (indoor temperature) are very close to the actual indoor temperature values. Figs. \[fig:proposed\] and \[fig:conv\] show the results of the proposed learning-based MPC and conventional MPC approaches on the building throughout one-year simulation. Comparing the power rate graphs and Table \[tab:Simulation results\] values using the two approaches, the proposed method decreased the cooling and heating power consumption by $40.56\%$ and $16.73 \%$, respectively. Furthermore, the deviations from the comfort level in the conventional approach is extremely higher compared to the proposed method. The zone temperature using the conventional MPC even violates the minimum comfort level.
CONCLUSIONS
===========
In this paper, a learning-based MPC strategy is introduced to control the thermal property of a four-zone office building. Predicting the building parameters is a crucial and challenging part of MPC since the building’s thermal model is nonlinear, associated with uncertainties, and strongly coupled. Thus, ANN is incorporated with the model- based control approach to address the mentioned issues. The occupancy profile predictions are generated through ANN, and then this data is fed into the model-based controller. Energy Plus software is used in this work to simulate a building with real materials and components, and to test the proposed approach on it. Results from the proposed learning-based approach showed significantly better performance, in maintaining the residents’ comfort and minimizing energy usage ($40.56 \%$ energy savings), compared to the conventional MPC. For future work, implementing learning-based control to consider the impact of occupants behavior, such as window opening, or the energy storage devices in the building management system will be considered.
![The heating/cooling power consumption rate and zone 1 temperature using the conventional MPC[]{data-label="fig:conv"}](zone1conventional.png){width="48.00000%"}
[99]{} Pérez-Lombard, Luis, José Ortiz, and Christine Pout. “A review on buildings energy consumption information." Energy and buildings 40, no. 3 (2008): 394-398. Gul, Mehreen S., and Sandhya Patidar. “Understanding the energy consumption and occupancy of a multi-purpose academic building." Energy and Buildings 87 (2015): 155-165. Serale, Gianluca, Massimo Fiorentini, Alfonso Capozzoli, Daniele Bernardini, and Alberto Bemporad. “Model predictive control (MPC) for enhancing building and HVAC system energy efficiency: Problem formulation, applications and opportunities." Energies 11, no. 3 (2018): 631. Zohrabi, Nasibeh, Sherif Abdelwahed, and Jian Shi. “Reconfiguration of MVDC shipboard power systems: A model predictive control approach." In 2017 IEEE Electric Ship Technologies Symposium (ESTS), pp. 253-258. IEEE, 2017. Eini, Roja, and Sherif Abdelwahed. “Urban Traffic Network Control in Smart Cities; a Distributed Model-based Control Approach." arXiv preprint arXiv:1905.09955 (2019). Morrissett, Adam, Roja Eini, Mostafa Zaman, Nasibeh Zohrabi, and Sherif Abdelwahed. “A Physical Testbed for Intelligent Transportation Systems." arXiv preprint arXiv:1907.12899 (2019). Klein J., “Machine learning perspectives for smart buildings: an overview," 2017. Pasandi, Hannaneh Barahouei, and Tamer Nadeem. “Challenges and Limitations in Automating the Design of MAC Protocols Using Machine-Learning." In 2019 International Conference on Artificial Intelligence in Information and Communication (ICAIIC), pp. 107-112. IEEE, 2019. Barahouei Pasandi, Hannaneh, and Tamer Nadeem. “Poster: Towards Self-Managing and Self-Adaptive Framework for Automating MAC Protocol Design in Wireless Networks." In Proceedings of the 20th International Workshop on Mobile Computing Systems and Applications, pp. 171-171. ACM, 2019. Moon, Jin Woo, and Jong-Jin Kim. “ANN-based thermal control models for residential buildings." Building and Environment 45, no. 7 (2010): 1612-1625. Ruano, Antonio E., Eduardo M. Crispim, Eusébio ZE Conceiçao, and Ma Manuela JR Lúcio. “Prediction of building’s temperature using neural networks models." Energy and Buildings 38, no. 6 (2006): 682-694. Lu, Tao, and Martti Viljanen. “Prediction of indoor temperature and relative humidity using neural network models: model comparison." Neural Computing and Applications 18, no. 4 (2009): 345. Diaconescu, Eugen. “The use of NARX neural networks to predict chaotic time series." Wseas Transactions on computer research 3, no. 3 (2008): 182-191. Eini, Roja, and Sherif Abdelwahed. “Distributed Model Predictive Control Based on Goal Coordination for Multi-Zone Building Temperature." In 2019 IEEE Green Technologies Conference (GreenTech), Lafayette, LA. 2019. Eini, Roja, Lauren Linkous, Nasibeh Zohrabi, and Sherif Abdelwahed. “A testbed for a smart building: design and implementation." In Proceedings of the Fourth Workshop on International Science of Smart City Operations and Platforms Engineering, pp. 1-6. ACM, 2019.
[^1]: $^{1}$Roja Eini is PhD student of Department of Electrical Engineering, Virginia Commonwealth University, Richmond, VA, USA [einir@vcu.edu]{}
[^2]: $^{2}$Sherif Abdelwahed is faculty of Department of Electrical Engineering, Virginia Commonwealth University, Richmond, VA, USA [sabdelwahed@vcu.edu]{}
|
---
abstract: 'We present recent lattice QCD results on nucleon form factors and N to $\Delta$ transition form factors. We predict the parity violating asymmetry in N to $\Delta$ and check the off-diagonal Goldberger-Treiman relation.'
address: |
Department of Physics, University of Cyprus,\
P.O. Box 20537, CY-1678 Nicosia, Cyprus\
E-mail: alexand@ucy.ac.cy
author:
- 'C. Alexandrou'
bibliography:
- 'ws-pro-sample.bib'
title: 'N and N to $\Delta$ transition form factors from Lattice QCD'
---
We present the evaluation, within lattice QCD, of fundamental physical quantities of the nucleon-$\Delta$ system. On the theoretical side, providing a complete set of form factors and coupling constants constitutes an important input for model builders and for fixing the parameters of chiral effective theories. On the experimental side, there is ongoing effort to measure accurately these quantities. Examples are the recent polarization experiments, of the electric, $G_E$, and magnetic, $G_M$, nucleon form factors, and the accurate measurements on the electric and scalar quadrupole multipoles and the magnetic dipole in N to $\Delta$ transition as well as the ongoing experiment to measure the parity violating asymmetry in N to $\Delta$. Using state-of-the-art-lattice techniques we obtain results with small statistical errors for pion masses, $m_\pi$, in the range 600-360 MeV. We use two flavors of dynamical Wilson fermions and domain wall valence quarks (DWF) on MILC configurations to study the role of pion cloud contributions. Results obtained with dynamical Wilson fermions and DWF are in agreement showing that lattice artifacts are under control.
In this work only the isovector nucleon form factors are evaluated since isoscalar contributions involved quark loops that are technically difficult to calculate. In Fig. \[fig:nucleonff\] we display the momentum dependence of the ratio of the isovector electric to magnetic form factors for the lightest pion mass namely 410 MeV in the quenched theory and 380 MeV for dynamical Wilson fermions [@NN]. The lattice results are in agreement but higher than experiment. Given the weak quark mass dependence of this ratio for the quark masses used in this work a linear extrapolation in $m_\pi^2$ to the physical limit fails to reproduce experiment. On the other hand, the isovector magnetic moments extracted from lattice results using a dipole Ansatz are well described by chiral effective theory [@chiral] that includes explicitly the $\Delta$. As can be seen in Fig. \[fig:nucleonff\] the extrapolated value of the magnetic moment is in agreement with experiment. This is consistent with the fact that lattice results are closer to experiment for the magnetic than for the electric form factor.
We evaluate the three electromagnetic Sachs and four axial Adler form factors for the N to $\Delta$ transition. We show in Fig. \[fig:asymmetry\] the the ratio $C_5^A/C_3^V$, which is the analogue of $g_A/g_V$ and determines, to a first approximation, the parity violating asymmetry [@axial] . The off-diagonal Goldberger-Treiman relation implies that the ratio $R_{GT}=\frac{f_\pi g_{\pi N\Delta}(Q^2)}{2M_N C_5^A(Q^2}$ is one, where $g_{\pi N\Delta}(Q^2)$ is determined from the matrix element of the pseudoscalar density $<\Delta^+|\bar{\psi}(x)\gamma_5\frac{\tau^3}{2}\psi(x)|p>$ and $f_\pi$ is the pion decay constant. As can be seen in Fig. \[fig:asymmetry\] this ratio approaches unity as $m_\pi$ decreases from $\sim 500$ MeV to $\sim 410 (380)$ MeV for quenched (dynamical) Wilson fermions.
[9]{} C. Alexandrou [*et al.*]{}, Phys. Rev. D [**74**]{} 034508 (2006). T. R. Hemmert and W. Weise, Eur. Phys. J. A [**15**]{},487 (2002). C. Alexandrou, hep-lat/0608025, C. Alexandrou [*et al.*]{}, hep-lat/0607030.
|
---
abstract: |
In the fragmentation of a color flux tube in high-energy $pp$ collisions or $e^+$-$e^-$ annihilations, the production of $q$-$\bar
q$ pairs along a color flux tube precedes the fragmentation of the tube. The local conservation laws in the production of these $q$-$\bar q$ pairs will lead to the correlations of adjacently produced hadrons. As a consequence, the fragmentation of a flux tube will yield a many-hadron correlation in the form of a chain of hadrons ordered in rapidity, with adjacent hadrons correlated in charges, flavor contents, and azimuthal angles. It will also lead to a two-hadron angular correlation between two hadrons with opposite charges or strangeness that is suppressed at $\Delta \phi\sim 0$ but enhanced at $\Delta \phi\sim \pi$, within a rapidity window $\Delta
y $$\sim$$1/(dN/dy)$.
address: 'Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA'
author:
- 'Cheuk-Yin Wong'
title: Signatures of Flux Tube Fragmentation and Strangeness Correlations in $pp$ Collisions
---
Introduction
============
The reaction mechanisms in $pp$ collisions provide insights in $AA$ collision at high energies. The importance of each reaction mechanism depends on the collision energy and the $p_T$ domain. The flux-tube fragmentation process [@Art74; @And79] is expected to dominate at low $p_T$ and low $\sqrt{s_{pp}}$, whereas the hard-scattering process [@Bla74; @Ang78] at high $p_{T}$ and high $\sqrt{s_{pp}}$. The two mechanisms cross-over at a certain transverse momentum $p_{Tb}(\sqrt{s_{pp}})$ that is a function of the collision energy, $\sqrt{s_{pp}}$. There are in addition other mechanisms such as the recombination of partons [@Hwa80] and the production of resonances.
It is desirable to establish, even if approximately, the boundary function $p_{Tb}(\sqrt{s_{pp}})$ that separates the region of flux-tube fragmentation dominance from the region of hard-scattering dominance. We need signatures for these two mechanisms to facilitate such a separation.
The signature for the hard scattering process is well known. It is given as a two-hadron $\Delta \phi-$$\Delta \eta$ angular correlation which shows a peak at $ (\Delta \phi,\Delta \eta)$$\sim$0 and a ridge along $\Delta \eta$ at $ \Delta \phi \sim \pi$. The peak at $\Delta\phi$$\sim$0 arises from the fragmentation of the jet associated with the trigger and the ridge at $\Delta\phi$$\sim$$\pi$ arises from the fragmentation of the other jet associated with colliding partons with unbalanced longitudinal momenta.
However, the signature for flux tube fragmentation has not been well studied. We would like to present here two signatures for flux-tube fragmentation [@Won15; @Won15a].
Many-hadron signatures of Flux Tube Fragmentation
=================================================
In the semi-classical description of the flux-tube fragmentation process [@Art74; @And79] for hadron production, the production of quark-antiquark pairs along a color flux tube precedes the fragmentation of the tube. The production of these quark-antiquark pairs must however obey conservation laws at the local production points. As a consequence, the produced $q$-$\bar q$ pairs will lead to correlations of adjacently produced hadrons, and the hadrons are ordered according to their rapidities along the tube. The rapidity-space-time ordering and the local conservation laws will yield a many-body correlation of the hadrons in charge, flavor, and momentum, which may provide vital information on space-time dynamics of quarks and hadrons in the flux-tube fragmentation process.
As an example, we examine the fragmentation of a flux tube with an invariant mass of 8.65 GeV, corresponding to the average invariant mass of one of the two flux tubes in a $pp$ collision at 17.3 GeV, with the flux tube formed by a quark of one proton with the diquark of the other proton. We carry out a Monte Carlo generation of hadrons in flux-tube fragmentation using the PYTHIA 6.4 program [@Sjo06]. An example of the produced hadrons involving the production of a pair of strange hadrons is shown in Fig. 1, with the production of 5 hadrons listed in Table I.
\[fig1\]
--------------------------- -------------- -------------- -------------- ---------------- -------------
$i$ 1 2 3 4 5
particle $\pi^0$ $K^+$ $ \bar K^0$ $\pi^-$ $p$
$ q_i$-$\bar q_i$ $u$-$\bar u$ $u$-$\bar s$ $s$-$\bar d$ $ d$ -$\bar u$ $u$ -$(ud)$
$y_i$ -1.55 - 1.15 - 0.75 0.27 1.78
$\phi_i$ 1.00 -2.01 1.44 -2.43 0.17
$\phi_{i}$$-$$\phi_{i+1}$
--------------------------- -------------- -------------- -------------- ---------------- -------------
: An example of primary hadrons $i$, their rapidities $y_i$, their azimuthal angles $\phi_i$ and their constituents $q_i$-$\bar q_i$ produced in the fragmentation of the $u$-$(ud)$ flux tube at an energy of $\sqrt{s}$=8.65 GeV obtained with PYTHIA 6.4 [@Sjo06].
In Table 1, the row of $q_i$-$\bar q_i$ shows that upon ordering the hadrons according to their repidities $y_i$ as in a chain, the flavors of the constituent antiquark $\bar q_i$ and the flavors of the neighboring constituent quark $q_{i+1}$ are correlated along the chain, on an event-by event basis. The row of $\phi_i$-$\phi_{i+1}$ of neighboring hadrons in Table I indicates that neighboring pairs of hadrons are azimuthally correlated, approximately in a back-to-back manner. The many-hadron signature requires the identification of all hadrons detected in the events. Predicted signature of this kind is yet to be observed.
Two-hadron angular correlation signature of flux-tube fragmentation
===================================================================
Another signature of the flux tube fragmentation utilizes the two-hadron angular correlations arising from the productions of $q$-$\bar q$ pairs. Because of local conservation laws, the production of $q$-$\bar q$ pairs will lead to correlations of adjacently produced hadrons. Adjacently produced hadrons however can be signaled by their rapidity difference $\Delta y$ falling within the window of $|\Delta y | $$\sim$$ 1/(dN/dy)$, on account of the space-time-rapidity ordering of produced mesons in a flux-tube fragmentation. Therefore, the local conservation laws of momentum, charge, and flavor will lead to a suppression of the angular correlation function $dN/(d\Delta \phi\, d\Delta y)$ for two hadrons with opposite charges or strangeness at $(\Delta \phi, \Delta y)$$
\sim$0, but an enhanced correlation on the back-to-back, away side at $\Delta \phi$$\sim$$ \pi$, within the window of $|\Delta y |$$\sim$$
1/(dN/dy)$. When we approximate the rapidity $y$ as the pseudorapidity $\eta$, the two-hadron angular correlations can be used as signatures for the fragmentation of a color flux tube as shown in Fig 2.
Comparison of theoretical angular correlations for two hadrons with unlike charges in Fig. 2(a) [@Won15] with experimental data from the STAR Collaboration in Fig. 2(b) [@Por05] indicates that in high-energy $pp$ collisions at $\sqrt{s_{pp}}=$200 GeV, the production of unlike-charge hadron pairs in the region of $p_T$$<$0.5 GeV/c are qualitatively consistent with the flux-tube fragmentation mechanism. However, the correlations for two hadrons with unlike-charges in the region with $p_T$$>$0.5 GeV/c exhibit a completely different pattern. Namely, they shows a peak at $(\Delta \phi, \Delta \eta)\sim$ 0, and a ridge along $\Delta\eta$ at $\Delta\phi\sim \pi$, which is a signature of the hard scattering process [@Por05]. This indicates that for $pp$ collisions energy at $\sqrt{s_{pp}}=200$ GeV, the boundary between the flux-tube fragmentation process and the hard-scattering process is $p_{Tb}=0.5$ GeV/c. At the LHC energy of $\sqrt{s}_{pp}=$5 TeV, the minimum bias data for the correlation of two hadrons with $p_T$$>$0.1 GeV/c show a pattern that appears to be a linear combination of the patterns for flux tube fragmentation and hard scattering [@CMS09]. Results in Fig. 2, together with the two hadron correlation data from lower energies [@Mak15] and higher energies [@CMS09] reveal that the $p_{Tb}$ boundary of separation moves to lower $p_T$ values as the collision energy increases.
Similarly, because of local conservation of strangeness in $q\bar q$ production, adjacently produced hadrons with opposite strangeness are correlated back-to-back in azimuthal angles.
=-11 =3.6in Adjacently produced mesons however can be signaled by their rapidity difference $\Delta y$ falling within the window of $|\Delta y
| $$\sim$$ 1/(dN/dy)$. The theoretical correlation function for two primary hadrons with opposite strangeness is shown in Fig. 3, with a suppression at $(\Delta \phi, \Delta \eta)\sim$ 0 and an enhancement at $\Delta \phi$$\sim$$ \pi$, within the window of $|\Delta y
|$$\sim$$ 1/(dN/dy)$.
=-3 =-7.4cm [[**Fig.3**]{} The correlation function for two hadrons with opposite strangeness in flux-tube fragmentation. ]{}
Conclusions and Discussions
===========================
In the fragmentation of a flux tube, the production of $q \bar q$ pairs obey local conservation laws and the hadrons follow space-time-rapidity ordering. As a consequence, a signature of the flux tube fragmentation consists of a chain of produced hadrons, correlated in rapidities, charges, flavors, and azimuthal angles. The observation of the chain of hadrons requires the identification of the produced hadrons which may be possible if the fraction of unobserved hadrons is small.
Another signature uses two-hadron angular correlations with opposite charges or strangeness, which exhibits a suppression at $(\Delta
y,\Delta \phi)$$\sim$0, but an enhancement at $\Delta
\phi$$\sim$$\pi$, with the window of $|\Delta y |$$\sim$$ 1/(dN/dy)$. It should be kept in mind however that resonance production and resonance decay into hadrons will exhibit angular correlations similar to the pattern of the flux-tube fragmentation. The resonance fraction give rise to complications and the two-hadron signature will work well if the resonance fraction is not dominant. Various estimates give the resonance fractions to be of order 10 to 30% for $pp$ collisions at $\sqrt{s_{pp}}=17.3$ GeV [@Won15a].
In flux-tube fragmentation, the production of two adjacent hadrons with like charges or the same strangeness is prohibited [@Won15]. Experimentally, two-hadron angular correlation of like charges with $\Delta \eta$$\sim$0 is not zero [@Por05; @Mak15] which indicates that there may be an additional mechanism for like charge production with $( \Delta \phi, \Delta
\eta)$$\sim 0$. Further theoretical search for the origin of the source of like charge and strangeness correlations at $( \Delta \phi,
\Delta \eta)$$\sim 0$ will be of interest.
[**Acknowledgments**]{} This work was supported in part by the Division of Nuclear Physics, U.S. Department of Energy, under Contract No. DE-AC05-00OR22725.
References {#references .unnumbered}
==========
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B. Andersson, G. Gustafson and C. Peterson, Z. Phys. C1, 105 (1979); B. Andersson, G. Gustafson and B. Söderberg, Z. Phys. C20, 317 (1983).
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R. J. Porter and T. A. Trainor, (STAR Collaboration), J. Phys. Conf. Ser., 98 (2005).
M. Maksiak (NA61/SHINE Collaboration), arXiv:1503.02470; M. Gazdzicki (NA61/SHINE Collaboration), EPJ Web Conf. [**95**]{}, 01005 (2015), arxiv:1412.4243.
CMS Collaboration, JHEP 1009, 091 (2010),\[arXiv:1009.4122\].
|
---
abstract: 'We study the complexity of (approximate) winner determination under the Monroe and Chamberlin–Courant multiwinner voting rules, which determine the set of representatives by optimizing the total (dis)satisfaction of the voters with their representatives. The total (dis)satisfaction is calculated either as the sum of individual (dis)satisfactions (the utilitarian case) or as the (dis)satisfaction of the worst off voter (the egalitarian case). We provide good approximation algorithms for the satisfaction-based utilitarian versions of the Monroe and Chamberlin–Courant rules, and inapproximability results for the dissatisfaction-based utilitarian versions of them and also for all egalitarian cases. Our algorithms are applicable and particularly appealing when voters submit truncated ballots. We provide experimental evaluation of the algorithms both on real-life preference-aggregation data and on synthetic data. These experiments show that our simple and fast algorithms can in many cases find near-perfect solutions.'
author:
- |
Piotr Skowron\
[University of Warsaw]{}\
[Warsaw, Poland]{}\
- |
Piotr Faliszewski\
[AGH University]{}\
[Krakow, Poland]{}\
- |
Arkadii Slinko\
[University of Auckland]{}\
[Auckland, New Zealand]{}\
bibliography:
- 'main.bib'
title: 'Achieving Fully Proportional Representation: Approximability Results[^1]'
---
Introduction {#sec::introduction}
============
We study the complexity of (approximate) winner determination under the Monroe [@monroeElection] and Chamberlin–Courant [@ccElection] multiwinner voting rules, which aim at selecting a group of candidates that best represent the voters. Multiwinner elections are important both for human societies (e.g., in indirect democracies for electing committees of representatives like parliaments) and for software multiagent systems (e.g., for recommendation systems [@budgetSocialChoice]), and thus it is important to have good multiwinner rules and good algorithms for them. The Monroe and Chamberlin–Courant rules are particularly appealing because they create an explicit (and, in some sense, optimal) connection between the elected committee members and the voters; each voter knows his or her representative and each committee member knows to whom he or she is accountable. In the context of recommendation systems this means that every selected item is personalized, i.e., recommended to a particular user. Moreover, the Monroe rule ensures the proportionality of the representation. We assume that $m$ candidates participate in the election and that the society consists of $n$ voters, who each rank the candidates, expressing their preferences about who they would like to see as their representative.
When choosing a $K$-member committee, the Monroe and Chamberlin–Courant rules work as follows. For each voter they assign a single candidate as their representative, respecting the following rules:
- altogether exactly $K$ candidates are assigned to the voters. For the Monroe rule, each candidate is assigned either to about $\frac{n}{K}$ voters or to none; for the Chamberlin–Courant rule there is no such restriction and each committee member might be representing a different number of voters. The committee should take this into account in its operation, i.e., by means of weighted voting.
- the candidates are selected and assigned to the voters optimally minimizing the total (societal) dissatisfaction or maximizing the total (societal) satisfaction.
The total (dis)satisfaction is calculated on the basis of individual (dis)satisfactions. We assume that there is a function $\alpha \colon {{{\mathbb{N}}}}\to {{{\mathbb{N}}}}$ such that $\alpha(i)$ measures how well a voter is represented by the candidate that this voter ranks as $i$’th best. The function $\alpha$ is the same for each voter. We can view $\alpha$ either as a *satisfaction function* (then it should be a decreasing one) or as a *dissatisfaction function* (then it should be an increasing one). For example, it is typical to use the Borda count scoring function whose $m$-candidate dissatisfaction variant is defined as $\alpha^m_{{{{{\mathrm{B, inc}}}}}} = i-1$, and whose satisfaction variant is $\alpha^m_{{{{{\mathrm{B, dec}}}}}} = m-i$. In the utilitarian variants of the rules, the assignment should maximize (minimize) the total satisfaction (dissatisfaction) calculated as the sum of the voters’ individual satisfactions (dissatisfactions) with their representatives. In the egalitarian variants, the assignment should maximize (minimize) the total satisfaction (dissatisfaction) calculated as the satisfaction (dissatisfaction) of the worst-off voter. The Monroe and Chamberlin–Courant rules create a useful connection between the voters and their representatives that makes it possible to achieve both candidates’ accountability to the voters, and proportional representation of voters’ views. Among common voting rules, the Monroe and Chamberlin–Courant rules seem to be unique in having *both* the accountability and the proportionality properties simultaneously. For example, First Past the Post system (where the voters are partitioned into districts with a separate single-winner Plurality election in each) can give very disproportionate results (forcing some of the voters to be represented by candidates they dislike). On the other side of the spectrum are the party-list systems, which achieve perfect proportionality. In those systems the voters vote for the parties, based on these votes each party receives some number of seats in the parliament, and then each party distributes the seats among its members (usually following a publicly available list of the party’s candidates). This makes the elected candidates feel more accountable to apparatchiks of their parties than to the voters. Somewhere between the First Past the Post system and the party-list systems, we have the single transferable vote rule (STV), but for STV it is difficult to tell which candidate represents which voters.
Unfortunately, the Monroe and Chamberlin–Courant rules have one crucial drawback that makes them impractical. It is ${{\mathrm{NP}}}$-hard to tell who the winners are! Specifically, ${{\mathrm{NP}}}$-hardness of winner determination under the Monroe and Chamberlin–Courant rules was shown by Procaccia et al. [@complexityProportionalRepr] and by Lu and Boutilier [@budgetSocialChoice]. Worse yet, the hardness holds even if various natural parameters of the election are small [@fullyProportionalRepr]. Rare easy cases include those, where the committee to be elected is small, or we consider the Chamberlin–Courant rule and the voters have single-peaked [@fullyProportionalRepr] or single-crossing preferences [@sko-yu-fal-elk:c:single-crossing-monroe-cc]. Lu and Boutilier [@budgetSocialChoice] proposed to use approximation algorithms and have given the first such algorithm for the Chamberlin–Courant system. Their procedure outputs an assignment that achieves no less than $1 - \frac{1}{e} \approx 0.63$ fraction of the optimal voter satisfaction. However, the approximation ratio $0.63$ here means that it is possible that, on average, each agent is represented by a candidate that this agent prefers to only about 63% of the candidates, even if there is a perfect solution that assigns each agent to their most preferred candidate. Such issues, however, would not occurr if we had a constant-factor approximation algorithm minimizing the total dissatisfaction. Indeed, if a perfect solution exists, then the optimal dissatisfaction is zero and a constant-factor approximation algorithm must also output this perfect solution.
The use of approximation algorithms in real-life applications requires some discussion. For example, their use is naturally justified in the context of recommendation systems. Here the strive for optimality is not crucial since a good but not optimal recommendation still has useful information and nobody would object if we replaced the exact recommendation with an approximate one (given that the exact one is hard to calculate). For example, Amazon.com may recommend you a book on gardening which may not be the best book for you on this topic, but still full of useful advice. For such situations, Herbert Simon [@sim:j:satisficing] used the term ‘satisficing,’ instead of optimizing, to explain the behavior of decision makers under circumstances in which an optimal solution cannot be easily determined. On page 129 he wrote: “Evidently, organisms adapt well enough to ÔsatisficeÕ; they do not, in general, ‘optimize’.” Effectively, what Simon says is that the use of approximation algorithms fits well with the human nature.
Still, the use of approximation algorithms in elections requires some care. It is conceivable that the electoral commission finds an allocation of voters to candidates with a certain value of (dis)satisfaction and one of the parties participating in the election finds an allocation with a better value. This can lead to a political deadlock. There are two ways of avoiding this. Firstly, an approximation algorithm can be fixed by law. In such a case, it becomes an acting voting rule and a new way to measure fairness in the society. Secondly, an electoral commission may calculate the allocation, but also publish the raw data and issue a call for submissions. If, within the period specified by law, nobody can produce a better allocation, then the committee goes ahead and announces the result. If someone produces a better allocation, then the electoral commission uses the latter one.
The use of approximation algorithms is even more natural in elections with partial ballots. Indeed, even if we use an exact algorithm to calculate the winners, the results will be approximate anyway since the voters provide us with approximations of their real preferences and not with their exact preferences.
Our Results {#sec:our}
-----------
In this paper we focus on approximation algorithms for winner determination under the Monroe and Chamberlin–Courant rules. Our first goal is to seek algorithms that find assignments for which the dissatisfaction of voters is within a fixed bound of the optimal one. Unfortunately, we have shown that under standard complexity-theoretic assumptions such algorithms do not exist. Nonetheless, we found good algorithms that maximize voter’s satisfaction. Specifically, we have obtained the following results:
1. The Monroe and Chamberlin–Courant rules are hard to approximate up to any constant factor for the dissatisfaction-based cases (both utilitarian and egalitarian ones; see Theorems \[theorem:noApprox1\], \[theorem:noApprox2\], \[theorem:noApprox3\] and \[theorem:noApprox4\]) and for the satisfaction-based egalitarian cases (see Theorems \[theorem:noApprox5\] and \[theorem:noApprox6\]).
2. For the satisfaction-based utilitarian framework we show the following. For the Monroe rule with the Borda scoring function we give a $(0.715-\epsilon)$-approximation algorithm (often, the ratio is much better; see Section \[sec:algorithms\]). In case of an arbitrary positional scoring function we give a ($1 - \frac{1}{e}$)-approximation algorithm (Theorem \[thm:gmMonroe\]). For the Chamberlin–Courant rule with the Borda scoring function we give a polynomial-time approximation scheme (that is, for each $\epsilon$, $0 < \epsilon < 1$, we have a polynomial-time $(1-\epsilon)$-approximation algorithm; see Theorem \[theorem:ptas\]).
3. We provide empirical evaluation of our algorithms for the satisfaction-based utilitarian framework, both on synthetic and real-life data. This evaluation shows that in practice our best algorithms achieve at least $0.9$ approximation ratios, and even better results are typical (see Section \[sec:experiments\]).
4. We show that our algorithms work very well in the setting where voters do not necessarily rank all the candidates, but only provide the so-called truncated ballots, in which they rank several most preferred candidates (usually at least three). We provide theoretical guarantees on the performance of our algorithms (Propositions \[prop:monTruncated\] and \[lemma:greedyCCTruncated\]) as well as empirical evaluation (see Section \[sec:truncated\]).
Our results show that, as long as one is willing to accept approximate solutions, it is possible to use the utilitarian variants of the Monroe and Chamberlin–Courant rules in practice. This view is justified both from the theoretical and from the empirical point of view. Due to our negative results, we did not perform empirical evaluation for the egalitarian variants of the rules, but we believe that this is an interesting future research direction.
Related Work {#sec:related}
------------
A large number of papers are related to our research in terms of methodology (the study of computational complexity and approximation algorithms for winner determination under various ${{\mathrm{NP}}}$-hard election rules), in terms of perspective and motivation (e.g., due to the resource allocation view of Monroe and Chamberlin–Courant rules that we take), and in terms of formal similarity (e.g., winner determination under the Chamberlin–Courant rule can be seen as a form of the facility location problem). Below we review this related literature.
There are several single-winner voting rules for which winner determination is known to be ${{\mathrm{NP}}}$-hard. These rules include, for example, Dodgson’s rule [@bar-tov-tri:j:who-won; @hem-hem-rot:j:dodgson; @bet-guo-nie:j:dodgson-parametrized], Young’s rule [@rot-spa-vog:j:young; @bet-guo-nie:j:dodgson-parametrized], and Kemeny’s rule [@bar-tov-tri:j:who-won; @hem-spa-vog:j:kemeny; @bet-fel-guo-nie-ros:j:fpt-kemeny-aaim]. For the single-transferable vote rule (STV), the winner determination problem becomes ${{\mathrm{NP}}}$-hard if we use the so-called parallel-universes tie-breaking [@con-rog-xia:c:mle]. Many of these hardness results hold even in the sense of parameterized complexity theory (however, there also is a number of fixed-parameter tractability results; see the references above for details).
These hardness results motivated the search for approximation algorithms. There are now very good approximation algorithms for Kemeny’s rule [@ail-cha-new:j:kemeny-approx; @cop-fla-rud:j:kemeny-approx; @ken-sch:c:kemeny-few-errors] and for Dodgson’s rule [@mcc-pri-sli:j:dodgson; @hem-hom:j:dodgson-greedy; @car-cov-fel-hom-kak-kar-pro-ros:j:dodgson; @fal-hem-hem:j:multimode; @car-kak-kar-pro:c:dodgson-acceptable]. In both cases the results are, in essence, optimal. For Kemeny’s rule there is a polynomial-time approximation scheme [@ken-sch:c:kemeny-few-errors] and for Dodgson’s rule the achieved approximation ratio is optimal under standard complexity-theoretic assumptions [@car-cov-fel-hom-kak-kar-pro-ros:j:dodgson] (unfortunately, the approximation ratio is not constant but depends logarithmically on the number of candidates). On the other hand, for Young’s rule it is known that no good approximation algorithms exist [@car-cov-fel-hom-kak-kar-pro-ros:j:dodgson].
The work of Caragiannis et al. [@car-kak-kar-pro:c:dodgson-acceptable] and of Faliszewski et al. [@fal-hem-hem:j:multimode] on approximate winner determination for Dodgson’s rule is particularly interesting from our perspective. In the former, the authors advocate treating approximation algorithms for Dodgson’s rule as voting rules in their own right and design them to have desirable properties. In the latter, the authors show that a well-established voting rule (so-called Maximin rule) is a reasonable (though not optimal) approximation of Dodgson’s rule. This perspective is important for anyone interested in using approximation algorithms for winner determination in elections (as might be the case for our algorithms for the Monroe and Chamberlin–Courant rules).
The hardness of the winner determination problem for the Monroe and Chamberlin–Courant rules have been considered in several papers. Procaccia, Rosenschein and Zohar [@complexityProportionalRepr] were the first to show the hardness of these two rules for the case of a particular approval-style dissatisfaction function. Their results were complemented by Lu and Boutilier [@budgetSocialChoice], Betzler, Slinko and Uhlmann [@fullyProportionalRepr], Yu, Chan, and Elkind [@yu-cha-elk:c:cc-sp-trees], Skowron et al. [@sko-yu-fal-elk:c:single-crossing-monroe-cc], and Skowron and Faliszewski [@sko-fal:t:max-cover]. These are showing the hardness in case of the Borda dissatisfaction function, obtain results on parameterized hardness of the two rules, and results on hardness (or easiness) for the cases where the profiles are single-peaked or single-crossing. Further, Lu and Boutilier [@budgetSocialChoice] initiated the study of approximability for the Chamberlin–Courant rule (and were the first to use satisfaction-based framework). Specifically, they gave the $(1-\frac{1}{e})$-approximation algorithm for the Chamberlin–Courant rule. The motivation of Lu and Boutilier was coming from the point of view of recommendation systems and, in that sense, our view of the rules is quite similar to theirs.
In this paper we take the view that the Monroe and Chamberlin–Courant rules are special cases of the following resource allocation problem. The alternatives are shareable resources, each with a certain capacity defined as the maximal number of agents that may share this resource. Each agent has preferences over the resources and is interested in getting exactly one. The goal is to select a predetermined number $K$ of resources and to find an optimal allocation of these resources (see Section \[sec:prelim\] for details). This provides a unified framework for the two rules and reveals the connection of proportional representation problem to other resource allocation problems. In particular, it closely resembles multi-unit resource allocation with single-unit demand [@ley-sho:b:multiagent-systems Chapter 11] (see also the work of Chevaleyre et al. [@Chevaleyre06issuesin] for a survey of the most fundamental issues in the multiagent resource allocation theory) and resource allocation with sharable indivisible goods [@Chevaleyre06issuesin; @AiriauEndrissAAMAS2010]. Below, we point out other connections of the Monroe and Chamberlin–Courant rules to several other problems.
Facility Location Problems.
: In the facility location problem, there are $n$ customers located in some area and an authority, say a city council, that wants to establish a fixed number $k$ of facilities to serve those customers. Customers incur certain costs (say transportation costs) of using the facilities. Further, setting up a facility costs as well (and this cost may depend on the facility’s location). The problem is to find $k$ locations for the facilities that would minimize the total (societal) cost. If these facilities have infinite capacities and can serve any number of customers, then each customer would use his/her most preferred (i.e., closest) facility and the problem is similar to finding the Chamberlin–Courant assignment. If the capacities of the facilities are finite and equal, the problem looks like finding an assignment in the Monroe rule. An essential difference between the two problems are the setup costs and the distance metric. The parameterized complexity of the Facility Location Problem was investigated in Fellows and Fornau [@FF11]. The papers of Procaccia et al. [@complexityProportionalRepr] and of Betzler et al. [@fullyProportionalRepr] contain a brief discussion of the connection between the Facility Location Problem and the winner determination problem under the Chamberlin–Courant rule.
Group Activity Selection Problem.
: In the group activity selection problem [@dar-elk-kur-lan-sch-woe:t:group-activity] we have a group of agents (say, conference attendees) and a set of activities (say, options that they have for a free afternoon such as a bus city tour or wine tasting). The agents express preferences regarding the activities and organisers try to allocate agents to activities to maximise their total satisfaction. If there are $m$ possible activities but only $k$ must be chosen by organisers, then we are in the Chamberline-Courant framework, if all activities can take all agents, and in the Monroe framework, if all activities have the same capacities. The difference is that those capacities may be different and also that in the Group Activity Selection Problem we may allow expression of more complicated preferences. For example, an agent may express the following preference “I like wine-tasting best provided that at most $10$ people participate in it, and otherwise I prefer a bus city tour provided that at least $15$ people participate, and otherwise I prefer to not take part in any activity”. The Group Activity Selection Problem is more general than the winner determination in the Monroe and Chamberline-Courant rules. Some hardness and easiness results for this problem were obtained in [@dar-elk-kur-lan-sch-woe:t:group-activity], but the investigation of this problem has only started.
Coalition Structure Generation.
: It is quite natural to view winner determination under the Monroe and Chamberlin–Courant rules as coalition structure generation problems in cooperative games. Here the voters are the players of a certain cooperative game, and the value of each “coalition” of voters is the (dis)satisfaction they derive from being assigned to a given candidate. The goal is to partition the set of the voters into $K$ disjoint coalitions (and assign distinct candidates to them) in a way that maximizes the sum of coalition values (or, in the egalitarian setting, the value of the worst off coalition). Formally, this is a very special case of coalition structure generation with externalities. Externalities come from the fact that no two coalitions can be assigned to the same candidate. (However, Monroe [@monroeElection] and Betzler et al. [@fullyProportionalRepr] mention variants of Monroe’s rule where each “coalition of voters” can be simply assigned to its most preferred candidate; winner determination in this case maps to a coalition structure generation problem without externalities.) It would be very interesting to compare the efficiency of algorithms developed for coalition structure generation with that of (exact) algorithms for the Monroe and Chamberlin–Courant rules.
The above connections show that, indeed, the complexity of winner determination under the Monroe and Chamberlin–Courant voting rules are interesting, can lead to progress in several other directions, and may have impact on other applications of artificial intelligence.
Preliminaries {#sec:prelim}
=============
We first define basic notions such as preference orders and positional scoring rules. Then we present our Resource Allocation Problem in full generality and discuss which restrictions of it correspond to the winner determination problem for the Monroe and Chamberlin–Courant voting rules. Finally, we briefly recall relevant notions regarding computational complexity.
**Preferences.**For each $n \in {{{\mathbb{N}}}}$, by $[n]$ we mean $\{1, \ldots, n\}$. We assume that there is a set $N = [n]$ of *agents* and a set $A
= \{a_{1}, \dots a_{m}\}$ of *alternatives*. Each alternative $a\in A$ has the *capacity* ${{{{\mathrm{cap}}}}}_a \in {{{\mathbb{N}}}}$, which gives the total number of agents that can be assigned to it. Further, each agent $i$ has a *preference order* $\succ_i$ over $A$, i.e., a strict linear order of the form $a_{\pi(1)} \succ_{i} a_{\pi(2)} \succ_{i} \dots
\succ_{i} a_{\pi(m)}$ for some permutation $\pi$ of $[m]$. For an alternative $a$, by ${{{{\mathrm{pos}}}}}_i(a)$ we mean the position of $a$ in the $i$’th agent’s preference order. For example, if $a$ is the most preferred alternative for $i$ then ${{{{\mathrm{pos}}}}}_i(a) = 1$, and if $a$ is the least preferred one then ${{{{\mathrm{pos}}}}}_i(a) = m$. A collection $V = (\succ_1,
\ldots, \succ_n)$ of agents’ preference orders is called a *preference profile*.
We will often include subsets of the alternatives in the descriptions of preference orders. For example, if $A$ is the set of alternatives and $B$ is some nonempty strict subset of $A$, then by $B \succ A-B$ we mean that for the preference order $\succ$ all alternatives in $B$ are preferred to those outside of $B$.
A *positional scoring function* (PSF) is a function $\alpha^m \colon
[m] \rightarrow {{{\mathbb{N}}}}$. A PSF $\alpha^m$ is an *increasing positional scoring function* (IPSF) if for each $i,j \in [m]$, if $i
< j$ then $\alpha^m(i) < \alpha^m(j)$. Analogously, a PSF $\alpha^m$ is a *decreasing positional scoring function* (DPSF) if for each $i,j \in [m]$, if $i < j$ then $\alpha^m(i) > \alpha^m(j)$.
Intuitively, if $\beta^m$ is an IPSF then $\beta^m(i)$ can represent the *dissatisfaction* that an agent suffers when assigned to an alternative that is ranked $i$’th in his or her preference order. Thus, we assume that for each IPSF $\beta^m$ it holds that $\beta^m(1) = 0$ (an agent is not dissatisfied by her top alternative). Similarly, a DPSF $\gamma^m$ measures an agent’s satisfaction and we assume that for each DPSF $\gamma^m$ it holds that $\gamma^m(m)=0$ (an agent is completely not satisfied being assigned his or her least desired alternative). Sometimes we write $\alpha$ instead of $\alpha^m$, when it cannot lead to a confusion.
We will often speak of families $\alpha$ of IPSFs (DPSFs) of the form $\alpha=(\alpha^m)_{m \in {{{\mathbb{N}}}}}$, where $\alpha^m \mbox{ is a PSF on }[m]$, such that:
1. For a family of IPSFs it holds that $ \alpha^{m+1}(i) = \alpha^m(i)$ for all $m \in {{{\mathbb{N}}}}$ and $i\in [m]$.
2. For a family of DPSFs it holds that $ \alpha^{m+1}(i+1) = \alpha^m(i)$ for all $m \in {{{\mathbb{N}}}}$ and $i\in [m]$.
In other words, we build our families of IPSFs (DPSFs) by appending (prepending) values to functions with smaller domains. To simplify notation, we will refer to such families of IPSFs (DPSFs) as *normal* IPSFs (normal DPSFs). We assume that each function $\alpha^m$ from a family can be computed in polynomial time with respect to $m$. Indeed, we are particularly interested in the Borda families of IPSFs and DPSFs defined by $\alpha^{m}_{{{{{\mathrm{B, inc}}}}}}(i) = i-1$ and $\alpha^{m}_{{{{{\mathrm{B, dec}}}}}}(i) = m - i$, respectively.
**Assignment functions.**A $K$-*assignment function* is any function $\Phi \colon N \rightarrow A$, such that $\|\Phi(N)\| \leq K$ (that is, it matches agents to at most $K$ alternatives), and such that for every alternative $a \in A$ we have that $\|\Phi^{-1}(a)\| \leq {{{{\mathrm{cap}}}}}_a$ (i.e., the number of agents assigned to $a$ does not exceed $a$’s capacity ${{{{\mathrm{cap}}}}}_a$).
We will also consider partial assignment functions. A partial $K$-assignment function is defined in the same way as a regular one, except that it may assign a null alternative, $\bot$, to some of the agents. It is convenient to think that for each agent $i$ we have ${{{{\mathrm{pos}}}}}_i(\bot) = m$. In general, it might be the case that a partial $K$-assignment function cannot be extended to a regular one. This may happen, for example, if the partial assignment function uses $K$ alternatives whose capacities sum to less than the total number of voters. However, in the context of Chamberlin–Courant and Monroe rules it is always possible to extend a partial $K$-assignment function to a regular one.
Given a normal IPSF (DPSF) $\alpha$, we may consider the following three functions, each assigning a positive integer to any assignment $\Phi$: $$\begin{aligned}
\ell_{1}^{\alpha}(\Phi) &= \sum_{i=1}^{n}\alpha({{{{\mathrm{pos}}}}}_{i}(\Phi(i))), \\
\ell_{\infty}^{\alpha}(\Phi) &= \mathrm{max}_{i = 1}^{n}\alpha({{{{\mathrm{pos}}}}}_{i}(\Phi(i))), \\
\ell_{\min}^{\alpha}(\Phi) &= \mathrm{min}_{i = 1}^{n}\alpha({{{{\mathrm{pos}}}}}_{i}(\Phi(i))) \textrm{.}\end{aligned}$$ These functions are built from individual dissatisfaction (satisfaction) functions, so that they can measure the quality of the assignment for the whole society. In the utilitarian framework the first one can be viewed as a *total (societal) dissatisfaction function* in the IPSF case and a *total (societal) satisfaction function* in the DPSF case. The second and the third can be used, respectively, as a total dissatisfaction and satisfaction functions for IPSF and DPSF cases in the egalitarian framework. We will omit the word total if no confusion may arise.
For each subset of the alternatives $S \subseteq A$ such that $\|S\|
\leq K$, we denote as $\Phi^S_{\alpha}$ the partial $K$-assignment that assigns agents only to the alternatives from $S$ and such that $\Phi^S_{\alpha}$ maximizes the utilitarian satisfaction $\ell_{1}^{\alpha}(\Phi^S_{\alpha})$. (We introduce this notation only for the utilitarian satisfaction-based setting because it is useful to express appropriate algorithms for this case; for other settings we have hardness results only and this notation would not be useful.)
**The Resource Allocation Problem.**Let us now define the resource allocation problem that forms the base of our study. This problem stipulates finding an optimal $K$-assignment function, where the optimality is relative to one of the total dissatisfaction or satisfaction functions that we have just introduced. The former is to be minimized and the latter is to be maximized.
\[def:assignment\] Let $\alpha$ be a normal IPSF. An instance of $\alpha$-<span style="font-variant:small-caps;">DU-Assignment</span> problem (i.e., of the disatisfaction-based utilitarian assignment problem) consists of a set of agents $N = [n]$, a set of alternatives $A = \{a_1, \ldots
a_m\}$, a preference profile $V$ of the agents, and a sequence $({{{{\mathrm{cap}}}}}_{a_1}, \ldots, {{{{\mathrm{cap}}}}}_{a_m})$ of alternatives’ capacities. We ask for an assignment function $\Phi$ such that: (1) $\|\Phi(N)\| \leq K$; (2) $ \|\Phi^{-1}(a) \| \leq {{{{\mathrm{cap}}}}}_{a}$ for all ${a \in A}$; and (3) $\ell_{1}^\alpha(\Phi)$ is minimized.
Problem $\alpha$-<span style="font-variant:small-caps;">SU-Assignment</span> (the satisfaction-based utilitarian assignment problem) is defined identically except that $\alpha$ is a normal DPSF and condition (3) is replaced with “($3'$) $\ell_{1}^\alpha(\Phi)$ is maximal.” If we replace $\ell^\alpha_1$ with $\ell_\infty^\alpha$ in $\alpha$-<span style="font-variant:small-caps;">DU-Assignment</span> then we obtain problem $\alpha$-<span style="font-variant:small-caps;">DE-Assignment</span>, i.e., the dissatisfaction-based egalitarian variant. If we replace $\ell^\alpha_1$ with $\ell_{\min}^\alpha$ in $\alpha$-<span style="font-variant:small-caps;">SU-Assignment</span> then we obtain problem $\alpha$-<span style="font-variant:small-caps;">SE-Assignment</span>, i.e., the satisfaction-based egalitarian variant.
Our four problems can be viewed as generalizations of the winner determination problem for the Monroe [@monroeElection] and Chamberlin–Courant [@ccElection] multiwinner voting systems (see the introduction for their definitions). To model the Monroe system, it suffices to set the capacity of each alternative to be $
\frac{\|N\|}{K}$ (for simplicity, throughout the paper we assume that $K$ divides $\|N\|$[^2]). We will refer to thus restricted variants of our problems as the <span style="font-variant:small-caps;">Monroe</span> variants. To represent the Chamberlin–Courant system, we set alternatives’ capacities to $\|N\|$. We will refer to the so-restricted variants of our problems as <span style="font-variant:small-caps;">CC</span> variants.
**Computational Issues**.For many normal IPSFs $\alpha$ and, in particular, for the Borda IPSF, even the above-mentioned restricted versions of the Resource Allocation Problem, namely, $\alpha$-<span style="font-variant:small-caps;">DU-Monroe</span>, $\alpha$-<span style="font-variant:small-caps;">DE-Monroe</span>, $\alpha$-<span style="font-variant:small-caps;">DU-CC</span>, and $\alpha$-<span style="font-variant:small-caps;">DE-CC</span> are ${{\mathrm{NP}}}$-complete [@fullyProportionalRepr; @complexityProportionalRepr] (the same holds for the satisfaction-based variants of the problems). Thus we seek approximate solutions.
Let $r$ be a real number such that $r \geq 1$ ($0 < r \leq 1$) and let $\alpha$ be a normal IPSF (a normal DPSF). An algorithm is an $r$-approximation algorithm for $\alpha$-<span style="font-variant:small-caps;">DU-Assignment</span> problem (for $\alpha$-<span style="font-variant:small-caps;">SU-Assignment</span> problem) if on each instance $I$ it returns a feasible assignment $\Phi$ such that $\ell_{1}^{\alpha}(\Phi) \leq r \cdot
{{{{\mathrm{OPT}}}}}$ (such that $\ell_{1}^{\alpha}(\Phi) \geq r \cdot
{{{{\mathrm{OPT}}}}}$), where ${{{{\mathrm{OPT}}}}}$ is the optimal total dissatisfaction (satisfaction) $\ell_{1}^{\alpha}(\Phi_{{{{\mathrm{OPT}}}}})$.
We define $r$-approximation algorithms for the egalitarian variants analogously. Lu and Boutilier [@budgetSocialChoice] gave a $(1 -
\frac{1}{e})$-approximation algorithm for the <span style="font-variant:small-caps;">SU-CC</span> family of problems.
Throughout this paper, we will consider each of the <span style="font-variant:small-caps;">Monroe</span> and <span style="font-variant:small-caps;">CC</span> variants of the problem and for each we will either prove inapproximability with respect to any constant $r$ (under standard complexity-theoretic assumptions) or we will present an approximation algorithm. In our inapproximability proofs, we will use the following two classic ${{\mathrm{NP}}}$-complete problems [@gar-joh:b:int].
An instance $I$ of <span style="font-variant:small-caps;">Set-Cover</span> consists of set $U = [n]$ (called the ground set), family ${{{\mathcal{F}}}}= \{F_{1}, F_{2}, \dots,
F_{m}\}$ of subsets of $U$, and positive integer $K$. We ask if there exists a set $I \subseteq [m]$ such that $\|I\|
\leq K$ and $\bigcup_{i\in I}F_i = U$.
<span style="font-variant:small-caps;">X3C</span> is a variant of <span style="font-variant:small-caps;">Set-Cover</span> where $\|U\|$ is divisible by $3$, each member of ${{{\mathcal{F}}}}$ has exactly three elements, and $K = \frac{\|U\|}{3}$.
<span style="font-variant:small-caps;">Set-Cover</span> remains ${{\mathrm{NP}}}$-complete even if we restrict each member of $U$ to be contained in at most two sets from ${{{\mathcal{F}}}}$ (it suffices to note that this restriction is satisfied by <span style="font-variant:small-caps;">Vertex-Cover</span>, which is a special case of <span style="font-variant:small-caps;">Set-Cover</span>). <span style="font-variant:small-caps;">X3C</span> remains ${{\mathrm{NP}}}$-complete even if we additionally assume that $n$ is divisible by $2$ and each member of $U$ appears in at most $3$ sets from ${{{\mathcal{F}}}}$ [@gar-joh:b:int].
We will also use results from the theory of parameterized complexity developed by Downey and Fellows [@DF99]. This theory allows to single out a particular parameter of the problem, say $k$, and analyze its ‘contribution’ to the overall complexity of the problem. An analogue of the class ${{\mathrm{P}}}$ here is the class ${{\mathrm{FPT}}}$ which is the class of problems that can be solved in time $f(k)n^{O(1)}$, where $n$ is the size of the input instance, and $f$ is some computable function (for a fixed $k$ everything gets polynomial). Parameterized complexity theory also operates with classes ${{\mathrm{W[1]}}}\subseteq {{\mathrm{W[2]}}}\subseteq \cdots$ which are believed to form a hierarchy of classes of *hard* problems (combined, they are analogous to the class ${{\mathrm{NP}}}$). It holds that ${{\mathrm{FPT}}}\subseteq {{\mathrm{W[1]}}}$, but it seems unlikely that ${{\mathrm{FPT}}}= {{\mathrm{W[1]}}}$, let alone ${{\mathrm{FPT}}}= {{\mathrm{W[2]}}}$. We point the reader to the books of Niedermeier [@nie:b:invitation-fpt] and Flum and Grohe [@flu-gro:b:parameterized-complexity] for detailed overviews of parametrized complexity theory. Interestingly, while both <span style="font-variant:small-caps;">Set-Cover</span> and <span style="font-variant:small-caps;">Vertex-Cover</span> are ${{\mathrm{NP}}}$-complete, the former is ${{\mathrm{W[2]}}}$-complete and the latter belongs to ${{\mathrm{FPT}}}$ (see, e.g., the book of Niedermeier [@nie:b:invitation-fpt] for these now-standard results and their history).
Hardness of Approximation {#sec:approximation}
=========================
We now present our inapproximability results for the Monroe and Chamberlin–Courant rules. Specifically, we show that there are no constant-factor approximation algorithms for the dissatisfaction-based variants of the rules (both utilitarian and egalitarian) and for the satisfaction-based egalitarian ones.
Naturally, these inapproximability results carry over to more general settings. For example, unless ${{\mathrm{P}}}= {{\mathrm{NP}}}$, there are no polynomial-time constant-factor approximation algorithms for the general dissatisfaction-based Resource Allocation Problem. On the other hand, our results do not preclude good satisfaction-based approximation algorithms for the utilitarian case and, indeed, in Section \[sec:algorithms\] we provide such algorithms.
\[theorem:noApprox1\] For each normal IPSF $\alpha$ and each constant factor $r > 1$, there is no polynomial-time $r$-approximation algorithm for $\alpha$-<span style="font-variant:small-caps;">DU-Monroe</span> unless ${{\mathrm{P}}}= {{\mathrm{NP}}}$.
Let us fix a normal IPSF $\alpha$ and let us assume, aiming at getting a contradiction, that there is some constant $r > 1$ and a polynomial-time $r$-approximation algorithm ${{{\mathcal{A}}}}$ for $\alpha$-<span style="font-variant:small-caps;">DU-Monroe</span>.
Let $I$ be an instance of <span style="font-variant:small-caps;">X3C</span> with ground set $U = [n]$ and family ${{{\mathcal{F}}}}= \{F_{1}, F_{2}, \dots, F_{m}\}$ of $3$-element subsets of $U$. Without loss of generality, we assume that $n$ is divisible by both $2$ and $3$ and that each member of $U$ appears in at most 3 sets from ${{{\mathcal{F}}}}$.
Using $I$, we build instance $I_{M}$ of $\alpha$-<span style="font-variant:small-caps;">DU-Monroe</span> as follows. We set $N = U$ (that is, the elements of the ground set are the agents) and we set $A
= A_1 \cup A_2$, where $A_1 = \{a_1, \ldots, a_m\}$ is a set of alternatives corresponding to the sets from the family ${{{\mathcal{F}}}}$ and $A_2$ is a set of dummy alternatives of cardinality $\|A_2\| = \frac{1}{2}{n^{2}r\cdot \alpha(3)}$, needed for the construction. We let $m' =
\|A_2\|$ and rename the alternatives in $A_2$ so that $A_2 =
\{b_1, \ldots, b_{m'}\}$. We set $K = \frac{n}{3}$.
We build agents’ preference orders using the following algorithm. For each $j \in N$, set $M_f(j) = \{ a_i \mid j \in F_i\}$ and $M_l
= \{ a_i \mid j \not\in F_i \}$. Set $m_f(j) = \|M_f(j)\|$ and $m_l(j) = \|M_l(j)\|$. As the frequency of the elements from $U$ is bounded by 3, we have $m_f(j) \leq 3$. For each agent $j$ we set his or her preference order to be of the form $M_f(j) \succ_j A_2 \succ_j
M_l(j)$, where the alternatives in $M_f(j)$ and $M_l(j)$ are ranked in an arbitrary way and the alternatives from $A_2$ are placed at positions $m_{f}(j) + 1, \dots, m_{f}(j) + m'$ in the way described below (see Figure \[fig:diag1\] for a high-level illustration of the construction).
![The alignment of the positions in the preference orders of the agents. The positions are numbered from the left to the right. The left wavy line shows the positions $m_{f}(\cdot)$, each no greater than $3$. The right wavy line shows the positions $m_{l}(\cdot)$, each higher than $nr \cdot \alpha(3)$. The alternatives from $A_{2}$ (positions of one such an alternative is illustrated with the circle) are placed only between the peripheral wavy lines. Each alternative from $A_{2}$ is placed on the left from the middle wavy line exactly 2 times, thus each such alternative is placed on the left from the right dashed line no more than $2$ times (exactly two times at the figure).[]{data-label="fig:diag1"}](diagram1)
We place the alternatives from $A_{2}$ in the preference orders of the agents in such a way that for each alternative $b_i \in A_{2}$ there are at most two agents that rank $b_i$ among their $nr \cdot
\alpha(3)$ top alternatives. The following construction achieves this effect. If $(i + j) \,\mathrm{mod}\, n < 2$, then alternative $b_{i}$ is placed at one of the positions $m_{f}(j) + 1,
\dots, m_{f}(j) + nr \cdot \alpha(3)$ in $j$’s preference order. Otherwise, $b_i$ is placed at a position with index higher than $m_{f}(j) + nr \cdot \alpha(3)$ (and, thus, at a position higher than $nr \cdot \alpha(3)$). This construction can be implemented because for each agent $j$ there are exactly $m' \cdot \frac{2}{n} = nr \cdot \alpha(3)$ alternatives $b_{i_{1}}, b_{i_{2}}, b_{i_{n \alpha(3) r}}$ such that $(i_{k} + j) \,\mathrm{mod}\, n < 2$.
Let $\Phi$ be an assignment computed by ${{{\mathcal{A}}}}$ on $I_{M}$. We will show that $\ell_{1}^{\alpha}(\Phi) \leq n \cdot \alpha(3) \cdot
r$ if and only if $I$ is a *yes*-instance of <span style="font-variant:small-caps;">X3C</span>.
($\Leftarrow$) If there exists a solution for $I$ (i.e., an exact cover of $U$ with $\frac{n}{3}$ sets from ${{{\mathcal{F}}}}$), then we can easily show an assignment in which each agent $j$ is assigned to an alternative from the top $m_{f}(j)$ positions of his or her preference order (namely, one that assigns each agent $j$ to the alternative $a_i \in A_1$ that corresponds to the set $F_i$, from the exact cover of $U$, that contains $j$). Thus, for the optimal assignment $\Phi_{{{{\mathrm{OPT}}}}}$ it holds that $\ell_{1}^{\alpha}(\Phi_{{{{\mathrm{OPT}}}}}) \leq
\alpha(3) \cdot n$. In consequence, $\mathcal{A}$ must return an assignment with the total dissatisfaction at most $nr \cdot \alpha(3)$.
($\Rightarrow$) Let us now consider the opposite direction. We assume that ${{{\mathcal{A}}}}$ found an assignment $\Phi$ such that $\ell_{1}^{\alpha}(\Phi) \leq nr \cdot \alpha(3)$ and we will show that $I$ is a *yes*-instance of <span style="font-variant:small-caps;">X3C</span>. Since we require each alternative to be assigned to either $0$ or $3$ agents, if some alternative $b_{i}$ from $A_2$ were assigned to some $3$ agents, at least one of them would rank $b_i$ at a position worse than $nr \cdot \alpha(3)$. This would mean that $\ell_{1}^{\alpha}(\Phi) \geq nr \cdot \alpha(3) +
1$. Analogously, no agent $j$ can be assigned to an alternative that is placed at one of the $m_l(j)$ bottom positions of $j$’s preference order. Thus, only the alternatives in $A_1$ have agents assigned to them and, further, if agents $x$, $y$, $z$, are assigned to some $a_i \in A_1$, then it holds that $F_i = \{x,y,z\}$ (we will call each set $F_i$ for which alternative $a_i$ is assigned to some agents $x,y,z$ *selected*). Since each agent is assigned to exactly one alternative, the selected sets are disjoint. Since the number of selected sets is $K = \frac{n}{3}$, it must be the case that the selected sets form an exact cover of $U$. Thus, $I$ is a *yes*-instance of <span style="font-variant:small-caps;">X3C</span>.
One may wonder if hardness of approximation for $\alpha$-<span style="font-variant:small-caps;">DU-Monroe</span> is not an artifact of the strict requirements regarding the number of chosen candidates. It turns out that unless ${{\mathrm{P}}}= {{\mathrm{NP}}}$, there is no $r$-$s$-approximation algorithm that finds an assignment with the following properties: (1) the aggregated dissatisfaction $\ell_{1}^{\alpha}(\Phi)$ is at most $r$ times higher than the optimal one, (2) the number of alternatives to which agents are assigned is at most $s K$ and (3) each selected alternative (the alternative that has agents assigned), is assigned to no more than $s \lceil \frac{n}{K}
\rceil$ and no less than $\frac{1}{s} \lceil \frac{n}{K}\rceil$ agents. (The proof is similar to the one used for Theorem \[theorem:noApprox1\].) Thus, in our further study we do not consider such relaxations of the problem.
\[theorem:noApprox2\] For each normal IPSF $\alpha$ and each constant $r > 1$, there is no polynomial-time $r$-approximation algorithm for $\alpha$-<span style="font-variant:small-caps;">DE-Monroe</span> unless ${{\mathrm{P}}}= {{\mathrm{NP}}}$.
The proof of Theorem \[theorem:noApprox1\] applies to this case as well. In fact, it even suffices to take $m' = \|A_2\| = \frac{1}{2}nr \cdot \alpha(3)$.
Results analogous to Theorems \[theorem:noApprox1\] and \[theorem:noApprox2\] hold for the <span style="font-variant:small-caps;">DU-CC</span> family of problems as well.
\[theorem:noApprox3\] For each normal IPSF $\alpha$ and each constant factor $r > 1$, there is no polynomial-time $r$-approximation algorithm for $\alpha$-<span style="font-variant:small-caps;">DU-CC</span> unless ${{\mathrm{P}}}= {{\mathrm{NP}}}$.
Let us fix a normal IPSF $\alpha$. For the sake of contradiction, let us assume that there is some constant $r > 1$, and a polynomial-time $r$-approximation algorithm ${{{\mathcal{A}}}}$ for $\alpha$-<span style="font-variant:small-caps;">DU-CC</span>. We will show that it is possible to use ${{{\mathcal{A}}}}$ to solve the ${{\mathrm{NP}}}$-complete <span style="font-variant:small-caps;">Vertex-Cover</span> problem.
Let $I = (U, {{{\mathcal{F}}}}, K)$ be an instance of <span style="font-variant:small-caps;">Vertex-Cover</span>, where $U = [n]$ is the ground set, ${{{\mathcal{F}}}}= \{F_1, \ldots, F_m\}$ is a family of subsets of $U$ (where each member of $U$ belongs to exactly two sets in ${{{\mathcal{F}}}}$), and $K$ is a positive integer.
Given $I$, we construct an instance $I_{CC}$ of $\alpha$-<span style="font-variant:small-caps;">DU-CC</span> as follows. The set of agents is $N = U$ and the set of alternatives is $A =
\bigcup_{j=1}^m A_j$, where each $A_j$ contains exactly $\alpha(2)
\cdot r \cdot n$ (unique) alternatives. Intuitively, for each $j\in [m]$, the alternatives in $A_j$ correspond to the set $F_j$. For each $A_j$, $1 \leq j \leq m$, we pick one alternative, which we denote $a_j$. For each agent $i \in N$, we set $i$’s preference order as follows: Let $F_j$ and $F_k$, $j < k$, be the two sets that contain $i$. Agent $i$’s preference order is of the form $a_j \succ_i a_k \succ_i A_k - \{a_k\} \succ_i A - (A_k \cup
\{a_j,a_k\})$ (a particular order of alternatives in the sets $A_k-\{a_k\}$ and $A - (A_k \cup \{a_j,a_k\})$ is irrelevant for the construction). We ask for an assignment of the agents to at most $K$ alternatives.
Let us consider a solution $\Phi$ returned by $\mathcal{A}$ on input $I_{CC}$. We claim that $\ell_{1}^{\alpha}(\Phi) \leq nr \cdot
\alpha(2)$ if and only if $I$ is a *yes*-instance of <span style="font-variant:small-caps;">Vertex-Cover</span>.
($\Leftarrow$) If $I$ is a *yes*-instance then, clearly, each agent $i$ can be assigned to one of the top two alternatives in his or her preference order (if there is a size-$K$ cover, then this assignment selects at most $K$ candidates). Thus the total dissatisfaction of an optimal assignment is at most $n\cdot \alpha(2)$. As a result, the solution $\Phi$ returned by ${{{\mathcal{A}}}}$ has total dissatisfaction at most $nr\cdot \alpha(2)$.
($\Rightarrow$) If $\mathcal{A}$ returns an assignment with total dissatisfaction no greater than $nr\cdot \alpha(2)$, then, by the construction of agents preference orders, we see that each agent $i$ was assigned to an alternative from a set $A_j$ such that $i \in F_j$. Since the assignment can use at most $K$ alternatives, this directly implies that there is a size-$K$ cover of $U$ with sets from ${{{\mathcal{F}}}}$.
\[theorem:noApprox4\] For each normal IPSF $\alpha$ and each constant factor $r > 1$, there is no polynomial-time $r$-approximation algorithm for $\alpha$-<span style="font-variant:small-caps;">DE-CC</span> unless ${{\mathrm{P}}}= {{\mathrm{NP}}}$.
The proof of Theorem \[theorem:noApprox3\] is applicable in this case as well. In fact, it even suffices to take the $m$ groups of alternatives, $A_1, \ldots, A_m$, to contain $\alpha(2) \cdot r$ alternatives each.
The above results show that approximating algorithms for finding the minimal dissatisfaction of agents is difficult. On the other hand, if we focus on agents’ total satisfaction then constant-factor approximation exist in many cases (see, e.g., the work of Lu and Boutilier [@budgetSocialChoice] and the next section). Yet, if we focus on the satisfaction of the least satisfied voter, there are no efficient constant-factor approximation algorithms for the Monroe and Chamberlin–Courant systems. (However, note that our result for the Monroe setting is more general than the result for the Chamberlin–Courant setting; the latter is for the Borda DPSF only.)
\[theorem:noApprox5\] For each normal DPSF $\alpha$ (where each entry is polynomially bounded in the number of alternatives) and each constant factor $r$, with $0 < r \leq 1$, there is no $r$-approximation algorithm for $\alpha$-<span style="font-variant:small-caps;">SE-Monroe</span> unless ${{\mathrm{P}}}= {{\mathrm{NP}}}$.
Let us fix a DPSF $\alpha=(\alpha^m)_{m\in {{{\mathbb{N}}}}}$, where each entry $\alpha^m$ is polynomially bounded in the number of alternatives $m$. For the sake of contradiction, let us assume that for some $r$, $0 < r \leq 1$, there is a polynomial-time $r$-approximation algorithm $\mathcal{A}$ for $\alpha$-<span style="font-variant:small-caps;">SE-Monroe</span>. We will show that the existence of this algorithm implies that <span style="font-variant:small-caps;">X3C</span> is solvable in polynomial time.
Let $I$ be an <span style="font-variant:small-caps;">X3C</span> instance with ground set $U = \{1, 2,
\dots, n\}$ and collection ${{{\mathcal{F}}}}= \{F_{1}, \dots, F_{m}\}$ of subsets of $U$. Each set in ${{{\mathcal{F}}}}$ has cardinality three. Further, without loss of generality, we can assume that $n$ is divisible by three and that each $i \in U$ appears in at most three sets from ${{{\mathcal{F}}}}$. Given $I$, we form an instance $I_M$ of $\alpha$-<span style="font-variant:small-caps;">SE-Monroe</span> as follows. Let $n'
= 3 \cdot (\alpha^{m+1}(1) \cdot \lceil \frac{1 - r}{r}\rceil +
3)$. The set $N$ of agents is partitioned into two subsets, $N_1$ and $N_2$. $N_1$ contains $n$ agents (intuitively, corresponding to the elements of the ground set $U$) and $N_2$ contains $n'$ agents (used to enforce certain properties of the solution). The set of alternatives $A$ is partitioned into two subsets, $A_1$ and $A_2$. We set $A_1 = \{a_1, \ldots, a_m\}$ (members of $A_1$ correspond to the sets in ${{{\mathcal{F}}}}$), and we set $A_2 = \{b_1, \ldots, b_{m'}\}$, where $m' = \frac{n'}{3}$.
For each $j$, $1 \leq j \leq n$, we set $M_f(j) = \{ a_i \mid j \in
F_i\}$. For each $j$, $1 \leq j \leq n$, we set the preference order of the $j$’th agent in $N_1$ to be of the form $$M_f(j) \succ A_2 \succ A_1 - M_f(j).$$ Note that by our assumptions, $\|M_f(j)\| \leq 3$. For each $j$, $1 \leq j \leq n'$, we set the preference order of the $j$’th agent in $N_2$ to be of the form $$b_{\left\lceil\frac{j}{3}\right\rceil} \succ A_2 -
\{b_{\left\lceil\frac{j}{3}\right\rceil}\} \succ A_1.$$ Note that each agent in $N_2$ ranks the alternatives from $A_1$ in positions $m'+1, \ldots, m'+m$. Finally, we set the number of candidates that can be selected to be $K =
\frac{n+n'}{3}$.
Now, consider the solution $\Phi$ returned by $\mathcal{A}$ on $I_{M}$. We will show that $\ell_{\infty}^{\alpha^{m +
m'}}(\Phi) \leq$ $r\alpha^{m + m'}(3)$ if and only if $I$ is a *yes*-instance of <span style="font-variant:small-caps;">X3C</span>.
($\Leftarrow$) If there exists an exact set cover of $U$ with sets from ${{{\mathcal{F}}}}$, then it is easy to construct a solution for $I_M$ where the satisfaction of each agent is greater or equal to $r\cdot\alpha^{m + m'}(3)$. Let $I \subseteq \{1, \ldots, m\}$ be a set such that $\bigcup_{i \in I}F_i = U$ and $\|I\| =
\frac{n}{3}$. We assign each agent $j$ from $N_1$ to the alternative $a_i$ such that (a) $i \in I$ and (b) $j \in F_i$, and we assign each agent from $N_2$ to his or her most preferred alternative. Thus, Algorithm ${{{\mathcal{A}}}}$ has to return an assignment with the minimal satisfaction greater or equal to $r\cdot\alpha^{m + m'}(3)$.
($\Rightarrow$) For the other direction, we first show that $r\cdot\alpha^{m + m'}(3) \geq \alpha^{m + m'}(m')$. Since DPSFs are strictly decreasing, it holds that: $$\label{eq:1}
r\cdot\alpha^{m + m'}(3) \geq r\cdot(\alpha^{m + m'}(m') + m' - 3).$$ Then, by the definition of DPSFs, it holds that: $$\label{eq:2}
\alpha^{m + m'}(m') = \alpha^{m + 1}(1).$$ Using the fact that $m' = (\alpha^{m+1}(1) \cdot \lceil
\frac{1 - r}{r}\rceil + 3)$ and using , we can transform inequality to obtain the following: $$\begin{aligned}
r\cdot\alpha^{m + m'}(3) &\geq r\cdot(\alpha^{m + m'}(m') + m' - 3) \\
& = r\cdot\left(\alpha^{m + m'}(m') + (\alpha^{m+1}(1) \cdot \left\lceil\frac{1 - r}{r}\right\rceil + 3) - 3\right)\\
& \geq r\cdot\alpha^{m + m'}(m') + (1-r)\cdot \alpha^{m+1}(1) \\
& = r\cdot\alpha^{m + m'}(m') + (1-r)\cdot \alpha^{m + m'}(m') = \alpha^{m + m'}(m').
\end{aligned}$$ This means that if the minimal satisfaction of an agent is at least $r\cdot\alpha^{m + m'}(3)$, then no agent was assigned to an alternative that he or she ranked beyond position $m'$. If some agent $j$ from $N_{1}$ were assigned to an alternative from $A_{2}$, then, by the pigeonhole principle, some agent from $N_{2}$ would be assigned to an alternative from $A_{1}$. However, each agent in $N_2$ ranks the alternatives from $A_1$ beyond position $m'$ and thus such an assignment is impossible. In consequence, it must be that each agent in $j$ was assigned to an alternative that corresponds to a set $F_{i}$ in ${{{\mathcal{F}}}}$ that contains $j$. Such an assignment directly leads to a solution for $I$.
Let us now move on to the case of <span style="font-variant:small-caps;">SE-CC</span> family of problems. Unfortunately, in this case our inapproximability argument holds for the case of Borda DPSF only (though we believe that it can be adapted to other DPSFs as well). Further, in our previous theorems we were showing that existence of a respective constant-factor approximation algorithm implies that ${{\mathrm{NP}}}$ collapses to ${{\mathrm{P}}}$. In the following theorem we will show a seemingly weaker collapse of ${{\mathrm{W[2]}}}$ to ${{\mathrm{FPT}}}$.
To prove hardness of approximation for $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SE-CC</span>, we first prove the following simple lemma.
\[lemma:coveringSubsets\] Let $K, p, l$ be three positive integers and let $X$ be a set of cardinality $lpK$. There exists a family ${{{\mathcal{S}}}}= \{S_1, \ldots,
S_{\binom{lK}{K}} \}$ of $pK$-element subsets of $X$ such that for each $K$-element subset $B$ of $X$, there is a set $S_i \in {{{\mathcal{S}}}}$ such that $B \subseteq S_i$.
Set $X' = [lK]$ and let $Y'$ be a family of all $K$-element subsets of $X'$. Replace each element $i$ of $X'$ with $p$ new elements (at the same time replacing $i$ with the same $p$ elements within each set in $Y'$ that contains $i$). As a result we obtain two new sets, $X$ and $Y$, that satisfy the statement of the theorem (up to the renaming of the elements).
\[theorem:noApprox6\] Let $\alpha^{m}_{{{{{\mathrm{B, dec}}}}}}$ be the Borda DPSF ($\alpha^{m}_{{{{{\mathrm{B, dec}}}}}}(i) = m - i$). For each constant factor $r$, $0 < r \leq 1$, there is no $r$-approximation algorithm for $\alpha^{m}_{{{{{\mathrm{B, dec}}}}}}$-<span style="font-variant:small-caps;">SE-CC</span> unless ${{\mathrm{FPT}}}= {{\mathrm{W[2]}}}$.
For the sake of contradiction, let us assume that there is some constant $r$, $0 < r \leq 1$, and a polynomial-time $r$-approximation algorithm ${{{\mathcal{A}}}}$ for $\alpha^{m}_{{{{{\mathrm{B, dec}}}}}}$-<span style="font-variant:small-caps;">SE-CC</span>. We will show that the existence of this algorithm implies that <span style="font-variant:small-caps;">Set-Cover</span> is fixed-parameter tractable for the parameter $K$ (since <span style="font-variant:small-caps;">Set-Cover</span> is known to be ${{\mathrm{W[2]}}}$-complete for this parameter, this will imply ${{\mathrm{FPT}}}={{\mathrm{W[2]}}}$).
Let $I$ be an instance of <span style="font-variant:small-caps;">Set-Cover</span> with ground set $U =
[n]$ and family ${{{\mathcal{F}}}}= \{F_{1}, F_{2}, \dots, F_{m}\}$ of subsets of $U$. Given $I$, we build an instance $I_{CC}$ of $\alpha^{m}_{{{{{\mathrm{B, dec}}}}}}$-<span style="font-variant:small-caps;">SE-CC</span> as follows. The set of agents $N$ consists of $n$ subsets of agents, $N_1, \ldots, N_n$, where each group $N_i$ contains exactly $n' =
\binom{\left\lceil \frac{2}{r} \right\rceil K}{K}$ agents. Intuitively, for each $i$, $1 \leq i \leq n$, the agents in the set $N_{i}$ correspond to the element $i$ in $U$. The set of alternatives $A$ is partitioned into two subsets, $A_1$ and $A_2$, such that: (1) $A_1 = \{a_1, \ldots, a_m\}$ is a set of alternatives corresponding to the sets from the family ${{{\mathcal{F}}}}$, and (2) $A_2$, $\|A_2\| = \left\lceil \frac{2}{r}\right\rceil \left\lceil
\frac{m(1 + r)}{K} \right\rceil K$, is a set of dummy alternatives needed for our construction. We set $m' = \|A\| = m +
\|A_2\|$.
Before we describe the preference orders of the agents in $N$, we form a family $R = \{r_1, \ldots, r_{n'}\}$ of preference orders over $A_2$ that satisfies the following condition: For each $K$-element subset $B$ of $A_2$, there exists $r_j$ in $R$ such that all members of $B$ are ranked among the bottom $\left\lceil \frac{m(1 +
r)}{K} \right\rceil K$ positions in $r_j$. By Lemma \[lemma:coveringSubsets\], such a construction is possible (it suffices to take $l = \left\lceil \frac{2}{r}\right\rceil$ and $p =
\left\lceil \frac{m(1 + r)}{K} \right\rceil$); further, the proof of the lemma provides an algorithmic way to construct $R$.
We form the preference orders of the agents as follows. For each $i$, $1 \leq i \leq n$, set $M_f(i) = \{ a_t \mid i \in F_t\}$. For each $i$, $1 \leq i \leq n$, and each $j$, $1 \leq j \leq n'$, the $j$’th agent from $N_i$ has preference order of the form: $$M_f(i) \succ r_j \succ A_1 - M_f(i)$$ (we pick any arbitrary, polynomial-time computable order of candidates within $M_f(i)$ and $M_l(i)$).
Let $\Phi$ be an assignment computed by ${{{\mathcal{A}}}}$ on $I_{M}$. We will show that $\ell_{\infty}^{\alpha^{m'}_{{{{{\mathrm{B, dec}}}}}}}(\Phi) \geq
r\cdot(m' - m)$ if and only if $I$ is a *yes*-instance of <span style="font-variant:small-caps;">Set-Cover</span>.
($\Leftarrow$) If there exists a solution for $I$ (i.e., a cover of $U$ with $K$ sets from ${{{\mathcal{F}}}}$), then we can easily show an assignment where each agent is assigned to an alternative that he or she ranks among the top $m$ positions (namely, for each $j$, $1 \leq
j \leq n$, we assign all the agents from the set $N_j$ to the alternative $a_i \in A_1$ such that $j \in F_i$ and $F_i$ belongs to the alleged $K$-element cover of $U$). Under this assignment, the least satisfied agent’s satisfaction is at least $m'-m$ and, thus, ${{{\mathcal{A}}}}$ has to return an assignment $\Phi$ where $\ell_{\infty}^{\alpha^{m'}_{{{{{\mathrm{B, dec}}}}}}}(\Phi) \geq r\cdot(m' -
m)$.
($\Rightarrow$) Let us now consider the opposite direction. We assume that ${{{\mathcal{A}}}}$ found an assignment $\Phi$ such that $\ell_{\infty}^{\alpha^{m}_{{{{{\mathrm{B, dec}}}}}}}(\Phi) \geq r\cdot(m' -
m)$ and we will show that $I$ is a *yes*-instance of <span style="font-variant:small-caps;">Set-Cover</span>. We claim that for each $i$, $1 \leq i \leq n $, at least one agent $j$ in $N_i$ were assigned to an alternative from $A_{1}$. If all the agents in $N_i$ were assigned to alternatives from $A_2$, then, by the construction of $R$, at least one of them would have been assigned to an alternative that he or she ranks at a position greater than $\|A_2\| - \left\lceil \frac{m(1 + r)}{K}\right\rceil
K = \left\lceil \frac{2}{r}\right\rceil \left\lceil \frac{m(1 + r)}{K}
\right\rceil K - \left\lceil \frac{m(1 + r)}{K}\right\rceil K$. For $x =
\left\lceil \frac{m(1 + r)}{K} \right\rceil K$ we have: $$\begin{aligned}
\left\lceil \frac{2}{r}\right\rceil x - x \geq m' - m'r +
mr
\end{aligned}$$ (we skip the straightforward calculation) and, thus, this agent would have been assigned to an alternative that he or she ranks at a position greater than $m' -
m'r + mr$. As a consequence, this agent’s satisfaction would be lower than $(m' - m)r$. Similarly, no agent from $N_{i}$ can be assigned to an alternative from $M_l(i)$. Thus, for each $i$, $1 \leq i \leq n$, there exists at least one agent $j \in N_{i}$ that is assigned to an alternative from $M_f(i)$. In consequence, the covering subfamily of ${{{\mathcal{F}}}}$ consists simply of those sets $F_k$, for which some agent is assigned to alternative $a_k \in A_1$.
The presented construction gives the exact algorithm for <span style="font-variant:small-caps;">Set-Cover</span> problem running in time $f(K)(n+m)^{O(1)}$, where $f(K)$ is polynomial in $\binom{\left\lceil \frac{2}{r}
\right\rceil}{K}$. The existence of such an algorithm means that <span style="font-variant:small-caps;">Set-Cover</span> is in ${{\mathrm{FPT}}}$. On the other hand, we know that <span style="font-variant:small-caps;">Set-Cover</span> is ${{\mathrm{W[2]}}}$-complete, and thus if ${{{\mathcal{A}}}}$ existed then ${{\mathrm{FPT}}}= {{\mathrm{W[2]}}}$ would hold.
Algorithms for the Utilitarian, Satisfaction-Based Cases {#sec:algorithms}
========================================================
We now turn to approximation algorithms for the Monroe and Chamberlin–Courant multiwinner voting rules in the satisfaction-based framework. Indeed, if one focuses on agents’ total satisfaction then it is possible to obtain high-quality approximation results. In particular, we show the first nontrivial (randomized) approximation algorithm for $\alpha_{{{{{\mathrm{B, dec}}}}}}$-<span style="font-variant:small-caps;">SU-Monroe</span>. We show that for each $\epsilon > 0$ we can provide a randomized polynomial-time algorithm that achieves $0.715 - \epsilon$ approximation ratio; the algorithm usually gives even better approximation guarantees. For the case of arbitrarily selected DPSF we show a $(1 - e^{-1})$-approximation algorithm. Finally, we show the first polynomial-time approximation scheme (PTAS) for $\alpha_{{{{{\mathrm{B, dec}}}}}}$-<span style="font-variant:small-caps;">SU-CC</span>. These results stand in sharp contrast to those from the previous section, where we have shown that approximation is hard for essentially all remaining variants of the problem.
The core difficulty in solving $\alpha$-<span style="font-variant:small-caps;">Monroe/CC-Assignment</span> problems lays in selecting the alternatives that should be assigned to the agents. Given a preference profile and a set $S$ of up to $K$ alternatives, using a standard network-flow argument, it is easy to find a (possibly partial) optimal assignment $\Phi^S_{\alpha}$ of the agents to the alternatives from $S$.
\[prop:assignment\] Let $\alpha$ be a normal DPSF, $N$ be a set of agents, $A$ be a set of alternatives (togehter with their capacities; perhaps represented implicitly as for the case of the Monroe and Chamberlin–Courant rules), $V$ be a preference profile of $N$ over $A$, and $S$ a $K$-element subset of $A$ (where $K$ divides $\|N\|$). Then there is a polynomial-time algorithm that computes a (possibly partial) optimal assignment $\Phi^S_{\alpha}$ of the agents to the alternatives from $S$.
Note that for the case of the Chamberlin–Courant rule the algorithm from the above proposition can be greatly simplified: To each voter we assign the candidate that he or she ranks highest among those from $S$. For the case of Monroe, unfortunately, we need the expensive network-flow-based approach. Nonetheless, Proposition \[prop:assignment\] allows us to focus on the issue of selecting the winning alternatives and not on the issue of matching them to the agents.
Below we describe our algorithms for $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-Monroe</span> and for $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-CC</span>. Formally speaking, every approximation algorithm for $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-Monroe</span> also gives feasible results for $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-CC</span>. However, some of our algorithms are particularly well-suited for both problems and some are tailored to only one of them. Thus, for each algorithm we clearly indicate if it is meant only for the case of Monroe, only for the case of CC, or if it naturally works for both systems.
Algorithm A (Monroe)
--------------------
Perhaps the most natural approach to solve $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-Monroe</span> is to build a solution iteratively: In each step we pick some not-yet-assigned alternative $a_i$ (using some criterion) and assign it to those $\lceil
\frac{N}{K} \rceil$ agents that (a) are not assigned to any other alternative yet, and (b) whose satisfaction of being matched with $a_i$ is maximal. It turns out that this idea, implemented formally as Algorithm A (see pseudo code in Figure \[alg:greedy\]), works very well in many cases. We provide a lower bound on the total satisfaction it guarantees in the next lemma. We remind the reader that the so-called $k$’th *harmonic number* $H_k = \sum_{i=1}^k\frac{1}{i}$ has asymptotics $H_k = \Theta(\log k)$.
$\Phi = \{\}$\
\[lemma:greedy\] Algorithm A is a polynomial-time $(1 -
\frac{K-1}{2(m-1)} - \frac{H_K}{K})$-approximation algorithm for $\alpha_{{{{{\mathrm{B, dec}}}}}}$-<span style="font-variant:small-caps;">SU-Monroe</span>.
Our algorithm explicitly computes an optimal solution when $K \leq 2$ so we assume that $K \geq 3$. Let us consider the situation in the algorithm after the $i$’th iteration of the outer loop (we have $i=0$ if no iteration has been executed yet). So far, the algorithm has picked $i$ alternatives and assigned them to $i\frac{n}{K}$ agents (recall that for simplicity we assume that $K$ divides $n$ evenly). Hence, each agent has $\lceil \frac{m-i}{K-i} \rceil$ unassigned alternatives among his or her $i+ \lceil \frac{m-i}{K-i} \rceil$ top-ranked alternatives. By pigeonhole principle, this means that there is an unassigned alternative $a_{\ell}$ who is ranked among top $i+ \lceil \frac{m-i}{K-i} \rceil$ positions by at least $\frac{n}{K}$ agents. To see this, note that there are $(n-i\frac{n}{K})\lceil \frac{m-i}{K-i} \rceil$ slots for unassigned alternatives among the top $i+ \lceil \frac{m-i}{K-i} \rceil$ positions in the preference orders of unassigned agents, and that there are $m-i$ unassigned alternatives. As a result, there must be an alternative $a_\ell$ for whom the number of agents that rank him or her among the top $i+ \lceil \frac{m-i}{K-i} \rceil$ positions is at least: $$\frac{1}{m-i}\left((n-i\frac{n}{K})\left\lceil \frac{m-i}{K-i} \right\rceil\right) \geq
\frac{n}{m-i}\left(\frac{K-i}{K}\right)\left(\frac{m-i}{K-i}\right)
=\frac{n}{K}.$$ In consequence, the $\lceil \frac{n}{K} \rceil$ agents assigned in the next step of the algorithm will have the total satisfaction at least $\lceil \frac{n}{K} \rceil \cdot (m - i - \lceil
\frac{m-i}{K-i} \rceil)$. Thus, summing over the $K$ iterations, the total satisfaction guaranteed by the assignment $\Phi$ computed by Algorithm A is at least the following value: (to derive the fifth line from the fourth one we note that $K(H_K-1) - H_K \geq 0$ when $K \geq 3$): $$\begin{aligned}
\ell_{1}^{\alpha_{b}}(\Phi) & \geq \sum_{i = 0}^{K-1} \frac{n}{K} \cdot \left(m - i - \lceil \frac{m-i}{K-i} \rceil\right) \\
& \geq \sum_{i = 0}^{K-1} \frac{n}{K} \cdot \left( m - i - \frac{m-i}{K-i} -1 \right) \\
& = \sum_{i = 1}^{K} \frac{n}{K} \cdot \left(m - i - \frac{m-1}{K-i+1} + \frac{i-2}{K-i+1} \right) \\
& = \frac{n}{K}\left( \frac{K(2m-K-1)}{2} -(m-1) H_K + K(H_K-1) - H_K \right) \\
& \geq \frac{n}{K}\left( \frac{K(2m-K-1)}{2} -(m-1) H_K \right) \\
& \geq (m-1)n \left( 1 - \frac{K-1}{2(m-1)} - \frac{H_K}{K} \right)
\end{aligned}$$ If each agent were assigned to his or her top alternative, the total satisfaction would be equal to $(m-1)n$. Thus we get the following bound: $$\begin{aligned}
\frac{\ell_{1}^{\alpha_{{{{{\mathrm{B, dec}}}}}}}(\Phi)}{{{{{\mathrm{OPT}}}}}} \leq 1 - \frac{K-1}{2(m-1)} - \frac{H_K}{K}. $$ This completes the proof.
Note that in the above proof we measure the quality of our assignment against, a perhaps-impossible, perfect solution, where each agent is assigned to his or her top alternative. This means that for relatively large $m$ and $K$, and small $\frac{K}{m}$ ratio, the algorithm can achieve a close-to-ideal solution irrespective of the voters’ preference orders. We believe that this is an argument in favor of using Monroe’s system in multiwinner elections. On the flip side, to obtain a better approximation ratio, we would have to use a more involved bound on the quality of the optimal solution. To see that this is the case, form an instance $I$ of $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-Monroe</span> with $n$ agents and $m$ alternatives, where all the agents have the same preference order, and where we seek to elect $K$ candidates (and where $K$ divides $n$). It is easy to see that each solution that assigns the $K$ universally top-ranked alternatives to the agents is optimal. Thus the total dissatisfaction of the agents in the optimal solution is: $$\begin{aligned}
\frac{n}{K}\left( (m-1) + \cdots + (m-K) \right) &= \frac{n}{K}
\left(\frac{K(2m-K-1)}{2}\right)
= n(m-1) \left( 1 - \frac{K-1}{2(m-1)} \right).\end{aligned}$$ By taking large enough $m$ and $K$ (even for a fixed value of $\frac{m}{K}$), the fraction $1 - \frac{K-1}{2(m-1)}$ can be arbitrarily close to the approximation ratio of our algorithm (the reasoning here is somewhat in the spirit of the idea of identifying maximally robust elections, as studied by Shiryaev, Yu, and Elkind [@shi-yu-elk:t:robust-winners]).
For small values of $K$, it is possible that the $\frac{H_K}{K}$ part of our approximation ratio would dominate the $\frac{K-1}{2(m-1)}$ part. In such cases we can use the result of Betzler et al. [@fullyProportionalRepr], who showed that for each fixed constant $K$, $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-Monroe</span> can be solved in polynomial time. Thus, for the finite number of cases where $\frac{H_K}{K}$ is too large, we can solve the problem optimally using their algorithm. In consequence, the quality of the solution produced by Algorithm A most strongly depends on the ratio $\frac{K-1}{2(m-1)}$. In most cases we can expect it to be small (for example, in Polish parliamentary elections $K = 460$ and $m \approx
6000$; in this case the greedy algorithm’s approximation ratio is about $0.96$).
Our algorithm has one more great advantage: Since it focuses on the top parts of voters’ preference orders, it can achieve very good results even if the voters submit so-called truncated ballots (that is, if they rank some of their top alternatives only). Below we present the formal analysis of the algorithm’s approximation ratio for this case. Unfortunately, we did not obtain a closed form formula and, instead, we present the guaranteed approximation ratio as a sum, in Proposition \[prop:monTruncated\] below. We also present the relation between the fraction of the top alternatives ranked by each of the voters and the approximation ratio for few values of $m$ and $K$ in Figure \[fig:monroe\_truncated\].
\[prop:monTruncated\] Let $P$ be the number of top positions in the agents’ preference orders that are known by the algorithm. In this case Algorithm A is a polynomial-time $r$-approximation algorithm for $\alpha_{{{{{\mathrm{B, dec}}}}}}$-<span style="font-variant:small-caps;">SU-Monroe</span>, where: $$\begin{aligned}
& r = \sum_{i = 0}^{K-1} \frac{1}{n(m-1)} \max(r(i), 0) \\
\text{and}\\
& r(i) = \left\{
\begin{array}{l l}
\frac{n}{K}(m - i - \frac{m-i}{K-i}) & \quad \text{if $\left(i + \frac{m-i}{K-i}\right) \leq P$,}\\\\
\frac{n}{K}\frac{(K-i)(m-i)}{4} & \quad \text{if $\left(i + \frac{m-i}{K-i}\right) > P$ and $(2P-m) \geq i \geq (K-2)$,} \\\\
\frac{n}{K}\frac{(m-P)(K-i)(P-i)}{m-i} & \quad \text{otherwise.}
\end{array} \right.
\end{aligned}$$
We use the same approach as in the proof of Lemma \[lemma:greedy\], except that we adjust our estimates of voters’ satisfaction. Consider a situation after some $i$’th iteration of the algorithm’s outer loop ($i=0$ if we are before the first iteration). If $i + \frac{m-i}{K-i}
\leq P$, then we can use the same lower bound for the satisfaction of the agents assigned in the $(i+1)$’th iteration as in the proof of Lemma \[lemma:greedy\]. That is, the agents assigned in the $(i+1)$’th iteration will have satisfaction at least $r_1(i) = \frac{n}{K} \cdot (m - i - \frac{m-i}{K-i})$.
For the case where $i + \frac{m-i}{K-i} > P$, the bound from Lemma \[lemma:greedy\] does not hold, but we can use a similar approach to find a different one. Let $P_x \leq P$ be some positive integer. We are interested in the number $x$ of not-yet assigned agents who rank some not-yet-selected alternative among their top $P_x$ positions (after the $i$’th iteration). Similarly as in the proof of Lemma \[lemma:greedy\], using the pigeonhole principle we note that: $$\begin{aligned}
x \geq \frac{1}{m-i} \left(n-i\frac{n}{K}\right)(P_x-i) =
\frac{n}{K} \cdot \frac{(K-i)(P_x-i)}{m-i} \textrm{.}
\end{aligned}$$ Thus, the satisfaction of the agents assigned in the $(i+1)$’th iteration is at least: $$\label{equ:sat-bound}
\min\left(x, \frac{n}{K}\right)(m - P_x) = \frac{n}{K} \cdot
(m-P_x) \min\left(\frac{(K-i)(P_x-i)}{m-i}, 1\right) \textrm{.}$$ The case $\frac{(K-i)(P_x-i)}{m-i} \geq 1$ (or, equivalently, $i +
\frac{m-i}{K-i} \leq P_x$) implies that $i + \frac{m-i}{K-i} \leq
P$ and for this case we lower-bound agents’ satisfaction by $r_1(i)$. For the case where $\frac{(K-i)(P_x-i)}{m-i} \leq 1$, i.e. where $i + \frac{m-i}{K-i} \geq P_x$, equation simplifies to: $$\begin{aligned}
\label{equ:sat-bound-2}
\frac{n}{K} \cdot (m-P_x) \cdot \frac{(K-i)(P_x-i)}{m-i} \textrm{.}
\end{aligned}$$ We use this estimate for the satisfaction of the agents assigned in the $(i+1)$’th iteration for the cases where (a) $i + \frac{m-i}{K-i} \geq \frac{m+i}{2}$ and (b) $\frac{m+i}{2} \leq P$ (or, equivalently, $(2P-m) \geq i \geq (K-2)$). In this case we estimate as follows: $$\begin{aligned}
\frac{n}{K} \cdot (m-P_x) \cdot \frac{(K-i)(P_x-i)}{m-i}
& \geq \frac{n}{K} \cdot (m-\frac{m+i}{2}) \cdot \frac{(K-i)(\frac{m+i}{2}-i)}{m-i}\\
& = \frac{n}{K} \cdot
\frac{(K-i)(m-i)^2}{4(m-i)} = \frac{n}{K} \cdot
\frac{(K-i)(m-i)}{4}\textrm{.}
\end{aligned}$$ For the remaining cases, we set $P_x = P$ and becomes: $$\begin{aligned}
\frac{n}{K} \cdot \frac{(m-P)(K-i)(P-i)}{m-i} \textrm{.}
\end{aligned}$$ Naturally, we replace our estimates by $0$ whenever they become negative.
To complete the proof, it suffice to, as in the proof of Lemma \[lemma:greedy\], note that $(m-1)n$ is an upper bound on the satisfaction achieved by the optimal solution.
{width="\textwidth"}
{width="\textwidth"}
For example, for the case of Polish parliamentary elections ($K=460$ and $m=6000$), to achieve $90\%$ of voters’ optimal satisfaction, each voter would have to rank about $8.7\%$ of the candidates.
Our results show that for most settings there is very little reason to ask the agents to rank all the alternatives. Using Proposition \[prop:monTruncated\], election designers can estimate how many alternatives the agents should rank to obtain a particular level of satisfaction and, since computing preference orders can be expensive for the agents, this way can save a large amount of effort.
Algorithm B (Monroe)
--------------------
There are simple ways in which we can improve the quality of the assignments produced by Algorithm A. For example, our Algorithm B first runs Algorithm A and then, using Proposition \[prop:assignment\], optimally reassigns the alternatives to the voters. As shown in Section \[sec:experiments\], this very simple trick turns out to noticeably improve the results of the algorithm in practice (and, of course, the theoretical approximation guarantees of Algorithm A carry over to Algorithm B).
Algorithm C (Monroe, CC)
------------------------
Algorithm C is a further heuristic improvement over Algorithm B. This time the idea is that instead of keeping only one partial function $\Phi$ that is iteratively extended up to the full assignment, we keep a list of up to $d$ partial assignment functions, where $d$ is a parameter of the algorithm. At each iteration, for each assignment function $\Phi$ among the $d$ stored ones and for each alternative $a$ that does not yet have agents assigned to by this $\Phi$, we compute an optimal extension of this $\Phi$ that assigns agents to $a$. As a result we obtain possibly more than $d$ (partial) assignment functions. For the next iteration we keep those $d$ of them that give highest satisfaction.
We provide pseudocode for Algorithm C in Figure \[alg:greedyImpr\]. If we take $d=1$, we obtain Algorithm B. If we also disregard the last two lines prior to returning the solution, we obtain Algorithm A.
Algorithm C can also be adapted for the Chamberlin–Courant rule. The only difference concerns creating the assignment functions: we replace the contents of the first foreach loop with the following code:
Note that, for the case of the Chamberlin–Courant rule, Algorithm C can also be seen as a generalization of Algorithm GM that we will discuss later in Section \[alg:gm\].
$Par = []$\
$Par$.push($\{\}$)\
Algorithm R (Monroe, CC)
------------------------
Algorithms A, B and C achieve very high approximation ratios for the cases where $K$ is small relative to $m$. For the remaining cases, where $K$ and $m$ are comparable, we can use a sampling-based randomized algorithm (denoted as Algorithm R) described below. We focus on the case of Monroe and we will briefly mention the case of CC at the end.
The idea of this algorithm is to randomly pick $K$ alternatives and match them optimally to the agents, using Proposition \[prop:assignment\]. Naturally, such an algorithm might be very unlucky and pick $K$ alternatives that all of the agents rank low. Yet, if $K$ is comparable to $m$ then it is likely that such a random sample would include a large chunk of some optimal solution. In the lemma below, we asses the expected satisfaction obtained with a single sampling step (relative to the satisfaction given by the optimal solution) and the probability that a single sampling step gives satisfaction close to the expected one. Naturally, in practice one should try several sampling steps and pick the one with the highest satisfaction.
\[lemma:randMonroe\] A single sampling step of the randomized algorithm for $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-Monroe</span> achieves expected approximation ratio of $\frac{1}{2}(1 + \frac{K}{m} - \frac{K^2}{m^2-m} +
\frac{K^3}{m^3-m^2})$. Let $p_{\epsilon}$ denote the probability that the relative deviation between the obtained total satisfaction and the expected total satisfaction is higher than $\epsilon$. Then for $K \geq 8$ we have $p_{\epsilon} \leq \exp \left(-
\frac{K\epsilon^2}{128} \right)$.
Let $N = [n]$ be the set of agents, $A = \{a_1, \ldots, a_m\}$ be the set of alternatives, and $V$ be the preference profile of the agents. Let us fix some optimal solution $\Phi_{{{{\mathrm{opt}}}}}$ and let $A_{{{{\mathrm{opt}}}}}$ be the set of alternatives assigned to the agents in this solution. For each $a_{i} \in A_{{{{{\mathrm{opt}}}}}}$, we write ${{{{\mathrm{sat}}}}}(a_{i})$ to denote the total satisfaction of the agents assigned to $a_{i}$ in $\Phi_{{{{\mathrm{opt}}}}}$. Naturally, we have $\sum_{a \in A_{{{{{\mathrm{opt}}}}}}} {{{{\mathrm{sat}}}}}(a) =
{{{{\mathrm{OPT}}}}}$. In a single sampling step, we choose uniformly at random a $K$-element subset $B$ of $A$. Then, we form a solution $\Phi_B$ by matching the alternatives in $B$ optimally to the agents (via Proposition \[prop:assignment\]). We write $K_{{{{{\mathrm{opt}}}}}}$ to denote the random variable equal to $\|A_{{{{\mathrm{opt}}}}}\cap B\|$, the number of sampled alternatives that belong to $A_{{{{{\mathrm{opt}}}}}}$. We define $p_{i} =
\Pr(K_{{{{{\mathrm{opt}}}}}} = i)$. For each $j \in \{1, \ldots, K\}$, we write $X_j$ to denote the random variable equal to the total satisfaction of the agents assigned to the $j$’th alternative from the sample. We claim that for each $i$, $0 \leq i \leq K$, it holds that: $$\begin{aligned}
{\mathop{\mathbb E}}\left(\sum_{j=1}^{K}X_{j} \,\Bigg|\, K_{{{{\mathrm{opt}}}}}= i\right) \geq \frac{i}{K}{{{{\mathrm{OPT}}}}}+ \frac{m-i-1}{2}
\cdot \left(n - i\frac{n}{K}\right).
\end{aligned}$$ Why is this so? Given a sample $B$ that contains $i$ members of $A_{{{{\mathrm{opt}}}}}$, our algorithm’s solution is at least as good as a solution that matches the alternatives from $B \cap A_{{{{\mathrm{opt}}}}}$ in the same way as $\Phi_{{{{\mathrm{opt}}}}}$, and the alternatives from $B - A_{{{{\mathrm{opt}}}}}$ in a random manner. Since $K_{{{{\mathrm{opt}}}}}= i$ and each $a_j \in A_{{{{\mathrm{opt}}}}}$ has equal probability of being in the sample, it is easy to see that the expected value of $\sum_{a_j \in B \cap A_{{{{\mathrm{opt}}}}}}{{{{\mathrm{sat}}}}}(a_j)$ is $
\frac{i}{K}{{{{\mathrm{OPT}}}}}$. After we allocate the agents from $B \cap A_{{{{\mathrm{opt}}}}}$, each of the remaining, unassigned agents has $m-i$ positions in his or her preference order where he ranks the agents from $A - A_{{{{\mathrm{opt}}}}}$. For each unassigned agents, the average score value associated with these positions is at least $\frac{m-i-1}{2}$ (this is so, because in the worst case the agent could rank the alternatives from $B \cap
A_{{{{\mathrm{opt}}}}}$ in the top $i$ positions). There are $(n - i\frac{n}{K})$ such not yet assigned agents and so the expected total satisfaction from assigning them randomly to the alternatives is $\frac{m-i-1}{2}
\cdot (n - i\frac{n}{K})$. This proves our bound on the expected satisfaction of a solution yielded by optimally matching a random sample of $K$ alternatives.
Since ${{{{\mathrm{OPT}}}}}$ is upper bounded by $(m-1)n$ (consider a possibly-nonexistent solution where every agent is assigned to his or her top preference), we get that: $${\mathop{\mathbb E}}\left(\sum_{j=1}^{K}X_{j} | K_{{{{\mathrm{opt}}}}}= i\right) \geq \frac{i}{K}{{{{\mathrm{OPT}}}}}+ \frac{m-i-1}{2(m-1)} \cdot \left(1 - \frac{i}{K}\right){{{{\mathrm{OPT}}}}}.$$ We can compute the unconditional expected satisfaction of $\Phi_B$ as follows: $$\begin{aligned}
{\mathop{\mathbb E}}\left(\sum_{j=1}^{K}X_{j}\right) & = \sum_{i=0}^{K}p_{i}{\mathop{\mathbb E}}\left(\sum_{j=1}^{K}X_{j} | K_{{{{\mathrm{opt}}}}}= i\right) \\
& \geq \sum_{i=0}^{K}p_{i}\left(\frac{i}{K}{{{{\mathrm{OPT}}}}}+ \frac{m-i-1}{2(m-1)}
\cdot \left(1 - \frac{i}{K}\right){{{{\mathrm{OPT}}}}}\right).
\end{aligned}$$ Since $\sum_{i=1}^{K}p_{i} \cdot i$ is the expected number of the alternatives in $A_{{{{{\mathrm{opt}}}}}}$, we have that $ \sum_{i=1}^{K}p_{i} \cdot
i = \frac{K^2}{m}$ (one can think of summing the expected values of $K$ indicator random variables; one for each element of $A_{{{{\mathrm{opt}}}}}$, taking the value $1$ if a given alternative is selected and taking the value $0$ otherwise). Further, from the generalized mean inequality we obtain $\sum_{i=1}^{K}p_{i} \cdot i^{2} \geq
\left(\frac{K^2}{m}\right)^{2}.$ In consequence, through routine calculation, we get that: $$\begin{aligned}
{\mathop{\mathbb E}}\left(\sum_{j=1}^{K}X_{j}\right)
& \geq \left(\frac{K}{m}{{{{\mathrm{OPT}}}}}+ \frac{m^2 - K^2 -m}{2m(m-1)} \cdot \left(1 - \frac{K}{m}\right){{{{\mathrm{OPT}}}}}\right) \\
& = \frac{{{{{\mathrm{OPT}}}}}}{2}\left(1 + \frac{K}{m} - \frac{K^2}{m^2-m} +
\frac{K^3}{m^3-m^2}\right).
\end{aligned}$$
It remains to assess the probability that the total satisfaction obtained through $\Phi_B$ is close to its expected value. Since $X_{j} \in \langle 0, \frac{(m-1)n}{K} \rangle$, from Hoeffding’s inequality we get: $$\begin{aligned}
p_{\epsilon} & = \Pr\left(\left|\sum_{j=1}^{K}X_{j} - {\mathop{\mathbb E}}(\sum_{j=1}^{K}X_{j})\right| \geq \epsilon {\mathop{\mathbb E}}(\sum_{j=1}^{K}X_{j})\right) \\
& \leq \exp \left(- \frac{2\epsilon^2 ({\mathop{\mathbb E}}(\sum_{j=1}^{K}X_{j}))^{2}}{K(\frac{(m-1)n}{K})^2}
\right) = \exp \left(- \frac{K\epsilon^2 ({\mathop{\mathbb E}}(\sum_{j=1}^{K}X_{j}))^{2}}{((m-1)n)^2} \right)
\end{aligned}$$ We note that since $\frac{K}{m}-\frac{K^2}{m^2-m} \geq 0$, our previous calculations show that ${\mathop{\mathbb E}}(\sum_{j=1}^{K}X_{j}) \geq
\frac{{{{{\mathrm{OPT}}}}}}{2}$. Further, for $K \geq 8$, Lemma \[lemma:greedy\] (and the fact that in its proof we upper-bound ${{{{\mathrm{OPT}}}}}$ to be $(m-1)n$) gives that ${{{{\mathrm{OPT}}}}}\geq \frac{mn}{8}$. Thus $p_{\epsilon} \leq \exp
\left(- \frac{K\epsilon^2}{128} \right)$. This completes the proof.
In the next theorem we will see that to have a high chance of obtaining a high quality assignment, we need to repeat the sampling step many times. Thus, for practical purposes, by Algorithm R we mean an algorithm that repreats the sampling process a given number of times (this parameter is given as input) and returns the best solution found (the assignment is created using Proposition \[prop:assignment\]).
The threshold for $\frac{K}{m}$, where the sampling step is (in expectation) better than the greedy algorithm is about 0.57. Thus, by combining the two algorithms, we can guarantee an expected approximation ratio of $0.715 - \epsilon$, for each fixed constant $\epsilon$. The pseudo-code of the combination of the two algorithms (Algorithm AR) is presented in Figure \[alg:combination\].
\[thm:combMonroe\] For each fixed $\epsilon$, Algorithm AR provides a $(0.715-\epsilon)$-approximate solution for the problem $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-Monroe</span> with probability $\lambda$ in time polynomial with respect to the input instance size and $-\log(1-\lambda)$.
Let $\epsilon$ be a fixed constant. We are given an instance $I$ of $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-Monroe</span>. If $m \leq 1 +
\frac{2}{\epsilon}$, we solve $I$ using a brute-force algorithm (note that in this case the number of alternatives is at most a fixed constant). Similarly, if $\frac{H_K}{K} \geq
\frac{\epsilon}{2}$ then we use the exact algorithm of Betzler et al. [@fullyProportionalRepr] for a fixed value of $K$ (note that in this case $K$ is no greater than a certain fixed constant). We do the same if $K \leq 8$.
On the other hand, if neither of the above conditions hold, we try both Algorithm A and a number of runs of the sampling-based algorithm. It is easy to check through routine calculation that if $\frac{H_K}{K} \leq \frac{\epsilon}{2}$ and $m >
1 + \frac{2}{\epsilon}$ then Algorithm A achieves approximation ratio no worse than $(1 -\frac{K}{2m} - \epsilon)$. We run the sampling-based algorithm $\frac{-512 \log (1 -
\lambda)}{K\epsilon^2}$ times. The probability that a single run fails to find a solution with approximation ratio at least $\frac{1}{2}(1 + \frac{K}{m} - \frac{K^2}{m^2-m} +
\frac{K^3}{m^3-m^2}) - \frac{\epsilon}{2}$ is $p_{\frac{\epsilon}{2}} \leq \exp \left(- \frac{K\epsilon^2}{4 \cdot
128} \right)$. Thus, the probability that at least one run will find a solution with at least this approximation ratio is at least: $$\begin{aligned}
1 - p_{\frac{\epsilon}{2}}^{\frac{-512 \log (1 - \lambda)}{K\epsilon^2}} = 1
- \exp \left(-\frac{K\epsilon^2}{4\cdot128} \cdot \frac{-512 \log
(1 - \lambda)}{K\epsilon^2} \right) = \lambda.
\end{aligned}$$ Since $m \leq 1 + \frac{2}{\epsilon}$, by routine calculation we see that the sampling-based algorithm with probability $\lambda$ finds a solution with approximation ratio at least $\frac{1}{2}(1 +
\frac{K}{m} - \frac{K^2}{m^2} + \frac{K^3}{m^3}) - \epsilon$. By solving the equality: $$\begin{aligned}
\frac{1}{2}\left(1 + \frac{K}{m} - \frac{K^2}{m^2} + \frac{K^3}{m^3}\right) =
1 -\frac{K}{2m}
\end{aligned}$$ we can find the value of $\frac{K}{m}$ for which the two algorithms give the same approximation ratio. By substituting $x =
\frac{K}{m}$ we get equality $1 + x - x^2 + x^3 = 2 - x$. One can calculate that this equality has a single solution within $\langle
0,1 \rangle$ and that this solution is $x \approx 0.57$. For this $x$ both algorithms guarantee approximation ratio of $0.715 -
\epsilon$. For $x < 0.57$ the deterministic algorithm guarantees a better approximation ratio and for $x > 0.57$, the randomized algorithm does better.
$\Phi_1 \leftarrow$ solution returned by Algorithm A\
$\Phi_2 \leftarrow$ run the sampling-based algorithm $\frac{-512 \log (1 - \lambda)}{K\epsilon^2}$ times; select the assignment of the best quality\
return the better assignment among $\Phi_1$ and $\Phi_2$
Let us now consider the case of CC. It is just as natural to try a sampling-based approach for solving $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-CC</span>, as we did for the Monroe variant. Indeed, as recently (and independently) observed by Oren [@ore:p:cc], this leads to a randomized algorithm with expected approximation ratio of $(1 -
\frac{1}{K+1})(1+\frac{1}{m})$. However, since we will later see an effective, deterministic, polynomial-time approximation scheme for $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-CC</span>, there is little reason to explore the sampling based approach.
Algorithm GM (Monroe, CC) {#alg:gm}
-------------------------
Algorithm GM (greedy marginal improvement) was introduced by Lu and Boutilier for the case of the Chamberlin–Courant rule. Here we generalize it to apply to Monroe’s rule as well, and we show that it is a $1-\frac{1}{e}$ approximation algorithm for $\alpha$-<span style="font-variant:small-caps;">SU-Monroe</span>. We point out that this approximation result for Monroe rule applies to all non-decreasing PSFs $\alpha$. For the Monroe rule, the algorithm can be viewed as an extension of Algorithm B.
The algorithm proceeds as follows. We start with an emtpy set $S$. Then we execute $K$ iterations. In each iteration we find an alternative $a$ that is not assigned to agents yet, and that maximizes the value $\Phi^{S \cup
\{a\}}_{\alpha}$. (A certain disadvantage of this algorithm for the case of Monroe is that it requires a large number of computations of $\Phi^S_{\alpha}$; since in Monroe’s rule each alternative can be assigned at most $\frac{n}{K}$ agents in the partial assignment $\Phi^S_{\alpha}$, computation of $\Phi^S_{\alpha}$ is a slow process based on min-cost/max-flow algorithm.) We provide the pseudocode for Algorithm GM in Figure \[alg:greedyOptImpr\].
$S \leftarrow \emptyset$\
\[thm:gmMonroe\] For any non-decreasing positional scoring function $\alpha$ Algorithm GM is an $(1 - \frac{1}{e})$-approximation algorithm for $\alpha$-<span style="font-variant:small-caps;">SU-Monroe</span>.
The proof follows by applying the powerful result of Nemhauser et al. [@submodular], which says that greedy algorithms achieve $1-\frac{1}{e}$ approximation ratio when used to optimize nondecreasing submodular functions (we explain these notions formally below). The main challenge in the proof is to define a function that, on one hand, satisfies the conditions of Nemhauser et al.’s result, and, on the other, models solutions for $\alpha$-<span style="font-variant:small-caps;">SU-Monroe</span>.
Let $A$ be a set of alternatives, $N = [n]$ be a set of agents with preferences over $A$, $\alpha$ be an $\|A\|$-candidate DPSF, and $K
\leq \|A\|$ be the number of representatives that we want to elect. We consider function $z: 2^{A} \rightarrow {{{\mathbb{N}}}}$ defined, for each set $S$, $S \subseteq A$ and $\|S\| \leq K$, as $z(S) =
\ell_{1}^{\alpha}(\Phi_{\alpha}^{S})$. Clearly, $z(S)$ is nondecreasing (that is, for each two sets $A$ and $B$, if $A
\subseteq B$ and $\|B\| \leq K$ then $z(A) \leq z(B)$. Since $\mathrm{argmax}_{S \subset A, \|S\| = K} z(S)$ is the set of winners under $\alpha$-Monroe and since Algorithm GM builds the solution iteratively by greedily extending initially empty set $S$ so that each iteration increases the value of $z(S)$ maximally, if $z$ were submodular then by the results of Nemhauser et al. [@submodular] we would get that Algorithm GM is a $(1-\frac{1}{e})$-approximation algorithm. Thus, our goal is to show that $z$ is submodular.
Formally, our goal is to show that for each two sets $S$ and $T$, $S
\subset T$, and each alternative $a \notin T$ it holds that $z(S
\cup \{a\}) - z(S) \geq z(T \cup \{a\}) - z(T)$ (this is the formal definition of submodularity). First, we introduce a notion that generalizes the notion of a partial set of winners $S$. Let $s: A
\rightarrow {{{\mathbb{N}}}}$ denote a function that assigns a capacity to each alternative (i.e., $s$ gives a bound on the number of agents that a given alternative can represent). Intuitively, each set $S$, $S \subseteq A$, corresponds to the capacity function that assigns $\lceil \frac{n}{k} \rceil$ to each alternative $a \in S$ and 0 to each $a \notin S$. Given a capacity function $s$, we define a partial solution $\Phi_{\alpha}^{s}$ to be one that maximizes the total satisfaction of the agents and that satisfies the new capacity constraints: $\forall_{a \in S} \|(\Phi_{\alpha}^{s})^{-1}(a)\| \leq
s(a)$. To simplify notation, we write $s \cup \{a\}$ to denote the function such that $(s \cup \{a\})(a) = s(a) + 1$ and $\forall_{a'
\in S \setminus\{a\}} (s \cup \{a\})(a') = s(a')$. (Analogously, we interpret $s \setminus \{a\}$ as subtracting one from the capacity for $a$; provided it is nonzero.) Also, by $s \leq t$ we mean that $\forall_{a \in A} s(a) \leq t(a)$. We extend our function $z$ to allow us to consider a subset of the agents only. For each subset $N'$ of the agents and each capacity function $s$, we define $z_{N'}(s)$ to be the satisfaction of the agents in $N'$ obtained under $\Phi_{\alpha}^{s}$. We will now prove a stronger variant of submodularity for our extended $z$. That is, we will show that for each two capacity functions $s$ and $t$ it holds that: $$\label{eq:submodularity}
s \leq t \Rightarrow z_{N}(s \cup \{a\}) - z_{N}(s) \geq z_{N}(t \cup \{a\}) - z_{N}(t).$$ Our proof is by induction on $N$. Clearly, Equation holds for $N' = \emptyset$. Now, assuming that Equation holds for every $N'
\subset N$ we will prove its correctness for $N$. Let $i$ denote an agent such that $\Phi_{\alpha}^{t \cup \{a\}}(i) = a$ (if there is no such agent then clearly the equation holds). Let $a_{s} =
\Phi_{\alpha}^{s}(i)$ and $a_{t} = \Phi_{\alpha}^{t}(i)$. We have: $$\begin{aligned}
z_{N}(t \cup \{a\}) - z_{N}(t) = \alpha({{{{\mathrm{pos}}}}}_i(a)) + z_{N \setminus \{i\}}(t) - \alpha({{{{\mathrm{pos}}}}}_i(a_{t})) - z_{N \setminus \{i\}}(t \setminus \{a_{t}\}).
\end{aligned}$$ We also have: $$\begin{aligned}
z_{N}(s \cup \{a\}) - z_{N}(s) \geq \alpha({{{{\mathrm{pos}}}}}_i(a)) + z_{N \setminus \{i\}}(s) - \alpha({{{{\mathrm{pos}}}}}_i(a_{s})) - z_{N \setminus \{i\}}(s \setminus \{a_{s}\}).
\end{aligned}$$ Since $\Phi_{\alpha}^{t}$ describes an optimal representation function under the capacity restrictions $t$, we have that: $$\begin{aligned}
\alpha({{{{\mathrm{pos}}}}}_i(a_t)) + z_{N \setminus \{i\}}(t \setminus a_{t})
\geq \alpha({{{{\mathrm{pos}}}}}_i(a_s)) + z_{N \setminus \{i\}}(t \setminus \{a_{s}\}).
\end{aligned}$$ Finally, from the inductive hypothesis for $N' = N \setminus \{i\}$ we have: $$\begin{aligned}
z_{N \setminus \{i\}}(s) - z_{N \setminus \{i\}}(s \setminus \{a_{s}\}) \geq
z_{N \setminus \{i\}}(t) - z_{N \setminus \{i\}}(t \setminus \{a_{s}\}).
\end{aligned}$$ By combining these inequalities we get: $$\begin{aligned}
z_{N}(s \cup \{a\}) - z_{N}(s) & \geq \alpha({{{{\mathrm{pos}}}}}_i(a)) + z_{N \setminus \{i\}}(s) - (\alpha({{{{\mathrm{pos}}}}}_i(a_{s})) + z_{N \setminus \{i\}}(s \setminus \{a_{s}\})) \\
& \geq \alpha({{{{\mathrm{pos}}}}}_i(a)) - \alpha({{{{\mathrm{pos}}}}}_i(a_{s})) + z_{N \setminus \{i\}}(t) - z_{N \setminus \{i\}}(t \setminus \{a_{s}\}) \\
& \geq \alpha({{{{\mathrm{pos}}}}}_i(a)) + z_{N \setminus \{i\}}(t) - \alpha({{{{\mathrm{pos}}}}}_i(a_t)) - z_{N \setminus \{i\}}(t \setminus \{a_{t}\}) \\
& = z_{N}(t \cup \{a\}) - z_{N}(t).
\end{aligned}$$ This completes the proof.
Formally speaking, Algorithm GM is never worse than Algorithm A. For Borda satisfaction function, it inherits the approximation guarantees from Algorithm A, and for other cases Theorem \[thm:gmMonroe\] guarantees approximation ratio $1-\frac{1}{e}$ (we do not know of any guarantees for Algorithm A for these cases). The comparison with Algorithms B and C is not nearly as easy. Algorithm GM is still likely better than them for satisfaction functions significantly different from Borda’s, but for the Borda case our experiments show that Algorithm GM is much slower than Algorithms B and C and obtains almost the same or slightly worse results (see Section \[sec:experiments\]).
Algorithm P (CC)
----------------
The idea of our algorithm (presented in Figure \[alg:greedy2\]) is to compute a certain value $x$ and to greedily compute an assignment that (approximately) maximizes the number of agents assigned to one of their top-$x$ alternatives.[^3] If after this process some agent has no alternative assigned, we assign him or her to his or her most preferred alternative from those already picked. Somewhat surprisingly, it turns out that this greedy strategy achieves high-quality results. (Recall that for nonnegative real numbers, Lambert’s W-function, ${{{{\mathrm{w}}}}}(x)$, is defined to be the solution of the equation $x = {{{{\mathrm{w}}}}}(x)e^{{{{{\mathrm{w}}}}}(x)}$.)
$\Phi = \{\}$\
$x = \lceil \frac{m{{{{\mathrm{w}}}}}(K)}{K} \rceil$\
\[lemma:greedyCC\] Algorithm P is a polynomial-time $(1 -
\frac{2{{{{\mathrm{w}}}}}(K)}{K})$-approximation algorithm for $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-CC</span>.
Let $x = \frac{m{{{{\mathrm{w}}}}}(K)}{K}$. We will first give an inductive proof that, for each $i$, $0 \leq i \leq K$, after the $i$’th iteration of the outer loop at most $n(1- \frac{w(K)}{K})^{i}$ agents are unassigned. Based on this observation, we will derive the approximation ratio of our algorithm.
For $i = 0$, the inductive hypothesis holds because $n(1-
\frac{{{{{\mathrm{w}}}}}(K)}{K})^{0} = n$. For each $i$, let $n_{i}$ denote the number of unassigned agents after the $i$’th iteration. Thus, after the $i$’th iteration there are $n_i$ unassigned agents, each with $x$ unassigned alternatives among his or her top-$x$ ranked alternatives. As a result, at least one unassigned alternative is present in at least $\frac{n_{i}x}{m-i}$ of top-$x$ positions of unassigned agents. This means that after the $(i+1)$’st iteration the number of unassigned agents is: $$\begin{aligned}
n_{i+1} \leq n_{i} - \frac{n_{i}x}{m-i} \leq n_{i}\left(1 -
\frac{x}{m}\right) = n_{i}\left(1 - \frac{{{{{\mathrm{w}}}}}(K)}{K}\right).
\end{aligned}$$ If for a given $i$ the inductive hypothesis holds, that is, if $n
_{i} \leq n\left(1- \frac{{{{{\mathrm{w}}}}}(K)}{K}\right)^{i}$, then: $$\begin{aligned}
n_{i+1} \leq n(1- \frac{{{{{\mathrm{w}}}}}(K)}{K})^{i}(1 -
\frac{{{{{\mathrm{w}}}}}(K)}{K}) = n\left(1- \frac{{{{{\mathrm{w}}}}}(K)}{K}\right)^{i+1}.
\end{aligned}$$ Thus the hypothesis holds and, as a result, we have that: $$\begin{aligned}
n_{k} \leq n\left(1- \frac{{{{{\mathrm{w}}}}}(K)}{K}\right)^{K} \leq
n\left(\frac{1}{e}\right)^{{{{{\mathrm{w}}}}}(K)} = \frac{n{{{{\mathrm{w}}}}}(K)}{K}.
\end{aligned}$$ Let $\Phi$ be the assignment computed by our algorithm. To compare it against the optimal solution, it suffices to observe that the optimal solution has the value of satisfaction of at most ${{{{\mathrm{OPT}}}}}\leq (m-1)n$, that each agent selected during the first $K$ steps has satisfaction at least $m-x = m-\frac{m{{{{\mathrm{w}}}}}(K)}{K}$, and that the agents not assigned within the first $K$ steps have satisfaction no worse than $0$. Thus it holds that: $$\begin{aligned}
\frac{\ell_{1}^{\alpha_{{{{{\mathrm{B, dec}}}}}}}(\Phi)}{{{{{\mathrm{OPT}}}}}} & \geq \frac{(n - \frac{n{{{{\mathrm{w}}}}}(K)}{K})(m - \frac{m{{{{\mathrm{w}}}}}(K)}{K})}{(m-1)n} \\
& \geq \left(1 - \frac{{{{{\mathrm{w}}}}}(K)}{K}\right)\left(1 - \frac{{{{{\mathrm{w}}}}}(K)}{K}\right) \geq 1 -
\frac{2{{{{\mathrm{w}}}}}(K)}{K}.
\end{aligned}$$ This completes the proof.
Since for each $\epsilon > 0$ there is a value $K_\epsilon$ such that for each $K > K_\epsilon$ it holds that $\frac{2{{{{\mathrm{w}}}}}(K)}{K} < \epsilon$, and $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-CC</span> problem can be solved optimally in polynomial time for each fixed constant $K$(see the work of Betzler et al. [@fullyProportionalRepr]), there is a polynomial-time approximation scheme (PTAS) for $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-CC</span> (i.e., a family of algorithms such that for each fixed $r$, $0 < r < 1$, there is a polynomial-time $r$-approximation algorithm for $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-CC</span> in the family; note that in PTASes we measure the running time by considering $r$ to be a fixed constant).
\[theorem:ptas\] There is a PTAS for $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-CC</span>.
The idea used in Algorithm P can also be used to address a generalized <span style="font-variant:small-caps;">SE-CC</span> problem. We can consider the following relaxation of <span style="font-variant:small-caps;">SE-CC</span>: Instead of requiring that each agent’s satisfaction is lower-bounded by some value, we ask that the satisfactions of a significant majority of the agents are lower-bounded by a given value. More formally, for a given constant $\delta$, we introduce an additional quality metric: $$\ell_{\min}^{\delta,\alpha}(\Phi) = \mathrm{max}_{N' \subseteq N:
\frac{||N|| - ||N'||}{||N||} \leq \delta}\mathrm{min}_{i \in
N'}\alpha(pos_{i}(\Phi(i))).$$ For a given $0 < \delta < 1$, by putting $x =
\frac{-m\ln(\delta)}{K}$, we get $(1 +
\frac{\ln(\delta)}{K})$-approximation algorithm for the $\ell_{\min}^{\delta,\alpha}(\Phi)$ metric.
Finally, we show that Algorithm P performs very well even if the voters cast truncated ballots. Proposition \[lemma:greedyCCTruncated\] gives the relation between the number of positions used by the algorithm and the approximation ratio. In Figure \[fig:cc\_truncated\] we show this relation for some values of the parameters $m$ and $K$.
\[lemma:greedyCCTruncated\] Let $Q$ be the number of top positions in the agents’ preference orders that are known by the algorithm ($Q \leq
\frac{m{{{{\mathrm{w}}}}}(K)}{K}$). Algorithm P that uses $x = Q$ instead of $x = \lceil \frac{m{{{{\mathrm{w}}}}}(K)}{K} \rceil$ is a polynomial-time $\left(\frac{m-Q}{m-1}(1 - e^{-\frac{QK}{m}})\right)$-approximation algorithm for $\alpha_{{{{\mathrm{B, dec}}}}}$-<span style="font-variant:small-caps;">SU-CC</span>.
Let $n_i$ denote the number of the agents not-yet-assigned until the $(i+1)$-th iteration of the algorithm. Using the same reasoning as in Lemma \[lemma:greedyCC\] we show that $n_i \leq n(1 -
\frac{Q}{m})^{i}$. As before, our proof proceeds by induction on $i$. It is evident that the hypothesis is correct for $i=0$. Now, assuming that $n_i \leq n(1 - \frac{Q}{m})^{i}$, we assess $n_{i+1}$ as follows: $$\begin{aligned}
n_{i+1} \leq n_{i} - \frac{n_{i}Q}{m-i} \leq n_{i}\left(1
-\frac{Q}{m}\right) \leq n\left(1 - \frac{Q}{m}\right)^{i+1}
\textrm{.}\end{aligned}$$ This proves the hypothesis. Thus, we can bound $n_K$: $$\begin{aligned}
n_{K} \leq n\left(1 - \frac{Q}{m}\right)^{K} \leq
n\left(\frac{1}{e}\right)^{\frac{QK}{m}} \textrm{.}\end{aligned}$$ This means that the satisfaction of the assignment $\Phi$ returned by our algorithm is at least: $$\begin{aligned}
\ell_{1}^{\alpha_{{{{{\mathrm{B, dec}}}}}}}(\Phi) \geq (n - n_K)(m - Q) \geq
n(m-Q)(1 - e^{-\frac{QK}{m}}) \textrm{.}\end{aligned}$$ In effect, it holds that: $$\begin{aligned}
\frac{\ell_{1}^{\alpha_{{{{{\mathrm{B, dec}}}}}}}(\Phi)}{{{{{\mathrm{OPT}}}}}} \geq \frac{n(m-Q)(1
- e^{-\frac{QK}{m}})}{n(m-1)} \geq \frac{m-Q}{m-1}\left(1 -
e^{-\frac{QK}{m}}\right)\textrm{.}\end{aligned}$$ This completes the proof.
{width="\textwidth"}
{width="\textwidth"}
For example, for Polish parliamentary elections ($K=460$, $m=6000$), it suffices that each voter ranks only $0.5\%$ of his or her top alternatives (that is, about $30$ alternatives) for the algorithm to find a solution with guaranteed satisfaction at least $90\%$ of the one (possibly infeasible) where every voter is assigned to his or her top alternative.
ILP Formulation (Monroe, CC)
----------------------------
To experimentally measure the quality of our approximation algorithms, we compare the results against optimal solutions that we obtain using integer linear programs (ILPs) that solve the Monroe and Chamberlin–Courant winner determination problem. An ILP for the Monroe rule was provided by Potthoff and Brams [@potthoff-brams], Lu and Boutilier [@budgetSocialChoice] adapted it also for the Chamberlin–Courant rule with arbitraty PSF $\alpha$. For the sake of completeness, below we recall the ILP whose optimal solutions correspond to $\alpha$-SU-Monroe winner sets for the given election (we also indicate which constraints to drop to obtain an ILP for finding $\alpha$-SU-CC winner sets):
1. For each $i$, $1 \leq i \leq n$, and each $j$, $1 \leq j \leq m$ we have a $0/1$ variable $a_{ij}$ indicating whether alternative $a_j$ represents agent $i$. For each $j$, $1 \leq j \leq m$, we have a $0/1$ variable $x_j$ indicating whether alternative $a_j$ is included in the set of winners.
2. Our goal is to maximize the value $\sum_{i =
1}^{n}\alpha(pos_{i}(a_j))a_{ij}$ subject to the following constraints:
1. For each $i$ and $j$, $1 \leq i \leq n, 1 \leq j \leq m$, $0
\leq a_{ij} \leq x_j$ (alaternative $a_j$ can represent agent $i$ only if $a_j$ belongs to the set of winners)
2. For each $i$, $1 \leq i \leq n$, $\sum_{1 \leq j \leq m}
a_{ij} = 1$ (every agent is represented by exactly one alternative).
3. \[item:bounding\] For each $j$, $1 \leq j \leq m$, $x_j
\lfloor \frac{n}{K}\rfloor \leq \sum_{1 \leq i \leq n} a_{ij}
\leq x_j \lceil \frac{n}{K} \rceil$ (each alternative either does not represent anyone or represents between $\lfloor \frac{n}{K}
\rfloor$ and $\lceil \frac{n}{K} \rceil$ agents; if we remove these constraints then we obtain an ILP for the Chamberlin-Courant rule).
4. $\sum_{j=1}^n x_j \leq K$ (there are exactly $K$ winners[^4]).
We used the GLPK 4.47 package (GNU Linear Programming Kit, version 4.47) to solve these ILPs, whenever it was possible to do so in reasonable time.
Empirical Evaluation of the Algorithms {#sec:experiments}
======================================
In this section we present the results of empirical evaluation of algorithms from Section \[sec:algorithms\]. In the experiments we evaluated versions of the randomized algorithms that use exactly 100 sampling steps. In all cases but one, we have used Borda PSF to measure voter satisfaction. In one case, with six candidates, we have used DPSF defined through vector $(3,3,3,2,1,0)$ (we made this choice due to the nature of the data set used; see discussion later).
We have conducted four sets of experiments. First, we have tested all our algorithms on relatively small elections (up to $10$ candidates, up to $100$ agents). In this case we were able to compare the solutions provided by our algorithms with the optimal ones. (To obtain the optimal solutions, we were using the ILP formulations and the GLPK’s ILP solver.) Thus we report the quality of our algorithms as the average of fractions ${C}/{C_{{{{{\mathrm{opt}}}}}}}$, where $C$ is the satisfaction obtained by a respective algorithm and $C_{{{{{\mathrm{opt}}}}}}$ is the satisfaction in the optimal solution. For each algorithm and data set, we also report the average fraction ${C}/{C_{{{{\mathrm{ideal}}}}}}$, where $C_{{{{\mathrm{ideal}}}}}$ is the satisfaction that the voters would have obtained if each of them were matched to his or her most preferred alternative. In our further experiments, where we considered larger elections, we were not able to compute optimal solutions, but fraction ${C}/{C_{{{{\mathrm{ideal}}}}}}$ gives a lower bound for ${C}/{C_{{{{{\mathrm{opt}}}}}}}$. We report this value for small elections so that we can see an example of the relation between ${C}/{C_{{{{{\mathrm{opt}}}}}}}$ and ${C}/{C_{{{{\mathrm{ideal}}}}}}$ and so that we can compare the results for small elections with the results for the larger ones. Further, for the case of Borda PSF the ${C}/{C_{{{\mathrm{ideal}}}}}$ fraction has a very natural interpretation: If its value for a given solution is $v$, then, on the average, in this solution each voter is matched to an alternative that he or she prefers to $(m-1)v$ alternatives.
In our second set of experiments, we have run our algorithms on large elections (thousands of agents, hundreds of alternatives), coming either from the NetFlix data set (see below) or generated by us using one of our models. Here we reported the average fraction ${C}/{C_{{{{\mathrm{ideal}}}}}}$ only. We have analyzed the quality of the solutions as a function of the number of agents, the number of candidates, and the relative number of winners (fraction $K/m$). (This last set of results is particularly interesting because in addition to measuring the quality of our algorithms, it allows one to asses the size of a committee one should seek if a given average satisfaction of agents is to be obtained).
In the third set of experiments, we have investigated the effect of submitting truncated ballots (i.e., preference orders where only some of the top alternatives are ranked). Specifically, we focused on the relation between the fraction of ranked alternatives and the approximation ratio of the algorithms. We run our experiments on relatively large instances describing agents’ preferences; thus, here as in the previous set of experiments, we used NetFlix data set and the synthetic data sets. We report the quality of the algorithms as the ratio ${C}/{C_{{{{\mathrm{ideal}}}}}}$.
In the fourth set of experiments we have measured running times of our algorithms and of the ILP solver. Even though all our algorithms (except for the ILP based ones) are polynomial-time, in practice some of them are too slow to be useful.
Experimental Data {#sec:DataSets}
-----------------
For the evaluation of the algorithms we have considered both real-life preference-aggregation data and synthetic data, generated according to a number of election models. The experitments reported in this paper predate the work of Mattei and Walsh [@mat-wal:c:preflib] on gathering a large collection of data sets with preference data, but we mention that the conference version of this paper contributed several data sets to their collection.
### Real-Life Data {#sec:real-data}
We have used real-life data regarding people’s preference on sushi types, movies, college courses, and competitors’ performance in figure-skating competitions. One of the major problems regarding real-life preference data is that either people express preferences over a very limited set of alternatives, or their preference orders are partial. To address the latter issue, for each such data set we complemented the partial orders to be total orders using the technique of Kamishima [@Kamishima:Nantonac]. (The idea is to complete each preference order based on those reported preference orders that appear to be similar.)
Some of our data sets contain a single profile, whereas the others contain multiple profiles. When preparing data for a given number $m$ of candidates and a given number $n$ of voters from a given data set, we used the following method: We first uniformly at random chose a profile within the data set, and then we randomly selected $n$ voters and $m$ candidates. We used preference orders of these $n$ voters restricted to these $m$ candidates.
**Sushi Preferneces.**We used the set of preferences regarding sushi types collected by Kamishima[@Kamishima:Nantonac].[^5] Kamishima has collected two sets of preferences, which we call <span style="font-variant:small-caps;">S1</span> and <span style="font-variant:small-caps;">S2</span>. Data set S1 contains complete rankings of $10$ alternatives collected from $5000$ voters. S2 contains partial rankings provided by $5000$ voters over a set of $100$ alternatives (each vote ranks $10$ alternatives). We used Kamishima [@Kamishima:Nantonac] technique to obtain total rankings.
**Movie Preferences.**Following Mattei et al. [@Mattei:Netflix], we have used the NetFlix data set[^6] of movie preferences (we call it <span style="font-variant:small-caps;">Mv</span>). NetFlix data set contains ratings collected from about $480$ thousand distinct users regarding $18$ thousand movies. The users rated movies by giving them a score between $1$ (bad) and $5$ (good). The set contains about $100$ million ratings. We have generated $50$ profiles using the following method: For each profile we have randomly selected $300$ movies, picked $10000$ users that ranked the highest number of the selected movies, and for each user we have extended his or her ratings to a complete preference order using the method of Kamishima [@Kamishima:Nantonac].
**Course Preferences.**Each year the students at the AGH University choose courses that they would like to attend. The students are offered a choice of six courses of which they have to attend three. Thus the students are asked to give an unordered set of their three top-preferred courses and a ranking of the remaining ones (in case too many students select a course, those with the highest GPA are enrolled and the remaining ones are moved to their less-preferred courses). In this data set, which we call <span style="font-variant:small-caps;">Cr</span>, we have $120$ voters (students) and $6$ alternatives (courses). However, due to the nature of the data, instead of using Borda count PSF as the satisfaction measure, we have used the vector $(3,3,3,2,1,0)$. Currently this data set is available as part of PrefLib [@mat-wal:c:preflib].
**Figure Skating.**This data set, which we call <span style="font-variant:small-caps;">Sk</span>, contains preferences of the judges over the performances in a figure-skating competitions. The data set contains $48$ profiles, each describing a single competition. Each profile contains preference orders of $9$ judges over about 20 participants. The competitions include European skating championships, Olympic Games, World Junior, and World Championships, all from 1998[^7]. (Note that while in figure skating judges provide numerical scores, this data set is preprocessed to contain preference orders.)
### Synthetic Data
For our tests, we have also used profiles generated using three well-known distributions of preference orders.
**Impartial Culture.**Under the impartial culture model of preferences (which we denote <span style="font-variant:small-caps;">IC</span>), for a given set $A$ of alternatives, each voter’s preference order is drawn uniformly at random from the set of all possible total orders over $A$. While not very realistic, profiles generated using impartial culture model are a standard testbed of election-related algorithms.
**Polya-Eggenberger Urn Model.**Following McCabe-Dansted and Slinko [@mcc-sli:j:experiments] and Walsh [@Walsh11], we have used the Polya-Eggenberger urn model [@bpublicchoice85] (which we denote <span style="font-variant:small-caps;">Ur</span>). In this model we generate votes as follows. We have a set $A$ of $m$ alternatives and an urn that initially contains all $m!$ preference orders over $A$. To generate a vote, we simply randomly pick one from the urn (this is our generated vote), and then—to simulate correlation between voters—we return $a$ copies of this vote to the urn. When generating an election with $m$ candidates using the urn model, we have set the parameter $a$ so that $\frac{a}{m!} = 0.05$ (Both McCabe-Dansted and Slinko [@mcc-sli:j:experiments] and Walsh [@Walsh11] call this parameter $b$; we mention that those authors use much higher values of $b$ but we felt that too high a value of $b$ leads to a much too strong correlation between votes).
**Generalized Mallow’s Model.**We refer to this data set as <span style="font-variant:small-caps;">Ml</span>. Let $\succ$ and $\succ'$ be two preference orders over some alternative set $A$. Kendal-Tau distance between $\succ$ and $\succ'$, denoted $d_{K}(\succ,\succ')$, is defined as the number of pairs of candidates $x, y \in A$ such that either $x \succ
y \land y \succ' x$ or $y \succ x \land x \succ' y$.
Under Mallow’s distribution of preferences [@mallowModel] we are given two parameters: A *center* preference order $\succ$ and a number $\phi$ between $0$ and $1$. The model says that the probability of generating preference order $\succ'$ is proportional to the value $\phi^{d_{K}(\succ,\succ')}$. To generate preference orders following Mallow’s distribution, we use the algorithm given by Lu and Boutilier [@mallowImplementation2011].
In our experiments, we have used a mixture of Mallow’s models. Let $A$ be a set of alternatives and let $\ell$ be a positive integer. This mixture model is parametrized by three vectors, $\Lambda =
(\lambda_{1}, \dots, \lambda_{\ell})$ (where each $\lambda_i$ is between $0$ and $1$, and $\sum_{i=1}^\ell\lambda_1=1$), $\Phi
= (\phi_{1}, \dots, \phi_{\ell})$ (where each $\phi_i$ is a number between $0$ and $1$), and $\Pi = (\succ_{1}, \ldots,
\succ_{\ell})$ (where each $\succ_i$ is a preference order over $A$). To generate a vote, we pick a random integer $i$, $1
\leq i \leq \ell$ (each $i$ is chosen with probability $\lambda_i$), and then generate the vote using Mallow’s model with parameters $(\succ_i,\phi_i)$.
For our experiments, we have used $a = 5$, and we have generated vectors $\Lambda$, $\Phi$, and $\Pi$ uniformly at random.
[|c|c|c|c|c|c||c|c|c|c|]{} & &\
& A & B & C & GM & R & C & GM & P & R\
<span style="font-variant:small-caps;">S1</span> & $0.94$ & $0.99$ & $\approx 1.0$ & $0.99$ & $0.99$ & $1.0$ & $\approx 1.0$ & $0.99$ & $0.99$\
<span style="font-variant:small-caps;">S2</span> & $0.95$ & $0.99$ & $1.0$ & $\approx 1.0$ & $0.99$ & $1.0$ & $\approx 1.0$ & $0.98$ & $0.99$\
<span style="font-variant:small-caps;">Mv</span> & $0.96$ & $\approx 1.0$ & $1.0$ & $\approx 1.0$ & $0.98$ & $1.0$ & $\approx 1.0$ & $0.96$ & $\approx 1.0$\
<span style="font-variant:small-caps;">Cr</span> & $0.98$ & $0.99$ & $1.0$ & $\approx 1.0$ & $0.99$ & $1.0$ & $\approx 1.0$ & $1.0$ & $\approx 1.0$\
<span style="font-variant:small-caps;">Sk</span> & $0.99$ & $\approx 1.0$ & $1.0$ & $\approx 1.0$ & $0.94$ & $1.0$ & $\approx 1.0$ & $0.85$ & $0.99$\
<span style="font-variant:small-caps;">IC</span> & $0.94$ & $0.99$ & $\approx 1.0$ & $0.99$ & $0.99$ & $1.0$ & $\approx 1.0$ & $0.99$ & $0.99$\
<span style="font-variant:small-caps;">Ml</span> & $0.94$ & $0.99$ & $1.0$ & $0.99$ & $0.99$ & $1.0$ & $\approx 1.0$ & $0.99$ & $0.99$\
<span style="font-variant:small-caps;">Ur</span> & $0.95$ & $0.99$ & $\approx 1.0$ & $0.99$ & $0.99$ & $1.0$ & $0.99$ & $0.97$ & $0.99$\
[|c|c|c|c|c|c||c|c|c|c|]{} & &\
& A & B & C & GM & R & C & GM & P & R\
<span style="font-variant:small-caps;">S1</span> & $0.95$ & $\approx 1.0$ & $1.0$ & $0.99$ & $0.99$ & $1.0$ & $\approx 1.0$ & $0.97$ & $0.99$\
<span style="font-variant:small-caps;">S2</span> & $0.94$ & $0.99$ & $\approx 1.0$ & $0.99$ & $0.99$ & $1.0$ & $\approx 1.0$ & $0.98$ & $\approx 1.0$\
<span style="font-variant:small-caps;">Mv</span> & $0.95$ & $0.99$ & $1.0$ & $\approx 1.0$ & $0.98$ & $1.0$ & $\approx 1.0$ & $0.97$ & $\approx 1.0$\
<span style="font-variant:small-caps;">Cr</span> & $0.96$ & $\approx 1.0$ & $1.0$ & $\approx 1.0$ & $0.99$ & $1.0$ & $1.0$ & $1.0$ & $1.00$\
<span style="font-variant:small-caps;">Sk</span> & $0.99$ & $\approx 1.0$ & $1.0$ & $\approx 1.0$ & $0.88$ & $1.0$ & $1.0$ & $0.91$ & $\approx 1.0$\
<span style="font-variant:small-caps;">IC</span> & $0.95$ & $0.99$ & $\approx 1.0$ & $0.99$ & $0.99$ & $1.0$ & $\approx 1.0$ & $0.99$ & $0.99$\
<span style="font-variant:small-caps;">Ml</span> & $0.95$ & $0.99$ & $\approx 1.0$ & $0.99$ & $0.99$ & $1.0$ & $\approx 1.0$ & $0.98$ & $0.99$\
<span style="font-variant:small-caps;">Ur</span> & $0.96$ & $0.99$ & $\approx 1.0$ & $0.99$ & $\approx 1.0$ & $1.0$ & $\approx 1.0$ & $0.96$ & $0.99$\
[|c|c|c|c|c|c||c|c|c|c|]{} & &\
& A & B & C & GM & R & C & GM & P & R\
<span style="font-variant:small-caps;">S1</span> & $0.85$ & $0.89$ & $0.9$ & $0.89$ & $0.89$ & $0.92$ & $0.89$ & $0.91$ & $0.92$\
<span style="font-variant:small-caps;">S2</span> & $0.85$ & $0.89$ & $0.89$ & $0.89$ & $0.89$ & $0.93$ & $0.9$ & $0.91$ & $0.92$\
<span style="font-variant:small-caps;">Mv</span> & $0.88$ & $0.92$ & $0.92$ & $0.92$ & $0.91$ & $0.97$ & $0.92$ & $0.93$ & $0.97$\
<span style="font-variant:small-caps;">Cr</span> & $0.94$ & $0.97$ & $0.96$ & $0.96$ & $0.96$ & $0.97$ & $0.97$ & $0.97$ & $0.97$\
<span style="font-variant:small-caps;">Sk</span> & $0.96$ & $0.96$ & $0.97$ & $0.97$ & $0.91$ & $1.0$ & $0.97$ & $0.82$ & $0.99$\
<span style="font-variant:small-caps;">IC</span> & $0.8$ & $0.84$ & $0.85$ & $0.84$ & $0.84$ & $0.85$ & $0.83$ & $0.84$ & $0.85$\
<span style="font-variant:small-caps;">Ml</span> & $0.83$ & $0.88$ & $0.88$ & $0.9$ & $0.88$ & $0.92$ & $0.90$ & $0.89$ & $0.94$\
<span style="font-variant:small-caps;">Ur</span> & $0.8$ & $0.85$ & $0.86$ & $0.87$ & $0.85$ & $0.9$ & $0.87$ & $0.87$ & $0.89$\
[|c|c|c|c|c|c||c|c|c|c|]{} & &\
& A & B & C & GM & R & C & GM & P & R\
S1 & $0.91$ & $0.96$ & $0.96$ & $0.95$ & $0.95$ & $0.98$ & $0.98$ & $0.96$ & $0.98$\
S2 & $0.88$ & $0.93$ & $0.93$ & $0.93$ & $0.93$ & $0.98$ & $0.98$ & $0.96$ & $0.98$\
Mv & $0.85$ & $0.89$ & $0.89$ & $0.89$ & $0.88$ & $0.99$ & $0.99$ & $0.97$ & $0.99$\
Cr & $0.95$ & $0.98$ & $0.99$ & $0.99$ & $0.98$ & $1.0$ & $1.0$ & $1.0$ & $1.0$\
Sk & $0.91$ & $0.92$ & $0.92$ & $0.92$ & $0.81$ & $1.0$ & $1.0$ & $0.91$ & $\approx 1.0$\
IC & $0.91$ & $0.95$ & $0.95$ & $0.94$ & $0.95$ & $0.96$ & $0.96$ & $0.95$ & $0.95$\
Ml & $0.89$ & $0.94$ & $0.94$ & $0.94$ & $0.93$ & $0.97$ & $0.98$ & $0.95$ & $0.98$\
Ur & $0.91$ & $0.95$ & $0.95$ & $0.94$ & $0.95$ & $0.98$ & $0.98$ & $0.94$ & $0.97$\
Evaluation on Small Instances
-----------------------------
We now present the results of our experiments on small elections. For each data set, we generated elections with the number of agents $n=100$ ($n=9$ for data set <span style="font-variant:small-caps;">Sk</span> because there are only $9$ voters there) and with the number of alternatives $m=10$ ($m=6$ for data set <span style="font-variant:small-caps;">Cr</span> because there are only $6$ alternatives there) using the method described in Section \[sec:real-data\] for the real-life data sets, and in the natural obvious way for synthetic data. For each algorithm and for each data set we ran $500$ experiments on different instances for $K=3$ (for the <span style="font-variant:small-caps;">Cr</span> data set we used $K=2$) and $500$ experiments for $K=6$ (for <span style="font-variant:small-caps;">Cr</span> we set $K=4$). For Algorithm $C$ (both for Monroe and for CC) we set the parameter $d$, describing the number of assignment functions computed in parallel, to $15$. The results (average fractions ${C}/{C_{{{{\mathrm{opt}}}}}}$ and ${C}/{C_{{{\mathrm{ideal}}}}}$) for $K=3$ are given in Tables \[table:qualityAlgs1\] and \[table:qualityAlgs3\]; the results for $K=6$ are given in Tables \[table:qualityAlgs2\] and \[table:qualityAlgs4\] (they are almost identical as for $K=3$). For each experiment in this section we also computed the standard deviation; it was always on the order of $0.01$. The results lead to the following conclusions:
1. For the case of Monroe, already Algorithm A obtains very good results, but nonetheless Algorithms B and C improve noticeably upon Algorithm A. In particular, Algorithm C (for $d=15$) obtains the highest satisfaction on all data sets and in almost all cases was able to find an optimal solution.
2. Both for Monroe and for CC, Algorithm R gives slightly worse solutions than Algorithm C.
3. The results do not seem to depend on the data sets used in the experiments (the only exception is Algorithm R for the Monroe system on data set <span style="font-variant:small-caps;">Sk</span>; however <span style="font-variant:small-caps;">Sk</span> has only 9 voters so it can be viewed as a border case).
Evaluation on Larger Instances
------------------------------
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For experiments on larger instances we needed data sets with at least $n = 10 000$ agents. Thus we used the NetFlix data set and synthetic data. (Additionally, we run the subset of experiments (for $n \leq 5000$) also for the <span style="font-variant:small-caps;">S2</span> data set.) For the Monroe rule we present results for Algorithm A, Algorithm C, and Algorithm R, and for the Chamberlin–Courant rule we present results for Algorithm C and Algorithm R. We limit the set of algorithms for the sake of the clarity of the presentation. For Monroe we chose Algorithm A because it is the simplest and the fastest one, Algorithm C because it is the best generalization of Algorithm A that we were able to run in reasonable time, and Algorithm R to compare a randomized algorithm to deterministic ones. For the Chamberlin–Courant rule we chose Algorithm C because it is, intuitively, the best one, and we chose Algorithm R for the same reason as in the case of Monroe.
First, for each data set and for each algorithm we fixed the value of $m$ and $K$ and for each $n$ ranging from $1000$ to $10000$ with the step of $1000$ we run $50$ experiments. We repeated this procedure for 4 different combinations of $m$ and $K$: ($m = 10$, $K = 3$), ($m =
10$, $K = 6$), ($m = 100$, $K = 30$) and ($m = 100$, $K = 60$). We measured the statistical correlation between the number of voters and the quality of the algorithms $C/C_{{{{\mathrm{ideal}}}}}$. The ANOVA test in most cases showed that there is no such correlation. The only exception was <span style="font-variant:small-caps;">S2</span> data set, for which we obtained an almost negligible correlation. For example, for ($m = 10, K = 3$) Algorithm $C$ under data set <span style="font-variant:small-caps;">S2</span> for Monroe’s system for $n = 5000$ gave $C/C_{{{{\mathrm{ideal}}}}} = 0.88$, while for $n = 100$ (in the previous section) we got $C/C_{{{{\mathrm{ideal}}}}} = 0.89$. Thus we conclude that in practice the number of agents has almost no influence on the quality of the results provided by our algorithms.
Next, we fixed the number of voters $n = 1000$ and the ratio $K/m =
0.3$, and for each $m$ ranging from $30$ to $300$ with the step of $30$ (naturally, as $m$ changed, so did $K$ to maintain the ratio $K/m$), we run 50 experiments. We repeated this procedure for $K/m =
0.6$. The relation between $m$ and $C/C_{{{{\mathrm{ideal}}}}}$ for <span style="font-variant:small-caps;">Mv</span> and <span style="font-variant:small-caps;">Ur</span>, under both the Monroe rule and the Chamberlin–Courant rule, is given in Figures \[fig:changing\_m\_monroe\] and \[fig:changing\_m\_cc\] (the results for $K/m = 0.6$ look similar).
Finally, we fixed $n = 1000$ and $m = 100$, and for each $K/m$ ranging from $0.1$ and $0.5$ with the step of $0.1$ we run $50$ experiments. The relation between the ratio $K/m$ and the quality $C/C_{{{{\mathrm{ideal}}}}}$ is presented in Figures \[fig:changing\_km\_monroe\] and \[fig:changing\_km\_cc\].
For the case of Chamberlin–Courant system, increasing the size of the committee to be elected improves overall agents’ satisfaction. Indeed, since there are no constraints on the number of agents matched to a given alternative, a larger committee means more opportunities to satisfy the agents. For the Monroe rule, a larger committee may lead to a lower total satisfaction. This happens if many agents like a particular alternative a lot, but only some of them can be matched to this alternative and others have to be matched to their less preferred ones. Nonetheless, we see that Algorithm C achieves $C/C_{{{{\mathrm{ideal}}}}} = 0.925$ even for $K/m = 0.5$ for the NetFlix data set.
Our conclusions from these experiments are the following. For the Monroe rule, even Algorithm A achieves very good results. However, Algorithm C consistently achieves better (indeed, almost perfect) ones. For the Chamberlin–Courant rule the randomized algorithm on some datasets performs better than the deterministic ones. However, even in such cases, the improvement over the Algorithm C is small.
Truncated ballots {#sec:truncated}
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The purpose of our third set of experiments was to see how our algorithm behave in practical settings with truncated ballotrs. We conducted this part of evaluation on relatively large instances, including $n=1000$ agents and up to $m=100$ alternatives. Thus, in this set of experiments, we used the same sets of data as in the previous subsection: the Netflix data set and the synthetic distributions. Similarly, we evaluated the same algorithms: Algorithm A, C, and R for the case of Monroe’s system, and Algorithm C, and R for the case of the Chamberlin–Courant system.
For each data set and for each algorithm we run experiments for 3 independent settings with different values of the parameters describing the elections: (1) $m=100$, $K=20$, (2) $m=100$, $K=10$, and (3) $m=20$, $K=4$. For each setting we run the experiments for the values of $P$ (the number of known positions) varying between 1 and $m$.
For each algorithm, data set, setting and each value of $P$ we run 50 independent experiments in the following way. From a data set we sampled a sub-profile of the appropriate size $n \times m$. We truncated this profile to the $P$ first positions. We run the algorithm for the truncated profile and calculated the quality ratio ${C}/{C_{{{{\mathrm{ideal}}}}}}$. When calculating ${C}/{C_{{{{\mathrm{ideal}}}}}}$ we assumed the worst case scenario, i.e., that the satisfaction of the agent from an alternative outside of his/her first $P$ positions is equal to 0. In other words, we used the positional scoring function described by the following vector: $\langle m-1, m-2, \dots, m-P, 0, \dots 0
\rangle$. Next, we averaged the values of ${C}/{C_{{{{\mathrm{ideal}}}}}}$ over all 50 experiments.
The relation between the percentage of the known positions in the preference profile and the average quality of the algorithm for the Monroe and Chamberlin–Courant systems are plotted in Figures \[fig:changing\_borda\_monroe\] and \[fig:changing\_borda\_cc\], respectively. We omit the plots for Mallow’s model, as in this case we obtained almost identical results as for the Urn model. We have the following conclusions.
1. All the algorithms require only small number of the top positions to achieve their best quality. Here, the deterministic algorithms are superior.
2. The small elections with synthetic distributions appear to be the worst case scenario—in such case we require the knowledge of about 40% of the top positions to obtain the highest approximation ratios of the algorithms. In the case of the NetFlix data set, even on small instances the deterministic algorithms require only about 8% of the top positions to get their best quality (however the quality is already high for 3-5% of the top positions). For the larger number of the alternatives, the algorithms do not require more than 3% of the top positions to reach their top results.
3. Algorithm C does not only give the best quality but it is also most immune to the lack of knowledge. These results are more evident for the case of the Monroe system.
Running time
------------
![The running time of the standard ILP solver for the Monroe and for the Chamberlin–Courant systems. For Monroe’s system, for [$K=9, m = 30$]{}, and for [$n \geq 200$]{} none of the single algorithm execution finished within 1 day.[]{data-label="fig:ilp_runtime"}](ilp_runtime){width="60.00000%"}
[|c|c|c|c|c||c|c|c|]{} & &\
& $n=$ & 2000 & 6000 & 10000 & 2000 & 6000 & 10000\
& A & $0.01$ & $0.03$ & $0.05$ & $0.01$ & $0.04$ & $0.07$\
& B & $0.08$ & $0.9$ & $2.3$ & $0.2$ & $1.4$ & $3.6$\
& C & $1.1$ & $8$ & $22$ & $2.1$ & $16$ & $37$\
& GM & $0.8$ & $7.3$ & $20$ & $1.9$ & $13$ & $52$\
& R & $7.6$ & $50$ & $180$ & $6.5$ & $52$ & $140$\
& C & $0.02$ & $0.07$ & $0.12$ & $0.05$ & $0.14$ & $0.26$\
& GM & $0.003$ & $0.009$ & $0.015$ & $0.003$ & $0.01$ & $0.018$\
& P & $0.009$ & $0.032$ & $0.05$ & $0.008$ & $0.02$ & $0.05$\
& R & $0.014$ & $0.04$ & $0.065$ & $0.02$ & $0.06$ & $0.11$\
\
& &\
& $n=$ & 2000 & 6000 & 10000 & 2000 & 6000 & 10000\
& A & $0.5$ & $1.6$ & $2.8$ & $0.9$ & $2.8$ & $4.9$\
& B & $0.8$ & $4$ & $9.5$ & $1.7$ & $8$ & $18$\
& C & $38$ & $140$ & $299$ & $64$ & $221$ & $419$\
& GM & $343$ & $2172$ & $5313$ & $929$ & $5107$ & $13420$\
& R & $41$ & $329$ & $830$ & $88$ & $608$ & $1661$\
& C & $4.3$ & $11$ & $19$ & $7.5$ & $19$ & $31$\
& GM & $0.06$ & $0.2$ & $0.4$ & $0.09$ & $0.3$ & $0.7$\
& P & $0.03$ & $0.1$ & $0.26$ & $0.03$ & $0.1$ & $0.2$\
& R & $0.06$ & $0.24$ & $0.45$ & $0.1$ & $0.4$ & $0.8$\
In our final set of experiments, we have measured running times of our algorithms on the data set <span style="font-variant:small-caps;">Mv</span>. We have used a machine with Intel Pentium Dual T2310 1.46GHz processor and 1.5GB of RAM. In Figure \[fig:ilp\_runtime\] we show the running times of the GLPK ILP solver for the Monroe and for Chamberlin–Courant rules. These running times are already large for small instances and they are increasing exponentially with the number of voters. For the Monroe rule, even for $K=9, m = 30, n=100$ some of the experiments timed out after 1 hour, and for $K=9, m = 30, n=200$ none of the experiments finished within one day. Thus we conclude that the real application of the ILP-based algorithm is very limited.
Example running times of the other algorithms for some combinations of $n$, $m$, and $K$ are presented in Table \[table:runningTimes\]. For the case of CC, essentially all the algorithms are very fast and the quality of computed solutions is the main criterion in choosing among them. For the case of Monroe, the situation is more complicated. While for small elections all the algorithms are practical, for elections with thousands of voters, using Algorithm GM becomes problematic. Indeed, even Algorithm C can be seen as a bit too slow if one expects immediate results. On the other hand, Algorithms A and B seem perfectly practical and, as we have seen in the previous experiments, give high-quality results.
Summary {#sec:conclusions}
=======
We have defined a certain resource allocation problem and have shown that it generalizes the problem of finding winners for the multiwinner voting rules of Monroe and of Chamberlin and Courant. Since it is known that the winners for these voting rules are hard to compute [@complexityProportionalRepr; @budgetSocialChoice; @fullyProportionalRepr; @sko-yu-fal-elk:c:single-crossing-monroe-cc; @sko-fal:t:max-cover], we focused on finding approximate solutions. We have shown that if we try to optimize agents’ dissatisfaction, then our problems are hard to approximate up to any constant factor. The same holds for the case where we focus on the satisfaction of the least satisfied agent. However, for the case of optimizing total satisfaction, we suggest good approximation algorithms. In particular, for the Monroe system we suggest a randomized algorithm that for the Borda score achieves an approximation ratio arbitrarily close to $0.715$ (and much better in many real-life settings), and ($1 - \frac{1}{e}$)-approximation algorithm for arbitrary positional scoring function. For the Chamberlin-Courant system, we have shown a polynomial-time approximation scheme (PTAS).
[p[0.2cm]{}|p[1.6cm]{}|p[1.6cm]{}|p[1.6cm]{}|p[1.6cm]{}|p[1.6cm]{}|p[2.0cm]{}|]{} & & &\
& dissat. & satisfaction & dissat. & satisfaction & dissat. & satisfaction\
& Inapprox. $\newline$ Theorem \[theorem:noApprox1\] & Good approx. & Inapprox. $\newline$ Theorem \[theorem:noApprox3\] & Good approx. & Inapprox. $\newline$ Theorem \[theorem:noApprox1\] Theorem \[theorem:noApprox3\] & Open problem\
& Inapprox. $\newline$ Theorem \[theorem:noApprox2\] & Inapprox. $\newline$ Theorem \[theorem:noApprox5\] & Inapprox. $\newline$ Theorem \[theorem:noApprox4\] & Inapprox. $\newline$ Theorem \[theorem:noApprox6\] & Inapprox. $\newline$ Theorem \[theorem:noApprox2\] Theorem \[theorem:noApprox4\] & Inapprox. $\newline$ Theorem \[theorem:noApprox5\] Theorem \[theorem:noApprox6\]\
Algorithm Approximation Runtime Reference
-- ----------- --------------------------------------------------------------- ------------------------------------------------------ ----------------------------------------
A $1 - \frac{K-1}{2(m-1)} - \frac{H_K}{K}$ $Kmn$ Lemma \[lemma:greedy\]
B as in Algorithm A $Kmn$$+$$O(\Phi^{S})$ Lemma \[lemma:greedy\]
C as in Algorithm A $dKmn$$+$$dO(\Phi^{S})$ Lemma \[lemma:greedy\]
GM as in Alg. A for Borda PSF; $1-\frac{1}{e}$ for others $KmO(\Phi^{S})$ Theorem \[thm:gmMonroe\]
R $\frac{1}{2}(1 + \frac{K}{m} - \frac{K^2m - K^{3}}{m^3-m^2})$ $\frac{|\log (1 -\lambda)|}{K\epsilon^2}O(\Phi^{S})$ Lemma \[lemma:randMonroe\]
AR $0.715$ $\max(\textrm{A}, \textrm{R})$ Theorem \[thm:combMonroe\]
Theorem \[theorem:ptas\]
P $1 - \frac{2{{{{\mathrm{w}}}}}(K)}{K}$ $nm{{{{\mathrm{w}}}}}(K)$ Lemma \[lemma:greedyCC\]
GM $1-\frac{1}{e}$ $Kmn$ Lu and Boutilier [@budgetSocialChoice]
C as in Algorithm GM $dKm(n$$+$$\log dm)$ Lu and Boutilier [@budgetSocialChoice]
R $(1-\frac{1}{K+1})(1 + \frac{1}{m})$ $\frac{|\log (1 -\lambda)|}{\epsilon^2}n$ Oren [@ore:p:cc]
: \[tab:algs\]A summary of the algorithms studied in this paper. The top of the table regards algorithms for Monroe’s rule and the bottom for the Chamberlin–Courant rule. In column “Approximation” we give currently known approximation ratio for the algorithm under Borda PSF, on profiles with $m$ candidates and where the goal is to select a committee of size $K$. Here, $O(\Phi^{S}) = O(n^2(K +
\mathrm{log}n))$ is the complexity of finding a partial representation function with the algorithm of Betzler et al. [@fullyProportionalRepr]. ${{{{\mathrm{w}}}}}(\cdot)$ denotes Lambert’s W-Function.
In Table \[table:summary\] we present the summary of our (in)approximability results. In Table \[tab:algs\] we present specific results regarding our approximation algorithms for the utilitarian satisfaction-based framework. In particular, the table clearly shows that for the case of Monroe, Algorithms B and C are not much slower than Algorithm A but offer a chance of improved peformance. Algorithm GM is intuitively even more appealing, but achieves this at the cost of high time complexity. For the case of Chamberlin-Courant rule, theoretical results suggest using Algorithm P (however, see below).
We have provided experimental evaluation of the algorithms for computing the winner sets both for the Monroe and Chamberlin–Courant rules . While finding solutions for these rules is computationally hard in the worst case, it turned out that in practice we can obtain very high quality solutions using very simple algorithms. Indeed, both for the Monroe and Chamberlin-Courant rules we recommend using Algorithm C (or Algorithm A on very large Monroe elections). Our experimental evaluation confirms that the algorithms work very well in case of truncated ballots. We believe that our results mean that (approximations of) the Monroe and Chamberlin–Courant rules can be used in practice.
Our work leads to a number of further research directions. First, it would be very interesting to find a better upper bound on the quality of solutions for the (satisfaction-based) Monroe and Chamberlin–Courant systems (with Borda PSF) than the simple $n(m-1)$ bound that we use (where $n$ is the number of voters and $m$ is the number of candidates). We use a different approach in our randomized algorithm, but it would be much more interesting to find a deterministic algorithm that beats the approximation ratios of our algorithms. One of the ways of seeking such a bound would be to consider Monroe’s rule with “exponential” Borda PSF, that is, with PSF of the form, e.g., $(2^{m-1}, 2^{m-2}, \ldots, 1)$. For such PSF our approach in the proof of Lemma \[lemma:greedy\] would not give satisfactory results and so one would be forced to seek different attacks. In a similar vein, it would be interesting to find out if there is a PTAS for Monroe’s system.
In our work, we have focused on PSFs that are strictly increasing/decreasing. It would also be interesting to study PSFs which increase/decrease but not strictly, that is allowing some equalities. We have started to work in this direction by considering the so-called $t$-approval PSF’s $\alpha_t$, which (in the satisfaction-based variant) are defined as follows: $\alpha_t(i) = 1$ if $i \leq t$ and otherwise $\alpha_t(i) = 0$. Results for this case for the Chamberlin–Courant rule are presented in the paper of Skowron and Faliszewski [@sko-fal:t:max-cover].
On a more practical side, it would be interesting to develop our study of truncated ballots. Our results show that we can obtain very high approximation ratios even when voters rank only relatively few of their top candidates. For example, to achieve 90% approximation ratio for the satisfaction-based Monroe system in Polish parliamentary election ($K=460, m=6000$), each voter should rank about $8.7\%$ of his or her most-preferred candidates. However, this is still over $500$ candidates. It is unrealistic to expect that the voters would be willing to rank this many candidates. Thus, how should one organize Monroe-based elections in practice, to balance the amount of effort required from the voters and the quality of the results?
Finally, going back to our general resource allocation problem, we note that we do not have any positive results for it (the negative results, of course, carry over from the more restrictive settings). Is it possible to obtain some good approximation algorithm for the resource allocation problem (in the utilitarian satisfaction-based setting) in full generality?
**Acknowledgements** This paper is based on two extended abstracts, presented at IJCAI-2013 and AAMAS-2012. We would like to thank the reviewers from these two conferences for very useful feedback. The authors were supported in part by Poland’s National Science Center grants UMO-2012/06/M/ST1/00358, DEC-2011/03/B/ST6/01393, and by AGH University of Science and Technology grant 11.11.120.865. Piotr Skowron was also supported by EU’s Human Capital Program “National PhD Programme in Mathematical Sciences” carried out at the University of Warsaw
.
[^1]: This paper combines and extends results presented at IJCAI-2013 (paper titled “Fully Proportional Representation as Resource Allocation: Approximability Results”; the paper contained most of our theoretical results) and at AAMAS-2012 (paper titled “Achieving Fully Proportional Representation is Easy in Practice”; the paper contained most of our experimental results).
[^2]: In general, this assumption is not as innocent as it may seem. Often dealing with cases there $K$ does not divide $\|N\|$ requires additional insights and care. However, for our algorithms and results, the assumption simiplifies notation and does not lead to obscuring any unexpected difficulties.
[^3]: This is very similar to the so-called MaxCover problem. Skowron and Faliszewski [@sko-fal:t:max-cover] have discussed the connection of MaxCover to the winner determination problem under the Chamberlin–Courant voting system (for approval-based satisfaction functions) and provided a number of FPT approximation schemes for it.
[^4]: For the Monroe framework inequality here is equivalent to equality. We use the inequality so that deleting constraints from item (\[item:bounding\]) leads to an ILP for the Chamberlin-Courant rule.
[^5]: The sushi data set is available under the following url: <http://www.kamishima.net/sushi/>
[^6]: http://www.netflixprize.com/
[^7]: This data set is available under the following url: <http://rangevoting.org/SkateData1998.txt>.
|
---
abstract: |
We present an analysis of the physical conditions in an extreme molecular cloud in the Antennae merging galaxies. This cloud has properties consistant with those required to form a globular cluster. We have obtained ALMA CO and 870$\mu$m observations of the Antennae galaxy system with $\sim 0''.5$ resolution. This cloud stands out in the data with a radius of $\lesssim 24$ pc and mass of $>5\times 10^6$ M$_\odot$. The cloud appears capable of forming a globular cluster, but the lack of associated thermal radio emission indicates that star formation has not yet altered the environment. The lack of thermal radio emission places the cloud in an early stage of evolution, which we expect to be short-lived ($\lesssim
1$ Myr) and thus rare. Given its mass and kinetic energy, for the cloud to be confined (as its appearance strongly suggests) it must be subject to an external pressure of P/$k_B \gtrsim
10^8$ K cm$^{-3}$ – 10,000 times higher than typical interstellar pressure. This would support theories that high pressures are required to form globular clusters and may explain why extreme environments like the Antennae are preferred environments for generating such objects. Given the cloud temperature of $\sim
25$ K, the internal pressure must be dominated by non-thermal processes, most likely turbulence. We expect the molecular cloud to collapse and begin star formation in $\lesssim 1$ Myr.
author:
- 'K. E. Johnson, A. K. Leroy, R. Indebetouw , C. L. Brogan, B. C. Whitmore, J. Hibbard, K. Sheth, A. Evans'
title: The Physical Conditions in a Pre Super Star Cluster Molecular Cloud in the Antennae Galaxies
---
Introduction
============
Globular Cluster Formation
--------------------------
Globular clusters are among the most ancient objects in the universe, often with ages $>12$ Gyr [@bolte95; @carretta00] and are common around massive galaxies in the universe today [@harris13]. The present-day abundance of globular clusters is remarkable given that that the fraction expected to survive $\sim 10$ Gyr is extremely small, potentially lower than 1% [@fall01; @whitmore07]. Thus, this extreme type of star formation may have been a critical mode in the early evolution of today’s massive galaxies.
Initial theories about globular cluster formation suggested that these objects were among the first to gravitationally collapse in the early universe [@peebles68]. Subsequent work, particularly after the launch of the Hubble Space Telescope, has demonstrated that clusters with extreme stellar densities often exceeding $\sim
10^4$ stars pc$^{-3}$ [e.g. @miocchi13] can still form in the universe today [@oconnell94] – the so-called “super star clusters” (SSCs). Since that time, numerous studies have indicated that the properties of SSCs are consistent with those expected of young globular clusters [e.g. @mclaughlin08].
A number of physical processes contribute to the destruction of clusters, including two-body relaxation, stellar mass loss and feedback, compressive and tidal shocks as clusters orbit their host galaxy, and tidal truncation. The extent to which each of these processes act on a specific cluster will depend on a variety of factors, including the orbital properties of the cluster [@gnedin97]. Indeed, the extent to which cluster disruption is mass-dependent is still debated [@fall09; @bastian12]. For unresolved clusters in galaxies outside the local group, there is typically limited (if any) dynamical information, and for clusters younger than a few Myr, little dynamical evolution will have taken place. For all of these reasons, it is not possible to say whether any particular SSC will survive for a Hubble time.
While there is no generally accepted definition of “Super Star Cluster”, here we adopt a definition based on a cluster having the [*potential*]{} to evolve into a globular cluster, regardless of whether or not it actually will do so over the following $\sim$ 10 Gyr. This requirement results in both mass and radius limits on the range of objects that can be considered as SSCs. Specifically, most present-day globular clusters have half-light radii of $< 10$ pc [although some have radii as large as $\sim
15$ pc, @vandenbergh91], and stellar masses of $\gtrsim
10^5$ M$_{\odot}$ [@harris94]. In addition, these clusters are expected to lose $\gtrsim 1/2$ of their mass due to dynamical effects over 10$^{10}$ years [@mclaughlin08], which suggests that to be a globular cluster progenitor, a young star cluster should have a mass of $\gtrsim 2\times 10^5$ M$_\odot$. If star formation efficiency is $\sim 20-50$% [@ashman01; @kroupa01], the initial molecular core from which the cluster is formed must have a mass of $\gtrsim
10^6$ M$_\odot$.
Optical techniques have been able to probe the evolution of SSCs (and presumably some future globular clusters) to ages as young as a few million years. Before this time, the clusters can be significantly shrouded by their birth material, limiting the usefulness of optical observations. Beginning in the late 1990’s efforts began to observe SSC evolution at even earlier ages ($\lesssim$ a few million years) by using radio observations to detect the free-free emission from the ionized gas around the cluster and internal to the cluster’s dust cocoon [@turner98; @kj99; @turner00; @beck00; @johnson01; @johnson02; @johnson03; @beck04; @turner04; @johnson04; @reines08; @johnson09; @tsai09; @aversa11; @kepley14]. We refer to these objects as “natal” SSCs, meaning that the clusters themselves have already formed, but they have not yet emerged from their birth material. Studies of natal clusters were able to place constraints on the relative lifetime of this enshrouded phase of SSC evolution to $\lesssim$ a million years and the gas density of the ionized hydrogen $n_e > 10^3$ cm$^{-3}$[@johnson03]. A large number of subsequent studies have now identified additional compact thermal radio sources in a number of galaxies, although their low detection rate supports their relatively short lifetime [@tsai09; @aversa11].
However, determining the physical conditions that give rise to SSCs (and their surviving descendants – globular clusters) has been mired in the fundamental difficulty that once an SSC is present in the molecular cloud, it will dramatically alter it. Thus in order to observationally probe the conditions capable of creating an SSC requires not only identifying molecular clouds that are compact (radii $\lesssim 25$ pc, see Section \[cloud\_size\]) and massive ($\gtrsim
10^6$ M$_{\odot}$), but also for which massive stars have not begun to disrupt the environment. Efforts to observe the actual formation of SSCs – before the star clusters have formed – requires high spatial resolution millimeter observations to determine the physical properties of the material from which the clusters will form. Such work has largely been stymied by the available observing facilities and limited to only the most nearby galaxies.
One example of a relatively nearby starburst system in which some progress has been made is M82. At 3.6 Mpc [@freedman94], relatively good linear resolution was achievable even before ALMA. This system was observed using the Owens Valley Radio Observatory (OVRO) in CO(2-1) with a linear resolution of 17 pc [@keto05]. While the compact molecular clouds observed in M82 are likely to be associated with early SSC evolution, multiwavelength observations suggest that these clouds have already begun star formation [@keto05], and have therefore disrupted their birth environment[^1].
This paper reports results from an ALMA Early Science project studying the Antennae galaxies. In a high resolution survey of CO emission from the Antennae [@whitmore14] the most immediately striking feature was a compact, high line width cloud with little associated star formation. This is coincident with a source identified by @herrera11 [@herrera12] as a potential proto-SSC using H$_2$ and earlier, lower resolution ALMA data. The strong compact H$_2$ emission appears to be due to warm (1700-2300 K) shocked gas. However, the size of the cold molecular component could only be constrained to $\lesssim 100$ pc, which precluded a conclusive identification.
The present paper characterizes this source, which we consider among the best candidates for a proto-SSC, and lays out the evidence for and against the source’s eventual evolution into a SSC or GC.
{width="7in"}
[cccc]{} CMA & 12CO1-0& $4.91''\times 3.15''$ & [@wilson00]\
SMA & 12CO3-2& $1.42''\times 1.12''$ & [@ueda12]\
SMA+PdBI & 12CO2-1& $3.3'' \times 1.5''$ & [@wei12]\
ALMA-SV & 12CO2-1& $1.68''\times 0.85''$ & [@espada12]\
ALMA-SV & 12CO3-2& $1.05''\times 0.60''$ & [@herrera12]\
ALMA-Cyc0 & 12CO3-2& $0.56''\times 0.43''$ & [this paper]{}\
ALMA Observations
=================
We obtained ALMA observations of the Antennae system in CO(3-2) and 870$\;\mu$m continuum with the goal of probing the conditions of cluster formation and early evolution; data calibration is discussed in detail in the overview paper [@whitmore14]. Briefly, the observations consisted of a 13-point mosaic, and were carried out in the “extended” configuration, with a maximum baseline of $\sim
400$ m and 5 km s$^{-1}$ spectral channels. The resulting rms was determined using line-free channels and found to be 3.3 mJy/beam. With an angular resolution FWHM of $0.''56 \times 0.''43$ ($59 \times
45$ pc), these observations are well-matched to the expected diameter of the precursor giant molecular clouds of $\lesssim 50$ pc (or a radius of 25 pc, see Section \[cloud\_size\]). For this paper, we also recalibrated and reimaged the SV data, as well as compared it to previous results to check for consistency. The SV CO(2-1) data used in this paper was taken with a beam size of $1.68''\times 0.85''$. The rms of the SV CO(2-1) observations was also measured using line-free channels and found to be 6.5 mJy/beam.
The ALMA observations enable the study of a compact and luminous source in the CO(3-2) data cube (see Figure \[plot\_zoom\_radio\]). This cloud is part of the super giant molecular cloud complex known as SGMC2 [@wilson00]; the full 3-D cube of SGMC2 is shown in Figure \[3D\]. The Antennae galaxy system was previously observed in both CO(2-1) and CO(3-2) by ALMA as part of the “science verification” (SV) process, which has already resulted in publications [see Table \[previous\_obs\], @espada12; @herrera12].
{width="7in"}
The specific source discussed here was singled out in the lower-resolution ALMA science verification data, with the CO(3-2) emission being coincident with strong H$_2$ emission – potentially indicating shocks due to infalling gas [@herrera12]. Even with lower resolution data ($\sim 100$ pc), it was speculated that this region might contain an SSC in the early stages of its evolution [@herrera12]. However, the spatial resolution of the SV CO(3-2) data is $\sim 100$ pc, or roughly twice that of the Cycle 0 data presented here. It is clear that in the SV data, the molecular cloud that is the subject of this paper is not resolved and is blended with other molecular material in the vicinity.
[cccccc]{} 12:01:54.73 & -18:52:53.2 & 1524$\pm 3$ & 52$\pm 5$ & 49$\pm 3$ & $0.66\pm0.12 \times 0.55\pm0.07$\
Cloud Analysis
--------------
Determining the properties of this cloud requires that it first be isolated from other emission in the region. As shown in Figures \[line\_profile\] and \[extracted\_cloud\], there is a redshifted secondary velocity component along this line of sight, that must be deconvolved from the primary source before analysis. We extract the primary cloud from the velocity cube for further analysis by creating a sub-cube around the cloud with dimensions of $\sim 1''\times 1'' \times 300$ km s$^{-1}$. The integrated intensity contours of the extracted primary cloud are shown in Figure \[plot\_zoom\_radio\], and the observed properties are listed in Table \[properties\]. We determine the cloud size by deconvolving the synthesized beam from the extracted source, which yields a half-light radius for the cloud of $\lesssim 24\pm 3$ pc ($\lesssim 0.23''$). The derived properties of the cloud are given in Table \[properties\]. As a sanity check, the properties of the primary cloud were also determined using the CPROPS program on the entire data cube (not exclusively the sub-cube) [@rosolowsky06], and the resulting parameters agree to within the uncertainties. Thus, by-hand measurement of the half-light size, automated Gaussian fits, and moment based measurements all yield roughly consistent sizes for our cloud. As this is a marginally resolved object with a clearly measurable line width methodological uncertainties do not overwhelm any of our conclusions. Throughout this text, we refer to the properties of the extracted primary cloud only, unless noted otherwise.
![The CO(3-2) and CO(1-0) spectra of a $\sim 1''\times 1''$ region around the candidate proto super star cluster molecular cloud taken from the full data cube (i.e. the primary cloud has not been extracted). The line emission indicates that there are two components along the line-of-sight that have different temperatures. Spectral profile of the CO(3-2) line from current work (ALMA Cycle 0 observations) and the CO(2-1) line from ALMA science verification [@espada12]. The CO(3-2) data is convolved to the CO(2-1) beam and both data sets are corrected for beam dilution. There is clearly more than one velocity component; in these convolved data, the secondary source was fit by a Gaussian and subtracted from the spectra. The CO(3-2)/CO(2-1) ratio is dramatically different in the two velocity components. \[line\_profile\]](Johnson_Extended1.png){width="1.0\columnwidth"}
Relative Astrometric Solutions
------------------------------
A comparison between the CO(3-2) emission and data at other wavelengths, requires an understanding of the relative astrometric accuracy. Based on the phase stability of the ALMA observations, we estimate the absolute astrometric accuracy to be better than $\sim
0.2''$. Centimeter observations from the VLA have an astrometric accuracy better than $\sim 0.1''$ [@brogan10], and therefore the 3.6 cm and CO(3-2) observations have a relative precision of better than the synthesized beam of the ALMA data, and we consider them to be astrometrically matched. We also register the astrometry of archival Hubble Space Telescope observations shown in Figure \[plot\_zoom\_radio\] by matching the Pa${\alpha}$ emission throughout the Antennae system to common features in the 3.6 cm emission.
Results
=======
Cloud Temperature and Optical Depth
-----------------------------------
We constrain the temperature and optical depth of the cloud using the CO(3-2) and CO(2-1) emission. We retrieved, recalibrated, and reimaged CO(2-1) observations from ALMA’s science verification period, shown over-plotted in Figure \[line\_profile\]. The CO(3-2) observations were convolved to the synthesized beam of the CO(2-1) observations and corrected for beam dilution using the Cycle 0 CO(3-2) source size, resulting in peak brightness temperatures of T$_{3-2}
= 17\pm3$ K and T$_{2-1} = 18\pm3$ K. The largest angular scale to which the CO(3-2) observations are sensitive is $\sim 6''$, and therefore we do not expect that any flux is resolved out on the size scales of interest here. The secondary component has a significantly lower CO(3-2)/CO(2-1) ratio, indicating much cooler gas than the primary cloud.
RADEX non-LTE modeling [@vandertak07] was used to analyze the CO(3-2) and CO(2-1) emission. The line intensities and their ratio were compared to a grid of RADEX models covering a range of values for kinetic temperature, CO column density, and H$_2$ volume density. The best-fit values result from a chi-squared minimization. Since the source is marginally resolved in CO(3-2), for CO(3-2) we set the beam filling fraction to 1, and for CO(2-1) we set it to the dilution factor, or the ratio of the CO(3-2) size to the CO(2-1) beamsize. The RADEX models indicate that these transitions are optically thick – the best fitting depth is $\tau\sim 3.5 \pm 0.5$, but the data do not rule out significantly higher values. The lines appear to be close to thermalized, with an excitation temperature within a degree of the kinetic temperature of 25$^{+10}_{-2}$ K; this temperature is on the upper end of the range of those found for dense molecular clouds in the Milky Way [@shirley13]. However, there is a degeneracy between the inferred temperature and density of the cloud, and the cloud could be warmer for densities $\lesssim 6\times
10^4$ cm$^{-3}$. In other words, the observed brightness could also be reproduced with a large column of subthermally excited, warm, relatively diffuse gas.
[ccccccccc]{}\[!b\] $< 53\times 41$ & $< 27\times 21^c$ & $29-85$ & $3.3-15$ & $> 2.8$ & $3.1-7.4$ & $23 - 35$ & $2 - 20 \times 10^{-21}$\
The temperature inferred here for the CO cloud is roughly 100$\times$ less than that inferred for the compact H$_2$ emission observed in this region of 1,700-2,300 K [@herrera11]. In addition, the H$_2$ FWHM line-width of $\sim 150$ km s$^{-1}$ [@herrera11] is significantly higher than the FWHM line-width of the CO emission measured here of 115 km s$^{-1}$ ($\sigma$ = 49 km s$^{-1}$). Therefore we infer that the origin of the CO and H$_2$ emission may not be identical. We speculate that the H$_2$ emission has a low filling factor, sampling only the most strongly shocked regions in and/or around the cloud.
![ These observations allow us to disentangle the molecular cloud associated with the secondary velocity component. Contours of the CO(3-2) moment 0 map of the molecular cloud extracted from the 3D data cube (0.4, 0.8, 1.6, 3.2 Jy beam$^{-1}$ km s$^{-1}$) overlaid on the color moment 0 map of the data cube with the primary cloud extracted. The cloud corresponding to the secondary velocity component can be seen in spatial projection with the primary cloud. \[extracted\_cloud\]](Johnson_Extended3.png){width="1.\columnwidth"}
Cloud Mass \[mass\]
-------------------
The mass of the cloud is estimated using four different methods, each subject to different caveats. First, given the source size, velocity dispersion of $\sigma_v = 49\pm 6$ km s$^{-1}$, and assuming an isothermal sphere we calculate the virial mass to be in the range of M$_{vir} = 2.9-8.5\times 10^7$ M$_{\odot}$, which is consistent with the virial mass estimated from ALMA SV observations of $\sim 5\times
10^{7}$ M$_{\odot}$ [@herrera12]. This mass estimate will only be valid if the cloud is in gravitational virial equilibrium; any additional velocity in the cloud will result in this method overestimating the mass. Given the complex dynamical structure of SGMCs, we treat the estimated virial mass as an [*upper*]{} limit.
For the second method we use the RADEX models of the CO(3-2) and CO(2-1) observations to determine the $\chi^2$ best fit column density of N$_{CO} \gtrsim 5\times 10^{18}$ cm$^{-2}$. If we assume an abundance ratio of $n_{H2}/n_{CO} = 2 \times 10^4$ [@blake87; @wilson92], this column density of CO corresponds to an H$_2$ mass of M$_{non-LTE} = 2.8 \times 10^6$ M$_{\odot}$. However, the $\chi^2$ values are shallow toward higher values of N$_{CO}$, and only weakly constrain the upper bound. The models also indicate that the CO(3-2) has an optical depth of $\tau \gtrsim 3$, and therefore this mass estimate is a [*lower*]{} limit.
The third method we employ assumes a conversion factor, X$_{CO}$. Values for X$_{CO}$ in starbursts are known to vary by at least a factor of four [@bolatto13], and thus the mass estimated using X$_{CO}$ should be regarded as uncertain by a corresponding factor. Here we adopt a “starburst” CO-to-H$_2$ conversion factor X$_{CO}
= 0.5\times 10^{20}$ cm$^{-2}$(K km s$^{-1}$)$^{-1}$ [@bolatto13]. If we were to adopt a “standard” X$_{CO} =
2\times 10^{20}$ cm$^{-2}$(K km s$^{-1}$)$^{-1}$, the resulting cloud mass would be a factor of four larger. We further assume that CO(3-2) is thermalized with respect to CO(1-0) given the brightness temperatures of T$_{3-2} = 17\pm3$ K and T$_{2-1} = 18\pm3$ K ; if the lines are not thermalized and CO(3-2) is relatively underpopulated, this mass estimate will be low. This method results in M$_{X_{CO}} =
3.3-15\times 10^6$ M$_{\odot}$.
The last method estimates the dust mass from the continuum at 870 $\mu$m. The continuum is detected at the 5.4 $\sigma$ level (see Figure \[continuum\]), with a peak brightness of $9.8\pm3.4 \times
10^{-4}$ Jy beam$^{-1}$. Assuming a dust emissivity of $\kappa = 0.9
\pm 0.13$ cm$^{2}$ g$^{-1}$ [@wilson08], optically-thin continuum, and a gas-to-dust ratio of 120$\pm 28$, and dust temperature of 20 K [@wilson08] this method results in a mass of M$_{cont} = 5\pm3 \times 10^6$ M$_{\odot}$. Based on the range of mass values determined above, we adopt a cloud mass of M$= 0.3 - 1.5
\times 10^7$ M$_{\odot}$. We note that the virial mass that would be inferred for this cloud appears to be a factor of 5-10$\times$ too high.
Given the estimated mass and size of this cloud, the resulting gas volume density is $\rho \gtrsim 100$ M$_{\odot}$ pc$^{-3}$. While there are molecular clouds found in the Milky Way with masses of $\sim
10^6$ M$_{\odot}$, their radii are $~2-4\times$ larger, resulting in significantly lower surface densities. Similarly massive clouds have also been identified in other nearby galaxies [@bolatto08; @meyer13]; their surface densities are also far lower (see Figure \[pressure\]).
![ Detecting the continuum emission associated with this source allows us to determine its dust mass. Contours of the CO(3-2) moment 0 map of the primary cloud overlaid on a color-scale image of the 870 $\mu$m continuum. \[continuum\]](Johnson_Extended4.png){width="1.\columnwidth"}
We estimate the mass of the stellar cluster that will potentially result from this molecular core by assuming a star formation efficiency (SFE, fraction of mass turned into stars over the lifetime of a cloud). The net efficiency can vary wildly, ranging from a few percent in the Milky Way [e.g. @lada03], to $>50\%$ in cluster-forming cores by basic boundedness arguments. If we adopt a SFE typical for clusters of $\epsilon = M_{stars}/(M_{stars} +
M_{gas})$ of $\sim 20\% - 50\%$ [@ashman01; @kroupa01], an initial cloud mass of $5\times 10^6 - 10^7$ M$_{\odot}$ would result in a cluster with a mass of $M_{star} = 1-5 \times 10^6$ M$_{\odot}$. This would be among the most massive SSCs that have formed in the Antennae if it forms a single cluster [@whitmore10]. Even if the star formation efficiency were as low as $\sim 5\%$, the resulting cluster would have a mass $> 2 \times 10^5$ M$_{\odot}$, still in the regime of super star cluster masses.
{width="1.5\columnwidth"}
Constraining the Ionizing Flux Associated with the Cloud
--------------------------------------------------------
In order to assess the extent to which star formation may have already affected the physical state of the ionized gas, we searched for ionizing flux potentially coming from stars within the molecular cloud. Figure \[plot\_zoom\_radio\] shows the Pa$\alpha$ emission in the region, and while there is diffuse emission associated with the SGMC in general, there is no discrete source associated with the molecular cloud in question. However, it is possible that given the embedded nature of this source, Pa$\alpha$ emission could suffer from significant extinction, and thus we also utilize radio observations to identify free-free emission that might be present.
We created two radio maps from archival 3.6 cm VLA observations (proposal codes AN079, AP478, AS796, AA301); one with a synthesized beam of $0.65''\times 0.42''$ to best match the beam of the ALMA CO(3-2) observations and resolve out diffuse emission, and a second with a synthesized beam of $1.12'' \times 0.85''$, which has greater sensitivity to extended emission. In the higher resolution map, there is no discrete source coincident with the molecular cloud discussed here, and the 5$\sigma$ detection threshold of the radio emission corresponds to an ionizing flux of N$_{Lyc} \approx 6\times
10^{50}$ s$^{-1}$. This is equivalent to $\sim 60$ O-type stars [@vacca96]. This limit is roughly three times lower than the previous limit placed on possible ionizing flux in this region of $2\times 10^{51}$ s$^{-1}$ using lower resolution 6 cm observations [@herrera11]. For comparison, the ionizing flux of 30 Dor has been estimated to be N$_{Lyc} \approx 4\times 10^{51}$ s$^{-1}$ [@crowther98].
In the lower resolution map, while there is no discrete source coincident with the molecular cloud, there is diffuse emission in the region. Without velocity information for this diffuse emission, it is not possible to disentangle potential line-of-sight confusion in this complex region. However, to estimate the possible contribution from stars within this molecular cloud to the diffuse emission, we first fit and subtract Gaussian profiles to the other dominate sources in SGMC2. After subtracting the emission due to nearby sources, we measure the flux density due to diffuse emission in the cloud aperture to be $0.03$ mJy, which corresponds to an ionizing flux of $1.1\times 10^{51}$ s$^{-1}$, which corresponds to $\sim 100$ O-type stars [@vacca96]. However, the diffuse morphology of this emission is not consistent with it coming from a single compact source.
Thus, while both the Pa${\alpha}$ and cm radio observations show diffuse emission associated with the SGMC2 structure, there is no discrete source associated with the compact cloud discussed here (Figure \[plot\_zoom\_radio\]). We constrain the possible ionizing flux that could be due to embedded stars in this cloud to be $\lesssim
60$ O-type stars. We can rule out an existing stellar cluster in this molecular core $\gtrsim 10^4$ M$\odot$ [@leitherer99], roughly two orders of magnitude smaller than the anticipated cluster mass (Section \[mass\]). Either star formation has not begun or it is so deeply embedded that its ionizing radiation is confined by gas continuing to accrete onto the protostars.
Determination of Cloud Pressure
-------------------------------
With a surface density of $\sim 4 \times 10^3$ M$_{\odot}$ pc$^{-2}$ and size-linewidth coefficient of $\sigma^2/R =
90$ km$^2$ s$^{-2}$ pc, the cloud is not consistent with being in either pure gravitational virial equilibrium or free-fall collapse [Figure \[pressure\], @heyer09]. Nevertheless, the morphology of the cloud indicates that gravity is playing a significant role (round, compact, and bright – making it stand out as a singular object in the data cube), suggesting that it is not a transient object. As illustrated in Figure \[pressure\], the observed line width value can be explained if the cloud is subject to external pressures of P/$k_B \sim 10^9$ K cm$^{-3}$, roughly five orders of magnitude higher than that typical in the interstellar medium of a galaxy [@jenkins83]. This is consistent with theoretical considerations that have argued SSC formation requires extreme pressures (P/$k_B \gtrsim 10^8$ K cm$^{-3}$) [@jog92; @elmegreen97; @ashman01]. This high pressure is also in accord with previous findings of compressive shocks in the overlap region [@wei12].
Expected Proto Super Star Cluster Cloud Size \[cloud\_size\]
------------------------------------------------------------
The expected physical size of a molecular cloud capable of forming an SSC can be estimated based on virial theorem arguments. Following previous work [@elmegreen89], the external cloud pressure $P_e$, cloud mass $M$, cloud radius $r$, and velocity dispersion $\sigma_v$ can be related by
$P_e = \dfrac{3 \Pi M \sigma_v^2}{4 \pi r^3}$,
where $\Pi$ is defined by $n_e = \Pi \langle n_e \rangle$, and here we adopt $\Pi = 0.5$. It has been estimated that SSC formation requires internal pressures of $P_0/k\gtrsim
10^{8}\;$K$\;$cm$^{-3}$ [@elmegreen97]. These high pressures inhibit the dispersal of the natal material and achieve sufficiently high star formation efficiencies in the cloud core. If we adopt a minimum mass of $M=10^6$ M$_{\odot}$ for a cloud capable of forming a SSC, the resulting cloud radius is $r\sim25$ pc. Likewise, for the velocity dispersion of $\sigma_v = 49$ km s$^{-1}$ observed for the cloud discussed here and the apparent external pressure of $P_0/k\gtrsim 10^{9}\;$K$\;$cm$^{-3}$ (see Figure \[pressure\]), the expected radius is $r\sim 25$ pc, which is within the uncertainty of the cloud half-light radius of $\lesssim 24 \pm 3$ pc extracted from these observations.
Discussion
==========
On the Origin of the High Pressure Inferred for the Molecular Cloud
-------------------------------------------------------------------
If the cloud is confined, as we expect from such a strong concentration of gas, then external pressure must play a key role. This high external pressure could result from the weight of the surrounding SGMC2 (if the SGMC is roughly in hydrostatic equilibrium) and/or large scale compressive shocks. The pressure generated by the surrounding molecular material can be estimated using P$_G/k_B \approx
1.5$ cm$^{-3}$ K (M$_{cloud}$/M$_{\odot}$)$^2$(r/pc)$^{-4}$ [@bertoldi92]. The SGMC2 region has an estimated total mass of $\sim 4\times 10^8$ M$_{\odot}$ [@wilson00] and radius of $\sim
400$ pc, resulting in a pressure from the overlying molecular material of P/$k_B\sim 10^7$ K cm$^{-3}$, which is at least an order of magnitude less than the internal pressure inferred for this cloud.
Given that this cloud is not only in the “overlap” region of the Antennae, but also appears to be at the nexus of two filaments of CO(3-2) emission that are suggestive of colliding flows [@whitmore14], a significant amount of external pressure could also be generated by ram pressure. This interpretation is consistent with previous work indicating strong H$_2$ line emission and an abrupt velocity gradient across this region [@herrera11; @herrera12]. However, the morphology of the cloud suggests an isotropic source of pressure, which is in tension with a ram pressure origin.
Timescales for Cloud Evolution and Star Formation \[timescales\]
----------------------------------------------------------------
The relevant timescales for this cloud to evolve are driven by the free-fall time and the crossing time. The compact size and marginally-resolved round morphology suggest that self-gravity has had a significant role in shaping the source, although that is not possible to conclusively demonstrate with the data in hand. The cloud must be largely supported by turbulence: given the inferred density of the cloud (n$\sim 10^3$ cm$^{-3}$), if the pressure were entirely thermal, it would require a gas temperature of $\sim 10^5$ K, which is not reasonable or consistent with these observations that indicate $T\sim 25$ K. We conclude that this cloud is most likely supported by turbulence. This conclusion is also supported by the virial mass being nearly an order of magnitude larger than the mass inferred from the dust continuum, indicating significant internal motion contributing to the observed line width.
Thus, we adopt the turbulent crossing time as the appropriate timescale for the evolution of this cloud and estimate it as $t_{cr} \sim D/\sigma_V \sim 1$ Myr, where $D$ is the diameter of the region. If self-gravity is [*not*]{} important, the cloud will disperse on this timescale – the turbulence will dissipate on this timescale in any case. If self-gravity is important (as argued above), the cloud will collapse on this timescale. The free-fall time of the cloud can be estimated as $t_{ff}=(3\pi/32~G~\rho)^{1/2} \approx 8 \times 10^5$ years. Given the strong associated H$_2$ emission [@herrera12] and line-width, which indicates internal velocities higher than virial equilibrium, it is plausible that this cloud has already begun free-fall collapse, in which case we are witnessing a very short-lived stage of cluster evolution. Given these arguments, we expect this cloud to collapse on timescales $\lesssim 1$ Myr.
Expected Number of Proto-SSC Molecular Clouds
---------------------------------------------
Given that this phase of SSC formation is expected to be extremely short-lived ($\lesssim 10^5 - 10^6$ yr), it is not surprising that this is [*the only*]{} clear example that we have found to date of a compact cloud [*without*]{} associated star formation that is sufficiently massive to form an SSC. Within $\sim 1$ Myr, we expect that this cloud will be associated with star formation, similar to the clouds observed in M82 by @keto05.
The estimated current star formation rate (SFR) for the Antennae is $\sim 7 - 20$ M$_{\odot}$ yr$^{-1}$ [@zhang01; @brandl09]. If cluster formation follows a power-law distribution of $dN/dM \propto M^{-2}$ [@zhang99] with lower and upper masses of $10^2$ and $10^7$ M$_{\odot}$, we expect $\sim 20$% of the stellar mass to be formed in clusters with M$> 10^6$ M$_{\odot}$. Thus we expect only a few SSCs masses of $>10^6$ M$_{\odot}$ to form every $\sim 5\times
10^5$ years. Optical studies indicate that there are six SSCs with ages $<10^7$ years and masses $>10^6$ M$_{\odot}$ [@whitmore10]. This suggests that a massive cluster is formed every $<$10$^7$/6 $\approx$ 1.7$\times$10$^6$ years, in agreement with the estimate derived above from the total SFR. The predicted number of pre-stellar molecular clouds capable of forming an SSC with mass $>
10^6$ M$_\odot$ can be generalized as, $$N_{\rm SSC-GMC}\,\simeq\,0.2 \times t_{collapse}\left(\frac{\rm SFR{(M_\odot yr^{-1})}}{\rm 10^6
M_{\odot} }\right)$$ where t$_{\rm collapse}$ is the timescale for cloud collapse (see Section \[timescales\]). Given a dynamical timescale for this cloud of $\sim 0.5-1$ Myr, we expect that if we observed the Antennae system at any point in its recent star forming history ($\sim
10^{7-8}$ yr), we would find at most one cloud of this evolutionary state and mass.
Implications for Globular Cluster Formation
-------------------------------------------
The physical properties of this cloud can provide insight into a mode of star formation that may have been dominant in the earlier universe, when globular clusters were formed prolifically. In particular, the pressure of P/$k_B\gtrsim 10^8$ K cm$^{-3}$ supports the hypothesis that such high pressures are necessary to form SSCs [@ashman01; @elmegreen02]. Based on what appears to be a nearly universal power-law distribution of cluster masses, numerous studies have suggested that the formation of SSCs is statistical in nature, resulting from a “size of sample” – galaxies with higher star formation rates will form more clusters overall, and the formation of SSCs results from populating the tail of the mass distribution [@fall12]. Based on these ALMA observations, we suggest an alternate interpretation: in a turbulent interstellar medium, the pressure distribution also has a power-law form, and the properties of clusters that form track the pressure distribution. Thus, while galaxies present a power-law distribution of cluster masses, only regions with sufficiently high pressure will be able to form the most massive SSCs. If pressures P/$k_B\gtrsim
10^8$ K cm$^{-3}$ are indeed required to form SSCs (and the surviving globular clusters), similarly high pressures must have been common during the peak of globular cluster formation a few billion years after the Big Bang.
We thank the anonymous referee for their many useful comments. This paper makes use of the following ALMA data : ADS/JAO.ALMA\#2011.0.00876.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. K.E.J. acknowledges support provided by the David and Lucile Packard Foundation through a Packard Fellowship.
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[^1]: A possible exception to this is a CO cloud located at 09h55m54.5s +69d40’50”, however the properties of this cloud are not provided by @keto05.
|
---
abstract: |
In this letter we will use higher-order supersymmetric quantum mechanics to obtain several families of complex solutions $g(x;a,b)$ of the Painlevé IV equation with real parameters $a,b$. We shall also study the algebraic structure, the eigenfunctions and the energy spectra of the corresponding non-hermitian Hamiltonians.\
[*Keywords:*]{} quantum mechanics, non-linear differential equations, Painlevé equations, complex potentials with real spectra
author:
- |
David Bermúdez[^1] and David J. Fernández C.[^2]\
[*Departamento de Física, Cinvestav, A.P. 14-740, 07000 México D.F., Mexico*]{}
title: |
Non-hermitian Hamiltonians and\
Painlevé IV equation with real parameters
---
Introduction
============
Since its birth, supersymmetric quantum mechanics (SUSY QM) catalyzed the study of exactly solvable Hamiltonians and gave a new insight into the algebraic structure characterizing these systems. Historically, the essence of SUSY QM was developed first as Darboux transformation in mathematical physics [@MS91] and as factorization method in quantum mechanics [@IH51].
On the other hand, there has been an increasing interest in the study of non-linear phenomena, which in many cases leads to the analysis of Painlevé equations [@VS93; @Adl94]. Although these were discovered from strictly mathematical considerations, nowadays they are widely used to describe several physical phenomena [@AC92]. In particular, the Painlevé IV equation ($P_{IV}$) is relevant in fluid mechanics, non-linear optics and quantum gravity [@Win92].
As it has been shown, there is a natural connection between quantum systems described by second-order polynomial Heisenberg algebras (PHA), whose Hamiltonians have the standard Schrödinger form and their differential ladder operators are of third order, and solutions $g(x;a,b)$ of $P_{IV}$ [@Adl94; @ARS80; @Fla80]. Moreover, these algebras can be realized by the $k$-th order SUSY partners $H_k$ of the harmonic oscillator Hamiltonian $H_0$, which leads to a simple method for generating solutions of $P_{IV}$, since the SUSY technique provides explicit expressions for the extremal states of $H_k$ and connecting formulae relating them with the corresponding $P_{IV}$ solutions [@ACIN00; @FNN04; @CFNN04; @MN08]. It is worth to note, however, that the need to avoid singularities in the new potential $V_k(x)$ and the requirement for the Hamiltonian $H_k$ to be hermitian lead to some restrictions [@BF11]: (i) first of all, the relevant transformation function has to be real, which implies that the associated factorization energy is real; (ii) as a consequence, the spectrum of $H_k$ consists of two independent physical ladders, an infinite one departing from $E_0 = 1/2$ (the ground state energy of $H_0$) plus a finite one with $k$ equidistant levels, all of which have to be placed below $E_0$. Regarding $P_{IV}$, these two restrictions imply that non-singular real solutions $g(x;a,b)$ can be obtained for certain real parameters $a,b$.
From the point of view of spectral design, it would be important to overcome restriction (ii) so that some (or all) steps of the finite ladder could be placed above $E_0$. In this way we would be able to manipulate not just the lowest part of the spectrum (as we did previously [@FNN04; @CFNN04; @BF11]), but also the excited state levels, which would endow us with new tools for spectral design. In this article we are going to show that this can be achieved if one relaxes restriction (i) as well, which will force us to use complex transformation functions (see [@ACDI99]) and will lead to the generation of complex solutions to $P_{IV}$.
This letter is organized as follows: in Section 2 we shall present the general framework of SUSY QM and PHA. In the next Section we will generate the complex solutions to $P_{IV}$, specifically, we shall analyze the domain of the parameter space $(a,b)$ for which we obtain real or complex solutions; then, in Section 4 we will study the eigenfunctions and the energy spectra of the non-hermitian Hamiltonians. We shall present our conclusions in Section 5.
General framework of SUSY QM and PHA
====================================
In the $k$-th order SUSY QM one typically starts from a given solvable Hamiltonian $$H_0 = -\frac12 \frac{d^2}{d x^2} + V_0(x),$$ and generates a chain of standard (first-order) intertwining relations [@AIS93; @MRO04; @Fer10] $$\begin{aligned}
H_j A_j^{+} & = A_j^{+} H_{j-1}, \quad H_{j-1}A_j^{-} = A_j^{-}H_j, \\
H_j & = -\frac12 \frac{d^2}{d x^2} + V_j(x),\\
A_j^{\pm} &= \frac{1}{\sqrt{2}}\left[\mp \frac{d}{d x} + \alpha_j(x,\epsilon_j)\right], \quad j = 1,\dots,k. \end{aligned}$$ Hence, the following equations must be satisfied $$\begin{aligned}
& \alpha_j'(x,\epsilon_j) + \alpha_j^2(x,\epsilon_j) = 2[V_{j-1}(x) - \epsilon_j], \label{rei} \\
& V_{j}(x) = V_{j-1}(x) - \alpha_j'(x,\epsilon_j). \label{npi}\end{aligned}$$ We are interested in the final Riccati solution $\alpha_{k}(x,\epsilon_{k})$, which turns out to be determined, either by $k$ solutions $\alpha_1(x,\epsilon_j)$ of the initial Riccati equation $$\alpha_1'(x,\epsilon_j) + \alpha_1^2(x,\epsilon_j) = 2 [V_0(x) - \epsilon_j], \quad j=1,\dots,k,$$ or by $k$ solutions $u_j$ of the associated Schrödinger equation $$H_0 u_j = - \frac12 u_j'' + V_0(x)u_j = \epsilon_j u_j, \quad j=1,\dots,k, \label{usch}$$ with $\alpha_1(x,\epsilon_j) = u_j'/u_j$.
Thus, there is a pair of $k$-th order operators interwining the initial $H_0$ and the final Hamiltonians $H_k$, namely, $$H_k B_k^{+} = B_k^{+} H_0, \quad H_0 B_k^{-} = B_k^{-} H_k,$$ where $$B_k^{+} = A_k^{+}\dots A_1^{+}, \quad B_k^{-} = A_1^{-}\dots A_k^{-}.$$
The normalized eigenfunctions $\psi_n^{(k)}$ of $H_k$, associated to the eigenvalues $E_n$, are proportional to the action of $B_k^{+}$ onto the corresponding ones of $H_0$ ($\psi_n$, $n=0,1,\dots$). Moreover, there are $k$ additional eigenstates $\psi_{\epsilon_j}^{(k)}$ associated to the eigenvalues $\epsilon_j$ ($j=1,\dots ,k$), which are simultaneously annihilated by $B_k^{-}$. Their corresponding explicit expressions are given by [@BF11; @FH99]: $$\begin{aligned}
\psi_n^{(k)} = \frac{B_k^{+}\psi_n}{[(E_n-\epsilon_1)\dots (E_n-\epsilon_k)]^{1/2}}, & \quad E_n, \label{psin}\\
\psi_{\epsilon_j}^{(k)} \propto \frac{W(u_1,\dots , u_{j-1},u_{j+1},\dots , u_k)}{W(u_1,\dots , u_k)}, & \quad \epsilon_j. \label{psie}\end{aligned}$$ Let us note that, in this formalism the obvious restriction $\epsilon_j < E_0=1/2$ naturally arises since we want to avoid singularities in $V_k(x)$.
On the other hand, a $m$-th order PHA is a deformation of the Heisenberg-Weyl algebra of kind [@CFNN04; @FH99]: $$\begin{aligned}
[H,L^\pm] &= \pm L^\pm , \\
[L^-,L^+] & \equiv Q_{m+1}(H+1) - Q_{m+1}(H) = P_m(H) , \\
Q_{m+1}(H) &= L^+ L^- = \prod\limits_{i=1}^{m+1} \left(H - \mathcal{E}_i\right) ,\end{aligned}$$ where $Q_{m+1}(x)$ is a $(m+1)$-th order polynomial in $x$, which implies that $P_m(x)$ is a polynomial of order $m$ in $x$ and $\mathcal{E}_i$ are the zeros of $Q_{m+1}(H)$, which correspond to the energies associated to the extremal states of $H$.
Now, let us take a look at the differential representation of the second-order PHA ($m=2$). Suppose that $L^+$ is a third-order differential ladder operator, chosen by simplicity as: $$\begin{aligned}
L^+ &= L_1^+ L_2^+ , \\
L_1^+ &= \frac{1}{\sqrt{2}}\left[-\frac{d}{d x} + f(x) \right], \\
L_2^+ &= \frac12\left[ \frac{d^2}{d x^2} + g(x)\frac{d}{d x} +
h(x)\right].\end{aligned}$$ These operators satisfy the following intertwining relationships: $$\begin{aligned}
HL_1^+ & = L_1^+ (H_{\rm a} + 1), \quad H_{\rm a} L_2^+ = L_2^+ H,\\
\Rightarrow \quad & [H,L^+] = L^+,\end{aligned}$$ where $H_{\rm a}$ is an auxiliary Schrödinger Hamiltonian. Using the standard equations for the first and second-order SUSY QM and decoupling the resulting system gives rise to $$\begin{aligned}
& f = x + g, \label{fdependg} \\
& h = - x^2 + \frac{g'}{2} - \frac{g^2}{2} - 2xg + a, \\
& V = \frac{x^2}2 - \frac{g'}2 + \frac{g^2}2 + x g + \mathcal{E}_1 -
\frac12 , \label{Vpivs}\end{aligned}$$ where $$g'' = \frac{g'^2}{2g} + \frac{3}{2} g^3 + 4xg^2 + 2\left(x^2 - a \right) g + \frac{b}{g}.$$ Note that this is the Painlevé IV equation ($P_{IV}$) with parameters $$a =\mathcal{E}_2 + \mathcal{E}_3-2\mathcal{E}_1 -1,\quad b = - 2(\mathcal{E}_2 - \mathcal{E}_3)^2.\label{abe}$$ Hence, if the three quantities $\mathcal{E}_i$ are real, we will obtain real parameters $a,b$ for the corresponding $P_{IV}$.
Let us note that the potential of equation (\[Vpivs\]) contains a harmonic oscillator term. In addition, three terms can be identified ($-g'/2 + g^2/2 + xg$) which in general lead to an anharmonicity in the potential and are completely determined by the solution to $P_{IV}$. As a consequence, one could say that the solution $g$ to $P_{IV}$ is the main responsible for the spectral differences which the Hamiltonian $H$ could have with respect to the harmonic oscillator (compare with [@DEK94]).
Complex solutions to $P_{IV}$ with real parameters
==================================================
It is well known that the first-order SUSY partner Hamiltonians of the harmonic oscillator are naturally described by second-order PHA, which are connected with $P_{IV}$, as we have shown in the previous section. Furthermore, there is a theorem stating the conditions for the hermitian higher-order SUSY partners Hamiltonians of the harmonic oscillator to have this kind of algebras (see [@BF11]). The main requirement is that the $k$ Schrödinger seed solutions have to be connected in the way $$\begin{aligned}
u_j=(a^{-})^{j-1}u_1,& \quad \label{us}\\
\epsilon_j=\epsilon_1-(j-1), & \quad j=1,\dots , k,\end{aligned}$$ where $a^{-}$ is the standard annihilation operator of $H_0$ so that the only free seed $u_1$ has to be a real solution of Eq. without zeros, associated to a real factorization energy $\epsilon_1$ such that $\epsilon_1<E_0=1/2$.
In this work we intend to overcome this restriction, although if we use the formalism as in [@BF11] with $\epsilon_1 > E_0$, we would obtain only singular SUSY transformations. In order to avoid this we will instead employ complex SUSY transformations. The simplest way to implement them is to use a complex linear combination of the two standard linearly independent real solutions which, up to an unessential factor, leads to the following complex solutions depending on a complex constant $\lambda + i \kappa$ ($\lambda, \kappa \in \mathbb{R}$) [@ACDI99]: $$u(x;\epsilon ) = e^{-x^2/2}\left[ {}_1F_1\left(\frac{1-2\epsilon}{4},\frac12;x^2\right)
+ x(\lambda + i\kappa)\, {}_1F_1\left(\frac{3-2\epsilon}{4},\frac32;x^2\right)\right], \label{u1}$$ where $_1F_1$ is the confluent hypergeometric (Kummer) function. The known results for the real case [@JR98] are obtained by making $\kappa=0$ and expressing $\lambda$ as $$\lambda= 2 \nu\frac{\Gamma(\frac{3 - 2\epsilon}{4})}{\Gamma(\frac{1-2\epsilon}{4})}, \label{nu}$$ with $\nu \in \mathbb{R}$.
Hence, through this formalism we will obtain the $k$-th order SUSY partner potential $V_k(x)$ of the harmonic oscillator and the corresponding $P_{IV}$ solution $g(x;\epsilon_1)$, both of which are complex, in the way $$\begin{aligned}
V_k(x) &= \frac{x^2}2 - \{\ln [W(u_1,\dots,u_k)]\}'' , \\
g(x;\epsilon_1) &= - x - \{\ln[\psi_{\mathcal{E}_1}(x)]\}'. \label{solg}\end{aligned}$$ Note that the extremal states of $H_{k}$ and their corresponding energies are given by $$\begin{aligned}
\psi_{\mathcal{E}_1} \propto \frac{W(u_1,\dots,u_{k-1})}{W(u_1,\dots,u_k)}, & \quad \mathcal{E}_1 = \epsilon_k = \epsilon_1 - (k - 1), \label{edo1}\\
\psi_{\mathcal{E}_2} \propto B_k^+ e^{-x^2/2}, & \quad \mathcal{E}_2 = \frac{1}{2}, \label{edo2}\\
\psi_{\mathcal{E}_3} \propto B_k^+ a^{+} u_1, & \quad \mathcal{E}_3 = \epsilon_1 + 1. \label{edo3}\end{aligned}$$ Recall that all the $u_j$ satisfy Eq. and $u_1$ corresponds to the general solution given in Eq. .
For $k=1$, the first-order SUSY transformation and Eq. lead to what is known as *one-parameter solutions* to $P_{IV}$, due to the restrictions imposed by Eq. onto the two parameters $a,b$ of $P_{IV}$ which makes them both depend on $\epsilon_1$ [@BCH95]. For this reason, this family of solutions cannot be found in any point of the parameter space $(a,b)$, but only in the subspace defined by the curve $\{\left( a(\epsilon_1), b(\epsilon_1)\right),\ \epsilon_1 \in \mathbb{R}\}$ consistent with Eqs. . Then, by increasing the order of the SUSY transformation to an arbitrary integer $k$, we will expand this subspace to obtain $k$ different families of one-parameter solutions. This procedure is analogous to iterated auto-Bäcklund transformations [@RS82]. Also note that by making cyclic permutations of the indices of the three energies $\mathcal{E}_i$ and the corresponding extremal states of Eqs. (\[edo1\]-\[edo3\]), we expand the solution families to three different sets, defined by $$\begin{aligned}
a_{1}=-\epsilon_1 + 2k -\frac{3}{2}, \quad & b_{1}=-2\left(\epsilon_1+\frac{1}{2}\right)^{2}, \label{ab1}\\
a_2= 2\epsilon_1 -k, \quad & b_2=-2k^2, \\
a_3=-\epsilon_1-k-\frac{3}{2}, \quad & b_3=-2\left(\epsilon_1 - k +\frac{1}{2}\right)^2,\end{aligned}$$ where we have added an index corresponding to the extremal state (Eqs. (\[edo1\]-\[edo3\])) taken to build up the $P_{IV}$ solution in Eq. . The first pair, $a_1,b_1$, can provide non-singular real or complex solutions, while the second and third ones can give just non-singular complex solutions, due to singularities in the real case. A part of the non-singular solution subspace for both real and complex cases is shown in Fig. \[parameterspace\]. One can check that those points which belong to two different sets have associated the same $P_{IV}$ solutions.
![Parameter space $(a,b)$ of the $P_{IV}$ solutions. The curves represent the solution subspace for non-singular real or complex (solid curves) and only complex (dashed curves) solutions.[]{data-label="parameterspace"}](figpspace.eps)
In turn, let us analyze some of the $P_{IV}$ solutions obtained by this method. The real solutions arise by taking $\kappa=0$, and expressing $\lambda$ as in Eq. with $\epsilon_1<1/2$. They can be classified into three relevant solution hierarchies, namely, confluent hypergeometric, complementary error and rational hierarchies. Let us note that the same set of real solutions to $P_{IV}$ can be obtained through inverse scattering techniques [@AC92] (compare the solutions of [@BCH95] with those of [@BF11]). In Fig. \[greal\], three real solutions to $P_{IV}$ are presented, which belong to the complementary error hierarchy.
![Some real solutions to $P_{IV}$, corresponding to $a_1=1$, $b_1=0$ ($k=1$, $\epsilon_1=-1/2$, $\nu=0.7$) (solid curve), $a_1=4$, $b_1=-2$ ($k=2$, $\epsilon_1=-3/2$, $\nu=0.5$) (dashed curve), and $a_1=7$, $b_1=-8$ ($k=3$, $\epsilon_1=-1/2$, $\nu=0.3$) (dotted curve).[]{data-label="greal"}](figgreal.eps)
Next, we study the complex solutions subspace, i.e. we allow now that $\epsilon_1 \geq 1/2$. The real and imaginary parts of the complex solutions $g(x;a,b)$ for two particular choices of real parameters $a,b$, which belong to different solution sets, are plotted in Fig. \[gcomplex\].
![Real (solid curve) and imaginary (dashed curve) parts of some complex solutions to $P_{IV}$. The upper plot corresponds to $a_2=12$, $b_2=-8$ ($k=2$, $\epsilon_1=7$, $\lambda=\kappa=1$) and the lower one to $a_3=-5$, $b_3=-8$ ($k=1$, $\epsilon_1=5/2$, $\lambda=\kappa=1$).[]{data-label="gcomplex"}](figgcomplex2.eps "fig:") ![Real (solid curve) and imaginary (dashed curve) parts of some complex solutions to $P_{IV}$. The upper plot corresponds to $a_2=12$, $b_2=-8$ ($k=2$, $\epsilon_1=7$, $\lambda=\kappa=1$) and the lower one to $a_3=-5$, $b_3=-8$ ($k=1$, $\epsilon_1=5/2$, $\lambda=\kappa=1$).[]{data-label="gcomplex"}](figgcomplex3.eps "fig:")
Note that, in general, $\psi_{\mathcal{E}_i}\neq 0\ \forall\ x \in \mathbb{R}$ , i.e., the solutions $g(x;a,b)$ are not singular. Moreover, both real and imaginary parts have an asymptotic null behaviour ($g\rightarrow 0$ as $|x|\rightarrow \infty$). This property becomes evident in Fig. \[gcomplex\], as well as in the parametric plot of the real and imaginary parts of $g(x;a,b)$ of Fig. \[complexpara\].
![Parametric plot of the real and imaginary parts of $g(x;a,b)$ for $a_1=-6$, $b_1=-2$ ($k=1$, $\epsilon_1=5/2$, $\lambda=1$, $\kappa=5$) and $|x| \leq 10$. For bigger values of $x$, the curve approaches the origin in both sides.[]{data-label="complexpara"}](figcomplexpara.eps)
Non-hermitian Hamiltonians
==========================
Let us analize the Hamiltonian $H_k$ obtained by the complex SUSY transformation. Note that the real case, which leads to hermitian Hamiltonians, has been studied previously [@AIS93; @Mie84], allowing to obtain some criteria related to the structure of the associated energy spectrum $\text{Sp}(H_k)$, the number of zeros of the eigenfunctions of $H_k$, and the way in which they are connected by the third-order ladder operators $L^{\pm}$. This action is in agreement with the fact that $\text{Sp}(H_k)$ consists of an infinite ladder plus a finite one: there are two extremal states (both annihilated by $L^{-}$) from which the two ladders start, one associated to $\epsilon_k$ and the other one to $E_0=1/2$; since the ladder starting from $\epsilon_k$ ends at $\epsilon_1$, the eigenfunction associated to $\epsilon_1$ is annihilated by $L^{+}$. The actions of $L^{\pm}$ onto any other eigenstate of $H_k$ are non-null, and connect only the eigenstates belonging to the same ladder.
As far as we know, complex SUSY transformations with real factorization energies were used for the first time by Andrianov et al. to obtain non-hermitian Hamiltonians with real spectra [@ACDI99]. These topics have been of great interest in the context of both parity-time (PT) symmetric Hamiltonians (see Bender et al. [@BB98]) and pseudo-hermitian Hamiltonians (see Mostazafadeh et al. [@MB04]). Next, we will examine the structure of non-hermitian SUSY generated Hamiltonians $H_k$.
First of all, the new Hamiltonians necessarily have complex eigenfunctions, although the associated eigenvalues are still real. In previous works, the factorization energy associated to the real transformation function $u_1$ was bounded, $\epsilon_1<E_0=1/2$. In this paper we are using complex transformation functions to be able to overcome this restriction and yet obtain non-singular solutions. This naturally leads to complex solutions to $P_{IV}$ generated through factorization energies which could be placed now above $E_0$. The resulting spectra for the non-hermitian Hamiltonians $H_k$ obey the same criteria as the real case, namely, they are composed of an infinite ladder plus a finite one, which now could be placed, either fully or partially, above $E_0$. The eigenfunctions associated to the energy levels of the original harmonic oscillator are given by Eq. and the ones associated to the new energy levels by Eq. , all of them square-integrable. A diagram of the described spectrum is shown in Fig. \[espectros\].
![Spectrum of the SUSY partner Hamiltonians $H_0$ (right) and $H_k$ (left) for $\epsilon_1>1/2$, $\epsilon_1\neq E_j$; Sp($H_k$) still contains one finite and one infinite ladders. The dark bars represent the original and mapped eigenstates of $H_0$ and $H_k$, while the light ones the $k$ new levels of $H_k$ introduced by the $k$-th order SUSY transformation. All of them have associated square-integrable eigenfunctions.[]{data-label="espectros"}](figespectros.eps)
The extremal states of the SUSY generated Hamiltonian $H_k$ are given by Eqs. (\[edo1\]-\[edo3\]). These are non-singular complex eigenfunctions of $H_k$ and, from their asymptotic behaviour, we conclude that those given by Eqs. (\[edo1\],\[edo2\]) are square-integrable. Note that in this case the oscillatory theorem does not hold anymore, neither for the real nor for the imaginary parts, although a related node structure emerges. The absolute value and the real and imaginary parts of $\psi_{\mathcal{E}_1}(x)$ for two particular cases are shown in Fig. \[waves\].
![Plot of the absolute value, the real, and the imaginary parts (solid, dashed and dotted lines, respectively) of the eigenfunction $\psi_{\mathcal{E}_1}(x)$ given by Eq. for the values $k=2$, $\epsilon_1=-1$, $\lambda=1$, $\kappa=1/2$ (up) and $k=2$, $\epsilon_1=4$, $\lambda=1$, $\kappa=6$ (down).[]{data-label="waves"}](figwave1.eps "fig:")\
![Plot of the absolute value, the real, and the imaginary parts (solid, dashed and dotted lines, respectively) of the eigenfunction $\psi_{\mathcal{E}_1}(x)$ given by Eq. for the values $k=2$, $\epsilon_1=-1$, $\lambda=1$, $\kappa=1/2$ (up) and $k=2$, $\epsilon_1=4$, $\lambda=1$, $\kappa=6$ (down).[]{data-label="waves"}](figwave2.eps "fig:")
On the other hand, complex transformations for $\epsilon_1=E_j$ are worth of a detailed study, namely, when the factorization energy $\epsilon_1$ belongs to the spectrum of the original harmonic oscillator Hamiltonian. For instance, let us consider a first-order SUSY transformation with $\epsilon_1=E_j$ and $u_1$ given by Eq. , i.e., $u_1$ is a complex linear combination of the eigenfunction $\psi_j$ of $H_0$ and the other linearly independent solution of the Schrödinger equation. It is straightforward to see that the action of the ladder operator $L^{-}=A_1^{+}a^{-}A_1^{-}$ is given by $$\begin{aligned}
E_{l},&\quad L^{-}(A_1^{+}\psi_{l}) \propto A_1^{+}\psi_{l-1},\\
E_j, &\quad L^{-}(A_1^{+}\psi_{j}) = 0,\\
E_{0}, &\quad L^{-}(A_1^{+}\psi_{0}) = 0,\end{aligned}$$ where $l \neq j$, $l \neq 0$, the shown energies correspond to the departure state, and we have used that $A_1^{+}\psi_j \propto 1/u_1 $. For $L^{+}=A_1^{+}a^{+}A_1^{-}$ we have $$\begin{aligned}
E_{l},&\quad L^{+}(A_1^{+}\psi_{l}) \propto \psi_{l+1},\\
E_j, &\quad L^{+}(A_1^{+}\psi_{j}) =0,\end{aligned}$$ which does not match with the established criteria for the non-singular real and complex cases with $\epsilon_1 \neq E_j$ since now it turns out that: $$\begin{aligned}
E_{j+1},&\quad L^{-}(A_1^{+}\psi_{j+1}) \propto A_1^{+}\psi_{j} \propto \frac{1}{u_1}\neq 0,\\
E_{j-1},&\quad L^{+}(A_1^{+}\psi_{j-1}) \propto A_1^{+}\psi_{j} \propto \frac{1}{u_1}\neq 0.\end{aligned}$$ The resulting Hamiltonian is isospectral to the harmonic oscillator but with a special algebraic structure because now one state (the one associated to $E_j$) is connected just in one way with the adjacent ones (associated to $E_j \pm 1$). A diagram representing this structure is shown in Fig. \[1susy\]. We are currently studying the $k$-th order case and expect to find the new criteria which will be valid for these special transformations.
![Spectra of the harmonic oscillator Hamiltonian $H_0$ and of its first-order SUSY partner $H_1$ when we use the factorization energy $\epsilon_1=E_j \in \text{Sp}(H_0)$. The level $E_j$ of $H_1$ is connected with its adjacent ones just in one way.[]{data-label="1susy"}](fig1susy.eps)
Conclusions
===========
In this letter, based on PHA and higher-order SUSY QM, we have introduced a method to obtain real and complex solutions $g(x;a,b)$ of the $P_{IV}$ with real parameters $a,b$. We have studied the properties of the resulting solutions, including the analysis of the subspace of the parameter space $(a,b)$ were non-singular real or complex solutions can be found. In addition, we have analyzed the algebras, the eigenfunctions and the spectra of the non-hermitian SUSY generated Hamiltonians.
Further investigation on the description of the analytic structure of the complex solutions and on the way in which the ladder operators $L^{\pm}$ map between different eigenstates and their corresponding energy levels is needed. Besides, we are looking for extensions of this technique to obtain $P_{IV}$ solutions associated to complex parameters $a,b$.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors acknowledge the support of Conacyt.
[99]{}
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[^1]: [*email:*]{} dbermudez@fis.cinvestav.mx
[^2]: [*email:*]{} david@fis.cinvestav.mx
|
---
abstract: 'An array of one-dimensional conductors coupled by transverse hopping and interaction is studied with the help of a variational wave function. This wave function is devised as to account for one-dimensional correlation effects. We show that under broad conditions our system possesses the superconducting ground state even if no attraction is present. The superconducting mechanism is of many-body nature and deviates substantially from BCS. The phase diagram of the model is mapped. It consists of two ordered phases competing against each other: density wave, spin or charge, and unconventional superconductivity. These phases are separated by the first order transition. The symmetry of the superconducting order parameter is a non-universal property. It depends on particulars of the Hamiltonian. Within the framework of our model possible choices are the triplet $f$-wave and the singlet $d_{xy}$-wave. Organic quasi-one-dimensional superconductors have similar phase diagram.'
author:
- 'A.V. Rozhkov'
title: ' Superconductivity without attraction in a quasi-one-dimensional metal '
---
Introduction
============
In this paper we will study a system of one-dimensional (1D) conductors arranged in a square lattice and coupled weakly in the transverse direction. The purpose of this work is to show that in a rather general situation such quasi-one-dimensional (Q1D) electron liquid with purely repulsive electron-electron interaction is either a superconductor, or an insulator with spin or charge density order. This is demonstrated with the help of a certain variational wave function which adequately captures 1D many-body effects.
The major issue in the description of the Q1D metal is the phenomenon of dimensional crossover. At high energy the system can be viewed as a collection of the Tomonaga-Luttinger (TL) liquids. However the TL liquid cannot support a physical electron as an elementary excitation. Thus, at low energy, where transverse single-electron hopping becomes important, it is necessary to abandon the TL notions and use the Fermi-liquid approach instead. Therefore, one is to stitch two different descriptions together to obtain complete picture.
From the technical point of view the source of the trouble is the conflict between the many-body TL correlations and transverse single-electron hopping, which is extremely difficult to handle within the framework of the 1D TL liquid [@boson].
A simple method for the crossover description was proposed in Ref.[@rozhkov]. The latter method is based on a variational wave function, whose generalized version we will use in this paper.
To provide an intuitive introduction to the approach of [@rozhkov] we briefly explain the structure of the variational wave function. Consider a 1D conductor described by the Tomonaga-Luttinger Hamiltonian. The ground state of this system is the ground state of all TL bosons, with every possible momenta $k_\|$ ($|k_\|| < \Lambda$, where $\Lambda$ is the cutoff of the theory). Let’s turn the transverse hopping on and couple $N_\perp$ of these conductors into three-dimensional array. In this situation the system will attempt to lower its ground state energy even further by taking advantage of the transverse hopping energy. However, in order to participate in hopping the bosons have to form many-body fermion-like excitations, which have finite overlap with the physical fermion.
To accommodate for the possibility of having two types of excitations, bosonic and fermionic, we device our variational state in the following fashion. We introduce intermediate cutoff $\tilde\Lambda<\Lambda$. All TL bosons, whose energy and momenta are high ($|k_\||>\tilde\Lambda$), remain in their ground states. The small momenta bosons ($|k_\||<\tilde\Lambda$) form fermion-like excitations, which are delocalized in transverse direction. To distinguish between the physical electrons and these fermionic excitations we will refer to the latter as quasiparticles. In other words, the wave function can be factorized into two parts. The high-energy part corresponds to the ground state of $|k_\||>\tilde\Lambda$ Tomonaga-Luttinger bosons, the low-energy part corresponds to the three-dimensional anisotropic Fermi liquid composed of the quasiparticles.
The variational energy is minimized by adjusting $\tilde \Lambda$. The energy of the quasiparticle transverse hopping is decreasing function of $\tilde \Lambda$. At the same time, the in-chain energy grows when $\tilde
\Lambda$ grows. The trade-off between the transverse kinetic energy and the in-chain potential energy determines the value of $\tilde \Lambda$.
If the optimal value of $\tilde \Lambda$ is non-zero, the low-energy excitations of the system are the quasiparticles. Properties of the fermionic quasiparticle state depend on quasiparticle effective Hamiltonian. It arises naturally after high-energy bosons are ‘integrated out’. In this effective Hamiltonian the anisotropy is insignificant. Due to this circumstance, a standard mean field theory can be used to map out the quasiparticle phase diagram. Since the physical electron and the quasiparticle have finite overlap, there is a direct correspondence between broken symmetry phases of the effective Hamiltonian and the physical system. We already mentioned that the possible phases of the system are spin-density wave (SDW), charge-density wave (CDW), and superconductivity. The mean-field treatment of the effective Hamiltonian was used in Ref. [@bour_caron] in order to demonstrate the stability of the superconductivity in Q1D metals.
It is remarkable that the superconductivity is stable in the system with no attraction. Superconductivity cannot be found if one apply mean-field approximation to the bare Hamiltonian. To discover the existence of the superconducting ground state the high-energy degrees of freedom must be properly accounted for.
Our mechanism is related to that of Kohn and Luttinger. It is known that classical Kohn-Luttinger mechanism gives extremely low transition temperature. Our system does not share this property. Due to Q1D nature of the system the superconducting coupling constant is not as minuscule as Kohn-Luttinger coupling constant. Consequently, the transition temperature in our model does not have to be unobservable small.
We will see that the phase diagram of our model is similar to the phase diagram of the organic superconductors: [*(i)*]{} when the nesting of the Fermi surface is good, the ground state is either SDW or CDW; [*(ii)*]{} under increased pressure the nesting is spoiled, the density wave becomes unstable, and it is replaced by the unconventional superconductivity; [*(iii)*]{} under even higher pressure the superconducting transition temperature vanishes, and the system shows no sign of the spontaneous symmetry breaking. This similarity suggests that the proposed mechanism may be relevant for these materials.
Yet, the purpose of this paper is not to model real-life systems. Indeed, assumptions made about Hamiltonian’s parameters may be too extreme for a real material. Rather, we want to demonstrate in a controllable way that the superconductivity in Q1D metals is a rather generic phenomena. Once this is done, the qualitative understanding developed in a specialized model may be applied to a more complicated situation, where analytical treatment is problematic.
The paper is organized as follows. In Sect. \[model\] we formulate our model. In Sect. \[MF\] we perform its mean-field analysis. The variational calculations are done in Sect. \[var\]. Different phases of the effective Hamiltonian (and the physical system) are mapped in Sect. \[phase\]. We discuss the derived results in Sect. \[disc\].
The system {#model}
==========
The Hamiltonian
----------------
We start our presentation by writing down the Hamiltonian for the array of coupled 1D conductors: $$\begin{aligned}
H&=&\int_0^L dx {\cal H},\label{H}\\
{\cal H}&=&\sum_{i} {\cal H}_i^{\rm 1D} + \sum_{i,j}
\left[ {\cal H}_{ij}^{\rm hop} + {\cal H}_{ij}^{\rho\rho}
%+ {\cal H}_{ij}^{SS}
\right] ,\end{aligned}$$ where the indices $i,j$ run over 1D conductors. In this paper we will adhere to the agreement of denoting the Hamiltonian densities with the calligraphic letters (e.g., ${\cal H}$) and full Hamiltonians with the italic letters (e.g., $H$).
In the above formula the Hamiltonian density ${\cal H}_i^{\rm 1D}$ contains the in-chain kinetic energy and interactions: $$\begin{aligned}
{\cal H}_i^{\rm 1D}
&=&
{\cal T}^{\rm 1D}_i
\left[
\psi^\dagger ,
\psi
\right]
+
{\cal V}^{\rm 1D}_i
\left[
\psi^\dagger ,
\psi
\right]
+
{\cal V}^{\rm 1D}_{{\rm bs},i}
\left[
\psi^\dagger ,
\psi
\right],
%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{H1D}%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
\\
%
{\cal T}_i^{\rm 1D}
&=&
- {i} v_{\rm F} \sum_{p\sigma}
p\/\/ \colon\!
\psi^\dagger_{p\sigma i}
(
\nabla
\psi^{\vphantom{\dagger}}_{p \sigma i}
)
\colon,
%%%%%%%%%%%%%%%%%%%%%%%
\label{T}
%%%%%%%%%%%%%%%%%%%%%%%
\\
%
{\cal V}^{\rm 1D}_i
&=&
g_2 \sum_{\sigma \sigma'}
\rho_{{\rm L}\sigma i} \rho_{{\rm R}\sigma' i}
+
g_{4} \left(
\rho_{{\rm L}\uparrow i}
\rho_{{\rm L}\downarrow i}
+
\rho_{{\rm R}\uparrow i}
\rho_{{\rm R}\downarrow i}
\right) ,
\label{V}
\\
{\cal V}^{\rm 1D}_{{\rm bs},i}
&=&
g_{\rm bs}
\rho_{2k_{\rm F} i}\rho_{-2k_{\rm F} i},
\label{BS}\end{aligned}$$ where the chirality index $p$ is equal to $+1$ ($p=-1$) for right-moving (left-moving) electrons. The subscript ‘bs’ stands for ‘backscattering’. The theory has an ultraviolet cutoff $\Lambda = \pi/a$. The symbol $\colon \ldots \colon$ denotes the normal order of the fermionic fields with respect to the non-interacting ground state. The Hamiltonian density ${\cal H}^{\rm 1D}_i$ is spin-rotationally invariant.
Different densities used in formulae above and throughout the paper are defined by the equations: $$\begin{aligned}
\rho_{p \sigma i}
&=&
\colon
\psi^\dagger_{p \sigma i}
\psi^{\vphantom{\dagger}}_{p \sigma i}
\colon,
\\
\rho_i
&=&
\sum_{p\sigma} \rho_{p\sigma i},
\\
\rho_{2k_{\rm F} i}
& = &
\sum_\sigma
\psi^\dagger_{{\rm R} \sigma i}
\psi^{\vphantom{\dagger}}_{{\rm L} \sigma i} ,
%\\
%\rho_{-2k_{\rm F} \sigma i}
%&=&
%\rho_{2k_{\rm F} \sigma i}^\dagger,
\\
\rho_{-2k_{\rm F} i}^{\vphantom{\dagger}}
&=&
\rho_{2k_{\rm F} i}^\dagger,
\\
%{\bf S}_{p i}
%&=&
%\sum_{\sigma \sigma'} \vec{\tau}_{\sigma \sigma'}
%\colon
% \psi^{\dagger}_{p \sigma i}
% \psi^{\vphantom{\dagger}}_{p \sigma' i}
%\colon ,
%\\
%{\bf S}_i
%& = &
%\sum_p {\bf S}_{pi},
%\\
{\bf S}_{2k_{\rm F} i}
&=&
\sum_{\sigma \sigma'} \vec{\tau}_{\sigma \sigma'}
\psi^{\dagger}_{{\rm R} \sigma i}
\psi^{\vphantom{\dagger}}_{{\rm L} \sigma' i} ,
\\
{\bf S}_{-2k_{\rm F}i}^{\vphantom{\dagger}}
&=&
{\bf S}_{2k_{\rm F}i}^{\dagger}, \end{aligned}$$ where $\vec{\tau}$ is the vector composed of the three Pauli matrices.
The coupling between the 1D conductors is described by the transverse terms: the single-electron hopping, $$\begin{aligned}
{\cal H}_{ij}^{\rm hop}
&=&
- t(i-j)
\sum_{p \sigma}
\left(
\psi^\dagger_{p\sigma i}
\psi^{\vphantom{\dagger}}_{p\sigma j}
+
{\rm H.c.}
\right), \end{aligned}$$ and the density-density interaction, $$\begin{aligned}
{\cal H}_{ij}^{\rho\rho}
&=&
g^\perp_0 (i-j)
\rho_{i} \rho_{j}
+
\label{perp}
\\
&&g_{2k_{\rm F}}^\perp (i-j)
\left(
\rho_{2k_{\rm F} i}
\rho_{-2k_{\rm F} j}
+ {\rm H.c.}
\right).
\nonumber \end{aligned}$$ We accept that all interactions are repulsive, weak, and that the in-chain interactions are stronger than the transverse interactions: $$\begin{aligned}
2 \pi v_{\rm F}
\gg
g_{2,4}
\gg
g_{\rm bs}
\gg
g^\perp_0
\gtrsim
g^\perp_{2 k_{\rm F}}
> 0,
%%%%%%%%%%%%%%%
\label{hier}%%%
%%%%%%%%%%%%%%%\end{aligned}$$ and the transverse hopping is small: $$\begin{aligned}
v_{\rm F} \Lambda \gg t.
%%%%%%%%%%%%%%%%%%%%%%%
\label{small_t}%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ The constraints on the Hamiltonian’s coefficients will be further discussed in Sect. \[var\], Sect. \[superconductivity\], and Sect. \[accuracy\].
Bosonized Hamiltonian
---------------------
In Sect. \[var\] we will need the bosonized version of Hamiltonian density ${\cal H}^{\rm 1D}$. The bosonic representation is based on the bosonization prescription for the electron field [@boson]: $$\begin{aligned}
\psi^\dagger_{p\sigma } (x)
&=&
(2\pi a)^{-1/2} \eta_{p\sigma}
{\rm e}^{{\rm i}\sqrt{2\pi} \varphi_{p\sigma}(x)},
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{bos}%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\varphi_{p\sigma}
&=&
\frac{1}{2}
\left(
\Theta_c + p\Phi_c + \sigma \Theta_s + p\sigma \Phi_s
\right).\end{aligned}$$ In the above formulae $\eta_{p\sigma}$ is Klein factor, $\Theta_{c,s}$ are the TL charge ([*c*]{}) and spin ([*s*]{}) boson fields, $\Phi_{c,s}$ are the dual fields. The chain indices $i,j$ are omitted in the expressions above. We will not show these indices explicitly in cases where such omissions do not introduce problems.
The bosonized one-chain Hamiltonian is: $$\begin{aligned}
{\cal H}^{\rm 1D}
\left[
\Theta,\Phi
\right]
=
{\cal H}_0^{\rm 1D}
\left[
\Theta,\Phi
\right]
+
{\cal V}_{\rm bs}^{\rm 1D}
[
\Theta, \Phi
], \label{H1D_bos}\end{aligned}$$ where ${\cal H}_0^{\rm 1D}$ is quadratic in the boson fields: $$\begin{aligned}
{\cal H}_0^{\rm 1D}
\left[
\Theta,\Phi
\right]
&=&
{\cal T}^{\rm 1D}
\left[
\Theta,\Phi
\right]
+
{\cal V}^{\rm 1D}
\left[
\Theta,\Phi
\right]
%%%%%%%%%%%%%%%%%%%%%%%%
\label{Hbos}%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\\
\nonumber
&=&
\frac{v_c}{2}
\left(
{\cal K}_c
\colon
\left(
\nabla \Theta_c
\right)^2
\colon
+
{\cal K}^{-1}_c
\colon
\left(
\nabla \Phi_c
\right)^2
\colon
\right)
\\
&&+
\frac{v_s}{2}
\left(
\colon
\left(
\nabla \Theta_s
\right)^2
\colon
+
\colon
\left(
\nabla \Phi_s
\right)^2
\colon
\right),
\nonumber \end{aligned}$$ while ${\cal V}^{\rm 1D}_{\rm bs}$ is not: $$\begin{aligned}
{\cal V}_{\rm bs}^{\rm 1D}
[
\Theta, \Phi
]
&=&
\frac{g_{\rm bs}}{2\pi^2 a^2}
\cos (
\sqrt{8\pi} \Phi_s
)
\quad\\
&&-
\frac{g_{\rm bs}}{2\pi}
\left[
\colon
\left(
\nabla \Phi_c
\right)^2
\colon
+
\colon
\left(
\nabla \Phi_s
\right)^2
\colon
\right].
\nonumber \end{aligned}$$ The symbol $\colon\ldots\colon$ denotes normal ordering of TL boson operators with respect to the non-interacting (${\cal K}_c = 1$, $v_{\rm s} = v_{\rm c}$, $g_{\rm bs} = 0$) bosonic ground state. The Tomonaga-Luttinger liquid parameters are given by the formulae: $$\begin{aligned}
{\cal K}_c
&=&
\sqrt{
\frac{2\pi v_{\rm F} + g_4 - 2g_2}
{2\pi v_{\rm F} + g_4 + 2g_2}
},
\\
%
v_c
&=&
\frac{1}{2\pi}
\sqrt{
\left( 2\pi v_{\rm F} + g_4 \right)^2 - 4g_2^2
},
\\
%
v_s
&=&
v_{\rm F} - \frac{g_4}{2\pi}.\end{aligned}$$ It is worth noting that $$\begin{aligned}
{\cal K}_c < 1,\end{aligned}$$ for repulsive interaction.
We will also need the expression: $$\begin{aligned}
%\rho_{2 k_{\rm F} \sigma}
\psi^\dagger_{{\rm R} \sigma }
\psi^{\vphantom{\dagger}}_{{\rm L} \sigma }
=
\frac{1}{2 \pi a}
{\rm e}^{
{\rm i}
\sqrt{2\pi}
(\Phi_c + \sigma \Phi_s)
},\end{aligned}$$ which gives operator $\psi^\dagger_{{\rm R} \sigma }
\psi^{\vphantom{\dagger}}_{{\rm L} \sigma }$ in terms of the TL bosons.
The mean-field approach {#MF}
=======================
Once the model is formulated, it is not difficult to analyze its mean-field phase diagram. Such analysis introduces serious qualitative errors. Yet, in order to appreciate fully the advantage of the many-body calculations proposed below the comparison with the mean-field results is very important.
From the outset we have to keep in mind that in our system several different symmetries might be broken. Thus, several order parameters should be taken into consideration: SDW, CDW, triplet and singlet superconductivity.
To perform the mean-field analysis we write the interaction terms as products of these order parameters. After that the order corresponding to the highest coupling constant and susceptibility is chosen.
CDW and SDW {#DW}
-----------
We start with the in-chain interaction (the biggest potential energy in the system): $$\begin{aligned}
%%%%%%%%%%%%%%%%%%%%%
\label{V_MF}%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%
{\cal V}^{\rm 1D}_i
+
{\cal V}^{\rm 1D}_{{\rm bs},i}
=
-
\left(
\frac{g_2}{2} - g_{\rm bs}
\right)
\rho_{{ 2k_{\rm F}}i}
\rho_{{ -2k_{\rm F}}i}
\\
-
\frac{g_2}{2}
{\bf S}_{{2k_{\rm F}}i}
\cdot
{\bf S}_{{ - 2k_{\rm F}}i}
+
\ldots,
\nonumber \end{aligned}$$ where $\ldots$ stand for $g_4$ term, which cannot be written as a product of two order parameters.
The transverse Hamiltonian may be expressed as a product of CDW and SDW order parameters: $$\begin{aligned}
\sum_{ij}
{\cal H}^{\rho \rho}_{ij}
%=
%\sum_{ij}
%g^\perp_0
%\rho_{i} \rho_{j}
%+
%g_{2k_{\rm F}}^\perp
%\left(
% \rho_{2k_{\rm F} i}
% \rho_{-2k_{\rm F} j}
% + {\rm H.c.}
%\right)
%%%%%%%%%%%%%%%%%%%%%%%%%
\label{Hrr_mf}%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
%\\
&=&
\sum_{ij}
g_{2k_{\rm F}}^\perp
\left(
\rho_{2k_{\rm F} i}
\rho_{-2k_{\rm F} j}
+ {\rm H.c.}
\right)
%\qquad\quad
\\
&-& g^\perp_0
\left(
\rho_{2k_{\rm F} ij}
\rho_{-2k_{\rm F} ij}
+
{\bf S}_{2k_{\rm F} ij}
\cdot
{\bf S}_{-2k_{\rm F} ij}
\right)
+
\ldots.
\nonumber \end{aligned}$$ The order parameter $\rho_{2k_{\rm F}ij}$ is equal to $\sum_\sigma
\psi^\dagger_{{\rm R} \sigma i}\psi^{\vphantom{\dagger}}_{{\rm L} \sigma j}$, and ${\bf S}_{2k_{\rm F}ij}$ is defined in a similar fashion. They are bond CDW and bond SDW. These types of order cannot take advantage of the in-chain interaction energy (the biggest interaction energy in the problem). Thus, they cannot compete against $\rho_{2k_{\rm F} i}$ and ${\bf S}_{2k_{\rm F}i}$. We will not study $\rho_{2 k_{\rm F} ij}$ and ${\bf S}_{2k_{\rm F}ij}$ anymore.
The non-interacting susceptibilities of SDW and CDW are equal to each other. Eqs. (\[V\_MF\]) and (\[hier\]) suggest that the SDW coupling constant is bigger than the CDW coupling constant: $$\begin{aligned}
g_{\rm SDW}
=
\frac{g_2}{2}
%+
%O(g^\perp)
>
g_{\rm CDW}
&=&
\frac{g_2}{2}
-
g_{\rm bs}
+
\frac{z^\perp g^\perp_{2k_{\rm F}}}{2},
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{dw_coupling}%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ where $z^\perp$ is the number of the nearest neighbours of a given chain. Thus, when the nesting is good, the mean-field analysis suggests that the ground state is SDW.
Superconducting orders
----------------------
Several sorts of the superconducting order parameter can be defined. They can be classified according to their spin and orbital symmetry. It is useful to define a $2\times 2$ matrix $\hat \Delta_{ij}$ with components: $$\begin{aligned}
%%%%%%%%%%%%%%%%%%%%%%%%
\label{sc_matrix}%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
(\hat \Delta_{ij})_{\sigma\sigma'}
=
\psi^\dagger_{{\rm L}\sigma i}
\psi^\dagger_{{\rm R}\sigma' j},\end{aligned}$$ and write $\hat \Delta_{ij}$ as a sum of three symmetric matrices ${ i}\vec{\tau}\tau^y$ and one antisymmetric matrix ${i} \tau^y$: $$\begin{aligned}
%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sc_order}%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
\hat \Delta_{ij}
=
\frac{1}{\sqrt{2}}
\left[
{\bf d}_{ij}
\cdot
({i}\vec{\tau}\tau^y)
+
\Delta_{ij}
{i} \tau^y
\right].\end{aligned}$$ The operator $\Delta_{ij}$ (${\bf d}_{ij}$) is the singlet (triplet) order parameter corresponding to a Cooper pair composed of two electrons one of which is on chain $i$ and the other is on chain $j$.
Furthermore, $\hat \Delta_{ij}$ may be symmetrized with respect to the chain indices as well: $$\begin{aligned}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sc_symm}%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hat \Delta^{s/a}_{ij}
=
\frac{1}{2}
\left(
\hat \Delta_{ij}
\pm
\hat \Delta_{ji}
\right).\end{aligned}$$ The superscript ‘s’ (‘a’) stands for ‘symmetric’ (‘antisymmetric’).
The operators $\Delta^{s/a}_{ij}$ and ${\bf d}^{s/a}_{ij}$ are defined in the same fashion. If $i=j$, the antisymmetric quantities are, obviously, zero.
As the following derivations show, all these variants of superconductivity are unstable at the mean-field level. The in-chain interaction energy can be expressed as: $$\begin{aligned}
{\cal V}_{i}^{{\rm 1D}}
+
{\cal V}_{{\rm bs},i}^{{\rm 1D}}
=
(g_{2} - g_{\rm bs})
\mathbf{d}_{ii}^{\vphantom\dagger}
\cdot
\mathbf{d}_{ii}^{\dagger}
\\
+
(g_{2} + g_{\rm bs})
\Delta_{ii}^{\vphantom\dagger}
\Delta_{ii}^{\dagger}
+
\ldots.
\nonumber \end{aligned}$$ For realistic interaction $g_2 > g_{\rm bs}$. Therefore, the one-chain order parameters ${\bf d}_{ii}$, $\Delta_{ii}$ are unstable.
The inter-chain interaction can be written as a bilinear of the superconducting order parameters ${\bf d}_{ij}^{s/a}$, $\Delta_{ij}^{s/a}$, $i \ne j$: $$\begin{aligned}
%%%%%%%%%%%%%%%%%%%%%%
\label{sc_coupling}%%%
%%%%%%%%%%%%%%%%%%%%%%
&&\sum_{ij} {\cal H}^{\rho\rho}_{ij}
=
\\
\nonumber
&&\qquad
\sum_{ij}
2 (g_0^\perp - g_{2 k_{\rm F}}^\perp)
\left[
\Delta_{ij}^{a}
(\Delta_{ij}^a)^\dagger
+
\mathbf{d}_{ij}^s
\cdot
(\mathbf{d}_{ij}^s)^{\dagger}
\right]
\\
&&\qquad
+
2 (g_0^\perp + g_{2 k_{\rm F}}^\perp)
\left[
\Delta_{ij}^{s}
(\Delta_{ij}^s)^\dagger
+
\mathbf{d}_{ij}^a
\cdot
(\mathbf{d}_{ij}^a)^{\dagger}
\right]
+ \ldots .
\nonumber \end{aligned}$$ For a realistic choice of the interaction constants: $$\begin{aligned}
g_{2k_{\rm F}}^\perp < g_0^\perp.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{bare_g}%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ Consequently, the two-chain order parameters are unstable, as well as their one-chain counterparts.
Mean-field phase diagram
------------------------
As a result of the above considerations the following mean-field phase diagram has emerged. If the nesting is good, the stable phase is SDW. It is characterized by the non-zero $\langle {\bf S}_{2k_{\rm F}i} \rangle$. The SDW state competes with the CDW state (non-zero $\langle \rho_{2k_{\rm F}i} \rangle$). SDW wins for it does not frustrate the backscattering interactions while CDW does \[see Eq.(\[V\_MF\])\].
In a system with poor nesting SDW becomes unstable [@sdw]. The mean-field theory predicts that such systems have no spontaneously broken symmetry.
This phase diagram will be corrected in a qualitative manner when the cooperative effects are accounted for. We will show that the many-body phenomena force the violation of Eq.(\[bare\_g\]), which makes the superconductivity stable in the systems with poor nesting. The same phenomena may lead to violation of inequality (\[dw\_coupling\]), inducing transition into CDW rather than SDW.
Variational procedure {#var}
=====================
In this section we develop the variational approach overcoming the deficiencies of the mean-field approximation.
To keep our discussion short, transparent, and intuitive we will assume that both backscattering and transverse interactions are zero: $g_{\rm bs} = 0$, $g^\perp_{0,{2k_{\rm F}}} = 0$. In such a situation the Hamiltonian is equal to: $$\begin{aligned}
{H}' = \sum_i {H}_{0i}^{\rm 1D} + \sum_{ij} {H}_{ij}^{\rm hop}.
%%%%%%%%%%%%%%%%%%%%%
\label{H'}%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ The first part of $H'$, the one-chain Hamiltonian $H_{0i}^{\rm 1D}$, is quadratic in terms of the TL bosons. The second part of $H'$, the transverse hopping $H^{\rm hop}$, is quadratic in terms of the physical fermion fields. Because of this circumstance, the variational derivations for $H'$ are simpler than for generic $H$. Yet, such derivations retain the most important features of the general case. This makes $H'$ an ideal object of initial investigation, which we extend later for the Hamiltonian with non-zero $g_{\rm bs}$ and $g^\perp_{0,{2 k_{\rm F}}}$.
Below the prime mark ($'$) will be used to distinguish between the most general Hamiltonian ${H}$, Eq.(\[H\]), and the special case ${H}'$, Eq.(\[H’\]). Likewise, the prime will decorate the objects associated with ${H}'$ (e.g., effective Hamiltonian ${H}^{{\rm eff}\prime}$, variational energy $E^{V\prime}$).
We first explain the heuristic idea behind our variational wave function. Let us think of our system in terms of the TL bosons. The bosonized version of ${\cal H}^{\rm 1D}_0$ is given by Eq. (\[H1D\_bos\]). However, the ground state $\left| 0_{\rm 1D} \right>$ of $H^{\rm 1D}_0$ is not a good approximation to the ground state of $H'$ for the finite-order perturbation theory in $t$ is not well-defined.
On the other hand, if we were to describe our system with the help of the bare electron degrees of freedom $\psi$, $\psi^\dagger$ we will not have problems to account for $H^{\rm hop}$. But, within the fermionic framework, the in-chain interaction energy is extremely difficult to handle.
To resolve this conflict we introduce the parameter $\tilde \Lambda < \Lambda$ and separate the total phase space of the model into two parts, the low-energy part (the degrees of freedom whose energy is smaller than $v_{\rm F} \tilde \Lambda$) and the high-energy part (the degrees of freedom whose energy is higher than $v_{\rm F} \tilde \Lambda$) [@rozhkov]. The high-energy part will be described in terms of the TL bosons, while the low-energy part will be described with the help of fermionic quasiparticles, which we will define below. The exact value of $\tilde \Lambda$ is to be found variationally, as a trade-off between the in-chain interaction and the transverse hopping.
The formal implementation of this approach goes as follows. First, the TL boson fields are split into two components: fast (with large momentum $k_\|$: $\Lambda>|k_\|| > \tilde\Lambda$) and slow (with small momentum $k_\|$: $|k_\|| < \tilde\Lambda$). The fast (slow) component will be marked by ‘$>$’ (‘$<$’) superscript: $$\begin{aligned}
\Theta_{c,s} (x)
&=&
\Theta_{c,s}^<(x) + \Theta_{c,s}^>(x)
\\
&=&
\sum_{|k_\|| < \tilde \Lambda}
\Theta_{c,s,k_\|}
{\rm e}^{{\rm i} k_\| x}
+
\sum_{\tilde \Lambda < |k_\|| < \Lambda}
\Theta_{c,s,k_\|}
{\rm e}^{{\rm i} k_\| x},
\nonumber
\\
%
\Phi_{c,s} (x)
&=&
\Phi_{c,s}^<(x) + \Phi_{c,s}^>(x)
\\
&=&
\sum_{|k_\|| < \tilde \Lambda}
\Phi_{c,s,k_\|}
{\rm e}^{{\rm i} k_\| x}
+
\sum_{\tilde \Lambda < |k_\|| < \Lambda}
\Phi_{c,s,k_\|}
{\rm e}^{{\rm i} k_\| x}.
\nonumber
%
%\varphi_{p\sigma} (x)
%&=&
%\varphi_{p\sigma}^<(x) + \varphi_{p\sigma}^>(x)
%\\
%&=&
%\sum_{|k_\|| < \tilde \Lambda}
% \varphi_{c,s,k_\|}
% {\rm e}^{{\rm i} k_\| x}
%+
%\sum_{\tilde \Lambda < |k_\|| < \Lambda}
% \varphi_{c,s,k_\|}
% {\rm e}^{{\rm i} k_\| x}.
%\nonumber \end{aligned}$$ This split of the bosonic degrees of freedom induces the split of the in-chain Hamiltonian density ${\cal H}_0^{\rm 1D}$: $$\begin{aligned}
{\cal H}^{\rm 1D}_0
\left[
\Theta,\Phi
\right]
=
{\cal H}^{\rm 1D}_0
\left[
\Theta^<, \Phi^<
\right]
+
{\cal H}^{\rm 1D}_0
\left[
\Theta^>,\Phi^>
\right].\end{aligned}$$ That is, the Hamiltonian $H_0^{\rm 1D}$, Eq.(\[Hbos\]), cleanly separates into two parts corresponding to fast and slow modes.
The quasiparticles $\Psi_{p\sigma}^\dagger(x)$ are defined with the help of Eq.(\[bos\]), in which $a$ is substituted by $\tilde a = \pi/\tilde\Lambda$ and the slow fields $\Theta_{c,s}^<$, $\Phi_{c,s}^<$ or $\varphi_{p\sigma}^<$ are placed instead of the bare fields $\Theta_{c,s}$, $\Phi_{c,s}$ or $\varphi_{p\sigma}$: $$\begin{aligned}
\Psi^\dagger_{p\sigma} (x) = (2\pi \tilde a)^{-1/2} \eta_{p\sigma}
{\rm e}^{{\rm i}\sqrt{2\pi} \varphi_{p\sigma}^<(x)}. \label{bosqp}\end{aligned}$$ Using the quasiparticle field $\Psi_{p\sigma}$ we refermionize ${\cal H}_0^{\rm 1D} [ \Theta^<, \Phi^< ]$: $$\begin{aligned}
{\cal H}_0^{\rm 1D}
=
{\cal H}_0^{\rm 1D}
\left[
\Psi^\dagger,
\Psi
\right]
+
{\cal H}_0^{\rm 1D}
\left[
\Theta^>,
\Phi^>
\right],
%%%%%%%%%%%%%%%%%%%
\label{H1Dmix}%%%%%
%%%%%%%%%%%%%%%%%%%
\\
{\cal H}_0^{\rm 1D}
\left[
\Psi^\dagger,
\Psi
\right]
=
{\cal T}^{\rm 1D}
\left[
\Psi^\dagger ,
\Psi
\right]
+
{\cal V}^{\rm 1D}
\left[
\Psi^\dagger ,
\Psi
\right],\end{aligned}$$ where ${\cal T}^{\rm 1D} [\Psi^\dagger, \Psi ]$ and ${\cal V}^{\rm 1D} [\Psi^\dagger, \Psi]$ are given by Eqs. (\[T\]) and (\[V\]).
The mixed representation of ${\cal H}_0^{\rm 1D}$, Eq.(\[H1Dmix\]), makes no sense in pure 1D problems since ${\cal H}_0^{\rm 1D}
\left[
\Psi^\dagger,
\Psi
\right]$ corresponds to an interacting 1D system, whose ground state and excitations have no simple representation in terms of $\Psi$’s. Indeed, our variational calculations will show that, if $t=0$, then $\tilde \Lambda = 0$. That is, no room for the quasiparticles is left in 1D situation. However, if $t \ne 0$, the quasiparticles delocalize in the transverse directions and lower the total energy of the system. In such a case $\tilde \Lambda$ does not have to be zero, as we will demonstrate.
The Hamiltonian density ${\cal H}^{\rm hop}$ can be easily expressed within the framework of the mixed quasiparticle-fast boson representation. One observes that the physical fermion is simply: $$\psi_{p\sigma}^\dagger
=
\sqrt{\tilde a/a}\Psi_{p\sigma}^\dagger
{\rm e}^{{\rm i}\sqrt{2\pi} \varphi_{p\sigma}^>},
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{physical}%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%$$ and that the fermionic and bosonic parts in this definition commute with each other. Therefore: $$\begin{aligned}
{\cal H}_{ij}^{\rm hop}
= -
\frac{\tilde a}{a}
t
\sum_{p \sigma}
\Psi^\dagger_{p\sigma i}
\Psi^{\vphantom{\dagger}}_{p\sigma j}
{\rm e}^
{
{\rm i} \sqrt{2\pi}
(
\varphi_{p\sigma i}^>-\varphi_{p\sigma j}^>
)
}
+
{\rm H.c.}
%%%%%%%%%%%%%%%
\label{Hhopmix}
%%%%%%%%%%%%%%%
%{\cal H}_{ij}^{\rho\rho} = g^\perp_0 \left(\tilde \rho_{i} \tilde \rho_{j}
%+ \frac{2}{\pi} \nabla \Phi_{ci}^> \nabla \Phi_{cj}^> \right) \\
%+ g_{2k_{\rm F}}^\perp \left(\frac{\tilde a}{a} \right)^2
%\left[ \left(
%\Psi_{{\rm L} \uparrow i}^\dagger
%\Psi_{{\rm R} \uparrow i}^{\vphantom{\dagger}}
%\Psi_{{\rm R} \uparrow j}^\dagger
%\Psi_{{\rm L} \uparrow j}^{\vphantom{\dagger}}
%{\rm e}^{i\sqrt{2\pi} ( \Phi_{si}^< - \Phi_{sj}^< )} +
%%\right.\\ +\left.
%\Psi_{{\rm L} \downarrow i}^\dagger
%\Psi_{{\rm R} \downarrow i}^{\vphantom{\dagger}}
%\Psi_{{\rm R} \downarrow j}^\dagger
%\Psi_{{\rm L} \downarrow j}^{\vphantom{\dagger}}
%{\rm e}^{-i\sqrt{2\pi} ( \Phi_{si}^< - \Phi_{sj}^< )} \right)
%{\rm e}^{i\sqrt{2\pi} ( \Phi_{ci}^< - \Phi_{cj}^< )}
%%\nonumber\\
%+ {\rm h.c.} \right], \nonumber
%\nonumber \end{aligned}$$ Eqs. (\[H1Dmix\]) and (\[Hhopmix\]) determine the form of the total Hamiltonian ${H}'$ in the mixed representation. Let us study this Hamiltonian.
The eigenenergies of the fast bosons are determined mostly by ${\cal H}_0^{\rm 1D} \left[\Theta^>,\Phi^>\right]$. These eigenenergies are bigger than $\sim v_{\rm F} \tilde \Lambda$. Small hopping term is only a correction to this quantity. Thus, we simply neglect contribution of ${H}^{\rm hop}$ to the high-energy sector’s properties and assume that all fast bosons are in the ground state $\left| 0_> \right>$ of the quadratic Hamiltonian: $$\begin{aligned}
H^>
=
\sum_i
\int {\cal H}_{0i}^{\rm 1D} \left[
\Theta^>, \Phi^>
\right]
dx.
%%%%%%%%%%%%%%%%%
\label{Hfast}%%%%
%%%%%%%%%%%%%%%%%\end{aligned}$$ When describing the quasiparticle state, we cannot neglect ${\cal H}^{\rm hop}$: the quasiparticles are low-lying excitations, and their energy may be arbitrary small. Thus, we construct our variational wave function as a product: $$\begin{aligned}
\left| {\rm var} \right>
=
\left| \{ \Psi \} \right>
\left| 0_> \right>,\end{aligned}$$ where $\left| \{ \Psi \} \right>$ is the unknown quasiparticle state. The variational energy is given by: $$\begin{aligned}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{EV'}%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E^{\rm V \prime}
=
\left< {\rm var} \right|
H'
\left| {\rm var} \right>
=
\left< \{\Psi\} \right|
H^{\rm eff\prime}
\left| \{\Psi \} \right>.\end{aligned}$$ This equation defines the effective quasiparticle Hamiltonian $H^{\rm
eff \prime}$ as a ‘partial average’ over the fast degrees of freedom: $$\begin{aligned}
H^{\rm eff \prime}
&=&
\left< 0_> \right|
H^\prime
\left| 0_> \right>
%%%%%%%%%%%%%%%%%%%%
\label{Heff'}%%%%%%%
%%%%%%%%%%%%%%%%%%%%
\\
\nonumber
&=&
H^{\rm 1D}_0 \left[
\Psi^\dagger ,
\Psi
\right]
+
\tilde H^{\rm hop}\left[
\Psi^\dagger ,
\Psi
\right]
+
\left< 0_> \right|
H^>
\left| 0_> \right>, \end{aligned}$$ where the last term is the $c$-number corresponding to the fast boson contribution to the variational energy, and the effective quasiparticle hopping in Eq.(\[Heff’\]) is defined by the formula: $$\begin{aligned}
\tilde {\cal H}_{ij}^{\rm hop}
&=&
- \tilde t
\sum_{p\sigma}
\Psi^\dagger_{p\sigma i}
\Psi^{\vphantom{\dagger}}_{p\sigma j}
+
{\rm H.c.} ,
\\
%
\tilde t
&=&
t \frac{\Lambda}{\tilde \Lambda}
\langle
{\rm e}^{{\rm i} \sqrt{2\pi} \varphi_{p\sigma }^>}
\rangle_>^2.
%%%%%%%%%%%%%%%%%%%%%%%%%
\label{t_tilde0}%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ The symbol $\langle \ldots \rangle_>$ is the short-hand notation for $\langle 0_>| \ldots |0_> \rangle$. The fast bosons introduce renormalization of the effective hopping of the quasiparticles. The expectation value in Eq.(\[t\_tilde0\]) is: $$\begin{aligned}
\langle
{\rm e}^{{\rm i} \sqrt{2\pi} \varphi_{p\sigma }^>}
\rangle_>
=
\left(
\frac{\tilde \Lambda}{\Lambda}
\right)^{({\cal K}_{\rm c} + {\cal K}_{\rm c}^{-1} + 2)/8}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{expectation}%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ To establish the above equality we must remember that $\left| 0_> \right>$ is the ground state of the quadratic Hamiltonian $H^>$. Thus: $$\begin{aligned}
&&\langle
{\rm e}^{{\rm i} \sqrt{2\pi} \varphi_{p\sigma }^>}
\rangle_>
=
{\rm e}^{-\pi \langle (\varphi_{p\sigma }^>)^2 \rangle_> },
\\
&&
\langle (\varphi_{p\sigma }^>)^2 \rangle_>
=
\\
&&\qquad
\frac{1}{4}
\left[
\langle (\Theta_{c}^>)^2 \rangle_>
+
\langle (\Phi_{c}^>)^2 \rangle_>
%\right.
%\\
%\left.
+
\langle (\Theta_{s}^>)^2 \rangle_>
+
\langle (\Phi_{s}^>)^2 \rangle_>
\right]
\nonumber\\
&&\qquad\qquad
=
\frac{1}{8\pi}
\left[
{\cal K}_c^{-1} + {\cal K}_c^{\vphantom{-1}} + 2
\right]
\ln \frac{\Lambda}{\tilde \Lambda}.
\nonumber \end{aligned}$$ Substituting Eq.(\[expectation\]) into Eq.(\[t\_tilde0\]) one finds: $$\begin{aligned}
\tilde t
=
t
\left(
\frac{\tilde \Lambda}{\Lambda}
\right)^
{({\cal K}_{\rm c} + {\cal K}_{\rm c}^{-1} - 2 )/4}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{t_tilde}%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ Assume now that the quasiparticle state $\left| \{ \Psi \} \right>$ is non-interacting fermion ground state. Then the variational energy may be expressed as follows: $$\begin{aligned}
E^{V\prime}/ L N_\perp = \varepsilon^{\rm 1D} + \varepsilon^{\rm F},
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{EV'_exp}%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ where $L$ is the length of the sample along the 1D conductors, $N_\perp$ is the number of these conductors; the one-dimensional contribution $\varepsilon^{\rm 1D}$ and the non-interacting fermion contribution $\varepsilon^{\rm F}$ are equal to: $$\begin{aligned}
\varepsilon^{\rm 1D}
=
\frac{v_{\rm c} \theta}{2\pi} \left(\tilde
\Lambda^2 - \Lambda^2 \right),
\\
\varepsilon^{\rm F}
=
- \frac{4}{\pi v_{\rm F}}\sum_i [ \tilde t(i)]^2
=
- \frac{4}{\pi v_{\rm F}}
\left(
\frac{\tilde\Lambda}{\Lambda}
\right)^{2\theta} \sum_i [ t(i)]^2,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{epsilonF}%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\theta
=
\frac{1}{4} \left(
{\cal K}_{\rm c}+ {\cal K}_{\rm c}^{-1} - 2
\right).\end{aligned}$$ Our expression for the fermion energy neglects all corrections coming from quasiparticle interaction and possible symmetry breaking since these are small.
It is convenient to define the characteristic transverse hopping energy as: $$\begin{aligned}
\bar t^2 = \sum_i [t(i)]^2,\end{aligned}$$ and the dimensionless ratio: $$\begin{aligned}
\zeta = \frac{\tilde \Lambda}{\Lambda} < 1.\end{aligned}$$ In terms of such quantities the variational energy is equal to: $$\begin{aligned}
E^{V\prime}/ L N_\perp = \frac{v_{\rm c} \theta}{2\pi} \Lambda^2
(\zeta^2 - 1) -
\frac{4}{\pi v_{\rm F}} \zeta^{2\theta} \bar t^2.\end{aligned}$$ Minimizing it with respect to $\zeta$ one finds that for small in-chain interactions ($\theta < 1$): $$\begin{aligned}
\zeta = \left( \frac{8 \bar t^2}{ v_{\rm c} v_{\rm F} \Lambda^2}
\right)^{1/(2-2\theta)}. \label{zeta}\end{aligned}$$ We see that, if $t=0$, the variational value of $\tilde \Lambda$ is zero. In other words, in pure 1D system the quasiparticles do not appear.
Another important result obtained from Eq.(\[zeta\]) is: $$\begin{aligned}
\tilde t \sim v_{\rm F} \tilde \Lambda.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{mf_cond}%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ This means that the anisotropy coefficient of the effective Hamiltonian is of order unity: $(\tilde t/ v_{\rm F} \tilde \Lambda) \sim 1$. Therefore, the mean-field treatment is appropriate for $H^{\rm eff\prime}$. The latter conclusion is crucial for it signifies the completion of our quest: the microscopic Hamiltonian $H'$, Eq.(\[H’\]), whose treatment is complicated by the presence of the 1D many-body effects, is replaced by the effective Hamiltonian $H^{\rm eff \prime}$, Eq. (\[Heff’\]), which can be studied with the help of the mundane mean-field approximation.
Finally, we must extend the derivation of the effective Hamiltonian to the situation of non-zero backscattering and transverse interactions. As with the case of ${H}'$, the effective Hamiltonian $H^{\rm eff}$ for the generic Hamiltonian $H$ is defined by the equation $ H^{\rm eff} = \left< H \right>_> $. It is straightforward to show that $H^{\rm eff}$ has the same form as $H$ but with certain renormalizations of the coupling constants: $$\begin{aligned}
\tilde g_2 = g_2,
\quad
\tilde g_4 = g_4,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{g24_tilde}%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\tilde g_{\rm bs} = g_{\rm bs},
\quad
\tilde g_0^\perp = g_0^\perp,
%\quad
%\tilde J_0^\perp = J_0^\perp,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{g_tilde}%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\tilde t
=
\zeta^\theta t,
\quad
\tilde g_{2k_{\rm F}}^\perp
=
\zeta^{{\cal K}_c - 1} g_{2k_{\rm F}}^\perp.
%\quad
%\tilde J_{2k_{\rm F}}^\perp
%=
%\zeta^{{\cal K}_c - 1} J_{2k_{\rm F}}^\perp.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{g_2kF_tilde}%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ The derivations of these expressions are similar to the derivation of Eq.(\[t\_tilde\]). For example, to calculate $\tilde g_{2k_{\rm F}}^\perp$ we must write: $$\begin{aligned}
&&
\langle
g_{2k_{\rm F}}^\perp
\rho_{2k_{\rm F}i} \rho_{-2k_{\rm F}j}
\rangle_>
\\
&&\qquad\qquad
=
g_{2k_{\rm F}}^\perp
\left(
\frac{\Lambda}{\tilde \Lambda}
\right)^2
%
\sum_{\sigma \sigma'}
\Psi^\dagger_{{\rm R} \sigma i}
\Psi^{\vphantom{\dagger}}_{{\rm L} \sigma i}
\Psi^\dagger_{{\rm L} \sigma' j}
\Psi^{\vphantom{\dagger}}_{{\rm R} \sigma' j}
\nonumber
\\
&&\qquad\qquad
\times
\langle
{\rm e}^
{
i\sqrt{2\pi}
\left[
(\Phi_{ci}^> - \Phi_{cj}^>)
+
(\sigma \Phi_{si}^> - \sigma' \Phi_{sj}^>)
\right]
}
\rangle_>
\nonumber
\\
&&\qquad\qquad
=
\tilde g_{2k_{\rm F}}^\perp
\sum_{\sigma \sigma'}
\Psi^\dagger_{{\rm R} \sigma i}
\Psi^{\vphantom{\dagger}}_{{\rm L} \sigma i}
\Psi^\dagger_{{\rm L} \sigma' j}
\Psi^{\vphantom{\dagger}}_{{\rm R} \sigma' j},
\nonumber\end{aligned}$$ where the effective coupling constant $\tilde g_{2k_{\rm F}}^\perp $ is given by the expression: $$\begin{aligned}
\tilde g_{2k_{\rm F}}^\perp
=
g_{2k_{\rm F}}^\perp
\left(
\frac{\Lambda}{\tilde \Lambda}
\right)^2
\langle
{\rm e}^{i\sqrt{2\pi}[
(\Phi_{ci}^> - \Phi_{cj}^>)
+ (\sigma \Phi_{si}^> - \sigma' \Phi_{sj}^>)
] }
\rangle_> .\end{aligned}$$ From this formula Eq.(\[g\_2kF\_tilde\]) for $\tilde g^\perp_{2k_{\rm F}}$ follows.
We want our effective Hamiltonian to be in the weak-coupling regime: when the coupling is weak, the kinetic energy of the quasiparticles dominates over their interaction, which justifies Eq.(\[epsilonF\]). Consequently, we need to impose a restriction on magnitude of the effective coupling constants. Thus, in addition to Eq.(\[hier\]) we require: $$\begin{aligned}
\tilde g^\perp_{2 k_{\rm F}} \ll 2 \pi \tilde v_{\rm F}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{geff_small}%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ Since $\tilde g_{2k_{\rm F}}^\perp =
g_{2k_{\rm F}}^\perp \zeta^{{\cal K}_c - 1}$, inequality (\[geff\_small\]) is equivalent to: $$\begin{aligned}
\frac{ {\bar t}} {v_{\rm F} \Lambda}
\gg
\left(
\frac{
g^\perp_{2k_{\rm F}}
}
{
v_{\rm F}
}
\right)^{(1 - \theta)/(1 - {\cal K}_c )}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{lower_b}%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ This gives the lower bound on the transverse hopping. In Sect. \[accuracy\] we will explain how this inequality should be modified in order to improve the accuracy of our method.
Keeping the above considerations in mind, one writes the equation for the effective Hamiltonian: $$\begin{aligned}
H^{\rm eff} = H^{\rm 1D} + \tilde H^{\rm hop} + \tilde H^{\rho \rho},
%\tilde H^{SS},\end{aligned}$$ where the tildes above $\tilde H^{{\rm hop}}$ and $\tilde H^{\rho\rho}$ signify that the coupling constants of these terms are renormalized according to Eqs. (\[g24\_tilde\]), (\[g\_tilde\]), and (\[g\_2kF\_tilde\]). The variational energy $E^V$ and $\zeta$ are given by Eqs.(\[EV’\_exp\]) and (\[zeta\]). The relation Eq.(\[mf\_cond\]) holds true for Hamiltonian $H^{\rm eff}$ implying the applicability of the mean-field approximation.
Our variational derivation is equivalent to the tree-level renormalization group (RG) result. Namely, one can execute the following sequence of transformations. Beginning with the Hamiltonian $H$ one bosonizes it to obtain the Tomonaga-Luttinger Hamiltonians for individual chains perturbed by the transverse interactions, transverse hopping, and in-chain backscattering. Because of the presence of relevant (in RG sense) operators the RG flow takes the Hamiltonian away from the Tomonaga-Luttinger fixed point. The flow must be stopped when the renormalized transverse hopping amplitude becomes of the order of the cutoff \[see Eq.(\[mf\_cond\])\]. At this point the bosonic Hamiltonian has to be refermionized. The resultant quasiparticle Hamiltonian coincides with $H^{\rm eff}$. The relationship between the variational approach and the RG procedure is shown on Fig.\[diag\] in a form of a commutative diagram.
This completes our derivation of the effective quasiparticle Hamiltonian and we are prepared to analyze the phase diagram of our system.
Phase diagram {#phase}
=============
How the phase diagram of the Hamiltonian $H$, Eq.(\[H\]), can be determined? It is essential to realize that the phase diagram of $H$ coincides with the phase diagram of $H^{\rm eff}$. Consider, for example, the anomalous expectation value $\langle \psi^\dagger_{{\rm L} \uparrow i}
\psi^\dagger_{{\rm R} \uparrow j} \rangle$. For such a quantity the following is correct: $$\begin{aligned}
\langle
\psi^\dagger_{{\rm L} \uparrow i}
\psi^\dagger_{{\rm R} \uparrow j}
\rangle
=
\left(
\frac{\Lambda}{\tilde \Lambda}
\right)
\langle
\Psi^\dagger_{{\rm L} \uparrow i}
\Psi^\dagger_{{\rm R} \uparrow j}
\rangle
\langle
{\rm e}^{i\sqrt{2\pi} \varphi_{p \sigma }^>}
\rangle_>^2.
%\langle
% {\rm e}^{i\sqrt{2\pi} \varphi_{{\rm R} \uparrow j}^>}
%\rangle_>.
%\nonumber\end{aligned}$$ Since the bosonic expectation value is non-zero, both $\langle \psi^\dagger_{{\rm L} \uparrow i}
\psi^\dagger_{{\rm R} \uparrow j}
\rangle $ and $\langle \Psi^\dagger_{{\rm L} \uparrow i}
\Psi^\dagger_{{\rm R} \uparrow j}
\rangle $ are either simultaneously zero or simultaneously non-zero. Same is true for other types of broken symmetries. This proves that the phase diagram of $H$ and the phase diagram of $H^{\rm eff}$ are identical. Since the properties of $H^{\rm eff}$ are accessible through the mean-field approximation, we are fully equipped to explore the model’s phase diagram.
Density waves
-------------
First, we consider the density wave phases. Both SDW and CDW have the same susceptibilities but different effective coupling constants: $$\begin{aligned}
\tilde g_{\rm SDW} = \frac{g_2}{2},
\\
\tilde g_{\rm CDW} = \frac{g_2}{2} - g_{\rm bs} +
\frac{z^\perp \tilde g^\perp_{2k_{\rm F}}}{2}.\end{aligned}$$ Due to strong renormalization of $\tilde g^\perp_{2k_{\rm F}}$, inequality Eq. (\[dw\_coupling\]), which is always satisfied for bare coupling constants, is not necessary fulfilled when the effective constants are compared. Therefore, depending on the microscopic details, the density wave phase could be of either nature. To be specific, we will study SDW below. The discussion for CDW is completely the same.
SDW in Q1D metal was thoroughly analyzed at the mean-field level in Ref. [@sdw]. We will follow this reference.
As we know, the stability of SDW depends crucially on the nesting of the Fermi surface. Shape of the Fermi surface is determined by the effective transverse hopping amplitudes $\tilde t (i)$. If one assume that the only non-zero hopping amplitude is the nearest neighbor amplitude $\tilde t_1$, then the resultant Fermi surface nests perfectly. In order to describe the Fermi surface with non-ideal nesting it is necessary to include at least the next-to-nearest neighbor hopping amplitude $\tilde t_2$. For such structure of hopping the SDW susceptibility is equal to: $$\chi_{\rm SDW} \approx
\frac{1}{\pi v_{\rm F}}
\times
\cases{
\ln\left( 2v_{\rm F}\tilde\Lambda/T \right),
& if $T>\tilde t_2=\zeta^\theta t_2$,\cr
\ln\left( 2v_{\rm F}\tilde\Lambda/ \tilde t_2\right),
& if $T<\tilde t_2=\zeta^\theta t_2$.
}
\label{chi}$$ The SDW transition temperature is derived by equating $(g_2/2) \chi_{\rm SDW}$ and unity. For $\tilde t_2 = 0$ it is: $$T_{\rm SDW}^{(0)}
\propto
v_{\rm F}\tilde\Lambda
\exp\left(
-2\pi v_{\rm F}/ g_2
\right).
\label{T0}$$ If $\tilde t_{2} > 0$ the transition temperature $T_{\rm SDW}$ becomes smaller then $T_{\rm SDW}^{(0)}$. It vanishes when $\tilde t_{2} \propto T_{\rm SDW}^{(0)}$. That is, exponentially small $\tilde t_2$ is enough to destroy SDW.
Superconductivity
-----------------
The destruction of the density wave does not automatically implies that the ground state becomes superconducting. By analogy with Eq.(\[sc\_coupling\]) we can write for the effective Hamiltonian: $$\begin{aligned}
&&\sum_{ij} \tilde {\cal H}_{ij}^{\rho\rho}
=
%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sc_eff}%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
&&\quad
\sum_{ij}
2
\left(
\tilde g_0^\perp - \tilde g_{2k_{\rm F}}^\perp
\right)
\left[
\tilde \Delta_{ij}^{a}
(\tilde \Delta_{ij}^{a})^\dagger
+
\tilde \mathbf{d}_{ij}^{s} \cdot
(\tilde \mathbf{d}_{ij}^{s})^{\dagger}
\right]
\nonumber
\\
&&\qquad
+
2 (\tilde g_0^\perp + \tilde g_{2 k_{\rm F}}^\perp)
\left[
\tilde \Delta_{ij}^{s}
(\tilde \Delta_{ij}^s)^\dagger
+
\tilde \mathbf{d}_{ij}^a
\cdot
(\tilde \mathbf{d}_{ij}^a)^{\dagger}
\right]
+ \ldots, \nonumber \end{aligned}$$ where order parameters $\tilde \Delta_{ij}^{s/a}$ and ${\bf d}_{ij}^{s/a}$ are defined by Eqs.(\[sc\_matrix\]), (\[sc\_order\]), and (\[sc\_symm\]), in which bare fermionic fields $\psi$ and $\psi^\dagger$ are replaced by the quasiparticle fields $\Psi$ and $\Psi^\dagger$.
From Eq.(\[sc\_eff\]) we see that the effective superconducting coupling constant $\tilde g_{\rm sc}$ is equal to: $$\begin{aligned}
\tilde g_{\rm sc}
=
2( \tilde g_{2k_{\rm F}}^\perp - \tilde g_0^\perp ).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{g_sc}%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ We conclude that the superconductivity is stable if: $$\begin{aligned}
\tilde g_{2k_{\rm F}}^\perp > \tilde g_0^\perp = g_0^\perp.
%%%%%%%%%%%%%%%%%%%%%%%%
\label{tilde_g}%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ At the same time one has to remember that for the [*bare*]{} coupling constants the inequality $g_{2k_{\rm F}}^\perp < g_0^\perp$ holds true \[see Eq.(\[bare\_g\])\]. Can both inequalities be satisfied at the same time? It is possible provided that the system is sufficiently anisotropic. Indeed, the inequalities (\[tilde\_g\]) and (\[bare\_g\]) are equivalent to: $$\begin{aligned}
\frac{8 {\bar t}^2} {v_c v_{\rm F} \Lambda^2}
<
\left(
\frac{g^\perp_{2k_{\rm F}}} {g_0^\perp}
\right)^{(2-2\theta)/(1 - {\cal K}_c )}
< 1.\end{aligned}$$ This is the necessary condition for the superconducting ground state. Similar condition was derived in [@rozhkov] for the spinless electrons. This inequality gives an upper bound on $t$. This bound will be discussed in Sect. \[accuracy\] in connection with the method’s dependability.
The final question is the type of the superconducting order realized in our system. As one can see from Eq.(\[sc\_eff\]) there are two candidates: singlet order parameter $\Delta_{ij}^a$ ($d_{xy}$-wave according to the accepted naming scheme [@review_RG1]) and triplet $\mathbf{d}^s_{ij}$ ($f$-wave). Both have the same coupling constant of $\tilde g_{\rm sc}$. This degeneracy cannot be lifted within our approach for we must include subtler effects into our consideration. We will argue below (Sect. \[symmetry\_subsection\]) that the answer is sensitive to microscopic details of the system. Therefore, in real materials either type of the superconductivity can be, in principle, realized.
Global phase diagram
--------------------
In this subsection we construct the global phase diagram of the system on the pressure-temperature plane.
The effect of the pressure on our Hamiltonian is twofold. First, it increases the next-to-nearest neighbor hopping amplitude $t_2$. Thus, the growth of the pressure spoils the nesting of the Fermi surface.
Second, it makes the system less anisotropic. This, in turn, leads to the reduction of the 1D renormalization of $\tilde g^\perp_{2k_{\rm F}}$ under increasing pressure. Therefore, one can say that $\tilde g^\perp_{2k_{\rm F}}$ is decreasing functions of pressure.
Consequently, at low pressure the nesting is good and the ground state is the density wave phase with the highest transition temperature possible: $T_{\rm SDW/CDW} = T^{(0)}_{\rm SDW/CDW}$. Under growing pressure the nesting property of the Fermi surface deteriorates and $T_{\rm SDW/CDW}$ becomes smaller.
The density wave transition temperature decays until some critical pressure $p_c$ at which it quickly goes to zero. At $p > p_c$ the subleading order, the superconductivity, is stabilized. The characteristic superconducting critical temperature is smaller than $T_{\rm SDW/CDW}^{(0)}$ for the density wave coupling constant is higher than that of the superconductivity. This is so because the density wave order benefits from the in-chain interaction $g_2 \rho_{\rm L} \rho_{\rm R} $ while the superconductivity cannot do this.
The superconducting order parameter is either triplet ($f$-wave) or singlet ($d_{xy}$-wave). The superconducting gap vanishes at four nodal lines on the Fermi surface.
Under even higher pressure $T_c \rightarrow 0$ for the system becomes less anisotropic and inequality (\[tilde\_g\]) becomes invalid.
The schematic diagram is shown on Fig.\[fig1\].
Discussion {#disc}
==========
This section is divided into four subsections. In subsection A we discuss the accuracy of our method.
In subsection B we speculate under what condition $d_{x^2 - y^2}$-wave superconductivity may be stabilized.
In subsection C we compare our approach with other theoretical methods available in the literature.
In subsection D our theoretical results are compared against published experimental data.
In subsection E we give our conclusions.
Accuracy of the variational approach {#accuracy}
------------------------------------
In general, variational approach is an uncontrollable approximation, and one may doubt our conclusions. Fortunately, the presented variational scheme is only a front for the tree-level RG transformation (see Fig. \[diag\]). Using RG notions, it is possible to prove rigorously that the superconductivity is stable at least in a certain parameter range. Since the stability of the superconducting phase depends on effective inter-chain interactions, we must show that the tree-level RG is enough to capture them adequately.
First, we must establish the structure of the tree-level RG flow. As implied by Eq.(\[hier\]), our model is near the Tomonaga-Luttinger fixed point ($g_{\rm bs} = g^\perp_{0,2 k_{\rm F}} = 0$, $t=0$). The fixed point Hamiltonian is perturbed by two relevant operators, $t$ and $g^\perp_{2 k_{\rm F}} $, and one marginal, $g_{\rm bs}$. We assume that the transverse hopping is the most relevant operator: at the dimensional crossover ($\tilde t \sim v_{\rm F} \tilde \Lambda$) inequality (\[geff\_small\]) is satisfied.
This tree-level picture disregards several corrections. The most well-known is $(-g_{\rm bs}^2)$ contribution to the RG equation for $g_{\rm bs}$. This term is of little immediate interest to us since it does not affect the inter-chain interactions.
We identify three terms, which amend the RG equations for inter-chain interaction constants. First, the backscattering contributes to the anomalous dimension of $g^\perp_{2 k_{\rm F}}$: the term proportional to $g_{\rm bs} g^\perp_{2 k_{\rm F}}$ enters the RG equation for $g^\perp_{2 k_{\rm F}}$. This correction may be neglected since the anomalous dimension is proportional to $g_2$, which is much larger than $g_{\rm bs}$ \[see (\[hier\])\].
Second, the in-chain interactions combined with the transverse hopping contribute a term of order $(g_{2,4}/v_{\rm F})^2 (t/v_{\rm F} \Lambda)^2$. This term corrects inter-chain couplings by the amount: $$\begin{aligned}
(\Delta g)_1 \sim
\int_0^{\ell^*} d\ell
\frac{
[g_{2,4} t(\ell)]^2
}
{
v_{\rm F}^3 \Lambda^2(\ell)
}
\sim
\frac{
(g_{2,4})^2
}
{
v_{\rm F}
},
\\
t (\ell) = t {\rm e}^{- \theta \ell},\end{aligned}$$ where $\ell$ denotes the scaling variable: $\Lambda(\ell) = \Lambda {\rm e}^{-\ell}$. The dimensional crossover occurs, and our our RG stops when $\ell$ reaches the value $\ell^* = \ln (\Lambda / \tilde \Lambda)$. At the crossover it is true: $t (\ell^*) / [ v_{\rm F} (\ell^*) \Lambda (\ell^*) ]
=
\tilde t / [v_{\rm F} \tilde \Lambda ] \sim 1$.
Third, the inter-chain interactions may contribute to the scaling equations additional terms of order $(g^\perp_{2 k_{\rm F}})^2$. Such term corrects inter-chain coupling constants by the amount: $$\begin{aligned}
(\Delta g)_2 \sim
\int_0^{\ell^*} d\ell
\frac{
[g^\perp_{2 k_{\rm F}} (\ell)]^2
}
{
v_{\rm F}
}
\sim
\frac{
(\tilde g^\perp_{2 k_{\rm F}})^2
}
{
g_{2}
},
\\
g^\perp_{2 k_{\rm F}} (\ell)
=
g^\perp_{2 k_{\rm F}} {\rm e}^{ - ( 1 - {\cal K}_c )\ell },
\quad
1 - {\cal K}_c \sim g_2/v_{\rm F}.\end{aligned}$$ Thus, the corrections to $\tilde g_{\rm sc}$ beyond the tree-level may be disregarded if $\tilde g^\perp_{2 k_{\rm F}}$ is much bigger than $(\Delta g)_{1,2}$. This condition is equivalent to: $$\begin{aligned}
(g_{2,4})^2 / v_{\rm F}
\ll
\tilde g^\perp_{2 k_{\rm F}}
\ll
g_{2}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{tilde_g2}%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ We already derived inequalities binding $\tilde g^\perp_{2 k_{\rm F}}$ \[see Eq.(\[geff\_small\]) and Eq.(\[tilde\_g\])\]. Since $v_{\rm F}$ is bigger than $g_2$, Eq.(\[geff\_small\]) gives less restrictive upper bound on $\tilde g^\perp_{2 k_{\rm F}}$ than Eq.(\[tilde\_g2\]). Therefore, if we want an assurance that our method does not lead us astray, we must abolish Eq.(\[geff\_small\]) and use Eq.(\[tilde\_g2\]) instead.
The situation with Eq.(\[tilde\_g\]) is somewhat more complicated: it is impossible to know, which quantity, $g_0^\perp$ or $(g_{2,4})^2 / v_{\rm F}$, is smaller. Thus, we define: $$\begin{aligned}
g_{\rm max} = \max \{
(g_{2,4})^2 / v_{\rm F},
g_0^\perp
\},
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{g_max}%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ and rewrite Eq.(\[tilde\_g2\]) in the form: $$\begin{aligned}
g_{\rm max} \ll \tilde g^\perp_{2 k_{\rm F}} \ll g_2.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{tilde_g3}%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ This inequality is self-consistent in the sense that $g_{\rm max} \ll g_2$ \[see Eq.(\[hier\])\]. It is convenient to cast Eq. (\[tilde\_g3\]) and Eq.(\[small\_t\]) as a constraint on the bare hopping amplitude: $$\begin{aligned}
\left(
\frac{
g^\perp_{2 k_{\rm F}}
}
{
g_2
}
\right)^\frac{1 - \theta}{1 - {\cal K}_c}
\ll
\frac{ t }{ v_{\rm F} \Lambda }
\ll
\left(
\frac{
g^\perp_{2 k_{\rm F}}
}
{
g_{\rm max}
}
\right)^\frac{1 - \theta}{1 - {\cal K}_c} \ll 1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{t<>}%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ If this inequality is satisfied, then the model’s phase diagram has a superconducting phase, and the superconductivity is not an artifact of the variational method. It is likely that some deviations from the constraints imposed by Eqs.(\[hier\]) and (\[t<>\]) are not deadly for superconductivity. Yet, they may affect the order parameter symmetry. This issue is discussed in the next subsection.
Symmetry of the superconducting order parameter {#symmetry_subsection}
-----------------------------------------------
We have seen that the symmetry of the order parameter cannot be unambiguously determined within the framework of our approximation: as Eq.(\[sc\_eff\]) suggests, both $f$-wave and $d_{xy}$-wave states have similar energies. Our method capture only gross features of the model, it is not delicate enough to calculate the superconducting coupling constant with higher accuracy. We can identify at least two mechanisms, which could lift the order parameter degeneracy. They work in opposite direction. Thus, the final outcome depends crucially on the minutiae of the microscopic model.
The mechanism promoting $f$-wave increases the coupling constant for this order parameter and decreases the $d_{xy}$-wave coupling constant. It operates in the following manner. The RG flow applied to our system generates a new spin-dependent transverse interaction: $$\begin{aligned}
\tilde {\cal H}_{ij}^{SS}
=
%J_0^\perp (i-j)
%{\bf S}_{ i} \cdot {\bf S}_{ j}
%+
%\\
\tilde J_{2k_{\rm F}}^\perp (i-j)
\left(
\tilde {\bf S}_{2k_{\rm F} i}
\cdot
\tilde {\bf S}_{-2k_{\rm F} j}
+
\text{H.c.}
\right).\end{aligned}$$ At the dimensional crossover ($\tilde t \sim \tilde v_{\rm F} \tilde \Lambda$) one has: $\tilde J_{{2 k_{\rm F}}} \sim g_{2,4}^2 / v_{\rm F}$. This estimate can be found in, e.g., Ref. [@bour_caron] \[see first row, second column of Table I where $t'_\perp \sim E_0 (l)$\]. The new term can be cast as: $$\begin{aligned}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sc_couplingSS}%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sum_{ij} \tilde {\cal H}_{ij}^{SS}
=
\sum_{ij}
- 2\tilde J_{2k_{\rm F}}^\perp
\left[
\tilde \mathbf{d}_{ij}^s
\cdot
(\tilde \mathbf{d}_{ij}^s)^{\dagger}
-3
\tilde \Delta_{ij}^{a}
(\tilde \Delta_{ij}^{a})^\dagger
\right]
\\
+
2 \tilde J_{2k_{\rm F}}^\perp
\left[
\tilde \mathbf{d}_{ij}^a
\cdot
(\tilde \mathbf{d}_{ij}^a)^{\dagger}
-3
\tilde \Delta_{ij}^{s}
(\tilde \Delta_{ij}^{s})^\dagger
\right].
\nonumber \end{aligned}$$ Thus, the $f$-wave coupling constant grows by $2 \tilde J^\perp_{2k_{\rm F}}$, and the $d_{xy}$-wave coupling constant decreases by $6 \tilde J^\perp_{2k_{\rm F}}$.
A factor in favor of the $d_{xy}$-wave superconductivity is the susceptibility. One can calculate two susceptibilities, $\chi^{\rm sc}_f$ and $\chi^{\rm sc}_d$, for two order parameters: $$\begin{aligned}
\chi^{\rm sc}_{f,d}
=
\frac{1}{2 \pi v_{\rm F}}
\ln \left(
\frac{\tilde v_{\rm F} \tilde \Lambda}{T}
\right)
+
C_{f,d},\end{aligned}$$ where $C_{f,d}$ are the non-universal constants. In other words, the divergent parts of both susceptibilities are identical, but the non-singular parts depend on the order parameter symmetry and the band structure. This happens because our two orders have different orbital structure ($f$-wave is symmetric with respect to inversion of the transverse coordinate, while $d_{xy}$-wave is antisymmetric).
Within the framework of our model (linear dispersion along the $x$-axis, square lattice, small $\tilde t_2$) we have $C_f < C_d$. Thus, the susceptibility of $d$-wave is higher.
The above analysis demonstrates that the symmetry of the order parameter is a non-universal property very sensitive to the microscopic details.
It is reasonable to ask if one can stabilize either of the remaining superconducting orders, ${\bf d}^a$ or $\Delta^s$, by modification of the model’s Hamiltonian. We can speculate that this might be possible provided that the spin-spin interaction is enhanced. Indeed, by examining Eq. (\[sc\_eff\]) and Eq. (\[sc\_couplingSS\]) one concludes that $\Delta^s$ ($d_{x^2-y^2}$-wave) could be non-zero if: $$\begin{aligned}
3 \tilde J^\perp_{{2 k_{\rm F}}} > \tilde g^\perp_{{2 k_{\rm F}}}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{J>g}%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ Such situation may be realized in a system with sufficiently large $g_{\rm bs}$ (to suppress CDW fluctuations) and sufficiently small bare values of $g_{2k_{\rm F}}^\perp$.
As for ${\bf d}^a$, it is always zero: the constants in front of ${\bf d}^a$ are strictly positive in both Eq.(\[sc\_eff\]) and Eq.(\[sc\_couplingSS\]).
Thus, we demonstrate that the Q1D metal allows for a broad class of superconducting orders. The choice between these orders depends on both the band structure and the interaction constants.
Other theoretical approaches
----------------------------
The root of the superconductivity in the real-life Q1D materials remains an unresolved issue. It is often suggested that the superconductivity in these compounds is not of phonon but rather of electron origin. There have been many attempts to construct a mechanism in line with this suggestion.
The theoretical literature on the subject can be split into two groups according to tools used. The studies employing the random phase approximation (RPA) or the fluctuation exchange approximation (FLEX) [@tanaka; @kuroki; @flex; @review_RPA] constitute the first group. The second group is made of the papers where RG [@dup; @nickel; @nickelII; @review_RG1; @review_RG2] is employed.
We have mentioned that our method is closely related to the RG transformation. Clearly, it will be interesting to compare our conclusions with the conclusions of other researchers who use similar strategies.
In Ref.[@dup; @nickel; @nickelII; @review_RG1; @review_RG2] the zero-temperature phase diagram of the Q1D metal was mapped with the help of a numerical implementation of the one-loop RG flow. The authors of the latter papers were found that, if the bare transverse interactions are zero or extremely small, the system undergoes a transition from the SDW phase to the superconducting phase with the order parameter $\Delta^s_{ij}$ ($d_{x^2-y^2}$-wave).
Furthermore, it was determined that, if the bare constants $g^\perp_{2k_{\rm F}}$ are sufficiently big, the transition is from the CDW phase into the superconducting phase with the $f$-wave order parameter $\mathbf{d}^s_{ij}$.
The results of these papers can be understood within the framework of our approach. In the limit where the only non-zero inter-chain term is the transverse hopping ($t \ne 0$, [@dup]), the RG flow generates both $\tilde g^\perp_{2 k_{\rm F}}$ and $\tilde J^\perp_{2 k_{\rm F}}$. These constants satisfy the relation Eq.(\[J>g\]). The mechanism behind this is described in the previous subsection.
As we pointed out, when Eq.(\[J>g\]) is valid, the most stable order parameter is $\Delta^s_{ij}$ ($d_{x^2-y^2}$-wave). Thus, our conclusions agrees with findings of Ref.[@dup].
The limit studied in [@nickel; @nickelII] is not compatible with our Eq.(\[hier\]). In the latter reference it was assumed that the in-chain backscattering is of the order of the in-chain forward scattering. Thus, we cannot apply our approach straightforwardly, but certain qualitative conclusions may be reached.
When bare $g^\perp_{2k_{\rm F}}$ is large, the effective coupling $\tilde J^\perp_{2k_{\rm F}}$ is small, and the effective coupling $\tilde g^\perp_{2k_{\rm F}}$ is large. The ground state of the system with good nesting is CDW. The destruction of the CDW phase takes place when the nesting becomes sufficiently poor. Once the CDW is gone, we find ourselves in a familiar situation where the stable superconducting order parameter is either $\mathbf{d}^s$ ($f$-wave) or $\Delta^a$ ($d_{xy}$-wave), consistent with $f$-wave found in [@nickel; @nickelII].
If we lower $g^\perp_{2k_{\rm F}}$ sufficiently, the stability of SDW state may be restored [@nickel; @nickelII]. The in-chain backscattering suppresses $\tilde g_{2 k_{\rm F}}^\perp$ and promotes $\tilde J^\perp_{2 k_{\rm F}}$, ultimately leading to inequality (\[J>g\]). In such a regime the most stable order parameter is $\Delta^s_{ij}$ ($d_{x^2-y^2}$-wave), which agrees with [@nickel; @nickelII].
The above argumentation lends additional support to the notion that the mechanism proposed in this paper is not an artifact of the variational approximation. It is also a convenient feature of our method that it is analytical and the results of other approaches can be understood within its framework.
Besides RG several authors use RPA or FLEX to determine the superconducting properties in the anisotropic Fermi systems [@tanaka; @kuroki; @flex; @review_RPA]. These approximations resemble the classical BCS scheme in which the phonons are replaced by boson-like excitations of some other kind. In the quoted papers the excitations mediating the attractive interaction between the electrons are spin-density and charge-density fluctuations.
The frameworks laid out by the RPA and FLEX schemes are very appealing and intuitive. They both predict that under certain condition the Q1D metal is an unconventional superconductor. There is, however, a weak point: both methods are unable to account for the peculiarities specific for 1D electron liquid. Such weakness artificially narrows the region of the parameter space where the superconductivity is stable.
Finally, the author recently developed a canonical transformation approach for 1D electron systems [@rozhkovII; @rozhkovIII; @rozhkovIV]. This method may be viewed as a generalization of the one discussed in this paper. The application of the canonical transformation method to the Q1D systems is in progress.
Experiment vs. theory
---------------------
The question remains if the model and the mechanism discussed above are of relevance to the Q1D superconductors, such as TMTSF and TMTTF [@review_RG1]. Of cause, the latter compounds have very complicated crystallographic structure: orthorhombic lattice, possibility of anion ordering, dimerization [@book]. Yet, one can hope that these difficulties are not of paramount importance as far as the superconducting mechanism is concerned.
If this hope is justified should be assessed by the mechanism’s ability to reproduce main features of the experimental data, at least qualitatively. We can look at the presented model with a good degree of optimism for it captures two most salient properties of the superconductivity in TMTSF/TMTTF.
The first of these two features is the common boundary shared by the superconducting and the SDW phases on the pressure-temperature phase diagram: the diagram of Fig.\[fig1\] is similar to the high-pressure part of the ‘universal’ phase diagram of the TMTSF/TMTTF compounds [@universal].
The second is the non-trivial orbital structure of the order parameter in the Q1D superconductors. There are numerous pieces of evidence in favor of the order parameter with zeros on the Fermi surface [@NMR; @NMR2; @imp; @field; @field2]. (However, there is a thermal transport measurement [@no_nodes] which contradicts to this picture.) The order parameters $\mathbf{d}^s_{ij}$ and $\Delta^{s,a}_{ij}$ are of this kind. Therefore, the predictions of our model is in qualitative agreement with the experiment.
Conclusions
-----------
We proposed the superconducting mechanism for the strongly anisotropic electron model without attractive interaction. We have shown that there is a region in the parameter space where the superconductivity is stable and shares a common boundary with SDW. The model supports two types of unconventional superconducting order parameter. Our mechanism may be relevant for the organic superconductors.
Acknowledgements
================
The author is grateful for the support provided by the Dynasty Foundation and by the RFBR grants No. 06-02-16691 and 06-02-91200.
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\[!t\]
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|
[**Comment on “Do Intradot Electron-Electron Interactions Induce Dephasing?”**]{}\
\
In a recent Letter, Jiang, Sun, Xie and Wang [@jiang] study transport through an interacting quantum dot embedded in one arm of an Aharonov-Bohm interferometer. Based on a theoretical analysis of the Aharonov-Bohm oscillation amplitude, Jiang [*et al.*]{} claim, contrary to earlier work by two of us [@kg], that at finite temperature the intradot interaction will [*not*]{} lead to any dephasing. Likewise, they claim that the theoretically predicted [@kg] and experimentally verified [@kobayashi] asymmetry of the Aharonov-Bohm oscillation amplitude is [*not*]{} associated with dephasing. In this Comment, we point out severe inconsistencies in the analysis of Ref. , and show that their conclusions are ill-founded.
Our main point is that the authors of Ref. employ an approximation scheme for the Green’s functions, which, [*by construction*]{}, is inappropriate to describe spin-flip-related dephasing [@kg] or any other inelastic process that is caused by intradot interaction. To evaluate the Green’s function of the quantum dot they consider the uncoupled dot, and then assign by hand constant widths to the bare many-body dot levels, as described in the paragraph below Eq. (3) of Ref. [@jiang]. This procedure results in the symmetrized Hartree approximation (a low-hierarchy variant of the equation-of-motion scheme). The latter is an effective [*single-particle approximation*]{} that oversimplifies the role of interaction. The failure of this scheme is most strikingly demonstrated in the first paragraph of the right column on page 3 of Ref. , where the authors’ “proof” of the nonexistence of dephasing relies on the relation $T_d = \Gamma_{11}^s \Gamma_{44}^d |\tilde G_{41}^r|^2$ for the transmission probability through the arm containing the quantum dot. This is equivalent to the relation $T=|t|^2$, valid for [*non-interacting*]{} systems, where $t$ is the transmission amplitude (proportional to the single-particle Green’s function involving one source- and one drain-electron operator). For [*interacting*]{} systems with inelastic channels, however, this relation is no longer valid, as has been shown in the literature [@mw; @kg].
To illustrate the importance of this point for the present context, we explicitly write down the cotunneling transmission of an electron at the Fermi energy through a single-level quantum dot with level energy $\epsilon$, tuned away from resonance, and charging energy $U\rightarrow \infty$. The correct employment of the many-body formalism [@mw; @kg] yields $$\label{general}
T_{\rm dot} = -\frac{2\Gamma_{\rm L}\Gamma_{\rm R}}
{\Gamma} {\rm Im} \, G_{\rm dot}^{\rm ret} \, ,$$ as opposed to the result $$\label{non}
T_{\rm dot}^{U=0} = \Gamma_{\rm L}\Gamma_{\rm R}
|G_{\rm dot}^{\rm ret}|^2 \, ,$$ that is valid for [*non-interacting*]{} electrons only. Here, $\Gamma_{\rm L}$ and $\Gamma_{\rm R}$ measure the coupling strengths between dot and the left and right lead, respectively, and $\Gamma=\Gamma_{\rm L}+\Gamma_{\rm R}$ is the broadening of the dot level. The [*correct*]{} cotunneling transmission [@kg] is thus obtained from Eq. (\[general\]) as $$\label{correct}
T_{\rm dot}^{\rm correct} =
\frac{\Gamma_{\rm L} \Gamma_{\rm R}}{\epsilon^2} \, ,$$ while within the Hartree approximation, Eq. (\[general\]) and (\[non\]) are equivalent and lead to the [*wrong*]{} result $$\label{wrong}
T_{\rm dot}^{\rm Hartree} = \frac{1}{[1+f(\epsilon)]^2} \cdot
\frac{\Gamma_{\rm L} \Gamma_{\rm R}} {\epsilon^2} \, ,$$ in order $\Gamma^2$. Equations (\[correct\]) and (\[wrong\]) differ by a factor $1/[1+f(\epsilon)]^2$, where $f(\epsilon)$ is the Fermi function. Repeating the above-mentioned discussion of the “decoherence rate” $r_T$ on page 3 of Ref. but with the correct expression for $T_{\rm dot}$ immediately leads to $r_T \rightarrow 1$ for $\epsilon > 0$ but $r_T \rightarrow 1/2$ for $\epsilon < 0$, indicating interaction-induced dephasing (and consequently asymmetry) for both a two-terminal and an open geometry. This is in accordance with Ref. . Qualitatively similar results hold at resonance.
There are several other inconsistencies in the paper. We give here some examples: (1) While $r_T$ and $r_G$ are claimed to be close to each other at low $k_B T$, $r_G$ clearly exceeds 1 near resonance (cf. Fig. 2 of Ref. ), invalidating it as a good measure of coherence or “visibility”. (2) Equation (4) of Ref. is wrong, as it relies on the single-particle formalism. (3) The quantity $\Delta G(\phi)$ should, by construction, vanish for $\phi=0$, which is clearly not the case in Fig. 2b of Ref. .
This work was supported by the Deutsche Forschungsgemeinschaft through SFB 491 and GRK 726, the US-Israel BSF, the ISF of the Israel Academy of Science, the Alexander von Humboldt foundation, the EC HPRN-CT-2002-00302-RTN, and the NSF grant DMR 0210575.
Jürgen König$^1$, Yuval Gefen$^2$, and Alessandro Silva$^{3}$
$^1$Institut für Theoretische Physik III\
Ruhr-Universität Bochum\
44780 Bochum, Germany
$^2$Department of Condensed Matter Physics\
The Weizmann Institute of Science\
76100 Rehovot, Israel
$^3$Center for Materials Theory\
Department of Physics and Astronomy\
Rutgers University, Piscataway, NJ 08854, USA
[PACS numbers: 73.23.Hk, 73.63.Kv, 73.40.Gk]{}
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|
---
abstract: 'We present a quantum dialogue protocol by using the Greenberger-Horne-Zeilinger (GHZ) state. In this paper, we point out that the ‘quantum dialogue’ communication scheme recently introduced by Nguyen can be eavesdropped on under an intercept-and-resend attack. We also give a revised control mode to detect this attack. Hence, within the present version two users can exchange their secret messages securely and simultaneously, and the efficiency of information transmission can be successfully increased.'
author:
- Yan
- 'Chang-Bao'
- 'Shou [^1]'
- 'Suc-Kyoung'
- 'Kyu-Hwang'
- 'Chung-In'
title: Quantum Dialogue by Using the GHZ State
---
Since the pioneering work of Bennett and Brassard published in 1984 [@BB84], different quantum key distribution protocols have been presented [@ABCHPRA02; @SJGPRA02; @XLGPRA02; @SPRA04; @WPRL04]. Different from the key distribution protocol, some quantum direct secure communication (QDSC) protocols have been shown recently [@BFPRL02; @DLLPRA03; @MZLCPL0518; @XFZJKPS05], which permit important messages to be communicated directly without first establishing a random key to encrypt them. However, these protocols [@BB84; @ABCHPRA02; @SJGPRA02; @XLGPRA02; @SPRA04; @WPRL04; @BFPRL02; @DLLPRA03; @MZLCPL0518; @XFZJKPS05] only permit messages to be transmitted from the sender (Alice) to the receiver (Bob). Two parties cannot simultaneously transmit their different secret messages to each other in only one quantum channel.
Very recently, Nguyen [@NBA04PLA] proposed an entanglement-based protocol, the so-called quantum dialogue, which would allow two people to exchange their messages simultaneously based on the well-known Bostroem-Felbinger [@BFPRL02] protocol. Man [*et al.*]{} [@MZLCPL05] proposed a modified protocol about the quantum dialogue. It is worth mentioning that the Bostroem-Felbinger protocol can be eavesdropped in some specific cases [@W03PRL; @ZMLPLA]. In this paper, we show that the quantum dialogue scheme [@NBA04PLA] is insecure under some eavesdropping attacks, and we present a new protocol to realize a quantum dialogue by using the GHZ state more securely. Thus, the two users can exchange their messages securely and simultaneously. The efficiency of information transmission is also successfully increased.
Now, let us review Nguyen’s quantum dialogue scheme. Suppose there are two users (say, Alice and Bob) and they want to transmit their messages to each other simultaneously. Bob first produces a large enough number of Einstein-Podolsky-Rosen (EPR) pairs, all in the state $$\label{e1}
|\Psi_{0,0}\rangle_{ht}=\frac{1}{\sqrt{2}}(|\uparrow\rangle_h|\downarrow\rangle_t+|\downarrow\rangle_h|\uparrow\rangle_t),$$ where [*h*]{} stands for “ [*home*]{}, ” [*t*]{} stands for “ [*travel*]{}, ” and $|\downarrow\rangle$ and $|\uparrow\rangle$ characterize two degrees of freedom of a qubit. Bob encodes his bits $(k_n, l_n)$ $(k_n, l_n\in\{0,1\})$ by applying an operation $C_{k_n,l_n}$ on the state $|\Psi_{0,0}\rangle_{ht}$, where $C_{0,0}$, $C_{0,1}$, $C_{1,0}$, and $C_{1,1}$ denote the Pauli matrices $I$, $\sigma_x$, $\sigma_y$, and $\sigma_z$, respectively. He keeps one qubit (home qubit) with him and sends another (travel qubit) to Alice. Then, Bob lets Alice know that he has sent a qubit. Alice tells Bob that she has received a qubit. Alice encodes her bits $(i_n,j_n)$ $(i_n, j_n \in\{0,1\})$ by performing an operation $C_{i_n,j_n}$ on the travel qubit; then, she sends it back to Bob. When Bob receives the encoded travel qubit, he performs a Bell basis measurement on the qubit pair and waits for Alice to tell him that it was a run in a message mode (MM) or in a control mode (CM). In a MM run, Bob decodes Alice’s bits and announces his Bell basis measurement result $(x_n,y_n)$ to let Alice decode his bits. In a CM run, Alice reveals her encoding value to Bob to check the security of their dialogue.
However, this security checking cannot detect Eve’s (eavesdropper’s) intercept-and-resend attack. Let us suppose Eve gets the qubit [*t*]{} and keeps it with her. Afterwards, she creates her own entangled pair in the same state as in Eq. (\[e1\]), i.e., Eve’s pair state is $$\label{e2}
|\Psi_{0,0}\rangle_{HT}=\frac{1}{\sqrt{2}}(|\uparrow\rangle_H|\downarrow\rangle_T+|\downarrow\rangle_H|\uparrow\rangle_T),$$ and sends her qubit [*T*]{} to Alice. Alice, taking [*T*]{} for [*t*]{}, encodes her bits by performing an appropriate transformation as described above and sends it back to Bob. Then, Eve intercepts the qubit [*T*]{} again and carries out a Bell basis measurement on the [*[HT]{}*]{}- pair to learn Alice’s secret bits. By the same Bell basis measurement, Eve knows the encoding transformation Alice performed on the qubit [*T*]{}. Eve then applies the same transformation on the qubit [*t*]{} she has kept and sends it back to Bob. After Bob announces publicly his Bell basis measurement result, Eve can deduce Bob’s bits. Clearly, Eve can eavesdrop completely on the contents of Bob and Alice’s dialogue. Even worse, Eve’s tampering is absolutely unnoticeable.
Now, we propose our new protocol to realize quantum dialogue by using the GHZ state in terms of the original scheme. However in our protocol the intercept-and-resend attack, as well as the other kinds of attacks presented by Nguyen [@NBA04PLA], can be detected. First, we write the eight GHZ state bases in two different bases as follows: $$\begin{aligned}
\label{e3}
&|\psi_{0,0;0,0}\rangle&=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)_{htp}\cr\cr&&=\frac{1}{2}[|+\rangle_h(|+\rangle_t|+\rangle_p+
|-\rangle_t|-\rangle_p)+|-\rangle_h(|+\rangle_t|-\rangle_p+|-\rangle_t|+\rangle_p)],\end{aligned}$$ $$\begin{aligned}
\label{e4}
&|\psi_{1,1;0,0}\rangle&=\frac{1}{\sqrt{2}}(|000\rangle-|111\rangle)_{htp}\cr\cr&&=\frac{1}{2}[|+\rangle_h(|-\rangle_t|+\rangle_p+
|+\rangle_t|-\rangle_p)+|-\rangle_h(|+\rangle_t|+\rangle_p+|-\rangle_t|-\rangle_p)],\end{aligned}$$ $$\begin{aligned}
\label{e5}
&|\psi_{0,1;0,0}\rangle&=\frac{1}{\sqrt{2}}(|100\rangle+|011\rangle)_{htp}\cr\cr&&=\frac{1}{2}[|+\rangle_h(|+\rangle_t|+\rangle_p+
|-\rangle_t|-\rangle_p)-|-\rangle_h(|+\rangle_t|-\rangle_p+|-\rangle_t|+\rangle_p)],\end{aligned}$$ $$\begin{aligned}
\label{e6}
&|\psi_{1,0;0,0}\rangle&=\frac{1}{\sqrt{2}}(|100\rangle-|011\rangle)_{htp}\cr\cr&&=\frac{1}{2}[|+\rangle_h(|+\rangle_t|-\rangle_p+
|-\rangle_t|+\rangle_p)-|-\rangle_h(|+\rangle_t|+\rangle_p+|-\rangle_t|-\rangle_p)],\end{aligned}$$ $$\begin{aligned}
\label{e7}
&|\psi_{0,0;0,1}\rangle&=\frac{1}{\sqrt{2}}(|010\rangle+|101\rangle)_{htp}\cr\cr&&=\frac{1}{2}[|+\rangle_h(|+\rangle_t|+\rangle_p-
|-\rangle_t|-\rangle_p)+|-\rangle_h(|+\rangle_t|-\rangle_p-|-\rangle_t|+\rangle_p)],\end{aligned}$$ $$\begin{aligned}
\label{e8}
&|\psi_{1,1;0,1}\rangle&=\frac{1}{\sqrt{2}}(|010\rangle-|101\rangle)_{htp}\cr\cr&&=\frac{1}{2}[|+\rangle_h(|+\rangle_t|-\rangle_p-
|-\rangle_t|+\rangle_p)+|-\rangle_h(|+\rangle_t|+\rangle_p-|-\rangle_t|-\rangle_p)],\end{aligned}$$ $$\begin{aligned}
\label{e9}
&|\psi_{0,1;0,1}\rangle&=\frac{1}{\sqrt{2}}(|110\rangle+|001\rangle)_{htp}\cr\cr&&=\frac{1}{2}[|+\rangle_h(|+\rangle_t|+\rangle_p-
|-\rangle_t|-\rangle_p)+|-\rangle_h(|-\rangle_t|+\rangle_p-|+\rangle_t|-\rangle_p)],\end{aligned}$$ $$\begin{aligned}
\label{e10}
&|\psi_{1,0;0,1}\rangle&=\frac{1}{\sqrt{2}}(|110\rangle-|001\rangle)_{htp}\cr\cr&&=\frac{1}{2}[|+\rangle_h(|+\rangle_t|-\rangle_p-
|-\rangle_t|+\rangle_p)+|-\rangle_h(|-\rangle_t|-\rangle_p-|+\rangle_t|+\rangle_p)],\end{aligned}$$ where $$\label{e11}
|+\rangle=\frac{1}{\sqrt{2}}\ (|0\rangle+|1\rangle),$$ $$\label{e12}
|-\rangle=\frac{1}{\sqrt{2}}\ (|0\rangle-|1\rangle),$$ [*h*]{} stands for “ [*home*]{}, ” [*t*]{} stands for “ [*travel*]{}, ” and [*p*]{} stands for “ [*post*]{} ”.
Suppose that Alice has a secret message consisting of $4N$ bits; i.e., Alice’s message $=\{(i_1, j_1, f_1, g_1), (i_2, j_2,
f_2, g_2),\ .\ .\ .\ ,(i_N, j_N, f_N, g_N) \}$, where $i_n, j_n,
g_n \in \{0, 1\}$, $f_n$=$0$. Bob has another secret message consisting of $4N$ bits, too; i.e., Bob’s message $=\{(k_1, l_1,
w_1, v_1), (k_2, l_2, w_2, v_2),\ .\ .\ .\ , (k_M, l_M, w_M,
v_M)\}$, where $k_n, l_n, v_n
\in \{0, 1\}$, $w_n$=$0$. For Bob and Alice, each of them can encode their own secret message (a, b, c, d) ($a, b, d \in [0, 1],c=0$) by performing the unitary operation $C_{a,b}^t \otimes C_{c,d}^p$ on the travelled qubits; $|\Psi_{a, b; c, d}\rangle=C_{a,b}^t \otimes
C_{c,d}^p
|\psi_{0,0;0,0}\rangle$,, where $C_{0,0}^t$, $C_{0,1}^t$, $C_{1,0}^t$, $C_{1,1}^t$, $C_{0,0}^p$, and $C_{0,1}^p$ denote $I^t$, $\sigma_x^t$, $\sigma_y^t$, $\sigma_z^t$, $I^p$, and $\sigma_x^p$, respectively.
(S1) To securely exchange their messages or, in other words, to carry out a secret dialogue, Bob first produces a large enough number of GHZ states, all in the state $|\psi_{0,0;0,0}\rangle$. Bob keeps particles [*$h$*]{} with him and chooses particles ([*$t$*]{}, [ *$p$*]{}) as encoding-decoding group particles. Bob and Alice arrange to only perform one of the eight operations on particles ([*$h$*]{}, [*$p$*]{}) as $$\label{e13}
\begin{array}{cccc}
U_0=I^t \otimes I^p,
\ \ U_1=\sigma_z^t \otimes I^p,
\ \ U_2=\sigma_x^t \otimes I^p,
\ \ U_3=i \sigma_y^t \otimes I^p,\cr\\
U_4=I^t \otimes \sigma_x^p,
\ \ U_5=\sigma_z^t \otimes \sigma_x^p,
\ \ U_6=\sigma_x^t \otimes \sigma_x^p,
\ \ U_7=i \sigma_y^t \otimes \sigma_x^p.
\end{array}$$
(S2) Bob encodes his bits $(k_n, l_n,w_n, v_n$) by applying $U_B=C_{k_n,l_n}^t \otimes C_{w_n, v_n}^p$$(B \in
\{0,1,2,3,4,5,6,7\})$ on the GHZ state $|\psi_{0,0;0,0}\rangle$. He keeps qubits [*$h$*]{} with him and sends qubits ([*${t}$*]{}, [*$p$*]{}) through two different quantum channels to Alice. Then, Bob lets Alice know that he has sent qubits ([*${t}$*]{}, [*$p$*]{}).
(S3) Alice confirms to Bob that she has received the qubits. Bob tells Alice which mode they are: the message mode (MM) run or the control mode (CM) run. If it is a CM (MM) run, the procedure goes to S4 (S5).
(S4) Alice selects randomly sufficiently large groups as checking groups and the groups leftover as encoding-decoding groups. She chooses randomly one of the two sets of measuring basis (MB), $\{|0\rangle, |1\rangle\}$ or $\{|+\rangle, |-\rangle\}$, by measuring her qubits to check for the intercept-and-resend attack. She announces her choices and her measurement outcomes. Bob performs his measurement under the same MB as that chosen by Alice on the corresponding photons in his checking groups. If their measurement outcomes coincide when both of them use the same basis according to Eq. [(\[e3\])$-$(\[e10\])]{}. There are no Eves in line. This communication continues. In this case, Alice and Bob continue to transmit the next bits. Otherwise, they have to discard their transmission and abort the communication.
(S5) Alice encodes her bits $(i_n, j_n, f_n, g_n$) by applying $U_A$=$C_{i_n,j_n}^t$$\otimes$ $C_{f_n, g_n}^p$$(A \in\{
0,1,2,3,4,5,6,7\})$ on the encoding-decoding groups. Then, Alice sends back qubits ([*${t}$*]{}, [*$p$*]{}) to Bob, and lets Bob know that.
(S6) Aware of Alice’s confirmation, Bob performs a GHZ state basis measurement on the three qubits. The result in state $|\Psi_{(x_n,y_n);(r_n, s_n)}\rangle$ $(x_n,y_n,r_n,s_n\in
\{0,1\})$ is $$\begin{aligned}
\label{e14}
&|\Psi_{(x_n,y_n);(r_n,s_n)}\rangle&=U_BU_A|\psi_{0,0;0,0}\rangle\cr\cr&&=
C_{k_n,l_n}^t \otimes C_{w_n, v_n}^p \otimes C_{i_n,j_n}^t\otimes
C_{f_n, g_n}^p |\psi_{0,0;0,0}\rangle\cr\cr&&= C_{i_n \oplus
k_n,j_n \oplus l_n}^t \otimes C_{f_n \oplus w_n,g_n \oplus
v_n}^p|\psi_{0,0;0,0}\rangle\cr\cr&&
=\phi_{i_n,j_n;k_n,l_n}\phi_{f_n,g_n;w_n,v_n}|\psi_{i_n \oplus
k_n, j_n \oplus l_n; f_n \oplus w_n, g_n \oplus v_n}\rangle,\end{aligned}$$ where $\oplus$ denotes an addition mod 2, $\phi_{i,j;k,l}$ and $\phi_{f,g;w,v}$ are phase factors ($\phi=1$ or $\pm i$ depending on the values of ($i,j,k,l$) and ($f,g,w,v$)).
(S7) With a certain possibility, Alice and Bob reveal some bits to check the entangle-and-measure attack when particles are travelling from Alice to Bob. If there are no attacks, Alice’s bits and Bob’s bits should have the deterministic correlation $(i_n=\mid x_n-k_n\mid$, $j_n=\mid y_n-l_n\mid$, $f_n=\mid r_n-w_n
\mid$, $g_n=\mid s_n-v_n\mid)$. The communication continues. Otherwise, Bob publicly tells Alice that Eve is in the line, and the communication is aborted.
(S8) Bob decodes Alice’s secret bits as $(i_n=\mid x_n-k_n\mid$, $j_n=\mid y_n-l_n\mid$, $f_n=\mid r_n-w_n \mid$, $g_n=\mid
s_n-v_n\mid)$, and he reads the 3 bits of information. If he publicly announces the values of $(x_n, y_n, r_n, s_n)$ , then Alice can also decode Bob’s secret bits as $(k_n=\mid
x_n-i_n\mid$, $l_n=\mid y_n-j_n\mid$, $w_n=\mid r_n-f_n\mid$, $v_n=\mid s_n-g_n\mid)$, and she read the 3 bits of information, too. Then the procedure goes to S2, and Bob and Alice continue to transmit their next bits. The dialogue has been successfully completed.
In some cases, Alice and Bob can take some bits as checking bits, and they can reveal the value of the checking bits to check for the entangle-and-measure attack [@NBA04PLA] when the two qubits are travelling from Alice to Bob; i.e., Bob takes $(k_n,
l_n)$ as secret bits and $(w_n, v_n)$ as checking bits. Alice takes $(i_n, j_n)$ as secret bits and $(f_n, g_n)$ as checking bits, and publicly reveals the value of the checking bits. Then, Bob checks for Eves: if both $f_n= \mid r_n-w_n\mid $ and $g_n=\mid s_n-v_n\mid$, Bob holds; there are no Eves measuring the two qubits ($t_n$, $p_n$). The communication is secure. Otherwise, Bob and Alice have to discard their transmission and abort the communication.
In conclusion, we have proposed a new quantum dialogue protocol for two legitimate parties to exchange their secret messages simultaneously by using the GHZ state. The result shows that, for such a GHZ state quantum channel, a quantum dialogue can be realized between two users successfully. Compared with previous schemes, there are some differences in scheme proposed in this paper. Firstly, the quantum channel is different. Secondly, because the encoding-decoding particles are sent with a large number of checking particles simultaneously through two different quantum channels, it is hard for Eve to catch the right two particles in one group of the encoding-decoding groups simultaneously. Thus, the total probability of an eavesdrop by Eve is very small. This scheme can also be generalized to the $N$-particle GHZ state system; Bob keeps particle 1 with him in each $N$-particle GHZ state and sends the remaining $N-1$ particles to Alice. Thus, $N-1$ channels are needed, so the total probability of an eavesdrop by Eve will go down with increasing number of channels. Thirdly, Alice and Bob can take some bits as checking bits and reveal the value of the checking bits to check for the entangle-and-measure attack when the qubits are sent from Alice to Bob, making sure the communication is secure. Fourthly, the protocol makes subtle use of superdense coding [@BWPRL92; @YWCRL04; @FSLZKBRS05] to double the quantum channel, and each party is able at the same time to send three secret bits information and to read three secret bits information. The efficiency of information transmission is successfully increased. The intercept-and-resend attack, as well as the entangle-and-measure attack, can be detected efficiently in our protocol. Though it is experimentally more difficult to prepare GHZ states as compared to EPR states, for such a GHZ state quantum channel, we can not only increase the effective information but also improve the security in our protocol, so our protocol is feasible.
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[^1]: E-mail: szhang@ybu.edu.cn
|
---
abstract: |
A perfect matching $M$ in an edge–colored complete bipartite graph $K_{n,n}$ is rainbow if no pair of edges in $M$ have the same color. We obtain asymptotic enumeration results for the number of rainbow matchings in terms of the maximum number of occurrences of a color. We also consider two natural models of random edge–colored $K_{n,n}$ and show that, if the number of colors is at least $n$, then there is ${{\it whp}}$ a random matching. This in particular shows that almost every square matrix of order $n$ in which every entry appears at most $n$ times has a Latin transversal.
[**Keywords:**]{} Rainbow Matchings, Latin Transversals, Random Edge–colorings.
author:
- Guillem Perarnau and Oriol Serra
bibliography:
- 'rainbow.bib'
title: 'Rainbow Matchings: existence and counting'
---
Introduction {#intro}
============
A subgraph $H$ of an edge–colored graph $G$ is [*rainbow*]{} if no color appear twice in $E(H)$. The study of rainbow subgraphs has a large literature; see e.g. [@ajmp2003; @fk2008; @KL2008; @lsww2010; @jw2007]. In this paper we deal with [*rainbow perfect matchings*]{} of edge–colored complete bipartite graphs $K_{n,n}$. These are equivalent to latin transversals in square matrices of order $n$, sets of $n$ pairwise distinct entries no two in the same row nor the same column. The following is a longstanding conjecture by Ryser [@r1967] on the existence of latin transversals in latin squares:
Every latin square of odd order admits a latin transversal.
For even size, there are some latin squares that have no rainbow matchings, such as the additive table of $\mathbb{Z}_{2n}$. Nevertheless, it was also conjectured (see e.g. [@s1975]) that every latin square of even size has a partial latin transversal of length $n-1$.
There are different approaches to address these conjectures. For instance, Hatami and Shor [@hs2008] proved that every latin square has a partial transversal of size $n-O(\log{n}^2)$. Another approach was given by Erd[ő]{}s and Spencer [@es1991] where they prove the following result:
\[thm:lopsi\] Let $A$ be square matrix of order $n$. If every entry in $A$ appears at most $\tfrac{n-1}{4e}$ times, then $A$ has a latin transversal.
In order to get this result the authors developed the Lopsided version of the Lovász Local Lemma. The main idea of this version is to set a different dependency graph called [*lopsidependency graph*]{}. In this graph edges may no longer represent dependencies and the hypothesis of the Local Lemma are replaced by a weaker assumption.
In this paper we address two problems related to rainbow matchings in edge–colored $K_{n,n}$: asymptotic enumeration and existence in random edge–colorings. Our edge–colorings are non necessarily proper and the results apply to proper edge–colorings as well.
Previous results on enumeration of latin transversals have been obtained by McKay, McLeod and Wanless [@mmw2006], where the authors give upper and lower bounds on the maximum number of transversals that a latin square can have. However there is still a large gap between these bounds. Here, we provide, under some hypothesis, upper and lower bounds for the probability that a random matching in an edge–colored $K_{n,n}$ is rainbow, that are asymptotically tight.
The bounds are obtained by techniques inspired by the framework devised by Lu and Székely [@ls2009] to obtain asymptotic enumeration results with the Local Lóvasz Lemma.
For an edge–coloring of the complete bipartite graph $K_{n,n}$, we let ${\mathcal{M}}$ denote the family of pairs of non incident edges that have the same color. Let $M$ be a random matching of $K_{n,n}$. For each $(e,f)\in{\mathcal{M}}$, we denote by $A_{ef}$ the event that the pair of edges $e,f$ belongs to $M$. This is the set of bad events in the sense that, if none of these events occur, then the matching $M$ is rainbow.
Therefore, given a set of bad events $A_1,\dots,A_m$, we consider the problem of estimating the probability of the event $\cap_{i=1}^m \overline{A_i}$. If the bad events are mutually independent, then the number of bad events that are satisfied follows a Poisson distribution with parameter $\mu=\sum_{i=1}^m \Pr(A_i)$. Hence, $$\Pr( \cap_{i=1}^m \overline{A_j}) = e^{-\mu}.$$
It is natural to expect a similar behaviour if the dependencies among the events are rare. This is known as the Poisson Paradigm (see e.g. [@as2008]). Our objective is to show that $$\Pr( \cap_{i=1}^m \overline{A_j}) \rightarrow e^{-\mu} \quad(n\rightarrow \infty).$$ Let $X_M$ denote the indicator variable that a random perfect matching $M$ is rainbow in a fixed edge–coloring of $K_{n,n}$. Let ${\mathcal M}$ denote the set of pairs of independent edges that have the same color (bad events.) Our first result is the following:
\[thm:given\] Fix an edge–coloring of $K_{n,n}$ such that no color appears more than $n/k$ times, where $k=k(n)$. Let $\mu=|{\mathcal{M}}|/n(n-1)$.
If $k\ge 12$ then there exist constants $c_1<1<c_2$ depending only in $k$, such that $$e^{-c_2\mu}\leq \Pr(X_M=1) \leq e^{-c_1\mu}.$$ In particular, if $k=\omega(1)$ then $$\Pr(X_M=1) = e^{-(1+o(1))\mu}.$$ Moreover, if $k=\omega(n^{1/2})$ then $$\Pr(X_M=1)=e^{-\mu}(1+o(1)).$$
In the proof of Theorem \[thm:given\] we obtain $c_1=1-2/k-12/k^2$ and $c_2=1+16/k$. Note that the probability of having a rainbow matching only depends on the number of bad events that the given coloring defines. Perhaps surprisingly, this probability does not depend on the structure of the set of bad events in the coloring.
The results in Theorem \[thm:given\] require the condition $k\ge 12$, which is one unit more than the one given by Erdős and Spencer [@es1991] for the existence of rainbow matchings. This prompts us to analyze the existence of rainbow matchings in [*random*]{} edge–colorings of $K_{n,n}$ in the more general setting when $k\ge 1$ (we can not use less than $n$ colors.) For the existence of rainbow matchings in random edge–colorings of $K_{n,n}$ we restrict ourselves to colorings with a fixed number $s=kn$ of colors. We define two natural random models that fit with this condition.
In the Uniform random model, ${\mathsf{URM}}$, each edge gets one of the $s$ colors independently and uniformly at random. In this model, every possible edge coloring with at most $s$ colors appears with the the same probability. In the Regular random model, ${\mathsf{RRM}}$, we choose an edge coloring uniformly at random among all the equitable edge colorings, where each color class has prescribed size $\tfrac{n}{k}$. Although they have the same expected behaviour, both models are interesting. A result analogous to the one in Theorem \[thm:given\] can be proven for these two models.
\[thm:random\] Let $c$ be a random edge coloring of $K_{n,n}$ in the model ${\mathsf{URM}}$ with $s\ge n$ colors. Then,
$$\Pr(X_M=1) = e^{-c(k)\mu}$$ where $\mu\sim \tfrac{n^2}{2s}$ and $$c(k)=2k\left( 1 - (k-1)\log
\left(\frac{k}{k-1}\right)\right)$$
Let $c$ be a random edge-coloring of $K_{n,n}$ in the ${\mathsf{RRM}}$ model with $s\ge n$ colors. Then $$\Pr(X_M=1) = e^{-(c(k)+o(1))\mu}$$
Observe that both models lead to similar results. In particular, if $k=\omega(1)$ $$\Pr(X_M=1) = e^{-(1+o(1))\mu}$$
The ${\mathsf{RRM}}$ behaves as expected since, as we have observed, just the number of bad events is relevant, and in this case it is approximately $\tfrac{n^3}{2k}$.
Since the colorings are random, we have a stronger concentration of the rainbow matching probability than in the case of fixed colorings. By using the random model ${\mathsf{URM}}$ we show that *with high probability* (${{\it whp}}$, meaning with probability tending to one as $n\rightarrow
\infty$), for any constant $k\ge 1$, every random coloring has a rainbow matching.
\[thm:whp\] Every random edge–coloring of $K_{n,n}$ in the ${\mathsf{URM}}$ with $s\ge n$ colors has ${{\it whp}}$ a rainbow matching.
To prove the Theorem \[thm:whp\] we use the second moment method on the random variable that counts the number of rainbow matchings in the ${\mathsf{URM}}$ model. Observe that the same result can be proven using the same idea for the ${\mathsf{RRM}}$ model.
The paper is organized as follows. In Section \[sec:asymp\] we provide a proof for Theorem \[thm:given\]. The random coloring models are defined in Section \[sec:random\], where we also prove Theorem \[thm:random\]. In the Subsection \[ssc:whp\] we display a prove for Theorem \[thm:whp\]. Finally on Section \[sec:open\] we discuss about open problems about rainbow matchings that arise from the paper.
Asymptotic enumeration {#sec:asymp}
======================
In this section we prove Theorem \[thm:given\]. When $k=\omega (1)$ for $n\to \infty$, it gives an asymptotically tight estimation of the probability that a random matching is rainbow. For constant $k$ the theorem provides exponential upper and lower bounds for this probability.
Lower bound
-----------
One of the standard tools to give a lower bound for $\Pr( \cap_{i=1}^m \overline{A_j})$ is the Local Lemma. In particular, as it is shown in [@es1991], it is convenient in our current setting to use the Lopsided version of it.
Given a set of events $A_1,\dots,A_m$, a graph $H$ with vertex set $V(H) = \{ 1,\ldots ,m\}$ is a *lopsidependency graph* for the events if, for each $i$ and each subset $S\subseteq
\{j \mid ij\not\in E(H),\, j\neq i\}$, we have $$\Pr(A_i\mid \cap_{j\in S} \overline{A_j})\leq \Pr(A_i).$$ Following Lu and Szekely [@ls2009], we adopt the more explanatory term [*negative dependency graph*]{} for this notion. We next recall the statement of the Lóvasz Local Lemma we will use. It includes an intermediate step, that appears in its proof, which will also be used later on.
\[lem:LLLL\] Let $\{A_1,\ldots ,A_m\}$ be events and let $H=(V,E)$ be a graph on $\{1,\ldots ,m\}$ such that, for each $i$ and each $S\subseteq
\{j \mid ij\not\in E(H),\, j\neq i\}$, $$\Pr (A_i|\cap_{j\in S}\overline{A_j})\le P(A_i).$$
Let $x_1,\ldots ,x_m\in (0,1)$. If, for each $i$, $$\label{eq:hyp}
\Pr (A_i)\le x_i\prod_{ij\in E(H)} (1-x_j),$$ then, for each $T\subset [m]$ we have $$\label{eq:step}
\Pr (A_i|\cap_{j\in T} \overline{A_j})\le x_i.$$ In particular, for each $S\subset [m]$ disjoint from $T$ we have $$\label{eq:step2}
\Pr (\cap_{i\in S}\overline{A_i}|\cap_{j\in T} \overline{A_j})\ge \prod_{i\in S}(1-x_i),$$ and $$\label{eq:lll}
\Pr (\cap_{j\in [m]} \overline{A_j})\ge \prod_{j\in [m]}(1-x_j).$$
Recall that ${\mathcal M}$ denotes the family of pairs of independent edges that have the same color and, for each such pair $\{e,f\}\in {\mathcal M}$, we denote by $A_{e,f}$ the event that the pair belongs to a perfect random matching $M$. We identify ${\mathcal{M}}$ with this set of events. We consider the following dependency graph:
\[def:h\] The rainbow dependency graph $H$ has the family ${\mathcal{M}}$ as vertex set. Two elements in ${\mathcal{M}}$ are adjacent in $H$ whenever the corresponding pairs of edges share some end vertex in $K_{n,n}$.
It is shown in Erdős and Spencer [@es1991] that the graph $H$ defined above is a negative dependency graph. The following lower bound can be obtained in a similar way to Lu and Szekely [@ls2009 Lemma 2]. Recall that we consider edge–colorings of $K_{n,n}$ in which each color appears at most $n/k$ times.
\[lem:lower\] With the above notations, if $k\ge 12$ then $$\Pr (\cap_{\{e,f\}\in{\mathcal{M}}}\overline{A_{e,f}})\ge e^{-(1+16/k)\mu},$$ where $\mu=\sum_{\{e,f\}\in {\mathcal{M}}}\Pr(A_{e,f})$.
In particular, if $k=k(n)=\omega (\sqrt{n})$, then $$\Pr (\cap_{\{e,f\}\in{\mathcal{M}}}\overline{A_{e,f}})\ge (1+o(1))e^{-\mu}.$$
Set ${\cal M}=\{A_1,\ldots ,A_m\}$. The size of ${\mathcal{M}}$ depends on the configuration of the colors in $E(K_{n,n})$. In the worst case all the colors appear repeated in exactly $n/k$ disjoint edges. Thus, $$|{\mathcal{M}}|\leq kn \binom{n/k}{2}\sim\frac{n^3}{2k}$$
Since we are taking a random perfect matching, $p=\Pr(A_{i})=\tfrac{1}{n(n-1)}$ for each $i$. Then $$\label{eq:mu}
\mu=\frac{|{\mathcal{M}}|}{n(n-1)}= \frac{1}{2}\left(1+\frac{1}{n-1}\right)\left(\frac{n}{k}-1\right) \le
\frac{n}{2k}$$ Set $t=4/k$. Since $n\ge k\ge 12$ we have $t\le 1/3$ and $p\le 1/35$. It can be checked that, for $4/n<t<7/50$ and $0<p<1/35$, we have $$p e^{(1+4t)t}<1-e^{-(1+4t)p}.$$ Choose $x_i$ in the interval $(p e^{(1+4t)t},1-e^{-(1+4t)p})$. For each $1\le i\le m$ we have $$\label{eq:xi}
\Pr (A_i)=p<x_ie^{-(1+4t)t}<x_i\prod_{ij\in E(H)}e^{-(1+4t)\Pr(A_j)}<x_i\prod_{ij\in E(H)}(1-x_j).$$ Thus, by Lemma \[lem:LLLL\], $$\Pr (\cap_{A_i\in {{\mathcal{M}}}} \overline{A_i})\ge \prod_{i=1}^m (1-x_i)\ge e^{-\left(1+16/k\right)\mu}.$$ This proves the first part of the Lemma. In particular, since $\mu\le n/2k$, $$\Pr (\cap_{A_i\in {{\mathcal{M}}}} \overline{A_i})\ge e^{-\mu}\left(1-\frac{16\mu}{k}\right)\ge e^{-\mu}\left(1-\frac{8n}{k^2}\right).$$ so that, if $k=k(n)=\omega (\sqrt{n})$, then $$\Pr (\cap_{A_i\in {{\mathcal{M}}}} \overline{A_i})\ge e^{-\mu}(1+o(1)).$$
Upper bound
-----------
Lu and Szekely [@ls2009] propose a new enumeration tool using the Local Lemma. Their objective is to find an upper bound for the non occurrence of rare events comparable with the Janson inequality. In order to adapt the Local Lemma, they define a new type of parametrized dependency graph: the *$\varepsilon$-near dependency graph*.
Let $A_1,\dots,A_m$ a set of events. A graph $H$ with vertex set $\{A_1,\ldots ,A_m\}$ is an $\varepsilon$-near-positive dependency graph ($\varepsilon$-NDG) if,
i) if $A_i\sim A_j$, then $\Pr(A_i\cap A_j)=0$.
ii) for any set $S\subseteq \{j:A_j\nsim A_i\}$ it holds $\Pr(A_i \mid \cap_{j\in S}
\overline{A_j})\geq (1-\varepsilon)\Pr(A_i)$.
Condition $i)$ implies that only incompatible events can be connected. Condition $ii)$ says that this set of non connected events can not shrink the probability of $A_i$ too much.
\[thm:ub\] Let $A_1 ,\dots, A_m$ be events with an $\varepsilon$–near–positive dependency graph $H$. Then we have, $$\Pr(\cap \overline{A_i})\leq \prod_{i} (1-(1-\varepsilon)\Pr(A_i)).$$
Observe that this upper bound gives an exponential upper bound of the form $e^{(1-\varepsilon)\mu}$.
Lu and Szekely [@ls2009] show also that an $\varepsilon$–near–positive dependency $H$ can be constructed using a family of matchings ${\mathcal{M}}$. Unfortunately the conditions of [@ls2009 Theorem 4] which would provide the upper bound in our case do not apply to our family ${\mathcal{M}}$ of matchings. We give instead a direct proof for the upper bound which is inspired by their approach.
\[lem:positive\] The graph $H$ is an $\varepsilon$–near-positive dependency graph with $\varepsilon=1- e^{-(2/k+32/k^2)}$.
Set ${\cal M}=\{A_1,\ldots ,A_m\}$. The graph $H$ clearly satisfies condition i) in the definition of $\varepsilon$-NDG. For condition ii) we want to show that, for each $i$ and each $T\subseteq
\{j \mid ij\not\in E(H),\, j\neq i\}$, we have the inequality $$\Pr (A_i|B)\ge (1-\epsilon)\Pr(A_i),$$ where $B=\cap_{j\in T}\overline{A_j}$. This is equivalent to show $$\Pr (B|A_i)\ge (1-\epsilon)\Pr(B).$$ Let $\{a_1,\ldots ,a_n\}$ and $\{ b_1,\ldots ,b_n\}$ be the vertices of the two stable sets of $K_{n,n}$. We may assume that $A_i$ consists of the two edges $a_{n-1}b_{n-1}, a_nb_n$. Then $\{A_j:j\in T\}$ consists of a set of $2$–matchings in $K_{n,n}-\{ a_{n-1},a_n,b_{n-1},b_n\}$. This is the complete bipartite graph $K_{n',n'}$, $n'=n-2$, with an edge–coloring in which each color appears at most $n'/k'$ times, where $k'= k(1+2/(n-2))$. Let us call $B'$ the event $B$ viewed in $K_{n'n'}$ (dashes in notation indicate changing the probability space from random matchings in $K_{n,n}$ to random matchings in $K_{n'n'}$), so that $$\label{eq:b'}
\Pr (B|A_i)=\Pr (B').$$ Let $C_{r,s}$ denote the $2$–matching $a_{n-1}b_r, a_nb_s$, where $r\neq s$. Define $T_{r,s}\subset T$ in such a way that $\{A_j:j\in T_{r,s}\}$ are the $2$–matchings in $\{A_j:j\in T\}$ which meet none of the two vertices $b_r, b_s$. Set $B_{r,s}=\cap_{j\in T_{r,s}}\overline{{A_j}}$. Let us show that $$\label{eq:induction}
\Pr(B)=\frac{1}{n(n-1)}\sum_{r\neq s}\Pr (B_{r,s}'),$$ where, as before, $B'_{r,s}$ denotes the event $B_{r,s}$ in the probability space of random matchings in $K_{n',n'}$.
We have $$\Pr(B)=\sum_{r\neq s}\Pr (B\cap C_{r,s})=\sum_{r\neq s}\Pr (B_{r,s}\cap C_{r,s}),$$ Note that, since none of the matchings involved in $B$ meets vertices in $\{ a_{n-1},a_n,b_{n-1},b_n\}$, we have, for all $r,s$, $r\neq s$, $$\Pr (B_{r,s}|C_{r,s})=\Pr (B_{r,s}|C_{n-1,n}).$$ Moreover, observe that $\Pr (B_{r,s}|C_{n-1,n})=\Pr (B'_{r,s})$. Therefore $$\Pr(B)=\sum_{r\neq s}\Pr (B_{r,s}| C_{r,s})\Pr (C_{r,s})=\frac{1}{n(n-1)}\sum_{r\neq s}\Pr (B_{r,s}|C_{n-1,n})= \frac{1}{n(n-1)}\sum_{r\neq s}\Pr (B_{r,s}'),$$ giving equality .
From inequality we know that $x'_j= 1- e^{-(1+16/k')p'}$ fulfills the hypothesis of the Local Lemma. We can now use the intermediate inequality of the Lemma with $S=T\setminus T_{r,s}$ to obtain $$\label{eq:step2b}
\Pr (B_{r,s}')= \frac{\Pr (B')}{\Pr(\cap_{j\in S}\overline{A_j})}\le \Pr (B')\prod_{j\in S}(1-x_j')^{-1}.$$ By combining with and we get
$$\label{eq:epsilon}
\Pr(B|A_i)\ge \Pr (B)\prod_{j\in S}(1-x_j').$$
Recall that $S=T\setminus T_{r,s}$ is the set of $2$-matchings in ${\mathcal{M}}'$ that are incident to $b_r$ or $b_s$. The size of this set can be bounded independently from $r$ and $s$ by $$|S|\leq 2 n'\left(\frac{n'}{k'}-1\right) \leq 2\frac{n^2}{k}$$ With our choice of $x'_j= 1- e^{-(1+16/k')p'}\le 1-e^{-(1+16/k)p}$ (where $p=1/n(n-1)$) we have $$\prod_{j\in S}(1-x_j')\ge e^{-(1+16/k)p|S|}\ge e^{-(2/k+32/k^2)}.$$ Therefore, by , $$\varepsilon = 1- e^{-(2/k+32/k^2)},$$ satisfies the conclusion of the Lemma.
Now we are able to prove Theorem \[thm:given\].
Set ${\cal M}=\{A_1,\ldots ,A_m\}$. By Lemma \[lem:positive\], the graph $H$ is an $\varepsilon$–near–positive dependency graph with $\varepsilon = 1- e^{-(2/k+32/k^2)}$. It follows from Theorem \[thm:ub\] that the probability of having a rainbow matching is upper bounded by $$\Pr(\cap_{i\in [m]} \overline{A_i}) \leq \prod_{i\in [m]}
\left(1-(1-\varepsilon)\Pr (A_i)\right) \le e^{-(1-\epsilon)\mu}.$$ By plugging in our value of $\varepsilon$ and by using $e^{-(2/k+32/k^2)}\ge 1-\frac{2}{k}-\frac{32}{k^2}$ we obtain $$\Pr(\cap_{i\in [m]} \overline{A_i}) \leq e^{-(1-2/k-12/k^2)\mu}.$$ Combining this upper bound with the lower bound obtained in Lemma \[lem:lower\] we obtain $$\exp \left\{-\left( 1+\frac{16}{k}\right) \mu\right\} \leq Pr(\cap \overline{A_i})\leq
\exp \left\{-\left( 1-\frac{2}{k}-\frac{12}{k^2}\right)\mu\right\}.$$ This proves the first part of the Theorem.
In particular, since $\mu\le n/2k$, if $k=\omega (n^{1/2})$ we get $$\Pr(\cap_{i\in [m]} \overline{A_i}) \leq e^{-\mu}(1+o(1)),$$ which matches the lower bound obtained in Lemma \[lem:lower\], thus proving the second part of the Theorem.
Random colorings {#sec:random}
================
In this section we will analyze the existence of rainbow matchings when the edge coloring of $K_{n,n}$ is given at random.
Recall that, in the uniform random model ${\mathsf{URM}}$, each edge of $K_{n,n}$ is given a color uniformly and independently chosen from a set $C$ with $s$ colors, i.e. every possible coloring with at most $s$ colors appears with the same probability.
In the regular random model ${\mathsf{RRM}}$ a coloring is chosen uniformly at random among all colorings of $E(K_{n,n})$ with equitable color classes of size $n^2/s$. In order to construct a coloring in the ${\mathsf{RRM}}$ we use a complete bipartite graph $H=(A,B)$, where $A$ contains $s$ blocks, each of size $n^2/s$, representing the colors and $B$ is the set of edges of $K_{n,n}$. Every perfect matching in $H$ gives an equitable coloring of $E(K_{n,n})$. Moreover, every equitable coloring of $E(K_{n,n})$ corresponds to the same number of perfect matchings. Therefore, by selecting a random perfect matching in $H$ with the uniform distribution, all equitable colorings have the same probability.
We established these two models since they simulate the worst situation in all the possible colorings admitted in Theorem \[thm:given\]: the probability for a matching of being rainbow only depends on the size of $|{\mathcal{M}}|$, and this set has its largest cardinality when there are few colors with a maximum number of occurrences. This means that we have $s=nk$ colors with $n/k$ occurrences each. Observe that in both models the expected size of each color class is also $n/k$, and in this sense, they are contiguous to the hypothesis of Theorem \[thm:given\]. One can draw an analogy between the ${\mathsf{URM}}$ and the Erdős-Rényi model $G(n,p)$ for random graphs, and also between the ${\mathsf{RRM}}$ and the regular random graph $G(n,d)$.
In the ${\mathsf{URM}}$ it is easy to compute the probability of having the rainbow property for a matching. Let $M$ be a random perfect matching, and $K_{n,n}$ provided with a random edge coloring with $s\ge n$ colors. For the indicator variable $X_M$ that $M$ is rainbow we have: $$\begin{aligned}
\Pr(X_M=1) & = & \frac{s}{s}\cdot\frac{s-1}{s}\cdot\frac{s-2}{s}\cdot\,\dots\,\cdot\frac{s-(n-1)}{s}
\nonumber\\
&=& \prod_{i=0}^{n-1} \left(1-\frac{i}{s}\right).\label{eq:URM}\end{aligned}$$
For $s=n$ we can get directly from $$\Pr(X_M=1)=\frac{n!}{n^n}\sim e^{-2\mu}.$$ Assume $s>n$. We have, for $0<x<1$, $$\label{eq:approx}
(1-x) = \exp\left\{ \log{(1-x)} \right\} $$ Therefore, $$\begin{aligned}
\Pr(X_M=1) & = & \prod_{i=1}^{n-1} \exp\left\{\log{\left(1-\frac{i}{s}\right)}\right\} \\
& = & \exp\left\{ \sum_{i=1}^{n-1} \log{\left(1-\frac{i}{s}\right)} \right\} \\
& \sim & \exp\left\{ \int_0^n \log{\left(1-\frac{x}{s}\right)} dx \right\}.\end{aligned}$$ We use $$\int_0^t \log\left(1-x\right)dx=(t-1)\log\left(1-t\right)-t$$ By writing $k=s/n$ $$\begin{aligned}
\Pr(X_M=1) & = & \exp\left\{ \left((k-1)\log{\left(\frac{k}{k-1}\right)}-1\right)n\right\}\\
& \sim & \exp\left\{ - 2k\left( 1 - (k-1)\log \left(\frac{k}{k-1}\right)\right) \mu\right\}.\end{aligned}$$ since $\mu\sim\tfrac{n}{2k}$.
It must be stressed that this result is consistent with the ones in Theorem \[thm:given\]. When $k=1$ we have $\Pr(X_M=1)= e^{-2\mu}$, while $\lim_{k\rightarrow \infty} \Pr(X_M=1)= e^{-\mu}$. Observe that, in this case, ${{\mathbb E}}(|{\mathcal{M}}|)$ is not exactly the same as for a given coloring. This is due to the variance on the number of occurrences of each color, but does not have a significant importance.
To study the property that a random selected matching is rainbow in the ${\mathsf{RRM}}$ we express the equitable edge colorings through permutations $\sigma\in Sym (n^2)$. Then, probability for a matching $M$ of being rainbow is,
$$\begin{aligned}
\Pr(X_M=1) & = &
\frac{n^2}{n^2}\cdot\frac{n^2-\tfrac{n^2}{s}}{n^2-1}\cdot\frac{n^2-2\tfrac{n^2}{s}}{n^2-2}
\cdot\,\dots\,\cdot\frac {
n^2-(n-1)\tfrac{n^2}{s}}{n^2-(n-1)} \\
&=&\prod_{i=0}^{n-1} \left(1-\frac{i(n^2-s)}{s(n^2-i)}\right)\\
&=& \exp \left\{\sum_{i=0}^{n-1} \log\left(1-\frac{i(n^2-s)}{s(n^2-i)}\right) \right\} \quad
\mbox{(by~\eqref{eq:approx})}\\
&\sim& \exp \left\{\int_{0}^{n} \log\left(1-\frac{x(n^2-s)}{s(n^2-x)}\right) dx \right\}.\end{aligned}$$
If $s=n$ we have $$\int_{0}^{n} \log\left(1-\frac{x(n-1)}{(n^2-x)}\right) dx = -n(n-1)\log{\left(\frac{n}{n-1}\right)},$$ which, by using the Taylor expansion of the logarithm, gives $$\Pr(X_M=1) = e^{-(1+o(1))2\mu}$$ In the case where $s>n$, and using $k=s/n$, we have $$\begin{aligned}
\int_{0}^{n} \log\left(1-\frac{x(n^2-s)}{s(n^2-x)}\right) dx &= &
\left((k-1)\log{\left(\frac{k}{k-1}\right)}-(n-k)\log{\left(\frac{n}{n-1}\right)}\right) n\\
&= & \left((k-1)\log{\left(\frac{k}{k-1}\right)} -1 +o(1)\right)n.\end{aligned}$$ Hence $$\begin{aligned}
\Pr(X_M=1) & = & \exp\left\{ - 2k\left(1- (k-1)\log \left(\frac{k}{k-1}\right) + o(1)\right)
\mu\right\}.\end{aligned}$$
Note that, for both models of random edge colorings, the probability that a fixed perfect matching is rainbow the same (up to a $o(1)$ term). Thus, in spite of being different models, the probability of having a rainbow matching is similar.
In general it is not true that $\Pr(X_M=1)=e^{-(1+o(1))\mu}$ but, if $k=\omega(1)$, then $\Pr(X_M=1)\rightarrow e^{-\mu}$ for both models since $$2k\left(1- (k-1)\log \left(\frac{k}{k-1}\right) \right) =
1+O\left(\frac{1}{k}\right)$$ This is natural since, when $k$ is large, the number of bad events decreases and the model behaves like in the case they were independent.
Observe that for the two random models we obtain the exact asymptotic value of the probability, while bounds provided by Theorem \[thm:given\] (when the size $|{\mathcal{M}}|$ of the set of bad events is maximum) are not sharp, although consistent with the values for the random models. Since the result proven for fixed colorings does only depend on the size of ${\mathcal{M}}$, the probability for the random model ${\mathsf{RRM}}$ should be exactly the same.
Existence of rainbow matchings {#ssc:whp}
==============================
The aim of this Section is to prove that ${{\it whp}}$ there exists a rainbow matching for a given random coloring of $E(K_{n,n})$ with $s\ge n$ colors. We only consider the ${\mathsf{URM}}$, but the results can be adapted to the ${\mathsf{RRM}}$. The number of rainbow matchings is counted by $X=\sum X_M$, which, according to Theorem \[thm:random\], has expected value $$\label{eq:expected}
\mathbb{E}(X)= n! \Pr(X_M=1)\sim n! \exp\left\{ - 2k\left( 1 - (k-1)\log \left(\frac{k}{k-1}\right)\right) \mu\right\}.$$ In order to have a rainbow matching we just need that $X\neq 0$.
Given two perfect matchings $M$ and $N$, the events that they are rainbow are positively correlated, $$\Pr(X_M=1\mid X_N=1)\geq \Pr(X_M=1).$$
To show that there exists some rainbow matching ${{\it whp}}$ we will use the second moment method. Let $X$ be a random variable with expected value $\mu$ and variance $\sigma^2$. Then, the Chebyshev inequality asserts that $$\label{eq:cheby}
\Pr(|X-\mu|> \alpha \sigma) \leq \frac{1}{\alpha^2}$$
In particular if $\alpha=\tfrac{\mu}{\sigma}$, $$\label{eq:cheby2}
\Pr(X=0)\leq \Pr(|X-\mu|>\mu) \leq \frac{\sigma^2}{\mu^2}$$
Observe that $X=0$ is equivalent to the non existence of any rainbow matching. Therefore, we need to compute $\sigma^2(X)$ and show that it is asymptotically smaller than ${{\mathbb E}}(X)^2$. Note that $$\mathbb{E}(X^2) = \sum_{M,N} \mathbb{E}(X_MX_N)$$
Let $M$ and $N$ two fixed matchings, then $$\mathbb{E}(X_MX_N)=\Pr(X_M)\Pr(X_{N\setminus M})$$
Given a fixed matching $M$ and a fixed intersection size $t$, we claim there are at most $e^{-1}\binom{n}{t}(n-t)!$ matchings $N$, such that $|M\cap N|=t$. There are $\binom{n}{t}$ ways of choosing which edges will be shared and once this edges have been fixed, at most $e^{-1}(n-t)!$ ways of completing the matching. Suppose that $\tau=\sigma_N\mid_{N\setminus M} \in {\mathcal{S}}_{n-t}$ is the permutation for extending the matching in the disjoint part. Since $N$ has intersection exactly $t$ with $M$, not any permutation is valid. We can assume ${\emph{wlog}}$ that $M$ is given by $\sigma_M=Id$, and therefore $\tau$ must be a derangement. Classical results state that the proportion of derangements in permutations of any length is at most $e^{-1}$. This concludes our claim.
Hence, $${{\mathbb E}}(X^2)= e^{-1}n! \sum_{t=0}^{n} \binom{n}{t}(n-t)! \Pr(X_M)\Pr(X_{N\setminus M})$$ Since $\sigma^2={{\mathbb E}}(X^2)-{{\mathbb E}}(X)^2$, $$\begin{aligned}
\frac{\sigma^2(X)}{{{\mathbb E}}(X)^2} &=& \frac{e^{-1}n! \sum_{t=0}^{n} \binom{n}{t}(n-t)!
\Pr(X_M)\Pr(X_{N\setminus M})}{(n! \Pr(X_M))^2} -1\\
& = & e^{-1}\sum_{t=0}^{n}\frac{1}{t!}\frac{\Pr(X_{N\setminus M})}{\Pr(X_M)}-1\\
$$
Given $X_M$, we know that the edges of $M\cap N$ are rainbow. In the remaining $n-t$ edges to color, we must avoid the $t$ colors that appear in $M\cap N$ $$\Pr(X_N\mid X_M) = \prod_{i=t}^{n-1} \left(1-\frac{i}{s}\right)$$
Then, $$f(s) = \sum_{t=0}^{n}\frac{1}{t!}\frac{\Pr(X_{N\setminus M})}{\Pr(X_M)} \sim
\sum_{t=0}^{n}\frac{1}{t!}e^{(1+O(1/k))\tfrac{t^2}{2s}} \sim
\sum_{t=0}^{\infty}\frac{1}{t!}e^{(1+O(1/k))\tfrac{t^2}
{2s}}$$
If the number of colors $s=\omega(1)$, then $f(s)\rightarrow e$. Observe that $s\geq n$. Otherwise, $\Pr(X_M)=0$ in the Equation .
Hence $\tfrac{\sigma^2}{\mu^2}\rightarrow 0$ and the theorem holds.
Actually, $$f(s)=\frac{1}{s}+O(s^{-2})$$
and Equation also provides an upper bound estimation for the probability $p$ that a random coloring has no rainbow matchings of the type $$p\leq (1+o(1))\frac{1}{n}$$ Observe that the proportion of Latin squares among the set of square matrices with $n$ symbols is of the order of $e^{-n^2}$ (see e.g. [@vw2001]), so that this estimation falls short to prove an asymptotic version of the original conjecture of Ryser.
Open Problems {#sec:open}
=============
On the ennumeration of Rainbow matchings, it would be interesting to prove exact upper and lower bounds for the case where the number of occurences of each color is at most $k$, with constant $k$. Theorem \[thm:given\] provides exponential upper and lower bounds as long as $k\geq 12$, but both are asymptotically equal if and only if $k=\omega(1)$.
A related problem is to improve the lower bound $k\geq 4e$ given by Erdős and Spencer [@es1991 Theorem 2] for the existence of rainbow matchings.
On the other hand, another really interesting problem is to prove that almost all latin squares have a latin transversal, i.e. the asymptotic version of the Ryser conjecture. We have stablished a probabilistic way to approach the problem. Unfortunately, as far as we know, there are no random models for latin squares. Some results on generating random latin squares can be found in [@mw1991; @jm1996]. Nevertheless, there are some almost sure results on Latin squares (see e.g.[@mw1999; @cgw2008]).
|
---
abstract: 'We calculate the triangular flow parameter $v_3$ of thermal photons from an event-by-event ideal hydrodynamic model for 0–40% central collisions of Pb nuclei at $\sqrt{s_{NN}}$=2.76 TeV at LHC. $v_3$ determined with respect to the participant plane (PP) is found to be non-zero, positive and its $p_T$ dependence is qualitatively similar to the elliptic flow parameter $v_2$(PP) of thermal photons in the range $1 \le p_T \le 6$ GeV/$c$. In the range $p_T \, \le $ 3 GeV/$c$, $v_3$(PP) is found to be about 50–75% of $v_2$(PP) and for $p_T \, >$ 3 GeV/$c$ the two anisotropy parameters become comparable. The local fluctuations in the initial density distribution as well as the initial global geometry of the produced matter in the event-by-event hydrodynamic framework are responsible for this substantial value of $v_3({\rm PP})$. However, as expected, the triangular flow parameter calculated with respect to the reaction plane $v_3$(RP) is found to be close to zero. We show that $v_3$(PP) strongly depends on the value of the fluctuation size scale $\sigma$ especially in the higher $p_T \, (\ge 3 {\rm GeV}/c)$ region where a larger value of $\sigma$ results in a smaller $v_3({\rm PP})$. In addition, the $v_3{\rm (PP)}$ is found to increase with the assumed formation time of the thermalized system.'
author:
- Rupa Chatterjee
- 'Dinesh K. Srivastava'
- Thorsten Renk
title: 'Triangular flow of thermal photons from an event-by-event hydrodynamic model for 2.76A TeV Pb+Pb collisions at LHC'
---
Introduction
============
The observation of collective flow and its description in terms of fluid dynamics is a cornerstone of contemporary understanding of the dynamics of ultrarelativistic heavy ion collisions in terms of the creation of thermalized QCD matter. In many recent studies it has been shown that fluid dynamics utilizing event-by-event (E-by-E) fluctuating initial conditions (IC) [@hannu; @pt; @scott; @hannah; @sorenson; @nex] can be successfully used to explain the large elliptic flow results for the most central collisions and also the significant triangular flow of charged particles at RHIC and LHC energies [@alver; @flow_phenix; @flow_lhc; @flow_atlas] which were underestimated previously by hydrodynamic with smooth initial density distribution. E-by-E hydrodynamics with fluctuating IC also explains the hardening of charged particle spectra at larger $p_T$ [@hannu; @hama], various structures observed in two particle correlations [@andrade], and it also has been used to constrain the viscosity over entropy ratio $\eta/s$ from simultaneous measurement of elliptic and triangular flow coefficients [@eta].
The thermal emission of photons is sensitive to the initial hot and dense stage of the expanding system [@phot] and thus photons are considered as one of the probes suitable to study IC fluctuations. It has been shown that fluctuations in the initial density distribution lead to a significant enhancement in the production of thermal photon compared to a smooth initial density distribution in ideal hydrodynamic calculations [@chre1]. Consequently a better agreement of the experimental direct photon spectrum is obtained in the region $p_T>$ 2 GeV/$c$ [@chre1] using thermal contribution from the fluctuating IC. The enhancement due to IC fluctuations is found to be more pronounced for peripheral collisions than for central collisions and is less pronounced at the LHC than at RHIC for similar centrality bins [@chre2]. We have also shown that the elliptic flow calculated using the E-by-E hydrodynamics is substantially larger compared to the results from a smooth initial state averaged profile in the region $p_T> 2$ GeV/$c$ [@chre3].
The success of fluid dynamics implies that the shape of initial spatial geometry or more precisely the initial spatial anisotropy of the overlapping zone between the two colliding nuclei leads to azimuthally anisotropic pressure gradients. As a result, an anisotropic momentum distribution of the final state particles is observed. The momentum anisotropies of the emitted particles are usually quantified by expanding the invariant particle distribution in transverse plane in terms of the Fourier decomposition $$\label{eq: v2}
\frac{dN}{d^2p_TdY} = \frac{1}{2\pi} \frac{dN}{ p_T dp_T dY}[1+ 2\, \sum_{n=1}^{\infty} v_n (p_T) \, \rm{cos} (n\phi)] \, .$$ Here $\phi$ is the azimuthal angle measured with respect to the reaction plane and various $v_n$ are called the anisotropic flow parameters. The elliptic flow parameter $v_2$ is a result of the almond shape of the initial geometry and is one of the key observables studied at the RHIC experiments [@fl2]. The significantly large $v_2$ measured at RHIC is considered as a sign of collectivity in the produced system.
The observation of significant triangular flow anisotropy $v_3$ of hadrons is attributed to the collision geometry fluctuations leading to a potential initial triangularity of the overlapping zone [@alver]. This is different from the case of $v_2$ where the global shape of the initial collision geometry already has an ellispoid shape which dominates over the local fluctuations. As a result, one finds large $v_2$ with respect to the reaction plane, whereas $v_3$ only takes a non-zero value when determined with respect to an E-by-E determined participant plane.
Triangular flow of photons is of particular interest as photons are emitted from every phase of the expanding system and at high $p_T$ predominantly reflect early time dynamics, hence they can be expected to particularly probe the initial conditions. A recent study [@v3_uli] has investigated triangular flow and higher flow harmonics of thermal photons from an E-by-E viscous hydrodynamic model with KLN and Glauber initial conditions and argued that this leads to a very sensitive measurement of viscosity.
There are three potential mechanisms by which the initial state might influence photon $v_3$: a) fluctuating IC lead to an overall triangular deformation of the matter distribution in the transverse plane which is mapped into a triangular flow patterm. This is the mechanism leading to hadron $v_3$, but since the global shape of the matter distribution can only be probed by late time dynamics when photon production above 1 GeV is suppressed by thermal factors, it is not evident that this mechanism also holds for e.m. emission. b) fluctuating IC lead to irregularly-shaped hotspots with copious photon production, and the early time evolution of such hotspots will likely lead to angular anisotropies in the photon emission. However, there is no obvious reason for this anisotropy to correlate with the hadronic $v_3$ event plane, and hence it can a priori be expected to average out in measurements unless a photon triangular event plane can be determined (which is however impossible in any real measurement due to the very limited number of high $p_T$ photons in any given event) c) hotspots in the fluctuating IC leads indirectly, e.g. by development of additional radial flow, to a magnification of the photon $v_3$. Interpreting any result requires to carefully disentangle these mechanisms.
In the present work we study the $p_T$ dependent triangular flow of thermal photons from fluctuating IC in detail for 0–40% central collision of Pb nuclei at LHC and the dependence of $v_3$ on the fluctuation size parameter and the initial formation time of the system and interpret out findings in the light of the three mechanisms outlined above.
Triangular flow of thermal photons from E-by-E hydrodynamic framework
=====================================================================
We use the E-by-E ideal hydrodynamic model framework developed in [@hannu] to calculate the triangular flow anisotropy of thermal photons at LHC energy. This (2+1) dimensional hydrodynamic model with fluctuating IC has been successfully used to reproduce the $p_T$ spectra and elliptic flow of hadrons at RHIC [@hannu]. It has also been used to calculate the spectra and elliptic flow of thermal photons at the RHIC and at the LHC energies [@chre1; @chre2; @chre3].
A Monte Carlo Glauber model is used for the initial state. The standard two-parameter Woods-Saxon nuclear density profile is used to randomly distribute the nucleons into the two colliding nuclei. Two nucleons from different nuclei are assumed to collide when the relation $d^2 < \frac{\sigma_{NN}}{\pi^2}$ is satisfied where, $d$ is the transverse distance between the colliding nucleons and $\sigma_{NN}$ is the inelastic nucleon nucleon cross-section. We take $\sigma_{NN}=$ 64 mb at LHC.
An entropy initialized wounded nucleon (sWN) profile is used where the initial entropy density is distributed around the collision participants (wounded nucleons) using a 2-dimensional Gaussian distribution function $$s(x,y) = \frac{K}{2 \pi \sigma^2} \sum_{i=1}^{\ N_{\rm WN}} \exp \Big( -\frac{(x-x_i)^2+(y-y_i)^2}{2 \sigma^2} \Big).
\label{eq:eps}$$ K is an overall constant in the Eq. above and the position of the $i$th nucleon in the transverse plane is denoted by ($x_i, y_i$). $\sigma$ is the most important parameter in the above equation which decides the granularity or the size of the initial density fluctuations. It is a free parameter and we use a default value of $\sigma=$ 0.4 fm as before [@hannu; @chre1; @chre2; @chre3]. We use an initial time $\tau_0=$ 0.14 fm/$c$ [@phe] at LHC from the EKRT minijet saturation model [@ekrt] as default value but later vary the value of $\sigma$ and $\tau_0$ from their default values to understand the effect of initial collision geometry and the hotspots on the triangular flow parameter better.
The temperature at freeze-out is taken as 160 MeV which reproduces the measured $p_T$ spectra of charges pions at LHC. 170 MeV is considered as the transition temperature from the plasma phase to hadronic phase and we use a lattice based equation of state [@eos] for our calculation.
We use complete leading order (LO) plasma rates from [@amy] and hadronic rates from [@trg] to calculate the triangular flow of thermal photons from the fluctuating IC. Next-to-leading order (NLO) plasma rates from [@nlo_thermal] are used to obtain the thermal photon spectra from smooth IC only. The total thermal emission is obtained by integrating the emission rates ($R=EdN/d^3pd^4x$) over the space-time history of the fireball as $$E \frac{dN}{d^3p}= \int d^4x \, R \left(E^*(x),T(x)\right).$$ Here T(x) is the local temperature and $E^* (x)$ = $p^\mu u_\mu (x)$, where $p^\mu$ is the four-momentum of the photons and $u_\mu$ is the local four-velocity of the flow field.
![(Color online) Thermal photon $p_T$ spectra considering complete leading order plasma rates [@amy] and next-to-leading order plasma rates [@nlo_thermal] for 2.76A TeV Pb+Pb collisions at LHC and for 0–40% centrality bin. ALICE direct photons spectrum [@alice] and next-to-leading order pQCD photons are also plotted for comparison.[]{data-label="spec_lhc"}](lhc_spec.eps){width="8.0"}
The triangular flow parameter $v_3$ is calculated with respect to the participant plane (PP) angle using the relation:
$$v_3^\gamma\{\text{PP}\}= \langle \cos (3(\phi - \psi_{3}^{\text{PP}})) \rangle_{\text{events}} \, ,$$
where the participant plane angle is defined as [@ndhh]
$$\psi_{3}^{\text{PP}} = \frac{1}{3} \arctan
\frac{\int \mathrm{d}x \mathrm{d}y \; r^3 \sin \left( 3\phi \right) \varepsilon\left( x,y,\tau _{0}\right) }
{ \int \mathrm{d}x \mathrm{d}y \; r^3 \cos \left( 3\phi \right) \varepsilon\left( x,y,\tau _{0}\right)} + \pi/3 \, .$$
Here $\varepsilon$ is the energy density, $r^{2}=x^{2}+y^{2}$, and $\phi$ is the azimuthal angle. The triangularity or the initial triangular eccentricity of the matter density is calculated using the relation
$$\epsilon_{3} = -\frac{\int \mathrm{d} x \mathrm{d} y \; r^{3} \cos \left[
3\left( \phi -\psi_{3}^{\text{PP}}\right) \right] \varepsilon \left(
x,y,\tau_{0}\right) } {\int \mathrm{d} x \mathrm{d} y \; r^{3} \varepsilon
\left( x,y,\tau _{0}\right) } \, .$$
Results
=======
Thermal photon $p_T$ spectra for 0–40% central collision of Pb nuclei at $\sqrt{s_{\rm NN}}$=2.76 TeV at LHC are shown in Figure \[spec\_lhc\]. Results from smooth IC (SIC) using complete LO plasma rates (blue dotted line) and NLO plasma rates (solid orange line) are compared. Here the smooth IC is obtained by taking initial state average of 1000 random events which essentially removes all the fluctuations in the initial density distribution. We see that the addition of NLO contribution to the complete LO rate increases the thermal photon production by 10–15% in the range $p_T <2$ GeV/$c$, whereas for $p_T>2$ GeV/$c$ LO and NLO spectra look quite similar and fall on top of each other. In addition, we observe that the additional NLO contribution to the thermal photon production is still not sufficient to match the experimental data in the range $p_T<$2 GeV/$c$.
As shown before [@chre2], thermal photons from fluctuating IC (FIC) along with the contribution from NLO pQCD [@ilkka] explain the direct photon $p_T$ spectrum measured by ALICE [@alice] well in the region $p_T>$2 GeV/$c$. The result from the fluctuating IC is obtained by taking final state average over the $p_T$ spectra from large number of random events.
The calculation of the elliptic and the triangular flow anisotropies of thermal photons from the E-by-E hydrodynamic model is numerically expensive process. The time taken by the NLO plasma rates to calculate the $p_T$ spectra is significantly larger than the time taken by the LO plasma rates and also the difference between the two rates is not significant for $p_T>$ 2 GeV/$c$. For this reason we consider it a good approximation to use the LO rates to calculate the flow anisotropies of thermal photons.
![(Color online) Triangular and elliptic flow of thermal photons with respect to RP and PP angles for 0–40% central collisions of Pb nuclei at LHC and for $\sigma=$0.4 fm.[]{data-label="v3_lhc"}](v3_lhc.eps){width="8.0"}
Figure \[v3\_lhc\] shows triangular flow parameter $v_3$ of thermal photons as a function of $p_T$ for 0–40% central collision of Pb nuclei at $\sqrt{s_{NN}}$=2.76 TeV at LHC. The $v_3$ calculated with respect to the participant plane angle (red solid line closed circle) as well as to the reaction plane (RP) (blue solid line closed squares) are shown for $\sigma$=0.4 fm. The elliptic flow parameters $v_2({\rm PP})$ and $v_2({\rm RP})$ calculated for the same centrality bin [@chre3] are shown as well for comparison. The $v_3$ results are obtained by averaging over the triangular flow results from 400 random events and we also show the standard errors on both the $v_3({\rm PP})$ and $v_3({\rm RP})$.
We see that $v_3({\rm PP})$ for thermal photons is positive and significant compared to the the elliptic flow results calculated for the same centrality bin in the region $1 \le p_T \le 6$ GeV/$c$. At $p_T=$ 1 GeV/$c$, the difference between $v_3({\rm PP})$ and $v_2({\rm PP})$ is maximum and $v_3({\rm PP})$ is almost half of the value of $v_2({\rm PP})$ at that $p_T$. The difference reduces more and more towards higher values of $p_T$. $v_3({\rm PP})$ is about 80% of the value of $v_2({\rm PP})$ at $p_T=$3 GeV/$c$ and at $p_T=5$ GeV/$c$, the difference between the two results is about 10–15%. Triangular flow calculated with respect to the reaction plane from individual events are found to be both positive and negative. As expected, the averaged $v_3({\rm RP})$ is zero within standard errors as shown in the Figure \[v3\_lhc\].
In order to understand the individual effects of global geometry and local fluctuations on the triangular flow results better we study $v_3$ of individual events by keeping the the number of wounded nucleons ($\rm N_{\rm WN}$) fixed. We take $\rm N_{\rm WN}=$ 200 and generate random events having different triangular flow eccentricity. The number of binary collisions $N_{\rm coll}$ also varies in an wide range in those events.
The variation of average transverse flow velocity with proper time for three different events with fixed $\rm N_{\rm WN}$ is shown in upper panel of Figure \[events\]. The $N_{\rm coll}$ values for Event 1, Event 2, and Event 3 are 537, 611 and 724 respectively and the initial triangular eccentricities of the events are 0.122, 0.325 and 0.177 respectively. We see that the average transverse flow velocity with time is largest for Event 3, the event with maximum $N_{\rm coll}$ and $\langle v_T \rangle$ is smallest for the Event 1.
![(Color online)\[F-3\] \[Upper panel\] Average transverse flow velocities for different events with fixed number of wounded nucleons and \[Lower panel\] $v_3({\rm PP})$ for the same events. []{data-label="events"}](vt.eps){width="8.0"}
![(Color online)\[F-3\] \[Upper panel\] Average transverse flow velocities for different events with fixed number of wounded nucleons and \[Lower panel\] $v_3({\rm PP})$ for the same events. []{data-label="events"}](v3_pp.eps){width="8.0"}
![(Color online) Triangular flow of thermal photons for 0–40% central collisions of Pb nuclei at LHC and for size parameter $\sigma=$ 0.4 and 0.8 fm. []{data-label="fig_sigma"}](v3_sigma.eps){width="8.0"}
The $v_3$ calculated with respect to the PP angle for the three events is shown in lower panel of Figure \[events\]. $v_3$ shows ordering similar to the average transverse flow velocity where, $v_3(\rm{PP}$) is largest for Event 3 and is smallest for Event 1 although, the triangular eccentricity is maximum for Event 2. This argues that indirect effects of fluctuations, such as the buildup of a strong flow field, contribute significantly to the observed result beyond leading to an overall triangular eccentricity. For hadrons a clear mapping between the initial eccentricity and $v_n$ has been observed [@ndhh], however this is not true for photons.
The fluctuation size scale $\sigma$ is an important parameter as both the geometry and the overal strength photon emission is strongly sensitive to its value. The initial density distribution becomes smoother for larger values of size parameter and in an earlier study [@chre3] we have shown that the elliptic flow parameter for $\sigma = $ 1.0 fm is quite similar to the elliptic flow parameter calculated using a smooth initial state averaged IC. Thus, it is crucial to study the dependence of photon $v_3$ on the value of $\sigma$. Figure \[fig\_sigma\] shows $v_3({\rm PP})$ and $v_3({\rm RP})$ as a function of $p_T$ for 0–40% central collisions of Pb nuclei at $\sqrt{s_{NN}}$=2.76 TeV at LHC and for two different $\sigma$ values. As shown in the figure, the value of $v_3({\rm PP})$ for $\sigma=$ 0.4 fm (red dashed line, open circles) is close to the $v_3({\rm PP})$ results for $\sigma=$ 0.8 fm (red solid line, closed circles) in the lower $p_T \, (\le 2 \, {\rm GeV}/$c$)$ region. However, for $p_T >$ 2 GeV/$c$, $v_3({\rm PP})$ is smaller for larger value of $\sigma$ and with increasing $p_T$, the $v_3({\rm PP})$ for $\sigma=$ 0.8 fm falls sharply compared to the flow result for $\sigma=$ 0.4 fm. The presence of the local fluctuations or the hotspots in the IC results in stronger radial flow velocity which allows to probe global geometry more efficiently and we see larger $v_3$ for the smaller value of $\sigma$.
One can expect even larger $v_3({\rm PP})$ (than the results shown in Figure \[v3\_lhc\]) when $\sigma$ is smaller than 0.4 fm. However, the flow anisotropy calculation become numerically expensive with smaller values of $\sigma$ and thus we can not show results for $\sigma <$ 0.4 fm.
As expected, the triangular flow parameter calculated with respect to the RP does not depend on the value of $\sigma$ and $v_3({\rm RP})$ close to zero for $\sigma=$ 0.8 fm (shown by black solid line, closed triangles).
Next we study the dependence of the triangular flow parameter on the initial formation time $\tau_0$ of the plasma. All the $v_3$ results shown till now are calculated for a very small $\tau_0=$0.14 fm which is taken from EKRT model [@ekrt] for most central collision of Pb nuclei at LHC. Now the formation time of the plasma can be larger for peripheral collisions than for central collisions [@chre2]. As we do not have the formation time at LHC for 0–40% centrality bin from EKRT model, we choose a sufficiently large $\tau_0=$ 0.6 fm/$c$ to start the hydrodynamic calculation in order to see the dependence of $v_3$ on the initial formation time of the plasma.
![(Color online) Triangular flow of thermal photons for 0–40% central collisions of Pb nuclei at LHC and for size parameter $\sigma=$ 0.4 and at $\tau_0$ values 0.14 and 0.6 fm/c.[]{data-label="fig_0.6"}](v3_0.6.eps){width="8.0"}
Figure \[fig\_0.6\] shows $v_3({\rm PP})$ as a function of $p_T$ for $\tau_0$ values 0.14 and 0.6 fm/$c$ and for $\sigma=$ 0.4 fm. It should be noted that the triangular flow parameter for two different $\tau_0$ values is calculated by keeping the total entropy of the system fixed.
The $v_3({\rm PP})$ for larger $\tau_0$ rises rapidly compared to the flow parameter calculated using smaller $\tau_0$ in the region $p_T \le$ 3 GeV/$c$. For $p_T>$ 3 GeV/$c$, $v_3$ for $\tau_0=$ 0.6 fm/$c$ does not change significantly with $p_T$ and becomes constant. This is contrary to the $v_3$ for $\tau_0=$ 0.14 fm/$c$ which drops with increasing $p_T$. The high $p_T$ photons are emitted early when the presence of local fluctuations in the IC is strong. However, these photons do not contribute significantly to the $v_3$ result and the small values of transverse flow velocity in the initial stage brings the $v_3$ down for $p_T >$ 3 GeV/$c$ when a smaller $\tau_0$ is considered. For $\tau_0=$ 0.6 fm/$c$, a large fraction of these high $p_T$ photons are not included in the calculation and as a result we get much larger $v_3$ [@cs].
Fig. \[fig\_0.6\] explicitly shows that the eccentricity of initial hotspots does not contribute to photon $v_3$ significantly as expected, only photons emitted somewhat later carry significant $v_3$. We can combine this with with Fig. \[F-3\] to conclude that the most important effect of initial functuations on thermal photon $v_3$ is indirect, i.e. the modification of the radial flow pattern which can transform even a small initial eccentricity into a large $v_3$.
Summary and conclusions
=======================
We calculate the triangular flow anisotropy of thermal photons from an E-by-E ideal hydrodynamic model. The complete leading order plasma rates and state of the art hadronic rates are used to calculate $v_3$ at LHC using suitable initial and final conditions. We show that the inclusion of NLO plasma rate to the complete LO rates does not change the spectra results significantly for $p_T \ge $ 2 GeV/$c$.
The flow parameter $v_3$ as a function of $p_T$ is calculated with respect to the reaction plane and participant plane angles for 2.76A TeV Pb+Pb collisions at LHC and for 0–40% central collisions. The value of $v_3(\rm PP)$ at $p_T=$ 1 GeV/$c$ is found to be about 50% smaller than the $v_2(\rm PP)$ calculated using same initial and final conditions. However, the two results become comparable for $p_T >$ 3 GeV/$c$. The triangular flow anisotropy calculated with respect to the RP is close to zero.
The $v_3(\rm PP)$ calculated using $\sigma=$ 0.8 fm is found to be similar to $v_3(\rm PP)$ for $\sigma$=0.4 fm in the region $p_T \le $2 GeV/$c$. For larger $p_T$ however, the $v_3$ for larger $\sigma$ is relatively smaller in magnitude. We see the variation of average transverse flow velocity with proper time and $v_3(\rm PP)$ as a function of $p_T$ as well for different events with same number of wounded nucleons. This is done in order to understand the role of $\langle v_T \rangle$ and $\epsilon_3$ in determining the triangular flow anisotropy parameter better. $v_3$ as a function of $p_T$ for two different values of initial formation time is also compared. A larger $\tau_0$ results in a much larger $v_3$ compared to the result calculated using a smaller $\tau_0$.
We find that photon $v_3$ probes the initial state geometry in an indirect way via the generation of additional transverse flow. The sensitivity to the fluctuation size scale as well as to the equilibration time offers useful constraints for a dynamical modeling of the pre-equilibrium phase.
RC gratefully acknowledge he financial support by the Dr. K. S. Krishnan Research Associateship from Variable Energy Cyclotron Centre, Department of Atomic Energy, Government of India. TR is supported by the Academy researcher program of the Academy of Finland, Project No. 130472. We thank Hannu Holopainen for providing us with the event-by-event hydrodynamic code and for many useful discussions. We also thank Ilkka Helenius for the NLO pQCD photon results and the ALICE collaboration for the direct photon spectrum for 0–40% central collisions of Pb nuclei at LHC. We acknowledge the computer facility of the CSC computer centre, Espoo.
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abstract: |
Let $G_{1}$ and $G_{2}$ be disjoint copies of a graph $G$, and let $g:V(G_{1})\rightarrow V(G_{2})$ be a function. A functigraph $F_{G}$ consists of the vertex set $V(G_{1})\cup
V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})\cup
\{uv:g(u)=v\}$. In this paper, we extend the study of the distinguishing number of a graph to its functigraph. We discuss the behavior of the distinguishing number in passing from $G$ to $F_{G}$ and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs.
address: 'Centre for advanced studies in Pure and Applied Mathematics, Bahauddin Zakariya University Multan, Pakistan Email: mfazil@bzu.edu.pk, mahru830@gmail.com, uali@bzu.edu.pk, imran.javaid@bzu.edu.pk'
author:
- 'Muhammad Fazil, Muhammad Murtaza, Usman Ali, Imran Javaid'
title: On the Distinguishing number of Functigraphs
---
Preliminaries
=============
Given a key ring of apparently identical keys to open different doors, how many colors are needed to identify them? This puzzle was given by Rubin [@frank; @rubin] in 1980 for the first time. In this puzzle, there is no need for coloring to be proper. Indeed, one cannot find a reason why adjacent keys must be assigned different colors, whereas in other problems like storing chemicals, scheduling meetings a proper coloring is needed, and one with a small number of colors is required.
From the inspiration of this puzzle, Albertson and Collins [@alb] introduced the concept of the distinguishing number of a graph as follows: A labeling $f: V(G)\rightarrow \{1,2,3,...,t\}$ is called a $t$-*distinguishing* if no non-trivial automorphism of a graph $G$ preserves the vertex labels. The *distinguishing number* of a graph $G$, denoted by $Dist(G)$, is the least integer $t$ such that $G$ has $t$-distinguishing labeling. For example, the distinguishing number of a complete graph $K_n$ is $n$, the distinguishing number of a path $P_n$ is $2$ and the distinguishing number of a cycle $C_{n},\ n\geq 6$ is $2$. For a graph $G$ of order $n$, $1\le Dist(G)\le n$ [@alb]. If $H$ is a subgraph of a graph $G$ such that automorphism group of $H$ is a subset of automorphism group of $G$, then $Dist(H)\le Dist(G)$.
Harary [@har] gave different methods (orienting some of the edges, coloring some of the vertices with one or more colors and same for the edges, labeling vertices or edges, adding or deleting vertices or edges) of destroying the symmetries of a graph. Collins and Trenk defined the distinguishing chromatic number in [@col] where they used proper $t$-distinguishing for vertex labeling. They have also given a comparison between the distinguishing number, the distinguishing chromatic number and the chromatic number for families like complete graphs, paths, cycles, Petersen graph and trees etc. Kalinowski and Pilsniak [@Pil] have defined similar graph parameters, the distinguishing index and the distinguishing chromatic index, they labeled edges instead of vertices. They have also given a comparison between the distinguishing number and the distinguishing index for a connected graph $G$ of order $n\ge 3$. Boutin [@bou] introduced the concept of determining sets. In [@albrt+bou+2], Albertson and Boutin proved that a graph is $t$-distinguishable if and only if it has a determining set that is $(t-1)$-distinguishable. They also proved that every Kneser graph $K_{n:k}$ with $n\ge 6$ and $k\ge 2$ is 2-distinguishable. A considerable literature has been developed in this area see [@albertson; @albrt+bou+1; @bog; @mch1; @mch2; @cheng; @kla1; @kla2; @tymoczko].
Unless otherwise specified, all the graphs $G$ considered in this paper are simple, non-trivial and connected. The *open neighborhood* of a vertex $u$ of $G$ is $N(u)=\{v\in
V(G):uv\in E(G)\}$ and the *closed neighborhood* of $u$ is $N(u)\cup \{u\}$. Two vertices $u,v$ are *adjacent twins* if $N[u]=N[v]$ and *non adjacent twins* if $N(u)=N(v)$. If $u,v$ are adjacent or non adjacent twins, then $u,v$ are *twins*. A set of vertices is called *twin-set* if every of its two vertices are twins. A graph $H$ is said to be a *subgraph* of a graph $G$ if $V(H) \subseteq V(G)$ and $E(H) \subseteq E(G)$. Let $S\subset V(G)$ be any subset of vertices of $G$. The *induced subgraph*, denoted by $<S>$, is the graph whose vertex set is $S$ and whose edge set is the set of all those edges in $E(G)$ which have both end vertices in $S$.
The idea of permutation graph was introduced by Chartrand and Harary [@char] for the first time. They defined the permutation graph as follows: a permutation graph consists of two identical disjoint copies of a graph $G$, say $G_1$ and $G_2$, along with $|V(G)|$ additional edges joining $V(G_1)$ and $V(G_2)$ according to a given permutation on $\{1, 2, . . . , |V(G)|\}$. Dorfler [@dor], introduced a mapping graph which consists of two disjoint identical copies of graph where the edges between the two vertex sets are specified by a function. The mapping graph was rediscovered and studied by Chen et al. [@yi], where it was called the functigraph. A functigraph is an extension of permutation graph. Formally the functigraph is defined as follows: Let $G_{1}$ and $G_{2}$ be disjoint copies of a connected graph $G$, and let $g:V(G_{1})\rightarrow V(G_{2})$ be a function. A *functigraph* $F_{G}$ of a graph $G$ consists of the vertex set $V(G_{1})\cup
V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})\cup
\{uv:g(u)=v\}$. Linda et al. [@kang; @kang1] and Kang et al. [@kang2] have studied the functigraph for some graph invariants like metric dimension, domination and zero forcing number. In [@fazil2], we have studied the fixing number of functigraph. The aim of this paper is to study the distinguishing number of functigraph.
Throughout the paper, we will denote the set of all automorphisms of a graph $G$ by $\Gamma(G)$, the functigraph of $G$ by $F_{G}$, $V(G_{1})=A$, $V(G_{2})=B$, $g:A\rightarrow B$ is a function, $g(V(G_{1}))= I$, $|g(V(G_{1}))|= |I|= s$.
This paper is organized as follows. In Section 2, we give sharp lower and upper bounds for distinguishing number of functigraph. This section also establishes the connections between the distinguishing number of graphs and their corresponding functigraphs in the form of realizable results. In Section 3, we provide the distinguishing number of functigraphs of complete graphs and join of path graphs. Some useful results related to these families have also been presented in this section.
Bounds and some realizable results
==================================
The sharp lower and upper bounds on the distinguishing number of functigraphs are given in the following result.
\[f5\]Let $G$ be a connected graph of order $n\geq 2$, then $$1\leq
Dist(F_{G})\leq Dist(G)+1.$$ Both bounds are sharp.
Obviously, $1\leq Dist(F_{G})$ by definition. Let $Dist(G)=t$ and $f$ be a $t$-distinguishing labeling for graph $G$. Also, let $u_i\in A$ and $v_i\in B$, $1\le i \le n$. We extend labeling $f$ to $F_G$ as: $f(u_i)=f(v_i)$ for all $1\le i \le n$. We have following two cases for $g$:
1. If $g$ is not bijective, then $f$ as defined earlier is a $t$-distinguishing labeling for $F_G$. Hence, $Dist(F_G)\le t$.
2. If $g$ is bijective, then $f$ as defined earlier destroys all non-trivial automorphisms of $F_G$ except the flipping of $G_1$ and $G_2$ in $F_G$, for some choices of $g$. Thus, $Dist(F_G)\le t+1$.
For the sharpness of the lower bound, take $G=P_{3}$ and $g:A\rightarrow B$, be a function such that $g(u_{i})= v_{1}, i=1,2$ and $g(u_{3})= v_{3}$. For the sharpness of the upper bound, take $G$ as rigid graph and $g$ as identity function.
Since at least $m$ colors are required to break all automorphisms of a twin set of cardinality $m$, so we have the following corollary.
\[rem3.3111\] Let $U_1,U_2,...,U_t$ be disjoint twin sets in a connected graph $G$ of order $n\geq 3$ and $m= max\{|U_i|: 1\leq i\leq t\}$,\
(i) $Dist(G)\ge m$.\
(ii) If $Dist(G)=m$, then $Dist(F_G)\le m$.
\[Lemma g constant\] Let $G$ be a connected graph of order $n\ge 2$ and $g$ be a constant function, then $Dist(F_G)=Dist(G)$.
Let $I=\{v\}\subset B$. Then $\Gamma(G)=\Gamma(< A\cup
\{v\}>)\subset \Gamma(F_G)$. Thus, vertices in $A\cup \{v\}$ are labeled by $Dist(G)$ colors. Since $g$ is a constant function, therefore all vertices in $V(F_G)\setminus \{A\cup\{v\}\}$ are not similar to any vertex in $A\cup\{v\}$ in functigraph $F_G$. Therefore, vertices in $V(F_G)\setminus\{A\cup\{v\}\}$ can also be labeled from these $Dist(G)$ colors. Hence, $Dist(F_G)=Dist(G)$.
\[rem3.31115\] Let $G$ be a connected graph and $Dist(F_G)=m_1$ if $g$ is constant and $Dist(F_G)=m_2$ if $g$ is not constant, then $m_1\geq m_2.$
A vertex $v$ of degree at least three in a connected graph $G$ is called a *major vertex*. Two paths rooted from the same major vertex and having the same length are called the *twin stems*.
We define a function $\psi: \mathbb{N} \setminus\{1\}\rightarrow
\mathbb{N}\setminus \{1\}$ as $\psi (m)=k$ where $k$ is the least number such that $m\le 2{k\choose2}+k$. For example, $\psi(19)=5$. Note that $\psi$ is well-defined.
\[LemmaTwinStem\] If a graph $G$ has $t\ge 2$ twin stems of length 2 rooted at same major vertex, then $Dist(G)\ge \psi(t)$.
Let $x\in V(G)$ be a major vertex and $xu_iu'_i$ where $1\leq i\leq
t$ are twin stems of length 2 attach with $x$. Let $H=<\{x,u_i,u'_i\}>$ and $k=\psi(t)$. We define a labeling $f:V(H)
\rightarrow \{1,2,...,k\}$ as: $$f(x)=k,$$ $$\label{array1}
f({u_i})=\left\{
\begin{array}{ll}
1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\,\,\,\,\,\,\,1\leq i\,\leq k\\
2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\,\,\,\,\,\,\,k+1\leq i\,\leq 2k\\
3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\,\,\,\,\,\,\,2k+1\leq i\,\leq 3k\\
\vdots \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vdots \\
k\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\,\,\,\,\,\,\,(k-1)k+1\leq i\,\leq k^2\\
\end{array}
\right.$$
$$\label{array2}
f({u'_i})=\left\{
\begin{array}{ll}
i\, \mbox{mod} (k)\,\,\,\,\,\,\,if \,\,\,\,\,\,\,\,\,\, 1\le i\,\mbox{mod} (k) \le k-1, & \\
k\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, i\,\mbox{mod}(k) =0,& \\
\end{array}
\right.$$
Using this labeling, one can see that $f$ is a $t$-distinguishing for $H$. Since permutations with repetition of $k$ colors, when 2 of them are taken at a time is equal to $2{k\choose 2}+k$, therefore at least $k$ colors are needed to label the vertices in $t$-stems. Hence, $k$ is the least integer for which $G$ has $k$-distinguishing labeling. Since $\Gamma(H)\subseteq \Gamma(G)$, therefore $Dist(G)\ge \psi(t)$.
![Graph with $Dist(G)= t = Dist(F_{G}).$[]{data-label="f1"}](DIS1.eps){width="10cm"}
\[Fix3\]
\[fff\] For any integer $t\geq 2$, there exists a connected graph $G$ and a function $g$ such that $Dist(G)=t=Dist(F_{G})$.
Construct the graph $G$ as follows: let $P_{(t-1)^{2}+1}: x_1x_2x_3...x_{(t-1)^{2}+1}$ be a path. Join $(t-1)^{2}+1$ twin stems $x_1u_iu'_i$ where $1\leq i\leq
(t-1)^{2}+1$ each of length two with vertex $x_1$ of $P_{(t-1)^{2}+1}$. This completes construction of $G$. We first show that $Dist(G)=t$. For $t=2$, we have two twin stems attach with $x_1$, and hence $Dist(G)=2$. For $t\ge 3$, we define a labeling $f: V(G)\rightarrow \{1,2,3,...,t\}$ as follows:\
$f(x_i)=t$, for all $i$, where $1\leq i \leq(t-1)^{2}+1.$ $$f({u_i})=\left\{
\begin{array}{ll}
1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\, 1\leq i\leq t-1, & \\
2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\, t\leq i\leq 2(t-1),& \\
3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\, 2t-1\leq i\leq
3(t-1),& \\
\vdots \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vdots \\
t-1\,\,\,\,\,\,\,\,\,\,if \,\,\,\, (t-1)(t-2)+1\leq i\leq
(t-1)^{2}, & \\
t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\, i=(t-1)^{2}+1.
\end{array}
\right.$$
$$f({u'_i})=\left\{
\begin{array}{ll}
i\,\ \mbox{mod}(t-1)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\,1\leq i\,\ \mbox{mod}(t-1)\leq t-2\,\,\,\, and \,\,\,\, i\ne(t-1)^{2}+1,& \\
t-1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\,i\,\ \mbox{mod}(t-1)=0,& \\
t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hskip 2.2cm if \,\,\,\,
i=(t-1)^{2}+1.
\end{array}
\right.$$
Using this labeling, one can see the unique automorphism preserving this labeling is the identity automorphism. Hence, $f$ is a $t$-distinguishing. Since permutation with repetition of $t-1$ colors, when 2 of them are taken at a time is $2{{t-1}\choose
2}+(t-1)$, therefore $(t-1)^2+1$ twin stems can be labeled by at least $t$-colors. Hence, $t$ is the least integer such that $G$ has $t$-distinguishing labeling. Now, we denote the corresponding vertices of $G_2$ as $v_{i}, v_{i}', y_i$ for all $i$, where $1\leq
i \leq(t-1)^{2}+1$ and construct a functigraph $F_{G}$ by defining $g:V(G_1)\rightarrow V(G_2)$ as follows: $g(u_i)=g(u'_i)=y_i$, for all $i$, where $1\leq i \leq (t-1)^{2}+1$ and $g(x_i)= g(y_i)$, for all $i$, where $1\leq i \leq (t-1)^{2}+1$ as shown in the Figure \[f1\]. Thus, $F_G$ has only symmetries of $(t-1)^{2}+1$ twin stems attach with $y_1$. Hence, $Dist(F_G)= t.$
Consider an integer $t\geq 4$. We construct graph $G$ similarly as in proof of Lemma \[fff\] by taking a path $P_{(t-3)^{2}+1}:x_1x_2...x_{(t-3)^{2}+1}$ and attach $(t-3)^{2}+1$ twin stems $x_1u_iu'_i$ where $1\leq i\leq (t-3)^{2}+1$ with any one of its end vertex say $x_1$. Using similar labeling and arguments as in proof of Lemma \[fff\] one can see that $f$ is $t-2$ distinguishing and $t-2$ is least integer such that $G$ has $t-2$ distinguishing labeling. Define functigraph $F_{G}$, where $g:
V(G_1)\rightarrow V(G_2)$ is defined by: $g(u_i)= g(u'_i)=y_i$, for all $i$, where $1\leq i \leq (t-3)^{2}+1$, $g(x_i)=v_i$, for all $i$, where $1\leq i \leq (t-3)^{2}-1$, $g(x_i)=y_i$, for all $i$, where $(t-3)^{2}\leq i \leq (t-3)^{2}+1$. From this construction, $F_G$ has only symmetries of $2$ twin stems attach with $y_1$, and hence $Dist(F_G)=2$. Thus, we have the following result which shows that $Dist(G)+Dist(F_{G})$ can be arbitrary large:
\[fffw1\] For any integer $t\geq 4$, there exists a connected graph $G$ and a function $g$ such that $Dist(G)+Dist(F_{G})=t$.
Consider $t\geq 3$. We construct graph $G$ similarly as in proof of Lemma \[fff\] by taking a path $P_{4(t-1)^{2}+1}$: $x_1x_2...x_{4(t-1)^{2}+1}$ and attach $4(t-1)^{2}+1$ twin stems $x_1u_iu'_i$, where $1\leq i\leq 4(t-1)^{2}+1$ with $x_1$. Using similar labeling and arguments as in proof of Lemma \[fff\] one can see that $f$ is $2t-1$ distinguishing and $2t-1$ is the least integer such that $G$ has $2t-1$ distinguishing labeling. Let us now define $g$ as $g(u_i)= g(u'_i)=y_i$, for all $i$, where $1\leq i\leq
4(t-1)^{2}+1$, $g(x_i)= v_i$, for all $i$, where $1\leq i\leq
3t^{2}-4t$ and $g(x_i)= y_i$, for all $i$, where $3t^{2}-4t+1\leq
i\leq 4(t-1)^{2}+1.$ Thus, $F_G$ has only symmetries of $(t-2)^{2}+1$ twin stems attach with $y_1$, and hence $Dist(F_G)=t-1$. After making this type of construction, we have the following result which shows that $Dist(G)-Dist(F_{G})$ can be arbitrary large:
\[fffw11\] For any integer $t\geq 3$, there exists a connected graph $G$ and a function $g$ such that $Dist(G)-Dist(F_{G})=t$.
The distinguishing number of functigraphs of some families of graphs
====================================================================
In this section, we discuss the distinguishing number of functigraphs on complete graphs, edge deletion graphs of complete graph and join of path graphs.
Let $G$ be the complete graph of order $n\ge3$ and $A$ and $B$ be its two copies. We use following terminology for $F_G$ in proof of Theorem \[f13\]: Let $I=\{v_1,v_2,...,v_s\}$ and $n_i=|\{u\in A:
g(u)= v_i\}|$ for all $i$, where $1\le i \le s$. Also, let $l=\mbox{max}\{n_i: 1\le i \le s\}$ and $m=|\{n_i: n_i=1, 1\le i \le
s \}|$. From the definitions of $l$ and $m$, we note that $2\le l
\le n-s+1$ and $0\le m \le s-1$.
Using function $\psi(m)$ as defined in previous section, we have following lemma:
\[CompleteFunctiIdentity\] Let $G$ be the complete graph of order $n\ge 3$ and $g$ be a bijective function, then $Dist(F_G)=\psi(n)$.
Let $A=\{u_1,u_2,...,u_n\}$ and $I=\{g(u_1),g(u_2),...,g(u_n)\}=B$. Also let $k=\psi(n)$. Let $f:V(F_G)\rightarrow \{1,2,...,k\}$ be a labeling in which $f(u_i)$ is defined as in equation (\[array1\]) and $f(g(u_i))$ as in equation (\[array2\]) in proof of Lemma \[LemmaTwinStem\]. Using this labeling one can see that $f$ is a $k$-distinguishing labeling for $F_G$. Since permutation with repetition of $k$ colors, when 2 of them are taken at a time is equal to $2{k\choose 2}+k$, therefore at least $k$ colors are needed to label the vertices in $F_G$. Hence, $k$ is the least integer for which $F_G$ has $k$-distinguishing labeling.
Let $G$ be a complete graph and let $g:A\rightarrow B$ be a function such that $2\le m\le s$. Without loss of generality assume $u_1,u_2,...,u_m\in A$ are those vertices of $A$ such that $g(u_i)\ne g (u_j)$ where $1\le i\ne j \le m$ in $B$. Also $(u_iu_j)(g(u_i)g(u_j))\in \Gamma (F_G)$ for all $i\ne j$ where $1\le i, j \le m$. By using similar labeling $f$ as defined in Lemma \[CompleteFunctiIdentity\], at least $\psi(m)$ color are needed to break these automorphism in $F_G$. Thus, we have following proposition:
\[PropSi(m)\] Let $G$ be a complete graph of order $n\ge 3$ and $g$ be a function such that $2\le m\le s$, then $Dist(F_G)\ge \psi(m)$.
The following result gives the distinguishing number of functigraphs of complete graphs.
\[f13\] Let $G=K_{n}$ be the complete graph of order $n\geq
3$, and let $1<s\leq n-1$, then $$Dist(F_{G})\in\{n-s,n-s+1,\psi(m)\}.$$
We discuss following cases for $l$:
1. If $l=n-s+1>2$, then $A$ contains $n-s+1$ twin vertices and $B$ contains $n-s$ twin vertices (except for $n=3,4$ where $B$ contains no twin vertices). Also, there are $m(=s-1)$ vertices in $A$ which have distinct images in $B$. These $m$ vertices and their distinct images are labeled by at least $\psi(m)$ colors (only 1 color if $m=1$) by Proposition \[PropSi(m)\]. Since $n-s+1$ is the largest among $n-s+1$, $n-s$ and $\psi(m)$. Thus, $n-s+1$ is the least number such that $F_G$ has $(n-s+1)$- distinguishing labeling. Thus, $Dist(F_G)=n-s+1$.
2. If $l=n-s+1=2$, then $\psi(m)\ge \mbox{max}\{n-s+1, n-s\}$, and hence $Dist(F_G)=\psi (m)$.
3. If $l<n-s$, then $B$ contains largest set of $n-s$ twin vertices in $F_G$. Also, there are $m (\le s-2)$ vertices in $A$ each of which has distinct image in $B$. Since $n-s\geq\psi(m)$, therefore $Dist(F_G)=n-s$.
4. If $l=n-s>2$, then both $A$ and $B$ contain largest set of $n-s$ twin vertices in $F_G$. Also, there are $m(=s-2)$ vertices in $A$ which have distinct images in $B$. Since $n-s\geq\psi(m)$, therefore $Dist(F_G)=n-s$.
5. If $l=n-s=2$, then we take two subcases:
1. If $1<s\leq \lfloor \frac{n}{2}\rfloor +1$, then both $A$ and $B$ contain largest set of $n-s$ twin vertices in $F_G$. Also, there are $m(=s-2)$ vertices in $A$ which have distinct images in $B$. Since $n-s\geq\psi(m)$ (if $\psi(m)$ exists), therefore $Dist(F_G)=n-s$.
2. If $\lfloor \frac{n}{2}\rfloor +1< s\leq n-1$, then $\psi(m)\ge \mbox{max}\{n-s+1, n-s\}$, and hence $Dist(F_G)=\psi(m)$.
Let $e^{\ast}$ be an edge of a connected graph $G$. Let $G-ie^{\ast}$ is the graph obtained by deleting $i$ edges from graph $G.$ A vertex $v$ of a graph $G$ is called *saturated* if it is adjacent to all other vertices of $G$.
We define a function $\phi:
\mathbb{N} \rightarrow \mathbb{N}\setminus \{1\}$ as $\phi (i)=k$, where $k$ is the least number such that $i\le {k\choose2}$. For instance, $\phi(32)=9$. Note that $\phi$ is well defined.
\[f1122\] Let $G$ be the complete graph of order $n\geq 5$ and $G_i=
G-ie^{\ast}$ for all $i$ where $1\leq i\leq\lfloor
\frac{n}{2}\rfloor$ and $e^{\ast}$ joins two saturated vertices of the graph $G$. If $g$ is a constant function, then $$Dist(F_{G_i})=max\{n-2i,\phi(i)\}.$$
On deleting $i$ edges $e^*$ from $G$, we have $n-2i$ saturated vertices and $i$ twin sets each of cardinality two. We will now show that exactly $\phi(i)$ colors are required to label vertices of all $i$ twin sets. We observe that, a vertex in a twin set can be mapped on any one vertex in any other twin set. Since two vertices in a twin set are labeled by a unique pair of colors out of $k\choose 2$ pairs of $k$ colors, therefore at least $k$ colors are required to label vertices of $i$ twin sets. Now, we discuss the following two cases for $\phi(i)$:
1. If $\phi(i)\le n-2i$, then number of colors required to label $n-2i$ saturated vertices is greater than or equal to number of colors required to label vertices of $i$ twin sets. Thus, we label $n-2i$ saturated vertices with exactly $n-2i$ colors and out of these $n-2i$ colors, $\phi(i)$ colors will be used to label vertices of $i$ twin sets.
2. If $\phi(i)> n-2i$, then number of colors required to label $n-2i$ saturated vertices is less than the number of colors required to label vertices of $i$ twin sets. Thus, we label vertices of $i$ twin sets with $\phi(i)$ colors and out of these $\phi(i)$ colors, $n-2i$ colors will be used to label saturated vertices in $G_i$.
If $g$ is constant, then by using same arguments as in the proof of Lemma \[Lemma g constant\], $Dist(F_{G_i})=Dist(G_i).$
Suppose that $G=(V_{1},E_{1})$ and $G^*=(V_{2},E_{2})$ be two graphs with disjoint vertex sets $V_{1}$ and $V_{2}$ and disjoint edge sets $E_{1}$ and $E_{2}$. The *join* of $G$ and $G^*$ is the graph $G+G^*$, in which $V(G+G^*)=V_{1}\cup V_{2}$ and $E(G+G^*)=E_{1}\cup
E_{2}\cup \{$ $uv$: $u\in V_1$, $v\in V_2\}$.
[@Alikhani] Let $G$ and $G^*$ be two connected graphs, then $Dist(G+G^*)\geq max\{Dist(G), Dist(G^*)\}.$
Let $P_n$ be a path graph of order $n\ge 2$, then for all $m,n\ge 2$ and $1<s<m+n$, $1\le Dist(F_{P_m+P_n})\le 3$.
Let $P_m: v_1,...,v_m$ and $P_n:u_1,...,u_n$. We discuss following cases for $m,n$.
1. If $m=2$ and $n=2$, then $P_2+P_2=K_4$, and hence $1\leq Dist(F_{K_4})\le 3$ by Theorem \[f13\].
2. If $m=2$ and $n=3$, then $P_2+P_3$ has 3 saturated vertices. Thus, $1\le Dist(F_{P_2+P_3})\le 4$ by Proposition \[f5\]. However, for all $s$ where $2 \le s \le 4$ and all possible definitions of $g$ in $F_{P_2+P_3}$, one can see $1 \le Dist(F_{P_2+P_3}) \le 3$.
3. If $m=3$ and $n=3$, then a labeling $f:V(P_3+P_3)\rightarrow \{1,2,3\}$ defined as:
$$f(x)=\left\{
\begin{array}{ll}
1\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\,\,\,\,\,\,\,\,\,\, x=v_1,v_2\\
2\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,\,\,\,\,\,\,\,\,\, x= v_3,u_3\\
3\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,\,\,\,\,\,\,\,\,\, x= u_1,u_2\\
\end{array} \right.$$
is a distinguishing labeling for $P_3+P_3$, and hence $Dist(P_3+P_3)= 3$. Thus, $1\le Dist(F_{P_3+P_3})\le 4$ by Proposition \[f5\]. However, for all $s$ where $2 \le s \le 5$ and all possible definitions of $g$ in $F_{P_3+P_3}$, one can see $1 \le
Dist(F_{P_3+P_3}) \le 3$.
4. If $m\ge 2$ and $n\ge 4$, then a labeling $f:V(P_m+P_n)\rightarrow \{1,2\}$ defined as:
$$f(x)=\left\{
\begin{array}{ll}
1\,\,\,\,\,\,\,\,\,\,\,\,\,\,if \,\,\,\,\,\,\,\,\,\,\,\,\, x=v_1,u_2,...,u_n\\
2\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,\,\,\,\,\,\,\,\,\, x=u_1,
v_2,...,v_m &
\end{array} \right.$$
is a distinguishing labeling for $P_m+P_n$, and hence $Dist(P_m+P_n)= 2$. Thus, result follows by Proposition \[f5\].
[999]{}
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|
---
abstract: 'Much of the past work in network analysis has focused on analyzing discrete graphs, where binary edges represent the “presence” or “absence” of a relationship. Since traditional network measures (e.g., betweenness centrality) utilize a discrete link structure, complex systems must be transformed to this representation in order to investigate network properties. However, in many domains there may be *uncertainty* about the relationship structure and any uncertainty information would be lost in translation to a discrete representation. Uncertainty may arise in domains where there is moderating link information that cannot be easily observed, i.e., links become inactive over time but may not be dropped or observed links may not always corresponds to a valid relationship. In order to represent and reason with these types of uncertainty, we move beyond the discrete graph framework and develop social network measures based on a [*probabilistic*]{} graph representation. More specifically, we develop measures of path length, betweenness centrality, and clustering coefficient—one set based on sampling and one based on probabilistic paths. We evaluate our methods on three real-world networks from Enron, Facebook, and DBLP, showing that our proposed methods more accurately capture salient effects without being susceptible to local noise, and that the resulting analysis produces a better understanding of the graph structure and the uncertainty resulting from its change over time.'
author:
- |
Joseph J. Pfeiffer, III\
Department of Computer Science\
Purdue University\
West Lafayette, IN 47907\
*jpfeiffer@purdue.edu* Jennifer Neville\
Departments of Computer Science and Statistics\
Purdue University\
West Lafayette, IN 47909\
*neville@cs.purdue.edu*
bibliography:
- 'Pfeiffer\_Neville.bib'
title: |
Methods to Determine Node Centrality and Clustering\
in Graphs with Uncertain Structure
---
Introduction
============
Much of the past work in network analysis has focused on analyzing discrete graphs, where entities are represented as nodes and binary edges represent the “presence” or “absence” of a relationship between entities. Complex systems of relationships are first transformed to a discrete graph representation (e.g., a friendship graph) and then the connectivity properties of these graphs are used to investigate and understand the characteristics of the system. For example, network measures such as the average shortest path length and clustering coefficient have been used to explore the properties of biological and information networks [@clusteringcoefficient; @Leskovec05graphsover], while measures such as centrality have been used for determining the most important and/or influential people in social networks [@freeman77centrality; @Bonacich:87]. The main limitation of measures defined for a discrete representation is that they cannot easily be applied to represent and reason about *uncertainty* in the link structure. Link uncertainty may arise in domains where graphs evolve over time, as links observed at a earlier time may no longer be present or active at the the time of analysis. For example in online social networks, users articulate “friendships" with other users and these links often persist over time, regardless of whether the friendship is maintained. This can result in uncertainty about whether an observed friendship link is still *active* at some later point in time. In addition, there may be uncertainty with respect to the [*strength*]{} of the articulated relationships [@relstrength], which can result in uncertainty about whether an observed relationship will be used to transmit information and/or influence. Furthermore, there are other network domains (e.g., gene/protein networks) where relationships can only be indirectly observed so there is uncertainty about whether an observed edge (e.g., protein interaction) actually indicates the presence of a valid relationship.
In this work, we formulate a probabilistic graph representation to analyze domains with these types of uncertainty and develop analogues for three standard discrete graph measures—average shortest path length, betweenness centrality, and clustering coefficient—in the probabilistic setting. Specifically, we use probabilities on graph edges to represent link uncertainty and consider the [*distribution*]{} of possible (discrete) graphs that they define, then we develop measures that consider the properties of the graph population defined by this distribution.
Our first set of measures compute [*expected*]{} values over the distribution of graphs, sampling a set of discrete graphs from this distribution in order to efficiently approximate the path length, centrality, and clustering measures. We then develop a second set of measures that can be directly computed from the probabilities, which removes the need for graph sampling. The second approach also affords us the opportunity to consider more than just shortest paths in the network. We note that previous focus on shortest paths is due in part to an implicit belief that short paths are more likely to result in successful transfer of information and/or influence between two nodes. This has led other works to generalize shortest paths to the probabilistic domain for their own purposes [@sampleprobpaths]. However, in a probabilistic framework we can also directly compute the likelihood of a path and consider the [*most probable*]{} paths, which are likely to facilitate information flow in the network.
With probabilistic paths, we also introduce a [*prior*]{} to incorporate the belief that the probability of successful information transfer is a function of path length—since the existence of a relationship does not necessarily mean that information/influence will be passed across the edge. This formulation, which models the likelihood of information spread throughout the graph, is consistent with the finding in [@Onnela2007TieStrength], which identified that constricting and relaxing the flow along the edges in the network was necessary to model the true patterns of information diffusion in an evolving communication graph.
We evaluate our measures on three real world networks: Enron email, Facebook micro communications, and DBLP coauthorships. In these datasets, the network transactions are each associated with timestamps (e.g., email date). Thus we are able to compute the local (node-level) and aggregate (graph-level) measures at multiple time steps, where at each time step $t$ we consider the network information available up to and including $t$. We compare against two different approaches that use the discrete representation: an *aggregate* approach, which unions all previous transactions (up to $t$) into a discrete graph, and a *slice* approach, where only transactions from a small window (i.e., $[t-\delta, t]$) are included in the discrete representation. For our methods, we estimate edge probabilities from the transactions observed up to $t$, weighting each transaction with an exponential decay function. Our analysis shows that our proposed methods more accurately capture the salient changes in graph structure compared to the discrete methods without being susceptible to local, temporal noise. Thus the resulting analysis produces a better understanding of the graph structure and its change over time.
Related Work
============
The notion of probabilistic graphs have been studied previously, notably by [@frankshortestpaths], [@probtraffic] and [@sampleprobpaths]. [@frankshortestpaths] showed how for graphs with probability distributions over the weights for each edge, Monte Carlo methods can be used to sample to determine the shortest path probabilities between the edges. [@probtraffic] then extends this to find the shortest weighted paths most likely to complete within a certain time constraint (e.g., the shortest distance across town in under half an hour). In [@sampleprobpaths], the most probable shortest paths are used to estimate the $k$-nearest neighbors in the graph for a particular node. Although [@sampleprobpaths] draws sample graphs based on [*likelihood*]{} (i.e., sampling each edge according to its probability), in their estimate of the shortest path distribution they weight each sample graph based on its probability, which is incorrect unless the samples are drawn uniformly at random from the distribution. In this work, we sample in the same manner as [@sampleprobpaths], but weight each sample uniformly in our expectation calculations—since, when the graphs are drawn from the distribution based on their likelihood, the graphs with higher likelihood are more likely to be sampled.
There has also been some recent work that has developed measures for time-evolving graphs, e.g., to identify the most central nodes throughout time [@tang-tempshortestpaths] and identify the edges that maximize communication over time [@vectorclocks]. However, these works fail to account for the uncertainty in both the link structure and the the communication across links (as users are unlikely to propagate all information across a single edge). Our use of a probabilistic graph framework and transmission prior address these two cases of uncertainty.
Sampling Probabilistic Graphs
=============================
Let $G=\left\langle V,E\right\rangle$, be a graph where $V$ is a collection of nodes and $E \in V \times V$ is the set of edges, or relationships, between the nodes. In order to represent and reason about relationship uncertainty, we associate each edge $e_{ij}$ (which connects node $v_{i}$ and $v_{j}$) with a probability $P(e_{ij})$. Then we can define $\mathcal{G}$ to be a distribution of discrete, unweighted graphs. Assuming independence among edges, the probability of a graph $G \in \mathcal{G}$ is: $P(G) = \prod_{e_{ij} \in E} P(e_{ij}) \prod_{e_{ij} \notin E} \left[1 - P(e_{ij}) \right]$. Since we have assumed edge independence, we can sample a graph $G_S$ from $\mathcal{G}$ by sampling edges independently according to their probabilities $P(e_{ij})$. Based on this, we can develop methods to compute the *expected* shortest path lengths, betweenness centrality rankings, and clustering coefficients using sampling.
#### Probabilistic Average Shortest Path Length
Let $\rho_{ij}=\{v_{k_1}, v_{k_2}, ..., v_{k_q}\}$ refer to a *path* of $q$ vertices connecting two vertices $v_i$ and $v_j$, i.e., $v_{k_1}=v_i$ and $v_{k_q}=v_j$, and from each vertex to the next there exists an edge: $e_{k_i k_{i+1}} \in E$ for $i=[1, q-1]$. Let $V(\rho_{ij})$ and $E(\rho_{ij})$ refer to the set of vertices and edges respectively, in the path and let $|\rho_{ij}| = |E(\rho_{ij})|$ refer to the [*length*]{} of the path. Assuming connected graphs, for every unweighted graph $G\!\!=\!\!\left\langle V,E\right\rangle \in \mathcal{G}$ there exists a shortest path $\rho_{ij}^{min}$ between every pair of nodes $v_i, v_j \in V$. Letting $\mbox{SP}_{ij} = |\rho_{ij}^{min}|$, we can then define the average shortest path length in $G$ as: $\overline{\mbox{SP}}(G) = \frac{1}{|V|\cdot(|V|-1)}{\sum_{i \in V} \sum_{j \in V; j\neq i} \mbox{SP}_{ij}}$.
Now, when there is uncertainty about the edges in G, we can compute the [*expected*]{} average shortest path length by considering the distribution of graphs $\mathcal{G}$. For any reasonable sized graph, the distribution $\mathcal{G}$ will be intractable to enumerate explicitly, so instead we sample from $\mathcal{G}$ to approximate the expected value. More specifically, we sample a graph $G_s$ by sampling edges uniformly at random according to their edge probabilities $P(e_{ij})$. Each graph that we sample in this manner has equal likelihood, thus we can draw $m$ sample graphs $G_S=\{G_1, ..., G_m\}$ and calculate the expected shortest path length with the following: $$\mathbb{E}_{\mathcal{G}}\!\left[\: \overline{\mbox{SP}} \: \right] =\sum_{G\in\mathcal{G}} \overline{\mbox{SP}}(G) \cdot P(G) \simeq\frac{1}{m}\sum_{m} \overline{\mbox{SP}}(G_m)$$
Since the sampled graphs are unweighted, it takes $O\left(\left|V\right|\left|E\right|\right)$ time to compute $\overline{\mbox{SP}}$ for each sample [@Brandes01FasterBC]. This results in an overall cost of $O\left(m\cdot\left|V\right|\left|E\right|\right)$ to compute $\mathbb{E}_{\mathcal{G}}\!\left[\: \overline{\mbox{SP}} \: \right]$.
#### Sampled Centrality
Betweenness centrality for a node $v_i$ is defined to be the number of shortest paths between other pairs of nodes which pass through $v_i$: $BC_i = | \{ \rho_{jk}^{min} \in G : v_i \in V(\rho_{jk}) \; \wedge \; i \neq j,k\} |$. Vertices that contribute to the existence of many shortest paths will have a higher BC score than other nodes that contribute to fewer shortest paths, thus BC is used a measure of importance or centrality in the network. It is difficult to directly compare BC values across graphs since the number of shortest paths varies with graph size and connectivity. Thus, typically analysis focuses on [*betweenness centrality rankings*]{} (BCR), where the nodes are ranked in descending order of their BC scores and the node with the highest BC score is given a BCR of 1.
As discussed above, we can compute the shortest paths for each unweighted graph $G\in\mathcal{G}$, then we can also compute the BCR values for each unweighted graph $G\in\mathcal{G}$. We denote $\mbox{BCR}_{i}(G)$ as the betweenness centrality ranking for node $v_{i}$ in $G$. Then we can approximate the expected BCR for each node by sampling a set of $m$ graphs from $\mathcal{G}$: $$\mathbb{E}_{\mathcal{G}}\!\left[ \mbox{BCR}_i \right] \simeq\frac{1}{m}\sum_{m} \mbox{BCR}_i(G_m)$$ Again, since the sampled graphs are unweighted, it takes $O\left(\left|V\right|\left|E\right|\right)$ time to compute the BCR for each sample [@Brandes01FasterBC], resulting in an overall cost of $O\left(m\cdot\left|V\right|\left|E\right|\right)$.
#### Sampled Clustering Coefficients
Clustering coefficient is a measure of how the nodes in a graph cluster together [@clusteringcoefficient]. For a node $v_i$ with $N_i\!=\!\{v_{j_1}, ..., v_{j_n}\}$ neighbors (e.g., $e_{i j_1}\!\!\! \in \!\!\! E$), its clustering coefficient is defined as $\mbox{CC}_i \!=\! \frac{1}{|N_i|(|N_i|-1)}\sum_{v_j \in N_i} \sum_{v_k \in N_i, k\neq j} \mathbb{I}_E(e_{jk})$, where $\mathbb{I}_E$ is an indicator function which returns 1 if $v_j$ is connected to $v_k$. CC can be thought of as the fraction connected pairs of neighbors of $v_i$. We denote $\mbox{CC}_{i}(G)$ as the clustering coefficient for node $v_{i}$ in graph $G$. Similar to paths, we can compute clustering coefficients for every graph $G \in \mathcal{G}$. Thus we can approximate the expected CC for each node by sampling a set of $m$ graphs from $\mathcal{G}$: $$\mathbb{E}_{\mathcal{G}}\!\left[ \mbox{CC}_i \right] \simeq\frac{1}{m}\sum_{m} \mbox{CC}_i(G_m)$$ Under the assumption that the maximum degree in the graph can be bounded by a fixed constant (which is typical for sparse social networks), we can compute the clustering coefficient for a single graph in $O(\left|V\right|)$ time (i.e., $O(1)$ for each node), which results in an overall cost of $O(m\cdot \left|V\right|)$.
Probabilistic Path Length
=========================
In the previous section, we discussed how to extend the discrete notions of shortest paths and centrality into a probabilistic graph framework via expected values, and we showed how to estimate approximate values using sampling. While our sampling-based measures are valid and give informative results (see section 6 for details), they have two limitations which restrict their applicability.
First, the effectiveness of the approximation depends on the number of samples from $\mathcal{G}$. We note that [@sampleprobpaths] used a Hoeffding Inequality to show that relatively few samples are needed to compute an accurate estimate of independent shortest paths in probabilistic graphs. However, since our the calculation of BCR is based on the joint occurrence of shortest paths in the graph, this bound will not hold for our measures.
Second, since the expectation is over possible worlds (i.e., $G \in \mathcal{G}$), the focus on shortest paths may no longer be the best way to capture node [*importance*]{}. We note that in the discrete framework, where all edges are equally likely, the use of shortest paths as a proxy for importance implies a prior belief that shorter paths are more likely to be used successfully to transfer information and/or influence in the network. In domains with link uncertainty, the flow of information/influence will depend on both the [*existence*]{} of paths in the network and the [*use*]{} of those paths for communication/transmission. In a probabilistic framework, we have an opportunity to explicitly incorporate the latter, by encoding our prior beliefs about transmission likelihood into measures of node importance. Furthermore, although a probabilistic representation enables analysis of more than just shortest paths, as we note above, even to capture shortest paths the sampling methods described previously may need many samples to accurately estimate the joint existence of shortest paths. Thus, a measure that explicitly uses the edge probabilities to calculate most *probable* paths may more accurately highlight nodes that serve to connect many parts of the network. We discuss each of these issues more below.
#### Most Probable Paths
To begin, we extend the notion of discrete paths to probabilistic paths in our framework. Specifically, we can calculate the probability of the existence of a path $\rho_{ij}$ as follows (again assuming edge independence): $P(\rho_{ij})=\prod_{e_{uv}\in E(\rho_{ij})}P(e_{uv})$. Using the path probabilities, we can now describe the notion of the [*most probable*]{} path. Given two nodes $v_i,v_j$, the most probable path path is simply the one with [*maximum likelihood*]{}: $ \rho_{ij}^{ML} = \arg\!\max P(\rho_{ij})$. We can compute the most likely paths in much the same way that shortest paths are computed on weighted discrete graphs, by applying Dijkstra’s shortest path algorithm, but instead of expanding on the shortest path, we expand the most probable path. Thus, all most probable paths can be calculated in $O\left(\left|V\right|\left|E\right|+\left|V\right|^{2}\log\left|V\right|\right)$.
#### Transmission Prior {#transmission-prior .unnumbered}
Previous focus on shortest paths for assessing centrality points to an implicit assumption that if an edge connects two nodes that it can be successfully used for transmission of information and/or influence in the network. Although there has been work both in maximizing the spread of information in a network through the use of central nodes [@boragatti-netflow; @betweennesscentralityrandomwalks] and in the study of information propagation through the use of transmission probabilities [@transmissionprob], there has been little prior work that has incorporated transmission probabilities into node centrality measures. Centrality measures based on random walks and eigenvectors [@betweennesscentralityrandomwalks] implicitly penalize longer paths as they consider [*all*]{} paths between nodes in the network. However, in our framework we can incorporate transmission probabilities to penalize the probabilities of longer paths in the graph, in order to more accurately capture the role nodes play in the spread of information across multiple paths in the network.
Consider the case where there is one path of nine people where each edge has high probability of existence (e.g., 0.95) and another path of three people where the edge probabilities are all moderate (e.g., 0.70), both ending at node $v$. Here, the longer path is more likely to exist than the shorter path, but in this example we are more interested in which path is used to transfer a virus to $v$. Even when an edge exists (i.e., the relationship is active), the virus will not be passed with certainty to the next node, thus the *transmission probability* is independent of the edge probability. Moreover, when the transmission probability is less than 1, it is more likely that the virus will be transmitted across the shorter path, since the longer path presents more opportunities for the virus to be dropped. This provides additional insight as to why shortest paths have always been considered important—there is generally a higher likelihood of transmission if it is passed through fewer nodes in the network.
To incorporate transmission likelihood into our probabilistic paths, we assign a probability $\beta$ of success for every step in a particular path—corresponding to the probability that information is transmitted across an edge and is received by the neighboring node. If we denote $l$ to be the length of a path $\rho$, and $s$ to be the number of successful transmissions along the path, we can use a binomial distribution to represent the transmission probability across $\rho$ with: $$\mbox{SBin}(s|\beta)=\mbox{Bin}(s=l|l,\beta)= \beta^{l}$$ Here SBin corresponds to the case where the transmission *always* succeeds (i.e., across all edges in $\rho$). Using this binomial distribution as a prior allows us to represent the expected probability of information spread in an intuitive manner, giving us a parameter $\beta$ which we can adjust to fit our expectations for the information spread in the graph. Note that setting $\beta=1$ is equivalent to the most probable paths discussed earlier. The prior effectively [*handicaps*]{} longer paths through the graph. Although, there is a correlation between shortest (certain) paths and handicapped (uncertain) paths, these formulations are *not* equivalent, since the latter produces a different set of paths when the shortest paths have low probability of existence.
#### ML Handicapped Paths {#ml-handicapped-paths .unnumbered}
Now that we have both the notion of a probabilistic path, and an appropriate prior for modeling the probability of information spreading along the edges in the path, we can formulate the *maximum likelihood handicapped path* between two nodes $v_{i}$ and $v_{j}$ to be: $$\rho_{ij}^{MLH}=\arg\!\max_{\rho_{ij}}\left[P(\rho_{ij})\cdot\mbox{SBin}(\:|\rho_{ij}|\: |\:\beta )\right]\label{eq:ml}$$ To compute the most likely handicapped (MLH) paths, we follow the same formulation as the most probable paths, keeping track of the path length and posterior at each point.
In the MLH formulation, probable paths are weighted by likelihood of transmission, thus nodes that lie on paths that are highly likely and relatively short, will have a high BC ranking. To calculate BCR ranking based on MLH paths, we can use a weighted betweenness centrality algorithm. Specifically, we modify Brandes’ algorithm [@Brandes01FasterBC] to start with the path that has the lowest probability of occurrence to be the one to backtrack from, enabling computation of the betweenness centrality in $O\left(\left|V\right|\left|E\right|+\left|V\right|^{2}\log\left|V\right|\right)$.
\[sec:Theory\]Comparison with Discrete Graphs
---------------------------------------------
The formulation of MLH Paths has inherent benefits, most notably with its direct connection to the previously well-studied notions of shortest paths and betweenness centrality in discrete graphs. In fact, we can view a discrete graph $G$ as being a special case of probabilistic graph with edge probabilities: $$P(e_{ij})=\begin{cases}
1 & \mbox{if an edge exists}\\
0 & \mbox{if the edge does not exist}\end{cases}\label{eq:probstatic}$$ We denote the distribution of graphs defined by these probabilities as $\mathcal{G}_1$. Note that the only graph in $\mathcal{G}_1$ with non-zero probability is $G$—since if an edge exists in a discrete graph, then it exists with complete certainty, likewise, if an edge is not present, we are certain it does not exist, thus $P(G)=1$.
For every pair of nodes $v_i$ and $v_j$, the shortest path in the discrete graph ($\rho_{ij} \in G$) is equal to the most probable path discovered by the MLH algorithm ($\rho_{ij}^{MLH} \in \mathcal{G}_1$), for $0<\beta<1$.
In $\mathcal{G}_1$ every $P(e_{ij})$ is either $1$ or $0$, thus every case where $P(\rho_{ij})>0$ is precisely $P(\rho_{ij})=1$. If we choose the shortest path from the discrete graph, it will have length $l^* = |\rho_{ij}|$, and the MLH probability for the same path will be $\beta^{l^*}$. Clearly, if a longer path were chosen by MLH, its probability would be less than $\beta^{l^*}$, and we know that no shorter paths exist—since all paths shorter than $\rho_{ij}$ would involve an edge than did not exist in $G$ and thus would have probability 0 .
The betweenness centrality using shortest paths on a discrete graph $G$ can be equivalently calculated with most probable handicapped paths over $\mathcal{G}_1$, where edge probabilities are defined by Equation \[eq:probstatic\].
This follows directly from Thm 1.
Probabilistic Clustering Coefficient
====================================
We now outline a probabilistic measure of clustering coefficient that can be computed without the need for sampling. If we assume independence between edges, the probability of a triangle’s existence is equal to the product of the probabilities of the three sides. The expected number of triangles is then the sum of the triangles probabilities that include a given node $v_{i}$. Denoting $\mbox{Tr}_{i}$ to be the expected triangles including $v_{i}$: $\mathbb{E}_{\mathcal{G}} \left[\mbox{Tr}_{i} \right]=\!\!\sum_{v_{j},v_{k}\in N_i,v_{j}\neq v_{k}}\!\! \left[P\left(e_{ij}\right)\cdot P\left(e_{ki}\right)\cdot P\left(e_{jk}\right) \right]$. Denoting $\mbox{Co}_{i}$ to be the expected *combinations* (i.e., coexisting pairs) of the neighbors of $v_{i}$, we then get: $\mathbb{E}_{\mathcal{G}}\left[\mbox{Co}_{i}\right]=\!\!\sum_{v_{j},v_{k}\in N_i,v_{j}\neq v_{k}}\!\!\left[P\left(e_{ij}\right)\cdot P\left(e_{ki}\right)\right]$. We can then define the probabilistic clustering coefficient to be the expectation of the ratio $\mbox{Tr}_{i}/\mbox{Co}_{i}$, and approximate it via a first order Taylor expansion [@taylorcite]:
$$\mbox{CC}_{i} =\mathbb{E}_{\mathcal{G}}\left[\frac{\mbox{Tr}_{i}}{\mbox{Co}_{i}}\right] \approx \frac{{\displaystyle \mathbb{E}_{\mathcal{G}}\left[\mbox{Tr}_{i}\right]}}{{\displaystyle \mathbb{E}_{\mathcal{G}}\left[\mbox{Co}_{i}\right]}}$$
Assuming again that the maximum degree in the graph can be bounded by a fixed constant, we can compute the probabilistic clustering coefficient in $O(\left|V\right|)$ time ($O(1)$ for each node). Additionally, the probabilistic approximation to the clustering coefficient shares connections with the traditional clustering coefficients on discrete graphs.
The probabilistic clustering coefficients computed in $\mathcal{G}_1$, with probabilities defined by \[eq:probstatic\] for a discrete graph $G$, are equal to the discrete clustering coefficients calculated on $G$.
Any triangle from $G$ has probability 1 in $\mathcal{G}_1$, while any non-triangle in $G$ clearly has probability 0. The same is true for the combinations of pairs of neighbors. As such, the sums of the numerators and denominators will be equal for both clustering coefficient.
Experiments
===========
To investigate the performance of our proposed MLH and sampling methods for average path length, betweenness centrality and clustering coefficient, we compare to traditional baseline social network measures on data from Enron, DBLP, and Facebook. These datasets all consist of time-stamped [*transactions*]{} among people (e.g., email, joint authorship). We will use the temporal activity information to derive probabilities for use in our methods, and evaluate our measures at multiple time steps to show the evolution of measures in the three datasets.
Datasets
--------
For our analysis we first use the Enron dataset [@enroncleaned]. The advantage to this dataset is that it allows us to understand the effects of our probabilistic measures because key events and central people have been well documented [@EnronTimeline]. We consider the subset of the data comprised of the emails sent between employees, resulting in a dataset with 50,572 emails among 151 employees.
Our second dataset is a sample from the DBLP computer science citation database. We considered the set of authors who had published more than 75 papers in the timeframe 1967-2006, and the coauthor relationships between them. The resulting subset of data consisted of 1,384 nodes, with 23,748 co-authors relationships.
Our third dataset is from the Purdue University Facebook network. Specifically we consider one year’s worth of wall-to-wall postings between users in the class of 2011 subnetwork. The sample has 2,648 nodes with 59,565 messages.
Methodology
-----------
We compare four network measures for each timestep $t$ in each dataset. When evaluating at time $t$, each method is able to utilize the graph edges that have occurred up to and including $t$. As baselines, we compare to (1) an *aggregate* method, which at a particular time $t$ computes standard measures for discrete graphs (e.g., BCR) on the union of edges that have occurred up to and including $t$, and (2) a time *slice* method, which again computes the standard measures, but only considers the set of edges that occur within the time window $[t-\delta, t]$. For the Enron and Facebook, we used $\delta=14$ days and for DBLP, we considered $\delta=1$ year.
We then compare to the sampling and MLH measures. For both the probabilistic methods, we need a measure of relationship strength to use as probabilities in our model. Although any notion of relationship strength can be substituted at this step, in this work we utilize a measure of relationship strength based on decayed message counts. More specifically, we define two separate and distinct notions of connection between nodes: *edges* and *messages*. We define an edge $e_{ij}$ to be the unobservable probabilistic connection between two nodes, indicating whether the nodes have an active relationship. This is in contrast to messages: a message $m_{ij}$ is a concrete and directly measurable communication between two nodes $v_{i}$ and $v_{j}$, such as a wall posting or email, occurring at a specific time, which we denote $t( m_{ij} )$. We define the probability of of nodes $v_{i}$ and $v_{j}$ having an *active* relationship at the current timestep $t_{now}$, based on observing a message at time $t(m_{ij})$, to be the exponential decay of a particular message: Note that the *scaling* parameter $\lambda$ refers to the adjustment of the basic time unit (e.g. 7 days to 1 week), not the *rate* parameter which defines the exponential probability density function, which in this case is $1$. This allows for assigning a probability of $1$ to the case when $t\left(m_{ij}\right) = t_{now}$, but it also assigns reasonable probabilities (i.e., slows the decay) for messages that happened in the recent past, which could still indicate active relationships.
Now, we assume we have $k$ messages between $v_i$ and $v_j$, and any of the messages $m_{ij}^1, \dots, m_{ij}^k$ can contribute to the relationship strength, which is defined to be 1 minus the probability that none of them contribute: $$\begin{aligned}
P\left(e_{ij}^{t}|m_{ij}^{1},\dots,m_{ij}^{k}\right)\!&\!=\!&\!1 -
\prod_{k}\left(1-\mbox{Exp}\left(m_{ij}^{k}|t_{now}\right)\right)\\\end{aligned}$$
In order to choose a scaling parameter $\lambda$ for the exponential decay, we measured the average correlation from the sampling method BCR against the time slice ranking and aggregate method for each Enron employee, for different values of $\lambda$ (see Figure \[fig:lambda-setting\].a). Note that a $\lambda$ close to 0 corresponds to ‘forgetting’ a transaction quickly and is highly correlated with the slice method, while a large $\lambda$ corresponds to ‘remembering’ a transaction for a long time, giving it high correlation with the aggregate method. In order to balance between short term change and long term trends we set $\lambda$ to a ‘middle ground’ with $\lambda=28$ days. This applies to both the Enron and Facebook datasets. For DBLP, where we evaluate yearly, $\lambda$ is set to $2$ years to keep the ratio between time slice and $\lambda$ consistent between Facebook, Enron, and DBLP.
In order to choose a value for the $\beta$ parameter in the MLH method, we measured the average correlation of the BCR from the MLH method and compared them to the sampling, aggregate, and slice rankings for different values of $\beta$. We can see in Figure \[fig:aug-24\].b that as long as $\beta$ is non-zero, it has minimal effect on the correlations. For the experiments reported in this paper, we set $\beta=.3$. Note that omission of the prior (i.e., $\beta=1$) in will make the MLH paths similar to the slice paths, with added paths between vertices which are disjoint in a particular time slice.
The final parameter setting is the number of samples to consider in each of sampling-based measures. Earlier we discussed how we are computing the joint instances of shortest paths, and that the bound by [@sampleprobpaths] does not hold. Due of this, we exploit the small size of the Enron dataset and take 10,000 samples; however, with the two larger graphs we use a smaller sample size of 200 in order to make the experiments tractable.
Method Correlations on Enron Data
---------------------------------
In order to illustrate the differences between the four methods, we analyze their respective BCR on the Enron data for the time window ending August 14$^{th}$, 2001. Figure \[fig:aug-24\].c-e shows the correlations of employee BCR across a pair of methods: points on the diagonal green line indicate ‘perfect’ correlation between the rankings of two methods.
Figure \[fig:aug-24\].c shows that the MLH method closely matches the sampling method, with only a few nodes varying from the diagonal. However, a large number of nodes that the sampling method determines to have high centrality are missed by the slice method, due to the slice’s inability to see transactions that occurred prior to the evaluation time window. Additionally, we note that August $14^{th}$, 2001 is relatively late in the Enron timeline, which results in the aggregate method having little correlation with the sampling method, since the more recent changes are washed out by past transactions in the aggregate approach.
Local Trend Analysis
--------------------
### Lay and Skilling
Here, we analyze two key figures at Enron: Kenneth Lay and Jeffery Skilling. These two were central to the Enron scandal—as first Lay, then Skilling, and then Lay again, assumed the position of CEO. We can analyze the BCR for Lay and Skilling during these transition periods, as we expect large changes to affect both of them.
The first event we consider (marked by a vertical red line in Figure \[fig:layskilling\]) is [*December 13$^{th}$ 2000*]{}, when it was announced that Skilling would assume the CEO position at Enron, with Lay retiring but remaining as a chairman [@EnronTimeline]. In Figure \[fig:layskilling\].a, both the sampling method and the MLH method identify a spike in BCR for both Lay and Skilling directly before the announcement. This is not surprising, as presumably Skilling and Lay were informing the other executives about the transition that was about to be announced.
The time slice method (\[fig:layskilling\].c) produces no change in Lay’s BCR, despite his central role in the transition. Skilling shows a few random spikes of BCR, which illustrates the variance associated with using the time slices. The aggregate model (\[fig:layskilling\].d) fails to reduce Skilling’s BCR to the expected levels following the announcement—this is fairly early in time and we are already seeing the aggregate method’s inability to track current events based on its union of all past transactions. Both the sampling method and the MLH methods capture this; MLH has him return to an extremely low centrality, while sampling has fairly low with some variance.
The second event we consider (marked by the 2nd vertical red line in Figure \[fig:layskilling\]) is [*August 14$^{th}$ 2001*]{}, when, seven months after initially taking the CEO position, Skilling approached Lay about resigning [@EnronTimeline]. During the entirety of Skilling’s tenure, we see that Lay has a slight effect on the sample rankings but is not what would be considered a ‘central’ node. Not surprisingly, Skilling has a fairly high centrality during his time as CEO; both the sampling method and MLH method capture this.
Prior to the announcement of Lay’s takeover as CEO, the slice method still had no weight on him, despite his previous involvement with the first transition. Also, we note that the sampling, MLH, and slice methods all agree that after Lay’s initial spike from the Skilling resignation, he resumes having a lower centrality, which the aggregate method misses. In general, the sampling method seems to mirror the slice method, albeit with less variance, but it not as smooth as the MLH method, indicating the utility of considering most probable paths.
### Kitchen and Lavorato
Next we analyze Louise Kitchen and John Lavorato, who were executives [@enroncleaned] for Enron Americas, which was the wholesale trading section of Enron [@EnronAmericas]. They are notable because of the extraordinarily high bonuses they received as Enron was being investigated, and were also found to have a high temporal betweenness centrality using the method defined by [@tang-tempshortestpaths]. We can see in Figure \[fig:kitchenlavorato\] (a,c,e) the rankings of Kitchen and Lavorato, and can see the benefit of using the probabilistic framework’s ability to key in on centralities at *specific* times, rather than using the temporal definition *through* time proposed by [@tang-tempshortestpaths]. We see that while Lavorato might have gotten a large bonus, he is *only* important during Skilling’s tenure as CEO; his centrality drops noticeably otherwise. On the other hand, Kitchen had extremely high rankings throughout.
Here, we see that the slice method exhibits high variability, especially with Kitchen, while the aggregate cannot recognize Lavorato’s lack of importance after Skilling’s departure. The MLH method is able to smoothly capture Kitchen’s centrality, while keeping Lavorato important solely during Skilling’s CEO tenure.
### Facebook Centrality
Unlike the Enron dataset, the Purdue Facebook dataset does not have well-established ground truths, where we can use the known characteristics and behaviors of particular nodes for evaluation. However, we can examine aspects of a few representative nodes to illustrate the problems that lie with usage of the aggregate or static methods. First, we can see from Figure \[fig:kitchenlavorato\].d that $v_a$ (red) has a consistently high ranking in the slice method, which the MLH method captures (\[fig:kitchenlavorato\]b). However, this person has a declining ranking in the aggregate method, as the aggregate is unable to capture current events—past information in the aggregate graph results in many paths that bypass $v_a$, missing this central node in later timesteps.
The next person we consider is denoted by $v_b$ (green). In \[fig:kitchenlavorato\].d, we can see that the slice method initially identifies this person as having high centrality, then their BCR bottoms out, and then peaks a few times again approximately midway through the timeline. The MLH method also initially identify $v_b$ as central, with a degradation over time. In contrast, the aggregate method fails to detect the inactivity later in the timeframe and continues to give $v_b$ a high centrality ranking throughout the entire time window.
The final person we consider is denoted by $v_c$ (blue) in Figure \[fig:kitchenlavorato\]. We can see in \[fig:kitchenlavorato\].d that the slice method exhibits large variability for $v_c$, but that there are many slices in the middle to end of the timeframe where the node is identified as highly central. The aggregate method is unaware of this activity and ranks $v_c$ at a relatively low level throughout the timeseries. In contrast, the MLH method is able to recognize the node’s growing importance as time evolves, and do so much more smoothly than the slice method (\[fig:kitchenlavorato\].d). In doing so, the MLH method can find instances of high centrality when both discrete methods fail.
Global Trend Analysis
---------------------
In Figure \[fig:avg\_CC\], we report the average path lengths for the various measures: MLH paths, probabilistic shortest paths, the aggregate shortest paths and the slice shortest paths. Additionally, we report the average sampled clustering coefficient, the clustering coefficient approximation, and the aggregate and slice discrete clustering coefficients. These are done for each of the three datasets through time, and we investigate changes in these global statistics to understand what, if any, changes occur with respect to the *small world* network structure of the data [@clusteringcoefficient].
In Figures \[fig:avg\_CC\].a,c,e, we show the clustering coefficients for each of the three datasets. The aggregate graph significantly overestimates the amount of *current* clustering in the graph, while the slice method is highly variable, especially for Enron. In general, both probabilistic measures are in between the two extremes, balancing the effects of recent data and decreasing the long term effect of past information, with the MLH performing similarly to the sampled clustering coefficient, and even better on DBLP, where sampling undercuts the clustering (likely due to small sample size).
Next, in Figures \[fig:avg\_CC\].b,d,f, we examine the *shrinking diameter* of these small world networks [@Leskovec05graphsover]. Here, the aggregate underestimates the path length at a current point in time. We can see that the most probable paths closely follows the sampling results, with both lying between the slice and aggregate measures while avoiding the variability of the slice method.
Conclusions
===========
In this paper we investigated the problem of calculating centrality and clustering in an uncertain network, and analyzed our methods using time evolving networks. We demonstrated the limitation of using an aggregate graph representation to capture uncertainty in the network structure due to changes over time, as well as the limitation of using a slice-based representation due to its extreme variability. We introduced sampling-based measures for average shortest path and betweenness centrality, as well as measures based on the most probable paths, which are more intuitive for capturing network flow. We also outlined exact methods for the computation of most probable paths (and by extention, most probable betweenness centrality), and incorporated the notion of transmission probability. Additionally, we developed a probabilistic clustering coefficient and gave a first order Taylor expansion approximation for computation.
We provided empirical evidence on the Enron, DBLP, and Facebook datasets showing the sampling and MLH’s intuitive centrality rankings for the Enron employees and Facebook members, as well as the global properties for all three. The probabilistic centrality and clustering formulations are inherently smoother than the measures computed from discretized time slices, however they can reason about [*likely*]{} change in graph structure due to changes over time, unlike the aggregate method, which includes all past information. We see the MLH formulation is smoother than the sampling method, indicating that the most probable paths through the graph may be more important to consider than shortest paths. Finally, we note that our experiments used a relatively simple estimate of relationship strength for the edge probabilities in the network. In future work we will investigate alternative formulations of edge uncertainty.
Acknowledgements
================
This material is based in part upon work supported by the Intelligence Advanced Research Projects Activity (IARPA) via Air Force Research Laboratory contract number FA8650-10-C-7060. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, AFRL or the U.S. Government. Pfeiffer is supported by a Purdue University Frederick N. Andrews Fellowship.
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---
abstract: 'We demonstrate the use of optimal control to design two entropy-manipulating quantum gates which are more complex than the corresponding, commonly used, gates, such as CNOT and Toffoli (CCNOT): A 2-qubit gate called PE (polarization exchange) and a 3-qubit gate called COMP (polarization compression) were designed using GRAPE, an optimal control algorithm. Both gates were designed for a three-spin system. Our design provided efficient and robust NMR radio frequency (RF) pulses for [${}^{13}\!{\mathrm{C}_{2}}$]{}-trichloroethylene (TCE), our chosen three-spin system. We then experimentally applied these two quantum gates onto TCE at the NMR lab. Such design of these gates and others could be relevant for near-future applications of quantum computing devices.'
address:
- 'Computer Science and Engineering, The Hebrew University of Jerusalem, Israel 91904, yosiat@cs.huji.ac.il'
- 'Computer Science Dept., Technion - Israel Institute of Technology, Technion City, Haifa, Israel 3200008, ye1@cs.technion.ac.il'
- 'Computer Science Dept., Technion - Israel Institute of Technology, Technion City, Haifa, Israel 3200008, talmo@cs.technion.ac.il'
- 'Computer Science Dept., Technion - Israel Institute of Technology, Technion City, Haifa, Israel 3200008, yossiv@cs.technion.ac.il'
author:
- 'ATIA, YOSI[^1]'
- 'ELIAS, YUVAL'
- 'MOR, TAL'
- 'WEINSTEIN, YOSSI'
title: Quantum Computing Gates via Optimal Control
---
Introduction
============
One of the major challenges in building a quantum computer is to coherently control a large quantum system well enough to perform an arbitrary quantum algorithm.[@DiVincenzo00] Nuclear magnetic resonance (NMR) offers an excellent test bed for developing techniques to control quantum systems.[@RLL09] Applying a quantum algorithm in NMR requires RF pulses that manipulate the spin system through the various steps of the algorithm.
In NMR, optimal control is used for designing and optimizing experiments for various applications such as imaging,[@CSNDMA+86; @MMSA+69; @RZ+96; @XKZML+08] liquid and solid state spectroscopy,[@Khaneja2005; @RKG+02; @KRLG+03; @KSBRKSN+04; @KVKGN+05; @VKKSN+05; @SIMPSON+08] quantum computation,[@ZYK+08; @KGB+02; @SSKG+05; @KHSYSG+07] dynamic nuclear polarization (DNP),[@MTZN+08; @PHKG+08; @HYRC+08; @Khaneja+07] and algorithmic cooling in the solid state.[@BLMR+08] Pulses in NMR can be designed by SIMPSON (see Ref. ), which is, originally, an NMR simulation program. SIMPSON was lately expanded to implement the optimal control algorithm GRAPE.[@Khaneja2005; @SIMPSON+08; @RNLKL08; @NKGK10; @RLL09; @RJ12; @CPB12; @FLL12]
In this work, we designed, using SIMPSON, a polarization exchange (PE) pulse, on a 3-qubit system. We then also implemented that designed pulse experimentally. A second pulse, compression (COMP), which requires more time hence is more challenging, was also designed and implemented on the same experimental system. These gates were chosen since they manipulate (redistribute) the entropy of two or more spins; PE is important in NMR in general, and COMP performs 3-bit reversible entropy compression.[@Sorensen89; @SV99]
These two gates and similar ones are building blocks of algorithmic cooling (see Refs. ), hence similar gates were already designed for that purpose. The first attempts were based on directly designing a unitary operation, with no numerical optimization. See for instance Ref. , for compression, and Ref. for polarization transfer (and heat-bath cooling beyond Shannon’s entropy bound). However, these initial attempts suffered from a large error in the experimentally implemented gates. The first full algorithmic cooling experiment in solid state NMR, described in Ref. , utilized numerical optimization to design COMP and SWAP pulses for [${}^{13}\!{\mathrm{C}}$]{}-labeled malonic acid. Later, GRAPE was applied to this system to obtain higher pulse fidelity, permitting four rounds of algorithmic cooling (beyond the polarization of the heat-bath).[@BLMR+08] Motivated by this result, the optimized pulses designed here have recently been combined to demonstrate algorithmic cooling in liquid state NMR[@AEMW14; @AtiaMSc] (based on the work presented here). GRAPE has also been used to design pulses for up to seven qubits in crotonic acid.[@RNLKL08; @FLL12]
The rest of this work consists of the following sections: Chapter \[chap:OC4QGate\] highlights the advantages of the state-to-state optimal control approach for the design of quantum gates. Chapter \[chap:materials-and-methods\] describes the experimental apparatus and the process of pulse design and analysis. The designed optimized pulses and the resulting spectra are described in Chapter \[chap:Results\]. We discuss the results and future prospects in Chapter \[chap:Discussion\].
Optimal Control as a Tool for Designing Quantum Gates {#chap:OC4QGate}
=====================================================
In this section we briefly clarify the advantages of state-to-state design of quantum gates by means of the well-established optimal control methodology. We focus on the gates relevant for this paper, polarization exchange and compression.
Numerical Optimization of the Quantum Gates {#sec:Numerical-Opt-QGates}
-------------------------------------------
As mentioned above, researchers originally tried to implement a chosen unitary gate directly, with no numerical optimization.[@CVS01; @POTENT] However, due to the need to compansate for undesired free evolution under Ising or Heisenberg spin-coupling and chemical shifts, the resulting pulse sequences were quite long and each pulse added a non-negligible noise. For example, the compression is supposed to enhance the polarization from 100% to 150%, and in Ref. the obtained result was only around 122%. Similarly, multiple PE gates were applied in Ref. , and one of them had a fidelity of around 70% only. Numerical optimization allowed better results (see Ref. ) but their design was rather complicated and a better method was required.[^2]
State-to-state optimal control method has the advantage of fully using the redundancy allowed in gate design. Instead of choosing a unitary transformation, the designer only tells the optimization software (SIMPSON) the initial state and the desired final state, while giving the software full freedom in searching for the optimal pulse. This freedom is then somewhat limited by fixing the duration of the pulse, and by demanding robustness of the pulse to small changes.
Designing polarization exchange and compression using the state to state approach {#sec:PE-COMP-Gates}
---------------------------------------------------------------------------------
A SWAP can be implemented via three controlled-NOT gates, in case these are easier to implement. In NMR, applying a specific gate to two spins while doing nothing to all other spins is highly challenging (because spin coupling and chemical shifts are continuously present).
Polarization Exchange (PE) “gate” is a variant of SWAP in which we only want to interchange the z-components of the spins without caring about the phases, thus we have some phase redundancy. For example, equation \[eq:SWAP-variant\] displays a variant of SWAP which does not perform a good swap of the states, but is perfect for PE: $$\begin{aligned}
\label{eq:SWAP-variant}
{\ensuremath{\left|00\right\rangle}} \rightarrow \phantom{-}{\ensuremath{\left|00\right\rangle}} \nonumber \\
{\ensuremath{\left|01\right\rangle}} \rightarrow \phantom{-}{\ensuremath{\left|10\right\rangle}} \nonumber \\
{\ensuremath{\left|10\right\rangle}} \rightarrow \phantom{-}{\ensuremath{\left|01\right\rangle}} \nonumber \\
{\ensuremath{\left|11\right\rangle}} \rightarrow -{\ensuremath{\left|11\right\rangle}}{}\end{aligned}$$ (note the phase in the last line). Thus, PE is actually not a single gate but a family of gates.
Another source of redundancy arises when two z-states have the same probability, for example if the density matrix is diagonal, such as $diag(3,1,1,-1)/4$ then any unitary transformation inside the subspace spanned by ${\ensuremath{\left|01\right\rangle}}$ and ${\ensuremath{\left|10\right\rangle}}$ does not disturb the PE, hence this gives another source of redundancy. This opportunity can appear anywhere during the pulse.
Any unitary operator $U$ (obtained by applying a fixed Hamiltonian for some time) can be replaced by a different trajectory, e.g., $U=U_1U_2$ even if $U_1$ is arbitrary. This is a source of redundancy which is independent of the above sources; for example, both SWAP and the variant of SWAP from eq. \[eq:SWAP-variant\] can be applied by an infinite variety of implementations.
Due to all these sources of redundancy the variety of potential pulses is expected to be quite vast, especially if we use the maximal incrementation allowed by the software. Thus there is a very good probability of finding robust pulses.
We want our pulse to be short, in order not to accumulate dissipation or errors. This limits the variety of redundant pulses the optimization can choose from; the redundancy increases with the number of increments.
Compression on three spins puts the maximal possible polarization on one of them. Similar to PE, compression is a family of gates, because there is a redundancy in terms of the remaining polarization on the other spins. Even if we limit that redundancy, by maximizing first the polarization on one spin, and then on the next (while keeping the maximum on the first one of course), etc., still there is redundancy in terms of the phase, trajectory and possibly identical probabilities of z-states. Here is a compression gate that we call COMP: ${\ensuremath{\left|011\right\rangle}} \leftrightarrow {\ensuremath{\left|100\right\rangle}}.$ See Ref. for explanation of the compression performed using this gate and similar ones.
Materials and Methods {#chap:materials-and-methods}
=====================
The experiments were performed on a Bruker Avance III 600 spectrometer using a standard 5 mm BBO probe. The sample was [${}^{13}\!{\mathrm{C}_{2}}$]{}-TCE with paramagnetic reagent Cr(acac)$_3$ for increased $T_1$ ratios (see Ref. ) in CDCl$_3$ (chloroform-d) solution, comprising two [${}^{13}\!{\mathrm{C}}$]{} and one [${}^{1}\!{\mathrm{H}}$]{} qubits (see Figure \[fig:TCE\_with\_spectrum\_ch3\]).
\[!h\] ![ Three spins in [${}^{13}\!{\mathrm{C}_{2}}$]{}-TCE were utilized in the experiment: H, C2, and C1. In the table, the chemical shifts relative to the transmitter frequency are in the diagonal elements, and the J-couplings are in the off diagonal elements. Note that the broadcasting frequency of the carbon channel is between the Larmor frequencies of C1 and C2. The carbon spectrum is at the bottom, the proton spectrum is in the small frame. Remark: The carbons have different T$_2^*$, hence their peaks have different heights, although their integrals, and therefore their polarizations, are equal up to an error of 2%.[]{data-label="fig:TCE_with_spectrum_ch3"}](TCE_with_spectrum600.png "fig:")
GRAPE - GRadient Ascent Pulse Engineering
-----------------------------------------
GRAPE[@Khaneja2005] is an optimal control algorithm which designs shaped pulses contemplating to apply a state to state transformation, or a given unitary propagator. A pulse with random shape (i.e., random amplitudes and phases) is generated, and then monotonically improved by gradient ascent[^3] in respect to a target function. Robustness of the pulses to experimental parameters such as RF or primary magnetic field inhomogeneities is supported inherently by the algorithm. In this work we used the GRAPE implementation in SIMPSON.[@SIMPSON+08]
The GRAPE state-to-state algorithm seeks the optimal amplitudes and phases of the RF fields that transform a given initial density matrix $\rho(0)=\rho_{0}$ sufficiently close to a desired target density matrix $\rho_{desired}$. For a specified pulse duration $T$, GRAPE attempts to increase the overlap between $\rho(T)$ and $\rho_{desired}$ beyond a given threshold.
The total energy of the pulse might be limited by hardware or the sample allowed temperatures. GRAPE can consider physical constraints such as the the maximum intensity of the RF transceiver, and can enable robustness to deviation e.g., in the RF pulse homogeneity. See table \[SensTable\].
Measurement of system parameters.
---------------------------------
SIMPSON requires the full Hamiltonian of the system, namely, the J-coupling constants and the chemical shifts of the spins. J-coupling constants were found directly from the spins’ excitation (90$^\circ$) spectra, i.e., by using Bruker **zg** pulse program. However, for the carbons the Larmor frequency difference is not much greater than the J-coupling, therefore it does not provide a good approximation for the chemical shift. We preferred to calculate their chemical shifts by matching the measured spectrum in the lab to the predicted spectrum of TCE using SIMPSON in simulation mode. After collecting the parameters measured, the rotating frame Hamiltonian used for the optimization is (coupling constants and chemical shifts are in Hz):
$$H=2\pi\hbar\left(541.7 I_z^{C1}-541.7 I_z^{C2}+200.8I_z^{C2}I_z^H+103.1I_z^{C1}I_z^{C2}+9I_z^{C1}I_z^{H}\right)$$
\[SensTable\]
Once a pulse is generated it is possible to see how sensitive it is to various parameters by evaluating the target function of the pulse with parameters deviating from the nominal.[^4] Knowing the sensitivity of a parameter is utilized for determining how much effort should be made in tuning or measuring it. The required robustness of the pulse to this parameter is also deduced from the sensitivity. Table \[SensTable\] summarizes the sensitivity to different parameters of the compression pulse, with expected nominal fidelity of 0.9985.
Table \[tbl:TCET1T2\] summarizes the T$_1$ and T$_2^*$ values of the spin system. T$_2^*$ was measured directly from the spectrum (single scan, line-width at half maximum). T$_1$ of each spin was measured in a different experiment, a few days earlier, by a standard inversion recovery method (Bruker pulse program: **t1ir**[^5]). Typically, 17 logarithmically even spaced delays were used, and T$_1$ was found by fitting the data to the expected exponential model.
\[tbl:TCET1T2\]
Pulse design\[sec:PulseDesignDetails\]
--------------------------------------
The PE pulse was generated by SIMPSON according to the following input parameters:
- **Initial state:** $I_z^{C1}+I_z^{C2}+4I_z^H$ $\propto$ diag$(6,-2,4,-4,4,-4,2,-6)$
- **Final state:** $I_z^{C1}+4I_z^{C2}+I_z^H$ $\propto$ diag$(6,4,-2,-4,4,2,-4,-6)$
- **Max RF:** 2 kHz
- **Duration:** 6 msec
The compression pulse was generated with the following SIMPSON parameters:
- **Initial state:** $I_z^{C1}+I_z^{C2}+I_z^H$ $\propto$ diag$(3,1,1,-1,1,-1,-1,-3)$
- **Final state:** $\frac{3}{2}I_z^{C1}+\frac{1}{2}I_z^{C2}+\frac{1}{2} I_z^H + 2I_z^HI_z^{C2}I_z^{C1} $ $\propto diag( 3, 1, 1, 1, -1, -1, -1, -3 )$
- **Max RF:** 2 kHz
- **Duration:** 13 msec
Notice that in both experiments the final and initial states are degenerate allowing redundancy of potential pulses.
The computation of the pulses required hundreds of iterations of the optimization, and hours or even days of computer time, depending on the complexity of the pulse and its constraints. In order to reduce the calculation time, the pulse was first optimized without RF robustness, to reach the vicinity of the efficient pulses. Then, a robustness constraint was added to the optimization, improving the pulse further (but vastly increasing the calculation time of each iteration).
Although infinite number of possible compression pulses maximize the polarization of C1, several considerations were made when selecting the final density matrix. It is desirable that at the end of the compression, C2 and H would retain some of their polarization, in order to reduce the reset times of the hydrogen in the next steps of algorithmic cooling. Non-required density matrix terms with long thermalization time were avoided as they don’t vanish before the next PE step, and may hinder the cooling. Note that the coupling constants determine the transition times, hence choosing a final density matrix sets a lower bound on the pulse length, indirectly affecting the efficiency. We chose a final density matrix which is identical to the result of applying 3-bit-compression (see Ref. ) to the initial density matrix.
SIMPSON does not take dissipation mechanisms into account. This is one of the reasons the observed efficiency of the designed pulses in the lab is lower than predicted by SIMPSON. The dissipation rates of a system depend on its state. For example, when a spin is on the x axis it will experience both $T_1$ and $T_2$ relaxation, while on the z axis it will experience only $T_1$ relaxation. Pulses which apply the same unitary operator might also evolve the system in different paths that will cause a different amount of dissipation, and will therefore have different fidelities. Therefore, the uncertainty of the pulse fidelity in the lab was dealt with by generating several pulses that apply the same operation (state to state), and picking the pulse with best fidelity in the lab. In future work, an analysis of dissipation processes can be done in order to find an optimal path. A better pulse could then be prepared by tailoring multiple pulses (generated by SIMPSON) that evolve the system through states in the optimal path.
Reference spectrum and power calibration
----------------------------------------
Before each lab session, two key procedures were performed: acquiring reference spectra and calibrating the power levels of the pulses.
The efficiency of a pulse or an algorithm is determined by comparing the resultant polarization of the spins with their equilibrium polarization. The polarization is proportional to the integral over the spin’s spectral lines. These spectra were displayed earlier in Figure \[fig:TCE\_with\_spectrum\_ch3\]. We acquired the reference spectra immediately after the AC experiments to avoid any drift effects.
The pulse generated by SIMPSON is given in nutation frequency units (Hz), which should be converted to the attenuation of the transmitter in the lab (dB). When the pulse is applied on two channels, both should be calibrated. The carbon channel was calibrated using a 5 msec selective pulse, generated by SIMPSON, which rotates C2 to the x axis while keeping C1 on the z axis. The pulse was transmitted in different power levels. The power level selected is the one that gave the minimal integral over C1, while the integral over C2 was close to the corresponding integral in the spectrum of a calibrated hard excitation (90$^\circ$) pulse. Knowing the carbon calibration, the power level of the hydrogen in the polarization exchange pulse was calibrated by finding the power which maximizes the polarization transferred to C2. The robustness of the pulses to RF inhomogeneity was observed during the power calibration. For instance, a deviation of $\pm0.5$ dB (i.e., $\pm5.9$%) from the optimal carbon and hydrogen power levels, reduced the final polarization of C2 by only 1.6% (from 3.8 to 3.74).
Results {#chap:Results}
=======
We measured a spectrum corresponding to the PE pulse (followed by a readout pulse) applied onto the equilibrium state. The spectrum clearly exhibits a polarization increase of C2: the measured polarization of C1 and C2 are 0.98$\pm$0.02 and 3.76$\pm$0.02 respectively. The transfer efficiency is therefore 94%, and the loss of C1 polarization is close to our prediction. The spectrum is displayed in Figure \[fig:PolarizationSwap600\].
![ The spectrum of the carbons after applying polarization exchange, followed by a non-selective 90$^\circ$ pulse. The integrals of C1 and C2 are $0.98\pm0.02$ and $3.76\pm0.02$ respectively, compared to a calibrated **zg** spectrum. []{data-label="fig:PolarizationSwap600"}](PolarizationSWAP+90_600.png)
Figure \[fig:comp\_amp\_phase\] shows the phase and amplitude of the part of the compression pulse applied to the carbon channel. Note that the rapid modulation of the amplitude and phase during the pulse are not always feasible by the hardware, thus reducing the pulse efficiency. The Fourier transform of the pulse’s x component is shown in figure \[fig:comp\_amp\_shape\]. It is difficult to understand from this image how the pulse addresses the two carbons.
\[H\] ![The shape of the compression pulse on the carbon channel. In the top graph, the vertical axis is the RF amplitude, limited by the maximal power output of the RF transmitter, and the horizontal axis is the time (the total duration of the pulse is 13 msec). The bottom graph shows phases of the same pulse (in degrees).[]{data-label="fig:comp_amp_phase"}](AmpCarbon.png "fig:") ![The shape of the compression pulse on the carbon channel. In the top graph, the vertical axis is the RF amplitude, limited by the maximal power output of the RF transmitter, and the horizontal axis is the time (the total duration of the pulse is 13 msec). The bottom graph shows phases of the same pulse (in degrees).[]{data-label="fig:comp_amp_phase"}](phaseCarbon.png "fig:")
\[hf\] ![Fourier transform of the x component of the compression pulse applied to the carbons. The spectral width is about 50 kHz.[]{data-label="fig:comp_amp_shape"}](xfftcarbon.png "fig:")
Figure \[fig:CompEq\] shows the spectrum of the carbons following the compression pulse and a readout pulse. The polarizations of C1 and C2 were 2.76$\pm$0.02 and -0.631$\pm$0.02 respectively, while the SIMPSON prediction was 3.00 and -0.73. The efficiency of the compression pulse, which refers to the acquired polarization of C1 was 92% with respect to the ideal case.
![ The spectrum of the carbons after applying a compression pulse, followed by a non-selective 90$^\circ$ pulse. The integrals of C1 and C2 are $2.76\pm0.02$ and $-0.631\pm0.02$ respectively, compared to a calibrated **zg** spectrum. []{data-label="fig:CompEq"}](CarbonsCompOnEq.png)
Discussion {#chap:Discussion}
==========
In this work we demonstrated the design and experimental application of two highly non-trivial quantum gates which are also important building blocks of algorithmic cooling (AC) – a method for extracting entropy from qubits into the environment.[@BMRVV02; @FLMR04] The implementation was done by using optimal control in liquid state NMR — the state-to-state mode of SIMPSON.
GRAPE has been used for NMRQC for the last few years. The SIMPSON implementation of GRAPE generated efficient and robust enough pulses. The parts missing from SIMPSON - the experimental limitations, as well as dissipation processes and numerical errors, cause a gap between the efficiency of the pulses as expected by the optimization and the observed efficiency. The following factors may explain the gap between SIMPSON’s predicted pulses efficiency ($>99\%$), and the efficiency observed in the lab which was between 92% to 94%.
1. Dephasing - throughout the evolution of the density matrix during the pulses, the system may experience different dephasing which depends on its instantaneous state. Assuming the dephasing time constant is the spins’ average T$_2^*$, 290 ms, the polarization lost during a 6 msec pulse and a 13 msec pulse is roughly 2% and 4% respectively. Although GRAPE can integrate dissipation processes into the state to state optimization, they are not supported by SIMPSON. A deeper analysis of the evolution of the density matrix is required for a better estimation.
2. The probe and the rest of the hardware has limited bandwidth, see for example Refs. , and therefore attenuates and distorts frequencies that are far from the carrier wave. In our case, the distance between the control points of the shaped pulses is roughly a few micro-seconds, which translates to a bandwidth of $\approx$1 MHz. Therefore, highly modulated pulses may reach lower efficiencies than smooth pulses.
3. The number of the pulse’s control points seems to affect the observed efficiency. The maximal number of point allowed by SIMPSON on our computer are 5000. If we could increase the number of control points then perhaps the efficiency would also increase.
Possible avenues for further improving pulse fidelity:
**Pulse design** - An alternative to SIMPSON is introduced in Ref. , an algorithm which generates smooth pulses, easier for the hardware to implement. Another alternative was recently introduced,[@2ndOGRAPE+11] an improved version of GRAPE which converges faster by utilizing a quasi-Newton method. This version (called BFGS-GRAPE) was implemented in Matlab within the Spinach package. The dephasing during the pulse transmission can be minimized by utilizing algorithms which take dephasing into account, or by giving higher priorities to trajectories of the density matrix with the least dephasing.
**Hardware** - The fidelity of the pulses can be improved by utilizing a feedback circuit, as described in Ref. , which reduces the fidelity degradation due to hardware bandwidth limitation.
Our work paves the way to implementing multiple rounds of algorithmic cooling in solution,[@AEMW14; @AtiaMSc] by concatenating PE, compression (such as those designed above) and reset steps. The current work could also be a step towards employing nuclear magnetic resonance quantum computing (NMRQC) and AC in biomedical [${}^{13}\!{\mathrm{C}}$]{}-magnetic resonance spectroscopy and imaging. AC, and other NMRQC tools, may enhance in-vivo spectroscopy of slow metabolic processes, particularly in the brain, whereby [${}^{13}\!{\mathrm{C}}$]{}-labeled metabolites (e.g., amino acids) are produced.[@EGMW11]
Acknowledgments
===============
This work was supported in part by the Wolfson Foundation, and by the Israeli MOD Research and Technology Unit. The work of T.M. was also supported in part by FQRNT through INTRIQ, and by NSERC.
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[^1]: Corresponding author.
[^2]: Related results were obtained for two-qubit gates involving superconducting qubits, using a numerical optimization approach based on simulated annealing.[@LCP08]
[^3]: Not to be confused with a magnetic field gradient used in NMR experiments.
[^4]: Sensitivity that originates from thermalization or dephasing will not be accurately estimated since SIMPSON does not take them into account. Also, sensitivity to a combination of variables was not tested.
[^5]: In **t1ir** (inversion recovery) sequence, a 180$^\circ$ pulse is applied, followed by a varying delay $\tau$, a 90$^\circ$ pulse and a data acquisition.
|
---
author:
- 'M. T. Beltrán, L. Olmi, R. Cesaroni, E. Schisano, D. Elia, S. Molinari, A. M. Di Giorgio, J. M. Kirk, J. C. Mottram, M. Pestalozzi, L. Testi,'
- 'M. A. Thompson'
date: 'Received date; accepted date'
title: 'A study of the high-mass star-forming region G29.96$-$0.02'
---
Introduction
============
The G29.96$-$0.02 star-forming region (hereafter G29-SFR), located at a distance of 6.2 kpc (Russeil et al. [@russeil11]), is a well-studied high-mass star-forming cloud which falls in one of the two Science Demonstration Phase (SDP) fields observed by the ESA [*Herschel*]{} Space Observatory (Pilbratt et al. [@pilbratt10]) for the [*Herschel*]{} Infrared GALactic plane survey (: Molinari et al. [@molinari10]). is a [*Herschel*]{} key project aimed at mapping the Galactic plane in five photometric bands (70, 160, 250, 350, and 500 $\mu$m). Figure \[ir\] shows the cloud as seen in different wavelengths, from 3.6 to 500 $\mu$m, by [*Spitzer*]{} and [*Herschel*]{}.
This cloud is dominated by IRAS 18434$-$0242, the brightest source from 24 to 500 $\mu$m (Fig. \[ir\]; Kirk et al. [@kirk10]), and one of the brightest radio and infrared sources in the Galaxy. This source is associated with a cometary region (hereafter G29-UC: Cesaroni et al. [@cesa94]; De Buizer et al. [@debuizer02]) and with a Hot Molecular Core (hereafter G29-HMC) located right in front of the cometary arc (Wood & Churchwell [@wood89]; Cesaroni et al. [@cesa94], [@cesa98]). The G29-HMC core, which has been mapped in several tracers (Cesaroni et al. [@cesa98]; Pratap et al. [@pratap99]; Maxia et al. [@maxia01]; Olmi et al. [@olmi03]; Beuther et al. [@beuther07]; Beltrán et al. [@beltran11]), shows a velocity gradient approximately along the east-west direction, which has been interpreted as rotation of a huge and massive toroid (4000 AU of radius and 88 $M_\odot$ at a distance of 6.2 kpc: Beltrán et al. [@beltran11]).
The G29-SFR cloud also contains a filament seen in absorption in the [*Spitzer*]{} images (Fig. \[ir\]) and in emission in the SCUBA Massive Pre-/Proto-cluster core Survey (SCAMPS: Thompson et al. [@thompson05]) at about $2'$ east of the G29-UC region (see [*Spitzer*]{} image at 8 $\mu$m in Fig. \[ir\]). This Infrared Dark Cloud (IRDC) has been extensively studied at high-angular resolution in dust continuum emission and NH$_2$D by Pillai et al. ([@pillai11]), who have resolved, with an angular resolution better than $5''$, the dust and line emission of the filament into multiple massive cores with low temperatures, $<20$ K, and a high degree of deuteration. These findings support the idea that this massive IRDC is in a very early stage of evolution, and could be in a pre-cluster phase. Only the brightest millimeter continuum core shows signs of high-mass star-formation activity, as indicated by the point source already visible at 24 $\mu$m that is driving a molecular outflow. That no active star formation has been detected in other parts of this IRDC (Pillai et al. [@pillai11]) supports the idea of this extincted filament being in a very early evolutionary phase.
As just seen, the G29-SFR cloud represents an ideal laboratory to study star formation because young stellar objects in different evolutionary stages and different masses are embedded in it. In this paper, we present a far-infrared (FIR) study of this cloud using the data in the 2 PACS and 3 SPIRE photometric bands, centered at 70, 160, 250, 350, and 500 $\mu$m. Our goal is to identify the FIR sources associated with this high-mass star-forming region and estimate their physical properties (mass, temperature, luminosity, and density) together with the Clump Mass Function (CMF) of the cloud. Combining the data with ${\it Spitzer}$ and radio continuum observations, we will investigate the evolutionary stage of the sources and their distribution in the cloud, and the physical parameters of the associated regions. Finally, we will derive the star formation efficiency and star formation rate in this cloud. This work complements the other wide-field studies carried out as part of the Hi-GAL SDP (e.g. Bally et al. [@bally10]; Battersby et al. [@battersby11]; Olmi et al. [@olmi13]).
Source selection
================
The first step to identify the Hi-GAL sources associated with the G29-SFR cloud is to define the limits of the molecular cloud. To study the distribution of the gas in the region we have used the $^{13}$CO () data of the Boston University–Five College Radio Astronomy Observatory Galactic Ring Survey (GRS: Jackson et al. [@jackson06]). Towards the direction of the G29-UC region, the $^{13}$CO () emission shows relatively narrow components at $\sim$8, 49, and 68 , and a much broader component from $\sim$90 to 110 . Taking into account that the systemic velocity of high-density tracers, such as NH$_3$ or CH$_3$CN, observed towards the G29-HMC core is $\sim$98–99 (Cesaroni et al. [@cesa98]; Beltrán et al. [@beltran11]), we selected the latter broad velocity component to determine the distribution of the gas in the cloud. The $^{13}$CO () emission has been averaged over the 95–105 velocity interval and compared with the Hi-GAL 250 $\mu$m emission. As one can see in Fig. \[co-sou\], the gas and dust emission are very well correlated. The G29-SFR cloud has been defined as the region contained approximately within the contour at 10-15$\%$ of the $^{13}$CO peak emission (5 K) and that at 7$\%$ of the 250 $\mu$m peak emission (36311 MJy/sr). Only the sources falling inside this region have been assigned to the G29-SFR cloud.
Source extraction
-----------------
The source extraction and brightness estimation techniques applied to the maps in this work are similar to the methods used during analysis of the BLAST05 (Chapin et al. [@chapin08]) and BLAST06 data (Netterfield et al. [@netterfield09]; Olmi et al. [@olmi09]). However, important modifications have been applied to adapt the technique to the SPIRE/PACS maps. The method used here defines in a consistent manner the region of emission of the [*same volume*]{} of gas/dust at different wavelengths, thus differing from the source grouping and band-merging procedures described by Molinari et al. ([@molinari11]) and Elia et al. ([@elia10]). Candidate sources are identified by finding peaks after a Mexican Hat Wavelet type convolution is applied to all five SPIRE/PACS maps. Initial candidate lists from 70, 160 and $250\,\mu$m are then found and fluxes at all three bands extracted by fitting a compact Gaussian profile to the source. Sources are not identified at 350 and $500\,\mu$m due to the greater source-source and source-background confusion resulting from the lower resolution, and also because these two SPIRE wavebands are in general more distant from the peak of the source Spectral Energy Distribution (SED). Each temporary source list at 70, 160 and $250\,\mu$m is then purged of overlapping sources and then all three lists are merged. After selecting the sources based on their integrated flux and allowed angular diameter, a final source catalog is generated. In the next stage, Gaussian profiles are fitted again to all SPIRE/PACS maps, including the 350 and $500\,\mu$m wavebands, using the size and location parameters determined at the shorter wavelengths during the previous steps (the size of the Gaussian is convolved to account for the differing beam sizes). Since the volume of emission is basically defined using the 250 $\mu$m band, this method does not fully exploit the higher angular resolution available at the shortest wavelengths. The interested reader can find more details in Olmi et al. ([@olmi13]).
The total number of sources associated with the G29-SFR cloud is 198. The position of the sources in equatorial and galactic coordinates, their fluxes in the 5 photometric Hi-GAL bands, and their possible association with MIPSGAL 24 $\mu$m sources are given in Table 1.
Analysis
========
Spectral Energy Distribution fitting {#sedfit}
------------------------------------
To estimate the dust temperature $T$, the mass $M_{\rm env}$, and the luminosity $L_{\rm bol}$, of the sources associated with the G29-SFR cloud, we fitted their observed SED with a modified blackbody of the form $B_\nu(T)(1-e^{-\tau_\nu})\Omega_S$, where $B_\nu(T)$ is the Planck function at a frequency $\nu$ for a dust temperature $T$, $\tau_\nu$ is the dust optical depth taken as $\tau_\nu\propto \nu^{\beta}$, where $\beta$ is the dust emissivity index, and $\Omega_S$ is the source solid angle. The source size $\theta$, which is not deconvolved, was estimated at 160 $\mu$m by the source extraction process (Olmi et al. [@olmi13]). The masses were calculated assuming a dust mass absorption coefficient of 0.5 cm$^2$/g at 1.3 mm (Kramer et al. [@kramer03]) and a gas-to-dust ratio of 100. To check whether the SED fitting improved, we searched for counterparts of the sources at shorter wavelengths in the MIPSGAL 24 $\mu$m catalog (Shenoy et al. [@shenoy12]). The method used to associate and MIPSGAL sources, which was based on both a positional and a color criteria, is described by Olmi et al. ([@olmi13]). For the remaining sources or those sources saturated at 24 $\mu$m, we searched for a counterpart in the Wide-field Infrared Survey Explorer (WISE) catalog at 22 $\mu$m (Wright et al. [@wright10]). To associate a WISE source to a source, we arbitrarily chose the closest WISE source located at $<12''$, the WISE angular resolution at 22 $\mu$m (Wright et al. [@wright10]). Finally, for the remaining sources or those sources saturated at 22 $\mu$m, we searched for a counterpart in the Midcourse Space Experiment (MSX) catalog at 21 $\mu$m (Price et al. [@price01]). In this case, we arbitrarily associated the closest MSX source located at $<18\farcs3$, the MSX angular resolution at 21 $\mu$m (Price et al. [@price01]). We found 103 MIPSGAL sources not saturated at 24 $\mu$m, 11 WISE sources not saturated at 22 $\mu$m, and 6 MSX sources associated with the ones. The SED fitting was performed using the 5 bands for 157 sources. For these 157 sources having a counterpart at shorter wavelengths, including the additional point in the SED did not improve the fitting. For 13 sources, only the 160, 250, 350, and 500 $\mu$m bands were used. For these sources, the flux at 160 $\mu$m, $S_{160 \mu m}$, was $\leq
S_{250 \mu m}$ and including the $S_{70 \mu m}$ in the SED clearly worsen the fit. This indicates that the 70 $\mu$m emission is likely tracing a different source component, more associated with the central stellar object, than that traced by the emission at 160 to 500 $\mu$m, more associated with the extended envelope surrounding the central source. The 5 bands plus the 21 $\mu$m band of MSX were used in the SED fitting for 4 sources. In these cases, $S_{70 \mu m} > S_{160 \mu m}$ and including the flux at 21 $\mu$m, which is smaller than that at 70 $\mu$m, clearly improved the fitting. For 6 sources, the 5 bands plus the 22 $\mu$m band of the WISE were used in the fitting. For these sources, $S_{70 \mu m} > S_{160 \mu m}$ and $S_{70 \mu m} >S_{22 \mu m}$, and again, including the flux at a shorter wavelength improved the fitting. Finally, for 18 sources, the 5 bands plus the 24 $\mu$m MIPSGAL band were used. For these sources, $S_{70 \mu m} > S_{160 \mu m}$ and $S_{70 \mu m} >S_{24 \mu m}$, and as in the previous cases, including the flux at a shorter wavelength improved the fitting. The MSX flux at 21 $\mu$m, the WISE flux at 22 $\mu$m, and the MIPSGAL flux at 24 $\mu$m used for the SED fitting of these 28 sources is given in Table 2. Table 3 shows the values of $\theta$, obtained from the source extraction process, of $\beta$, $T$, $M_{\rm env}$, and $L_{\rm bol}$, obtained from the SED fitting, and of the surface density $\Sigma$, for the 198 sources. The surface density was calculated following the expression $\Sigma=M_{\rm env}/(\pi\times R^2)$, where the radius of the sources $R$ was obtained from their sizes, $\theta$, and following the expression $R=\theta/2\times d$, where $d$ is the distance to the G29-SFR cloud.
---------------------------------------------- ----------------------- ------------- ---------------------
Radius (pc) 0.36 (0.36) 0.34 (0.35) 0.37 (0.38)
Mass ($M_\odot$) 379 (115) 435 (172) 340 (86)
Surface density (gcm$^{-2}$) 0.24 (0.06) 0.27 (0.1) 0.22 (0.04)
Temperature (K) 29 (25) 22 (22) 33 (30)
Luminosity ($L_\odot$) $6.2\times10^3$ (470) 706 (247) $1\times10^4$ (713)
Luminosity-to-mass ratio ($L_\odot/M_\odot$) 23 (5) 6 (2) 34 (10)
---------------------------------------------- ----------------------- ------------- ---------------------
The source associated with the G29-UC region and G29-HMC core {#hmc}
--------------------------------------------------------------
Figure \[plateau\] shows an overlay of the continuum emission at 2.7 and 1.4 mm obtained with the IRAM-Plateau de Bure interferometer (PdBI) (Beltrán et al. [@beltran11]) on the maps at 70 and 160 $\mu$m towards the position of the G29-UC region and G29-HMC core. The source in our catalog is \#242. As seen in Table 1, this is the brightest source in all 5 bands. At 2.7 and 1.4 mm, the G29-UC region is outlined by continuum emission showing a cometary arc shape, while the G29-HMC core emission is visible westwards in front of the arc. The emission of the G29-HMC core is better resolved at 1.4 mm, where it shows a flattened structure. The peak of the 1.4 mm continuum emission (Beltrán et al. [@beltran11]), indicated with a white cross in Fig. \[plateau\], coincides with the G29-HMC core. As one can see in this figure, at 70 $\mu$m, the emission seems to be mainly associated with the G29-HMC core. In fact, the peak of the 70 $\mu$m emission coincides with that of the 1.4 mm continuum emission. At 160 $\mu$m, the emission also seems to be more associated with the HMC than with the region, although in this case the peak of the emission is located towards the north of the G29-HMC core. The angular resolution of the emission at 250, 350, and 500 $\mu$m is not enough to properly study with which component, the HMC or the region, this sub-millimeter emission is associated.
![SED of the source \#242, associated with the G29-UC region and G29-HMC core. The black circles and black square show the and MSX at 21 $\mu$m data, respectively, with error bars. The black solid line represents the best-fit modified blackbody.[]{data-label="sed"}](242-sedfit.eps){width="8cm"}
![Histograms of some parameters of the sources detected towards G29: [*a)*]{} radius of the sources; [*b)*]{} mass; [*c)*]{} H$_2$ surface density; [*d)*]{} temperature; [*e)*]{} luminosity; and [*f)*]{} luminosity-to-mass ratio. The dotted line in panel $c$ indicates the minimum surface density needed to form massive stars according to theory (Butler & Tan [@butler12]).[]{data-label="histo1"}](histo_1_paper.eps){width="8.5cm"}
From the SED fitting (Fig. \[sed\]), we derived a mass of $\sim$2880 $M_\odot$ for source \#242, for a dust temperature of 77 K, the highest of the sources in the G29-SFR cloud, a size of $\sim$$19''$, and a dust emissivity index of 0.8. The surface density is 2.3 gcm$^{-2}$, well above the theoretical threshold of 1 gcm$^{-2}$ (Krumholz & McKee [@krumholz08]) necessary for high-mass star formation to occur. The luminosity of this source is $\sim$$8\times10^5$ $L_\odot$ and is the highest in the whole cloud. Kirk et al. ([@kirk10]) constructed the SED of this source by using the SPIRE Fourier Transform Spectrometer data from 190 to 670 $\mu$m and archival data from 2.4 to 1.3 mm (see their Fig. 1). From the SED fitting, these authors obtained a temperature of $\sim$80 K, in agreement with our value, and a dust emissivity index of $\sim$1.7, twice the one that we obtained. The dust luminosity integrated under the fitted modified blackbody in the range 2–2000 $\mu$m is $1.6\times10^6$ $L_\odot$, assuming a distance of 8.9 kpc. The luminosity would be $\sim$$8\times10^5$ $L_\odot$ for a distance of 6.2 kpc, in agreement with our estimated $L_{\rm bol}$. As for the mass, Kirk et al. ([@kirk10]) estimate a mass of 1500 $M_\odot$, assuming a distance of 8.9 kpc, using the fitted dust temperature and the SCUBA 850 $\mu$m flux density (Thompson et al. [@thompson06]). The mass would be $\sim$730 $M_\odot$ for a distance of 6.2 kpc. This value is a factor $\sim$4 smaller than the one that we obtained from the SED fitting. Besides the different method used to estimate the mass, this difference could be accounted for, in part, by the different opacity coefficient (0.01 cm$^2$/g at 850 $\mu$m) and dust emissivity index ($\beta$=1.7) used by these authors.
Source physical parameters {#histo-sect}
--------------------------
Figure \[histo1\] shows the distribution of radii, masses, surface densities, temperatures, luminosities and luminosity-to-mass ratios of the sources. Table \[taverage\] shows the mean and median values for the same physical quantities. Beltrán et al. ([@beltran06]) observed a sample of southern hemisphere high-mass protostellar candidates at 1.2 mm with the SEST antenna. In the following we will confront the physical parameters obtained for the sources in the G29-SFR cloud with those of Beltrán et al. ([@beltran06]) because them carried out a detailed comparison of the values of their sources with those estimated in other millimeter continuum surveys. The mean and median values of 0.36 pc for the radius of the sources associated with the G29-SFR cloud suggest that these sources are probably clumps (e.g. Giannini et al. [@giannini12]) that will not form individual stars but multiple star systems or star clusters. Unfortunately, the [*Herschel*]{} observations do not have enough spatial resolution to resolve these clumps into individual cores or stars. These values of the radius are consistent with the mean and median values of 0.25 and 0.2 pc found by Beltrán et al. ([@beltran06]). The mean and median values of the mass are also consistent with the mean and median values of 320 and 102 $M_\odot$ found by Beltrán et al. ([@beltran06]) for their sample, and indicates that the sources associated with the G29-SFR cloud and detected by ${\it Herschel}$ are mostly massive objects. The mean temperature is in agreement with the mean temperature of 28 K found by Beltrán et al. ([@beltran06]), and with the value of 32 K found by Molinari et al. ([@molinari00]) for a sample of luminous high-mass protostellar candidates in the northern hemisphere.
The average and median values of the surface density, of 0.24 and 0.06 gcm$^{-2}$, are similar to the mean and median values of 0.4 and 0.14 gcm$^{-2}$ estimated by Beltrán et al. ([@beltran06]). These values are slightly lower than the minimum surface density needed, according to theory (Krumholz & McKee [@krumholz08]), to form massive stars. In a recent work, Butler & Tan ([@butler12]) find typical mass surface densities of 0.15 gcm$^{-2}$ for cores, and of 0.3 gcm$^{-2}$ for clumps in infrared dark clouds, some of which are likely to form massive stars. Butler & Tan ([@butler12]) consider the cores as structures of about 100 $M_\odot$ embedded in clumps. These cores, which are virialized and in approximate pressure equilibrium with the surrounding clump environment, are undergoing global collapse to feed a central accretion disk. On the other hand, the clump is defined as the gas cloud that fragments to form a star cluster. These authors propose that fragmentation in these clumps could be inhibited by magnetic fields rather than radiative heating and that the initial conditions of local massive star formation in the Galaxy may be better characterized by surface density values of $\sim$0.2 gcm$^{-2}$ rather than 1 gcm$^{-2}$. This would imply smaller accretion rates and longer formation timescales ($> 10^5$ yr) for massive stars than those predicted my McKee & Tan ([@mckee03]).
The mean luminosity estimated, $6.2\times10^3$ $L_\odot$, would correspond to a main-sequence star of spectral type B1 following Table 1 of Mottram et al. ([@mottram11]), and thus, it also indicates that the sources associated with the G29-SFR cloud are mostly high-mass sources. Note, however, that this value is an order of magnitude smaller than the average value of $6.7\times10^4$ $L_\odot$ obtained by Beltrán et al. ([@beltran06]) for a sample of massive protostellar candidates. This is not surprising, taking into account that the bolometric luminosities calculated by these authors are to be considered upper limits because estimated from the IRAS flux densities. The IRAS beam is so large ($\sim$2$'$) that when integrating the flux density for a single protostellar candidate, there might be an important contribution not only from other sources that may fall into such a large beam, but also from inter-clump diffuse emission. The latter contribution is subtracted out when doing the source extraction but is included if one simulates what would be seen with a larger beam like that of IRAS.
The luminosity-to-mass ratio, $L_{\rm bol}/M_{\rm env}$, is an important parameter for establishing the age of a source. This ratio is expected to increase with time as more gas is incorporated into the star that becomes more luminous. The mean and median $L_{\rm bol}/M_{\rm env}$ values for the sources in the G29-SFR cloud are 23 and 5 $L_\odot/M_\odot$, respectively, which are significantly lower than the average and median values of 99 $L_\odot/M_\odot$ obtained by Beltrán et al. ([@beltran06]). However, as already mentioned, this discrepancy could be due to the fact that the bolometric luminosities of the sources in the Beltrán et al. sample are likely upper limits because they were estimated from the IRAS fluxes.
Centimeter emission associated with the G29-SFR cloud {#cm}
-----------------------------------------------------
Figure \[nvss\] shows a zoom-in towards the central region of the G29-SFR cloud. In this figure, the 20 cm emission of the Multi-Array Galactic Plane Imaging Survey (MAGPIS: Helfand et al. [@helfand06]) is overlaid on the Hi-GAL 70 $\mu$m emission. The angular resolution of both sets of data is similar, which makes the comparison straightforward. The positions of the ten 21-cm sources associated with the cloud from the NRAO/VLA Sky Survey (NVSS: Condon et al. [@condon98]) catalog at 1.4 GHz are also indicated in the figure. NVSS source \#1 is associated with the G29-UC region and with source \#242. Table \[tnvss\] gives the coordinates, flux densities at 21 cm, and major and minor axes of the NVSS sources after deconvolving with the restoring beam of 45$''$ of the NVSS images. The source fluxes have been obtained from the NVSS catalog instead of estimating them directly from the MAGPIS map at 20 cm because of the better surface brightness sensitivity of NVSS. The deconvolved sizes for five out of ten sources are upper limits, which indicates that either the source is unresolved or that the emission is too large to properly fit it with just one Gaussian (Condon et al. [@condon98]). The latter is the case for NVSS source \#10, which, as seen in Fig. \[nvss\], is very extended and with a very low-level emission, and therefore difficult to fit with a Gaussian. Figure \[temp-cm\] shows the 20 cm continuum emission overlaid on the 70–350 $\mu$m color temperature, $T_{(70-350)}$, map. To calculate the color-color temperature map, we first smoothed the 70 $\mu$m map (that with the highest angular resolution: $9\farcs2$) to the resolution of the 350 $\mu$m map ($25''$), and then reprojected both maps to the same pixel and map size. These two wavelengths happen to bracket the peak of the SED and are hence most sensitive to temperature changes. Clearly, the colour temperature is a proxy for the dust temperature, but may differ significantly from the temperature estimate obtained by fitting the whole SED with a modified blackbody. As seen in Fig. \[temp-cm\], the positions of the NVSS sources, except for the very diffuse NVSS sources \#3 and 10, coincide with local maxima of the color-color temperature.
------- ------------- -------------- ------- ---------- ----------
1$^a$ 18 46 04.09 $-$2 39 19.1 2.38 26.6 20.4
2 18 46 17.11 $-$2 36 30.0 0.025 $<$26.3 $<$18.4
3 18 46 21.24 $-$2 38 20.3 0.035 63.5 31.2
4 18 46 09.83 $-$2 41 34.9 3.06 49.3 44.7
5 18 46 06.69 $-$2 42 20.8 1.55 127.3 50.1
6 18 46 00.89 $-$2 41 57.4 0.137 $<$14.2 $<$14.1
7 18 45 54.17 $-$2 42 39.0 0.435 35.6 24.5
8 18 46 03.00 $-$2 45 41.0 0.097 $<$80.3 $<$21.6
9 18 46 01.67 $-$2 46 01.6 0.052 $<$15.5 $<$15.3
10 18 46 10.25 $-$2 49 12.3 0.010 $<$125.9 $<$125.0
------- ------------- -------------- ------- ---------- ----------
\
$^a$ G29-UC and source \#242.
------- --------- -------- --------- ---------- ------ --------- ------ --
1$^a$ 0.35 274 2028 19 84 9.0 O6
2 $<$0.33 $>$32 $>$226 $>$0.23 0.88 $<$0.85 B0
3 0.67 11 93 0.08 1.2 2.9 B0
4 0.71 865 803 6 108 29 O5
5 1.2 152 258 1 55 46 O6.5
6 $<$0.21 $>$426 $>$1027 $>$3 4.8 $<$1.0 O9.5
7 0.44 311 607 2 15 5.5 O8.5
8 $<$0.63 $>$35 $>$171 $>$0.24 3.4 $<$4.3 09.5
9 $<$0.23 $>$136 $>$557 $>$0.96 1.8 $<$0.71 B0
10 $<$1.9 $>$0.4 $>$ 10 $>$0.003 0.35 $<$7.2 B0.5
------- --------- -------- --------- ---------- ------ --------- ------ --
\
$^a$ G29-UC and source \#242.
As seen in Fig. \[nvss\], the centimeter emission is well correlated with the 70 $\mu$m emission, even at the low level emission. Note how both the centimeter and the FIR emission trace the arcs seen eastwards of NVSS sources \#4 and 5. These arcs are shock fronts where hydrogen is ionized, and gives rise to the radio continuum. It is also possible that important shock gas coolants like the \[OI 63 $\mu$m\] line could be in part contaminating the PACS70 $\mu$m emission. The fact that the centimeter emission is so extended and well correlated with the dust emission would suggest that it is associated with a group of regions that are ionizing and disrupting the cloud. Assuming that the centimeter continuum emission comes from homogeneous optically thin regions, we calculated the physical parameters of the 10 NVSS sources (using the formalism of Mezger & Henderson [@mezger67] and Rubin [@rubin68]) and list them in Table \[tparam\]. Column 1 gives the NVSS number of the source (Table \[tnvss\]), column 2 the spatial radius $R$ of the region, which was determined from the deconvolved source size (Table \[tnvss\]), column 3 the source averaged brightness $T_B$, column 4 the electron density $n_e$, column 5 the emission measure $EM$, column 6 the number of Lyman-continuum photons per second $N_{\rm Ly}$, column 7 the mass of ionized gas $M_{\rm
ion}$, which was calculated assuming a spherical homogeneous distribution, and column 8 the spectral type of the ionizing source. The spectral type was computed from the estimated $N_{\rm Ly}$ and using the tables of Davies et al. ([@davies11]) and Mottram et al. ([@mottram11]), which are for Zero Age Main Sequence (ZAMS) stars. Note that if the 21 cm emission is optically thick, then $T_B$, $n_e$, $EM$, $N_{\rm
Ly}$, $M_{\rm ion}$, and therefore, the spectral type should be considered as lower limits. For the sources with upper limits for the deconvolved sizes (Table \[tnvss\]), $R$ and $M_{\rm ion}$ should be taken as upper limits, while $T_B$, $n_e$ and $EM$ as lower limits. As seen in Table \[tparam\], most of the sources are early B or late O types. However, the cloud would also contain 3 sources, with one of them being the G29-UC region (NVSS source \#1), with spectral types O5–O6.5. Therefore, it is possible that these massive sources, with their strong winds and radiation pressure, are disrupting and shaping the cloud. This effect may contribute to underestimate the number of ionizing photons and, in turn, the luminosities of the stars. Note that sources \#4 and 5, located at the head of the large arc-like structure seen towards the east, have spectral types O5 and O6.5, respectively.
Physical parameters as a function of the distance to the NVSS sources in the G29-SFR cloud
------------------------------------------------------------------------------------------
To check whether there is a variation of the source physical parameters as a function of the distance to the most massive sources in the G29-SFR cloud, we plotted the distribution of masses, surface densities, luminosities, temperatures, and luminosity-to-mass ratios of the sources as a function of the distance to the NVSS sources \#1, 4, 5, 6, and 8 (Fig. \[dg29\]). The NVSS sources \#2, 3, 7, and 10 have not been taken into account because are located close to the border of the cloud. NVSS source \#9 is located very close to NVSS source \#8, and therefore, the distributions should be very similar. The data have been binned in intervals of $\sim$80$''$. For NVSS source \#1 (G29-UC), only one source (\#242) is found in the first interval of $\sim$80$''$, which means that the first point in the plots takes into account only the physical parameters of this source.
The physical parameters of NVSS source \#1 (G29-UC) and its immediate surroundings have the highest values of all the centimeter sources in the G29-SFR cloud. This is evident in Fig. \[dg29\] when comparing NVSS source \#1 to sources \#4, 5, 6, and 8, but it is also true for the rest of NVSS sources not shown in this plot. Given the large error bars in Fig. \[dg29\], one can only see a marginal trend of the mass and surface density, which seem to decrease with the distance from NVSS source \#1. The surface density is above the minimum value of 0.2 gcm$^{-2}$ needed to form massive stars according to theory (Butler & Tan [@butler12]), up to a distance of $\sim$150$''$ from the G29-UC region. A similar marginal trend is seen for NVSS source \#4. Although in this case, the decrease in $M_{\rm env}$ is even less obvious, and the surface density is slightly above 0.2 gcm$^{-2}$ only in a small region ($\lesssim80''$) surrounding the source. Regarding the luminosity, temperature and luminosity-to-mass ratio, again, the highest values are found towards the NVSS source \#1 (G29-UC).
![Distributions of mass, surface density, temperature, luminosity, and luminosity-to-mass ratio as a function of the distance to the NVSS sources in the G29-SFR cloud. The NVSS number (Tables \[tnvss\] and \[tparam\]) is indicated in the lefthand upper corner of the upper panels. The dotted line in the surface density distributions indicates the minimum value needed needed to form massive stars according to theory (Butler & Tan [@butler12]). The first bin contains only one point and, thus, the standard deviation is zero.[]{data-label="dg29"}](distances_uchii.eps){width="8.5cm"}
The fact that the most massive and luminous sources in the cloud are located close to the strongest source in the G29-SFR cloud (\#242 or G29-UC) suggest that there is a privileged area for massive star formation in the cloud. Based on the central location of the G29-UC region inside the G29-SFR cloud (Fig. \[co-sou\]), this indicates that high-mass stars form preferentially at the center of the cloud, as expected. An inhomogeneous density distribution of the cloud, with higher density towards the center of the cloud (maybe already present as an initial condition), could be responsible for this source distribution. This is consistent with the findings of most millimeter continuum surveys.
24$\mu$m-dark versus 24$\mu$m-bright sources {#preproto}
--------------------------------------------
Because star formation does not occur simultaneously all over a cloud, one would expect to find young stellar objects in different evolutionary stages associated with the G29-SFR cloud. To search for differences in the evolutionary stage of the sources, we cross-correlated our sources with the [*Spitzer*]{} MIPSGAL 24 $\mu$m catalog. The last column in Table 1 indicates whether a source is associated or not with 24 $\mu$m emission. Obviously, we counted as associated those sources saturated at 24 $\mu$m, like for example the source \#242 (G29-UC). Based on this association, we divided the sources into two groups: those without a 24 $\mu$m counterpart, that we call 24$\mu$m-dark, and those with a 24 $\mu$m counterpart, that we call 24$\mu$m-bright. The former are expected to be the youngest sources in the cloud. As a result of this cross-correlation we discovered 81 sources not associated with 24 $\mu$m emission and 117 sources associated with it. As shown in Fig. \[co-sou\], both kind of Young Stellar Objects (YSOs) are uniformly distributed over the cloud.
![Histogram of the \[70–160\] color for 24 $\mu$m-bright ([*solid line*]{}) and 24 $\mu$m-dark ([*dashed line*]{}) sources.[]{data-label="70-160"}](histo_colors.eps){width="8.5cm"}
All the sources in our sample have been selected to be detected in all 5 photometric bands. Therefore, by definition, all the sources have been detected at 70 $\mu$m, which would suggest that most of them, if not all, are protostellar. However, this does not mean that there are no prestellar sources in the G29-SFR cloud (see Pillai et al. [@pillai11]). The analysis of the sources not detected at 70 $\mu$m, and likely prestellar, although being highly interesting, goes beyond the scope of the present study. To check whether 24$\mu$m-dark and 24$\mu$m-bright sources show any difference in their 70 $\mu$m fluxes, we plotted and histogram of the \[70–160\] color for both kind of sources (Fig. \[70-160\]). As seen in this figure, the \[70–160\] color of 24$\mu$m-dark sources is clearly smaller than those of the 24$\mu$m-bright ones. This indicates that the possible different evolutionary phase of the sources is also supported by the data.
Figure \[mips\] shows the distribution of radii, masses, surface densities, temperatures, luminosities and luminosity-to-mass ratios for 24$\mu$m-dark and 24$\mu$m-bright sources. Table \[taverage\] shows the mean and median values for the same physical quantities. One sees that the distributions of the two types of objects are different. A closer inspection of the data using the Kolmogorov-Smirnov (KS) statistical test shows that, except for the radius distributions, the probability of the mass, surface density, temperature, luminosity, and luminosity-to-mass ratio distributions being the same for 24$\mu$m-dark and 24$\mu$m-bright sources is very low ($P$ $\lesssim$ 0.004). Therefore, the physical properties of the two groups are statistically different. The temperature, luminosity, and, in particular, the luminosity-to-mass ratio are smaller for the 24$\mu$m-dark than for the 24$\mu$m-bright objects, while the mass and the surface density are higher. That $T_d$, $L_{\rm bol}$, and $L_{\rm
bol}/M_{\rm env}$ are smaller for 24$\mu$m-dark than for 24$\mu$m-bright sources is consistent with the former being in an earlier evolutionary phase. Figure \[mips\] also shows that a relatively large number of 24$\mu$m-dark and 24$\mu$m-bright sources have surface densities high enough to form massive stars according to theory (Butler & Tan [@butler12]).
![Histograms of [*a)*]{} radius of the sources; [*b)*]{} mass; [*c)*]{} H$_2$ surface density; [*d)*]{} temperature; [*e)*]{} luminosity; and [*f)*]{} luminosity-to-mass ratio, for 24 $\mu$m-bright ([*solid line*]{}) and 24 $\mu$m-dark ([*dashed line*]{}) sources. The dotted line in panel $c$ indicates the minimum surface density needed to form massive stars according to theory (Butler & Tan [@butler12]).[]{data-label="mips"}](histo_mipsgal_paper.eps){width="9cm"}
The most significant difference between the two groups is found in the value of $L_{\rm bol}/M_{\rm env}$. In fact, the mean and median value of $L_{\rm bol}/M_{\rm env}$ is $\sim$6 and $\sim$5 times lower for the 24 $\mu$m-dark sources compared to the 24 $\mu$m-bright ones, which supports our assumption that the sources not associated with 24 $\mu$m emission are in an earlier evolutionary phase.
![Distributions of [*a)*]{} radius of the sources; [*b)*]{} mass; [*c)*]{} H$_2$ surface density; [*d)*]{} temperature; [*e)*]{} luminosity; and [*f)*]{} luminosity-to-mass ratio as a function of the distance to the G29-UC region for 24 $\mu$m-bright ([*filled circles*]{}) and 24 $\mu$m-dark ([*open red circles*]{}) sources. The dotted line in the surface density distributions indicates the minimum value needed to form massive stars according to theory (Butler & Tan [@butler12]). The first bin contains only one point and, thus, the standard deviation is zero.[]{data-label="dg29-pre"}](plots_mipsgal_distance.ps){width="13.5cm"}
We also investigated whether the radii, masses, surface densities, temperatures, luminosities and luminosity-to-mass ratios of the two types of sources show any correlation as a function of the distance to the G29-UC region. Figure \[dg29-pre\] indicates that both groups show the same trends, that is, the mass, surface density, and luminosity of the sources marginally decrease when moving away from the G29-UC region, while the size, temperature and luminosity-to-mass ratio, except for the high values close to the G29-UC region, do no significantly change.
Discussion
==========
Evolutionary phase of the sources {#evol}
---------------------------------
To investigate the stability of the sources, we calculated their Jeans masses, $M_J$, and virial masses, $M_{\rm virial}$. $M_J$ was calculated following the expression $M_J = [T/10\,{\rm K}]^{3/2}\times[n_{\rm H_2}/1\times10^4\,{\rm
cm^{-3}}]^{-1/2}$, where the dust temperature $T$ was obtained from the SED fitting and the H$_2$ volume density $n_{\rm H_2}$ was calculated assuming that the sources have spherical symmetry (the size of the sources is that obtained from the source extraction process). $M_{\rm virial}$ was estimated from the line width, $\Delta V$, of $^{13}$CO (1–0) towards the position of each source following the expression of MacLaren et al. ([@macLaren88]), $M_{\rm virial} =
0.509\times d\times \theta\times \Delta V^2$, where $d$ is the distance in kpc, $\theta$ is the size of the source in arcsec obtained from the source extraction process, and $\Delta V$ is in . The choice of $^{13}$CO (1–0) to estimate the virial masses, which could be partially optically thick and therefore overestimate the line width, is based on the fact that it is the only molecular tracer covering the whole cloud. To have an idea of how large the overestimate of the line widths could be, we checked the value towards source \#242 (associated with the G29-UC region and G29-HMC core), which has been extensively observed in different molecular tracers. The line width estimated with $^{13}$CO is 6.2 and is similar to the values of $\sim$5.5 estimated in CS (5–4) and (7–6), and HCO$^+$ (3–2) with the JCMT and the IRAM 30-m telescopes (Olmi et al. [@olmi99]; Churchwell et al. [@churchwell10]). $M_{\rm virial}$ depends on the density profile, and for a power-law density distribution of the type $\rho \propto r^{-p}$, the virial mass should be multiplied by a factor $3(5-2p)/5(3-p)$, which is $\leq 1$ for $p< 3$. Thus, the values estimated should be taken as upper limits.
![Histogram of [*a)*]{} the mass-to-Jeans mass ratio and [*b)*]{} mass-to-virial mass ratio for 24 $\mu$m-bright ([*solid line*]{}) and 24 $\mu$m-dark ([*dashed line*]{}) sources. The dotted vertical line indicates [*a)*]{} $M_{\rm env}=M_J$ and [*b)*]{} $M_{\rm
env}=M_{\rm virial}$.[]{data-label="jeans"}](mjeans_mvirial_mipsgal.eps){width="7.5cm"}
Figure \[jeans\] shows the $M_{\rm env}$–$M_J$ ratio and the $M_{\rm
env}$–$M_{\rm virial}$ ratio for all the sources, 24 $\mu$m-bright and 24 $\mu$m-dark. As seen in this plot, almost all the sources have masses well above $M_J$. In particular, 90% of the 24 $\mu$m-bright sources and 96% of the 24 $\mu$m-dark ones have masses well above $M_J$. In fact, the mean and median values of the $M_{\rm env}$–$M_J$ ratio are 296 and 14 for 24 $\mu$m-bright sources, and 735 and 86 for 24 $\mu$m-dark sources. This indicates that most of the sources in the G29-SFR cloud would be gravitationally supercritical if only supported by thermal pressure, in which case, they should be collapsing. The $M_{\rm env}$–$M_{\rm virial}$ ratio confirms that an additional supporting agent, such as turbulence, is likely acting against gravity in these sources, because only 5% (6 out of 117 sources) of the 24 $\mu$m-bright sources and 7% (6 out of 81 sources) of the 24 $\mu$m-dark ones have masses above the virial mass. The mean and median values of the $M_{\rm env}$–$M_{\rm virial}$ ratio are 0.2 and 0.07 for 24 $\mu$m-bright sources, and 0.3 and 0.1 for 24 $\mu$m-dark sources.
This result seems to be in contrast with the results of other studies of high-mass star-forming clumps, where the mass of the clumps is found to be larger than the virial mass (e.g. Hofner et al. [@hofner00]; Fontani et al. [@fontani02]). López-Sepulcre et al. ([@lopez10]) findings for a sample of 29 IR-bright and 19 IR-dark high-mass cluster-forming clumps are similar to ours, although on average their objects are closer to virial equilibrium. What are the sources of uncertainty in our estimate of the $M_{\rm
env}$–$M_{\rm virial}$ ratio? The major problem is that very likely the $^{13}$CO emission is not tracing the same volume of gas as the 1.2 mm continuum emission. This means that the $^{13}$CO line width may not be representative of the gas contributing to $M_{\rm env}$. However, to allow for a mean value of $M_{\rm env}/M_{\rm virial}$$=1$, one should shift the distributions in Fig. 12b by an order of magnitude, which implies a decrease of the line width by a factor $\sim$3. This seems too much, as observations of different tracers with different resolutions in high-mass star forming regions reveal changes by only a few km/s, for line widths of several km/s. Another source of error could be the temperature estimate, which enters almost linearly into the calculation of $M_{\rm env}$. It is thus difficult to believe that this effect may contribute by more than 20–30%, by far less than the factor 10 required to match $M_{\rm
env}$ to $M_{\rm virial}$. Finally, density gradients may affect the estimate of $M_{\rm virial}$. Assuming a power-law density profile as steep as $\rho\propto R^{-2}$, with $R$ radius of the clump, our values of $M_{\rm virial}$ should decrease only by a factor 0.6 (see MacLaren et al. 1988), still not sufficient to justify the observed ratio $M_{\rm env}$–$M_{\rm virial}$.
We conclude that none of the previous effects can explain the distributions in Fig. 12b. However, it is possible that [*all*]{} of them contribute to the result. While this is certainly possible for a limited number of sources (especially those with $M_{\rm env}/M_{\rm virial}\lesssim 1$), it seems likely that $M_{\rm env}/M_{\rm virial}$ is indeed $<$1 for the majority of the objects.
Assuming that this is the case, it is interesting to note that of the 36 sources located at $\lesssim 4'$ of the G29-UC region, 14% (5 sources including source \#242: G29-UC) have $M_{\rm env} > M_{\rm virial}$. On the other hand, of the remaining 162 sources, which are located at $> 4'$, only 4% (7 sources), have $M_{\rm env} > M_{\rm virial}$. Despite the poor statistics, this result seems to suggest that the sources that should be undergoing collapse and forming stars are preferentially concentrated towards the dominant source in the G29-UC cloud.
![$L_{\rm bol}$–$M_{\rm env}$ ([*upper panel*]{}) and $L_{\rm bol}$/$M_{\rm env}$–$M_{\rm env}$ ([*lower panel*]{}) plots for 24 $\mu$m-bright ([*red diamonds*]{}) and 24 $\mu$m-dark ([*green diamonds*]{}) sources in the G29-SFR cloud. Black lines represent the evolutionary tracks of Molinari et al. ([@molinari08]) (see § \[evol\]). The different models are for different initial masses of 80, 140, 350, 700 and 1500 $M_\odot$ (from left to right).[]{data-label="tracks"}](newmol-tracks-pre-proto.eps){width="9cm"}
The fact that the sources associated with the G29-SFR cloud appear to be in different evolutionary stages is also suggested by the association or not with [*Spitzer*]{} 24 $\mu$m emission, as already discussed in § \[preproto\].
To check the validity of this evolutionary phase difference for the sources in the G29-SFR cloud, we decided to use the evolutionary sequence tool of Molinari et al. ([@molinari08]). These authors have developed an empirical model to describe the pre-main sequence evolution of YSOs in the high-mass regime based on an $L_{\rm bol}$–$M_{\rm env}$ diagram, where $L_{\rm bol}$ is the bolometric luminosity of the sources, and $M_{\rm env}$ the total envelope mass. Based on the model of collapse in turbulence supported cores of McKee & Tan ([@mckee03]), which describes the free-fall accretion of material onto a central source as a time-dependent process, Molinari et al. ([@molinari08]) have constructed evolutionary tracks in the $L_{\rm bol}$–$M_{\rm env}$ diagram. According to this evolutionary sequence, sources in different phases should occupy different regions of the $L_{\rm bol}$–$M_{\rm env}$ diagram. For the high-mass regime, the bolometric luminosity of a YSO evolving towards the ZAMS increases by several orders of magnitude during the accretion phase. Therefore, one would expect 24 $\mu$m-dark sources to have a lower $L_{\rm bol}$ than the 24 $\mu$m-bright ones for similar $M_{\rm env}$. Elia et al. ([@elia10]) prefer to use the $L_{\rm bol}/M_{\rm env}$ ratio versus $M_{\rm env}$ as a diagnostic, based on the fact that an earlier evolutionary stage source should have smaller $L_{\rm bol}/M_{\rm env}$ ratio than more evolved ones.
As seen in Fig. \[tracks\], 24 $\mu$m-dark and 24 $\mu$m-bright sources occupy different regions of the $L_{\rm bol}$–$M_{\rm env}$ and $L_{\rm bol}$/$M_{\rm
env}$–$M_{\rm env}$ diagrams, with 24 $\mu$m-dark sources having lower $L_{\rm
bol}$ and $L_{\rm bol}$/$M_{\rm env}$ for similar $M_{\rm env}$, as expected. This confirms that the sources not associated with 24 $\mu$m emission are indeed in an earlier evolutionary phase than those associated. In fact, almost all the 24 $\mu$m-dark sources occupy a lower part of the accretion phase of the Molinari et al. evolutionary tracks, while the 24 $\mu$m-bright ones are located closer to the ZAMS, as indicated by the end of the ascending tracks.
Embedded population in the G29-SFR cloud
----------------------------------------
As discussed in the previous section, most of the sources in the G29-SFR cloud seem to be in the main accretion pre-main sequence phase or early ZAMS phase (Fig. \[tracks\]). This seems to indicate that the population in the G29-SFR cloud, mostly massive sources, should be highly embedded. In a recent work, Faimali et al. ([@faimali12]) analyze Hi-GAL data on another massive star-forming region G305 and propose a far-IR color criterion to select massive embedded sources. According to these authors, the \[70–500\] and the \[160–350\] colors should be most sensitive to the embedded population. Based on the fact that the embedded massive protostars in G305, associated with typical signposts of massive star formation such as free-free emission, water and/or methanol masers, and 24 $\mu$m emission, are confined to an area of $L_{\rm bol}$-color plots, these authors propose that embedded massive star-forming sources, both prestellar and protostellar, should have \[70–500\]$\geq1$ and \[160–350\]$\geq1.6$ for $L_{\rm bol}>10^3\,L_\odot$. To check whether these selection criteria for embedded massive sources are valid for our sources, we plot the luminosity versus color in Fig. \[l-color\]. The distribution of sources is very similar to that found by Faimali et al. ([@faimali12]) for the sources in G305. In the G29-SFR cloud, we found 46 24 $\mu$m-bright and 7 24 $\mu$m-dark sources that satisfy the criterion for embedded massive star candidates, a number similar to that found by Faimali et al. ([@faimali12]) in G305. This would indicate that only $\sim$27% of the population in the G29-SFR cloud would be embedded massive star candidates. However, as previously mentioned, most of the sources in the G29-SFR cloud seem to be pre-main sequence sources in the main accretion phase or early ZAMS phase, and therefore, embedded. One possible explanation for this discrepancy could be that most of these sources in the G29-SFR cloud have $L_{\rm bol}<$10$^3\,L_\odot$, and therefore lie by definition outside the selection criterion area. However, by doing this, the selection criterion would miss those young massive embedded protostars in a very early evolutionary phase that have not yet reached their final luminosity (see Fig. \[tracks\]).
![Luminosity–color plots for the 24 $\mu$m-bright ([*red diamonds*]{}) and 24 $\mu$m-dark ([*green diamonds*]{}) sources in the G29-SFR cloud. The empty red and green diamonds indicate sources with $M_{\rm env}< 100\, M_\odot$. Dashed lines indicate the threshold of the area defined by Faimali et al. ([@faimali12]) for selecting embedded massive YSOs, at a luminosity $>10^3\,L_\odot$.[]{data-label="l-color"}](L-color-100mass.eps){width="8cm"}
The second problem with the Faimali et al. ([@faimali12]) selection criterion is that, as shown in Fig. \[l-color\], there are a few sources that are clearly not massive ($M_{\rm env}< 100\, M_\odot$) and have $L_{\rm
bol}>$10$^3\,L_\odot$ (see Fig. \[l-color\]) that would fall inside the massive embedded population area. If we lower the limit to $M_{\rm env}< 50 \,
M_\odot$, there are still 8 24 $\mu$m-bright sources that would satisfy the criterion. Therefore, all this suggests that the far-IR color selection criterion for embedded massive YSOs of Faimali et al. ([@faimali12]) cannot be applied in all the massive star forming regions.
The star formation efficiency and rate
--------------------------------------
Observations of OB associations and Giant Molecular Clouds indicate that the overall star formation efficiency, SFE=$M_{\rm stars}/(M_{\rm stars}+M_{\rm
cloud})$, is very low, $\sim$3–4% (Evans & Lada [@evans91]; Lada [@lada99]). To estimate the SFE in the G29-SFR cloud, we first need the total gass mass of the cloud, $M_{\rm cloud}$, and the mass of the stars, $M_{\rm stars}$. The former can be estimated from the data, while $M_{\rm
stars}$ can be calculated by assuming that the emission in the G29-SFR cloud is consistent with that of a stellar cluster. To check this, we calculated the Lyman continuum, $N_{\rm Ly}$, of the cloud by measuring the radio flux at 20 cm, and compared this value with the bolometric luminosity, $L_{\rm bol}$, of the cloud. $L_{\rm bol}$ was calculated integrating the emission, inside the same area used to estimate the centimeter flux, in the 5 bands and fitting the SED with a modified blackbody. The total radio flux at centimeter wavelengths is 30.6 Jy, which corresponds to $N_{\rm Ly}$=$1.08\times10^{50}$ s$^{-1}$. The total $L_{\rm bol}$ is $2.2\times10^{6}\,L_\odot$. For comparison, the sum of $L_{\rm
bol}$ of all the sources that fall inside the area used to estimate the radio flux at 20 cm is $1.1\times10^{6}\,L_\odot$. These values are consistent with the expected $N_{\rm Ly}$ and $L_{\rm bol}$ of a stellar cluster according to the simulations of a large collection ($10^6$) of clusters with sizes ranging from 5 to 500000 stars each (L. Testi, private communication; see Sánchez-Monge et al. [@sanchez12] for a description of the cluster generation). For each cluster simulated, the total mass, bolometric luminosity, maximum stellar mass and integrated Lyman continuum are computed. For a bolometric luminosity of $2.2\times10^{6}\,L_\odot$, 90% of the simulated clusters have a total stellar mass $M_{\rm stars}$ between 600 and 4170 $M_\odot$. The total gas mass of the cloud, estimated by fitting a modified blackbody to the integrated emission of the cloud, inside the same area used to estimate the radio flux at 20 cm, at the Herschel wavelengths, is $8\times10^{4}\,M_\odot$. Therefore, the overall SFE of the G29-SFR cloud ranges from 0.7 to 5%, as low as that estimated in other molecular clouds (Evans & Lada [@evans91]). For comparison, the sum of the masses of all the sources that fall inside the area used to estimate the centimeter flux is slightly smaller $3\times10^{4}\,M_\odot$, and the SFE slightly higher, from 2 to 12%.
{width="14cm"}
The star formation rate of the cloud can be estimated as SFR=($M_{\rm
cloud}\times$SFE)/$t$, where $t$ is the star formation timescale needed for the protostars to reach the ZAMS. To compare our study of the G29-SFR cloud with that of Faimali et al. ([@faimali12]), we assume the same timescale of 0.5 Myr used by these authors, which is based on a steady-state star formation model (Offner & McKee [@offner11]). The SFR obtained for the G29-SFR cloud ranges from 0.001 to 0.008 $M_\odot$yr$^{-1}$. These values are smaller than those of 0.01–0.02 $M_\odot$yr$^{-1}$ estimated by Faimali et al. ([@faimali12]) for the G305 cloud, but consistent with the values of $\sim$0.0002–0.001 $M_\odot$yr$^{-1}$ estimated by Veneziani et al. ([@veneziani12]) for the whole $l=30\degr$ SDP field, and with the SFRs of $\sim$0.0005 to $\sim$0.008 $M_\odot$yr$^{-1}$ estimated for Galactic regions by Chomiuk & Povich ([@chomiuk11]). The fact that the SFR of the Milky Way is of about 2 $M_\odot$yr$^{-1}$ (Chomiuk & Povich [@chomiuk11]), indicates that hundreds to a few thousands of molecular clouds similar to the G29-SFR cloud are needed to account for the Galactic star formation rate.
The clump mass function {#ms}
-----------------------
Figure \[imf\] shows the mass spectrum of the sources in the G29-SFR cloud. Olmi et al. ([@olmi13]) have analyzed the whole $l=30\degr$ SDP field and estimated a statistical mass completeness limit, from the 160 $\mu$m maps at the 80% confidence level, of 73 $M_\odot$ for a temperature of 20 K, a dust mass absorption coefficient $\kappa_{0} = 11$cm$^2$g$^{-1}$, evaluated at $\nu_0=c/250\,\mu$m, and a gas-to-dust ratio of 100 (Martin et al. [@martin12]), a dust emissivity index of 2, and a median distance for the whole field of 7.6 kpc. Assuming a distance of 6.2 Kpc for the G29-SFR cloud, the mass completeness limit is of $\sim$49 $M_\odot$.
If the source mass distribution can be represented by a power law of the type $dN/dM \propto M_{\rm env}^{-\alpha}$, then the histogram of the mass spectrum can be fitted with a straight line of slope $-\alpha$. The solid line in the figures corresponds to $\alpha=2.35$, i.e., the Salpeter ([@salpeter55]) Initial Mass Function (IMF), and the dashed line to $\alpha=1.70$, corresponding to the mass function of molecular clouds derived from gas, mainly CO, observations (e.g. Kramer et al. [@kramer98]). The dotted line corresponds to the best-fit power-law index of $\alpha=2.15\pm0.30$ obtained with a procedure that implements both the discrete and continuous maximum likelihood estimator for fitting the power-law distribution to data, along with a goodness-of-fit based approach to estimating the lower cutoff of the data (see Clauset et al. [@clauset09] and Olmi et al. [@olmi13] for a detailed description of this method). This lower cutoff will be indicated here as $M_{\rm inf}$, which will thus represent the value below which the behavior of the distribution departs from a power-law. Following Clauset et al. ([@clauset09]), we have chosen the value of $M_{\rm inf}$ that makes the probability distributions of the measured data and the best-fit power-law model as similar as possible above $M_{\rm inf}$. In order to quantify the difference between these probability distributions, the Kolmogorov-Smirnov statistics is used. The value of $M_{\rm inf}$ for the sources in the G29-SFR cloud is $\sim$$300\pm130~M_\odot$. This is well above the mass completeness limit. The right panel shows the normalized cumulative mass distribution of the 58 sources with masses above $M_{\rm inf}$.
The best-fit power-law index $\alpha$ of 2.15 obtained for the G29-SFR cloud is the same obtained by Olmi et al. ([@olmi13]) for the whole $l=30\degr$ SDP field. $M_{\rm inf}$ is consistent within the errors with the value of $200\pm79~M_\odot$ obtained for the whole field. The power-law index is also consistent with the value of 2.20 obtained by the same authors for $l=59\degr$, the second SDP field. $M_{\rm inf}$ for this field, $7.3\pm2.2~M_\odot$, is much lower than the value of $\sim$$300~M_\odot$ estimated for the G29-SFR cloud, but this is not surprising taking into account that the $l=59\degr$ region contains mostly low- to intermediate-mass sources (the median mass for this field is of about 2.1 $M_\odot$: Olmi et al. [@olmi13]). These values of the power-law index $\alpha$ agree with the typical values found by Swift & Beaumont ([@swift10]), for CMFs of both low- and high-mass star-forming regions. This suggests that from the shape of the CMF it is not possible to foresee a different evolution towards the IMF for high- and low-mass star-forming clumps (Olmi et al. [@olmi13]).
The value of $\alpha=2.15\pm0.30$ is also consistent within the errors with the value of 2.35 of the stellar IMF (Salpeter [@salpeter55]). The observational similarity between the CMF and the IMF, first noted by Motte et al. ([@motte98]) for the low-mass star-forming region $\rho$ Ophiuchi, has been since then observed in many other low-mass star-forming regions (e.g. Simpson et al.[@simpson08] and references therein). This similar behavior has inspired the idea that gravitational fragmentation plays a key role in determining the final mass of the stars, that is, the IMF, in clustered regions (Motte et al. [@motte98]). That the CMF of high-mass star-forming regions mimics the stellar IMF (this work; Beltrán et al. [@beltran06]) seems to suggest that also in this case, the fragmentation of massive clumps may determine the IMF and the masses of the final stars. In other words, the processes that determine the clump mass spectrum might be self-similar across a broad range of clump and parent cloud masses.
Conclusions
===========
We have conducted a far-infrared (FIR) study of the G29-SFR cloud using the data at 70, 160, 250, 350, and 500 $\mu$m aimed at identifying the sources associated with this high-mass star-forming region and estimate their physical properties.
A total of 198 sources have been detected in all 5 bands. The mean and median values of their physical properties are 0.36 and 0.36 pc for the radius, 379 and 115 $M_\odot$ for the mass, 0.24 and 0.06 gcm$^{-2}$ for the surface density, 29 and 25 K for the temperature, $6.2\times10^3$ and 470 $L_\odot$ for the luminosity, and 23 and 5 $L_\odot/M_\odot$ for the luminosity-to-mass ratio.
The G29-SFR cloud is associated with 10 NVSS sources and with extended centimeter continuum emission well correlated with the 70 $\mu$m emission. This suggests that the cloud would contain a group of regions that are ionizing and disrupting the cloud. Assuming that the centimeter continuum emission comes from homogeneous optically thin regions, we estimated that most of the NVSS sources would be early B or late O types. The cloud would also contain 3 sources, with one of them being that associated with the G29-UC region, with spectral types O5–O6.5. The study of the distribution of masses, surface densities, luminosities, temperatures, and luminosity-to-mass ratios of the sources as a function of the distance to the NVSS sources indicates that the most massive and luminous sources in the cloud are located close to the G29-UC region. This could suggest that there is a privileged area for massive star formation towards the center of the G29-SFR cloud.
There are 117 sources associated with 24 $\mu$m emission, called 24 $\mu$m-bright, and 87 sources not associated, called 24 $\mu$m-dark. Both groups are uniformly distributed over the cloud. The radius of 24 $\mu$m-dark and 24 $\mu$m-bright sources is similar, the temperature and luminosity are smaller for the 24 $\mu$m-dark than for the 24 $\mu$m-bright objects, and the mass and surface density are higher. The luminosity-to-mass ratio is $\sim$5–6 times lower for 24 $\mu$m-dark sources. The 24 $\mu$m-dark and 24 $\mu$m-bright sources occupy different regions of the $L_{\rm bol}$–$M_{\rm env}$ and $L_{\rm
bol}/M_{\rm env}$–$M_{\rm env}$ diagrams, with the 24 $\mu$m-dark sources having lower $L_{\rm bol}$ and $L_{\rm bol}/M_{\rm env}$ for similar $M_{\rm env}$, as expected. All this suggests that the sources not associated with 24 $\mu$m emission are in an earlier evolutionary phase than those associated. This is supported by the fact that the \[70–160\] color of 24 $\mu$m-dark sources is clearly smaller than that of the 24 $\mu$m-bright ones.
Almost all the sources in the G29-SFR cloud have masses well above the Jeans mass and would be gravitationally supercritical if only supported by thermal pressure. However, only $\sim$6% of the sources have masses above the virial mass, which confirms that an additional supporting agent, such as turbulence, might be acting against gravity in these sources. The percentage of sources with masses larger than the virial mass is clearly higher for those located at $\lesssim 4'$ of the G29-UC region. This suggests that the sources that should be undergoing collapse and forming stars are preferentially concentrated towards the dominant source in the cloud.
The overall SFE of the G29-SFR cloud ranges from 0.7 to 5%, and it is as low as that estimated in other molecular clouds. The SFR ranges from 0.001 to 0.008 $M_\odot$yr$^{-1}$ and is consistent with the values estimated for Galactic regions. To account for the SFR of 2 $M_\odot$yr$^{-1}$ of the Milky Way, hundreds to a few thousands of molecular clouds similar to the G29-SFR cloud would be needed.
The mass spectrum of the sources with masses above $300~M_\odot$, well above the completeness limit, can be well-fitted with a power law of slope $\alpha=2.15\pm0.30$, consistent with the values obtained by Olmi et al. ([@olmi13]) for the whole $l=30\degr$, associated with high-mass star formation, and $l=59\degr$, associated with low- to intermediate-mass star formation, SDP fields. The observational similarity of the CMF for low- and high-mass star-forming regions suggests that from the CMF itself is not possible to predict a different evolution of the clumps towards the IMF. The fact that the CMF of the G29-SFR cloud mimics, within the errors, the stellar IMF suggests a self-similar process which determines the shape of the mass spectrum over a broad range of masses, from stellar to cluster size scales.
It is a pleasure to thank Annie Zavagno for critically reading the manuscript. Hi-GAL data processing and analysis has been possible thanks to the Italian Space Agency support via contract I/038/080/0. SPIRE has been developed by a consortium of institutes led by Cardiff Univ. (UK) and including: Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA). PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF-IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESAPRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain). This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This research made use of data products from the Midcourse Space Experiment. Processing of the data was funded by the Ballistic Missile Defense Organization with additional support from NASA Office of Space Science. This research has also made use of 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.
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& & & & & & & & & &\
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& & & & & & & & & &\
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\
1 &18 46 06.05 &$-$2 41 18.3 &29.93 &$-$0.04 & 55$\pm$9 & 58$\pm$8 & 38$\pm$4 & 25$\pm$3 & 7.5$\pm$1.1 &N\
2 &18 45 51.92 &$-$2 42 23.8 &29.89 &$+$0.00 & 99$\pm$12 & 156$\pm$17 & 95$\pm$11 & 46$\pm$6 & 17$\pm$2 &Y\
4 &18 46 11.67 &$-$2 38 37.7 &29.98 &$-$0.04 & 0.59$\pm$0.11 & 24$\pm$5 & 36$\pm$8 & 24$\pm$5 & 14$\pm$3 &N\
5 &18 45 59.57 &$-$2 43 10.4 &29.89 &$-$0.03 & 0.70$\pm$0.13 & 19$\pm$5 & 14$\pm$3 & 9.8$\pm$2.2 & 2.8$\pm$0.7 &N\
6 &18 45 57.49 &$-$2 44 04.1 &29.87 &$-$0.03 & 0.62$\pm$0.11 & 15$\pm$4 & 16$\pm$3 & 10$\pm$2 & 4.3$\pm$1.0 &N\
7 &18 46 05.63 &$-$2 44 32.8 &29.88 &$-$0.06 & 0.72$\pm$0.13 & 14$\pm$4 & 12$\pm$3 & 6.6$\pm$1.7 & 2.2$\pm$0.6 &N\
8 &18 46 06.38 &$-$2 44 45.9 &29.88 &$-$0.07 & 0.83$\pm$0.15 & 12$\pm$1 & 8.2$\pm$0.9 & 4.8$\pm$0.7 & 2.0$\pm$0.3 &N\
9 &18 46 25.08 &$-$2 37 52.2 &30.02 &$-$0.08 & 0.34$\pm$0.06 & 4.1$\pm$0.6 & 6.2$\pm$0.8 & 4.3$\pm$0.6 & 1.7$\pm$0.3 &N\
11 &18 46 26.19 &$-$2 37 03.1 &30.03 &$-$0.08 & 1.9$\pm$0.3 & 6.5$\pm$0.8 & 4.7$\pm$0.6 & 2.0$\pm$0.3 & 0.7$\pm$0.1 &N\
12 &18 45 46.50 &$-$2 33 14.1 &30.01 &$+$0.09 & 0.36$\pm$0.07 & 3.8$\pm$1.2 & 9.5$\pm$1.2 & 6.4$\pm$0.9 & 2.8$\pm$0.4 &N\
13 &18 45 48.53 &$-$2 37 40.9 &29.95 &$+$0.05 & 0.48$\pm$0.09 & 8.9$\pm$1.3 & 12$\pm$2 & 7.2$\pm$1.1 & 3.0$\pm$0.5 &N\
14 &18 46 07.43 &$-$2 34 06.0 &30.04 &$+$0.01 & 3.8$\pm$0.43 & 10$\pm$1 & 8.6$\pm$1.0 & 4.7$\pm$0.6 & 1.8$\pm$0.2 &Y\
16 &18 46 41.10 &$-$2 36 28.9 &30.07 &$-$0.13 & 0.88$\pm$0.17 & 5.5$\pm$0.6 & 6.1$\pm$0.7 & 3.8$\pm$0.5 & 1.8$\pm$0.3 &N\
17 &18 46 40.03 &$-$2 41 49.3 &29.99 &$-$0.17 & 2.7$\pm$0.3 & 8.9$\pm$1.0 & 11$\pm$1 & 7.9$\pm$1.0 & 4.0$\pm$0.6 &Y\
18 &18 46 41.04 &$-$2 37 05.8 &30.06 &$-$0.14 & 0.05$\pm$0.05 & 2.8$\pm$1.8 & 3.8$\pm$2.3 & 2.6$\pm$1.6 & 1.4$\pm$0.8 &N\
19 &18 46 42.12 &$-$2 37 28.4 &30.06 &$-$0.14 & 0.71$\pm$0.13 & 6.1$\pm$0.8 & 5.7$\pm$0.7 & 3.0$\pm$0.4 & 1.2$\pm$0.2 &N\
20 &18 46 40.78 &$-$2 39 44.2 &30.02 &$-$0.16 & 0.42$\pm$0.08 & 3.5$\pm$0.4 & 6.4$\pm$0.7 & 4.5$\pm$0.6 & 2.1$\pm$0.3 &N\
21 &18 46 39.64 &$-$2 35 14.3 &30.08 &$-$0.12 & 0.85$\pm$0.15 & 9.6$\pm$1.1 & 6.0$\pm$0.7 & 2.7$\pm$0.4 & 1.1$\pm$0.2 &N\
22 &18 46 42.42 &$-$2 40 07.0 &30.02 &$-$0.16 & 1.8$\pm$0.25 & 7.7$\pm$0.9 & 5.7$\pm$0.6 & 2.5$\pm$0.3 & 0.9$\pm$0.1 &N\
24 &18 46 16.79 &$-$2 34 56.4 &30.05 &$-$0.03 & 0.85$\pm$0.11 & 4.8$\pm$0.6 & 5.3$\pm$0.6 & 3.2$\pm$0.4 & 0.7$\pm$0.1 &Y\
25 &18 46 31.69 &$-$2 37 13.6 &30.04 &$-$0.10 & 0.64$\pm$0.12 & 6.9$\pm$0.8 & 7.1$\pm$0.8 & 4.0$\pm$0.5 & 1.6$\pm$0.2 &N\
26 &18 46 15.19 &$-$2 34 27.6 &30.05 &$-$0.02 & 0.48$\pm$0.09 & 3.6$\pm$0.4 & 6.1$\pm$0.7 & 3.7$\pm$0.5 & 1.7$\pm$0.2 &N\
27 &18 46 40.23 &$-$2 34 35.7 &30.10 &$-$0.11 & 1.4$\pm$0.2 & 7.2$\pm$1.0 & 5.9$\pm$0.8 & 3.0$\pm$0.4 & 1.3$\pm$0.2 &N\
31 &18 46 36.02 &$-$2 42 40.3 &29.97 &$-$0.16 & 0.38$\pm$0.07 & 3.5$\pm$0.4 & 6.9$\pm$0.8 & 5.3$\pm$0.7 & 2.9$\pm$0.4 &N\
33 &18 45 43.88 &$-$2 37 55.5 &29.94 &$+$0.07 & 2.3$\pm$0.3 & 7.3$\pm$0.821 & 5.6$\pm$0.6 & 2.5$\pm$0.3 & 0.7$\pm$0.1 &Y\
34 &18 46 27.23 &$-$2 34 06.7 &30.08 &$-$0.06 & 0.71$\pm$0.13 & 5.8$\pm$0.7 & 5.3$\pm$0.6 & 2.8$\pm$0.4 & 1.0$\pm$0.1 &N\
35 &18 45 49.20 &$-$2 35 18.6 &29.99 &$+$0.07 & 1.8$\pm$0.2 & 6.7$\pm$0.7 & 5.1$\pm$0.6 & 2.2$\pm$0.3 & 0.8$\pm$0.1 &N\
37 &18 45 47.36 &$-$2 36 05.6 &29.97 &$+$0.07 & 1.1$\pm$0.2 & 6.3$\pm$1.1 & 5.0$\pm$0.9 & 2.9$\pm$0.5 & 1.2$\pm$0.2 &Y\
38 &18 46 52.32 &$-$2 39 16.4 &30.05 &$-$0.20 & 2.0$\pm$0.3 & 5.2$\pm$0.6 & 3.8$\pm$0.4 & 1.8$\pm$0.2 & 0.6$\pm$0.1 &N\
40 &18 45 48.46 &$-$2 34 47.7 &29.99 &$+$0.08 & 2.2$\pm$0.4 & 5.1$\pm$0.9 & 3.8$\pm$0.6 & 1.9$\pm$0.3 & 0.7$\pm$0.1 &N\
42 &18 46 22.46 &$-$2 34 06.8 &30.07 &$-$0.05 & 2.1$\pm$0.3 & 5.5$\pm$0.6 & 3.9$\pm$0.5 & 1.8$\pm$0.2 & 0.6$\pm$0.1 &Y\
44 &18 46 19.25 &$-$2 46 40.4 &29.88 &$-$0.13 & 3.4$\pm$0.5 & 7.3$\pm$0.9 & 5.0$\pm$0.6 & 3.2$\pm$0.4 & 1.4$\pm$0.2 &N\
45 &18 45 59.56 &$-$2 49 42.1 &29.79 &$-$0.08 & 0.54$\pm$0.10 & 6.1$\pm$0.7 & 7.7$\pm$0.9 & 4.5$\pm$0.6 & 1.8$\pm$0.2 &N\
48 &18 46 16.81 &$-$2 33 47.2 &30.06 &$-$0.02 & 3.3$\pm$0.4 & 5.8$\pm$0.6 & 3.5$\pm$0.4 & 1.6$\pm$0.2 & 0.5$\pm$0.1 &N\
49 &18 46 55.28 &$-$2 36 33.1 &30.10 &$-$0.19 & 1.8$\pm$0.3 & 5.1$\pm$0.6 & 4.1$\pm$0.5 & 2.0$\pm$0.3 & 0.8$\pm$0.1 &N\
54 &18 46 28.85 &$-$2 31 14.1 &30.12 &$-$0.05 & 2.0$\pm$0.2 & 3.5$\pm$0.4 & 2.7$\pm$0.3 & 1.1$\pm$0.1 & 0.2$\pm$0.03 &N\
55 &18 46 58.48 &$-$2 35 22.8 &30.12 &$-$0.19 & 0.67$\pm$0.12 & 4.1$\pm$0.5 & 4.4$\pm$0.5 & 2.6$\pm$0.3 & 1.2$\pm$0.2 &N\
61 &18 46 49.82 &$-$2 33 42.6 &30.13 &$-$0.14 & 0.45$\pm$0.11 & 6.7$\pm$0.7 & 5.1$\pm$0.6 & 2.6$\pm$0.3 & 1.0$\pm$0.1 &N\
62 &18 46 28.18 &$-$2 50 00.2 &29.84 &$-$0.19 & 1.0$\pm$0.1 & 4.0$\pm$0.4 & 3.5$\pm$0.4 & 1.9$\pm$0.2 & 0.9$\pm$0.1 &N\
64 &18 46 37.41 &$-$2 45 28.8 &29.93 &$-$0.19 & 1.1$\pm$0.1 & 5.0$\pm$0.6 & 4.6$\pm$0.5 & 2.6$\pm$0.3 & 1.0$\pm$0.1 &Y\
65 &18 45 56.00 &$-$2 49 46.6 &29.79 &$-$0.07 & 2.3$\pm$0.3 & 6.4$\pm$0.7 & 4.4$\pm$0.5 & 2.0$\pm$0.3 & 0.7$\pm$0.1 &Y\
66 &18 46 28.51 &$-$2 49 15.7 &29.86 &$-$0.18 & 1.4$\pm$0.2 & 3.9$\pm$0.5 & 3.3$\pm$0.4 & 1.8$\pm$0.2 & 0.8$\pm$0.1 &N\
69 &18 46 17.94 &$-$2 51 46.3 &29.80 &$-$0.16 & 6.8$\pm$0.8 & 9.6$\pm$1.1 & 5.5$\pm$0.6 & 2.5$\pm$0.3 & 0.9$\pm$0.1 &Y\
70 &18 46 27.79 &$-$2 51 34.9 &29.82 &$-$0.20 & 0.32$\pm$0.06 & 3.1$\pm$0.4 & 7.7$\pm$0.9 & 6.2$\pm$0.8 & 3.3$\pm$0.5 &N\
73 &18 46 27.25 &$-$2 50 50.9 &29.83 &$-$0.19 & 0.42$\pm$0.08 & 2.1$\pm$0.3 & 4.7$\pm$0.5 & 3.5$\pm$0.5 & 2.0$\pm$0.3 &N\
74 &18 45 53.61 &$-$2 49 27.9 &29.79 &$-$0.06 & 0.28$\pm$0.19 & 7.5$\pm$0.9 & 5.5$\pm$0.6 & 2.6$\pm$0.3 & 0.9$\pm$0.1 &N\
75 &18 46 29.85 &$-$2 45 18.8 &29.92 &$-$0.16 & 0.39$\pm$0.13 & 4.3$\pm$0.5 & 3.1$\pm$0.3 & 1.4$\pm$0.2 & 0.4$\pm$0.1 &N\
77 &18 46 23.22 &$-$2 50 44.3 &29.82 &$-$0.18 & 1.9$\pm$0.3 & 4.0$\pm$0.5 & 3.3$\pm$0.4 & 1.6$\pm$0.2 & 0.6$\pm$0.1 &N\
79 &18 46 12.13 &$-$2 51 47.1 &29.79 &$-$0.14 & 1.7$\pm$0.2 & 8.1$\pm$0.9 & 5.9$\pm$0.7 & 2.7$\pm$0.4 & 0.7$\pm$0.1 &Y\
81 &18 46 40.58 &$-$2 45 45.8 &29.93 &$-$0.20 & 2.8$\pm$0.4 & 5.5$\pm$0.7 & 3.8$\pm$0.5 & 2.0$\pm$0.3 & 0.8$\pm$0.1 &Y\
83 &18 47 00.90 &$-$2 35 54.5 &30.12 &$-$0.20 & 0.63$\pm$0.11 & 2.8$\pm$0.3 & 3.1$\pm$0.3 & 1.9$\pm$0.3 & 0.8$\pm$0.1 &N\
86 &18 46 36.87 &$-$2 46 22.0 &29.91 &$-$0.19 & 0.13$\pm$0.05 & 2.0$\pm$0.4 & 2.1$\pm$0.4 & 1.1$\pm$0.2 & 0.3$\pm$0.1 &N\
87 &18 46 34.97 &$-$2 45 54.8 &29.92 &$-$0.18 & 0.89$\pm$0.14 & 3.8$\pm$0.5 & 3.3$\pm$0.4 & 1.6$\pm$0.2 & 0.5$\pm$0.1 &N\
88 &18 46 46.84 &$-$2 43 32.0 &29.98 &$-$0.21 & 0.89$\pm$0.16 & 5.8$\pm$0.6 & 4.6$\pm$0.5 & 2.3$\pm$0.3 & 0.7$\pm$0.1 &N\
90 &18 46 27.24 &$-$2 47 28.4 &29.88 &$-$0.16 & 1.4$\pm$0.2 & 4.3$\pm$0.6 & 3.8$\pm$0.5 & 2.0$\pm$0.3 & 0.7$\pm$0.1 &Y\
98 &18 45 35.30 &$-$2 38 33.8 &29.91 &$+$0.10 & 1.6$\pm$0.2 & 11$\pm$1 & 10$\pm$1 & 6.7$\pm$0.9 & 3.4$\pm$0.5 &N\
99 &18 45 37.76 &$-$2 35 56.5 &29.96 &$+$0.11 & 2.7$\pm$0.3 & 3.5$\pm$0.4 & 3.2$\pm$0.4 & 2.0$\pm$0.3 & 1.1$\pm$0.1 &N\
100 &18 46 44.22 &$-$2 43 44.2 &29.97 &$-$0.20 & 2.6$\pm$0.5 & 3.6$\pm$0.6 & 2.0$\pm$0.4 & 0.7$\pm$0.1 & 0.09$\pm$0.02 &N\
104 &18 46 17.83 &$-$2 29 50.3 &30.12 &$+$0.00 & 0.70$\pm$0.11 & 5.3$\pm$0.6 & 3.8$\pm$0.4 & 1.9$\pm$0.2 & 0.6$\pm$0.1 &Y\
109 &18 45 36.99 &$-$2 41 25.1 &29.87 &$+$0.07 & 0.97$\pm$0.18 & 4.6$\pm$0.6 & 3.5$\pm$0.4 & 1.6$\pm$0.2 & 0.6$\pm$0.1 &N\
111 &18 45 35.83 &$-$2 40 00.6 &29.89 &$+$0.08 & 0.70$\pm$0.13 & 3.7$\pm$0.6 & 3.5$\pm$0.6 & 1.9$\pm$0.3 & 0.8$\pm$0.1 &N\
113 &18 45 31.58 &$-$2 39 33.8 &29.89 &$+$0.10 & 0.81$\pm$0.15 & 2.0$\pm$0.7 & 2.4$\pm$0.8 & 1.8$\pm$0.6 & 0.8$\pm$0.3 &N\
122 &18 46 09.87 &$-$2 41 08.1 &29.94 &$-$0.05 & 187$\pm$21 & 219$\pm$25 & 117$\pm$13 & 44$\pm$6 & 15$\pm$2 &Y\
123 &18 46 05.00 &$-$2 42 23.6 &29.91 &$-$0.04 & 46$\pm$8 & 210$\pm$24 & 171$\pm$19 & 97$\pm$13 & 38$\pm$5 &Y\
124 &18 46 08.76 &$-$2 42 01.8 &29.93 &$-$0.05 & 39$\pm$12 & 77$\pm$22 & 57$\pm$16 & 28$\pm$8 & 6$\pm$2 &Y\
125 &18 46 11.87 &$-$2 41 30.7 &29.94 &$-$0.06 & 184$\pm$27 & 162$\pm$22 & 79$\pm$11 & 40$\pm$6 & 20$\pm$3 &Y\
126 &18 46 12.87 &$-$2 38 58.3 &29.98 &$-$0.05 & 36$\pm$4 & 128$\pm$14 & 111$\pm$12 & 66$\pm$9 & 29$\pm$4 &Y\
127 &18 46 00.41 &$-$2 41 14.9 &29.92 &$-$0.02 & 122$\pm$15 & 150$\pm$17 & 85$\pm$10 & 37$\pm$5 & 12$\pm$2 &N\
129 &18 45 59.01 &$-$2 41 10.1 &29.92 &$-$0.01 & 25$\pm$4 & 63$\pm$9 & 62$\pm$9 & 34$\pm$5 & 15$\pm$2 &N\
130 &18 46 12.92 &$-$2 39 29.6 &29.97 &$-$0.05 & 0.65$\pm$0.12 & 64$\pm$7 & 92$\pm$10 & 56$\pm$7 & 25$\pm$4 &N\
131 &18 46 06.45 &$-$2 37 49.2 &29.98 &$-$0.01 & 0.65$\pm$0.12 & 36$\pm$2 & 39$\pm$29 & 27$\pm$20 & 11$\pm$8 &N\
132 &18 46 10.96 &$-$2 43 28.2 &29.91 &$-$0.07 & 52$\pm$8 & 37$\pm$5 & 19$\pm$3 & 7.9$\pm$1.2 & 1.7$\pm$0.3 &Y\
133 &18 46 13.16 &$-$2 36 35.6 &30.01 &$-$0.03 & 0.68$\pm$0.12 & 33$\pm$8 & 28$\pm$6 & 17$\pm$3.9 & 6.9$\pm$1.7 &N\
134 &18 45 58.73 &$-$2 40 32.7 &29.93 &$-$0.01 & 0.58$\pm$0.10 & 28$\pm$3 & 42$\pm$5 & 28$\pm$3.6 & 15$\pm$2 &N\
135 &18 46 13.04 &$-$2 43 37.9 &29.91 &$-$0.08 & 26$\pm$21 & 8.4$\pm$7 & 3.0$\pm$2.5 & 1.4$\pm$1.1 & 0.09$\pm$0.02 &Y\
136 &18 46 13.66 &$-$2 37 29.1 &30.00 &$-$0.04 & 0.51$\pm$0.09 & 12$\pm$3 & 14$\pm$2 & 8.6$\pm$1.6 & 2.9$\pm$0.6 &N\
137 &18 45 55.11 &$-$2 39 19.5 &29.94 &$+$0.02 & 18$\pm$2 & 61$\pm$7 & 46$\pm$5 & 22$\pm$2.8 & 6.9$\pm$1.0 &Y\
138 &18 46 17.21 &$-$2 38 17.4 &30.00 &$-$0.06 & 7.4$\pm$1.1 & 15$\pm$2 & 12$\pm$2 & 6.0$\pm$0.9 & 0.88$\pm$0.21 &Y\
139 &18 46 23.72 &$-$2 41 01.0 &29.97 &$-$0.10 & 23$\pm$3 & 23$\pm$3 & 15$\pm$2 & 7.2$\pm$0.9 & 1.7$\pm$0.2 &Y\
141 &18 46 07.15 &$-$2 44 58.5 &29.88 &$-$0.07 & 5.5$\pm$2.7 & 5.5$\pm$2.6 & 2.6$\pm$1.2 & 1.1$\pm$0.6 & 0.19$\pm$0.16 &N\
142 &18 45 52.10 &$-$2 43 46.4 &29.87 &$-$0.01 & 0.65$\pm$0.12 & 20$\pm$3 & 21$\pm$2 & 9.8$\pm$1.3 & 3.0$\pm$0.4 &N\
143 &18 46 17.64 &$-$2 38 06.9 &30.00 &$-$0.06 & 2.7$\pm$1.8 & 1.8$\pm$1.3 & 0.89$\pm$0.79 & 0.10$\pm$0.02 & 0.09$\pm$0.02 &Y\
144 &18 46 22.17 &$-$2 37 04.6 &30.02 &$-$0.07 & 11$\pm$2 & 9.8$\pm$2.2 & 7.2$\pm$1.8 & 3.7$\pm$1.1 & 1.4$\pm$0.5 &N\
147 &18 46 20.96 &$-$2 38 57.5 &30.00 &$-$0.08 & 16$\pm$2 & 11$\pm$1 & 5.9$\pm$0.7 & 1.8$\pm$0.2 & 0.33$\pm$0.06 &Y\
148 &18 45 54.33 &$-$2 38 21.9 &29.95 &$+$0.03 & 2.6$\pm$0.4 & 30$\pm$3 & 27$\pm$3 & 14$\pm$2 & 6.6$\pm$0.9 &Y\
149 &18 45 53.99 &$-$2 38 52.9 &29.94 &$+$0.02 & 3.6$\pm$0.5 & 30$\pm$4 & 25$\pm$3 & 13$\pm$2 & 5.4$\pm$0.8 &N\
150 &18 45 55.02 &$-$2 45 59.7 &29.84 &$-$0.03 & 1.9$\pm$0.4 & 30$\pm$4 & 31$\pm$4 & 17$\pm$2 & 5.7$\pm$0.8 &N\
151 &18 45 54.60 &$-$2 45 42.7 &29.84 &$-$0.03 & 0.69$\pm$0.13 & 11$\pm$2 & 13$\pm$2 & 9.3$\pm$1.3 & 5.1$\pm$0.8 &N\
152 &18 46 03.88 &$-$2 48 31.2 &29.82 &$-$0.09 & 4.2$\pm$0.8 & 32$\pm$4 & 22$\pm$2 & 10$\pm$1 & 3.0$\pm$0.4 &Y\
153 &18 46 01.99 &$-$2 35 29.2 &30.01 &$+$0.02 & 23$\pm$3 & 23$\pm$3 & 13$\pm$1 & 5.2$\pm$0.7 & 1.5$\pm$0.2 &Y\
155 &18 45 46.45 &$-$2 42 47.2 &29.87 &$+$0.02 & 1.4$\pm$0.7 & 11$\pm$2 & 11$\pm$2 & 9.2$\pm$2.0 & 4.2$\pm$0.9 &N\
159 &18 45 47.89 &$-$2 44 39.4 &29.85 &$+$0.00 & 2.8$\pm$0.4 & 27$\pm$3 & 30$\pm$3 & 19$\pm$2 & 7.8$\pm$1.1 &Y\
160 &18 45 55.86 &$-$2 37 23.3 &29.97 &$+$0.03 & 10$\pm$1 & 17$\pm$2 & 11$\pm$1 & 4.0$\pm$0.5 & 0.57$\pm$0.10 &Y\
161 &18 45 53.42 &$-$2 45 27.1 &29.85 &$-$0.02 & 0.87$\pm$0.16 & 11$\pm$2 & 9.5$\pm$1.3 & 4.4$\pm$0.7 & 0.81$\pm$0.17 &N\
162 &18 46 15.53 &$-$2 44 18.5 &29.90 &$-$0.10 & 4.8$\pm$1.4 & 9.8$\pm$1 & 9.6$\pm$1.2 & 5.7$\pm$0.8 & 2.8$\pm$0.4 &N\
163 &18 46 01.89 &$-$2 47 00.7 &29.84 &$-$0.07 & 0.65$\pm$0.12 & 10$\pm$8 & 10$\pm$8 & 5.8$\pm$4.4 & 2.4$\pm$1.8 &N\
164 &18 45 51.09 &$-$2 44 30.4 &29.86 &$-$0.01 & 0.65$\pm$0.12 & 6.1$\pm$1.5 & 7.2$\pm$1.7 & 4.7$\pm$1.2 & 2.8$\pm$0.7 &N\
165 &18 46 08.89 &$-$2 35 16.5 &30.03 &$-$0.00 & 5.7$\pm$0.7 & 12$\pm$1 & 12$\pm$1 & 7.0$\pm$0.9 & 2.4$\pm$0.3 &Y\
167 &18 46 11.40 &$-$2 48 24.1 &29.84 &$-$0.11 & 10$\pm$1 & 12$\pm$2 & 7.9$\pm$1.0 & 3.2$\pm$0.5 & 1.0$\pm$0.2 &Y\
168 &18 46 01.66 &$-$2 47 49.7 &29.83 &$-$0.07 & 3.5$\pm$0.6 & 22$\pm$2 & 17$\pm$2 & 9.0$\pm$1.2 & 3.3$\pm$0.5 &Y\
169 &18 45 44.69 &$-$2 42 24.9 &29.87 &$+$0.03 & 5.0$\pm$0.7 & 9.6$\pm$1.1 & 7.0$\pm$0.8 & 3.7$\pm$0.6 & 2.5$\pm$0.4 &Y\
170 &18 46 29.94 &$-$2 36 27.1 &30.05 &$-$0.09 & 4.5$\pm$0.6 & 16$\pm$2 & 12$\pm$1 & 6.0$\pm$0.8 & 2.2$\pm$0.3 &Y\
171 &18 45 59.89 &$-$2 47 25.5 &29.83 &$-$0.06 & 0.59$\pm$0.11 & 13$\pm$1 & 20$\pm$2 & 12$\pm$2 & 4.4$\pm$0.6 &N\
172 &18 45 42.87 &$-$2 42 53.5 &29.86 &$+$0.03 & 4.3$\pm$0.6 & 21$\pm$3 & 27$\pm$3 & 19$\pm$3 & 12$\pm$2 &Y\
175 &18 46 31.77 &$-$2 39 33.6 &30.01 &$-$0.12 & 0.56$\pm$0.10 & 7.6$\pm$0.9 & 12$\pm$1 & 9.7$\pm$1.3 & 5.8$\pm$0.8 &N\
177 &18 46 05.18 &$-$2 30 09.6 &30.09 &$+$0.05 & 11$\pm$1 & 26$\pm$3 & 18$\pm$2 & 8.9$\pm$1.2 & 3.6$\pm$0.5 &Y\
178 &18 46 47.22 &$-$2 39 36.4 &30.03 &$-$0.18 & 3.7$\pm$0.5 & 17$\pm$2 & 14$\pm$2 & 6.8$\pm$0.9 & 2.9$\pm$0.4 &Y\
179 &18 46 42.73 &$-$2 35 41.6 &30.08 &$-$0.13 & 6.1$\pm$0.7 & 6.5$\pm$0.7 & 3.6$\pm$0.4 & 1.7$\pm$0.2 & 0.71$\pm$0.11 &Y\
180 &18 46 50.21 &$-$2 41 36.5 &30.01 &$-$0.21 & 7.6$\pm$0.8 & 14$\pm$2 & 8.5$\pm$0.9 & 3.8$\pm$0.5 & 1.3$\pm$0.2 &Y\
182 &18 45 56.20 &$-$2 47 13.3 &29.82 &$-$0.05 & 1.6$\pm$0.4 & 27$\pm$3 & 23$\pm$3 & 12$\pm$2 & 4.2$\pm$0.6 &Y\
184 &18 45 44.88 &$-$2 43 31.6 &29.86 &$+$0.02 & 1.6$\pm$0.5 & 14$\pm$2 & 17$\pm$2 & 12$\pm$2 & 7.1$\pm$1.1 &Y\
185 &18 46 23.02 &$-$2 43 49.3 &29.93 &$-$0.12 & 4.3$\pm$0.6 & 7.5$\pm$0.9 & 4.8$\pm$0.5 & 1.9$\pm$0.2 & 0.64$\pm$0.09 &Y\
186 &18 46 15.38 &$-$2 49 44.0 &29.82 &$-$0.14 & 4.0$\pm$0.6 & 8.9$\pm$1.0 & 4.6$\pm$0.5 & 1.9$\pm$0.2 & 0.77$\pm$0.11 &Y\
187 &18 45 49.75 &$-$2 32 48.2 &30.03 &$+$0.09 & 5.2$\pm$0.6 & 19$\pm$2 & 14$\pm$2 & 5.6$\pm$0.7 & 2.2$\pm$0.3 &Y\
188 &18 46 46.27 &$-$2 36 20.0 &30.08 &$-$0.15 & 2.1$\pm$0.3 & 6.1$\pm$0.7 & 4.0$\pm$0.5 & 1.7$\pm$0.2 & 0.62$\pm$0.09 &Y\
189 &18 46 46.57 &$-$2 35 42.9 &30.09 &$-$0.15 & 6.6$\pm$0.8 & 8.7$\pm$1.1 & 5.5$\pm$0.7 & 2.8$\pm$0.4 & 1.3$\pm$0.2 &Y\
191 &18 46 48.73 &$-$2 40 17.9 &30.03 &$-$0.19 & 3.8$\pm$0.4 & 12$\pm$1 & 8.8$\pm$1.0 & 4.0$\pm$0.5 & 1.3$\pm$0.2 &N\
192 &18 46 42.78 &$-$2 38 49.1 &30.04 &$-$0.16 & 0.95$\pm$0.16 & 9.0$\pm$1.1 & 6.6$\pm$0.8 & 2.9$\pm$0.4 & 0.79$\pm$0.13 &Y\
193 &18 45 50.00 &$-$2 30 49.7 &30.06 &$+$0.10 & 0.54$\pm$0.10 & 10$\pm$2 & 13$\pm$3 & 9.8$\pm$2.0 & 4.7$\pm$1.0 &N\
194 &18 45 45.93 &$-$2 36 36.7 &29.96 &$+$0.08 & 15$\pm$2 & 18$\pm$2 & 12$\pm$1 & 5.8$\pm$0.7 & 2.6$\pm$0.4 &Y\
195 &18 46 31.55 &$-$2 32 34.5 &30.11 &$-$0.07 & 0.30$\pm$0.08 & 5.1$\pm$0.6 & 7.7$\pm$0.9 & 6.4$\pm$0.9 & 3.3$\pm$0.5 &Y\
196 &18 45 47.56 &$-$2 37 22.7 &29.95 &$+$0.06 & 3.1$\pm$0.4 & 15$\pm$2 & 11$\pm$1 & 5.6$\pm$0.7 & 2.2$\pm$0.3 &Y\
197 &18 46 52.33 &$-$2 40 02.5 &30.04 &$-$0.20 & 9.9$\pm$1.2 & 12$\pm$1 & 6.7$\pm$0.8 & 2.3$\pm$0.3 & 0.31$\pm$0.06 &Y\
198 &18 45 50.16 &$-$2 35 01.3 &29.99 &$+$0.07 & 9.3$\pm$1.0 & 14$\pm$2 & 7.0$\pm$0.8 & 2.8$\pm$0.4 & 0.85$\pm$0.12 &Y\
199 &18 46 48.81 &$-$2 41 17.3 &30.01 &$-$0.20 & 12$\pm$1 & 19$\pm$2 & 10$\pm$1 & 3.9$\pm$0.5 & 1.0$\pm$0.1 &Y\
200 &18 46 10.48 &$-$2 46 23.5 &29.86 &$-$0.09 & 0.67$\pm$0.12 & 101$\pm$1 & 9.7$\pm$1.1 & 5.2$\pm$0.7 & 2.0$\pm$0.3 &N\
201 &18 46 33.82 &$-$2 34 23.3 &30.09 &$-$0.09 & 5.4$\pm$0.6 & 8.9$\pm$1.0 & 5.1$\pm$0.6 & 2.1$\pm$0.3 & 0.77$\pm$0.11 &Y\
202 &18 46 45.27 &$-$2 39 16.9 &30.04 &$-$0.17 & 2.3$\pm$0.4 & 7.2$\pm$1.1 & 4.4$\pm$0.7 & 1.7$\pm$0.3 & 0.46$\pm$0.09 &Y\
203 &18 45 45.77 &$-$2 39 00.4 &29.93 &$+$0.05 & 1.5$\pm$0.2 & 12$\pm$1 & 12$\pm$1 & 7.1$\pm$0.9 & 3.0$\pm$0.4 &Y\
205 &18 46 52.16 &$-$2 42 07.0 &30.01 &$-$0.22 & 6.9$\pm$0.8 & 13$\pm$1 & 8.5$\pm$0.9 & 3.7$\pm$0.5 & 1.2$\pm$0.2 &Y\
207 &18 46 14.92 &$-$2 50 17.6 &29.81 &$-$0.14 & 13$\pm$2 & 9.2$\pm$1.3 & 4.6$\pm$0.6 & 1.8$\pm$0.3 & 0.61$\pm$0.10 &Y\
208 &18 45 59.55 &$-$2 48 26.6 &29.81 &$-$0.07 & 0.89$\pm$0.16 & 6.6$\pm$0.8 & 4.7$\pm$0.5 & 1.8$\pm$0.3 & 0.42$\pm$0.08 &N\
209 &18 45 42.44 &$-$2 31 26.2 &30.03 &$+$0.12 & 0.52$\pm$0.09 & 14$\pm$3 & 23$\pm$5 & 16$\pm$4 & 7.8$\pm$2.0 &N\
210 &18 46 49.13 &$-$2 38 04.4 &30.06 &$-$0.17 & 9.5$\pm$1.1 & 9.3$\pm$1.0 & 5.0$\pm$0.6 & 2.0$\pm$0.3 & 0.47$\pm$0.07 &Y\
212 &18 45 50.40 &$-$2 47 58.4 &29.80 &$-$0.03 & 1.3$\pm$0.2 & 10$\pm$1 & 12$\pm$1 & 8$\pm$1 & 3.6$\pm$0.5 &N\
213 &18 46 51.47 &$-$2 38 02.6 &30.07 &$-$0.18 & 6.2$\pm$0.7 & 11$\pm$1 & 6.8$\pm$0.8 & 2.7$\pm$0.4 & 0.90$\pm$0.13 &Y\
214 &18 45 43.98 &$-$2 45 11.7 &29.83 &$+$0.01 & 5.3$\pm$0.6 & 21$\pm$2 & 23$\pm$3 & 13$\pm$2 & 5.0$\pm$0.7 &N\
215 &18 45 56.64 &$-$2 34 45.2 &30.01 &$+$0.05 & 1.1$\pm$0.1 & 7.5$\pm$0.9 & 9.0$\pm$1.0 & 5.8$\pm$0.8 & 2.7$\pm$0.4 &Y\
217 &18 45 53.82 &$-$2 46 55.7 &29.82 &$-$0.04 & 0.79$\pm$0.14 & 13$\pm$1 & 10$\pm$1 & 4.8$\pm$0.6 & 1.9$\pm$0.3 &N\
219 &18 46 01.96 &$-$2 30 47.6 &30.08 &$+$0.06 & 0.29$\pm$0.21 & 6.9$\pm$0.9 & 5.2$\pm$0.7 & 2.5$\pm$0.4 & 0.69$\pm$0.12 &Y\
220 &18 46 14.12 &$-$2 32 10.8 &30.08 &$+$0.00 & 4.0$\pm$0.4 & 10$\pm$1 & 7.0$\pm$0.8 & 3.1$\pm$0.4 & 1.2$\pm$0.2 &Y\
221 &18 46 10.14 &$-$2 51 21.1 &29.79 &$-$0.13 & 3.3$\pm$0.4 & 12$\pm$1 & 7.7$\pm$0.9 & 3.2$\pm$0.4 & 0.97$\pm$0.14 &Y\
228 &18 45 50.23 &$-$2 48 25.2 &29.80 &$-$0.03 & 0.49$\pm$0.15 & 10$\pm$1 & 9.9$\pm$1.1 & 4.8$\pm$0.6 & 1.6$\pm$0.2 &Y\
232 &18 45 35.61 &$-$2 39 04.1 &29.91 &$+$0.09 & 2.0$\pm$0.3 & 11$\pm$1 & 9.2$\pm$1.0 & 4.4$\pm$0.6 & 1.7$\pm$0.2 &N\
242 &18 46 03.84 &$-$2 39 21.2 &29.96 &$-$0.02 & 7235$\pm$809 &1810$\pm$202 & 498$\pm$56 &348$\pm$45 &105$\pm$15 &Y\
243 &18 45 59.45 &$-$2 45 05.8 &29.86 &$-$0.04 & 552$\pm$62 & 228$\pm$25 & 115$\pm$13 & 56$\pm$7 & 20$\pm$3 &Y\
245 &18 46 11.25 &$-$2 41 56.2 &29.93 &$-$0.06 & 605$\pm$68 & 338$\pm$38 & 184$\pm$21 & 90$\pm$12 & 28$\pm$4 &Y\
247 &18 46 17.08 &$-$2 36 43.5 &30.02 &$-$0.05 & 601$\pm$67 & 280$\pm$31 & 142$\pm$16 & 65$\pm$8 & 25$\pm$3 &Y\
251 &18 45 54.67 &$-$2 42 53.2 &29.86 &$-$0.01 & 222$\pm$25 & 76$\pm$8 & 40$\pm$4 & 15$\pm$2 & 3.2$\pm$0.5 &Y\
253 &18 46 01.75 &$-$2 45 27.7 &29.86 &$-$0.05 & 352$\pm$39 & 186$\pm$21 & 91$\pm$10 & 36$\pm$5 & 13$\pm$2 &Y\
254 &18 46 07.24 &$-$2 42 20.7 &29.92 &$-$0.05 & 99$\pm$14 & 173$\pm$24 & 116$\pm$16 & 49$\pm$7 & 16$\pm$3 &Y\
257 &18 46 06.94 &$-$2 42 58.6 &29.91 &$-$0.06 & 142$\pm$34 & 97$\pm$24 & 50$\pm$12 & 25$\pm$6 & 7.0$\pm$1.8 &Y\
258 &18 45 55.72 &$-$2 42 31.1 &29.89 &$-$0.01 & 177$\pm$22 & 92$\pm$12 & 48$\pm$6 & 25$\pm$4 & 11$\pm$2 &Y\
259 &18 46 07.95 &$-$2 43 23.8 &29.90 &$-$0.06 & 175$\pm$20 & 78$\pm$10 & 43$\pm$5 & 18$\pm$3 & 6.2$\pm$0.9 &Y\
262 &18 46 04.86 &$-$2 42 44.1 &29.91 &$-$0.05 & 81$\pm$10 & 106$\pm$12 & 58$\pm$7 & 19$\pm$2 & 6.5$\pm$0.9 &Y\
268 &18 45 45.64 &$-$2 31 52.3 &30.03 &$+$0.11 & 98$\pm$11 & 48$\pm$5 & 26$\pm$3 & 13$\pm$2 & 7.1$\pm$1.0 &Y\
269 &18 45 44.56 &$-$2 32 18.4 &30.02 &$+$0.11 & 215$\pm$24 & 88$\pm$10 & 41$\pm$5 & 16$\pm$2 & 4.5$\pm$0.6 &Y\
270 &18 46 09.73 &$-$2 43 41.7 &29.90 &$-$0.07 & 116$\pm$14 & 60$\pm$7 & 26$\pm$3 & 10$\pm$1 & 2.6$\pm$0.4 &N\
274 &18 46 08.27 &$-$2 48 04.0 &29.83 &$-$0.10 & 104$\pm$12 & 44$\pm$5 & 26$\pm$3 & 11$\pm$1 & 4.2$\pm$0.6 &N\
275 &18 45 44.00 &$-$2 32 00.3 &30.03 &$+$0.11 & 123$\pm$14 & 61$\pm$7 & 30$\pm$3 & 13$\pm$22 & 4.1$\pm$0.6 &Y\
276 &18 46 08.37 &$-$2 47 45.6 &29.84 &$-$0.10 & 19$\pm$2 & 16$\pm$2 & 13$\pm$2 & 9.0$\pm$1.2 & 3.9$\pm$0.6 &Y\
278 &18 46 35.43 &$-$2 40 34.6 &30.00 &$-$0.14 & 33$\pm$4 & 36$\pm$4 & 26$\pm$3 & 15$\pm$2 & 6.5$\pm$0.9 &Y\
280 &18 46 26.29 &$-$2 40 55.9 &29.98 &$-$0.11 & 62$\pm$7 & 46$\pm$5 & 23$\pm$3 & 8.9$\pm$1.2 & 2.2$\pm$0.3 &Y\
281 &18 46 01.29 &$-$2 46 23.4 &29.85 &$-$0.06 & 63$\pm$7 & 56$\pm$7 & 36$\pm$4 & 17$\pm$2 & 5.0$\pm$0.7 &Y\
282 &18 45 51.24 &$-$2 30 17.7 &30.01 &$+$0.10 & 120$\pm$13 & 50$\pm$6 & 24$\pm$3 & 9.5$\pm$1.2 & 3.6$\pm$0.5 &Y\
285 &18 46 06.28 &$-$2 30 13.5 &30.10 &$+$0.04 & 24$\pm$3 & 19$\pm$2 & 11$\pm$1 & 5.3$\pm$0.7 & 2.2$\pm$0.3 &Y\
286 &18 45 59.66 &$-$2 29 09.3 &30.10 &$+$0.08 & 21$\pm$2 & 32$\pm$4 & 26$\pm$3 & 14$\pm$2 & 5.8$\pm$0.8 &Y\
288 &18 46 23.02 &$-$2 43 05.6 &29.94 &$-$0.12 & 46$\pm$5 & 33$\pm$4 & 17$\pm$2 & 6.8$\pm$0.9 & 2.4$\pm$0.3 &Y\
289 &18 46 22.69 &$-$2 40 12.0 &29.98 &$-$0.09 & 34$\pm$4 & 34$\pm$4 & 19$\pm$2 & 8.2$\pm$1.1 & 2.7$\pm$0.4 &Y\
291 &18 46 11.54 &$-$2 44 15.3 &29.90 &$-$0.08 & 26$\pm$3 & 9.0$\pm$1.5 & 4.0$\pm$0.8 & 1.4$\pm$0.4 & 0.14$\pm$0.12 &Y\
292 &18 46 14.13 &$-$2 43 28.8 &29.91 &$-$0.09 & 10$\pm$2 & 9.4$\pm$1.9 & 4.3$\pm$0.9 & 1.9$\pm$0.4 & 0.39$\pm$0.12 &Y\
293 &18 46 21.84 &$-$2 40 30.7 &29.97 &$-$0.09 & 35$\pm$4 & 25$\pm$3 & 13$\pm$1 & 4.5$\pm$0.6 & 0.95$\pm$0.17 &Y\
294 &18 46 25.27 &$-$2 40 35.7 &29.98 &$-$0.11 & 21$\pm$2 & 18$\pm$2 & 12$\pm$1 & 5.9$\pm$0.8 & 2.6$\pm$0.4 &Y\
301 &18 46 05.74 &$-$2 48 28.1 &29.82 &$-$0.09 & 30$\pm$3 & 28$\pm$3 & 22$\pm$2 & 12$\pm$2 & 4.7$\pm$0.7 &N\
305 &18 46 17.13 &$-$2 48 57.3 &29.84 &$-$0.14 & 24$\pm$3 & 17$\pm$2 & 9.3$\pm$1.0 & 4.3$\pm$0.6 & 1.7$\pm$0.2 &Y\
306 &18 46 15.61 &$-$2 49 21.4 &29.83 &$-$0.14 & 12$\pm$1 & 12$\pm$1 & 6.8$\pm$0.8 & 2.6$\pm$0.3 & 0.70$\pm$0.10 &Y\
307 &18 46 22.52 &$-$2 41 50.2 &29.95 &$-$0.10 & 10$\pm$2 & 8.3$\pm$1.8 & 5.1$\pm$1.1 & 2.3$\pm$0.5 & 0.79$\pm$0.19 &Y\
309 &18 46 07.27 &$-$2 48 57.0 &29.82 &$-$0.10 & 16$\pm$2 & 8.2$\pm$0.9 & 4.5$\pm$0.6 & 1.9$\pm$0.3 & 1.0$\pm$0.1 &N\
311 &18 46 03.84 &$-$2 36 31.1 &30.00 &$+$0.01 & 10$\pm$1 & 11$\pm$1 & 11$\pm$1 & 7.1$\pm$0.9 & 2.7$\pm$0.4 &Y\
318 &18 46 05.61 &$-$2 35 17.5 &30.02 &$+$0.01 & 12$\pm$1 & 14$\pm$2 & 11$\pm$1 & 5.9$\pm$0.8 & 1.7$\pm$0.2 &Y\
323 &18 46 03.43 &$-$2 35 20.1 &30.01 &$+$0.02 & 22$\pm$2 & 15$\pm$2 & 5.7$\pm$0.7 & 1.1$\pm$0.2 & 0.36$\pm$0.09 &Y\
324 &18 45 57.37 &$-$2 36 50.2 &29.98 &$+$0.03 & 21$\pm$3 & 18$\pm$2 & 8.2$\pm$0.9 & 2.8$\pm$0.4 & 0.53$\pm$0.10 &Y\
325 &18 45 56.12 &$-$2 48 38.9 &29.80 &$-$0.06 & 33$\pm$4 & 24$\pm$3 & 12$\pm$1 & 4.6$\pm$0.6 & 1.3$\pm$0.2 &Y\
326 &18 46 27.13 &$-$2 39 48.3 &29.99 &$-$0.11 & 6.3$\pm$0.8 & 4.6$\pm$0.6 & 2.6$\pm$0.4 & 0.9$\pm$0.2 & 0.08$\pm$0.02 &Y\
327 &18 46 19.36 &$-$2 45 06.4 &29.90 &$-$0.12 & 9.3$\pm$1.1 & 6.1$\pm$0.7 & 2.9$\pm$0.4 & 1.4$\pm$0.2 & 0.45$\pm$0.08 &Y\
329 &18 46 05.85 &$-$2 30 33.3 &30.09 &$+$0.04 & 25$\pm$3 & 23$\pm$3 & 12$\pm$1 & 4.4$\pm$0.6 & 1.1$\pm$0.2 &Y\
332 &18 46 21.60 &$-$2 50 08.4 &29.83 &$-$0.16 & 9.5$\pm$1.1 & 4.4$\pm$0.5 & 3.4$\pm$0.4 & 1.5$\pm$0.2 & 0.45$\pm$0.07 &Y\
335 &18 45 56.04 &$-$2 37 47.5 &29.96 &$+$0.02 & 8.7$\pm$1.2 & 7.4$\pm$1.2 & 3.3$\pm$0.6 & 0.6$\pm$0.2 & 0.08$\pm$0.02 &Y\
338 &18 46 49.17 &$-$2 36 09.5 &30.09 &$-$0.16 & 11$\pm$1 & 14$\pm$2 & 7.6$\pm$0.8 & 3.2$\pm$0.4 & 1.1$\pm$0.2 &Y\
339 &18 46 20.46 &$-$2 46 34.1 &29.88 &$-$0.13 & 7.9$\pm$0.9 & 8.0$\pm$0.9 & 4.9$\pm$0.6 & 2.3$\pm$0.3 & 1.2$\pm$0.2 &Y\
340 &18 46 40.22 &$-$2 38 11.5 &30.04 &$-$0.14 & 6.2$\pm$0.7 & 8.8$\pm$1.0 & 6.8$\pm$0.8 & 3.5$\pm$0.5 & 1.4$\pm$0.2 &Y\
341 &18 46 40.16 &$-$2 35 34.1 &30.08 &$-$0.12 & 13$\pm$1 & 9.4$\pm$1.1 & 3.2$\pm$0.4 & 1.0$\pm$0.1 & 0.31$\pm$0.05 &Y\
342 &18 45 42.77 &$-$2 37 22.4 &29.95 &$+$0.08 & 21$\pm$2 & 17$\pm$2 & 8.3$\pm$0.9 & 3.2$\pm$0.4 & 0.94$\pm$0.13 &Y\
343 &18 46 15.14 &$-$2 51 14.3 &29.80 &$-$0.15 & 5.9$\pm$0.7 & 9.1$\pm$1.0 & 6.6$\pm$0.7 & 3.3$\pm$0.4 & 1.2$\pm$0.17 &Y\
344 &18 46 25.04 &$-$2 48 46.4 &29.86 &$-$0.17 & 8.0$\pm$0.9 & 5.1$\pm$0.6 & 2.6$\pm$0.3 & 1.2$\pm$0.2 & 0.51$\pm$0.07 &Y\
349 &18 46 49.98 &$-$2 42 49.3 &29.99 &$-$0.21 & 13$\pm$1 & 4.7$\pm$0.5 & 2.0$\pm$0.3 & 0.96$\pm$0.15 & 0.35$\pm$0.06 &Y\
352 &18 46 13.30 &$-$2 32 35.8 &30.07 &$+$0.00 & 2.4$\pm$0.3 & 4.4$\pm$0.5 & 3.3$\pm$0.4 & 1.6$\pm$0.2 & 0.46$\pm$0.06 &Y\
357 &18 45 58.60 &$-$2 35 02.0 &30.01 &$+$0.04 &0.31$\pm$0.09 & 2.8$\pm$0.7 & 5.9$\pm$1.3 & 5.5$\pm$1.2 & 2.9$\pm$0.7 &Y\
\[tflux\]
----- --------- ---- -------
132 MIPSGAL 24 0.44
135 MIPSGAL 24 0.27
143 MIPSGAL 24 0.024
147 MIPSGAL 24 0.047
242 MSX 21 1340
243 MSX 21 19
245 MSX 21 26
247 WISE 22 16
251 MIPSGAL 24 4.6
253 WISE 22 9.9
257 WISE 22 4.3
258 MIPSGAL 24 1.5
259 MIPSGAL 24 1.2
268 MIPSGAL 24 2.0
269 MIPSGAL 24 3.9
270 WISE 22 1.3
274 WISE 22 2.7
275 MIPSGAL 24 1.02
282 MIPSGAL 24 1.1
285 MIPSGAL 24 0.74
288 MIPSGAL 24 0.52
291 MIPSGAL 24 0.021
305 MIPSGAL 24 0.31
309 WISE 22 0.49
327 MIPSGAL 24 0.20
342 MIPSGAL 24 0.16
344 MSX 21 8.0
349 MIPSGAL 24 0.39
----- --------- ---- -------
\
$^a$ The number corresponds to the identification number (Table 1).
\[tsed\]
[lcccccc]{}
\
&& & & & &\
& & & & & &\
\
\
\
&& & & & &\
& & & & & &\
\
1 & 0.8 &19.44 & 41.9 & 645 & 0.50 & 3823\
2 & 1.4 &27.48 & 27.4 & 966 & 0.38 & 7577\
4 & 1.2 &21.81 & 15.8 &2426 & 1.5 & 705\
5 & 2.2 &18.75 & 15.8 & 216 & 0.18 & 438\
6 & 1.8 &18.30 & 15.8 & 457 & 0.40 & 427\
7 & 2.2 &17.67 & 15.8 & 172 & 0.16 & 363\
8 & 1.4 &18.91 & 18.7 & 257 & 0.21 & 272\
9 & 2.0 &18.19 & 12.9 & 272 & 0.24 & 111\
11 & 1.8 &23.06 & 21.6 & 41 & 0.02 & 233\
12 & 2.6 &22.26 & 10.0 & 513 & 0.31 & 123\
13 & 1.0 &25.56 & 18.7 & 543 & 0.25 & 253\
14 & 0.8 &26.24 & 27.4 & 204 & 0.09 & 400\
16 & 0.6 &23.91 & 24.5 & 323 & 0.17 & 188\
17 & 0.4 &27.48 & 27.4 & 724 & 0.28 & 405\
18 & 1.2 &17.28 & 15.8 & 257 & 0.25 & 75\
19 & 1.6 &24.07 & 18.7 & 115 & 0.06 & 185\
20 & 1.8 &26.33 & 12.9 & 363 & 0.15 & 103\
21 & 1.8 &27.48 & 18.7 & 91 & 0.04 & 226\
22 & 1.6 &26.38 & 21.6 & 68 & 0.03 & 247\
24 & 1.8 &24.39 & 18.7 & 72 & 0.04 & 180\
25 & 1.4 &26.15 & 21.6 & 172 & 0.07 & 197\
26 & 2.0 &24.27 & 12.9 & 257 & 0.13 & 105\
27 & 1.2 &23.59 & 24.5 & 122 & 0.06 & 232\
31 & 2.6 &21.48 & 10.0 & 431 & 0.28 & 104\
33 & 1.8 &23.80 & 21.6 & 48 & 0.03 & 277\
34 & 1.6 &26.87 & 18.7 & 108 & 0.04 & 175\
35 & 1.8 &24.00 & 24.5 & 36 & 0.02 & 243\
37 & 1.2 &19.97 & 21.6 & 129 & 0.10 & 190\
38 & 1.4 &23.16 & 24.5 & 46 & 0.03 & 208\
40 & 1.2 &18.57 & 27.4 & 54 & 0.05 & 222\
42 & 1.4 &26.38 & 24.5 & 46 & 0.02 & 208\
44 & 0.4 &25.37 & 33.2 & 172 & 0.08 & 315\
45 & 1.2 &23.28 & 21.6 & 243 & 0.13 & 193\
48 & 1.4 &22.49 & 27.4 & 32 & 0.02 & 269\
49 & 1.0 &20.18 & 27.4 & 77 & 0.06 & 201\
54 & 2.4 &22.40 & 21.6 & 7.7 & 0.005& 183\
55 & 2.4 &21.59 & 12.9 & 108 & 0.07 & 96\
61 & 2.4 &27.20 & 15.8 & 54 & 0.02 & 176\
62 & 0.6 &23.19 & 27.4 & 122 & 0.07 & 136\
64 & 1.0 &19.60 & 24.5 & 122 & 0.09 & 170\
65 & 1.4 &24.80 & 24.5 & 54 & 0.03 & 247\
66 & 0.8 &24.30 & 27.4 & 91 & 0.05 & 158\
69 & 1.2 &25.05 & 30.3 & 58 & 0.03 & 494\
70 & 2.4 &24.43 & 10.0 & 645 & 0.32 & 111\
73 & 2.6 &24.41 & 10.0 & 305 & 0.15 & 70\
74 & 2.2 &24.87 & 15.8 & 72 & 0.03 & 152\
75 & 2.6 &23.23 & 15.8 & 23 & 0.01 & 114\
77 & 1.0 &21.61 & 27.4 & 58 & 0.04 & 181\
79 & 2.0 &27.48 & 24.5 & 32 & 0.01 & 287\
81 & 0.8 &21.63 & 30.3 & 77 & 0.05 & 244\
83 & 2.4 &23.39 & 12.9 & 77 & 0.04 & 67\
86 & 2.2 &12.81 & 15.8 & 27 & 0.05 & 58\
87 & 1.6 &25.50 & 21.6 & 38 & 0.02 & 139\
88 & 2.0 &27.48 & 18.7 & 48 & 0.02 & 187\
90 & 1.2 &27.11 & 24.5 & 61 & 0.02 & 173\
98 & 0.8 &27.48 & 24.5 & 513 & 0.20 & 336\
99 & 0.4 &22.96 & 33.2 & 108 & 0.06 & 199\
100 & 2.4 &16.04 & 24.5 & 3.8 & 0.004 & 206\
104 & 2.0 &25.49 & 18.7 & 41 & 0.02 & 157\
109 & 1.6 &19.07 & 21.6 & 43 & 0.04 & 156\
111 & 1.2 &21.09 & 21.6 & 86 & 0.06 & 127\
113 & 2.0 &19.03 & 12.9 & 122 & 0.10 & 50\
122 & 1.8 &26.27 & 27.4 & 575 & 0.25 & 12985\
123 & 1.4 &24.86 & 21.6 &3234 & 1.54 & 7446\
124 & 2.6 &13.86 & 39.0 & 102 & 0.16 & 3544\
125 & 0.6 &23.61 & 41.9 &1288 & 0.68 & 11232\
126 & 0.6 &19.55 & 27.4 &3844 & 3.0 & 4647\
127 & 1.4 &28.26 & 30.3 & 645 & 0.24 & 8845\
129 & 0.4 &21.03 & 30.3 &2161 & 1.4 & 2651\
130 & 2.2 &26.26 & 12.9 &2883 & 1.2 & 1739\
131 & 1.4 &20.69 & 15.8 &2041 & 1.4 & 821\
132 & 1.2 &24.30 & 41.9 & 81 & 0.04 & 3765\
133 & 1.8 &19.34 & 15.8 & 767 & 0.60 & 677\
134 & 1.2 &21.90 & 15.8 &2883 & 1.8 & 798\
135 & 2.0 &18.32 & 39.0 & 4.3 & 0.004 & 1369\
136 & 1.8 &21.60 & 15.8 & 363 & 0.23 & 339\
137 & 1.8 &28.28 & 21.6 & 431 & 0.16 & 2338\
138 & 2.2 &21.19 & 21.6 & 48 & 0.03 & 713\
139 & 1.4 &24.27 & 36.1 & 86 & 0.04 & 1555\
141 & 2.4 &12.37 & 24.5 & 6.5 & 0.01 & 345\
142 & 2.0 &27.48 & 15.8 & 343 & 0.13 & 461\
143 & 1.0 &14.80 & 44.8 & 3.8 & 0.005 & 140\
144 & 0.6 &26.62 & 39.0 & 115 & 0.05 & 720\
147 & 1.6 &24.34 & 36.1 & 16 & 0.008 & 1041\
148 & 1.4 &26.16 & 18.7 & 767 & 0.33 & 811\
149 & 1.8 &22.43 & 18.7 & 384 & 0.23 & 919\
150 & 2.2 &25.70 & 15.8 & 431 & 0.19 & 915\
151 & 1.6 &28.24 & 21.6 & 609 & 0.23 & 312\
152 & 2.2 &30.00 & 18.7 & 162 & 0.05 & 930\
153 & 1.6 &30.00 & 36.1 & 51 & 0.02 & 1528\
155 & 0.4 &24.92 & 24.5 &1147 & 0.54 & 337\
159 & 1.4 &26.74 & 18.7 & 861 & 0.35 & 885\
160 & 2.6 &25.77 & 30.3 & 12 & 0.005 & 894\
161 & 2.6 &23.14 & 21.6 & 32 & 0.02 & 237\
162 & 1.6 &24.09 & 21.6 & 243 & 0.12 & 258\
163 & 2.0 &16.97 & 21.6 & 136 & 0.14 & 265\
164 & 0.6 &17.39 & 21.6 & 484 & 0.47 & 200\
165 & 0.8 &24.47 & 30.3 & 272 & 0.13 & 594\
167 & 1.6 &23.25 & 27.4 & 51 & 0.03 & 702\
168 & 1.4 &30.00 & 21.6 & 288 & 0.09 & 661\
169 & 0.4 &17.80 & 33.2 & 243 & 0.23 & 445\
170 & 1.4 &25.20 & 27.4 & 153 & 0.07 & 563\
171 & 1.6 &21.54 & 15.8 & 645 & 0.41 & 405\
172 & 0.4 &25.67 & 27.4 &5432 & 2.4 & 860\
175 & 0.4 &17.39 & 21.6 &2426 & 2.4 & 279\
177 & 1.0 &30.00 & 27.4 & 323 & 0.11 & 1017\
178 & 1.4 &25.93 & 21.6 & 243 & 0.11 & 559\
179 & 1.0 &22.50 & 36.1 & 38 & 0.02 & 425\
180 & 1.4 &22.70 & 30.3 & 72 & 0.04 & 617\
182 & 2.2 &29.25 & 15.8 & 323 & 0.11 & 686\
184 & 0.4 &28.02 & 24.5 &2426 & 0.91 & 469\
185 & 1.8 &24.95 & 24.5 & 31 & 0.01 & 363\
186 & 1.8 &27.68 & 27.4 & 27 & 0.01 & 362\
187 & 1.8 &30.00 & 27.4 & 97 & 0.03 & 639\
188 & 1.4 &24.14 & 24.5 & 48 & 0.02 & 220\
189 & 0.6 &24.57 & 36.1 & 108 & 0.05 & 476\
191 & 1.8 &25.75 & 21.6 & 81 & 0.04 & 466\
192 & 2.0 &24.94 & 18.7 & 61 & 0.03 & 234\
193 & 2.2 &16.71 & 21.6 & 288 & 0.30 & 314\
194 & 0.6 &28.03 & 39.0 & 216 & 0.08 & 1094\
195 & 0.6 &18.93 & 18.7 & 861 & 0.71 & 184\
196 & 1.4 &30.00 & 21.6 & 204 & 0.07 & 470\
197 & 2.6 &27.99 & 30.3 & 6 & 0.002 & 724\
198 & 2.0 &28.68 & 24.5 & 34 & 0.01 & 668\
199 & 2.0 &29.81 & 24.5 & 46 & 0.02 & 891\
200 & 2.2 &26.78 & 21.6 & 91 & 0.04 & 260\
201 & 1.4 &30.00 & 30.3 & 41 & 0.01 & 422\
202 & 2.0 &24.03 & 21.6 & 27 & 0.01 & 249\
203 & 1.0 &25.12 & 21.6 & 384 & 0.18 & 368\
205 & 1.4 &30.00 & 30.3 & 68 & 0.02 & 596\
207 & 1.4 &25.40 & 36.1 & 24 & 0.01 & 694\
208 & 2.2 &22.35 & 18.7 & 29 & 0.02 & 173\
209 & 0.4 &23.35 & 18.7 &2883 & 1.6 & 423\
210 & 2.0 &29.79 & 27.4 & 16 & 0.005& 618\
212 & 0.8 &17.82 & 21.6 & 543 & 0.50 & 340\
213 & 1.8 &29.56 & 24.5 & 43 & 0.02 & 513\
214 & 2.6 &29.64 & 12.9 & 407 & 0.14 & 456\
215 & 0.8 &25.20 & 21.6 & 431 & 0.20 & 270\
217 & 2.4 &30.00 & 15.8 & 102 & 0.03 & 331\
219 & 2.4 &28.04 & 15.8 & 48 & 0.02 & 154\
220 & 1.4 &30.00 & 24.5 & 86 & 0.03 & 391\
221 & 2.0 &28.99 & 21.6 & 48 & 0.02 & 441\
228 & 2.2 &27.80 & 15.8 & 129 & 0.05 & 267\
232 & 1.6 &30.00 & 24.5 & 108 & 0.04 & 344\
242 & 0.8 &19.39 & 76.7 &2883 & 2.3 &791095\
243 & 1.4 &14.84 & 70.9 & 431 & 0.58 & 39363\
245 & 2.6 &23.75 & 73.8 & 229 & 0.12 & 47243\
247 & 0.6 &23.65 & 56.4 &1215 & 0.64 & 41592\
251 & 1.6 &21.38 & 56.4 & 72 & 0.05 & 13062\
253 & 1.8 &27.42 & 62.2 & 193 & 0.08 & 23156\
254 & 1.4 &22.55 & 27.4 &1023 & 0.59 & 7999\
257 & 0.6 &19.57 & 59.3 & 457 & 0.35 & 11617\
258 & 0.8 &21.06 & 44.8 & 513 & 0.34 & 10648\
259 & 1.2 &23.91 & 41.9 & 229 & 0.12 & 10559\
262 & 2.4 &22.82 & 44.8 & 77 & 0.04 & 6097\
268 & 0.4 &16.48 & 53.5 & 384 & 0.42 & 5751\
269 & 1.4 &26.91 & 44.8 & 129 & 0.05 & 15111\
270 & 1.4 &25.58 & 53.5 & 72 & 0.03 & 8826\
274 & 0.6 &25.40 & 56.4 & 216 & 0.10 & 7396\
275 & 1.2 &29.45 & 41.9 & 162 & 0.06 & 7511\
276 & 0.4 &11.75 & 39.0 & 323 & 0.69 & 1204\
278 & 0.4 &13.71 & 39.0 & 609 & 0.96 & 2268\
280 & 1.8 &22.85 & 41.9 & 54 & 0.03 & 3690\
281 & 1.4 &27.87 & 41.9 & 204 & 0.08 & 4078\
282 & 1.0 &27.69 & 44.8 & 162 & 0.06 & 5954\
285 & 0.4 &15.01 & 53.5 & 144 & 0.19 & 1966\
286 & 0.6 &14.34 & 36.1 & 645 & 0.92 & 1637\
288 & 0.8 &23.29 & 47.7 & 144 & 0.08 & 3018\
289 & 1.2 &26.84 & 39.0 & 136 & 0.06 & 2195\
291 & 2.6 &24.19 & 30.3 & 3.8 & 0.002& 1390\
292 & 2.0 &18.77 & 27.4 & 15 & 0.01 & 584\
293 & 2.2 &27.86 & 27.4 & 31 & 0.01 & 1965\
294 & 0.6 &23.29 & 41.9 & 193 & 0.11 & 1375\
301 & 0.4 &30.00 & 41.9 & 513 & 0.17 & 1957\
305 & 0.6 &30.00 & 47.7 & 102 & 0.03 & 1619\
306 & 1.6 &23.90 & 30.3 & 32 & 0.02 & 779\
307 & 1.0 &21.63 & 36.1 & 48 & 0.03 & 604\
309 & 0.4 &24.86 & 59.3 & 51 & 0.02 & 1206\
311 & 0.4 &13.35 & 36.1 & 323 & 0.53 & 717\
318 & 0.8 &15.52 & 36.1 & 153 & 0.19 & 875\
323 & 2.6 &27.19 & 27.4 & 6.5 & 0.003& 1195\
324 & 2.6 &24.65 & 24.5 & 14 & 0.007& 1224\
325 & 1.6 &27.05 & 39.0 & 41 & 0.02 & 1978\
326 & 2.6 &24.05 & 24.5 & 3.8 & 0.002& 344\
327 & 0.6 &22.58 & 50.6 & 31 & 0.02 & 634\
329 & 2.0 &26.47 & 27.4 & 41 & 0.02 & 1554\
332 & 0.8 &26.97 & 44.8 & 27 & 0.01 & 565\
335 & 2.4 &26.01 & 27.4 & 3.8 & 0.002& 421\
338 & 1.4 &30.00 & 33.2 & 51 & 0.02 & 778\
339 & 0.6 &25.09 & 39.0 & 81 & 0.04 & 509\
340 & 0.8 &21.85 & 33.2 & 115 & 0.07 & 471\
341 & 2.2 &30.00 & 33.2 & 5.8 & 0.002& 688\
342 & 1.0 &25.91 & 41.9 & 58 & 0.03 & 1512\
343 & 1.0 &26.35 & 30.3 & 91 & 0.04 & 474\
344 & 0.8 &25.27 & 44.8 & 26 & 0.01 & 505\
349 & 1.0 &30.00 & 50.6 & 12 & 0.004& 821\
352 & 1.4 &17.62 & 27.4 & 32 & 0.03 & 213\
357 & 0.4 & 9.66 & 21.6 & 683 & 2.2 & 140\
|
---
abstract: 'We present high-resolution spectroscopic measurements of the abundances of the $\alpha$ element titanium (Ti) and s-process elements yttrium (Y) and lanthanum (La) for 59 candidate M giant members of the Sagittarius (Sgr) dwarf spheroidal (dSph) + tidal tail system pre-selected on the basis of position and radial velocity. As expected, the majority of these stars show peculiar abundance patterns compared to those of nominal Milky Way stars, but as a group the stars form a coherent picture of chemical enrichment of the Sgr dSph from \[Fe/H\] = -1.4 to solar abundance. This sample of spectra provides the largest number of Ti, La and Y abundances yet measured for a dSph, and spans metallicities not typically probed by studies of the other, generally more metal-poor Milky Way (MW) satellites. On the other hand, the overall \[Ti/Fe\], \[Y/Fe\], \[La/Fe\] and \[La/Y\] patterns with \[Fe/H\] of the Sgr stream plus Sgr core do, for the most part, resemble those seen in the Large Magellanic Cloud (LMC) and other dSphs, only shifted by $\Delta$\[Fe/H\]$\sim$+0.4 from the LMC and by $\sim$+1 dex from the other dSphs; these relative shifts reflect the faster and/or more efficient chemical evolution of Sgr compared to the other satellites, and show that Sgr has had an enrichment history more like the LMC than the other dSphs. By tracking the evolution of the abundance patterns along the Sgr stream we can follow the time variation of the chemical make-up of dSph stars donated to the Galactic halo by Sgr. This evolution demonstrates that while the bulk of the stars currently in the Sgr dSph are quite unlike those of the Galactic halo, an increasing number of stars farther along the Sgr stream have abundances like Milky Way halo stars, a trend that shows clearly how the Galactic halo could have been contributed by present day satellite galaxies even if the [*present*]{} chemistry of those satellites is now different from typical halo field stars. Finally, we analyze the chemical abundances of a moving group of M giants among the Sgr leading arm stars at the North Galactic Cap, but having radial velocities unlike the infalling Sgr leading arm debris there. Through use of “chemical fingerprinting”, we conclude that these mostly receding northern hemisphere M giants also are Sgr stars, likely [*trailing arm*]{} debris overlapping the Sgr leading arm in the north.'
author:
- |
Mei-Yin Chou, Katia Cunha, Steven R. Majewski, Verne V. Smith,\
Richard J. Patterson, David Mart[í]{}nez-Delgado and Doug Geisler
title: |
A Two Micron All-Sky Survey View of the Sagittarius Dwarf Galaxy:\
VI. s-Process and Titanium Abundance Variations Along the Sagittarius Stream
---
Introduction
============
Though it is now clear that accretion of dwarf galaxies likely played a prominent role in creating the Milky Way’s (MW) stellar halo (Searle & Zinn 1978), with strong observational evidence (e.g., Majewski 1993; Majewski, Munn & Hawley 1996; Bell et al. 2008) and a theoretical backing by $\Lambda$CDM models (e.g., Bullock & Johnston 2005; Robertson et al. 2005; Abadi et al.2006; Font et al. 2006a), it is often highlighted that the chemical abundance patterns of current MW satellites are rather different than those of halo field stars, which typically show significantly higher \[$\alpha$/Fe\] than MW dSph stars at the same \[Fe/H\] (e.g., Fulbright 2002; Shetrone et al. 2003; Tolstoy et al. 2003; Venn et al. 2004; Geisler et al. 2005). The reason for these differences remains a matter of speculation. One interpretation is that the dwarf systems that contributed the bulk of the halo were more massive, Magellanic Cloud-sized systems that were accreted and destroyed very early on, so that the chemistry of the accreted stars was necessarily dominated only by enrichment from Type II supernovae (SN II)(Robertson et al. 2005; Font et al. 2006a). Alternatively (or in addition), if current dwarf satellites have experienced both prolonged chemical evolution and tidal disruption, this will naturally lead to evolution in the types of stars that satellites contribute to the Galactic halo (Majewski et al. 2000, 2002).
In principle, one can test the bridge from dwarf galaxy chemistry to halo star chemistry [*directly*]{} if one can identify the stars already contributed by the dwarf galaxy to the halo. Perhaps the easiest way to do this is by exploring elemental abundances along the tidal tails of disrupting dwarf galaxies. Presently, the best known example of a tidally disrupting Milky Way satellite is the Sagittarius (Sgr) dwarf spheroidal (dSph) galaxy, a system that is especially interesting for understanding the issues raised above because the proximity of its core and tails make it particularly accessible to high resolution spectroscopic study, and because the tails produced by its steady assimilation into the MW have been tracked over a substantial length (e.g., Ibata et al. 2001; Majewski et al. 2003, hereafter Paper I; Belokurov et al. 2007) corresponding to several gigayears of tidal disruption (Law et al. 2005, “Paper IV” hereafter). Thus the extensive Sgr tidal stream can provide a key link between dwarf galaxies, tidal disruption and capture, and the chemical evolution and origin of the MW halo.
Exploration of the chemistry of any dwarf galaxy can be used to study star formation environments that are quite distinct from those in the MW, but Sgr provides an interesting case study for chemical evolution in its own right. It is the most luminous and massive of the current MW dSph systems, and has a history punctuated by a number of star formation episodes (Sarajedini & Layden 1995; Layden & Sarajedini 2000; Siegel et al. 2007). This active star formation history quickly elevated Sgr’s mean metallicity to almost solar by about 5 Gyr ago (Bellazzini et al.2006; Siegel et al. 2007), making it the dSph with the most metal-rich stellar populations known associated with the Milky Way. Because Sgr has been tidally disrupting for at least 2.5-3.0 Gyr (Paper IV), in principle this means that Sgr could have contributed stars with a wide metallicity range to the Galactic halo.
This has been confirmed by the recent analysis of Chou et al. (2007, hereafter Paper V), who measured \[Fe/H\] ranging from -1.4 dex to 0.0 dex for stars identified spatially, dynamically and by spectral type (i.e. the M giants characteristic of the Sgr dSph) with the Sgr stream. Interestingly, the Paper V analysis revealed a strong metallicity gradient along the Sgr stream — proof of a time dependence in the enrichment level of the stars donated to the halo by Sgr, and evidence for preferential tidal stripping of metal poor stars, which has led to divergent metallicity distribution functions (MDFs) between lost and retained Sgr stars. A similar metallicity gradient in the Sgr leading and trailing streams has also been seen by the high resolution analysis of M giants by Monaco et al. (2007). Because stars are typically stripped from the outer parts of dSphs, the changing MDF along the Sgr stream suggests a strong metallicity gradient within the original Sgr system.
Sgr is also chemically interesting because its abundance patterns reveal that while it is undersolar and depleted with respect to the MW pattern for its $\alpha$-, odd-$Z$ and iron peak-elements among its \[Fe/H\] $\gtrsim$ -1 populations, Sgr’s more metal-poor populations seem to have \[$\alpha$/Fe\] that [*do*]{} resemble those of the Galactic halo (Smecker-Hane & McWilliam 2002, Bonifacio et al. 2004, Monaco et al. 2005b, Sbordone et al.2007). Presently these findings are based on only about 6 Sgr stars having both \[Fe/H\] $< -1$ and measured \[$\alpha$/Fe\] from all of the above studies, as well as a handful of stars from M54 (possibly the nucleus of Sgr — see discussions by Ibata et al.1994; Sarajedini et al. 1995; Da Costa & Armandroff 1995; Bassino & Muzzio 1995; Layden & Sarajedini 2000; Paper I, but also cf. Monaco et al. 2005a) in the study by Brown et al.(1999). However, if true, the trend may not be unique to Sgr: Abundance patterns (e.g., \[$\alpha$/Fe\]) for some of the very most metal-poor stars in other dSphs also seem to overlap those of halo stars of the same metallicity (Shetrone et al. 2003; Geisler et al. 2005; Tolstoy 2005). Thus, if [*these*]{} dSph systems experienced tidal disruption, the few very metal poor stars they now hold with MW-like abundance patterns may only represent the residue of a formerly much larger metal-poor population that may have been predominantly stripped from the satellites over their lifetime (see discussion of this in the case of the Carina dSph in Majewski et al. 2002 and Muñoz et al. 2006). Such a scenario could explain the possible origins of the abundance dichotomy between the present dSphs and the Galactic halo.
In this paper we continue our detailed exploration of Sgr stream stars from Paper V with a focus on the chemical trends of the $\alpha$-element titanium (Ti) and the s-process elements yttrium (Y) and lanthanum (La) along the stream. Our study is based on the largest sample of high resolution spectra yet obtained of Sgr stars, and includes a significant number for stars with \[Fe/H\] $<
-0.9$ (§2-3). Collectively, and when combined with additional data from the literature, these stars present a much more complete picture of the chemical abundance patterns for Sgr stars, including the first view of Sgr s-process abundances for \[Fe/H\] $<-0.9$. In §4 we show that, as seen before with smaller samples, the Sgr \[$\alpha$/Fe\] abundances are enhanced and MW-like among the more metal-poor stars, but we also show, for the first time that this trend also arises among the s-process elements we explore. Because of the overall variation of the MDF reported in Paper V, there are increasing numbers of stars with “MW-like” abundances with distance along the stream away from the Sgr core. Thus, the Sgr stream provides a direct connection between the unusual, non-MW-like abundance patterns in the Sgr core and stars with halo-like abundance patterns contributed by Sgr several Gyr ago. In this way, Sgr provides a counter-example of the Robertson et al. (2005) and Font et al. (2006a) hypothesis that the $\alpha$-enhanced stars of the halo were deposited there long ago.
In §5 we also discuss the implications of the Sgr abundance patterns for the chemical evolution of this dSph. By comparison of the Sgr abundance patterns to those of other MW satellites, we show that Sgr more closely resembles the Large Magellanic Cloud in its chemical evolution than the other dSph-type systems.
Finally, throughout this paper we simultaneously analyze the chemical abundances of a moving group of M giants among the Sgr leading arm stars at the North Galactic Cap but having different radial velocities than the infalling Sgr leading arm debris there. In Paper V we showed these stars to have a very similar MDF to leading Sgr arm stars stripped several Gyr ago, a result expected if this moving group were constituted by Sgr stars stripped at about the same time. In a demonstration of the technique of “chemical fingerprinting”, we conclude from their peculiar Ti, Y and La patterns, which also match those of the extreme Sgr leading arm, that most of these receding northern hemisphere M giants could represent Sgr [*trailing arm*]{} stars in the northern hemisphere.
Observations
============
Our analysis here makes use of the same spectra described in Paper V for six M giant stars in the Sgr core, thirty candidate stars in the Sgr leading arm north of the Galactic plane, ten in the Sgr leading arm but south of the Galactic plane, and thirteen stars in the North Galactic Cap (NGC) moving group, which have velocities much higher and mostly opposite those of the infalling Sgr leading arm there. We selected the first three groups of these stars to be likely members of the Sgr stream not only by their spatial distribution, but also based on their radial velocities, which are appropriate for the Sgr stream at these positions based on Sgr debris models (Paper IV; Figs. 1 and 2 in Paper V). The Sgr core, leading arm north and leading arm south groups represent a dynamical sequence from possibly still bound stars to those stars stripped from Sgr several Gyr ago, respectively. We have argued in Paper V, and will again here (§4), that the NGC moving group stars look to be former Sgr members with chemistry resembling the leading arm south stars; given their positions, velocities and inferred dynamical age, these are most probably Sgr trailing arm stars in the northern hemisphere.
The stars in our sample were observed with the 6.5-m Clay telescope at Las Campanas Observatory and the MIKE spectrograph at $R\sim$19,000, the 4-m Mayall telescope echelle at $R\sim$35,000 at Kitt Peak, and the 3.5-m TNG telescope and SARG spectrograph operating at $R\sim$46,000 in the Canary Islands [^1]. The data reduction followed standard procedures and is described in Paper V. Figure 1 shows examples of portions of the spectra from each instrument including the two Ti lines used for our analysis here. Further details of the observations and the positions, photometry and velocities for the program stars can be found in Paper V (with the latter data summarized in Table 1 of that paper).
Derivation of Abundances
========================
The required input parameters for the abundance analysis are effective temperature $T_{\rm eff}$, surface gravity (usually parameterized as log $g$), and metallicity. The details on the determination of the effective temperature, surface gravity, and iron abundances for our target stars are given in Paper V. The model atmospheres adopted in the analysis were interpolated from Kurucz (1994) grids[^2] and are the same as those used in the analysis in Paper V. Abundances in this study were derived from the LTE code MOOG (Sneden 1973) along with the adopted model atmospheres from Paper V.
We measure the equivalent widths (EWs) of eleven lines (used to derive the \[Fe/H\] presented in Paper V), two lines and one line in a particular part of the spectrum that is relatively free from TiO and other molecular contamination and that was previously investigated by Smith & Lambert (1985; 1986; 1990 — hereafter “S&L”) in their spectroscopic exploration of M giants. These particular elements were chosen not only because they have well-defined, measurable spectral lines, but also because they show distinct abundance ratios (relative to Fe) in many dwarf galaxies when compared to the MW. This circumstance has been found in the core of Sgr itself (Bonifacio et al. 2004; Monaco et al. 2005b; Sbordone et al. 2007), as well as in other dSphs (Shetrone et al. 2003; Geisler et al. 2005) and the Large Magellanic Cloud (LMC; Smith et al. 2002; Johnson et al. 2006; Pomp[é]{}ia et al. 2008; Mucciarelli et al. 2008).
The $gf$-values for and were determined by measuring their equivalent widths in the solar flux atlas of Kurucz et al. (1984) and varying the $gf$-values for each line in order to match the solar titanium and yttrium abundances of $A$(Ti)=4.90 and $A$(Y)=2.21 (Asplund, Grevesse, & Sauval 2005); the adopted solar $gf$-values are listed in Table 1. The measured EWs of the and lines for each of our Sgr spectra are given in Table 2. We also include the EW’s measured for several standard stars, which we have analyzed for a comparison and as a control sample.
We analyzed via spectral synthesis analysis because this line is affected by hyperfine splitting. An example of the spectral synthesis for this line is shown in Figure 2. The line we are interested in is from angular momentum $J=3$ to $J=3$, with nuclear spin $I=7/2$. The lower energy for this transition is 1016.10 cm$^{-1}$, and the higher energy is 14375.17 cm$^{-1}$. There are nineteen hyperfine splitting lines for this transition, and the splitting constants are $A=3.38$ and $B=0.84$ for the lower energy level (Lawler et al. 2001). However, the $A$ and $B$ constants for the higher level are unknown. We used various $A$ and $B$ to create a synthetic spectrum and fit it to a very high-resolution Arcturus spectrum. The best-fit $A$ is -30. $B$ is a secondary parameter so it actually doesn’t affect the spectrum too much; we have adopted $B=-0.5$ here.
The derived abundance results are summarized in Table 3. For each star, the columns give the derived effective temperature using the Houdashelt et al. (2000) color-temperature relation applied to the 2MASS $(J-K_s)_o$ color, and the derived values of the surface gravity ($\log{g}$), microturbulence ($\xi$), abundance $A$(X), and abundance ratios \[Fe/H\] or \[X/H\] for each element X as well as the standard deviation in the abundance determinations. The details for how we derive the atmospheric parameters ($T_{\rm
eff}$, $\log{g}$, \[Fe/H\], $\xi$) can be found in Paper V. The standard deviation represents the line to line scatter (for Ti and Y) and different estimates of the continuum level (for La). We measured two lines in one order, and two EW measurements of the same line in two adjacent orders. For , we have three different abundance measurements from different continuum level adjustments, and give the resulting average abundance and standard deviation of those values.
We also have measured abundances for the stars Arcturus, $\beta$ Peg, $\beta$ And, $\rho$ Per, and HD 146051, which are nearby K (Arcturus) and M giants that provide a control sample for our abundance work. In Table 4 we summarize literature values (S&L; McWilliam & Rich 1994; Smith et al. 2000) for the atmospheric parameters and abundances for relevant chemical elements of these control sample stars for comparison to our own derived values (no such data are available for HD 146051). Because the references listed adopt somewhat different solar abundances scales, we list the absolute abundances, $A$(X), rather than abundance ratios in order to facilitate a comparison. We note that S&L used the K giant $\alpha$ Tau as a reference star in their analysis. In order to compute absolute abundances for the control stars we derive abundances for $\alpha$ Tau using the equivalent width measurements in S&L and the model atmospheres and $gf$ values adopted in this study. The derived abundances for $\alpha$ Tau are $A$(Fe)=7.52, $A$(Ti)=4.94 and $A$(Y)=2.32. We used these values and the relative abundance ratios \[X/H\] in S&L to get the absolute abundances of $\beta$ Peg, $\beta$ And and $\rho$ Per listed in Table 4. As can be seen, the derived abundances for Arcturus and $\rho$ Per in this work agree with the literature values, within the stated errors and with the standard deviation of the differences less than 0.2 dex. Our derived values of $\beta$ Peg and $\beta$ And are lower than those in S&L. This is due largely to differences in the adopted stellar atmosphere parameters.
To explore the sensitivity in the derived abundances to changes in stellar parameters we varied $T_{\rm eff}$, $\log{g}$ and microturbulence for the control stars and tabulate the abundance changes in Table 5. This table shows the sensitivities of the , , and lines corresponding to changes in $T_{\rm eff}$ by +100 K, $\log{g}$ by +0.2 dex (where $g$ is measured in ${\rm cm\, s^{-2}}$), and $\xi$ by +0.2 ${\rm km\,s^{-1}}$, respectively.
With these dependencies in hand, we derive the abundances for the M giant comparison stars adopting the previously published stellar parameters from S&L. The resulting abundances are provided as separate entries in Table 4, and indicate general agreement, within calculated uncertainties, for A(Fe) values when similar atmospheric values are adopted for the M giant comparison stars. This exercise also demonstrates how the derived $A$(Y) values much more closely match the previously derived values for these stars when we adopt similar stellar parameters. The lower $A$(Y) we find in this paper for the control stars arises primarily from our derivation of lower $\log{g}$ for these stars. As previously mentioned in Paper V, the S&L studies predated the availability of Hipparcos parallaxes, and therefore they adopted higher $\log{g}$ values from less accurate absolute magnitudes.
Finally, our analysis shows that, in general, we tend to find $A$(Ti) slightly lower by about 0.2-0.3 dex compared to S&L, even when similar atmospheric parameters are adopted for the M giants. Note however that different sets of Ti lines have been used in the two studies and that different families of model atmospheres can also account for some of the abundance differences. In addition, when comparing the derived abundances with the previously published results from S&L one has to keep in mind that S&L computed only relative abundances using $\alpha$ Tau as a reference star with the underlying assumption that its abundance distribution is approximately solar. Indeed, Kovács (1983) conducted a detailed absolute abundance analysis for $\alpha$ Tau and validated this assumption at a level of roughly $\pm 0.2$ dex. Absolute abundances computed from the relative abundances published in S&L will carry the uncertainty in the underlying $\alpha$ Tau abundance distribution.
Further circumstantial evidence that our Ti abundances are more reliable comes from a comparison of our derived \[Ti/Fe\] for the control stars to the \[Ti/Fe\] trend of other disk stars, as shown in the top panel of Figure 3 (where the control sample stars are shown as brown stars); as may be seen, were we to shift our Ti abundances several 0.1 dex higher to make them more consistent with the S&L values, all five of the control stars (including HD146051 now) would have \[Ti/Fe\] above the mean for disk stars of similar \[Fe/H\], and in some cases anomalously so. In addition, as we will show in Figures 3 and 10 below, a several tenths of a dex offset in $A$(Ti) would make the already high \[Ti/Fe\] abundances found for metal poor Sgr stars compared to other dSphs and the LMC even more extreme.
Derived Abundance Patterns\
and Variation Along the Sgr Stream
==================================
Abundance Differences Compared to Milky Way Field Stars
-------------------------------------------------------
To verify whether our sample is indeed dominated by members of the Sgr stream, rather than random MW field stars, we appeal to the well-known abundance pattern differences between the MW and dSph satellites in general, and between the MW and the Sgr dSph in particular. In principle, one might expect that stars recently stripped from dSph systems to bear similar peculiar chemical hallmarks as the stars they left behind in the dSph core. The promise of “chemical fingerprinting” stars to their birth systems has long been discussed (Freeman & Bland-Hawthorne 2002; De Silva et al. 2007), but has yet to be used much in practice. A side benefit of our study is that it lends itself to a direct test of the viability of chemical fingerprinting in a well-defined, fairly controlled context (i.e., testing a sample of stars specifically selected to have been born in one particular system – Sgr — against contamination from stars from another system — the MW). In addition, in §4.2 we [*apply*]{} chemical fingerprinting to test the notion that the NGC group of stars may be from Sgr.
It has long been observed that present dSph stars are typically underabundant in \[$\alpha$/Fe\] compared to Milky Way stars at the same \[Fe/H\], presumably a result of the much slower enrichment history of these smaller systems, which allows the products of Type Ia supernovae (SN Ia) (including much of the iron) to be introduced at lower overall metallicities. The same underabundance trend is found for various light s-process elements, like yttrium, which are thought to be converted to heavier s-process elements (like lanthanum) by high neutron exposure in low mass asymptotic giant branch (AGB) stars; thus, low Y and high La abundances are also an indication of a slow enrichment star formation history, at least compared to the Milky Way.
Detailed abundance studies of the Sgr core have shown similar overall trends in abundances patterns as other dSphs, but also some trends apparently characteristic of the Sgr system itself. For example, Smecker-Hane & McWilliam (2002) and McWilliam & Smecker-Hane (2005) found that high metallicity (\[Fe/H\] $>$ -1) Sgr stars show extraordinarily enhanced heavy s-process (e.g., La) abundances, while at the same time these stars have low abundances of Mn and Cu. Similar chemical trends are also found in the LMC (Johnson et al. 2006; Pomp[é]{}ia et al. 2008). Since manganese and copper yields from SN II decrease, relative to iron, in lower metallicity supernovae, the low values of \[Mn/Fe\] and \[Cu/Fe\] could be the result of nucleosynthesis from low-metallicity SN II, while the large values of \[La/Fe\] are the product of the s-process yields from low-metallicity, low-mass AGB stars. These patterns can arise from intense star-formation bursts, along with loss of some SN II ejecta via galactic winds, followed by long quiescent periods in the dSph. After long periods of time (several Gyr), low-mass, low-metallicity AGB stars eventually add significant amounts of ejecta into the interstellar medium (ISM) and leave their signature on the highest metallicity stars.
Figures 3-6 show the distributions of \[Ti/Fe\], \[Y/Fe\], \[La/Fe\], and \[La/Y\] as a function of \[Fe/H\] for all of our targeted stars (middle panels), for those of MW and other dSph stars (top panels), and for our targeted stars superposed on those of LMC stars (bottom panels). The MW abundance distributions have been taken from Gratton & Sneden (1994), Fulbright (2000), Johnson (2002) and Reddy et al. (2003). The other dSph data are from Shetrone et al. (2001; 2003), Sadakane et al. (2004) and Geisler et al. (2005). The LMC data are from Johnson et al. (2006), Pomp[é]{}ia et al. (2008) and Mucciarelli et al. (2008).
It is immediately obvious from inspection of Figures 3-6 that the bulk of our sample stars do not share the same chemical abundance patterns as Milky Way stars. To quantify this assessment, we fit linear trends to the Milky Way distributions (the lines shown in Figs. 3-6), and determine the dispersions around those trends for Galactic stars: 0.15 dex for \[Y/Fe\] and 0.12 dex for \[La/Fe\]. For the \[Ti/Fe\] versus \[Fe/H\] distribution we fit two linear trends on either side of the apparent Milky Way transition at \[Fe/H\]=-0.7, and find dispersions of 0.10 and 0.05 dex to either side of that break, respectively. (These measured dispersions represent both the intrinsic dispersion of abundances as well as measurement errors from the various surveys of MW stars.) For each of our M giants we determine the number of standard deviations, $N_\sigma$, away from the Milky Way mean trend that star is at its \[Fe/H\] in each of the distributions shown (\[Ti/Fe\], \[Y/Fe\] and \[La/Fe\] as a function of \[Fe/H\], Fig. 7). These deviation measures are tabulated in columns 2-4 of Table 6. We also derive an average deviation for all three trends (column 6 of Table 6). These deviations allow us to characterize how “Milky Way-like” each star is; we adopt as a definition of “Milky Way-like” those stars that always lie within 1.5$\sigma$ of the mean MW abundance trends. By this definition, only four stars from the 30 leading arm north sample, no stars in the leading arm south sample, and one star from the 13 NGC sample have abundance patterns approximating “MW-like”; these stars are designated with overlying “cross” symbols in Figure 7 (as well as in Figure 9, described below). For comparison, by our adopted definition of “MW-like” we would also classify four of 27 of the bona fide Sgr stars from Monaco et al. and Sbordone et al. as “MW-like”. This simple analysis attests to the true peculiarity of the stars in our sample by a Galactic standard, and that there is likely little contamination by MW stars.
Note that in Paper V we had divided the Sgr leading arm north group into a “best” (the fainter and farther stars that are most likely to be in the Sgr leading arm) and a “less certain” subsample. The latter included stars with brighter magnitudes ($K_{s,0} < 7.5$) that we considered the most susceptible to contamination by Galactic (thick disk) stars as well as those stars closest to the Galactic bulge. We showed in Paper V (see, e.g., Fig. 9 of that paper) that the metallicity distributions of the “best” and “less certain” samples were similar in shape, spread and mean values. Further analysis of the relative abundance patterns between these two subsamples here also reveals no obvious distinctions: Figure 8 shows that there is no apparent difference in the overall chemical patterns between the less certain and best LN subsamples, even though the latter are at projected distances more commensurate with those predicted by the Paper IV model for the Sgr leading arm in the Northern Hemisphere and observed for other proposed Sgr leading arm tracers (e.g., the K/M giants and blue horizontal-branch stars (BHBs) from SDSS/SEGUE spectra by Yanny et al. 2009, and the RR Lyrae stars (RRLSs) with large negative Galactic Standard of Rest velocities ($v_{GSR}$) in the Virgo stellar stream (VSS) region likely to be Sgr stars by Prior et al. 2009b). Therefore, we continue to see no evidence that the two LN subsamples are stars of a different origin and, given the other evidence presented here that they are all most likely to be Sgr stream stars, we continue to consider them together as the single “Sgr leading arm north sample" in this paper.[^3]
Figure 9, which shows the distribution of \[Y/Fe\] versus \[Ti/Fe\], further demonstrates the true distinction between our sample stars and those in the standard Galactic populations; stars in our Sgr sample are shown color-coded by their Sgr system grouping in the top panel. We also compare the MW stars with those of other dSph and LMC stars on the middle and bottom panels, respectively. Figure 9 shows a striking segregation of stars by their parent system (particularly by \[Y/Fe\]), and one that further illustrates the differences in chemical evolution between stars we have selected to be Sgr-members and the nominal MW populations (see §5). In the top panel of Figure 9 we have marked with large crosses those stars that were deemed most MW-like by the above $\sigma$ analysis; as might be expected, these tend to lie closest to the V-shaped distribution of MW stars, but even then not in the main locus of MW stars, but, rather, skirting it. Figure 9 further reinforces the conclusion that the sample of M giants we have observed should have very little contamination by nominal MW field stars, and this includes stars we have selected to lie in the NGC group (see §4-2). Furthermore, we notice that stars from the other dSphs as well as the LMC sit in the bottom left part of Figure 9 (middle and bottom panels), so that they share a similar — but not exactly the same — distribution as our Sgr stars. It is worth noting that about a third of the stars selected to be from the Sgr system (including one or two Sgr core stars) have higher \[Ti/Fe\] than the bulk of the dSph and LMC stars (a feature seen also in Fig. 3). Nevertheless, since about 2/3 of the “Sgr system” stars do share the same chemical patterns as other dwarf galaxy systems in Figure 9, we can not rule out the possibility that a small number of these particular Sgr stars are from other dwarf satellite debris. We discuss this point further in §4.3.
The observed deviation of our selected Sgr stream stars away from MW abundance trends as a function of \[Fe/H\] is illuminating. Deviation trends can be used as a quantitative chemical marker, and combinations of abundance patterns for different elements are thought to provide unique, or at least fairly distinctive, chemical fingerprints that can be used to identify the star formation sites (e.g., the parent galaxies) of specific stars. For example, inspection of the \[Ti/Fe\] vs \[Fe/H\] trends shown in Figure 3 (top and middle panels) and Figure 7 reveal clear differences between our sample of likely Sgr stars and MW stars: While there is a general agreement in the \[Ti/Fe\] levels for the MW and Sgr stars at the lowest metallicities (\[Fe/H\] $\lesssim$-1.0) — where both the Sgr and MW stars are equally enhanced in \[$\alpha$/Fe\], a feature that betrays the signature of prevalent SN II enrichment — the differences in abundance trends increase rapidly above \[Fe/H\] $\sim -1.0$, with the great majority of Sgr stream stars falling below the MW trend. The latter trend reflects the lower star formation efficiency and/or slower chemical enrichment (and, thus, greater relative SN Ia yields) of Sgr stars relative to those of the MW (see §5).
Meanwhile, the trend for \[Y/Fe\] shows the Sgr stars to be underabundant with respect to the MW and subsolar over the whole range of sampled \[Fe/H\] (Figs. 4 and 7). On the other hand, \[La/Fe\] is primarily below the MW trend until \[Fe/H\] $\sim -0.5$, when it quickly rises well above the MW level. This trend was previously found by Smecker-Hane & McWilliam (2002). The relative abundances of heavy to light s-process elements shown by \[La/Y\] (Fig. 6) accentuate the differences between Sgr and MW stars at all metallicities. We further interpret the meaning of these various trends in §5.
Chemical Trends Along the Sgr Stream and Chemical Fingerprinting of the\
North Galactic Cap Stellar Group
------------------------------------------------------------------------
Presently, the only known dSph satellite of the MW known to contain a significant population of M giants is the Sgr system. While the Magellanic Clouds contain large numbers of M giants, the analyses undertaken in Paper I and elsewhere have revealed no evidence of any M giant tidal structures from either of the Magellanic Clouds. At low Galactic latitudes M giants are found associated with the Monoceros/Galactic Anticenter Stellar Structure (GASS, Rocha-Pinto et al. 2003), but it is still not clear whether this structure is a tidal stream or a part of the MW disk (e.g., Martin et al. 2004b; Momany et al. 2004, 2006; López-Corredoira et al. 2007). The only other MW substructure that is certainly a tidal debris remnant and known to contain M giant stars is the more distant, Triangulum-Andromeda (TriAnd) star cloud (Rocha-Pinto et al. 2004), also at low Galactic latitudes ($-40 ^{\circ}< b < -20 ^{\circ}$). For higher Galactic latitudes, the analysis of M giant distributions in Paper I showed that a major fraction ($\gtrsim 75\%$) of M giants found in the halo away from the Galactic plane lie along the Sgr orbital plane; since overall the presence of M giants in other MW dSphs or substructures is relatively rare, the concentration of high latitude Galactic M giants along the Sgr plane suggests that the vast majority of these stars were indeed contributed by the Sgr dSph.
This proposition received further support with subsequent study of the radial velocities of these M giants, which show most of them also to lie in correlated trends of velocity with Sgr orbital plane longitude (Majewski et al. 2004, “Paper II” hereafter; Paper IV) corresponding to the Doppler motion of the most recently stripped (e.g., within the last 1-2 Gyr) Sgr debris as it wraps around the Galactic center (see Paper IV). Recently, Yanny et al. (2009) have shown that our adopted radial velocity (RV) sequence for Sgr stars in the northern sky (e.g., as shown in Figure 2 of Paper V) is well matched by SDSS/SEGUE-derived radial velocities for candidate Sgr K/M giants and blue horizontal-branch stars (BHBs) along the leading arm. This correspondence lends further credence to the notion that most of our RV-selected Sgr leading arm stars are indeed from the Sgr dSph.
However, not [*all*]{} M giants in the Sgr plane are found to lie along the primary velocity trends of this most recently lost debris. In particular, a number of radial velocity “outliers” have been found in the southern Galactic hemisphere, mostly in the Sgr longitudinal range $\Lambda_{\odot} = 20-90^{\circ}$ (Paper II). These particular stars have been associated with older parts of the Sgr leading arm that have already passed below the Galactic plane and that form a new loop around the Galactic plane (Paper IV); a fraction of these stars constitute our “Leading Arm South (LS)” sample in Paper V, where we show them to have a more metal-poor distribution overall than the dynamically younger, “Leading Arm North (LN)” sample. Figures 3-7 in this paper show these LS stars to also have abundance patterns indicating that they are not only dynamically older Sgr stream stars (i.e. stripped from Sgr longer ago), but also likely to have actually formed earlier than the LN stars on average. Specifically, the LS stars show a higher mean \[Ti/Fe\] level (0.26 dex) than that of either the Sgr core ($-0.05$ dex) or LN (0.13 dex) samples, which demonstrates that, on average, the LS stars were formed when enrichment in Sgr was still dominated by SN II. This difference in \[Ti/Fe\] trends between the LS and LN is consistent with the age-metallicity relation of Sgr (Siegel et al. 2007), which shows that stars of the mean metallicity of the LS (\[Fe/H\] $=-1.1$) were formed on average $\sim$11 Gyr ago, only $\sim$1-2 Gyr after the oldest known populations in Sgr and before significant numbers of SN Ia progenitors (mass transfer binaries) from those earliest populations could have evolved to supernova stage and introduced significant iron yields into the Sgr ISM. On the other hand, Siegel et al.’s age-metallicity relation suggests that stars of the mean metallicity of the LN, \[Fe/H\] $\sim -0.7$, were formed about 7.5 Gyr ago, or well after the onset of routine SN Ia enrichment. As pointed out in Paper V, that the dynamical age difference between the LN and LS stars is only about 0.8 Gyr whereas the mean metallicity difference between these samples is $\Delta$\[Fe/H\] $\sim -0.4$ dex (and that both of these differ significantly in mean metallicity from that of the core stars, at $<$\[Fe/H\]$>=-0.4$) suggests that the Sgr progenitor must have had a significant radial metallicity gradient before disruption. The age-metallicity relation of Siegel et al. implies that there must have been a significant mean age gradient as well.
A second group of M giant stars that do not lie along the primary velocity sequences of more recently stripped Sgr stream stars are those we have called the North Galactic Cap (NGC) group, which have velocities more positive than the LN stars in the same part of the sky (Paper II). In Paper V we showed that our sample of these stars has a similar metallicity distribution to that of the LS sample, and, by reference to the Paper IV model, we concluded by their position, velocity and metallicity distribution that the NGC group might be the trailing arm counterpart above the Galactic plane to the Leading Arm South group below the Galactic plane — that is, the two groups are Sgr debris of the same dynamical age. Figures 3-6 and Figure 9 further support this notion by showing that the abundance patterns of the NGC group very closely align with those of the LS. Not only does this comparison chemically fingerprint the NGC stars as likely Sgr debris, but it places these stars into the Sgr disruption dynamical time sequence, and in a place in that sequence that makes sense within the context of the Paper IV model for trailing arm debris in the North Galactic Hemisphere that was stripped at the same time as the LS stars, i.e. roughly 3 Gyr ago (see green debris in Fig. 1 of Paper IV or Fig. 1 of Paper V). By tagging the NGC group of stars to the Sgr trailing arm through its abundance patterns we demonstrate here one of the earliest direct applications of the concept of “chemical fingerprinting”.
Caveats and Alternative Scenarios
---------------------------------
Despite the above conclusions regarding the association of all four of our target subsamples to the Sgr system, the recent discovery of the “Virgo stellar stream” (VSS, Duffau et al. 2006; Vivas et al. 2008) and “Virgo overdensity” (VOD, Juri[ć]{} et al. 2008) shows these features to span a wide range of Northern Galactic Hemisphere sky (over 1000 deg$^{2}$ in the case of the VOD; Juri[ć]{} et al. 2008), and it is worth evaluating whether any of our subsamples — namely the Sgr LN stars and the NGC groups, which are also in the Northern Galactic Hemisphere — could be related to these extensive Virgo structures. Both the VSS and VOD have been explained in terms of debris from dSph mergers with the MW, so could account for high northern latitude stars with non-MW-like abundance patterns. We compare five properties of the LN and NGC groups against those of the VSS and VOD to demonstrate that there is unlikely a connection between the latter two halo substructures and the LN and NGC group stars.
\(1) [*Distances:*]{} The distance of the VSS is $\sim19$ kpc (Duffau et al. 2006; Newberg et al. 2007; Prior et al. 2009), while the VOD is estimated to have a distance of $\sim$6-20 kpc (Juri[ć]{} et al. 2008; Vivas et al. 2008; Keller et al. 2009). Using the \[Fe/H\] values measured for the M giants in Paper V, we have used the corresponding isochrones from Marigo et al. (2008) and the observed, dereddened $K_s$ magnitudes to estimate photometric parallax distances to each of our stars. This exercise is one fraught with large uncertainties, since for each star we have to assume an age (which can be roughly deduced from the age-metallicity relation, e.g., that in Siegel et al. 2007), a mean \[$\alpha$/Fe\] (which may or may not be well traced by \[Ti/Fe\]), and whether a particular star is on the first or second ascent giant branches; nevertheless we attempted the exercise to get a rough idea of whether the stars could be at the distances of the VSS or VOD.
Based on the rough photometric parallaxes we find that all of the LN and NGC stars are closer than the VSS, whereas a large number of the LN and NGC stars are in the range of distance quoted above for the VOD. Fortunately, other properties of these stellar systems give more definitive discrimination between them (see below).
\(2) [*Sky Positions:*]{} The VSS is a narrow structure currently mapped from $l = 279^{\circ}$ to $317^{\circ}$ and $b =
60^{\circ}$ to $63^{\circ}$ (Duffau et al. 2006); even extrapolating that swath along the same great circle, the VSS hardly intersects the region of the sky covered by the NGC group sample, which, in any case, is much more broadly dispersed than the relatively narrowly confined, $\sim 3^{\circ}$-wide VSS. The positions of only two LN stars overlap the extrapolated swath of the VSS, but the velocities of these two stars (stars 1236549$-$002941 and 1319368$-$000817) grossly mismatch that expected for the VSS at their positions.
On the other hand, the VOD [*is*]{} much more broadly distributed on the sky, covering around 1000 deg$^2$ with a center at $(l,b)\sim (300^{\circ},65^{\circ})$; however, the area of the sky covered by this excess (Juri[ć]{} et al. 2008) is very different than the areas covered by either the LN or NGC samples. Only a mere seven stars from the LN and one star from the NGC group subsamples even lie in the most liberal definition of the angular extent of the VOD.
\(3) [*Velocities:*]{} A number of VSS stars have had spectra taken that show them to have $v_{GSR}$ spanning $\sim100-130$ ${\rm
km\,s^{-1}}$ (Duffau et al. 2006; Newberg et al. 2007; Prior et al. 2009). The only available velocity information on the VOD comes from the QUEST RR Lyrae star survey (Vivas et al. 2008) combined with a sample of BHBs from the SDSS survey (Sirko et al. 2004). Vivas et al. find that there are three moving groups in the VOD region, with average $v_{GSR}=+215$, $-49$ and $-171 {\rm
km\,s^{-1}}$, respectively. These authors suggest that the VOD actually consists of the VSS plus other halo substructures. The stated $v_{GSR}$ range of the VSS has the opposite sign as most of the Sgr LN stars, but does cross the range of $v_{GSR}$ spanned by our NGC group M giants. However, no more than three of our NGC sample stars, and five of the LN sample stars barely match [*both*]{} the position and velocity distribution of either the VSS or VOD.
\(4) [*Metallicities:*]{} The M giants in our Sgr sample stretch broadly from solar metallicity down to the the lowest metallicity where M spectral type giants can form (\[Fe/H\]$\sim-1.4$), but there is no correspondingly high metallicity population that has been identified with the VSS and VOD thus far. The metallicity of VSS RR Lyrae stars is $\sim -1.86$ to $-1.95$ (Duffau et al. 2006; Prior et al. 2009), and VOD main sequence stars are at $\sim -2.0$ (An et al. 2009). Since the metallicity of RR Lyraes can extend to solar metallicity (as seen, for example, in the local MW disk; Layden 1994), the lack of metal-rich RRLSs found in the VOS or VVS suggests that the stellar populations in the VSS/VOD are metal-poor in the whole, and even slightly more metal-poor than the RR Lyrae stars found among the Sgr tidal debris (which have \[Fe/H\]$\sim-1.7$ to $-1.8$; Vivas et al. 2005, Prior et al. 2009b, Starkenburg et al. 2009).
\(5) [*Evidence for M giant populations:*]{} Of course, the very existence of M giants in a stellar population is determined by its metallicity, with only relatively metal-rich populations making first ascent giant branch stars that late in spectral type. There is no evidence for M giant stars distributed kinematically and positionally as counterparts to the other VSS/VOD tracers. If either the VSS or VOD structures contained an M giant population, one might expect at least a concentration of those M giants at the densest, “core” parts of those structures as traced by other stellar types; yet the analysis of 2MASS M giants in Paper I, as well as more recently by Sharma et al. (2009, in prep.), reveals no concentration of M giants either centered on the VOD or following the VSS. On the other hand, that the Sgr stream [*does*]{} contain M giants, that the M giants in the Galactic halo are almost entirely concentrated along the Sgr orbital plane (except at the very lowest latitudes, where Monoceros/GASS and TriAnd contribute), and that the M giants we have selected are generally matched in position on the sky with known Sgr features makes it far more likely that the LN and NGC group stars are associated with Sgr than the VSS or VOD.
On the other hand, there does remain an alternative scenario to that we have adopted as most likely here (namely that all of our subsamples are related to the Sgr system) that is presently very difficult to discriminate against. It has recently been proposed that all or some satellites of the MW may have been accreted as one or more groups of galaxies (Li & Helmi 2008; D’Onghia & Lake 2008; Metz et al. 2009). For example, D’Onghia & Lake (2008) proposed that an “LMC group” — composed of the LMC, Small Magellanic Cloud (SMC), Sgr and other satellite galaxies — were originally part of a collection of galaxies once bound to each other and that later fell into the MW together. This hypothesis can explain the planer orbital configuration of some dSphs in the MW halo (e.g., Kunkel 1979; Lyden-Bell 1982; Majewski 1994; Palma et al. 2002; Metz et al. 2008), and could explain the planar configurations of tidal debris, even if that debris ultimately derived from different satellites. With slight differences in initial orbits of the parent dwarf galaxies, that debris today might have differing orientations and velocities, despite being in nearly the same plane and having the same general direction of angular momentum; in principle, therefore, one could explain our various M giant samples lying in a largely planar distirbution as deriving from different parent accreted dwarf galaxies, but perhaps parents that fell into the MW together.
One motivation for proposing such an “infalling group” scenario in the specific case of the Sgr system is that it might provide an origin for the recently discovered, but not yet definitively-explained bifurcation of the northern Sgr stream leading arm as seen in the SDSS imaging (Belokurov et al. 2006; Fellhauer et al. 2006).[^4] On the other hand, Yanny et al. (2009) find not only the metallicities, but the velocities and distances to be indistinguishable in the two pieces of the bifurcated SDSS Sgr feature. This suggests a strong coherence in star formation and chemical enrichment histories, at least for those stars along the bifurcated, Northern Hemisphere Sgr arm — and, were such a coherence between two parent systems to exist, it would be very hard at present to distinguish this from a single parent origin (i.e. the known Sgr core) for all of the debris. At present, in the absence of evidence supporting multiple parent systems producing multiple M giant substructures along the same plane in the sky, we prefer the most straightforward interpretation for the connection of the LN and LS groups as parts of a single Sgr leading arm, as well as the identification of the NGC group with (a diffuse distribution of) the Sgr trailing arm.
In conclusion, detailed chemical abundance analysis of our various samples demonstrates that these stars are by and large dSph-like and with little contamination by the nominal Milky Way populations, halo or disk. Because the leading arm stars were also pre-selected to be in the Sgr stream and to follow the expected velocity trends for Sgr debris, because they form a clear and logical chemical sequence, and because no evidence for other M giant tidal debris from any other satellite is found to intersect the Sgr stream in relevant parts of the sky, and, furthermore, because all evidence continues to support that the bulk of all high latitude M giants have been contributed from the Sgr system, we conclude that the vast majority of our leading arm stars must be from the Sgr dSph. Since the NGC sample stars share similar chemical patterns of Sgr LS stars, we conclude that they are likely Sgr stars as well, but this hypothesis requires them to be from the trailing arm debris due to their peculiar velocities. Therefore, we propose that all of our M giant samples, taken as a group, together paint a chemical portrait of the Sgr dwarf as it appeared $\sim3$ Gyr ago.
Finally, we close this discussion of caveats with a warning to the reader that although we believe the LN, LS and even the NGC group stars to be primarily constituted by stars stripped from the Sgr dSph system, these samples represent highly biased sets of such stars by distance: In the present high resolution spectroscopic study we selected targets that we expected to be Sgr stream members but that were also bright enough to allow echelle spectroscopy to adequate $S/N$ for chemical abundance analysis. Moreover, we selected the target stars for this study (and that in Paper V) from an input catalog — those M giants that had radial velocities from medium resolution spectroscopy (e.g., Paper II, Paper IV) — that itself shares some degree of the same bias. Given this strong prejudice for the closest, brightest Sgr members we could find, one should exercise due caution in how these stars are used. For example, one should not use the stars in Table 3 to constrain the mean distances of different parts of the Sgr stream since those mean distances no doubt lie beyond the stars presented here; the latter are likely several or more $\sigma$ outliers in the distance distribution of Sgr stream stars.
Global Chemical Evolution of the Sgr System
===========================================
We can take advantage of our entire Sgr high resolution core + tidal tail sample to build the most complete (in terms of metallicity span) chemical portrait of the Sgr progenitor yet assembled; this portrait is summarized by the middle panels of Figures 3-6 as well as Figure 9. Chemical evolution is driven by the cycle of nucleosynthesis and subsequent transfer of nucleosynthetic products into the environment for incorporation into future generations of stars. Chemical evolution models guide the interpretation of observed patterns in a galactic system such as those demonstrated in Figures 3-6 and 9.
\[Ti/Fe\]
---------
For example, as mentioned above, $\alpha$ elements are mainly produced by SN II while iron is synthesized largely by SN Ia, whose progenitors have a lifetime of $\sim1$ Gyr. Therefore, \[$\alpha$/Fe\] is high for early chemical enrichment in a stellar system and then declines as SN Ia “turn-on". Titanium acts mainly as an $\alpha$ element, and from its trend with \[Fe/H\] we can infer how far chemical evolution proceeded in the first $\sim1$ Gyr of the system’s life, an indication of the initial star formation rate (SFR). Figure 3 shows that the downturn in \[Ti/Fe\] happens at around \[Fe/H\] $=-0.9$ for Sgr, which is only a few tenths of a dex lower than the transition seen in the MW field star population; this suggests a lower early SFR in Sgr than for the MW field population (Monaco et al. 2005b; Sbordone et al. 2007). Alternatively, Lanfranchi et al. (2006) argue that initially Sgr may have started out with a high star formation efficiency — at least higher than that of other dSphs — but the resulting intense galactic wind acted to substantially squelch the SFR. Lanfranchi et al. claim that most of the observed Sgr stars formed after the beginning of the wind, which explains why they have lower \[$\alpha$/Fe\] than MW stars.
Figure 3 also shows the \[Ti/Fe\] trends for the LMC and other dSphs, which are very different from the trends for the MW and the metal-poor Sgr stars. The chemical differences among these various systems are due to their unique star formation histories (Venn et al. 2004; Johnson et al. 2006). As may be seen, in the LMC and other dSphs, the \[Ti/Fe\] is low over all \[Fe/H\] probed, due to a lower early SFR, which is the result of low star formation efficiencies and high galactic winds (Geisler et al. 2007), and also probably because of fewer high-mass SN II in these small systems (Woosley & Weaver 1995; Tolstoy et al. 2003; Pomp[é]{}ia et al. 2008).
We also find the metal-poor Sgr stars (with \[Fe/H\] $\lesssim-1.2$) to have high Ti abundances, similar to the trend for the MW. This implies that the early chemical composition of Sgr was more like the MW (Shetrone 2004) than the LMC and other dSphs; from a similar comparison of \[$\alpha$/Fe\] abundances of low \[Fe/H\], Monaco et al. (2005b) conclude that the Sgr progenitor was probably a relatively large, star forming, gas-rich object. On the other hand, Monaco et al. (2005b) also suggest Sgr should have had a different subsequent chemical evolution from the MW and other Local Group galaxies due to the strong and disruptive dynamical interactions this system has clearly had with the MW; these interactions in a gas-rich system can typically trigger star formation activity (e.g., Kravtsov et al. 2004; Zaritsky & Harris 2004).
s-process Elements
------------------
The s-process elements are thought primarily to occur during thermal pulses in the intershell convection zone in low mass AGB stars. The neutron flux per seed nucleus is roughly inversely proportional to the metallicity of the AGB. Therefore, at low metallicities AGB stars produce heavier s-process elements like La more efficiently than lighter species like Y, all of the way up to the formation of the heaviest s-process element, Pb, in the most metal-poor environments (see the review by Busso et al. 1999 and 2004). Figures 5-6 show upturns in \[La/Fe\] and \[La/Y\] for Sgr at \[Fe/H\] $\sim-0.5$. Since \[La/Y\] is enhanced in metal-rich Sgr stars, the high ratio of heavy to light s-process elements (\[hs/ls\]) among the metal-rich stars indicates a strong contribution from low-metallicity AGB progenitors (Smecker-Hane & McWilliam 2002), and a slower SFR of Sgr than the MW, so that the low-metallicity AGB yields have enough time to contaminate the ISM (Venn et al. 2004; Pomp[é]{}ia et al. 2008). On the other hand, Sgr and LMC stars at lower metallicity show similar trends to MW stars, particularly in Y and La; this indicates that these stars might be among the first ones formed because they were not yet contaminated by the AGB yields (Johnson et al. 2006; Mucciarelli et al. 2008). Moreover, models for the evolution of old, gas-poor dSphs (e.g., Sculptor) predict subsolar Y over all \[Fe/H\] probed, and an upturn in La abundance at \[Fe/H\] $\sim-1.6$, due to the appearance of the products of metal-poor AGB progenitors in the ISM (Fenner et al. 2006; Gibson 2007). The patterns of La in Sgr basically agree with the prediction of the gas-poor dSphs, except the upturn in Sgr occurs at higher \[Fe/H\]. That suggests a higher early SFR of Sgr than other dSphs, and the high \[hs/ls\] can be seen as another “clock” of star formation, in addition to the \[$\alpha$/Fe\] ratio.
Relative Galaxy Star Formation Rates
------------------------------------
Figures 3-6 have demonstrated abundance trends for Sgr that distinguish it from the MW and other MW satellites. Several of these differences relate to the onset of specific chemical chronometers, such as the contribution to the gaseous environment of the yields from SN Ia (seen as a decrease in \[Ti/Fe\]) and metal-poor AGB (seen as an increase in \[La/Y\], for example), where the \[Fe/H\] corresponding to the chemical signature is correlated to the SFR prior to the creation of that signature. A comparison of the above two chronometers (\[Ti/Fe\] and \[La/Y\]) suggests a natural sequence in relative SFRs from the other MW dSph systems (lowest SFRs), to the LMC (a modest SFR), and Sgr (the highest SFR among the MW satellites), based solely on the positions of the transitions in chemical properties in \[Fe/H\].
To emphasize the point that the chemical histories of the MW satellites likely differ primarily as a function of their SFRs, we modify the comparisons of the chemical properties of Sgr to those of the LMC and other dSphs as shown in Figures 3-6 by shifting the distributions of the dSph ensemble and the LMC by $\Delta$\[Fe/H\] = +1 and +0.4, respectively (these particular values were selected by eye for illustrative purposes only). The results, shown in Figure 10, reveal a much closer agreement in the overall shapes of the abundance trends shown, suggesting even the possibility of a “universal” chemical enrichment pattern among MW satellites, with the primary difference being the \[Fe/H\] placement of the pattern, which itself is a function of the early SFR.
It is immediately obvious from Figures 3-6 and Figure 10 that Sgr much more closely matches the overall chemical evolution of the LMC than it does other MW dSphs. Indeed, their similar chemistries suggest that it may be reasonable to consider the LMC — a late type, dwarf spiral or irregular galaxy — as a more appropriate paradigm for the pre-interaction state of Sgr than is provided by the other dSphs, even though Sgr morphologically resembles the other dSphs now in terms of its more regular structure and lack of current star formation or gas. Sgr’s present morphological difference with the LMC is easily accounted for by the fact that the LMC, which apparently has just fallen into the MW environment for the first time (Kallivayalil et al. 2006; Besla et al. 2007; Piatek et al. 2008), has not experienced the tidal battering that the MW-bound Sgr has experienced for at least the past few Gyr. It is well known that severe tidal encounters such as has been experienced by Sgr can “tidally stir” dwarf irregular systems into dSphs (Mayer et al. 2001; Skillman et al. 2003; Klimentowski et al. 2007). Evidence that as recently as several Gyr ago Sgr was actively forming stars can be found in its color-magnitude diagram, which shows evidence for populations as young as 2 Gyr old or even younger (Sarajedini & Layden 1995; Layden & Sarajedini 2000; Siegel et al. 2007).
Typically SFRs and chemical enrichment are thought to be driven by the mass of a system. The mass-metallicity relation of dSphs has been investigated by, for example, Yoshii & Arimoto (1987) and Tamura et al. (2001), who find that metallicity is roughly logarithmically proportional to the mass of these satellite galaxies. Because Sgr is presently dominated by stellar populations with higher metallicity, that it is more massive than the other MW dSphs is consistent with the notion of a mass-metallicity relation. Curiously, however, Sgr is more chemically evolved than the LMC, despite the fact that the mass of Sgr, at least that estimated the system had several Gyr ago by Paper IV of $\sim 2-5 \times 10^8$ M$_{\odot}$, is less than that of the LMC ($\sim 1-2 \times 10^{10}$ M$_{\odot}$; Gardiner & Noguchi 1996; van der Marel et al. 2002). Certainly star formation efficiency may also be a significant driver in the rate of chemical evolution (Lanfranchi et al. 2006, 2007), but an expected consistency with mass-metallicity relations, and the possibility of a universal enrichment pattern (e.g., Fig. 10), may hint that Sgr was actually once significantly more massive than even the more recent, high estimates of its former mass, perhaps with a mass comparable to or larger than the LMC. If so, this would imply a much longer tidal stripping history for Sgr than has been previously observed (e.g., Paper I; Belokurov et al. 2006) and modeled (Paper IV; Fellhauer et al. 2006), perhaps with much of that mass lost as pure dark matter (to explain the lack of obvious stars from earlier orbits than has heretofore been observed).
Summary
=======
Chemical abundances in stars are fossil records of the enrichment history of a galaxy and their distinctive patterns provide signatures that, if not uniquely branding its stars, at least allow us a means to test whether particular stars are likely to be associated with that system based on whether their chemistry fits into its overall abundance patterns. We have applied this test here not only to demonstrate the likely high purity of the spatially- and kinematically-selected sample of M giant Sgr tidal tail star candidates used in Paper V to show the existence of a strong metallicity gradient along the Sgr tails, but also to prove the likely Sgr-origin of the somewhat mysterious “North Galactic Cap moving group” also discussed in that previous contribution. We conclude that most of the NGC moving group stars are probably from the old trailing debris arm of Sgr (their MDF also supports this conclusion, see Paper V). Thus the present paper demonstrates the applicability of “chemical fingerprinting”, a technique long discussed as one of the potentially valuable future tools of stellar populations research.
The evolving chemical content of a galaxy depends on many variables — such as the stellar initial mass function and the SFR — that lead to the specific distribution of chemical patterns among the stars in the system. Tidal stripping can also shape the [*observed*]{} abundance patterns of the stars in a galaxy by preferentially removing certain populations, particularly if the system has spatial variations in metallicity and/or an age-metallicity relation over timescales overlapping the period during which the tidal loss of stars occurs. Both of these situations are at play in the Sgr system. Only by surveying both the stars lost from as well as remaining in a galaxy like Sgr do we have hope of recovering an unbiased view of its chemical history. We have explored the chemical abundance patterns in stars along the Sgr tidal tails, which, when combined with data on stars in the Sgr core, afford us the most complete view of the original chemical abundance distributions of this system to date; but even then, our view here is likely to be rather incomplete and still tells us only part of the story. For example, we still do not have good information on the chemical properties of the oldest, most metal-poor Sgr populations.
In stars associated with Sgr we have found lower values of \[Ti/Fe\] at \[Fe/H\] $> -1$ than for MW stars at the same \[Fe/H\] (Fig. 3). And while the MW exhibits an apparent transition of \[Ti/Fe\] to solar-like levels at \[Fe/H\] $\sim -0.7$, such a transition (and to even lower \[Ti/Fe\] levels) happens for Sgr stars at \[Fe/H\] $\sim
-1$. This shows that Sgr had a slower SFR than the MW, like other Galactic satellites. We also find the Sgr stars to have subsolar abundances of the light s-process element yttrium at all \[Fe/H\] probed, while the heavy s-process element lanthanum is enhanced relative to the MW for \[Fe/H\] $\gtrsim -0.5$ (Figs. 4-6). This indicates the importance of low-metallicity AGB nucleosynthesis in the metal-rich Sgr stars (Smecker-Hane & McWilliam 2002) and is another signature of a slower SFR than occurred in the MW.
We also find that although the LMC and other dSphs exhibit similar chemical pattern trends as Sgr, these patterns exhibit their significant abundance transitions at lower \[Fe/H\] than for Sgr. Such differences suggest a faster enrichment and more rapid star formation history in Sgr relative to those in the LMC and other dSphs. After applying a hypothetical shift in \[Fe/H\] of +1 dex in the abundance patterns of other dSphs and +0.4 dex for the LMC we find that the chemical patterns of Sgr, LMC and other dSphs strongly resemble each other (Fig. 10), which suggests the possibility of a universal chemical enrichment progression among MW satellites that differs only by the SFRs; these SFRs, in turn, are likely correlated to the original masses of these systems. The relative shifts suggest that the relative SFR of these galaxies are, in order from slowest to fastest: the other MW dSphs, the LMC, Sgr and the MW respectively. Based on their close chemical similarities we suggest that Sgr was probably formerly more similar in nature to the present morphology and structure of the LMC than to those of other MW dSphs. The wide ranges of metallicities in Sgr suggests a large age spread, and implies a long duration of star formation in its central regions (Smecker-Hane & McWilliam 2002). This agrees with the discovery of quite young populations in the Sgr core (Siegel et al. 2007), which shows that Sgr has had even relatively recent star formation activity, like the LMC.
The color-magnitude distribution of Sgr stars demonstrates that its star formation history was highly variable, including some fairly well-defined “bursts” (Layden & Sarajedini 2000; Bellazzini et al. 1999; Monaco et al. 2002; Siegel et al.2007). These produced populations with different, but overlapping radial density profiles in the progenitor satellite, and likely a strong internal metallicity gradient. In Paper V we found a strong metallicity gradient along the Sgr tidal arms, with the stars in these arms increasingly more metal-poor with angular separation (i.e. orbital phase difference) from the core. By comparing this gradient to models of the timescale for the disruption that produced these tails (Paper IV) we argued in Paper V that Sgr must have experienced a quite rapid change in its binding energy over the past several Gyr, even if the Sgr progenitor had one of the steepest metallicity gradients among the known MW dSphs. These multiple population bursts also created Sgr’s unique chemical patterns, especially at higher metallicities, compared to those of the MW (as well as to those of other dSphs). Nevertheless, while the abundance patterns of stars presently in the Sgr core differ greatly from those of MW stars of the same metallicity, in this paper we have found that the more metal-poor stars in the Sgr tail actually have abundance patterns that more closely resemble those of MW stars at their respective \[Fe/H\]. The Sgr example vividly demonstrates that while the current populations of stars in dSph satellites are indeed chemically differentiated from the MW field population and that “one could not build the MW halo from the [*present*]{} MW satellites” (as emphasized by, e.g., Unavane et al. 1996), this point is not very relevant because it is still possible that the MW halo field population could have derived from the [*stripped off*]{} populations of these very same satellites. Majewski et al. (2002) and Muñoz et al. (2006) have previously made the same point using the Carina dSph example.
The Sgr system can be seen as an evolutionary bridge from dSphs to the MW in galaxy evolution. On the one hand, recent models (Robertson et al. 2005; Font et al. 2006a,b) predict that the local halo assembled rapidly — before SN Ia had time to occur — with the early accretion and dissipation of a few massive satellites to produce the high values of \[$\alpha$/Fe\] at low \[Fe/H\] seen among halo field stars. Font et al. suggest the MW satellites accreted $\sim$9 Gyr ago have all been disrupted completely, while the surviving satelliites of today were only recently accreted into the MW system within the past few Gyr on relatively circular orbits. The implication is that these satellite systems have yet to contribute stars to the halo. On the other hand, Sgr clearly provides an example of an ongoing merger event that is not only contributing stars to the halo, but some low metallicity stars of high \[$\alpha$/Fe\]; it may therefore be seen as a counterexample to the ”early accretion” hypothesis. However, Font et al. point out that none of the present MW satellites are presently located in the inner halo except Sgr and because of this Sgr may be an exceptional case. But the recent work on the Ursa Minor (Muñoz et al. 2005), Leo I (Sohn et al. 2007; Mateo et al. 2008) and Carina dSphs (Muñoz et al. 2006, 2008) yielding evidence for tidally stripped stars from these systems suggests that the present satellites may also have already contributed stars to the MW halo. This, coupled with the finding that some of the MW dSph stars at the lowest metallicities are indeed $\alpha$-enhanced (Shetrone et al. 2003; Geisler et al. 2005; Tolstoy 2005), suggests that Sgr may not be the only current contributor of such stars; at the very least these systems are likely to be [*future*]{} contributors of these stars, which shows that they did not all originate in early accretions.
The results of this paper would be considerably strengthened by a careful survey of the chemical abundance patterns of Sgr trailing arm stars. The significant overlap in orbital phase position along the Sgr leading arm (see Fig. 1 in Paper IV) “fuzzes out” the resolution of the dynamical stripping time. In contrast to the stronger phase mixing in the leading arm, the dynamics of the longer trailing arm demonstrate much better energy sorting of the debris, so that position along the trailing tail is much better correlated to the amount of time since a star was stripped from the Sgr core. Further scrutiny of cleanly isolated trailing arm stars may reveal even more clear chemical trends with dynamical age. We will investigate this in future work.
M.-Y.C. and S.R.M. acknowledge support from NSF grants AST-0307851 and AST-0807945. This project was also supported by the [*SIM Lite*]{} key project [*Taking Measure of the Milky Way*]{} under NASA/JPL contract 1228235. VVS and KC also thank support from the NSF via grant AST-0646790. D.G. gratefully acknowledges support from the Chilean [*Centro de Astrofísica*]{} FONDAP No. 15010003 and the Chilean Centro de Excelencia en Astrofísica y Tecnologías Afines (CATA).
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[lccc]{}
& 7489.572 & 2.249 & 2.339E-01\
& 7496.120 & 2.240 & 8.770E-02\
& 7450.320 & 1.740 & 1.202E-01\
& 7483.234 & 0.126 & 2.080E-04\
& 7483.237 & 0.126 & 4.160E-04\
& 7483.259 & 0.126 & 2.080E-04\
& 7483.262 & 0.126 & 4.160E-04\
& 7483.267 & 0.126 & 6.240E-04\
& 7483.304 & 0.126 & 4.160E-04\
& 7483.308 & 0.126 & 6.240E-04\
& 7483.315 & 0.126 & 8.320E-04\
& 7483.367 & 0.126 & 6.240E-04\
& 7483.374 & 0.126 & 8.320E-04\
& 7483.382 & 0.126 & 1.039E-03\
& 7483.449 & 0.126 & 8.320E-04\
& 7483.458 & 0.126 & 1.039E-03\
& 7483.468 & 0.126 & 1.247E-03\
& 7483.550 & 0.126 & 1.039E-03\
& 7483.561 & 0.126 & 1.247E-03\
& 7483.573 & 0.126 & 1.455E-03\
& 7483.670 & 0.126 & 1.247E-03\
& 7483.682 & 0.126 & 1.455E-03\
[lccc]{}
Sgr(core) & & &\
$1849222-293217$ & & 193.5 & 38.3\
$1853333-320146$ & & & 14.1\
$1854283-295740$ & 147.1 & 124.8 & 35.1\
$1855341-302055$ & 134.0 & 95.8 & 38.9\
$1855556-293316$ & & & 28.5\
$1902135-313030$ & 64.9 & 49.7 & 34.8\
\
Sgr (north leading arm)& & &\
$0919216+202305$ & 105.8 & 91.0 & 12.0\
$0925364+213807$ & 113.0 & 109.1 & 37.0\
$1034395+245206$ & 121.5 & 66.8 & 72.0\
$1100516+130216$ & 74.3 & 57.1 & 23.7\
$1101112+191311$ & 117.9 & 84.3 & 32.0\
$1111493+063915$ & 115.6 & 84.8 & 24.4\
$1112480+013211$ & 112.0 & 91.0 & 40.0\
$1114573-215126$ & 112.0 & 113.0 & 16.0\
$1116118-333057$ & & & 39.0\
$1128316-031647$ & 122.3 & 97.5 & 37.8\
$1135388-022602$ & 132.1 & 113.5 & 24.7\
$1140226-192500$ & 113.7 & 98.2 & 24.0\
$1208101-090753$ & 113.3 & 84.8 & 18.8\
$1223590-073028$ & 111.3 & 103.8 & 18.0\
$1224255-061852$ & 121.0 & 89.2 & 33.8\
$1227367-031834$ & 114.0 & 93.7 & 33.0\
$1236549-002941$ & 126.1 & 108.6 & 36.0\
$1249078+084455$ & 100.6 & 95.0 & 34.0\
$1318500+061112$ & 92.3 & 78.8 & 22.0\
$1319368-000817$ & 109.5 & 105.6 & 24.0\
$1330472-211847$ & 113.0 & 84.2 & 24.0\
$1334532+042053$ & 100.5 & 96.2 & 22.0\
$1348366+220101$ & 99.1 & 75.4 & 19.8\
$1407060+063311$ & 115.4 & 109.5 & 22.3\
$1411221-061013$ & 106.9 & 104.2 & 26.0\
$1435018+070827$ & 108.7 & 88.9 & 21.7\
$1450544+244357$ & 74.0 & 54.9 & 20.0\
$1456137+151112$ & 117.0 & 88.2 & 52.3\
$1512142-075250$ & 100.8 & 77.5 & 26.0\
$1538472+494218$ & 131.5 & 123.8 & 26.0\
\
Sgr(south leading arm) & & &\
$2031334-324453$ & & 66.2 & 19.2\
$2037196-291738$ & 131.6 & 111.1 & 19.0\
$2046335-283547$ & & 63.3 & 12.8\
$2050020-345336$ & 99.6 & 79.5 & 24.3\
$2105585-275602$ & 129.9 & 82.0 & 24.5\
$2114412-301256$ & 126.4 & 100.2 & 40.8\
$2130445-210034$ & 87.7 & 51.6 & 21.4\
$2135183-203457$ & 167.0 & 134.6 & 35.5\
$2154471-224050$ & 127.4 & 123.6 & 31.9\
$2226328-340408$ & 53.7 & 38.8 & 20.5\
\
NGP & & &\
$1033045+491604$ & 89.4 & 60.4 & 44.4\
$1041479+294917$ & 111.1 & 66.1 & 34.7\
$1051302+004400$ & 92.0 & 67.5 & 23.4\
$1115376+000800$ & 116.3 & 92.1 & 21.5\
$1214190+071358$ & 49.9 & 42.3 & 13.0\
$1257013+260046$ & 92.3 & 73.2 & 23.0\
$1343047+221636$ & 102.5 & 75.0 & 24.0\
$1412161+294303$ & 119.7 & 103.2 & 40.0\
$1424425+414932$ & 111.1 & 100.6 & 24.0\
$1429456+230043$ & 88.5 & 72.7 & 18.0\
$1513011+222640$ & 73.6 & 69.0 & 22.0\
$1536502+580017$ & 95.1 & 82.8 & 26.0\
$1545189+291310$ & 64.5 & 53.9 & 13.0\
\
Standard Stars & & &\
Arcturus & 74.2 & 52.9 & 27.9\
$\beta$ Peg & 131.0 & 106.0 & 34.0\
$\beta$ And & 120.7 & 103.0 & 47.7\
$\rho$ Per & 127.7 & 111.4 & 40.4\
HD146051(mike) & 129.1 & 100.8 & 43.6\
HD146051(kpno) & 124.9 & 95.7 & 45.8\
[lccccccccccccccc]{}
Sun & & & & 7.45 & & & 4.90 & & & 2.21 & & & 1.13 & &\
\
Sgr(core) & & & & & & & & & & & & & & &\
$1849222-293217$ & 3850 & 0.9 & 2.43 & 7.24 & -0.21 & 0.10 & 5.54 & 0.85 & & 1.70 & -0.30 & & 1.51 & 0.59 & 0.02\
$1853333-320146$ & 3750 & 0.7 & 2.60 & 7.15 & -0.30 & 0.14 & CR & & & 1.10 & -0.81 & & 1.38 & 0.55 & 0.01\
$1854283-295740$ & 3750 & 0.0(-) & 3.21 & 6.48 & -0.97 & 0.06 & 4.20 & 0.27 & 0.17 & 0.98 & -0.26 & & 0.65 & 0.49 & 0.02\
$1855341-302055$ & 3800 & 1.0 & 1.84 & 7.47 & 0.02 & 0.08 & 4.66 & -0.32 & 0.09 & 1.89 & -0.34 & & 1.36 & 0.21 & 0.02\
$1855556-293316$ & 3700 & 0.5 & 2.36 & 7.18 & -0.27 & 0.07 & CR & & & 1.37 & -0.57 & & 1.37 & 0.51 & 0.03\
$1902135-313030$ & 3750 & 0.0(-) & 1.04 & 6.41 & -1.04 & 0.11 & 3.80 & -0.06 & 0.10 & 1.18 & 0.01 & & CR & &\
\
Sgr (north leading arm)& & & & & & & & & & & & & & &\
$0919216+202305$ & 3700 & 0.25 & 1.47 & 6.82 & -0.63 & 0.08 & 4.29 & 0.02 & 0.12 & 0.75 & -0.83 & 0.06 & 0.14 & -0.36 & 0.00\
$0925364+213807$ & 3600 & 0.5 & 1.29 & 7.22 & -0.23 & 0.07 & 4.70 & 0.03 & 0.24 & 1.76 & -0.22 & 0.07 & CR & &\
$1034395+245206$ & 3700 & 0.25 & 1.45 & 6.80 & -0.65 & 0.09 & 4.25 & 0.00 & 0.33 & 2.08 & 0.52 & 0.12 & CR & &\
$1100516+130216$ & 3800 & 0.0 & 1.35 & 6.39 & -1.06 & 0.08 & 3.93 & 0.09 & 0.11 & 0.82 & -0.33 & 0.13 & 0.06 & -0.01 & 0.03\
$1101112+191311$ & 3700 & 0.8 & 1.51 & 7.47 & 0.02 & 0.09 & 4.53 & -0.39 & 0.11 & 1.78 & -0.45 & 0.09 & 1.15 & 0.00 & 0.04\
$1111493+063915$ & 3600 & 0.0(-) & 1.71 & 6.75 & -0.70 & 0.09 & 4.08 & -0.12 & 0.02 & 1.03 & -0.48 & & 0.05 & -0.38 & 0.04\
$1112480+013211$ & 3800 & 0.5 & 1.60 & 6.97 & -0.48 & 0.12 & 4.42 & 0.00 & 0.07 & 1.55 & -0.18 & 0.02 & 0.44 & -0.21 & 0.02\
$1114573-215126$ & 3550 & 0.0(-) & 1.33 & 6.64 & -0.81 & 0.07 & 4.42 & 0.33 & 0.30 & 0.81 & -0.59 & 0.05 & CR & &\
$1116118-333057$ & 3650 & 0.0(-) & 1.39 & 6.32 & -1.13 & 0.05 & CR & & & 0.19 & 0.19 & & CR & &\
$1128316-031647$ & 3700 & 0.9 & 1.64 & 7.41 & -0.04 & 0.05 & 4.61 & -0.25 & 0.01 & 1.90 & -0.27 & & 1.10 & 0.01 & 0.06\
$1135388-022602$ & 3700 & 0.9 & 1.23 & 7.45 & 0.00 & 0.07 & 5.15 & 0.25 & 0.02 & 1.66 & -0.55 & & 0.75 & -0.38 & 0.12\
$1140226-192500$ & 3800 & 0.6 & 1.16 & 7.07 & -0.38 & 0.05 & 4.78 & 0.26 & 0.08 & 1.31 & -0.52 & 0.08 & 0.53 & -0.22 & 0.05\
$1208101-090753$ & 3750 & 0.0(-) & 1.82 & 6.46 & -0.99 & 0.07 & 4.16 & 0.25 & 0.04 & 0.69 & -0.53 & & -0.16 & -0.30 & 0.05\
$1223590-073028$ & 3600 & 0.0 & 1.50 & 6.73 & -0.72 & 0.08 & 4.29 & 0.11 & 0.20 & 0.88 & -0.61 & 0.02 & CR & &\
$1224255-061852$ & 3750 & 0.3 & 1.71 & 6.80 & -0.65 & 0.08 & 4.33 & 0.08 & 0.03 & 1.30 & -0.26 & & 0.19 & -0.29 & 0.06\
$1227367-031834$ & 3850 & 0.5 & 1.68 & 6.90 & -0.55 & 0.08 & 4.47 & 0.12 & 0.09 & 1.36 & -0.30 & 0.02 & 0.37 & -0.21 & 0.02\
$1236549-002941$ & 3750 & 0.5 & 1.33 & 7.06 & -0.39 & 0.10 & 4.80 & 0.29 & 0.05 & 1.53 & -0.28 & 0.03 & CR & &\
$1249078+084455$ & 3800 & 0.3 & 1.52 & 6.78 & -0.67 & 0.10 & 4.37 & 0.14 & 0.22 & 1.31 & -0.23 & 0.13 & 0.38 & -0.08 & 0.08\
$1318500+061112$ & 3850 & 0.4 & 1.67 & 6.68 & -0.77 & 0.10 & 4.21 & 0.08 & 0.16 & 1.01 & -0.43 & 0.14 & CR & &\
$1319368-000817$ & 3500 & 0.0(-) & 1.64 & 6.86 & -0.59 & 0.08 & 4.16 & -0.15 & 0.25 & 1.09 & -0.53 & 0.02 & CR & &\
$1330472-211847$ & 3850 & 0.0(-) & 1.75 & 6.35 & -1.10 & 0.08 & 4.34 & 0.54 & 0.03 & 0.76 & -0.35 & 0.09 & CR & &\
$1334532+042053$ & 3700 & 0.25 & 1.49 & 6.83 & -0.62 & 0.06 & 4.28 & 0.00 & 0.24 & 1.05 & -0.54 & 0.15 & 0.29 & -0.22 & 0.03\
$1348366+220101$ & 3800 & 0.1 & 1.49 & 6.63 & -0.82 & 0.09 & 4.20 & 0.12 & 0.05 & 0.80 & -0.59 & & -0.01 & -0.32 & 0.12\
$1407060+063311$ & 3700 & 0.0(-) & 1.78 & 6.50 & -0.95 & 0.09 & 4.29 & 0.34 & 0.23 & 0.84 & -0.42 & & -0.06 & -0.24 & 0.05\
$1411221-061013$ & 3700 & 0.25 & 1.51 & 6.89 & -0.56 & 0.06 & 4.38 & 0.04 & 0.26 & 1.18 & -0.47 & 0.04 & 0.10 & -0.47 & 0.05\
$1435018+070827$ & 3700 & 0.0(-) & 1.61 & 6.53 & -0.92 & 0.09 & 4.18 & 0.20 & 0.09 & 0.83 & -0.46 & & -0.19 & -0.40 & 0.14\
$1450544+244357$ & 3800 & 0.0 & 1.63 & 6.37 & -1.08 & 0.08 & 3.86 & 0.04 & 0.11 & 0.70 & -0.43 & 0.19 & -0.10 & -0.15 & 0.01\
$1456137+151112$ & 3750 & 0.0(-) & 1.71 & 6.47 & -0.98 & 0.08 & 4.26 & 0.34 & 0.02 & 1.38 & 0.15 & & -0.07 & -0.19 & 0.10\
$1512142-075250$ & 3700 & 0.0(-) & 1.26 & 6.48 & -0.97 & 0.08 & 4.19 & 0.26 & 0.01 & 0.97 & -0.27 & & CR & &\
$1538472+494218$ & 3600 & 0.0(-) & 1.52 & 6.39 & -1.06 & 0.08 & 4.55 & 0.71 & 0.20 & 0.99 & -0.16 & 0.01 & -0.20 & -0.27 & 0.06\
\
Sgr(south leading arm) & & & & & & & & & & & & & & &\
$2031334-324453$ & 3800 & 0.0(-) & 2.67 & 6.13 & -1.32 & 0.11 & 3.97 & 0.39 & & 0.54 & -0.35 & & CR & &\
$2037196-291738$ & 3700 & 0.0 & 2.32 & 6.75 & -0.70 & 0.09 & 4.21 & 0.01 & 0.14 & 0.79 & -0.72 & & 0.39 & -0.04 & 0.02\
$2046335-283547$ & 3750 & 0.0(-) & 2.56 & 6.19 & -1.26 & 0.06 & 3.86 & 0.22 & & 0.42 & -0.53 & & CR & &\
$2050020-345336$ & 3800 & 0.0(-) & 2.12 & 6.41 & -1.04 & 0.10 & 4.06 & 0.20 & 0.14 & 0.79 & -0.38 & & -0.04 & -0.13 & 0.02\
$2105585-275602$ & 3700 & 0.0(-) & 2.13 & 6.49 & -0.96 & 0.10 & 4.09 & 0.15 & 0.08 & 0.87 & -0.38 & & CR & &\
$2114412-301256$ & 3750 & 0.0(-) & 2.06 & 6.30 & -1.15 & 0.10 & 4.25 & 0.50 & 0.08 & 1.12 & 0.06 & & 0.21 & 0.23 & 0.01\
$2130445-210034$ & 3750 & 0.0(-) & 2.62 & 6.10 & -1.35 & 0.10 & 3.71 & 0.16 & 0.04 & 0.63 & -0.23 & & 0.04 & 0.26 & 0.03\
$2135183-203457$ & 3700 & 0.0(-) & 2.30 & 6.55 & -0.90 & 0.13 & 4.52 & 0.52 & 0.3 & 1.08 & -0.23 & & 0.04 & -0.19 & 0.01\
$2154471-224050$ & 3800 & 0.1 & 2.14 & 6.54 & -0.91 & 0.06 & 4.44 & 0.45 & 0.26 & 1.00 & -0.30 & & 0.19 & -0.03 & 0.04\
$2226328-340408$ & 3800 & 0.0(-) & 1.93 & 6.11 & -1.34 & 0.09 & 3.58 & 0.02 & 0.15 & 0.61 & -0.26 & & -0.47 & -0.26 & 0.01\
\
NGP & & & & & & & & & & & & & & &\
$1033045+491604$ & 3800 & 0.3 & 1.56 & 6.70 & -0.75 & 0.09 & 4.01 & -0.14 & 0.00 & 1.47 & 0.01 & 0.14 & 0.46 & 0.08 & 0.01\
$1041479+294917$ & 3800 & 0.0(-) & 1.56 & 6.21 & -1.24 & 0.10 & 4.20 & 0.54 & 0.15 & 1.01 & 0.04 & & 0.04 & 0.15 & 0.01\
$1051302+004400$ & 3800 & 0.0(-) & 1.65 & 6.07 & -1.38 & 0.06 & 4.05 & 0.53 & 0.06 & 0.70 & -0.13 & & -0.32 & -0.07 &\
$1115376+000800$ & 3800 & 0.0 & 1.56 & 6.49 & -0.96 & 0.10 & 4.42 & 0.48 & 0.04 & 0.76 & -0.49 & & -0.06 & -0.23 & 0.09\
$1214190+071358$ & 3750 & 0.0(-) & 1.73 & 6.32 & -1.13 & 0.10 & 3.51 & -0.26 & 0.21 & 0.45 & -0.63 & 0.18 & CR & &\
$1257013+260046$ & 3750 & 0.0(-) & 1.68 & 6.49 & -0.96 & 0.09 & 4.01 & 0.07 & 0.11 & 0.81 & -0.44 & 0.22 & 0.01 & -0.16 & 0.04\
$1343047+221636$ & 3750 & 0.0(-) & 2.26 & 6.37 & -1.08 & 0.10 & 3.94 & 0.12 & 0.09 & 0.81 & -0.32 & 0.02 & -0.03 & -0.08 & 0.03\
$1412161+294303$ & 3800 & 0.6 & 1.76 & 7.02 & -0.43 & 0.07 & 4.50 & 0.03 & 0.12 & 1.59 & -0.19 & 0.05 & 0.49 & -0.21 & 0.06\
$1424425+414932$ & 3700 & 0.1 & 1.59 & 6.73 & -0.72 & 0.07 & 4.31 & 0.13 & 0.17 & 1.01 & -0.48 & 0.06 & 0.10 & -0.31 & 0.02\
$1429456+230043$ & 3750 & 0.0(-) & 1.88 & 6.48 & -0.97 & 0.11 & 3.93 & 0.00 & 0.16 & 0.66 & -0.58 & 0.09 & 0.04 & -0.12 & 0.01\
$1513011+222640$ & 3700 & 0.0(-) & 1.17 & 6.60 & -0.85 & 0.10 & 3.94 & -0.11 & 0.24 & 0.91 & -0.45 & 0.08 & -0.12 & -0.40 & 0.02\
$1536502+580017$ & 3750 & 0.0(-) & 1.53 & 6.61 & -0.84 & 0.10 & 4.14 & 0.08 & 0.17 & 0.94 & -0.43 & 0.08 & -0.07 & -0.36 & 0.01\
$1545189+291310$ & 3850 & 0.1 & 1.52 & 6.45 & -1.00 & 0.09 & 3.96 & 0.06 & 0.07 & 0.51 & -0.70 & 0.04 & -0.18 & -0.31 & 0.03\
\
Standard Stars & & & & & & & & & & & & & & &\
Arcturus & 4250 & 1.4 & 1.66 & 6.74 & -0.71 & 0.10 & 4.54 & 0.35 & 0.07 & 1.44 & -0.06 & 0.09 & 0.34 & -0.08 & 0.07\
$\beta$ Peg & 3750 & 0.6 & 1.73 & 6.98 & -0.47 & 0.08 & 4.59 & 0.16 & 0.02 & 1.54 & -0.20 & 0.02 & 0.57 & -0.09 & 0.01\
$\beta$ And & 3850 & 0.9 & 1.96 & 7.12 & -0.33 & 0.06 & 4.55 & -0.02 & 0.12 & 1.89 & 0.01 & 0.00 & 0.79 & -0.01 & 0.02\
$\rho$ Per & 3650 & 0.7 & 1.35 & 7.36 & -0.09 & 0.15 & 4.93 & 0.12 & 0.06 & 1.97 & -0.15 & 0.10 & 0.86 & -0.18 & 0.03\
HD146051(mike) & 4000 & 1.0 & 2.07 & 6.98 & -0.47 & 0.08 & 4.70 & 0.27 & 0.05 & 1.69 & -0.05 & & 0.75 & 0.09 & 0.02\
HD146051(kpno) & 4000 & 1.0 & 1.81 & 7.06 & -0.39 & 0.08 & 4.74 & 0.23 & 0.01 & 1.76 & -0.06 & 0.06 & 0.80 & 0.06 & 0.02\
HD146051 (average)& 4000 & 1.0 & & 7.02 & -0.43 & & 4.72 & 0.25 & & 1.73 & -0.06 & & 0.78 & 0.08 &\
[lccccccc]{}
Arcturus & & & & & & &\
This Work & 4250 & 1.4 & 1.66 & 6.74$\pm$0.10 & 4.54$\pm$0.07 & 1.44$\pm$0.09 & 0.34$\pm$0.07\
McWilliam & Rich 1994 & 4280 & 1.3 & 1.4 & 6.98$\pm$0.18 & 4.77$\pm$0.13 & 1.28$\pm$0.30 & 0.53$\pm$0.01\
Smith et al. 2000 & 4300 & 1.7 & 1.6 & 6.78$\pm$0.11 & 4.64$\pm$0.12 & 1.4$\pm$0.15 & 0.62\
\
$\beta$ Peg & & & & & & &\
This Work & 3750 & 0.6 & 1.73 & 6.98$\pm$0.08 & 4.59$\pm$0.02 & 1.54$\pm$0.02 & 0.57$\pm$0.01\
Smith & Lambert 1985 & 3600 & 1.2 & 2.0 & 7.41$\pm$0.13 & 4.87$\pm$0.10 & 2.19 &\
This Work with S&L Parameters & 3600 & 1.2 & 2.0 & 7.32$\pm$0.10 & 4.59$\pm$0.04 & 2.00$\pm$0.02 &\
\
$\beta$ And & & & & & & &\
This Work & 3850 & 0.9 & 1.96 & 7.12$\pm$0.06 & 4.55$\pm$0.12 & 1.89$\pm$0.01 & 0.79$\pm$0.02\
Smith & Lambert 1985 & 3800 & 1.6 & 2.1 & 7.42$\pm$0.11 & 4.87$\pm$0.12 & 2.35 &\
This Work with S&L Parameters & 3800 & 1.6 & 2.1 & 7.38$\pm$0.07 & 4.62$\pm$0.13 & 2.29$\pm$0.00 &\
\
$\rho$ Per & & & & & & &\
This Work & 3650 & 0.7 & 1.35 & 7.36$\pm$0.15 & 4.93$\pm$0.06 & 1.97$\pm$0.10 & 0.86$\pm$0.03\
Smith & Lambert 1986 & 3500 & 0.8 & 1.8 & 7.57$\pm$0.17 & 4.86$\pm$0.14 & 2.00 &\
This Work with S&L Parameters & 3500 & 0.8 & 1.8 & 7.30$\pm$0.20 & 4.58$\pm$0.12 & 2.00$\pm$0.09 &\
[lccc]{}
$\beta$ Peg & & &\
$\Delta A$(Fe) & $-0.06$ & $+0.08$ & $-0.09$\
$\Delta A$(Ti) & $+0.10$ & $+0.03$ & $-0.10$\
$\Delta A$(Y) & $-0.03$ & $+0.09$ & $-0.02$\
$\Delta A$(La) & $+0.03$ & $+0.08$ & $+0.00$\
\
$\beta$ And & & &\
$\Delta A$(Fe) & $-0.07$ & $+0.07$ & $-0.09$\
$\Delta A$(Ti) & $+0.10$ & $+0.03$ & $-0.08$\
$\Delta A$(Y) & $-0.04$ & $+0.09$ & $-0.02$\
$\Delta A$(La) & $+0.01$ & $+0.08$ & $-0.01$\
\
$\rho$ Per & & &\
$\Delta A$(Fe) & $-0.12$ & $+0.06$ & $-0.09$\
$\Delta A$(Ti) & $+0.04$ & $+0.04$ & $-0.15$\
$\Delta A$(Y) & $-0.07$ & $+0.08$ & $-0.04$\
$\Delta A$(La) & $+0.00$ & $+0.09$ & $+0.00$\
[lccccc]{}
Sgr(core) & & & & &\
$1849222-293217$ & 15.6 & -2.3 & 7.8 & 25.7 & 8.6\
$1853333-320146$ & & -5.7 & 6.8 & 12.5 & 6.3\
$1854283-295740$ & 0.1 & -1.4 & 5.1 & 6.6 & 2.2\
$1855341-302055$ & -5.5 & -2.7 & 3.7 & 11.9 & 4.0\
$1855556-293316$ & & -4.1 & 6.0 & 10.1 & 5.1\
$1902135-313030$ & -3.1 & 0.5 & & 3.6 & 1.8\
\
Sgr (north leading arm)& & & & &\
$0919216+202305$ & -0.9 & -5.5 & -1.9 & 8.3 & 2.8\
$0925364+213807$ & -0.3 & -1.7 & & 2.0 & 1.0\
$1034395+245206$ & -1.3 & 3.7 & & 5.0 & 2.5\
$1100516+130216$ & -1.7 & -1.8 & 0.5 & 4.0 & 1.3\
$1101112+191311$ & -6.8 & -3.5 & 1.8 & 12.1 & 4.0\
$1111493+063915$ & -3.7 & -3.1 & -2.2 & 9.0 & 3.0\
$1112480+013211$ & -0.9 & -1.2 & -0.5 & 2.6 & 0.9\
$1114573-215126$ & 0.6 & -3.8 & & 4.4 & 2.2\
$1116118-333057$ & & 1.8 & & 1.8 & 1.8\
$1128316-031647$ & -4.4 & -2.2 & 2.0 & 8.6 & 2.9\
$1135388-022602$ & 5.0 & -4.1 & -1.3 & 10.4 & 3.5\
$1140226-192500$ & 4.2 & -3.6 & -0.5 & 8.3 & 2.8\
$1208101-090753$ & -0.1 & -3.2 & -1.8 & 5.1 & 1.7\
$1223590-073028$ & -1.5 & -4.0 & & 5.5 & 2.8\
$1224255-061852$ & 0.2 & -1.6 & -1.3 & 3.1 & 1.0\
$1227367-031834$ & 1.2 & -2.0 & -0.4 & 3.6 & 1.2\
$1236549-002941$ & 4.8 & -2.0 & & 6.8 & 3.4\
$1249078+084455$ & 1.3 & -1.4 & 0.5 & 3.2 & 1.1\
$1318500+061112$ & -1.8 & -2.7 & & 4.5 & 2.3\
$1319368-000817$ & -3.9 & -3.5 & & 7.4 & 3.7\
$1330472-211847$ & 2.7 & -1.9 & & 4.6 & 2.3\
$1334532+042053$ & -1.2 & -3.6 & -0.7 & 5.5 & 1.8\
$1348366+220101$ & -1.4 & -3.8 & -1.8 & 7.0 & 2.3\
$1407060+063311$ & 0.7 & -2.5 & -1.3 & 4.5 & 1.5\
$1411221-061013$ & -0.3 & -3.1 & -2.8 & 6.2 & 2.1\
$1435018+070827$ & -0.6 & -2.8 & -3.0 & 6.4 & 2.1\
$1450544+244357$ & -2.2 & -2.5 & -0.6 & 5.3 & 1.8\
$1456137+151112$ & 0.7 & 1.4 & -1.1 & 3.2 & 1.1\
$1512142-075250$ & 0.0 & -1.5 & & 1.5 & 0.8\
$1538472+494218$ & 4.3 & -0.6 & -1.6 & 6.5 & 2.2\
\
Sgr(south leading arm) & & & & &\
$2031334-324453$ & 1.2 & -1.7 & & 2.9 & 1.5\
$2037196-291738$ & -1.2 & -4.7 & 0.8 & 6.7 & 2.2\
$2046335-283547$ & -0.4 & -3.0 & & 3.4 & 1.7\
$2050020-345336$ & -0.6 & -2.2 & -0.3 & 3.1 & 1.0\
$2105585-275602$ & -1.1 & -2.2 & & 3.3 & 1.7\
$2114412-301256$ & 2.3 & 0.9 & 2.7 & 5.9 & 2.0\
$2130445-210034$ & -1.0 & -0.9 & 2.7 & 4.6 & 1.5\
$2135183-203457$ & 2.5 & -1.2 & -0.7 & 4.4 & 1.5\
$2154471-224050$ & 1.8 & -1.7 & 0.7 & 4.2 & 1.4\
$2226328-340408$ & -2.4 & -1.1 & -1.8 & 5.3 & 1.8\
\
NGP & & & & &\
$1033045+491604$ & -3.9 & 0.3 & 1.8 & 6.1 & 2.0\
$1041479+294917$ & 2.7 & 0.9 & 1.6 & 5.2 & 1.7\
$1051302+004400$ & 2.6 & -0.2 & -1.5 & 4.3 & 1.4\
$1115376+000800$ & 2.1 & -3.0 & -1.1 & 6.2 & 2.1\
$1214190+071358$ & -5.1 & -3.8 & & 8.9 & 4.5\
$1257013+260046$ & -1.9 & -2.6 & -0.5 & 5.0 & 1.7\
$1343047+221636$ & -1.4 & -1.7 & 0.0 & 3.1 & 1.0\
$1412161+294303$ & -0.2 & -1.3 & -0.4 & 1.9 & 0.6\
$1424425+414932$ & -1.3 & -3.1 & -1.6 & 6.0 & 2.0\
$1429456+230043$ & -2.6 & -3.6 & -0.2 & 6.4 & 2.1\
$1513011+222640$ & -3.6 & -2.8 & -2.5 & 8.9 & 3.0\
$1536502+580017$ & -1.8 & -2.7 & -1.9 & 6.4 & 2.1\
$1545189+291310$ & -2.0 & -4.4 & -1.9 & 8.3 & 2.8\
![ Distribution of \[Y/Fe\] versus \[Ti/Fe\] for our stars ([*top panel*]{}), stars in other dSphs ([*middle panel*]{}) and stars in the LMC ([*bottom panel*]{}) — with all stars color coded as in the previous figures — compared with Milky Way stars. The latter lie within a well-defined, V-shaped distribution near the top of each panel. ](f9.eps)
[^1]: Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundacion Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias.
[^2]: From http://kurucz.harvard.edu/grids.html.
[^3]: It is perhaps worth mentioning that present Sgr stream models, such as that presented in Paper IV, show that the leading arm may wrap a second time around the Galactic center — and with the debris in the second leading arm wrap lying closer to the Sun in the Northern Hemisphere than the debris in the first wrap. This is a potential source of contamination of the LN sample by further extensions of itself; however, if the trend between the LN and LS stars is extrapolated, stars in a second leading arm debris wrap should be even more metal-poor on average than the LS sample. That there is no clear distinction in metallicities between the closer and farther LN stars — while the LS stars [*are*]{} more metal poor on average — suggests that this “self-contamination” of the LN sample by debris from the second wrap of the Sgr leading arm may not be significant.
[^4]: We point out that Fellhauer et al. (2006) attribute one each of the two Sgr structures in the SDSS Northern Galactic Hemisphere data to the leading and trailing arms of the Sgr stream, respectively. But this explanation is not consistent with the predictions of the distances of these structures by the Paper IV model, which was designed to constrain not only the positions of Sgr data on the sky (the main criteria used by the Fellhauer et al. 2006 Sgr model), but the extant Sgr stream velocity data as well. Furthermore, Yanny et al. (2009) suggest that the two branches might be from debris that was stripped at similar times, due to the similar velocities, metallicities, and relative densities of K/M giant, BHB, and F-turnoff stars.
|
---
abstract: 'This paper deals with the complexity of strings, which play an important role in biology (nucleotid sequences), information theory and computer science \[1,2,4\]. The $d$-complexity of a string is defined as the number of its distinct $d$-substrings given in Definition 1. The case $d=1$ is studied in detail.'
author:
- ZOLTÁN KÁSA
title: '**On the $d$-complexity of strings**'
---
[*Faculty of Mathematics and Informatics, Babeş-Bolyai University,\
RO-3400 Cluj, str. Kogălniceanu 1, Romania,\
E-mail: kasa@cs.ubbcluj.ro*]{}
Introduction
============
Let $ X$ be an alphabet, and $X^k$ the set of all strings of length $k$ over $X.$ The $i$ consecutive appearance of a letter $a$ in a string will be denoted by $a^i$. If $i = 0$ then this means the absence of the corresponding letter. The definitions are from \[2\].
Let $d$, $k$ and $s$ be positive integers, [**p**]{} $ = x_1x_2\cdots x_k \in X^k.$ A $d$-substring of [**p**]{} is defined as [**q**]{} = $ x_{i_1}x_{i_2}\cdots x_{i_s}$ where$i_1 \ge 1$,$ 1 \le i_{j+1} - i_j \le d,$ for $\;\; j = 1,2,\cdots , s-1,$ $i_s \le k.$
The $d$-complexity ${\bf K}_d({\bf p})$ of the string ${\bf p}$ is the number of all distinct $d$-substrings of ${\bf p}.$
[*Example.*]{} Let ${\it X}$ be the English alphabet and ${\bf p}$ = ISIS. In this string there are two 2-substrings of length 1 (I, S), four 2-substrings of length 2 (IS, II, SI, SS), four 2-substrings of length 3 (ISI, ISS, IIS, SIS), and a single one of length 4 (ISIS). Then ${\bf K}_{2}({\bf p}) = 2 + 4 + 4 + 1 = 11.$
In the case of strings of length $k$, consisting of different symbols, the $d$-complexity will be denoted by $N(k,d)$. For any $k \ge 1$ and ${\bf p} \in X^k$ we have $\; k \le {\bf K}_1({\bf p}) \le \displaystyle\frac{k(k+1)}{2}.\;$ If $|X| \ge 2,\;\; k\ge 1, \;\; d \ge 1$ and ${\bf p} \in X^k$ then $ k \le {\bf K}_d({\bf p}) \le 2^{k}-1.$ If [**p**]{} is a string, consisting of different symbols, and $ d$ a positive integer, then $ a_{i,d}({\bf p})$ will denote the number of $d$-substrings of [**p**]{} which terminate in the position $ i$. If $k \ge 1$ and ${\bf p} \in X^k$ consists of different symbols, then for $ \;\;i = 1,2, \ldots ,k$
$$a_{i,d}({\bf p}) = 1 + a_{i-1,d}({\bf p}) + a_{i-2,d}({\bf p}) + \ldots
+ a_{i-d,d}({\bf p}),\hspace*{0.5 cm}$$
Computing the value of N(k,d)
=============================
The $d$-complexity of a string with different symbols can be obtained by the formula $$N(k,d) =\sum_{i=1}^{k}a_{i,d}({\bf p})$$ where [**p**]{} is any string of $k$ different symbols. Because of (1) we can write in the case of $ d \ge 2$ $$a_{i,d} + \frac{1}{d-1} = \left(a_{i-1,d} + \frac{1}{d-1}\right) + \cdots
+ \left(a_{i-d,d} + \frac{1}{d-1} \right).$$ Let be $$b_{i,d}=a_{i,d}+\frac{1}{d-1},\hspace*{0.5 cm}{\rm and}\hspace*{0.5 cm}
c_{i,d}=(d-1)b_{i,d}$$ then $$c_{i,d} = c_{i-1,d} + c_{ i-2,d}+\ldots + c_{i-d,d}$$ and the sequence $c_{i,d}$ is one of Fibonacci-type. For any $d$ we have $a_{1,d} = 1$ and from this $c_{1,d} = d$ results. Therefore the numbers $ c_{i,d}$ are defined by the following recurrence equations:
$ c_{n,d} = c_{n-1,d} + c_{n-2,d} + \ldots + c_{n-d,d} $ for $\;\;n > 0,\hspace*{2 cm} \;$
$ c_{n,d}=1$ for $\;\; n \le 0.\hspace*{2 cm} \; $
These numbers can be generated by the following generating function:
$$\begin{aligned}
F_d(z)&=&\sum_{n\ge 0}^{}{c_{n,d}z^n}=
\frac{1+(d-2)z - z^2 - \cdots - z^d}{1-2z+z^{d+1}}\nonumber \\
&=& \frac{1+(d-3)z - (d-1)z^2 +z^{d+1} }{(1-z) (1-2z+z^{d+1})}\nonumber\end{aligned}$$
The $d$-complexity $N(k,d)$ can be expressed with these numbers $ c_{n,d}$ by the following formula: $$N(k,d) = \frac{1}{d-1}\left(\sum_{i=1}^{k}{c_{i,d}-k}\right), \;\;\;
\hspace*{0.5 cm}\hbox{for} \;\; d > 1$$ and $$N(k,1) = \frac{k(k+1)}{2}$$ or $$N(k,d) = N(k-1,d) +\frac{1}{d-1} ( c_{k,d}- 1), \qquad \hbox{for}\;\;\;
d > 1, \;\; k > 1.$$ If $d = 2$ then $$F_2(z) =\frac{1-z^2}{1-2z+z^3} = \frac{1+z}{1-z-z^2}=\frac{F(z)}{z}+F(z)$$ where $F(z)$ is the generating function of the Fibonacci numbers $ F_n$ (with $F_0= 0,\;\; F_1 = 1$). Then, from this formula we have $$c_{n,2} = F_{n+1} +F_{ n} = F_{n+2}$$ and $$N(k,2) = \sum_{i=1}^{k}{F_{i+2}}-k = F_{k+4}-k -3$$ Taking into account the formula for $F_n$ we have $$N (k,2) = \left\lfloor \frac{1}{\sqrt{5}}\left(
\frac{1+\sqrt{5}}{2}\right)^{k+4}+\frac{1}{2}\right\rfloor -k-3$$ which can be approximated by $$\lfloor 3.0652475\cdot (1.6180339)^k + 0.5\rfloor - k - 3.$$ Table 1 lists the values of $N(k,d)$ for $k \le 10$ and $d \le 10.$
[k]{} $\left\backslash ^d\right.$ 1 2 3 4 5 6 7 8 9 10
----------------------------------- ---- ----- ----- ----- ----- ------ ------ ------ ------ ------
1 1 1 1 1 1 1 1 1 1 1
2 3 3 3 3 3 3 3 3 3 3
3 6 7 7 7 7 7 7 7 7 7
4 10 14 15 15 15 15 15 15 15 15
5 15 26 30 31 31 31 31 31 31 31
6 21 46 58 62 63 63 63 63 63 63
7 28 79 110 122 126 127 127 127 127 127
8 36 133 206 238 250 254 255 255 255 255
9 45 221 383 464 494 506 510 511 511 511
10 55 364 709 894 974 1006 1018 1022 1023 1023
From the definition of the $d$-substrings follows that $$N(k,d) = N(k,d+1), \qquad \hbox{for}\quad d\ge k-1$$ but $$N(k,k-1) = 2^k - 1$$ and then $$N(k,d) = 2^k - 1, \qquad \hbox{for any}\quad d \ge k-1.$$ The following proposition gives the value of $N(k,d)$ in almost all cases:
[*\[3\].*]{} For $k \ge 2d-2$ we have $$N(k,k-d) = 2^k - (d-2)\cdot 2^{d-1} - 2.$$
The main step in the proof is based on the formula
$$N(k,k-d-1) = N(k,k-d) - d\cdot 2^{{\rm d-1}}.$$
The value of $N(k,d)$ can be also obtained by computing the number of sequences of length $ k$ of $0's$ and $1's$, with no more than $ d-1$ adjacent zeros. In such a sequence one 1 represents the presence, one 0 does the absence of a letter of the string in a given $d$-substring. Let $b_{\rm k,d}$ denote the number of $k$-length sequences of zeros and ones, in which the first and last position is 1, and the number of adjacent zeros is at most $d-1.$ Then easily can be proved that
$b_{k,d} = b_{k-1,d} + b_{k-2,d} + \ldots +b_{k-d,d},$ for $ \;\;k > 1,$ $\; $
$ b_{1,d} = 1,$
$ b_{k,d} = 0$, for all $k \le 0,$ $\;$
because any such sequence of length $ k-i$ ($i=1,2,...,d$) can be continued in order to obtain a similar sequence of length $k$ in only one way (by adding a sequence of the form $0^{i-1}1$ on the right). For $b_{k,d}$ the following formula also can be derived: $$b_{k,d} = 2b_{k-1,d} - b_{k-1-d,d}.$$ If we add one 1 or 0 in a internal position (e.g in the $(k-2)^{th})$ of each $ b_{k-1,d}$ sequences, then we obtain $2b_{k-1,d}$ sequences of length $k$, but between these $ b_{k-1-d,d}$ sequences will have $d$ adjacent zeros.
The generating function corresponding to $b_{n,d}$ is $$B_{d}(z) = \sum_{n\ge 0}^{}{b_{n,d}z^n} =
\frac{z}{1-z \cdots - z^d}= \frac{z(1-z)}{1-2z+z^{d+1}}.$$ Adding zeros on the left and/or on the right to these sequences, we can obtain the number $N(k,d)$, as the number of all these sequences. Thus $$N(k,d) = b_{k,d} + 2b_{k-1,d} + 3b_{k-2,d} + \cdots + kb_{1,d}.$$ ($i$ zeros can be added in $i+1$ ways to these sequences: $0$ on the left and $i$ on the right, $1$ on the left and $i-1$ on the right, and so on).
From the above formula, the generating function corresponding to the complexities $N(k,d)$ can be obtained as a product of the two generating functions $B_d(z)$ and $A(z) = \sum_{n\ge 0}^{}{nz^n}= 1/(1-z)^2$, thus: $$N_d(z)=\sum_{n\ge 0}^{}{N(n,d)z^n} = \frac{z}{(1-z)(1-2z+z^{d+1})}.$$
The 1-complexity
================
We shall use the term [*complexity*]{} instead of the [*1-complexity*]{} and the notation [**K**]{}([**p**]{}) instead of ${\bf K}_{1}({\bf p})$. A $k$-length string [**p**]{} over an $n$-letter alphabet has maximal complexity if
$${\bf K}({\bf p}) = \sum_{i=1}^{k}{ \min ( n^i, k-i+1)}.$$ In the following we give some results which can be proved immediately (in all cases ${\bf p} \in X^k)$:
$a)\;\;\;k \le {\bf K}({\bf p}) \le \displaystyle\frac{k(k+1)}{2}$.
$b)\;\;\; {\rm For \;a \;trivial\; string }\;\; {\bf p} = a^k,\quad
{\bf K}({\bf p}) = k.$
$c)\;\;\; {\rm If } \;\; x_{k} \neq x_{i} \;{\rm for }\;
i = 1,2,\cdots ,k-1,$ then $${\bf K}(x_1 x_2\cdots x_k) = k +{\bf K}(x_1 x_2\cdots x_{k-1})$$.
$d)\;\;\; {\rm If\; {\bf p} \;is \;not\; a\; trivial\; string,\; then}
\quad 2k-1 \le {\bf K}({\bf p}) \le \displaystyle\frac{k(k+1)}{2}. $
$ e) \;\;\;{\rm If} \;{\bf p} = a^{i-1}ba^{k-i} \;{\rm for\; a \;fixed}
\;i \;\;(1 \le i \le \lfloor k /2 \rfloor)$ then $${\bf K}({\bf p}) = (i+1) k - i^2.$$
$ f)\;\;\; {\rm If\; {\bf p}\; has\; at \;least \; \ell\; different\;
letters \;then} \quad {\bf K}({\bf p}) \ge k\ell
-\displaystyle\frac{\ell (\ell -1)}{2}$.
(For the string $ a_1 a_ 2 \cdots a_{l-1}b^{k-l}$ with $a_i\neq a_j$ for $i\neq j$, and $a_i\neq b$ we have equality in the above formula).
$ g)\;\;\; {\rm If \;{\bf p}} \; \in X^k, {\bf q} \in Y^m \;{\rm and}
\; X \cap Y = \emptyset $ then $${\bf K}({\bf pq}) = {\bf K}({\bf p}) + {\bf K}({\bf q}) + km.$$
$ h)\;\;\; {\rm If}\; {\bf p}$ has only different letters then
([**p**]{}) =$\displaystyle\frac{k(k+1)}{2}, $
${\bf K}({\bf pp}^R) =2k^2,\;\;$ where [**p**]{}$^R$ is the reverse string of [**p**]{},
${\bf K}({\bf p}^n) =
\displaystyle\frac{k(k+1)}{2}+(n-1)k^2,\;\;$ where ${\bf p}^n$ is [**p**]{} concatenated $n$ times.
$i)\;\;\; {\bf K}(x_1 x_2 \cdots x_k x_1 x_2\cdots x_n) =
\displaystyle\frac{k(k+1)}{2} + nk$ for $1\le n\le k$, $ x_i\neq x_j$ for $i\neq j.$
There arise the following two problems:
1\. [*Find a minimal length string with a given complexity.*]{}
This problem always has solution. (If the complexity is $C$, then in the worst case the string consisting of $C$ identical letters represents a trivial solution).
2\. [*Find a $k$-length string with a given complexity, if it exists.*]{}
These problems can be solved by a [*branch-and-bound*]{}-type algorithm. We shall construct a tree in which each node is a string. The root is a letter of the alphabet. Each node (i.e. each string) is obtained from its parent node by adding a new letter of the alphabet. The contruction will be continued at a node if its complexity is less than the given complexity, or in the case of the second problem only if its length is also less than $k.$ This algorithm can be improved by omitting some branches, which do not produce essentially new strings, e.g. if we have a four letter alphabet, then the strings $abd$ and $abc$ are isomorphic (differ only some letters, but not the form). This can be given by the following recursive algorithm. Let $a_1,a_2,\cdots ,a_n$ be the letters of the alphabet, $k$ the length and $C$ the desired complexity. The symbol “+” will denote the concatenation of a string with a letter, $|w|$ the length of string $ w$. The algorithm starts with [*generate(“$a_1$”)*]{}.
Of course, if $C < k$ or $ C > k(k+1)/2$ or doesn’t exist a string with the desired complexity and length, then this algorithm produces nothing. To solve the first problem, we omit the restriction on length in the above algorithm.
If there is always a string with a given complexity, the question is: there exists a nontrivial string with a given complexity or not? (A nontrivial string contains at least two different letters). The answer is yes, except some cases.
If $C$ is a natural number different from 1, 2 and 4, then there exists a nontrivial string of complexity equal to $C.$
[*Proof*]{}. To prove this proposition we give the complexity of the following $k$-length strings:
$ {\bf K}(a^{k-1}b) = 2k-1$ for $ k \ge 1\hspace*{1cm}\;$
$ {\bf K}(ab^{k-3}aa) = 4k-8$ for $k \ge 4\hspace*{1cm}\;$
$ {\bf K}(abcd^{k-3}) = 4k-6$ for $k \ge 3\hspace*{1cm}\;$
These can be proved immediately from the definition of the complexity.
1\. If $C$ is odd then we can write $ C = 2k-1$ for a given $k.$ From this $k =(C+1)/2$ results, and the string $a^{k-1}b$, has complexity $C.$
2\. If $C$ is even, then $C = 2\ell $.
2.1. If $\ell = 2h$, then $4k-8 =C$ gives $4k-8 = 4h$, and from this $k=h+2$ results. The string $ ab^{k-3}aa$ has complexity $C.$
2.2. If $\ell = 2h+1$ then $4k-6 = C$ gives $4k-6 = 4h+2,$ and from this $k=h+2$ results. The string $abcd^{k-3}$ has complexity $C.\;$
In the proof we have used more than two letters in a string only in the case of the numbers of the form $4h+2$ (case 2.2 above). The new question is, if there exist always nontrivial strings formed only of two letters with a given complexity. The answer is yes anew. We must prove this only for the numbers of the form $4h+2.$ If $C = 4h+2$ and $C \ge 34,$ we use the followings:
${\bf K}(ab^{k-7}abbabb)= 8k-46,$ for $k \ge 10,\hspace*{1cm}$
${\bf K}(ab^{k-7}ababba) = 8k-42,$ for $k \ge 10.\hspace*{1cm}$
If $h = 2s$, then $8k-46 = 4h+2$ gives $k = s+6,$ and the string $ab^{k-7}abbabb$ has complexity $4h+2.$
If $h = 2s+1,$ then $8k-42 = 4h+2$ gives $k = s+6,$ and the string $ab^{k-7}ababba$ has complexity $4h+2.$ For $C < 34$ only 14, 26 and 30 are feasible. The string $ab^4a$ has complexity 14, $ab^6a$ complexity 26, and $ab^5aba$ complexity 30. Easily can be proved, using a tree like in the above algorithm, that for 6, 10, 18 and 22 such strings does not exist. Then the following is true.
If $C$ is a natural number different from 1, 2, 4, 6, 10, 18 and 22, then there exists a nontrivial string formed only of two letters, with the given complexity $C.$
In relation with the second problem a new one arises: How many strings of length $k$ and complexity $C$ there exist? For small $k$ this problem can be studied exhaustively. Let $X$ be of $k$ letters, and let us consider all strings of length $k$ over $X.$ By a computer program we have got Table 2, which contains the frequency of strings with a given length and complexity.
------------------------------------------------------------------------
------------------ --- --- -- ------------------ --- --- ---- ---
[*length=2*]{} [*length=3*]{}
[*complexity*]{} 2 3 [*complexity*]{} 3 4 5 6
[*frequency*]{} 2 2 [*frequency*]{} 3 0 18 6
------------------ --- --- -- ------------------ --- --- ---- ---
------------------ --- --- --- ---- ---- ----- ----
[*length=4*]{}
[*complexity*]{} 4 5 6 7 8 9 10
[*frequency* ]{} 4 0 0 36 48 144 24
------------------ --- --- --- ---- ---- ----- ----
------------------ --- --- --- --- ---- ---- ----- ----- ------ ------ -----
[*length=5* ]{}
[*complexity*]{} 5 6 7 8 9 10 11 12 13 14 15
[*frequency* ]{} 5 0 0 0 60 0 200 400 1140 1200 120
------------------ --- --- --- --- ---- ---- ----- ----- ------ ------ -----
[lrrrrrrrrr]{} [*length=6*]{} & & & & & & & & &\
[*complexity*]{} & 6 & 7 & 8& 9 & 10 & 11 & 12 & 13\
[*frequency*]{} & 6 & 0 & 0 & 0 & 0 & 90 & 0 & 0\
\
& 14 & 15 &16 &17 &18 &19 & 20 & 21 &\
& 300 &990 &270&5400&8280 &19800&10800& 720 &\
------------------------------------------------------------------------
**Table 2.**
Let $|X|=k$ and let $f_k(C)$ denote the frequency of the $k$-length strings over $X$ having a complexity $ C$. Then the following proposition is true.
$f_k(C) = 0$ if $C<k$ or $ C>\displaystyle\frac{k(k+1)}{2}, $\
$f_k(k) = k$,\
$f_k(2k-1) = 3k(k-1),$\
$f_k\left( \displaystyle\frac{k(k+1)}{2} -1\right) =
\displaystyle\frac{k(k-1)k!}{2}, $\
$f_k\left( \displaystyle\frac{k(k+1)}{2} \right) =
k! $
[*Proof.*]{} The first two and the last ones are evident. Let us prove the third. If the complexity of a $k$-length string is $2k-1,$ then it must contain exactly two substrings of length $1,2,\cdots ,k-1$, and only one of the length $k$, and must be formed of two letters. (If it contains 3 letters than the complexity is $\ge 3k-3$, see the property $f).$ ) In this case the 2-lentgh substrings can be only $aa, ab$ or $aa, ba$ or $ab, ba$, and with these only strings of the form $a^{k-1}b$, $ba^{k-1}$ and $(ab)^{k/2}$ (if $k$ is even) or $(ab)^{(k-1)/2}a$ (if $k$ is odd) can be generated. In every case the two letters can be chosen in $k(k-1)$ ways, and because of the three above possibility $f_k(2k-1)=3k(k-1)$.
The last but one comes from the following: $k$ letters can form $k!$ different $k$-length strings of maximal complexity, and the complexity of such a string can be diminished by one if we replace a letter by another already being present in that string. We can choose a position for one already given in $k(k-1)$ ways, and because of the symmetry of the letters in these positions, the number of new strings is $k!k(k-1)/2$.
As regards the distribution of the frequaency $0$, we can prove the following.
If $C=k+1, k+2, \cdots , 2k-2,\;\;$ then $\; f_k(C)=0.$
If $\;C\;=\;2k, 2k+1, \cdots , 3k-5,\;\;\;$ then $\; f_k(C)=0.\;$
[*Proof.* ]{} The complexity of the trivial $k$-length string is $k$, and this contains only one letter $k$ times. If in such a string we replace one or more letters by a new one, the number of substrings of any length, except the whole string, will increase by at least one. Then the complexity will be at least $2k-1$, and there are no strings with complexity between $k$ and $2k-1$. To prove the second formula, we use the following, easy to see assertion: [*if a k-length string has n i-length substrings, then it has at least [min]{}(n,k-i+1) $\;$ (i+1)-length substrings.*]{}
By replacing a letter with a new one in the strings of complexity $2k-1$, we obtain at least complexity $3k-3$. If we replace one $a$ (or more) with one $b$ (or more), or inversely, but not to obtain a trivial string, and keeping the length, the number of 2-length substrings will increase by 3, and by the above assertion will increase the number of $3-, 4-, \cdots ,
(k-2)-$length substrings. Then the complexity will be at least $2+3(k-3)+2+1$ which is $3k-4. \blacksquare$
Strings of length $k$ may have complexity between $k$ and $k(k+1)/2$. Let us denote by $b_k$ the least number for which
$f_k(C) \ne 0\;\;$ for all $C$ with $\;\;b_k \le C \le
\displaystyle\frac{k(k+1)}{2}$.
The number $b_k$ exists for any $k$ (in the worst case it may be equal to $k(k+1)/2$). In the Table 2 we can see that $b_3=5$, $b_4=7$, $b_5=11$ and $b_6=14.$
We give the following conjecture:
[Conjecture.]{} If $k= \displaystyle\frac{\ell (\ell +1)}{2}+2+i$, where $\ell \ge 2$ and $0\le i\le \ell $ then $$b_k = \frac{\ell (\ell^2 -1)}{2}+3\ell+2+i(\ell+1).\;\; \blacksquare$$
We can easily see that $f_k(b_k) \ne 0$ for $k\ge 5$, because of ${\bf K}(ab^{k-\ell}ab^{\ell -2})$ $=b_k$.
Conclusions {#conclusions .unnumbered}
===========
We have studied the $d$-complexity of strings, which is defined as the number of all distinct $d$-substrings of it. The concept of the $d$-substring is a generalization of that of the substring: not only a contiguous part of a string can be chosen as substring, but parts which have distance between them no greater than $d$. The $d$-complexity of strings with different letters only, can be computed by a Fibonacci-type sequence. Proposition 1 gives a formula for this complexity in almost all cases.
The 1-complexity is studied in detail. In propositions 2 and 3 we prove that, except some cases, a string with a given complexity can be associated to any natural number. The frequency of strings with a given complexity is also considered. It is conjectured that if we consider strings of length $k$, there exists a value between $k$ and $k(k+1)/2$ from which 0 frequency no more exists.
W. Ebeling, R. Feistel, Physik der Selbstorganisation und Evolution, Akademie-Verlag, Berlin, 1982.
A. Iványi, On the d-complexity of words, [*Annales Univ. Sci.Budapest. Sect. Comput.*]{} [**8**]{} (1987) 69-90.
Z. Kása, Computing the d-complexity of words by Fibonacci-like sequences, [*Studia Univ. Babeş-Bolyai, Math.*]{} [**35**]{}, 3 (1990) 49-53.
N. Vörös, On the complexity of symbol sequences, in: [*Conference of Young Programmers and Mathematicians*]{} (ed. A. Iványi) Eötvös Loránd University, Budapest, 1984, 43-50.
|
---
abstract: 'In this paper, we present a class of finite volume schemes for incompressible flow problems. The unknowns are collocated at the center of the control volumes, and the stability of the schemes is obtained by adding to the mass balance stabilization terms involving the pressure jumps across the edges of the mesh.'
author:
- 'R. Eymard [^1]'
- 'R. Herbin [^2]'
- 'J.-C. Latché'
- 'B. Piar [^3]'
title: A class of collocated finite volume schemes for incompressible flow problems
---
Incompressible flows, Stokes problem, Navier-Stokes equations, Finite Volumes
35Q30, 65M12, 76D05, 76D07, 76M12
Introduction
============
The use of collocated finite volumes for fluid flow problems is appealing for several reasons. Among them, let us mention a very inexpensive assembling step (in particular compared to finite elements, because there is no numerical integration to perform) and the possibility to use, at least to some extent, general unstructured meshes with a low complexity of the data structure (compared with staggered schemes) suitable for the implementation of adaptative mesh refinement strategies. These features make collocated finite volumes attractive for industrial problems, and they are widely used in Computational Fluid Dynamics, either in proprietary or in commercial (FLUENT, CFX, …) codes.
However, when applied to incompressible flow problems, cell-centered collocated finite volumes suffer from a lack of coercivity, which is usually handled by a stabilization technique initially proposed by Rhie and Chow [@rhi-83-num], and further developed in subsequent works. We present here an alternative strategy, based on the addition of “pressure-laplacian-like” stabilization terms in the mass balance ([[*i.e.*]{}]{} the continuity constraint) equation. In contrast to the case of stabilizations [*à la*]{} Rhie and Chow, for this class of schemes, we are able to prove the stability and convergence of most variants for steady or evolution Stokes or Navier-Stokes equations; optimal ([[*i.e.*]{}]{} first order in energy norms) error bounds are also provided for the Stokes problem [@eym-06-sta; @eym-07-conv; @eym-08-conv]. Numerical tests for these schemes for a variety of flow problems can be found in [@che-06-num; @eym-07-new; @eym-07-sta; @che-08-col].
In this paper, we restrict the exposition to the stationary Stokes problem:
$$\begin{aligned}
&
-\Delta {{\boldsymbol u}}+ {{\boldsymbol \nabla}}p = {{\boldsymbol f}}&& \hspace{-20ex} \textrm{ in } \Omega, \hspace{20ex}
\label{qdm} \\ &
{{\rm div}}({{\boldsymbol u}}) = 0 \; && \hspace{-20ex} \textrm{ in } \Omega,
\label{mass} \\ & \displaystyle
{{\boldsymbol u}}= 0 && \hspace{-20ex} \textrm{ on } \partial \Omega,
\label{BC}\end{aligned}$$
\[pbcont\_s\]
where $\Omega$ is a polygonal domain of ${\mathbb{R}}^2$, $\partial \Omega$ is the boundary of $\Omega$, ${{\boldsymbol u}}$ stands for the velocity, $p$ for the pressure, the mean value of which is supposed to be zero, and ${{\boldsymbol f}}$ is a forcing term.
The stability of the Stokes problem may be readily proved in two steps. First, multiplying Equation by the unknown ${{\boldsymbol u}}$ and integrating over $\Omega$ yields: $$\int_\Omega |{{\boldsymbol \nabla}}{{\boldsymbol u}}|^2 {\, {\rm d}{{\boldsymbol x}}}+ \int_\Omega {{\boldsymbol \nabla}}p \cdot {{\boldsymbol u}}{\, {\rm d}{{\boldsymbol x}}}=\int_\Omega {{\boldsymbol f}}\cdot {{\boldsymbol u}}{\, {\rm d}{{\boldsymbol x}}}.$$ Integrating by parts and using the boundary condition , we get: $$\int_\Omega {{\boldsymbol \nabla}}p \cdot {{\boldsymbol u}}{\, {\rm d}{{\boldsymbol x}}}=-\int_\Omega p \, {{\rm div}}{{\boldsymbol u}}{\, {\rm d}{{\boldsymbol x}}},
\label{div-grad}$$ so this therm vanishes by , and we obtain a control of the velocity ${{\boldsymbol u}}$ in ${{\rm H}^{1}}(\Omega)^2$ provided that the forcing term ${{\boldsymbol f}}$ be regular enough, say ${{\boldsymbol f}}\in {{\rm L}^{2}}(\Omega)^2$, which is more than needed in the continuous case but will make the finite volume scheme easier to write. To obtain a control on the pressure, we use a classical result which is a consequence of a lemma due to Nečas: $$\forall q \in {{\rm L}^{2}}_0(\Omega), \mbox{ there exists } {{\boldsymbol v}}\in {{\rm H}^{1}}_0(\Omega)^d \mbox{ such that }
\left| \begin{array}{l}
{{\rm div}}{{\boldsymbol v}}= q \mbox{ a.e. in } \Omega,
\\[1ex]
{\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol v}}|\hspace{-.1em}|_{{{\rm H}^{1}}(\Omega)^d}\hspace{.2em}} \leq c {\hspace{.2em}|\hspace{-.1em}| q |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}},
\end{array}\right.
\label{necas}$$ where ${{\rm L}^{2}}_0(\Omega)$ stands for the subspace of ${{\rm L}^{2}}(\Omega)$ of zero mean value functions and the real number $c$ only depends on $\Omega$. Choosing ${{\boldsymbol v}}$ satisfying this relation for $q=p$, multiplying by ${{\boldsymbol v}}$ and using the estimate for ${{\boldsymbol u}}$ yields a bound for ${\hspace{.2em}|\hspace{-.1em}| p |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}}$.
From this computation, we conclude that the stability of the Stokes problem stems from three basic arguments: $(i)$ the coercivity of the diffusion operator, $(ii)$ the duality of the ${{\boldsymbol \nabla}}$ and ${{\rm div}}$ operators with respect to the ${{\rm L}^{2}}$ inner-product, $(iii)$ the stability of the gradient operator. In this paper, we show how to build collocated finite volume schemes satisfying $(i)$ and $(ii)$, and how to circumvent the fact that the property $(iii)$ is not satisfied. The presentation is organized as follows. In a first part, we derive the different variants of the proposed schemes for a model problem, namely choosing for the computational domain $\Omega$ the unit square and for the mesh a uniform grid. In a second part, we briefly discuss how to extend these schemes to general domains and meshes.
A model problem: solving the Stokes system with structured two-dimensional grids
================================================================================
In this section, we restrict the presentation to the solution of Problem with $\Omega=(0,1)\times(0,1)$ using a structured uniform grid, as sketched on Figure \[fig:mesh\_and\_checkerboard\]. We first describe the discretization, then we present the possible schemes and discuss their stability features.
Discrete spaces
---------------
We suppose given a uniform structured mesh ${{\mathcal T}}$ (with step $h$) of $\Omega$, and denote ${{\mathcal E}}$ (resp. ${{\mathcal E}_{{\rm int}}}$, ${{\mathcal E}_{{\rm ext}}}$) the set of edges (resp. internal edges, external edges) of the mesh. For any two neighbouring control volumes $K$ and $L$ of ${{\mathcal T}}$, we denote by $K|L$ the common edge of $K$ and $L$, and by ${{\boldsymbol n}}_{K|L}$ the normal vector to $K|L$ oriented from $K$ to $L$ (so ${{\boldsymbol n}}_{K|L}=-{{\boldsymbol n}}_{L|K}$). If ${\sigma}\in {{\mathcal E}_{{\rm ext}}}$, ${{\boldsymbol n}}_{\sigma}$ stands for the normal vector to ${\sigma}$ outward $\Omega$. For any control volume $K\in{{\mathcal T}}$, we denote by ${{\mathcal E}}(K)$ the set of edges of $K$. Let ${{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\subset {{\rm L}^{2}}(\Omega)$ be the set of functions which are piecewise constant over each control volume. For any $v \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$ and $K \in {{\mathcal T}}$, we denote by $v_K$ the value of $v$ over $K$. We define a discrete inner product for the functions of ${{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$ as follows: $$\begin{array}{l}
\forall v \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}},\ \forall w \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}},
\\[1ex] \displaystyle \hspace{10ex}
(v,w)_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}=
\hspace{-2ex} \sum_{\begin{array}{c}\scriptstyle {\sigma}\in {{\mathcal E}_{{\rm int}}}\\[-1ex] \scriptstyle ({\sigma}=K|L) \end{array}} (v_K-v_L)\,(w_K-w_L)
+ \hspace{-2ex} \sum_{\begin{array}{c}\scriptstyle {\sigma}\in {{\mathcal E}_{{\rm ext}}}\\[-1ex] \scriptstyle ({\sigma}\in {{\mathcal E}}(K)) \end{array}} 2\, v_K \, w_K.
\end{array}
\label{disc_prod}$$ This inner product is associated to the norm defined by ${\hspace{.2em}|\hspace{-.1em}| v |\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}}=(v,v)_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}$, $\forall v \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$. This inner product and this norm plays at the discrete level the same role as (and, to some extent, are consistent with) the ${{\rm H}^{1}}$ inner product and norm in the continuous case; they will be referred to hereafter as the discrete ${{\rm H}^{1}}$ inner product and norm. The discrete ${{\rm H}^{1}}$ norm is known to control the ${{\rm L}^{2}}$ norm [@eym-00-fin] ([[*i.e.*]{}]{} there exists a real number $c$ only depending on $\Omega$ and not on the mesh such that the following discrete Poincaré relation holds: $\forall v \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}},\ {\hspace{.2em}|\hspace{-.1em}| v |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}}\leq c {\hspace{.2em}|\hspace{-.1em}| v |\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}$). These definitions naturally extend to vector-valued functions by, $\forall {{\boldsymbol v}}=({{\boldsymbol v}}^{(1)},{{\boldsymbol v}}^{(2)})$ and ${{\boldsymbol w}}=({{\boldsymbol w}}^{(1)},{{\boldsymbol w}}^{(2)}) \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2$, $({{\boldsymbol v}},{{\boldsymbol w}})_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}=({{\boldsymbol v}}^{(1)},{{\boldsymbol w}}^{(1)})_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}+({{\boldsymbol v}}^{(2)},{{\boldsymbol w}}^{(2)})_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}$ and $ {\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol v}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}} = {\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol v}}^{(1)} |\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}} + {\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol v}}^{(2)} |\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}}$. The following inner product and seminorm will also be used hereafter: $$\forall p \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}},\ \forall q \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}, \quad
[p,q]_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}=
\hspace{-2ex} \sum_{\begin{array}{c}\scriptstyle {\sigma}\in {{\mathcal E}_{{\rm int}}}\\[-1ex] \scriptstyle ({\sigma}=K|L) \end{array}} (p_K-p_L)\,(q_K-q_L),
\quad
{\hspace{.2em}| q |_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}}=[q,q]_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}.
\label{disc_snorm}$$
The natural scheme
------------------
Integrating the relations of over each control volume $K$ of the mesh yields: $$\begin{array}{l} \displaystyle
\int_{\partial K} -{{\boldsymbol \nabla}}{{\boldsymbol u}}\cdot {{\boldsymbol n}}_{\partial K} {\, {\rm d}\sigma}+ \int_{\partial K} p\, {{\boldsymbol n}}_{\partial K} {\, {\rm d}\sigma}= \int_K {{\boldsymbol f}}{\, {\rm d}{{\boldsymbol x}}},
\\[2ex] \displaystyle
\int_{\partial K} {{\boldsymbol u}}\cdot {{\boldsymbol n}}_{\partial K} {\, {\rm d}\sigma}=0,
\end{array}$$ where $\partial K$ stands for the boundary of $K$ and ${{\boldsymbol n}}_{\partial K}$ for the normal vector to $\partial K$ outward $K$. The natural scheme for the solution of Problem thus consists in searching $({{\boldsymbol u}},p)\in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2 \times \bar {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$ such that, $\forall K \in {{\mathcal T}}$:
$$\begin{aligned}
& \displaystyle
(-\Delta_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}})_K + ({{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}p)_K = {{\boldsymbol f}}_K,
\label{nat_qdm} \\[1ex] & \displaystyle
({{\rm div}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}})_K=0,
\label{nat_mass} \end{aligned}$$
\[nat\_scheme\]
where $\bar {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$ stands for the space of functions of ${{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$ with zero mean value, ${{\boldsymbol f}}_K$ is the mean value of ${{\boldsymbol f}}$ over $K$ and
$$\begin{aligned}
& \displaystyle
(-\Delta_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}})_K = \frac 1 {h^2} \sum_{{\sigma}=K|L} ({{\boldsymbol u}}_K -{{\boldsymbol u}}_L) + \frac 2 {h^2} \sum_{{\sigma}\in {{\mathcal E}}(K)\cap {{\mathcal E}_{{\rm ext}}}} {{\boldsymbol u}}_K,
\label{disc_lapl} \\ & \displaystyle
({{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}p)_K = \frac 1 {h^2} \sum_{{\sigma}=K|L} h\, \frac{p_L+p_K} 2\,{{\boldsymbol n}}_{\sigma}+ \frac 1 {h^2} \sum_{{\sigma}\in {{\mathcal E}}(K)\cap {{\mathcal E}_{{\rm ext}}}} h \, p_K \,{{\boldsymbol n}}_{\sigma},
\label{disc_grad} \\ & \displaystyle
({{\rm div}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}})_K=\frac 1 {h^2} \sum_{{\sigma}=K|L} h \, \frac{{{\boldsymbol u}}_K +{{\boldsymbol u}}_L} 2 \cdot {{\boldsymbol n}}_{\sigma}.
\label{disc div} \end{aligned}$$
\[disc\_ops\]
Since $\forall K \in {{\mathcal T}}$, $\sum_{{\sigma}\in {{\mathcal E}}(K)} h \, {{\boldsymbol n}}_{\sigma}=0$, we have $2\, h\, ({{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}p)_K = \sum_{{\sigma}=K|L} (p_L-p_K)\ {{\boldsymbol n}}_{\sigma}$, and thus, reordering the summations, we get for any pressure $q \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$ and velocity ${{\boldsymbol v}}\in ({{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}})^2$: $$\begin{array}{l} \displaystyle
\int_\Omega {{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}q \cdot {{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}= \sum_{K\in{{\mathcal T}}} {{\boldsymbol v}}_K \cdot
\left[ \sum_{{\sigma}=K|L} h \, \frac{q_L-q_K} 2\,{{\boldsymbol n}}_{\sigma}\right]=
\\ \hspace{25ex} \displaystyle
-\sum_{K\in{{\mathcal T}}} q_K \left[\sum_{{\sigma}=K|L} h \, \frac{{{\boldsymbol v}}_K +{{\boldsymbol v}}_L} 2 \cdot {{\boldsymbol n}}_{\sigma}\right]
=-\int_\Omega q\ {{\rm div}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}},
\end{array}$$ which shows that the discrete gradient and divergence operators are transposed operators with respect to the ${{\rm L}^{2}}$ inner product, [[*i.e.*]{}]{} that the stability property $(ii)$ indeed is verified by this scheme. We then remark that, reordering the summations: $$\int_\Omega -\Delta_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}}\cdot {{\boldsymbol u}}{\, {\rm d}{{\boldsymbol x}}}={\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol u}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}},$$ which shows that a discrete equivalent of property $(i)$ is also verified by the scheme. Mimicking the computation in the continuous case, [[*i.e.*]{}]{} multiplying by ${{\boldsymbol u}}_K$, reordering the summation, using and the discrete Poincaré estimate, we thus get a bound for ${{\boldsymbol u}}$ in the discrete ${{\rm H}^{1}}$ norm. To show the stability of the scheme in natural energy norms, the next step would be to control the ${{\rm L}^{2}}$ norm of the pressure through its gradient; unfortunately, the following result shows that it is not possible, at least not uniformly with respect to $h$.
(0,0)(8.5,4.3) (0,-0.5)[ (1,1)(2,1)(2,2)(1,2) (2,1)(3,1)(3,2)(2,2) (3,1)(4,1)(4,2)(3,2) (1,2)(2,2)(2,3)(1,3) (2,2)(3,2)(3,3)(2,3) (3,2)(4,2)(4,3)(3,3) (1,3)(2,3)(2,4)(1,4) (2,3)(3,3)(3,4)(2,4) (3,3)(4,3)(4,4)(3,4) (0.5,1)(4.5,1) (0.5,3)(4.5,3) (1,0.5)(1,4.5) (3,0.5)(3,4.5) (0.5,2)(4.5,2) (0.5,4)(4.5,4) (2,0.5)(2,4.5) (4,0.5)(4,4.5) (1,4.6)(2,4.6) (1.4,4.8)[$h$]{} (0.4,3.)(0.4,4.) (0.,3.3)[$h$]{} (5.5,2)(6.,2)(6.,2.5)(5.5,2.5) (5.5,2)(6.,2)(6.,2.5)(5.5,2.5)(5.5,2) (6.2,2.1)[$p_{\rm cb}=1$]{} (5.5,3)(6.,3)(6.,3.5)(5.5,3.5) (5.5,3)(6.,3)(6.,3.5)(5.5,3.5)(5.5,3) (6.2,3.1)[$p_{\rm cb}=-1$]{} ]{}
We associate to each $K\in{{\mathcal T}}$ its row number $i$ and column number $j$, and define the particular pressure field $p_{\rm cb}$ by $(p_{\rm cb})_K=(-1)^{i+j}$ (see Figure \[fig:mesh\_and\_checkerboard\]). Then the following estimate holds with a real number $c$ independent of $h$: $$\forall {{\boldsymbol v}}\in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2, \quad \int_\Omega {{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}\, p_{\rm cb} \cdot {{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}\leq c\, h {\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol v}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}.$$
Since, for any pair of neighbouring control volumes $K$ and $L$, $(p_{\rm cb})_K+(p_{\rm cb})_L=0 $, we have, $\forall K \in {{\mathcal T}}$, $h\, ({{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}\, p_{\rm cb})_K= \sum_{{\sigma}\in {{\mathcal E}}(K)\cap {{\mathcal E}_{{\rm ext}}}} p_K \,{{\boldsymbol n}}_{\sigma}$. Thus, $\forall {{\boldsymbol v}}\in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2$: $$\int_\Omega {{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}\, p_{\rm cb} \cdot {{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}=
\sum_{\begin{array}{c}\scriptstyle {\sigma}\in {{\mathcal E}_{{\rm ext}}}\\[-1ex] \scriptstyle ({\sigma}\in {{\mathcal E}}(K)) \end{array}}
h\ p_K \,{{\boldsymbol n}}_{\sigma}\cdot {{\boldsymbol v}}_K.$$ By the Cauchy-Schwarz inequality, we obtain: $$\int_\Omega {{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}\, p_{\rm cb} \cdot {{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}\leq
\ \Bigl[
\sum_{\begin{array}{c}\scriptstyle {\sigma}\in {{\mathcal E}_{{\rm ext}}}\\[-1ex] \scriptstyle ({\sigma}\in {{\mathcal E}}(K)) \end{array}}
h^2 p_K^2
\Bigr]^{1/2}
\ \Bigl[
\sum_{\begin{array}{c}\scriptstyle {\sigma}\in {{\mathcal E}_{{\rm ext}}}\\[-1ex] \scriptstyle ({\sigma}\in {{\mathcal E}}(K)) \end{array}}
|{{\boldsymbol v}}_K|^2
\Bigr]^{1/2},$$ which concludes the proof since, in the first term, $p_K^2=1$ and so this term is bounded by $4\,h$ and the second one is controlled by ${\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol v}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}$.
A first stabilization
---------------------
The basic idea governing the construction of the first stabilized scheme proposed here is to take benefit of the following partial stability result for the discrete gradient.
There exists two positive real numbers $c_1$ and $c_2$ independent of $h$ such that, $\forall q \in \bar {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$, one can find ${{\boldsymbol v}}\in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2$ satisfying: $${\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol v}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}=1 \mbox{ and }
\int_\Omega {{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}q \cdot {{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}\geq
c_1 {\hspace{.2em}|\hspace{-.1em}| q |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}} - c_2 \, h {\hspace{.2em}| q |_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}.$$ \[stab\_grad\]
Let $q$ be a function of $\bar {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$. The idea of this proof is rather natural: let $\tilde {{\boldsymbol v}}\in {{\rm H}^{1}}_0(\Omega)^2$ be a function such that holds, and let us choose an interpolate of $\tilde {{\boldsymbol v}}$, say ${{\boldsymbol v}}\in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2$, as test function. For ${\sigma}\in {{\mathcal E}_{{\rm int}}}$, ${\sigma}=K|L$, we denote by ${{\boldsymbol v}}_{\sigma}$ the mean value of $\tilde {{\boldsymbol v}}$ over ${\sigma}$ and define $\delta {{\boldsymbol v}}_{\sigma}=({{\boldsymbol v}}_K+{{\boldsymbol v}}_L)/2-{{\boldsymbol v}}_{\sigma}$. We suppose that the interpolation operator is stable, in the sense that ${\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol v}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}} \leq c {\hspace{.2em}|\hspace{-.1em}| \tilde {{\boldsymbol v}}|\hspace{-.1em}|_{{{\rm H}^{1}}(\Omega)^2}\hspace{.2em}}$ with $c$ independent of $h$, and is such that $h^2 |\delta {{\boldsymbol v}}_{\sigma}|^2 \leq c\,h^2 {\hspace{.2em}|\hspace{-.1em}| \tilde {{\boldsymbol v}}|\hspace{-.1em}|_{{{\rm H}^{1}}(K \cup L)^2}^2\hspace{.2em}}$. Such an interpolation operator is given by instance by simply taking for ${{\boldsymbol v}}_K$ the mean value of $\tilde {{\boldsymbol v}}$ over $K$ [@eym-08-conv]. Thanks to the choice of $\tilde {{\boldsymbol v}}$ and $\delta {{\boldsymbol v}}_{\sigma}$, we have: $$\int_\Omega q\ {{\rm div}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}= {\hspace{.2em}|\hspace{-.1em}| q |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}^2\hspace{.2em}}
+ \sum_{\begin{array}{c}\scriptstyle {\sigma}\in {{\mathcal E}_{{\rm int}}}\\[-1ex] \scriptstyle ({\sigma}=K|L) \end{array}} h\,(q_K-q_L) \,\delta {{\boldsymbol v}}_{\sigma}\cdot {{\boldsymbol n}}_{\sigma}.$$ By the Cauchy-Schwarz inequality, we get for the last term $T$: $$T \leq
\Bigl[
\sum_{\begin{array}{c}\scriptstyle {\sigma}\in {{\mathcal E}_{{\rm int}}}\\[-1ex] \scriptstyle ({\sigma}=K|L) \end{array}}
(q_K -q_L)^2
\Bigr]^{1/2}
\ \Bigl[
\sum_{\begin{array}{c}\scriptstyle {\sigma}\in {{\mathcal E}_{{\rm int}}}\\[-1ex] \scriptstyle ({\sigma}=K|L) \end{array}}
h^2 \, |\delta {{\boldsymbol v}}_{\sigma}|^2
\Bigr]^{1/2}.$$ The first term is exactly ${\hspace{.2em}| q |_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}$ and the second one, thanks to the approximation property of the interpolation operator, is controlled by $h {\hspace{.2em}|\hspace{-.1em}| \tilde {{\boldsymbol v}}|\hspace{-.1em}|_{{{\rm H}^{1}}(\Omega)^2}\hspace{.2em}}$, itself bounded by $h {\hspace{.2em}|\hspace{-.1em}| q |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}}$ thanks to . To conclude the proof, it only remains to normalize ${{\boldsymbol v}}$ ([[*i.e.*]{}]{} to use ${{\boldsymbol v}}/ {\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol v}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}$ instead of ${{\boldsymbol v}}$) and invoke the stability of the interpolation operator.
This suggests for a stabilized scheme to search for $({{\boldsymbol u}},p) \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2 \times \bar {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$ such that, $\forall K \in {{\mathcal T}}$:
$$\begin{aligned}
& \displaystyle
(-\Delta_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}})_K + ({{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}p)_K = {{\boldsymbol f}}_K,
\label{BP_qdm} \\[1ex] & \displaystyle
({{\rm div}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}})_K + \lambda\, h^2 \, (-\Delta_S p)_K=0,
\label{BP_mass} \end{aligned}$$
\[BP\_scheme\]
with $\lambda>0$ and $\displaystyle (-\Delta_S p)_K=\frac 1 {h^2} \sum_{{\sigma}=K|L} (p_K -p_L)$.
The stabilization term introduced in the mass balance may be seen as a finite volume analogue of the classical so-called Brezzi-Pitkäranta stabilization [@bre-84-sta] usual in the finite element context.
The scheme admits a unique solution and is stable in natural energy norms, [[*i.e.*]{}]{} there exists a real number $c$ independent of $h$ such that the solution $({{\boldsymbol u}},p)$ of satisfies: $${\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol u}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}+{\hspace{.2em}|\hspace{-.1em}| p |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}} \leq c {\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol f}}|\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)^2}\hspace{.2em}}.$$
Multiplying by $h^2\,{{\boldsymbol u}}_K$ and by $h^2 \, p_K$ and summing over $K \in {{\mathcal T}}$, we now obtain, thanks to the duality of the discrete gradient and divergence operators: $${\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol u}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}} + \lambda\, h^2 {\hspace{.2em}| p |_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}} \leq c.$$ We now choose ${{\boldsymbol v}}$ to satisfy Lemma \[stab\_grad\] with $q=p$. Multiplying by $ h^2 \, {{\boldsymbol v}}_K$ and summing over $K \in {{\mathcal T}}$, we get: $${\hspace{.2em}|\hspace{-.1em}| p |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}} \leq c \Bigl[ h {\hspace{.2em}| p |_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}} + ({{\boldsymbol u}},{{\boldsymbol v}})_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}+ \Bigl| \int_\Omega {{\boldsymbol f}}\cdot {{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}\Bigr| \Bigr],$$ which yields a control on ${\hspace{.2em}|\hspace{-.1em}| p |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}}$ and concludes the proof.
A second stabilization and an [*inf-sup*]{} stability result
------------------------------------------------------------
Let us now suppose that the integer number $1/h$ is even. In this case, the mesh may be partitionned in square $2 \times 2$ patches of control volumes, which are called hereafter “clusters”. The set of internal edges of the mesh ${{\mathcal E}_{{\rm int}}}$ similarly decomposes into two subsets, ${{\mathcal E}_{{\rm int}}}={{\mathcal E}_{{\rm int}}^{\hspace{.1em}{{\scalebox{0.6}{$\square$}}}}}\cup {{\mathcal E}_{{\rm int}}^{\hspace{.1em}{{\scalebox{0.6}{$+$}}}}}$, the first one (${{\mathcal E}_{{\rm int}}^{\hspace{.1em}{{\scalebox{0.6}{$\square$}}}}}$) containing the edges separating two control volumes of two different clusters, the second one (${{\mathcal E}_{{\rm int}}^{\hspace{.1em}{{\scalebox{0.6}{$+$}}}}}$) containing the edges separating two control volumes of a same cluster. For $q\in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$,${\hspace{.2em}| q |_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}$ can accordingly be split in two parts, ${\hspace{.2em}| q |_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}}={\hspace{.2em}| q |_{{{\scalebox{0.6}{$\square$}}}}^2\hspace{.2em}}+{\hspace{.2em}| q |_{{{\scalebox{0.6}{$+$}}}}^2\hspace{.2em}}$ with: $${\hspace{.2em}| q |_{{{\scalebox{0.6}{$\square$}}}}^2\hspace{.2em}}=\sum_{\begin{array}{c}\scriptstyle {\sigma}\in {{\mathcal E}_{{\rm int}}^{\hspace{.1em}{{\scalebox{0.6}{$\square$}}}}}\\[-0.5ex] \scriptstyle ({\sigma}=K|L) \end{array}} (q_K-q_L)^2
\quad \mbox{and}\quad
{\hspace{.2em}| q |_{{{\scalebox{0.6}{$+$}}}}^2\hspace{.2em}}=\sum_{\begin{array}{c}\scriptstyle {\sigma}\in {{\mathcal E}_{{\rm int}}^{\hspace{.1em}{{\scalebox{0.6}{$+$}}}}}\\[-0.5ex] \scriptstyle ({\sigma}=K|L) \end{array}} (q_K-q_L)^2.$$ We have the following weak stability result.
There exists a positive real number $c$ independent of $h$ such that, $\forall q \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$, one can find ${{\boldsymbol v}}\in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2$ satisfying: $${\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol v}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}=1 \mbox{ and }
\int_\Omega {{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}q \cdot {{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}\geq c\, h\, \bigl[ {\hspace{.2em}| q |_{{{\scalebox{0.6}{$\square$}}}}\hspace{.2em}} - {\hspace{.2em}| q |_{{{\scalebox{0.6}{$+$}}}}\hspace{.2em}} \bigr].$$ \[stab\_grad\_2\]
(0,0)(8.5,4.3) (0,-0.5)[ (1,1)(3,1)(3,3)(1,3) (2,2)(3,2)(3,3)(2,3) (2.35,2.4)[$K$]{} (1.35,2.4)[$L_{{\scalebox{0.6}{$+$}}}$]{} (3.35,2.4)[$L_{{\scalebox{0.6}{$\square$}}}$]{} (2.35,1.4)[$M_{{\scalebox{0.6}{$+$}}}$]{} (2.35,3.4)[$M_{{\scalebox{0.6}{$\square$}}}$]{} (0.5,1)(4.5,1) (0.5,3)(4.5,3) (1,0.5)(1,4.5) (3,0.5)(3,4.5) (0.5,2)(4.5,2) (0.5,4)(4.5,4) (2,0.5)(2,4.5) (4,0.5)(4,4.5) (5.5,2)(6.,2)(6.,2.5)(5.5,2.5)(5.5,2) (6.1,2.1)[cluster]{} (5.5,3)(6.,3)(6.,3.5)(5.5,3.5)(5.5,3) (6.1,3.1)[control volume]{} ]{}
Let $q \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$ be given, and be such that ${\hspace{.2em}| q |_{{{\scalebox{0.6}{$\square$}}}}\hspace{.2em}} \geq {\hspace{.2em}| q |_{{{\scalebox{0.6}{$+$}}}}\hspace{.2em}}$ (otherwise, the result if the lemma is trivial). Let $K$ be a control volume of ${{\mathcal T}}$ and $L_{{\scalebox{0.6}{$\square$}}}$, $M_{{\scalebox{0.6}{$\square$}}}$, $L_{{\scalebox{0.6}{$+$}}}$ and $M_{{\scalebox{0.6}{$+$}}}$ be its 4 adjacent control volumes, as sketched on figure \[fig:cluster\]. We define ${{\boldsymbol v}}_K$ by: $${{\boldsymbol v}}_K=\begin{bmatrix} q_{L_{{\scalebox{0.6}{$\square$}}}} - q_K \\ q_{M_{{\scalebox{0.6}{$\square$}}}} - q_K \end{bmatrix}.$$ We have: $$\begin{array}{l} \displaystyle
h^2\ ({{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}q )_K \cdot {{\boldsymbol v}}_K= \frac h 2\,\Bigl[ (q_{L_{{\scalebox{0.6}{$\square$}}}} - q_K)^2 + (q_{M_{{\scalebox{0.6}{$\square$}}}} - q_K)^2
\\ \hspace{20ex}
+ (q_{L_{{\scalebox{0.6}{$\square$}}}} - q_K)(q_{L_{{\scalebox{0.6}{$+$}}}} - q_K)+ (q_{M_{{\scalebox{0.6}{$\square$}}}} - q_K)(q_{M_{{\scalebox{0.6}{$+$}}}} - q_K)\Bigr],
\end{array}$$ and thus, by Young’s inequality: $$h^2\ ({{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}q )_K \cdot {{\boldsymbol v}}_K \geq \frac h 4\,\Bigl[ (q_{L_{{\scalebox{0.6}{$\square$}}}} - q_K)^2 + (q_{M_{{\scalebox{0.6}{$\square$}}}} - q_K)^2
-(q_{L_{{\scalebox{0.6}{$+$}}}} - q_K)^2 - (q_{M_{{\scalebox{0.6}{$+$}}}} - q_K)^2 \Bigr].$$ Summing over the control volumes, we get: $$\int_\Omega {{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}q \cdot {{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}\geq \frac h 2 \, \bigl[ {\hspace{.2em}| q |_{{{\scalebox{0.6}{$\square$}}}}^2\hspace{.2em}} - {\hspace{.2em}| q |_{{{\scalebox{0.6}{$+$}}}}^2\hspace{.2em}} \bigr].$$ On the other hand, from the expression of ${{\boldsymbol v}}$, we deduce that ${\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol v}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}} \leq c_3 {\hspace{.2em}| q |_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}}$ with $c_3$ independent of $h$, and the conclusion follows by normalizing ${{\boldsymbol v}}$ and using the fact that, $\forall a,\, b \geq 0,\ a \geq b,\ a^2 - b^2 \geq (a-b) (a^2+b^2)^{1/2}$.
Let us now consider the following scheme, which consists in searching for $({{\boldsymbol u}},p)\in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2 \times \bar {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$ such that, $\forall K \in {{\mathcal T}}$:
$$\begin{aligned}
& \displaystyle
(-\Delta_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}})_K + ({{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}p)_K = {{\boldsymbol f}}_K,
\label{cluster_qdm} \\[1ex] & \displaystyle
({{\rm div}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}})_K + \lambda\, h^2\, (-\Delta_{S,{{\scalebox{0.6}{$+$}}}}\, p)_K=0,
\label{cluster_mass} \end{aligned}$$
\[cluster\_scheme\]
with $\lambda>0$ and $\displaystyle (-\Delta_{S,{{\scalebox{0.6}{$+$}}}}\, p)_K=\frac 1 {h^2} \sum_{{\sigma}=K|L,\ {\sigma}\in{{\mathcal E}_{{\rm int}}^{\hspace{.1em}{{\scalebox{0.6}{$+$}}}}}} (p_K -p_L)$.
The stabilization involved in may be seen as a finite volume analogue of the so-called “local jump stabilization” introduced in [@sil-90-sta; @kec-92-ana].
The scheme admits a unique solution and is stable in natural energy norms, [[*i.e.*]{}]{} there exists a real number $c$ independent of $h$ such that the solution $({{\boldsymbol u}},p)$ of satisfies: $${\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol u}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}+{\hspace{.2em}|\hspace{-.1em}| p |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}} \leq c {\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol f}}|\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)^2}\hspace{.2em}}.$$
Multiplying by $h^2\,{{\boldsymbol u}}_K$ and by $h^2 \, p_K$ and summing over $K \in {{\mathcal T}}$, we obtain, thanks to the duality of the discrete gradient and divergence operators: $${\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol u}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}} + \lambda \, h^2 {\hspace{.2em}| p |_{{{\scalebox{0.6}{$+$}}}}^2\hspace{.2em}} \leq c.$$ We choose ${{\boldsymbol v}}$ to satisfy Lemma \[stab\_grad\_2\] with $q=p$. Multiplying by $h^2\,{{\boldsymbol v}}_K$ and summing over $K \in {{\mathcal T}}$, we get: $$h {\hspace{.2em}| p |_{{{\scalebox{0.6}{$\square$}}}}\hspace{.2em}} \leq c \Bigl[ h {\hspace{.2em}| p |_{{{\scalebox{0.6}{$+$}}}}\hspace{.2em}} + ({{\boldsymbol u}},{{\boldsymbol v}})_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}+ \Bigl| \int_\Omega {{\boldsymbol f}}\cdot {{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}\Bigr| \Bigr],$$ which gives a control on $h {\hspace{.2em}| p |_{{{\scalebox{0.6}{$\square$}}}}\hspace{.2em}}$ and thus on $h {\hspace{.2em}| p |_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}$. The conclusion now follows as for the precedent scheme. We now choose ${{\boldsymbol v}}$ to satisfy Lemma \[stab\_grad\] with $q=p$, multiply by $h^2\,{{\boldsymbol v}}_K$ and sum over $K \in {{\mathcal T}}$ to get $${\hspace{.2em}|\hspace{-.1em}| p |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}} \leq c \Bigl[ h {\hspace{.2em}| p |_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2\hspace{.2em}} + ({{\boldsymbol u}},{{\boldsymbol v}})_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}+ \Bigl| \int_\Omega {{\boldsymbol f}}\cdot {{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}\Bigr| \Bigr],$$ which yields a control on ${\hspace{.2em}|\hspace{-.1em}| p |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}}$.
Let $\bar {{\cal X}_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^{\hspace{.1em}{{\scalebox{0.6}{$\square$}}}}}\subset {{\rm L}^{2}}_O(\Omega)$ be the space of constant by cluster and zero mean value functions. Combining lemmata \[stab\_grad\] and \[stab\_grad\_2\], we obtain that, $\forall q \in \bar {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$, ${\hspace{.2em}|\hspace{-.1em}| q |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}}$ may be controlled by the gradient of $q$ up to $h {\hspace{.2em}| p |_{{{\scalebox{0.6}{$\square$}}}}\hspace{.2em}}$. Since this latter quantity vanishes for any function of $\bar {{\cal X}_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^{\hspace{.1em}{{\scalebox{0.6}{$\square$}}}}}$, we have the following discrete [*inf-sup*]{} stability result.
The pair of spaces ${{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2 \times \bar {{\cal X}_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^{\hspace{.1em}{{\scalebox{0.6}{$\square$}}}}}$ is inf-sup stable, in the sense that there exists a positive real number $c$ independent of $h$ such that, $\forall q \in \bar {{\cal X}_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^{\hspace{.1em}{{\scalebox{0.6}{$\square$}}}}}$, there exists ${{\boldsymbol v}}\in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2$ satisfying: $${\hspace{.2em}|\hspace{-.1em}| {{\boldsymbol v}}|\hspace{-.1em}|_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}\hspace{.2em}}=1 \quad \mbox{ and } \quad
\int_\Omega {{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}q \cdot {{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}\geq c {\hspace{.2em}|\hspace{-.1em}| q |\hspace{-.1em}|_{{{\rm L}^{2}}(\Omega)}\hspace{.2em}}.$$ \[inf-sup-clust\]
The pair ${{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2 \times \bar {{\cal X}_{{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^{\hspace{.1em}{{\scalebox{0.6}{$\square$}}}}}$ thus could be used instead of ${{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2 \times \bar {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$, and the stabilization consequently dropped. However, from our practice, the second choice is slightly more accurate; it is also easier to implement, since the velocity and the pressure are approximated by the same discrete space. Note however that recovering a pressure constant by cluster is exactly what happens when the parameter $\lambda$ is large; the accuracy of the scheme then can be expected to be very robust with respect to the value of $\lambda$, which is the main interest of this second stabilization with respect to the first one. Finally, in the context of transient problems, making use of the [*inf-sup*]{} stable alternative could be interesting to implement pressure correction schemes, in which stabilizations are difficult to insert.
[The proposed schemes and may be recast under a “discrete variational form”. For instance, may be written as follows: $$\begin{array}{ll}\displaystyle
({{\boldsymbol u}},{{\boldsymbol v}})_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}+ \int_\Omega p \, {{\rm div}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}}= \int_\Omega {{\boldsymbol f}}\cdot {{\boldsymbol v}}{\, {\rm d}{{\boldsymbol x}}},
\quad &
\forall {{\boldsymbol v}}\in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2;
\\[2ex] \displaystyle
\int_\Omega q \, {{\rm div}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}}{\, {\rm d}{{\boldsymbol x}}}+ \lambda \, h^2 \, [p,q]_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}= 0,
&
\forall q \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}.
\end{array}$$ Indeed, the equations of may be recovered from this formulation by choosing for the test functions the characteristic functions of the control volumes. This variational formulation is used for the extension of the schemes to more general meshes, in particular by changing the form of the discrete ${{\rm H}^{1}}$ inner product (see section \[sec:suschi\]). ]{}\[var\_form\]
Generalizations
===============
In this section, we turn to the case where $\Omega$ is a polygonal bounded domain of ${\mathbb{R}}^2$. Since the main arguments necessary for the generalization of the schemes described above stem for error estimates, we first address this issue; then two specific cases are treated.
Convergence issues
------------------
The error analysis briefly presented here relies of the arguments developed in [@eym-00-fin] for the analysis of schemes for elliptic problems. We consider the scheme for which consists in searching $({{\boldsymbol u}},p)\in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}^2 \times \bar {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$ such that, $\forall K \in {{\mathcal T}}$:
$$\begin{aligned}
& \displaystyle
(-\Delta_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}})_K + ({{\boldsymbol \nabla}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}p)_K = {{\boldsymbol f}}_K,
\label{gene_qdm} \\[1ex] & \displaystyle
({{\rm div}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}})_K + (T_S)_K=0,
\label{gene_mass} \end{aligned}$$
\[gene\_scheme\]
where $T_S$ stands for a possible stabilization term, the discrete Laplace operator $\Delta_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}$ and divergence ${{\rm div}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}$ read: $$(-\Delta_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}})_K = \frac 1 {|K|} \sum_{{\sigma}\in{{\mathcal E}}(K)} F_{\sigma}({{\boldsymbol u}}), \quad
({{\rm div}}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}{{\boldsymbol u}})_K= \frac 1 {|K|} \sum_{{\sigma}=K|L} G_{\sigma}({{\boldsymbol u}}),$$ and the numerical fluxes $F_{\sigma}({{\boldsymbol u}})$ and $G_{\sigma}({{\boldsymbol u}})$ are functions of the mesh and the value of the unknown ${{\boldsymbol u}}$ in the control volumes located “near” the edge ${\sigma}$.
We define a set of points $({{\boldsymbol x}}_K)_{K \in {{\mathcal T}}}$ such that, for any control volume $K \in {{\mathcal T}}$, the point ${{\boldsymbol x}}_K$ lies inside $K$. Then let ${r_{\hspace{-0.1em} {{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}}$ be the interpolation operator which associates to any function $u \in {{\rm C}^{0}}(\Omega)$ the function ${r_{\hspace{-0.1em} {{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}}u \in {{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$ by $\forall K \in {{\mathcal T}},\ ({r_{\hspace{-0.1em} {{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}}u)_K = u({{\boldsymbol x}}_K)$. We make the following consistency assumptions:
--------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$(H_c)$ For ${\sigma}\in{{\mathcal E}_{{\rm int}}}$, the fluxes $F_{\sigma}$ and $G_{\sigma}$ are consistent up to the second order, in the sense that, for any affine vector-valued function polynomial ${{\boldsymbol \varphi}}$: $$F_{\sigma}({r_{\hspace{-0.1em} {{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}}{{\boldsymbol \varphi}})=\int_{\sigma}{{\boldsymbol \nabla}}{{\boldsymbol \varphi}}\cdot {{\boldsymbol n}}_{\sigma}{\, {\rm d}\sigma}, \quad
G_{\sigma}({r_{\hspace{-0.1em} {{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}}{{\boldsymbol \varphi}})=\int_{\sigma}{{\boldsymbol \varphi}}\cdot {{\boldsymbol n}}_{\sigma}{\, {\rm d}\sigma}.$$ For ${\sigma}\in{{\mathcal E}_{{\rm ext}}}$, $F_{\sigma}$ satisfies the same consistency relation supposing that ${{\boldsymbol \varphi}}$ vanishes on ${\sigma}$ and $G_{\sigma}$ vanishes.
--------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Together with the fact that the scheme is stable in the discrete energy norms, which implies that the assumptions $(i)$ (coercivity of the diffusion term) and $(ii)$ (duality of the discrete gradient and divergence operator) hold, this consistency assumption $(H_c)$ is central for proving first order error estimates (in energy norms) for the Brezzi-Pitkäranta stabilization [@eym-06-sta] and the stabilization by clusters [@eym-08-conv].
Non-uniform structured grids
----------------------------
We now examine the consequences of these consistency requirements when $\Omega$ is still $(0,1)\times(0,1)$ and the grid is still structured but no-longer uniform. Let $K$ and $L$ be two adjacent control volumes separated by the edge ${\sigma}$, $h_K^\perp$ (resp. $h_L^\perp$) be the length of $K$ (resp. $L$) in the direction perpendicular to ${\sigma}$. The natural choice for ${{\boldsymbol x}}_K$ (resp. ${{\boldsymbol x}}_L$) is the mass center of $K$ (resp. $L$), and, in this condition, the discretization for $G_{\sigma}$ must be: $$G_{\sigma}= |{\sigma}|\ \Bigl[ \frac{h^\perp_L}{h^\perp_K +h^\perp_L}\ {{\boldsymbol u}}_K + \frac{h^\perp_K}{h^\perp_K +h^\perp_L}\ {{\boldsymbol u}}_L \Bigr] \cdot {{\boldsymbol n}}_{\sigma}.$$ Imposing to the discrete gradient operator to be the transposed of the divergence with respect to the ${{\rm L}^{2}}$ inner product, we obtain that the flux associated to the gradient of the pressure through ${\sigma}$, let say $H_{\sigma}$, reads: $$H_{\sigma}=|{\sigma}|\ \Bigl[ \frac{h^\perp_K}{h^\perp_K +h^\perp_L}\ p_K + \frac{h^\perp_L}{h^\perp_K +h^\perp_L}\ p_L \Bigr] \ {{\boldsymbol n}}_{\sigma},$$ which is not the standard (and only a first order) interpolation.
(0,0)(7,3) (0.5,1)(6.5,1) (0.5,2)(6.5,2) (1,0.5)(1,2.5) (3,0.5)(3,2.5) (6,0.5)(6,2.5) (1.1,1.7)[$K$]{} (5.65,1.7)[$L$]{} (3.1,1.7)[${\sigma}$]{} (2,1.5) (2.1,1.2)[${{\boldsymbol x}}_K$]{} (4.5,1.5) (4.6,1.2)[${{\boldsymbol x}}_L$]{} (1,0.4)(2.95,0.4) (1.9,-0.1)[$h^\perp_K$]{} (3.05,0.4)(6,0.4) (4.4,-0.1)[$h^\perp_L$]{}
General grids {#sec:suschi}
-------------
A scheme for general grids, including grids involving hanging nodes, is presented in [@eym-07-new]. This scheme may work with the Brezzi-Pitkäranta stabilization or with a stabilization by cluster. For this generalization, two new ingredients, in particular, are necessary:
- The definition of a diffusion operator. This is performed using a variational approach with a modified form for the inner product $(\cdot,\cdot)_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}$, as mentioned in Remark \[var\_form\].
- A suitable definition for the clusters, which are seen as patches of elements satisfying the following general condition: $$\forall K \in {{\mathcal T}}\mbox{ such that } {{\mathcal N}}_K \not \subset {{\mathcal G}}_K,\quad
\inf_{(a_L) \subset {\mathbb{R}}} \frac
{\displaystyle \Bigl| \sum_{L\in {{\mathcal N}}_K \setminus {{\mathcal G}}_K} a_L\ {{\boldsymbol n}}_{K|L}\ \Bigr|^2}
{\displaystyle \sum_{L\in {{\mathcal N}}_K \setminus {{\mathcal G}}_K} a_L^2}
\geq c >0,
\label{cond_cluster}$$ where, $\forall K \in {{\mathcal T}}$, ${{\mathcal N}}_K$ is the set of the neighbours of $K$ ([[*i.e.*]{}]{} the control volumes sharing an edge with $K$) and ${{\mathcal G}}_K$ is the cluster containing $K$. The condition is exactly the condition which allows to control the ${\hspace{.2em}|\hspace{-.1em}| \cdot |\hspace{-.1em}|_{{{\scalebox{0.6}{$\square$}}}}\hspace{.2em}}$ norm of a function of ${{\cal X}_{{\raisebox{-0.1em}{\scalebox{0.6}{${{\mathcal T}}$}}}}}$ by its gradient, as in lemma \[stab\_grad\_2\]. Considering now a family of meshes, this control will be uniform if the real number $c$ does not depend on the considered mesh, and Relation thus acts as a regularity criterion for the meshes.
[20]{}
, [*On the Stabilization of Finite Element Approximations of the [S]{}tokes Equations*]{}, in Efficient Solution of Elliptic Systems, W. Hackbusch ed., Vieweg, 1984, pp. 11–19.
, [*Numerical results using a colocated finite-volume scheme on unstructured grids for incompressible fluid flows*]{}, Numer. Heat Tranf. B-Fundam., 49 (2006), pp. 259–276.
, [*Collocated finite volume schemes for the simulation of natural convective flows on unstructured meshes*]{}, Int. J. Numer. Methods Fluids, 56 (2008), pp. 2045–2068.
, [*Finite Volume Methods*]{}, in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds., Vol. VII, North Holland, 2000, pp. 713–1020.
, [*On a stabilized colocated finite volume scheme for the [S]{}tokes problem*]{}, Math. Model. Numer. Anal., 40 (2006), pp. 501–528.
, [*A new colocated finite volume scheme for the incompressible [N]{}avier-[S]{}tokes equations on general non-matching grids*]{}, Comptes Rendus Math., 344 (2007), pp. 659–662.
, [*Convergence analysis of a colocated finite volume scheme for the incompressible [N]{}avier-[S]{}tokes equations on general 2[D]{} or 3[D]{} meshes*]{}, SIAM J. Numer. Anal., 45 (2007), pp. 1–36.
, [*On the stability of colocated clustered finite volume simplicial discretizations for the 2[D]{} [S]{}tokes problem*]{}, Calcolo, 44 (2007), pp. 219–234.
, [*Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows*]{}, submitted (2008).
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, [*Numerical study of the turbulent flow past an airfoil with trailing edge separation*]{}, AIAA J., 21 (1983), pp. 1525–1532.
, [*Stabilised bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the [S]{}tokes problem*]{}, Comput. Meth. Appl. Mech. Eng., 79 (1990), pp. 71–86.
[^1]: Université de Marne-la-Vallée, France ([eymard@univ-mlv.fr]{})
[^2]: Université de Provence, France ([herbin@latp.univ-mrs.fr]{})
[^3]: Institut de Radioprotection et de Sûreté Nucléaire (IRSN), France ([@irsn.fr]{})
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---
abstract: '$^{75}$As NMR investigation of a single crystal of superconducting LiFeAs is presented. The Knight shift and the *in situ* ac susceptibility measurements as a function of temperature and external field are indicative of two superconducting (SC) transition temperatures, each of which is associated with its own upper critical field. Strikingly, the Knight shift maintains its normal state value over a temperature range in the SC state before it drops abruptly being consistent with spin-singlet pairing. Together with our previous NMR study, the anomalous SC state featured by the constant Knight shift is attributed to the extremely sensitive SC properties of LiFeAs, probably stemming from its proximity to a critical instability.'
author:
- 'S.-H. Baek'
- 'L. Harnagea'
- 'S. Wurmehl'
- 'B. Büchner'
- 'H.-J. Grafe'
bibliography:
- 'mybib.bib'
title: 'Anomalous superconducting state in LiFeAs implied by the $^{75}$As Knight shift measurement'
---
It is commonly argued that superconductivity in iron pnictides is driven by the antiferromagnetic (AFM) spin fluctuations which are associated with nesting between the hole and electron Fermi surface pockets, although the SC gap symmetry seems to vary among the materials from nodal to nodeless.[@lumsden10a; @stewart11; @chubukov12] An exception in this general picture is LiFeAs that is superconducting as is, without any signature of nesting and static magnetism, yet with rather high $T_c\sim18$ K.[@morozov10; @borisenko10; @pitcher10]
While the absence of nesting and static magnetism in LiFeAs[@borisenko10] might support a non-magnetic origin for the SC pairing, such as phonons[@kordyuk11] or orbital fluctuations,[@kontani10] AFM spin fluctuations remain a strong candidate that is responsible for the SC pairing,[@jeglic10; @platt11; @taylor11; @hajiri12] e.g., by recovering the nesting condition by the magnetic response shifting.[@qureshi12] If this is indeed the case, it would strengthen the belief that AFM spin fluctuations are fundamental to the superconductivity of iron-pnictides. On the other hand, spin-triplet pairing which is driven by ferromagnetic spin fluctuations originating from strong Hund coupling was also suggested,[@brydon11] being followed by some experimental supports.[@pramanik11; @haenke12; @baek12; @li13] Such debates about the pairing mechanism in LiFeAs may imply that the nature of superconductivity in this material is different from other iron-pnictide families, and it was suggested that close proximity of the system to a strong magnetic instability may effect the unusual sensitivity of the SC properties.[@baek12; @brydon11]
In an effort to confirm the underlying instability and to uncover its nature, we carried out $^{75}$As nuclear magnetic resonance (NMR) in a single crystal of LiFeAs chosen from a different batch than those used in our previous NMR study,[@baek12] focusing on the low temperature range near and below $T_c\sim 18$ K. While the *in situ* ac susceptibility and the NMR signal intensity confirm the bulk $T_c$, the Knight shift remains constant down to a temperature at which it drops sharply. Although the constant Knight shift behavior in the SC state is not easily reproducible in other single crystals, our data suggest that an anomalous superconducting state where the Knight shift does not change could be stabilized. We discuss that LiFeAs is very close to a critical instability which affects the SC state particularly near the region of the normal/superconducting boundary.
The single crystal of LiFeAs was grown by a self-flux method as described in Ref. . Due to the sensitivity of the sample to air and moisture, the sample was carefully sealed into a quartz tube filled with Ar gas for NMR measurements. The sealed sample was mounted on a goniometer for an accurate alignment of the sample along the external field. $^{75}$As ($I=3/2$) nuclear magnetic resonance (NMR) experiments were performed in the range of temperature 3.6 — 25 K and external field 0 — 16 T. We also carried out $^{75}$As nuclear quadrupole resonance (NQR) to determine the quadrupole frequency $\nu_Q$. The NQR spectrum shows a width of 75 kHz at 20 K, which is much narrower than $\sim 170$ kHz in a powder sample[@li10] and thus indicates a sign of good chemical homogeneity of the sample.
The Knight shift $\mathcal{K}$, i.e., the local static spin susceptibility, was measured from $^{75}$As NMR central line at various external fields applied parallel and perpendicular to the crystallographic $c$ axis. The large quadrupole frequency $\nu_Q=21.08$ MHz of the $^{75}$As ($I=3/2$), which is almost $T$-independent in the low temperature range investigated, shifts the central transition of the $^{75}$As by the second order quadrupole effect given by $\Delta\nu=3\nu_Q^2/16\omega_n(1-\cos^2\theta)(1-9\cos^2\theta)$ for $I=3/2$ where $\omega_n$ is the unshifted Larmor frequency and $\theta$ is the angle between the external field $H$ and the $c$ axis. The Knight shift shown in Fig. 1 was obtained by subtracting $\Delta\nu$ from the total shift of the central line. The SC transition temperature $T_c$ was identified from a sudden drop of $\mathcal{K}$ for a given external field $H$, which indicates spin singlet Cooper pairing. Whereas this behavior seems consistent with previous other NMR studies in this compound,[@li10; @jeglic10] we find that $T_c(H)$, particularly for $H \parallel c$, are much lower than values reported in literature.[@lee10a; @heyer11; @kurita11; @cho11; @khim11] At 8.5 T for $H\parallel c$, for example, $\mathcal{K}$ does not drop down even at 3.6 K, indicating that $T_c\le3.5$ K is significantly lower than an expected value ($>10$ K).[@lee10a; @heyer11; @kurita11; @cho11; @khim11]
![\[fig:knight\] Knight shift ($\mathcal{K}$) of $^{75}$As as a function of temperature and field for two field orientations. A second order quadrupole correction was made for $H \perp c$. Superconducting transition temperature for each field was determined from the sharp drop of $\mathcal{K}$ as denoted by arrows. ](knight.eps){width="\linewidth"}
In order to confirm the transition temperature, we measured the *in situ* ac susceptibility $\chi_\text{ac}$ using the NMR radio frequency (rf) circuit. In the SC state, the Meissner effect induces the change of impedance and thus the tuning frequency of the rf circuit. Therefore, the onset of superconductivity could be detected by monitoring $\chi_\text{ac}$ as a function of temperature. Fig. 2 shows $\chi_\text{ac}(T)$ measured at various external fields $H$. Here we define $T_c$ as a temperature where $\chi_\text{ac}$ reaches 10% of the full drop to the low temperature plateau at each field, which are denoted by down arrows. Clearly $T_{c}$ detected by $\chi_\text{ac}$ is much higher than that obtained by the Knight shift measurements for each field (up arrows). Note that, at 8.5 T parallel to the $c$ axis, a clear onset was observed at $11$ K by $\chi_\text{ac}$ which is compatible with values reported thus far,[@lee10a; @heyer11; @kurita11; @cho11; @khim11] in stark contrast to the absence of the Knight shift anomaly down to 3.6 K. It may be worthwhile to note that $\chi_\text{ac}$ displays a small but noticeable anomalous change in its slope at $T_c$ obtained by $\mathcal{K}$.
As the two experimental methods seem to distinguish different onset temperatures of the SC transition, here we define the two onset temperatures obtained by $\chi_\text{ac}$ and $\mathcal{K}$ by [$T_c^\text{ac}$]{} and [$T_c^{\mathcal{K}}$]{}, respectively. While a sharp drop of $\mathcal{K}$ is usually a good indication of spin singlet superconductivity, $\chi_\text{ac}$ alone is not sufficient in general to verify a bulk $T_c$, because other non-superconducting effects might alter the temperature dependence of $\chi_\text{ac}$. To check the validity of [$T_c^\text{ac}$]{}, we carefully examined the temperature evolution of the $^{75}$As spectra. In the SC state, the signal intensity should decrease due to supercurrents which reduce the sample volume that can be penetrated by the rf field, and therefore it could be another good probe for detecting the onset of bulk superconductivity. Fig. 3 shows the $^{75}$As NMR spectrum as a function of temperature measured at 8.5 T, where the Boltzmann correction was made by multiplying $T$ for each spectrum. For $H\perp c$, the signal intensity starts to decrease at $\sim 15$ K, which agrees with [$T_c^\text{ac}$]{} determined from $\chi_\text{ac}$, as shown in Fig. 3(c). The agreement of the signal intensity with $\chi_\text{ac}$ in their temperature dependences was also confirmed for $H \parallel c$ \[see Fig. 3(b) and (c)\]. Note that the anisotropy of $\chi_\text{ac}$ below [$T_c^\text{ac}$]{} remarkably coincides with that of the signal intensity. For direct comparison, $\mathcal{K}$ is shown in the upper panel of Fig. 3(c).
![\[fig:chiac\] *In situ* ac susceptibility $\chi_\text{ac}$ measured in the NMR tank circuit as a function of temperature and external field. The transition temperature [$T_c^\text{ac}$]{} (down arrows) is considerably higher than [$T_c^{\mathcal{K}}$]{} (up arrows) determined by the Knight shift measurements. Data for $H \parallel c$ are shown as solid lines (no arrows for clarity). ](chiac.eps){width="\linewidth"}
We emphasize that $T_c^\text{ac}(H)$ indeed confirms bulk superconductivity which has been unanimously proven in our single crystals by numerous other measurements including dc magnetic susceptibility,[@morozov10] specific heat,[@stockert11] resistivity,[@heyer11] angle-resolved photoemission spectroscopy (ARPES),[@borisenko10; @kordyuk11; @borisenko12] neutron,[@qureshi12] and scanning tunneling spectroscopy (STS),[@haenke12] as well as by theoretical supports.[@lankau10; @knolle12] In particular, note that the specific heat measured in our single crystals manifests the bulk $T_c\sim 9$ K in a field of 9 T applied along the $c$ axis[@stockert11] that is comparable to [$T_c^\text{ac}$]{} at 8.5 T, while the Knight shift remains constant down to 3.6 K at 8.5 T (see Fig. 3). Furthermore, $T_c^\text{ac}(H)$ and the related $H_{c2}$ are in satisfactory agreement with the results measured in other samples by different groups.[@kurita11; @cho11; @khim11] Therefore, we conclude that [$T_c^\text{ac}$]{} is equivalent to the onset temperature of the *bulk* Meissner effect which modifies the signal intensity and $\chi_\text{ac}$ simultaneously.
Further analysis of the Knight shift, the signal intensity, and the ac susceptibility obtained at various external fields reveals quite different field dependence of [$T_c^{\mathcal{K}}$]{} and [$T_c^\text{ac}$]{}. (For raw $^{75}$As spectra at external fields other than 8.5 T, see supplemental material.[@supple]) The resulting $H$-$T$ phase diagram is presented in Fig. 4. We find that the $H$-dependence of [$T_c^\text{ac}$]{} is in qualitative agreement with other studies.[@heyer11; @khim11; @kurita11] For example, the data from Khim et al.[@khim11] are compatible with [$T_c^\text{ac}$]{} data in the $H$-dependence as well as the anisotropy.
![\[fig:sp\_comp\]Temperature dependence of $^{75}$As NMR central line at $H=8.5$ T for (a) $H\perp c$ and (b) $H \parallel c$. (c) Signal intensity and Knight shift versus temperature at 8.5 T. Temperature dependence of signal intensity for both $H \perp c$ and $H\parallel c$ agrees well with that of $\chi_\text{ac}$, indicating that [$T_c^\text{ac}$]{} represents the onset of screening due to superconductivity. The Knight shift, however, reveals [$T_c^{\mathcal{K}}$]{} which is significantly lower than [$T_c^\text{ac}$]{}. $\mathcal{K}_\parallel$ was offset vertically for comparison. ](sp_comp.eps){width="\linewidth"}
In contrast, the $H$-dependence of [$T_c^{\mathcal{K}}$]{} is very different from that of [$T_c^\text{ac}$]{}, other than its much lower values. For $H \perp c$, while [$T_c^\text{ac}$]{} exhibits almost a linear $H$-dependence up to 16 T, [$T_c^{\mathcal{K}}$]{} does not decrease linearly with increasing $H$. Consequently, the difference $T_c^\text{ac}-T_c^\mathcal{K}$ becomes larger at higher fields. This trend is more pronounced for $H \parallel c$. Note that the estimated $H$-dependence of [$T_c^{\mathcal{K}}$]{} for $H\parallel c$ (dashed line in Fig. 4) agrees with the absence of [$T_c^{\mathcal{K}}$]{} at 8.5 T down to 3.6 K.
![\[fig:phasedia\] $H$-$T$ phase diagram in LiFeAs. Two onset temperatures [$T_c^\text{ac}$]{} and [$T_c^{\mathcal{K}}$]{} were obtained by the ac susceptibility and the Knight shift, respectively. Data from Khim et al.[@khim11] and Li et al.[@li13] are shown for comparison. The onset temperatures of the paramagnetic irreversibility from Li et al. fall between [$T_c^\text{ac}$]{} and [$T_c^{\mathcal{K}}$]{} line, while $T_c (H\parallel c)$ in Ref. determined from resistivity (not shown for clarity) almost coincides with $T_c^\text{ac} (H)$. Note that shades of blue and red are applicable only to the case of $H \perp c$ and thus some care is needed to compare the data for $H\parallel c$.](phasedia.eps){width="\linewidth"}
Our experimental results naturally raise important questions. Does $\mathcal{K}$, i.e., the intrinsic spin susceptibility, remain unchanged across [$T_c^\text{ac}$]{} but drop below [$T_c^{\mathcal{K}}$]{}? Do the two seemingly distinguishable SC states above and below [$T_c^{\mathcal{K}}$]{}occur in a single phase? It should be emphasized that only one phase must be present in the normal state above [$T_c^\text{ac}$]{}, because NMR and NQR spectra exhibit very sharp single lines, and their signal intensities are well conserved at all temperatures investigated. Although inhomogeneous superconductivity is extremely unlikely due to bulk superconductivity in our single crystals, here we discuss the possibility that the two SC transitions result from *phase segregation* in bulk form below [$T_c^\text{ac}$]{}, i.e., a partial volume fraction of the sample (region I) becomes superconducting at [$T_c^\text{ac}$]{} first, and the rest of the sample (region II) remains normal down to [$T_c^{\mathcal{K}}$]{} but undergoes the SC transition at [$T_c^{\mathcal{K}}$]{}.
If phase segregation takes place at [$T_c^\text{ac}$]{}, the otherwise single spectrum would be segregated into two parts arising from SC region I and normal region II, respectively. In this case, the unchanged Knight shift of the “total” spectrum between [$T_c^{\mathcal{K}}$]{}and [$T_c^\text{ac}$]{} could be realized *only* either (i) if the SC transition in region I is extremely sharp so that the decreasing Knight shift is not detected, or (ii) if triplet superconductivity occurs in region I so that $\mathcal{K}$ of region I is still the same as that of the normal region II. The consequence of case (i) should be an almost discontinuous change of the signal intensity just below [$T_c^\text{ac}$]{}. On the contrary, we find that the $T$-dependence of the signal intensity shows a gradual change over a temperature range \[see Fig. 3(c)\], ruling out this scenario. Similarly, case (ii) is also ruled out as following. Since singlet superconductivity occurs at [$T_c^{\mathcal{K}}$]{}, we should have two different SC pairing states below [$T_c^{\mathcal{K}}$]{}. Since $\mathcal{K}$ from region II decreases while $\mathcal{K}$ from region I remains constant, two NMR lines or noticeable broadening below [$T_c^{\mathcal{K}}$]{} should be observed. As shown in Fig. 3(a), however, the well-defined single line at all temperatures is inconsistent with this scenario. Also by a close inspection of the $T$-dependence of $^{75}$As spectrum at other external fields,[@supple] the phase segregation scenario turns out highly improbable. Hence, we reach the remarkable conclusion that $\mathcal{K}$ is indeed a constant in the SC state between [$T_c^\text{ac}$]{} and [$T_c^{\mathcal{K}}$]{}, suggesting that the anomalous SC state may change at [$T_c^{\mathcal{K}}$]{} to a somewhat “normal” SC state with singlet-pairing symmetry.
*A priori*, the unchanged Knight shift through [$T_c^\text{ac}$]{} contrasts with spin-singlet pairing, because it implies that the spin degree of freedom of electrons does not vanish in the SC state. Surprisingly, another signature of the possible unusual SC state in LiFeAs was also verified independently in a different single crystal by recent magnetometry measurements[@li13] which report a paramagnetic (PM) response within the SC state at high fields. The onset temperature of PM irreversibility $T_\text{irr}$ as a function of $H \parallel c$ is located between the [$T_c^\text{ac}$]{} and [$T_c^{\mathcal{K}}$]{} lines (see Fig. 4), whereas $T_c$ determined from resistivity is well consistent with $T_c^\text{ac}(H \parallel c)$. The anomalous PM response in the SC state, which is ascribed to the triplet component induced by high fields,[@li13] is indeed in excellent agreement with the nonvanishing spin susceptibility in the SC state revealed by the constant Knight shift. Note that, since $T_\text{irr}$ is a crossover temperature rather than a measure of the actual transition, $T_\text{irr}>T_{c}^\mathcal{K}$ is very reasonable. Therefore, combining our NMR results and Ref. , we interpret that both the constant Knight shift and the PM response observed in the similar region of the phase diagram are signs of spin-triplet pairing that could perhaps be stabilized under certain conditions, which may be a realization of the theoretical prediction that a spin-triplet could occur in iron-pnictides depending on various parameters such as Hund coupling and onsite Coulomb repulsion.[@daghofer08; @nicholson11; @brydon11]
Interestingly, the PM irreversibility at high fields was not reproduced in other samples, being attributed to the extreme sensitivity of samples[@li13] which was already proposed in our previous NMR study.[@baek12] Such a difficult reproducibility of the constant Knight shift behavior as well as of the PM response in Ref. , together with the extreme sensitivity of the SC properties demonstrated in our previous NMR study,[@baek12] suggests that the triplet-like anomalous SC state is unstable in nature, being susceptible to even a tiny off-stoichiometry. This is also consistent with a large variation of $T_c$ from 15.5 K to 18 K which has been found in a recent transport study,[@rullier12] although all the measured samples show the lowest residual-resistivity ratios and thus appear to be of high quality.
Here we argue that the peculiar sensitivity of LiFeAs could be a natural consequence of the close proximity to a critical instability, near which the unusual SC state could emerge. Hence, as long as the off-stoichiometry is small (i.e., the sample quality is pure enough), the effect of the instability would persist especially at high temperatures/fields causing $T_c (H=0)$ and $H_{c2} (T=0)$ very much sample-dependent, whether or not the unusual SC state is actually stabilized. Note that this picture indeed accounts well for the non-trivial large variation of $T_c$ and $H_{c2}$ reported so far in LiFeAs[@heyer11; @kurita11; @cho11; @khim11; @rullier12] and the persisting 2D superconducting fluctuations up to 1.4$T_c$.[@rullier12] Furthermore, given the possible realization of the anomalous SC state which differs from the usual spin-singlet state, contradicting experimental results regarding the pairing symmetries in LiFeAs[@haenke12; @hashimoto12; @jang12] may be reconciled with each other, in terms of the closeness to a critical ferromagnetic instability.
This work has been supported by the Deutsche Forschungsgemeinschaft through SPP1458 (Grant No. GR3330/2 and BE1749/13) and through Emmy Noether Programme WU595/3-1.
{width="\textwidth"}
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abstract: 'The 2-m robotic Liverpool Telescope reacted promptly to the gamma–ray burst GRB 050502a discovered by [*INTEGRAL*]{} and started observing 3 min after the onset of the GRB. The automatic identification of a bright afterglow of $r''\sim15.8$ triggered for the first time an observation sequence in the $BVr''i''$ filters during the first hour after a GRB. Observations continued for $\sim$1 day using the [*RoboNet-1.0*]{} network of 2-m robotic telescopes. The light curve in all filters can be described by a simple power law with index of $1.2\pm0.1$. We find evidence for a bump rising at $t\sim0.02$ days in all filters. From the spectrum and the light curve we investigate different interpretative scenarios and we find possible evidence for a uniform circumburst medium with clumps in density, as in the case of GRB 021004. Other interpretations of such bumps, such as the effect of energy injection through refreshed shocks or the result of a variable energy profile, are less favored. The optical afterglow of GRB 050502a is likely to be the result of slow electron cooling with the optical bands lying between the synchrotron peak frequency and the cooling frequency.'
author:
- |
C. Guidorzi, A. Monfardini, A. Gomboc, C. G. Mundell, I. A. Steele, D. Carter,\
M. F. Bode, R. J. Smith, C. J. Mottram, M. J. Burgdorf
- 'N. R. Tanvir'
- 'N. Masetti'
- 'E. Pian'
nocite: '[@Masetti00]'
title: |
The Early ($<$1 hr) Multi–Color Afterglow of GRB 050502a:\
Possible Evidence for a Uniform Medium with Density Clumps
---
Introduction
============
Although a considerable number of Gamma–Ray Bursts (GRBs) have detected optical counterparts, there are still few with optical afterglow measurements within minutes of the gamma rays: Figure \[fig:all\] shows the early light curves (unfiltered, $R$ and $V$) for all of these. The early afterglow is particularly interesting as it carries information about the immediate surroundings of the GRB progenitor, concerning either the circumburst medium or the interaction between shells and the ISM in the fireball scenario. For two GRBs, an optical flash was detected simultaneously with the gamma rays: GRB 990123 and GRB 041219a: the former has been interpreted as the signature of a reverse shock [@Akerlof99], while for the latter a correlation between the gamma–ray and optical radiation light curves seems to favor a common origin [@Vestrand05]. These early afterglows show considerable variety: e.g., in the case of GRB 030418 the optical emission was found to rise for the first 600 s, slowly vary for 1400 s and then faded as a power law. This was interpreted as due to the variable extinction by the local circumburst medium [@Rykoff04]. In the cases of GRB 990123 and GRB 021211, the early light curve is described by a power law whose index varies from $\sim 2$ to $\sim 1$ a few min after the GRB: at 0.5 min and 2.7 min in the rest frame, respectively [@Holland04]. This has been interpreted as due to the transition between reverse and forward shocks.
GRB 021004, one of the best observed GRBs in optical [@Holland03; @Fynbo05; @deUgarte05], exhibited a number of bumps in its light curve, with all but the first bump being detected from radio to U band. Different interpretations have been suggested to explain the light curve features: @Lazzati02 modeled it using a variable density profile, most likely a uniform medium with clumps with density variations of the order of $\Delta n/n\sim10$ and size of $10^6$ cm. Other authors [@Nakar03; @Bjornsson04; @deUgarte05] account for the bumps with episodes of energy injections when inner shells catch up with the afterglow shock at late times. In addition, @Nakar03 show that the bumps could be also explained by a variable energy profile that is angularly-dependent on jet structure (“patchy shell” model).
In this Letter, we report the robotic detection and automatic identification of GRB 050502a using the 2-m Liverpool Telescope (LT) located in La Palma, Canary Islands: these observations represent one of the first observations of a multi–color light curve in the first hour since the burst. In addition, we report on late follow–up observations performed with LT and the 2-m Faulkes Telescope North (FTN) located at Maui, Hawaii, both members of the [*RoboNet-1.0*]{} consortium[^1] [@Gomboc05a].
Observations and Results
========================
On 2005 May 02 [*INTEGRAL*]{} detected GRB 050502a at 02:13:57 UT and determined its position at $\alpha$=13:29:45.4 and $\delta$=+42:40:26.8 (J2000) with an error radius of 2 arcmin (90% C.L.) [@Gotz05_a]. The GRB had a duration of 20 s. In the 20–200 keV band it had a peak flux of $2\times10^{-7}$ erg cm$^{-2}$ s$^{-1}$ and a fluence of $1.4\times10^{-6}$ erg cm$^{-2}$ [@Gotz05_b], thus ranking among faint/intermediate fluence GRBs. ROTSE–IIIb started observing at 23.3 s after the GRB and detected a 14.3-mag (unfiltered) unknown fading source at $\alpha$=13:29:46.3 and $\delta$=+42:40:27.7 (J2000) ($l=98^{\circ}.76$, $b=+72^{\circ}.61$) [@Yost05]. @Prochaska05 acquired a spectrum with Keck–I 3.5 hr after the GRB and identified a strong absorption feature, which they interpret as SiII1260 at redshift $z=3.793$.
The LT responded robotically to the [*INTEGRAL*]{} alert and started observing 3 min after the GRB onset (2.5 min after the notice time). Independently of ROTSE–IIIb it detected a bright fading source not present in the USNO–B1.0, 2MASS and GSC 2.3 catalogs, with a position consistent with that of the optical transient (OT) of ROTSE–IIIb [@Gomboc05b]. The automatic identification of the bright and rapidly-fading OT by the LT GRB robotic pipeline (see @Gomboc05c for technical details) resulted in the automatic triggering of a multi–color imaging sequence that provided light curves in $BVr'i'$ filters from 3 min to 1 hr after the GRB onset. The robotic follow–up with LT ended after the first hour. Subsequent follow–up observations were triggered manually on both the LT and FTN (Table \[tab:obs\]). Magnitudes in $r'$ and $i'$ have been calibrated using the SDSS DR3 photometric database[^2]. We obtained a consistent calibration using Landolt standard field stars [@Landolt92], for which @Smith02 provide SDSS calibration. For the $B$ and $V$ filters, we calibrated with Landolt standard field stars. The zero-points were stable during the night and fully consistent with the photometric values. This is also confirmed by the Carlsberg Meridian Telescope at La Palma[^3]. Finally we corrected for the airmass and Galactic extinction. The Galactic extinction [@Schlegel98] towards GRB 050502a is low: $A_V=0.03$. We evaluated the extinction in the other filters following @Cardelli89: $A_B=0.04$, $A_{r'}=0.03$ and $A_{i'}=0.02$. Magnitudes have been converted into flux densities $F_{\nu}$ (mJy) following @Fukugita95.
Figure \[fig:LC\] shows the multi–color light curve acquired by the LT during the first hour and the later points with both LT and FTN. An achromatic bump rising at $t\sim0.02$ d is evident. Fitting each light curve with a power law of the form $F\propto t^{-\alpha}$, and excluding points $0.02$ d$<t<0.2$ d, we obtain power–law indices consistent across all bands: $\alpha_B=1.20\pm0.04$, $\alpha_V=1.16\pm0.06$, $\alpha_{r'}=1.19\pm0.04$, $\alpha_{i'}=1.16\pm0.03$. By fitting only the $r'$ points obtained during the detection mode within 3.8 min of the GRB onset time, we get a power–law index of $\alpha_{r',\textrm{early}}=1.3\pm0.1$, consistent with the slopes reported above.
Figure \[fig:SED\] shows the rest–frame Spectral Energy Distribution (SED) at two epochs: before the bump ($t=0.004$ d), where no strong evidence for significant color change is observed (see Fig. \[fig:LC\]), and at the bump ($t=0.035$ d). Optical fluxes have been obtained by interpolation. During the bump, a linear interpolation between consecutive points has been adopted, considering that the variability timescales are much larger than the time difference between the pairs of data points used for interpolation. Moreover, we back-extrapolated to $t=0.004$ d a [*Swift*]{} X–ray upper limit determined around 1.3 d [@Hurkett05], assuming a power–law decay, $F_X\propto t^{-\alpha_X}$, and two different slopes: i) $\alpha_X=\alpha_X^{(1)}=1.45$ (solid arrow in Fig. \[fig:SED\]); ii) $\alpha_X=\alpha_X^{(2)}=0.95$ (dashed arrow in Fig. \[fig:SED\]). The reasons for these choices are clarified in Sec. \[sec:disc\]. In case (i) the power–law index between optical and X–rays must be: $\beta_{OX}>0.7$; in case (ii) it must be: $\beta_{OX}>1.1$. However a word of caution is needed, particularly because we know from the [*Swift*]{} observation that during the first few hundred seconds the early X–ray afterglows can be characterized by a steep decline followed by a shallower decay [@Tagliaferri05]. The back-extrapolation for the radio upper limits provided by @vanderHorst05 between 0.6 d and 1.1 d is much more difficult, given that in general the behavior of the early radio afterglow is likely to be very different from the optical one. Hereafter, we do not consider these radio limits.
We note a possible marginal reddening of the spectrum at the time of the bump (see bottom panel of the inset in Fig. \[fig:SED\]), albeit not statistically significant: the flux ratio between the bump and the pre-bump epochs does not vary significantly for different optical bands (see also GRB 000301C, Masetti et al. 2000). Due to the high $z$, the Lyman-$\alpha$ forest suppresses both $B$ and $V$ band fluxes. This accounts for the unusually-steep SED in the optical: by fitting all the four points with a power law, $F\propto \nu^{-\beta}$, the index is around $\beta=2.8\pm0.8$ with a poor $\chi^2$ ($\chi^2/{\textrm dof}=116/2$). However, if we assume a standard value of $\beta=0.8$ (see Sec. \[sec:disc\]), we find that the flux deficiency at high $\nu$ can be ascribed to the Lyman-$\alpha$ forest (see the top panel of the Inset in Fig. \[fig:SED\]).
Discussion {#sec:disc}
==========
The reality of the bump we find in the light curve at $t\sim0.02$ d is also supported by a rebrightening observed in the IR [@Blake05]: initially they observed a decay of 1.1 mag in the $J$ band between 47 min and 94 min (corresponding to a power–law decay index of $\alpha=1.5$, no error reported), followed by a rebrightening of $\Delta J\sim0.1$ between 94 min (0.065 d) and 121 min (0.084 d). In addition to our measurements, Fig. \[fig:LC\] also shows two unfiltered points by ROTSE–IIIb [@Yost05] and two other $R$ measures reported by @Mirabal05, which we converted to $r'$ assuming $0.3<R-I<0.6$ (no uncertainty was reported, so we assumed the systematic of 0.3 of the USNO–B1.0 magnitudes, as they calibrated with a USNO–B1.0 field star). In particular, the latter points seem to confirm the presence of the bump in $r'$, despite the large uncertainties. @Durig05 report unfiltered observations of the bump. Since the conversion of unfiltered to standard magnitudes requires some assumptions and implies large uncertainties, we are not as confident about the proper intercalibration of those converted magnitudes and our data as we are at earlier epochs, when the decay is simply monotonic. Therefore, lacking a comparison dataset of unfiltered data covering both the monotonic early decay and the bump, we have not included @Durig05 data in Fig. \[fig:LC\].
Following @Lazzati02, if we interpret the bump as due to density variations of the ISM, this is possible only if the observation occurred at a frequency $\nu=\nu_O$ (let $\nu_O$ be the frequency of our optical bands) below the cooling break $\nu_c$ and above the peak synchrotron frequency $\nu_m$: $\nu_m<\nu<\nu_c$. In the following we consider the two cases of uniform ISM and wind environment, respectively.
In the case of uniform ISM, the expected power–law index of the light curve is $\alpha=3(p-1)/4$, where $p$ is the electron energy distribution index [@Sari98]. From our measure of $\alpha=1.2\pm0.1$ we derive $p=2.6\pm0.1$. We also note that when $\nu_c$ crosses the optical band we should expect a steepening in the light curve of $\Delta\alpha=0.25$. Since we do not find evidence for this before $t<1$ d, the only possibility is that $\nu_O<\nu_c$ at least until $t\sim$1 d. The energy spectrum at frequency $\nu_m<\nu<\nu_c$ is a power law with index $\beta=(p-1)/2$, i.e. $\beta=0.8\pm0.05$. Figure \[fig:SED\] shows that this is consistent with our result. The cooling break $\nu_c$ must lie between the optical band $\nu_O$ and the X–ray $\nu_X$: $\nu_O<\nu_c<\nu_X$. The power–law index of the spectrum between $\nu_c$ and $\nu_X$ is expected to be $\beta_{cX}=p/2=1.3\pm0.05$. The X–ray power–law decay index, $\alpha_X$, is expected to be: $\alpha_X=3(p-1)/4$ ($\nu_c>\nu_X$), $\alpha_X=(3p-2)/4$ after $\nu_c$ has crossed the X–ray band ($\nu_c<\nu_X$), thus experiencing a steepening of $\Delta\alpha_X=0.25$. As this is expected to occur soon after the GRB, it is sensible to back-extrapolate the X–ray upper limit assuming for most of the time $\alpha_X=(3p-2)/4=1.45$. From Fig. \[fig:SED\], as long as we assume the validity of the X–ray upper limit back-extrapolated to $t=0.004$ d assuming $\alpha_X=1.45$ (solid arrow), we find that the shallowest power–law index allowed between optical and X–rays is $\beta_{OX}>0.7$ . Thus, this is consistent with a broken power law with power–law indices from $0.8$ to $1.3$. In summary, we conclude that the case of a uniform ISM is fully consistent with our observations.
In the case of wind environment and $p<2$ we must use the relation $\alpha=(p+8)/8$ by @Dai01 for $\nu_m<\nu<\nu_c$, which yields $p=1.6\pm0.8$. The case of $p>2$ is incompatible with the data: from the relation $\alpha=(3p-1)/4$ by @Chevalier99 we derive a value of $p=1.9\pm0.1$. From $\beta_{mc}=(p-1)/2$ and $\beta_{cX}=p/2$, holding for $\nu_m<\nu<\nu_c$ and for $\nu_c<\nu<\nu_X$, respectively, we derive: $\beta_{mc}=0.3\pm0.4$ and $\beta_{cX}=0.8\pm0.4$. Concerning the back-extrapolation of the X–ray upper limit, $\alpha_X$ is expected to be: $\alpha_X=(p+8)/8$ ($\nu_c>\nu_X$), $\alpha_X=(p+6)/8$ after $\nu_c$ has crossed the X–ray band ($\nu_c<\nu_X$), thus experiencing a steepening of $\Delta\alpha_X=0.25$. For the same reason as in the previous case, it is reasonable to assume $\alpha_X=(p+6)/8=0.95$ for most of the time. The consequent limit on the spectrum is $\beta_{OX}>1.1$ (dashed arrow in Fig. \[fig:SED\]). This is compatible only with $\beta_{cX}$. Furthermore, $\nu_c$ should be very close to the optical bands: this implies that during our observation $\nu_c$ should cross the optical bands, producing a slope change in the power–law decay of $\Delta\alpha=0.25$, which is not observed. If we assume that $\nu_c>\nu_X$ for most of the time between $t=0.004$ d and the epoch of the X–ray observation ($\sim1.33$ d), we derive the X–ray upper limit assuming $\alpha_X=(p+8)/8=1.2$, yielding $\beta_{OX}>0.9$, which is not consistent with $\beta_{OX}=\beta_{mc}=0.3\pm0.4$.
In contrast to GRBs 990123 and 021211, we find no evidence for a change in the temporal slope within the first few minutes of the onset of GRB 050502a, ruling out a transition from reverse to forward shock emission at this time. In GRB 050502a the bump rises at $\sim$6 min after the GRB in the rest-frame, to be compared with 0.5 min and 2.7 min of GRB 990123 and GRB 021211, respectively, when the above transition between reverse and forward shocks is supposed to occur. Should GRB 050502a have exhibited a similar transition, we should have detected it before the bump. We conclude that, despite the fact that a wind environment cannot be ruled out, the uniform ISM with clumps in density seems to better account for our observations.
The interpretation of the bump as the result of a refreshed shock catching up with the afterglow front shock seems more problematic, even if it cannot be ruled out. In fact, according to the original refreshed-shocks scenario [@Kumar00; @Granot03], we should expect that the duration $\Delta t$ of the bump is comparable with its start time: $\Delta t\approx t$. In the case of GRB 050502a our measures and those by @Mirabal05 show that, in spite of the uncertainty, $\Delta t\approx0.2$ d and $t\sim0.02$ d. Following @Kumar00, the impact between the two shells should produce a forward shock in the outer shell responsible for the bump and a reverse shock propagating in the inner shell. If $E_1$ and $E_2$ are the energy of the outer and inner shells, respectively, the increase in the emission due to the forward shock is expected to be $f=(1+E_2/E_1)^{(p+3)/4}$. From Fig. \[fig:LC\] we measure a flux increase of $10^{\Delta m/2.5}\sim1.6$ ($\Delta m\sim0.5$); from $p=2.6$ we obtain $E_2/E_1\sim0.4$. The spectrum at the bump is expected to have two peaks: the lower $\nu$ peak is due to the reverse shock in the inner shell and its frequency should be $\sim7\gamma_{0i}^2(E_2/E_1)^{1.1} \simeq 64(\gamma_{0i}/5)^2$ times lower than the peak frequency of the outer shell, i.e. $\nu_m$, which we know is below the optical bands at the time of the bump ($\gamma_{0i}$ is the Lorentz factor of the outer shell at the time of impact). The increase of emission at this frequency due to the inner shell is expected to be a factor $\sim8(\gamma_{0i}E_2/E_1)^{5/3} \simeq 25(\gamma_{0i}/5)^{5/3}$. Thus, the bump should have been more evident at low frequency: $\nu_m/64/(\gamma_{0i}/5)^2<\nu_O$, i.e. IR or radio. Unfortunately, the lack of early radio observations prevents this prediction from being tested. In the $J$-band @Blake05 report a rebrightening of $\sim0.1$ mag, which however seems smaller than that observed by us in the optical. Moreover, according to @Blake05 the $J$-band rebrightening occurs between 0.065 d and 0.084 d, i.e. later than 0.02 d of the optical bands.
In conclusion, although the refreshed-shock scenario cannot be completely ruled out due to the lack of early radio observations, our observations appear to be more difficult to reconcile with its predictions than with those of the variable density environment.
CG and AG acknowledge their Marie Curie Fellowships from the European Commission. CGM acknowledges financial support from the Royal Society. AM acknowledges financial support from PPARC. MFB is supported by a PPARC Senior Fellowship. The Liverpool Telescope is operated on the island of La Palma by Liverpool John Moores University at the Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. The Faulkes Telescope North is operated with support from the Dill Faulkes Educational Trust.
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[llrrrl]{} LT & SDSS-R & 3.1 & 10 & $15.67\pm0.03$ & detection mode\
LT & SDSS-R & 3.5 & 10 & $15.80\pm0.03$ & detection mode\
LT & SDSS-R & 3.8 & 10 & $15.96\pm0.03$ & detection mode\
LT & Bessell-B & 5.4 & 30 & $18.25\pm0.08$ & multi-color sequence\
LT & Bessell-V & 6.7 & 30 & $17.35\pm0.04$ & multi-color sequence\
LT & SDSS-R & 8.1 & 30 & $17.00\pm0.03$ & multi-color sequence\
LT & SDSS-I & 9.5 & 30 & $16.94\pm0.04$ & multi-color sequence\
LT & Bessell-B & 10.8 & 60 & $19.10\pm0.08$ & multi-color sequence\
LT & Bessell-V & 12.6 & 60 & $18.01\pm0.04$ & multi-color sequence\
LT & SDSS-R & 14.3 & 60 & $17.69\pm0.04$ & multi-color sequence\
LT & SDSS-I & 16.1 & 60 & $17.64\pm0.04$ & multi-color sequence\
LT & Bessell-B & 17.8 & 120 & $19.81\pm0.09$ & multi-color sequence\
LT & Bessell-V & 20.6 & 120 & $18.70\pm0.05$ & multi-color sequence\
LT & SDSS-R & 23.4 & 120 & $18.29\pm0.04$ & multi-color sequence\
LT & SDSS-I & 26.2 & 120 & $18.21\pm0.10$ & multi-color sequence\
LT & Bessell-B & 29.1 & 180 & $20.12\pm0.10$ & multi-color sequence\
LT & Bessell-V & 32.9 & 180 & $19.05\pm0.08$ & multi-color sequence\
LT & SDSS-R & 36.6 & 180 & $18.69\pm0.06$ & multi-color sequence\
LT & SDSS-I & 40.4 & 180 & $18.51\pm0.10$ & multi-color sequence\
LT & Bessell-B & 44.2 & 120 & $20.72\pm0.18$ & multi-color sequence\
LT & Bessell-V & 47.0 & 120 & $19.48\pm0.09$ & multi-color sequence\
LT & SDSS-R & 49.8 & 120 & $18.84\pm0.07$ & multi-color sequence\
LT & SDSS-I & 52.6 & 120 & $18.50\pm0.11$ & multi-color sequence\
LT & Bessell-B & 55.3 & 180 & $21.00\pm0.20$ & multi-color sequence\
LT & Bessell-V & 59.1 & 180 & $19.70\pm0.12$ & multi-color sequence\
FTN& Bessell-R & 348 & 4x200& $21.6\pm0.2$ & late follow-up\
FTN& Bessell-V & 370 & 6x200& $22.6\pm0.2$ & late follow-up\
FTN& Bessell-R & 620 & 4x200& $22.4\pm0.5$ & late follow-up\
FTN& SDSS-I & 690 & 4x200& $22.3\pm0.4$ & late follow-up\
LT & SDSS-R & 1340 & 24x150& $24.0\pm0.6$ & late follow-up\
.
[^1]: Funded by UK PPARC through a consortium of 10 UK universities.
[^2]: http://cas.sdss.org/astro/en/tools/chart/navi.asp
[^3]: http://www.ast.cam.ac.uk/$\sim$dwe/SRF/camc\_extinction.html
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---
abstract: 'We study theoretically the behavior of a class of hydrodynamic dipoles. This study is motivated by recent experiments on synthetic and biological swimmers in microfluidic *Hele-Shaw* type geometries. Under such confinement, a swimmer’s hydrodynamic signature is that of a potential source dipole, and the long-range interactions among swimmers are obtained from the superposition of dipole singularities. Here, we recall the equations governing the positions and orientations of interacting asymmetric swimmers in doubly-periodic domains, and focus on the dynamics of swimmer pairs. We obtain two families of ‘relative equilibria’-type solutions that correspond to pursuit and synchronization of the two swimmers, respectively. Interestingly, the pursuit mode is stable for large tail swimmers whereas the synchronization mode is stable for large head swimmers. These results have profound implications on the collective behavior reported in several recent studies on populations of confined microswimmers.'
author:
- |
Eva Kanso and Alan Cheng Hou Tsang\
[Aerospace and Mechanical Engineering, University of Southern California]{}\
[854 Downey Way, Los Angeles, CA 90089-1191]{}\
bibliography:
- 'reference\_arXiv.bib'
title: '**Pursuit and Synchronization in Hydrodynamic Dipoles**'
---
Introduction
============
Active systems, i.e., systems driven internally by self-propelled individual units, often exhibit rich collective behavior at the system’s scale; a scale that is typically several orders of magnitude larger than the scale of the individual unit. Such collective behavior naturally arises in disparate biological systems, from schools of fish [@couzin:jtb2002a] to suspensions of motile bacteria [@cisneros:ef2007a] and assemblages of sub-cellular extracts [@sanchez:n2012a]. It also emerges in inanimate systems such as driven and self-propelled droplets and reactive colloids [@schaller:n2010a; @bricard:n2013a], and provide an attractive paradigm for reconfigurable smart materials [@Richtering2006a] and biomedical devices [@Jackson2013a].
The question of how these highly-coordinated collective motions arise from piecewise interactions among individual units has been the subject of intense research in the past few years. A well-studied example is the behavior of self-propelled particles in a viscous fluid, [@saintillan:prl2008a]. Most of this work has focused on the instabilities and spatiotemporal fluctuations in three-dimensional (3D) systems. However, motivated by recent technological advances in producing and manipulating large ensembles of particles in microfluidic devices [@bricard:n2013a; @beatus:pr2012a; @desreumaux:prl2013a], attention began to shift to the collective dynamics of particles confined in quasi two-dimensional (2D) geometries. Geometric confinement changes drastically the nature of the hydrodynamic interactions among particles, [@beatus:pr2012a]. The long-ranged hydrodynamic interactions in 3D are driven by the force dipoles exerted by self-propelled particles on the fluid medium, [@saintillan:prl2008a]. In quasi-2D geometries, the solid walls screen the force dipole contribution, making it subdominant in comparison with the potential dipole arising from incompressibility, [@desreumaux:prl2013a]. As a result, the long-range interactions among swimmers can be obtained from the superposition of dipole singularities.
In this paper, we revisit the hydrodynamic dipole model proposed by Brotto *et al.* [@brotto:prl2013a] for asymmetric dumbbell swimmers in confined Hele-Shaw type geometries. The head-tail asymmetry causes a given swimmer to reorient, not only in response to the flow gradient as anticipated by Jeffery’s equation, but also in response to the flow velocity itself. This result is rooted in the fact that the lubrication forces between the swimmer and the solid walls hinder its advection by the fluid, inducing unequal translational motility coefficients at the swimmer’s head and tail. In [@brotto:prl2013a], Brotto *et al.* derived a kinetic theory-type model for a population of interacting swimmers and predicted a novel long-wave linear instability that leads to the emergence of large-scale directed motion and polarization in isotropic populations of confined large head swimmers. Lefauve and Saintillan [@lefauve:pre2014a] and Tsang and Kanso [@tsang:pre2014a] used numerical simulations to explore the implications of these instabilities on the collective behavior in finite-sized populations of interacting swimmers.
The present paper examines the detailed dynamics of a pair of asymmetric (head-tail) swimmers in doubly-periodic domains, where the orientation dynamics is dominated by the flow field itself, thus neglecting reorientation in response to the flow gradient as done in [@brotto:prl2013a; @lefauve:pre2014a]. We particularly focus on a special class of solutions where the two swimmers move with constant speed and at constant orientation. We find two families of these “equilibrium-like" solutions: (1) both swimmers swim side by side in a synchronized way; and (2) one swimmer tailgates the other. We analyze their stabilities and find that they depend on the details of the head-tail asymmetry. We conclude this work by discussing the significance of these results to the behavior of populations of swimmers.
Note that a dynamical theory of dipole interactions has also been pursued in two additional contexts. One motivation stems from the desire to obtain low-order representations of two-dimensional, inviscid and incompressible fluids in terms of interacting particles such as point vortices and point dipoles, see, e.g., [@kulik:tmp2010a; @smith:pnp2011a; @smith:rcd2013a; @newton:dcds2005a; @yanovsky:pl2009a]. A shortcoming of these models is that the dipole’s self-propelled speed is ill-defined; thus, a dipole, unless properly desingularized, induces infinite velocity through its center. Another motivation for dipole models that is closer to the focus of this paper grew out of efforts to examine the role of hydrodynamic coupling in fish schooling. It is a well-known result in fluid dynamics that the leading order flow of a self-propelled body is . Kanso and co-workers proposed a finite dipole dynamical system that captures the far-field hydrodynamic interactions [@tchieu:prsa2012a; @tsang:jnls2013a]. Each dipole consists of a pair of equal and opposite strength point vortices separated by a finite constant distance. By construction, the self-propelled speed is well-defined. as opposed to reorienting in response to the flow gradient along the dipole’s direction predicted by Jeffery’s equation for slender bodies in viscous fluids. Kanso and Tsang [@kanso:fdr2014a] presented a unified framework for deriving two point dipole models: a dipole consistent with the finite dipole model, appropriate for bluff bodies (fish) in potential flows, and another consistent with Jeffery’s equation for slender bodies and equivalent to the microswimmer model employed in [@desreumaux:epje2012a; @brotto:prl2013a]. They further showed that, in unbounded domains, dipole pairs can synchronize their motions for a range of initial conditions; however, the details of the synchronized motion differ between the two models.
Problem Formulation {#sec:model}
===================
#### Microswimmer model.
Consider a microswimmer composed of two connected disks of radii $R_{tail}$ and $R_{head}$ located at $z_{tail}$ and $z_{head}$ respectively, where $z = x+ {\mathrm{i}}y$ is the complex coordinate (${\mathrm{i}}= \sqrt{-1}$). Assume the two disks are connected by a frictionless rod of length $\ell$. The equations of motion for the swimmer’s tail ($z_{tail}$) and head ($z_{head}$) can be written in complex notation as (see [@brotto:prl2013a]) $$\begin{split}
\label{eq:eomtail}
\dot{\bar{z}}_{tail} & = U_o{\mathrm{e}^{ -{\mathrm{i}}\alpha_o }}+\mu_{tail}\bar{w}(z_{tail})+\lambda_{tail} {\mathrm{e}^{ -{\mathrm{i}}\alpha_o }}, \\[2ex]
\dot{\bar{z}}_{head} & = U_o{\mathrm{e}^{ -{\mathrm{i}}\alpha_o }}+\mu_{head}\bar{w}(z_{head})-\lambda_{head} {\mathrm{e}^{ -{\mathrm{i}}\alpha_o }}.
\end{split}$$ Here, $U_o$ is the swimmer’s self-propelled velocity, $\alpha_o$ its orientation angle, and $w(z)$ is the velocity field of the ambient fluid. The bar notation denotes the complex conjugate, $\bar{z} = x - i y$. The coefficients $\mu_{tail}$, $\mu_{head}$ are the translational mobility coefficients whereas $\lambda_{tail}$, $\lambda_{head}$ are unknown Lagrange multipliers that enforce the constraint $| z_{head} - z_{tail}| = \ell$. In particular, the translational mobility coefficients $\mu_{tail}$ and $\mu_{head}$ arise from the balance of hydrodynamic drag and wall friction acting on the tail and head, and are decreasing functions of $R_{tail}$ and $R_{head}$ respectively, with values less than 1, see, e.g. [@desreumaux:epje2012a; @brotto:prl2013a].
We define the hydrodynamic center of the swimmer to be $z_o = (\lambda_{tail}z_{head} + \lambda_{head}z_{tail})/(\lambda_{tail}+\lambda_{head})$. Our goal is to rewrite the equations of motion in terms of the swimmer’s hydrodynamic center $z_o$ and orientation $\alpha_o$. Let $\ell \gg R_{tail}, R_{head}$, and use Taylor series expansion to expand the flow velocity at the tail and head $$\begin{split}
\label{eq:velocitytail}
{\bar{w}}(z_{tail}) & = \bar{w}(z_o)+ {\mathrm{e}^{ {\mathrm{i}}\alpha_o }}\frac{\lambda_{tail}}{\lambda_{tail}+\lambda_{head}} \left. \frac{d\bar{w}}{dz} \right|_{z_o}+\ldots \\
{\bar{w}}(z_{head}) & = \bar{w}(z_o)+{\mathrm{e}^{ {\mathrm{i}}\alpha_o }}\frac{\lambda_{head}}{\lambda_{tail}+\lambda_{head}} \left. \frac{d\bar{w} }{dz}\right|_{z_o}+\ldots .
\end{split}$$ Substitute into to get that the equation governing the translational motion of the swimmer’s center $$\label{eq:eomcenter}
\begin{split}
\dot{\bar{z}}_o= U_o{\mathrm{e}^{ - {\mathrm{i}}\alpha_o }}+\mu \bar{w}(z_o) ,
\end{split}$$ where $\mu = {(\lambda_{head}\mu_{tail}+\lambda_{tail}\mu_{head})}/{(\lambda_{head}+\lambda_{tail})}$. To obtain the equation governing the rotational motion of the swimmer, note that, by definition, $\ell{\mathrm{e}^{ {\mathrm{i}}\alpha_o }} = z_{head} - z_{tail}$, which gives, upon differentiating both sides with respect to time and further simplifications, $$\label{eq:eomori}
\dot{\alpha}_o = {\mathrm{Re} \left[ \frac{(\dot{\bar{z}}_{head}- \dot{\bar{z}}_{tail}) {\mathrm{i}}{\mathrm{e}^{ {\mathrm{i}}\alpha_o }}}{\ell} \right]}.$$ Here, Re denotes the real part of the expression in bracket. Now substitute and into to get $$\label{eq:eomorientation}
\dot{\alpha}_o = {\mathrm{Re} \left[ \nu_1 \frac{d {\overline{w}}}{dz} {\mathrm{i}}{\mathrm{e}^{ 2 {\mathrm{i}}\alpha_o }}+\nu_2 {\overline{w}} {\mathrm{i}}{\mathrm{e}^{ {\mathrm{i}}\alpha_o }} \right]}.$$ where $\dfrac{d {\overline{w}}}{dz} $ and $ {\overline{w}}$ are evaluated at $z_o$ and the constant parameters $\nu_1$ and $\nu_2$ are given by $$\label{eq:coeff}
\begin{split}
\nu_1 = \frac{(\lambda_{head}\mu_{head}+\lambda_{tail}\mu_{tail})}{\lambda_{head}+\lambda_{tail}}, \qquad
\nu_2 = (\mu_{head}-\mu_{tail})/\ell.
\end{split}$$ The sign of $\nu_2$ dictates how the swimmer orients in local flow: it aligns to the local flow when $\nu_2>0$, that is, for large tail swimmers for which $\mu_{head}-\mu_{tail}>0$ (because $\mu_{head/tail}$ is a decreasing function of $R_{head/tail}$, [@beatus:pr2012a]), and opposite to the local flow when $\nu_2 <0$ , that is, for large head swimmers for which $\mu_{head}-\mu_{tail}<0$.
#### Hydrodynamic interactions of multiple microswimmers.
Consider the interaction of multiple microswimmers in an unbounded fluid domain. By virtue of and , the dynamics of $N$ swimmers, all having the same self-propelled velocity $U$, can be expressed in concise complex notation $$\label{eq:eom}
\begin{split}
\dot{{\overline{z}}}_n & = U {\mathrm{e}^{ -{\mathrm{i}}\alpha_n }}+ \mu {\overline{w}}(z_n), \\[1ex]
\dot{\alpha}_n & = {\mathrm{Re} \left[ \nu_1 \frac{d {\overline{w}}}{dz} {\mathrm{i}}{\mathrm{e}^{ 2 {\mathrm{i}}\alpha_n }}+\nu_2 {\overline{w}} {\mathrm{i}}{\mathrm{e}^{ {\mathrm{i}}\alpha_n }} \right]}.
\end{split}$$ Here, $z_n$ and $\alpha_n$ denote the position and orientation of each swimmer ($n=1,\ldots, N$). To close the model, one needs to obtain an expression for the fluid velocity field $w(z)$. Recalling that each swimmer induces a far-field velocity which is that of a potential source dipole [@beatus:pr2012a], the far-field flow of a microswimmer $j$ located at $z_j = x_j + {\mathrm{i}}y_j$ and oriented at an arbitrary angle $\alpha_j$ can be described by the complex velocity ${\overline{w}}(z)=u_x-i u_y=\sigma {\mathrm{e}^{ {\mathrm{i}}\alpha_j }}/(z-z_j)^2$, where $\sigma$ is the dipole strength. Note that $\sigma=R^2 U$, where $R$ is the effective radius of the swimmer. A microswimmer $n$ responds to the flow induced by all microswimmers in the fluid domain, namely, $$\label{eq:velocityDipole}
{\overline{w}}(z_n)= \sum_{\stackrel{j\neq n}{j=1}}^N \sigma \dfrac{{\mathrm{e}^{ {\mathrm{i}}\alpha_j }}}{(z_n-z_j)^2}.$$
#### Microswimmers in doubly-periodic domains.
When the swimmers are placed in a doubly-periodic domain, one needs to take into account, not only the velocity field induced by the swimmers themselves but also the effect of their image system. A given swimmer $n$ has a doubly-infinite set of images. Thus, evaluating $w(z)$ requires the evaluation of conditionally-convergent, doubly-infinite sums of terms that decay as $1/|z|^2$. These sums are evaluated using an approximate numerical approach in [@lefauve:pre2014a]. In [@tsang:pre2014a], we offered a closed-form analytic expression for these infinite sums in terms of the Weierstrass elliptic function, namely, $$\label{eq:velocityDipolePeriodic}
{\overline{w}}(z) = \sum_{n=1}^N\sigma \rho( z-z_n;\omega_1,\omega_2){\mathrm{e}^{ {\mathrm{i}}\alpha_n }}.$$ The Weierstrass elliptic function $\rho(z)$ is given by $\rho\left(z;\omega_{1},\omega_{2}\right)=\frac{1}{z^2}+\sum_{k,l} \left(\frac{1}{(z-\Omega_{kl})^2} -\frac{1}{\Omega_{kl}^{2}}\right)$, with $\Omega_{kl}=2k\omega_{1}+2l\omega_{2}$, $k,l\in \mathbb{Z} \!-\!\{0\}$, and $\omega_{1}$ and $\omega_{2}$ being the half-periods of the doubly-periodic domain. This function has infinite numbers of double pole singularities located at $z=0$ and $z=\Omega_{kl}$, corresponding to the $1/|z|^2$ singularities induced by the potential dipoles. Equations and form a closed system for $N$ swimmers in a doubly-periodic domain.
We conclude by writing the system of equations and in dimensionless form using the swimmers radius $R$ as a length scale and $R/U$ as a time scale. That is, we introduce the dimensionless spatial variable $\tilde{z}=z/R$ and time variable $\tilde{t}=t U/R$. We then drop the tilde notation assuming all variables are non-dimensional. Equations and have the same form but the parameters $U$ and $\sigma$ are now normalized to one, that is, $U=1$ and $\sigma=1$. The parameter values $\mu$, $\nu_1$ and $\nu_2$ are also non-dimensional.
Pursuit and synchronization {#sec:results}
===========================
We consider two microswimmers in a doubly-periodic domain, and focus on their dynamic response when $\nu_1=0$, that is, when their alignment with the flow gradient is negligible. In this case, the orientation dynamics is dominated by alignment with the flow due to head-tail hydrodynamic asymmetry.
![ (a) Two modes of solutions are obtained from : a pursuit mode (blue) where the swimmers trail one another and a synchronization mode (red) where they swim side by side. The shown solid curves correspond to $\omega_1=-{\mathrm{i}}\omega_2=5$ and the dashed lines to $\omega_1=-{\mathrm{i}}\omega_2=10$. The separation distance $c$ is set to $c=4$. (b) Summary of the stability analysis for these two modes.[]{data-label="fig:twodipolesratio"}](ratio_2.eps){width="\textwidth"}
#### Periodic solutions and relative equilibria.
We look for special solutions where the two swimmers move at the same velocity and orientation for all time. That is, we look for solutions where $\dot{z}_1 = \dot{z}_2=$ constant and $\dot{\alpha}_1 = \dot{\alpha}_2 = 0$. To obtain the initial conditions that lead to this behavior, it is convenient to rewrite the equations of motion (\[eq:eom\],\[eq:velocityDipolePeriodic\]) in terms of the reduced coordinate $z_1-z_2$ which we set to $z_1-z_2= \beta = c {\mathrm{e}^{ {\mathrm{i}}\theta }}$ (see inset of Figure \[fig:twodipolesratio\](a)). To this end, one gets $$\begin{split}
\label{eq:eomtwodipoles}
\dot{\bar{\beta}} & = {\mathrm{e}^{ - {\mathrm{i}}\alpha_1 }} - {\mathrm{e}^{ - {\mathrm{i}}\alpha_2 }}+\mu \rho(\beta) ( {\mathrm{e}^{ {\mathrm{i}}\alpha_2 }} - {\mathrm{e}^{ {\mathrm{i}}\alpha_1 }}), \\[2ex]
\dot{\alpha}_1 & = \dot{\alpha}_2= \nu_2 \text{Re}[{\mathrm{i}}{\mathrm{e}^{ {\mathrm{i}}(\alpha_1+\alpha_2) }} \rho(\beta)].
\end{split}$$ The translation equation for $\dot{\beta}$ is identically zero when $\alpha_1 =\alpha_2=\alpha$. Whereas to guarantee $\dot{\alpha}_1 = \dot{\alpha}_2 = 0$, one must satisfy the condition $$\begin{split}
\label{eq:condition}
\left\{ \begin{array}{l}
{\text{Re}[\rho(c {\mathrm{e}^{ - {\mathrm{i}}\theta }})]}=0 \textrm{ \ therefore \ } \alpha = \dfrac{\pi}{4}, \dfrac{3\pi}{4} \textrm{\ \ and \ } (\alpha,\theta) = \left\{ \begin{array}{l} (\alpha,\alpha) \\
(\alpha,\alpha+ \pi/2) \end{array} \right. \\
\textrm{or} \\[2ex]
\dfrac{\text{Im}[\rho(c {\mathrm{e}^{ - {\mathrm{i}}\theta }})]}{\text{Re}[\rho(c {\mathrm{e}^{ - {\mathrm{i}}\theta }})]} = -\tan{2 \alpha}, \qquad {\text{Re}[\rho(c {\mathrm{e}^{ - {\mathrm{i}}\theta }})]} \neq 0,
\end{array} \right.
\end{split}$$ . Namely, the five solutions given by $\alpha = 0, \dfrac{\pi}{4}, \dfrac{\pi}{2}, \dfrac{3\pi}{4}, \pi$ and $\theta = \alpha$ correspond to the two dipoles moving parallel to each other in a “pursuit" mode (blue arrows), whereas the five solutions given by $\alpha = 0, \dfrac{\pi}{4}, \dfrac{\pi}{2}, \dfrac{3\pi}{4}, \pi$ and $\theta = \alpha + \pi/2$ correspond to the two dipoles moving in tandem in a “synchronized" mode (red arrows).
![Aperiodic behavior of two dipoles in doubly-periodic domain. The parameter values are $\alpha=\pi/3$, $z_1(0)=-2\exp(i\theta)$, $z_2(0)=2\exp(i\theta)$, while $\theta$ is obtained by numerically solving the first condition in . The positions of the dipoles are marked by ‘$\times$’ at $t=0$ and by ‘$o$’ at the end of the integration. As time progresses, the two trajectories densely fill the whole domain.[]{data-label="fig:aperiodic"}](aperiodic.eps){width="75.00000%"}
The second set of solutions, that is, the values of $(\alpha, \theta)$ for which ${\text{Im}[\rho(c {\mathrm{e}^{ - {\mathrm{i}}\theta }})]}/{\text{Re}[\rho(c {\mathrm{e}^{ - {\mathrm{i}}\theta }})]} = -\tan{2 \alpha}$ and ${\text{Re}[\rho(c {\mathrm{e}^{ - {\mathrm{i}}\theta }})]} \neq 0$, are not analytically available and need to be computed numerically. Figure \[fig:twodipolesratio\](a) shows the values of $(\alpha,\theta)$ that satisfy these conditions – clearly, two branches of solutions are obtained. These solutions depend implicitly on the domain size $(\omega_1,\omega_2)$ and on $c$, the separation distance between the two swimmers. In other words, for a choice of domain size and separation distance $c$, $(\alpha,\theta)$ are computed accordingly such that the two dipoles move at the same constant velocity and orientation for all time. The two branches shown in Figure \[fig:twodipolesratio\](a) correspond to two modes of behavior: a pursuit mode where one swimmer trails the other, and a synchronization mode where the two swimmers move side by side. These solutions, while they correspond to the dipoles moving at constant velocity and orientation, can exhibit two distinct types of dynamical behavior due to the doubly-periodic nature of the domain, namely, they could lead to aperiodic and periodic motion of the dipoles. Aperiodic motion refers to the case where the paths of the dipoles densely fill the whole domain, . This seems to be the generic behavior for arbitrary initial conditions. Periodic behavior refers to trajectories that satisfy the condition $$\label{eq:onedipoleperiodic}
\bar{z}_1(T)= \bar{z}_1(0) + 2 p \omega_1 + 2 q \omega_2, \qquad
\bar{z}_2(T)= \bar{z}_2(0) + 2 p \omega_1 + 2 q \omega_2$$ where $p$ and $q$ are integers and $T$ is the period of the motion. This amount to the additional condition $$\label{eq:onedipoleperiodic2}
\alpha_1(0)= \alpha_2(0) = \tan^{-1}(\frac{q}{p}).$$ The ratio of $q/p$ indicates the ratio of the number of times the dipole crosses the $y$ and $x$ axes in one period $T$. Figure \[fig:twodipolesperiodic\] depicts the periodic behavior of two dipoles in pursuit and synchronization modes for $q/p = 3$.
![ (a) Pursuit and (b) synchronization in two dipoles undergoing periodic motion. The parameter values are $\alpha=\tan^{-1}(3)$, $z_1(0)=-2\exp(i\theta)$, $z_2(0)=2\exp(i\theta)$, while $\theta$ is obtained by numerically solving the first condition in . The positions of the dipoles are marked by ‘$\times$’ at $t=0$ and by ‘$o$’ at the end of the integration time $t = 80$.[]{data-label="fig:twodipolesperiodic"}](twodipolesperiodic.eps){width="75.00000%"}
![Pursuit and synchronization modes as attracting modes. Two dipoles hone in on quasi periodic trajectories where: (a) one dipole is in pursuit of the other for $\nu_2=0.5$. (b) the two dipoles synchronize and move along parallel trajectories for $\nu_2=-0.5$. The initial conditions are $z_1(0)=\alpha_1(0)=\alpha_2(0)=0$ while $z_2(0)=1.5+1{\mathrm{i}}$ in (a) and $z_2(0)=2+1{\mathrm{i}}$ in (b).[]{data-label="fig:attractor"}](twodipolesperiodic_perturb.eps){width="75.00000%"}
#### Stability analysis.
We analyze the linear stability of the pursuit and synchronization modes by considering small perturbations $\delta \beta = \delta \beta_x + {\mathrm{i}}\delta \beta_y$, $\delta \alpha_1$ and $\delta \alpha_2$ about $\beta = c {\mathrm{e}^{ {\mathrm{i}}\theta }}$ ($\beta_x = c \cos\theta$, and $\beta_y= c \sin\theta$) and $\alpha_1 = \alpha_2 = \alpha$, with $(\alpha,\theta)$ satisfying . We linearize equations accordingly. The linearized equations can be written in matrix form as follows: $$\label{eq:perturbtwodipoles}
\dfrac{d}{dt}\left(\begin{array}{c} \delta \beta_x \\ \delta \beta_y \\ \delta \alpha_1 \\ \delta \alpha_2 \end{array}\right) = M \left(\begin{array}{c} \delta \beta_x \\ \delta \beta_y \\ \delta \alpha_1 \\ \delta \alpha_2 \end{array} \right),$$ where the Jacobian matrix $M$ is given by $$\label{eq:jacobian}
M = \left(\begin{array}{cccc} 0 & 0 & -\sin\alpha-\mu\text{Re}[{\mathrm{i}}{\mathrm{e}^{ {\mathrm{i}}\alpha }} \rho(\beta)] & \sin\alpha+\mu\text{Re}[{\mathrm{i}}{\mathrm{e}^{ {\mathrm{i}}\alpha }} \rho(\beta)] \\
0 & 0 & -\cos\alpha-\mu\text{Im}[{\mathrm{i}}{\mathrm{e}^{ {\mathrm{i}}\alpha }} \rho(\beta)] & \cos\alpha+\mu\text{Im}[{\mathrm{i}}{\mathrm{e}^{ {\mathrm{i}}\alpha }} \rho(\beta)]\\
\nu_2 \text{Re}[{\mathrm{i}}{\mathrm{e}^{ {\mathrm{i}}2 \alpha }} \rho^{\prime}(\beta)] & -\nu_2 \text{Re}[{\mathrm{e}^{ {\mathrm{i}}2 \alpha }} \rho^{\prime}(\beta)]
& -\nu_2 \text{Re}[{\mathrm{e}^{ {\mathrm{i}}2 \alpha }} \rho(\beta)] & -\nu_2 \text{Re}[ {\mathrm{e}^{ {\mathrm{i}}2 \alpha }} \rho(\beta)] \\
\nu_2 \text{Re}[{\mathrm{i}}{\mathrm{e}^{ {\mathrm{i}}2 \alpha }} \rho^{\prime}(\beta)] & -\nu_2 \text{Re}[{\mathrm{e}^{ {\mathrm{i}}2 \alpha }} \rho^{\prime}(\beta)]
& -\nu_2 \text{Re}[{\mathrm{e}^{ {\mathrm{i}}2 \alpha }} \rho(\beta)] & -\nu_2 \text{Re}[ {\mathrm{e}^{ {\mathrm{i}}2 \alpha }} \rho(\beta)]\end{array} \right).$$ We compute the eigenvalues numerically and find that, for large tail swimmers $\nu_2>0$, the pursuit mode is stable, whereas for large head swimmers $\nu_2<0$, the synchronization mode is stable. Our findings are summarized in Figure \[fig:twodipolesratio\](b).
We test our results numerically by integrating the nonlinear equations (\[eq:eom\], \[eq:velocityDipolePeriodic\]) for arbitrary choices of initial conditions. Interestingly, the pursuit and synchronization modes seem to be globally attracting modes in the case of large tail and large head swimmers, respectively. Figure \[fig:attractor\](a) shows a depiction of two large-tail swimmers honing in on quasi-periodic pursuit trajectories, while (b) depicts two large-head swimmers synchronizing their motion in finite time to swim side by side.
#### The limit of unbounded domain.
We conclude this section by noting that in the limit of infinite domain, the solutions of the doubly-periodic system (\[eq:eom\], \[eq:velocityDipolePeriodic\]) converge to the relative equilibria of the unbounded system (\[eq:eom\], \[eq:velocityDipole\]). In the unbounded system, the relative equilibria can be obtained either by symmetry arguments or by analytical manipulation of the equations of motion. Namely, one has two families of relative equilibria $\theta=\alpha$ and $\theta=\alpha+\pi/2$ which correspond to pursuit and synchronization trajectories, respectively. The convergence of the solutions in to these solutions is relatively fast, as indicated in Figure \[fig:twodipolesratio\](a). As $\omega_1$, $\omega_2 \rightarrow \infty$, the Jacobian matrix $M$ converges to $$\label{eq:jacobian2}
M_{\infty} = \left(\begin{array}{cccc} 0 & 0 & -\sin\alpha+\dfrac{\mu}{c^2} \sin(\alpha-2 \theta) & \sin\alpha-\dfrac{\mu}{c^2} \sin(\alpha-2 \theta)\\[1ex]
0 & 0 & -\cos\alpha- \dfrac{\mu}{c^2} \cos(\alpha-2 \theta) & \cos\alpha+ \dfrac{\mu}{c^2} \cos(\alpha-2 \theta)\\[1ex]
\dfrac{2\nu_2}{c^3} \sin(2\alpha-3 \theta)& \dfrac{2 \nu_2}{c^3}\cos(2\alpha-3 \theta)
& - \dfrac{\nu_2}{c^2} \cos(2\alpha-2 \theta) & - \dfrac{\nu_2}{c^2} \cos(2\alpha-2 \theta) \\[1ex]
\dfrac{2\nu_2}{c^3} \sin(2\alpha-3 \theta) & \dfrac{2\nu_2}{c^3} \cos(2\alpha-3 \theta)
& - \dfrac{\nu_2}{c^2} \cos(2\alpha-2 \theta) & - \dfrac{\nu_2}{c^2} \cos(2\alpha-2 \theta) \end{array} \right).$$ The corresponding eigenvalues are $[0$, $0$, $0$, $\mp 2 \nu_2 /c^2]$. The eigenvalue $-2\nu_2/c^2$ corresponds to the pursuit mode where $\alpha = \theta$, whereas $+2\nu_2/c^2$ corresponds to the synchronization mode. This means that, for large tail swimmers with $\nu_2>0$, the pursuit mode is linearly stable and the synchronization mode is unstable, whereas for large head swimmers when $\nu_2>0$, the opposite is true, thus confirming the results obtained above for finite-sized doubly-periodic domains.
Discussion {#sec:discussion}
==========
![Emergent collective behavior in large tail swimmers (top row) and large head swimmers (bottom row) starting from a uniform isotropic distribution. Large tail swimmers tend to tailgate each other, thus forming active lanes, while large head swimmers tend to form stationary clusters. Parameter values are $\mu = 0.9$, $\nu_2 = 1$ (top row) and $\nu_2=-0.5$ (bottom row) in a total population of $400$ swimmers.[]{data-label="fig:populations"}](largetail.eps){width="95.00000%"}
We revisited the hydrodynamic dipole model governing the interaction of asymmetric microswimmers in Hele-Shaw confinement, [@brotto:prl2013a]. Following [@tsang:jnls2013a; @tsang:pre2014a], we obtained a closed-form expression for the velocity field induced by the swimmers and their image system in doubly-periodic domains. We treated in details the dynamics of two interacting swimmers, and found two special solutions that correspond to pursuit and synchronization of the two dipoles. The pursuit mode is stable and attracting for large tail swimmers while the synchronization mode is stable and attracting for large head swimmers. By attracting, we mean that, starting from arbitrary initial conditions, large tail swimmers tend to tailgate each other while large head swimmers tend to synchronize their motion in finite time to swim side by side. These results are particularly interesting in light of the collective behavior reported in [@lefauve:pre2014a; @tsang:pre2014a] on populations of such swimmers. In these works, large tail swimmers were observed to “develop active lanes" [@lefauve:pre2014a] and “tail-gate each other" [@tsang:pre2014a], as shown in Figure \[fig:populations\](top row), which suggests that the pursuit mode remains stable as the system size increases.
Populations of large head swimmers were shown to form heavily polarized sharp density waves in [@lefauve:pre2014a], consistent with predictions based on linear stability analysis of a kinetic-type continuum model [@brotto:prl2013a]. One could conjecture that the synchronization mode observed here in pairs of large head swimmers may be responsible for the global polarization observed in [@lefauve:pre2014a]. However, this thinking is too simplistic. The emergence of global polarization patterns in finite size populations is not intuitive given the nature of dipolar interactions among the swimmers. Further, these polarized density waves were not observed in the detailed parametric study reported in [@tsang:pre2014a Figure 7]. Instead, [@tsang:pre2014a] reported, in agreement with unpublished results by Levaufe and Saintillan, that large head swimmers tend to form stationary clusters (see Figure \[fig:populations\](bottom row)), which are not predicted by the linear stability analysis of [@brotto:prl2013a]. All this is to say that the global patterns of the finite size systems in [@lefauve:pre2014a] and [@tsang:pre2014a] are in agreement, except for the global polarization pattern. This inconsistency may be due to differences in the system size – thousands of particles in [@lefauve:pre2014a] versus hundreds in Figure \[fig:populations\] and in [@tsang:pre2014a] – or to differences in the details of the numerical implementation. In [@lefauve:pre2014a], the point dipole model is desingularized and hydrodynamic interactions are approximated for fast computations, whereas [@tsang:pre2014a] use a local repulsion potential for collision avoidance and accurately account for hydrodynamic interactions and the doubly-infinite image system. Irrespective of the reason, the results reported in this study suggest that the global polarization mode in large head swimmers is not “robust" to system perturbances, whereas the pursuit mode in large tail swimmers is.
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abstract: |
In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with at least five invariant lines, including the line at infinity and including multiplicities. For each orbit we exhibit its configuration. We characterize in terms of algebraic invariants and comitants and also geometrically, using divisors of the complex projective plane, the class of quadratic differential systems with at least five invariant lines. These conditions are such that no matter how a system may be presented, one can verify by using them whether the system has or does not have at least five invariant lines and to check to which orbit (or family of orbits) it belongs.
[**Keywords:**]{} quadratic differential system, Poincaré compactification, algebraic invariant curve, algebraic affine invariant, configuration of invariant lines.
author:
- |
Dana SCHLOMIUK[^1]\
[*Département de Mathématiques et de Statistiques*]{}\
[*Université de Montréal*]{}\
[*C.P. 6128, succ. Centre-ville Montréal, QC H3C 3J7 Canada*]{}\
E-mail: dasch@dms.umontreal.ca
- |
Nicolae VULPE[^2]\
[*Institute of Mathematics and Computer Science*]{}\
[*Academy of Science of Moldova* ]{}\
[*str. Academiei 5, Chişinău MD-2028, Moldova*]{}\
E-mail: nvulpe@math.md
title: Planar quadratic vector fields with invariant lines of total multiplicity at least five
---
Introduction
=============
We consider here real planar differential systems of the form \[il1\] (S)= p(x,y), = q(x,y), where $p,\,q\in {\mathbb{R}}[x,y]$, i.e. $p,\ q$ are polynomials in $x,\ y$ over ${\mathbb{R}}$, and their associated vector fields D=p(x,y) + q(x,y). \[il2\] Each such system generates a complex differential vector field when the variables range over ${\mathbb{C}}$. To the complex systems we can apply the work of Darboux on integrability via invariant algebraic curves (cf.[@Darb]). For a brief introduction to the work of Darboux we refer to the survey article [@Dana1]. Some applications of the work of Darboux in connection with the problem of the center are given in [@Dana2].
For the system (\[il1\]) we can use the following definition. \[df1\] An affine algebraic invariant curve of a polynomial system [(\[il1\])]{} (or an algebraic particular integral) is a curve $f(x,y)=0$ where $f\in {\mathbb{C}}[x,y]$, $\deg(f)\ge1$, such that there exists $k(x,y)\in
{\mathbb{C}}[x,y]$ satisfying $\tilde Df=fk$ in ${\mathbb{C}}[x,y]$. We call $k$ the cofactor of $f$ with respect to the system.
Poincaré was the first to appreciate the work of Darboux [@Darb], which it called “admirable” (see [@Po2]) and inspired by Darboux’s work, Poincaré wrote two articles [@Po3],[@Po4] where he also stated a problem still open today.
With this brilliant work Darboux open up a whole new area of investigations where one studies how the presence of particular algebraic integrals impacts on global properties of the systems, for example on global integrability. In recent years there has been a surge in activity in this area of research and this article is part of a growing literature in the subject. In particular we mention here [@Cris_Llib], [@Cris_Llib_Pant] and the work of C. Christopher, J.V. Perreira and J. Llibre on the notion of multiplicity of an invariant algebraic curve of a differential system [@Cris_Llib_Per].
In this article, which is based on [@Dana_Vlp1], we study systematically the simplest kind of such a structure, i.e. quadratic systems (\[il1\]) possessing invariant lines. Some references on this topic are: [@Sib3; @Druzhkova; @Art_Llib2; @Lyubim1; @Lyubim2; @Popa_Sib1; @Popa2; @Sokulski; @ZX; @Lib_Vul].
To a line $f(x,y)=ux+vy+w=0$ we associate its projective completion $F(X,Y,Z)=uX+vY+wZ=0$ under the embedding ${\mathbb{C}}^2\hookrightarrow {{\mathbb P}}_2({\mathbb{C}})$, $(x,y)\mapsto [x:y:1]$. The line $Z=0$ is called the line at infinity of the system (\[il1\]). It follows from the work of Darboux that each system of differential equations of the form (\[il1\]) yields a differential equation on the complex projective plane which is the compactification of the complex system (\[il1\]) on ${{\mathbb P}}_2({\mathbb{C}})$ (cf. Section \[proj\_eq\]). The line $Z=0$ is an invariant manifold of this complex differential equation. Let us denote by [[**Q**S****]{}]{}&=& { S | S (p(x,y),q(x,y))=1\
((p(x,y)),(p(x,y)))=2 .};\
[[**Q**S**L******]{}]{}&=& {S[[**Q**S****]{}]{}| \
.}. For the multiplicity of the line at infinity the reader is refereed to Section \[proj\_eq\].
We shall call *degenerate quadratic differential system* a system (\[il1\]) with $\deg \gcd(p,q)\ge1$ and $\max\big(\deg(p),\deg(q)\big)=2$. To a system [(\[il1\])]{} in ${{\bf Q\bf S}}$ we can associate a point in $ {\mathbb{R}}^{12}$, the ordered tuple of the coefficients of $p(x,y)$, $q(x,y)$ and this correspondence is an injection \[bigection\] :[[**Q**S****]{}]{}&\^[12]{}\
S & = [B]{}(S) The topology of ${\mathbb{R}}^{12}$ yields an induced topology on [**QS**]{}.
\[def:multipl\] We say that an invariant straight line ${\cal
L}(x,y)=ux+vy+w=0$, $(u,v)\ne(0,0)$, $(u,v,w)\in {\mathbb{C}}^3$ for a quadratic vector field $\tilde D$ has multiplicity $m$ if there exists a sequence of real quadratic vector fields $\tilde D_k$ converging to $\tilde D$, such that each $\tilde D_k$ has $m$ distinct (complex) invariant straight lines ${\cal
L}^1_k=0,\ldots, {\cal L}^m_k=0$, converging to ${\cal L}=0$ as $k\to\infty$ (with the topology of their coefficients), and this does not occur for $m+1$.
\[pr:m\_il\] [[@Art_Llib2]]{} The maximum number of invariant lines (including the line at infinity and including multiplicities) which a quadratic system could have is six.
We call configuration of invariant lines of a system $(S)$ in ${{\bf Q\bf S\bf L}}$ the set of all its invariant lines (real or/and complex), each endowed with its own multiplicity and together with all the real singular points of $(S)$ located on these lines, each one endowed with its own multiplicity.
We associate to each system in [**QSL**]{} its configuration of invariant lines. In analogous manner to how we view the phase portraits of the systems on the Poincaré disc (see for example, [@Lib_DS]), we can also view the configurations of real lines on the disc. To help imagining the full configurations, we complete the picture by drawing dashed lines whenever these are complex.
On the class of quadratic systems acts the group of real affine transformations and time rescaling. Since quadratic systems depend on 12 parameters and since this group depends on 7 parameters, the class of quadratic systems modulo this group action, actually depends on five parameters.
It is clear that the configuration of invariant lines of a system is an affine invariant. The notion of multiplicity defined by Definition \[def:multipl\] is invariant under the group action, i.e. if a quadratic system $S$ has an invariant line $l$ of multiplicity $m$, then each system $\tilde S$ in the orbit of $S$ under the group action has an invariant line $l~$ of the same multiplicity $m$.
In this article we shall consider the case when the system (\[il1\]) has at least five invariant lines considered with their multiplicities.
The problems which we solve in this article are the following:
I\) Construct a system of representatives of the orbits of systems with at least five invariant lines, including the line at infinity and including multiplicities. For each orbit exhibit its configuration.
II\) Characterize in terms of algebraic invariants and comitants and also geometrically, using divisors or zero-cycles of the complex projective plane, the class of quadratic differential systems with at least five invariant lines. These conditions should be such that no matter how a system may be presented to us, we should be able to verify by using them whether the system has or does not have at least five invariant lines and to check to which orbit or perhaps family of orbits it belongs.
Our main results are formulated in Theorems 5.1 and 6.1. Theorem 5.1 gives a total of 11 distinct orbits of systems with a configuration with exactly six invariant lines including the line at infinity and including multiplicities. Theorem 6.1 gives a system of representatives for 17 distinct orbits of systems with exactly five invariant lines including the line at infinity and including multiplicities. Furthermore we give a complete list of representatives of the remaining orbits which are classified in 13 one-parameter families. We characterize each one of these 13 families in terms of algebraic invariants and comitants and geometrically. As the calculation of invariants and comitants can be implemented on a computer, this verification can be done by a computer.
All quadratic systems with at least five invariant lines including the line at infinity and including multiplicities are algebraically integrable, i.e. they all have the rational first integrals and the phase portraits of these systems can easily be drawn. We leave the discussion of issues related to integrability, as well as the drawing of the phase portraits of the systems we consider here, in a follow up paper of this work.
The invariants and comitants of differential equations used in the classification theorems (Theorems \[th\_mil\_6\] and \[th\_mil\_5\]) are obtained following the theory established by K.Sibirsky and his disciples (cf. [@Sib1], [@Sib2], [@Vlp1], [@Popa4]).
Differential equations in ${{\mathbb P}}_2({\mathbb{C}})$ of first degree and first order and their invariant projective curves {#proj_eq}
================================================================================================================================
In [@Darb] Darboux considered differential equations of first degree and first order of the complex projective plane. These are equations of the form |
[ccc]{} L & M & N\
X & Y & Z\
dX & dY & dZ\
|=0 where $L$, $M$, $N$ are homogeneous polynomials of the same degree $m$. These are called equations in Clebsch form $(CF)$. [^3]
We remark that we can have an infinity of such equations yielding the same integral curves. Indeed, for any ordered triple $L,M,N$ of homogeneous polynomials of the same degree $m$ and for any homogeneous polynomial $A$ of degree $m-1$, the $(CF)$-equation corresponding to \[equiv\] L’=L+AX,M’=M+AY,N’=N+AZ has the same integral curves as the equation $(CF)$. Two equations $(CF)$ determined by polynomials $L,M,N$ and $L',M',N'$ satisfying are said to be equivalent. \[th:Darb1\] Let $L,\,M,\,N$ be homogeneous polynomials of the same degree $m$ over ${\mathbb{C}}$. Then there exists a unique $A$, more precisely $$A=-\frac{1}{m+2}\Big(\frac{\partial L }{\partial X}+\frac{\partial
M }{\partial Y}+\frac{\partial N }{\partial Z}\Big)$$ such that if $L',M',N'$ satisfy (\[equiv\]) for this $A$ then $$\frac{\partial L' }{\partial X}+\frac{\partial M' }{\partial Y}+\frac{\partial
N' }{\partial Z}\equiv0.$$ \[th:Darb2\] Every equation (CF) with $m=\deg(L)=\\deg(M)=\deg(N)$ is equivalent to an equation \[ABC\] AdX +BdY+CdZ=0 where $A,\ B,\ C$ are homogeneous polynomials of degree $m+1$ subject to the identity \[ABC=0\] AX +BY+CZ=0 \[def:Darb\] An algebraic invariant curve for an equation $(CF)$ is a projective curve $F(X,Y,Z)=0$ where $F$ is a homogeneous polynomial over ${\mathbb{C}}$ such that $F\mid
\hat DF$ where $\hat D$ is the differential operator $$\hat D=L\frac{\partial }{\partial X}+M\frac{\partial }{\partial
Y}+N\frac{\partial }{\partial Z}$$ i.e. $\exists K\in {\mathbb{C}}[X,Y,Z]$ such that $\hat DF=FK$. $K$ is called the [*cofactor*]{} of $F$ with respect to the equation $(CF)$.
We now show that this definition is in agreement with Definition \[df1\], i.e. it includes as a particular Definition \[df1\].
To a system (\[il1\]) we can associate an equation (\[ABC\]) subject to the identity (\[ABC=0\]). We first associate to the systems (\[il1\]) the differential form $$\omega_1=q(x,y)dx-p(x,y)dy$$ and its associated differential equation $\omega_1=0$.
We consider the map $j: {\mathbb{C}}^3 \setminus \{Z = 0\} \to {\mathbb{C}}^2$, given by $i(X,Y,Z)= (X/Z,Y/Z)=(x,y)$ and suppose that $\max\big(
\deg(p),\deg(q)\big)= m>0$. Since $x=X/Z$ and $y=Y/Z$ we have: $$dx= (ZdX-XdZ)/Z^2 \ , \qquad dy= (ZdY-YdZ)/Z^2 \ ,$$ the pull–back form $j^*(\omega_1)$ has poles at $Z=0$ and its associated equation $j^*(\omega_1)=0$ can be written as $$j^*(\omega_1)= q(X/Z,Y/Z) (ZdX-XdZ)/Z^2 - p(X/Z,Y/Z) (ZdY-YdZ)/Z^2
= 0 .$$ Then the $1$–form $\omega= Z^{m+2} j^*(\omega_1)$ in ${\mathbb{C}}^3\setminus \{Z\ne 0\}$ has homogeneous polynomial coefficients of degree $m+1$, and for $Z\ne0$ the equations $\omega=0$ and $j^*(\omega_1)=0$ have the same solutions. Therefore the differential equation $\omega=0$ can be written as (\[ABC\]) where A(X,Y,Z) &=& Z Q(X,Y,Z)= Z\^[m+1]{} q(X/Z,Y/Z),\
B(X,Y,Z) &=& -Z P(X,Y,Z)=-Z\^[m+1]{} p(X/Z,Y/Z), \[24\]\
C(X,Y,Z) &=& Y P(X,Y,Z)- X Q(X,Y,Z) and $P(X,Y,Z)=Z^m p(X/Z,Y/Z),\quad Q(X,Y,Z)=Z^m q(X/Z,Y/Z)$. Clearly $A$, $B$ and $C$ are homogeneous polynomials of degree $m+1$ satisfying (\[ABC=0\]).
The equation (\[ABC\]) becomes in this case $$P(YdZ-ZdY) +Q(ZdX-XdZ)=0$$ or equivalently $$\label{CFp}
\left|\begin{array}{ccc}
P& Q & 0\\[0mm]
X & Y & Z\\
dX & dY & dZ\\
\end{array}\right|=0.$$ We observe that $Z=0$ is an algebraic invariant curve of this equation according to Definition \[def:Darb\], with cofactor . We shall also say that $Z=0$ is an invariant line for the systems (\[il1\]).
To an affine algebraic curve $f(x,y)=0$, $\deg f=n$, we can associate its projective completion $F(X,Y,Z)=0$ where $F(Z,Y,Z)=Z^{n}f(X/Z,Y/Z)$. From the indicated above the correspondence between a system (\[il1\]) and equation (\[CFp\]) follows the next proposition. \[pr:f=0\] Let $f=0$ ($\deg f=n$) be an invariant algebraic curve of (\[il1\]) according to Definition \[df1\], with cofactor $k(x,y)$. Then its associated projective completion $F(X,Y,Z)=0$ where $F(Z,Y,Z)=Z^{n}f(X/Z,Y/Z)$ is an invariant algebraic curve according to Definition \[def:Darb\] for the equation (\[CFp\]), with cofactor $K(X,Y,Z)=Z^{m-1}k(X/Z,Y/Z)$. Conversely, starting now with an equation in Clebsch form $(CF)$ we can consider its restriction on the affine chart $Z=1$ and associate a differential system: \[syst:Z=1\] |
[ccc]{} L & M & N\
X & Y & Z\
dX & dY & dZ\
|=0 (M -yN)dx-(L -xN)dy=0 { x=L -xN\
y= M -yN,. where $\hat L=L(x,y,1)$, $\hat M=M(x,y,1)$, $\hat N=N(x,y,1)$. The following proposition follows easily by using Euler’s formula $XF'_X+YF'_Y+ZF'_Z= nF$ for a homogeneous polynomial $F(Z,Y,Z)$ of degree n. Let $F(Z,Y,Z)=0$ ($\deg F=n$) be an invariant algebraic curve (according to Definition \[def:Darb\]) for the equation $(CF)$ with cofactor $K(X,Y,Z)$, such that $Z\nmid F$. Then $f(x,y)=F(x,y,1)=0$ is an invariant affine algebraic curve (according to Definition \[df1\]) of the differential system in corresponding to $(CF)$, with cofactor $k(x,y)=K(x,y,1)-nN(x,y,1)$. \[def:Z-mult\] We say that $Z=0$ is an invariant line of multiplicity $m$ for a system $(S)$ of the form (\[il1\]) if and only if there exists a sequence of systems $(S_i)$ of the form (\[il1\]) such that $(S_i)$ tend to $(S)$ when $i\to \infty$ and the systems $(S_i)$ have $m-1$ distinct invariant affine lines ${\cal
L}^j_i=u^j_ix+v^j_iy+w^j_i=0$, $(u^j_i,v^j_i)\ne(0,0)$, $(u^j_i,v^j_i,w^j_i )\in {\mathbb{C}}^3$ $(j=1,\ldots,m-1)$ such thatfor every $ j$$\displaystyle{\lim_{i\to\infty}(u^j_i,v^j_i,w^j_i)=(0,0,1)}$.
Divisors associated to invariant lines configurations {#divisors}
=======================================================
Consider real differential systems of the form: \[2l1\] (S){ &=p\_0+ p\_1(x,y)+p\_2(x,y)p(x,y),\
&=q\_0+ q\_1(x,y)+q\_2(x,y)q(x,y) . with $$\bal
&p_0=a_{00},\quad p_1(x,y)= a_{10}x+ a_{01}y,\quad p_2(x,y)=
a_{20}x^2 +2
a_{11}xy + a_{02}y^2,\\
&q_0=b_{00},\quad q_1(x,y)= b_{10}x+ b_{01}y,\quad\ q_2(x,y)=
b_{20}x^2 +2
b_{11}xy + b_{02}y^2.\\
\eal$$ Let $
a=(a_{00},a_{10},a_{01},a_{20},a_{11},a_{02},b_{00},b_{10},b_{01},b_{20},b_{11},b_{02})$ be the 12-tuple of the coefficients of system and denote ${\mathbb{R}}[
a,x,y]={\mathbb{R}}[a_{00},a_{10},a_{01},a_{20},a_{11},a_{02},b_{00},b_{10},b_{01},b_{20},b_{11},b_{02},x,y]$. Let us denote by ${\mbox{\boldmath $a$}}=({\mbox{\boldmath $a$}}_{00},{\mbox{\boldmath $a$}}_{10}\ldots,{\mbox{\boldmath $b$}}_{02})$ a point in ${\mathbb{R}}^{12}$. Each particular system yields an ordered 12-tuple ${\mbox{\boldmath $a$}}$ of its coefficients. Let $$\bal
P(X,Y,Z)=& p_0({\mbox{\boldmath $a$}})Z^2+ p_1({\mbox{\boldmath $a$}},X,Y)Z+\,p_2({\mbox{\boldmath $a$}},X,Y)=0,\\
Q(X,Y,Z)=& q_0({\mbox{\boldmath $a$}})Z^2+ q_1({\mbox{\boldmath $a$}},X,Y)Z+\,q_2({\mbox{\boldmath $a$}},X,Y)=0.\\
\eal$$ We denote $\quad \sigma(P,Q)= \{w\in{{\mathbb P}}_2({\mathbb{C}})\ |\ P(w)= Q(w)=0\}$.
\[df3\_1a\] We consider formal expressions of the form ${{\bf D}}= \sum
n(w)w$ where $w \in{{\mathbb P}}_2({\mathbb{C}})$ or $w$ is an irreducible curve of ${{\mathbb P}}_2({\mathbb{C}})$ and $n(w)$ is an integer and only a finite number of $n(w)$ are not zero. Such an expression will be called a zero-cycle of ${{\mathbb P}}_2({\mathbb{C}})$ (respectively a divisor of $Z=0$ or a divisor of ${{\mathbb P}}_2({\mathbb{C}})$ ) if $w \in{{\mathbb P}}_2({\mathbb{C}})$ (respectively, $w$ belongs to the line $Z=0$, or $w$ is an irreducible curve of ${{\mathbb P}}_2({\mathbb{C}})$). We call degree of the expression ${{\bf D}}$ the integer $\deg({{\bf D}}) = \sum n(w)$. We call support of ${{\bf D}}$ the set ${\mbox{\rm Supp\,}}({{\bf D}})$ of points $w$ such that $n(w)\ne0$. In this section, for systems we shall assume the conditions $\max(\deg(p),\deg(q))=2$ and $\gcd(p,q)=1$. Let $C(X,Y,Z)=YP(X,Y,Z) - XQ(X,Y,Z)$. [[**D**]{}]{}\_[\_S]{}(P,Q) &=& \_[w(P,Q)]{}I\_w(P,Q)w;\
[[**D**]{}]{}\_[\_S]{}(C,Z) &=& \_[w{ Z = 0}]{}I\_w(C,Z)w ZC(X,Y,Z);\
[[**D**]{}]{}\_[\_S]{}(P,Q;Z) &=& \_[w{ Z = 0}]{}I\_w(P,Q)w;\
[[**D**]{}]{}\_[\_S]{}(P,Q,Z) &=& \_[w{ Z = 0}]{}(I\_w(C,Z),I\_w(P,Q))w,\
where $I_w(F,G)$ is the intersection number (see, [@Fult]) of the curves defined by homogeneous polynomials $F,\ G\in
{\mathbb{C}}[X,Y,Z]$ and $\deg(F),\deg(G)\ge1$.
\[invar1\] n\_[\_]{}\^[\^]{} =&\# { wSupp [[**D**]{}]{}\_[\_S]{}(C,Z)| w[[P]{}]{}\_2()};\
d\_\^[\^]{} =&[[**D**]{}]{}\_[\_S]{}(P,Q;Z).
A complex projective line $uX+vY+wZ=0$ is invariant for the system $(S)$ if either it coincides with $Z=0$ or it is the projective completion of an invariant affine line $ux+vy+w=0$.
Let $S\in {{\bf Q\bf S\bf L}}$. Let us denote $$\bal
{{\bf I\bf L}}(S)=&\left\{\ \ l\ \ \left|\ba{ll} & l\ \hbox {is a line in ${{\mathbb P}}_2({\mathbb{C}})$ such }\\
& \hbox{that}\ l\ \hbox{is invariant for}\ (S)\\
\ea\ \right\}\right.;\\
M(l)=& \ \hbox{the multiplicity of the invariant line $l$ of
$(S)$}.
\eal$$ We note that the line $l_\infty: Z=0$ is included in ${{\bf I\bf L}}(S)$ for any $S\in {{\bf Q\bf S\bf L}}$. Let $l_i\,:\ f_i(x,y)=0$, $i=1,\ldots,k$, be all the distinct invariant affine lines (real or complex) of a system $S\in {{\bf Q\bf S\bf L}}$. Let $l'_i\,:\ {\cal F}_i(X,Y,Z)=0 $ be the complex projective completion of $l_i$. We denote && [G]{}: \_[i]{}[F]{}\_i(X,Y,Z)Z=0; Sing [G]{}={w| w [G]{}};\
&&(w)= . [[**D**]{}]{}\_[\_[[**I**L****]{}]{}]{}(S)&=&\_[l[[**I**L****]{}]{}(S)]{} M(l)l,(S)[[**Q**S**L******]{}]{};\
Supp [[**D**]{}]{}\_[\_[[**I**L****]{}]{}]{}(S)& =& { l | l[[**I**L****]{}]{}(S)}. \[invar\] M\_[\_[[**I**L****]{}]{}]{}=&[[**D**]{}]{}\_[\_[[**I**L****]{}]{}]{}(S);\
N\_[\_]{} =&\# Supp [[**D**]{}]{}\_[\_[[**I**L****]{}]{}]{};\
N\_[\_]{} =&\# { lSupp [[**D**]{}]{}\_[\_[[**I**L****]{}]{}]{}|l[[P]{}]{}\_2() };\
n\^[\^]{}\_[\_[[G]{},]{}]{}=&\# {Supp [[**D**]{}]{}\_[\_S]{}(P,Q) | \[0pt\]\[0pt\][$\big|_{{\mathbb{R}}^2}$]{}};\
d\^[\^]{}\_[\_[[G]{},]{}]{}=&\_[\[0pt\]\[0pt\][$|_{{\mathbb{R}}^2}$]{}]{}I\_(P,Q);\
m\_[\_[G]{}]{}=& {() | Sing [G]{}\[0pt\]\[0pt\][$|_{{\mathbb{C}}^2}$]{}};\
m\^[\^]{}\_[\_[G]{}]{}=& {() | Sing [ G]{}\[0pt\]\[0pt\][$|_{{\mathbb{R}}^2}$]{}}.
The main $T$-comitants associated to configurations of invariant lines {#T-comit}
======================================================================
On the set $\widehat{{\bf Q\bf S}}$ of all differential systems of the form acts the group $Aff(2,{\mathbb{R}})$ of affine transformation on the plane. Indeed for every $g\in Aff(2,{\mathbb{R}})$, $g:\ {\mathbb{R}}^{2}\longrightarrow {\mathbb{R}}^{2}$ we have: $$g:\ \left(\ba{c} \tilde x\\ \tilde y \ea\right) =
M\left(\ba{c} x\\ y \ea\right) +B;\qquad
g^{-1}:\ \left(\ba{c} x\\ y \ea\right) =
M^{-1}\left(\ba{c}\tilde x\\\tilde y \ea\right) -M^{-1}B,$$ where $M=|| M_{ij} || $ is a $2\times 2$ nonsingular matrix, $B$ is a $2\times 1$ matrix over ${\mathbb{R}}$. For every $S\in \widehat{{\bf Q\bf S}}$ we can form its transformed system $\tilde S=g S$: $$\frac{\partial\tilde x}{\partial t} =\tilde p(\tilde x,\tilde y),\qquad\quad
\frac{\partial\tilde y}{\partial t} =\tilde q(\tilde x,\tilde y),\eqno(\tilde S)$$ where $$\left(\ba{c} \tilde p(\tilde x,\tilde y)\\ \tilde q(\tilde x,\tilde y)\ea\right) =
M\left(\ba{c} (p\,{\mbox{\footnotesize$\circ$}}\, {g^{-1}})(\tilde x,\tilde y)\\
(q\,{\mbox{\footnotesize$\circ$}}\, {g^{-1}})(\tilde x,\tilde y) \ea\right).$$ The map &&\
Aff(2,)[[**Q**S****]{}]{} && [[**Q**S****]{}]{}\
(g, S) && S=gS\
&&\
verifies the axioms for a left group action. For every subgroup $G\subseteq Aff(2,{\mathbb{R}})$ we have an induced action of $G$ on $\widehat{{\bf Q\bf S}}$. We can identify the set $\widehat{{\bf Q\bf S}}$ of systems with ${\mathbb{R}}^{12}$ via the map $\widehat{{\bf Q\bf S}}$ $\longrightarrow {\mathbb{R}}^{12}$ which associates to each system the 12-tuple ${\mbox{\boldmath $a$}}=({\mbox{\boldmath $a$}}_{00},{\mbox{\boldmath $a$}}_{10}\ldots,{\mbox{\boldmath $b$}}_{02})$ of its coefficients.
The action of $Aff(2,{\mathbb{R}})$ on $\widehat{{\bf Q\bf S}}$ yields an action of this group on ${\mathbb{R}}^{12}$. For every $ g\in Aff(2,{\mathbb{R}})$ let $r_g:\
{\mathbb{R}}^{12}\longrightarrow {\mathbb{R}}^{12}$, $r_g({\mbox{\boldmath $a$}})=\tilde {\mbox{\boldmath $a$}}$ where $\tilde {\mbox{\boldmath $a$}}$ is the 12-tuple of coefficients of $\tilde S$. We know that $r_g$ is linear and that the map $r:\ Aff(2,{\mathbb{R}})\longrightarrow GL(12,{\mathbb{R}})$ thus obtained is a group homomorphism. For every subgroup $G$ of $Aff(2,{\mathbb{R}})$, $r$ induces a representation of $G$ onto a subgroup $\cal G$ of $GL(12,{\mathbb{R}})$. \[def:comit\] A polynomial $U(
a\,,x,y)\in {\mathbb{R}}[a,x,y]$ is called a comitant of systems with respect to a subgroup $G$ of $Aff(2,{\mathbb{R}})$, if there exists $\chi\in {\bf{Z}}$ such that for every $({\frak g},\, {\mbox{\boldmath $a$}})\in G\times{\mathbb{R}}^{12}\ $ and for every $(x,y)\in {\mathbb{R}}^2$ the following relation holds: $$U(r_{\frak g}({\mbox{\boldmath $a$}}) ,\ {\frak g}(x,y)\,)\equiv\ (\det\,{\frak
g})^{-\chi}\, U( {\mbox{\boldmath $a$}},x,y),$$ where $\det {\frak g}=\det M$. If the polynomial $U$ does not explicitly depend on $x$ and $y$ then it is called invariant. The number $ \chi\in {\bf{Z}}$ is called the weight of the comitant $ U( a ,x,y)$. If $G=GL(2,{\mathbb{R}})$ (or $G=Aff(2,{\mathbb{R}})$) then the comitant $U( a
,x,y)$ of systems is called $GL$-comitant (respectively, affine comitant). \[def\_G\] A subset $X\subset {\mathbb{R}}^{12}$ will be called $G$-invariant, if for every $ {\frak g}\in G$ we have . Let us consider the polynomials C\_i(a,x,y)&=&yp\_i(a,x,y)-xq\_i(a,x,y)\[a,x,y\], i=0,1,2,\
D\_i(a,x,y)&=&p\_i(a,x,y)+ q\_i(a,x,y)\[a,x,y\], i=1,2. As it was shown in [@Sib1] the polynomials \[C\_i:D\_i\] { C\_0(a,x,y),C\_1(a,x,y),C\_2(a,x,y),D\_1(a), D\_2(a,x,y) } of degree one in the coefficients of systems are $GL$-comitants of these systems. Let $f,$ $g\in$ ${\mathbb{R}}[a,x,y]$ and \[trsv\] (f,g)\^[(k)]{}= \_[h=0]{}\^k (-1)\^h [kh]{} . $(f,g)^{(k)}\in {\mathbb{R}}[a,x,y] $ is called the transvectant of index $k$ of $(f,g)$ (cf. [[@Gr_Yng], [@Olver]]{})
\[th:Vlp\] [[@Vlp1]]{} Any $GL$-comitant of systems can be constructed from the elements of the set by using the operations: $+,\, -,\,\times,$ and by applying the differential operation $(f,g)^{(k)}$.
Let $T(2,{\mathbb{R}})$ be the subgroup of $Aff(2,{\mathbb{R}})$ formed by translations. Consider the linear representation of $T(2,{\mathbb{R}})$ into its corresponding subgroup ${\cal T}\subset GL(12,{\mathbb{R}})$, i.e. for every $ \tau\in T(2,{\mathbb{R}})$, $\tau:\ x=\tilde x+\alpha, y=\tilde
y+\beta$ we consider as above $r_\tau:\ {\mathbb{R}}^{12}\longrightarrow
{\mathbb{R}}^{12}$. \[def:T-com\] Consider a polynomial$U(a,x,y)=\sum_{j=0}^{d} U_{i}( a
)x^{d-j}y^j\in {\mathbb{R}}[a,x,y]$ which is a $GL$-comitant of systems . We say that this polynomial is a $T$-comitant of systems if for every $(\tau,\, {\mbox{\boldmath $a$}})\in T(2,{\mathbb{R}})\times {\mathbb{R}}^{12}$ $U_j(r_\tau({\mbox{\boldmath $a$}}))\, =\, U_j( {\mbox{\boldmath $a$}})$, $\forall$ $j=0,1,\ldots,d$. Consider $s$ polynomials$U_i(a,x,y)=\sum_{j=0}^{d_i} U_{ij}( a )x^{d_i-j}y^j\in
{\mathbb{R}}[a,x,y]$, $i=1,\ldots,s$and assume that the polynomials $U_i$ are $GL$-comitants of systems where $d_i$ denotes the degree of the binary form $U_i( a,x,y)$ in $x$ and $y$ with coefficients in ${\mathbb{R}}[a]$. We denote by $${\cal U}=\left\{\,U_{ij}( a )\in {\mathbb{R}}[a]\ |\ i=1,\ldots,s,\
j=0,1,\ldots,d_i\,\right\},$$ the set of the coefficients in ${\mathbb{R}}[a]$ of the $GL$-comitants $U_i( a ,x,y)$, $i=1,\ldots,s$ and by $V(\cal U)$ its zero set: $$V({\cal U})=\left\{\,{\mbox{\boldmath $a$}}\in {\mathbb{R}}^{12}\ |\
U_{ij}({\mbox{\boldmath $a$}})=0, \ \forall\ U_{ij}( a )\in \cal U\,\right\}.$$ Let $U_1,U_2,\ldots, U_s$ be $GL$-comitants of systems and homogeneous polynomials in the coefficients of these systems. A $GL$-comitant $U(a ,x,y)$ of systems is called a conditional $\ T$-comitant (or $CT$-comitant) modulo $\left<U_1,U_2,...,U_s\right>$ (i.e. modulo the ideal generated by $U_{ij}(a)$ $(i=1,\ldots,s; j=0,1,\ldots,
d_i)$ in the ring ${\mathbb{R}}[a]$) if the following two conditions are satisfied:
\(i) the algebraic subset $V({\cal U})\subset {\mathbb{R}}^{12}$ is $Aff(2,{\mathbb{R}})$-invariant (see Definition \[def\_G\]);
\(ii) for every ($\tau,\ {\mbox{\boldmath $a$}})\in T(2,{\mathbb{R}})\times V(\cal U)$ we have $ U(r_\tau( {\mbox{\boldmath $a$}}) ,\ \tilde x,\,\tilde y)=
U( {\mbox{\boldmath $a$}},\ \tilde x,\,\tilde y)\
\mbox{in}\ {\mathbb{R}}[\tilde x,\,\tilde y]. $ A polynomial $U(a ,x,y)\in {\mathbb{R}}[a ,x,y]$, homogeneous of even degree in $x$, $y$ has well determined sign on with respect to $x,\,y$ if for every $ {\mbox{\boldmath $a$}}\in V$, the binary form $u(x,y)=U({\mbox{\boldmath $a$}},x,y)$ yields a function of constant sign on ${\mathbb{R}}^2\setminus\{u=0\}$. We draw the attention to the fact, that if a $CT$-comitant $U( a ,x,y)$ of systems of even weight is a binary form of even degree in $x$ and $y$ and of even degree in $a$ and also has well determined sign on some $Aff(2,{\mathbb{R}})$-invariant algebraic subset $V$, then this sign is conserved after an affine transformation of the plane $x,y$ and time rescaling. We now construct polynomials $D(a,x,y)$ and $H(a,x,y)$ which will be shown in Lemma \[Table:Propreties\] to be $T$-comitants. \[not:1\] Consider the polynomial $\Phi_{\alpha,\beta}=\alpha P+\beta Q\in
{\mathbb{R}}[a,X,Y,Z,\alpha,\beta]$ where $P=Z^2p(X/Z,Y/Z),$ $Q=Z^2q(X/Z,Y/Z)$, $p,$ $q\in {\mathbb{R}}[a,x,y]$ and $\max
(\deg_{(x,y)}p,\deg_{(x,y)}q)=2$. Then $$\bal
\Phi_{\alpha,\beta}=&\ c_{11}(a,\alpha,\beta)X^2 +2
c_{12}(a,\alpha,\beta)XY
+ c_{22}(a,\alpha,\beta)Y^2+
2c_{13}(a,\alpha,\beta)XZ\\ &
+2c_{23}(a,\alpha,\beta)YZ
+c_{a,33}(\alpha,\beta)Z^2,\qquad
\Delta(a,\alpha,\beta) =\ \det\left|\left|c_{ij}(a,\alpha,\beta)
\right|\right|_{i,j\in\{1,2,3\}},\\
& D(a,x,y) = 4\Delta(a,-y,x),\qquad H(a,x,y) =
4\big[\det\left|\left|c_{ij}(a,-y,x)
\right|\right|_{i,j\in\{1,2\}}\big].\\
\eal$$ \[pr:D\_C\] Consider $m\le3$ distinct directions in the affine plane, where by direction we mean a point $(u,v)\in
{\mathbb{C}}^2\setminus(0,0)$. For the existence of an invariant straight line of a system $S$ of coefficients ${\mbox{\boldmath $a$}}$ corresponding to each one of these directions it is necessary that there exist $m$ distinct common factors of the polynomials $C_2({\mbox{\boldmath $a$}},x,y)$ and $D({\mbox{\boldmath $a$}},x,y)$ over ${\mathbb{C}}$. [[*Proof:*]{} ]{}Suppose that ${\cal L}(x,y)\equiv ux+vy+w=0$ is an invariant line for a quadratic system corresponding to ${\mbox{\boldmath $a$}}\in {\mathbb{R}}^{12}$. Then we must have $r,s,t\in {\mathbb{C}}$ such that \[2l2\] p(x,y)+ q(x,y)=[L]{}(x,y)(rx+sy+t). Hence $$up(x,y)+v q(x,y)=(ux+vy+w)(rx+sy+t).$$ So $\Phi_{u,v}({\mbox{\boldmath $a$}},x,y)=0$ is a reducible conic which occurs if and only if the respective determinant $\Delta({\mbox{\boldmath $a$}},u,v)=0$. But $4\Delta({\mbox{\boldmath $a$}},u,v)=D({\mbox{\boldmath $a$}},-v,u)=0$. The point at infinity of ${\cal
L}=0$ is $[-v:u:0]$ and so $C_2({\mbox{\boldmath $a$}},-v,u)=0$. Hence, the two homogeneous polynomials of degree 3 in $x$, $y$ must have the common factor $ux+vy$.
------------------------------------------------------------------------
\[rm:H\] Consider two parallel invariant affine lines , $(u,v)\ne(0,0)$, ${\cal
L}_i(x,y)\in {\mathbb{C}}[x,y],\ (i=1,2)$ of a quadratic system $S$ of coefficients ${\mbox{\boldmath $a$}}$. Then , i.e. the T-comitant $H(a,x,y)$ can be used for determining the directions of parallel invariant lines of systems . Indeed, according to (\[2l2\]) from the hypothesis we must have $$up(x,y)+ vq(x,y)=(ux+vy+w_1)(ux+vy+w_2).$$ Therefore for the quadratic form in $x$ and $y$: $F_2({\mbox{\boldmath $a$}},x,y)
=up_2({\mbox{\boldmath $a$}},x,y)+vq_2({\mbox{\boldmath $a$}},x,y)$ we obtain $F_2=(ux+vy)^2$ and hence ${\mbox{\rm Discriminant\,}}(F_2)=0$. Then calculations yield: ${\mbox{\rm Discriminant\,}}(F_2({\mbox{\boldmath $a$}},x,y))=-H({\mbox{\boldmath $a$}},-v,u)$ and hence $H({\mbox{\boldmath $a$}},-v,u)=0$.
We construct the following polynomials which will be shown in Lemma \[Table:Propreties\] to be $T$-comitants: \[not1\] \[Comit:Bi\] &B\_3(a,x,y)=(C\_2,D)\^[(1)]{}=Jacob( C\_2,D),\
&B\_2(a,x,y)=(B\_3,B\_3)\^[(2)]{} - 6B\_3(C\_2,D)\^[(3)]{},\
&B\_1(a)=\_x( C\_2,D)/y\^9=-2\^[-9]{}3\^[-8]{}(B\_2,B\_3)\^[(4)]{}.\
\[pr:BGI\] Suppose $\tilde d=\deg \gcd\left(C_2({\mbox{\boldmath $a$}},x,y),D({\mbox{\boldmath $a$}},x,y)\right)$. Then: $$\bal
\tilde d=0\ &\ \Leftrightarrow\ \ B_1({\mbox{\boldmath $a$}})\ne0;\\
\tilde d=1\ &\ \Leftrightarrow\ \ B_1({\mbox{\boldmath $a$}})=0,\ B_2({\mbox{\boldmath $a$}},x,y)\ne0;\\
\tilde d=2\ &\ \Leftrightarrow\ \ B_2({\mbox{\boldmath $a$}},x,y)=0,\ B_3({\mbox{\boldmath $a$}},x,y)\ne0;\\
\tilde d=3\ &\ \Leftrightarrow\ \ B_3({\mbox{\boldmath $a$}},x,y)=0.
\eal$$ [[*Proof:*]{} ]{}Since the polynomial $B_3(a)$ is the Jacobian of the cubic binary forms $C_2(a,x,y)$ and $D(a,x,y)$ we conclude that $\tilde d=3$ if and only if $B_3({\mbox{\boldmath $a$}},x,y)=0$. We assume $B_3({\mbox{\boldmath $a$}},x,y)\ne0$ (i.e. $\tilde d\le2$) and consider the two subcases: $B_2({\mbox{\boldmath $a$}},x,y)=0$ and $B_2({\mbox{\boldmath $a$}},x,y)\ne0$.
[**1)**]{} Assuming that $B_2({\mbox{\boldmath $a$}},x,y)=0$ then $\tilde d=2$. Indeed, suppose $\tilde d<2$. From the condition $B_2=0$ yields $B_1=0$ and since the polynomial $B_1(a)$ is the resultant of the binary forms $C_2(a,x,y)$ and $D(a,x,y)$ we get $\tilde d=1$, i.e. these polynomials have a common linear factor $ax+by$. We may assume $b=0$ (the case $b\ne0$ can be reduced to this one via the transformation $x_1=ax+b$, $y_1=x$). Then $$C_2=x(a_1x^2+b_1xy +c_1y^2)\equiv x\tilde A(x,y),\quad
D=x(a_2x^2+b_2xy +c_2y^2)\equiv x\tilde B(x,y).$$ Considering , calculations yield $\quad
B_2({\mbox{\boldmath $a$}},x,y)= 3x^4\cdot {\mbox{\rm Res\,}}_x (\tilde A,\tilde B)/y^4
$and we obtain a contradiction: since $B_2=0$ according to [@Walker] (see Theorem 10.7 on page 29) the polynomials $\tilde A$ and $\tilde B$ have a common nonconstant factor, i.e. $\tilde d>1$. Conversely, suppose that $\tilde d=2$. Then clearly we have $$C_2=(ax+by)\tilde C,\qquad D=(cx+dy)\tilde C$$ and taking into account calculations yield $B_2=0$.
[**2)**]{} Let us assume now that the condition $B_2({\mbox{\boldmath $a$}},x,y)\ne0$ holds. Then $\tilde d\le1$ and since the polynomial $B_1(a)$ is the resultant of the binary forms $C_2(a,x,y)$ and $D(a,x,y)$ we get $\tilde d=1$ if and only if $B_1(a)=0$.
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From the Propositions \[pr:D\_C\] and \[pr:BGI\] the next result follows: \[lm:BGI\] For the existence of an invariant straight line in one (respectively 2 or 3 distinct ) directions in the affine plane it is necessary that $B_1=0$ (respectively $B_2=0$ or $B_3=0$).
Let us apply a translation $x=x'+x_0$, $y=y'+y_0$ to the polynomials $p(a,x,y)$ and $q(a,x,y)$. We obtain $ \tilde p(\tilde
a(a,x_0,y_0),x',y')=p(a, x'+x_0, y'+y_0),$ $\quad \tilde q(\tilde
a(a,x_0,y_0),x',y')=q(a, x'+x_0, y'+y_0).$ Let us construct the following polynomials $$\bal
\Gamma_i(a,x_0,y_0)& \equiv {\mbox{\rm Res\,}}_{x'}
\Big(C_i\big(\tilde a(a,x_0,y_0),x',y'\big),C_0\big(\tilde
a(a,x_0,y_0),x',y'\big)\Big)/(y')^{i+1},\\
& \Gamma_i(a,x_0,y_0) \in {\mathbb{R}}[a,x_0,y_0],\ (i=1,2).\\
\eal$$ \[not2\] \[2l4a\] \_i(a,x,y)=.\_i(a,x\_0,y\_0)|\_[{x\_0=x, y\_0=y}]{}\[a,x,y\] (i=1,2). \[obs\_\] It can easily be checked using the Definition \[def:comit\] that the constructed polynomials $\tilde{\cal
E}_1(a,x,y)$ and $\tilde{\cal E}_2(a,x,y) $ are affine comitants of systems and are homogeneous polynomials in coefficients $a_{00},\ldots, b_{02}$ and non-homogeneous in $x,y$ and $\
\deg_{a} \tilde{\cal E}_1=3,\ \deg_{\,(x,y)} \tilde{\cal
E}_1=5,\ \
\deg_{a} \tilde{\cal E}_2=4,\ \deg_{\,(x,y)} \tilde{\cal E}_2=6.
$ \[GCD:Ei\] Let ${\cal E}_i(a,X,Y,Z)$ $(i=1,2)$ be the homogenization of $\tilde{\cal E}_i(a,x,y)$, i.e. $${\cal E}_1(a,X,Y,Z)=Z^5\tilde {\cal E}_1(a,X/Z,Y/Z),\qquad {\cal
E}_2(a,X,Y,Z)=Z^6\tilde {\cal E}_1(a,X/Z,Y/Z)$$ and $ \qquad
{\cal H}(a,X,Y,Z)=\gcd\Big({\cal E}_1(a,X,Y,Z),\
{\cal E}_2(a,X,Y,Z)\Big).
$ In what follows we shall examine the geometrical meaning of these affine comitants. We shall prove the following theorem: \[theor:E1,E2\] The straight line ${\cal L}(x,y)\equiv ux+vy+w=0$ $u,v,w\in {\mathbb{C}}$, $(u,v)\ne(0,0)$ is an invariant line for a system in ${{\bf Q\bf S}}$ corresponding to a point ${\mbox{\boldmath $a$}}\in{\mathbb{R}}^{12}$ if and only if the polynomial ${\cal L}$ is a common factor of the polynomials $\tilde{\cal E}_1({\mbox{\boldmath $a$}},x,y)$ and $\tilde{\cal E}_2({\mbox{\boldmath $a$}},x,y)$ over ${\mathbb{C}}$, i.e. $$\tilde{\cal E}_i({\mbox{\boldmath $a$}},x,y)=(ux+vy+w)\widetilde W_i(x,y)\in
{\mathbb{C}}[x,y]\quad (i=1,2).$$
To prove this Theorem we first prove the following lemma: \[lem:C0,C1,C2\] The straight line $\tilde {\cal L}(x,y)\equiv ux+vy=0$ is an invariant line of a system of coefficients ${\mbox{\boldmath $a$}}$ with ${\mbox{\boldmath $a$}}_{00}^2+{\mbox{\boldmath $b$}}_{00}^2\ne0$ if and only if $
C_0({\mbox{\boldmath $a$}},-v,u)=0,$ $C_1({\mbox{\boldmath $a$}},-v,u)=0,$ and $ C_2({\mbox{\boldmath $a$}},-v,u)=0$. These condition are equivalent to the following ones: \_x(C\_0(a,x,y),C\_1(a,x,y))/y\^2|\_[()]{}=0=\_x(C\_0(a,x,y),C\_2(a,x,y))/y\^3|\_[()]{}. \[2l3\] [[*Proof:*]{} ]{}According to Definition \[df1\] the line $\tilde {\cal L}(x,y)$=0 is a particular algebraic integral for a system if and only if the identity (\[2l2\]) holds for this system and this line. So in this case $$u(p_0({\mbox{\boldmath $a$}})+ p_1({\mbox{\boldmath $a$}},x,y)+p_2({\mbox{\boldmath $a$}},x,y))+v(q_0({\mbox{\boldmath $a$}})+q_1({\mbox{\boldmath $a$}},x,y)+q_2({\mbox{\boldmath $a$}},x,y)) = (ux+vy)(S_0+S_1(x,y)),$$ for some $S_0\in{\mathbb{C}}$ and $S_1\in{\mathbb{C}}[x,y]$. Herein we obtain: &(i) & up\_0()+vq\_0()=0;\
&(ii) & up\_1(,x,y)+vq\_1(,x,y)=(ux+vy)S\_0();\
&(iii) & up\_2(,x,y)+vq\_2(,x,y)=(ux+vy)S\_1(,x,y). We observe that, if $x=-v$ and $y=u$ then the left-hand sides of $(i)$, $(ii)$ and $(iii)$ become $C_0({\mbox{\boldmath $a$}},-v,u)$,$C_1({\mbox{\boldmath $a$}},-v,u)$ and $C_2({\mbox{\boldmath $a$}},-v,u)$, respectively. At the same time the right-hand sides of these identities vanish. Therefore the following equations are obtained: C\_0(,-v,u)=0, C\_1(,-v,u)=0, C\_2(,-v,u)=0.\[2l4\] As the degree of $C_0(a,x,y)$ is one, the relations (\[2l4\]) hold.
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*Proof of the Theorem \[theor:E1,E2\]:* Consider the straight line ${\cal L}(x,y)=0$. Let $(x_0,y_0)\in {\mathbb{R}}^2$ be any fixed non-singular point of the systems (i.e. $p(x_0,y_0)^2+q(x_0,y_0)^2\ne0$) which lies on the line ${\cal L}(x,y)=0$, i.e. $ux_0+vy_0+w=0$. Let $\tau_0$ be the translation $x=x'+x_0$, $y=y'+y_0$, $\tau_0(x',y')=(x,y)$. Then $${\cal L}(x,y)= {\cal L}(x'+x_0,y'+y_0)=ux'+vy'\equiv \tilde{\cal
L}(x',y')$$ and consider the line $ux'+vy'=0$. By Lemma \[lem:C0,C1,C2\] the straight line $\tilde {\cal L}(x',y')=0$ will be an invariant line of systems ($\ref{2l1}{}^{\tau_0}$) if and only if the conditions (\[2l3\]) are satisfied for these systems, i.e. $
\Gamma_1({\mbox{\boldmath $a$}},x_0, y_0))=\Gamma_2({\mbox{\boldmath $a$}},x_0,y_0)=0
$ for each point $(x_0,y_0)$ situated on the line ${\cal L}(x,y)\equiv
ux+vy+w=0$, since the relation $ux_0+vy_0+w=0$ is satisfied.
Thus we have$ \Gamma_i({\mbox{\boldmath $a$}},x_0, y_0)= (ux_0+vy_0+w)\tilde
\Gamma_i(a,x_0, y_0) \ \ (i=1,2)$.Taking into account the notations (\[2l4a\]) we conclude that the statement of Theorem \[theor:E1,E2\] is true.
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We now consider the possibility for a straight line to be a multiple invariant line.
\[lm3\] If ${\cal L}(x,y)\equiv ux+vy+w=0$, $u,v,w\in {\mathbb{C}}$, $(u,v)\ne(0,0)$ is an invariant straight line of multiplicity $k$ for a quadratic system then $[{\cal L}(x,y)]^k\mid \gcd(\tilde {\cal
E}_1,\tilde {\cal E}_2)$, i.e. there exist $W_i({\mbox{\boldmath $a$}},x,y)\in
{\mathbb{C}}[x,y]$ $(i=1,2)$ such that \_i(,x,y)= (ux+vy+w)\^k W\_i(,x,y),i=1,2. \[2l5\] [[*Proof:*]{} ]{}Suppose that line ${\cal L}(x,y)\equiv ux+vy+w=0$ is an invariant line of multiplicity $k$ for a system which corresponds to point ${\mbox{\boldmath $a$}}\in {\mathbb{R}}^{12}$. Let us denote by ${\mbox{\boldmath $a$}}_\varepsilon\in
{\mathbb{R}}^{12}$ the point corresponding to the perturbed system $(\ref{2l1}_\varepsilon)$, which has $k$ distinct invariant lines: ${\cal L}_{i\varepsilon}(x,y)$ $(i=1,2,...k)$.
According to Theorem \[theor:E1,E2\] for systems $(\ref{2l1}_\varepsilon)$ the following relations are valid: $$\tilde {\cal E}_{j\varepsilon}({\mbox{\boldmath $a$}}_\varepsilon,x,y) = {\cal
L}_{1\varepsilon}\cdot {\cal L}_{2\varepsilon}...\cdot {\cal
L}_{k\varepsilon}\widetilde W_j({\mbox{\boldmath $a$}}_\varepsilon,x,y),\quad j=1,2,$$ and according to Definition \[def:multipl\] when perturbation $\varepsilon\to 0$ then $ {\cal L}_{i\varepsilon}(x,y) \to {\cal
L}(x,y),\ \forall i=1,..k. $ At the same time $
\tilde {\cal E}_{j\varepsilon} \to \tilde {\cal E}_j = {\cal L}(x,y)^k W_j,\quad
j=1,2.$ Lemma \[lm3\] is proved.
------------------------------------------------------------------------
\[Mult:Z=0\] If the line $l_\infty:Z=0$ is of multiplicity $k>1$ then $Z^{k-1}\mid \gcd({\cal E}_1, {\cal E}_2)$. Indeed, suppose that the line $l_\infty:Z=0$ is of multiplicity $k>1$ for a system $S$ which corresponds to a point ${\mbox{\boldmath $a$}}\in
{\mathbb{R}}^{12}$. Then by Definition \[def:Z-mult\] there exist a perturbed system $S_\varepsilon $ corresponding to the point ${\mbox{\boldmath $a$}}_\varepsilon\in {\mathbb{R}}^{12}$ which has $k-1$ distinct invariant affine straight lines: ${\cal
L}_{i\varepsilon}(x,y)=u_{i\varepsilon} x+ v_{i\varepsilon} y+
w_{i\varepsilon}$, $(u_{i\varepsilon},v_{i\varepsilon})\ne(0,0)$, $(u_{i\varepsilon},v_{i\varepsilon},w_{i\varepsilon})\in{\mathbb{C}}^3$ $(i=1,2,...k-1)$ such that for every $i$: $\displaystyle{\lim_{\varepsilon\to 0}
(u_{i\varepsilon},v_{i\varepsilon},w_{i\varepsilon})=(0,0,1)}.$
By Lemma \[lm3\] each of the $k-1$ affine lines ${\cal
L}_{i\varepsilon}$ must be a factor of the polynomial ${\cal
H}({\mbox{\boldmath $a$}}_\varepsilon,X,Y,Z)=\gcd\left({\cal
E}_1({\mbox{\boldmath $a$}}_\varepsilon,X,Y,Z),{\cal
E}_2({\mbox{\boldmath $a$}}_\varepsilon,X,Y,Z)\right)$. Therefore we conclude that for the system $S$ we have $ Z^{k-1}\mid {\cal H}({\mbox{\boldmath $a$}},X,Y,Z)$.
As a next step we shall determine necessary conditions for the existence of parallel invariant lines. Let us consider the following $GL$-comitants of systems : \[not3\] $$\ba{ll}
M(a,x,y) = 2\,{\mbox{\rm Hess\,}}\big(C_2(x,y)\big), &
\eta(a) = {\mbox{\rm Discriminant\,}}\big(C_2(x,y)\big),\\
K(a,x,y) = {\mbox{\rm Jacob\,}}\big(p_2(x,y),q_2(x,y)\big),\qquad &
\mu(a) = {\mbox{\rm Discriminant\,}}\big(K(a,x,y)\big),\\
N(a,x,y) = K(a,x,y) + H(a,x,y), &
\theta(a) = {\mbox{\rm Discriminant\,}}\big(N(a,x,y)\big),
\ea$$ the geometrical meaning of which is revealed in the next 3 lemmas below. \[lem:K,mu\] Let $S\in {{\bf Q\bf S}}$ and let ${\mbox{\boldmath $a$}}\in {\mathbb{R}}^{12}$ be its 12-tuple of coefficients. The common points of $P=0$ and $Q=0$ on the line $Z=0$ are given by the common linear factors over ${\mathbb{C}}$ of $p_2$ and $q_2$. This yields the geometrical meaning of the T-comitants $\mu(a)$ and $K(a,x,y)$: (p\_2(x,y),q\_2(x,y)) ={
[lcl]{} 0 & iff &()0;\
1 & iff &()=0, K(,x,y)=0;\
2 & iff &K(,x,y)=0.
. The proof follows from the fact that $K$ is the Jacobian of $p_2(x,y)$ and $q_2(x,y)$ (i.e. $p_2$ and $q_2$ are proportional if and only if $K({\mbox{\boldmath $a$}},x,y)=0$ in ${\mathbb{R}}[x,y]$) and $\mu={\mbox{\rm Res\,}}_x(p_2,q_2)/y^4$.
We shall prove the following assertion: \[lm4\] A necessary condition for the existence of one couple (respectively, two couples) of parallel invariant straight lines of a systems corresponding to ${\mbox{\boldmath $a$}}\in{\mathbb{R}}^{12}$ is the condition $\theta({\mbox{\boldmath $a$}}) =0$ (respectively, $N({\mbox{\boldmath $a$}},x,y)=0$). [[*Proof:*]{} ]{}Let$ {\cal L}_i(x,y)\equiv ux+vy+w_i=0$, $(u,v)\ne(0,0)$, $(u,v,w_i)\in{\mathbb{C}}^3$ $(i=1,2) $ be two distinct $(w_1\ne w_2)$ parallel invariant lines for a quadratic system . Then by (\[2l2\]) we have $$up(x,y)+vq(x,y)=\xi(ux+vy+w_1)(ux+vy+w_2)$$ and via a time rescaling we may assume $\xi$= 1. Therefore for the quadratic homogeneities we obtain (ua\_[20]{} +vb\_[20]{} )x\^2+2(ua\_[11]{} +vb\_[11]{})xy +(ua\_[02]{} +vb\_[02]{})y\^2= (u x +v y)\^2, \[(2l6\] and hence, for the existence of parallel invariant lines the solvability of the following systems of quadratic equations with respect to parameters $u$ and $v$ is necessary: \[2l7\] (A\_1) u a\_[20]{} + v b\_[20]{}= u\^2; (A\_2) ua\_[11]{} + vb\_[11]{} = uv; (A\_3) ua\_[02]{} + vb\_[02]{} = v\^2. Without loss of generality we may consider $uv\ne0$, otherwise a rotation of phase plane can be done. We now consider $vA_1 - u
A_2$ and $uA_3 - v A_2$: $$\bal
&vA_1 - u A_2: \quad & -a_{11}u^2 + (a_{20}-b_{11})uv + b_{20}v^2=0, \\
& uA_3 - v A_2: &\quad a_{02}u^2 + (b_{02}-a_{11})uv -
b_{11}v^2=0.
\eal$$ Let $F_1(u,v)$ and $F_2(u,v)$ be the left hand sides of the above equations. Clearly, for the existence of two directions $(u_1,v_1)$ and $(u_2,v_2)$ such that in each of them there are 2 parallel invariant straight lines of a system it is necessary that the ${\rm rank}(U)=1$, where $$U=\left(\ba{ccc} -a_{11} &\ a_{20}\!-\!b_{11}\ & b_{20} \\
a_{02} &\ b_{02}\!-\!a_{11}\ & -b_{11} \\
\ea\right).$$ Hence, it is necessary $$\tilde A=\left|\!\!\ba{cc} -a_{11} &\ a_{20}\!-\!b_{11}\ \\
a_{02} &\ b_{02}\!-\!a_{11}\ \\
\ea\!\!\right|=0,\qquad
\tilde B=\left|\!\!\ba{cc} -a_{11} &\ b_{20} \\
a_{02} &\ -b_{11} \\
\ea\!\!\right|=0,\qquad
\tilde C=\left|\!\!\ba{cc} a_{20}\!-\!b_{11}\ & b_{20} \\
b_{02}\!-\!a_{11}\ & -b_{11} \\
\ea\!\!\right|=0.$$ Since the resultant of the binary forms $F_1(u,v)$ and $F_2(u,v)$ is $ {\mbox{\rm Res\,}}_u(F_1,F_2)/v^4=\tilde B^2-\tilde A\tilde C, $ we conclude that for the existence of one couple of parallel invariant lines it is necessary that $\tilde B^2-\tilde A\tilde
C=0$. On the other hand calculations yield$
N({\mbox{\boldmath $a$}},x,y)= \tilde Cx^2 + 2 \tilde B xy + \tilde Ay^2,$ $ \theta = 4({\tilde B}^2-\tilde A\tilde C)$ and this completes the proof of lemma.
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\[lm\_3:2\] The type of the divisor $D_S(C,Z)$ for systems (\[il1\]) is determined by the corresponding conditions indicated in Table 1, where we write $w_1^c+w_2^c+w_3$ if two of the points, i.e. $w_1^c, w_2^c$, are complex but not real. Moreover, for each type of the divisor $D_S(C,Z)$ given by Table 1 the quadratic systems (\[il1\]) can be brought via a real linear transformation to one of the following canonical systems $({{\bf S}}_{I})-({{\bf S}}_{V})$ corresponding to their behavior at infinity.
[|c|c|c|c|]{}\
\[0pt\]\[0pt\][Case]{} & \[0pt\]\[0pt\][Type of $D_S(C,Z)$]{} & Necessary and sufficient\
& & conditions on the comitants\
------------------------------------------------------------------------
$1$ & $w_1+w_2+w_3 $ & $\eta>0 $\
------------------------------------------------------------------------
$2$ & $w_1^c+w_2^c+w_3 $ & $\eta<0$\
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$3$ & $2w_1+w_2 $ & $\eta=0,\quad M\ne0$\
------------------------------------------------------------------------
$4$ & $3w $ & $ M=0,\quad C_2\ne0$\
------------------------------------------------------------------------
$5$ & $D_S(C,Z)$ undefined & $ C_2=0$\
$$\bal
&\left\{\ba{rcl}
\displaystyle \frac{dx}{dt}&=&k+cx+dy+gx^2+(h-1)xy,\\[2mm]
\displaystyle \frac{dy}{dt}&=& l+ex+fy+(g-1)xy+hy^2;
\ea\right. &\qquad ({{\bf S}}_I)\\[3mm]
&\left\{\ba{rcl}
\displaystyle \frac{dx}{dt}&=&k+cx+dy+gx^2+(h+1)xy,\\[2mm]
\displaystyle \frac{dy}{dt}&=& l+ex+fy-x^2+gxy+hy^2;
\ea\right.& \hspace{2cm}({{\bf S}}_{I\!I})\\[3mm]
&\left\{\ba{rcl}
\displaystyle \frac{dx}{dt}&=&k+cx+dy+gx^2+hxy,\\[2mm]
\displaystyle \frac{dy}{dt}&=& l+ex+fy+(g-1)xy+hy^2;
\ea\right.&\qquad ({{\bf S}}_{I\!I\!I})\\[3mm]
&\left\{\ba{rcl}
\displaystyle \frac{dx}{dt}&=&k+cx+dy+gx^2+hxy,\\[2mm]
\displaystyle \frac{dy}{dt}&=& l+ex+fy-x^2+gxy+hy^2,
\ea\right.&\qquad ({{\bf S}}_{I\!V})\\[3mm]
&\left\{\ba{rcl}
\displaystyle \frac{dx}{dt}&=&k+cx+dy+x^2,\\[2mm]
\displaystyle \frac{dy}{dt}&=& l+ex+fy+xy.
\ea\right.&\qquad ({{\bf S}}_{V})
\eal$$
[[*Proof:*]{} ]{}The Table 1 follows easily from the definitions of $\eta(a)$ and $
M(a,x,y)$ in Notation \[not3\]. It is well known that a cubic binary form in $x,y$ over ${\mathbb{R}}$ can be brought via a real linear transformation of the plane $(x,y)$: ${\frak g}(x,y) = (\tilde x,\tilde y)$ to one of the following four canonical forms forms \[can\_forms\] I. y(x-y); II. x(x\^2+y\^2); III. x\^2y; IV.x\^3; V. 0. Let us consider a system (\[il1\]) corresponding to a point ${\mbox{\boldmath $a$}}\in{\mathbb{R}}^{12}$ and let us consider the $GL$-comitant ${C_2}( {\mbox{\boldmath $a$}},x,y)=yp_2({\mbox{\boldmath $a$}},x,y)-xq_2({\mbox{\boldmath $a$}},x,y)$ simply as a cubic binary form in $x$ and $y$. Then the transformed binary form $g{C_2}(
{\mbox{\boldmath $a$}},x,y)=C_2({\mbox{\boldmath $a$}},{\frak g}^{-1}(\tilde x,\tilde y))$ is one of the canonical forms corresponding to cases indicated in Table 1.
On the other hand, according to the Definition \[def:comit\] of a $GL$-comitant, for ${C_2}({\mbox{\boldmath $a$}},x,y)$ whose weight $\chi=-1$, we have for the same linear transformation ${\frak g}\in GL(2,{\mathbb{R}})$ $$C_2(r_{\frak g}({\mbox{\boldmath $a$}}),\, {\frak g}(x,y))= \det(g)\, C_2( {\mbox{\boldmath $a$}},\, x,y).$$ Using ${\frak g}(x,y) = (\tilde x,\tilde y)$ we obtain $
C_2(r_{\frak g}({\mbox{\boldmath $a$}}),\, \tilde x,\tilde y)=\det(g) C_2( {\mbox{\boldmath $a$}},\,
{\frak g}^{-1}(\tilde x,\tilde y)),$ where we may assume $\det(g)=1$ via the rescaling: $\tilde x\to \tilde x/\det(g)$, $\tilde y\to\tilde y/\det(g)$. Thus, recalling that $$p_2(\tilde x,\tilde y)=\tilde a_{20}\tilde x^2+2\tilde
a_{11}\tilde x\tilde y+\tilde a_{02}\tilde y^2,\qquad q_2(\tilde
x,\tilde y)=\tilde b_{20}\tilde x^2+2\tilde b_{11}\tilde x\tilde
y+\tilde b_{02}\tilde y^2,$$ for the first canonical form in (\[can\_forms\]) we have $$C_2(\tilde {\mbox{\boldmath $a$}},\tilde x,\tilde y)=-\tilde b_{20}\tilde x^3+(\tilde
a_{20}-2\tilde b_{11})\tilde x^2\tilde y+(2\tilde a_{11}-\tilde
b_{02})\tilde x\tilde y^2+\tilde a_{02}\tilde y^3= \tilde x\tilde
y(\tilde x-\tilde y).$$ Identifying the coefficients of the above identity we get the canonical form $({{\bf S}}_{I})$.
Analogously for the cases $II,\ III$ and $IV$ we obtain the canonical form $({{\bf S}}_{I\!I})$, $({{\bf S}}_{I\!I\!I})$ and $({{\bf S}}_{I\!V})$ associated to the respective polynomials in (\[can\_forms\]).
Let us consider the case $V$, i.e. $C_2({\mbox{\boldmath $a$}},x,y)=0$ in ${\mathbb{R}}[x,y]$. Then we obtain the systems $$\frac{dx}{dt}=k+cx+dy+gx^2+hxy,\quad
\frac{dy}{dt}= l+ex+fy+gxy+hy^2$$ with $g^2+h^2\ne0$. By interchanging $x$ and $y$ we may assume $g\ne0$ and then via the linear transformation $\tilde x=g x+h y,$ $\tilde
y=y$ we obtain the systems $(S_{V})$.
------------------------------------------------------------------------
In order to determine the existence of a common factor of the polynomials ${\cal E}_1({\mbox{\boldmath $a$}},X,Y,Z)$ and ${\cal E}_2({\mbox{\boldmath $a$}},X,Y,Z)$ we shall use the notion of the resultant of two polynomials with respect to a given indeterminate (see for instance, [@Walker]).
Let us consider two polynomials $f,g\in R[x_1,x_2,\ldots,x_r]$ where $R$ is a unique factorization domain. Then we can regard the polynomials $f$ and $g$ as polynomials in $x_r$ over the ring ${\cal R}= R[x_1,x_2,\ldots,x_{r-1}]$, i.e. $$\bal
&f(x_1,x_2,\ldots,x_r)=a_0+a_1x_r+\ldots+a_nx_r^n,\\
&g(x_1,x_2,\ldots,x_r)=b_0+a_1x_r+\ldots+b_mx_r^m \quad
a_i,b_i\in{\cal R}.
\eal$$ \[Trudi:2\][[@Walker]]{} Assuming $n,m>0$, $a_nb_m\ne0$ the resultant ${\mbox{\rm Res\,}}_{x_r}(f,g)$ of the polynomials $f$ and $g$ with respect to $x_r$ is a polynomial in $R[x_1,x_2,\ldots,x_{r-1}]$ which is zero if and only if $f$ and $g$ have a common factor involving $x_r$. We also shall use the following remark: \[rem:transf\] Assume $s,\, \gamma\in {\mathbb{R}}$, $\gamma>0$. Then the transformation $x=\gamma^{s}x_1$, $y=\gamma^{s}y_1$ and does not change the coefficients of the quadratic part of a quadratic system, whereas each coefficient of the linear (respectively constant ) part will be multiplied by $\gamma^{-s}$ (respectively by $\gamma^{-2s}$).
The configurations of invariant lines of quadratic\
differential systems with $ M_{{}_{{\bf I\bf L}}}=6$ {#Sec:m_il:6}
=====================================================
We denote by ${{\bf Q\bf S\bf L}}_{\bf6}$ the class of all quadratic differential systems with $p,$ $q$ relatively prime $((p,q)=1)$, $Z\nmid C$ and possessing a configuration of 6 invariant straight lines including the line at infinity and including possible multiplicities. \[lm\_3:1\] For a quadratic system $S$ in ${{\bf Q\bf S\bf L}}_{\bf6}$ the conditions $N({\mbox{\boldmath $a$}},x,y)=0$ and $B_3({\mbox{\boldmath $a$}},x,y)=0$ in ${\mathbb{R}}[x,y]$, are satisfied. [[*Proof:*]{} ]{}Indeed, if for a system the condition $M_{{}_{{\bf I\bf L}}}=6$ is satisfied, then taking into account the Definition \[def:multipl\] we conclude that there exists a perturbation of the coefficients of the system within the class of quadratic systems such that the perturbed systems has $6$ distinct invariant lines (real or complex, including the line $Z=0$). Hence, the perturbed systems must possess 2 couples of parallel lines with distinct directions and an additional line in a third direction. Then, by continuity and according to Lemma \[lm4\] and Corollary \[lm:BGI\] we have $B_3({\mbox{\boldmath $a$}},x,y)=0$ and $N({\mbox{\boldmath $a$}},x,y)=0$.
------------------------------------------------------------------------
By Theorem \[theor:E1,E2\] and Lemma \[lm3\] we obtain the following result: \[gcd:5\] If $M_{{}_{{\bf I\bf L}}}=6$ then$\deg\gcd\big({\cal
E}_1,({\mbox{\boldmath $a$}},X,Y,Z), {\cal E}_2({\mbox{\boldmath $a$}},X,Y,Z)\big)=5$, i.e. ${\cal E}_1\mid {\cal E}_2$.
\[th\_mil\_6\] (i) The class ${{\bf Q\bf S\bf L}}_{\bf6}$ splits into 11 distinct subclasses indicated in [**Diagram 1**]{} with the corresponding Configurations 6.1-6.11 where the complex invariant straight lines are indicated by dashed lines. If an invariant straight line has multiplicity $k>1$, then the number $k$ appears near the corresponding straight line and this line is in bold face. We indicate next to the real singular points their multiplicities as follows: $\left(I_w(p,q)\right)$ if $w$ is a finite singularity, $\left(I_w(C,Z),\ I_w(P,Q)\right)$ if $w$ is an infinite singularity with $I_w(P,Q)\ne0$ and $\left(I_w(C,Z)\right)$ if $w$ is an infinite singularity with $I_w(P,Q)=0$.
\(ii) We consider the orbits of the class ${{\bf Q\bf S\bf L}}_{\bf6}$ under the action of the real affine group and time rescaling. The systems [*(VI.1)*]{} up to [*(VI.11)*]{} from the Table 2 form a system of representatives of these orbits under this action. A differential system $(S)$ in ${{\bf Q\bf S\bf L}}_{\bf6}$ is in the orbit of a system belonging to $(VI.i)$ if and only if $B_3({\mbox{\boldmath $a$}},x,y)=0=N({\mbox{\boldmath $a$}},x,y)$ and the corresponding conditions in the middle column (where the polynomials $H_i$ $(i=1,2,3)$ and $N_j$ $(j=1,\ldots,4)$ are $CT$-comitants to be introduced below) is verified for this system $(S)$. The conditions indicated in the middle column are affinely invariant.
Wherever we have a case with invariant straight lines of multiplicity greater than one, we indicate the corresponding perturbations proving this in the Table 3.
**Diagram 1 $(M_{{}_{{\bf I\bf L}}}=6)$**
[|l|c|c|]{}\
\[0pt\]\[0pt\][Orbit representative]{} & Necessary and sufficient & \[0pt\]\[0pt\][Configuration]{}\
& conditions &\
([*VI.1*]{}):$\ba{l} \dot x=x^2-1,\
\dot y=y^2-1 \ea$ & $\ba{l}\eta>0,\ H_1>0 \ea $ & [*Config. 6.1*]{}\
([*VI.2*]{}):$\ba{l} \dot x= x^2+1,\
\dot y= y^2+1 \ea$ & $\ba{l}\eta>0,\ H_1<0 \ea\ $ & [*Config. 6.2*]{}\
([*VI.3*]{}):$\ba{l} \dot x= 2xy,\
\dot y=y^2-x^2-1\!\! \ea$ & $ \ba{l}\eta<0,\ H_1<0 \ea $ & [*Config. 6.3*]{}\
([*VI.4*]{}):$\ba{l} \dot x= 2xy,\
\dot y= y^2-x^2+ 1\!\! \ea$ & $ \ba{l}\eta<0,\ H_1>0 \ea $ & [*Config. 6.4*]{}\
([*VI.5*]{}):$\ba{l} \dot x= x^2,\
\dot y= y^2 \ea$ & $ \ba{l}\eta>0,\ H_1=0 \ea $ & [*Config. 6.5*]{}\
([*VI.6*]{}):$\ba{l} \dot x= 2xy,\
\dot y= y^2-x^2 \ea$ & $ \ba{l}\eta<0,\ H_1=0 \ea $ & [*Config. 6.6*]{}\
([*VI.7*]{}):$\ba{l} \dot x=x^2-1,\
\dot y= 2y \ea$ & $ MD\ne0, \eta\!=\!H\!=\!N_1\!=\!N_2=0 $ & [*Config. 6.7* ]{}\
([*VI.8*]{}):$\ba{l} \dot x=1 - x^2,\
\dot y= -2xy \ea$ & $ MH\ne0,\eta=H_2=0, H_3>0 $ & [*Config. 6.8*]{}\
([*VI.9*]{}):$\ba{l} \dot x=-1 - x^2,\
\dot y= -2xy \ea$ & $ MH\ne0, \eta=H_2=0, H_3<0 $ & [*Config. 6.9*]{}\
():$\ba{l} \dot x= x^2,\
\dot y= 1 \ea $ & $ M\!\ne\!0,\eta\!= \!H\!=\!D\!=\!N_1\!=\!N_2\!=\!0 $ & [*Config. 6.10*]{}\
():$\ba{l} \dot x= x,\
\dot y= y-x^2 \ea $ & $ \ba{l}\eta= M= N_3=N_4=0\ea $ & [*Config. 6.11*]{}\
[|l|l|]{}\
Perturbations&Invariant straight lines\
([*VI.5${}_\varepsilon$*]{}) : $\ba{l} \dot x=
x^2-\varepsilon^2,\
\dot y= y^2-\varepsilon^2 \ea$ & $ \ba{l} x=\pm \varepsilon,\ y=\pm \varepsilon,\ y=x \ea $\
([*VI.6${}_\varepsilon$*]{}) : $\ba{l} \dot x= 2xy,\
\dot y=\varepsilon^2 -x^2+ y^2 \ea $ & $ \ba{l} x=0,\ y\pm ix=\varepsilon,\ y\pm ix=-\varepsilon \ea $\
([*VI.7${}_\varepsilon$*]{}) : $\ba{l} \dot x=-1+ x^2,\
\dot y= 2y(\varepsilon y+1) \ea$ & $ \ba{l} x=\pm 1,\, \varepsilon
y=-1,\, y=0,\, x-2 \varepsilon y=1\!\!\ea $\
([*VI.8${}_\varepsilon$*]{}) : $\ba{l} \dot x=1 - x^2,\
\dot y= -2xy-\varepsilon y^2 \ea$& $ \ba{l} y=0,\ x=\pm 1, \
x+ \varepsilon y=\pm 1\ea $\
([*VI.9${}_\varepsilon$*]{}) : $\ba{l} \dot x=-1 - x^2,\
\dot y= -2xy-\varepsilon y^2 \ea$& $ \ba{l} y=0,\ x=\pm i, \
x+ \varepsilon y=\pm i\ea $\
([*VI.10${}_\varepsilon$*]{}): $\bigg\{\ba{l} \dot
x=(1-\varepsilon)^2x^2-\varepsilon^2,\\
\dot y= \left (2\,\varepsilon ^2 y+1\right )
\left (2\,\varepsilon y+1\right ) \ea $ & $ \ba{l}
(1-\varepsilon)x=\pm
\varepsilon,\, 2\varepsilon y=-1,\, 2\varepsilon^2 y=-1,\!\!\!\\
(\varepsilon -1)^2 x-4\varepsilon^3 y-\varepsilon (\varepsilon +1)=0\ea $\
([*VI.11${}_\varepsilon$*]{}): $\bigg\{\ba{l} \dot
x=x+\varepsilon x^2,\\
\dot y= y-x^2-2\varepsilon xy-2\varepsilon^2y^2 \ea $ & $ \ba{l}
x=0,\ \varepsilon x+1=0,\ x+ \varepsilon y=0,\\
x+ 2\varepsilon y=0,\ \varepsilon x+2\varepsilon^2y-1=0\ea $\
[*Proof of the Theorem \[th\_mil\_6\]*]{}: According to Table 1 we shall consider the subcases corresponding to distinct types of the divisor $D_S(C,Z)$). Since we only discuss the case $C_2\ne0$, in what follows it suffices to consider only the canonical forms $({{\bf S}}_I)$ to $({{\bf S}}_{I\,V})$. The idea of the proof is to perform a case by case discussion for each one of these canonical forms, for which according to Lemma \[lm\_3:1\] the conditions $B_3=0=N$ must be fulfilled. These conditions yield specific conditions on the parameters. The discussion proceeds further by breaking these cases in more subcases determined by more restrictions on the parameters. Finally we construct new invariants or T-comitants which put these conditions in invariant form.
For constructing the invariant polynomials included in the statement of Theorem \[th\_mil\_6\] we shall use the $T$-comitants $D(a,x,y)$ and $H(a,x,y)$ indicated before as well as the (\[C\_i:D\_i\]).
Systems with the divisor $D_S(C,Z)=1\cdot w_1+1\cdot w_2+1\cdot w_3$
--------------------------------------------------------------------
For this case we shall later need the following polynomial which is shown to be an affine invariant in Lemma \[Table:Propreties\]. \[not\_H1\] Let us denote$ H_1(a)= -\big((C_2,C_2)^{(2)},C_2)^{(1)},D\big)^{(3)}. $ According to Lemma \[lm\_3:2\] a system with this type of divisor can be brought by linear transformations to the canonical form $({{\bf S}}_I)$ for which we have: N(,x,y)&=& (g\^2-1)x\^2+2(g-1)(h-1)xy+(h\^2-1)y\^2. Hence the condition $N=0$ yields $(g-1)(h-1)=g^2-1=h^2-1=0$ and we obtain 3 possibilities: $(a)\ g=1=h;$ $(b)\ g=1=-h;$ $(c)\ g=-1=-h$. The cases $(b)$ and $(c)$ can be brought by linear transformations to the case $(a)$. Hence the resulting polynomials are: $p_2(x,y)=x^2$ and $q_2(x,y)=y^2$. Then the term in $x$ of the first equation and the term in $y$ in the second equation can be eliminated via a translation. Thus we obtain the systems \[s4.1\] x=k + dy + x\^2,y=l + ex +y\^2 for which we have $ B_3= 3[-e^2x^4+2e^2x^3y +4(l-k)x^2y^2
-2d^2xy^3+d^2y^4]. $ Hence, the condition $B_3=0$ yields $d=e=k-l=0$ and we get the systems of the form: \[CF\_1\] x=l + x\^2,y=l + y\^2. By Remark \[rem:transf\] ( $\gamma=|l|,\ s=1/2$) for systems (\[CF\_1\]) we can consider $l\in \{-1,0,1\}$. Clearly these systems possess the invariant straight lines $x=\pm\sqrt{-l}$, $y=\pm\sqrt{-l}$, $y=x$. Therefore, we obtain Config. 6.1 (respectively, Config. 6.2) for $l<0$ (respectively, for $l>0$) and Config. 6.5 for $l=0$. For systems (\[CF\_1\]) the affine invariant $H_1({\mbox{\boldmath $a$}})=-2^93^3l$ and, hence, ${\mbox{\rm sign\,}}(l)=-{\mbox{\rm sign\,}}(H_1({\mbox{\boldmath $a$}}))$.
Systems with the divisor $D_S(C,Z)=1\cdot w^c_1+1\cdot w^c_2+1\cdot w_3$
------------------------------------------------------------------------
In this case by Lemma \[lm\_3:2\] the systems can be brought by linear transformations to the canonical form $({{\bf S}}_{I\!I})$ for which we have: N(,x,y)&=&(g\^2-2h+2)x\^2 +2g(h+1)xy+(h\^2-1)y\^2. Hence the condition $N=0$ yields $g=h-1=0$ and we may consider $c=d=0$ due to the translation $x=x_1-d/2$, $y=y_1-c/2$. We thus obtain the systems \[S2:N=0\] x= k +2xy, y= l + ex +fy -x\^2 + y\^2 for which$
B_3=6\,\left[(ef-2k)x^4+(f^2-e^2)x^3y-(4k+ef)x^2y^2-2ky^4\right].
$ Hence, the condition $B_3=0$ yields $k=e=f=0$ and we obtain the following form \[CF\_3\] x= 2xy, y= l -x\^2 + y\^2 where $l\in \{-1,0,1\}$ by the Remark \[rem:transf\] ( $\gamma=|l|,\
s=1/2$). It is not difficult to convince ourselves that these systems possess as invariant straight lines the components over ${\mathbb{C}}$ of: $$x=0,\quad x^2+2\,i\, xy-y^2-l=0,\quad x^2-2\,i\,xy-y^2-l=0,$$ with the intersection points:$
p_{1,2}=(0,\pm\sqrt{-l}),\ \ p_{3,4}=(\pm\sqrt{l},0).
$ On the other hand for systems (\[CF\_3\]) we have $ H_1= 2^{10}\,3^2\,l.$ Therefore, if $H_1\ne0$ we get Config. 6.3 for $H_1<0$ and Config. 6.4 for $H_1>0$, whereas for $H_1=0$ we obtain Config. 6.6.
Systems with the divisor $D_S(C,Z)=2\cdot w_1+1\cdot w_2$
---------------------------------------------------------
For this case we shall later need the following polynomials which are shown to be $CT$-comitants in Lemma \[Table:Propreties\]. \[not:H2,3-N1,2\] Let us denote $$\bal
&H_2(a,x,y)=(C_1,\ 2H\!-\!N)^{(1)}\!-\!2D_1N,\quad
N_1(a,x,y)=C_1(C_2,C_2)^{(2)} \!-\!2C_2(C_1,C_2)^{(2)},\\
&H_3(a,x,y)=(C_2,D)^{(2)}, \qquad
N_2(a,x,y)=D_1(C_1,C_2)^{(2)}\!-\!\Big((C_2,C_2)^{(2)},C_0\Big)^{(1)}.
\eal$$ We are in the case of the canonical form $({{\bf S}}_{I\!I\!I})$ for which we have: \[eq\_H\] N(a,x,y)&=(g\^2-1)x\^2 +2h(g-1)xy+h\^2y\^2,\
H(a,x,y)&=-(g-1)\^2x\^2 -2h(g+1)xy-h\^2y\^2.\
The condition $N=0$ yields $h=g^2-1=0$ and we shall examine two subcases: and $H(a,x,y)=0$.
### The case $H(a,x,y)\not=0$
In this case for $h=0$ we have $H(a,x,y)=-(g-1)^2x^2\not=0$ and hence the condition $N=0$ yields $g=-1$. Moreover, for systems $({{\bf S}}_{I\!I\!I})$ we can consider $e=f=0$ due to the translation of the origin of coordinates to the point $(f/2,e/2)$. Thus, the systems $({{\bf S}}_{I\!I\!I})$ can be brought to the form \[S3\_NM\_\] x =k +cx +dy -x\^2, y =l - 2xy, for which$
B_3= 6x(-2lx^3 +cd xy^2+d^2y^3).
$So, the condition $B_3=0$ yields $l=d=0$ and we obtain the systems \[s4.4\] x =k +cx -x\^2, y = - 2xy with $k\ne0$ (otherwise systems (\[s4.4\]) become degenerate).
So far we have only used the necessary conditions $N=0$ and $B_3=0$ for this particular case. These are not sufficient for having 6 invariant lines. According to Lemma \[gcd:5\] we must have ${\cal E}_1\mid {\cal E}_2$ (see Notation \[GCD:Ei\]). Calculations yield : $${\cal E}_1= (kZ^2-X^2){\cal H},\quad
{\cal E}_2 =X(X^2-cXZ-kZ^2){\cal H},
\quad {\cal H}=\gcd\left({\cal E}_1,{\cal E}_2\right)=
2Y\left(kZ^2+cXZ-X^2\right).$$ Since $k\ne0$ according to Lemma \[Trudi:2\] we obtain the condition $ {\mbox{\rm Res\,}}_Z( {\cal E}_1/{\cal H},\ {\cal E}_2/{\cal
H})=-c^2kX^6\equiv0 $ must hold. This yields $c=0$ and the systems (\[s4.4\]) become \[CF\_7\] x =k - x\^2, y = - 2xy. By Remark \[rem:transf\] ($\gamma=|k|,\ s=1/2$) we may assume $k\in\{-1,1\}$.
For the systems (\[CF\_7\]) we have $ {\cal H}= \gcd\left({\cal
E}_1,{\cal E}_2\right)=2Y\left(kZ^2-X^2\right)^2$ and according to Lemma \[lm3\] each one of the two invariant lines $x=\pm\sqrt{k}$ of the systems (\[CF\_7\]) could be of the multiplicity two. And they are indeed of multiplicity two as it is shown by the perturbations ([*VI.8${}_\varepsilon$*]{}) (for $k=1$) and ([*VI.9${}_\varepsilon$*]{}) (for $k=-1$) from Table 3. Thus, we obtain Config. 6.8 for $k=1$ and Config. 6.9 for $k=-1$.
On the other hand for the systems (\[CF\_7\]) we have $
H_2 = 16cx^2,$ $H_3 =32kx^2.$ Hence the $T$-comitants $H_2 $ and $H_3 $ capture exactly the conditions $c=0$ and $k>0$ or $c=0$ and $k<0$ and this leads to the corresponding conditions in Table 2.
### The case $H(a,x,y)=0$
According to (\[eq\_H\]) the conditions $N=H=0$ yield $h=0,$ $g=1$ and translating the origin of coordinates to the point $(-c/2,0)$ the systems $({{\bf S}}_{I\!I\!I})$ can be brought to the form \[S3\_NH\_0\] x=k+dy+ x\^2,y= l+ex+fy. For these systems we have$
B_3= 6dxy^2(fx-dy)
$ and the condition $B_3=0$ yields $d=0$. So, we obtain the systems \[s4.5\] x=k+ x\^2,y= l+ex+fy for which we have $D({\mbox{\boldmath $a$}},x,y)=-f^2x^2y$. [**1)**]{} If $D\ne0$ then $f\ne0$ and by Remark \[rem:transf\] ($\gamma=f/2,\ s=1$) we can consider $f=2$. Then via the translation we may assume $l=0$ and we obtain the systems \[s4.6\] x=k+ x\^2,y= ex+2y, for which calculations yield \[val:Ei\] &[E]{}\_1= , \_2=3(e X + 2Y)(X\^2 + kZ\^2)[H]{},\
& [H]{}= 2Z(X\^2+kZ\^2),\_Y([E]{}\_1/[H]{}, [E]{}\_2/[H]{})=-2e(X\^2 + kZ\^2)\^2. Hence for ${\cal E}_1\mid {\cal E}_2$ the condition ${\mbox{\rm Res\,}}_Y({\cal E}_1/{\cal H},\ {\cal E}_2/{\cal H})=0$ must be fulfilled in ${\mathbb{R}}[X,Z]$. This yields $e=0$ and then we obtain:$ {\mbox{\rm Res\,}}_X\Big( ({\cal E}_1/{\cal H})|_{e=0},({\cal E}_2/{\cal
H})|_{e=0}\Big)= 32(k+1)Y^3Z^2=0$. Hence $k+1=0$ and for $e=k+1=0$ we obtain the system \[CF\_9\] x= x\^2-1,y= 2y, for which ${\cal H}=\gcd\left({\cal E}_1,{\cal
E}_2\right)=YZ(X-Z)^2(X+Z)$. This system possesses the invariant affine lines $
x=\pm1,\quad y=0.
$ Moreover, taking into account the polynomial ${\cal H}$, by Lemma \[lm3\] and Corollary \[Mult:Z=0\] the line $x=1$ as well as the line $l_\infty:Z=0$ could be of multiplicity two. This is confirmed by the perturbations ([*VI.7${}_\varepsilon$*]{}) from Table 3. Since this system possesses only two finite singularities $(\pm1,0)$ which are simple, we conclude that the configuration of the invariant lines of the system (\[CF\_9\]) is Config. 6.7.
It remains to observe that the conditions $e=0=k+1$ are equivalent to $N_1=N_2=0$, as for systems (\[s4.5\]) we have $ N_1=
8e\,x^4,\quad N_2=16(k+1)x $.
[**2)**]{} The condition $D=0$ implies $f=0$ and we obtain the systems \[s4.7\] x=k+ x\^2,y= l+ex. Calculations yield: \[val:Eia\] &[E]{}\_1= ,\_2=(e X + l Z)(X\^2 + kZ\^2)[H]{}, where $ {\cal H}= Z\left(X^2+kZ^2\right).$ Hence for ${\cal
E}_1\mid {\cal E}_2$ according to Lemma \[Trudi:2\] at least one of the following conditions must hold: $${\mbox{\rm Res\,}}_X({\cal E}_1/{\cal H},\ {\cal E}_2/{\cal H})=-4\, ek(e^2k +
l^2)^2Z^6=0,\quad {\mbox{\rm Res\,}}_Z({\cal E}_1/{\cal H},\ {\cal E}_2/{\cal
H})=-4\, ek(e^2k + l^2)^2X^6=0,$$ and we obtain that either $ek=0$ or $e^2k + l^2=0$. Since the second case yields a degenerate system we obtain the necessary condition $ek=0$. It is easy to observe that for $e^2+k^2\ne0$ we obtain ${\cal E}_1\nmid {\cal E}_2$. Therefore $k=e=0$ (then $l\ne0$) and via the additional rescaling $y\to l\,y$ we obtain the system: \[CF\_10\] x= x\^2,y= 1 for which ${\cal H}=\gcd\left({\cal E}_1,{\cal
E}_2\right)=X^3Z^2$. By Lemma \[lm3\] and Corollary \[Mult:Z=0\] the line $x=0$ as well as the line $Z=0$ could be of multiplicity three. This is confirmed by the perturbations ([*VI.10${}_\varepsilon$*]{}) from Table 3. It remains to note that for systems (\[s4.7\]) we obtain $ N_1=8e\,x^4,$ $
N_2=16kx $ and, hence in this case we obtain Config. 6.10 if and only if
Systems with the divisor $D_S(C,Z)=3\cdot w$
--------------------------------------------
For this case we shall later need the following polynomials which are shown to be $CT$-comitants in Lemma \[Table:Propreties\]. \[not\_HC3\] Let us denote$
N_3= \left(C_2,C_1\right)^{(1)},\ \
N_4= 4\left(C_2,C_0\right)^{(1)} - 3C_1D_1.$ We are in the case of the canonical form $({{\bf S}}_{I\!V})$ for which we have: $$N=(g^2-2h)x^2+2ghxy+h^2y^2.$$ So, the condition $N=0$ yields $h=g=0$ and due to the translation $x=x_1+e/2,$ $y=y_1$ we may assume $e=0$. Hence the systems $({{\bf S}}_{I\!V})$ become \[S4\_N0\] x=k+cx+dy,y= l+fy-x\^2, for which$
B_3= 6dx^3(fx-dy).
$The condition $B_3=0$ yields $d=0$ and we shall examine the systems of the form \[s4.8\] x=k+cx,y= l+fy-x\^2. Calculations yield \[val\_Eib\] &[E]{}\_1= ,\
&[E]{}\_2=Z(c X + k Z)\^2[H]{},= Z\^2(cX+kZ).\
Since the polynomial ${\cal E}_2/{\cal H}$ depends only on $X$ and $Z$ for ${\cal E}_1\mid {\cal E}_2$ the following condition must hold: $f(c-f)=0$. We claim that for $f=0$ we cannot have ${\cal
E}_1\mid {\cal E}_2$. Indeed, assuming $f=0$ we obtain the quadratic form ${\cal E}_1/{\cal H}=cX^2+2kXZ+ clZ^2$ in $X$ and $Z$, which must divide $Z(cX+kZ)^2$. This clearly implies that the discriminant of this form must be zero, i.e. $4(k^2-c^2l)=0$. However this leads to degenerate systems.
Therefore we must have $c-f=0$ and for the systems (\[s4.8\]) with $f=c$ calculations yield: $ {\cal E}_1=X\,\tilde{\cal H},$ $
{\cal E}_2=Z(cX+kZ)\,\tilde{\cal H}, $ where $\tilde{\cal H}=Z^2(c
X + k Z)^2$. Therefore ${\cal E}_1\mid {\cal E}_2$ if and only if $k=0$ and we obtain the systems$ \dot x= cx,\quad \dot y=
l+cy-x^2.$with $c\ne0$. We may assume $c=1$ by Remark \[rem:transf\] ( $\gamma=c,\ s=1$) and via the translation of the origin of coordinates to the point $(0,-l)$ we obtain $l=0$. This leads to the following system \[CF\_11\] x= x,y= y -x\^2, with ${\cal H}=\gcd\left({\cal E}_1,{\cal E}_2\right)=X^3Z^2$ and by Lemma \[lm3\] and Corollary \[Mult:Z=0\] each one of the invariant lines $x=0$ and $Z=0$ is of multiplicity 3. This is confirmed by the perturbed systems ([*VI.11${}_\varepsilon$*]{}) from Table 3. On the other hand for systems (\[s4.8\])$
N_3=3(c-f)x^3,\qquad N_4=3x[4kx+(f^2-c^2)y]
$ and hence, the conditions $c-f=k=0$ are equivalent to $N_3=N_4=0$. Taking into account the existence of the simple singular point $(0,0)$ placed on the line $x=0$ we obtain Config. 6.11.
All the cases in Theorem \[th\_mil\_6\] are thus examined. To finish the proof of the Theorem \[th\_mil\_6\] it remains to show that the conditions occurring in the middle column of Table 2 are affinely invariant. This follows from the proof of Lemma \[Table:Propreties\].
------------------------------------------------------------------------
The configurations of invariant lines of quadratic differential systems with $M_{{}_{{\bf I\bf L}}}=5$ {#Sec:m_il:5}
========================================================================================================
We denote by ${{\bf Q\bf S\bf L}}_{\bf5}$ the class of all quadratic differential systems with $p,$ $q$ relatively prime $((p,q)=1)$, $Z\nmid C$ and possessing a configuration of five invariant straight lines including the line at infinity and including possible multiplicities. \[lm\_NB2\] If for a quadratic system $(S)$ $M_{{}_{{\bf I\bf L}}}=5$, then for this system one of the two following conditions are satisfied:\
$$\bal
&(i)\quad N({\mbox{\boldmath $a$}},x,y)=0=B_2({\mbox{\boldmath $a$}},x,y)\ \mbox{in}\ {\mathbb{R}}[x,y];\quad (ii)\quad
\theta({\mbox{\boldmath $a$}})=0= B_3({\mbox{\boldmath $a$}},x,y) \ \mbox{in}\ {\mathbb{R}}[x,y].
\eal$$ [[*Proof:*]{} ]{}Indeed, if for a system the condition $M_{{}_{{\bf I\bf L}}}=5$ is satisfied then taking into account the Definition \[def:multipl\] we conclude that there exists a perturbation of the coefficients of the system within the class of quadratic systems such that the perturbed systems have five distinct invariant lines (real or imaginary, including the line $Z=0$). Hence, the perturbed systems must possess either 2 couples of parallel lines with distinct directions or one couple of parallel lines and 2 additional lines with distinct directions. Then, by continuity and according to Lemma \[lm:BGI\] and Corollary \[lm4\] we respectively have either the conditions $(i)$ or $(ii)$.
------------------------------------------------------------------------
By Theorem \[theor:E1,E2\] and Lemmas \[lm3\] and \[lm\_3:1\] we obtain the following result: \[gcd:4\] (a) If for a system $(S)$ of coefficients ${\mbox{\boldmath $a$}}\in {\mathbb{R}}^{12}$,$M_{{}_{{\bf I\bf L}}}=5$ then\
$\deg\gcd\big({\cal
E}_1({\mbox{\boldmath $a$}},X,Y,Z), {\cal E}_2({\mbox{\boldmath $a$}},X,Y,Z)\big)=4;$ (b) If $N({\mbox{\boldmath $a$}},x,y)\not\equiv0$ then $M_{{}_{{\bf I\bf L}}}\le5$. \[th\_mil\_5\] (i) The class ${{\bf Q\bf S\bf L}}_{\bf5}$ splits into 30 distinct subclasses indicated in [**Diagram 2**]{} with the corresponding Configurations 5.1-5.30 where the complex invariant straight lines are indicated by dashed lines. If an invariant straight line has multiplicity $k>1$, then the number $k$ appears near the corresponding straight line and this line in bold face. We indicate next to the singular points their multiplicities as follows: $\left(I_w(p,q)\right)$ if $w$ is a finite singularity, $\left(I_w(C,Z),\ I_w(P,Q)\right)$ if $w$ is an infinite singularity with $I_w(P,Q)\ne0$ and $\left(I_w(C,Z)\right)$ if $w$ is an infinite singularity with $I_w(P,Q)=0$.
\(ii) We consider the orbits of the class ${{\bf Q\bf S\bf L}}_{\bf5}$ under the action of the real affine group and time rescaling. The systems [*(V.1)*]{} up to [*(V.30)*]{} from the Table 4 form a system of representatives of these orbits under this action. A differential system $(S)$ in ${{\bf Q\bf S\bf L}}_{\bf5}$ is in the orbit of a system belonging to $(V.i)$ if and only if the corresponding conditions in the middle column (where the polynomials $H_i$ $(i=7,\ldots,
11)$ and $N_j$ $(j=5,6)$ are $CT$-comitants to be introduced below) are verified for this system $(S)$. The conditions indicated in the middle column are affinely invariant.
Wherever we have a case with invariant straight lines of multiplicity greater than one, we indicate the corresponding perturbations in the Table 5. \[rem:H>0\] We observe that in the middle column of the Table 5 (and of the Table 2) there occur conditions of the form $\mathcal{M}(a,x,y)=0$ in ${\mathbb{R}}[x,y]$ or of the form $\mathcal{M}(a,x,y)>0$ (or $<0$), where $\mathcal{M}(a,x,y)$ is a homogeneous polynomial in $a$ and separately in $x$ an $y$, which is a $CT$-comitant. All polynomials occurring in conditions of the second type are of even weight, of even degree in $a_{00},\ldots,b_{02}$ and have a well determined sign on the corresponding variety indicated in the Lemma \[Table:Propreties\].
**Diagram 2 $(M_{{}_{{\bf I\bf L}}}=5)$**
**Diagram 2 $(M_{{}_{{\bf I\bf L}}}=5)$ [*(continued)*]{}**
**Diagram 2 $(M_{{}_{{\bf I\bf L}}}=5)$ [*(continued)*]{}**
[|l|c|c|]{}\
\[0pt\]\[0pt\][Orbit representative]{} & Necessary and sufficient & \[0pt\]\[0pt\][Configuration]{}\
& conditions &\
------------------------------------------------------------------------
([*V.1*]{}) $\left\{\!\!\ba{l} \dot x=(x+1)(gx+1), \\[-0.8mm]
\dot y=(g-1)xy+y^2, \\[-0.8mm]
\hspace{8mm} g(g^2-1)\ne0 \ea\!\!\right. $ & $ \ba{c}\eta>0,\ B_3=\theta=0,\\
N\ne0,\, \mu\ne0,\, H_1\ne0\ea $ & Config. 5.1\
------------------------------------------------------------------------
([*V.2*]{}) $\left\{\!\!\ba{l} \dot x=g(x^2-4),\ g\ne0\\[-0.8mm]
\dot y=(g^2-\!4)\!+\!(g^2+\!4)x \\[-0.8mm]
\hspace{7mm} -x^2+gxy-y^2 \ea\!\!\!\!\right. $ & $ \ba{c}\eta<0,\, B_3=\theta=0,\\
N\ne0,\, \mu\ne0,\, H_1\ne0\ea $ & Config. 5.2\
([*V.3*]{}) $\bigg\{\!\!\ba{l} \dot x=-1+x^2,\\
\dot y=g(y^2-1),\ g\ne0 \ea $ & $ \ba{c}\eta>0,\, B_2=N=0,\, B_3\ne0,\\
H_1>0, \ H_4=0,\ H_5>0 \ea $ & Config. 5.3\
([*V.4*]{}) $\bigg\{\!\!\ba{l} \dot x=-1+x^2,\\
\dot y=g(1+y^2),\ \ g\ne0 \ea $ & $ \ba{c}\eta>0,\, B_2=N=0,\, B_3\ne0,\\
H_4=0,\ H_5<0 \ea $ & Config. 5.4\
([*V.5*]{}) $\bigg\{\!\!\ba{l} \dot x=1+x^2,\\
\dot y=g(1+y^2),\ \ g\ne0 \ea $ & $ \ba{c}\eta>0,\, B_2=N=0,\, B_3\ne0,\\
H_1<0,\ H_4=0,\ H_5>0 \ea $ & Config. 5.5\
([*V.6*]{}) $\bigg\{\!\!\ba{l} \dot x= 1+2xy,\\[-0.8mm]
\dot y=g-x^2+y^2,\ \ g\in{\mathbb{R}}\ea $ & $ \ba{c}\eta<0,\, B_3\ne0,\,
B_2=N=0 \ea $ & Config. 5.6\
([*V.7*]{}) $\bigg\{\!\!\ba{l} \dot x= 1+x,\\[-0.8mm]
\dot y=-xy+y^2 \ea $ & $ \ba{c}\eta>0,\, B_3=\theta=0,\\[-0.8mm]
N\ne0,\, \mu= H_6\!=\!0\ea $ & $ \ba{l} \mbox{Config.\ 5.7} \ea $\
([*V.8*]{}) $\bigg\{\!\!\ba{l} \dot x= gx^2,\ \ g(g^2-1)\ne0\\
\dot y=(g-1)xy+y^2 \ea $ & $ \ba{c} \eta>0,\, B_3=\theta=0,\\
N\ne0,\, \mu\ne0,\, H_1=0 \ea $ & $ \ba{l} \mbox{Config.\ 5.8}\ea $\
([*V.9*]{}) $\bigg\{\!\!\ba{l} \dot x= 2x,\\[-0.8mm]
\dot y=1-x^2-y^2 \ea $ & $ \ba{c}\eta<0,\, B_3=\theta=0,\\[-0.8mm]
N\ne0, \, \mu= H_6\!=\!0\ea $ & $ \ba{l} \mbox{Config.\ 5.9}\ea $\
([*V.10*]{}) $\bigg\{\!\!\ba{l} \dot x= gx^2,\ \ g\ne0 \\
\dot y=-x^2+gxy-y^2 \ea $ & $ \ba{c} \eta<0,\, B_3=\theta=0,\\
N\ne0,\, \mu\ne0,\, H_1=0 \ea $ & $ \ba{l} \mbox{Config.\ 5.10}\ea $\
([*V.11*]{}) $\bigg\{\!\!\ba{l} \dot x=x^2+xy,\\
\dot y=y+y^2 \ea\!\! $ & $\!\! \ba{c} \eta=0,\,M\ne0,\, B_3=\theta=0,\\
\mu\ne0,\, N\ne0,\,D\ne0 \ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.11}\ea $\
([*V.12*]{}) $\bigg\{\!\!\ba{l} \dot x=-1+x^2,\\
\dot y=y^2 \ea $ & $\!\! \ba{l} \eta>0,\, B_2=N=0,\,
B_3\ne0,\\ H_1>0,\,
H_4= H_5=0 \ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.12}\ea $\
([*V.13*]{}) $\bigg\{\!\!\ba{l} \dot x= g(x^2-1),\\
\dot y=2y,\ \ g(g^2-1)\ne0 \ea $ & $\!\! \ba{c} \eta=0,\,M\ne0, B_3=N=0,\\
H\!=\!N_1\!=\!0,\, N_2D\ne0,\, N_5>0 \ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.13}\ea $\
([*V.14*]{}) $\bigg\{\!\!\ba{l} \dot x=(x+1)(gx+1),\\
\dot y=(g\!-\!1)xy, \
g(g^2\!-\!1)\!\ne\!0 \ea\!\! $ & $\!\! \ba{c} \eta=0,\,M\ne0, B_3=\theta=0,\\
NK\ne0,\ \mu=H_6=0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.14}\ea $\
([*V.15*]{}) $\bigg\{\!\!\ba{l} \dot x= g(x^2+1),\\
\dot y=2y,\ \ \ g\ne0 \ea $ & $\!\! \ba{c} \eta=0,\,M\ne0, B_3=N=0,\\
H\!=\!N_1\!=\!0,\, N_2D\ne0,\, N_5<0 \ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.15}\ea $\
[|l|c|c|]{}\
\[0pt\]\[0pt\][Orbit representative]{} & Necessary and sufficient & \[0pt\]\[0pt\][Configuration]{}\
& conditions &\
([*V.16*]{}) $\bigg\{\!\!\ba{l} \dot x=1+x^2,\\
\dot y=y^2 \ea $ & $\!\! \ba{c} \eta>0,\, B_2=N=0,\,
B_3\ne0,\\ H_1<0,\,
H_4= H_5=0 \ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.16}\ea $\
([*V.17*]{}) $\bigg\{\!\!\ba{l} \dot x= x^2,\\
\dot y=2y \ea\!\! $ & $\!\! \ba{c} \eta=0,\,M\ne0, B_3=N=0,\\
H\!=\!N_1\!=\!N_5\!=\!0,\,
N_2D\ne0 \ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.17}\ea $\
([*V.18*]{}) $\bigg\{\!\!\ba{l} \dot x=1+x,\\
\dot y=-xy \ea\!\! $ & $\!\! \ba{c} \eta=0,\,M\ne0, B_3=\theta=0,\\
N\!\ne\!0,\, \mu\!=\!K\!=\!H_6\!=\!0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.18}\ea $\
([*V.19*]{}) $\bigg\{\!\!\ba{l} \dot x=x^2+xy,\\
\dot y=y^2\ea\!\! $ & $\!\! \ba{c} \eta=0,\,M\ne0, B_3=\theta=0,\\
\mu\ne0,\, N\ne0,\,D=0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.19}\ea $\
([*V.20*]{}) $\bigg\{\!\!\ba{l} \dot x=-1+x^2,\\
\dot y=1 \ea\!\! $ & $\!\! \ba{c} \eta\!=\!0,\,M\!\ne\!0, B_3\!=\!N\!=\!H\!=\!0,\\
D=N_1=0, N_2\ne0, N_5>0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.20}\ea $\
([*V.21*]{}) $\bigg\{\!\!\ba{l} \dot x=-1+x^2,\\
\dot y=x+2y \ea\!\! $ & $\!\! \ba{c} \eta=0,\,M\ne0, B_3=N=0,\\
H\!=\!N_2\!=\!0, D\ne0,\,N_1\ne0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.21}\ea $\
([*V.22*]{}) $\bigg\{\!\!\ba{l} \dot x= 1-x^2,\\
\dot y=1-2xy \ea\!\! $ & $\!\! \ba{c} \eta=0,\,M\ne0, B_2=N=0,\\
B_3\ne0,\,H_2=0,\,H_3>0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.22}\ea $\
([*V.23*]{}) $\bigg\{\!\!\ba{l}\dot x=-1+x^2,\\
\dot y=-3+y-x^2+xy \ea\!\! $ & $\!\! \ba{c} \eta=M=0,\, N\ne0,\\
B_3=\theta= N_6=0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.23} \ea $\
([*V.24*]{}) $\bigg\{\!\!\ba{l} \dot x= 1+x^2,\\
\dot y=1 \ea\!\! $ & $\!\! \ba{c} \eta\!=\!0,\,M\!\ne\!0, B_3\!=\!N\!=\!H\!=\!0,\\
D=N_1=0, N_2\ne0, N_5<0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.24}\ea $\
([*V.25*]{}) $\bigg\{\!\!\ba{l}\dot x= -1-x^2,\\
\dot y=1-2xy \ea\!\! $ & $\!\! \ba{c} \eta=0,\,M\ne0, B_2=N=0,\\
B_3\ne0,\,H_2=0,\,H_3<0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.25}\ea $\
([*V.26*]{}) $\bigg\{\!\!\ba{l}\dot x=g-x,\ \ g\in\{0,1\}, \\
\dot y=y-x^2 \ea\!\! $ & $\!\! \ba{c} \eta=M=0,\, N_3\ne0,\\
B_3=N= D_1=0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.26}\ea $\
([*V.27*]{}) $\bigg\{\!\!\ba{l}\dot x=1+x, \\
\dot y=y-x^2 \ea\!\! $ & $\!\! \ba{c} \eta=M=0,\, \,N_4\ne0,\\
B_3\!=\!N\!=\! N_3\!=\!0,\,D_1\ne0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.27} \ea $\
([*V.28*]{}) $\bigg\{\!\!\ba{l} \dot x= x^2,\\
\dot y=1+x \ea\!\! $ & $\!\! \ba{c} \eta=0,\,M\ne0, B_3=N=0,\\
H=D=N_2=0, N_1\ne0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.28}\ea $\
([*V.29*]{}) $\bigg\{\!\!\ba{l}\dot x=-x^2,\\
\dot y=1-2xy \ea\!\! $ & $\!\! \ba{c} \eta=0,\,M\ne0,\,
B_2=N=0,\\ B_3\ne0,\, H_2=H_3=0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.29}\ea $\
([*V.30*]{}) $\bigg\{\!\!\ba{l}\dot x=1,\ \ g\in\{-1,0,1\}, \\
\dot y=g-x^2 \ea\!\! $ & $\!\! \ba{c} \eta=M=0,\, N_4\ne0,\\
B_3=N = N_3=D_1 =0
\ea\!\! $ & $ \ba{l} \mbox{Config.\ 5.30} \ea $\
[|l|l|]{}\
Perturbations& Invariant straight lines\
([*V.7${}_\varepsilon$*]{}):$\ba{l} \dot x= (x+1)(\varepsilon
x+1),\ \dot y= (\varepsilon-1)xy+y^2\!\! \ea$ & $ \ba{l}y=0,\, x=-1,\, y-x=1,\, \varepsilon x=-1 \ea $\
([*V.8${}_\varepsilon$*]{}):$\ba{l} \dot x=
(x+\varepsilon)(gx+\varepsilon),\ \dot y= (g-1)xy+y^2\!\! \ea$ & $ \ba{l}y=0,\, x=-\varepsilon,\, y-x=\varepsilon,\, gx=-\varepsilon \ea $\
([*V.9${}_\varepsilon$*]{}): $\Big\{\ba{l} \dot
x=2x(\varepsilon x+1),\\[-1.4mm] \dot y= 1+2\varepsilon x -x^2+2\varepsilon xy-y^2\!\! \ea$ & $ \ba{l}x=0,\ \varepsilon x=-1,\ y\pm i x+1=0 \ea $\
(): $\Big\{\ba{l} \dot
x=4g\varepsilon^2+\varepsilon(g^2+4)x+gx^2,\\[-1.1mm]
\dot y= \varepsilon^2(4-g^2)-x^2+g xy-y^2\!\! \ea$ & $ \ba{l}x=-g\varepsilon,\ gx=-4\varepsilon,\\[-1.1mm] x+g\varepsilon=\pm i(y+2\varepsilon) \ea $\
():$\ba{l} \dot
x=\varepsilon x+x^2+(1+\varepsilon)xy,\ \dot y= y+ y^2\!\! \ea$ & $ \ba{l} x=0,\, y=-1,\, y=0,\, x+ \varepsilon y=-\varepsilon \ea $\
(): $\ba{l} \dot x=x^2-1,\
\dot y= y^2-\varepsilon^2 \ea$ & $ \ba{l} x=\pm 1,\ y=\pm \varepsilon \ea $\
(): $\ba{l} \dot
x=g(x^2-1),\ \dot y= 2y(\varepsilon y+1) \ea$ & $ \ba{l}y=0,\ x=\pm1,\ \varepsilon y=-1 \ea $\
():$\ba{l} \dot
x=(x\!+\!1)(g x\!+\!1),\, \dot y= (g\!-\!1)xy\!-\!\varepsilon
y^2\!\! \ea$ & $ \ba{l} x\!=\!-1,\, g x\!=\!-1,\, y\!=\!0,\, x\!+\! \varepsilon y\!=\!-1 \!\ea $\
(): $\ba{l} \dot
x=g(x^2+1),\ \dot y= 2y(\varepsilon y+1) \ea$ & $ \ba{l}y=0,\ x=\pm i,\ \varepsilon y=-1 \ea $\
(): $\ba{l} \dot x=x^2+1,\
\dot y= y^2-\varepsilon^2 \ea$ & $ \ba{l} x=\pm i,\ y=\pm \varepsilon \ea $\
():$\ba{l} \dot
x=x^2-\varepsilon^2 ,\ \dot y= 2y(\varepsilon y+1) \ea$ & $ \ba{l}y=0,\ x=\pm\varepsilon,\ \varepsilon y=-1 \ea $\
():$\ba{l} \dot
x=(x\!+\!1)(\varepsilon x\!+\!1),\, \dot y=
(\varepsilon\!-\!1)xy\!-\!\varepsilon y^2\!\! \ea$ & $ \ba{l} x\!=\!-1,\, \varepsilon x\!=\!-1,\, y\!=\!0,\, x\!+\! \varepsilon y\!=\!-1 \!\ea $\
(): $\ba{l} \dot
x=\varepsilon^2 x+ x^2+(1+\varepsilon)xy,\ \dot y= \varepsilon y+
y^2\!\! \ea$ & $ \ba{l} x=0,\, y=0,\, y=-\varepsilon,\, x+ \varepsilon y=-\varepsilon^2 \!\!\ea $\
(): $\ba{l} \dot x=x^2-1,\
\dot y= 1-\varepsilon^2 y^2\ea$ & $ \ba{l} x=\pm 1,\ \varepsilon y=\pm 1 \ea $\
(): $\Big\{\ba{l} \dot
x=(x+1)(x+4\varepsilon x-1),\\[-1.1mm]
\dot y= (x+2y)(1+4\varepsilon y)\!\! \ea$ & $ \ba{l} x=-1,\ x(1+4\varepsilon)=1,\\[-1.1mm] 4\, \varepsilon y=-1,\ x- 8\, \varepsilon y=1 \ea $\
(): $\ba{l} \dot x=1-x^2,\
\dot y= 1-2xy-\varepsilon y^2 \ea$ & $ \ba{l} x=\pm\, 1,\ x+ \varepsilon y=\pm\sqrt{1+\varepsilon} \ea $\
(): $\Bigg\{\ba{l} \dot
x=(1+\varepsilon)(x-1+2\varepsilon)(x+1-2\varepsilon),\\[-1.1mm]
\dot y= (4\varepsilon^2-3)+(1+2\varepsilon)y-x^2\\[-1.1mm]
\qquad +(1-2\varepsilon)xy-2\varepsilon^2 y^2\!\!
\ea$ & $ \ba{l} x=\pm(1-2\varepsilon),\ x+\varepsilon y=1,\\ x+ 2\varepsilon y=1+2\varepsilon \ea $\
(): $\ba{l} \dot x=x^2+1,\
\dot y= 1-\varepsilon^2 y^2\ea$ & $ \ba{l} x=\pm i,\ \varepsilon y=\pm 1 \ea $\
(): $\ba{l} \dot x=-1-x^2,\
\dot y= 1-2xy-\varepsilon y^2 \ea$ & $ \ba{l} x=\pm\, i,\ x+ \varepsilon y=\pm\, i\sqrt{1-\varepsilon} \ea $\
(): $\Bigg\{\ba{l} \dot
x=g+(2g\varepsilon-1)x-2\varepsilon x^2,\\[-1.1mm]
\dot y= (2g\varepsilon+1)y-x^2-6g\varepsilon^2xy\\[-1.1mm]
\qquad +3\varepsilon^2(1+2g\varepsilon-3g^2\varepsilon^2) y^2\!\!
\ea$ & $ \ba{l} x=g,\ 3\varepsilon[x+\varepsilon(3g\varepsilon+1)y]=-1,\\
2\varepsilon x=-1,\ \varepsilon[x+3\varepsilon(g\varepsilon-1)y]=1 \ea $\
(): $\Big\{\ba{l} \dot
x=1+ x+ \varepsilon x^2,\\[-1.1mm]
\dot y= y-x^2 -2\varepsilon xy-2\varepsilon^2y^2\!\! \ea$ & $ \ba{l}1+x+ \varepsilon^2x^2=0,\\[-1.1mm] \varepsilon(x+2\varepsilon y)^2-(x+2\varepsilon y)=1 \ea $\
(): $\Bigg\{\ba{l} \dot
x=(\varepsilon-1)\varepsilon^2+2\varepsilon^3 x\\[-1.1mm]
\qquad+(1-\varepsilon)(1-2\varepsilon+3\varepsilon^2)x^2,\\[-1.1mm]
\dot y= (1-\varepsilon)(2\varepsilon^2y+1)(x+2\varepsilon y+1)
\!\!
\ea$ & $ \ba{l}(\varepsilon-1)x=\varepsilon,\ 2\varepsilon^2y=-1,\\[-1.1mm]
(1-2\varepsilon+3\varepsilon^2)x=\varepsilon(1-\varepsilon),\\[-1.1mm]
(\varepsilon -1)^2x-4\varepsilon^3 y=\varepsilon(\varepsilon+1) \ea $\
(): $\ba{l} \dot
x=\varepsilon^2-x^2,\ \dot y= 1-2xy-\varepsilon y^2 \ea$ & $ \ba{l} x=\pm\, \varepsilon,\ x+ \varepsilon y=\pm\sqrt{\varepsilon^2+\varepsilon} \ea $\
(): $\Big\{\ba{l} \dot
x=1+ \varepsilon x+ \varepsilon^3 x^2,\\[-1.1mm]
\dot y= g+ \varepsilon y-x^2 -2\varepsilon^3
xy-2\varepsilon^6y^2\!\!
\ea$ & $ \ba{l}1+\varepsilon x+ \varepsilon^3x^2=0,\\[-1.1mm]
\varepsilon^3(x\!+\!2\varepsilon^3 y)^2\!-\!\varepsilon(x\!+\!2\varepsilon^3 y)\!=\!1\!+\!2g\varepsilon^3\!\!\! \ea $\
*Proof of Theorem \[th\_mil\_5\]:* Since we only discuss the case $C_2\ne0$, in what follows it suffices to consider only the canonical forms $({{\bf S}}_I)$ to $({{\bf S}}_{I\,V})$. The idea of the proof is the same as in the proof of the Theorem \[th\_mil\_6\]. We shall perform a case by case discussion for each one of these canonical forms, for which according to Lemma \[lm\_NB2\] we must examine two subcases: $(i)\ N=B_2=0$ and $ (ii)\ N\ne0$, $\theta=B_3=0$. Each one of these conditions yields specific conditions on the parameters. The discussion proceeds further by breaking these cases in more subcases determined by more restrictions on the parameters. Finally we construct invariants or T-comitants which put these conditions in invariant form.
Systems with the divisor $D_S(C,Z)=1\cdot w_1+1\cdot w_2+1\cdot w_3$
--------------------------------------------------------------------
For this case we shall later need the following polynomials which are shown to be $T$-comitants in Lemma \[Table:Propreties\]. \[not\_H4,5\] Let us denote $$\bal
&H_4(a)=\big((C_2,D)^{(2)},(C_2,D_2)^{(1)}\big)^{(2)},\
H_6(a,x,y)= 16N^2(C_2,D)^{(2)}+ H_2^2(C_2,C_2)^{(2)},\\
&H_5(a)=\big((C_2,C_2)^{(2)},(D,D)^{(2)}\big)^{(2)}+
8\big((C_2,D)^{(2)},(D,D_2)^{(1)}\big)^{(2)}.
\eal$$
### The case $N=0=B_2$
It was shown above (see page ) that the systems $(S_I)$ with $N({\mbox{\boldmath $a$}},x,y)=0$ can be brought by an affine transformation to the systems (\[s4.1\]) for which we have &&B\_2=648. Hence the condition $B_2=0$ yields $de=e(4k-4l-e^2)=d(4k-4l+d^2)=0$. According to Lemma \[gcd:4\], in order to have $M_{{}_{{\bf I\bf L}}}=5$ we must satisfy the condition $\
\deg\gcd({\cal E}_1, {\cal E}_2)=4. $ We claim, that for this it is necessary that $d=e=0$. Indeed, let us suppose, that $de=0$ but $d^2+e^2\ne0$. Then by interchanging $x$ and $y$ we may assume $d=0,\ e=2$ via Remark \[rem:transf\] ($\gamma=e/2,\ s=1$). Then we obtain the systems \[ss2\_1\] x=k + x\^2,y=l +2x + y\^2, for which the condition $B_2=2^73^4(k-l-1)x^4=0$ yields $k=l+1$. Then for the systems (\[ss2\_1\]) with $k=l+1$ we obtain $$\bal
&{\cal E}_1= -2\big[Y^2 - YZ + Z( X + Z + l Z)\big]{\cal H},\quad
{\cal E}_2=-(X + Y - Z)\big[Y^2 + Z(2 X + l Z)\big]{\cal H},\\
\eal$$ where $ {\cal H}= (Y-X+Z)(X^2+Z^2+lZ^2)$. Thus, $\deg{\cal H}=3$ and we shall show that for all values given to the parameter $l$ the degree of $\gcd({\cal E}_1,{\cal E}_2)$ remains three. Indeed, since the common factor of the polynomials ${\cal E}_1/{\cal H}$ and ${\cal E}_2/{\cal H}$ must depend on $Y$, according to Lemma \[Trudi:2\] it is sufficient to observe that $ {\mbox{\rm Res\,}}_Y({\cal
E}_1/{\cal H},\ {\cal E}_2/{\cal H})=-8 Z^2[X^2 + (1 +
l)Z^2]^2\ne0$. This proves our claim and hence, the condition $d=e=0$ must hold. Since for systems (\[s4.1\]) we have $H_4=96(d^2+e^2)$ this condition is equivalent to $H_4=0$.
Assuming $H_4=0$ (i.e. $d=e=0$) the systems (\[s4.1\]) become \[2CF\_1\] x=k + x\^2,y=l + y\^2, and calculations yield $\quad
{\cal E}_1= 2(X - Y)\,{\cal H},\quad
{\cal E}_2= \big[X^2 - Y^2 + (k - l)Z^2\big]{\cal H},$ where${\cal H}= (X^2 + k Z)(Y^2 + l Z^2). $ Hence by Theorem \[theor:E1,E2\] each system in the family possesses four invariant affine lines which means that for these systems $M_{{}_{{\bf I\bf L}}}\ge5$. We observe that to have an additional common factor of ${\cal E}_1$ and ${\cal E}_2$ it is necessary and sufficient that $k-l=0$. So, to have $M_{{}_{{\bf I\bf L}}}=5$ the condition $k-l\ne0$ must be satisfied. This condition is equivalent to $B_3\ne0$, since for the systems (\[2CF\_1\]) we have $
B_3=4(l-k)x^2y^2$.
Systems (\[2CF\_1\]) possess the invariant lines, components of the conics: $x^2+k=0,$ $ y^2+l=0, $ and then we obtain the following configurations of invariant straight lines ([**Diagram 2**]{}):\
$(i)$ Config. 5.3 for $k<0$ and $l<0$; $(ii)$ Config. 5.4 for $kl<0$; $(iii)$Config. 5.5 for $k>0$ and $l>0$; $(iv)$ Config. 5.12 for $kl=0$ and $k+l<0$; $(v)$ Config. 5.16 for $kl=0$ and $k+l>0$.
On the other hand for the systems (\[2CF\_1\]) we have:$
H_1= -2^73^2\,(k+l), \ \ H_5= 2^{11}3\,kl.
$Herein we conclude, that these two $T$-comitants capture in invariant form exactly the conditions for distinguishing the Configurations 5.3–5.5, 5.12 and 5.16 as it is indicated in Table 4.
We observe, that if for the systems (\[2CF\_1\]) the condition $H_5\le0$ holds (i.e. $kl\le0$) then interchanging $x$ and $y$ we may assume $l\ge0$. Moreover, by Remark \[rem:transf\] ( $\gamma=|k|,\ s=1/2$) we may assume $k\in\{-1,1\}$. We also note that for $l<0$ (respectively, $l>0$) we may set $l=-g^2$ (respectively, $l=g^2$) and due to the substitution $y\to gy$ we obtain the canonical system ([*V.3*]{}) (respectively, ([*V.4*]{}) and ([*V.5*]{}) ) from Table 4.
### The case $N\ne0$, $\theta=0=B_3$
For the canonical systems $({{\bf S}}_I)$ we calculate $
\theta= -8(h-1)(g-1)(g+h).
$ Hence the condition $\theta=0$ yields $(h-1)(g-1)(g+h)=0$ and without loss of generality we can consider $h=1$. Indeed, if $g=1$ (respectively, $g+h=0$) we can apply the linear transformation which will replace the straight line $x=0$ with $
y=0$ (respectively, $x=0 $ with $y=x$) reducing this case to $h=1$. Assume $h=1$. Then $N=(g^2-1)x^2\ne0$ and we may assume $e=f=0$ via a translation. Thus the systems $({{\bf S}}_I)$ become \[s1\_kap\] x=k + cx+dy +g x\^2, y=l + (g-1)xy + y\^2 and calculations yield $ \mu =\ 32g^2$ and $B_3 =
-3l(g-1)^2x^4+6l(g-1)^2x^3y-6d^2gxy^3+3d^2gy^4\
+3\left[(4gl-k(g+1)^2+c^2+cd-cdg\right]x^2y^2.$The condition $B_3$=0 implies $dg=0$. We shall examine two subcases: $\mu\ne0$ and $\mu=0$. [**The subcase $\mu\ne0$.**]{} In this case we obtain $g\ne0$, and from $g-1\ne0$ the condition $B_3=0$ for the systems (\[s1\_kap\]) yields $d= l= c^2-k(g+1)^2=0$. Since $N\ne0$ then $g+1\ne0$ and we may set $c=u(g+1) $ where $u$ is a new parameter. Then $k=u^2$ and we obtain the systems\
\[s1\_kap\_\] x=u\^2 + u(g+1)x + gx\^2, y=(g-1)xy +y\^2, for which $H_1=576u^2(g-1)^2$.
[**1)**]{} If $ H_1\ne0$ then $u\ne0$ and we may assume $u=1$ via Remark \[rem:transf\] ($\gamma=u$, $s=1$). This leads to the systems \[2CF\_18\] x=(x+1)(gx+1), y=(g-1)xy +y\^2, for which $g(g^2-1)\ne0$ and calculations yield: \[GCD:1\] [H]{}=([E]{}\_1,[E]{}\_2)= Y (Y-X - Z) (X + Z) (g X + Z). Hence, $\deg {\cal H}=4$. By hypothesis $N\ne0$ and hence, according to Lemma \[gcd:4\] for every $ g$ such that $g(g^2-1)\ne0$, $M_{{}_{{\bf I\bf L}}}\le5$. By Theorem \[theor:E1,E2\], from (\[GCD:1\]) the systems (\[2CF\_18\]) possess the following four distinct invariant affine lines: $
y=0,\ \ x+1=0,\ \ x-y+1=0,\ \ gx+1=0.
$ Thus we obtain the Config. 5.1. [**2)**]{} For $ H_1=0$ we have $u=0$ and the systems (\[s1\_kap\_\]) become \[2CF\_20\] x= gx\^2, y=(g-1)xy +y\^2, with $g(g^2-1)\ne0$ and we calculate: $
{\cal H}=\gcd\left({\cal E}_1,{\cal E}_2\right)= g X^2Y(X - Y).$ Hence $\deg {\cal H}=4$ and we obtain $M_{{}_{{\bf I\bf L}}}\ge5$. Since $N\ne0$ by Lemma \[gcd:4\], $M_{{}_{{\bf I\bf L}}}$ cannot be equal to 6. The systems (\[2CF\_20\]) possess the invariant lines $x=0,$ $y=0$ and $x=y$. Moreover, according to Lemma \[lm3\] the line $x=0$ could be of multiplicity two and the perturbations ([*V.8${}_\varepsilon$*]{}) from Table 5 show this. Hence, for $H_1=0$ we obtain Config. 5.8. [**The subcase $\mu=0$.**]{} The condition $\mu =32g^2=0$ yields $g=0$, and for the systems (\[s1\_kap\]) the condition $B_3=0$ yields $g=l=c(c+d)-k=0$. Thus, $g=l=0$, $k=c(c+d)\ne0$, otherwise we get degenerate systems (\[s1\_kap\]). Hence, we may assume $c=1$ via Remark \[rem:transf\] ($\gamma=c$, $s=1$) and we obtain the systems \[S\_mu0\] x=d+1 +x+dy, y=-xy +y\^2. Calculations yield: $$\bal
& {\cal E}_1= \left[-X^2 + 2 X Y + d(Y + Z)^2 + Z(2 Y +
Z)\right]{\cal H},\qquad {\cal H}= Y Z (X - Y + Z + d Z),\\
&{\cal E}_2= (Y-X) (Y + Z) \big[X + Z + d (Y + Z)\big]{\cal
H},\quad {\mbox{\rm Res\,}}_X({\cal E}_1/{\cal H},\ {\cal E}_2/{\cal H})=-9d(d +
1)^2(Y + Z)^6.
\eal$$ Hence $\deg\,{\cal H}=\gcd\left({\cal E}_1,{\cal E}_2\right)=3$ and the condition on the parameter $d$ so as to have an additional common factor of ${\cal E}_1$, ${\cal E}_2$, according to Lemma \[Trudi:2\] is ${\mbox{\rm Res\,}}_X({\cal E}_1/{\cal H},\ {\cal
E}_2/{\cal H})\equiv0$. Since $d+1\ne0$ (otherwise we get the degenerate system ) this condition yields $d=0$. Then we obtain the following system \[2CF\_22\] x=1 +x, y=-xy +y\^2 for which ${\cal H}=\gcd\left({\cal E}_1,{\cal
E}_2\right)=YZ(X+1)(X-Y+Z)$. We observe that this system possesses the invariant affine straight lines: $
y=0,\ \ x+1=0,\ \ x-y+1=0.$ Taking into account that $Z\mid{\cal H}$, we have by Corollary \[Mult:Z=0\] that the line $Z=0$ could be of multiplicity two. This is confirmed by the perturbations ([*V.7${}_\varepsilon$*]{}) from Table 5. On the other hand for the systems (\[S\_mu0\]) calculations yields $H_6=128dx^2(x^2-xy+y^2)(x^2-2xy-dy^2)$. Hence, the conditions $g=0=d$ are equivalent to $\mu=0$ and $H_6=0$. In this case we obtain Config. 5.7.
Systems with the divisor $D_S(C,Z)=1\cdot w_1^c+1\cdot w_2^c+1\cdot w_3$
------------------------------------------------------------------------
We are in the case of the canonical form $({{\bf S}}_{I\!I})$.
### The case $N=0=B_2$
It was shown above (see page ) that the systems $({{\bf S}}_{I\!I})$ with $N({\mbox{\boldmath $a$}},x,y)=0$ can be brought by an affine transformation to the systems (\[S2:N=0\]) for which we have && B\_2=648x\^4-16k(e\^2-f\^2)xy(x\^2-y\^2) -48efkx\^2y\^2 +8efky\^4,\
&& B\_3=6. If $B_3=0$ then $k=e=f=0$ and we obtain the systems (\[CF\_3\]) for which $M_{{}_{{\bf I\bf L}}}=6$ (see page ). Hence $B_3\ne0$ and this implies $k\ne0$, otherwise from $B_2=0$ we obtain again $e=f=0$. Therefore $k\ne0$ and we may consider $k>0$ via the change $x\to -x$ and by Remark \[rem:transf\] ($\gamma=k,\ s=1/2$) we may assume $k=1$. Then the condition $B_2=0$ yields $e=f=0$ and we obtain the systems \[2CF\_4\] x= 1 +2xy, y= l -x\^2 + y\^2 which possess the invariant lines$ y+ix=\pm
\sqrt{-l-i}$,$y-ix=\pm\sqrt{i-l}.$This leads to the Config. 5.6.
### The case $N\ne0,$ $\theta=B_3=0$
For the systems $({{\bf S}}_{I\!I})$ we calculate\
$$\theta= 8(h+1)[(h-1)^2+g^2],\quad N=(g^2-2h+2)x^2 +2g(h+1)xy+(h^2-1)y^2$$ and hence by $N\ne0$, the condition $\theta=0$ yields $h=-1$. Then we may assume $f=0$ due to a translation and the systems $({{\bf S}}_{I\!I})$ become \[S\_IIa\] x=k+cx+dy+gx\^2,y= l+ex-x\^2+gxy-y\^2. For these systems calculations yield Coefficient$[B_3,\
y^4]=-3d^2g $ and $ \mu= 32\,g^2$. We shall examine two subcases: $\mu\ne0$ and $\mu=0$. [**The subcase $\mu\ne0$**]{}. This yields $g\ne0$ and then the condition $B_3=0$ implies $d=0$. Moreover, we may assume $c=0$ via the translation of the origin of coordinates to the point $(-c/(2g),-c/4)$. Thus, the systems (\[S\_IIa\]) become $$\dot x= k +2gx^2,\quad
\dot y= l +ex -x^2 +2gxy -y^2,$$ for which we calculate $\ B_3 =
3\left[k(4-g^2)-4gl\right]x^2(x^2-y^2)+
6\left[l(4-g^2)+4gk+e^2\right]x^3y.
$ Hence, the condition $B_3=0$ yields the following linear system of equations with respect to parameters $k$ and $l$:\
$$k(4-g^2)-4gl=0, \quad 4gk+l(4-g^2)+e^2=0.$$ Setting $e=u(g^2+4)$ ( $u$ is a new parameter) we have the following solution of this system: $k=-4gu^2$, $l=(g^2-4)u^2$. Thus we obtain the systems \[S\_II\_kap0\] x= -4gu\^2 + gx\^2, y= (g\^2-4)u\^2 +u(g\^2+4)x -x\^2 +g xy -y\^2, for which $H_1= -2^{12}3^2u^2g^2$. [**1)**]{} If $H_1\ne0$ we have $u\ne0$ and we can assume $u=1$ via the Remark \[rem:transf\] ($\gamma=u,$ $s=1$). Hence the systems (\[S\_II\_kap0\]) become \[2CF\_19\] x= g(x\^2-4), y= (g\^2-4) +(g\^2+4)x -x\^2 +gxy -y\^2. and calculations yield:$\quad {\cal H}= g(X - 2 Z)(X + 2
Z)(X^2+Y^2-4 X Z + 2\,g Y Z + 4Z^2 + g^2 Z^2). $ Hence, $M_{{}_{{\bf I\bf L}}}\ge5$ and since $N\ne0$ by Lemma \[gcd:4\] $M_{{}_{{\bf I\bf L}}}$ cannot be equal to 6. By Theorem \[theor:E1,E2\] the systems (\[2CF\_19\]) possess the following four distinct invariant straight lines: $y-ix+g+2i=0,$ $ y+ix-2i+g=0, $ $ x=\pm2.$ Thus we obtain the Config. 5.2. [**2)**]{} For $H_1=0$ we have $u=0$ and the systems (\[S\_II\_kap0\]) become \[2CF\_21\] x= gx\^2, y= -x\^2 +gxy -y\^2, with $g\ne0$. We calculate $ {\cal H}=\gcd\left({\cal
E}_1,{\cal E}_2\right)= g X^2(X^2 + Y^2) $ and hence $\deg
{\cal H}=4$. Since $N\ne0$ by Lemma \[gcd:4\] we obtain that $M_{{}_{{\bf I\bf L}}}$ equals exactly 5. The systems (\[2CF\_20\]) possess the following invariant straight lines: $ x=0,\ \ y=\pm ix$ and the line $x=0$ could be of multiplicity two. This is confirmed by the perturbations ([*V.10${}_\varepsilon$*]{}) from Table 5. Thus we get Config. 5.10. [**The subcase $\mu=0$.**]{} Then we obtain $g=0$ and we may assume $e=0$ via a translation. Therefore the systems (\[S\_IIa\]) become $\quad
\dot x= k+cx+dy,\quad \dot y= l -x^2 -y^2
\quad $ and we calculate$ B_3= 12kx^4 +
6(4l-c^2-d^2)x^3y-12kx^2y^2. $ Hence the condition $B_3=0$ yields $k=4l-c^2-d^2=0$. We replace $c$ by $2c$ and $d$ by $2d$ and then we obtain $l=c^2+d^2$. This leads to the systems: \[S\_II\_2\] x= 2cx+2dy,y= c\^2+d\^2 -x\^2 -y\^2 for which calculations yield: $$\bal
&{\cal E}_1= \left[d X^2 - 2\,c XY - d Y^2 + 2\,
(c^2+d^2)XZ +d(c^2+d^2) Z^2\right]{\cal H},\\
&{\cal E}_2=(c X + d Y)[X^2 + Y^2 + 2\,d X Z -
2\,cY Z + (c^2 + d^2) Z^2]\,{\cal H},\\
& {\cal H}=2 Z[X^2 + Y^2 - 2\,d X Z + 2\,c Y Z + (c^2 + d^2) Z^2].
\eal$$ Thus, $\deg{\cal H}=3$ and we need an additional factor of ${\cal
E}_1$ and ${\cal E}_2$. Since $c^2+d^2\ne0$ we observe that such common factor of the polynomials ${\cal E}_1/{\cal H}$ and ${\cal
E}_2/{\cal H}$ must depend on $X$. Hence, by Lemma \[Trudi:2\] the following condition must hold: $$\bal
{\mbox{\rm Res\,}}_X({\cal E}_1/{\cal H},\ {\cal E}_2/{\cal H})&=4\,d(c^2 +
d^2)^2(Y - c Z)^6=0.
\eal$$ Therefore the condition $d=0$ must be satisfied and then $c\ne0$ (otherwise we get the degenerate system ). On the other hand for the systems (\[S\_II\_2\]) we have $H_6=-2^{13}dx^3(3x^2-y^2)(dx^2-2cxy-dy^2)$ and hence the condition $d=0$ is are equivalent to $H_6=0$. We may assume $c=1$ via the Remark \[rem:transf\] ($\gamma=c$, $s=1$) and then we obtain the system \[2CF\_23\] x= 2x,y= 1 -x\^2 -y\^2. For this system we calculate $\quad {\cal H}=\gcd\left({\cal
E}_1,{\cal E}_2\right)=4\, XZ(X^2 + Y^2 + 2Y Z + Z^2) $ and according to Theorem \[theor:E1,E2\] the system (\[2CF\_23\]) possess the invariant affine lines: $ x=0$ and $y\pm ix +1=0 $.Moreover, by Corollary \[Mult:Z=0\] the line $l_\infty: Z=0$ could be of multiplicity two. This is confirmed by the perturbations ([ *V.9${}_\varepsilon$*]{}) from Table 5. Therefore we obtain the Config. 5.9.
Systems with the divisor $D_S(C,Z)=2\cdot w_1+1\cdot w_2$
---------------------------------------------------------
We are in the case of the canonical form $({{\bf S}}_{I\!I\!I})$. For this case we shall later need the following polynomial which is shown to be a $CT$-comitant in Lemma \[Table:Propreties\]. \[not\_N5\] Let us denote$N_5({\mbox{\boldmath $a$}},x,y)=
\big((D_2,C_1)^{(1)} + D_1D_2\big)^2
-4\big(C_2,C_2\big)^{(2)}\big(C_0,D_2\big)^{(1)}$.
### The case $N=0=B_2$
It was previously shown (see page ) that to examine the systems $({{\bf S}}_{I\!I\!I})$ with $N({\mbox{\boldmath $a$}},x,y)=0$ we have to consider two subcases: $H({\mbox{\boldmath $a$}},x,y)\ne0$ and $H({\mbox{\boldmath $a$}},x,y)=0$. [**The subcase $H({\mbox{\boldmath $a$}},x,y)\ne0.$**]{} In this case the systems $({{\bf S}}_{I\!I\!I})$ with $N=0$ can be brought by an affine transformation to the systems (\[S3\_NM\_\]) (see page ) for which we have: $\ B_2= -648d(8clx^4+
16dlx^3y+ d^3 y^4).$ Therefore, the condition $B_2=0$ yields $d=0$ and we obtain the systems \[S3\_B2\] x =k +cx -x\^2, y =l - 2xy, for which calculations yield: $\
{\cal E}_1=\left[-2 X^2 Y + Z^2(2 k Y + c l Z)\right] {\cal H},
$${\cal H}=(kZ^2+cXZ-X^2),$ $
{\cal E}_2= (X^2 - c X Z - k Z^2)(2 X Y - l Z^2) {\cal H}.$ Thus, $\deg{\cal H}=2$ and to have $M_{{}_{{\bf I\bf L}}}=5$ the polynomials ${\cal E}_1/{\cal H}$ and ${\cal E}_2/{\cal H}$ must have a common factor of degree two. We observe, that this common factor necessarily depends on $X$ and hence by Lemma \[Trudi:2\] the following condition must hold: $${\mbox{\rm Res\,}}_X({\cal E}_1/{\cal H},\ {\cal E}_2/{\cal H})=-2 c^2 Y Z^6(4k
Y^2 + 2cl YZ - l^2 Z^2)^2\equiv0.$$ Herein we obtain either $c=0$ or $k=l=0$, but the second case yields degenerate systems. So, we assume $c=0$ and then for the systems (\[S3\_B2\]) we obtain $\quad {\cal E}_1= 2Y\,{\cal H},$ $\ {\cal H}= (kZ^2-X^2)^2,$ $ {\cal E}_2= (-6 X Y + 3 l
Z^2)\,{\cal H}.$ We observe, that $\deg\,{\cal H}=4$ and that the polynomials ${\cal E}_1$ and ${\cal E}_2$ do not have an additional common factor if and only if $l\ne0$. This condition is equivalent to $B_3\ne0$, since for the systems (\[S3\_B2\]) we have $B_3=-12lx^4$. By $l\ne0$ we may consider $l=1$ via the rescaling $y\to ly$ and we obtain the systems \[2CF\_7\] x =k -x\^2, y =1 - 2xy. Moreover, due to the re-scaling $x\to|k|^{1/2}x$, $y\to|k|^{-1/2}y$ and $t\to|k|^{-1/2}t$ (for $k\ne0$) we may assume $k\in \{-1,0,1\}$. These systems possess two invariant lines $x=\pm\sqrt{k}$. By Lemma \[lm3\] for $k\ne0$ each one of these lines could be of multiplicity two and for $k=0$ the invariant line $x=0$ of the system (\[2CF\_7\]) is of the multiplicity four. This is confirmed by the perturbations ([*V.22${}_\varepsilon$*]{}) (respectively, ([*V.25${}_\varepsilon$*]{}); ([*V.29${}_\varepsilon$*]{})) from Table 5 for $k=1$ (respectively, $k=-1$; $k=0$). Thus, we obtain Config. 5.22 (respectively, Config. 5.25; Config. 5.29).
On the other hand for the systems (\[S3\_B2\]) calculations yield: $H_2= 16cx^2$ and $H_3= 32kx^2$. Hence, these $T$-comitants capture exactly the conditions $c=0$ and $k<0$ (respectively $c=0$, $k=0$ or $c=0$, $k>0$) . It remains to observe, that the condition $B_3\ne0$ implies $H\ne0$, since for $H=0$ the condition $B_2=0$ implies $B_3=0$ (see the subcase $H(a,x,y)=0$ below). [**The subcase $H(a,x,y)=0$.**]{} It is previously shown (see page ) that if $N=H=0$ then the systems $({{\bf S}}_{I\!I\!I})$ can be brought by an affine transformation to the systems (\[S3\_NH\_0\]). For these systems we have $\ B_2=
-648d^4y^4,$ $ B_3= 6dxy^2(fx-dy) $ and hence the condition $B_2=0$ yields $d=0$. Therefore the conditions $B_2=0$ and $B_3=0$ are equivalent and since for any quadratic system the condition $B_3=0$ implies $B_2=0$ (see the formulas on page ), we shall use in this case the condition $B_3=0$.
Assuming $d=0$ we obtain the systems (\[s4.5\]) for which $D(x,y)=-f^2x^2y$ and we shall consider two subcases: $D\ne0$ and $D=0$. [**1)**]{} For $D\not=0$ the systems (\[s4.5\]) can be brought by an affine transformation to the systems (\[s4.6\]) and calculations yield the values (\[val:Ei\]) of the affine comitants ${\cal E}_1$ and ${\cal E}_2$. We observe that $\deg\,{\cal H}=3$ and taking into account that the polynomials ${\cal E}_1/{\cal H}$ and ${\cal E}_2/{\cal H}$ cannot have the common factor $Z$, to have an additional factor of these polynomials according to Lemma \[Trudi:2\] at least one of the following two conditions must hold: $\ {\mbox{\rm Res\,}}_X({\cal E}_1/{\cal
H},\, {\cal E}_2/{\cal H})= -36 e(k+1)(4 Y^2 Z + e^2k Z^3)^2=0,\ $ $ {\mbox{\rm Res\,}}_Y({\cal E}_1/{\cal H},\, {\cal E}_1/{\cal H})=-6e(X^2 +
kZ^2)^2=0. $ Thus we obtain either the condition $e=0$ or $k=-1$.On the other hand for systems (\[s4.6\]) we obtain $N_1=8ex^4$ and $N_2=16(k+1)x$ and we shall consider two subcases: $N_1=0$ and $N_1\ne0,$ $N_2=0$.
[**1a)**]{} Assume $N_1=0$. Then $e=0$ and the systems (\[s4.6\]) become \[2CF\_10\] x=k+ x\^2,y= 2y. Calculations yield: $ \
{\cal E}_1=2(X-Z)\,{\cal H},\ $ $
{\cal E}_2= 3(X^2 + k Z^2)\,{\cal H},\ $ $ {\cal H}=4YZ(kZ^2+X^2),
$ $ {\mbox{\rm Res\,}}_X({\cal E}_1/{\cal H},\ {\cal E}_2/{\cal
H})=12(k+1)Z^2.$ Hence $\deg{\cal H}=4$ and we observe that in order not to have an additional common factor of the polynomials ${\cal E}_1$ and ${\cal E}_2$ we must have $k+1\ne0$ (i.e. $N_2\ne0$). The systems (\[2CF\_10\]) possess the invariant affine lines $y=0,\ x=\pm\sqrt{-k} $. According to Corollary \[Mult:Z=0\] the line $l_\infty: Z=0$ could be of multiplicity 2 and the line $x=0$ also could be of multiplicity 2 in the case when $k=0$. Since for systems (\[s4.6\]) we have $N_5=-64kx^2$, we obtain Config. 5.11 for $N_5>0$, Config. 5.15 for $N_5<0$ and Config. 5.17 for $N_5=0$.
Note that for $k<0$ (respectively, $k>0$) one can set $k=-g^2$ (respectively, $k=g^2$) and due to the substitution $x\to gx$ we obtain the canonical system ([*V.13*]{}) (respectively, ([*V.15*]{})) from Table 4. It remains to observe that the perturbations ([*V.13${}_\varepsilon$*]{}), ([*V.15${}_\varepsilon$*]{}) and ([*V.17${}_\varepsilon$*]{}) from Table 5 confirm the validity of the Config. 5.13, 5.15 and 5.17, respectively.
[**1b)**]{} For $N_1\ne0$, $N_2=0$ we have $e\ne0$, $k=-1$ and then for the systems (\[s4.6\]) calculations yield: $\ {\cal E}_1=
\left[4 Y + e (X - Z)\right]{\cal H}, $ $
{\cal E}_2= (ce X + 2 Y) (X + Z)\,{\cal H}, $ $
{\cal H}= Z(X+Z)(X-Z)^2, $ $
{\mbox{\rm Res\,}}_Y({\cal E}_1/{\cal H},\ {\cal E}_2/{\cal H})=-2 e(X + Z)^2.
$ Hence $\deg{\cal H}=\deg\gcd\left({\cal E}_1,{\cal
E}_2\right)= 4$ and since $N_1\ne0$ (i.e. $e\ne0$) the polynomials ${\cal E}_1$ and ${\cal E}_2$ could not have an additional common factor. Assuming $e=1$ via the rescaling $y\to
ey$ the systems (\[s4.6\]) become \[2CF\_13\] x=-1+ x\^2,y= x+2y. This system possesses the invariant lines $x=\pm1$. According to Lemma \[lm3\] and Corollary \[Mult:Z=0\] the line $x=1$ as well as the line $Z=0$ could be of multiplicity two. This is confirmed by the perturbations ([*V.21${}_\varepsilon$*]{}) from Table 5. Thus for $N_1\ne0$ and $N_2=0$ we obtain Config. 5.21. [**2)**]{} If $D=0$ then we have $f=0$ and the systems (\[s4.5\]) become the systems (\[s4.7\]) (see page ) for which calculations yield the corresponding expressions (\[val:Eia\]) for the affine comitants ${\cal E}_1$ and ${\cal E}_2$. As $\deg\,{\cal H}=3$ we need an additional common factor of ${\cal E}_1$ and ${\cal E}_2$. Taking into account that these polynomials depend only on $X$ and $Z$, according to Lemma \[Trudi:2\] at least one of the following two conditions must hold: $${\mbox{\rm Res\,}}_X({\cal E}_1/{\cal H},\ {\cal E}_2/{\cal H})=-4\,ek(e^2 k +
l^2)^2 Z^6=0,\ \,{\mbox{\rm Res\,}}_Z({\cal E}_1/{\cal H},{\cal E}_2/{\cal
H})=-4\,ek(e^2 k + l^2)^2 X^6=0.$$ Hence we obtain either $ek=0$ or $e^2 k + l^2=0$. Since the second condition leads to degenerate systems, we must examine the conditions $e=0$ and $k=0$. For systems (\[s4.7\]) we have $N_1=8ex^4$ and $N_2=16kx$ and we shall consider two subcases: $N_1=0$ and $N_1\ne0$, $N_2=0$.
[**2a)**]{} Assume $N_1=0$. Then $e=0$ and the systems (\[s4.7\]) become $ \ \dot x=k+ x^2,\ \dot y= l.\ $ Calculations yield $\
{\cal E}_1= X\,{\cal H}, \ $ $ \
{\cal E}_2= (X^2 + k Z^2)\,{\cal H},\ $ $\
{\cal H}= \,l Z^2(X^2 + k Z^2),\ $ Therefore $\deg\,{\cal H}=4$ and the polynomials ${\cal E}_1$ and ${\cal E}_2$ could not have an additional common factor if and only if $k\ne0$ (i.e. $N_2\ne0$). Since $l\ne0$ (otherwise we get degenerate systems) after the rescaling $x\to |k|^{1/2} x$, $
y\to l|k|^{-1/2}y$ and $ t\to|k|^{-1/2}t$ we get the systems \[2CF\_14\] x=k+ x\^2,y= 1 with $k\in\{-1,1\}$. The systems (\[2CF\_14\]) possess two invariant affine lines: $x=\pm\sqrt{-k}$ which are distinct due to the condition $k\ne0$. Moreover, by Corollary \[Mult:Z=0\] the line $Z=0$ could be of multiplicity three. This is confirmed by the perturbations ([*V.20${}_\varepsilon$*]{}) (for $k<0$) and ([*V.24${}_\varepsilon$*]{}) (for $k>0$) from Table 5. On the other hand for systems (\[2CF\_14\]) we have $N_5=-64kx^2$. Therefore, for $N_1=0$ and $N_2\ne0$ we obtain the Config. 5.20 if $N_5>0$ and the Config. 5.24 if $N_5<0$.
[**2b)**]{} Let $N_1\ne0$, $N_2=0$. In this case we have $e\ne0$, $k=0$ and systems (\[s4.7\]) become \[S\_III\_a\] x= x\^2,y= l+ex. For the systems (\[S\_III\_a\]) calculations yield: $\ {\cal
E}_1=(e X + 2 l Z)\,{\cal H},\ $ $\
{\cal E}_2= X (e X + l Z)\,{\cal H},\ $ $
{\cal H}= X^3Z.$ Therefore $\deg\,{\cal H}=4$ and since $l\ne0$ (otherwise we get the degenerate systems (\[S\_III\_a\])) and $e\ne0$ ( $N_1\ne0$) we conclude that the polynomials ${\cal E}_1$ and ${\cal E}_2$ could not have an additional common factor, i.e. each non-degenerate system of the family (\[S\_III\_a\]) belongs to ${{\bf Q\bf S\bf L}}_{\bf5}$. Via the rescaling $x\to le^{-1}x$, $ y\to e\,y $ and $ t\to el^{-1}t$ systems (\[S\_III\_a\]) become \[2CF\_28\] x= x\^2,y= 1+x. This system possesses the invariant line $x=0$. Taking into account the polynomial ${\cal H}$, by Lemma \[lm3\] and Corollary \[Mult:Z=0\] we obtain that the line $x=0$ could be of multiplicity three whereas the line $Z=0$ could be of multiplicity two. This is confirmed by the perturbations ([*V.28${}_\varepsilon$*]{}) from Table 5. Thus we obtain the Config. 5.28.
### The case $N\ne0,$ $\theta=0=B_3$
Since for the systems $({{\bf S}}_{I\!I\!I})$ we have \[S\_III\_1\] = -8h\^2(g-1),=32gh\^2,N= (g\^2-1)x\^2+2h(g-1)xy+h\^2y\^2 we shall consider two cases: $\mu\ne0$ and $\mu=0$. [**The subcase $\mu\ne0$**]{}. Then $gh\ne0$ and the condition $\theta=0$ yields $g=1$. Then the systems $({{\bf S}}_{I\!I\!I})$ with $g=1$ by the transformation $\ x\to\ x -d/h,$ $ y\to (hy+
2d-ch)/h^2 $ will be brought to the systems: $ \dot x=k+ x^2
+xy,\ \dot y= l+ex+fy+y^2, $ for which $ B_3=
-3e^2x^4+(3l-12k)x^2y^2-6kxy^3. $ Hence the condition $B_3=0$ yields $e=k=l=0$ and we obtain the system \[2CF\_24\] x= x\^2 +xy,y= fy+y\^2, for which we can assume $f\in\{0,1\}$ via Remark \[rem:transf\] ($\gamma=f,\ s=1$). For the systems (\[2CF\_24\]) calculations yield:$
{\cal H}=\gcd\left({\cal E}_1,{\cal E}_2\right)= X^2Y(Y+fZ).
$ Hence, $\deg\,{\cal H}=4$, i.e $M_{{}_{{\bf I\bf L}}}\ge5$ and since $N\ne0$ by Lemma \[gcd:4\] we have $M_{{}_{{\bf I\bf L}}}<6$. By Theorem \[theor:E1,E2\] the systems (\[2CF\_24\]) possess the invariant lines $x=0,$ $y=0$ and $y+f=0$. Moreover, according to Lemma \[lm3\] the line $x=0$ of could be of the multiplicity two, and the lines $y=0$ and $y=-f$ are distinct if and only if $f\ne0$. Since for the systems (\[2CF\_24\]) we have $D= -f^2x^2y$, the condition $f\ne0$ can be expressed by using this $T$-comitant.
If $D\ne0$ (then $f=1$) the perturbed systems ([*V.11${}_\varepsilon$*]{}) from Table 5 show that the invariant line $x=0$ is double one. Thus, for $D\ne0$ we obtain Config. 5.11.
Assume $D=0$. Then $f=0$ and the invariant line $x=0$ as well as the line $y=0$ is of multiplicity two. This is confirmed by the perturbed systems ([*V.19${}_\varepsilon$*]{}) from Table 5. Therefore the case $D=0$ leads to the Config. 5.19. [**The subcase $\mu=0$.**]{} In this case from (\[S\_III\_1\]) we obtain $h=0$ and the condition $N\ne0$ yields $g^2-1\ne0$. Then the systems $({{\bf S}}_{I\!I\!I})$ with $h=0$ will be brought via the translation $ x\to x+f/(1-g),$ $ y\to y+e/(1-g)$ to the systems: \[S\_III\_2\] x=k+ cx +dy +gx\^2,y= l+(g-1)xy, for which $\ B_3=
-3\,l\,(g-1)^2x^4-3\,c\,d(g-1)x^2y^2-6\,d^2g\,xy^3. $ Hence, as $N\ne0$ the condition $B_3=0$ yields $l=c\,d=d\,g=0$. We claim that if $d\ne0$ then for the systems (\[S\_III\_2\]) we have $M_{{}_{{\bf I\bf L}}}<5$. Indeed, suppose $d\ne0$. Hence the condition $B_3=0$ yields $l=c=g=0$. Thus we obtain the systems $\ \dot x=k+ dy,\ \dot y= -xy,$ for which calculations yield: $${\cal E}_1=(-k X^2 + d^2 Y^2 + 2 d k Y Z + k^2 Z^2)\,{\cal
H},\quad
{\cal E}_2= -X (d Y + k Z)^2{\cal H},\quad {\cal H}= YZ^2.$$ Thus, $\deg\,{\cal H}=3$ and since $d\ne0$ to have an additional common factor of ${\cal E}_1$ and ${\cal E}_2$ by Lemma \[Trudi:2\] the following condition must hold: $\ {\mbox{\rm Res\,}}_Y({\cal
E}_1/{\cal H},\ {\cal E}_2/{\cal H})= d^4 k^2 X^6=0.$ Therefore, $dk=0$ and since $d\ne0$ we obtain $k=0$. However this condition leads to the degenerate systems. Our claim is proved.
Let us assume $d=0$. Then the condition $B_3=0$ yields $l=0$ and the systems (\[S\_III\_2\]) become \[S\_III\_3\] x=k+ cx +gx\^2,y= (g-1)xy. Calculations yield: $\
{\cal E}_1=(X^2 - k Z^2)\,{\cal H},\ $ $
{\cal E}_2= X(g X^2 + c X Z + k Z^2)\,{\cal H},\ $ $ {\mbox{\rm Res\,}}_X({\cal
E}_1/{\cal H},\ {\cal E}_2/{\cal H})=k^2[c^2 - k(1 + g)^2] Z^6 $,where $ {\cal H}= (g-1)Y(g X^2 + c X Z + k Z^2). $Hence $\deg {\cal H}=3$ and we need an additional common factor of ${\cal E}_1$ and ${\cal E}_2$. For this, according to Lemma \[Trudi:2\], the condition ${\mbox{\rm Res\,}}_X({\cal E}_1/{\cal H},\ {\cal
E}_2/{\cal H})=0$ is necessary, i.e. $k\left[(c^2-k(g+1)^2\right]=0$. As $k\ne0$ (otherwise we get degenerate systems) we obtain the condition .
Assume $c^2=k(g+1)^2$. Since $N\ne0$ (i.e. $g+1\ne0$) we may set $c=u(g+1)$, where $u$ is a new parameter. Then $k=u^2\ne0$ and via the Remark \[rem:transf\] ($\gamma=u$, $s=1$) the systems (\[S\_III\_3\]) will be brought to the form: \[2CF\_26\] x=1+ (g+1)x +gx\^2,y= (g-1)xy. For systems (\[2CF\_26\]) we obtain $\
{\cal H}=\gcd\left({\cal E}_1,{\cal E}_2\right)= 2(g-1) Y(X + Z)^2(gX + Z).
$ Hence $\deg\,{\cal H}=4$, i.e. $M_{{}_{{\bf I\bf L}}}\ge5$ and since $N\ne0$ by Lemma \[gcd:4\], $M_{{}_{{\bf I\bf L}}}\ne6$ for any system (\[2CF\_26\]).
On the other hand the conditions $d=0$ and $c^2-k(g-1)^2=0$ are equivalent to $B_3=H_6=0$. Indeed, the condition $B_3=0$ implies $dg=0$ for system (see above) and then $\quad
H_6=64(g-1)^2x^4\big[2(g-1)^2[k(g+1)^2-c^2]x^2-5cdxy-2d^2y^2\big].
$ Hence as $N\ne0$ (i.e. $g-1\ne0$) the condition $H_6=0$ yields $d=c^2-k(g-1)^2=0$.
We observe, that $Z\mid {\cal H}$ if and only if $g=0$. So, since by $N\ne0$ the condition $g=0$ is equivalent to $K=2g(g-1)x^2=0$, we shall examine two subcases: $K\ne0$ and $K=0$.
[**1)**]{} If $K\ne0$ then $g\ne0$ and according to Theorem \[theor:E1,E2\] the systems (\[2CF\_26\]) possess the following invariant affine straight lines: $
y=0,\ x+1=0,\ gx+1=0.
$ By $g-1\ne0$ the invariant line $gx+1=0$ cannot coincide with $x=-1$. Moreover, the line $x=-1$ could be of multiplicity two and this is confirmed by the perturbations ([*V.14${}_\varepsilon$*]{}) from Table 5. Thus for $K\ne0$ we obtain the Config. 5.14. [**2)**]{} For $K=0$ we obtain $g=0$ and then the line $l_\infty:\ Z=0$ appears as a component of a conic in the pencil of conics corresponding to systems (\[2CF\_26\]). Hence, the invariant line $x+1=0$ as well as the line $Z=0$ is of multiplicity two, as is shown by the perturbations ([*V.18${}_\varepsilon$*]{}) from Table 5. Thus for $K=0$ we obtain the Config. 5.18.
Systems with the divisor $D_S(C ,Z)=3\cdot w$
---------------------------------------------
We are in the case of the canonical form $({{\bf S}}_{I\!V})$ and we shall later need the following polynomial which is shown to be a $CT$-comitant in Lemma \[Table:Propreties\]. \[not\_N7\] Let us denote$ N_6({\mbox{\boldmath $a$}},x,y)=
8D+C_2\left[8(C_0,D_2)^{(1)}-3(C_1,C_1)^{(2)}+2D_1^2\right]. $
### The case $N=0=B_2$
It was previously shown (see page ) that for $N({\mbox{\boldmath $a$}},x,y)=0$ we have to examine the systems (\[S4\_N0\]) for which we have $\
B_2= -648\,d^4 x^4,$ $ B_3= 6dx^3(fx-dy).
$ Thus the condition $B_2=0$ is equivalent to $B_3=0$ and this yields $d=0$. Then we obtain the systems (\[s4.8\]) for which the expressions of the affine comitants ${\cal E}_1$ and ${\cal
E}_2$ are given in (\[val\_Eib\]). We observe that $\deg\,{\cal
H}=3$ and we need to have an additional common factor of ${\cal
E}_1$ and ${\cal E}_2$. Since the polynomial ${\cal E}_2/{\cal
H}$ does not depend of $Y$, to have such a common factor by Lemma \[Trudi:2\] at least one of two following conditions must hold: && \_X([E]{}\_1/[H]{}, [E]{}\_2/[H]{})= (c - f)\^2 (c\^2fY - k\^2Z + c\^2l Z)\^2Z\^4=0;\
&& \_Z([E]{}\_1/[H]{}, [E]{}\_2/[H]{})=(c - f)\^2(c + f) X\^4(k\^2 X - c\^2l X + cf k Y)\^2=0. Hence, we obtain either $(c-f)(c+f)=0$ or $k=cl=cf=0$, however the second case leads to the degenerate systems (\[s4.8\]). On the other hand for systems (\[s4.8\]) we obtain $\
N_3= 3(c-f)x^3,$ $ D_1=c+f.$ So, the condition $(c-f)(c+f)=0$ is equivalent to $N_3D_1=0$ and we shall consider two subcases: $N_3=0$ and $N_3\ne0$, $D_1=0$. [**The subcase $N_3=0$**]{}. Then $f=c$ and we get the systems: \[s4.8:f=c\] x=k+ cx,y= l+cy-x\^2 for which calculations yield: $\
{\cal E}_1=2\,X\,{\cal H},$$
{\cal E}_2=Z (c X + k Z)\,{\cal H}, $$ {\cal H}=
Z^2(c X + k Z)^2.
$ So $\deg\,{\cal H}=4$. It is easy to observe that the polynomials ${\cal E}_1$ and ${\cal E}_2$ do not have an additional common factor if and only if $k\ne0$ and that the polynomial ${\cal H}$ has a factor of multiplicity four if $c=0$. On the other hand for systems (\[s4.8:f=c\]) $D_1= 2c$ and $N_4({\mbox{\boldmath $a$}},x,y)=12kx^2$. Hence the conditions $c=0$ and $k\ne0$ can be expressed by using the $CT$-comitants $D_1$ and $N_4$. So, we shall consider two cases: $D_1\ne0$ and $D_1=0$. [**1)**]{} Assume $D_1\ne0$. Then $c\ne0$ and the systems (\[s4.8:f=c\]) can be brought by the affine transformation $\
x=c^{-1}k x_1,\quad y=c^{-3}k^2 y_1-c^{-1}l,\quad t=c^{-1}t_1
$ to the system \[2CF\_16\] x=1+ x,y= y-x\^2 for which ${\cal H}=\gcd\left({\cal E}_1,{\cal
E}_2\right)=Z^2(X+Z)^2$. By Lemma \[lm3\] and Corollary \[Mult:Z=0\] the line $x=-1$ could be of multiplicity 2, whereas the line $Z=0$ could be of multiplicity 3. This is confirmed by the perturbations ([*V.27${}_\varepsilon$*]{}) from Table 5. Thus, for $N_3=0,\ N_4\ne0$ and $D_1\ne0$ we obtain the Config. 5.27. [**2)**]{} If $D_1=0$ for systems (\[s4.8:f=c\]) we have $c=0$ and since $k\ne0$ we may consider $k=1$ via the rescaling $x\to kx$ and $y\to k^2y$. Thus we obtain the systems \[2CF\_17\] x=1,y= l-x\^2, and we can assume $l\in \{-1,0,1\}$ via the rescaling $x\to|l|^{1/2}x$, $y\to|l|^{3/2}y$ and $t\to|l|^{1/2}t$. For the systems (\[2CF\_17\]) we have ${\cal H}=\gcd\left({\cal
E}_1,{\cal E}_2\right)=Z^4$ and hence by Corollary \[Mult:Z=0\] the line $Z=0$ could be of multiplicity 5. This is confirmed by the perturbation ([*V.30${}_\varepsilon$*]{}) from Table 5. Thus, we obtain the Config. 5.30. [**The subcase $N_3\ne0$, $D_1=0$**]{}. These conditions yield $f=-c\ne0$ and we may assume $c=-1$ via the Remark \[rem:transf\] ($\gamma=-c,$ $s=1$). Then systems (\[s4.8\]) will be brought by a translation to the systems: \[2CF\_30\] x=k-x,y= y-x\^2, in which we can assume $k\in\{0,1\}$ due to the rescaling $x\to
kx$ and $y\to k^2y$. For systems (\[2CF\_30\]) we calculate: $\
{\cal E}_1=2( k X - Y)\,{\cal H},\ $ $
{\cal E}_2=(X - k Z)^2{\cal H},\ $ $
{\cal H}= Z^3(k Z-X),\ $ i.e. $\deg\,{\cal H}=4$. We observe that the polynomials ${\cal E}_1/{\cal H}$ and ${\cal E}_1/{\cal H}$ cannot have a common factor. By Corollary \[Mult:Z=0\] the line $Z=0$ could be of multiplicity four. This is confirmed by the perturbations ([*V.26${}_\varepsilon$*]{})from Table 5. Thus, for $D_1=0$ and $N_3\ne0$ we obtain the Config. 5.26.
### The case $N\ne0$, $\theta=0=B_3$
For the systems $({{\bf S}}_{I\!V})$ we calculate: $\
\theta= 8h^3,$ $
N= (g^2-2h)x^2+2ghxy +h^2y^2. $ From $\theta=0,$ $N\ne0$ we obtain $h=0$, $g\ne0$ and we may assume $g=1$ via the rescaling $x\to g^{-1}x$, $y\to g^{-2}y$. Then the systems $({{\bf S}}_{I\!V})$ with $h=0$ and $g=1$ will be brought by the translation $x\to
x-c/2$, to the systems: $\quad \dot x=k+ dy+x^2,\ \ \dot y =l + fy -x^2 +xy. $For these systems we have $\ B_3=3(2df-k-f^2)x^4-
6d(d-4f)x^3y-9d^2x^2y^2. $ Hence the condition $B_3=0$ yields $d=0,$ $k=-f^2$ and we obtain the systems \[S\_IV\_1\] x=-f\^2 + x\^2,y =l + fy -x\^2 +xy, for which calculations yield: $\ {\cal E}_1=(X^2 + 2f X Z + l
Z^2)\,{\cal H},\quad
{\cal E}_2=3(X - f Z)(X + f Z)^2{\cal H},
$ where ${\cal H}= 2(X - fZ)^2(X + fZ)$. Hence $\deg\,{\cal
H}=3$ and to have an additional common factor of ${\cal E}_1$ and ${\cal E}_2$, according to Lemma \[Trudi:2\], the condition $ {\mbox{\rm Res\,}}_X({\cal E}_1/{\cal H},\ {\cal E}_2/{\cal H})= 9(f^2 -
l)^2(3f^2 + l)Z^6=0$ must hold. We observe that for $l=f^2$ the systems (\[S\_IV\_1\]) become degenerate. Therefore $l=-3f^2$ and since $f\ne0$ (otherwise we get the degenerate system) we may assume $f=1$ via Remark \[rem:transf\] ($\gamma=f$, $s=1$). Thus we obtain the canonical system: \[2CF\_29\] x=-1 + x\^2,y =-3 + y -x\^2 +xy and calculations yield ${\cal H}=\gcd\left({\cal E}_1,{\cal
E}_2\right)= 2(X - Z)^3(X + Z)$. Therefore, according to Theorem \[theor:E1,E2\], the system (\[2CF\_29\]) possesses the invariant straight lines $x=1$ and $x=-1$ and the line $x=1$ could be of multiplicity three. This is confirmed by the perturbations ([*V.23${}_\varepsilon$*]{}) from Table 5. Thus we obtain Config. 5.23. Since for systems (\[S\_IV\_1\]) we have $N_6= 8(l+3f^2)x^3$, the condition $l+3f^2=0$ is equivalent to $N_6=0$.
All the cases in Theorem \[th\_mil\_5\] are thus examined. To finish the proof of the Theorem \[th\_mil\_5\] it remains to show that the conditions occurring in the middle column of Table 4 are affinely invariant. This follows from the proof of Lemma \[Table:Propreties\].
------------------------------------------------------------------------
\[Table:Propreties\] The polynomials which are used in Theorems \[th\_mil\_6\] or \[th\_mil\_5\] have the properties indicated in the Table 6. In the last column are indicated the algebraic sets on which the $GL$-comitants on the left are $CT$-comitants.
[|c|c|c|c|c|c|]{}\
\[0pt\]\[0pt\][Case]{} & \[0pt\]\[0pt\][$GL$-comitants]{} & & \[0pt\]\[0pt\][Weight]{} & Algebraic subset\
& & $\ \ a\ \ $ & $\!x$ and $y\!$ & & $V(*)$\
------------------------------------------------------------------------
$1$ & $\eta(a)$, $\mu(a)$, $\theta(a)$ & $4$ & $0$ & $ 2$ & $V(0)$\
------------------------------------------------------------------------
$2$ & $C_2(a,x,y)$ & $1$ & $3$ & $-1$ & $V(0)$\
------------------------------------------------------------------------
$3$ & $H(a,x,y),\ K(a,x,y)$ & $2$ & $2$ & $ 0$ & $V(0)$\
------------------------------------------------------------------------
$4$ & $M(a,x,y),\ N(a,x,y)$ & $2$ & $2$ & $ 0$ & $V(0)$\
------------------------------------------------------------------------
$5$ & $D(a,x,y)$ & $3$ & $3$ & $-1$ & $V(0)$\
------------------------------------------------------------------------
$6$ & $B_1(a)$ & $12$ & $0$ & $3$ & $V(0)$\
------------------------------------------------------------------------
$7$ & $B_2(a,x,y)$ & $8$ & $4$ & $0$ & $V(0)$\
------------------------------------------------------------------------
$8$ & $B_3(a,x,y)$ & $4$ & $4$ & $-1$ & $V(0)$\
------------------------------------------------------------------------
$9$ & $H_1(a)$ & $6$ & $0$ & $2$ & $V(0)$\
------------------------------------------------------------------------
$10$ & $H_2(a,x,y))$ & $3$ & $2$ & $0$ & $V(0)$\
------------------------------------------------------------------------
$11$ & $H_3(a,x,y)$ & $4$ & $2$ & $0$ & $V(0)$\
------------------------------------------------------------------------
$12$ & $H_4(a)$ & $6$ & $0$ & $2$ & $V(0)$\
------------------------------------------------------------------------
$13$ & $H_5(a)$ & $8$ & $0$ & $2$ & $V(0)$\
------------------------------------------------------------------------
$14$ & $H_6(a,x,y))$ & $8$ & $6$ & $0$ & $V(0)$\
------------------------------------------------------------------------
$15$ & $N_1(a,x,y)$ & $3$ & $4$ & $-1$ & $V(\eta,H)$\
------------------------------------------------------------------------
$16$ & $N_2(a,x,y)$ & $3$ & $1$ & $0$ & $V(\eta,H,B_3)$\
------------------------------------------------------------------------
$17$ & $N_3(a,x,y)$ & $2$ & $3$ & $-1$ & $V(M,N)$\
------------------------------------------------------------------------
$18$ & $N_4(a,x,y)$ & $2$ & $2$ & $-1$ & $V(M,N,N_3)$\
------------------------------------------------------------------------
$19$ & $N_5(a,x,y)$ & $4$ & $2$ & $0$ & $V(\eta,H,B_3)$\
------------------------------------------------------------------------
$20$ & $N_6(a,x,y)$ & $3$ & $3$ & $-1$ & $V(M,\theta,B_3)$\
------------------------------------------------------------------------
$21$ & $D_1(a)$ & $1$ & $0$ & $0$ & $V(M,N)$\
[[*Proof:*]{} ]{}[*I. Cases 1,…,14*]{}. Let us consider the action of the translation group $T(2,{\mathbb{R}})$ on systems in $\widehat{{\bf Q\bf S}}$. It $\tau\in T(2,{\mathbb{R}})$, i.e. $\tau:\ x =
\tilde x+x_0,$ $y=\tilde y+y_0$ and $S_{{\mbox{\boldmath $a$}}}$ is a system in $\widehat{{\bf Q\bf S}}$ of coefficients ${\mbox{\boldmath $a$}}\in {\mathbb{R}}^{12}$, then applying this action to $S_{{\mbox{\boldmath $a$}}}$ we obtain the system $S_{\tilde {\mbox{\boldmath $a$}}}$ of coefficients $\tilde{\mbox{\boldmath $a$}}\in {\mathbb{R}}^{12}$, i.e. $$S_{\tilde{\mbox{\boldmath $a$}}}:\quad \left\{\ba{l}\dot {\tilde x}=
P({\mbox{\boldmath $a$}},x_0,y_0)+P_x({\mbox{\boldmath $a$}},x_0,y_0)\tilde x
+P_y({\mbox{\boldmath $a$}},x_0,y_0)\tilde y+p_2({\mbox{\boldmath $a$}},\tilde x,\tilde y), \\
\dot {\tilde y}= Q({\mbox{\boldmath $a$}},x_0,y_0)+Q_x({\mbox{\boldmath $a$}},x_0,y_0)\tilde
x+Q_y({\mbox{\boldmath $a$}},x_0,y_0)\tilde y+q_2({\mbox{\boldmath $a$}},\tilde x,\tilde y).\ea\right.$$ Then calculations yield: $$\bal
& U (\tilde{\mbox{\boldmath $a$}})=U ({\mbox{\boldmath $a$}})\quad
\text{for each}\quad U\in \{\eta,\mu,\theta,B_1,H_1,H_4,H_5\}, \\
& W (\tilde{\mbox{\boldmath $a$}},\tilde x,\tilde y)=W ({\mbox{\boldmath $a$}},\tilde x,\tilde y)\quad
\text{for each}\quad W\in \{C_2,K,H,M,N,D,B_2,B_3,H_2,H_3,H_6\}. \\
\eal$$ Since this holds for every ${\mbox{\boldmath $a$}}\in{\mathbb{R}}^{12}$, according to Definition \[def:T-com\] we conclude that the GL- comitants indicated in the lines 1–15 of Table 6 are $T$-comitants for systems .
[*II. Cases 15,…,21*]{}. [**1)**]{} We consider firstly the $GL$-comitants $N_1(a,x,y)$, $N_2(a,x,y)$ and $N_5(a,x,y)$ which according to Tables 2 and 4 were used only when the conditions $\eta=0=H$ are satisfied. According to Lemma \[lm\_3:2\] for $\eta=0$ there correspond three canonical forms: $({{\bf S}}_{I\!I})$, $({{\bf S}}_{I\!I\!I})$ and $({{\bf S}}_{V})$. Since for the systems $({{\bf S}}_{V})$ we have $H=-x^2\ne0$, we need to consider the following cases: $(i)$ $\eta=0$ and $M\ne0$;$(ii)$ $\eta=0$ and $M=0$ and $C_2\ne0$.
$(i)$ For $\eta=0$ and $M\ne0$ we are in the class of systems $({{\bf S}}_{I\!I\!I})$, for which the condition $H=-(g-1)^2x^2-2h(g+1)xy-hy^2=0$ yields $h=g-1=0$ and this leads to systems (see page ). On the other hand for any system corresponding to a point $\tilde{\mbox{\boldmath $a$}}\in
{\mathbb{R}}^{12}$ in the orbit under the translation group action of a system calculations yield: $$\bal
& N_1(\tilde {\mbox{\boldmath $a$}},\tilde x,\tilde y)=8\tilde x^2(e\tilde
x^2-2d\tilde y^2),\quad N_2(\tilde {\mbox{\boldmath $a$}},\tilde x,\tilde
y)=4(f^2+4k)\tilde x + 4df \tilde y+8d (x_0\tilde y +2y_0\tilde x),\\
& N_5(\tilde {\mbox{\boldmath $a$}},\tilde x,\tilde y)=-16(4k \tilde x^2 - d^2
\tilde y^2) + 64d\tilde x (x_0\tilde y -y_0\tilde x),\quad B_3(\tilde {\mbox{\boldmath $a$}},\tilde x,\tilde y)
= 6d\tilde x\tilde y^2(f\tilde x-d\tilde y).
\eal$$ Hence the value of $N_1$ does not depend of the vector defining the translation and for $B_3=0$ the same occurs for $N_2$ and $N_5$. Therefore we conclude that for $M\ne0$ the polynomial $N_1$ is a $CT$-comitant modulo $\langle\eta,H\rangle$, whereas the polynomials $N_2$ and $N_5$ are $CT$-comitants modulo $\langle\eta,H,B_3\rangle$.
$(ii)$ Assume now that $M=0$ and $C_2\ne0$. Then we are in the class of systems $({{\bf S}}_{I\!V})$, for which the condition $H=-(g^2+4h)x^2-2ghxy-hy^2=0$ yields $g=h=0$. In this case using an additional translation (see page ) we obtain the systems . Then for any system corresponding to a point $\tilde{\mbox{\boldmath $a$}}\in {\mathbb{R}}^{12}$ in the orbit under the translation group action of a system calculations yield: $\
N_1(\tilde {\mbox{\boldmath $a$}},\tilde x,\tilde y)=-24 d\tilde x^4,\quad N_2(\tilde
{\mbox{\boldmath $a$}},\tilde x,\tilde y)=12d(c+f)\tilde x,\quad N_5(\tilde
{\mbox{\boldmath $a$}},\tilde x,\tilde y)=0. $ Since the condition $M=0$ implies $\eta=0$, considering the case $(i)$ above we conclude that independently of either $M\ne0$ or $M=0$, the $GL$-comitant $N_1$ is a $CT$-comitant modulo $\langle\eta,H\rangle$ and $N_2$ and $N_5$ are $CT$-comitants modulo $\langle\eta,H,B_3\rangle$.
[**2)**]{} Let us now consider the $GL$-comitants $N_3(a,x,y)$, $N_4(a,x,y)$, $N_6(a,x,y)$ and $D_1(a)$. According to Tables 2 and 4 the polynomials $N_3$, $N_4$ and $D_1$ (respectively $N_6$) were used only when the conditions $M=N=0$ (respectively $M=\theta=0,N\ne0$) are satisfied. In both cases we are in the class of systems $({{\bf S}}_{I\!V})$ and we shall consider the two subcases: $N=0$ and $N\ne0$, $\theta=0$.
$(i)$ If for the system $({{\bf S}}_{I\!V})$ the condition $N=0$ is fulfilled then as it was shown on the page we obtain systems . Then for any system corresponding to a point $\tilde{\mbox{\boldmath $a$}}\in {\mathbb{R}}^{12}$ in the orbit under the translation group action of a system calculations yield: $$\bal
& N_3(\tilde {\mbox{\boldmath $a$}},\tilde x,\tilde y)=3(c-f)\tilde x^3+2d\tilde x^2\tilde y,\quad
B_3(\tilde {\mbox{\boldmath $a$}},\tilde x,\tilde y)=6d\tilde x^3(f\tilde x-d\tilde y),\\
& N_4(\tilde {\mbox{\boldmath $a$}},\tilde x,\tilde y)=12 k\tilde x^2 +
3(f^2-c^2)\tilde x\tilde y
-3d(c+f)\tilde y^2+6\tilde x^2[(c-f)x_0+2dy_0],\\
& N_6(\tilde {\mbox{\boldmath $a$}},\tilde x,\tilde y)=8c(c-f)\tilde x^3 + 16 df
\tilde x^2 \tilde y-8d^2\tilde x\tilde y^2-48d x_0\tilde x^3,\quad
D_1(\tilde {\mbox{\boldmath $a$}})=c+f.
\eal$$ These relations show us that: $(\alpha)$ the $GL$-comitants $N_3$ and $D_1$ are $CT$-comitants modulo $\langle M,N\rangle$; $(\beta)$ the $GL$-comitant $N_4$ is a $CT$-comitant modulo $\langle M,N,N_3\rangle$; $(\gamma)$ the $GL$-comitant $N_6$ is a $CT$-comitant modulo $\langle M,N,B_3\rangle$.
$(ii)$ Assume that for the system $({{\bf S}}_{I\!V})$ the conditions $\theta=0$ and $N\ne0$ are fulfilled. As it was shown on the page for $B_3=0$ we obtain systems . For any system corresponding to a point $\tilde{\mbox{\boldmath $a$}}\in {\mathbb{R}}^{12}$ in the orbit under the translation group action of a system we have $N_6(\tilde {\mbox{\boldmath $a$}},\tilde
x,\tilde y)=8(l+3f^2)\tilde x^3$. Therefore, since the condition $N=0$ implies $\theta=0$, considering the case $(i)$ above we conclude that independently of either $N\ne0$ or $N=0$, the $GL$-comitant $N_6$ is a $CT$-comitant modulo $\langle M,\theta,B_3\rangle$.
The Table 6 shows us that all the conditions indicated in the middle column of Tables 2 and 4 are affinely invariant. Indeed, the $CT$-comitants $N_i$, $i=1,\ldots,...,6$ and $D_1$ are used in Table 2 only for the varieties indicated in the last column of Table 6. This complete the proof of the Theorems \[th\_mil\_6\] and \[th\_mil\_5\].
------------------------------------------------------------------------
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[^1]: Work supported by NSERC and by the Quebec Education Ministry
[^2]: Partially supported by NSERC
[^3]: Darboux used the notion of Clebsch connex to define them.
|
---
abstract: 'A symbolic calculus for a pseudo-differential operators acting on sections of a homogeneous vector bundle over a compact homogeneous space $G/H$ with compact $G$ and $H$ is developed. We realize the symbol of a pseudo-differential operator as a linear operator acting on corresponding irreducible unitary representations of $H$ valued in the algebra $C^\infty(G)$ of smooth functions. We write down how left invariant vector fields of $SU(2)$ act on the sections of homogeneous vector bundles associated to the fibration ${\mathbb{T}}\hookrightarrow SU(2)\rightarrow{\mathbb{C}}P^1$, which is known as the Hopf fibration. Lastly, we outline how functional calculus of a pseudo-differential operator can be computed using our calculus.'
address: |
Institute of Mathematic\
Polish Academy of Sciences\
Warszawa\
Poland
author:
- Mitsuru Wilson
bibliography:
- 'References\_symbolic.bib'
title: 'Global symbolic calculus of pseudo-differential operators on homogeneous vector bundles'
---
**Introduction**
================
The resolvent operator $(A - \lambda)^{-1}$ plays a central role in the global analysis on a compact manifold $M$ associated with a linear elliptic differential operator $A$. In particular, one can easily obtain the corresponding heat operator $e^{-tA}$ for $t\in{\mathbb{R}}_+$ from the resolvent. Detailed knowledge of the terms in the asymptotic expansions of the integral kernels of the resolvent and the heat operator is of great value in calculating the asymptotic of eigenvalues [@gilkey2018invariance; @widom1984spectral], scalar curvature and Ricci curvature [@floricel2016ricci; @gilkey2018invariance] and indices of Fredholm operators [@atiyah1973heat; @getzler1983pseudodifferential]. In mathematical physics, they are used in quantum field theory [@dewitt1988dynamical].
The intrinsic symbolic calculus of pseudo-differential operators pioneered by Widom [@widom1980complete] and received contributions [@SFGK1988resolvent]. Its importance lies in the generality of the operators to which it may be applied. Although Widom’s approach can be applied to general smooth manifolds, Ruzhansky shed light on another approach to constructing intrinsic symbolic calculus for compact Lie groups [@ruzhansky2008pseudo; @ruzhansky2009pseudo; @ruzhansky2010quantization] wherein examples of toruses and $SU(2)$ are explicitly computed [@ruzhansky2008pseudo; @ruzhansky2010quantization; @ruzhansky2013global]. The basic idea of Ruzhansky was the use of the representation theory of compact Lie groups to decompose smooth functions on the compact Lie group into Fourier series and developed the symbolic calculus for pseudo-differential operators which act on smooth functions. Subsequently, parametrices for pseudo-differential operators were computed in [@ruzhansky2009pseudo] and a version of the local index theorem was pursued in [@cardona2019index].
Smooth functions on a compact Lie group can be viewed as smooth sections of the trivial line bundle. This paper outlines how to generalize the development thereafter [@ruzhansky2009pseudo] to operators acting on sections of a homogeneous vector bundle .
For a compact manifold $M$, the set of Hörmander class pseudo-differential operators on $M$ is denoted by $\Psi^m(M)$. These operators are defined to be the class of operators in $\Psi^m({\mathbb{R}}^n)$ in all local coordinate. Operators in $\Psi^m({\mathbb{R}}^n)$ are characterized by the symbols satisfying $$\label{eq:Hormander.pseudo.differential.operator}
\left|\partial_\xi^\alpha\partial_x^\beta p(\xi, x)\right| \leq C(1+|\xi|)^{m - |\alpha|}$$ for all multi-indices $\alpha,~\beta$ and all $\xi,~x\in{\mathbb{R}}^n$. Moreover, the definition of Hörmander class pseudo-differential operators extend naturally to fibre-preserving operators on sections of vector bundles, which satisfy in every locally trivial coordinate. This class of operators is studied extensively, for example, in [@hormander1971fourier] by Hörmander himself. In this paper, we study classical pseudo-differential operators on homogeneous vector bundles, although much of the theory can be proved for the general Hörmander class pseudo-differential operators. Classical pseudo-differential operators are Hörmander class pseudo-differential operators whose symbol can be written of the form $$p(\xi, x) \sim \sum_{j\geq0}p_{m-j}(\xi, x)$$ where each $p_k\in\Psi^k({\mathbb{R}}^n)$ for each coordinate. Although their theory is mathematically intriguing, much of the differential geometric aspects are missing. This paper will explore more geometric approach than that of Ruzhansky. Homogeneous vector bundles are quite important in mathematics. For instance, in [@bott1957homogeneous; @bott1965index], invariant differential operators on homogeneous vector bundles have been studied extensively and the their indices were computed in terms of the unitary representations.
In Section \[sec:pseudo.differential.calculus.homogeneous.bundles\], we first review the basic structure of homogeneous vector bundles. Moreover, we define the symbol of an operator, obtain an alternative expression for the symbol and compute an exact decomposition for the space of sections of a homogeneous vector bundle and express each section as a series indexed by the unitary dual of of $G$ (not the structure group $K\subset G$).
The most important development in Section \[sec:pseudo.differential.calculus.homogeneous.bundles\] is the composition formula ; in order to obtain this formula, we also give the difference operator construction in the homogeneous vector bundle case.
In Section \[sec:parametrix\], we provide the parametrix formula for a classical pseudo-differential operator such that the symbol of the highest order is invertible.
We present an example using the fibration ${\mathbb{T}}\hookrightarrow SU(2)\rightarrow{\mathbb{C}}P^1$ in Section \[sec:example.su(2).u(2)\]. There, we compute how the left invariant vector fields act on the homogeneous vector bundles associated to the representations of $\hat{\mathbb{T}}={\mathbb{Z}}$ and their symbols.
Finally, we outline how to deploy functional calculus for operators with well defined parametrix for some parameter in Section \[sec:remarks\].
**Acknowledgement** {#acknowledgement .unnumbered}
===================
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 691246. I thank Piotr Hajac for some fruitful discussions and supporting this project as the principal grant holder. I would also like to extend my appreciation to Alexander Gorokhovsky for some useful discussions directly leading to the completion of this project.
**Pseudo-differential calculus on homogeneous vector bundles** {#sec:pseudo.differential.calculus.homogeneous.bundles}
==============================================================
Symbolic calculus for pseudo-differential operators that act on smooth functions of homogeneous spaces was outlined also in [@ruzhansky2009pseudo]. A next generalization of this result is the symbolic calculus for pseudo-differential operators acting on sections of a homogeneous vector bundle. A section of a vector bundle $\pi:E\rightarrow M$ of rank $r$ over a smooth manifold $M$ is a function $s:M\rightarrow E$ such that $\pi(s(x))=x$ for all $x\in M$. We denote by $\Gamma(E)$ the space of all smooth sections.
Harmonic analysis on homogeneous vector bundle {#subsec:homogeneous.vector.bundle}
----------------------------------------------
We give a basic construction that describes all homogeneous vector bundles over $M = G/K$ where $G$ is a Lie group and $K$ is a closed subgroup. A vector bundle $E$ over $M$ is called a homogeneous vector bundle if $G$ acts on $E$ from the left such that $$gE_x = E_{gx}~\mathrm{for}~x\in M,~g\in G$$ and the mapping from $E_x$ to $E_{gx}$ induced by $g$ is linear for $g\in G$ and $x\in M$.
Let $(\tau,E_0)$ be a finite dimensional representation of $K$. $K$ acts on $G\times E_0$ by $(g,v)\cdot k = (gk,\tau(k)^{-1}v)$. Then, $$E = G\times_\tau E_0 := (G\times E_0)^K$$ is isomorphic to $E$, and offers another description of a homogeneous vector bundle. We also say that $E$ is a homogeneous vector bundle associated to the representation $E_0$.
Let $G$ be a compact Lie group and let $K$ be a closed subgroup of $G$. We assume that $G/K$ is orientable. Let $E$ be a homogeneous vector bundle over $M=G/K$ associated to an irreducible unitary representation $E_0$ of $K$. Let $G$ act on $\Gamma(E)$ by $g\cdot s(x) = gs(g^{-1}x)$ for $g\in G$, $s\in\Gamma(E)$, and $x\in M$. Then, the $G$-action on $\Gamma(E)$ extends to a unitary representation on the Hilbert space $L^2(E)$, which is obtained from $\Gamma(E)$ by completion with respect to a $G$-invariant Hermitian inner product. Such inner product can be obtained easily. Let $\langle~,~\rangle$ be a $K$-invariant Hermitian inner product on $E_0$. Then if $s_1,~s_2\in\Gamma(E)$, we may think of them as maps from $G$ to $E_0$ such that $k^{-1}\cdot s_j(g) = s_j(gk)$ so that $(s_1,s_2) := \int_G\langle s_1,s_2\rangle dg$ defines a Hermitian inner product on $\Gamma(E)$. The action of $G$ with this inner product becomes unitary.
$L^2(E)$ is unitarily equivalent to $$\begin{aligned}
\label{eq:unitary.equivalence}
L^2(G, E_0)^\tau := \left\{f\in L^2(G)\otimes E_0: \tau(k)^{-1}f(g) = f(gk) \right\}\end{aligned}$$ with the $G$ action given by $g_0\cdot f(g) := f(g^{-1}_0g)$. This unitary equivalence is given simply by $$\begin{aligned}
A : L^2(E) & \longrightarrow L^2(G, E_0)^\tau\\
f & \mapsto A(f)(g) = g^{-1} f(gx),\end{aligned}$$
In fact, if $f:G\rightarrow {\mathbb{C}}^r$, then the Fourier transform extends to each component $f^\alpha(g)$ of $f(g)$ as a complex function on $G$ to express $f$ as a Fourier series $$\begin{aligned}
f^\alpha(g) = \sum_{\lambda\in\widehat G}\dim\lambda{{\operatorname{Tr}}}(\lambda(g)\widehat {f^\alpha}(\lambda)).\end{aligned}$$
This can be generalized to functions on $G$ valued in $E_0$ by taking components relative to an ordered orthonormal basis $\{e_k\}$ with respect to the Hermitian inner product $\langle~,~\rangle$. Let $\{e_k^*\}$ be the corresponding dual basis of $E^*_0$ and suppose $f\in L^2(G, E_0)^\tau$. Then, $x\mapsto e_k^*(f(x))$ is a complex valued function on $G$ and
$$\begin{aligned}
\label{eq:vector.valued.fourier.series}
f(x) = \sum_{k=1}^{\dim E_0} e_k^*(f(x))e_k = \sum_{k=1}^{\dim E_0}\sum_{\lambda\in\widehat G}\dim\lambda{{\operatorname{Tr}}}(\lambda(g)\widehat {e_k^*(f)}(\lambda))e_k .\end{aligned}$$
Let $F\rightarrow G/K$ be another homogeneous vector bundle associated to an irreducible unitary representation $F_0$ of $K$, $\{f_b\}$ be an orthonormal basis of $F_0$, and suppose $\dim F_0 = s$. We define a continuous linear operator $$A:\Gamma(E)\longrightarrow\Gamma(F)$$ to be a pseudo-differential operator. If, in addition, $A$ satisfies $g\cdot A(f(x)) = A(g\cdot f(x))$, then it is called an invariant pseudo-differential operator.
For $i = 0, 1$ we denote by $\pi_i: M \times M \rightarrow M$ the projections $(x_0, x_1) \rightarrow x_i$. Given complex vector bundles $E_i\rightarrow M$, $i = 0, 1$, we define the vector bundle $E_0\boxtimes E_1 \rightarrow M\times M$ by $E_0\boxtimes E_1 := \pi_0^*E_0\otimes\pi_1^*E_1$.
We have the following theorem.
Let $M$ be a manifold and $E,~F\rightarrow M$ vector bundles, and let $${\operatorname{Hom}}(E, F)\rightarrow M\times M$$ be the bundle whose fiber at $(x, y)\in M\times M$ is $Hom(E_x, F_y)$. If $$A : \Gamma_c(E) \longrightarrow\Gamma(F)$$ is a continuous linear mapping, there exists [*S*chwartz kernel distribution]{} $\mathcal K_A \in\mathcal{D}'(\Gamma({\operatorname{Hom}}(E,F)))$ of $A$ such that $$\langle \psi, A(\phi)\rangle
= \langle \psi\boxtimes\phi, \mathcal K_A\rangle$$ where $$\psi\boxtimes\phi\in C^\infty(M\times M),~~~(\psi\boxtimes\phi)(x,y)=\psi(x)\phi(y).$$
A proof is contained, for example, in [@hormander1971fourier Section 5.2]. Interpreted as a distribution, we can write the action of the operator $A$ on a section $s\in \Gamma_c(E)$ as $$\begin{aligned}
A(s)(y) = \int_MK_A(x ,y)s(x)dx.\end{aligned}$$
Let $\lambda : G \rightarrow U\left(\mathcal H_\lambda \right)$ be an irreducible unitary representation, $E$ and $F$ homogeneous vector bundles associated to irreducible unitary representations $E_0$ and $F_0$ of a closed subgroup $K\subset G$, respectively. The symbol of a continuous linear operator $A :\Gamma(E) \rightarrow\Gamma(F)$ at $x \in G$ and $\lambda \in \operatorname{Rep}(G)$ is defined as $$\begin{aligned}
\label{formula:symbol.definition}
\sigma_A(\lambda, x) :=\widehat k_x(\lambda) \in \operatorname{End}\left(\mathcal{H}_{\lambda}\right)\otimes{\operatorname{Hom}}(E_x,F_x)\end{aligned}$$ where $k_y(x) = K_A(x, y)$ is the Schwartz kernel of $A$. Hence, $$\sigma_A(\lambda, x) = \int_G K_A(x, y) \lambda(x)^* dx$$ in the sense of distributions, and the Schwartz kernel can be regained from the symbol as well: $$K_A(x, y) = \sum_{\lambda \in \widehat G} \dim(\lambda)
{\operatorname{Tr}}\left(\lambda(y)\otimes \sigma_A(\lambda, x)\right)$$ where this equality is interpreted in the sense of distribution and the trace is taken over the indices of $\lambda$ and $\lambda(y)\otimes \sigma_A(\lambda, x)$ is interpreted as $\lambda(y)$ is multiplied on each $\operatorname{End}\left(\mathcal{H}_{\lambda}\right)$-component of $\sigma_A(\lambda, x)$. The following proposition shows that operator $A$ can be represented by its symbol.
Let $\sigma_A$ be the symbol of a continuous linear operator $A : \Gamma(E) \rightarrow \Gamma(F)$. Then
$$\begin{aligned}
\label{eq:representation.of.pseudo}
A s(x) = \sum_{\lambda \in \widehat G} \dim\lambda {\operatorname{Tr}}\left(\lambda(x)\otimes \sigma_A(\lambda,x) (\widehat s (\lambda))\right)\end{aligned}$$
for every $s \in \Gamma(E)$ and $x \in G$.
By , $s$ can be represented as a $K$-invariant $E_0$-valued function on $G$, and $\sigma_A(\lambda,x)\in \operatorname{End}\left(\mathcal{H}_{\lambda}\right)\otimes{\operatorname{Hom}}(E_x,F_x)$ acts on $\widehat s (\lambda)\in \operatorname{End}\left(\mathcal{H}_{\lambda}\right)\otimes E_0$. The case in which $E_0={\mathbb{C}}$, the trivial $K$-representation, with $K=\{e\}$, the trivial group, has been treated in [@ruzhansky2008pseudo Theorem 2.4].
Again, since $\Phi(\cdot):=\sigma_A(\lambda,\cdot)$ is an element in $\operatorname{End}\left(\mathcal{H}_{\lambda}\right)\otimes\Gamma({\operatorname{Hom}}(E,F))$, repeating the argument used for , $$\begin{aligned}
\Phi\in\operatorname{End}\left(\mathcal{H}_{\lambda}\right)\otimes \big(C(G)\otimes{\operatorname{Hom}}(E_0,F_0)\big)^\tau.\end{aligned}$$ Since ${\operatorname{Hom}}(E_0,F_0)$ is a finite dimensional vector space, the analysis in [@ruzhansky2008pseudo Theorem 2.4] would be practically unaffected. This completes the proof.
For a symbol $\sigma_A$, the corresponding operator $A$ defined by will be also denoted by $\operatorname{Op}(\sigma_A)$. The operator defined by formula will be called the pseudo-differential operator associated to the symbol $\sigma_A$.
Note again that, by , $\sigma_A$ carries two sets of indices from $\operatorname{Hom}\big(E_x,F_x\big)$ and $\operatorname{End}(\mathcal H_\lambda)$. Elements in ${\operatorname{Hom}}(E_x,F_x)$ can be represented as matrices relative to bases of $E_x$ and $F_x$. The following is a straightforward generalization of [@ruzhansky2009pseudo Theorem 10.4.6].
Suppose $\{e_a\}$ and $\{f_b\}$ be orthonormal bases of $E_0$ and $F_0$, respectively. Let $\sigma_A(\lambda,x)$ be the symbol of a continuous operator $A:\Gamma(E)\rightarrow \Gamma(F)$. Then, $$\begin{aligned}
\label{formula:symbol}
\sigma_A(\lambda, x)_{ab} = \big(\left(\lambda(x)^*\otimes f^*_b\right)A\left(\lambda(x)\otimes e_a\right)\big).\end{aligned}$$
For each $a$ and $b$, and $m$ and $n$, $$\begin{aligned}
\sum_{\alpha=1}^{\dim\lambda}\lambda_{\alpha m}^*(x) f_b^*A(\lambda_{\alpha n} e_a)
&=\sum_{k=1}^{\dim\lambda}\lambda_{\alpha m}^*(x) f_b^*\sum_{\eta\in\hat{G}}\dim\eta{\operatorname{Tr}}(\eta(x)\sigma_A(\eta,x)(\widehat{\lambda_{\alpha n}}(\eta)e_a))\\
&=\sum_{k=1}^{\dim\lambda}\lambda_{\alpha m}^*(x)f_b^*\sum_{\eta\in\hat{G}}\dim\eta\sum_{i,j,\ell}\eta_{ij}(x)\sigma_A(\eta_{j\ell},x)\left(\widehat{\lambda_{kn}}(\eta)_{\ell i}e_a\right)\\
&=\sum_{k,j}\lambda_{\alpha m}^*(x)f_b^*\left(\dim\eta~\lambda_{kj}(x)\sigma_A(\lambda_{jn},x)(e_a)\frac{1}{\dim\eta}\right)\\
&=\sum_{k,j}\lambda_{\alpha m}^*(x)\lambda_{kj}(x)f_b^*\left(\sigma_A(\lambda_{jn},x)(e_a)\right)\\
&=f_b^*\left(\sigma_A(\lambda_{mn},x)(e_a)\right)\\
&=\sigma_A(\lambda_{mn},x)_{ab}\end{aligned}$$ where $\sigma_A(\lambda_{mn},x)$ simply means $m,n$ component of the $\operatorname{End}(\mathcal H_\lambda)$-indices.
The representation of the symbol is relative to the choices of bases. The change of symbol under changes of bases is given simply by the change of basis theorem in linear algebra. For a completion, we state this fact with our notations.
Suppose $\{e_a\}$ and $\{f_b\}$ be orthonormal bases of $E_0$ and $F_0$, respectively, and $\{e_a'\}$ and $\{f_b'\}$ be other orthonormal bases. Let $\sigma_A(\lambda,x)$ be the symbol of a continuous operator $A:\Gamma(E)\rightarrow \Gamma(F)$ relative to the first set of first bases and $\sigma_A'(\lambda,x)$ be the symbol of $A$ relative to the second set of bases. Then, $$\begin{aligned}
\label{formula:change.of.basis}
\sigma_A'(\lambda, x) = \left(1\otimes V\right)\circ \sigma_A(\lambda, x)\circ\left(1\otimes U^*\right)\end{aligned}$$ where $U:E_0\rightarrow E_0$ and $V:F_0\rightarrow F_0$ are the change of basis linear transformations $e_a\mapsto e'_a$ and $f_b\mapsto f'_b$, respectively.
Note that $\Gamma\left(E\right)$ carries induced action of its Lie algebra $\mathfrak{g}$ of $G$ given by the usual formula: $$\begin{aligned}
X\cdot s(x) := \frac{d}{dt}\Big|_{t=0}\exp(tX)s(\exp(-tX)x),\qquad x\in G,~X\in\mathfrak{g},~s\in \Gamma\left(E\right).\end{aligned}$$
If the identification in is used, then the action of $X\in\mathfrak g$ on $f\in\big(L^2(G)\otimes E_0\big)^K$ is $$\begin{aligned}
X\cdot f(x) := \frac{d}{dt}\Big|_{t=0}f(\exp(-tX)x).\end{aligned}$$
This action of the Lie algebra $\mathfrak g$ extends to an action of the universal enveloping algebra $U(\mathfrak g)$ of $\mathfrak g$. Consider the action of the operator $$\label{eq:laplacian}
\mathcal L := - X_1^2 - \ldots-X_{\dim G}^2$$ where $\left\{X_1, \ldots, X_{\dim G}\right\}$ is an orthonormal basis of $\mathfrak g$. $\mathcal L$ does not depend on a choice of orthonormal bases and $\mathcal L(\lambda_{jk}(x)) = c_\lambda\lambda_{jk}(x)$ for some $c_\lambda\in{\mathbb{C}}$, which depends only on the unitary irreducible representation $\lambda\in\widehat G$. We denote by $\langle\lambda\rangle:=\left(1 + |c_\lambda|^2\right)^{1/2}$.
Denote $\Xi=(I + \mathcal L)^{1/2}$. Then, $\Xi^s\in\Gamma\big(\operatorname{End}(E)\big)$ and $\Xi^s\in\mathcal D'\left(\Gamma\left(\operatorname{End}(E)\right)\right)$ for every $s\in{\mathbb{R}}$. Let us define $$\begin{aligned}
\label{eq:sobolev.space}
\langle f, g\rangle_s := \left(\Xi^s f, \Xi^s g\right)_{L^2(E)}\end{aligned}$$ for $f, g \in\Gamma(E)$. The completion of $\Gamma(E)$ with respect to the norm $f \mapsto\Vert f\Vert_s=\langle f, f\rangle_s^{1/2}$ lends us a definition of the Sobolev space $H^s(E)$ of order $s \in{\mathbb{R}}$. It is easy to check that the operator $\Xi^r$ defines an isomorphism $H^s(E) \rightarrow H^{s - r}(E)$ for every $r, s \in {\mathbb{R}}$.
In view of the identification , it can be proved that $H^s(E)$ is unitarily equivalent to $(H^s(G)\otimes E_0)^K$. Note that $\Xi^s\in\Psi^s(E)$. Using this identification,
Let $A$ be a pseudo-differential operator, $X\in\mathfrak g$ and $\phi\in C^\infty(G)$. Then, $$\begin{aligned}
\sigma_{\phi A}(\lambda, x) & = \phi(x) \sigma_A(\lambda, x) \\
\sigma_{X\circ A}(\lambda, x) &=\sigma_X(\lambda, x) \sigma_A(\lambda, x)+\left(X\sigma_A\right)(\lambda, x) \end{aligned}$$
Suppose $A$ is an operator and $\phi\in C^\infty(G)$, $e_1,\ldots,e_{\dim E_0}$ and $f_1,\ldots,f_{\dim F_0}$ be bases of $E_0$ and $F_0$. Then, since $$\begin{aligned}
\lambda^*(x)\otimes f^*_b\left(\phi(x) A \lambda(x)\otimes e_a\right)
& = \phi(x)\lambda^*(x)\otimes f^*_b\left(A \lambda(x)\otimes e_a\right),
\end{aligned}$$ $\sigma_{\phi(x)A}(\lambda,x) = \phi(x)\sigma_A(\lambda,x)$.
Since $X\big(\lambda(x)\otimes\sigma_A(\lambda,x)\big) = X\big(\lambda(x)\big)\otimes\sigma_A(\lambda,x) + \lambda(x)\otimes X\big(\sigma_A(\lambda,x)\big)$, $$\begin{aligned}
X \circ A f(x)
& = X \sum_{\lambda \in \widehat G} \dim\lambda {\operatorname{Tr}}\left(\lambda(x)\otimes\sigma_A(\lambda, x) \widehat f(\lambda)\right) \\
& = \sum_{\lambda\in \widehat G} \dim\lambda {\operatorname{Tr}}\left(\left(X \lambda\right)(x)\otimes\sigma_A(\lambda, x) \widehat f(\lambda)\right) \\
& \qquad+\sum_{\lambda\in \widehat G} \dim\lambda {\operatorname{Tr}}\left(\lambda(x)\otimes X\left(\sigma_A(\lambda, x)\right) \widehat f(\lambda)\right) \\
& = \sum_{\lambda\in \widehat G} \dim\lambda {\operatorname{Tr}}\left(\lambda(x)\lambda^*(x)X\left(\lambda\right)(x)\otimes\sigma_A(\lambda, x) \widehat f(\lambda)\right) \\
& \qquad+\sum_{\lambda\in \widehat G} \dim\lambda {\operatorname{Tr}}\left(\lambda(x)\otimes X\left(\sigma_A(\lambda, x)\right) \widehat f(\lambda)\right)
\end{aligned}$$
Suppose $G\subset U(n)$ is a connected compact matrix Lie group. Then, the symbol $\sigma_{X}$ of a left invariant vector field $X\in\mathfrak g$ can be computed relatively simply. Let $\exp:\mathfrak g\rightarrow G$ be the exponential map and $\lambda:G\rightarrow$U$(n)$ a representation so that $$\begin{aligned}
\nonumber
\lambda(g)^*X\left(\lambda(g)\right) & = \frac{d}{dt}\Big\vert_{t=0}\lambda(g)^*\lambda\left(g\exp(tX)\right)\\
\nonumber
& = \frac{d}{dt}\Big\vert_{t=0}\lambda(g)^*\lambda\left(g\right)\lambda\left(\exp(tX)\right)\\
\nonumber
& = \frac{d}{dt}\Big\vert_{t=0}\lambda\left(\exp(tX)\right)\\
\nonumber
& = \frac{d}{dt}\Big\vert_{t=0}\exp\left(t(\lambda_*X)\right)\\
\label{eq:symbol.of.vector.fields}
& = \lambda_*X
\end{aligned}$$ at the identity of $e\in G$. In fact, $\lambda(e)=I_{\dim\lambda}$, so the symbol is essentially $X$ put in block diagonal form. In particular, if $G=SU(2)$, the inclusion $\lambda:SU(2)\rightarrow U(2)$ is the unique fundamental representation with $V_\lambda = {\mathbb{C}}^2$. In this case, a basis of $\mathfrak{su}(2)$ is given by the Pauli matrices $$\begin{aligned}
\label{eq:pauli.matrices}
H =
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}, \qquad
X =
\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}, \qquad
Y =
\begin{pmatrix}
0 & -i\\
i & 0
\end{pmatrix}\end{aligned}$$ and a generic element of $SU(2)$ can be written as $$\begin{pmatrix}
\alpha & -\bar \beta\\
\beta & \bar\alpha
\end{pmatrix},\qquad \vert\alpha\vert^2 + \vert\beta\vert^2 = 1.$$ Thus, $\sigma_A(\lambda,x) = A$ for all $A\in\mathfrak{su}(2)$. In general, the representation of $SU(2)$ is given by the symmetric tensor power of ${\mathbb{C}}^2$, so it suffices to compute the symbol of left invariant vector fields at the fundamental representation. That is, $$\begin{aligned}
\label{eq:representation.of.su(2)}
\sigma_A(\operatorname{Sym}^k(\lambda),x) = \operatorname{Sym}^k(A)\end{aligned}$$ where $\operatorname{Sym}^k(\lambda):SU(2) \rightarrow U\big(\operatorname{End}(\operatorname{Sym}^k({\mathbb{C}}^2))\big)$ is the $k$th symmetric tensor power representation, which is known to be irreducible unitary and these are all the irreducible unitary representations of $SU(2)$.
From the general representation theoretic viewpoint, every irreducible unitary representation of a connected compact Lie group is contained in the tensor products of the fundamental representations. We will use this example in Section \[sec:example.su(2).u(2)\].
As had been studied in literature [@SFGK1988resolvent; @widom1980complete], the formula that highlights the study of pseudo-differential calculus is the composition formula for the symbols. Let $H_0$ be yet another irreducible unitary representation of $K$ and $H$ the associated vector bundle. Consider $$\begin{aligned}
\Gamma(E)\overset{A}{\longrightarrow}\Gamma(F)\overset{B}{\longrightarrow} \Gamma(H)\end{aligned}$$ where the domains and the ranges of $A$ and $B$ makes sense. It is natural to question how to express the symbol $\sigma_{AB}$ of the composed operator $AB$ in terms of individual symbols $\sigma_A$ and $\sigma_{B}$. In Widom’s formulation, the symbol of an operator is some function in two continuous variables and the symbol of the composition of two operators is expressed as a linear combination of products of derivatives of each symbol. However, in our setting, there is no clear notion of differentiability in the representation variable. We formulate what it means to differentiate with respect to the representation variables in the next subsection.
Difference operators
--------------------
First, we recall some results from [@fischer2015intrinsic]. A symbol can be viewed as a collection $\sigma = \{\sigma(\lambda,x)\in\operatorname{End}(\mathcal{H}_{\lambda})\otimes{\operatorname{Hom}}(E_x,F_x): \lambda \in \widehat{G},x\in G\}$. Moreover, a norm on symbol can be defined using the Hilbert-Schmit inner product:
$$\begin{aligned}
\langle\sigma,\tau\rangle = {\operatorname{Tr}}\left(\tau^*\sigma\right)\end{aligned}$$
where ${\operatorname{Tr}}(\cdot)$ above is the matrix trace. The operator norm $\Vert\cdot\Vert_{op}$ of an $m\times n$ matrix $A$ is defined as $$\label{eq:operator.norm.of.symbols}
\|A\|_{op} :=\sup\left\{\|A x\| : x \in {\mathbb{C}}^n,\|x\|\leq 1\right\}.$$
Note that each $\psi\in C^\infty(G)$ defines a left convolution $L(\psi)$ and a right convolution $R(\psi)$, which act on on $\Gamma(E)$: $$\begin{aligned}
L(\psi)(f)(x) = \psi*f(x) := \int_G \psi(y)f(y^{-1}x)dy
$$ and $$\begin{aligned}
R(\psi)(f)(x) = f*\psi(x):= \int_G f(xy) \psi(y) dy.\end{aligned}$$
It is easy to show that $$\|L(\psi)\|_{B\left(L^2(E)\right)}
= \|R(\psi)\|_{B\left(L^2(E)\right)}
= \sup _{\lambda \in \operatorname{Rep}(G)}\|\widehat{\psi}(\lambda)\|_{op}.$$
On the other hand, the operator associated with $\sigma$ is the operator $\operatorname{Op}(\sigma)$ defined on $L^2(E)$ by $$\begin{aligned}
\operatorname{Op}(\sigma)\left(\phi(x)\right) = \sum_{\lambda \in \widehat G} \dim\lambda{\operatorname{Tr}}(\lambda(x) \sigma(\lambda, x) \widehat{\phi}(\lambda)), \quad \phi \in L^2(E), x \in G.\end{aligned}$$
We also recall the notion of difference operators and of classes of symbols in [@fischer2015intrinsic], which is an analogue of which is a refinement of its initial discovery made in [@ruzhansky2008pseudo]. For each $\lambda, \mu \in \operatorname{Rep}(G)$ and $\sigma \in \Sigma(G)$, we define the linear mapping $\triangle_\lambda \sigma(\mu)$ on $\mathcal{H}_\lambda\otimes\mathcal{H}_{\mu}$ by $$\begin{aligned}
\triangle_{\lambda} \sigma(\mu) :=\sigma(\lambda \otimes \mu) - \sigma\left(I_{\mathcal H_\lambda} \otimes \mu\right).\end{aligned}$$
We also define the iterated difference operators as follows. For any $a \in {\mathbb{N}}$ and $\tau=\left(\lambda_1, \ldots, \lambda_a\right) \in \operatorname{Rep}(G)^a$, we write $$\begin{aligned}
\triangle^\tau :=\triangle_{\lambda_1} \ldots \triangle_{\lambda_a}.\end{aligned}$$
If $\xi \in \operatorname{Rep}(G)$ and $\sigma \in \Sigma$, then $\triangle^\tau\sigma(\xi,x)$ is an element of $$\begin{aligned}
\label{space:endmorphism.tensor.representations}
\operatorname{End}\left(\mathcal H_\xi^{\otimes\tau}\right)\otimes&{\operatorname{Hom}}(E_x,F_x) \\
\nonumber
&:= \operatorname{End}\left(\mathcal H_{\lambda_1} \otimes \ldots \otimes \mathcal H_{\lambda_a} \otimes \mathcal H_\xi \right)\otimes {\operatorname{Hom}}(E_x,F_x).\end{aligned}$$
Let $m \in {\mathbb{R}}$. The set $S^m(E,F)$ is the space of all the symbols $\sigma = \{\sigma(\lambda, x)\in \operatorname{End}\left(\mathcal H_\lambda\right)\otimes {\operatorname{Hom}}(E_x,F_x):(\lambda, x) \in \widehat G \times G\}$, which are smooth $x$ such that for each $\tau \in \operatorname{Rep}(G)^a$ and $X \in \operatorname{Diff^k}(G)$ there exists $C>0$ satisfying $$\begin{aligned}
\forall(\lambda, x) \in\widehat G\times G \qquad\left\Vert X \triangle^\tau\sigma(\lambda, x)\right\Vert_{op}\leq C\langle\lambda\rangle^{\frac{m - a}{2}}.\end{aligned}$$ The norm $\Vert\cdot\Vert_{op}$ is in the sense of the operator norm as in . We say that a symbol is smoothing when it is in $$\begin{aligned}
S^{-\infty}(E,F)=\cap_{m \in \mathbb{R}} S^{m}(E,F).\end{aligned}$$
Now we are ready to move onto the composition formula in the next subsection.
Composition formula
-------------------
In general, we can prove the following composition theorem.
\[theorem:composition.formula\] Let $m_1, m_2 \in {\mathbb{R}}$ and $\rho>\delta \geq 0$. Let $E_0$, $F_0$ and $H_0$ be irreducible unitary representations of $K$ and $E$, $F$ and $H$ be corresponding associated vector bundles to the principal bundle $G\rightarrow G/K$. Suppose $$\begin{aligned}
A:\Gamma(E)\longrightarrow \Gamma(F)\end{aligned}$$ and $$\begin{aligned}
B:\Gamma(F)\longrightarrow\Gamma(H)\end{aligned}$$ are continuous linear maps such that $A\left(\Gamma(E)\right) \subset \Gamma(F)$ with symbols $\sigma_A$ and $\sigma_B$ and they satisfy $$\begin{aligned}
\left\|\triangle_\lambda^\alpha\sigma_A(\lambda, x)\right\|_{op} & \leq C_\alpha\langle\lambda\rangle^{m_1 - \rho|\alpha|} \\
\left\|X^\beta\sigma_B(\lambda, x)\right\|_{op} & \leq C_\beta\langle\lambda\rangle^{m_2 + \delta|\beta|}.\end{aligned}$$ for all multi-indices $\alpha$ and $\beta$, uniformly in $x\in G$ and $\lambda\in\widehat G$. Then, $$\begin{aligned}
\label{formula:composition}
\sigma_{A B}(x, \lambda) \sim \sum_{\alpha \geq 0} \frac{1}{\alpha !}\left(\triangle_{\lambda}^\alpha \sigma_A\right)(x, \lambda) X^{(\alpha)} \sigma_B(x, \lambda).\end{aligned}$$
In view of [@ruzhansky2008pseudo Theorem 8.3], it suffices to reduce this case to the case of scalar functions. This is achieved by representing $f\in \Gamma(E)$ as an element in $E_0$-valued function on $G$. Again, the the scalar valued (trivial line bundle) case of this formula had been proved in [@ruzhansky2008pseudo Theorem 8.3], which proves it for each component of $f$. Since the fibre $E_0$ is finite dimensional, that result extends trivially.
Theorem \[theorem:composition.formula\] is a slight improvement of the treatment in [@ruzhansky2009pseudo Chapter 13] for the case of scalar valued functions on homogeneous spaces because our case for $H=\{e\}$ and $E_0={\mathbb{C}}$ the trivial representation would reduce to their case.
Sobolev spaces
--------------
Recall that Sobolev spaces were defined in Section \[subsec:homogeneous.vector.bundle\] by completion of $\Gamma(E)$ with the inner product in .
\[theorem:boundedness.in.sobolev.space\] Let $G$ be a compact Lie group and $A:\Gamma(E)\rightarrow\Gamma(F)$ be an operator with symbol $\sigma_A$. Suppose that there are constants $m, C_\alpha\in{\mathbb{R}}$ such that $$\begin{aligned}
\label{inequality:sobolev.space}
\left\Vert X^\alpha\sigma_A(\lambda, x)\right\Vert\leq C_\alpha\langle\lambda\rangle^m\end{aligned}$$ holds for all $x \in G$, $\lambda\in$Rep$(G)$ and all multi-indices $\alpha$ where $X^\alpha=X_1^{\alpha_1} \cdots X_{\dim G}^{\alpha_{\dim G}}$ is as in . Then, $A$ is a bounded linear operator $$\begin{aligned}
H^s(E) \longrightarrow H^{s - m}(F)\end{aligned}$$ for all $s \in {\mathbb{R}}$.
The case of $H=\{e\}$ and $E_0=F_0={\mathbb{C}}$ was shown in [@ruzhansky2013global Theorem 3.2]. Suppose $\{e_k\}$ and $\{f_\ell\}$ are bases of $E_0$ and $F_0$, respectively, with corresponding dual bases $\{e^*_k\}$ and $\{f^*_\ell\}$. Note that $B(e^*_a(f(x))):=f_b^*\circ A(e^*_a(f(x))e_a)$ defines a linear operator, which satisfies . Thus, it defines a bounded linear operator from $H^s(G)\rightarrow H^{s-m}$ [@ruzhansky2013global]. Using the identification of the Sobolev spaces and to the linear combination using the operator $B$, it can be seen that $$\begin{aligned}
A : (H^s(G)\otimes E_0)^K \longrightarrow (H^{s-m}(G)\otimes F_0)^K\end{aligned}$$ is bounded.
Let $m \in {\mathbb{R}}$. We denote $\sigma_A\in\Sigma_0^m(E,F)$ if the singular support of the map $y \mapsto K_A(x, y)$ is in $\{e\}$ and if $$\begin{aligned}
\left\Vert\triangle_\lambda^\alpha X^\beta\sigma_A(\lambda, x)\right\Vert_{op}
\leq C_{A \alpha \beta m}\langle\lambda\rangle^{m-|\alpha|}\end{aligned}$$ for all $x \in G$, all multi-indices $\alpha, \beta$ and all $\lambda\in$Rep$(G)$. Moreover, we say that $\sigma_A\in\Sigma_{k+1}^m(E,F)$ if and only if $$\begin{aligned}
\sigma_A \in\Sigma_k^m(E,F) \\
\sigma_{\partial_j}\sigma_A - \sigma_A\sigma_{\partial_j}\in\Sigma_k^m(E,F)\\
\left(\triangle_\lambda^\gamma \sigma_A\right) \sigma_{\partial_j} \in \Sigma_k^{m+1-|\gamma|}(E,F) $$ for all $|\gamma|>0$ and $1 \leq j \leq \dim(G)$. Let $$\begin{aligned}
\Sigma^m(E,F) :=\bigcap_{k=0}^{\infty} \Sigma_k^m(E,F)\end{aligned}$$
Suppose $G$ is a compact Lie group with a closed subgroup $H$ and $m \in {\mathbb{R}}$. If $E$ and $F$ are homogeneous vector bundles over $G/H$ associated to irreducible representations $E_0$ and $F_0$, then, $A \in \Psi^m(E,F)$ if and only if $\sigma_A\in\Sigma^m(E,F)$.
By repeating the argument of Theorem \[theorem:boundedness.in.sobolev.space\], this theorem reduces to Theorem 10.9.6 in [@ruzhansky2009pseudo].
**Application and parametrics** {#sec:parametrix}
===============================
In this section, we are concerned with the case $E=F$, so we set $\Psi^m(E):=\Psi^m(E,E)$
Let $A\in \Psi^m(E)$ and $z\in{\mathbb{C}}$. $B_z \in\Psi^{-m}(E)$ will be called a resolvent parametric of $A$ if it is a parametric of $A - z$ (that is, $B_z(A - z)$ differs from the identity up to a smoothing operator.
In this section, we present an explicit computation of the asymptotic expansion of the intrinsic symbol of a resolvent parametric of $A$ in terms of the representation of $G$ and the terms in the asymptotic expansion of the intrinsic symbol of $A$.
\[theorem:parametrix\] Let $\sigma_{A_j} \in \Sigma^{m - j}(E)$, and set $$\begin{aligned}
\sigma_A(\lambda, x) \sim \sum_{j=0}^\infty\sigma_{A_j}(\lambda, x).\end{aligned}$$ Assume that $\sigma_{A_0}(\lambda, x) = \sigma_{B_0}(\lambda, x)^{-1}$ is an invertible matrix for every $x\in G$ and $\lambda\in$Rep$(G)$, and that $B_0=\mathrm{Op}\left(\sigma_{B_0}\right) \in \Psi^{- m}(E)$. Then, there exists $\sigma_B \in \Sigma^{-m}(E)$ such that $I-B A$ and $I-A B$ are smoothing operators. Moreover, $$\begin{aligned}
\sigma_B(\lambda, x) \sim \sum_{k=0}^\infty\sigma_{B_k}(\lambda, x)\end{aligned}$$ where the operators $B_k \in \Psi^{-m-k}(E)$ are determined by the recursion $$\begin{aligned}
\label{eq:composition.formula}
\sigma_{B_N}(\lambda, x) = - \sigma_{B_0}(\lambda, x) \sum_{k=0}^{N - 1} \sum_{j=0}^{N-k} \sum_{|\gamma|=N-j-k} \frac{1}{\gamma !}\triangle_{\lambda}^{\gamma} \sigma_{B_k}(\lambda, x) X^{\gamma} \sigma_{A_j}(\lambda, x).\end{aligned}$$
If $\sigma_I\sim\sigma_{BA}$ for some $\sigma_B\sim\sum_{k=0}^\infty\sigma_{B_k}$, then by Theorem \[theorem:composition.formula\] we have $$\begin{aligned}
\begin{split}
I_{\dim\lambda}\otimes I_{\dim E_0} = \sigma_I(\lambda,x) & \sim \sigma_{BA}(\lambda,x) \\
& \sim \sum_{\gamma \geq 0} \frac{1}{\gamma !}\left(\triangle_\lambda^\gamma\sigma_B(\lambda,x)\right)X^\gamma\sigma_A(\lambda,x) \\
& \sim \sum_{\gamma \geq 0} \frac{1}{\gamma !}\left(\triangle_\lambda^\gamma\sum_{k=0}^\infty\sigma_{B_k}(\lambda,x)\right)X^\gamma\sum_{j=0}^{\infty} \sigma_{A_j}(\lambda,x)
\end{split}\end{aligned}$$ We now find $\sigma_{B_k}$. Note that $I_{\dim\lambda}\otimes I_{\dim E_0} = \sigma_{B_0}(\lambda,x)\sigma_{A_0}(\lambda,x)$, and for $|\gamma|\geq1$ we may suppose that $$\begin{aligned}
\sum_{|\gamma|=N-j-k} \frac{1}{\gamma !}\left(\triangle_{\lambda}^{\gamma} \sigma_{B_k}(\lambda, x)\right) X^\gamma \sigma_{A_j}(\lambda, x) = 0.\end{aligned}$$ Then, provides the solution to these equations, and it can be easily verified that $\sigma_{B_N}\in\Sigma^{-m-N}(G)$ by induction on $N$ after noting that $B_0\in\Psi^{-m}(E)$. Thus, $\sigma_B\sim\sum_{k=0}^\infty\sigma_{B_k}$. Finally, notice that $\sigma_{I_{\dim\lambda}\otimes I_{\dim E_0}} \sim\sigma_{B A}$
**Example: the fiberation $SU(2)\rightarrow U(2)$** {#sec:example.su(2).u(2)}
===================================================
We now present examples of symbols as aforementioned in Section \[subsec:homogeneous.vector.bundle\]. For example, suppose $K = {\mathbb{T}}\subset SU(2) = G$ and $E_n = {\mathbb{C}}^{\otimes n}\in\widehat{{\mathbb{T}}} \cong{\mathbb{Z}}$. In fact, $E_n \cong \{e^{2\pi in t}:t\in{\mathbb{R}}\}$. Note that $G/K={\mathbb{C}}P^1 = S^2$. Then, the homogeneous vector bundles associated to these representations are all line bundles $\mathcal O(n) := G\times_K{\mathbb{C}}^{\otimes n}\in\widehat{{\mathbb{T}}} \cong{\mathbb{Z}}$ and $\mathcal O(-n) := \mathcal(n)^*$. More concretely, the sections $\Gamma\left(\mathcal O(n)\right)$ of the bundle $O(n)$ are given by the functions $f:G\rightarrow E_n$ such that $$\begin{aligned}
f
\begin{pmatrix}
e^{2\pi i t}\alpha & -e^{-2\pi i t}\bar \beta\\
e^{2\pi i t}\beta & e^{-2\pi i t}\bar\alpha
\end{pmatrix}
& =
e^{2\pi in t}f
\begin{pmatrix}
\alpha & -\bar \beta\\
\beta & \bar\alpha
\end{pmatrix}\end{aligned}$$ In other words, $$\begin{aligned}
f(e^{2\pi it}\alpha, \beta, e^{-2\pi it}\bar\alpha,\bar\beta)
=
e^{2\pi int}
f(\alpha, \beta, \bar\alpha,\bar\beta)\end{aligned}$$ for all $t\in{\mathbb{R}}/{\mathbb{Z}}$. That is, $f(\alpha, \beta, \bar\alpha,\bar\beta) = \alpha^n\sum c_{\ell m p}(\alpha\bar\alpha)^\ell(\alpha\bar\beta)^m(\bar\alpha\beta)^p$. The sum $\sum c_{\ell m p}(\alpha\bar\alpha)^\ell(\alpha\bar\beta)^m(\bar\alpha\beta)^p$ corresponds to functions in $C^\infty(G/K)$.
We now compute the action of $\mathfrak{su}(2)$ on the sections of $O(n)$. Let $\left\{ H, X,Y\right\}$ be the basis used in . Here, $\exp(tA) = e^{2\pi itA} := \sum \frac{(2\pi itA)^k}{k!}$ for all $A\in\mathfrak{su}(2)$. Their exponential is given by $$\begin{aligned}
\exp(tH) = &
\begin{pmatrix}
e^{2\pi it} & 0\\
0 & e^{-2\pi it}
\end{pmatrix},\quad
\exp(tX) =
\begin{pmatrix}
\cos(2\pi t) & \sin(2\pi t)\\
\sin(2\pi t) & \cos(2\pi t)
\end{pmatrix}\\
&\exp(tY) =
\begin{pmatrix}
\cos(2\pi t) & \sin(2\pi t)\\
-\sin(2\pi t) & \cos(2\pi t)
\end{pmatrix}
.\end{aligned}$$ We can compute their actions on the sections. In fact, it is enough to compute their actions on $\alpha^n$, so let $f(\alpha,\bar\alpha,\beta,\bar\beta)$. $$\begin{aligned}
\begin{split}
H\cdot f(\alpha,\bar\alpha,\beta,\bar\beta) & = \frac{d}{dt}\Big\vert_{t=0}f(e^{2\pi it}\alpha,e^{-2\pi it}\bar\alpha,\beta,\bar\beta)\\
& = \frac{d}{dt}\Big\vert_{t=0} e^{ - 2\pi int} f(\alpha,\bar\alpha,\beta,\bar\beta) \\
& = ( - 2\pi n~i) f(\alpha,\bar\alpha,\beta,\bar\beta)
\end{split}
\end{aligned}$$ $$\begin{aligned}
\begin{split}
X\cdot f(\alpha,\bar\alpha,\beta,\bar\beta) & = (2\pi i) f_*(\frac{d}{dt})_{(\bar\beta,\beta,\bar\alpha,\alpha)}
\end{split}
\end{aligned}$$ and $$\begin{aligned}
\begin{split}
Y\cdot f(\alpha,\bar\alpha,\beta,\bar\beta) & = (2\pi i) f_*(\frac{d}{dt})_{(\bar\beta,\beta,\bar\alpha,-\alpha)}\\
\end{split}
\end{aligned}$$ and the symbol $\sigma(\ell,x)$ of $A\in\mathfrak{su}(2)$ is given by $$\begin{aligned}
\sigma_A(\xi_\ell,x) & = \left(\xi_\ell\right)_*A\\
& = A^{\otimes (2\ell+1)} \end{aligned}$$ where $\xi_\ell:=\operatorname{Sym}^{(2\ell + 1)}(\lambda)$ in , $\ell\in\frac12{\mathbb{N}}$. With the above notation, then, the action of a vector vector field $A\in\mathfrak{su}(2)$ on the sections of the homogeneous vector bundle associated to the representation $E_n\in\widehat{\mathbb{T}}={\mathbb{Z}}$ can be written as $$\begin{aligned}
\label{formula:symbol.of.vector.fields}
A\cdot f(\alpha, \beta, \bar\alpha, \bar\beta) = \sum_{\ell\in\frac{1}{2}{\mathbb{N}}}(2\ell + 1){\operatorname{Tr}}\left(\xi_\ell(\alpha, \beta, \bar\alpha, \bar\beta)
A^{\otimes (2\ell + 1)}\widehat f(\ell)\right)\end{aligned}$$ for $f\in\Gamma(\mathcal O(n))$ and the Haar measure on $SU(2)$ is given by $$f\mapsto \frac1{2\pi^2}\int_{0}^1\int_0^1\int_{0}^{\pi/2} f(\alpha,\bar\alpha,\beta,\bar\beta)\sin\eta\cos\eta d\eta d\xi_1 d\xi_2$$ where we identified $$\begin{aligned}
\alpha = e^{2\pi i\xi_1}\sin\eta\\
\beta = e^{2\pi i\xi_2}\cos\eta.\end{aligned}$$
**Concluding remarks and functional calculus** {#sec:remarks}
==============================================
The main aim of this section is to provide a guideline of a derivation of the asymptotic expansions. Note that each summand in defines bilinear maps $P_j(\sigma,\tau)$ for $\sigma\in\Sigma^m(E)$ and $\tau\in\Sigma^{m'}(E)$ $$P_j:\Sigma^m(E)\times\Sigma^{m'}(E)\longrightarrow\Sigma^{m+m' - j}(E)$$ where $P(\sigma,\tau) := \sum_{j\leq m}P_j(\sigma,\tau)$ is the composition of symbols operators corresponding to $\sigma$ and $\tau$, respectively, computed using .
Suppose $\sigma\in \Sigma^m(E)$ such that $\sigma^{-1}\in\Sigma^{m'}(E)$ whereby $\sigma^{-1}$ denotes the inverse of $\sigma$ mod $\Sigma^{-\infty}$. Define $$\begin{aligned}
\label{formula:inverse}
Q_0(\sigma) = \sigma^{-1},\quad Q_j(\sigma) = - \sigma^{-1} \sum_{r=1}^j P_r(\sigma, Q_{j-r}(\sigma)).\end{aligned}$$ Induction shows that if $m + m'<1$, then $$Q_j(\sigma)\in\Sigma^{m' - j + j(m+m')}(E)$$ and $$Q(\sigma) : = \sum_{j\geq0}Q_j(\sigma)$$ defines an element of $\Sigma^{m'}(E)$. Moreover, and that $\sum_{r+s\leq N}P_r(\sigma, Q_s(\sigma)) = 1$ implies that $$\begin{aligned}
P(\sigma,Q(\sigma)) = 1.\end{aligned}$$ This proves that if $A\in\Psi^m(E)$ and $\sigma_A^{-1}\in\Sigma^{m'}(E)$ with $m + m'<1$, then $A$ has a right inverse modulo $\Psi^{-\infty}(E)$, which belongs to $\Psi^{m'}(E)$ whose symbol is $Q(\sigma_A)$. However, a similar argument shows that $A$ also has a left inverse. Thus, we have proved the following improvement of Theorem \[theorem:parametrix\].
If $A\in\Psi^m(E)$ and $\sigma^{-1}_A \in \Sigma^{m'}(E)$ with $m + m' <1$, then $A$ has a two-sided inverse in $\Psi^{m'}(E)$ whose symbol is $Q(\sigma_A)$ modulo $\Sigma^{-\infty}(E)$.
Using the above theorem, we expect that for an operator $A$ in the above class of operators whose symbol $\sigma_A$ has purely discrete spectrum ${\operatorname{Spec}}(\sigma_A)$ and an analytic function $f$ on ${\operatorname{Spec}}(\sigma_A)$, $$\begin{aligned}
\label{equation:cauchy.integration}
\sigma_{f(A)} & = - \frac1{2\pi i}\int_\gamma f(\xi)Q(\sigma - \xi)d\xi\\
\nonumber
& = f(\sigma) + \sum_{k=1}^\infty\sum_{l=2}^{2k} \frac{(-1)^{k+1}}{2\pi i}\int_\gamma f(\xi)Q_{k,l}(\sigma - \xi)d\xi\end{aligned}$$
Since $(\sigma - z)^{-1}$ can be expressed as $$\begin{aligned}
(\sigma - z)^{-1} = \sum_{z_\alpha\in{\operatorname{Spec}}(\sigma_A)}\sigma_\alpha(z_\alpha - z)^{-1}.\end{aligned}$$ Then, $$\begin{aligned}
Q_{k,l}(\sigma - z) = \sum_{\alpha_0,\alpha_1,\ldots,\alpha_k}\prod_{j=0}^k(z_{\alpha_j} - z)^{-1}Q_{k,l}(\sigma:\sigma_{\alpha_0},\ldots,\sigma_{\alpha_l})\end{aligned}$$ where $Q_{k,l}(\sigma:\sigma_{\alpha_0},\ldots,\sigma_{\alpha_k})$ means the $j$th $\sigma^{-1}$ term in $Q_{k,l}(\sigma)$ is replaced by $\sigma_{\alpha_j}$. We can evaluate the integrals in . $$\begin{aligned}
\label{formula:cauchy.integral.2}
\frac{(-1)^{k+1}}{2\pi i}\int_\gamma f(\xi)\prod_{j=0}^k(\zeta_j-\xi)^{-1}d\xi = \sum_{l=0}^kf(\zeta_l)\prod_{a\neq l}(\zeta_a - \zeta_l)^{-1}\end{aligned}$$ for distinct $\zeta_l$. If $\zeta_l$ are not distinct, the above summation can be modified by replacing $f$ by its derivatives. Thus, we denote by $$\begin{aligned}
\frac1{k!}f^{(k)}(\zeta_0,\ldots,\zeta_k). $$ With this notation, the integral in becomes $$\begin{aligned}
\label{eq:funtional.calculus}
f(\sigma) + \sum_{k=1}^\infty\sum_{l=2}^{2k} \frac1{l!}f^{(l)}(\zeta_0,\ldots,\zeta_l) Q_{k,l}(\sigma:\sigma_{\alpha_0},\ldots,\sigma_{\alpha_l})\end{aligned}$$
Indeed, if $f$ is analytic and $A\in\Psi^0(E)$, then a modification of [@widom1980complete Theorem 4.1] shows that $f(A)\in\Psi^0(E)$ and $f(\sigma_A)=\sigma_{f(A)}$ where the symbol $\sigma_{f(A)}$ is defined by the formula . However, the computation of explicit formula is very difficult, unless $f$ and $A$ are known explicitly.
|
---
abstract: 'Recently [@Townsend1], an extension of the topologically massive gravity (TMG) in $2+1$ dimensions, dubbed as minimal massive gravity (MMG), which is free of the bulk-boundary unitarity clash that inflicts the former theory and all the other known three dimensional theories, was found. Field equations of MMG differ from those of TMG at quadratic terms in the curvature that do not come from the variation of an action depending on the metric alone. Here we show that MMG is a unique theory and there does not exist a deformation of TMG or MMG at the cubic and quartic order (and beyond) in the curvature that is consistent at the level of the field equations. The only extension of TMG with the desired bulk and boundary properties having a single massive degree of freedom is MMG.'
author:
- 'Emel Altas, Bayram Tekin'
title: 'Holographically Viable Extensions of Topologically Massive and Minimal Massive Gravity ?'
---
INTRODUCTION
============
One of the most promising approaches to a quantum theory of gravity is via the anti-de Sitter (AdS)- conformal field theory (CFT) [@maldacena] correspondence where there is a boundary field theory dual to the bulk gravity. In $2+1$ dimensions, where gravity is somewhat less complicated, this idea has been vigorously pursued in many different works. Einstein’s gravity with a cosmological constant in $2+1$ dimensions is locally trivial with no propagating degrees of freedom; therefore to study a dynamical theory which might mimic realistic gravity and teach us something about four-dimensional quantum gravity, the next option is to consider the parity-noninvariant topologically massive gravity (TMG) which has a single massive graviton [@Deser1]. TMG with a cosmological constant has two copies of Virasoro algebra, as its asymptotic symmetry algebra, in the two-dimensional boundary of $AdS_{3}$. In TMG, unitarity of the putative boundary CFT is in conflict with the unitarity of the bulk theory except, ostensibly, at the chiral point where the problematic negative central charge of the boundary field theory vanishes and the other central charge is positive [@strom]. But, exactly at this point in the parameter space of couplings, there arise solutions with asymptotically non-AdS (logarithmic) behavior which cannot be eliminated from the spectrum on solid physical grounds except with ad hoc strong boundary conditions [@carlip; @grumiller; @gribet]. So apparently, TMG by itself does not have a unitary dual CFT in asymptotically AdS spacetimes, and hence most probably is not viable as a quantum theory (at least in the sense of AdS/CFT correspondence).
Another dynamical theory, new massive gravity (NMG) [@nmg], a judiciously chosen quadratic extension of Einstein’s gravity and with two helicity-2 (albeit massive) degrees of freedom closer to the four-dimensional gravity, also has the bulk-boundary unitarity clash and hence does not posses the expected holographic description. Unfortunately, healthy deformations of NMG in the bulk, such as the cubic, quartic [@sinha] or infinite order ones [@binmg; @binmgc; @paulos] also suffer from boundary nonunitarity and so probably lack a CFT dual.
With all these negative results, there seems to be an apparent impasse: Einstein’s gravity has a healthy boundary structure [@brown] but suffers from bulk triviality and all the locally nontrivial theories seem to suffer from bad boundary behavior and so one may wonder if is it not possible to construct a dynamical theory of gravity in 2+1 dimensions that is unitary both in the bulk and on the boundary. It turns out that one can actually construct such a theory [@Townsend1] once one gives up the condition that the theory comes from the variation of an action which is purely defined in terms of the metric. (There does exist an action in the first order, that is the dreibein and the spin-connection formulation [@Townsend1; @Baykal]). Equations without a proper Lagrangian formulation are not unheard of in macroscopic physics, but clearly this is a rather novel idea in microscopic phenomena. But (quantum) gravity is so elusive that one must try many different routes to get a possible understanding of it. In [@Townsend1] keeping the bulk properties of TMG intact, a theory with an improved boundary behavior was formulated in terms of consistent field equations. Namely, the field equations do not have a Bianchi identity for generic smooth metrics, but they do satisfy a Bianchi identity for all solutions of the theory. Therefore the theory is consistent as a classical one and can also be studied as a quantum theory. Its bulk and boundary unitarity and chiral version and conserved charges were constructed in [@Alishahiha:2014dma; @Tekin:2014jna]. TMG’s deformation with two helicity-2 degrees of freedom was constructed in [@mmg2] called MMG$_{2}$, which also has unitary bulk and boundary properties for a large class of spacetimes.
The fact that MMG has these remarkable properties which the other three-dimensional theories lack, begs the question if the theory is unique or if it is part of a large class of theories that are defined by consistent field equations but do not come from the variation of an action. In this current work, we show several things. First we prove that at the quadratic order in the curvature MMG is the only possible deformation of TMG. Our proof will make the rather “magical” appearance of the on-TMG-shell conserved $J$-tensor more intuitive. Then we move on to the cubic and quartic orders in the curvature and show in detail that there does not exist a deformation of TMG or MMG with a single massive degree of freedom. The Schouten identities satisfied by the powers of the Ricci tensor guarantee that no new algebraically independent rank -2 tensors built with the powers of the curvature arise beyond the quadratic terms and hence the proof is valid for all theories based on the powers of the curvature and not its derivatives. \[Note that if derivatives of the curvature are introduced the problem turns into a separate one, diverging from the idea of extending the single massive degree of freedom theories. MMG$_{2}$ discussed above is an example of that.\]
Our construction here basically answers the following problem. Let ${\cal E}_{\mu\nu}=0$ be the field equations $${\cal E}_{\mu\nu}=\sigma G_{\mu\nu}+\Lambda_0 g_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}+\gamma Y_{\mu\nu} =0,
\label{denklem_esas}$$ where the Einstein and Cotton tensors read, respectively, as $$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R, \hskip 0.7 cm
C_{\mu\nu}=\eta_{\mu}\thinspace^{\alpha\beta}\nabla_{\alpha}S_{\beta \nu},$$ and $S_{\mu \nu} \equiv R_{\mu \nu} -\frac{1}{4} g_{\mu \nu}R$ is the Schouten tensor. The completely antisymmetric tensor is defined in terms of the completely symmetric symbols as $\eta^{\nu\rho\sigma}\equiv\epsilon^{\nu\rho\sigma}/\sqrt{-\det g}$. The main question is to find all the possible $Y^{\mu\nu}$ tensors which satisfy the on-shell conservation: Namely we demand the on-shell Bianchi identity : $$\nabla_{\mu}\varepsilon^{\mu\nu}=\gamma\nabla_{\mu}Y^{\mu\nu}=0. \hskip 0.5 cm (\mbox{on shell})$$ Let us study the problem order by order in the powers of curvature.
$R^2$-extensions
=================
Let us assume that one has the most general quadratic tensor as $$Y^{\mu\nu} \equiv a {\cal{S}}_2^{\mu \nu} +b g^{\mu\nu}{\cal S}_2 +c S_{\mu\nu}S +d g_{\mu\nu}S^{2},$$ where, not to clutter the notation, we have defined ${\cal{S}}_2^{\mu \nu} \equiv S^\mu_\rho S^{\rho \nu}$ and ${\cal{S}}_2 \equiv S_{\mu \nu} S^{\mu \nu}$ which will come in handy when more powers of the tensors are constructed. The trace and divergence are, respectively, $$Y = (a + 3 b) {\cal{S}}_2+(c+ 3 d)S^{2},$$ $$\begin{split}
\nabla_{\mu}Y^{\mu\nu}=\Big ((a+c)S_{\rho}\thinspace^{\nu}
+(2d+c)S\delta_{\rho}^{\nu}\Big)\nabla^{\rho}S+S^{\mu\rho}\Big (a\nabla_{\mu}S_{\rho}\thinspace^{\nu}+2b\nabla^{\nu}S_{\mu\rho}\Big ).
\end{split}$$ For this vector to vanish on the TMG mass shell, we must turn the last part to a Cotton tensor, which is possible only if $a=-2b$ yielding $$\nabla_{\mu}Y^{\mu\nu}=\Big ((a+c)S_{\rho}\thinspace^{\nu}+(2d+c)S\delta_{\rho}^{\nu}\Big )\nabla^{\rho}S+a\eta_{\lambda}\thinspace^{\nu\mu}S_\mu\,^\rho C_{\rho}\thinspace^{\lambda},
\label{Ydenk}$$and reducing the trace to $$Y = b{\cal{S}}_2 +(c+ 3 d)S^{2}.
\label{trace2}$$ Here the discussion bifurcates: If $b\ne 0$, then the modified theory does not preserve TMG’s property that all solutions have $\nabla_\mu R=0$. On the other hand, if $b=0$, the modified theory keeps this property of TMG intact. But in the latter case, no new theories arise beyond TMG since one has theories are $Y_{\mu\nu} = c S_{\mu\nu}S +d g_{\mu\nu}S^{2} $, which for constant $S$ simply gives a shift of TMG parameters. So we assume $b \ne 0$. In this case, in (\[Ydenk\]), the term with the Cotton tensor vanishes onshell (\[denklem\_esas\]). The first term in (\[Ydenk\]) does not vanish unless one sets $$a+c=0, \hskip 1 cm 2d+c=0,$$ which reduces, after fixing the overall coefficient as choose $a=-1$, the $Y_{\mu \nu}$-tensor to the $J$-tensor found in [@Townsend1] as $$J^{\mu\nu}=-{\cal{S}}_2^{\mu \nu} +\frac{1}{2} g^{\mu\nu}{\cal S}_2 +S^{\mu\nu}S-\frac{1}{2}g^{\mu\nu}S^{2},$$ which can be recast as $J^{\mu\nu}\equiv \frac{1}{2}\eta^{\mu\rho\sigma}\eta^{\nu\tau\eta}S_{\rho\tau}S_{\sigma\eta}$ which has the following interesting properties. Its trace is given as $$J = \frac{1}{2} \Big ({\cal S}_2- S^2 \Big ),$$ which is nothing but the the quadratic part of NMG, the theory that defines a massive spin-2 particle with two helicities. Quite remarkably, as noted in [@mmg2], the variation of the quadratic part of NMG splits into two parts as $$\delta_g \int \sqrt{-g}\, d^3 x \, J \equiv J_{ \mu \nu} + H_{\mu \nu},
\label{action}$$ where the $H$-tensor is $$H_{ \mu \nu} \equiv \frac{1}{2}\eta_{\mu}\,^{ \alpha\beta}\nabla_{\alpha} C_{\beta \nu} + \frac{1}{2}\eta_{\nu}\,^{ \alpha\beta}\nabla_{\alpha} C_{\beta \mu}.$$ Clearly one has $ \nabla_\mu H^{\mu \nu} = - \nabla_\mu J^{\mu \nu} = \eta^{\nu\alpha\beta}S_{\alpha \sigma} C_{\beta}\,^\sigma $, and so it follows from (\[action\]) that $J$- and $H$-tensors are not separately automatically covariantly conserved. But when the $J$-tensor is augmented to TMG equations, one gets a consistent, on-shell, conservation, which can also be coupled to matter consistently, albeit in a rather complicated way [@Arv]. We would like to note the following observation: The $H$-tensor, when looked at with closer scrutiny, is nothing but the three-dimensional version of the Bach tensor $B_{\mu \nu}$ that measures whether the spacetime is a conformally Einstein manifold or not in four dimensions. Namely, in $n$ dimensions, the Bach tensor is given as $$B_{\mu \nu} = \nabla ^\alpha \nabla^\beta W_{ \mu \alpha \nu \beta } + \frac{1}{2} R^{ \alpha \beta } W_{\mu \alpha \nu \beta } ,$$ which in this form does not allow a three-dimensional analog since the Weyl-tensor ($W_{ \mu \alpha \nu \beta }$) vanishes identically. But an equivalent form of the Bach tensor is $$B_{\mu \nu} = \frac{1}{2} \nabla^\alpha C_{\alpha \mu \nu } + \frac{1}{2} R_{ \alpha \beta } W_\mu\,^{\alpha}\,_\nu\,^\beta ,$$ where $ C_{\alpha \mu \nu }$ is the three index Cotton tensor that serves as a “potential” to the Weyl-tensor and is defined in any dimension as $$C_{\alpha \mu \nu } = \nabla_{\alpha} R_{\mu \nu} - \nabla_{\mu} R_{\alpha \nu} - \frac{1}{ 2( n-1)} \Big ( g_{ \mu \nu} \nabla_{\alpha } R- g_{ \alpha \nu} \nabla_{\mu } R \Big ).
\label{3indexcot}$$ In three dimensions, one has $$C_{\mu \nu} = \frac{1}{2} \eta_{\mu}\,^{ \alpha \beta} C_{ \alpha \beta \nu},
\label{2indexcot}$$ and hence follows the equivalence of the $H$- tensor and the three-dimensional version of the Bach tensor. This is a rather unexpected result which says that when restricted to the conformally Einstein (or conformally flat, which are the same in three dimensions ) metrics, the quadratic part of NMG reduces to that of MMG. The quadratic part of NMG, without the Einstein term, was studied in [@deser_prl] as a separate model. Note that in four dimensions the Bach tensor is divergence free but not so in other dimensions, including three dimensions.
Before we move on to the higher powers, let us give a rederivation of the uniqueness of MMG .
UNIQUENESS OF MMG
=================
Suppose $X_{\mu\nu}$ is a symmetric and divergence-free ($\nabla_{\mu} X^{\mu\nu}=0$) tensor, coming from the variation of an action purely based on the metric. For the $X$-tensor to be divergence free, the action has to be diffeomorphism invariant at least up to a boundary term as in the case of TMG. We shall denote the traces without and index as $X\equiv X_{\mu\nu}g^{\mu\nu}$. Using this tensor, we can build a symmetric two-tensor quadratic in our given tensor as $$Y^{\mu\nu}\equiv\frac{1}{2}\,\eta^{\mu\rho\sigma}\eta^{\nu\tau\eta}\widetilde{X}_{\rho\tau}\widetilde{X}_{\sigma\eta}\,,
\label{Y_tensor}$$ where $\widetilde{X}_{\sigma\eta}=X_{\sigma\eta}+ a\ g_{\sigma\eta}X$ with $a$ real number for now. Note that with just one single parameter, the above $Y$-tensor is in a specific form: The most general quadratic form reads $$Y_{\mu\nu}\equiv X_{\mu}^{\rho}X_{\rho\nu}+c_1 g_{\mu\nu}X_{\rho\sigma}X^{\rho\sigma}+c_2 X_{\mu\nu}X +c_3 g_{\mu\nu}X^{2}.
\label{J-tensor}$$ But taking this second form simply extends the length of the following computations eventually resulting to the same conclusion. Using $$\eta^{\mu\sigma\rho}\eta_{\nu\alpha\beta}=-\delta^{\mu}{_{\nu}}\Big(\delta^{\sigma}{_{\alpha}}\delta^{\rho}{_{\beta}}-\delta^{\sigma}{_{\beta}}\delta^{\rho}{_{\alpha}}\Big)+\delta^{\mu}{_{\alpha}}\Big(\delta^{\sigma}{_{\nu}}\delta^{\rho}{_{\beta}}-\delta^{\sigma}{_{\beta}}\delta^{\rho}{_{\nu}}\Big)-\delta^{\mu}{_{\beta}}\Big(\delta^{\sigma}{_{\nu}}\delta^{\rho}{_{\alpha}}-\delta^{\sigma}{_{\alpha}}\delta^{\rho}{_{\nu}}\Big),\label{identity}$$ it is easy to show that (\[Y\_tensor\]) has all the required tensor structures in it. Hence, we shall start with it. Then one has $\widetilde{X}=(1+3a)X$, and the divergence of $\widetilde{X}^{\sigma\eta}$ reads $$\nabla_{\sigma}\widetilde{X}^{\sigma\eta}=\frac{a}{(1+3a)}\nabla^{\eta}\widetilde{X}.$$ Using this, one can compute the divergence of $Y^{\mu\nu}$ as $$\nabla_{\mu}Y^{\mu\nu} = \,\eta^{\mu\rho\sigma}\eta^{\nu\tau\eta}\widetilde{X}_{\rho\tau}\nabla_{\mu}\widetilde{X}_{\sigma\eta}
\equiv \eta^{\nu\eta\tau}\widetilde{X}_{\rho\tau}{Z}_{\eta}~^{\rho}.$$ where we defined a new tensor $Z^{\mu\nu}$ as $$Z^{\mu\nu}=\eta^{\mu\alpha\beta}\nabla_{\alpha}\widetilde{X}_{\beta}~^{\nu}.$$ Let us now check the properties of the $Z$-tensor. It is traceless, but it is not automatically symmetric as can be seen from $$\eta_{\mu\nu\sigma}Z^{\mu\nu}=-\nabla_{\nu}\widetilde{X}_{\sigma}~^{\nu}+\nabla_{\sigma}\widetilde{X}.$$ But with the choice $a=-\frac{1}{2}$ , $Z^{\mu\nu}$ becomes symmetric. So we make this choice which yields $\widetilde{X}_{\sigma\eta}=X_{\sigma\eta}-\frac{1}{2}g_{\sigma\eta}X$ and $\widetilde{X}=-\frac{1}{2}X$ . With these, one can compute the divergence of $Z_{\mu\nu}$ as $$\nabla_{\mu}Z^{\mu\nu}=\eta^{\nu\alpha\beta}R_{\alpha\lambda}\widetilde{X}_{\beta}~^{\lambda}=\eta^{\nu\alpha\beta}G_{\alpha\lambda}\widetilde{X}_{\beta}~^{\lambda}=\eta^{\nu\alpha\beta}S_{\alpha\lambda}\widetilde{X}_{\beta}~^{\lambda},$$ which is clearly nice as we have started to see the tensors related to the metric, [*i.e.*]{} the Einstein or the Schouten tensor. The last equation vanishes without the use of any field equation (which we have not yet introduced) if $\widetilde{X}_{\beta}~^{\lambda}$ is of the form $$\widetilde{X}_{\beta}~^{\lambda}=a_{0}\delta_{\beta}~^{\lambda}+a_{1}S_{\beta}~^{\lambda}+a_{2}{{\cal{S}}_2}_{\beta}~^{\lambda}+a_{3}{{\cal{S}}_3}_{\beta}~^{\lambda}+a_{4}{{\cal{S}}_4}_{\beta}~^{\lambda}+ \sum_{i=5}^\infty a_i {{\cal{S}}_i}_{\beta}~^{\lambda}.$$ Note that we do not introduce any derivative terms, as they will bring in extra propagating degrees of freedom when we build our field equations. We have separated the powers beyond 4 as they will not yield independent two-tensor structures, due to the Schouten identities, as shown below. And moreover, for this section, let us stay at the quadratic order and deal with the cubic and quartic order terms in the next section. So $\widetilde{X}$ reads $$\widetilde{X}=3a_{0}+a_{1}S+a_{2}{\cal{S}}_2.$$ From $X_{\sigma\eta}=\widetilde{X}_{\sigma\eta}-g_{\sigma\eta}\widetilde{X}$, one obtains the ${X}$-tensor as $$X_{\sigma\eta}=-2g_{\sigma\eta}a_{0}+a_{1}(S_{\sigma\eta}-g_{\sigma\eta}S)+a_{2}\big ( {{\cal S}_2}_{\sigma\eta}-g_{\sigma\eta}{\cal{S}}_2 \big ),$$ or in terms of Einstein tensor, one has $$X_{\sigma\eta}=-2g_{\sigma\eta}a_{0}+a_{1}G_{\sigma\eta}+a_{2} \big (G_{\sigma}\ ^{\mu}G_{\mu\eta}+\frac{R}{2}G_{\sigma\eta}+\frac{R^{2}}{8}g_{\sigma\eta}-g_{\sigma\eta}G_{\mu\nu}^{2} \big ).$$ We assumed that the covariant divergence of $X^{\mu \nu}$ vanishes which is possible if and only if $a_2=0$. Then the $Z^{\mu\nu}$ reads $$Z^{\mu\nu}=a_{1}C^{\mu\nu},$$ which leads to $$\nabla_{\mu}Y^{\mu\nu}=\,\eta^{\nu\eta\tau}\widetilde{X}_{\rho\tau}{Z}_{\eta}~^{\rho} = a_1 \eta^{\nu\eta\tau} ( a_0 g_{ \rho \tau} + a_1 S_{\rho \tau} ) C_\eta\,^\rho,$$ which vanishes on shell for the field equations $$C_{\mu \nu} = c_1 g_{\mu \nu} + c_2 S_{\mu \nu} + c_3 Y_{\mu \nu},$$ which is just MMG with $ Y^{\mu \nu} = J^{\mu \nu}$ proving the uniqueness of the theory at the quadratic order. Let us now move on to the cubic and quartic powers.
$R^{3}$ and $R^{4 }$ extensions ?
=================================
$R^{3}$ extension
------------------
Suppose we have the following deformation of TMG and MMG, $$\sigma G_{\mu\nu}+\Lambda g_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}+\gamma_1 J_{\mu\nu} + \gamma_2 {\cal{ K}}_{\mu\nu}=0,
\label{fieldeqn3}$$ with the most general two-tensor ${\cal{ K}}_{\mu\nu}$ built from the powers of the Ricci tensor and not from its derivatives. (Needless to say, since the Ricci and Riemann tensors are double duals of each other in three dimensions, one does not consider the Riemann tensor.) Therefore, one has the following tensor: $$\begin{split}
{\cal{ K}}^{\mu\nu}\equiv a_{1} {\cal{R}}_3^{\mu\nu} +a_{2}g^{\mu\nu} {\cal{R}}_3
+a_{3}R {\cal{R}}_2^{\mu\nu} +a_{4}R^{\mu\nu} {\cal{R}}_2+a_{5}g^{\mu\nu}R{\cal{R}}_2
+a_{6}R^{\mu\nu}R^{2}+a_{7}g^{\mu\nu}R^{3}.
\end{split}$$ We should note that one can eliminate one of the terms since not all of these, ostensibly, algebraically independent terms are actually independent. The quickest way to see this is to use the Cayley-Hamilton theorem. At the end of this discussion, we shall make use of this theorem, but for now, let us proceed with this form of the ${\cal{ K}}$-tensor. Its trace is reads $$\begin{split}
{\cal{ K}}= ( a_{1}+ 3 a_2){\cal{R}}_3
+( a_3 + a_{4} + 3 a_5) R{\cal{R}}_2
+(a_{6}+ 3 a_7) R^{3} ,
\end{split}$$ and its covariant-divergence can be computed as $$\begin{split}
\nabla_{\mu}{\cal{ K}}^{\mu\nu}=
\nabla_{\mu}R \Bigg ( (a_{1}+a_{3}) {\cal{R}}_2^{\mu\nu}+ \Big (\frac{3a_{3}}{4}+2a_{6}+\frac{a_{4}}{2}\Big)RR^{\mu\nu}
+\Big (\frac{a_{4}}{2}+a_{5}+\frac{3a_{2}}{4} \Big ) g_{\mu \nu} {\cal{R}}_2\\+\Big(\frac{a_{5}}{2}+\frac{a_{6}}{2}+3a_{7}\Big) g_{\mu \nu} R^{2} \Bigg )
+{\cal{R}}_2^{\alpha\rho} \Big(3a_{2}\nabla^{\nu}S_{\rho\alpha}+a_{1}\nabla_{\alpha}S_\rho\,^\nu \Big) +R^{\mu\nu}R_{\alpha\beta} \Big (a_{1}\nabla_{\alpha}S_{\beta\mu}+2a_{4}\nabla_{\mu}S_{\beta\alpha}\Big) \\
+RR^{\mu\rho}\Big (a_{3}\nabla_{\mu}S_{\rho}^{\nu}+2a_{5}\nabla^{\nu}S_{\mu\rho}\Big).
\end{split}$$ For this divergence to vanish on shell of the theory (\[fieldeqn3\]), one must see the appearance of the Cotton, Einstein or the $J$-tensors. For this purpose, one should turn the last three terms into the Cotton tensors and the terms multiplying $\nabla_\mu R$ to the $J$-tensor. These, respectively, can be achieved if one sets $$a_1 + 2 a_4=0, \hskip 0,8 cm a_1 + 3a_2 =0, \hskip 0,8 cm a_3 + 2a_5 =0,$$ and $$\begin{split}
a_{1}+a_{3}= k, \hskip 0.4 cm a_{3}+ \frac{8}{3} a_6 + \frac{2}{3} a_4 =- k, \hskip 0.3 cm a_{4}+2a_{5} + \frac{3}{2} a_2=-k \hskip 0.3 cm
a_{6}+a_{5}+6a_{7}= \frac{5}{8}k .
\end{split}$$ These reduce the divergence of the ${\cal{ K}}^{\mu\nu}$-tensor to $$\begin{split}\nabla_{\mu}{\cal{ K}}^{\mu\nu}= k J^{\mu \nu} \nabla_{\mu}R + a_1 \eta_{\lambda}~^{\nu\alpha}R_{\alpha\beta}R^{\beta}~_{\rho}C^{\lambda\rho}+a_1 \eta_{\lambda\mu\alpha}R^{\mu\nu}R_{\alpha\beta}C^{\lambda\beta}+ a_3 \eta_{\lambda\mu}~^{\nu}RR^{\mu\rho}C^{\lambda}~_{\rho},
\label{div-K}
\end{split}$$ where we have also made use of (\[3indexcot\]) and (\[2indexcot\]). The penultimate term vanishes due to symmetries, and one can combine the remaining two terms that have Cotton tensor using the three-dimensional identity valid for any vector $\xi_\mu$, $$\eta^{\lambda\nu\alpha}\xi^{\rho}=g^{\lambda\rho}\eta^{\beta\nu\alpha}\xi_{\beta}+g^{\nu\rho}\eta^{\lambda\beta\alpha}\xi_{\beta}+g^{\alpha\rho}\eta^{\lambda\nu\beta}\xi_{\beta},$$ as $$\eta_{\lambda}\thinspace^{\nu\alpha}R^{\beta}\thinspace_{\rho}=\delta_{\lambda}^{\beta}\eta^{\sigma\nu\alpha}R_{\sigma\rho}
+g^{\nu\beta}\eta_{\lambda}\thinspace^{\sigma\alpha}R_{\sigma\rho}+g^{\alpha\beta}\eta_{\lambda}\thinspace^{\nu\sigma}R_{\sigma\rho},$$ to arrive at $$\begin{split}\nabla_{\mu}{\cal{ K}}^{\mu\nu}= k \Bigg ( J^{\mu \nu} \nabla_{\mu}R + \eta_{\lambda\mu}~^{\nu}RR^{\mu\rho}C^{\lambda}~_{\rho} \Bigg).
\label{div-K2}
\end{split}$$ The trace reduces to a simple expression in terms of the trace of the $J$-tensor as $${\cal{ K}}= -\frac{k}{2} R \Big ( R_{\alpha\beta}^{2}- \frac{3 }{8} R^{2} \Big ) = - k R J.
\label{trace}$$ The last two equations are all we need to find the $K$-tensor that could possibly vanish on the TMG or MMG shell. It is important to realize that the number $k$ plays a crucial role here. If $k=0$, then clearly [*without* ]{} using the field equations. ${\cal{ K}}^{\mu\nu}$ is conserved, and it is traceless. Explicitly one has $$\begin{split}
{\cal{ K}}^{\mu\nu}={\cal{R}}_3^{\mu\nu}-\frac{1}{3}g^{\mu\nu}{\cal{R}}_3-R{\cal{R}}_2^{\mu\nu}-\frac{1}{2}R^{\mu\nu}{\cal{R}}_2 +\frac{1}{2}g^{\mu\nu}R{\cal{R}}_2+\frac{1}{2}R^{\mu\nu}R^{2}-\frac{1}{6}g^{\mu\nu}R^{3}.
\end{split}$$ But this is a red herring: as the Cayley-Hamilton theorem shows, this tensor is identically zero. Now consider a $3\times 3$ matrix $A$; then this matrix satisfies the same equation as its eigenvalues : $$A^{3}-(\mbox{Tr}A)A^{2}+\frac{1}{2}\left[(\mbox{Tr}A)^{2}-\mbox{Tr}(A^{2})\right]A-\mbox{det}(A)I_{3}=0.$$ Taking the trace of this equation, one has the determinant in terms of traces as $$\mbox{det} A=\frac{1}{6}\left[(\mbox{Tr}A)^{3}-3\mbox{Tr}(A^{2})(\mbox{Tr}A)+2\mbox{Tr}(A^{3})\right].$$ These two equations for the matrix $A = (R^\mu_\nu)$ yield ${\cal{ K}}_{\mu\nu}=0$. The second option is to consider $ k \ne 0$, and then in (\[div-K2\]), the second term vanishes both on the TMG and MMG mass shell, but the first term does not vanish. One could ask whether the theory, as in TMG, requires $\nabla_\mu R = 0$, which is not so,as is clear from the trace equation (\[trace\]). Hence, there does not exist a nontrivial tensor cubic in the curvature that could be used to deform TMG or MMG while keeping its single particle content intact.
$R^{4}$ extension
------------------
The most general two-tensor built with the powers of the Ricci tensor is $$\begin{split}
{\cal{ L}}^{\mu\nu}=a_{1}{\cal R}_4^{\mu \nu}+a_{2}{\cal R}_2^{\mu \nu}{\cal R}_2 +a_{3}R{\cal R}_3^{\mu \nu}+a_{4}R^{2}{\cal R}_2^{\mu \nu}
+a_{5}R^{\mu\nu}R^{3}+a_{6}R^{\mu\nu}{\cal R}_3\\
+a_{7}R^{\mu\nu}R{\cal R}_2+a_{8}g^{\mu\nu}{\cal R}_4
+a_{9}g^{\mu\nu}R{\cal R}_3+a_{10}g^{\mu\nu}R^2{\cal R}_2+a_{11}g^{\mu\nu}R^{4}+a_{12}g^{\mu\nu}{\cal R}_2^2.
\end{split}$$ Due to Schouten identity and the fact that $\cal{K}^{\mu \nu}$ is zero, not all terms are linearly independent in this tensor, but we shall work with this general form and eliminate the dependent terms later. Then its divergence follows as $$\begin{split}
\nabla_\mu {\cal{ L}}^{\mu\nu}=\Bigg ((\frac{5}{4}a_{1}+a_{3}){\cal R}_3^{\mu \nu}+(\frac{3}{4}a_{2}+\frac{3}{4}a_{6}+a_{7})R^{\mu\nu}{\cal R}_2^{\mu \nu}
+(a_{3}+2a_{4}+\frac{1}{2}a_{2})R{\cal R}_2^{\mu \nu} \\
+(\frac{3}{4}a_{4}+\frac{1}{2}a_{7}+3a_{5})R^{2}R^{\mu\nu}+(\frac{1}{2}a_{5}+4a_{11}+\frac{1}{2}a_{10})g^{\mu\nu}R^{3}+\\(\frac{1}{2}a_{6}+a_{9}+a_{8})g^{\mu\nu}{\cal R}_3
+(\frac{1}{2}a_{7}+a_{12}+2a_{10}+\frac{3}{4}a_{9})g^{\mu\nu}R{\cal R}_2 \Bigg )\nabla_{\mu}R\\
+R^{\mu\alpha}{\cal R}_2(a_{2}\nabla_{\mu}S_{\alpha}~^{\nu}+4a_{12}\nabla^{\nu}S_{\mu\alpha})+R{\cal R}_2^\mu\,_\beta (a_{3}\nabla_{\mu}S^{\beta\nu}+3a_{9}\nabla^{\nu}S_{\mu}~^{\beta})\\
+RR^{\mu\alpha}R^{\beta\nu}(a_{3}\nabla_{\mu}S_{\alpha\beta}+2a_{7}\nabla_{\beta}S_{\alpha\mu})+R^{2}R^{\mu\alpha}(a_{4}\nabla_{\mu}S_{\alpha}~^{\nu}+2a_{10}\nabla^{\nu}S_{\mu\alpha})\\+{\cal R}_2^\mu\,_\beta R_{\rho}~^{\nu}(a_{1}\nabla_{\mu}S^{\beta\rho}+3a_{6}\nabla^{\rho}S_{\mu}~^{\beta})
+{\cal R}_3^{\mu \rho}(a_{1}\nabla_{\mu}S_{\rho}~^{\nu}+4a_{8}\nabla^{\nu}S_{\mu\rho})\\+R^{\mu\alpha}{\cal R}_2^{\nu \beta} (a_{1}\nabla_{\mu}S_{\alpha\beta}+2a_{2}\nabla_{\beta}S_{\alpha\mu}).
\end{split}$$ The terms in the last four lines can be written in terms of the Cotton tensor only if the numerical parameters are related as $$a_{2}=-\frac{a_{1}}{2},\,\,\,\,a_{6}=-\frac{a_{1}}{3},\,\,\,\, a_{7}=-\frac{a_{3}}{2},~a_{8}=-\frac{a_{1}}{4},~ a_{9}=-\frac{a_{3}}{3},~ a_{10}=-\frac{a_{4}}{2},~a_{12}=\frac{a_{1}}{8}.$$ The terms multiplying the derivative of the curvature scalar gives rise to the $J$-tensor when the parameters are tuned as $$\frac{5a_{1}}{4}+a_{3}=- k \hskip 0.3 cm a_{3}- \frac{1}{4} a_1 +2 a_4 = \frac{3k}{4}, \hskip 0.3 cm - \frac{1}{4} a_3+3a_{5} + \frac{3}{4} a_4=-\frac{5k}{16} , \hskip 0.3 cm
-\frac{1}{4}a_{4}+\frac{1}{2}a_{5}+4a_{11}= \frac{17}{192}k .$$ This linear equation set is solved for all $a_{i}$ in terms of $a_1$ and $k$, upon use of which one arrives at the divergence as $$\begin{split}
\nabla_\mu {\cal L}^{\mu\nu}=k\Bigg (R^{\mu\alpha}J_{\alpha}~^{\nu}-\frac{1}{3}g^{\mu\nu}(R^{\alpha\beta}J_{\alpha\beta}-\frac{1}{8}RJ)\Bigg )\nabla_{\mu}R+\Bigg (\frac{1}{2}a_{1} { \cal R}_2+(\frac{k}{8}+\frac{a_{1}}{2})R^{2}\Bigg )\nabla_{\mu}J^{\mu\nu}\\-a_{1}RR^{\mu\nu}\nabla_{\alpha}J^{\alpha}_{\mu}+a_1\eta_{k}~^{\nu}~_{\mu}{\cal R}_3^{\mu\rho} C^{k}~_{\rho},
\label{div-L}
\end{split}$$ and the trace as $${\cal L}=\frac{k}{8}R^{2}J.$$ From the trace, we learn that in general the curvature scalar will not be constant since the $J$-tensor has the square of the Ricci tensor in it, and hence we must set $k=0$ for the first term in the divergence to vanish since the term in the parentheses is not generically zero. The other terms in (\[div-L\]) vanish on shell. Once again, we seem to have gotten an on shell-conserved tensor, but it turns out that this tensor given as $$\begin{split}
{\cal{ L}}^{\mu\nu}={\cal R}_4^{\mu \nu} -\frac{1}{2}{\cal R}_2^{\mu \nu}{\cal R}_2 - \frac{5}{4}R{\cal R}_3^{\mu \nu}+ \frac{3}{4}R^{2}{\cal R}_2^{\mu \nu}
-\frac{7}{24}R^{\mu\nu}R^{3}- \frac{1}{3}R^{\mu\nu}{\cal R}_3\\
+\frac{5}{8}R^{\mu\nu}R{\cal R}_2-\frac{1}{4}g^{\mu\nu}{\cal R}_4
+\frac{5}{12}g^{\mu\nu}R{\cal R}_3-\frac{3}{8}g^{\mu\nu}R^2{\cal R}_2-\frac{1}{12}g^{\mu\nu}R^{4}+\frac{1}{8}g^{\mu\nu}{\cal R}_2^2.
\end{split}$$ is identically zero, if one uses the fact that $R^{\mu \rho } {\cal K}^\mu\,_\rho =0$ which yields $$\begin{split}
{\cal R}_4^{\mu \nu }= \frac{1}{3}R^{\mu\nu}{\cal{R}}_3+R{\cal{R}}_3^{\mu\nu}+\frac{1}{2}{\cal R}_2^{\mu\nu}{\cal{R}}_2 -\frac{1}{2}R^{\mu\nu}R{\cal{R}}_2-\frac{1}{2}{\cal R}_2^{\mu\nu}R^{2}+\frac{1}{6}R^{\mu\nu}R^{3},
\end{split}$$ and its trace $${\cal R}_4=\frac{4}{3}R{\cal R}_3+\frac{1}{2}{\cal{R}}_2^2-R^{2}{\cal{R}}_2+\frac{1}{6}R^{4}.
\label{tracer4}$$ Since ${\cal L}^{\mu\nu}=0$ identically, there are no nontrivial quartic extensions of TMG and MMG. Beyond the quartic order, it is easy to show that all the possible rank-2 tensors built form the powers of the curvature can be written in terms of the lower order ones [@gurses]. To see this, let us denote the traceless Ricci tensor as $ \tilde{R}_{\mu\nu}$; then one has $$\delta_{[\mu_{1}\mu_{2}\mu_{3}\mu_{4}]}^{\nu_1\nu_{2}\nu_3\nu_{4}}\widetilde{R}_{\nu_{1}}^{\mu_{2}}\widetilde{R}_{\nu_{2}}^{\mu_{3}}\widetilde{R}_{\nu_{3}}^{\mu_{4}}\widetilde{R}_{\nu_{4}}^{\mu_{1}}=\frac{1}{4}\widetilde{\cal{R}}_{4}-\frac{1}{8}\widetilde{\cal{R}}_{2}^2=0,$$ where the bracket represents the total antisymmetrization. The result is just the same as (\[tracer4\]) written in the traceless tensors. The more important object is the rank-2 tensor $$\delta_{[\mu_{1}\mu_{2}\mu_{3}\mu_{4}]}^{\nu_1\nu_{2}\nu_3\nu_{4}}\widetilde{R}_{\nu_{1}}^{\mu_{2}}\widetilde{R}_{\nu_{2}}^{\mu_{3}}\widetilde{R}_{\nu_{3}}^{\mu_{4}}\widetilde{R}_{\nu_{4}}^{\mu}\widetilde{R}_{\nu}^{\mu_{1}}=\frac{1}{4} ({\widetilde{\cal{R}}_{5}})_\nu^\mu-\frac{1}{8} \widetilde{\cal{R}}_{2} ({\widetilde{\cal{R}}_{3}})_\nu^\mu
-\frac{1}{12}\widetilde{\cal{R}}_{3} ({\widetilde{\cal{R}}_{2}})_\nu^\mu=0,$$ which proves the claim. Therefore, there does not exist a nontirvial on TMG-shell conserved rank-2 tensor beyond the quadratic one already found in [@Townsend1].
Conclusions
===========
In this work, we have made an exhaustive search of possible deformations of the topologically massive gravity beyond the minimal massive gravity, with the condition that the single massive degree of freedom is intact, and have shown that no such deformations exist. Minimal massive gravity is a rather unique theory improving the boundary behavior of TMG while keeping its bulk properties intact. Therefore it is a candidate model which might have a dual unitary boundary conformal field theory unlike the other three-dimensional gravity theories. The model has been subject to recent works both in terms of classical solutions and in terms of semiclassical analysis besides the ones we quoted before in [@Arv2; @Altas; @setare; @gaston; @Yekta; @deger; @Ali]. With this work, we have also shown that it is highly difficult to construct on-shell conserved rank-2 tensors in three dimensions, a question which needs to be studied in higher dimensions. It would also be of some interest to extend these models to the ones with two massive degrees of freedom, extending the work initiated in [@mmg2].
The work of B.T. was supported by TUBITAK Grant No. 113F155.
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---
abstract: 'There is a subset of computational problems that are computable in polynomial time for which an existing algorithm may not complete due to a lack of high performance technology on a mission field. We define a subclass of deterministic polynomial time complexity class called mission class, as many polynomial problems are not computable in mission time. By focusing on such subclass of languages in the context for successful military applications, we also discuss their computational and communicational constraints. We investigate feasible (non)linear models that will minimize energy and maximize memory, efficiency, and computational power, and also provide an approximate solution obtained within a pre-determined length of computation time using limited resources so that an optimal solution to a language could be determined.'
author:
- 'Venkat R. Dasari Mee Seong Im Billy Geerhart'
bibliography:
- 'im-dasari.bib'
title: Complexity and mission computability of adaptive computing systems
---
Introduction {#sec:intro}
============
Given each mission has a computational requirement that must be satisfied, a lot of effort is expended to find computationally efficient algorithms. One gauge of efficiency is based on the computational complexity of an algorithm which is generally expressed in terms of time complexity that describes how long it takes for an algorithm to compute an answer using limited number of resources. The time complexity will affect the efficiency of applications using those algorithms. In mission-oriented tactical environments, computational efficiency in terms of flops/watt and computational speeds to match mission requirements is a very important factor in determining the fitness of an application for mission deployment. The requirements include computational efficiency of each platform in regards to available resources and constraints. Network specific constraints also need to be taken into consideration when assessing the efficiency of distributed computation. Furthermore, the complexity of input tasks and the computational decision-making requirements will increase, having variable computational cost on each platform, and often these functions need to be optimized[@fazel2005network; @lee2005non; @nygren2010akamai]. The type of optimization algorithms will also have an effect on the time to compute. For instance, optimizations can be linear or nonlinear, and nonlinear optimization can be further classified into concave and convex optimizations based on the complexity (see Fig. \[fig:optimization\]). Linear optimization functions run faster than nonlinear optimization functions as linear programming isolates computation to just the vertices of the available parameter space while nonlinear optimization must continuously explore the parameter space. In addition to the complexity of the computational tasks, resource constraints will severely affect the ability of tactical computing platforms to complete assigned tasks in the desired time. Some of the mission critical applications demand well-defined execution times. Mission optimized computations combined with automated intelligence have been shown to be efficient and will achieve the desired results in mission time[@spang2017mon; @capacity-bounds-multipath-networks; @Afergan:2005:EPB:1251522.1251523]. The mission requirements define the need for optimized algorithms, which are a subset of the deterministic polynomial time complexity class $P$ of problems, where we define it as the $M$ class for mission ready algorithms.
[1.0]{} ![Optimization problems of different degree of complexity are described above. Computational decisions that depend on complex optimization algorithms take longer time to compute, affecting the mission effectiveness.[]{data-label="fig:optimization"}](linearOpt "fig:"){width="1.0\linewidth"}
[1.0]{} ![Optimization problems of different degree of complexity are described above. Computational decisions that depend on complex optimization algorithms take longer time to compute, affecting the mission effectiveness.[]{data-label="fig:optimization"}](nonlinearHills "fig:"){width="1.0\linewidth"}
Context-aware adaptive computational frameworks are remarkably flexible to support the class $M\subseteq P$ of languages using heterogeneous underlying hardware implementation details from multiple applications using non-intrusive methods under resource constrained tactical environments. Mission optimized adaptive computational framework will improve the execution efficiency of the mission applications. Adaptive computing framework will be highly useful in deploying intelligent tactical computing platforms in smart cities that present complex environments with constrained resources and heterogeneity with optimal efficiency. It will also enable the computing platforms to adapt to the communication related constraints like bandwidth and network reachability. In tactical environments, adaptive computing must make decisions regarding local or remote computing for solving a computational problem presented to it. Computational offloading is influenced by not only the computational complexity of the problem, but it also depends upon the network resource constraints like the link quality and communication cost for the offloading.
With military missions, efficiency of computing platforms are affected by resource constraints in the field. Availability of the limited amount of energy and battery power, the timing of communication lines for crucial decision-making, and the efficiency of available central processing power (CPU) that are accessible in real time are immensely crucial aspects in order to carry out a successful computation required by the mission. By focusing on military operations, we will analyze to optimize necessary computations in a mission where computations are computable in polynomial time and investigate when such computations can be optimized, reduced in the usage of CPU power, and successfully executed within mission time. Such study will require us to develop a heterogeneous (non)linear platform to reduce current state-of-the-art computational complexity to the complexity sufficient for military applications.
In this manuscript, we study a new class of polynomial time computational complexity called $M$ to satisfy the mission requirements in order to understand the computability of a given algorithm in mission time. Mission times can vary based on the mission objectives but deployability of a given algorithm is tied to its ability to complete its computation in that particular mission time. In the section on computational-complexity, we give a brief background on computational complexity, while in the section on adaptive mission computation, we define this new $M$ class of algorithms and we specifically look at how algorithms can be optimized to be placed from the $P$ class of algorithms into the $M$ class by using an adaptive computing framework. Any adaptive computing framework must consider the constraints which we outline in the section regarding the effect of resources on computational efficiency. Next we apply those constraints towards a constraints-aware distributed computing framework, and then we give an example where computational jobs assigned to a cluster of local machines may fail and how the distributed computing platform reacts to these failures. In the final section, we summarize our construction of the mission class of problems and how focusing on this subset will enhance the performance of tactical mobile computing platforms.
Computational Complexity {#section:computational-complexity}
========================
Computational complexity has a direct impact on mission ready algorithms. For example, RSA encryption is simple if given the public key, but RSA decryption is difficult without knowledge of the private key. As such, asymmetric encryption can be performed easily in the field, but mounting a brute force attack on an adversary using RSA encryption is not even considered in real-time applications. This variation in complexity is why we group problems into two types: class $P$, and class $NP$[@bossaerts2017computational].
We denote $P$ as the class of questions for which some algorithm can provide an answer, and thus solve the language, in polynomial time. So in the mathematics literature, we often refer to $P$ as [*deterministic polynomial time complexity class*]{}. The class $P$ of problems are decidable in polynomial time on a deterministic single-tape Turing machine: $P=\bigcup_{k\geq 0}\text{Time}(n^k)$, and such problems are simple for computers to solve, all within a reasonable amount of time. One example includes determining whether or not a word $w$ is a member of the language $L=\{0^k1^k :k\geq 0\}$. Another example includes the problem to determine whether a directed path exists from vertex $s$ to vertex $t$ in a directed graph, i.e., given a directed graph $G$, define $$\begin{aligned}
\begin{split}
&\text{PATH}(G,s,t) :=\\
& \{ \langle G,s,t\rangle : G \text{ has a directed path from }s \mbox{ to }t \}.
\end{split}\end{aligned}$$ This problem is known as [*directed $s$-$t$ connectivity*]{}, and it is a classical result that $\text{PATH}$ is indeed in the class $P$[@barnes1998sublinear]. In fact, reasonable deterministic computational models are polynomially equivalent, i.e., one such model can simulate another model with only a polynomial increase in running time.
We denote $NP$ as the complexity class for which an algorithm can provide a solution in polynomial time with a non-deterministic Turing machine. We thus refer to $NP$ as [*nondeterministic polynomial time complexity class*]{}. Such questions are ones with solutions that can be [*verified*]{} in polynomial time using a deterministic Turing machine. By definition, $NP$ is the class of languages that are decidable in polynomial time on a nondeterministic Turing machine, i.e., $NP=\bigcup_{k\geq 0}\text{NTime}(n^k)$, where $$\begin{aligned}
\begin{split}
&\text{NTime}(t(n)) := \{ L : L\mbox{ is a language decided by an}\\
& \mathcal{O}(t(n))\mbox{-time nondeterministic Turing machine}\}.
\end{split}\end{aligned}$$
In fact, many questions in the class $P$ can be changed ever so slightly to then be placed in the class $NP$. For example, the previous directed path question that connects two vertices can be placed into the $NP$ class by instead considering the question of whether or not a Hamiltonian path connects two vertices, where a [*Hamiltonian path*]{} in a directed graph $G$ is a directed path that passes through each vertex exactly once. That is, given a directed graph $G$, we define a [*Hamiltonian path*]{} as $$\begin{aligned}
\begin{split}
&\text{HamPATH}(G,s,t) = \{ \langle G,s,t\rangle: G \mbox{ is a directed}\\
& \mbox{graph with a Hamiltonian path from }s \mbox{ to }t \}.
\end{split}\end{aligned}$$
One can easily obtain an [*exponential*]{} time algorithm for the $\text{HamPATH}$ problem by a brute-force approach that checks all possible permutations of vertices (if there are $n$ vertices, then there are $n!$ permutations to check, and we only need to verify that a potential path is Hamiltonian). Although it is well-known that $\text{HamPATH}$ is in $NP$,[@gurevich1987expected; @itai1982hamilton] it remains an open problem on determining whether or not $\text{HamPATH}$ is actually solvable in polynomial time. We thus have the containment $P\subseteq NP$ of classes of languages.
![We say $P$ (deterministic polynomial time complexity class) is the class of languages for which some algorithm can solve the question in polynomial time while $NP$ (nondeterministic polynomial time complexity class) is the class of languages for which an algorithm may be very difficult to find, but if provided an answer, then it can be [*verified*]{} in polynomial time. The subclass $M$ consists of functions in $P$ for which adequate computational resources may not be available in the tactical environment to complete mission computation but for which a significant proportion of the language can be completed within the confinement of limited resources, such as technology and time, so that the validity of a solution for the language may correctly be deduced decisively.[]{data-label="fig:mission_comp"}](mission-computability)
In the study of computability, $P=NP$ problem is one of the very important and interesting open problems in mathematics and in theoretical computer science with deep ramifications in cryptography[@baker1975relativizations; @marks2016universality], algorithms[@cook2006p; @yan2002number; @shor2004progress], artificial intelligence[@cook2006p; @buss2012towards], game theory[@sahni1974computationally], economics[@vives1984duopoly], to name a few. Intuitively, the $P=NP$ conjecture is an investigation of whether every problem whose solution can be quickly verified in polynomial time can also be solved fairly quickly in polynomial time. So it remains an open problem to show whether or not the following containment $NP\stackrel{?}{\subseteq} P$ of languages holds.
There is however a class of (optimization) problems that lie in the class $NP$, so a polynomial time reduction to their complexity may be difficult to construct with current tools.
Now, suppose a full high performance computing (HPC) may not be available on the mission field but for which a significant proportion of the language being computed within a confined time frame is sufficient to determine the validity of the language, and thus critical decisions may need to be immediately determined on site. Because of the importance and a critical significance of such problems, we define a subclass of problems in $P$ in the section on adaptive mission computation.
Adaptive Mission Computation {#section:related-work}
============================
With adaptive computing in mind [@pozueco2013adaptable; @urbieta2017adaptive], the class $M$ of functions can be described as having the ability to manage time constraint and the ability to complete a computation in the class $P$ as we minimize the usage of limited HPC resources, such as energy, time, and memory, while simultaneously attempt to maximize computational power (CPU) and efficiency to effectively work around computational constraints for a heterogeneous computational platform for the class $M$ of languages. The timing of such computations may be optimized by using a convex or concave (non)linear platform, and by investigating how computations perform in a mission, the limits of computational complexity and computational capacity may be precisely described.
We are interested in a mission-focused problem $\mathcal{P}$, which is a relation from a set $I$ of instances (input) to a set $S$ of solutions, where deterministic and approximation algorithms exist, i.e., $\mathcal{P}\subseteq I\times S$, $\mathcal{P}\in P$, and $\mathcal{P}$ has an algorithm that consistently returns a feasible, approximated solution, which is characterized by its distance from its value to the optimal solution. Thus the class $M$ is defined as the following:
\[defn:mission-computable-class\] Given the polynomial time complexity class $$P =\{ \mathcal{P}: \forall x\in I\: \exists \: y \in S \ni (x,y)\in \mathcal{P}\},$$ we define [*mission computable polynomial time complexity class*]{} as $$M =\{ \mathcal{P}\in P: \forall x\in I \: \exists \mbox{ approx. soln. }
\widetilde{y} \ni (x,\widetilde{y})\in \mathcal{P}\}.$$
Note that the approximated solution $\widetilde{y}$ does not need to be in the solution set $S$, but given any $\varepsilon > 0$, $\widetilde{y}$ must satisfy $d(y,\widetilde{y})< \varepsilon$, where $d$ is an intrinsic notion of a distance between the two solutions $y$ and $\widetilde{y}$, which depends on the problem being considered.
Our class $M$ of computational complexity is aimed at solving computational problems in mission time and it is restricted to adaptive computing framework (see Fig. \[fig:mission\_comp\]).
Before we further discuss approximated solutions, we will give some basic conditions about $\mathcal{P}$. In order to solve the language $\mathcal{P}$, one needs to recognize if $(x,y)\in \mathcal{P}$. One then needs to construct (using a deterministic algorithm with polynomial time complexity) that for each $x$, there exists $y$ such that $(x,y)\in \mathcal{P}$. Finally, we need to optimize the problem $\mathcal{P}$ in such a way that for each instance $x$, find the best (and efficient) solution $\widetilde{y}$ such that $(x,\widetilde{y})\in \mathcal{P}$.
A feasible solution is an approximate solution, and such solutions are classified by the value of its distance from the ideal (optimal) solution. Thus, the ratio of the rough solution to the optimal solution is determined by the input size, the growth of the performance algorithm, and time limitations. Some current approximation algorithms include linear programming, dynamic programming, local search, randomized algorithm, and heuristic algorithm. Linear programming formulates the language as a linear model, dynamic programming constructs a solution from optimal solutions to sub-problems of the original language, a local search algorithm looks for a better neighboring solution when given a solution (this algorithm [*deforms*]{} an input until an improved and preferred optimal solution is found), randomized algorithm embeds and executes a random decision generator, and heuristic algorithm is encoded by exploratory learning strategies that offer no guarantee of a more suitable solution.
The mission class $M$ is the subclass of optimization problems that are solvable by a polynomial time and approximate algorithm, in a finite sequence of steps (whose order is bounded by pre-determined complexity), that computes a comparable result when given an instance from the set of inputs. The algorithm cost for the number of operations (time complexity) and the storage space (space complexity) are in the order of polynomial time $\text{Time}(n^k)$ for some $k$.
Effect of Resources on Computational Efficiency {#section:resource-effects}
===============================================
For smooth operations, adaptive computations require allocation algorithms and technology congestion protocols using behavior models along with network efficiency fairness characterization. These algorithms need to make an optimal decision which can be characterized as an optimization problem that is either linear or nonlinear. As such, objective functions together with a set of constraint inequalities are often used to broaden the scope for a successful military operation. Computational constraints like power, memory, size, storage and CPU influence the performance of a computational platform in contested and congested environments, and are driving the need for a constraints-aware adaptive computation framework. Such a framework will change the computational behaviors of the platforms in response to available resources and the complexity of the input computational tasks. For example, a computational problem can be solved in a distributed manner in order to optimize available computational resources among different computational platforms. However, when a local computing platform is incapable of performing a required computation due to lack of resources or due to lack of required software, the computations are offloaded to a remote computer capable of computing the problem and provide the solution. The network related constraints like signal strength, bandwidth and energy required for offloading are important factors in determining if a remote computation makes sense. Any adaptive computing framework will need to take these factors into consideration when making decisions.
Constraints-aware distributed computing {#section:computing-with-constraints}
=======================================
One foundation for adaptive computing is a constraints-aware distributed computing algorithm. Given too many jobs assigned to an array of cores, the algorithm can be programmed to minimize the number of failed jobs; such an algorithm is inherently designed to allow for failure, but it is this failure that allows for feedback to the calling applications. For example, a security camera might be running at $60$ Hz, but the image analysis at $60$ Hz would consume more than the local resources. The application dedicated to analyzing the images would give a time-to-complete restriction on each frame, but a local optimizer would determine that most of the frames will fail the time restrictions and terminate most jobs immediately while informing the calling application about the dropped jobs.
The logic for a single machine with a single core can be programmed using integer programming. Given we want to maximize the total computations done on a single machine, our decision variables can be an ordered list that is represented as a binary matrix $b_{i,j}$ (see Eq. \[eq:decisionVariables\]), where $i$ represents the $i$-th job that will be computed in order on the host machine, while $j$ represents whether or not the $j$-th image is assigned to the $i$-th job. Given the decision variables, the objective function can be used to maximize the computations done on the local machine by doing a weighted sum (see the first term of Eq. \[eq:objectiveFunction\]). Constraints are added to prevent multiple images being assigned to the same job as in Eq. \[eq:constraint1\] or to prevent the same image being applied to multiple jobs as in Eq. \[eq:constraint1b\], while the time constraints in Eq. \[eq:constraint2\] are put in place to prevent images from being scheduled too far into the future on the local machine. Integer programming must work within the constraints. However, we expect some jobs to fail to be assigned, so a null image with zero computation time and zero constraints is assigned to the $0$-th image. Furthermore, a compact list with all null jobs appearing at the end is preferable to limit degeneracy, so the objective function has an additional term for the zeroth null image appearing later in the list (see the second term of Eq. \[eq:objectiveFunction\]). Feeding the variables, objective function, and constraints into an integer optimizer will show which jobs should be abandoned on the local machine. These abandoned jobs can then be scheduled on a distributed computing platform.
$$\begin{aligned}
\label{eq:decisionVariables}
\begin{split}
i&=\text{job index}, \hspace{4mm}
j=\text{image index},\\
b_{i,j}&=
\begin{cases}
0 &\text{if $j$-th image not assigned to job $i$}, \\
1 & \text{if $j$-th image assigned to job $i$},
\end{cases}
\end{split}\end{aligned}$$
$$\begin{aligned}
\label{eq:helperFunctions}
\begin{split}
T_j&=\text{analysis time for the $j$-th image}, \\
R_j&=\text{time the $j$-th image must be processed},
\end{split}\end{aligned}$$
objective function: $$\label{eq:objectiveFunction}
\sum_{i,j\neq0}b_{i,j}\cdot T_j \text{ }+10^{-5}\cdot\sum_i i\cdot b_{i,0},$$
constraint: one image per job $$\label{eq:constraint1}
\forall i \hspace{4mm} \text{ } \sum_{j} b_{i,j}=1,$$
constraint: each image is processed at most once $$\label{eq:constraint1b}
\forall j\neq 0 \hspace{4mm} \text{ } \sum_{i} b_{i,j}\leq 1,$$
constraint: image processed on time: $$\label{eq:constraint2}
\forall i \text{ } \hspace{4mm} \sum_{k\leq i,j} b_{k,j}\cdot T_j\leq \sum_j b_{i,j}\cdot R_{j}.$$
Once the abandoned jobs are handed to a distributed computing platform, a global optimizer can determine which jobs should be scheduled or rescheduled. The distributed computing platform is really just a bunch of local machines connected together with each machine competing for the same resources, so again each application must be willing to accept that some jobs will be dropped. This means each machine must throttle its own computational usage through tagging its own jobs with the appropriate priority level. In the scenario of image analysis, every tenth frame can be tagged as a high priority job relative to the calling machine. Although the local optimizer outlined previously is fairly trivial, the same framework can be expanded to include a number of cores ($c$) and a number of machines ($m$). The decision variables are again just an expansion of the binary matrix, from $b_{i,j} $ to $b_{m,c,i,j}$. Extra constraints can be added to satisfy transfer times, power used, or even the amount of random-access memory (RAM) used. The objective function can be tweaked to emphasize particular features such as computation time or energyConsumed/FLOP.
The common theme between the local optimizer and the global optimizer is that the computational size must be varied to fit the computational resources. This is just a foundation for adaptive computing as the next step is to vary the allocation of resources to competing applications. For instance image analysis across multiple security cameras can be considered a single image analysis application, while another application could be attempting to apply machine learning to detect hostile agents from the data provided by the security cameras. The overall goal is to maximize mission effectiveness by varying the resources to each application as the needs change. For instance, image analysis in hostile zones should be given priority access to local resources, while the machine learning project should be given loose timing constraints to allow the jobs to be scheduled on HPC machines located far away from the hostile zone.
Discussion {#section:discussion}
==========
Tactical computing platforms are mostly mobile with limited computing and communication resources. Data processing and problem solving tasks are time sensitive and their speed depends upon the available computational resources. In order to accomplish mission computation goals, the platforms and the algorithms must be optimized to the mission requirements.
We have described a new class of computational complexity class $M$ that is a subset of the polynomial time class $P$ in order to address a class of problems that needs to be computed in mission time. Mission times are determined by context in which the computation is used and the completion time of that task to determine the usefulness of the computation. As described in the section on adaptive mission computation, all the polynomial class of computational tasks that can be completed in mission time will fall into the computational complexity class $M$, and they are mission ready. Polynomial problems that cannot be computed in mission time will require additional optimization until they satisfy mission requirements.
Defining a new computational complexity class will allow us to define the computational requirements for any applications and algorithms to be mission ready. Algorithms that can process input data in polynomial time might still fail to complete when deployed in tactical environments due to resource constraints. Reducing such a $P$ class of algorithms to $M$ class through optimization and heuristics will enable us to optimize a given set for $P$ class of problems to tactical environments.
In our future work, we will test a variety of mission deployable applications to determine their mission readiness without additional optimization and adjustments to their code.
Acknowledgements
================
This work is supported by a research collaboration between the U.S. Army Research Laboratory and U.S. Military Academy. M.S.I. is partially supported by National Research Council Research Associateship Programs.
|
---
abstract: 'The unexpected features of the two-stream instability in electrostatic quantum plasmas are interpreted in terms of the coupling of approximate fast and slow waves. This is accomplished through the factorization of the dispersion relation into different sectors having positive or negative energy. Therefore, the concept of negative and positive energy waves is useful not only for classical, but for quantum plasmas as well. We discuss the limitations of the quantum two-stream model in view of the weak coupling assumption.'
author:
- 'F. Haas'
- 'A. Bret'
- 'P. K. Shukla'
title: 'Physical interpretation of the quantum two-stream instability'
---
Introduction
============
Since it has been discussed for the first time in the framework of the quantum hydrodynamical model [@Haas], the quantum two-stream instability has attracted considerable attention in the literature. The reason for this is that it is a benchmark displaying many of the particularities of quantum plasmas, including a new unstable branch of the dispersion relation for large wave-numbers and almost stationary, quasineutral, nonlinear oscillations [@Haas] without analog in classical plasmas. Later on, a kinetic (Wigner-Poisson) treatment of the problem [@Anderson] showed that temperature effects can suppress the quantum instability. The quantum fluid equations have been further applied to several quantum streaming instability problems, like the three-stream quantum plasmas [@Haas2], the quantum dusty plasmas [@Ali], the electron-positron-ion quantum plasmas [@Mushtaq] or the magnetized multi-stream instability [@Ren]. The hydrodynamic formalism has also been applied to the quantum filamentation instability, with [@Bret] or without [@Bret2] magnetization.
The purpose of the present contribution is to provide an intuitive explanation of the quantum two-stream instability, in terms of the coupling of electrostatic modes with distinct energy contents. Indeed, literature still lacks a physical (“with the hands") understanding of the quantum streaming instabilities. In addition to analytic and numeric approaches, the physical interpretation of quantum plasma effects is a most welcome extension, helping to describe the unexpected features of these problems. As will be shown in the sequel, it is possible to identify approximate positive and negative energy modes in electrostatic two-stream quantum plasmas. Moreover, the energy exchange between these waves provide the clue for the stability analysis in such systems.
In addition, we also examine the range of validity of the quantum two-stream model. As shown in Section IV, the quantum parameter $H$, which is the ratio between the plasmon energy and the kinetic energy of each beam, should be small in order to fulfill the collisionless assumption. As far as we know, a clear statement about the smallness of $H$ is missing. Nevertheless, the more intriguing aspects of the quantum two-stream instability appear for small $H$, where the so-called quantum instability branch for large wave-numbers appear [@Haas]. The present contribution gives a physical justification for this exotic unstable quantum branch in terms of negative energy waves. Finally, the fact that a large quantum parameter tends to enhance collisions is a reason more to develop a (still lacking) efficient collisional quantum plasma theory.
Negative energy modes are a well-known tool for the analysis of streaming instabilities in classical systems [@Sturrock; @Akhiezer]. In this work we show that the same ideas are also useful to quantum plasmas. The basic heuristic concept of negative energy wave is as follows. Consider, for definiteness, coherent wave motion taking place in some dispersive medium crossed by a number of streaming particles. When the absolute value of the wave’s phase velocity is slightly smaller than the speed of some of the other streams, the oscillation mode is referred to as a negative energy wave. The mechanism of the classical two-stream instability can be explained in terms of energy transfer from a positive energy Langmuir wave taking place in one beam, to a negative energy Langmuir wave taking place in the second beam. Naturally in quantum plasma physics such a classical picture of resonant wave–particle interaction should be taken with caution. However, we will show that negative energy modes can still be identified in terms of macroscopic (fluid) quantum plasma equations for the two-stream instability.
This work is organized as follow. In Section II, we review the basic features of the quantum two-stream instability, and of the electric field energy density in a weakly dissipative dielectric medium. In Section III, fast and slow approximate waves in two-stream quantum plasmas are identified. The coupling of these waves is then analyzed in detail, in order to understand why some wave-numbers are stable and others are not, when varying the strength of quantum effects. A detailed account on the limitations of the quantum two-stream model for large quantum parameters $H$ due to the neglect of collisions is included in Section IV. Section V is reserved to the conclusions.
Time-averaged energy density of electrostatic oscillations
==========================================================
Consider two counter propagating electron beams with equal equilibrium number densities $n_1 = n_2 = n_0/2$, and equilibrium velocities $u_1 = - u_2 = u_0 > 0$, in presence of an immobile ionic background of particle density $n_0$. Following the same notation of Ref. [@Haas], it is convenient to normalize the frequencies to the plasma frequency $\omega_p$ and the wave-numbers to $\omega_p/u_0$. Using the two-stream quantum hydrodynamic model [@Haas], a dielectric function $\varepsilon = 1 - F(\Omega)$ is derived, where the characteristic function $F$ is $$\label{e1}
F(\Omega) = \frac{1}{2}\left[\frac{1}{(\Omega+K)^2- H^2\,K^4/4} + \frac{1}{(\Omega-K)^2- H^2\,K^4/4}\right] \,,$$ in terms of a non-dimensional quantum parameter $H = \hbar\omega_{p}/mu_{0}^2$, where $\hbar$ is Planck’s constant over $2\pi$ and $m$ is the electron mass.
The dispersion relation $\varepsilon = 0$ is a second-order polynomial equation for $\Omega^2$, with solutions $\Omega^2 = \Omega_{\pm}^{2}(K)$, where $$\begin{aligned}
\label{e2}
\Omega_+ &=& \frac{1}{2}\left[2+4K^2+H^2\,K^4 + 2\sqrt{1+8K^2+4H^2\,K^6}\,\right]^{1/2} \,,\\
\label{e3}
\Omega_- &=& \frac{1}{2}\left[2+4K^2+H^2\,K^4 - 2\sqrt{1+8K^2+4H^2\,K^6}\,\right]^{1/2} \,,\end{aligned}$$ corresponding to four possible branches of the eigen-frequency $\Omega$ as a function of the wave-number $K$. We use parity properties to restrict the analysis to positive $K$ and $\Omega$ values. As detailed in Ref. [@Haas], when $0 < H < 1$ there is instability provided $K < K_A$ (semiclassical branch) or $K_B < K < K_C$ (quantum branch), where $K_A < K_B < K_C$ are given by $$\label{e4}
K_A = \frac{\left[2-2\sqrt{1-H^2}\,\right]^{1/2}}{H} \,,\quad K_B = \frac{\left[2+2\sqrt{1-H^2}\,\right]^{1/2}}{H} \,,\quad K_C = \frac{2}{H} \,.$$ On the other hand, when $H \ge 1$ the instability condition is just $K < K_C$ (see Fig. 1 of Ref. [@Haas]).
In order to evaluate the energy content of each wave mode, it is necessary to consider the time-averaged energy density $<W_{e}>$ for longitudinal plasma oscillations, which is $$\label{e5}
<W_e> = \frac{\varepsilon_0}{4}\,\frac{\partial(\Omega \,\varepsilon_{h})}{\partial\Omega}\,|E_1|^2 \,,$$ where $\varepsilon_0$ is the vacuum permittivity, $\varepsilon_h$ is the Hermitian part of the dielectric function and $E_1$ is the amplitude of the electric field perturbation. Since the underlying model is dissipation-free, one has $\varepsilon = \varepsilon_h$. Eq. (\[e5\]) is still valid in our quantum plasma system, since it is derived from the Maxwell’s equations only. Quantum effects are contained in the modified dielectric function. Moreover, $<W_e>$ includes both the contributions due to the electrostatic energy and the “acoustic energy", defined as the kinetic energy associated to the coherent particle wave motion [@Stix]. The expression for $<W_e>$ can equally be deduced from a generalized Poynting theorem, like for the classical two-stream instability [@LashmoreDavies].
Proceeding from Eqs. (\[e1\]) and (\[e5\]), one get $\partial(\Omega \,\epsilon_{h})/\partial\Omega \sim \psi(\Omega)$, omitting a complicated positive factor, where $$\label{e6}
\psi(\Omega) \equiv - 6\, K^4 + H^2\, K^6 + \frac{H^2\, K^8}{8} + K^2 \,(4-H^2 K^2)\,\Omega^2 + 2\, \Omega^4 \,.$$ Considering $\Omega = \Omega_{+}(K)$ from Eq. (\[e2\]), one finds, $$\label{e7}
\psi(\Omega_{+}) = 1 + 8\,K^2 + 4\,H^2\,K^6 + (1+4\,K^2)\,\sqrt{1+8\,K^2+4\,H^2\,K^6} \, > 0 \,,$$ so that this mode is always a [*positive energy*]{} wave.
On the other hand, taking $\Omega = \Omega_{-}(K)$ from Eq. (\[e3\]) gives, $$\label{e8}
\psi(\Omega_{-}) = 1 + 8\,K^2 + 4\,H^2\,K^6 - (1+4\,K^2)\,\sqrt{1+8\,K^2+4\,H^2\,K^6} \,.$$ From Eqs. (\[e4\]) and (\[e8\]), it can be shown that $\psi(\Omega_{-}) < 0$ if and only if $K < K_C$. Therefore, if $K > K_C$, the mode $\Omega = \Omega_{-}(K)$ is a stable [*positive energy*]{} mode; if $K < K_C$, the unstable modes with $K_B<K<K_C$ (see Eq. \[e4\]) and $\Omega = \Omega_{-}(K)$ are [*negative energy*]{} waves. Actually, they correspond to an absolute instability as they are of the form $\Omega = i\gamma$, for real $\gamma > 0$. The stable mode $\Omega = \Omega_{-}(K)$ for $K_A < K < K_B$, which exists only for $H < 1$, has negative energy. Finally, one has $\psi(\Omega_{-}) = 0$ for the marginally stable wave-number $K = K_C$.
Further insight can be gained analyzing the characteristic function $F(\Omega)$ given by Eq. (\[e1\]). It has vertical asymptotes at $\Omega = \pm \Omega_{>}$ and $\Omega = \pm \Omega_{<}$, where $\Omega_{>} = K + H\,K^2/2$ and $\Omega_{<} = K - H\,K^2/2$. Since the dispersion relation is quadratic with real coefficients for $\Omega^2$, stability is assured when the graph of $F(\Omega)$ intercepts the horizontal line $F = 1$ four times. Actually, the case $K = K_C$ is special because the quartic equation for $\Omega$ degenerates into a quadratic one, which can be shown to correspond always to stable oscillations. Figure 1 shows a typical unstable case when $K < K_A$, for $K = 0.9, H = 0.1$.
{width="70.00000%"}
On the other hand, Figure 2 explains why the wave-numbers satisfying $K > K_C$ are stable, since the graph of the characteristic function always intercepts the horizontal line $F = 1$ four times.
{width="70.00000%"}
Fast and slow approximate modes in electrostatic two-stream quantum plasmas
===========================================================================
Figure 3 shows the dispersion curves for $H < 1$. Since the dispersion relation is an algebraic equation with real coefficients, it is possible to follow Sturrock rules [@Sturrock; @Akhiezer], identifying several stable/unstable zones in the $(\Omega, K)$ space. Here we consider real wave-numbers and don’t analyze the amplification problem. Curve 1 is a positive energy mode parametrized by $\Omega = \Omega_{+}(K)$ given by Eq. (\[e2\]). Curves 2 and 3 are both described by $\Omega = \Omega_{-}(K)$ given by Eq. (\[e3\]). However, according to the preceding analysis, curve 2 is a negative energy mode, while curve 3 is a positive energy mode. The coupling of these waves gives rise to the purely quantum (absolute) instability for large wave-numbers, $K_B < K < K_C$ in Fig. 3.
{width="70.00000%"}
As apparent from Fig. (3), it is not possible to have an exact coupling between the negative and positive energy waves described by curves 2 and 3 respectively, since they have no intersection. Nevertheless, from the structure of the graphics, one might suspect that some approximate coupling can take place. To give support to this conjecture, a useful approach is to identify approximate negative and positive energy modes corresponding to the exact waves in an appropriate limit (e.g. for sufficiently large wave-numbers). Hence we follow the style of Ref. [@LashmoreDavies] and write the dispersion relation under the factorized form $$\begin{aligned}
\label{e9}
\Bigl[\Omega &-& K - \frac{1}{\sqrt{2}}\,\left(1+\frac{H^2 K^4}{2}\right)^{1/2}\Bigr]\,\Bigl[\Omega - K + \frac{1}{\sqrt{2}}\,\left(1+\frac{H^2 K^4}{2}\right)^{1/2}\Bigr] \\
&\times& \Bigl[\Omega + K - \frac{1}{\sqrt{2}}\,\left(1+\frac{H^2 K^4}{2}\right)^{1/2}\Bigr]\,\Bigl[\Omega + K + \frac{1}{\sqrt{2}}\,\left(1+\frac{H^2 K^4}{2}\right)^{1/2}\Bigr] \nonumber = \frac{1}{4},\end{aligned}$$ where two fast $$\begin{aligned}
\label{e10a}
\Omega \simeq \Omega_{f,r} \equiv K + \frac{1}{\sqrt{2}}\,\bigl(1+\frac{H^2 K^4}{2}\bigr)^{1/2} \,,\\
\label{e10b}
\Omega \simeq \Omega_{f,l} \equiv - K + \frac{1}{\sqrt{2}}\,\bigl(1+\frac{H^2 K^4}{2}\bigr)^{1/2},\end{aligned}$$ and two slow $$\begin{aligned}
\label{e11a}
\Omega \simeq \Omega_{s,r} \equiv K - \frac{1}{\sqrt{2}}\,\bigl(1+\frac{H^2 K^4}{2}\bigr)^{1/2} \,,\\
\label{e11b}
\Omega \simeq \Omega_{s,l} \equiv - K - \frac{1}{\sqrt{2}}\,\bigl(1+\frac{H^2 K^4}{2}\bigr)^{1/2} \,,\end{aligned}$$ approximate space-charge modes can be identified. The subscripts $r$ and $l$ refer to quantum plasma longitudinal oscillations on the rightward and leftward beams, respectively. Indeed, in the reference frame of the beam moving to the right, the Doppler-shifted linear terms in $K$ would disappear in Eqs. (\[e10a\]) and (\[e11a\]). Restoring momentarily physical units, we would then have $\omega^2 = \omega_{b}^2 + \hbar^2\,k^4/(4\,m^2)$, which is the quantum modified Langmuir dispersion relation. Here, $\omega$ is the wave-frequency, $k$ is the wave-number and $\omega_b = \omega_{p}/\sqrt{2}$ is the beam’s plasma frequency. Moreover, using Eqs. (\[e2\]–\[e3\]) it can be proved that $\Omega_+ \simeq \Omega_{f,r} \simeq K + H\,K^2/2$ and $- \Omega_- \simeq \Omega_{s,r} \simeq K - H\,K^2/2$ for large $K$, so that the fast and slow waves are the asymptotic forms of exact branches of the dispersion relation. It is in this phase velocity context that the $(f,s)$ subscripts refers to “fast space-charge wave" or “slow space-charge wave", when focusing on a particular beam [@Briggs]. Similar considerations apply to the leftward beam: $\Omega_- \simeq \Omega_{f,l} \simeq -K + H\,K^2/2$ and $- \Omega_+ \simeq \Omega_{s,l} \simeq - K - H\,K^2/2$ for large $K$.
Calculating the time-averaged energy density $<W_{e}>$ for electrostatic oscillations using Eq. (\[e5\]), it is directly verified that $\Omega_{f,r}$ and $\Omega_{s,l}$ are positive energy modes for all $K$. On the other hand, $\Omega_{f,l}$ and $\Omega_{s,r}$ can have negative energy content depending on special conditions. As for the classical two-stream instability [@Briggs], instability is expected when a wave on a beam has free energy to drive an instability in the counter propagating beam. Inspecting Eqs. (\[e10a\]–\[e11b\]) shows that the only possible couplings are $\Omega_{f,l} = \Omega_{s,r}$ or, reversing directions, $\Omega_{f,r} = \Omega_{s,l}$. Focusing on $K > 0$ and positive frequencies, instability is expected when the fast positive energy wave of the rightward beam couples to a negative energy wave on the leftward beam. It is found that the slow wave $\Omega_{s,r}$ is a negative energy mode provided $K < K_C$ (assuming non-negative $\Omega_{s,r}$). Hence the matching condition is $\Omega_{f,l} = \Omega_{s,r}$, or $$\label{e12}
\frac{1}{\sqrt{2}}\,\left(1+\frac{H^2\,K^4}{2}\right)^{1/2} = K \,,$$ as illustrated in Fig. 4, which shows intersection of the fast leftward and slow rightward modes for $H = 0.1$. As in the classical case, the coupling occurs for ${\rm Re}(\Omega) = 0$, in analogy with ideal MHD instabilities [@LashmoreDavies].
{width="70.00000%"}
In the context of this interpretation, the wave-numbers $K_m$ satisfying the coupling condition (\[e12\]) correspond to maximal instability growth-rate. For $H \neq 0$, they are given by $$\label{e13}
K^{2}_m = \frac{2}{H^2}\left[1 \pm \sqrt{1-\frac{H^2}{2}}\,\right] \,.$$ Notice that in the classical case where $H = 0$, Eq. (\[e12\]) yields only the solution $K_{m}^2 = 1/2$.
Taking the plus sign in Eq. (\[e13\]), one finds $$\label{e14}
K_m \equiv K_{m,q} = \frac{\sqrt{2}}{H}\left[1 + \sqrt{1-\frac{H^2}{2}}\,\right]^{1/2} \,.$$ Assuming $H \ll 1$, we get to leading order, $$\label{e15}
K_{m,q} = K_C - \frac{H}{8} + O(H^3) > K_B \simeq K_C - \frac{H}{4} + O(H^3) \,,$$ where $K_B$ and $K_C$ are defined in Eq. (\[e4\]). Therefore, $K_B < K_{m,q} < K_C$, which is exactly what should be expected for an instability arising from the coupling of the positive energy mode shown in curve 3 and the negative energy mode shown in curve 2 of Figure 3. This explains the physical origin of the purely quantum instability described in Ref. [@Haas], occurring for large wave-numbers, when $H < 1$. For a fixed value of $H < 1$, there is a good agreement between $K_{m,q}$ and the exact wave-number for fastest growing quantum instability. The discrepancy comes from the fact that Eqs. (\[e10a\]–\[e11b\]) show just approximate modes.
On the other hand, taking the minus sign in Eq. (\[e13\]) gives $$\label{e16}
K_m \equiv K_{m,c} = \frac{\sqrt{2}}{H}\left[1 - \sqrt{1-\frac{H^2}{2}}\,\right]^{1/2} \,.$$ We get, to leading order, $$\label{e17}
K_{m,c} = \frac{1}{\sqrt{2}} + \frac{H^2}{16\sqrt{2}} + O(H^4) < K_A \simeq 1 + \frac{H^2}{8} + O(H^4) \,,$$ where $K_A$ is defined in Eq. (\[e4\]). The wave-number $K_{m,c}$ can be interpreted as the semiclassical branch, since it corresponds to the exact classical wave-number for maximal instability, $K_c = 1/\sqrt{2}$. Moreover $K_{m,c} < K_A$ corresponds to the coupling of the positive (curve 1) and negative energy (curve 2) branches in Figure 3. Finally, we found a satisfactory agreement between the exact and approximate values of the wave-number for maximal growth-rate, at a fixed $H$.
When $H \geq 1$, the elliptic-like branch of Figure 3 disappears. The instability for $K < K_C$ and $H \geq 1$ can also be understood in terms of negative energy modes. However, we omit these considerations since these parameter ranges are outside the scope of the quantum two-stream model. Indeed, as shown in the next Section, $H$ should be small to avoid the strong coupling regime.
Necessary validity condition for the cold quantum two-stream model
==================================================================
In a cold two-stream plasma model, be classical or quantum, it is implicit that each beam has a well-defined identity, which is reasonable only if the respective velocity dispersions are much smaller than the beam velocities. It is relevant to analyze the implications of this point in view of the collisionless hypothesis underlying the original equations [@Haas].
Consider first the case of classical statistics, which applies for sufficiently small particle densities, or large temperatures. In this case, the beam velocity dispersion is the thermal velocity $v_{th}$, so that the present model assumes $u_0 \gg v_{th}$, which is equivalent to a cold beam approximation. Moreover, the thermal velocity of the beams is a free parameter, independent of the particle density. It can be verified that $g_C = e^2 n_{0}^{1/3}/(\varepsilon_{0}\,m\,v_{th}^2) \ll 1$ (small classical coupling constant), together with $H \geq 1$, implies some relativistic velocities which are out of the scope of the model. Therefore, $H\ll 1$ is necessary to support the collisionless hypothesis.
Let us now rule out the possibility of a large quantum parameters for dense degenerate plasmas a well. In this case, due to Pauli blocking, the system is all the more ideal that the density is large, and it is appropriate [@Landau; @Manfredi] to define the quantum coupling parameter $g_Q = e^2 n_{0}^{1/3}/(\varepsilon_{0}\,T_{F}) \sim (\hbar\,\omega_{p}/(m\,u_{F}^2))^2$. Here, $T_F$ and $u_F$ are the electronic Fermi temperature and velocity, respectively. In the degenerate case, $u_F$ measures the velocity spread of each beam, and increases with the density. Hence, the quantum two-stream model assumes $u_0 \gg u_F$, imposing a bound on the electron particle densities. However, for such small densities one has $H \ll \hbar\,\omega_{p}/(m\,u_{F}^2) \sim g_{Q}^2 \ll 1$. Here again, the collisionless and the well-defined two-stream hypothesis, demand $H\ll 1$. The present work provide physical arguments to understand the quantum instability branch for small $H$ and large wave-numbers, where the weak coupling regime apply. In addition, large quantum parameters are admissible in the context of a quantum Dawson $N-$stream model [@Haas; @Dawson], in which case the beam velocities need not to be large in comparison to the Fermi velocities.
Conclusion
==========
The main results of this work stem from the factorized form of the dispersion relation, as expressed in Eq. (\[e9\]). This allows to identify fast and slow waves propagating in both the positive and the negative directions. The quantum two-stream instability grows when the free energy available in a positive energy wave carried by one beam, is transferred to a negative energy wave carried by the other beam. The coupling of these waves and their energy exchange has been discussed in terms of their electrostatic energy density, giving rise to stable or unstable scenarios. While the mathematical techniques for the quantum plasma problem are the same as for the classical two-stream plasma, the quantum dispersion relation is more subtle as evidenced by Fig. (\[figure3\]). The above analysis can in principle be pursued on similar problems such as the quantum beam-plasma instability and on theories with a discrete structure such as the quantum multi-stream model.
.5cm [**Acknowledgments**]{}\
This research was financially supported by the Alexander von Humboldt Foundation, by projects FIS 2006-05389 of the Spanish Ministerio de Educación y Ciencia and PAI-05-045 of the Consejería de Educación y Ciencia de la Junta de Comunidades de Castilla-La Mancha. We acknowledge the anonymous referee for his constructive questions, in particular for pointing out the model’s applicability condition on the parameter $H$.
[10]{} F. Haas, G. Manfredi and M. Feix, Phys. Rev. E [**62**]{}, 2763 (2000). D. Anderson, B. Hall, M. Lisak and M. Marklund, Phys. Rev. E [**65**]{}, 046417 (2002). F. Haas, G. Manfredi and J. Goedert, Braz. J. Phys. [**33**]{}, 128 (2003). S. Ali and P. K. Shukla, Eur. Phys. J. D [**41**]{}, 319 (2007). A. Mushtaq and R. Khan, Phys. Scr. [**78**]{}, 015501 (2008). H. J. Ren, Z. W. Wu, J. T. Cao and P. K. Chu, Phys. Lett. A [**372**]{}, 2676 (2008). A. Bret, Phys. Plasmas [**15**]{}, 022109 (2008). A. Bret, Phys. Plasmas [**14**]{}, 084503 (2007). P. A. Sturrock, Phys. Rev. [**117**]{}, 1426 (1960). A. I. Akhiezer, I. A. Akhiezer, R. V. Polovin, A. G. Sitenko and K. N. Stepanov, Plasma Electrodynamics (vol. 1, Linear Theory), Pergamon Press, Oxford, 1975. T. H. Stix, Waves in Plasmas, Springer Verlag, New York, 1992. C. N. Lashmore-Davies, Phys. of Plasmas [**14**]{}, 092101 (2007). R. J. Briggs, in Advances in Plasma Physics (vol. 4), ed. A. Simon and W. B. Thompson, John Wiley & Sons, New York, 1971. L. D. Landau and E. M. Lifshitz, [*Statistical Physics*]{}, part 1, Butterworth-Heinemann, Oxford, 1980. See Eq. (2.12) of G. Manfredi, Fields Inst. Commun. [**46**]{}, 263 (2005). J. Dawson, Phys. Fluids [**4**]{}, 869 (1961).
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author:
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Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371\
E-mail:
title: Index and overlap construction for staggered fermions
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Introduction
============
Staggered fermions had long been perceived as disadvantaged compared to Wilson fermions regarding the index theorem connection between (would-be) zero-modes and gauge field topology. For Wilson fermions, the would-be zero-modes can be identified as eigenmodes with low-lying real eigenvalues; these can be assigned chirality $\pm1$ according to the sign of $\psi^{\dagger}\gamma_5\psi$, thereby determining an integer-valued index which coincides with the topological charge of the background lattice gauge field in accordance with the index theorem when the gauge field is not too rough [@SV; @Itoh; @overlap(H)]. It coincides with the index obtained from the exact chiral zero-modes of the overlap Dirac operator [@Neu]. In contrast, for staggered fermions, no way to identify the would-be zero-modes was known. They appeared to be mixed in with the other low-lying modes (all having purely imaginary eigenvalues) [@SV; @Damgaard-Heller] and only separating out close to the continuum limit [@Davies(index)]. It seemed that, away from the continuum limit, the best one could have was a field-theoretic definition of the staggered fermion index [@SV]. The latter had the disadvantages of being non-integer, requiring a renormalization depending on the whole ensemble of lattice gauge fields, and being significantly less capable than the Wilson fermion index of maintaining the index theorem in rougher backgrounds [@SV].
Recently the consensus viewpoint described above was found to be incorrect: Staggered fermions do have identifiable would-be zero-modes away from the continuum limit, with identifiable chiralities and integer-valued index satisfying the index theorem when the lattice gauge field is not too rough [@DA(sindex)]. The would-be zero-modes, chiralities and index can be identified in a [*spectral flow*]{} approach based on a new hermitian version of the staggered Dirac operator, paralleling the spectral flow approach to the index for usual Wilson fermions [@Itoh; @overlap(H)].
Further developments along this line have led to a new version of overlap fermions built from staggered fermions in place of Wilson fermions [@DA(pairs)]. The construction has the remarkable feature of reducing the 4 fermion flavors described by the staggered fermion to 2 flavors for the staggered overlap fermion. It turns out that underlying this construction is a new Wilson-type fermion, obtained by adding a Wilson-type term to the staggered fermion, which gives masses $\sim 1/a$ to 2 of the flavors while leaving the remaining 2 flavors massless. Other Wilson-type terms are also possible; another one which reduces the flavors from 4 to 1 was subsequently proposed in [@Hoelbling]. Numerical investigations of the 2-flavor staggered overlap fermion have been reported in [@Forcrand(POS)]. The methods of [@DA(sindex); @DA(pairs)] were later applied to naive fermions and minimally doubled fermions [@Creutz1].
A posteriori, these results and constructions can superficially seem quite straightforward. But a priori the odds were very much against any of this working out in a sensible way. There were a number of surprises and unexpected aspects, and these will be highlighted in the present review.
Would-be zero-modes and index of the staggered Dirac operator
=============================================================
In the continuum setting, the spectral flow perspective on the index of the Dirac operator $D$ arises by considering the eigenvalues $\{\lambda(m)\}$ of the hermitian operator H(m)=\_5(D-m) \[3\] The spectral flow is defined as the net number of eigenvalues $\lambda(m)$ of $H(m)$ that cross the origin, counted with sign $\pm$ depending on the slope of the crossing, as $m$ is varied over some range. It can be shown that the spectral flow of $H(m)$ comes entirely from eigenvalue crossings at $m=0$ and equals minus the index of $D$.
In the lattice setting with Wilson fermions, the spectral flow perspective [@Itoh; @overlap(H)] is based on the hermitian lattice analogue of (\[3\]): H\_W(m)=\_5(D\_W-m) \[6\] where $D_W$ is the Wilson Dirac operator. The eigenvalue crossings of $H_W(m)$ are in one-to-one correspondence with [*real*]{} eigenvalues of $D_W$, and the index of $D_W$ (obtained from the would-be zero-modes, i.e. the eigenmodes with low-lying real eigenvalues) coincides with minus the spectral flow of the low-lying eigenvalue crossings of $H_W(m)$. Numerical results illustrating this can be found, e.g., in [@Itoh]. An illustration in the $d\!=\!2$ case is given in Fig.1 where the eigenvalues of $H_W(m)$ are plotted as functions of $m$.
In the case of staggered fermions, the staggered Dirac operator $D_{st}$ is anti-hermitian and therefore all its eigenvalues are purely imaginary. Hence the identification of would-be zero modes and index in the Wilson case does not carry over to the staggered case: there are no real eigenvalues, and in fact the staggered analogue of (\[6\]), $\Gamma_5(D_{st}-m)$, is not even hermitian. The lack of any obvious way to distinguish the would-be zero-modes from the other low-lying eigenmodes of $D_{st}$ gave rise to the consensus viewpoint that staggered fermions are disadvantaged in this regard relative to Wilson fermions.
However, it turns out that there is an alternative spectral flow approach in the staggered case [@DA(sindex)]. Note that in the continuum setting, instead of (\[3\]) one can just as well use the hermitian operator $H(m)=iD-m\gamma_5$ for the spectral flow perspective on the index. But now the staggered analogue, H\_[st]{}(m)=iD\_[st]{}-m\_5 \[7\] is also hermitian and so its spectral flow can be considered as well. Here $\Gamma_5$ is the analogue of $\gamma_5$ in the staggered formulation; it is hermitian and corresponds up to $O(a^2)$ discretization errors to $\gamma_5\otimes{\bf 1}$ in the spin$\otimes$flavor interpretation [@GS]. Since $H_{st}(0)=iD_{st}$, the would-be zero-modes of $D_{st}$ are able to be identified as the eigenmodes with eigenvalues $-i\lambda=-i\lambda(0)$ for which the associated flow $\lambda(m)$ crosses zero at a low-lying value of $m$. Furthermore, the sign of the slope of the crossing is minus the chirality of the would-be zero-mode, and hence the index is minus the spectral flow of $H_{st}(m)$ coming from the crossings at low-lying values of $m$. See [@DA(sindex)] for the details of this identification.
This way of identifying the would-be zero-modes of $D_{st}$ from the low-lying eigenvalue crossings of $H_{st}(m)$ relies on an implicit assumption, namely that there is a clear separation between the low-lying and high-lying crossings. Actually, there is no a priori reason to believe that this assumption is true, even in smooth gauge field backgrounds or in the free field case. In fact one would expect that it is not true. The clear separation between low-lying and high-lying crossings in the Wilson case (as seen in Fig.1) relies crucially on the property $\gamma_5^2={\bf 1}$. But the staggered version $\Gamma_5$ does not have this property. The eigenvalues of $\Gamma_5$ are not $\pm1$ but are distributed throughout the interval $[-1,1]$. E.g. $0$ is an eigenvalue of $\Gamma_5$ in the free field case; this can be seen from the free field momentum representation of $\Gamma_5^2$ which is $\prod_{\nu}\cos^2(p_{\nu})$. In light of this one would expect that, even in the free field case, the eigenvalue crossings of $H_{st}(m)$ will be an arbitrary mess with no clear separation into low-lying and high-lying crossings.
The first and biggest surprise in all this – a miracle in fact – is that, contrary to expectations, the spectral flow of $H_{st}(m)$ does have a clear separation between low-lying and high-lying eigenvalue crossings, at least when the gauge field is not too rough. In fact in the free field case there are no high-lying crossings at all, cf. the bound (\[bound\]) below. Fig.2 shows the spectral flow in a moderately roughened U(1) background with topological charge $Q=1$ on a 2-dimensional lattice. Now there are high-lying crossings, but they are clearly separated from the low-lying ones, so the would-be zero-modes of $D_{st}$, their chiralities, and index, can be unambiguously identified.
The absence of high-lying eigenvalue crossings for $H_{st}(m)$ in the free field case can be seen analytically as follows. A simple calculation of $H_{st}(m)^2$ in the free field momentum representation gives \_[st]{}(m)\^2=\_\^2(p\_)+m\^2\_\^2(p\_) \[8\] Set $s_{\mu}=\sin(p_{\mu})$, $c_{\nu}=\cos(p_{\nu})$. Then, in the case of 2 spacetime dimensions, starting from $\hat{H}_{st}(m)^2=s_1^2+s_2^2+m^2(1-s_1^2)(1-s_2^2)$, we find \_[st]{}(m)\^2=m\^2+(1-m\^2)(s\_1\^2+s\_2\^2)+m\^2s\_1\^2s\_2\^2 m\^2 \[9\] and \_[st]{}(m)\^2=1+s\_1\^2s\_2\^2+(m\^2-1)(1-s\_1\^2)(1-s\_2\^2) 1 \[10\] Note that both of these bounds are saturated. Identical bounds can be derived in the $d=4$ case, although the derivations are more complicated. Hence in the free field case, for both $d=2$ and $d=4$ dimensions (and probably also for higher dimensions), we have H\_[st]{}(m)\_[free]{}\^2 {
[ll]{} m\^2 &\
1 &\
. \[bound\] This bound has a generalization to the case of gauge fields satisfying an “admissibility condition” on the plaquettes that implies a separation between low-lying and high-lying eigenvalue crossings for $H_{st}(m)$ when the $\epsilon$ in the condition is sufficiently small – see [@DA(sindex)] for details. Thus the situation is analogous to the Wilson case where the admissibility condition guarantees the separation between low-lying and high-lying crossings for the hermitian Wilson operator $H_W(m)$ [@L-Neu(bound)].
Comparing Fig.’s 1 and 2 we see that the form of the spectral flow of $H_{st}(m)$ is very different from the Wilson case, and the separation between the low-lying and high-lying crossings is much larger. However, this is not one of the surprises alluded to in the abstract. Instead, the surprise here is that there is no surprise – the staggered spectral flow has the same form as in the Wilson case once the correct interpretation of the hermitian staggered operator $H_{st}(m)$ is identified. It turns out that the parameter $m$ in the staggered case should be identified not with the corresponding $m$ in $H_W(m)$ but with the Wilson parameter $r$ in the Wilson case. This will be explained further below; see Fig.3.
Since the staggered fermion in $d$ dimensions describes $2^{d/2}$ flavors, the index theorem in this case should be $\mbox{index}(D_{st})=2^{d/2}(-1)^{d/2}Q$. This is confirmed by numerical results in smooth enough backgrounds. E.g. in Fig.2 the two positive slope low-lying crossings in the $Q=1$ background in 2 dimensions imply that the index is $-2$ in accordance with the index theorem. The eigenvalues of $D_{st}$ in this background correspond to $m=0$ in Fig.2. The would-be zero-modes of $D_{st}$ can be identified as the two eigenmodes with the low-lying eigenvalues belonging to the two eigenvalue flows that cross the origin.
![\[pos2eps\] Eigenvalue flow of $H_{st}(m)$ in the same $Q\!=\!1$ background as Fig. 1.](pos1.eps){width="2.5in"}
![\[pos2eps\] Eigenvalue flow of $H_{st}(m)$ in the same $Q\!=\!1$ background as Fig. 1.](pos2.eps){width="2.5in"}
Staggered overlap construction
==============================
In the Wilson case, the spectral flow perspective on the index leads to (D\_W)= [-]{} =(D\_[ov]{}) \[10a\] for any $m_0$ in the region between where the low-lying and high-lying eigenvalue crossings of $H_W(m)$ occur (e.g. $m_0=1/a$) [@overlap(H); @Neu]. Here $D_{ov}=\frac{1}{a}\Big(1+\gamma_5\frac{H_W(m_0)}{\sqrt{H_W(m_0)^2}}\Big)$ is the overlap Dirac operator [@Neu]. The intimate connection between the Wilson index, the hermitian operator $H_W(m)$ and the overlap Dirac operator suggests there may exist a staggered version of the overlap Dirac operator connected to the staggered index and staggered hermitian operator $H_{st}(m)$ discussed above. But there is a problem: The connections and properties of the overlap Dirac operator in the Wilson case rely crucially on the property $\gamma_5^2={\bf 1}$. (E.g. without it the GW relation would not hold and $D_{ov}$ would not have exact zero-modes.) Therefore the natural replacement $\gamma_5\to\Gamma_5$ is not possible when constructing the overlap operator in the staggered case since $\Gamma_5^2\ne{\bf 1}$. Attempts to construct different versions of $\gamma_5$ in the staggered case which do satisfy $\gamma_5^2={\bf 1}$ invariably lead to unnatural, problematic operators which violate either lattice rotation invariance or gauge invariance [@DA(unpublished)].
This “$\Gamma_5^2\ne{\bf 1}$ problem” initially appears insurmountable, but the second main surprise in all this is that it does have a solution [@DA(pairs)]. The theoretical idea behind the solution is as follows. In the staggered setting there is a naturally arising operator which squares to the identity, namely $\Gamma_{55}$, acting on the staggered fermion fields by $\Gamma_{55}\chi(x)=(-1)^{n_1+\dots+n_d}\chi(x)$. It has the spin$\otimes$flavor interpretation $\gamma_5\otimes\gamma_5$, which is not what we want. But if the staggered overlap construction can be set up such that the physical flavors are those with positive flavor-chirality under ${\bf 1}\otimes\gamma_5$ then $\gamma_5\otimes\gamma_5$ will be the same as $\gamma_5\otimes{\bf 1}$ on the physical flavors, and then $\Gamma_{55}$ may be used for the role of $\gamma_5$ in the staggered overlap construction.
In fact this can be achieved simply by replacing $\gamma_5\to\Gamma_{55}$ and $H_W \to H_{st}$ in the overlap formula for $D_{ov}$. The key observation is that in this construction we have $\Gamma_{55}H_{st}(m_0)=i\Gamma_{55}D_{st}-m_0\Gamma_{55}\Gamma_5$ and $\Gamma_{55}\Gamma_5$ has the spin$\otimes$flavor interpretation $(\gamma_5\otimes\gamma_5)(\gamma_5\otimes{\bf 1})
={\bf 1}\otimes\gamma_5$ up to $O(a^2)$ effects. Thus the 2 fermion flavors with positive flavor-chirality get a negative mass from the $-m_0\Gamma_{55}\Gamma_5$ term, and hence become massless flavors of the staggered overlap fermion, while the 2 flavors with negative flavor-chirality get positive masses from this term and hence become heavy, decoupling flavors of the staggered overlap fermion with masses $\sim 1/a$, just like the ”doubler” species in the usual overlap fermion construction. Note that the exact [*flavored*]{} chiral symmetry $\{D_{st}\,,\Gamma_{55}\}=0$ of the staggered fermion hereby becomes an exact [*unflavored*]{} GW chiral symmetry $\{D_{sov}\,,\Gamma_{55}\}=aD_{sov}\Gamma_{55}D_{sov}$ of the resulting staggered overlap Dirac operator $D_{sov}$. Moreover, a staggered version of the index relations (\[10a\]) holds [@DA(sindex); @DA(pairs)]: $\frac{1}{2}\mbox{index}(D_{st})=
-\frac{1}{2}\mbox{Tr}\frac{H_{st}(m_0)}{\sqrt{H_{st}(m_0)^2}}
=\mbox{index}(D_{sov})$. The factor $\frac{1}{2}$ multiplying index$(D_{st})$ reflects the reduction from 4 to 2 flavors in the staggered overlap fermion.
The interpretation of the staggered overlap fermion becomes more straightforward if we change the hermitian staggered operator by $H_{st}(m)\;\to\;\Gamma_{55}D_{st}-m\Gamma_5
=\Gamma_{55}(D_{st}-m\Gamma_{55}\Gamma_5)$. As mentioned in [@DA(pairs)], this operator is closely related to, and has the same eigenvalue spectrum as, the previous operator $iD_{st}-m\Gamma_5$. Everything in the preceding continues to hold with this new $H_{st}(m)$. The staggered overlap Dirac operator takes a more recognizable form though: it can now be written as D\_[sov]{}=(1+(D\_[st]{}-m\_0\_[55]{}\_5) ) \[10e\] From this we see that underlying the staggered overlap construction is a new [*staggered version of Wilson fermions*]{} with the Dirac operator D\_[sW]{}=D\_[st]{}+W\_[st]{},W\_[st]{}=(1-\_[55]{}\_5). \[10f\] The “Wilson term” $W_{st}$ decouples the negative flavor-chirality modes by giving them mass $2r/a$ while keeping the two positive flavor-chirality modes as the physical modes. Hence $D_{sW}$ describes two physical quark flavors on which $\Gamma_5=\Gamma_{55}$ up to $O(a)$ effects. It has the $\Gamma_{55}$ hermiticity $D_{sW}^{\dagger}=\Gamma_{55}D_{sW}\Gamma_{55}$. A 2-flavor overlap fermion can then be obtained by taking $D_{sW}-m$ with $m=\frac{r\rho}{a}\,$, $\rho\in(0,2)$ as the kernel in the usual overlap construction. For $\frac{r}{a}=m_0$ and $\rho=1$ this is precisely the 2-flavor staggered overlap Dirac operator $D_{sov}$ obtained above in (\[10e\]). But now we see that it can be generalized to any $\rho\in(0,2)$. Furthermore, the role of the parameter $m_0$ in the staggered overlap construction is hereby clarified: it is analogous to the Wilson parameter in the usual overlap construction.
The general staggered overlap operator can also be expressed as $D_{sov}=\frac{1}{a}(1+\Gamma_{55}\frac{H_{sW}(\rho)}{\sqrt{H_{sW}(\rho)^2}})$ where $H_{sW}$ is another hermitian staggered operator given by[^1] H\_[sW]{}(m)=\_[55]{}(D\_[sW]{}-m)=\_[55]{}D\_[st]{}-\_5+(1-m)\_[55]{} \[10g\] This is the true analogue of the hermitian Wilson operator $H_W(m)=\gamma_5(D_W-m)$, and its spectral flow has a similar form (Fig.4).[^2] On the other hand, the spectral flow of the hermitian Wilson operator as a function of the Wilson parameter $r$, with fixed $m=1$ (Fig.3), has a similar form to the spectral flow of our previous hermitian staggered operator $H_{st}(m)$ as a function of $m$ (Fig.2) as anticipated.[^3]
![\[pos4eps\] Eigenvalue flow of $H_{sW}(m)$ in the same $Q\!=\!1$ background as Fig.s 1,2,3.](pos3.eps){width="2.5in"}
![\[pos4eps\] Eigenvalue flow of $H_{sW}(m)$ in the same $Q\!=\!1$ background as Fig.s 1,2,3.](pos4.eps){width="2.5in"}
To summarize, three new lattice fermion formulations have been introduced: staggered versions of Wilson fermions, domain wall fermions[^4] and overlap fermions. The relation of ordinary staggered fermions to these staggered-based formulations is analogous to the relation of naive fermions to the Wilson-based formulations. The crucial property of the “Wilson term” in the staggered version of Wilson fermions is that it splits the 4 flavors into pairs with positive and negative flavor-chirality, giving a mass $\sim 1/a$ to the latter while leaving the former pair as the massless physical fermions. This allows $\Gamma_{55}\sim\gamma_5\otimes\gamma_5$ to be used in place of $\Gamma_5$, thus overcoming the “$\Gamma_5^2\ne{\bf 1}$ problem”. The staggered versions of domain wall and overlap fermions can then be constructed simply by replacing $\gamma_5\to\Gamma_{55}$ and $D_W\to D_{sW}$ in the usual formulations.
Each of the staggered-based formulations is a new alternative and competitor to the corresponding Wilson-based formulation, and should be more computationally efficient since the constructions start from staggered rather than naive fermions. However, the gain in efficiency is reduced because the staggered “Wilson term” is less local; it involves $\Gamma_5$ which is a 4-link operator [@Forcrand(POS)]. This reduction in efficiency may possibly be ameliorated by smearing the links [@Hoelbling].
Much remains to be done to clarify the theoretical and practical properties of the new staggered-based formulations. They break some of the symmetries of the original staggered fermion and the consequences of this need to be investigated and clarified. This may be done in lattice perturbation theory and also non-perturbatively by numerical lattice QCD calculations. E.g. it should be checked for the staggered Wilson fermion that a chiral (massless) limit can be reached by tuning the bare quark mass.[^5] For the staggered domain wall fermion and staggered overlap fermion it should be checked that the lattice QCD theory has an approximately and exactly massless phase, respectively. Work on this is currently underway.
[99]{}
J. Smit and J.C. Vink, Nucl. Phys. B 286, 485 (1987) S. Itoh, Y. Iwasaki and T. Yoshié, Phys. Rev. D 36, 527 (1987) R. Narayanan and H. Neuberger, Nucl. Phys. B 443, 305 (1995) H. Neuberger, Phys. Lett. B 417, 141 (1998); [*ibid*]{} 427, 353 (1998) P.H. Damgaard, U.M. Heller, R. Niclasen and K. Rummukainen, Phys. Rev. D 61:014501 (1999) E. Follana, A. Hart and C.T.H. Davies, Phys. Rev. Lett. 93:241601 (2004) D.H. Adams, Phys. Rev. Lett. 104:141602 (2010) D.H. Adams, arXiv:1008.2833 C. Hoelbling, Phys. Lett. B 696, 422 (2011) Ph. de Forcrand, A. Kurkela and M. Panero, in proceedings of [*Lattice 2010*]{} \[arXiv:1102.1000\] M. Creutz, T. Kimura and T. Misumi, JHEP 1012:041 (2010) M.F.L. Golterman and J. Smit, Nucl. Phys. B 245, 61 (1984) P. Hernandez, K. Jansen and M. Lüscher, Nucl. Phys. B 552,363 (1999); H. Neuberger, Phys. Rev. D 61:085015, 2000 D.H. Adams, unpublished calculations
M. Creutz, T. Kimura and T. Misumi, arXiv:1101.4239
[^1]: We set $r=1$ and use lattice units to get the second equality.
[^2]: In 2 dimensions the staggered Wilson fermion has one physical flavor and one doubler whereas the usual Wilson fermion has three doublers. This explains why there is one high-lying eigenvalue crossing in Fig.4 and three high-lying crossings in Fig.1.
[^3]: Note from (\[10g\]) that the flow parameter $m$ of $H_{sW}(m)$ is multiplied onto $\Gamma_{55}$, whereas for $H_{st}(m)$ it is multiplied onto $\Gamma_5$. Thus the gauge field topology can be probed by a hermitian operator whose varying term is the flavor non-singlet $m\Gamma_{55}$, contrary to an incorrect claim in [@Forcrand(POS)].
[^4]: Staggered domain wall fermions were not discussed here; see [@DA(pairs)].
[^5]: A step in this direction was made in [@Creutz2].
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abstract: 'Thermal physics is a core course requirement for most physics degrees and encompasses both thermodynamics and statistical mechanics content. However, the primary content foci of thermal physics courses vary across universities. This variation can make creation of materials or assessment tools for thermal physics difficult. To determine the scope and content variability of thermal physics courses across institutions, we distributed a survey to over 140 institutions to determine content priorities from faculty and instructors who have taught upper-division thermodynamics and/or statistical mechanics. We present results from the survey, which highlight key similarities and differences in thermal physics content coverage across institutions. Though we see variations in content coverage, we found 9 key topical areas covered by all respondents in their upper-division thermal physics courses. We discuss implications of these findings for the development of instructional tools and assessments that are useful to the widest range of institutions and physics instructors.'
author:
- 'Katherine D. Rainey'
- 'Bethany R. Wilcox'
bibliography:
- '2019\_PERC-refs.bib'
title: 'Faculty survey on upper-division thermal physics content coverage'
---
\[sec:intro\]Introduction
=========================
Thermal physics, which includes both thermodynamics and statistical mechanics, is a core course required for attaining a physics bachelors’ degree at most institutions. However, anecdotally the material covered in thermal physics courses often varies between instructors and across institutions. This content variability poses a significant challenge in development of standardized thermal physics assessments and teaching tools that can be utilized by a wide range of instructors. Though there is a body of research surrounding student understanding of thermal physics concepts [@dreyfus2015resource], less is known about the breadth of topics covered in upper-division thermal physics courses.
Here, we present findings from a survey distributed with the purpose of soliciting instructor priorities in upper-division thermal physics as a part of a broader research effort to develop a standardized upper-division thermal physics assessment. Findings may lay an important foundation for other researchers interested in developing course materials and assessments for thermal physics, and inform instructors in defining course objectives and content-foci for their thermal physics courses.
In this paper, we begin by describing the process of constructing and distributing the survey (Sec. \[sec:methods\]). Then, we present results of the survey (Sec. \[sec:results\]), including general course information, key concepts covered, and valued scientific practices. We also consider response consistency between survey responses and submitted syllabi, followed by an analysis of content variability across institutions. We conclude with a short consideration of implications of the survey and future directions (Sec. \[sec:conclusion\]).
\[sec:methods\] Methods
=======================
The faculty survey was designed to solicit key information about thermal physics courses, such as content covered, general course structure and emphasis (thermodynamics, statistical mechanics, or both), and needs or interest in an upper-division thermal physics assessment. This section describes methods for developing and distributing the survey with an emphasis on creating a format that was accessible and relatively short in duration, while still soliciting sufficient information. [**Survey Development:**]{} Prior to constructing the survey, a focus group was conducted with four experts, all with experience teaching thermal physics and researching student difficulties in thermal physics. The focus group solicited expert perspectives surrounding upper-division thermal physics, including textbooks, content coverage, learning goals, and existing thermodynamics assessments. Outcomes from the focus group informed several questions included on the survey. For example, participants discussed notational conventions as one major challenge for a thermal physics assessment (e.g. the sign convention of work). To address this concern, one question on the survey solicited specific notational issues worth considering in development of a thermal physics assessment. Additionally, textbooks brought up during the focus group comprised the list of textbook options provided on the survey.
To faciliate ease of responses, the survey was a primarily multiple-response format with only a select set of questions being free-response. Thus, one of the first steps in survey development was determining which options to provide for various multiple-response questions. We began by investigating the scope of thermal physics in texts; we analyzed six thermal physics texts brought up during the focus group [@baierlein1999thermal; @carter2001csthermo; @kittel1980thermal; @salinger1975thermo; @schroeder1999thermal; @zemansky10020heat] for key content coverage. This process involved reviewing each text and identifying topical areas for each based on chapter titles, section headings, and emphasized key terms. Based on the frequency of topics appearing across the different texts, we classified topical areas into *core topics* and *supporting topics*. To put these into an accessible form for use in the survey, topics were sorted and condensed into 29 core topics, most with roughly 4 supporting topics (see Table \[table:contentfreq\]). For example, the core topic of “thermodynamic laws” had four supporting topics: 0th law, 1st law, 2nd law, 3rd law. Some core topics had no supporting topics (e.g. semiconductors) while some had as many as seven (e.g. energy and thermodynamic potentials); the one exception to this was statistical mechanics, which had 14 supporting topics.
In addition to focusing on content, and in response to recent calls in science education literature for more consideration of scientific practices in course materials, assessment, and instruction [@national2012framework], the survey also solicited information on the scientific practices valued by respondents in their thermal physics courses. The list of scientific practices provided on the survey was pulled from the Next Generation Science Standards (NGSS) list of science and engineering practices [@NGSS2013practices]. In their list, the NGSS combined similar practices together (e.g. developing and using models); however, in upper-division courses, it is less clear that all paired practices would be targeted together. Thus, to collect more specific data about individual practices, paired NGSS practices were split into separate categories. For example, “developing and using models” was split into “developing models” and “using models” for the survey.
The survey was administered through the survey platform Qualtrics and hosted by the University of Colorado Boulder (CU). The survey was divided into 4 major sections: (1) general course information, (2) content coverage, (3) scientific practices, and (4) interest in, and concerns about, an upper-division thermal physics assessment. Respondents also had the option to identify their institution and submit their course syllabus. Additionally, gender and racial identity information were collected at the end of the survey.
After initial construction of the survey, we solicited feedback from CU physics faculty who were familiar with teaching upper-division thermal physics. Based on these discussions, and informed by the frequency of topical areas appearing across the six different analyzed texts, we grouped the core topics into two categories: assumed core topics and other core topics. Assumed core topics are topics that one might expect are covered in every thermal physics course: energy and thermodynamic potentials; engines and refrigerators; entropy; equilibrium; monatomic gases; heat; temperature; thermodynamic laws; and work. The survey presented these assumed core topics at the beginning of Section (2) of the survey, with their supporting topics shown on the same page. A free-response textbox followed these assumptions to allow respondents to indicate disagreement with the assumptions made. All other core topics were provided on the following page of the survey without their supporting topics displayed. After selecting from the list of other core topics, associated supporting topics for each of the selected core topics were displayed on the following page. This conditional formatting was motivated by the desire to reduce respondent fatigue due to survey length.
[**Survey Distribution:**]{} To ensure the information collected was reflective of a broad range of institutions, we collected contact information for a large variety of physics degree-granting institutions, including minority serving institutions (MSIs) and women’s colleges, for use in distributing the survey. Institutions were identified using the American Physical Society’s “Top Educators” lists [@APSTopEd], each of which identifies 16-20 institutions with the highest average number of physics bachelors’ degrees awarded by the institution per year. We also utilized the overall and underrepresented minority (URM) lists for Ph.D.-granting, MS-granting, and BS-granting institutions. Beyond that, we used the American Physical Society’s MSIs list [@MSI2017], which included a list of Historically Black Colleges and Universities, Black-serving insitutions, and Hispanic-serving institutions, to identify all other physics-degree-granting MSIs not on the “Top Educators” lists; the MSI list included institutions with both large and small physics departments. We also identified women’s colleges with the “Women in Physics” report produced by the American Institute of Physics [@women2000AIP]. We note that other small physics departments (e.g. those that are not Top Educators, or at MSIs or women’s colleges) were not targeted in the initial distribution of the survey, but will be targeted in the broader project moving forward.
After identifying institutions, we obtained contact information of department chairs from physics department websites. We then emailed the survey solicitation to the department chairs, with a specific request for the email to be forwarded to all faculty within their department who were currently teaching or had previously taught upper-division thermal physics. In addition to department chairs, the research team solicited the help of their professional contacts at different institutions to take the survey or forward it to faculty in their department.
\[sec:results\]Results
======================
The survey was open for response collection for three and a half months. During this time, 59 respondents fully completed the survey while 2 completed all of the survey except questions regarding scientific practices and assessment. Only responses that completed the sections with core topics and supporting topics and beyond were used for analysis. We do not report response rate, as it is unclear how many people recieved the solicitation forwarded from their department chairs.
![Highest physics degree offered by Minority-Serving Institution (MSI) or Women’s College classification. Bachelor’s degrees (BS), Master’s degrees (MS), and PhDs are indicated. \[MSI\]](MSI_inst_2){width="\linewidth"}
Racial demographics of respondents included Asian (16%, N=9), Black/African American (2%, N=1), Caucasian (74%, N=43), and Hispanic (2%, N=1); no other racial identities were indicated and 7% (N=4) preferred not to answer. Additionally, 83% (N=48) of respondents were men and 14% (N=8) were women (no other gender identities were indicated); 3% (N=2) preferred not to provide their gender. Three respondents did not provide any demographic information.
----------------------------------- ----- --------------------------------- ----- ----------------------------------- ----
Energy & Thermodynamic Potentials Equilibrium Statistical Mechanics 92
*Chemical Potential* 93 *Thermal Equilibrium* 98 Processes 89
*Energy Sources* 49 *Stable & Unstable Equilibrium* 41 Diatomic Gases 84
*Enthalpy* 89 Heat Fermions 84
*Equipartition* 95 *Heat Capacity* 100 Blackbody Radiation 82
*Free Energy (Gibbs & Helmholtz)* 95 *Heat Transfer* 72 Bosons 80
*Internal Energy* 100 *Latent Heat* 90 Phases 79
*Maxwell’s Relations* 77 Temperature Kinetic Theory 75
Engines & Refrigerators *Absolute Zero* 98 Quantum Phenomenon 75
*Heat Engines* 93 *Negative Temperature* 69 Pressure Diagrams 72
*Refrigerators* 82 *Thermodynamic Temperature* 89 Scaling 71
Entropy *Temperature Measurement* 59 Magnetism 64
*Boltzmann’s Law* 90 Thermodynamic Laws Chemical Reactions 54
*dS=dQ/T* 93 *0th* 89 Conduction, Convection, Radiation 53
*Entropy & Information* 57 *1st* 100 Solids 51
*TS Diagrams* 71 *2nd* 100 Pure Substances 49
Gases *3rd* 89 Diffusion 46
*Ideal Gas Law* 100 Work Cooling Techniques 31
*Mixtures of Gases* 57 *Mechanical* 98 Fluids 20
*van der Waals Interactions* 71 *Path dependence* 84 Semiconductors 12
----------------------------------- ----- --------------------------------- ----- ----------------------------------- ----
\[table:contentfreq\]
We collected institutional information, including selectivity, research activity, student population, and highest physics degree offered via the Carnegie Classifications [@Carnegie2018] and institutions’ physics department websites. From the Carnegie Classifications, we identified 70% (N=34) of identifiable institutions as being selective or more selective with regards to admissions practices, while 31% (N=15) are considered “inclusive” institutions. Additionally, 18 schools are classified as having high or very high research activity.
Overall, we identified 52 unique institutions from the survey, 28 of which were MSIs and/or women’s colleges; one institution could not be identified and one was not in the Carnegie Classifications database. Figure \[MSI\] presents institution type by highest physics degree offered and MSI/women’s college classification. In a few cases, (N=7) institutions were represented by 2-3 responses; it was evident from submitted syllabi and individual item reponses that these were submitted by different people.
[**Course Information:**]{} We asked respondents if their course focused on thermodynamics, statistical mechanics, or both (thermal physics); 97% (N=59) selected thermal physics and the remaining 3% (N=2) of responses were split evenly between thermodynamics and statistical mechanics. Most institutions reported one semester of thermal physics (79%, N=48); some reported two quarters (10%, N=6) or two semesters (8%, N=5), while a small minority reported one quarter (3%, N=2). The student population was composed of mostly juniors (N=41) and seniors (N=39), though some (N=12) reported sophomores in the course as well.
The majority of respondents (72%, N=44) reported using *An Introduction to Thermal Physics* by Daniel V. Schroeder [@schroeder1999thermal]. *Thermal Physics* by Charles Kittel and Herbert Kroemer [@kittel1980thermal] was the second most frequently cited text (16%, N=10). All other texts appeared at a frequency of 7% or below. Most of the instructors (74%, N=45) teach with the assumption that their students have little to no prior exposure to thermal physics content. Some (N=19) expected familiarity with topics such as energy, heat, the first and second laws of thermodynamics, and the ideal gas law. A few (N=7) said they expect thermal physics exposure from the introductory physics sequence, though several noted that thermal physics is only covered for a few weeks, and sometimes not at all, in that sequence.
These data show most institutions require one semester of thermal physics, most instructors use Schroeder’s text [@schroeder1999thermal], and many instructors assume their students have no prior exposure to thermal physics content. These results suggest two implications for PER: (1) development of Schroeder-based thermal physics assessments and materials could serve many instructors and institutions, though would still exclude the sizable population of instructors and institutions who do not use that text; and (2) pretest administration of an upper-division thermal physics assessment may not produce meaningful measurements of student understanding of thermal physics content prior to taking the course.
[**Key Topical Areas:**]{} Table \[table:contentfreq\] shows frequency of assumed supporting topics and other core topics. All assumed core topics (see Section \[sec:methods\]) appeared at a frequency of 100%; these frequencies are not reported in Table \[table:contentfreq\]. Frequency of supporting topics is given relative to the number of times the corresponding core topic was selected; the frequency of core topics is given relative to the total number of valid responses. We present frequencies of all other core topics, but do not present their 56 associated supporting topics or their frequencies due to space limitations.
Four respondents reported teaching thermal physics but did not select statistical mechanics as a core topic. This result may be due to statistical mechanics being covered in their course but not seen as a core focus by the respondent; we note one of these respondents mentioned statistical distribution functions in a textbox but did not select statistical mechanics as a core topic.
These results are relevant for researchers interested in materials and assessment development in upper-division thermal physics, and can be used to guide content-foci for those endeavors such that they serve a wide range of instructors and institutions.
[**Scientific Practices:**]{} Of the 16 practices presented on the survey, three appeared at a frequency of over 85%: using mathematical thinking (98%, N=58), asking questions (95%, N=56), and using models (86%, N=51). Review of syllabi indicates the practice of “asking questions” may have been misinterpreted; the NGSS practice refers to asking scientific questions (namely for scientific investigations), but we suspect respondents may have interpreted this practice as referring to asking questions about content during class or office hours. The next most frequently appearing practices were constructing explanations (70%, N=41), communicating information (64%, N=38), and computational thinking (61%, N=36). The remaining 10 practices appeared at a frequency of 56% or less.
These results highlight at most three scientific practices that stand out as valued by nearly all thermal physics instructors in our sample and demonstrate many other scientific practices are less of a universal focus for thermal physics courses at the upper-division level. Thus, researchers should pay particular attention to including opportunities for students to demonstrate and develop the practices of using models and using mathematical thinking in thermal physics-oriented materials and assessments.
[**Response Consistency:**]{} As a verification of the survey data, we checked for consistency between survey responses and submitted syllabi for the 39 responses that provided a syllabus. We looked at key topics on syllabi and compared with the associated survey response to ensure topics appearing on the syllabus also appeared on the survey response. No core or supporting topics had more than 3 discrepancies when comparing between survey responses and the 39 syllabi. Discrepancies could be due to the amount of focus placed on those topics in the course. For example, Bose-Einstein condensates may appear on the syllabus but may not be seen as a major content focus for the instructor when completing the survey, resulting in a discrepancy between their syllabus and response. Some topics, such as large systems (N=10), interacting systems (N=8), and Boltzmann and/or quantum statistics (N=9), appeared in syllabi but did not appear as explicitly named core or supporting topics on the survey. However, those who included topics such as these on their syllabus selected other topics on the survey that encompass or require the same idea, such as multiplicity, thermal equilibrium, and statistical mechanics. Canonical ensembles (N=11) and thermodynamic identities (N=6) were the other most common topics that appeared on syllabi but were not provided as options on the survey.
This analysis shows that the survey reliably captured the scope of content coverage for most survey responses without large discrepancies.
[**Content Variability:**]{} To investigate the claim of content variation across upper-division thermal physics courses, we examined survey responses to see how many topics were selected by all instructors. We looked at the three groups of topics laid out in Table \[table:contentfreq\]: assumed core topics, assumed core topics’ supporting topics, and other core topics.
We found that 9/9 (100%) of assumed core topics, 5/32 (16%) of assumed supporting topics, and 0/20 (0%) of other core topics were selected by all respondents. When repeated with institutions with multiple responses (e.g. different instructors at the same institution), we saw an average of 72% of assumed supporting topics and 20% of other core topics chosen by all respondents at a given institution.
These results support the anecdotal claim that upper-division thermal physics content coverage varies both across institutions and between instructors at the same institution (though to a lesser extent). It also makes the case, however, that there are some topics, namely our assumed core topics, that all or most instructors prioritize in their upper-division thermal physics courses.
\[sec:conclusion\]Conclusions
=============================
Our data suggest important considerations for researchers and instructors interested in curricular materials and assessment development for upper-division thermal physics. Despite the demonstrated content variability within thermal physics, our results point to content-foci, scientific practices, and reference texts that can act as baselines for materials that can serve a broad range of institutions and instructors. The results presented here will lay the groundwork for development of an upper-division thermal physics assessment. In order for this assessment to be useful broadly, we carefully and deliberately collected data from institutions that serve a wide range of student populations. We recommend other researchers interested in making widely-available upper-division materials utilize similar methods in collecting input from a wide range of institutions to inform their work. Results from this survey can inform upper-division thermal physics investigations in PER and the methodology can be reproduced for investigation of the scope of other upper-division physics courses.
|
---
abstract: 'Students acquire knowledge as they interact with a variety of learning materials, such as video lectures, problems, and discussions. Modeling student knowledge at each point during their learning period and understanding the contribution of each learning material to student knowledge are essential for detecting students’ knowledge gaps and recommending learning materials to them. Current student knowledge modeling techniques mostly rely on one type of learning material, mainly problems, to model student knowledge growth. These approaches ignore the fact that students also learn from other types of material. In this paper, we propose a student knowledge model that can capture knowledge growth as a result of learning from a diverse set of learning resource types while unveiling the association between the learning materials of different types. Our multi-view knowledge model (MVKM) incorporates a flexible knowledge increase objective on top of a multi-view tensor factorization to capture occasional forgetting while representing student knowledge and learning material concepts in a lower-dimensional latent space. We evaluate our model in different experiments to show that it can accurately predict students’ future performance, differentiate between knowledge gain in different student groups and concepts, and unveil hidden similarities across learning materials of different types.'
author:
- |
Siqian Zhao[^1]\
\
\
\
Chunpai Wang\
\
\
\
Shaghayegh Sahebi\
\
\
\
bibliography:
- 'sigproc.bib'
title: Modeling Knowledge Acquisition from Multiple Learning Resource Types
---
Introduction {#sec:intro}
============
Related Work {#sec:related}
============
Multi-View Knowledge Modeling
=============================
\[sec:mvrbtf\]
Experiments {#sec:experiments}
===========
Student Performance Prediction
------------------------------
\[sec:experiments:performance\]
Student Knowledge Modeling {#sec:experiments:knowledge}
--------------------------
Learning Resource Modeling {#sec:experiments:resource}
--------------------------
Conclusions
===========
\[sec:conclusions\]
Acknowledgments
===============
This paper is based upon work supported by the National Science Foundation under Grant No. 1755910.
[^1]: First two authors contributed equally to this work.
|
---
abstract: 'We describe a method for initializing characteristic evolutions of the Einstein equations using a linearized solution corresponding to purely outgoing radiation. This allows for a more consistent application of the characteristic (null cone) techniques for invariantly determining the gravitational radiation content of numerical simulations. In addition, we are able to identify the [*ingoing*]{} radiation contained in the characteristic initial data, as well as in the initial data of the 3+1 simulation. We find that each component leads to a small but long lasting (several hundred mass scales) transient in the measured outgoing gravitational waves.'
address:
- ' Department of Mathematics, Rhodes University, Grahamstown, 6139 South Africa '
- ' Departament de Física, Universitat de les Illes Balears, Palma de Mallorca, E-07122, Spain '
- ' Theoretical Astrophysics Including Relativity, California Institute of Technology, Pasadena, CA 91125, USA '
author:
- Nigel Bishop
- Denis Pollney
- Christian Reisswig
bibliography:
- 'references.bib'
- 'add\_refs.bib'
title: Initial data transients in binary black hole evolutions
---
Introduction {#sec:introduction}
============
It is well known in numerical relativity that current practice for the setting of initial data introduces spurious radiation into the system, in both the 3+1 and the characteristic approaches. The error in the initial data leads to an initial burst of spurious “junk” radiation that results from solving the constraint equations on a single hypersurface, without knowing the past history of the radiation content. Common practice regards the signal as physical only after it has settled down following the burst arising from the initial data solution. While it is straightforward to handle the junk radiation in this way, a more serious issue is whether the spurious radiation content of the initial data leads to longer-term transients in the wave signal. This question has been considered before, but previous work on the long-term effect of the initial data is limited [@Bishop05; @Kelly2007; @Sperhake2007; @Lovelace2009].
Characteristic extraction is a method of invariantly measuring the gravitational wave emission of an isolated source by transporting the data to null infinity (${\ensuremath{\mathcal{J}^+}}$) using a null formulation of the Einstein equations [@Bishop98b; @Babiuc:2005pg; @Babiuc:2009; @Reisswig:2009us; @Reisswig:2009rx; @Babiuc:2010ze]. Initial data is needed on a null cone in the far field region, say $r> 100M$. Previous work has taken the simplistic approach of setting the null shear $J=0$ everywhere, although a recent investigation sets $J$ by the condition that the Newman-Penrose quantity $\psi_0=0$ [@Babiuc:2010ze]. Setting shear-free initial data is not necessarily incorrect physically – for example in the Schwarschild geometry in natural coordinates $J=0$ everywhere, and it is possible to construct a radiating solution with $J=0$ everywhere at a specific time. However in the generic case, a radiating solution has $J\ne 0$, and imposing $J=0$ in effect means that the outgoing radiation implied by the boundary data must be matched at the initial time by ingoing radiation. In principle, the spurious incoming content of $J=0$ data extends out towards infinity, and so could contaminate the entire evolution. However, previous results comparing characteristic extraction with conventional finite radius extrapolation indicate that it is at most a minor effect [@Reisswig:2009us; @Reisswig:2009rx].
Since the characteristic initial data is needed only in the far field region, linearized theory provides a suitable approximation. That is, it is possible to construct initial data that, at the linearized approximation, represents the physical situation of a gravitational field with purely outgoing radiation produced by sources in a central region [@Bishop-2005b]. We first solve the case of two equal mass objects in circular orbit around each other in a Minkowski background as a model analytic problem.
We then develop and implement a procedure to calculate characteristic metric data that contains purely outgoing radiation. This is done within the context of characteristic extraction, so that initial data on the null cone is constructed that is compatible with given data on a worldtube $\Gamma$ at constant radius. We are then able to compare the waveforms computed by characteristic extraction using as initial data (a) $J=0$, and (b) the linearized solution. We find that while the dominant gravitational wave modes are largely unaffected by the choice of initial $J$, a small residual difference is visible between the two approaches, and can take several hundred $M$ to be damped below other effects.
Any mis-match between the linearized solution and the actual data is an indication of an ingoing radiation content. Now, [*on the worldtube $\Gamma$*]{}, the characteristic metric data is determined entirely by the 3+1 data so that any ingoing radiation can be traced back to an ingoing radiation content in the conformally flat 3+1 initial data. In this way we show that the 3+1 initial data contains a component of ingoing radiation which results in a long-lasting transient.
The plan of the paper is as follows. Sec. \[s-rev\] introduces the notation, and reviews results needed from previous work. Sec. \[s-2=mS\] applies linearized characteristic theory to calculate metric data for two equal mass non-spinning black holes in circular orbit around each other. Sec. \[s-cotm\] describes the method to construct metric data everywhere from data on a worldtube. Sec. \[s-res\] describes our numerical results, which are obtained within the context of a binary black hole inspiral and merger. The paper ends with Sec. \[s-conc\], Discussion and Conclusion.
Review of results needed from other work {#s-rev}
========================================
The Bondi-Sachs metric
----------------------
The formalism for the numerical evolution of Einstein’s equations, in null cone coordinates, is well known [@Bondi62; @Isaacson83; @Bishop96; @Bishop97b; @Gomez01; @Bishop99]. For the sake of completeness, we give a summary of those aspects of the formalism that will be used here. We start with coordinates based upon a family of outgoing null hypersurfaces. Let $u$ label these hypersurfaces, $x^A$ $(A=2,3)$ label the null rays, and $r$ be a surface area coordinate. In the resulting $x^\alpha=(u,r,x^A)$ coordinates, the metric takes the Bondi-Sachs form [@Bondi62; @Sachs62] $$\begin{aligned}
ds^2 &=& -\left(e^{2\beta}(1 + W_c r) -r^2h_{AB}U^AU^B\right)du^2
\nonumber \\
& &- 2e^{2\beta}dudr -2r^2 h_{AB}U^Bdudx^A
+ r^2h_{AB}dx^Adx^B,
\label{eq:bmet}\end{aligned}$$ where $h^{AB}h_{BC}=\delta^A_C$ and $det(h_{AB})=det(q_{AB})$, with $q_{AB}$ a metric representing a unit 2-sphere; $W_c$ is a normalized variable used in the code, related to the usual Bondi-Sachs variable $V$ by $V=r+W_c r^2$. As discussed in more detail below, we represent $q_{AB}$ by means of a complex dyad $q_A$. Then, for an arbitrary Bondi-Sachs metric, $h_{AB}$ can be represented by its dyad component $$J=h_{AB}q^Aq^B/2,$$ with the spherically symmetric case characterized by $J=0$. We also introduce the fields $$K=\sqrt{1+J \bar{J}},\qquad
U=U^Aq_A,$$ as well as the (complex differential) eth operators $\eth$ and $\bar \eth$ [@Gomez97].
In the Bondi-Sachs framework, Einstein’s equations $R_{\alpha\beta}=8\pi(T_{\alpha\beta} -\frac{1}{2}g_{\alpha\beta}T)$ are classified as: hypersurface equations – $R_{11},q^AR_{1A},h^{AB}R_{AB}$ – forming a hierarchical set for $\beta,U$ and $W_c$; evolution equation $q^Aq^B R_{AB}$ for $J$; and constraints $R_{0\alpha}$. An evolution problem is normally formulated in the region of spacetime between a timelike or null worldtube $\Gamma$ and future null infinity (${\mathcal J}^+$), with (free) initial data $J$ given on $u=0$, and with boundary data for $\beta,U,U_{,r},W_c,J$ satisfying the constraints given on the inner worldtube. We extend the computational grid to [$\mathcal{J}^+$]{}by compactifying the radial coordinate $r$ by means of a transformation $$r \rightarrow x=\frac{r}{r+r_{\rm wt}}.$$ In characteristic coordinates, the Einstein equations remain regular at ${\mathcal J}^+$ under such a transformation.
The free initial data for $J$ essentially determines the ingoing radiation content at the beginning of the evolution. For the case of binary black hole evolutions in a 3+1 formalism, the initial Cauchy data is determined by solving the Hamiltonian and momentum constraints, usually under the assumption of conformal flatness. Compatible null data initial solutions are not known, and so we must choose an [*ansatz*]{} for $J$ which is approximately compatible. Previous work [@Reisswig:2009us; @Reisswig:2009rx] has simply set $J=0$. In Section \[s-2=mS\], we propose a refinement whereby $J$ is set according to a linearized solution which is determined by the Cauchy initial data solution.
The spin-weighted formalism and the $\eth$ operator
---------------------------------------------------
A complex dyad $q_A$ is a 2-vector whose real and imaginary parts are unit vectors that are orthogonal to each other. Further, $q_A$ represents the metric, and has the properties $$q^A q_A = 0, \qquad q^A \bar q_A = 2,\qquad
q_{AB}=\frac{1}{2}(q_A\bar{q}_B+\bar{q}_A q_B).
\label{eq:diad.norm.def}$$ Note that $q_A$ is not unique, up to a unitary factor: if $q_A$ represents a given 2-metric, then so does $q^\prime_A=e^{i\alpha}q_A$. Thus, considerations of simplicity are used in deciding the precise form of dyad to represent a particular 2-metric.
Having defined a dyad, we may construct complex quantities representing angular tensor components on the sphere, for example $X_1=T_A q^A$, $X_2=T^{AB} q_A \bar{q}_B$, $X_3=T^{AB}_C \bar{q}_A\bar{q}_B\bar{q}^C$. Each object has no free (angular) indices, and has associated with it a spin-weight $s$ defined as the number of $q$ factors less the number of $\bar{q}$ factors in its definition. For example, $s(X_1)=1,s(X_2)=0,s(X_3)=-3$, and, in general, $s(X)=-s(\bar{X})$. We define derivative operators $\eth$ and $\bar{\eth}$ acting on a quantity $V$ with spin-weight $s$ $$\eth V=q^A \partial_A V + s \Upsilon V,\qquad
\bar{\eth} V=\bar{q}^A \partial_A V - s \bar{\Upsilon} V$$ where the spin-weights of $\eth V$ and $\bar{\eth} V$ are $s+1$ and $s-1$, respectively, and where $$\Upsilon=-\frac{1}{2} q^A\bar{q}^B\nabla_A q_B.
\label{e-G}$$ Some commonly used dyad quantities are $$\begin{array}{rll}
&{\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}\mbox{Spherical polars} &\mbox{Stereographic} \\
\hline
{\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}ds^2=& d\theta^2+sin^2\theta d\phi^2& 4(dq^2+dp^2)/(1+q^2+p^2)^2\\
{\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}q^A=&(1,i\sin\theta)& \frac{1}{2}(1+q^2+p^2)(1,i) \\
{\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}\Upsilon=&-\cot\theta &q+ip.
\end{array}$$
The spin-weights of the quantities used in the Bondi-Sachs metric are $$\begin{array}{lll}
s(W_c)=s(\beta)=0,\qquad & s(J)=2,\qquad & s(\bar{J})=-2, \\
s(K)=0, \qquad & s(U)=1,\qquad & s(\bar{U})=-1.
\end{array}$$
We will be using spin-weighted spherical harmonics [@Newman-Penrose-1966; @Goldberg:1967] ${}_s Y_{\ell m}$, where the suffix ${}_s$ denotes the spin-weight, and in the case $s=0$ the $s$ will be omitted i.e. $Y_{\ell m}={}_0 Y_{\ell m}$. It is convenient to make use of the formalism described in [@Zlochower03; @Bishop-2005b], and have basis functions whose spin-weight 0 components are purely real; following [@Bishop-2005b], these are denoted as ${}_sZ_{\ell m}$. Note that the effect of the $\eth$ operator acting on $Z_{\ell m}$ is $$\begin{aligned}
\eth Z_{\ell m} & = & \sqrt{\ell(\ell+1)}\;{}_1Z_{\ell m}, \\
\eth^2 Z_{\ell m} & = &\sqrt{(\ell -1)\ell(\ell+1)(\ell+2)}\;{}_2Z_{\ell m}.\end{aligned}$$
Solutions to the linearized Einstein equations
----------------------------------------------
Ref. [@Bishop-2005b] (see also [@Reisswig:2006]) obtained solutions to the linearized Einstein equations in Bondi-Sachs form using the ansatz F(u,r,x\^A)=(f\_[,m]{}(r)(iu))\_sZ\_[,m]{} for a metric coefficient $F$ with spin-weight $s$. Here, we need the results for linearization about a Minkowski background, in which the spacetime is vacuum everywhere except on a spherical shell at $r=r_0$. Strictly speaking, we should be performing the linearization about a Schwarzschild (or even Kerr) background rather than about Minkowski. In the Kerr case it is not known how to do so, and in the Schwarzschild case the difference is that, in Eq. (\[e-j0\]), the $1/r^3$ term is replaced by a term whose leading-order behaviour is also $1/r^3$ but which is not representable analytically. We consider the lowest order case $\ell=2$ and in the exterior of the shell ([*i.e., *]{}$r>r_0$), and describe that part of the solution that represents purely outgoing gavitational radiation. \[e-lin-22\] \_[2,]{}(r)=b\_1 \[e-b1\] j\_[2,]{}(r)=(12 b\_1+6ic\_1+i\^3 c\_2) ++ \[e-j0\] u\_[2,]{}(r)&=&( ++\
&&--) w\_[2,]{}(r)&=& r\^2\
&&+r +2\^2c\_2\
&&-- \[e-w0\] The solution is determined by setting the constant (real valued) parameters $b_1$, $c_1$ and $c_2$. The gravitational news corresponding to this solution is given by [$\mathcal{N}$]{}=(n\_[2,]{}(iu))\_2Z\_[2,m]{} n\_[2,]{}=-i\^3 c\_2 We will also need the solution in the case $\nu=0,\ell=2$ in the exterior region $r>r_0$ \_[2,0]{}(r)=b\_0 \[e-b20\] j\_[2,0]{}(r)=( + + ) u\_[2,0]{}(r)=( + - ) \[e-U20\] w\_[2,0]{}(r)=-2 b\_0r - . \[e-w20\] in terms of the additional parameters $b_0$, $c_3$ and $c_4$.
Black hole binaries in circular orbit: Solution in the linearized limit {#s-2=mS}
=======================================================================
The linearized solution described in the previous section will be used to set initial data on the null cone. We seek a solution which corresponds roughly to the source of the gravitational radiation which we will eventually measure, namely a binary black hole system. In order to be able to apply the linearized theory, we model each black hole as having a matter density that is described by a Dirac-$\delta$ function whose location moves uniformly around a spherical shell. More precisely, the matter density $\rho$ in the spacetime is $$\rho=\frac{M}{r_0^2}\delta(r-r_0) \delta(\theta-\pi/2) \left(
\delta(\phi - \nu u) +\delta(\phi -\nu u -\pi) \right),$$ with respect to Bondi-Sachs coordinates $(u,r,\theta,\phi)$. The mass of each black hole is $M$, the circular orbit has radius $r_0$, and the black holes move with angular velocity $\nu$. We next express $\rho$ in terms of spherical harmonics =\_[,m]{} (\_[,m]{} (|m| iu)) Z\_[,m]{}, and apply the usual procedure, that is multiplication by $Z^*_{\ell,m}$ followed by integration over the sphere, to determine the coefficients $\rho_{\ell,m}$. We find that, for $\ell\le 2$, the only nonzero coefficients are
[ll]{} \_[0,0]{}=(r-r\_0),& \_[2,0]{}=-(r-r\_0),\
\_[2,2]{}=(r-r\_0), & \_[2,-2]{}=-i(r-r\_0).
In linearized form, the $R_{11}$ Einstein equation is \_[,r]{}=2r v\_1\^2, \[e-R11\] where $v_1$ is the covariant component of velocity in the $r$-direction. Imposing the gauge condition that the coordinates should be such that, on the worldline of the origin, the metric takes Minkowski form, it follows that $\beta=0$ there and consequently at all points within $r<r_0$. Expanding $\beta$ in terms of spherical harmonics =\_[,m]{} (b\_[,m]{} (|m| iu)) Z\_[,m]{}, and integrating Eq. (\[e-R11\]), we find the coefficients $b_{\ell,m}$ for $r>r_0$, $$\begin{array}{ll}
b_{0,0} = \frac{2 M v_1^2}{r_0}\sqrt{\pi},\; \qquad&
b_{2,0} = -\frac{M v_1^2}{r_0}\sqrt{5\pi},\; \\
b_{2,2} = \frac{M v_1^2}{r_0}\sqrt{15\pi},\; &
b_{2,-2} = -i\frac{M v_1^2}{r_0}\sqrt{15\pi}.
\end{array}$$
The determination of the remaining metric coefficients depends on the value of $(\ell,m)$. The case $(0,0)$ is straightforward, and we find for $r>r_0$ J=0,U=0,=,W\_c=-+.
The case $(2,0)$ uses the results for a static shell on a Minkowski background in [@Bishop-2005b]. We solve the jump conditions across the shell for the various metric quantities [^1]. The result is, in the interior, b\_[2,0]{}=0,j\_[2,0]{}(r)=-, and in the exterior $$\begin{aligned}
b_{2,0} & = & -\frac{M v_1^2 \sqrt{5\pi}}{r_0}, \\
j_{2,0}(r) & = & \frac{4M v_1^2 \sqrt{30\pi}}{3r_0}\left(-1+\frac{r_0}{r}
- \frac{r_0^3}{5r^3}\right).\end{aligned}$$ \[e-bj20\]
The cases $(2,\pm 2)$ use the results for a dynamic shell on a Minkowski background in [@Bishop-2005b] [^2]. The script constructs the general solution inside and outside the shell $r=r_0$, and uses the constraint equations, as well as regularity conditions at the origin and at infinity, to eliminate some of the unknown coefficients. It imposes the jump conditions at the shell, and finds a unique solution for the remaining unknowns. The result in the exterior is & = & (b\_[2,2]{}(2iu)) Z\_[2,2]{} + (-i b\_[2,2]{}(2iu)) Z\_[2,-2]{}\
J & = &(j\_[2,2]{}(r)(2iu)) \_2Z\_[2,2]{} + (-i j\_[2,2]{}(r)(2iu)) \_2Z\_[2,-2]{} \[e-j22\] where b\_[2,2]{}=, \[e-b0\] and $j_{2,2}(r)$ takes the form given in Eq. (\[e-j0\]). The gravitational news is [$\mathcal{N}$]{}& = &(-i(2)\^3 c\_2 (i2u))\_2Z\_[2,2]{}\
&&+(-(2)\^3 c\_2 (i2u))\_2Z\_[2,-2]{}. \[e-N\] Although the coefficient $c_2$ is complicated, it can be expressed to leading order in $r_0 \nu$ c\_2= r\_0\^3. \[e-c1c2\] It follows that, again to leading order in $r_0\nu$, the news is [$\mathcal{N}$]{}&=&M v\_1\^2 r\_0\^2\^3 16\
&&((-i(2iu))\_2Z\_[2,2]{}+(-(2iu)) \_2Z\_[2,-2]{}) from which it is easy to deduce, via the Bondi relation, that the rate of energy loss of the system is =-M\^2v\_1\^4r\_0\^4\^6. In the limit of a low velocity circular orbit, $v_1=1$, $\nu^2=M/(4r_0^3)$, the above formula reduces to =-, which is identical to that found from the standard quadrupole formula [@Misner73].
Constructing the metric from data on a worldtube {#s-cotm}
================================================
In characteristic extraction, the Cauchy evolution provides the characteristic metric variables $\beta, J, U$ and $W_c$ on the worldtube $\Gamma$, decomposed into spherical harmonics ${}_sY_{\ell, m}$, at every time step. In this section, we develop a method to find coefficients of the linearized solutions that provide a fit to the actual numerical data at the worldtube (to linear order and excluding incoming radiation). Then we use the linearized solutions with the coefficients just found to predict $J$ everywhere at some chosen time $u$, and in this way provide initial data for a numerical characteristic evolution. We restrict attention to the dominant modes ${}_sY_{2,2}, {}_sY_{2,-2}, {}_sY_{2,0}$. The method uses a Fourier decomposition in the time domain, and works well when the data is approximately sinusoidal, with amplitude and frequency varying slowly. Accordingly, for the binary black hole computation, the method is applied over a time domain that excludes both the junk radiation and the merger.
A metric variable $A$ may be written as A=a\_[Y,2]{} \_sY\_[2,2]{} + a\_[Y,0]{} \_sY\_[2,0]{} + a\_[Y,2]{}\^\* \_sY\_[2,-2]{} where ${}^*$ denotes the complex conjugate. The relationship between the coefficients of ${}_sY_{2,2}$ and ${}_sY_{2,-2}$ follows theoretically from the requirement that the spin weight 0 metric components must be real; and further the metric data has been checked to confirm that it does satisfy the relationship. Transforming to the ${}_sZ_{\ell, m}$ basis, we find A=a\_[Z,2]{} \_sZ\_[2,2]{} +a\_[Z,-2]{} \_sZ\_[2,-2]{} +a\_[Z,0]{}\_sZ\_[2,0]{}, where a\_[Z,2]{} = (a\_[Y,2]{}),a\_[Z,-2]{} =- (a\_[Y,2]{}),a\_[Z,0]{} = a\_[Y,0]{} \[e-A\] so that the metric data on $\Gamma$ can be re-expressed as coefficients of ${}_sZ_{2,2}$, ${}_sZ_{2,-2}$ and ${}_sZ_{2,0}$ at discrete time values. Although the data is oscillatory in time, it is not at constant frequency but is a superposition of multiple solutions with different frequencies. The linearized solutions behave as $e^{i\nu u}$ for fixed $\nu$, so for the theory to be applicable the next step is to decompose the metric data into a superposition of constant frequency components. This is achieved by making a discrete fast Fourier transform of each metric coefficient a\_[Z,2,k]{}=\_[j=1]{}\^L a\_[Z,2]{}(u\_j)() \[e-aZ2\] where there are $L$ data points over the time interval $(u_1,u_L)$. The frequency $\nu$ is related to $k$ by =. We found that $J$ at ${\ensuremath{\mathcal{J}^+}}$ from the linearized solutions provides a smoother fit to the actual data if high frequencies are eliminated (compare Sec. \[s-res\], Fig. \[f-J22scri\]), and so we undertake further processing only for $k\le L_1$ (with $1<L_1\ll L$); the setting of the Fourier coefficients for $k>L_1$ is described later.
In the case $k> 1$, for each value of $k$ Eqs. (\[e-b1\]) to (\[e-w0\]) evaluated at the worldtube are four equations for the three unknowns $b_{1,k}, c_{1,k}, c_{2,k}$. Such an over-determined system can be tackled by a least-squares-fit algorithm, or alternatively by ignoring one of the equations so making the system uniquely determined. We found that the reconstructed linearized solution gave a better fit to the actual data at ${\ensuremath{\mathcal{J}^+}}$ in the case that Eq. (\[e-w0\]) for $W_c$ was ignored. This then means that a comparison between the actual and reconstructed data for $W_c$ at the worldtube provides an indication of the error, which is expected because of (a) incoming radiation in the initial Cauchy data, (b) Fourier transform effects, and (c) other effects. We discuss item (b) at the end of this section, and items (a) and (c) in the next section.
In the case $k=1$, $\nu=0$, and Eqs. (\[e-b20\]) to (\[e-w20\]) are four equations for the constants $b_0, c_3, c_4$. We solved four equations for three unknowns using a least-squares-fit algorithm, because this approach led to the reconstructed data having a better fit to the actual data than in the case that Eq. (\[e-w20\]) was ignored.
In this way, for a given spherical harmonic say $Z_{2,2}$, we obtain values for the constants of the frequencies represented by $k=1\cdots
L_1$. Now, our purpose is to use the worldtube data to estimate $J$ off the worldtube. From Eq. (\[e-j0\]) we can write j\_[2,k]{}(r)=d\_[0,k]{}++ where for $2\le k\le L_1$ d\_[0,k]{}&=&(12b\_[1,k]{}+6ic\_[1,k]{}+i\^3c\_[2,k]{}),\
d\_[1,k]{}&=&2 c\_[1,k]{},\
d\_[2,k]{}&=& c\_[1,k]{}, and for $k=1$ d\_[0,1]{}=,d\_[1,1]{}=d\_[2,1]{}=2. We then apply the inverse discrete Fourier transform to find d\_0(u)=\_[k=1]{}\^[L]{} d\_[0,k]{}( ), where $d_{0,k}=0$ for $L_1+1\le k\le L-L_1+1$, and d\_[0,L-k+1]{}=d\_[0,k+1]{}\^\*k=1(L\_1-1). \[e-d0L\] Eq. (\[e-d0L\]) follows from the condition that $d_0(u)$ (and all other coefficients in the time domain) are real. The functions $d_1(u)$ and $d_2(u)$ are found in a similar way, and so we are able to find the coefficient of $J(u,r)$ of a given spherical harmonic, say ${}_2Z_{2,2}$. Repeating the calculation for the other spherical harmonics ${}_2Z_{2,-2}$ and ${}_2Z_{2,0}$ leads to a prediction of $J(u,r,x^A)$ to lowest order $\ell=2$.
The coefficient of ${}_2Z_{2,0}$ is not oscillatory, but can rather be described as slowly varying. While in principle, such behaviour can be represented by a Fourier decomposition, we found that a better fit was obtained by regarding the solution as almost constant and solving Eqs. (\[e-b20\]) to (\[e-U20\]) at each time step, with Eq. (\[e-w20\]) used as a measure of the error [^3].
We investigated the possibility of errors introduced in the Fourier transform and inverse transform process by comparing *on the worldtube* the actual and reconstructed values of $\beta$, $J$ and $U$, because the construction is such that they should be identical[^4]. The following comparison is for the R100 case as specified in the next section. We found that there was essentially no difference between the original and reconstructed data, apart from minor variations over about the first and last $\pm 30M$ of the time interval (of total duration $1290M$), presumably caused by the cut-off of high frequencies. This test was performed for both $L_1=50$ and $L_1=100$ with no visible difference seen in the graphs, indicating that the precise way in which high frequencies are removed is not important.
Numerical results {#s-res}
=================
[lccc]{} & Worldtube location & Initial time & Initial data\
------------------------------------------------------------------------
Data set & $R_\Gamma [M]$ & $u_0 [M]$ & $J$\
------------------------------------------------------------------------
J0-R100-u0 & $100$ & $0$ & $J=0$\
J0-R250-u0 & $250$ & $0$ & $J=0$\
J0-R100-u450 & $100$ & $450$ & $J=0$\
J0-R250-u900 & $100$ & $900$ & $J=0$\
Jlin-R100-u450 & $100$ & $450$ & $J=J_{\rm lin}$\
Jlin-R250-u900 & $250$ & $900$ & $J=J_{\rm lin}$\
We apply the method outlined in the previous section to the problem of measuring gravitational waves from binary black hole simulations. Following our implementation of characteristic extraction [@Reisswig:2009us; @Reisswig:2009rx], we first evolve a spacetime with a 3+1 (Cauchy) evolution code, recording metric data on a world tube, $\Gamma$, of fixed coordinate radius. This data is subsequently used as inner boundary data for a null-cone evolution of the Einstein equations, which transports the data to ${\ensuremath{\mathcal{J}^+}}$, where the gravitational waves are measured. The linearized solution allows us to specify data for $J$ on the initial null cone which is compatible (to the linear level) with the data on $\Gamma$ and thus, importantly, to the Cauchy 3+1 spacetime.
As a fiducial test case, we return to the well-studied model of an 8-orbit binary system with equal mass non-spinning black holes carried out in [@Reisswig:2009us; @Reisswig:2009rx]. For the Cauchy evolution, we use the `Llama` multipatch code described in [@Pollney:2009ut; @Pollney:2009yz]. We output metric data on two worldtubes located at $R_\Gamma=100M$ and $R_\Gamma=250M$ that are used as inner boundary data for a subsequent characteristic evolution. The waveforms at ${\ensuremath{\mathcal{J}^+}}$ should be independent of the worldtube location. Thus, evolutions from different worldtube locations help us validate our results. Table \[table:models\], summarizes the various characteristic evolutions that we have performed, all of which are based on the same Cauchy data, but with different characteristic initial data, $J$, and starting points in Bondi time, $u_0$.
The first approach follows the original prescription laid out in [@Reisswig:2009us; @Reisswig:2009rx]. The characteristic evolution is started coincident with the first available Cauchy data, at coordinate time $t_0=u_0=0$. The characteristic variable $J$ is initialized by the shear-free solution $J=0$. Since we are beginning the characteristic evolution from the initial Cauchy slice, these models include the spurious junk radiation contained in the conformally flat constraint solution. Models J0-R100-u0 and J0-R250-u0 listed in Table \[table:models\] follow this prescription, using data from the worldtubes at $R_\Gamma=100M$ and $R_\Gamma=250M$, respectively.
It is interesting to compare the results of the fully nonlinear characteristic Einstein evolution with the corresponding linearized solution. Fig. \[f-J22scri\] plots the $(\ell,m)=(2,2)$ spherical harmonic modes of $J_\mathrm{num}$, computed by the null Einstein evolution J0-R100-u0. The linearized solution $J_\mathrm{lin}$ is computed using linearly reconstructed worldtube data according to the prescription in Sections \[s-2=mS\] and \[s-cotm\]. We compute the linearized solution from the boundary data at $\Gamma$ after the initial data junk radiation has passed the worldtube radius $R_\Gamma$, at around $u=150M$, when the system has settled to the expected binary black hole inspiral pattern compatible with the solution construction. The upper panel of Fig. \[f-J22scri\] plots the real and imaginary parts of $J$, evaluated at ${\ensuremath{\mathcal{J}^+}}$. The center panel plots the amplitudes of $J$, while the bottom panel shows the relative difference between the linearly estimated $J_\mathrm{lin}$ and the fully relativistic result $J_\mathrm{num}$ (model J0-R100-u0). The linearized $J_{\rm lin}$ and numerically evolved $J_{\rm num}$ differ initially, but eventually, after around $u=450M$, differ by less that $1\%$, which remains consistent for the bulk of the time.
We first discuss whether the difference between the solutions can be due to effects other than ingoing radiation; such effects could include (a) nonlinearity, or (b) the linearization background being Minkowski rather than Schwarzschild. Looking at Figs. \[f-J22scri\] and \[f-W22scri\], we see that to lowest order the metric components are slowly varying sinusoidal functions; this statement also applies to the other metric components (graphs not shown). We would therefore expect that the magnitude of effects (a) and (b) would be roughly constant, if not with some increase at later times as merger is approached. Indeed for nonlinear effects we can look at the Einstein equation for $R_{11}$, which is $\beta_{,r}=0$ to linear order, with the actual value being an indication of the magnitude of nonlinearity, and we find that this quantity does slowly increase with time. However, Figs. \[f-J22scri\] and \[f-W22scri\] show that the difference between the solutions, until about $450M$, [*is getting smaller*]{}. This indicates that the effect is independent of non-linearities and present already at the linear level. The linearized solution assumes that the radiation is *purely outgoing*, whereas the actual data may contain incoming modes originating in either the characteristic or Cauchy initial data – both options being possible since $J$ at ${\ensuremath{\mathcal{J}^+}}$ is influenced by both data sets. Thus, the explanation for Fig. \[f-J22scri\] is that it reflects the slow decay of the effect of *incoming modes* in the initial data, until saturated by other factors (such as non-linear effects).
![Components of the ${}_2Y_{2,2}$ mode of $J$ at ${\ensuremath{\mathcal{J}^+}}$ estimated from boundary data at $R_\Gamma=100M$ once using linearized solutions, and once using the full non-linear characteristic evolution J0-R100-u0. The numerical solution is denoted by $J_\mathrm{num}$, while the reconstructed linear solution is $J_\mathrm{lin}$. The linearized solution makes use of data starting from a time after which the initial burst of junk radiation has left the system. The two solutions agree reasonably well only after a time $t_2$ Eq. (\[eq:incoming-time\]), a time after which the incoming radiation content of the Cauchy initial data has essentially settled to zero.[]{data-label="f-J22scri"}](diff_J_scri)
A similar effect is seen in the characteristic variable $W_c$, related to the Newtonian potential, plotted in Fig. \[f-W22scri\]. In this case, however, there is clarity about the source of the incoming radiation: it must be in the Cauchy data. This is because in characteristic extraction, the characteristic metric at the worldtube is determined entirely by the Cauchy data. Again, the lower panel shows an approximately exponential decay in the differences, until around $u=400M$. The residual steady state differences result from other effects, which gradually increase with the strength of the gravitational radiation towards the binary merger.
![ Components of the ${}_2Y_{2,2}$ mode of $W_c$ at the worldtube $R_\Gamma=100M$ once using the linearly reconstructed data, and once using the original data as obtained from the Cauchy data. The numerically obtained Cauchy data is denoted by $W_{c,\mathrm{num}}$, while the reconstructed data is $W_{c,\mathrm{lin}}$. The reconstructed data makes use of the original Cauchy data starting from a time after which the initial burst of junk radiation has left the system. The numerical and reconstructed data agree reasonably well only after a time $t_2$ Eq. (\[eq:incoming-time\]), a time after which the incoming radiation content of the Cauchy initial data has essentially settled to zero.[]{data-label="f-W22scri"}](diff_W_wt)
The findings above indicate that a physically expected purely outgoing inspiral radiation pattern is only present after some time $$u > u_{\rm incoming}\,,
\label{eq:incoming-time}$$ where $u_{\rm incoming}$ is the length of a time interval until the incoming radiation content of the Cauchy initial data has settled to a negligible amount at the given worldtube location. Hence, in order to construct physically meaningful and consistent initial data via the outgoing linearized solution, it is optimal to begin the solution at a time $u_0>u_{\rm incoming}$ after which both the junk *and* incoming radiation content of the initial data have subsided. The results of Figs. \[f-J22scri\] and \[f-W22scri\] suggest that the linearized solution provides a reasonably good approximation to the data $u\approx 450M$, at which time the system has settled into an outgoing radiative solution.
For instance, in model Jlin-R100-u450, the exact worldtube location is $R_\Gamma=100.8492$. Allowing $50M$ for the visible outgoing junk radiation to pass, we set the time range over which we use the boundary data for linearized solution construction to $(u_0,u_f)=(150.192, 1439.856)$, which includes the inspiral, but not the merger and ringdown. The time increment in is given by $du=0.144$, thus comprising 8967 data points. Referring to Eq. (\[e-aZ2\]), we have $L=8957$, $L_1=100$. In order to determine $J_{\rm lin}$ at the initial time $u_0$, we use the general form of the linearized solution, [Eq. (\[e-lin-22\])]{}: $$\begin{aligned}
J&=&\left(e_0+\frac{e_1}{r}+\frac{e_2}{r^3}\right)\,{}_2Y_{2,2}
+\left(e_0+\frac{e_1}{r}+\frac{e_2}{r^3}\right)^*\,{}_2Y_{2,-2}
\nonumber
\\ &&+\left(e_3+\frac{e_4}{r}+\frac{e_5}{r^3}\right)\,{}_2Y_{2,0}
\label{e-Jlin}\end{aligned}$$ The coefficients are determined by comparing with the worldtube data. We perform a Fourier transform on the numerically determined worldtube variables over the time interval $(u_0,u_f)$. The spectrum determines the constants $b_1$, $c_1$ and $c_2$ of Eqs. \[e-lin-22\] at each fixed frequency $\nu$. These values are transformed back to the time domain, and evaluated at $t=450M$ in order to determine the coefficients of [Eq. (\[e-Jlin\])]{}: $$\begin{array}{ll}
e_0 = (4.5217 + 3.7702 i)\times 10^{-4},\quad &
e_1 = -0.04578 - 0.17159 i, \\
e_2 = 12.582 + 42.500 i, &
e_3 = -1.4788\times 10^{-4}, \\
e_4 = 0.020365, &
e_5 = -35.563.
\end{array}
\label{e-450}$$
A goal of this paper is to investigate, within the context of characteristic extraction, the effect of the initial data on the calculation of the gravitational news. To this end, we compare waveforms at ${\ensuremath{\mathcal{J}^+}}$ from two characteristic evolutions based on the same Cauchy boundary data, but different initial data constructions: Jlin-R100-u450 and J0-R100-u450. Both evolutions use boundary data from $R_\Gamma=100M$ and begin at the initial time $u_0=t_2=450M$, which was determined above to be a point where the linearized solution is well-matched to the nonlinear evolution. The model Jlin-R100-u450 uses the initial data determined by the linearized solutions, Eqs. (\[e-Jlin\], \[e-450\]). In contrast, J0-R100-u450 simply sets $J=0$, corresponding to the original prescription of [@Reisswig:2009us; @Reisswig:2009rx]. Fig. \[f-diff\_amp\_vs\_time\] plots the Bondi news at ${\ensuremath{\mathcal{J}^+}}$ as computed from both evolutions, denoted by ${\ensuremath{\mathcal{N}}}^{450}_{\rm lin}$ and ${\ensuremath{\mathcal{N}}}^{450}_0$ for the linearized and $J=0$ initial data runs, respectively. Whereas the phase of ${\ensuremath{\mathcal{N}}}$ shows very little difference between the runs (middle panel), the amplitude shows visible oscillations for the ${\ensuremath{\mathcal{N}}}^{450}_0$ evolution (upper panel and inset). The waveforms agree to within $1\%$ only after a time $u=400M$ (which must be added to the $u_0=450M$ starting point of the simulation). Therefore, the influence of the $J=0$ [*ansatz*]{} for the characteristic initial data has a notable impact over an extended time.
We note, however, that the original prescription of [@Reisswig:2009us; @Reisswig:2009rx] used characteristic initial data $J=0$ at the initial Cauchy time $u_0=t_0=0$ (that is, including the junk radiation). At that time, the shear-free approximation $J=0$, for the characteristic initial data is compatible with the Cauchy initial data solution.
Hence, we also compare the waveforms of model Jlin-R100-u450 against those of model J0-R100-u0, where the latter model uses $J=0$ initial data at time $u_0=0$. The results are plotted in Fig. \[f-diff\_amp\_vs\_time-original\] with the J0-R100-u0 results labeled by ${\ensuremath{\mathcal{N}}}^0_0$. We still observe an oscillation in the amplitude in ${\ensuremath{\mathcal{N}}}_0$, though it is drastically reduced compared to the ${\ensuremath{\mathcal{N}}}^{450}_0$ of model J0-R100-u450 shown in Fig. \[f-diff\_amp\_vs\_time\]. The relative errors in amplitude, plotted in the bottom panel of Fig. \[f-diff\_amp\_vs\_time-original\], are well below $1\%$ over the entire evolution, and the total dephasing is smaller than $\Delta \phi = 0.04 \rm{rad}$. We note that the differences are larger than the systematic error reported in [@Reisswig:2009us; @Reisswig:2009rx]. In that work, the error estimate refers to the difference between evolutions starting from the same initial data but different worldtube locations, at a fixed resolution (though the results converge to the same waveform as the resolution is increased). Our results indicate that at typical current resolutions, the choice of characteristic initial data has a larger influence on the simulation error than the worldtube location. The initial burst of junk as well as spurious incoming transients can alter the measured waveforms by an amount that is of the order of the discretization error of current numerical relativity codes (e.g. [@Hannam:2009hh_pbl] and references therein), over a period of several hundred $M$.
![Time domain differences between the ${\ensuremath{\mathcal{N}}}_0^{450}$ and ${\ensuremath{\mathcal{N}}}_\mathrm{lin}^{450}$ waveforms of models J0-R100-u450 and Jlin-R100-u450, computed from worldtube location $R_\Gamma=100M$ for which the characteristic runs have been initialized by $J=0$ and linearized solutions at a time $u_0=450M$, respectively. The top panel plots the wave amplitude, the middle shows the phase, and the lower plots the differences between the two solutions. The ${\ensuremath{\mathcal{N}}}_0^450$ data shows notable oscillations in amplitude at early times (inset), which decay exponentially with time. The waveforms have been aligned at the amplitude peak.[]{data-label="f-diff_amp_vs_time"}](diff_amp_vs_time)
![Time domain differences between the ${\ensuremath{\mathcal{N}}}_0^0$ (starting off at $u_0=t_0=0$) and ${\ensuremath{\mathcal{N}}}_\mathrm{lin}^{450}$ (starting off at $u_0=450$) waveforms of models J0-R100-u0 and Jlin-R100-u450, computed from worldtube location $R_\Gamma=100M$ for which the characteristic runs have been initialized by $J=0$ and linearized solutions respectively. The top panel plots the wave amplitude, the middle shows the phase, and the lower plots the differences between the two solutions. The ${\ensuremath{\mathcal{N}}}_0^0$ data shows slight oscillations in amplitude at early times (inset), which decay exponentially with time. The waveforms have been aligned at the amplitude peak.[]{data-label="f-diff_amp_vs_time-original"}](diff_amp_vs_time_std)
The numerical tests described so far were with the extraction radius $R_\Gamma=100M$. All these runs were repeated with the extraction radius re-set to $R_\Gamma=250M$. The results are qualitatively similar to the $R_\Gamma=100M$ case, and the details are not presented here. One interesting feature was the behaviour of $W_c$ at the worldtube, [*i.e., *]{}the analogy to Fig. \[f-W22scri\]: it took until about $900M$ until the decay in the difference between the linearized and actual data was saturated. This is surprising since one would expect that the radiation that passed $R_\Gamma=100M$ at about $u=450M$ would pass $R_\Gamma=250M$ at $u=600M$. One possible explanation is that the saturation is due to nonlinear effects, and since they are somewhat weaker at $R_\Gamma=250M$, saturation takes longer. Furthermore, the boundary data amplitudes are one order of magnitude smaller at $R_\Gamma=250M$ than those at $R_\Gamma=100M$, and thus potentially more sensitive to incoming modes. Further studies would be required to fully understand the nature of these effects.
Discussion and Conclusion {#s-conc}
=========================
Characteristic evolutions provide a means of determining radiated gravitational energy which is free from the ambiguities associated with local measures, namely non-linear effects in the near-zone, as well as ambiguities due to gauge and extrapolations to infinity. The principal remaining issue has been to specify appropriate initial data on the null cone, compatible with the data on the worldtube and the Cauchy 3+1 spacetime. The strong correlation between finite radius results and characteristic extraction, as well the invariance of the results on the worldtube location observed in [@Reisswig:2009us; @Reisswig:2009rx] suggests that even the simplest initial data [*ansatz*]{}, $J=0$, can provide results which are accurate enough for astrophysical estimates, provided the Cauchy 3+1 spacetime is essentially free of radiation at the initial characteristic time. However, in scenarios where strong outgoing radiation is present during the initial characteristic time, $J=0$ data effectively represents incoming radiation, and may alter significantly the evolution of the wave signal towards ${\ensuremath{\mathcal{J}^+}}$.
The gravitational wave solution developed in Sec. \[s-2=mS\] provides initial data $J=J_{\rm lin}$ which is compatible, to the linearized level, with outgoing radiation from a binary system. As such, it provides a more physically motivated starting point than the shear-free, $J=0$, alternative. Importantly, we find that the evolutions which take place from either $J=J_{\rm lin}$ initialized at a time $u_0=450M$ when outgoing radiation is present at the worldtube, and $J=0$ at the initial time $u_0=t_0=0$ when the Cauchy 3+1 spacetime is conformally flat are very similar (compare Fig. \[f-diff\_amp\_vs\_time-original\]), and indeed very similar to the purely 3+1 result which can be obtained by polynomial extrapolating finite radius measurements. That is, for simple choices of initial $J$, the physical conclusions are not altered dramatically, provided the Cauchy 3+1 spacetime does not contain strong amounts of radiation at the worldtube location during the initial characteristic time $u_0$.
On the other hand, we have demonstrated that the choice of characteristic initial data does result in a small but measurable difference, which decays at a slow exponential rate over a time period of several hundred $M$. We see this transient in the graphs of gravitational news exhibited in Fig. \[f-diff\_amp\_vs\_time\]. Since the linearized initial data contains only an outgoing mode, we conclude that the shear-free characteristic initial data contains [*incoming*]{} radiation. While this is expected, it is interesting that it takes so much time for the effect to decay away.
This study was designed to assess the long-term effect of characteristic initial data, but it also provides information about the incoming radiation in 3+1 initial data. The point is that the characteristic data on the worldtube is determined entirely by the 3+1 data, and the extent to which this data does not fit the linearized solution is a measure of its incoming radiation content. By construction, the quantities $\beta, U$ and $J$ in the linearized solution must fit the data, with the difference in $W_c$ being an indication of incoming radiation in the 3+1 evolution. Since $W_c$ is not a gauge invariant quantity, it is not possible to make a quantitative statement about the magnitude of the incoming radiation, but Fig. \[f-W22scri\] indicates that it takes until at least $\pm 400M$ until it is possible to neglect the effect of incoming radiation in the region $r<100M$.
The effect of incoming radiation in both 3+1 and characteristic initial data decays exponentially, but even so it has a more long-term impact than the outgoing junk radiation which passes the $r=100M$ worldtube radius by approximately $u=150M$. The effect of incoming radiation has not received much attention in previous literature dealing with the problem of initial data construction. It has potential significance for the construction of high-accuracy gravitational-wave templates. Further investigations may also explain properties of the incoming radiation content, giving further insight to the peeling property of more complex spacetimes other than Kerr. In particular, it will be important to determine, using a gauge invariant measure, the long-term effect of incoming radiation in 3+1 initial data sets.
The results suggest some promising avenues for future study. Current methods (“characteristic extraction”) transport data in one direction (from the Cauchy to the characteristic code). Great efficiencies would be possible if the coupling were also carried out in the other direction, so that the characteristic evolution would provide outer boundary data for the Cauchy evolution. The linearized wave solution provides a simple recipe for isolating ingoing vs. outgoing modes on the characteristic grid, and may be useful in designing a method for stably transporting data in both directions across the world tube interface.
The authors would like to thank Sascha Husa and C. D. Ott for helpful discussions. The authors have enjoyed the hospitality of Rhodes University, the Max-Planck-Institut für Gravitationsphysik, Caltech and Universitat de les Illes Balears during the course of this work. This work was supported by the National Research Foundation, South Africa, Bundesministerium für Bildung und Forschung, and the National Science Foundation under grant numbers AST-0855535 and OCI-0905046. CR’s travel was supported by C. D. Ott. DP has been supported by grants CSD2007-00042 and FPA-2007-60220 of the Spanish Ministry of Science. Computations were performed on the NSF Teragrid (allocation TG-MCA02N014 and TG-PHY100033), the LONI network (`www.loni.org`) under allocation `loni_numrel05`, the Barcelona Supercomputing Center, and on the Caltech compute cluster “Zwicky” (NSF MRI award No. PHY-0960291).
References {#references .unnumbered}
==========
[^1]: Maple script for this purpose (`nu0_regular_0.map` with output in `nu0_regular_0.out`) are provided in the supplementary data.
[^2]: The calculation is provided in the Maple script `regular_0.map` with the output in `regular_0.out` in the supplementary data.
[^3]: The Matlab scripts `ft_wt_driver.m, FT_WT.m` and `nu0.m` are provided in the supplementary data.
[^4]: Note that $W_c$ is not necessarily identical, since there are may be differences due to nonlinearities and incoming radiation.
|
---
abstract: 'In this paper, we investigate detectability and identifiability of attacks on linear dynamical systems that are subjected to external disturbances. We generalize a concept for a security index, which was previously introduced for static systems. The generalized index exactly quantifies the resources necessary for targeted attacks to be undetectable and unidentifiable in the presence of disturbances. This information is useful for both risk assessment and for the design of anomaly detectors. Finally, we show how techniques from the fault detection literature can be used to decouple disturbances and to identify attacks, under certain sparsity constraints.'
author:
- 'Henrik Sandberg and André M.H. Teixeira [^1] [^2] [^3]'
title: '**From Control System Security Indices to Attack Identifiability** '
---
Introduction {#sec:intro}
============
As modern control systems increasingly rely on information and communication technology (ICT) infrastructures to exchange real-time measurements and actuator signals, their exposure to malicious cyber threats also grows: each measurement and actuator signal may be compromised and altered by a skillful cyber adversary. Therefore, cyber security and resilience with respect to attacks are important properties of modern control systems that are tightly coupled to ICT infrastructures.
Some of the main challenges in designing cyber-secure control systems are related to: analyzing the risk of cyber attacks; devising protection mechanisms to prevent and remove high-risk threats; and also to timely detect and mitigate on-going attacks. While the first two challenges relate to conventional ICT cyber security approaches ([[*i.e.,* ]{}]{}risk management [@teixeira+15]), the third approach is closely related to the well-known control field of fault diagnosis. Although both relate to detecting anomalies, there exist subtle differences between classical fault diagnosis and attack detection in cyber security. Classical control-theoretic approaches to anomaly detection ([[*e.g.,* ]{}]{}fault detection, isolation, and identification) typically deal with independent disturbances and faults; thus they typically do not consider possibly colluding malicious cyber attacks, which may even attempt to hide the attacks by mimicking physical disturbances and faults. In fact, this paper addresses the latter scenario, discussing detectability conditions of sparse attacks that may be masked by plausible disturbances, and connecting the results to fundamental limitations well-known in the controls literature, in terms of fault detection and identification [@ding08] and input reconstruction [@patton+99].
The topic of cyber-secure control systems has been receiving increasing attention recently. An overview of existing cyber threats and vulnerabilities in networked control systems is presented in [@cardenas+08; @Teixeira_Automatica2015]. Rational adversary models are highlighted as one of the key items in security for control systems, thus making adversaries endowed with intelligence and intent, as opposed to faults. Therefore, these adversaries may exploit existing vulnerabilities and limitations in the traditional anomaly detection mechanisms and remain undetected, or indistinguishable from disturbances and process noise. In fact, [@pasqualetti+13] uses such fundamental limitations to characterize a set of undetectable attack policies for networked systems modeled by differential-algebraic equations. Related undetectable attack policies were also considered in [@Teixeira_Automatica2015; @smith+15]. A common thread within these approaches is that undetectable attacks are constrained to be entirely decoupled from the anomaly detector’s output.
Detectability conditions of undetectable false-data injection attacks to control systems are closely examined in [@Teixeira_Allerton2012], where it is shown that mismatches between the system’s and the attack policy’s initial conditions may lead to detectable attacks. Additionally, modifications to the system dynamics, input, and output matrices that reveal stealthy data attacks were also characterized.
Other work has analyzed undetectable attacks with respect to the amount of effort they require, [[*i.e.,* ]{}]{}the number of attack signals that must be injected by the adversary to remain undetected. As discussed in [@teixeira+15], such analysis provides insight into the likelihood of such attacks occurring, which is a core component of determining the risk ([[*i.e.,* ]{}]{}impact and likelihood) of such threat scenarios.
For static systems, [@sandberg+10] first proposed a security index for measurement attacks, which corresponds to the minimum number of measurements that need to be corrupted as to ensure undetectability. The computation of the security index involves solving an NP-hard problem, in general, which has later been investigated by [@sou+12; @hendrickx+14; @kosut14; @yamaguchi+15]. Under certain structures of the problem, this work proposed efficient algorithms to compute the security index in polynomial time.
Related problems have been investigated for dynamical systems. The work in [@fawzi+14] characterizes the number of corrupted sensor channels that cannot be detected during a finite time-interval. For sensor attacks that can be detected, a resilient state estimation scheme inspired by compressed sensing is proposed. The work in [@sundaram+11] explored the notion of strong observability to characterize the conditions for which the initial state can be recovered under the presence of sparse unknown input signals. For sensor attack scenarios, [@chen+15] determines the smallest number of sensors needed for undetectable attacks. The notion of security index for dynamical system under sensor and actuator attacks was also extended to dynamical systems at steady-state and for finite-time intervals in [@teixeira+13].
This work investigates the notion of security index for dynamical systems under both attacks and disturbances. In particular, we consider the case where attacks are said to be undetectable if they can be masked (explained) by a disturbance signal. The formulation of the security index is related to well-known limitations in the fault detection literature, and the complexity of computing these indices for special cases is discussed and related to the literature. For detectable attacks, the concept of identifiable attacks is defined, as well as a weaker notion of identifiability where only certain entries of the attack signal can be uniquely determined. Connections between these definitions and the security index are investigated, based on which attacks with sufficiently high sparsity are shown to be identifiable. Finally, for identifiable attacks, an attack reconstruction procedure is proposed.
The outline of this paper is as follows. The dynamical system under the influence of disturbances and attacks is described in Section \[sec:prel\], where undetectable attacks, potentially masked by disturbances, are defined. Section \[sec:index\] formulates the security index for dynamical systems under the influence of both disturbances and attacks, and discusses important special cases and their connection to the literature. The role of security indices in (possibly partial) attack identification under disturbances is examined in Section \[sec:identification\], whereas concluding remarks are given in Section \[sec:conclusion\].
### Notation {#notation .unnumbered}
For a set $I$, $|I|$ denotes its cardinality. For a vector $a\in\mathbb{C}^m$, we denote its $i$-th element by $a_i$. By $a^i \in \mathbb{C}^m$, we mean a vector whose $i$-th element is non-zero, [[*i.e.,* ]{}]{}$a_i \neq 0$, and the other elements are arbitrary. The support of $a\in\mathbb{C}^m$, $\text{supp}(a)$, is the set of indices $i$ where $a_i \neq 0$, and $\|a\|_0:=|\text{supp}(a)| $ is the number of non-zero elements in $a$. Similar notations are used for discrete-time signals $a$, where $a(k)\in\mathbb{C}^m$, $k=0,1,2,\ldots$ A discrete-time linear system $G\in\mathcal{R}_p^{p\times m}(z)$ has a rational proper transfer matrix $G(z)$ of dimension $p \times m$. We also define ${{\mathrm{normalrank}\,}}[G(z)] := \max_z {{\mathrm{rank}\,}}[G(z)]$.
Preliminaries {#sec:prel}
=============
Let us consider the discrete-time system $y=G_d d + G_a a$, $G_d\in\mathcal{R}_p^{p\times o}(z)$ and $G_a\in\mathcal{R}_p^{p\times m}(z)$, with a realization $$\begin{aligned}
x(k+1) & = Ax(k) + B_d d(k) + B_a a(k) \\
y(k) & = Cx(k) + D_d d(k) + D_a a(k),
\end{aligned}
\label{eq:G}$$ for times $k =0,1,2\ldots$ Here $x(k)\in {{\mathbb{R}}}^n$ is the state vector, $d(k)\in{{\mathbb{R}}}^o$ are unknown disturbance (or fault) signals, $a(k)\in{{\mathbb{R}}}^{m}$ are potential attack signals, and $y(k)\in{{\mathbb{R}}}^{p}$ are the measurements available to the operator of the system. Additionally, we assume to have distinct measurement, attack, and disturbances signals, in the sense that $$\begin{aligned}
{{\mathrm{rank}\,}}\begin{bmatrix}
B_d \\
D_d
\end{bmatrix} &= o,\quad
{{\mathrm{rank}\,}}\begin{bmatrix}
B_a \\
D_a
\end{bmatrix} = m, \quad
{{\mathrm{rank}\,}}\left[C\right] = p.
\end{aligned}$$ It turns out that the value of the initial state, $x(0)$, is important in the following, but initially we will let it be a free variable.
The system model is similar to those studied in the fault detection and diagnosis literature, see, [[*e.g.,* ]{}]{}[@patton+99; @ding08]. The signals $d$ and $a$ represent different types of anomalies that can occur in the system, although of different nature. We next want to determine when we can detect and distinguish between these anomalies. We could think of $d$ as natural disturbances, or faults, that are to be expected, and that have no malicious intent. They could represent measurement and process noise, for example. One important aspect is that a malicious attacker could use such disturbances to hide his or her attack $a$ from being seen in the output $y$. We will typically let $d$ be a free variable, where the only available knowledge about the disturbance signals amounts to their signature matrices, $B_d$ and $D_d$. Thus, to ensure robustness with respect to disturbances, an anomaly detection algorithm wishing to detect potential attack signals must be designed so that it is decoupled from $B_d$ and $D_d$. Under this disturbance model, we check whether a disturbance exists that will “mask” the attack. If this is the case, the operator is not able to distinguish between attacks and disturbances, and cannot conclude whether an attack is present, or not.
Naturally, several other disturbance models exist, such as assuming known upper bounds on the disturbance signal’s energy or instantaneous peak, or constraining the disturbance to belong to a given class of signals, [[*e.g.,* ]{}]{}constant or sinusoidal signals. In particular, the results in this paper can be straightforwardly extended to disturbances modeled as the output of an autonomous discrete-time system $$\begin{aligned}
x_d(k+1) & = A_d x_d(k) \\
d(k) & = C_d x_d(k),
\end{aligned}
\label{eq:Disturbance}$$ which is parametrized by a free initial condition $x_d(0)$.
The attack can potentially occur in $m$ different locations in the system ($a(k)\in\mathbb{R}^m$), and we will be concerned about the possibility for the operator with access to the above model and the signal $y$ to detect an attack signal $a\neq 0$. We make the following definitions to formalize these ideas.
An attack signal $a$ is *persistent* when $a(k) \not \rightarrow 0$ as $k\rightarrow \infty$.
In this paper, we are mainly concerned with persistent attacks, since they have non-vanishing impact.
A (persistent) attack signal $a$ is
- *undetectable* if there exists a simultaneous (masking) disturbance signal $d$ and initial state $x(0)$ such that $y(k)=0$, $k\geq 0$;
- *asymptotically undetectable* if there exists a simultaneous (masking) disturbance signal $d$ and initial state $x(0)$ such that $y(k)\rightarrow 0$, $k\rightarrow \infty$.
Note that the definition of undetectable attacks is the same as in [@pasqualetti+13], if we assume there are no disturbances in the system . The reason for calling the disturbance “masking” comes from linearity of the system: If $0=G_a a + G_d d$, then clearly $y=G_a a = -G_d d$, and it is impossible to in the output distinguish between the undetectable attack and the masking disturbance, if they occur by themselves without the other.
We will next be interested in quantifying the minimal resources needed by the attacker to achieve undetectability, when he or she want to target a specific attack element $a_i$, $i\in\{1,\ldots,m\}$. Hence, we will search for sparse signals $a^i$ satisfying the above conditions.
The Dynamical Security Index {#sec:index}
============================
For an attack signal $a$ to be undetectable, we need to ensure there exists a masking disturbance $d$ and an initial state $x(0)$ resulting in zero output. Existence of such a signal can easily be checked by considering the matrix pencil (the Rosenbrock system matrix) $$P(z) = \begin{bmatrix} A-zI & B_d & B_a \\ C & D_d & D_a \end{bmatrix},$$ see [@ZDG96]. An attack signal $a(k)=z_0^k a_0$, $a_0\in\mathbb{C}^m$, $z_0\in\mathbb{C}$, is undetectable iff there exists $x_0\in\mathbb{C}^n$ and $d_0\in\mathbb{C}^o$ such that $$P(z_0)\begin{bmatrix} x_0 \\ d_0 \\ a_0\end{bmatrix} = 0, \label{eq:z0}$$ [[*i.e.,* ]{}]{}$P(z_0)$ should not have full column rank. The undetectable attack is also persistent iff $|z_0|\geq 1$.
Note that, if the initial state $x(0)\neq x_0$, the attack signal $a(k)=z_0^k a_0$ may actually be detectable. Following the analysis in [@Teixeira_Allerton2012], if $A$ is Schur ($\rho(A)<1$), the attack signal is only *asymptotically* undetectable, since there will be a vanishing transient visible in the output. This transient can be made arbitrarily small by the attacker choosing $a_0$ small. Hence, the difference between asymptotically undetectable and undetectable attacks may not be very large in practice.
If the attacker would like to target the element $i$, [[*i.e.,* ]{}]{}$a_i\neq 0$, and remain undetected, he or she needs to find a vector $a_0^i\in\mathbb{C}^m$ satisfying . In general, this may require the attacker to target several elements $a_j$, $j\neq i$. To measure the minimal number of elements required to achieve this, we introduce the following *security index* $\alpha_i$, which generalizes a concept first introduced for non-dynamical systems in [@sandberg+10]: $$\begin{aligned}
\alpha_i := & \min_{|z_0|\geq 1,x_0,d_0,a_0^i} && \|a_0^i\|_0 \\
& \mathrm{subject \, to} & & P(z_0)\begin{bmatrix} x_0 \\ d_0 \\ a_0^i\end{bmatrix} = 0.
\end{aligned}
\label{eq:alpha}$$ Note that for all $i$ it holds $\alpha_i \geq 1$, and if there is no feasible solution, we define $\alpha_i = +\infty.$ Note also that this is a combinatorial optimization problem, because of the objective function $\|a_0^i\|_0$, and in general is hard to solve [@hendrickx+14]. However, in several cases of interest, it has a simple solution, as discussed below.
We can think of the signals $a^i(k)= z_0^k a_0^i$ resulting from as the sparsest possible persistent undetectable attacks against an element $i$. These signals should be of interest to both the operator and the attacker, in the sense that they show how the attacker can modify the solutions to the system equations , without modifying the measurable output $y$. Also, if the number $\alpha_i$ is large, it indicates that it will require significant coordinated resources by the attacker to accomplish undetectable attacks against $a_i$. An operator can thus use the index in performing a quantitative risk assessment, as illustrated in, [[*e.g.,* ]{}]{}[@teixeira+15]. The index $\alpha_i$ also has implications for the possibility of the operator to reconstruct (“identify”) a detectable attack $a^i$, as will be further explored in Section \[sec:identification\].
There are some concepts in the literature that are similar to $\alpha_i$ above. In power system observability analysis, a related concept is that of critical $k$-tuples, see, [[*e.g.,* ]{}]{}[@sou+12]. For sensor attack scenarios, [@chen+15] determines the smallest number of sensors needed for undetectable attacks. There are also close connections to the spark of a matrix, used in compressed sensing, see, [[*e.g.,* ]{}]{}[@donoho+03]. Also, in [@teixeira+13], an optimization problem related to was studied. Some further connections are made in the special cases considered in the following subsections.
Critical Attack Signals ($\alpha_i = 1$)
----------------------------------------
A particularly serious situation is when $\alpha_i=1$, since the attacker then can target element $i$ undetected without the need to access any other resources. Let us denote $$\begin{aligned}
P_i(z) & = \begin{bmatrix} A-zI & B_d & B_{a,i} \\ C & D_d & D_{a,i} \end{bmatrix} \in\mathbb{C}^{(n+p)\times (n+o+1)}, \\
P_d(z) & = \begin{bmatrix} A-zI & B_d \\ C & D_d \end{bmatrix} \in\mathbb{C}^{(n+p)\times (n+o)},
\end{aligned}$$ where $B_{a,i},D_{a,i}$ are the $i$-th columns of $B_a,D_a$. If there is a $z_0\in\mathbb{C}$, $|z_0|\geq 1$, such that $${{\mathrm{rank}\,}}[P_d(z_0)] = {{\mathrm{rank}\,}}[P_i(z_0)],$$ then $\alpha_i= 1$. An even more serious situation occurs when $${{\mathrm{normalrank}\,}}[P_d(z)] = {{\mathrm{normalrank}\,}}[P_i(z)].
\label{eq:seriousrank}$$ If this easily checked condition is fulfilled, it is possible to find an undetectable attack signal $a^i(k)=z_0^k a_0^i $ of cardinality one, using any complex frequency $z_0$.
Note that holds when there are many disturbances in relation to the number of available measurements, [[*i.e.,* ]{}]{}$o\geq p$.
Transmission Zeros
------------------
If the Rosenbrock system matrix $P(z)$ has full column normal rank and the realization is minimal, the only solutions to that exist correspond to the system’s finite number of transmission zeros, see, [[*e.g.,* ]{}]{}[@ZDG96]. Hence, to find $\alpha_i$ we only need to inspect the corresponding system zero directions. When the zero directions are all one-dimensional, the computation of $\alpha_i$ becomes especially simple. Generically, one would expect the zero directions to be one-dimension, but there are several interesting situations where this is not the case, as we shall see below (although these will be invariant zeros, and not transmission zeros).
Sensor Attacks {#sec:sensattack}
--------------
The situation where the system is subjected to sensor attacks have received particular interest in the literature, see, [[*e.g.,* ]{}]{}[@fawzi+14; @chen+15; @lee+15]. In this case we have $B_d=B_a=0$, and in we only need to consider $z_0\in\{\lambda_1(A),\ldots,\lambda_n(A)\}$, [[*i.e.,* ]{}]{}the eigenvalues of $A$, where $x_0$ are eigenvectors of $A$. If the eigenvalues are simple, the eigenspace corresponding to each eigenvalue is one-dimensional, and again the computation of $\alpha_i$ is simplified.
As a further special case, suppose all sensors are potentially attackable and there are no disturbances, and so $D_a=I_p$ and $D_d=0$. Also suppose that the operator has high redundancy in the system in the sense that the realization is observable using any one of the outputs $y_i$, $i\in\{1,\ldots,p\}$, by itself. Considering the PBH test [@ZDG96], this means that any one of the eigenmodes $z_0^kx_0$ is visible in all the sensors, and all elements in the vector $C x_0$ are non-zero. Thus an undetectable attack ($Cx_0+a_0^i=0)$ must involve all the signals in $a$, and for all $i$ the security index must be $\alpha_i = m = p$ (or $\alpha_i = +\infty$ if $A$ is Schur). Hence, one way to make undetectable attacks hard is to install many redundant sensors, each of which with the individual power to observe the entire system state with little noise, which is in agreement with [@fawzi+14].
Sensor Attacks for Static Systems
---------------------------------
If we assume $A=I$, $B_d=B_a=D_d=0$ (only sensors attacked), we have essentially recovered the original security index $\alpha_i$, as defined in [@sandberg+10]. The motivation for the index there was to quantify the vulnerability of power system state estimators to false data injection attacks. Note that because $A=I$ and $B_d=B_a=0$, this problem only concerns systems in steady-state. Perhaps one would think that this makes the problem easier, but in fact the problem can be significantly harder in practice. This is because the dimension of the eigenspace corresponding to the sole eigenvalue is of dimension $n$, and not one-dimensional as is frequently the case in the previous examples. Intuitively, one can understand this since the attacker here has no constraints in time to fulfill and thus has many more options for being undetectable. This fact together with the potentially high dimension $n$ in a power system has spurred several investigations on the efficient computation of $\alpha_i$. The problem in general is NP-hard [@hendrickx+14], but in the examples deriving from power systems the matrix $C$ has a useful structure that can be exploited. In particular, [@hendrickx+14; @kosut14; @yamaguchi+15] show how max-flow min-cut algorithms can be used to solve the problem in polynomial time. Under slightly different assumptions on the structure of $C$, [@sou+13] shows how $\ell_1$-relaxation can also exactly solve the problem in polynomial time.
Attack Identification and Decoupling {#sec:identification}
====================================
In this section, we turn to the related problem of attack identification, which concerns the possibilities to reconstruct elements of an attack signal from the measured output.
Attack Identification
---------------------
To formalize the attack identification problem, the following definitions are made.
A (persistent) attack signal $a$ is
- *identifiable* if for all attack signals $\tilde a\neq a$, and all corresponding disturbances $d$ and $\tilde d$ and initial states $x(0)$ and $\tilde x(0)$, we have $\tilde y \neq y$;
- *asymptotically* *identifiable* if for all attack signals $\tilde a(k)\not\rightarrow a(k)$, and all corresponding disturbances $d$ and $\tilde d$ and initial states $x(0)$ and $\tilde x(0)$, we have $\tilde y(k)\not \rightarrow y(k)$, as $k\rightarrow \infty$.
Identifiable attack signals $a$ excite the output $y$ in a unique way that cannot be achieved by any other attack $\tilde a$. This is equivalent to the system possessing a certain left inverse, as will be explored in Section \[sec:decouple\]. Note that identifiability of $a$ is a much stronger requirement than detectability of $a$ (which means that the attack $a$ is such that $y\neq 0$ for all disturbances $d$ and initial states $x(0)$). Since identifiability is such a strong requirement, we will also be interested in the following weaker notion.
A (persistent) attack signal $a$ is
- $i$-*identifiable* if for all attack signals $\tilde a$ with $\tilde a_i\neq a_i$, and all corresponding disturbances $d$ and $\tilde d$ and initial states $x(0)$ and $\tilde x(0)$, we have $\tilde y \neq y$;
- *asymptotically* $i$-*identifiable* if for all attack signals $\tilde a$ with $\tilde a_i(k)\not\rightarrow a_i(k)$, and all corresponding disturbances $d$ and $\tilde d$ and initial states $x(0)$ and $\tilde x(0)$, we have $\tilde y(k)\not \rightarrow y(k)$, as $k\rightarrow \infty$.
This notion is weaker than identifiability since an attack $a$ can be $i$-identifiable even if there exists another attack $\tilde a \neq a$, with $a_i=\tilde a_i$, such that $y=\tilde y$. Hence, $i$-identifiability concerns only the sensitivity of the output $y$ with respect to the $i$-th element in $a$. Identifiability is therefore the same as $i$-identifiability for all $i$. Interestingly, there is a tight connection between detectability, identifiability, and the previously introduced security index.
Suppose that the initial state $x(0)$ is unknown to the operator (and can take any value), and that the attacker can manipulate at most $q$ attack elements simultaneously ($\|a\|_0\leq q$).
- There exists persistent undetectable attacks $a^i$ iff $q\geq \alpha_i$;
- All persistent attacks are $i$-identifiable iff $q < \alpha_i/2$;
- All persistent attacks are identifiable iff $q < \min_i \alpha_i/2$.
\[thm:alpha\]
(i): Follows directly from the definition of $\alpha_i$, where we pick $x(0)=x_0$. (ii): Consider first two attacks $a$ and $\tilde a$, both of cardinality $q<\alpha_i/2$, such that $a_i\neq \tilde a_i$. Let $y=G_d d +G_a a$ and $\tilde y = G_d \tilde d + G_a \tilde a$ and suppose that $y = \tilde y$, in contradiction to the theorem. This implies that $0 = G_d (d-\tilde d)+ G_a (a-\tilde a)$. Since $a_i\neq \tilde a_i$, the attack signal $a-\tilde a$ would constitute an undetectable attack against element $i$. Furthermore, the cardinality of this signal is strictly smaller than $\alpha_i/2 + \alpha_i/2$, which is a contradiction to the optimality of security index $\alpha_i$. Hence, we must have $\tilde y \neq y$, and the attack $a$ is $i$-identifiable. Conversely, assume that $q\geq \alpha_i/2$ and let us construct two attacks $a$ and $\tilde a$ that are not $i$-identifiable. Suppose first that $\alpha_i$ is even and that $q=\alpha_i/2$. There exists an undetectable attack $a^\star$, targeting element $i$, with support in an index set $I$, $|I|=\alpha_i$. Thus $0 = G_d d^\star + G_a a^\star$. Let us split $I$ into two disjoint sets, $J$ and $K$ of equal size, $I=J\cup K$, $|J|=|K|=\alpha_i/2$. In a corresponding manner we can make the split $a^\star = a - \tilde a$, where $a$ and $\tilde a$ have support in $J$ and $K$, respectively. It is now clear that $0 \neq y = G_d d^\star + G_a a = G_a \tilde a = \tilde y$, and since $a_i\neq \tilde a_i$ this is an example of a non $i$-identifiable attack $a$. A similar argument can be applied when $\alpha_i$ is odd, concluding the proof. (iii): Follows since identifiability is the same as $i$-identifiability for all $i$.
In some cases it may be more realistic to assume that the operator actually knows the initial state of the system . We can then state the following corollary to the above theorem, which applies in the asymptotic limit when $k\rightarrow \infty$.
Suppose that $A$ is Schur, that the initial state $x(0)$ is known to the operator, and that the attacker can manipulate at most $q$ attack elements simultaneously ($\|a\|_0\leq q$).
- There exists persistent asymptotically undetectable attacks $a^i$ iff $q\geq \alpha_i$;
- All persistent attacks are asymptotically $i$-identifiable iff $q < \alpha_i/2$;
- All persistent attacks are asymptotically identifiable iff $q < \min_i \alpha_i/2$.
\[cor:q\]
The only difference to the proof of Theorem \[thm:alpha\] is that we need to add a transient term $y_\text{trans}(k) = CA^{k}(x(0)-x_0)$ to all outputs, see [@Teixeira_Allerton2012]. Here $x_0$ is an initial state rendering the relevant attack undetectable. Since $\rho(A)<1$ by assumption, this term decays to zero exponentially and the asymptotic results follow.
We note that other papers have previously pointed out the connection between detectability and identifiability of attacks, see, [[*e.g.,* ]{}]{}[@pasqualetti+13]. The main contribution here is to introduce $i$-identifiability and show the relation to the security index $\alpha_i$. As an example, assume that $\alpha_1 = 1$, $\alpha_2 = 3$, and that $q=1$. Then there will exist attacks against $a_1$ that are not visible in $y$, but all attacks against $a_2$ will not only be visible but also identifiable through $y$. How to possibly conduct the identification is discussed next.
Decoupling the Attacks from the Disturbances {#sec:decouple}
--------------------------------------------
To identify attacks $a$ in the output $y$, there are several useful results in the fault detection literature, see, [[*e.g.,* ]{}]{}[@patton+99; @ding08]. In particular, we will use a result on the existence of decoupling filters, which isolate the influence of the attack from that of the disturbance. A key result towards identification is the existence of certain left inverses.
\[def:leftinv\] Consider the linear system $y=Gu$ with $m$ inputs, $p$ outputs, and with realization $$\begin{aligned}
x(k+1) & = Ax(k)+Bu(k) \\
y(k) & = Cx(k) + Du(k).
\end{aligned}$$ Then $G$ has a *left inverse* when $y(k)=0$, $k\geq 0$, implies that $u(k)=0$, $k \geq 0$, provided $x(0)=0$.
The following condition for existence of a left inverse is well known, see, [[*e.g.,* ]{}]{}[@moylan77; @hou+98].
\[lem:leftinv\] A linear system $G\in\mathcal{R}_p^{p\times m}(z)$ has a *left inverse* iff ${{\mathrm{normalrank}\,}}G(z)=m$.
From fault detection [@ding08], it is known that if $G_d,G_a\in \mathcal{R}_{p}(z)$ and $$\begin{aligned}
{{\mathrm{normalrank}\,}}[G_d(z)] & = m', \\ {{\mathrm{normalrank}\,}}[G_d(z)\,\, G_a(z)] & = m'+m'',
\end{aligned}
\label{eq:rankcond}$$ then there exists a post-filter $R\in \mathcal{R}_{p}^{p\times p}(z)$ (of full normal rank) such that we can decouple the effects of the attacks and the disturbances in the following way: $$\begin{bmatrix} r \\ y' \end{bmatrix} =R(G_d d+ G_a a) = \begin{bmatrix} 0 & \Delta \\ G_d' & G_a' \end{bmatrix} \begin{bmatrix} d \\ a \end{bmatrix},
\label{eq:decouple}$$ where ${{\mathrm{normalrank}\,}}[G_d'(z)]= {{\mathrm{normalrank}\,}}[G_d'(z) \, G'_a(z)] = m'$ and ${{\mathrm{normalrank}\,}}[\Delta(z)]=m''$. Note that if all attacks are undetectable in the sense of , then $m''=0$, and $\Delta$ will be the empty matrix. On the other hand, if for some $i$, $\alpha_i>1$, then $m''>0$ and there is a non-trivial system $\Delta$. The residual signal $r$ is only influenced by the attack $a$, and we can use it to detect and potentially identify $a$. Notice that for all attacks $a$ there exists a disturbance $d$ such that $0=y'=G_d'd+G_a'a$, so that $r=\Delta a$ is the only reliable source of information in regards to $a$. We have the following proposition on the relation between the measured output $y$ and the filtered version $r$.
Let the initial state of the decoupling filter $R$ be chosen to $x_R(0)=0$. Suppose the initial state $x(0)$ is unknown to the operator (and can take any value), and that the attacker can manipulate at most $q$ attack elements simultaneously ($\|a\|_0\leq q$).
- There exists persistent undetectable attacks $a^i$ in the signal $r$ iff $q\geq \alpha_i$;
- All persistent attacks are $i$-identifiable in the signal $r$ iff $q < \alpha_i/2$;
- All persistent attacks are identifiable in the signal $r$ iff $q < \min_i \alpha_i/2$.
\[prop:R\]
Recalling that $R$ has full normal rank, we can use Lemma \[lem:leftinv\] and Definition \[def:leftinv\] to conclude that $Ry=0$ is equivalent to $y=0$. Since there is always a $d$ such that $y'=0$ in , the undetectability and identifiability properties of $a$ in $y=G_d d+G_a a$ must carry over to the relation $r=\Delta a$, to which we can apply Theorem \[thm:alpha\].
If we suppose that $q<\min_i \alpha_i/2$, all persistent attacks are identifiable. A procedure to identify $a$ could include the following steps (we leave the details for future work): First apply the post-filter $R$ to $y$ to obtain the relation $r = \Delta a$. The initial state $x(0)$ is unknown, and could cause a non-zero transient in $r$ even in the absence of an attack $a$. However, the dynamics of the transients are known, and can be filtered out from $r$ to obtain a new transient-free residual $r'$. The signal $r'$ is identically zero if $r$ can be completely explained by a transient $y_\text{trans}(k) = CA^{k}x(0)$. Undetectable attacks could also be also “hiding” in the transient, and by forming $r'$ the visible effects of such possible attacks also disappear. However, since we know that $q<\min_i \alpha_i$, there are no such persistent attacks affecting $a$, and so to identify $a$ we can equally well use the relation $r' = \Delta a$, where the initial state of $\Delta$ is zero, $x_\Delta(0)=0$. To find $a$, we can form the systems $\Delta_I:=[\Delta_i]_{i\in I}$ out of the columns $\Delta_i$ of $\Delta$, for all subsets $|I|\leq q$, $I\subseteq \{1,\ldots,m\}$. Since all attacks are identifiable, these $\Delta_I$ are left invertible, and give each rise to an attack estimation $\hat a_I$. From identifiability of $a$ it follows that any estimate $\hat a_I$ satisfying $r'=\Delta \hat a_I$ is actually equal to the real persistent attack $a$, which concludes the procedure.
Note that the real bottleneck here is the number of systems $\Delta_I$ that need to be formed and inverted. The problem is in fact essentially the same as in compressed sensing, see, [[*e.g.,* ]{}]{}[@donoho+03]. Finally, we remark that the procedure can be modified to handle attacks that are only $i$-identifiable, but the estimates $\hat a_I$ will then only necessarily correctly identify element $a_i$.
Conclusion {#sec:conclusion}
==========
In this paper, we have studied detectability and identifiability of attacks on dynamical systems that are also subjected to disturbances. For this purpose, we generalized the concept of security index, which was previously introduced for static systems in [@sandberg+10]. In particular, the index exactly quantifies the resources necessary for targeted attacks to be undetectable and unidentifiable in the presence of disturbances. Such information is relevant for both risk assessment and for the design of anomaly detectors. We also discussed how these concepts relate to recent other work on attack detection and identification. Finally, we showed how techniques from the fault detection literature can be exploited to identify attacks under certain sparsity constraints.
[^1]: This work was supported in part by the Swedish Research Council (grant 2013-5523), the Swedish Civil Contingencies Agency through the CERCES project, and the EU 7th Framework Programme (FP7/2007-2013, grant agreement n$^\circ$ 608224).
[^2]: H. Sandberg is with the Department of Automatic Control, School of Electrical Engineering, KTH Royal Institute of Technology, Stockholm, Sweden. Email: [hsan@kth.se]{}
[^3]: A.M.H. Teixeira is with the Faculty of Technology, Policy and Management, Delft University of Technology, Delft, the Netherlands. Email: [andre.teixeira@tudelft.nl]{}
|
---
abstract: 'We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump Lèvy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior $C^{1,\alpha}$ regularity for general fully nonlinear integro-differential equations. Our estimates remain uniform as the degree of the equation approaches two, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations.'
author:
- Luis Caffarelli and Luis Silvestre
bibliography:
- 'nonl.bib'
title: 'Regularity theory for fully nonlinear integro-differential equations'
---
Introduction
============
Integro-differential equations appear naturally when studying discontinuous stochastic processes. The generator of an $n$-dimensional Lèvy process is given by an operator with the general form $$\label{e:fulloperator}
Lu(x) = \sum_{ij} a_{ij} \partial_{ij} u + \sum_i b_i \partial_i u + \int_{{\mathbb R}^n} (u(x+y)- u(x) - {\nabla}u(x) \cdot y) \ \chi_{B_1}(y) {\; \mathrm{d}}\mu(y).$$ The first term corresponds to the diffusion, the second to the drift, and the third to the jump part. In this paper we focus on the equations that we obtain when we consider purely jump processes; processes without diffusion or drift part. The operators have the general form $$\label{e:integral}
Lu(x) = \int_{{\mathbb R}^n} (u(x+y) - u(x) - {\nabla}u(x) \cdot y \ \chi_{B_1}(y)) {\; \mathrm{d}}\mu(y).$$ where $\mu$ is a measure such that $\int_{{\mathbb R}^n} \frac{|y|^2}{1+|y|^2} {\; \mathrm{d}}\mu(y) < +\infty$.
The value of $L u(x)$ is well defined as long as $u$ is bounded in ${\mathbb R}^n$ and $C^{1,1}$ at $x$. These concepts will be made more precise later.
The operator $L$ described above is a linear integro-differential operator. In this paper we want to obtain results for nonlinear equations. We obtain this kind of equations in stochastic control problems [@So]. If in a stochastic game a player is allowed to choose from different strategies at every step in order to maximize the expected value of some function at the first exit point of a domain, a convex nonlinear equation emerges $$\label{e:mL}
Iu(x) = \sup_\alpha L_\alpha u(x)$$
In a competitive game with two or more players, more complicated equations appear. We can obtain equations of the type $$\label{e:mmL}
Iu(x) = \inf_\beta \sup_\alpha L_{\alpha \beta} u(x)$$
The difference between and is convexity. Alternatively, also an operator like $Iu(x) = \sup_\alpha \inf_\beta L_{\alpha \beta} u(x)$ can be considered. A characteristic property of these operators is that $$\label{e:general}
\inf_{\alpha \beta} L_{\alpha \beta} v(x) \leq I (u+v) (x) - Iu (x) \leq \sup_{\alpha \beta} L_{\alpha \beta} v(x)$$
A more general and better description of the nonlinear operators we want to deal with is the operators $I$ for which holds for some family of linear integro-differential operators $L_{\alpha \beta}$. The idea is that an estimate on $I(u+v) - Iu$ by a suitable extremal operator can be a replacement for the concept of ellipticity. Indeed, if we consider the extremal Pucci operators [@CC], $M^+_{\lambda,\Lambda}$ and $M^-_{\lambda,\Lambda}$, and we have $M^-_{\lambda,\Lambda} v(x) \leq I(u+v) - Iu \leq M^+_{\lambda,\Lambda} v(x)$, then it is easy to see that $I$ must be an elliptic second order differential operator. If instead we compare with suitable nonlocal extremal operators, we will have a concept of ellipticity for nonlocal equations. We will give a precise definition in section \[s:maximal\] (Definition \[d:axiomatic\]).
We now explain the natural Dirichlet problem for a nonlocal operator. Let $\Omega$ be an open domain in ${\mathbb R}^n$. We are given a function $g$ defined in ${\mathbb R}^n \setminus \Omega$, which is the boundary condition. We look for a function $u$ such that $$\begin{aligned}
Iu(x) &= 0 && \text{for every } x \in \Omega \\
u(x) &= g(x) &&\text{for } x \in {\mathbb R}^n \setminus \Omega\end{aligned}$$
Notice that the boundary condition is given in the whole complement of $\Omega$ and not only ${\partial}\Omega$. This is because of the nonlocal character of the operator $I$. From the stochastic point of view, it corresponds to the fact that a discontinuous Lèvy process can exit the domain $\Omega$ for the first time jumping to any point in ${\mathbb R}^n \setminus \Omega$.
In this paper we will focus mainly in the regularity properties of solutions to an equation $Iu=0$. We will briefly present a very general comparison principle from which existence of solutions can be obtained in smooth domains. In order to obtain regularity results, we must assume some *nice* behavior of the measures $\mu$. Basically, our assumption is that they are symmetric, absolutely continuous and not too degenerate. To fix ideas, we can think of integro-differential operators with a kernel comparable with the respective kernel of the fractional laplacian $-(-{\triangle})^{\sigma/2}$. In this respect, the theory we develop can be understood as a theory of viscosity solutions for fully nonlinear equations of fractional order.
In this paper we would like to quickly present the necessary definitions and then prove some regularity estimates. Our results in this paper are
- A comparison principle for a general nonlinear integro-differential equation.
- A nonlocal version of the Alexandroff-Backelman-Pucci estimate.
- The Harnack inequality for integro-differential equations with kernels that are comparable with the ones of the fractional laplacian but can be very discontinuous.
- A Hölder regularity result for the same class of equations as the Harnack inequality.
- A $C^{1,\alpha}$ regularity result for a large class of nonlinear integro-differential equations.
Even though there are some known results about Harnack inequalities and Hölder estimates for integro-differential equations with either analytical proofs [@S1] or probabilistic proofs [@BK], [@BK2], [@BL], [@SV], the estimates in all these previous results blow up as the order of the equation approaches $2$. In this way, they do not generalize to elliptic differential equations. We provide estimates that remain uniform in the degree and therefore make the theory of integro-differential equations and elliptic differential equations appear somewhat unified. Consequently, our proofs are more involved than the ones in the bibliography.
In this paper we only consider nonlinear operators that are translation invariant. The *variable coefficient* case will be considered in future work. I future papers, we are also planning to address the problem of the interior regularity of the integro-differential Hamilton-Jacobi-Bellman equation. This refers to the equation involving a convex nonlocal operator like . In that case we obtain an analogue of the Evans-Krylov theorem proving that the solutions to the equation have enough regularity to be classical solutions.
The structure of the paper is as follows. After this introduction, the second section presents the appropriate definitions of subsolution and supersolution of an integro-differential equation in the viscosity sense. In our definition we allow any kind of discontinuities outside of the domain of the equations. In the third section we give the general description of the elliptic nonlocal equations that we want to study. We define a nonlocal elliptic operator by comparing its increments with a suitable maximal operator. This definition is more general than . In the fourth section we study the stability of our definitions. A comparison principle is proven in section five under very mild assumptions. Next, in section six we show how to obtain an elliptic partial differential equation as a limit of integro-differential equations. We believe one of the most nontrivial results in the paper is the nonlocal ABP estimate developed in section seven. In sections eight and nine we construct a special function and prove some pointwise estimates that will help in proving the Harnack inequality and Hölder estimates in sections ten and eleven. In section twelve we show the $C^{1,\alpha}$ estimates. And finally in section thirteen we show how to generalize our previous results when our operators have truncated kernels. This last section is important for applications since very often the kernels of an integro differential equation are comparable to the ones of the fractional laplacian only in a neighborhood of the origin.
Definitions {#s:definitions}
===========
As we mention in the introduction, equation was given in too much generality for our purposes. We will restrict our attention to the operators where $\mu$ is given by a symmetric kernel $K$. It takes the form $$\label{e:linear0}
Lu(x) = \mathrm{PV} \int_{{\mathbb R}^n} (u(x+y) - u(x)) K(y) {\; \mathrm{d}}y \ .$$
The kernel $K$ must be a positive function, satisfy $K(y) = K(-y)$, and also $$\label{e:minimumassumptionforlinear}
\int_{{\mathbb R}^n} \frac{|y|^2}{|y|^2+1} K(y) {\; \mathrm{d}}y < +\infty$$ It is not necessary to subtract the term $-{\nabla}u(x) \cdot y \chi_{B_1}$ if we think of the integral in the principal value sense. Alternatively, due to the symmetry of the kernel $K$, the operator can also be written as $$Lu(x) = \frac{1}{2} \int_{{\mathbb R}^n} (u(x+y) + u(x-y) - 2u(x)) K(y) {\; \mathrm{d}}y \ .$$
In order to simplify the notation, we will write ${\delta}(u,x,y) := u(x+y) + u(x-y) - 2u(x)$. The expression for $L$ can be written shortly as $$\label{e:linear}
Lu(x) = \int_{{\mathbb R}^n} {\delta}(u,x,y) K(y) {\; \mathrm{d}}y \ .$$ for some kernel $K$ (which would be half of the one of ). We will alternate from writing the operators in the form or whenever it is convenient.
The nonlinear integro-differential operators that arise in stochastic control have the form where we think that for each $L_{\alpha \beta}$ we have a kernel $K_{\alpha \beta}$ so that $L_{\alpha \beta}$ has the form . We will define a more general form for nonlinear integro-differential operators in section \[s:maximal\].
The minimum assumption in order to have $Iu$ well defined is that every kernel $K_{\alpha \beta}$ must satisfy in a uniform way. More precisely $$\label{e:minimumassumption}
\text{if } K(y) := \sup_{\alpha \beta} K_{\alpha \beta} (y) \ \text{ then } \int_{{\mathbb R}^n} \frac{|y|^2}{|y|^2+1} \ K (y) \ {\; \mathrm{d}}y < +\infty$$
The value of $Iu$ can be evaluated in the classical sense if $u \in C^{1,1}$. If we want to evaluate the value of $Iu (x)$ at one point $x$ only, we need $u$ to be punctually $C^{1,1}$ in the sense of the following definition.
A function $\varphi$ is said to be $C^{1,1}$ at the point $x$, and we write $u \in C^{1,1}(x)$, if there is a vector $v \in {\mathbb R}^n$ and a number $M>0$ such that $$|\varphi(x+y) - \varphi(x) - v \cdot y| \leq M |y|^2 \qquad \text{for $|y|$ small enough.}$$
We give a definition of viscosity sub- and super-solutions for integro-differential equations by evaluating the operators in $C^{1,1}$ test functions that *touch* the function $u$ from either above or below. Often for nonlocal equations the definition is given by test functions that remain on one side of $u$ in the whole space ${\mathbb R}^n$. We take a sligtly different approach. We consider a test function $\varphi$ that touches $u$ at a point $x$ and remains on one side of $u$ but it is only defined locally, in a neighborhood $N$ of $x$. Then we complete $\varphi$ with the tail of $u$ to evaluate the integrals . We do this in order to allow arbitrary discontiuities in the function $u$ outside of the domain $\Omega$ where it may be a solution of the equation.
\[d:viscositysolutions\] A function $u :{\mathbb R}^n \to {\mathbb R}$, upper (lower) semi continuous in $\overline \Omega$, is said to be a subsolution (supersolution) to $Iu = f$, and we write $Iu\geq f$ ($Iu \leq f$), if every time all the following happen
- $x$ is any point in $\Omega$.
- $N$ is a neighborhood of $x$ in $\Omega$.
- $\varphi$ is some $C^2$ function in $\overline N$.
- $\varphi(x) = u(x)$.
- $\varphi(y) > u(y)$ ($\varphi(y) < u(y)$) for every $y \in N \setminus \{x\}$.
Then if we let $$v := \begin{cases}
\varphi &\text{in } N \\
u &\text{in } {\mathbb R}^n \setminus N \ ,
\end{cases}$$ we have $Iv(x) \geq f(x)$ ($Iv(x) \leq f(x)$).
A solution is a function $u$ which is both a subsolution and a supersolution.
Note that Definition \[d:viscositysolutions\] is essentially the same as Definition 2 in [@BI].
For the set of test functions, we could also use a function $\varphi$ that is $C^{1,1}$ only at the contact point $x$. This is a larger set of test functions, so a priori it may provide a stronger concept of solution. In section \[s:stability\] we will show that the two approaches are actually equivalent.
Usually the nonlocal operators $I$ allow some growth at infinity. If the value of $Iu(x)$ is well defined every time $u \in C^{1,1}(x)$ and $u \in L^1\left({\mathbb R}^n, w \right)$ for some weight $w$ that is locally bounded, then the above definition would apply for semicontinuous functions in $\overline \Omega$ that are in $L^1({\mathbb R}^n, w)$ but not necesarily bounded. In most cases, our regularity results in this paper can be extended to the unbounded case by truncating the function and adding an error term in the right hand side.
Maximal operators {#s:maximal}
=================
In and we consider the supremum or an *inf-sup* of a collection of linear operators. Let us consider a collection of linear operators ${\mathcal{L}}$ that includes all of them. The maximal and a minimal operator respect to ${\mathcal{L}}$ are defined as: $$\begin{aligned}
{\mathrm{M}^+_\mathcal{L}}v(x) &= \sup_{L \in {\mathcal{L}}} L u(x) \\
{\mathrm{M}^-_\mathcal{L}}v(x) &= \inf_{L \in {\mathcal{L}}} L u(x) .\end{aligned}$$
For example, an important class that we will use for regularity results is given by the class ${\mathcal{L}}_0$ of operators $L$ of the form \[e:linear\] with $$\label{e:uniformellipticity}
(2 - \sigma) \frac{\lambda}{|y|^{n+\sigma}}\leq K(y) \leq (2 - \sigma) \frac{\Lambda}{|y|^{n+\sigma}} \, ,$$ then ${\mathrm{M}^+}_{{\mathcal{L}}_0}$ and ${\mathrm{M}^-}_{{\mathcal{L}}_0}$ take a very simple form: $$\begin{aligned}
{\mathrm{M}^+}_{{\mathcal{L}}_0} v(x) &= (2 - \sigma) \int_{{\mathbb R}^n} \frac{\Lambda {\delta}(v,x,y)^+ - \lambda {\delta}(v,x,y)^-}{|y|^{n+\sigma}} {\; \mathrm{d}}y \label{e:Mp}\\
{\mathrm{M}^-}_{{\mathcal{L}}_0} v(x) &= (2 - \sigma) \int_{{\mathbb R}^n} \frac{\lambda {\delta}(v,x,y)^+ - \Lambda {\delta}(v,x,y)^-}{|y|^{n+\sigma}} {\; \mathrm{d}}y \ . \label{e:Mm} \end{aligned}$$ We will use these maximal operators to obtain regularity estimates. The factor $(2-\sigma)$ is important when $\sigma \to 2$. We need such factor if we want to obtain second order differential equations as limits of integro-differential equations. In terms of the regularity, we need the factor $(2-\sigma)$ for the estimates not to blow up as $\sigma \to 2$.
Another interesting class is given when the kernels have the form $$K(y) = (2-\sigma) \frac{ y^t A y}{|y|^{n+2+\sigma}} \ ,$$ for symmetric matrices $A$ such that $\lambda I \leq A \leq \Lambda I$. This is a smaller class than the ${\mathcal{L}}_0$ above if we choose the respective constants $\lambda$ and $\Lambda$ accordingly, but it is all we need to recover the classical Pucci extremal operators [@CC] as $\sigma \to 2$.
Let $K(x)$ be the suppremum of $K_\alpha(x)$ where $K_\alpha$ are all the kernels of all operators $L \in {\mathcal{L}}$. As a replacement for , for any class ${\mathcal{L}}$ we will assume $$\label{e:minimumassumptionforclass}
\int_{{\mathbb R}^n} \frac{|y|^2}{|y|^2+1} \ K (y) \ {\; \mathrm{d}}y < +\infty$$
Using the extremal operators, we give a general definition of ellipticity for nonlocal equations. The following is the kind of operators for which the results in this paper apply.
\[d:axiomatic\] Let ${\mathcal{L}}$ be a class of linear integro differential operators. We always assume . An elliptic operator $I$ respect to ${\mathcal{L}}$ is an operator with the following properties:
- If $u$ is any bounded function, $Iu(x)$ is well defined every time $u \in C^{1,1}(x)$.
- If $u$ is $C^2$ in some open set $\Omega$, then $Iu(x)$ is a continuous function in $\Omega$.
- If $u$ and $v$ are bounded functions $C^{1,1}(x)$, then $$\label{eq:puccicontinuous}
{\mathrm{M}^-_\mathcal{L}}(u-v)(x) \leq Iu(x) - Iv(x) \leq {\mathrm{M}^+_\mathcal{L}}(u-v)(x)$$
Definition \[d:viscositysolutions\] applies for the general nonlocal elliptic operators of Definition \[d:axiomatic\] *mutatis mutandis*.
Definition \[d:axiomatic\] may apply to operators $I$ whether or not they are translation invariant. However, in this paper we will only focus on the translation invariant case. In other words, for all nonlinear operators $I$ in this paper we assume that $\tau_z I u = I (\tau_z u)$, where $\tau_z$ is the translation operator $\tau_z u(x) := u(x-z)$.
We will show that any operator of the form is elliptic with respect to any class that contains all the operators $L_{\alpha \beta}$ as long as condition is satisfied (Lemma \[l:puccicontinuous\] and Lemma \[l:c11\]). However the Definition \[d:axiomatic\] allows a richer class of equations. For example we can consider an operator $I$ given by $$I u (x) = \int_{{\mathbb R}^n} \frac{G(u(x+y) - u(x))}{|y|^{n+\sigma}} {\; \mathrm{d}}\sigma$$ for any monotone Lipschitz function $G$ such that $G(0)=0$. This operator $I$ would be elliptic with respect to the class ${\mathcal{L}}_0$.
\[l:puccicontinuous\] Let $I$ be an operator like in and ${\mathcal{L}}$ be any collection of integro-differential operators. Assume every $L_{\alpha \beta}$ belongs to the class ${\mathcal{L}}$. Then for every $u,v \in C^{1,1}(x)$ we have $${\mathrm{M}^-_\mathcal{L}}(u-v) (x) \leq Iu(x) - Iu(x) \leq {\mathrm{M}^+_\mathcal{L}}(u-v)(x)$$
Since $u \in C^{1,1}(x)$, $L_{\alpha \beta} u(x)$ is defined classically for any $L_{\alpha \beta}$. Let’s assume first that $I$ is convex. We have $$Iu(x) = \sup_\alpha L_\alpha u(x) \ .$$ Thus, for every ${\varepsilon}>0$, there is an $\alpha_1$ and an $\alpha_2$ such that $$\begin{aligned}
Iu(x) - L_{\alpha_1} u(x) &< {\varepsilon}\\
Iv(x) - L_{\alpha_2} v(x) &< {\varepsilon}.\end{aligned}$$ Thus we have $$\begin{aligned}
L_{\alpha_2} u(x) - L_{\alpha_2} v(x) - {\varepsilon}&\leq Iu(x) - Iv(x) \leq L_{\alpha_1} u(x) - L_{\alpha_1} v(x) + {\varepsilon}\\
{\mathrm{M}^-_\mathcal{L}}(u-v)(x) - {\varepsilon}&\leq Iu(x) - Iv(x)\leq {\mathrm{M}^+_\mathcal{L}}(u-v) (x) + {\varepsilon}\ .\end{aligned}$$
Since we can take ${\varepsilon}$ as small as we want, we obtain ${\mathrm{M}^-_\mathcal{L}}(u-v)(x) \leq Iu(x) - Iv(x)\leq {\mathrm{M}^+_\mathcal{L}}(u-v) (x)$ for every convex $I$.
For the nonconvex case, we can write $$Iu(x) = \inf_\beta \sup_\alpha L_{\alpha \beta} u(x) = \inf_\beta I_\beta u(x),$$ where the $I_\beta$ is the convex operator given by $I_\beta u(x) = \sup_\alpha L_{\alpha \beta} u(x)$. Now a similar idea applies.
For every ${\varepsilon}>0$, there is an $\beta_1$ and an $\beta_2$ such that $$\begin{aligned}
Iu(x) - I_{\beta_1} u(x) &< {\varepsilon}\\
Iv(x) - I_{\beta_2} v(x) &< {\varepsilon}.\end{aligned}$$ Thus we have $$\begin{aligned}
I_{\beta_1} u(x) - I_{\beta_1} v(x) - {\varepsilon}&\leq Iu(x) - Iv(x) \leq I_{\beta_1} u(x) - I_{\beta_1} v(x) + {\varepsilon}\\
{\mathrm{M}^-_\mathcal{L}}(u-v)(x) - {\varepsilon}&\leq Iu(x) - Iv(x)\leq {\mathrm{M}^+_\mathcal{L}}(u-v) (x) + {\varepsilon}\ .\end{aligned}$$
Taking ${\varepsilon}\to 0$, we obtain ${\mathrm{M}^-_\mathcal{L}}(u-v)(x) \leq Iu(x) - Iv(x)\leq {\mathrm{M}^+_\mathcal{L}}(u-v) (x)$ for any $I$ of the form .
The family of operators that satisfy the condition have another very curious property. Definition \[d:viscositysolutions\] is made so that we never have to evaluate the operator $I$ in the original function $u$. Every time we touch $u$ with a smooth function $\varphi$ from above, we construct a test function $v \in C^{1,1}(x)$ to evaluate $I$. It is somewhat surprising that if $I$ is any nonlinear operator $I$ that is an $\inf \sup$ (or a $\sup \inf$) of linear operators that satisfy , then this turn out to be unnecessary, since $I$ can be evaluated classically in $u$ at those points $x$ where $u$ can be touched by above with a paraboloid. This is explained in the next lemma.
\[l:classic2\] Let $I$ be an operator like in so that for every $K_{\alpha \beta}$ the equation holds.
If we have a subsolution, $Iu \geq f$ in $\Omega$ and $\varphi$ is a $C^2$ function that touches $u$ from above at a point $x \in \Omega$, then $I u(x)$ is defined in the classical sense and $I u(x) \geq f(x)$.
For any $r>0$, we define $$v_r = \begin{cases}
\varphi & \text{in } B_r \\
u & \text{in } {\mathbb R}^n \setminus B_r \ ,
\end{cases}$$ and we have ${\mathrm{M}^+}v_r(x) \geq I v_r (x) \geq f(x)$. Thus $$(2-\sigma) \int {\delta}(v_r,x,y)^+ \frac{\Lambda}{|y|^{n+\sigma}} - {\delta}(v_r,x,y)^- \frac{\lambda}{|y|^{n+\sigma}} {\; \mathrm{d}}y \geq f(x)$$
Since $\varphi$ touches $u$ from above at $x$, for any $y \in {\mathbb R}^n$, ${\delta}(v_r,x,y) \geq {\delta}(u,x,y)$. Since $v_r \in C^{1,1}(x)$, $|{\delta}(v_r,x,y)|/|y|^{n+\sigma}$ is integrable, and then so is ${\delta}(u,x,y)^+ /|y|^{n+\sigma}$.
We have $$(2-\sigma) \int {\delta}(v_r,x,y)^- \frac{\lambda}{|y|^{n+\sigma}} {\; \mathrm{d}}y \leq (2-\sigma) \int {\delta}(v_r,x,y)^+ \frac{\Lambda}{|y|^{n+\sigma}} {\; \mathrm{d}}y - f(x)$$
Since $\varphi$ touches $u$ from above at $x$, ${\delta}(v_r,x,y)$ will decrease as $r$ decreases. Therefore, for every $r < r_0$ $$\label{e:ii22}
(2-\sigma) \int_{{\mathbb R}^n} {\delta}(v_r,x,y)^- \frac{\lambda}{|y|^{n+\sigma}} {\; \mathrm{d}}y \leq (2-\sigma) \int_{{\mathbb R}^n} {\delta}(v_{r_0},x,y)^+ \frac{\Lambda}{|y|^{n+\sigma}} {\; \mathrm{d}}y - f(x)$$
But ${\delta}(v_r,x,y)^-$ is monotone increasing as $r$ decreases, and it converges to ${\delta}(u,x,y)^-$ as $r \to 0$. From monotone convergence theorem $$\lim_{r \to 0} (2-\sigma) \int_{{\mathbb R}^n} {\delta}(v_r,x,y)^- \frac{\lambda}{|y|^{n+\sigma}} {\; \mathrm{d}}y = (2-\sigma) \int_{{\mathbb R}^n} {\delta}(u,x,y)^- \frac{\lambda}{|y|^{n+\sigma}} {\; \mathrm{d}}y \ .$$ And from , the integrals are uniformly bounded and thus $$(2-\sigma) \int_{{\mathbb R}^n} {\delta}(u,x,y)^- \frac{\lambda}{|y|^{n+\sigma}} {\; \mathrm{d}}y \leq (2-\sigma) \int_{{\mathbb R}^n} {\delta}(v_{r_0},x,y)^+ \frac{\Lambda}{|y|^{n+\sigma}} {\; \mathrm{d}}y - f(x) < +\infty$$
Therefore, ${\delta}(u,x,y)/|y|^{n+\sigma}$ is integrable, and $L_{\alpha \beta} u$ is well defined in the classical sense for any $\alpha$ and $\beta$. Thus, $Iu(x)$ is computable in the classical sense. The difference ${\delta}(v_r-u,x,y)/|y|^{n+\sigma}$ is monotone decreasing as $r \searrow 0$, converges to zero, and it is bounded by the integrable function ${\delta}(v_{r_0}-u,x,y)/|y|^{n+\sigma}$. We can pass to the limit in the following expression: $$\begin{aligned}
\lim_{r \to 0} {\mathrm{M}^+}(v_r - u) (x) &= \lim_{r \to 0} (2-\sigma) \int {\delta}(v_r - u,x,y)^+ \frac{\Lambda}{|y|^{n+\sigma}} {\; \mathrm{d}}y \\
&= 0\end{aligned}$$
Now we use Lemma \[l:puccicontinuous\] to conclude $$I u(x) \geq I v_r (x) + {\mathrm{M}^-}(u-v_r) = f(x) - {\mathrm{M}^+}(v_r - u) \to f(x)$$
So $I u(x) \geq f(x)$.
Lemma \[l:classic2\] is convenient to make proofs involving ${\mathrm{M}^+}$ and ${\mathrm{M}^-}$ because it allows to deal with viscosity solutions almost as if they were classical solutions. It is not clear to what other types of nonlinear operators a result like Lemma \[l:classic2\] would extend.
Stability properties {#s:stability}
====================
In this section we show a few technichal properties of the operators $I$ like . First that if $u \in C^{1,1}(\Omega)$ then $Iu$ is continuous in $\Omega$. As it was mentioned in the previous sections, it is necessary to justify that the operators of the form satisfy the conditions of Definition \[d:axiomatic\]. Next, we will show that our notion of viscosity solutions allows to touch with solutions that are only punctually $C^{1,1}$ instead of $C^2$ in a neighborhood of the point. Then we will show the important stability property of Definition \[d:viscositysolutions\]. Namely we show that if a sequence of subsolutions (or supersolutions) in $\Omega$ converges in a suitable way on any compact set in ${\mathbb R}^n$, then the limit is also a subsolution (or supersolution).
We start with a technichal lemma.
\[l:realanalysis\] Let $f \in L^\infty({\mathbb R}^n)$ and $g_\alpha$ be a family of functions so that $|g_\alpha(x)| \leq g(x)$ for some $L^1$ function $g$. Then the family $f \ast g_\alpha$ is equicontinuous in every compact set.
Let $K$ be a compact set in ${\mathbb R}^n$. Let ${\varepsilon}>0$. Since $g \in L^1$, we can pick a large $R$ so that $K \subset B_R$ and $${\left\Vertf\right\Vert}_{L^\infty} \left( \int_{{\mathbb R}^n \setminus B_R(x)} g(y) {\; \mathrm{d}}y \right) \leq {\varepsilon}/8$$ for any $x \in K$. We write $f = f_1 + f_2$, where $f_1 = f \chi_{B_{2R}}$ and $f_2 = f \chi_{{\mathbb R}^n \setminus B_{2R}}$. From the above inequality, we have $|f_2 \ast g_\alpha| \leq {\varepsilon}/8$ in $K$.
Since $g \in L^1$, there is a $\delta_0 > 0$ so that $$\label{e:s1}
\int_A g(x) {\; \mathrm{d}}x < \frac{{\varepsilon}}{16 {\left\Vertf\right\Vert}_{L^\infty}} \qquad \text{ for any set } |A| < \delta_0$$
Let $\eta_t$ be a standard mollifier with compact support. We have $f_1 \ast \eta_t \to f_1$ a.e. (in every Lebesgue point of $f_1$). Recall that the support of $f_1$ is in $B_R$. For $t$ large, $f_1 \ast \eta_t=0$ ouside $B_{4R}$. By Egorov’s theorem, there is a set $A \subset B_{4R}$ such that $$\begin{aligned}
&|A| < \delta_0 \label{e:s11} \\
&f_1 \ast \eta_t \to f_1 \qquad \text{uniformly in } {\mathbb R}^n \setminus A\end{aligned}$$
In particular, there is a $\tilde f_1 = f_1 \ast \eta_{t_0}$ such that $|f_1 - \tilde f_1| < \frac{{\varepsilon}} { 8 {\left\Vertg\right\Vert}_{L^1}}$ in ${\mathbb R}^n \setminus A$. We have $${\left\Vert(f_1-\tilde f_1) (1-\chi_A) \ast g_\alpha\right\Vert}_{L^\infty} \leq {\left\Vert(f_1-\tilde f_1) (1-\chi_A)\right\Vert}_{L^\infty} {\left\Vertg_\alpha\right\Vert}_{L^1} < \frac{{\varepsilon}}{8}$$
On the other hand, from and , we also get $$\label{e:s2}
{\left\Vert(f_1-\tilde f_1) \chi_A \ast g_\alpha\right\Vert}_{L^\infty} < \frac{{\varepsilon}}{8}$$
Since $\tilde f_1$ is continuous and ${\left\Vertg_\alpha\right\Vert}_{L^1}$ is bounded, the family $\tilde f_1 \ast g_\alpha$ is equicontinuous. There is a $\delta>0$ so that $|\tilde f_1 \ast g_\alpha(x) - \tilde f_1 \ast g_\alpha(y)| < {\varepsilon}/4$ every time $|x-y| < \delta$. Moreover $$\begin{aligned}
|f \ast g_\alpha(x) - f \ast g_\alpha(y)| &\leq |\tilde f_1 \ast g_\alpha(x) - \tilde f_1 \ast g_\alpha(y)| + |(f_1-\tilde f_1) \ast g_\alpha(x) - (f_1-\tilde f_1) \ast g_\alpha(y)| \\
&\qquad + |f_2 \ast g_\alpha(x) - f_2 \ast g_\alpha(y)| \\
&\leq {\varepsilon}/4 + |(f_1-\tilde f_1)\chi_A \ast g_\alpha(x)| + |(f_1-\tilde f_1)\chi_A \ast g_\alpha(y)| \\
&\phantom{ \leq {\varepsilon}/4 + } + |(f_1-\tilde f_1)(1-\chi_A) \ast g_\alpha(x)| + |(f_1-\tilde f_1)(1-\chi_A) \ast g_\alpha(y)| \\
&\phantom{ \leq {\varepsilon}/4 + } + |f_2 \ast g_\alpha(x)| + |f_2 \ast g_\alpha(y)|\\
&\leq {\varepsilon}\end{aligned}$$ for any $\alpha$ and every time $|x-y| < \delta$.
\[l:c11\] Let $I$ be an operator like in , assuming only . Let $v$ be a bounded function in ${\mathbb R}^n$ and $C^{1,1}$ in some set $\Omega$. Then $Iv$ is continuous in $\Omega$.
We must prove the $L_{\alpha \beta} v$ in are equicontinuous. Like in , we write $K = \sup_{\alpha \beta} K_{\alpha \beta}$.
Let ${\varepsilon}>0$ and $x_0 \in \Omega$. Since $v$ is $C^{1,1}$ in $\Omega$, there is a constant $C$ so that $$|{\delta}(v,x,y)| < C|y|^2 \qquad \text{if } x \in \Omega \text{ and } |y| < {\mathrm{dist}}(x, {\partial}\Omega)$$
Let $r>0$ such that $$\int_{B_r} C|y|^2 K(y) {\; \mathrm{d}}y < {\varepsilon}/3$$
We have $$\begin{aligned}
L_{\alpha \beta} v(x) &= \int_{{\mathbb R}^n} {\delta}(v,x,y) K_{\alpha \beta} (y) {\; \mathrm{d}}y \\
&= \int_{B_r} {\delta}(v,x,y) K_{\alpha \beta} (y) {\; \mathrm{d}}y + \int_{{\mathbb R}^n \setminus B_r} {\delta}(v,x,y) K_{\alpha \beta} (y) {\; \mathrm{d}}y \\
&=: w_1(x) + w_2(x)\end{aligned}$$ where $$|w_1| = {\left\vert\int_{B_r} {\delta}(v,x,y) K_{\alpha \beta} (y) {\; \mathrm{d}}y\right\vert} \leq \int_{B_r} C|y|^2 K(y) {\; \mathrm{d}}y < {\varepsilon}/3$$ and $$\begin{aligned}
w_2 &= \int_{{\mathbb R}^n \setminus B_r} (v(x+y) + v(x-y) - 2v(x)) K_{\alpha \beta} (y) {\; \mathrm{d}}y \\
&= v \ast g_{\alpha \beta} + v \ast \hat g_{\alpha \beta} - 2\left( \int g_{\alpha \beta} {\; \mathrm{d}}y \right) v\end{aligned}$$ where $g_{\alpha \beta}(y) = \chi_{{\mathbb R}^n \setminus B_r}(y) K_{\alpha \beta}(y)$ and $\hat g_{\alpha \beta}(y) = g_{\alpha \beta}(-y)$. For any $\alpha$ and $\beta$, $g_{\alpha \beta} \leq \chi_{{\mathbb R}^n \setminus B_r} K$, which is in $L^1$. From Lemma \[l:realanalysis\], $w_2$ is equicontinuous. So there is a $\delta>0$ such that $$|w_2(x) - w_2(x_0)| < {\varepsilon}/3 \qquad \text{if } |x-x_0|<\delta$$
Therefore $$L_{\alpha \beta} v(x) - L_{\alpha \beta} v(x_0)| \leq |w_1(x)|+|w_1(x_0)|+|w_2(x)-w_2(x_0)| < {\varepsilon}$$ uniformly in $\alpha$ and $\beta$. Thus $|Iv(x) - Iv(x_0)| < {\varepsilon}$ every time $|x-x_0|<\delta$.
When we gave the definition of viscosity solutions in section \[s:definitions\], we used $C^2$ test functions. Now we show that it is equivalent to use punctually $C^{1,1}$ functions.
\[l:tc11\] Let $I$ be elliptic respect to some class ${\mathcal{L}}$ in the sense of Definition \[d:axiomatic\]. Let $u : {\mathbb R}^n \to {\mathbb R}$ be an upper semicontinuous function such that $I u \geq 0$ in $\Omega$ in the viscosity sense. Let $\varphi : {\mathbb R}^n \to {\mathbb R}$ be a bounded function, punctually $C^{1,1}$ at a point $x \in \Omega$. Assume $\varphi$ touches $u$ from above at $x$. Then $I\varphi (x)$ is defined in the classical sense and $I \varphi(x) \geq f(x)$.
Since $\varphi$ is $C^{1,1}$, the expression is clearly integrable for every $\alpha$ and $\beta$ and $I\varphi(x)$ is defined classically.
Also because $\varphi$ is $C^{1,1}$, there is a quadratic polynomial $q$ touching $\varphi$ from above at $x$. Let $$v_r(x) = \begin{cases}
q & \text{in } B_r \\
u & \text{in } {\mathbb R}^n \setminus B_r \ .
\end{cases}$$
Since $I u \geq f$ in $\Omega$ in the viscosity sense then $I v_r (x) \geq f(x)$ with $I v_r(x)$ well defined. Moreover let $$u_r(x) = \begin{cases}
q & \text{in } B_r \\
\varphi & \text{in } {\mathbb R}^n \setminus B_r \ .
\end{cases}$$ we have $$\begin{aligned}
I \varphi(x) &\geq I u_r(x) + {\mathrm{M}^-_\mathcal{L}}(\varphi-u_r)(x) \geq I u_r(x) && \text{since $\varphi-u_r$ has a minimum at $x$}\\
& \geq I v_r(x) + {\mathrm{M}^-_\mathcal{L}}(u_r - v_r) (x) \\
& \geq f(x) + {\mathrm{M}^-_\mathcal{L}}(u_r - v_r) (x) \\
& \geq f(x) + \int_{B_r} {\delta}(q-\varphi,x,y)^- K(y) {\; \mathrm{d}}y && \text{where $K$ is the one from \eqref{e:minimumassumptionforclass}} \\
& \geq f(x) - C \int_{B_r} |y|^2 K(y) {\; \mathrm{d}}y \ .\end{aligned}$$
Since $|y|^2 K(y)$ is integrable in a neighborhood of the origin, the expression $$\int_{B_r} C |y|^2 K(y) {\; \mathrm{d}}y$$ goes to zero as $r \to 0$. Thus, for any ${\varepsilon}>0$, we can find a small $r$ so that $$I \varphi(x) \geq f(x) - {\varepsilon}\ .$$ Therefore $I \varphi(x) \geq f(x)$.
One of the most useful properties of viscosity solutions is their stability under uniform limits on compact sets. We will prove a slightly stronger result. We show that the notion of viscosity supersolution is stable with respect to the natural limits for lower semicontinuous functions. This type of limit is well known and usually called $\Gamma$-limit.
A sequence of lower semicontinuous functions $u_k$ $\Gamma$-converges to $u$ in a set $\Omega$ if the two following conditions hold
- For every sequence $x_k \to x$ in $\Omega$, $\liminf_{k \to \infty} u_k(x_k) \geq u(x)$.
- For every $x \in \Omega$, there is a sequence $x_k \to x$ in $\Omega$ such that $\limsup_{k \to \infty} u_k(x_k) = u(x)$.
Naturally, a uniformly convergent sequence $u_k$ would also converge in the $\Gamma$ sense. An important property of $\Gamma$-limits is that if $u_k$ $\Gamma$-converges to $u$, and $u$ has a strict local minimum at $x$, then $u_k$ will have a local minimum at $x_k$ for a sequence $x_k \to x$.
\[l:stability\] Let $I$ be elliptic in the sense of Definition \[d:axiomatic\] and $u_k$ be a sequence of functions that are bounded in ${\mathbb R}^n$ and lower semicontinuous in $\Omega$ such that
- $I u_k \leq f_k$ in $\Omega$.
- $u_k \to u$ in the $\Gamma$ sense in $\Omega$.
- $u_k \to u$ a.e. in ${\mathbb R}^n$.
- $f_k \to f$ locally uniformly in $\Omega$.
Then $Iu \leq f$ in $\Omega$.
Let $\varphi$ be a test function from below for $u$ touching at a point $x$ in a neighborhood $N$.
Since $u_k$ $\Gamma$-converges to $u$ in $\Omega$, for large $n$, we can find $x_k$ and $\delta_k$ such that $\varphi+d_k$ touches $u_k$ at $x_k$. Moreover $x_k \to x$ and $d_k \to 0$ as $k \to + \infty$.
Since $I u_k \leq f_k$, if we let $$v_k = \begin{cases}
\varphi + d_k & \text{in } N \\
u_k & \text{in } {\mathbb R}^n \setminus N \ ,
\end{cases}$$ we have $I v_k (x_k) \leq f_k(x_k)$.
Let $z \in N$ be such that ${\mathrm{dist}}(z,{\partial}N) > \rho >0$. We have $$\begin{aligned}
{\left\vertI v_k (z) - I v(z)\right\vert} &\leq \max\left( |{\mathrm{M}^+_\mathcal{L}}(v_k-v)(z)|, |{\mathrm{M}^+_\mathcal{L}}(v-v_k)(z)| \right) \\
&\leq \sup_{L \in {\mathcal{L}}} {\left\vert L (v_k - v)(z) \right\vert} \\
&\leq \int_{{\mathbb R}^n} |{\delta}(v_k-v,z,y)| K(y) {\; \mathrm{d}}y \\
&\leq \int_{{\mathbb R}^n \setminus B_\rho} |{\delta}(v_k-v,z,y)| K(y) {\; \mathrm{d}}y\end{aligned}$$
The sequence $v_k$ is bounded and ${\delta}(v_k-v,z,y)$ converges to zero almost everywhere. Since $K \in L^1({\mathbb R}^n \setminus B_\rho)$, we can use dominated convergence theorem to show that the above expression goes to zero as $k \to +\infty$. Moreover the convergence is uniform in $z$. We obtain $I v_k \to I v$ locally uniformly in $N$.
From Definition \[d:axiomatic\], we have that $Iv$ is continuous in $N$. We now compute $$|Iv_k(x_k) - I v(x)| \leq |Iv_k(x_k) - Iv(x_k)| + |Iv(x_k) - Iv(x)| \to 0 \ .$$
So $Iv_k(x_k)$ converges to $Iv(x)$, as $k \to +\infty$. Since $x_k \to x$ and $f_k \to f$ locally uniformly, we also have $f_k(x_k) \to f(x)$, which finally implies $Iv(x) \leq f(x)$.
In the previous lemma we showed the stability of supersolutions under $\Gamma$ limits. Naturally, we also have the corresponding result for subsolutions. In that case we would consider the natural limit in the space of upper semicontinuous functions which is the same as the $\Gamma$-convergence of $-u_k$ to $-u$. As a corollary, we obtain the stability under uniform limits.
Let $I$ be elliptic in the sense of Definition \[d:axiomatic\] and $u_k$ be a sequence of functions that are bounded in ${\mathbb R}^n$ and continuous in $\Omega$ such that
- $I u_k = f_k$ in $\Omega$.
- $u_k \to u$ locally uniformly in $\Omega$.
- $u_k \to u$ a.e. in ${\mathbb R}^n$.
- $f_k \to f$ locally uniformly in $\Omega$.
Then $Iu = f$ in $\Omega$.
*$\Gamma$-convergence* was introduced by De Giorgi in the framework of variational analysis to study convergence of sequences of functionals in Banach spaces. Here we are using the same notion of convergence for functions in ${\mathbb R}^n$. This type of limit usually appears in viscosity solution theory in one form or another, even though the term $\Gamma$-convergence is rarely used.
Comparison principle
====================
The comparison principle for viscosity solutions that we present here follows very standard ideas in the subject. It originated from the idea of Jensen [@J] of sup and inf-convolutions. The method has been succesfully applied to integro-differential equations already [@Sayah]. In [@BI] a very general proof was given where the solutions are allowed to have an arbitrary growth at infinity. Our definitions do not quite fit into the previous framework mainly because we consider the general class of operators given by Definition \[d:axiomatic\] and we allow discontinuities outside of the domain of the equation $\Omega$. However, with small modifications, the same techniques can be adapted to our equations. We sketch the important ideas to prove the comparison principle in this section. There are two things that make the proof simpler than usual and are worth to be pointed out. One is the fact that in this paper we are only considering translation invariant equations. The other is that our operators are purely integro-differential (like instead of ) and they are well defined each time the functions are punctually $C^{1,1}$, which is very convenient to simplify the proof of Lemma \[l:Sclass\].
The result of this section that is important for the regularity theory is Theorem \[t:Sclass\], since we are going to apply it in section \[s:c1a\] to incremental quotients of a solution to an equation.
In order to have a comparison principle for a nonlinear operator $I$, we need to impose a minimal ellipticity condition to our collection of linear operators ${\mathcal{L}}$. The following assumption will suffice.
\[a:e1\] There is a constant $R_0 \geq 1$ so that for every $R>R_0$, there exists a $\delta>0$ (that could depend on $R$) such that for any operator $L$ in ${\mathcal{L}}$, we have that $L \varphi > \delta$ in $B_R$, where $\varphi$ is given by $$\varphi(x) = \min( R^3 , |x|^2)$$
In later sections we will need stronger assumptions to prove further regularity properties of the solutions. But for the comparison principle Assumption \[a:e1\] is enough. Note that assumption \[a:e1\] is very mild. It just says that given the particular function $\min( R^3 , |x|^2)$, the value of the operator will be strictly positive in $B_R$, but it does not require any unifom estimate on how that happens. If the operators $L \in {\mathcal{L}}$ are scale invariant, it justs means that when we apply them to $\min(1,|x|^2)$ they are strictly positive in some neighborhood of the origin.
\[t:comparison\] Let ${\mathcal{L}}$ be some class satisfying Assumption \[a:e1\]. Let $I$ be elliptic respect to ${\mathcal{L}}$ in the sense of definition \[d:axiomatic\]. Let $\Omega$ be a bounded open set, and $u$ and $v$ be two functions such that
- $u,v$ are bounded in ${\mathbb R}^n$.
- $u$ is lower-semicontinuous at every point in $\overline \Omega$.
- $v$ is upper-semicontinuous at every point in $\overline \Omega$.
- $Iu \geq f$ and $Iv \leq f$ in $\Omega$.
- $u \leq v$ in ${\mathbb R}^n \setminus \Omega$.
Then $u \leq v$ in $\Omega$.
By $u$ being lower-semicontinuous at every point in $\overline \Omega$, we mean that $u$ is semicontinuous in $\overline \Omega$ with respect to ${\mathbb R}^n$. The same applies for the function $v$.
We will use the usual idea of sup- and inf-convolutions in order to prove comparison. We start by defining these concepts
Given an upper semicontinuous function $u$, the sup-convolution approximation $u^{\varepsilon}$ is given by $$u^{\varepsilon}(x) = \sup_y u(x+y) - \frac{|y|^2}{{\varepsilon}}$$ On the other hand, if $u$ is lower semicontinuous, the inf-convolution $u_{\varepsilon}$ is given by $$u_{\varepsilon}(x) = \inf_y u(x+y) + \frac{|y|^2}{{\varepsilon}}$$
Notice that $u^{\varepsilon}\geq u$ and $u_{\varepsilon}\leq u$. Note also that $u^{\varepsilon}$ is a supremum of translations of $u$ and $u_{\varepsilon}$ is an infimum of translations of $u$.
The following two propositions are very standard, so we skip their proofs
\[p:gammaforinfconv\] If $u$ is bounded and lower-semicontinuous in ${\mathbb R}^n$ then $u_{\varepsilon}$ $\Gamma$-converges to $u$.
If $u$ is bounded and upper-semicontinuous in ${\mathbb R}^n$ then $-u^{\varepsilon}$ $\Gamma$-converges to $-u$.
\[p:supconvE\] If $Iu \geq f$ then $I u^{\varepsilon}\geq f - d_{\varepsilon}$. And if $Iv \leq f$ then $I v_{\varepsilon}\leq f + d_{\varepsilon}$, where $d_{\varepsilon}\to 0$ as ${\varepsilon}\to 0$ and depends on the modulus of continuity of $f$.
Proposition \[p:gammaforinfconv\] is a straightforward generalization of the fact that $u^{\varepsilon}\to u$ locally uniformly if $u$ is continuous.
\[l:uc11\] Let $u : {\mathbb R}^n \to {\mathbb R}$ be a lower semicontinuous function in ${\mathbb R}^n$ such that $I u \leq 0$ in $\Omega$ in the viscosity sense. Let $x$ be a point in $\Omega$ so that $u \in C^{1,1}(x)$. Then $Iu(x)$ is defined in the classical sense and $Iu(x) \leq 0$.
Use $u$ as a test function for itself with Lemma \[l:tc11\].
\[l:Sclass\] Let $I$ be elliptic in the sense of Definition \[d:axiomatic\]. Let $u$ and $v$ be two bounded functions such that
- $u$ is upper-semicontinuous and $v$ is lower-semicontinuous in ${\mathbb R}^n$.
- $Iu \geq f$ and $Iv \leq g$ in the viscosity sense in $\Omega$.
Then ${\mathrm{M}^+_\mathcal{L}}(u-v) \geq f-g$ in $\Omega$ in the viscosity sense.
By Proposition \[p:supconvE\], we have that also $I u^{\varepsilon}\geq f-d_{\varepsilon}$ and $I v_{\varepsilon}\leq g+d_{\varepsilon}$. Moreover $-u^{\varepsilon}\to -u$ and $v_{\varepsilon}\to v$ in the $\Gamma$ sense. By the stability of viscosity solutions under $\Gamma$ limits and since $d_{\varepsilon}\to 0$, it is enough to show that ${\mathrm{M}^+_\mathcal{L}}(u^{\varepsilon}- v_{\varepsilon}) \geq f - g - 2 d_{\varepsilon}$ in $\Omega$ for every ${\varepsilon}> 0$.
Let $\varphi$ be a $C^2$ function touching $(u^{\varepsilon}- v_{\varepsilon})$ by above at the point $x$. Note that for any ${\varepsilon}>0$, both functions $u^{\varepsilon}$ and $- v_{\varepsilon}$ are semiconvex, which means that for each of them there is a paraboloid touching it from below at every point $x$. If a $C^2$ function touches $(u^{\varepsilon}- v_{\varepsilon})$ by above at the point $x$, then both $u^{\varepsilon}$ and $-v_{\varepsilon}$ must be $C^{1,1}(x)$. But by Lemma \[l:tc11\] and Lemma \[l:puccicontinuous\], this means that we can evaluate $Iu^{\varepsilon}(x)$ and $Iv_{\varepsilon}(x)$ in the classical sense and $${\mathrm{M}^+_\mathcal{L}}(u^{\varepsilon}- v_{\varepsilon})(x) \geq I u^{\varepsilon}(x) - I v_{\varepsilon}(x) \geq f - g - 2 d_{\varepsilon}$$ which clearly implies that also ${\mathrm{M}^+_\mathcal{L}}\varphi(x) \geq f - g - 2 d_{\varepsilon}$ since $\varphi$ touches $u^{\varepsilon}- v_{\varepsilon}$ by above. Thus ${\mathrm{M}^+_\mathcal{L}}(u^{\varepsilon}- v_{\varepsilon}) \geq f-g - 2d_{\varepsilon}$ in $\Omega$ in the viscosity sense.
Taking ${\varepsilon}\to 0$ and using Lemma \[l:stability\] we finish the proof.
The result of Lemma \[l:Sclass\] is almost the result we need to prove the comparison principle, except that we want to allow functions $u$ and $v$ that are discontinuous outside of the domain $\Omega$. We fix this last detail in the following theorem.
\[t:Sclass\] Let $I$ be elliptic in the sense of Definition \[d:axiomatic\]. Let $u$ and $v$ be two bounded functions in ${\mathbb R}^n$ such that
- $u$ is upper-semicontinuous and $v$ is lower-semicontinuous in $\overline \Omega$
- $Iu \geq f$ and $Iv \leq g$ in the viscosity sense in $\Omega$.
Then ${\mathrm{M}^+_\mathcal{L}}(u-v) \geq f-g$ in $\Omega$ in the viscosity sense.
First we will show that there exist two sequences $u_k$ and $v_k$, lower and upper semicontinuous respectively, such that
- $u_k = u$ in $\overline \Omega$ for every $n$.
- $v_k = v$ in $\overline \Omega$ for every $n$.
- $u_k \to u$ and $v_k \to v$ a.e. in ${\mathbb R}^n \setminus \overline \Omega$.
- $I u_k \geq f_k$ and $I v_k \leq g_k$ with $f_k \to f$ and $g_k \to g$ locally uniformly in $\Omega$.
It is clear that we can find two sequences $u_k$ and $v_k$ satisfying the first three items above by doing a standard mollification of $u$ and $v$ away from $\Omega$ and then *filling the gap* in a semicontinuous way. What we will show is that then there are functions $f_k$ and $g_k$ for which the fourth item also holds.
The function $u_k-u$ vanishes in $\Omega$ and thus ${\mathrm{M}^-_\mathcal{L}}(u_k - u)$ is defined in the classical sense in $\Omega$. Moreover $${\mathrm{M}^-}_{{\mathcal{L}}} (u_k - u) (x) \geq -2 \int_{{\mathbb R}^n \setminus B_{{\mathrm{dist}}(x,{\partial}\Omega)}(x)} |u_k(x+y) - u(x+y)| K(y) {\; \mathrm{d}}y =: h_k(x)$$
Note that $h_k$ is continuous in $\Omega$ and by dominated convergence $h_k \to 0$ locally uniformly in $\Omega$ as $k \to \infty$.
Let $\varphi$ be function touching globally $u_k$ by above at a point $x$, assuming only that $\varphi \in C^{1,1}(x)$. Then also $\varphi + u - u_k \in C^{1,1}(x)$. But $\varphi + u - u_k$ touches $u$ from above at $x$, so by Lemma \[l:tc11\] $I(\varphi + u - u_k)(x) \geq f(x)$. But now $$I\varphi(x) \geq I(\varphi + u - u_k)(x) + {\mathrm{M}^-_\mathcal{L}}(u - u_k)(x) \geq f(x) + h_k(x)$$ so we prove the fourth item above for $u_k$ by choosing $f_k = f + h_k$. Similarly we prove it for $v_k$.
Now that we have such sequences $u_k$ and $v_k$ we apply Lemma \[l:stability\] and finish the proof.
\[l:maximumprinciple\] Let $u$ be a bounded function, upper-semicontinuous at every point in $\overline \Omega$, such that ${\mathrm{M}^+_\mathcal{L}}u \geq 0$ in the viscosity sense in $\Omega$. Then $\sup_{\Omega} u \leq \sup_{{\mathbb R}^n \setminus \Omega} u$.
Let us choose $R>R_0$ large enough so that $\Omega \subset B_R$. For any ${\varepsilon}>0$, let $\varphi_M$ be the function $$\varphi_M(x) = M + {\varepsilon}\left( 1 - \min(R^3,|x|^2) \right) \ .$$
Note that $M \leq \varphi_M(x) \leq M+{\varepsilon}$ for every $x \in {\mathbb R}^n$. Also, by Assumption \[a:e1\], there is a $\delta>0$ such that ${\mathrm{M}^+_\mathcal{L}}\varphi_M(x) \leq -{\varepsilon}\delta $ for any $x \in B_R$.
Let $M_0$ be the smallest value of $M$ for which $\varphi_M \geq u$ in ${\mathbb R}^n$. We will show that $M_0 \leq \sup_{{\mathbb R}^n \setminus \Omega} u$. Otherwise, if $M_0 > \sup_{{\mathbb R}^n \setminus \Omega} u$, there must be a point $x_0 \in \Omega$ for which $u(x_0) = \varphi_{M_0} u(x_0)$. But in that case $\varphi_{M_0}$ would touch $u$ by above at $x_0 \in \Omega$ and by the definition of ${\mathrm{M}^+_\mathcal{L}}u \geq 0$ in the viscosity sense we would have that ${\mathrm{M}^+_\mathcal{L}}\varphi_{M_0} \geq 0$ arriving to a contradiction. Therefore, for every $x \in {\mathbb R}^n$, we have $$\begin{aligned}
u(x) &\leq \varphi_{M_0} (x) \\
&\leq M_0 + {\varepsilon}\\
&\leq \sup_{{\mathbb R}^n \setminus \Omega} u + {\varepsilon}\end{aligned}$$
We finish the proof by making ${\varepsilon}\to 0$.
By theorem \[t:Sclass\], ${\mathrm{M}^+_\mathcal{L}}(u-v) \geq 0$ in $\Omega$. Then Lemma \[l:maximumprinciple\] says that $\sup_{\Omega} (u-v) \leq \sup_{{\mathbb R}^n \setminus \Omega} (u-v)$, which finishes the proof.
Once we have the comparison principle for semicontinuous sub and supersolutions, existence of the solution of the Dirichlet problem follows using the Perron’s method [@I] as long as we can construct suitable barriers.
Second order elliptic equations {#s:secondorder}
===============================
It is well known that $$\lim_{\sigma \to 2} \int_{{\mathbb R}^n} \frac{c_n (2-\sigma)}{|y|^{n+\sigma}} {\delta}(u,x,y) {\; \mathrm{d}}y = \lim_{\sigma \to 2} -(-{\triangle})^{\sigma/2}u(x) = {\triangle}u(x)$$
With a simple change of variables $z=Ay$, we arrive to the following identity $$\label{e:i1}
\lim_{\sigma \to 2} \int_{{\mathbb R}^n} \frac{c_n (2-\sigma)}{\det A |A^{-1} z|^{n+\sigma}} {\delta}(u,x,z) {\; \mathrm{d}}z = \sum a_{ij} u_{ij}(x)$$ where $\{a_{ij}\}$ are the entries of $AA^t$.
This means that we can recover any linear second order elliptic operator as a limit of integro-differential ones like . Moreover let us say we have a fully nonlinear operator of the form $F(D^2 u)$. Let us assume the function $F$ is Lipschitz and monotone in the space of symmetric matrices. Then $F$ can be written as $$F(M) = \inf_\alpha \sup_\beta \sum a_{ij}^{\alpha \beta} M_{ij}$$ for some collection of positive matrices $\{a_{ij}^{\alpha \beta}\} = A_{\alpha\beta}A_{\alpha\beta}^t$. Thus any elliptic fully nonlinear operator can be recovered as a limit of integro-differential operators as $$F(D^2 u) = \lim_{\sigma \to 2} \left( \inf_\alpha \sup_\beta \int \frac{c_n (2-\sigma)}{\det A_{\alpha\beta} |A_{\alpha\beta}^{-1} z|^{n+\sigma}} {\delta}(u,x,z) {\; \mathrm{d}}z \right)$$ as long as the limit commutes with the operations of infimum and supremum. That is going to be the case every time the convergence is uniform in $\alpha$ and $\beta$ which is the case for example if the matrices $A_{\alpha\beta}$ are uniformly elliptic.
Another posibility is to take a family $A_{\alpha \beta}$ so that $$F(D^2 u) = \lim_{\sigma \to 2} \left( \inf_\alpha \sup_\beta \int \frac{{\delta}(u,x,A_{\alpha \beta} y)}{|y|^{n+\sigma}} {\; \mathrm{d}}y \right) \ .$$
Note that we can also consider operators of the form $$I u(x) := (2-\sigma) \int \frac{1}{|y|^{n+\sigma-2}} G \left( \frac{{\delta}(u,x,y)}{|y|^2} , y \right) {\; \mathrm{d}}y$$ with $G(d,y)$ being an arbitrary function, lipschitz and monotone in $d$, such that $G(0,y)=0$. This suggests an unusual family of second order nonlinear equations: for $P$ a quadratic polynomial $$F(D^2 P) = \int_{S^1} G(P(\sigma),\sigma) {\; \mathrm{d}}\sigma \ .$$
A nonlocal ABP estimate. {#s:abp}
========================
The Alexandroff-Backelman-Pucci (ABP) estimate is a key ingredient in the proof of Harnack inequality by Krylov and Sofonov. It is the relation that allows us to pass from an estimate in measure, to a pointwise estimate. In this section we obtain an estimate for integro-differential equations that converges to the ABP estimate as $\sigma$ approaches $2$. In a later section, we will use this nonlocal version of the ABP theorem to prove the Harnack inequality for $\sigma$ close to $2$.
In this and the next few sections we will consider the class ${\mathcal{L}}_0$ defined by the condition \[e:uniformellipticity\]. We write ${\mathrm{M}^+}$ and ${\mathrm{M}^-}$ to denote ${\mathrm{M}^+}_{{\mathcal{L}}_0}$ and ${\mathrm{M}^-}_{{\mathcal{L}}_0}$.
Let $u$ be a function that is not positive outside the ball $B_1$. Consider its concave envelope $\Gamma$ in $B_3$ defined as $$\Gamma(x) := \begin{cases}
\min {\left\{p(x) : \text{for all planes } p>u \text{ in } B_2\right\}} & \text{in } B_3 \\
0 & \text{in } {\mathbb R}^n \setminus B_3
\end{cases}$$
\[l:abp2\] Let $u \leq 0$ in ${\mathbb R}^n \setminus B_1$. Let $\Gamma$ be its concave envelope in $B_3$. Assume ${\mathrm{M}^+}u(x) \geq -f(x)$ in $B_1$. Let ${\rho_0}= 1/(8\sqrt{n})$, $r_k = {\rho_0}2^{-\frac{1}{2-\sigma}-k}$ and $R_k(x) = B_{r_k}(x) \setminus B_{r_{k+1}}(x)$.
There is a constant $C_0$ depending only on $n$, $\lambda$ and $\Lambda$ (but not on $\sigma$) such that for any $x \in \{u = \Gamma\}$ and any $M>0$, there is a $k$ such that $$\label{e:abp1}
|R_k(x) \cap \{u(y) < u(x) + (y-x)\cdot {\nabla}\Gamma(x) - M r_k^2 \}| \leq C_0 \frac{f(x)}{M} |R_k(x)|$$ where ${\nabla}\Gamma$ stands for any element of the superdifferential of $\Gamma$ at $x$, which will coincide with its gradient, and also the gradient of $u$, when these functions are differentiable.
Since $u$ can be touched by a plane from above at $x$, from Lemma \[l:classic2\], ${\mathrm{M}^+}u(x)$ is defined classically and we have $${\mathrm{M}^+}u(x) = (2-\sigma) \int_{{\mathbb R}^n} \frac{\Lambda {\delta}^+ - \lambda {\delta}^-}{|y|^{n+\sigma}} {\; \mathrm{d} x}\ .$$
Recall ${\delta}= {\delta}(u,x,y) := u(x+y) + u(x-y) - 2 u(x)$.
Note that if both $x+y \in B_3$ and $x-y \in B_3$ then ${\delta}(u,x,y) \leq 0$, since $u(x) = \Gamma(x) = p(x)$ for some plane $p$ that remains above $u$ in the whole ball $B_3$. Moreover, if either $x+y \notin B_3$ or $x-y \notin B_3$, then both $x+y$ and $x-y$ are not in $B_1$, so $u(x+y)\leq 0$ and $u(x-y)\leq 0$. Therefore, in any case ${\delta}(u,x,y) \leq 0$. Thus we have $$\begin{aligned}
-f(x) &\leq {\mathrm{M}^+}u(x) = (2-\sigma) \int_{{\mathbb R}^n} \frac{- \lambda {\delta}^-}{|y|^{n+\sigma}} {\; \mathrm{d}}y \\
&\leq (2-\sigma) \int_{B_{r_0}(x)} \frac{- \lambda {\delta}^-}{|y|^{n+\sigma}} {\; \mathrm{d}}y\end{aligned}$$ where $r_0 = {\rho_0}2^{-\frac{1}{2-\sigma}}$.
Splitting the integral in the rings $R_k$ and reorganizing terms we obtain $$f(x) \geq (2-\sigma) \lambda \sum_{k=0}^\infty \int_{R_k(x)} \frac{{\delta}^-}{|y|^{n+\sigma}} {\; \mathrm{d} x}$$
Let us assume that equation does not hold. We will arrive to a contradiction. We can use the oposite of to estimate each integral in the terms of the previous equation. $$\begin{aligned}
f(x) &\geq (2-\sigma) \lambda \sum_{k=0}^\infty \int_{R_k(x)} \frac{{\delta}^-}{|y|^{n+\sigma}} {\; \mathrm{d} x}\\
&\geq c (2-\sigma) \sum_{k=0}^\infty M \frac{r_k^2}{r_k^\sigma} C_0 \frac{f(x)}{M} \\
&\geq c (2-\sigma) \frac{{\rho_0}^2}{1-2^{-(2-\sigma)}} C_0 f(x) \\
&\geq c C_0 f(x)\end{aligned}$$ where the last inequality holds because $(2-\sigma) \frac{1}{1-2^{-(2-\sigma)}}$ remains bounded below for $\sigma \in (0,2)$. By choosing $C_0$ large enough, we obtain a contradiction.
Note that Lemma \[l:abp2\] implies that if ${\mathrm{M}^+}u (x) \geq g(x)$ then $u(x) \neq \Gamma(x)$ at every point where $g(x)>0$.
Lemma \[l:abp2\] would hold for any particular choice of $\rho_0$ (modifying $C_0$ accordingly). The particular choice $\rho_0=1/8\sqrt{n}$ is convenient for the proofs in section 9 later in this paper.
\[l:abp1\] Let $\Gamma$ be a concave function in $B_r$. Assume that for a small ${\varepsilon}$ $$| \{ y : \Gamma(y) < \Gamma(x) + (y-x) \cdot {\nabla}\Gamma(x) - h \} \cap (B_r \setminus B_{r/2})| \leq {\varepsilon}|B_r \setminus B_{r/2}|$$ then $\Gamma(y) \geq \Gamma(x) + (y-x) \cdot {\nabla}\Gamma(x) - h$ in the whole ball $B_{r/2}$.
Let $y \in B_{r/2}$. There are two points $y_1$, $y_2$ in $B_r \setminus B_{r/2}$ such that
1. $y = (y_1+y_2)/2$.
2. $|y_1-x|=|y_2-x|=\frac 34 r$.
Let us consider the balls $B_1 = B_{r/4}(y_1)$ and $B_2 = B_{r/4}(y_2)$ (See Figure \[f:b1b2\]). They are symmetric respect to $y$ and they are completely contained in $B_r \setminus B_{r/2}$. If ${\varepsilon}$ is small enough, there will be two points $z_1 \in B_1$ and $z_2 \in B_2$ so that
1. $y = (z_1+z_2)/2$
2. $\Gamma(z_1) \geq \Gamma(x) + (z_1-x) \cdot {\nabla}\Gamma(x) - h$
3. $\Gamma(z_2) \geq \Gamma(x) + (z_2-x) \cdot {\nabla}\Gamma(x) - h$
and by the concavity of $\Gamma$ we finish the proof since $\Gamma(y) \geq (\Gamma(z_1)+\Gamma(z_2))/2$.
\[c:abp3\] For any ${\varepsilon}_0>0$ there is a constant $C$ such that for any function $u$ with the same hypothesis as in Lemma \[l:abp2\], there is an $r \in (0,{\rho_0}2^{-\frac{1}{2-\sigma}})$ such that: $$\begin{aligned}
&\frac{{\left\vert{\left\{y \in B_r \setminus B_{r/2}(x) : u(y) < u(x) + (y-x) \cdot {\nabla}\Gamma(x) - C f(x) r^2\right\}}\right\vert}}{ |B_r(x) \setminus B_{r/2}(x)|} \leq {\varepsilon}_0 . \label{e:abp31}\\
&|{\nabla}\Gamma (B_{r/4}(x))| \leq C f(x)^n |B_{r/4}(x)| \label{e:abp32}\end{aligned}$$ Recall ${\rho_0}= 1/(8\sqrt{n})$.
From Lemma \[l:abp2\] we have right away by choosing $M = C f(x)/{\varepsilon}_0$. Equation follows then as a consequence of Lemma \[l:abp1\] and concavity.
\[t:abp\] Let $u$ and $\Gamma$ be functions as in Lemma \[l:abp2\]. There is a finite family of (open) cubes $Q_j$ ($j=1, \dots, m$) with diameters $d_j$ such that
1. Any two cubes $Q_i$ and $Q_j$ in the family do not intersect.
2. $\{u = \Gamma\} \subset \bigcup_{j=1}^m \overline Q_j$.
3. $\{u = \Gamma\} \cap \overline Q_j \neq \emptyset$ for any $Q_j$.
4. $d_j \leq {\rho_0}2^{\frac{-1}{2-\sigma}}$, where ${\rho_0}= 1/(8\sqrt{n})$.
5. $|{\nabla}\Gamma(\overline Q_j)| \leq C(\max_{\overline Q_j} f )^n |Q_j|$.
6. $|\{y \in 4 \sqrt{n} Q_j : u(y) > \Gamma(y) - C (\max_{\overline Q_j} f ) d_j^2\}| \geq \mu |Q_j|$.
The constants $C>0$ and $\mu>0$ depend on $n$, $\Lambda$ and $\lambda$ (but not on $\sigma$).
In order to obtain such family we start by covering $B_1$ with a tiling of cubes of diameter ${\rho_0}2^{\frac{-1}{2-\sigma}}$. We discard all those that do not intersect $\{u = \Gamma\}$. Whenever a cube does not satisfy (e) and (f), we split it into $2^n$ cubes of half diameter and discard those whose closure does not intersect $\{u = \Gamma\}$. The problem is to prove that eventually all cubes satisfy (e) and (f) and this process finishes after a finite number of steps.
Let us assume the process does not finish in a finite number of steps. We assume it produces an infinite sequence nested of cubes. The intersection of their closures will be a point $x_0$. Since all of them intersect the contact set $\{u = \Gamma\}$, which is a closed set, then $u(x_0) = \Gamma(x_0)$. We will now find a contradiction by showing that eventually one of these cubes containing $x_0$ will not split.
Given ${\varepsilon}_0>0$, by Corollary \[c:abp3\], there is a radius $r$ with $0<r<{\rho_0}2^{\frac{-1}{2-\sigma}}$ such that $$\begin{aligned}
&\frac{{\left\vert{\left\{y \in B_r(x_0) \setminus B_{r/2}(x_0) : u(y) < u(x_0) + (y-x_0) \cdot {\nabla}\Gamma(x_0) - C f(x_0) r^2\right\}}\right\vert}}{ |B_r(x_0) \setminus B_{r/2}(x_0)|} \leq {\varepsilon}_0. \label{e:a1}\\
&|{\nabla}\Gamma (B_{r/4}(x_0))| \leq C f(x_0)^n |B_{r/4}(x_0)| \label{e:a2}\end{aligned}$$
There is a cube $Q_j$, with $x_0 \in \overline Q_j$, with diameter $d_j$, such that $r/4 < d_j < r/2$. Therefore (see Figure \[f:qandb\]) $$\begin{aligned}
B_{r/2}(x_0) &\supset \overline Q_j \\
B_{r}(x_0) &\subset 4 \sqrt{n} Q_j\end{aligned}$$
Recall that in $B_2$, $\Gamma(y) \leq u(x_0) + (y-x_0) \cdot {\nabla}\Gamma(x_0)$ simply because $\Gamma$ is concave and $\Gamma(x_0) = u(x_0)$. Using and that $d_j$ and $r$ are comparable, we get $$\begin{aligned}
|\{ y \in 4 \sqrt{n} Q_j &: u(y) \geq \Gamma(y) - C (\max_{\overline Q_j} f) d_j^2 \} | \geq \\
&\geq {\left\vert{\left\{y \in 4\sqrt{n} Q_j : u(y) \geq u(x_0) + (y-x_0) \cdot {\nabla}\Gamma(x_0) - C f(x_0) r^2 \right\}}\right\vert} \\
&\geq (1-{\varepsilon}_0) |B_r(x_0) \setminus B_{r/2}(x_0)| \geq \mu |Q_j|\end{aligned}$$
Thus (f) follows. Moreover, since $\overline Q_j \subset B_r$, also (e) holds for $Q_j$. Therefore $Q_j$ would not be split and the process must stop.
Note that the upper bound for the diameters ${\rho_0}2^\frac{-1}{2-\sigma}$ becomes very small as $\sigma$ is close to $2$. If we add $\sum |{\nabla}\Gamma(Q_j)|$ and let $\sigma \to 2$, we obtain the classical Alexandroff estimate as the limit of the Riemann sums. For each $\sigma>0$ we have $$|{\nabla}\Gamma(\{u = \Gamma\})| \leq \sum_j C(\max_{\overline Q_j} f^+ )^n |Q_j| \ .$$ As $\sigma \to 2$, the cube covering of $\{u=\Gamma\}$ becomes thinner and the above becomes the integral $$|{\nabla}\Gamma(\{u = \Gamma\})| \leq C \int_{\{u=\Gamma\}} f^+(x)^n {\; \mathrm{d}}x \ .$$
A special function {#s:specialfunction}
==================
In this section we only construct a special function that is a subsolution of a minimal equation outside a small ball. The importance of this function is that it is strictly positive in a larger ball and we will use that fact in a later section to prove the Harnack inequality.
\[l:negp\] There is a $p>0$ and $\sigma_0 \in (0,2)$ such that the function $$f(x) = \min( 2^p , |x|^{-p} )$$ is a subsolution to $$\label{e:t1}
{\mathrm{M}^-}f(x) \geq 0$$ for every $\sigma_0<\sigma<2$ and $|x|>1$.
It is enough to show for $x = e_1 = (1,0,\dots,0)$. For every other $x$ such that $|x|=1$, the relation follows by rotation. If $|x|>1$, we can consider the function $\tilde f(y) = |x|^p f(|x|y) \geq f(y)$, thus ${\mathrm{M}^-}f(x) = C {\mathrm{M}^-}\tilde f(x/|x|) \geq C {\mathrm{M}^-}f(x/|x|) > 0$.
Let $x = e_1 = (1,0,\dots,0)$. We use the following elementary relations that hold for any $a>b>0$ and $q>0$, $$\begin{aligned}
(a+b)^{-q} &\geq a^{-q} (1 - q \frac b a) \\
(a+b)^{-q} + (a-b)^{-q} &\geq 2 a^{-q} + \frac{1}{2} q (q+1) b^2 a^{-q-2}\end{aligned}$$ then for $|y|<1/2$, $$\begin{aligned}
{\delta}&= |x+y|^{-p} + |x-y|^{-p} - 2|x|^{-p} \\
&= (1 + |y|^2 + 2 y_1)^{-p/2}+(1 + |y|^2 - 2 y_1)^{-p/2}-2 \\
&\geq 2 (1+|y|^2)^{-p/2} + \frac{1}{2}p(p+2) y_1^2 (1 + |y|^2)^{-p/2-2} - 2 \\
&\geq p \left( -|y|^2 + \frac{1}{2}(p+2) y_1^2 - \frac{1}{4}(p+2)(p+4) y_1^2 |y|^2 \right)\end{aligned}$$
We choose $p$ large such that $$\label{e:r1}
\frac{1}{2} (p+2) \lambda \int_{{\partial}B_1} y_1^2 {\; \mathrm{d}}\sigma(y) - \Lambda |{\partial}B_1| = \delta_0 > 0$$
We use the above relation to bound the part of the integral in the definition of ${\mathrm{M}^-}$ for which $y$ stays in a small ball $B_r$ (with $r<1/2$). We estimate ${\mathrm{M}^-}f(e_1)$. $$\begin{aligned}
{\mathrm{M}^-}f(e_1) &= (2-\sigma) \int_{B_r} \frac{\lambda {\delta}^+ - \Lambda {\delta}^-}{|y|^{n+\sigma}} {\; \mathrm{d}}y + (2-\sigma) \int_{{\mathbb R}^n \setminus B_r} \frac{\lambda {\delta}^+ - \Lambda {\delta}^-}{|y|^{n+\sigma}} {\; \mathrm{d}}y \\
&\geq (2-\sigma) C \int_0^r \frac{ \lambda p \delta_0 s^2 - \frac{1}{4} p(p+2)(p+4) C \Lambda s^4}{s^{1+\sigma}} {\; \mathrm{d}}s - (2-\sigma) \int_{{\mathbb R}^n \setminus B_r} \Lambda \frac{2^p}{|y|^{n+\sigma}} {\; \mathrm{d}}y \\
&\geq c r^{2-\sigma} p \delta_0 - p (p+2)(p+4) C \frac{2-\sigma}{4-\sigma} r^{4-\sigma} - \frac{2-\sigma}{\sigma} C 2^{p+1} r^{-\sigma}\end{aligned}$$ where we used to bound the first integral and that $0 \leq f(x) \leq 2^p$ to bound the second integral. Now we choose (and fix) $r \in (0,1/2)$ small, and then take $\sigma_0$ close enough to $2$, so that if $2>\sigma>\sigma_0$, the factor $(2-\sigma)$ makes the second and third terms small enough so that we get $${\mathrm{M}^-}f(e_1) \geq \frac{c r^{2-\sigma} p \delta_0 }{2} > 0$$ which finishes the proof.
\[c:negp\] Given any $\sigma_0 \in (0,2)$, there is a $p>0$ and $\delta$ such that the function $$f(x) = \min( \delta^{-p} , |x|^{-p} )$$ is a subsolution to $${\mathrm{M}^-}f(x) \geq 0$$ for every $\sigma_0<\sigma<2$ and $|x|>1$.
The only difference with Lemma \[l:negp\] is that now we are given the value of $\sigma_0$ beforehand. Let $\sigma_1$ and $p_0$ be the $\sigma_0$ and $p$ of Lemma \[l:negp\]. So we know that for $\sigma > \sigma_1$, the result of the Corollary holds if $\delta = 1/2$ and $p=p_0$. If we take $\delta<1/2$ we are only making the function larger away from $x$, so the result will still hold for $\sigma > \sigma_1$. Now we will pick $\delta$ smaller so that the result also holds for $\sigma_0 < \sigma \leq \sigma_1$.
The key is that if $p \geq n$, $|x|^{-p}$ is not integrable around the origin. So we take $p = \max(p_0,n)$. Now, let $x = e_1$ as in the proof of lemma \[l:negp\]. Assume $\sigma_0 < \sigma \leq \sigma_1$. We write $$\begin{aligned}
{\mathrm{M}^-}f(e_1) &= (2-\sigma) \int_{{\mathbb R}^n} \frac{\lambda {\delta}^+}{|y|^{n+\sigma}} {\; \mathrm{d}}y - (2-\sigma) \int_{{\mathbb R}^n} \frac{\Lambda {\delta}^-}{|y|^{n+\sigma}} {\; \mathrm{d}}y \\
&=: I_1 + I_2\end{aligned}$$ where $I_1$ and $I_2$ represent the two terms in the right hand side above. Since $\sigma > \sigma_0$, $f \in C^2(x)$ and $f$ is bounded below, we have $I_2 \geq -C$ for some constant $C$ depending on $\sigma_0$, $\lambda$, $\Lambda$ and dimension. On the other hand, since $\sigma \leq \sigma_1$ and $(|x+y|^{-p}+|x-y|^{-p}-|x|^{-p})^+$ is not integrable, if we choose $\delta$ small enough we can make $I_1$ be as large as we wish. In particular, we can choose $\delta$ such that $I_1 > C > -I_2$, thus ${\mathrm{M}^-}f(e_1) > 0$.
\[c:phi\] Given any $\sigma_0 \in (0,2)$, there is a function $\Phi$ such that
- $\Phi$ is continuous in ${\mathbb R}^n$.
- $\Phi(x) = 0$ for $x$ outside $B_{2\sqrt{n}}$.
- $\Phi(x) > 2$ for $x \in Q_3$.
- ${\mathrm{M}^-}\Phi > -\psi(x)$ in ${\mathbb R}^n$ for some positive function $\psi(x)$ supported in $\overline B_{1/4}$.
for every $\sigma>\sigma_0$.
Let $p$ and $\delta$ be as in Corollary \[c:negp\]. We consider $$\Phi = c \begin{cases}
0 &\text{in } {\mathbb R}^n \setminus B_{2\sqrt{n}} \\
|x|^{-p} - (2\sqrt{n})^{-p} &\text{in } B_{2\sqrt{n}} \setminus B_\delta \\
q &\text{in } B_\delta
\end{cases}$$ where $q$ is a quadratic paraboloid chosen so that $\Phi$ is $C^{1,1}$ accross ${\partial}B_\delta$. We choose the constant $c$ so that $\Phi(x) > 2$ for $x \in Q_3$ (Recall $Q_3 \subset B_{3\sqrt{n}/2} \subset B_{2\sqrt{n}}$). Since $\Phi \in C^{1,1}(B_{2\sqrt{n}})$, ${\mathrm{M}^-}\Phi$ is continuous in $B_{2\sqrt{n}}$ and from Corollary \[c:negp\], ${\mathrm{M}^-}\Phi \geq 0$ outside $B_{1/4}$.
Point estimates
===============
The main ingredient in the proof of Harnack inequality, as shown in [@CC], is a lemma that links a pointwise estimate with an estimate in measure. The corresponding lemma in our context is the following.
\[l:keylemma\] Let $\sigma>\sigma_0>0$. There exist constants ${\varepsilon}_0>0$, $0<\mu<1$ and $M>1$ (depending only on $\sigma_0$, $\lambda$, $\Lambda$ and dimension) such that if
- $u \geq 0$ in ${\mathbb R}^n$.
- $\inf_{Q_3} u \leq 1$.
- ${\mathrm{M}^-}u \leq {\varepsilon}_0$ in $Q_{4\sqrt{n}}$.
then $ |\{u \leq M \} \cap Q_1 | > \mu $.
By $Q_r(x)$ we mean the open cube $\{ y : |y_j - x_j| \leq r/2 \text{ for every j} \}$, and $Q_r := Q_r(0)$. We will also use the following notation for dilations: if $Q = Q_r(x)$, then $\lambda Q := Q_{\lambda r}(x)$.
If we assume $\sigma \leq \sigma_1 < 2$, there is a simpler proof of Lemma \[l:keylemma\] using the ideas from [@S1]. The result here is more involved because we want an estimate that remains uniform as $\sigma \to 2$.
Consider $v := \Phi - u$, where $\Phi$ is the special function constructed in Corollary \[c:phi\]. We want to apply Theorem \[t:abp\] (rescaled) to $v$. Note that ${\mathrm{M}^+}v \geq {\mathrm{M}^-}\Phi - {\mathrm{M}^-}(u) \geq -\psi-{\varepsilon}_0$. Let $\Gamma$ be the concave envelope of $v$ in $B_{6\sqrt{n}}$.
Let $Q_j$ be the family of cubes given by Theorem \[t:abp\]. We have $$\begin{aligned}
\max v &\leq C | {\nabla}\Gamma(B_{2 \sqrt{n}}) |^{1/n} \leq \left( \sum_j |{\nabla}\Gamma(\overline Q_j)| \right)^{1/n} \\
& \leq \left( C \sum_j ( \max_{Q_j} (\psi+{\varepsilon}_0)^+ )^n |Q_j| \right)^{1/n} \\
& \leq C {\varepsilon}_0 + C \left( \sum_j (\max_{Q_j} \psi^+)^n |Q_j| \right)^{1/n}\end{aligned}$$
However, since $\max_{Q_3} u \leq 1$ and $\min_{Q_3} \Phi \geq 2$, then $\max v \geq 1$ and we have $$1 \leq C {\varepsilon}_0 + C \left( \sum_j (\max_{Q_j} \psi^+)^n |Q_j| \right)^{1/n}$$
If we choose ${\varepsilon}_0$ small enough, this will imply $$\frac{1}{2} \leq C \left( \sum_j (\max_{Q_j} \psi^+)^n |Q_j| \right)^{1/n}$$
Recall that $\psi$ is supported in $\overline B_{1/4}$ and it is bounded, thus: $$\frac 12 \leq C \left( \sum_{Q_j \cap B_{1/4} \neq \emptyset} |Q_j| \right)^{1/n}$$
Which provides a bound below for the sum of the volumes of the cubes $Q_j$ that intersect $B_{1/4}$. $$\label{e:h1}
\sum_{Q_j \cap B_{1/4} \neq \emptyset} |Q_j| \geq c$$
The diameters of all cubes $Q_j$ are bounded by ${\rho_0}2^{\frac{-1}{2-\sigma}}$, which is always smaller than ${\rho_0}= 1/(8 \sqrt{n})$. Therefore, every time $Q_j$ intersects $B_{1/4}$, the cube $4\sqrt{n}Q_j$ will be contained in $B_{1/2}$.
Let $M_0 := \min_{B_{1/2}} \Phi$. By Theorem \[t:abp\], we have $$\label{e:h2}
|\{x \in 4 \sqrt{n} Q_j : v(x) \geq \Gamma(x) - C d_j^2 \} | \geq c |Q_j|$$ and $C d_j^2<C\rho_0^2$.
Let us consider the cubes $4 \sqrt{n} Q_j$, for every cube $Q_j$ that intersects $B_{1/4}$. It provides an open cover of the union of the corresponding cubes $\overline Q_j$ and it is contained in $B_{1/2}$. We take a subcover with finite overlapping that also covers the union of the original $\overline Q_j$. Combining with we obtain $$| \{ x \in B_{1/2} : v(x) \geq \Gamma(x) - C \rho_0^2 \} | \geq c$$ Then $$| \{ x \in B_{1/2} : u(x) \leq M_0 + C \rho_0^2 \} | \geq c$$
Let $M = M_0 + C \rho_0^2$. Since $B_{1/2} \subset Q_1$, we have $$| \{ x \in Q_1 : u(x) \leq M \} | \geq c$$ which finishes the proof.
Lemma \[l:keylemma\] is the key to the proof of Harnack inequality. The following Lemma is a consequence of Lemma \[l:keylemma\] as it is shown in Lemma 4.6 in [@CC]. We have intentionally written Lemma \[l:keylemma\] and the following one identical to their corresponding versions in [@CC].
Let $u$ be as in lemma \[l:keylemma\]. Then $$| \{ u > M^k \} \cap Q_1 | \leq (1-\mu)^k$$ for $k = 1,2,3,\dots$, where $M$ and $\mu$ are as in Lemma \[l:keylemma\].
As a consequence, we have that $$|\{u \geq t\} \cap Q_1| \leq d t^{-{\varepsilon}} \qquad \forall t > 0$$ where $d$ and ${\varepsilon}$ are positive universal constants.
By a standard covering argument we obtain the following theorem.
\[t:wharnack0\] Let $u \geq 0$ in ${\mathbb R}^n$, $u(0) \leq 1$, and ${\mathrm{M}^-}u \leq {\varepsilon}_0$ in $B_{2}$ (supersolution). Assume $\sigma \geq \sigma_0$ for some $\sigma_0>0$. Then $$|\{ u > t \} \cap B_1 | \leq C t^{-{\varepsilon}} \qquad \text{for every $t>0$.}$$ where the constant $C$ depends on $\lambda$, $\Lambda$, $n$ and $\sigma_0$.
Scaling the above theorem we obtain the following version.
\[t:wharnack\] Let $u \geq 0$ in ${\mathbb R}^n$ and ${\mathrm{M}^-}u \leq C_0$ in $B_{2r}$ (supersolution). Assume $\sigma \geq \sigma_0$ for some $\sigma_0>0$. Then $$|\{ u > t \} \cap B_r | \leq C r^n (u(0)+C_0 r^\sigma)^{\varepsilon}t^{-{\varepsilon}} \qquad \text{for every $t$.}$$ where the constant $C$ depends on $\lambda$, $\Lambda$, $n$ and $\sigma_0$.
For second order equations, Theorems \[t:wharnack0\] and \[t:wharnack\] are referred in the literature as $u$ being in $L^{\varepsilon}$ (See [@CC]).
Harnack inequality
==================
Harnack inequality is a very important tool in analysis. In this section we obtain a version for integro-differential equations. Our estimate depends only on a lower bound $\sigma \geq \sigma_0>0$ but it remains uniform as $\sigma \to 2$. In that respect, we can consider this estimate as a generalization of Krylov-Safonov Harnack inequality.
This section is not needed for the rest of the paper because we will prove our regularity results using Theorem \[t:wharnack\] only. A reader interested only in the regularity results can skip this section.
\[t:harnack\] Let $u \geq 0$ in ${\mathbb R}^n$, ${\mathrm{M}^-}u \leq C_0$ and ${\mathrm{M}^+}u \geq -C_0$ in $B_2$. Assume $\sigma \geq \sigma_0$ for some $\sigma_0>0$. Then $u(x) \leq C (u(0)+C_0)$ for every $x \in B_{1/2}$.
Dividing by $u(0)+C_0$, it is enough to consider $u(0) \leq 1$ and $C_0=1$.
Let ${\varepsilon}>0$ be the one from Theorem \[t:wharnack\]. Let $\gamma = n/{\varepsilon}$. Let us consider the minimum value of $t$ such that $$u(x) \leq h_t(x) := t (1-|x|)^{-\gamma} \text { for every } x \in B_1.$$ There must be an $x_0 \in B_1$ such that $u(x_0) = h_t(x_0)$, otherwise we could make $t$ smaller. Let $d = (1-|x_0|)$ be the distance from $x_0$ to ${\partial}B_1$.
For $r=d/2$, we want to estimate the portion of the ball $B_r(x_0)$ covered by $\{u < u(x_0)/2\}$ and by $\{u > u(x_0)/2\}$. We will show that $t$ cannot be too large. In this way we obtain the result of the theorem, since the upper bound $t<C$ implies that $u(x) < C (1-|x|)^{-\gamma}$.
Let us first consider $A := \{u > u(x_0)/2\}$. By the $L^{\varepsilon}$ estimate (Theorem \[t:wharnack0\]) we have $$\begin{aligned}
|A \cap B_1| &\leq C {\left\vert\frac{2}{u(x_0)}\right\vert}^{{\varepsilon}} \\
&\leq C t^{-{\varepsilon}} d^n\end{aligned}$$
Whereas $|B_r| = C d^n$, so if $t$ is large, $A$ can cover only a small portion of $B_r(x_0)$ at most. $$\label{e:a0}
{\left\vert \{u>u(x_0)/2\} \cap B_r(x_0) \right\vert} \leq C t^{-{\varepsilon}} {\left\vertB_r\right\vert}$$
In order to get a contradiction, we will show that ${\left\vert\{u<u(x_0)/2\} \cap B_r(x_0)\right\vert} \leq (1-\delta) B_r$ for a positive constant $\delta$ independent of $t$.
We estimate ${\left\vert\{u<u(x_0)/2\} \cap B_{\theta r} (x_0)\right\vert}$ for $\theta>0$ small. For every $x \in B_{\theta r} (x_0)$ we have $u(x) \leq h_t(x) \leq (d-\theta d/2)^{-\gamma} \leq u(x_0) (1-\theta/2)^{-\gamma}$, with $(1-\theta/2)^{-\gamma}$ close to one.
Let us consider $$v(x) = (1-\theta/2)^{-\gamma} u(x_0) - u(x)$$ so that $v \geq 0$ in $B_{\theta r}$, and also ${\mathrm{M}^-}v \leq 1$ since ${\mathrm{M}^+}u \geq -1$. We would want to apply Theorem \[t:wharnack\] to $v$. The only problem is that $v$ is not positive in the whole domain but only on $B_{\theta r}$. In order to apply Theorem \[t:wharnack\] we have to consider $w=v^+$ instead, and estimate the change in the right hand side due to the truncation error.
We want to find an upper bound for ${\mathrm{M}^-}w = {\mathrm{M}^-}v^+$ instead of ${\mathrm{M}^-}v$. We know that $${\mathrm{M}^-}v(x) = (2-\sigma) \int_{{\mathbb R}^n} \frac{\lambda {\delta}(v,x,y)^+ - \Lambda {\delta}(v,x,y)^-}{|y|^{n+\sigma}} {\; \mathrm{d} x}\leq 1.$$
Therefore $$\begin{aligned}
{\mathrm{M}^-}w &= (2-\sigma) \int_{{\mathbb R}^n} \frac{\lambda {\delta}(w,x,y)^+ - \Lambda {\delta}(w,x,y)^-}{|y|^{n+\sigma}} {\; \mathrm{d} x}\\
& \leq 1 + (2-\sigma) \int_{{\mathbb R}^n \cap \{v(x+y)<0\}} -\Lambda \frac{v(x+y)}{|y|^{n+\sigma}} {\; \mathrm{d} x}\\
& \leq 1 + (2-\sigma) \int_{{\mathbb R}^n \setminus B_{\theta r}(x_0)} \Lambda \frac{(u(x+y)-(1-\theta/2)^{-\gamma} u(x_0))^+}{|y|^{n+\sigma}} {\; \mathrm{d} x}\label{e:aa1}\end{aligned}$$
Notice that the restriction $u \geq 0$ does not provide an upper bound for this last expression. We must obtain it in a different way.
Let us consider the largest value $\tau>0$ such that $u(x) \geq g_\tau := \tau (1-|4x|^2)$. There must be a point $x_1 \in B_{1/4}$ such that $u(x_1) = \tau (1-|4x_1|^2)$. The value of $\tau$ cannot be larger than $1$ since $u(0) \leq 1$. Thus we have the upper bound $$\begin{aligned}
(2-\sigma) &\int_{{\mathbb R}^n} \frac{{\delta}(u,x_1,y)^-}{|y|^{n+\sigma}} {\; \mathrm{d} x}\\
&\leq (2-\sigma) \int_{{\mathbb R}^n} \frac{{\delta}(g_\tau,x_1,y)^-}{|y|^{n+\sigma}} {\; \mathrm{d} x}\leq C\end{aligned}$$ for a constant $C$ that is independent of $\sigma$.
Since ${\mathrm{M}^-}u(x_1) \leq 1$, then $$(2-\sigma) \int_{{\mathbb R}^n} \frac{{\delta}(u,x_1,y)^+}{|y|^{n+\sigma}} {\; \mathrm{d} x}\leq C \ .$$
In particular since $u(x_1) \leq 1$ and $u(x_1-y) \geq 0$, $$(2-\sigma) \int_{{\mathbb R}^n} \frac{(u(x_1+y)-2)^+}{|y|^{n+\sigma}} {\; \mathrm{d} x}\leq C \ .$$
We can use the inequality above to estimate . We can assume $u(x_0) > 2$, since otherwise $t$ would not be large.
$$\begin{aligned}
(2-\sigma) &\int_{{\mathbb R}^n \setminus B_{\theta r}(x_0)} \Lambda \frac{(u(x+y)-(1-\theta/2)^{-\gamma} u(x_0))^+}{|y|^{n+\sigma}} {\; \mathrm{d} x}\\
&\leq (2-\sigma) \int_{{\mathbb R}^n \setminus B_{\theta r}(x_0)} \Lambda \frac{(u(x_1+y+x-x_1)-(1-\theta/2)^{-\gamma} u(x_0))^+}{|y+x-x_1|^{n+\sigma}} \ \frac{|y+x-x_1|^{n+\sigma}}{|y|^{n+\sigma}} {\; \mathrm{d}}y \\
&\leq C (\theta r)^{-n-\sigma}\end{aligned}$$
So finally we obtain $${\mathrm{M}^-}w \leq C (\theta r)^{-n-\sigma}$$
Now we can apply Theorem \[t:wharnack\] to $w$ in $B_{\theta r}$. Recall $w(x_0) =((1-\theta/2)^{-\gamma}-1)u(x_0)$, we have $$\begin{aligned}
\left\vert {\left\{ u < \frac{u(x_0)}{2} \right\}} \cap B_{\frac{\theta r}{2}} \right\vert &= | \{ w > u(x_0) ((1-\theta/2)^{-\gamma}-1/2) \} \cap B_{\theta r /2} | \\
\leq C (\theta r)^n &\left( ((1-\theta/2)^{-\gamma}-1 )u(x_0)+C (\theta r)^{-n-\sigma} (r\theta)^\sigma \right)^{\varepsilon}\left( u(x_0) ((1-\theta/2)^{-\gamma}- \frac 1 2 ) \right)^{-{\varepsilon}} \\
\leq C (\theta r)^n &\left( ((1-\theta/2)^{-\gamma}-1)^{\varepsilon}+ \theta^{-n {\varepsilon}} t^{-{\varepsilon}} \right)\end{aligned}$$
Now let us choose $\theta>0$ so that the first term is small: $$C (\theta r)^n ((1-\theta/2)^{-\gamma}-1)^{\varepsilon}\leq \frac{1}{4} {\left\vertB_{\theta r/2}\right\vert} \ .$$
Notice that the choice of $\theta$ is independent of $t$. For this fixed value of $\theta$ we observe that if $t$ is large enough, we will also have $$C (\theta r)^n \theta^{-n {\varepsilon}} t^{-{\varepsilon}} \leq \frac{1}{4} {\left\vertB_{\theta r/2}\right\vert}$$ and therefore $$| \{ u < u(x_0)/2 \} \cap B_{\theta r /2} | \leq \frac{1}{2} {\left\vertB_{\theta r/2}\right\vert}$$ which implies that for $t$ large $$| \{ u > u(x_0)/2 \} \cap B_{\theta r /2} | \geq c {\left\vertB_r\right\vert} \ .$$ But this contradicts . Therefore $t$ cannot be large and we finish the proof.
Hölder estimates.
=================
The purpose of this section is to prove the following Hölder regularity result.
\[t:ca\] Let $\sigma>\sigma_0$ for some $\sigma_0>0$. Let $u$ be bounded function in ${\mathbb R}^n$, such that $$\begin{aligned}
{\mathrm{M}^+}u &\geq -C_0 \qquad \text{in } B_1 \\
{\mathrm{M}^-}u &\leq C_0 \qquad \text{in } B_1\end{aligned}$$ then there is an $\alpha > 0$ (depending only on $\lambda$, $\Lambda$, $n$ and $\sigma_0$) such that $u \in C^\alpha(B_{1/2})$ and $$u_{C^\alpha(B_{1/2})} \leq C \big( \sup_{{\mathbb R}^n} |u| + C_0 \big)$$ for some constant $C>0$.
Even though this result could be obtained as a consequence of the Harnack inequality, we will prove it using only Theorem \[t:wharnack\]. We do it in this way because it looks potentially simpler to generalize since we proved the Harnack inequality (Theorem \[t:harnack\]) using Theorem \[t:wharnack\].
Theorem \[t:ca\] follows from the following Lemma by a simple scaling.
Let $\sigma>\sigma_0$ for some $\sigma_0>0$. Let $u$ be a function such that $$\begin{aligned}
-1/2 &\leq u \leq 1/2 && \text{in } {\mathbb R}^n \\
{\mathrm{M}^+}u &\geq -{\varepsilon}_0 && \text{in } B_1 \\
{\mathrm{M}^-}u &\leq {\varepsilon}_0 && \text{in } B_1\end{aligned}$$ then there is an $\alpha > 0$ (depending only on $\lambda$, $\Lambda$, $n$ and $\sigma_0$) such that $u \in C^\alpha$ at the origin. More precisely $$|u(x) - u(0)| \leq C |x|^\alpha$$ for some constant $C$.
We will show that there exists sequences $m_k$ and $M_k$ such that $m_k \leq u \leq M_k$ in $B_{4^{-k}}$ and $$\label{e:al}
M_k - m_k = 4^{-\alpha k}$$ so that the theorem holds with $C= 4^\alpha$.
For $k=0$ we choose $m_0=-1/2$ and $M_0=1/2$. By assumption we have $m_0 \leq u \leq M_0$ in the whole space ${\mathbb R}^n$. We want to construct the sequences $M_k$ and $m_k$ by induction.
Assume we have the sequences up to $m_k$ and $M_k$. We want to show we can continue the sequences by finding $m_{k+1}$ and $M_{k+1}$.
In the ball $B_{4^{-k-1}}$, either $u \geq (M_k + m_k)/2$ in at least half of the points (in measure), or $u \leq (M_k + m_k)/2$ in at least half of the points. Let us say that ${\left\vert \{u \geq (M_k + m_k)/2 \} \cap B_{4^{-k-1}} \right\vert} \geq {\left\vertB_{4^{-k-1}}\right\vert}/2$.
Consider $$v(x) := \frac{u(4^{-k} x) - m_k}{(M_k-m_k)/2}$$ so that $v(x) \geq 0$ in $B_1$ and ${\left\vert \{v \geq 1 \} \cap B_{1/4} \right\vert} \geq {\left\vertB_{1/4}\right\vert}/2$. Moreover, since ${\mathrm{M}^-}u \leq {\varepsilon}_0$ in $B_1$, $${\mathrm{M}^-}v \leq \frac{4^{-k \sigma} {\varepsilon}_0}{(M_k-m_k)/2} = {\varepsilon}_0 4^{k (\sigma-\alpha)} \leq {\varepsilon}_0$$ if $\alpha$ is chosen less than $\sigma$.
From the inductive hypothesis, for any $j \geq 1$, we have $$v \geq \frac{(m_{k-j}-m_k)}{(M_k-m_k)/2} \geq \frac{(m_{k-j}- M_{k-j} + M_k - m_k)}{(M_k-m_k)/2} \geq -2 \cdot 4^{\alpha j}+2 \geq 2 (1 - 4^{\alpha j}) \qquad \text{in } B_{2^j}$$
Therefore $v(x) \geq -2 (|4x|^\alpha-1)$ outside $B_1$. If we let $w(x) = \max(v,0)$, then ${\mathrm{M}^-}w \leq {\mathrm{M}^-}v + {\varepsilon}_0$ in $B_{3/4}$ if $\alpha$ is small enough. We still have ${\left\vert \{w \geq 1 \} \cap B_1 \right\vert} \geq {\left\vertB_1\right\vert}/2$. Given any point $x \in B_{1/4}$, can can apply Theorem \[t:wharnack\] in $B_{1/2}(x)$ to obtain $$C(w(x) + 2 {\varepsilon}_0)^{\varepsilon}\geq |\{w>1\} \cap B_{1/2}(x)| \geq \frac{1}{2} |B_{1/4}| \ .$$
If we have chosen ${\varepsilon}_0$ small, this implies that $w \geq \theta$ in $B_{1/4}$ for some $\theta>0$. Thus if we let $M_{k+1} = M_k$ and $m_{k+1} = m_k + \theta (M_k-m_k)/2$ we have $m_{k+1} \leq u \leq M_{k+1}$ in $B_{2^{k+1}}$. Moreover $M_{k+1} - m_{k+1} = (1 - \theta/2) 2^{-\alpha k}$. So we must choose $\alpha$ and $\theta$ small and so that $(1 - \theta/2) = 4^{-\alpha}$ and we obtain $M_{k+1} - m_{k+1} = 4^{-\alpha (k+1)}$
On the other hand, if ${\left\vert \{u \leq (M_k + m_k)/2 \} \cap B_{4^{-k}} \right\vert} \geq {\left\vertB_{4^{-k}}\right\vert}/2$, we define $$v(x) := \frac{M_k - u(4^{-k} x)}{(M_k-m_k)/2}$$ and continue in the same way using that ${\mathrm{M}^+}u \geq -{\varepsilon}_0$.
$C^{1+\alpha}$ estimates. {#s:c1a}
=========================
In this section we prove an interior $C^{1,\alpha}$ regularity result for the solutions to a general class of fully nonlinear integro-differential equations. The idea of the proof is to use the Hölder estimates of Theorem \[t:ca\] to incremental quotients of the solution. There is a difficulty in that we have no uniform bound in $L^\infty$ for the incremental quotients outside of the domain. This becomes an issue since we are dealing with nonlocal equations. The way we solve it is by assuming some extra regularity of the family of integral operators ${\mathcal{L}}$. The extra assumption, compared to the assumptions for Hölder regularity , is a modulus of continuity of $K$ in measure, so as to make sure that far away oscillations tend to cancel out.
Given $\rho_0>0$, we define the class ${\mathcal{L}}_1$ by the operators $L$ with kernels $K$ such that $$\begin{aligned}
(2-\sigma)\frac{\lambda}{|y|^{n+\sigma}} &\leq K(y) \leq (2-\sigma)\frac{\Lambda}{|y|^{n+\sigma}} \label{e:lic1a1}\\
\int_{{\mathbb R}^n \setminus B_{\rho_0}} \frac{|K(y)-K(y-h)|}{|h|} {\; \mathrm{d}}y &\leq C \qquad \text{every time $|h|<\frac {\rho_0} 2$} \label{e:lic1a2}
\end{aligned}$$
A simple condition for to hold would be that $|{\nabla}K(y)| \leq \frac{\Lambda}{|y|^{1+n+\sigma}}$.
In the following theorem we give interior $C^{1,\alpha}$ estimates for fully nonlinear elliptic equations.
\[t:c1a\] Assume $\sigma>\sigma_0$. There is a $\rho_0>0$ (depending on $\lambda$, $\Lambda$, $\sigma_0$ and $n$) so that if $I$ is a nonlocal elliptic operator with respect to ${\mathcal{L}}_1$ in the sense of Definition \[d:axiomatic\] and $u$ is a bounded function such that $I u = 0$ in $B_1$, then there is a universal (depends only on $\lambda$, $\Lambda$, $n$ and $\sigma_0$) $\alpha>0$ such that $u \in C^{1+\alpha}(B_{1/2})$ and $$u_{C^{1+\alpha}(B_{1/2})} \leq C\left( \sup_{{\mathbb R}^n} |u|+|I0| \right)$$ for some constant $C>0$ (where by $I0$ we mean the value we obtain when we apply $I$ to the function that is constant equal to zero). The constant $C$ depends on $\lambda$, $\Lambda$, $\sigma_0$, $n$ and the constant in .
Because of the assumption , the class ${\mathcal{L}}_1$ is included in ${\mathcal{L}}_0$ given by \[e:uniformellipticity\]. Since $I u = 0$ in $B_1$, in particular ${\mathrm{M}^+}u \geq Iu - I0 = -I0$ and also ${\mathrm{M}^-}u \leq I0$ in $B_1$, and therefore by Theorem \[t:ca\] we have $u \in C^\alpha (B_{1-\delta})$ for any $\delta > 0$ with ${\left\Vertu\right\Vert}_{C^\alpha} \leq C ( \sup |u|+|I0|)$.
Now we want to improve the obtained regularity iteratively by applying Theorem \[t:ca\] again until we obtain Lipschitz regularity in a finite number of steps.
Assume we have proved that $u \in C^\beta(B_r)$ for some $\beta>0$ and $1/2 < r <1$. We want to apply Theorem \[t:ca\] for the difference quotient $$w^h = \frac{ u(x+h) - u(x) }{|h|^\beta}$$ to obtain $u \in C^{\beta+\alpha}(B_{r-\delta})$. By Theorem \[t:Sclass\], ${\mathrm{M}^+}_{{\mathcal{L}}_1} w^h \geq 0$ and ${\mathrm{M}^-}_{{\mathcal{L}}_1} w^h \leq 0$ in $B_r$. In particular ${\mathrm{M}^+}w^h \geq 0$ and ${\mathrm{M}^-}w^h \leq 0$ in $B_r$.
The function $w^h$ is uniformly bounded in $B_r$ because $u \in C^\beta(B_r)$. Outside $B_r$ the function $w^h$ is not uniformly bounded, so we cannot apply Theorem \[t:ca\] immediately. However, $w^h$ has oscillations that cause cancellations in the integrals because of our assumption .
Let $\eta$ be a smooth cutoff function supported in $B_r$ such that $\eta \equiv 1$ in $B_{r - \delta/4}$, where $\delta$ is some small positive number that will be determined later.
Let us write $w^h = w^h_1 + w^h_2$, where $$\begin{aligned}
w^h_1 &= \frac{ \eta u (x+h) - \eta u(x) }{|x|^\beta} \\
w^h_2 &= \frac{ (1-\eta) u (x+h) - (1-\eta) u(x) }{|x|^\beta} \\\end{aligned}$$
Let $x \in B_{r/2}$ and $|h|<\delta/16$. In this case $(1-\eta)u(x) = (1-\eta) u (x+h) = 0$ and $w^h(x) = w^h_1(x)$. We have to show that $w^h_1 \in C^{\beta+\alpha}(B_{r-\delta})$.
We have $$\begin{aligned}
{\mathrm{M}^+}w^h_1 \geq {\mathrm{M}^+}_{{\mathcal{L}}_1} w^h_1 = {\mathrm{M}^+}_{{\mathcal{L}}_1} (w^h - w^h_2) \geq 0 - {\mathrm{M}^+}_{{\mathcal{L}}_1} w^h_2 \ , \\
{\mathrm{M}^-}w^h_1 \leq {\mathrm{M}^-}_{{\mathcal{L}}_1} w^h_1 = {\mathrm{M}^-}_{{\mathcal{L}}_1} (w^h - w^h_2) \leq 0 - {\mathrm{M}^-}_{{\mathcal{L}}_1} w^h_2 \ .\end{aligned}$$ In order to apply Theorem \[t:ca\], we will show that $|{\mathrm{M}^+}_{{\mathcal{L}}_1} w^h_2|$ and $|{\mathrm{M}^-}_{{\mathcal{L}}_1} w^h_2|$ are bounded in $B_{r-\delta/2}$ by $C \sup|u|$ for some universal constant $C$. We must show those inequalities for any operator $L \in {\mathcal{L}}_1$.
Since $(1-\eta)u(x) = (1-\eta) u (x+h) = 0$. $w^h(x) = w^h_1(x)$, we have the expression $$L w^h_2 = \int_{{\mathbb R}^n} \frac{ (1-\eta)u(x+y+h) - (1-\eta)u(x+y)}{|h|^\beta} K(y) {\; \mathrm{d}}y$$ and we notice that both terms $(1-\eta)u(x+y+h) = (1-\eta)u(x+y) = 0$ for $|y| < \delta/8$. We take $\rho_0 = \delta/4$, therefore we can integrate by parts the incremental quotient to obtain $$\begin{aligned}
|L w^h_2| &= {\left\vert \ \int_{{\mathbb R}^n} (1-\eta)u(x+y) \frac{ K(y) - K(y-h) }{|h|^\beta} {\; \mathrm{d}}y\right\vert} \\
&\leq \int_{{\mathbb R}^n} |(1-\eta)u(x+y)| |h|^{1-\beta} \frac{ |K(y) - K(y-h)| }{|h|} {\; \mathrm{d}}y \qquad \text{using \eqref{e:lic1a2}} \\
&\leq |h|^{1-\beta} \int_{{\mathbb R}^n \setminus B_{\delta/4}} \frac{ |K(y) - K(y-h)| }{|h|} {\; \mathrm{d}}y \ \sup_{{\mathbb R}^n} |u| \\
&\leq C |h|^{1-\beta} |u| \leq C \sup_{{\mathbb R}^n} |u|\end{aligned}$$
So, we have obtained ${\mathrm{M}^+}w^h_1 \geq -C \sup |u|$ and ${\mathrm{M}^-}w^h_1 \leq C \sup |u|$ in $B_{r-\delta/2}$ for $|h|<\delta/16$. We can apply theorem \[t:ca\] to get that $w^h_1$ (and thus also $w^h$) is uniformly $C^\alpha$ in $B_{r-\delta}$. By the standard telescopic sum argument [@CC], this implies that $u \in C^{\alpha+\beta}(B_{r-\delta})$.
Iterating the above argument, we obtain that $u$ is Lipschitz in $[1/\alpha]$ steps. Then, for any unit vector $e$, we use the same reasoning for the incremental quotients $$w^h = \frac{u(x+he) - u(x)}{h}$$ to conclude that $u \in C^{1,\alpha}$ in a smaller ball. If we choose the constant $\delta$ appropriately, we get $u \in C^{1,\alpha}(B_{1/2})$
Note that the value of $\rho_0$ in Theorem \[t:c1a\] is not scale invariant. If we want to scale the estimate to apply it to a function $u$ such that $Iu=0$ in $B_r$, then we also have to multiply the value of $\rho_0$ times $r$.
Note that the family ${\mathcal{L}}$ given by the operators $L$ with the form $$Lu(x) = \int_{{\mathbb R}^n} \frac{c_n (2-\sigma)}{\det A |A^{-1} z|^{n+\sigma}} {\delta}(u,x,z) {\; \mathrm{d}}z$$ satisfies the conditions and . Thus, from the arguments in section \[s:secondorder\] and Theorem \[t:c1a\], we reconver the $C^{1,\alpha}$ estimates for fully nonlinear elliptic equations.
Truncated kernels. {#s:truncated}
==================
For applications, it is important to be able to deal with integro-differential operators whose kernels do not satisfy in the whole space ${\mathbb R}^n$ but only in a neighborhood of the origin. For example we want to be able to deal with the operators related to truncated $\alpha$-stable Levy processes. In this section we extend our regularity resuls for this kind of operators.
We consider the following class ${\mathcal{L}}$. We say that an operator $L$ belongs to ${\mathcal{L}}$ if its corresponding kernel $K$ has the form $$K(y) = K_1(y) + K_2(y) \geq 0 \ .$$ Where $$(2-\sigma) \frac{\lambda}{|x|^{n+\sigma}} \leq K_1(y) \leq (2-\sigma) \frac{\Lambda}{|x|^{n+\sigma}}$$ and $K_2 \in L^1({\mathbb R}^n)$ with ${\left\VertK_2\right\Vert}_{L^1} \leq \kappa$.
In this class ${\mathcal{L}}$ we can consider kernels that are comparable to $|y|^{-n-\sigma}$ near the origin but decay exponentially at infinity, or even become zero outside some ball. For example $$\begin{aligned}
K(y) &= \frac{1}{|y|^{n+\sigma}} e^{-|y|^2} \: \text{or} \\
K(y) &= \frac{a(y)}{|y|^{n+\sigma}} \chi_{B_1}(y) \: \text{ where } \lambda \leq a(y) \leq \Lambda \ . \\\end{aligned}$$
This class ${\mathcal{L}}$ is larger than the class ${\mathcal{L}}_0$ in . However, in the following lemma we show that the extremal operators ${\mathrm{M}^+}_{\mathcal{L}}$ and ${\mathrm{M}^-}_{\mathcal{L}}$ are controlled by the corresponding extremal operators of ${\mathcal{L}}_0$, ${\mathrm{M}^+}$ and ${\mathrm{M}^-}$, plus the $L^\infty$ norm of $u$.
Let $u$ be a bounded function in ${\mathbb R}^n$ and $C^{1,1}$ at the point $x$. Then $$\begin{aligned}
{\mathrm{M}^-_\mathcal{L}}u(x) &\geq {\mathrm{M}^-}u(x) - 4 \kappa {\left\Vertu\right\Vert}_{L^\infty} \\
{\mathrm{M}^+_\mathcal{L}}u(x) &\leq {\mathrm{M}^+}u(x) + 4 \kappa {\left\Vertu\right\Vert}_{L^\infty}\end{aligned}$$
All we have to do is show that for each $L \in {\mathcal{L}}$, he have $Lu(x) \geq {\mathrm{M}^-}u(x) - \kappa \inf_{{\mathbb R}^n} u$ and $Lu(x) \leq {\mathrm{M}^+}u(x) + \kappa \sup_{{\mathbb R}^n} u$.
We have $$\begin{aligned}
Lu &= \int {\delta}(u,x,y) (K_1(y)+K_2(y)) {\; \mathrm{d}}y \\
&= \int {\delta}(u,x,y) K_1(y) {\; \mathrm{d}}y + \int {\delta}(u,x,y) K_2(y) {\; \mathrm{d}}y \\
&\geq {\mathrm{M}^-}u(x) + \int (u(x+y) + u(x-y) - 2u(x)) K_y(y) {\; \mathrm{d}}y \\
&\geq {\mathrm{M}^-}u(x) - 4 {\left\Vertu\right\Vert}_{L^\infty} {\left\VertK_2\right\Vert}_{L^1} = {\mathrm{M}^-}u(x) - 4 \kappa {\left\Vertu\right\Vert}_{L^\infty}\end{aligned}$$
In a similar way the inequality for ${\mathrm{M}^+_\mathcal{L}}u(x)$ follows.
\[c:mlp\] If $u$ is bounded in ${\mathbb R}^n$ and in an open set $\Omega$, ${\mathrm{M}^+_\mathcal{L}}u \geq -C$ and ${\mathrm{M}^-_\mathcal{L}}u \leq C$, then $$\begin{aligned}
{\mathrm{M}^+}u &\geq -C - 4 \kappa {\left\Vertu\right\Vert}_{L^\infty} \\
{\mathrm{M}^-}u &\leq C + 4 \kappa {\left\Vertu\right\Vert}_{L^\infty} \ .\end{aligned}$$
\[t:ca2\] Let $\sigma>\sigma_0$ for some $\sigma_0>0$. Let $u$ be bounded function in ${\mathbb R}^n$, such that $$\begin{aligned}
{\mathrm{M}^+_\mathcal{L}}u &\geq -C_0 \qquad \text{in } B_1 \\
{\mathrm{M}^-_\mathcal{L}}u &\leq C_0 \qquad \text{in } B_1\end{aligned}$$ then there is an $\alpha>0$ (depending only on $\lambda$, $\Lambda$, $n$ and $\sigma_0$) such that $u \in C^\alpha(B_{1/2})$ and $$u_{C^\alpha(B_{1/2})} \leq C \big( {\left\Vertu\right\Vert}_{L^\infty} + C_0 \big)$$ for some constant $C>0$ that depends on $\lambda$, $\Lambda$, $n$ and $\sigma_0$ and $\kappa$.
Form Corollary \[c:mlp\] $$\begin{aligned}
{\mathrm{M}^+}u &\geq -C_0 - 4 \kappa {\left\Vertu\right\Vert}_{L^\infty} \\
{\mathrm{M}^-}u &\leq C_0 + 4 \kappa {\left\Vertu\right\Vert}_{L^\infty} \ .\end{aligned}$$
Then, from Theorem \[t:ca\] $$\begin{aligned}
u_{C^\alpha(B_{1/2})} &\leq C \big( {\left\Vertu\right\Vert}_{L^\infty} + C_0 + 4 \kappa {\left\Vertu\right\Vert}_{L^\infty} \big) \\
&\leq \tilde C \big( {\left\Vertu\right\Vert}_{L^\infty} + C_0 \big) \ .\end{aligned}$$
If we use Theorem \[t:ca2\] instead of Theorem \[t:ca\] in the proof of Theorem \[t:c1a\], we obtain a $C^{1,\alpha}$ result for a class ${\mathcal{L}}$ that includes kernels with exponential decay or compact support.
\[t:genc1a\] Let ${\mathcal{L}}$ be the class of operators with kernels $K$ such that $$\begin{aligned}
\int_{{\mathbb R}^n \setminus B_{\rho_0}} \frac{|K(y)-K(y-h)|}{|h|} {\; \mathrm{d}}y &\leq C \qquad \text{every time $|h|<\frac {\rho_0} 2$} \label{e:prevcond}\\
K &= K_1 + K_2 \\
(2-\sigma)\frac{\lambda}{|y|^{n+\sigma}} &\leq K_1(y) \leq (2-\sigma)\frac{\Lambda}{|y|^{n+\sigma}} \\
{\left\VertK_2\right\Vert}_{L^1} &\leq \kappa\end{aligned}$$
There is a $\rho_0>0$ so that if $I$ be a nonlocal elliptic operator in the sense of Definition \[d:axiomatic\] and $u$ is a bounded function such that $I u = 0$ in $B_1$ then there is an $\alpha>0$ (depending only on $\lambda$, $\Lambda$, $n$ and $\sigma$) such that $u \in C^{1+\alpha}(B_{1/2})$ and $$u_{C^{1+\alpha}(B_{1/2})} \leq C \left( \sup_{{\mathbb R}^n} |u| + |I0| \right)$$ for some constant $C>0$.
We can prove Theorem \[t:ca2\] because in our $C^\alpha$ estimates we allow a bounded right hand side. Theorem \[t:genc1a\] would be more general if the inequality was required with $K_1$ instead of $K$. In order to prove such result we would need to have $C^{1,\alpha}$ estimates like the ones of Theorem \[t:c1a\] with a nonzero right hand side. This type of results is well known for elliptic partial differential equations [@C2] and we are planning to extend it to nonlocal equations in future work.
It is not hard to check that if the assumption involved $K_1$ instead of $K$, then the class ${\mathcal{L}}$ above would be the same as the larger class ${\mathcal{L}}_0$ of and Theorem \[t:genc1a\] would apply to a very large family of operators.
|
---
author:
- |
Otkrist Gupta\
[otkrist@mit.edu]{}
- |
Dan Raviv\
[raviv@mit.edu]{}\
Massachusetts Institute of Technology\
Cambridge, MA\
- |
Ramesh Raskar\
[raskar@media.mit.edu]{}
bibliography:
- 'video\_expression\_paper.bib'
title: 'Multi-velocity neural networks for gesture recognition in videos'
---
Introduction
============
Nonverbal communication is a key factor in the interaction between individuals, replacing or amplifying spoken words. Our body language, voice pitch, intonation and volume, movement of our pupils or our chronemics choices are just a few examples emphasizing the richness of human communication skills [@mehrabian1974approach]. A special subset of nonverbal interactions, explored in this paper, is based on facial expressions. Their perception initiates rapid cognitive processes in the brain and have both communicative and reflexive components [@frith2009role]. They assist in verbal communication by providing context to what we are saying, making their recognition important for studying social interactions.
{width="0.95\linewidth"}
We use computers on a daily basis, interacting with artificial agents in increasing number of tasks. Advances in speech and natural language processing have presented us with personalized smart agents [@siewiorek2008application; @freed2008radar], while examining our facial gestures has provided a richer set of tools for improving human computer interaction [@cowie2001emotion]. In the last two decades we have seen several attempts to automate the way computers respond towards human emotions [@klein1999computer; @cerezo2007interactive; @andre2000exploiting], where the ultimate goal is to create humanoid robots which can blend in the environment [@brown2014meet].
Unfortunately this task is hard to solve, and current state-of-the-art results are far from satisfactory. Recent advances in machine learning have shown that if we provide a neural network with enough samples, it can learn very complex structures [@hinton2006fast]. Today hard tasks in computer vision, such as labeling images, recognizing objects and faces or classifying videos have become a feasible task for computers which can now provide competitive results to humans and sometimes even outperform them [@ranzato2011deep; @taigman2014deepface].
In this paper we focus on facial gesture recognition from videos using deep neural networks. We tackle the problem of a small size labeled dataset, and present a new layer which compensates for velocity changes in the time domain. We compare our methods to multiple techniques and datasets, as well as presenting our own collected data, and we report state-of-the-art results in almost every category. We summarize our contributions by:
Contributions
-------------
1. 2. 3. We report state of the art results for video gesture recognition using spatio-temporal convolutional neural networks.
4. We introduce a new topology and protocol for semi-supervised learning, where the number of labeled data points is only a fraction of the entire dataset.
Related Work
============
Machine learning techniques such as Support Vector Machines have been used for facial expression recognition given the movement of facial fiducial points [@kotsia2007facial; @michel2003real; @shan2005robust; @dhall2011emotion] achieving real time performance [@ren2014face]. Many of these techniques involve a pipeline with multiple phases - face detection and alignment, feature extraction/landmark localization and classification as the final step. Other interesting approaches [@chen20153d; @walecki2015variable; @presti2015using; @vieriu2015facial] we should mention are based on temporal features [@liu2014learning; @wang2013capturing], and multiple kernels [@liu2014combining], action units [@zhao2015joint; @senechal2015facial], as well as emotion recognition from speech [@nwe2003speech; @schuller2004speech]. We will compare our method against some of those approaches in section \[experimentsandresults\].
Recently, deep neural nets have been shown to perform well on classification tasks on images and videos, outperforming most traditional learning systems. One of the most interesting results was presented three years back on a large scale dataset (LSVRC 2011), where a deep convolutional net outperformed all other methods by far [@krizhevsky2012imagenet]. With advances in convolutional neural nets, we have seen neural nets applied to video classification [@karpathy2014large; @tran2014learning] and even facial expression recognition [@abidin2012neural; @gargesha2002facial] but these networks were not deep enough or used other feature extraction techniques like PCA or Fisherface.
Training a neural net normally requires a large labeled dataset which is hard to obtain using reasonable resources. Providing high quality results when only a small part of the data is labeled is an interesting problem referred to as semi-supervised learning. In [@lee2013pseudo] the authors pre-trained the system using pseudo labels, while in [@weston2012deep; @kingma2014semi] they embedded the data in a low dimensional space. Very recently superior results have been shown [@liu2014facial; @kahou2013combining; @jung2015joint; @he2015multimodal; @kahou2015emonets] using deep neural nets to combine labels and un-labeled data in the same package. In this paper we follow those guidelines and train from start-to-end a hybrid system composed of autoencoders for unlabeled data and additional loss function for the classification tasks.
Method
======
{width="\linewidth"}
We propose a semi-supervised approach using a deep neural network, by combining an autoencoder with a classification loss function, and training both of them in parallel. The input for the first layer is a short sequence of facial gestures composed of 9 frames cropped to $145 \times 145$ pixels window. The loss function is evaluated by combining a predictive loss from 7 different pre-labeled gestures (for the labeled part of the dataset), and autoencoder Euclidean loss for the entire (labeled and un-labeled) collection. The weights of each layer are dynamically altered such that the importance of the autoencoder loss decreases with relation to the predictive loss as the training progresses. While generating the data, we use Viola and Jones face detection [@viola2004robust] for cropping the faces. We use slow fusion based convolutional neural network with convolutions in both space and time (see figure \[fig:completearch\] for a detailed overview).
Action autoencoder {#subsectionautoencoder}
------------------
Our action autoencoder consists of convolutional autoencoder for learning deep features and reducing the dimensionality of the data. We use convolutional filters with weight sharing in the first 6 layers followed by 2 fully connected layers. This network is similar to Imagenet [@krizhevsky2012imagenet] but accepts inputs of size $145 \times 145 \times 9$ as an input. Using shorthand notation the full architecture can be written as $C(96,11,3)-N-C(256,5,2)-N-C(384,3,2)-N-FC(4096)-FC(4096)-DC(96,11,3)-N-DC(256,5,2)-N-DC(384,3,2)$, where $C(n,f,s)$ stands for convolution layers with $n$ filters of size $f$ and stride $s$. $DC(n,f,s)$ stands for deconvolution layers with $n$ deconvolving filters of size $f$ and stride $s$. $FC(n)$ stands for fully connected layers with $n$ nodes and $N$ stands for local response normalization layers. We extend the convolution layers in time and use slow fusion model [@karpathy2014large] which slowly combines temporal information in successive layers. The first convolution filters have size 3 and stride 2 in time domain, the next layer has size 2 and stride 2 and the third layer combines all temporal features. The deconvolution layers are extended in time as well and reverse the slow fusion generating temporal features successively (see figure \[fig:autoencoder\]).
Multi-Velocity Encoders {#subsectionmultivelocity}
-----------------------
One of the main challenges in action recognition is related to assigning similar classification to objects at different velocities. In this work we propose to learn the velocity of the sequence in parallel to its classification by adaptive temporal interpolation. Our multi-velocity autoencoder consists of 3 action autoencoders combined together to access temporal features for different velocities. We achieve this by adding a convolution layer as the first layer which uses cubic b-spline interpolation to *slow down* the video and generate intermediate frames. Piece-wise cubic b-spline interpolation is preferred over polynomial techniques as it can minimize interpolation error for fewer points and lower degree polynomials [@hou1978cubic]. For initialization a sampling factor of $1$, $2/3$ and $1/3$ is chosen, which is later refined as a part of the learning.
{width="0.95\linewidth"}
{width="0.95\linewidth"}
Next we show how to generate the required weights for interpolation and encode them as a neural network layer. Cubic b-splines are continuous *piecewise-polynomial* functions containing polynomials of degree $3$ or less. A cubic b-spline spanning $n+1$ points comprises of $n$ cubic polynomials $\left( \mathbf{S_n}(x)^{N}_{n=1} \right)$ which can be uniquely defined using $4n$ coefficients. These coefficients can be recovered by applying linear constraints arising from continuity and differentiability of the function on the break points (or *knots*). We represent input video at each pixel as a function of time and use cubic b-splines to approximate intermediate values. We represent intermediate polynomials between $n+1$ frames as a coefficient vector $\bar{p}$ containing coefficients for all $n$ polynomials.
Let $\bar{x},\bar{y}$ be the frame numbers and pixel values known to us at the different frames, these frames are obvious choices for break points as we try approximating space between frames using b-spline curves. Cubic b-spline coefficients $\bar{p}$ for each pixel can be generated by solving a linear equation ${\mathbf{A}}\bar{p} = {\mathbf{T}}\bar{y}$ as shown in the appendix. Here both ${\mathbf{A}}$ and ${\mathbf{T}}$ depend only on frame numbers ($\bar{x}$) and are independent of pixel values ($\bar{y}$) or pixel coordinates.
Let $u_o$ be one of new points in time where we want to interpolate a video frame, we can compute it now by selecting $k^{th}$ consecutive frames containing $u_o$, choosing piecewise polynomial contained in-between these frames $\left(\mathbf{S_k}(x)\right)$ and evaluating it at $u_o$. We can write this as a dot product between coefficients and input $\bar{r} \cdot \bar{p}$, where $u_o$ lies between $k^{th}$ consecutive frames and $\bar{r}$ is as defined below in :
$$\label{eqn:singleinterpolant}
\bar{[r_i]} =\begin{cases}
u_o^{i-4k} & \text{if } x_{{\lfloori/4\rfloor}} \leq u_o \leq x_{{\lfloori/4\rfloor}+1} \\
0 & o/w
\end{cases}$$
Extending to several temporal positions; Let $\bar{u} = [u_i]$ be locations in time where frames needs to be interpolated, we compute them as a matrix vector product ${\mathbf{R}}\bar{p}$. Here each row of ${\mathbf{R}} = [r_{j,i}]$ is computed using a shifted version of the equation given above, specifically: $$\label{eqn:interpolantmatrix}
[r_{j,i}] =\begin{cases}
(u_j - j)^{i-4k} & \text{if } x_{{\lfloori/4\rfloor}} < u_j < x_{{\lfloori/4\rfloor}+1} \\
0 & o/w
\end{cases}$$
Equation shows how interpolated pixel values $\mathbf{F}(\bar{u})$ are linearly related to b-spline coefficients ${\mathbf{R}}\bar{p}$. Solving for $\bar{p}$ using $\bar{p} = {\mathbf{A^{-1}}}{\mathbf{T}}\bar{y}$ we infer that $\mathbf{F}(\bar{u}) = {\mathbf{R}}{\mathbf{A^{-1}}}{\mathbf{T}}\bar{y}$. The new velocity layer weights are initialized by computing matrix ${\mathbf{R}}{\mathbf{A^{-1}}}{\mathbf{T}}$ which is independent of pixel values $\bar{y}$ and their spatial locations. We can represent this matrix as a *caffe* convolution layer with shared weights, which contains $n$ filters of size $1 \times 1 \times n$ applied to all frames of video. We use algorithm \[bsplinealgo\] to create 3 different weight matrices which interpolate sampling factors of $1$, $2/3$ and $1/3$.
[**Input:**]{}[ Frame numbers $\bar{x}$, new temporal locations $\bar{u}$]{}\
[**Output:**]{}[ Caffe Weight Matrix ${\mathbf{W}}$]{}
$nSplines \gets length(\bar{x}) - 1$ $p \gets 4i$ ${\mathbf{T_{p,i}}} \gets 1$ ${\mathbf{T_{p+1,i+1}}} \gets 1$ $s \gets p - 4h$ ${\mathbf{A_{p+h,p:p+3}}} \gets [h^3, h^2, h, 1]$ ${\mathbf{A_{i+2,s+4:s+8}}} \gets -1^{h+1}[3h^2, 2h, 1, 0] $ ${\mathbf{A_{i+3,s+4:s+8}}} \gets -1^{h+1}[6h, 2, 0, 0] $ ${\mathbf{A_{i+3,i-4:i+3}}} \gets [6, 0, 0, 0, -6, 0, 0, 0] $ ${\mathbf{A_{i+4,0:7}}} \gets [6, 0, 0, 0, -6, 0, 0, 0] $ $p \gets find(\bar{x},{\lfloor\bar{u}(i)\rfloor})$ ${\mathbf{R_{i,4p+h}}} \gets (\bar{u}(i) - \bar{x}(p))^h$ ${\mathbf{W}} \gets {\mathbf{RA^{-1}T}}$\
${\mathbf{W}}$
Semi-Supervised Learner
-----------------------
One of the main challenges we face today for training deep neural networks is the need for large labeled datasets. The richness of data is probably one of the main reasons why neural nets report such impressive predictive results in almost every field, but it is also extremely hard to collect and label such datasets. In Semi-supervised paradigm, we assume that only a part of the data is labeled, yet we wish to utilize the knowledge hidden within the entire set. Here we combine the action autoencoder convolution layers with a softmax loss function for the labeled set. The classifier neural net is inspired by Imagenet [@krizhevsky2012imagenet], with additional fully connected layers which are shared with the autoenoder, to generate deeper classification features from the latter. The full architecture of the predictor is $C(96,11,3)-N-C(256,5,2)-N-C(384,3,2)-N-FC(4096)-FC(8192)-FC(4096)-FC(512)-FC(8)$ with softmax layers in the end for label classification. Please refer to section \[subsectionautoencoder\] for explanation of the architecture shorthand.
The protocol we suggest for training the net is as important as the topology itself. We begin by training the autoencoder as a sole learner from the outer layer to the inner ones. Meaning, we adaptively add layers to the autoencoder, train the neural net, and use the produced weights as initialization for the next step. This is one of the traditional approaches used to train autoencoders [@hinton2006fast; @carreira2005contrastive]. Next, we use the weights for initialization of the semi-supervised net, allowing the entire net to fine tune. A key factor in training is the learning rate of the two matched learners. We begin the training using a higher learning rate for the autoencoder (with predictor layers staying fixed using zero learning rate) and end the process with increased importance to the labeled loss function. While training on the labeled data, ratio between the two varies from a factor of $10^{3}$ to a factor of $10^5$ favoring the loss layer.
### Multi-Velocity Semi-Supervised Learner
![Results from reconstruction using multi velocity encoders, bottom 3 images are output from autoencoder ensemble. (a) Input video sequence. (b) Reconstruction using encoder with sampling factor of $1/3$. (c) Reconstruction using sampling factor of $2/3$. (d) Reconstruction at original velocity.[]{data-label="fig:multivelocity_autoencoder_results"}](images/multivelocity.png){width="\linewidth"}
Finally we attach the new proposed Multi-Velocity layers as the first structure of the semi-supervised neural net. Each sub-structure (See Figure \[fig:completearch\]), has its own autoencoder, all of which are concatenated after the inner most convolution layer into a feature vector (size $12288$), later used by the labeled loss function. The learner loss function can be expressed as a weighted sum of autoencoder and predictor loss given in equation \[predictorloss\] below.
$$\label{predictorloss}
L = \alpha\sum_v{||\bar{x}-\bar{x}_v||} - \beta\sum_j{y_j log \left(\frac{e^{o_j}}{\sum_k{e^{o_k}}}\right)}$$
Here $\bar{x}, \bar{x}_v$ are autoencoder inputs and outputs, $y_j$ are the input labels and $o_j$ is the outputs from predictor layer. $\sum_v{||\bar{x}-\bar{x}_v||}$ is the combined Euclidean loss across three multi-velocity encoders and $- \sum_j{y_j log \left(\frac{e^{o_j}}{\sum_k{e^{o_k}}}\right)}$ is softmax loss [@bengio2005convex]. While training using labeled data, the loss coefficient $\beta$ is selected to keep softmax loss an order of magnitude higher than the Euclidean loss. Loss coefficient $\alpha$ is adjusted as softmax loss goes down to continue training predictor layers, without overfitting autoencoder layers. Notice that we use two coefficients for the energy function and not just controlling the ratio between the two since the back-propagation algorithm has its own additional parameters.
Datasets
========
In order to evaluate the proposed architecture we use two known datasets from literature as well as present two additional datasets collected by us; The first dataset contains more than 160 million images combined into 6.5 million short (25 frames) clips, used by us to train our autoencoders. The second dataset is comprised of 2777 short clips labeled for seven emotions. In the following section we elaborate on the four datasets.
Autoencoder dataset
-------------------
In order to train very deep neural nets we must obtain a huge collection of data. Here we collected 6.5 million video clips containing 25 frames each, adding up to more than 162 million face images. We used viola-jones face detector to find and segment out the faces. Next, we localized landmarks for each frame using a deformable model for the face [@asthana2014incremental] and detected the facial pose by fitting a 3D model to the landmarks. This process allowed us to restrict the dataset to videos which contain faces tilted less than 30 degrees and remove any faces looking sideways.
In order to extract only meaningful video clips we removed clips with static gestures or those where the faces were rapidly altering, either due to some high speed movement or simply due to appearance of a different face. We achieved this by blurring the clips and calculating the difference between consecutive frames.
The raw videos were taken from public sources such as CNN, MSNBC, FOX and CSPAN. To our knowledge this is the biggest facial dataset reported in literature, and we plan to make it public.
Asevo dataset
-------------
In order to collect and label our own gestures we developed a video recording and annotation tools. We developed the application using python based OpenCV and captured the clips using Logitech C920 HD camera. The database contains facial clips from 160 subjects both male and female, where gestures were artificially generated according to a specific request, or genuinely given due to a shown stimulus. We collected a total of 2777 clips out of which 1745 were captured after providing the stimulus while 1032 were generated artificially. To create natural facial expressions we selected a bank of YouTube videos for each facial expression and showed them to subjects, capturing their reaction to the visual stimulus. We quantitatively summarize this dataset in table \[table:Asevodistribution\], where posed clips refers to the artificially generated expressions and non-posed to the stimulus activation procedure.
[0.95]{}[|l|>X|>X|>X|]{}
Emotion & Posed & Non-Posed & Cumulative\
Anger & 132 & 318 & 450\
Sadness & 118 & 148 & 266\
Contempt & 153 & 301 & 454\
Fear & 137 & 96 & 233\
Surprise & 188 & 232 & 420\
Joy & 172 & 503 & 675\
Disgust & 132 & 147 & 279\
Total & 1032 & 1745 & 2777\
Cohn Kanade Dataset
-------------------
The Cohn Kanade Dataset [@lucey2010extended] is one of the most popular datasets used for facial expression recognition. The dataset contains 593 sequences out of which 327 are labeled for 7 emotions. Along with posed facial expressions, the dataset also contains non-posed smile expressions. However the dataset lacks depth in having other non-posed expressions and is not extensive as Asevo dataset in capturing naturally expressed emotions. Each video clip contains facial expression going from baseline neutral to peak of expressed emotion.
MMI Dataset
-----------
MMI facial expression dataset [@pantic2005web] is an ongoing effort for representing both posed and non posed facial expressions. The dataset has total 2894 video clips out of which 197 have been labeled for six basic emotions. MMI originally contained only posed facial expressions and recently was extended to contain induced happiness, disgust and surprise [@valstar2010induced]. Each video clip in MMI contains people going from neutral to peak and then back to neutral facial expression.
Experiments and Results {#experimentsandresults}
=======================
Video autoencoder
-----------------
Our first experiment shows qualitatively results of a single video autoencoder. We use $145 \times 145 \times 9$ clips as input, where the spatial resolution received by downsampling all clips to that single size using bspline interpolation, and $9$ frames are extracted from the clip by using every third frame. We use *caffe* [@jia2014caffe] to train the system. In practice we convert each video clip into a image strip containing consecutive frames placed horizontally and use *caffe “imagedata”, “split”* and *“concat”* layers for video data input.
We minimize contrastive divergence [@carreira2005contrastive] to train autoencoder layers *successively*. We train the first 4 beginning and end layers by creating an intermediate neural network $(C(96,11,3)-N-C(256,5,2)-N-DC(256,5,2)-N-DC(384,3,2))$ and training it on facial video clips. We then train third convolution and deconvolution layer by initializing weights from previously trained neural net and fixing the weights for first 4 beginning and end layers. We fine tune all layers once the neural net weights have converged. We repeat the process for fourth fully connected layer to generate deep features.
Please refer to figure \[fig:autoencoder\_results\] to see results from neural net based reconstruction using different number of layers.
Multi-velocity video autoencoder
--------------------------------
Multi velocity semi-supervised learner comprises of an array of three independent autoencoders and a predictor net. We initialize the autoencoders using the weights from the video autoencoder and add a convolution layer as described in section \[subsectionmultivelocity\]. We fine tune the multi-velocity layers by creating 3 datasets containing video clips at different velocities. We achieve that by selecting every third frame to create set 1 *(speed = 3x)*, selecting every second frame to generate set 2 *(speed = 2x)* and taking first 9 frames for set 3 *(speed = 1x)*. The weights from this step are used for initialization of our multi-velocity predictor which described next.
Multi-velocity predictor
------------------------
For training, testing and validation we divide each dataset into 3 parts randomly. We select 50% inputs for training, 30% of dataset for testing and use 20% of dataset for validation. After the dataset was split, we further increased the size of the training dataset by shifting each video along both axes, rotating images and taking their mirror.
We train our proposed semi-supervised learner and the multi-velocity semi-supervised learner on the three datasets (MMI, CK and Asevo), and compare our results against multiple kernel methods [@liu2014combining] and expression-lets base approaches [@liu2014learning]. We used sources downloaded from *Visual Information Processing and Learning Resources* [@vipl] as a reference to compare to our methods. Note that we made the same data partitioning scheme (train, validation, test) for all methods to show a fair comparison.
We outperform all the methods compared on all the datasets used, by a substantial gap, in almost all cases. We summarize our findings in Table \[table:predictor\_results\], and show confusion matrices per facial expression in Tables \[table:cfckplus\] and \[table:cfAsevo\]. [For baseline comparison against other deep neural architectures, we compare our methods against [@krizhevsky2012imagenet] and GoogleNet [@szegedy2015going]. We further verified our results against prior state of the art methods discussed in [@liu2014deeply] by performing **10 fold cross validation**. On MMI we get 66.15 (vs 63.4) % and on CK+ we get 94.18 (vs 92.4) %, making our method **state of the art** for face expression recognition.]{}
Discussion and Future Work
==========================
This paper presents a learning strategy for large datasets with a dramatically lower number of labeled points, in addition to new layers carefully designed to improve recognition in multi-velocity setup. We currently trim the videos to the facial window using Viola and Jones face detection, and focus solely on frontal views. Recognition in-the-wild still remains a challenge with a known low success rate. We believe that given a large and rich dataset this problem would be feasible to solve in our system, and we plan to explore that in the future.
We introduced a new layer, which adaptively resamples the videos, achieving a multi-velocity invariant learning procedure. Inserting invariants into a learning process is a research direction that we must push forward. Today training of deep neural network is still time consuming, where huge clusters are being heavily used on reasonably large datasets. We are already reaching the time-space limit of this process, and better/smarter approaches need to be considered for advancement. Our multi-velocity setup is one approach for reducing the need for data in multiple velocities, while other invariants should be explored in future work.
Conclusions
===========
In this paper we introduced a new topology and learning protocol for semi-supervised convolutional neural networks on video sequences. We further developed a multi-velocity layer based on temporal resampling which was tuned as part of the learning procedure on an enormous collected facial dataset. We report state-of-the-art results on our own data and on public available datasets.
Appendix A: Equations for B spline Interpolation {#appendixbspline .unnumbered}
================================================
Let $\{x_k,f(x_k)\}^N_{k=0}$ be $N+1$ observations of a function $f$. Cubic spline is defined as a set of polynomials $\mathbf{S_n}(x)^{N-1}_{n=0}$ with coefficients $p_{n,i}$ which approximate $f$ as follows
$$\label{splineexp}
\begin{aligned}
\mathbf{S}(x) \hspace{0.35em}= \hspace{0.35em} \mathbf{S_n}(x) \hspace{0.35em}= \hspace{0.35em} p_{n,0} \hspace{0.35em}+ \hspace{0.35em} p_{n,1}(x_n-x_k) \hspace{0.35em}+ \hspace{0.35em} \\
p_{n,2}(x_n-x_k)^2 \hspace{0.35em}+ \hspace{0.35em} p_{n,3}(x_n-x_k)^3,
\end{aligned}$$
where $x_{k}<x_n<x_{k+1}$. We need at least $4N$ constraints to recover $p_{n,i}$ uniquely. We can generate $4N - 2$ constraints by fixing the values of polynomials at the boundaries and assuming first and second derivatives of adjacent polynomials coincide at the boundaries as well. We add additional constraints assuming that the curve is *natural* [@chung1979spline] and has zero derivative at boundaries. The coefficients $p_{k,n}$ are constrained by:
\[splinecons\]
& (x) = f(x\_k) &k {0..N}\
& (x) - (x\_[k+1]{}) = 0 &k {0..N-2}\
& (x) - (x\_[k+1]{}) = 0 &k {0..N-2}\
& (x) - (x\_[k+1]{})= 0&k {0..N-2}\
& (x) = (x) = 0
Let ${\mathbf{A}}$ denote matrix representing the constraints on spline polynomial coefficients and $\bar{p}$ represent coefficients as described in equation \[splinecons\]. Let $\bar{y}$ be the vector of function values $f(x_k), k \in \{0..N\} $ known to us. Right side of \[splinecons\] can be written as a product of matrix ${\mathbf{T}}$ with vector $\bar{y}$, where ${\mathbf{T}}$ is a binary matrix. Then ${\mathbf{A}}\bar{p} = {\mathbf{T}}\bar{y}$ or $\bar{p} = {\mathbf{A}}^{-1}{\mathbf{T}}\bar{y}$.
|
---
abstract: 'We consider a SDE with a smooth multiplicative non-degenerate noise and a possibly unbounded Hölder continuous drift term. We prove existence of a global flow of diffeomorphisms by means of a special transformation of the drift of Itô-Tanaka type. The proof requires non-standard elliptic estimates in Hölder spaces. As an application of the stochastic flow, we obtain a Bismut-Elworthy-Li type formula for the first derivatives of the associated diffusion semigroup.'
author:
- |
F. Flandoli$^{1}$, M. Gubinelli$^{2}$, E. Priola$^{3}$\
\
\
title: Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift
---
Introduction
============
In this paper we study the existence of a global stochastic flow of diffeomorphisms for the following stochastic differential equation in $\mathbb{R}^{d}$ $$dX_{t}^{x}=b\left( X_{t}^{x}\right)
dt+\sum_{i=1}^{k}\sigma_{i}\left( X_{t}^{x}\right) dW_{t}^{i},\quad
t\geq0,\quad X_{0}^{x}=x, \label{SDE}$$ where $W_{t}=(W_{t}^{1},...,W_{t}^{k})$ is a standard Brownian motion in ${\mathbb{R}}^{k}$. We assume that the diffusion coefficients $\sigma_i : \RR^d \to \RR^d$, $ i=1,\dots,k$, are smooth and non-degenerate and we allow the drift term $b: \RR^d \to \RR^d $ to be unbounded and Hölder continuous.
Following a common language, we say that equation (\[SDE\]) is *weakly complete* if there exists a unique global strong solution for every $x\in\mathbb{R}^{d}$, and that it is *strongly complete* if there exists a global stochastic flow of homeomorphisms. If the coefficients $b$ and $\sigma_{i}$ are globally Lipschitz, then one has strong completeness (see [@K] and [@K1]).
Weak completeness is true under much weaker assumptions: for instance, when the coefficients $b$ and $\sigma_{i}$ are locally Lipschitz continuous and have at most linear growth. In dimension one, these assumptions also imply strong completeness (see [@K] and [@K1]) but in dimension larger than one there are counterexamples, from [@LS], even in the case of smooth bounded coefficients. These examples indicate that some form of global control at infinity on the increments of the coefficients is necessary. For (at least) locally Lipschitz coefficients, there are indeed positive results of strong completeness (see [@FIZ], [@L], [@MS]).
Strong completeness for non-locally Lipschitz coefficients can be established replacing the global Lipschitz condition on the coefficients with global log-Lipschitz type conditions (see [@RZ], [@Za1], [@FZ], [@FL]). Such log-Lipschitz conditions are stronger than the Hölder continuity.
Many papers prove weak completeness for SDEs with non-locally Lipschitz continuous coefficients assuming a non-degenerate diffusion matrix $\sigma$. First papers in this direction were [@Zv] and [@V] in which the method of the so called Zvonkin’s transformation was introduced. More recent papers dealing with such approach are [@GM], [@Kry-Ro], [@Za], [@Za2] (see also the references therein). In the case of non-degenerate additive noise and time dependent drift $b$, the most advanced result (but see also the 1-dimensional results reported in [@RY]) is [@Kry-Ro]; in such paper it is shown that it is sufficient to assume that $b\in L^{q}\left(
0,T;L^{p}_{loc}\left( \mathbb{R}^{d}\right) \right) $ with $\frac{d}{p}+\frac{2}{q}<1$, $p \ge2$ and $q>2$, plus a non-explosion condition, to get weak completeness. This result has been generalized in [@Za] to cover also the case in which $\sigma$ is variable, time-dependent and non-degenerate. We do not know about strong completeness under such weak assumptions.
The contribution of the present paper is to prove strong completeness for SDEs with “locally uniformly $\theta$-Hölder continuous” drift $b$, for some $\theta\in\left( 0,1\right) $ (see ), removing boundedness of $b$ or additional regularity assumed in previous works. Also, we allow non-degenerate, bounded and $C^{3}_b (\RR^d, \RR^d) $-diffusion coefficients $(\sigma_i)_{i=1,\dots,k}$. We point out that our result seems to be new even in the case of constant and non-degenerate $(\sigma_i)_{i=1,\dots,k}$.
In spite of the fact that $b$ is not even differentiable, under the previous assumptions, we construct a stochastic flow of $C^{1}$-diffeomorphisms (see Theorem \[th:flow1\]) using the approach of [@FGP] rather than the Zvonkin’s transformation method used in the above mentioned works on strong completeness (we compare the two methods in Section 3).
In [@FGP] in order to study a linear stochastic transport equation with a [*bounded vector field*]{} $\tilde b (t,x)$ which is Hölder continuous in $x$, uniformly in time, we have showed that if in $\sigma= (\sigma_i)$ is constant and non-degenerate and $b = \tilde b$, then there exists a stochastic flow of $C^{1}$-diffeomorphisms. This result can be extended without difficulties to the case in which $\sigma$ is not constant, bounded, non-degenerate, and time-dependent (see [@Za2] where this case is investigated by the Zvonkin’s transformation or Remark \[vi\] where we show such result following the approach of [@FGP]).
In the present situation, since our $b$ is [*unbounded,*]{} we need new global regularity results in Hölder spaces for the solution $u$ of the elliptic equation $$\lambda u (x) - \frac{1}{2} \mathrm{Tr} (a(x) D^{2}u (x)) - b(x)\cdot D u(x)=
b(x),\;\;\; x \in\mathbb{R}^{d},$$ to be interpreted componentwise, where $\lambda>0$ is large enough, $a(x) = \sigma(x) \sigma^{*} (x)$ ($\sigma^{*}(x)$ denotes the adjoint matrix of $\sigma(x)$). The study of this equation will be the subject of Section 2 of the present paper. The required estimates are not covered by recent papers dealing with elliptic and parabolic equations with unbounded coefficients (compare with [@Ce], [@BL], [@KP] and the references therein). To obtain such result we prove a crucial Lemma \[semi\] concerning estimates on the derivatives of the associated diffusion semigroup when it is applied to *unbounded* functions $f$; in its proof we also use an argument from the proof of [@Pr2 Theorem 3.3]. In Remark \[fine\] we show a possible extension of our Theorem \[th:flow1\] to the case in which $b$ and $\sigma$ are time-dependent.
We finish the paper by showing that a Bismut-Elworthy-Li formula holds for the diffusion semigroup associated to (see Theorem \[bismut\]). Under the poor regularity of $b$ assumed here, this result is new. Bismut-Elworthy-Li formula requires a suitable form of differentiability of the solution of with respect to the initial condition $x$; we have this result as a byproduct of our Theorem \[th:flow1\] on existence of a differentiable stochastic flow.
#### Notations and assumption
The euclidean norm in any $\mathbb{R}^{k}$, $k\geq1$, will be denoted by $|\cdot|$ and its inner product by $\cdot$ or $\langle\cdot,\cdot\rangle$. For $\theta\in(0,1)$, we define the set $C_{}^{\theta}(\mathbb{R}^{d};\mathbb{R}^{k})$, $k,\,d\geq1$, as set of all vector-fields $f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{k}$ for which$$\label{hol}
\lbrack f]_{\theta}:=\sup_{x\neq y\in\mathbb{R}^{d},\left|
x-y\right| \leq 1}\frac{|f(x)-f(y)|}{|x-y|^{\theta}}<\infty.$$ These are the [*“locally uniformly $\theta$-Hölder continuous”*]{} vector fields mentioned in the introduction. The function $f\left( x\right) =|x|^{\theta}$ is a classical example. We let $$\label{vai}
[ f]_{\theta,1}:=\sup_{x\neq
y\in\mathbb{R}^{d}}\frac{|f(x)-f(y)|}{(|x-y|^{\theta
}\vee|x-y|)}<\infty,$$ where $a\vee b=\max(a,b)$, for $a,b\in\mathbb{R}$. By a simple argument we have $ [f]_\theta \le [f]_{\theta,1} \le 2 [f]_{\theta}
$, so in particular functions in $ C_{}^{\theta}(\mathbb{R}^{d};\mathbb{R}^{k})$ have [*at most linear growth.*]{} The set $C_{}^{\theta}(\mathbb{R}^{d};\mathbb{R}^{k})$ becomes a Banach space with respect to the norm $$\Vert f\Vert_{\theta}=\big\|\,{(1+|\cdot|)^{-1}}\,{f(\cdot)}\big\|_{0}+[f]_{\theta},$$ where $\Vert\cdot\Vert_{0}$ denotes the supremum norm over $\mathbb{R}^{d}$. We say that $f\in
C_{}^{n+\theta}(\mathbb{R}^{d};\mathbb{R}^{k})$, $n \ge 1$, if $f\in
C_{}^{\theta}(\mathbb{R}^{d};\mathbb{R}^{k})$ and moreover, for all $i=1,\dots,n,$ the Fréchet derivatives $D^{i}f$ are *bounded* and $\theta$-Hölder continuous. Define the corresponding norm as $$\Vert f\Vert_{n+\theta}=\Vert f\Vert_{\theta }+\sum_{i=1}^{n}\Vert
D^{i}f\Vert_{0}+[D^{n}f]_{\theta}.$$ If $\mathbb{R}^{k}=\mathbb{R}$, we simply write $C_{}^{n+\theta}(\mathbb{R}^{d})$ instead of $C_{}^{n+\theta}(\mathbb{R}^{d};\mathbb{R})$, $n\geq0$. $C_{b}^{n+\theta}(\mathbb{R}^{d};\mathbb{R}^{k})$ is the subspace of $C_{}^{n+\theta}(\mathbb{R}^{d};\mathbb{R}^{k})$, consisting of all bounded functions of $C_{}^{n+\theta}(\mathbb{R}^{d};\mathbb{R}^{k})$. In particular, $C^{\theta}_b(\mathbb{R}^{d})$ is the usual Banach space of all real bounded and $\theta$-Hölder continuous functions on $\RR^d$ (cf. [@Kr]). $C_{b}^{n}(\mathbb{R}^{d};\mathbb{R}^{k})$ is the space of all bounded functions from $\RR^d$ into $\RR^k$ having also bounded derivatives up to the order $n \ge 1$ and we set $C_{b}^{n}(\mathbb{R}^{d};\mathbb{R}) = C_{b}^{n}(\mathbb{R}^{d})$. Finally, we say that $f:\mathbb{R} ^{d}\rightarrow
\mathbb{R}^{d}$ is of class $C^{n,\alpha }$, $n \ge 1 $, $\alpha \in (0,1)$, if $f$ is continuous on $\mathbb{R}^{d}$, $n$-times differentiable and the derivatives up to the order $
n$ are $\alpha$-Hölder continuous on each compact set of $\RR^d$.
Throughout the paper we will assume a fixed stochastic basis with a $d$-dimensional Brownian motion $\left( \Omega,\left( \mathcal{F}{}_{t}\right) ,{}\mathcal{F},P,\left( W_{t}\right) \right) $ to be given. Denote by $\mathcal{F}_{s,t}$ the completed $\sigma$-algebra generated by $W_{u}-W_{r}$, $s\leq r\leq u\leq t$, for each $0\le s<t$.
On equation , we will consider the following assumptions.
\[hy1\] There exists ${\theta}\in(0,1)$ such that $b\in
C_{}^{\theta }(\mathbb{R}^{d};\mathbb{R}^{d})$.
\[hy2\] The diffusions coefficients $\sigma_{i}: \mathbb{R}^d \to \mathbb{R}^d$, $i=1, \ldots, k$, are bounded functions of class $C^{3 }_b (\mathbb{R}^d, \mathbb{R}^d).$
\[hy3\] Consider the $d \times k$ matrix $\sigma(x) =
(\sigma_i(x))$, and its adjoint matrix $\sigma^*(x)$, $x \in \mathbb{R}^d$; we assume that, for any $x \in \mathbb{R}^d$, there exists the inverse of $a(x)= \sigma (x)\sigma^* (x)$ and $$\label{si}\| a^{-1}\|_0 = \sup_{x \in \mathbb{R}^d}
\| a^{-1} (x) \| < \infty$$
($\| a^{-1} (x) \|$ denotes the Hilbert-Schmidt norm of the $d \times d$ symmetric matrix $a^{-1} (x)).$
Regularity results for the associated elliptic problem
=======================================================
Estimates on the derivatives of the diffusion semigroup
-------------------------------------------------------
Here, we consider the SDE , assuming that $\sigma$ satisfies Hypotheses \[hy2\] and \[hy3\] and imposing in addition that $$\label{hyy}
b \in C^3(\mathbb{R}^d;\mathbb{R}^d) \;\;
\text{with all bounded derivatives
up to the third order.}$$ Clearly this is stronger than Hypothesis \[hy1\] but $b$ is not assumed to be bounded.
Let $(P_t)$ be the corresponding diffusion semigroup, i.e., for any $g: \mathbb{R}^d \to \mathbb{R}$ Borel and bounded, $$P_t g (x) = \mathbb{E} [g(X_t^x)],\;\;\; x \in\mathbb{R}^d,\;\; t
\ge 0,$$ where $(X_t^x)$ is the unique strong solution to under .
In our next result, we will prove estimates on the spatial derivatives of $P_t f$, $t>0,$ assuming that $f \in C^{\theta} (\mathbb{R}^d)$. To this purpose, we will use the so-called Bismut-Elworthy-Li formula (see ) for the spatial derivatives of $P_t f$ (cf. [@EL]).
Let us comment on such formula. Probabilistic formulae for the spatial derivatives of Markov semigroups have been much studied for different classes of degenerate and non-degenerate diffusion processes even with jumps (see [@B], [@KS], [@EL], [@DZ1], [@Ce], [@Fu] [@Pr1], [@Pr2], [@Za2] and the references therein). The martingale approach of [@EL] mainly works for non-degenerate semigroups (but see also [@Ce Chapter 3] and [@Za2]); it has been also used for some infinite dimensional diffusion processes (see [@DZ1] and [@Ce]). On the other hand, in case of degenerate diffusion semigroups, more complicate formulae for the derivatives can be established by Malliavin Calculus (see [@B], [@KS], [@Fu] and [@Pr1]). Some applications to Mathematical Finance are given in [@F].
The next lemma is of independent interest since the function $f$ in is not assumed to be bounded (compare with [@Ce Chapter 1] and [@BL Chapter 6]).
\[semi\] Assume Hypotheses \[hy2\] and \[hy3\] and condition . There exist constants $c_j>0$, $M_j>0$, $j=1, 2,3$ ($c_j$ and $M_j$ depends on $\theta$, $\| a^{-1}\|_0$, $d,$ $\| \sigma\|_0$ and on the supremum norms of derivatives of $\sigma$ and $b$ up to the order $j$), such that, for any $f \in C^{\theta} (\mathbb{R}^d)$, $t>0$, it holds $$\label{gra}
\| D^j P_t f \|_0 \le M_j [f]_{\theta } \,
\frac{e^{c_j \, t}}{t^{(j -\theta)/2}},\;\;\; t>0, \;\; \text{for}
\;\; j=1, 2,3.$$
[*I Step.*]{} First note that $\mathbb{E}
[\sup_{t \in [0,T]}
|X_t^x|^{q}] \le C_T(1 + |x|^q)$, for any $T>0$, $x \in
\mathbb{R}^d$, $q \ge 1$ (see, for instance, [@K Chapter II]).
It is also known that, for any $t \ge 0$, the mapping: $$\begin{aligned}
\label{frec}
x \mapsto
X_t^x \;\;
\text{is three times Fr\'echet differentiable from $\RR^d$ into
$L^2 (\Omega)$}\end{aligned}$$ (see [@Ce Section 1.3] which contains a more general result). Let us write the Fréchet derivatives: $$\eta_t (x, h) = D_x (X_t^x) [h], \;\; \xi_t (x,h,k) =
D_x^2 (X_t^x) [h, k],\;\; \psi_t (x,h,k,l) =
D_x^3 (X_t^x) [h, k,l],$$ for any $x,h,k,l \in \RR^d$. These derivatives satisfy suitable stochastic variation equations (see [@K Chapter II]). We only write down the variation equation for $\eta_t = \eta_t (x,h)$: $$d \eta_t = Db (X_t^x) \eta_t + D \sigma (X_t^x) \eta_t
dW_t,\;\;\; \eta_0 =h.$$ Using standard estimates, based on the Burkholder inequality, we get that, for any $p \ge 1$, that there exist positive constants $C$ and $c$ (depending on $p$, $\| Db\|_0$ and $\| D \sigma \|_0$) such that, for any $x \in \RR^d$, $h \in \RR^d$, $$\label{lip}
\E |\eta_t (x,h)|^p \le C |h|^p e^{c t},\;\; t \ge 0.$$ In a similar way, using the second and third variation equations, we obtain the estimates: $$\label{lip1}
\E |\xi_t (x,h,k)|^p \le C_2 |h|^p |k|^p e^{\hat c_2 t},\;\;$$ $$\E |\psi_t (x,h,k,l)|^p \le C_3 |h|^p |k|^p |l|^p \, e^{\hat
c_3
t},\;\;
t \ge 0,$$ for any $x, h,k,l \in \RR^d$ (with positive constants $C_i $ and $\hat c_i$ which depend on $p$ and on the supremum norms of the derivatives of $b$ and $\sigma$ up to the order $i$, $i=2,3$).
Arguing similarly to [@Ce Section 1.5] one can prove that, for any $f \in C^{\theta } (\mathbb{R}^d) $, $t>0$, the map: $x
\mapsto P_t f (x)$ is differentiable on $\mathbb{R}^d$ and, moreover, we have the following Bismut-Elworthy-Li formula: $$\label{bism}
\langle D P_t f(x), h \rangle =
\mathbb{E} \Big [ f(X_t^x) \, J^1 (t,x,h) \Big], \;\;
x,\, h \in \mathbb{R}^d, \; t>0,
\; \mbox{where}$$ $$J^1 (t,x,h)= \frac{1}{t}
\int_0^t \langle \sigma^{*} (X_s^x) \, a^{-1}(X_s^x)
\, \eta_s (x,h) , dW_s
\rangle.$$ Note that formula is first proved for bounded $f\in C^2_b (\mathbb{R}^d)$. Then a straightforward approximation argument shows that holds even for (a possibly unbounded) $f \in C^{\theta}
(\mathbb{R}^d)$. However, to be precise, in [@Ce], it is assumed that $\sigma(x)$ is an invertible $d \times d $ matrix and so the expression of $J^1$ in [@Ce Section 1.5] contains $\sigma^{-1} (X_s^x)$ instead of our $\sigma^{*} (X_s^x) \,
a^{-1}(X_s^x)$. We briefly explain why holds following the proof of [@PZ Theorem 5.1]. We only discuss the crucial point of the argument which is needed to get when $f\in C^2_b (\mathbb{R}^d)$. One has by the Itô formula $$f(X_t^x) = P_t f(x) +
\int_0^t \langle DP_{t-s}f(X_s^x) ,
\sigma (X_s^x) dW_s \rangle.$$ Multiplying both terms of the identity by the martingale $$K_t =
\int_0^t \langle \sigma^{*} (X_s^x) \, a^{-1}(X_s^x)
\, \eta_s (x,h) , dW_s \rangle,$$ and taking the expectation, one arrives at $$\mathbb{E} [f(X_t^x) K_t] = \int_0^t \mathbb{E} [\langle DP_{t-s}f(X_s^x)
, \eta_s (x,h) \rangle ] ds =
t \langle DP_t f(x), h\rangle.$$ Thus is proved.
Now [*the problem is to show that, for $ f \in
C^{\theta}
(\mathbb{R}^d)$, $t>0$, the map: $x \mapsto \langle D P_t f(x), h \rangle$ is a bounded function (we cannot use as in [@Ce] the boundedness of $f$).*]{}
By using , we get easily that there exist $C_1 >0$ depending on $\| a^{-1}\|_0$, $ \|Db \|_0$ and $\| D \sigma\|_0$ such that $$\label{f4}
\mathbb{E} |J^1 (t,x,h)|^2 \le
\frac{C_1 e^{C_1 t}}{t} |h|^2,\;\;\; t > 0.$$ Now we prove the crucial estimate of the first derivative in . We use an argument from the proof of [@Pr2 Theorem 3.3]. Introduce the deterministic process $$Y_t^x = x + \int_0^t b(Y_s^x)ds,\;\; t \ge 0, \; x \in
\mathbb{R}^d,$$ which solves $ \dot {Y^x_t } = b(Y^x_t), \;\;\;
Y^x_0 = x. $ Using that $\sigma$ is bounded and applying the Gronwall lemma, we find, for any $q \ge 1$, $$\label{f9}
\mathbb{E} |X_t^x - Y_t^x |^q \le M t^{q/2} \,
e^{c_1 t },\;\;\; t \ge 0, \; x \in
\mathbb{R}^d,$$ where $M$ depends on $\| \sigma\|_0$ and $q$ and $c_1 $ on $\|
Db\|_0$ and $q$. Since $$\mathbb{E} \big [ f(Y_t^x) \, J^1 (t,x,h) ]
= f(Y_t^x) \langle D (P_t 1)(x), h \rangle
=0,\;\; t>0,\;\; h \in \mathbb{R}^d,
\; x \in \mathbb{R}^d,$$ we have (see also ) $$| \langle D P_t f(x), h \rangle| = \Big|
\mathbb{E} \big [ (f(X_t^x)- f(Y_t^x)) \, J^1 (t,x,h) \big] \Big|$$ $$\le 2[f]_{\theta} \, \mathbb{E} \big[ (|X_t^x - Y_t^x |^{\theta} \,
\vee |X_t^x - Y_t^x |^{}) \, \,
|J^1 (t,x,h)| \big]$$ $$\label{dfr} \le 2[f]_{\theta} \, \big(\mathbb{E} \big[
|X_t^x- Y_t^x |^{2\theta} \vee |X_t^x - Y_t^x |^{2} \big]
\big)^{1/2} \,
(\mathbb{E} |J^1 (t,x,h)|^2)^{1/2},$$ $t>0$. Using that $a \vee b \le a + b$, $a, b \ge 0$, and the previous estimates and , we find $$\begin{aligned}
\label{s7}
| \langle D P_t f(x), h \rangle| \le C''
[f]_{\theta}(t^{\theta/2} + t^{1/2})
\frac{e^{c'' t}}{ t^{1/2}}|h| \le
[f]_{\theta} \frac{C' e^{c' t}}{ t^{1/2 - \theta/2}} |h|, \;\;
t>0,\; x \in \mathbb{R}^d,\end{aligned}$$ where $C'$ and $c$ depend on $ \| \sigma \|_0,$ $\|
a^{-1} \|_0$, $\| D \sigma \|_0$, $\| D b \|_0$ and $\theta$.
Let us consider the remaining estimates in . We have, using the semigroup law, $P_t f = P_{t/2}
(P_{t/2} f)$ and so (cf. [@Ce formula (1.5.2)]), for any $x,\, h, \, k \in \RR^d,$ $ t>0,$ $$\langle D^2 (P_t f) (x) k, h \rangle
=
D_k \Big( \mathbb{E} \Big [ (P_{t/2}f)(X_{t/2}^{(\cdot)})
\, J^1 (t/2,(\cdot),h) \Big] \Big)(x)$$ $$= \mathbb{E} \Big [ \langle D P_{t/2}f(X_{t/2}^{x}), \eta_{t/2}
(x, k) \rangle
\, J^1 (t/2,x,h) \Big] \, +
\, \mathbb{E} \Big [ P_{t/2}f(X_{t/2}^{x})
\, D_k J^1 (t/2,x,h) \Big]$$$$= \Gamma_1(t,x) + \Gamma_2(t,x),$$ where $D_k$ denotes the directional derivative along the vector $k$ (indeed, for any fixed $t>0$ and $h \in \RR^d$, the mapping: $x \mapsto J^1 (t/2,x,h)$ is Fréchet differentiable from $\RR^d$ into $L^2 (\Omega)$; this follows easily, using , , and ). We have $$D_k J^1 (t/2,x,h)= \frac{2}{t}
\int_0^{t/2} \langle D\sigma^{*} (X_s^x)[\eta_s (x,k) ]
\, a^{-1}(X_s^x)
\, \eta_s (x,h) , dW_s
\rangle$$ $$- \frac{2}{t}
\int_0^{t/2} \langle \sigma^{*} (X_s^x)
\, a^{-1}(X_s^x) \, Da (X_s^x) [\eta_s (x,k) ] \,
a^{-1}(X_s^x)
\, \eta_s (x,h) , dW_s
\rangle$$$$+\frac{2}{t}
\int_0^{t/2} \langle \sigma^{*} (X_s^x)
\, a^{-1}(X_s^x)
\, \xi_s (x,h,k) , dW_s
\rangle.$$ Using the Schwarz inequality, and $$\sup_{x \in \mathbb{R}^d} (\E| \langle
D P_{t/2} f(X^x_{t/2}), \eta_{t/2} (x,k)
\rangle |^2)^{1/2}
\le [f]_{\theta} \frac{C'' e^{c' \theta t}}{ t^{1/2 - \theta/2}}
|k|,$$ we get immediately $ |\Gamma_1(t,x) | \le M [f]_{\theta } \,
\frac{e^{c \, t}}{t^{(2 -\theta)/2}}|h| |k|$, $t>0$, $x \in \RR^d$. To estimate $\Gamma_2$, first note that, by taking $f=1$, $$0= \langle D^2 (P_t 1) (x) k, h \rangle = 0 +
\mathbb{E} \big [
\, D_k J^1 (t/2,x,h) \big],$$ for any $x, h, k \in \RR^d$. We find (arguing similarly to ) $$\Gamma_2(t,x)=
\mathbb{E} \Big [ \big(P_{t/2}f(X_{t/2}^{x}) -
P_{t/2}f(Y_{t/2}^{x}) \big)
\, D_k J^1 (t/2,x,h) \Big].$$ Since $$|P_{s}f(x) - P_{s}f(y)| \le \E |f (X_{s}^{x}) -
f(X_{s}^{y})| \le 2 [f]_{\theta}
\mathbb{E} \big[ (|X_s^x - X_s^y |^{\theta} \,
+ |X_s^x - X_s^y |^{})$$ $$\le 2 [f]_{\theta} M (|x-y|^{\theta} + |x-y|)
e^{c_1 s },\;\;\; s \ge 0, \;\; x, \, y \in
\mathbb{R}^d,$$ we find, for any $x \in \RR^d$, $t>0,$ $$|\Gamma_2(t,x)| \le 2 M
e^{c_1 t/2 } [f]_{\theta} \, \mathbb{E} \big[ (|X_{t/2}^x - Y_{t/2}^x
|^{\theta} \,
+ |X_{t/2}^x - Y_{t/2}^x |^{}) \, \,
|D_k J^1 (t/2,x,h)| \big].$$ $$\le 2 M
e^{c_1 t/2 } [f]_{\theta} \,
\big(\mathbb{E} \big[
|X_{t/2}^x- Y_{t/2}^x |^{2\theta} + |X_{t/2}^x - Y_{t/2}^x |^{2}
\big] \big)^{1/2} \,
(\mathbb{E} |D_k J^1 (t/2,x,h)|^2)^{1/2}$$ $$\le [f]_{\theta} \frac{C_1 e^{c_1 t}}{ t^{1/2 - \theta/2}}
|k||h|,$$ where $C_1$ and $c_1$ depends on $ \| \sigma \|_0,$ $\|
a^{-1} \|_0$, $\| D \sigma \|_0$, $\| D^2 \sigma \|_0$ $\| D b \|_0$, $\| D^2 b \|_0$ and $\theta$. We have so obtained estimate in corresponding to $j =2$.
The estimate for $j =3$ follows in a similar way.
The main regularity result
----------------------------
With respect to the previous section, here we consider the elliptic operator $$L u(x) = \frac{1}{2} Tr (a(x) D^2u (x)) + b(x)\cdot D u(x),
\;\;\; x \in \mathbb{R}^d,$$ with $a(x) = \sigma(x) \sigma^*(x)$, assuming Hypotheses \[hy1\], \[hy2\] and \[hy3\].
The next result provides new estimates for $L$ in Hölder spaces. These estimates are not covered by recent papers dealing with elliptic and parabolic equations with unbounded coefficients, due to the fact that in our case also [*$f$ can be unbounded*]{} (compare with [@Ce], [@BL], [@KP] and the references therein).
\[bbo\] Let $\theta \in (0,1)$. For any $\theta' \in
(0,\theta)$, there exists $\lambda_0>0$ (depending on $\theta, \theta ' , d,$ $[b]_{\theta}$, $\| \sigma \|_0$, $\| a^{-1}\|_0$, $\| D^k \sigma\|_0,$ $k=1, 2, 3$) such that, for $\lambda \ge \lambda_0$, for any $f \in C^{\theta} (\mathbb{R}^d)$, the equation $$\label{u}
\lambda u - L u = f$$ admits a unique classical solution $u = u_{\lambda}\in C^{2+ \theta'} (\mathbb{R}^d)$ for which $$\label{sh} \| u\|_{2+ \theta \, '} = \| u(\cdot) \, (1+
|\cdot|)^{-1} \|_0 +
\|Du\|_0 + \|D^2u\|_0 + [D^2 u ]_{\theta\, '} \le
C(\lambda) \| f \|_{\theta }$$ with $C(\lambda)$ (independent on $u$ and $f$) such that $C(\lambda)$ $\to 0$ as $\lambda \to + \infty$.
Uniqueness can be proved by the following argument (cf. [@Kry2 page 606]). Consider $\eta (x) = \sqrt{1+ |x|^2}$, $x \in \mathbb{R}^d$.
Defining $u =v \eta $, we obtain an elliptic equation for the bounded function $v$, i.e., $$\lambda v(x) - \frac{1}{2} Tr (a(x) D^2v (x)) -
(b(x) + \frac{a(x) D \eta(x)}{\eta(x)}) \cdot D v(x)$$ $$\label{v}
- \Big(\frac{1}{2} \frac{ Tr(a(x) D^2 \eta(x))}{\eta(x)} +
b(x) \cdot \frac{ D \eta(x)}{\eta(x)} \Big) v(x) = \frac{f(x)}
{\eta(x)},$$ $x \in \mathbb{R}^d$. Note that $v$ has first and second bounded derivatives. For $\lambda$ large enough (depending on $\| \sigma\|_0$ and $\| \frac{b}{\eta}\|_0$), uniqueness of $v$ follows by the classical maximum principle.
Now we divide the rest of the proof in some steps.
[**Step I.**]{} We assume [*in addition that $b
\in C^3 (\mathbb{R}^d, \mathbb{R}^d)$ and has all bounded derivatives up to the third order*]{} (but it is not necessarily bounded). We prove that, for sufficiently large $\lambda>0$, there exists a unique solution $u = u_{\lambda}
\in C^{2+ \theta } (\mathbb{R}^d)$ to the equation $$\lambda u - L u = f \in C^{\theta} (\mathbb{R}^d).$$ Moreover there exists $C$ (independent on $u$ and $f$) such that $$\label{gh}
\| u \|_{2+ \theta} \le C \| f \|_{\theta}.$$ Estimates are new Schauder estimates since $f$ is not assumed to be bounded (compare with [@Ce] and [@BL])
We consider the function $$\label{fr}
u (x) = \int_0^{\infty} e^{- \lambda t } \mathbb{E} [f(X_t^x)] dt =
\int_0^{\infty} e^{- \lambda t } P_t f(x)dt, \;\; x \in \mathbb{R}^d,$$ where $(X_t^x)$ is the solution of and show that, for $\lambda$ large enough, $u$ is a $C^{2+ \theta}(\mathbb{R}^d)$-solution to our PDE.
Using that $\mathbb{E} |X_t^x - X_t^y| \le C e^{Ct} |x-y| $, $t \ge 0$, $x , y \in \mathbb{R}^d$, we find $$|u(x) - u(y)| \le c [f]_{\theta,1} \, (|x-y|^{\theta} \vee
|x-y|),\;\; x , y \in \mathbb{R}^d,$$ and also $\|u(\cdot) \, (1+ |\cdot |)^{-1} \|_0 \le C
\|f(\cdot) \, (1+ |\cdot |)^{-1} \|_0 $, for $\lambda$ large enough.
By Lemma \[semi\] we get, for $\lambda$ large enough, $$\|Du \|_0 + \|D^2 u \|_0 \le C [f]_{\theta}.$$ To estimate the second derivatives of $u$, we proceed as in [@Pr2 Theorem 4.2]. We have, for any $x, y \in \mathbb{R}^d$ with $|x-y| \le 1$, $$|D^2u(x) - D^2u(y)| = \int_0^{|x-y|^2} e^{- \lambda t } |D^2P_t f(x)
- D^2P_t f(y)| dt$$ $$+ \int_{|x-y|^2}^{\infty} e^{- \lambda t } |D^2P_t f(x) - D^2P_t
f(y)| dt$$ $$\le c'' |x-y|^{\theta} [f]_{\theta} +\,
C|x - y| \, [f]_{\theta} \, \int_{|x-y|^2}^{\infty} e^{-
\lambda t } \, \frac{e^{c t}}{t^{(3 -\theta)/2}} \; dt \le c'
[f]_{\theta} |x-y|^{\theta}.$$ It remains to check that $u$ is a solution. This is not difficult thanks to Lemma \[semi\] (see, for instance, [@Ce Chapter 1] or argue as in [@Pr2 Theorem 4.1]).
[**Step II.**]{} Under the assumptions of Step I, for any $\alpha \in (0, \theta)$, we have $$\label{gh1}
\| u \|_{2+ \alpha } \le C(\lambda) \| f \|_{\theta},$$ with $C(\lambda) \to 0 $, as $\lambda \to + \infty$. This is clear if we replace $\| u\|_{2+ \alpha}$ with $\displaystyle{ \| u(\cdot) \, (1+
|\cdot|)^{-1} \|_0 }$ $+
\|Du\|_0 + \|D^2 u \|_{0}$. Therefore, we only consider $[D^2 u]_{\alpha}.$
Combining the interpolatory estimate: $[v]_{\alpha} \le
C \| v\|_0^{1 - {\alpha}}$ $\|Dv\|^{\alpha}_0$, $v \in C^{1}_b (\mathbb{R}^d)$ (where $C =C(d)$, see [@Kr Section 3.2]) with estimates of Lemma \[semi\] corresponding to $j=2,3$, we find, for any $t>0$, $$[D^2 P_t f]_{\alpha} \le C \|D^2 P_t f \|_0^{1-\alpha} \,
\|D^3 P_t f \|_0^{\alpha} \le
C_4 [f]_{\theta} \, \frac{e^{c_4 t}}{t^{\gamma}},$$ with $\gamma = \frac{ 2 - \theta + \alpha}{2 } <1$ (since $\alpha < \theta$). It follows $$[D^2 u ]_\alpha \le C_4 [f]_{\theta}
\int_0^{+\infty} \frac{e^{(c_4 -\lambda) t}} { t^{\gamma}
} dt \le C_5 [f]_{\theta} (\lambda- c_4)^{\gamma-1}.$$ The assertion is proved.
[**Step III.**]{} We require that $b \in C^{\theta} (\mathbb{R}^d, \mathbb{R}^d)$ as in Hypothesis \[hy1\] and prove the following a-priori estimates: if $\lambda $ is large enough and $u \in C^{2+ \theta' }(\mathbb{R}^d)$, $0 < \theta' < \theta $, is a solution to $
\lambda u - L u = f \in C^{\theta}(\mathbb{R}^d)$, then $$\label{sh1}
\| u(\cdot) \, (1+
|\cdot|)^{-1} \|_0 + \| D u\|_{0} + \| D^2
u \|_{\theta'}
\le K(\lambda) \| f \|_{\theta},$$ with $K(\lambda ) \to 0$, as $\lambda \to
+\infty.$
To prove the estimate we introduce $\rho \in C_0^{\infty}(\mathbb{R}^d)$, $0
\le \rho \le 1 $, $\rho(x) = \rho(-x)$, for any $x \in
\mathbb{R}^d,$ $\int \rho(x)\, dx =1$. Moreover, $b * \rho$ indicates $b$ convoluted with $\rho$.
Write $ \lambda u(x) -
\frac{1}{2} Tr (a(x) D^2u (x)) - (b * \rho) (x)\cdot D u(x) =
f(x) +
\langle \big( b - (b* \rho) \big)(x), Du(x)\rangle
$. It is easy to see that $b * \rho$ (even if it can be unbounded) is a $C^{\infty}-$function with all bounded derivatives. Moreover, there exists $C = C (\theta, D \rho, D^2 \rho,
D^3 \rho)>0$ such that $$\begin{aligned}
\label{roo}
\| D^k (b* \rho) \|_0 \le C [b]_{\theta},\;\;\; k=1,2,3.\end{aligned}$$ The function $b - (b* \rho)$ is bounded and we have $$\| b - (b* \rho)\|_0 \le C [b]_{\theta}.$$ It follows that $ b - (b* \rho) \in C^{\theta}_b (\mathbb{R}^d,
\mathbb{R}^d)$. Applying Step II, we find that $$\label{df}
\| u\|_{2+ \theta '}
\le C(\lambda) \| f\|_{\theta} +
C (\lambda)\big \| \langle b - (b* \rho), Du\rangle
\big \|_{\theta}$$ with $C(\lambda) \to 0$. Using that $$\| \langle b - (b* \rho), Du\rangle
\|_{\theta} \le c [ b ]_{\theta} \|Du
\|_0
+ c [ b ]_{\theta} \| Du\|_{\theta}
\le c [b]_{\theta} \| u\|_{2 + \theta \, '},$$ for some constant $c$ depending on $\theta$, we rewrite : $$\| u\|_{2+ \theta'}
\le C(\lambda) \| f\|_{\theta} +
C (\lambda) c [b]_{\theta } \| u\|_{2 + \theta \, '}.$$ Choosing $\lambda_0 >0$ such that $C(\lambda) < \frac{1}{c \,
[b]_{\theta}}$, for $\lambda \ge \lambda_0$, we find, with $u =
u_{\lambda}$ $$\label{apri}
(1 - C (\lambda) c_{} [b]_{\theta} )\,
\| u\|_{2+ \theta \, '}
\le C(\lambda) \| f\|_{\theta}.$$ Defining $K(\lambda ) = \frac{C(\lambda ) }{1 - C (\lambda) c_{}
[b]_{\theta}
} $, we get the assertion.
[**Step IV.**]{} We show that for $\lambda \ge
\lambda_0 $ (see Step III) there exists a classical solution $u = u_{\lambda }
\in C^{2 + \theta '} (\mathbb{R}^d)$ to . This assertion will conclude the proof.
We fix $\lambda \ge \lambda_0$. To prove the result, we will use the continuity method. To this purpose, using the test function $\rho$ of Step III, we consider: $$\label{drr}
\lambda u(x) - \frac{1}{2} Tr (a(x) D^2u (x)) - (1 - \delta)
(b
* \rho) (x) \cdot D u(x) - \delta b(x)\cdot D u(x) = f(x),$$ $x \in \mathbb{R}^d,$ where $\delta \in [0,1]$ is a parameter. Let us define $$\Gamma = \{ \delta \in [0,1] \, :\, \mbox{eq. \eqref{drr}
has a unique solution $u =u_{\delta}
\in C^{2+ \theta '}(\mathbb{R}^d)$},
\mbox{ for any} \;
f \in C^{\theta}(\mathbb{R}^d)\}.$$ $\Gamma $ is not empty since $0 \in \Gamma $ by Step I. Let us fix $\delta_0 \in \Gamma$ and rewrite equation corresponding to an arbitrary $\delta \in [0,1]$ as $$\lambda u(x) - \frac{1}{2} Tr (a(x) D^2u (x)) - (1 - \delta_0)
(b* \rho) (x) \cdot D u(x) - \delta_0 b(x)\cdot D u(x)$$ $$= f(x) + [\delta - \delta_0] \, ( b - \, b* \rho )(x) \cdot D
u(x).$$ Introduce the operator ${\cal T} : C^{2+ \theta ' }(\mathbb{R}^d)
\to
C^{2+ \theta ' }(\mathbb{R}^d)$. For any $v \in C^{2+ \theta ' }(\mathbb{R}^d)
$, ${\cal T} v =u $ is the (unique) $C^{2+ \theta ' }(\mathbb{R}^d)$-function which solves $$\lambda u(x) - \frac{1}{2} Tr (a(x) D^2u (x)) - (1 - \delta_0)
(b* \rho) (x) \cdot D u(x) - \delta_0 b(x)\cdot D u(x)$$ $$= f(x) + [\delta - \delta_0] \, ( b - \, b* \rho )(x) \cdot D
v(x).$$ Using the a-priori estimates , we get that $$\| {\cal T} v - {\cal T} w \|_{2+ \theta'} \le 2 K(\lambda)
|\delta - \delta_0| \, [b]_{\theta} \,\| v - w \|_{2 + \theta '},
\;\; \; v , w \in C^{2+ \theta ' }(\mathbb{R}^d).$$ Choosing $|\delta - \delta_0|$ small enough, the operator $\cal T$ becomes a contraction on $C^{2+ \theta ' }(\mathbb{R}^d)$ and it has a unique fixed point which is the solution to . Therefore for $|\delta - \delta_0|$ small enough, we have that $\delta \in \Gamma$. A compacteness argument shows that $\Gamma = [0,1]$. The assertion is proved.
Differentiable stochastic flow
===============================
[Given ]{} $x\in{\mathbb{R}}^{d}$, consider the stochastic differential equation in ${\mathbb{R}}^{d}$ : $$dX_{t}=b\left( X_{t}\right) dt + \sigma (X_t) dW_{t},\quad \quad
X_{s}=x,\;\;\; t \ge s \ge 0. \label{SDE1}$$ As already mentioned our key result is the existence of a *differentiable* stochastic flow $(x,s,t)\mapsto\varphi_{s,t}(x)$ for equation (\[SDE1\]). Recall the relevant definition from H. Kunita [@K]:
A *stochastic flow of diffeomorphisms* (resp. *of class* $C^{1,\alpha}$) on the stochastic basis $\left( \Omega,\left( \mathcal{F}{}_{t}\right)
,{}\mathcal{F},P,\left( W_{t}\right) \right) $ associated to equation (\[SDE1\]) is a map $(s,t,x,\omega)\mapsto\phi_{s,t}(x)\left( \omega\right) $, defined for $0\leq s\leq t $, $x\in{\mathbb{R}}^{d}$, $\omega\in \Omega$ with values in ${\mathbb{R}}^{d}$, such that
- given any $s \ge 0 $, $x\in{\mathbb{R}}^{d}$, the process $X^{s,x}=\left( X_{t}^{s,x}\left( \omega\right) ,t\ge s
,\omega\in\Omega\right) $ defined as $X_{t}^{s,x}=\phi_{s,t}(x)$ is a continuous $\mathcal{F}_{s,t}$-measurable solution of equation (\[SDE1\]);
- $P$-a.s., for all $0\leq s\leq t $, $\phi_{s,t}$ is a diffeomorphism and the functions $\phi_{s,t}(x)$, $\phi_{s,t}^{-1}(x)$, $D\phi_{s,t}(x)$, $D\phi_{s,t}^{-1}(x)$ are continuous in $(s,t,x)$ (resp. of class $C^{1,\alpha}$ in $x$ uniformly in $(s,t)$, for $0\leq s\leq t \le
T$, with $T>0$);
- $P$-a.s., $\phi_{s,t}(x)
=\phi_{u,t}(\phi_{s,u}(x))$, for all $0\leq s\leq u\leq t $, $x\in{\mathbb{R}}^{d}$, and $\phi_{s,s}(x)=x$.
Starting from the work of Zvonkin, an important approach to the analysis of SDEs with non-regular drift is based on the transformation $\Psi _{t}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$, solution of the vector valued equation $$\frac{\partial\Psi_{t}}{\partial t}+L\Psi_{t}=0\text{ on }\left[
0,T\right] ,\quad\Psi_{T}\left( x\right) =x$$ where $\Psi_{t}(x)= \Psi(t,x)$ and $\left[ 0,T\right] $ is a time interval where the SDE is considered. At time $T$, the solution is an isomorphism by definition; one has to prove suitable regularity and invertibility of $\Psi_{t}$ for $t\in\left[
0,T\right] $. Then $Y_{t}:=\Psi_{t}\left( X_{t}\right) $ satisfies$$dY_{t}=D\Psi_{t}\left( \Psi_{t}^{-1}\left( Y_{t}\right) \right)
\sigma\left( \Psi_{t}^{-1}\left( Y_{t}\right) \right) dW_{t}.$$ The irregular drift has been removed. This approach, although successful (see [@Ba], [@GM], [@Kry-Ro], [@Za], [@Za2]), raises two delicate questions: i) one has to deal with unbounded initial conditions; ii) one has to prove some form of invertibility.
We propose a variant, based on the same operator $L$ but on the vector valued equation $$\lambda\psi-L\psi=b$$ (under other assumptions one can treat also the time-dependent case through the parabolic equation $\lambda\psi_{t}-\frac{\partial\psi_{t}}{\partial t}-L\psi_{t}=b$, see [@FGP]). We find it more tractable than the case of unbounded initial condition; and we translate the difficult invertibility issue in the smallness of the gradient of the solution, obtained by means of a large $\lambda$. When the gradient of $\psi$ is less than one, the function $\Psi\left( x\right)
=x+\psi\left( x\right) $ is invertible and the process $Y_{t}:=\Psi\left( X_{t}\right) $ satisfies$$dY_{t}= D\Psi\left( \Psi^{-1}\left( Y_{t}\right) \right)
\sigma\left( \Psi^{-1}\left( Y_{t}\right) \right)
dW_{t}+\lambda \psi\left( \Psi^{-1}\left( Y_{t}\right) \right) dt .$$ So, at the end, the transformed equation has the same degree of difficulty as in the case of the Zvonkin’s transformation.
\[th:flow1\] Assume Hypotheses \[hy1\], \[hy2\], \[hy3\] and fix any $\theta '' \in (0, \theta)$. Then we have the following facts:
- (pathwise uniqueness) For every $s\ge 0 $, $x\in{\mathbb{R}}^{d}$, the stochastic equation (\[SDE1\]) has a unique continuous adapted solution $X^{s,x}=\left( X_{t}^{s,x}\big(
\omega\right) ,t\ge s,$ $\omega\in\Omega\big) $.
- (differentiable flow) There exists a stochastic flow $\phi= (\phi_{s,t})$ of diffeomorphisms for equation (\[SDE1\]). The flow is also of class $C^{1,{\theta}^{''}}$.
- (stability) Let $(b^{n})\subset
C^{{\theta}}(\mathbb{R}^d, \mathbb{R}^d)$ and let $(\phi^{n})$ be the corresponding stochastic flows. Assume that there exists $b \in C^{{\theta}}(\mathbb{R}^d,
\mathbb{R}^d)$ such that $b_n - b \in C^{{\theta}}_b(\mathbb{R}^d,
\mathbb{R}^d)$, $n \ge 1$, and $\|b - b_n \|_{C^{{\theta}}_b}
\to 0$ as $n \to \infty$. If $\phi$ is the flow associated to $b$, then, for any $p\geq 1$, $T> 0$, $$\label{stability1}
\lim_{n\rightarrow\infty}\sup_{x\in{\mathbb{R}}^{d}} \sup_{0 \le
s\le T} E[ \sup_{u \in [s,T]}
\frac{|\phi_{s,u}^{n}(x)-\phi_{s,u}(x)|^{p}}{(1+|x|)^p}]=0.$$$$\sup_{n\in \mathbb{N}}\sup_{x\in {\mathbb{R}}^{d}}\sup_{0\leq s\leq
T}E[\sup_{u\in \lbrack s,T]}\Vert D\phi _{s,u}^{n}(x)\Vert
^{p}]<\infty , \label{bound}$$ $$\label{stability2}
\lim_{n\rightarrow\infty}\sup_{x\in{\mathbb{R}}^{d}}\sup_{0\leq
s\leq T} E[ \sup_{u \in [s,T]} \Vert
D\phi_{s,u}^{n}(x)-D\phi_{s,u}(x)\Vert^{p}]=0.$$ ($\Vert\cdot\Vert$ denotes the Hilbert-Schmidt norm).
**Step 1** (auxiliary elliptic systems). Let us choose $\theta '$ such that $0< \theta ''
< \theta' < \theta $.
For a fixed $\lambda \ge \lambda_0 >0$ (see Theorem \[bbo\]) we consider the unique classical solution $\psi = \psi_{\lambda} \in C^{2+ \theta ' }
(\mathbb{R}^d, \mathbb{R}^d)$ to the elliptic system $$\label{pde}
\lambda\psi_{\lambda } - L\psi_{\lambda} =b,\;\;\;$$ where $$Lu(x)=\frac{1}{2} Tr ( \sigma(x) \sigma^*(x) D^2 u(x) ) +
b(x)\cdot D u(x),$$ for any smooth function $u: {\mathbb{R}}
^{d}\rightarrow{\mathbb{R}}^{d}$ (clearly has to be interpreted componentwise).
Define $$\Psi_{\lambda}(x)=x+\psi_{\lambda}(x).$$
Similarly to [@FGP Lemma 8] we have
\[diff\] For $\lambda $ large enough, such that $\Vert D \psi_{\lambda}\Vert_{0} <1$ (see Theorem \[bbo\]), the following statements hold:
[[ ]{}]{} (i) $\Psi_{\lambda}$ has bounded first and spatial derivatives and moreover the second (Fréchet) derivative $D^2_x
\Psi_{\lambda}$ is globally $\theta '$-Hölder continuous.
[[ ]{}]{} (ii) $\Psi_{\lambda}$ is a $C^2$-diffeomorphism of $\mathbb{R}^d$.
[[ ]{}]{} (iii) $\Psi_{\lambda}^{-1}$ has bounded first and second derivatives and moreover $$\begin{aligned}
\label{gra1}
D\Psi _{\lambda }^{-1}(y)= \sum_{k\geq 0}\big(-D\psi _{\lambda }
(\Psi _{\lambda }^{-1}(y))\big)^{k},\;\;\;y\in \mathbb{R}^{d}.\end{aligned}$$
[*In the sequel we will use a value of $\lambda$ for which Lemma \[diff\] holds and simply write $\psi$ and $\Psi$ for $\psi_{\lambda}$ and $\Psi_{\lambda}$.*]{}
**Step 2** (conjugated SDE). Define $$\widetilde{b}(y)= \lambda\psi(\Psi^{-1}(y)),\quad\widetilde{\sigma
}(y)=D\Psi(\Psi^{-1}(y)) \, \sigma ( \Psi^{-1}(y) )$$ and consider, for every $s\ge 0 $ and $y\in{\mathbb{R}}^{d}$, the SDE $$Y_{t}=y+\int_{s}^{t}\tilde{\sigma}(Y_{u})dW_{u}+
\int_{s}^{t}\widetilde {b}(Y_{u})du,\qquad t\ge s. \label{conjugated
SDE}$$ This equation is equivalent to equation (\[SDE1\]), in the following sense. If $X_{t}$ is a solution to , then $Y_{t}=\Psi(X_{t})$ verifies equation (\[conjugated SDE\]) with $y=\Psi(x)$: it is sufficient to apply Itô formula to $\Psi(X_{t})$ and use equation (\[pde\]).
Viceversa, given a solution $Y_{t}$ of equation (\[conjugated SDE\]), let $X_{t}=\Psi^{-1}(Y_{t})$, then it is possible to prove by direct application of Itô formula that $X_{t}$ is a solution of with $x=\Psi^{-1}(y)$. This is not very important since below we will obtain this fact indirectly.
**Step 3** (proof of (i) and (ii)). We have clearly $\widetilde{b}$ and $\widetilde{\sigma}\in C^{1+{\theta'}}$ (with first order derivatives bounded and in $C^{\theta'}_b)$ so that, in particular, they are Lipschitz continuous.
By classical results (see [@K Chapter 2]) this implies existence and uniqueness of a strong solution $Y$ of equation (\[conjugated SDE\]) and even the existence of a $C^{1,\theta^{''}}$ stochastic flow of diffeomorphisms $\varphi_{s,t}$ associated to equation (\[conjugated SDE\]).
The uniqueness of $Y$ implies the pathwise uniqueness of solutions of the original SDE since two solutions $X,\tilde{X}$ give rise to two processes $Y_{t}=\Psi (X_{t})$ and $\tilde{Y}_{t}=\Psi (\tilde{X}_{t})$ solving , then $Y=\tilde{Y}$ and then necessarily $X=\tilde{X}$. By the Yamada-Watanabe theorem pathwise uniqueness together with weak existence (which is a direct consequence of the Girsanov formula) gives the existence of the (unique) solution $(X_{t}^{x})_{t\geq s}$ of eq. starting from $x$ at time $s$. Moreover setting $$\phi
_{s,t}=\Psi^{-1}\circ \varphi
_{s,t}\circ \Psi$$ we realize that $\phi _{s,t}$ is the flow of (in the sense that $X_{t}^{x}=\phi _{s,t}(x)$, $P$-a.s.).
**Step 4.** (proof of (iii)). Let $\psi ^{n}$ and $\psi$ be the solutions in $C^{2+\theta' }({\mathbb{R}}^{d};{\mathbb{R}}^{d})$ respectively of the elliptic problem associated to $b_{n}$ and to $b \in C^{\theta }({\mathbb{R}}^{d};{ \mathbb{R}}^{d})$. Notice that we can make a choice of $\lambda $ independent of $n$. We write $$\lambda \left( \psi ^{n}-\psi \right)-L\left( \psi ^{n}-\psi \right)
=\left( b^{n}-b\right) +\left( b^{n}-b\right) \cdot D\psi ^{n},
\;\;\; n \ge 1.$$ By Theorem \[bbo\] we have $\sup_{n \ge 1}\| \psi_n\|_{C^{2+\theta' }} \le C < \infty$. Since $b - b_n$ is a bounded function, by the classical maximum principle (see [@Kr]) we infer also that $\psi - \psi_n$ is a bounded function on $\RR^d$ and $$\begin{aligned}
\label{max}
\| \psi - \psi_n \|_0 \le \frac{C +1}{\lambda}
\| b - b_n \|_0,\;\;\; n \ge 1.
\end{aligned}$$ It follows that $\psi - \psi_n \in
C^{2 + \theta' }_b({\mathbb{R}}^{d};{ \mathbb{R}}^{d})$ and $\| \psi - \psi_n \|_{C^{2 + \theta' }_b}$ $\to 0$ as $n \to
\infty$.
Fix $p \ge 1$ and consider the flows $\varphi
_{s,t}^{n}=\Psi^{n}\circ \phi _{s,t}^{n}\circ (\Psi^{n})^{-1}$ which satisfy $$\varphi_{s,t}^{n}(y)=y+\int_{s}^{t}\widetilde{b}_{}^{n}\circ\varphi
_{s,u}^{n}(y)du+\int_{s}^{t}\widetilde{\sigma}_{}^{n}\circ\varphi
_{s,u}^{n}(y)\cdot dW_{u},$$ We have $\widetilde{\sigma }^{n}\rightarrow \widetilde{\sigma }$ and $\widetilde{b}^{n}\rightarrow \widetilde{b}$, as $n \to \infty$, in $C^{1+\theta' }({\mathbb{R}}^{d};{\mathbb{R}}^{d\times k}) $ and $C^{1+\theta' }({\mathbb{R}}^{d};{\mathbb{R}}^{d})$, respectively. By standard arguments, using the Gronwall lemma, the Doob inequality and the Burkholder inequality (compare, for instance, with the proof of [@K Theorem II.2.1]) we obtain the analog of for the auxiliary flows $\varphi
_{s,t}^{n}$ and $\varphi _{s,t}$: $$\label{stability1-bis}
\lim_{n\rightarrow\infty}\sup_{x\in{\mathbb{R}}^{d}} \sup_{0 \le
s\le T} E[ \sup_{u \in [s,T]}
\frac{|\varphi_{s,u}^{n}(x)-\varphi_{s,u}(x)|^{p}}{(1+|x|)^p}]=0.$$ We can also prove the inequality $$\label{stability20-bis}
\sup_{n\in \mathbb{N}}\sup_{x\in {\mathbb{R}}^{d}}\sup_{0\leq s\leq
T}E[\sup_{u\in \lbrack s,T]}\Vert D\varphi _{s,u}^{n}(x)\Vert
^{p}]<\infty ,$$ for $D\varphi _{s,t}^{n}(y)$, using the fact that the stochastic equation for $D\varphi _{s,t}^{n}(y)$ has the identity as initial condition and random coefficients $D\widetilde{b}^{n}\left( \phi
_{s,u}^{n}\right) $ and $D\widetilde{\sigma }^{n}\left( \phi
_{s,u}^{n}\right) $ which are uniformly bounded functions (since $\|
D \widetilde{b}^{n} \|_0$ $ +\| D\widetilde{\sigma }^{n} \|_0 \le
C$, uniformly in $n$).
To prove is then enough to estimate $D\phi _{s,u}^{n}$ using , the uniform boundedness of the derivatives of $\Psi^{n}$ and its inverse (note that the uniform boundedness of the $D(\Psi^{n})^{-1}$ can be proved by ).
To prove we remark that to estimate the difference $\varphi _{s,t}^{n}(\Psi^n(x))-\varphi _{s,t}(\Psi(x))$ we can split it as $ \varphi _{s,t}^{n}(\Psi^{n}(
x))-\varphi_{s,t}(\Psi^{n}( x)) +\varphi_{s,t}(\Psi^{n}( x))-\varphi
_{s,t}(\Psi( x)) . $ The two differences can then be controlled by $$\mathbb{E}[\sup_{s\le u \le T}|\varphi _{s,u}^{n} (\Psi^{n}(
x))-\varphi_{s,u} (\Psi^{n}( x))|^p]\le a_n \,
(1+|\Psi^{n}(
x)|)^p\le
a_n \, (1+| x|)^p,$$ (where $a_n = \sup_{x\in{\mathbb{R}}^{d}} \sup_{0 \le s\le T} E[
\sup_{u \in [s,T]}
\frac{|\varphi_{s,u}^{n}(x)-\varphi_{s,u}(x)|^{p}}{(1+|x|)^p}]$ and $\lim_{n \to \infty} a_n =0$) and by $$\mathbb{E}[\sup_{s\le u \le T}|\varphi_{s,u}(\Psi^{n}( x))-\varphi
_{s,u}(\Psi( x))|^p]\le \sup_{z\in\mathbb{R}^d}\mathbb{E}[\sup_{s\le u \le T}\|D\varphi_{s,u}(z)\|^p] |\Psi^n(x)-\Psi(x)|^p$$ $$\le C \|\Psi^n-\Psi\|_{0}^p,$$ with $\lim_{n \to \infty} \|\Psi^n-\Psi\|_{0}
= \lim_{n \to \infty} \|\psi^n-\psi\|_{0}
=0$ (see ).
Finally, one has to check that $ (\Psi^{n})^{-1}$ converges to $\Psi^{-1}$ in the supremum norm. This follows from the inequality $$\sup_{y \in \RR^d}|(\Psi^{n})^{-1} (y) - \Psi^{-1}(y)|
\le \sup_{x \in
\RR^d}
| (\Psi^{n})^{-1}(\Psi^{n} (x) ) - \Psi^{-1}(\Psi^n (x) )|$$$$\le \sup_{x \in \RR^d}
| \Psi^{-1}(\Psi^{n} (x) ) - \Psi^{-1}(\Psi (x) )|
\le \| D \Psi^{-1} \|_0
\, \| \Psi - \Psi^n \|_0,$$ which tends to 0, as $n \to \infty$.
Arguing as in the proof of [@K Theorem II.3.1], we get the following linear equation for the derivative $D\phi _{s,t}(x)$ $$\label{dove}
\begin{split}
\lbrack D\Psi_{}(\phi_{s,t}(x))] D\phi_{s,t}(x) =D\Psi_{}(x)
+\int_{s}
^{t}[D^{2}\Psi_{} (\phi_{s,u}(x))]D\phi_{s,u}(x)
\, \sigma (\phi_{s,u}(x)) dW_{u}\\
+ \int_{s} ^{t} D\Psi_{} (\phi_{s,u}(x))
\, [D \sigma (\phi_{s,u}(x))] D\phi_{s,u}(x) dW_{u}
-\lambda\int_{s}^{t}[D\psi_{}(\phi_{s,u}(x))]D\phi_{s,u}(x)du,
\end{split}$$ $0\leq s\leq t\leq T$, $x\in \mathbb{R}^{d}$. From the fact that $ \lim_{n \to \infty} \| \psi ^{n} - \psi \|_{C^{2+\theta' }_b} =0 $ together with and , we finally obtain $$\lim_{n\rightarrow \infty }\sup_{x\in {\mathbb{R}}^{d}}\sup_{0\leq
s\leq T}E[\sup_{u\in \lbrack s,T]}\Vert D\phi _{s,u}^{n}(x)-D\phi
_{s,u}(x)\Vert ^{p}]=0, \label{stima1}$$ which concludes the proof.
We consider now two possible extensions of Theorem \[th:flow1\] to the case when coefficients $b$ and $\sigma_i$ are time-dependent continuous functions defined on $[0,T] \times \RR^d$, i.e., we are dealing with $$\label{fy}
dX_{t}^{x}=b\left( t, X_{t}^{x}\right)
dt+\sum_{i=1}^{k}\sigma_{i}\left(t, X_{t}^{x}\right) dW_{t}^{i},
\;\;\; t\in\left[ 0,T\right] ,\quad
X_{0}=x.$$
\[vi\]
*Let us treat the case in which also $b$ is [*bounded.*]{} Following [@FGP], an analogous of our Theorem \[th:flow1\] holds for if we require that $b$ and $\sigma_i$ are continuous and bounded functions such that $$\sup_{t \in [0,T]} (\| b(t , \cdot) \|_{C^{\theta}_b}
+ \, \| \sigma_i (t ,
\cdot) \|_{C^{1+ \theta}_b} ) < \infty,\;\;\; i=1, \ldots, k,$$ and, moreover (as in Hypothesis \[hy3\]) we assume that $ \sigma (t,x)$ is uniformly non-degenerate, i.e., there exists the inverse of $a(t,x)= \sigma (t,x)\sigma^* (t,x)$, for any $t \in [0,T]$, $x \in \RR^d$, and $$\begin{aligned}
\label{f7}
\| a^{-1}\|_{0} = \sup_{x \in \mathbb{R}^d, \, t \in [0,T]}
\| a^{-1} (t,x) \| < \infty.\end{aligned}$$ To prove Theorem \[th:flow1\] under these hypotheses, one can follow the proof of the analogous result proved in [@FGP]. We only give a sketch of the argument.*
First note that [@FGP Theorem 2] remains the same even with the previous non-constant $\sigma = (\sigma_i)$ (indeed it is a special case of a result in [@KP]). Then [@FGP Lemma 4] is true with $\sigma$ in by the following rescaling argument. Consider $\lambda \ge 1$ and $$\partial_{t}u_{\lambda}+ L u_{\lambda} -
\lambda u_{\lambda}=f\;\;\; \text{in} \;\; [0,\infty)\times
{\mathbb{R}}^{d},$$ where $L $ is the Kolmogorov operator associated to the SDE, i.e., $$L =
\frac{1}{2} \mathrm{Tr} [a(t,x) D^2 u(t,x)]
+ b (t,x) \cdot D
u(t,x)$$ (here $( \sigma (t,x) \sigma^*(t,x)) = a(t,x)$ and $D$ and $D^2$ denote spatial derivatives). Define a function $v$ on $[0,\infty)\times {\mathbb{R}}^{d}$ such that $
v(\lambda t, \sqrt{\lambda}\, x) = u_{\lambda} (t,x)$, $t \ge
0, $ $x \in \mathbb{R}^d.
$ It is easy to see that, for any $s \ge 0$, $y \in \mathbb{R}^d$, $$\partial_{s} v (s,y) + \mathrm{Tr} [
a \Big (\frac{s}{\lambda}, \frac{y}{\sqrt{\lambda}} \Big) D^2
v(s,y)] + \frac{1}{ \sqrt{\lambda}} b \Big ( \frac{s}{\lambda},
\frac{y}{\sqrt{\lambda}} \Big) \cdot Dv(s,y)
- v (s,y) =
\frac{1}{\lambda}{f \Big( \frac{s}{\lambda},
\frac{y}{\sqrt{\lambda}} \Big)}.$$ Now the spatial Hölder seminorms of $ (s, y ) \mapsto
a(\frac{s}{\lambda}, \frac{y}{\sqrt{\lambda}})$ and $(s, y ) \mapsto
b( \frac{s}{\lambda}, \frac{y}{\sqrt{\lambda}}) $ are clearly independent on $\lambda \ge 1$ and on $s \ge 0$. By [@KP Theorem 2.4], we deduce in particular, for any $\lambda \ge 1$, $$\sup_{s \ge 0} \|D v (s, \cdot) \|_{0}
\le \frac{C}{\lambda} \sup_{s \ge 0}\| f(s, \cdot) \|_{\theta },$$ where $C$ is independent of $\lambda$. It follows the assertion of [@FGP Lemma 6] since $$\sup_{t \ge 0} \|D u_{\lambda} (t, \cdot) \|_{0}
= \sqrt{\lambda}\sup_{s \ge 0} \|D v (s, \cdot) \|_{0}
\le \frac{C}{\sqrt{\lambda} } \sup_{s \ge 0}\| f(s, \cdot)
\|_{\theta }.$$ The proof of [@FGP Theorem 5] (which deals with the stochastic flow) remains true even with $\sigma$ in by a straightforward modification.
\[fine\]
*An analogous of Theorem \[th:flow1\] holds for requiring that Hypotheses \[hy1\], \[hy2\] and \[hy3\] are satisfied “uniformly in time”.*
One assumes that $b$ and $\sigma_i$ are continuous functions defined on $[0,T] \times \RR^d$, $i=1, \ldots, k$. Moreover, there exists ${\theta}\in(0,1)$ such that $b(t, \cdot )\in C_{}^{\theta }(\mathbb{R}^{d};\mathbb{R}^{d})$, $t \in [0,T]$, and $\sup_{t \in [0,T]}
\|b(t, \cdot ) \|_{C_{}^{\theta }(\mathbb{R}^{d}, \RR^d)} < \infty$. In addition, $\sigma_i(t, \cdot ) \in C^{3 }_b (\mathbb{R}^d,
\mathbb{R}^d) $, $t \in [0,T]$, $$\sup_{t \in [0,T]}
\| \sigma_i (t, \cdot) \|_{C^{3 }_b (\mathbb{R}^d,
\mathbb{R}^d) } < \infty,$$ $i =1, \ldots, k$, and one requires that condition holds. Theorem \[th:flow1\] under these assumptions may be established by adapting the (time-independent) proof given in the present paper. However, the complete argument, even if it does not present special difficulties, is considerably longer (for instance, one has to prove the analogous of the Bismut-Elworthy-Li formula in the time-dependent case).
We close the section by an application of the stochastic flow. We obtain a Bismut-Elworthy-Li type formula for the derivative of the diffusion semigroup $(P_t)$ associated to (compare with [@B] and [@EL]). It seems the first time that such formula is given for diffusion semigroups associated to SDEs with coefficients which are not locally Lipschitz.
\[bismut\] Let $f :\mathbb{R}^d \to \mathbb{R} $ be uniformly continuous and bounded. For any $x, \, h \in
\mathbb{R}^d$, we have (cf. ) $$D_h P_t f (x) =
\frac{1}{t} \mathbb{E} [ f(\phi_t(x))
\, \int_0^t \langle (\sigma^{*}a^{-1})
(\phi_u(x)) D_h \phi_u (x) ,
dW_u \rangle
],\;\;\; t>0,\; x \in \mathbb{R}^d,$$ where $\langle D P_t f(x), h \rangle = D_h P_t f (x)$ and $D
\phi_u(x) $ solves with $s=0$ (we set $\phi_u(x)= \phi_{0,u}(x)$).
We prove the formula when $f \in C^{\infty}_b (\mathbb{R}^d)$. Indeed, then, by a straightforward uniform approximation of $f$, one can obtain the formula in the general case.
Let $\vartheta :{\mathbb{ R}}^{d}\rightarrow {\mathbb{R}}$ be a smooth test function such that $0\leq
\vartheta (x)\leq 1$, $x\in {\mathbb{R}}^{d}$, $\vartheta (x)=\vartheta (-x)$, $\int_{{\mathbb{R}}^{d}}\vartheta (x)dx=1$, $\mathrm{supp}\,(\vartheta )\subset $ $B(0,2)$, $\vartheta (x)=1$ when $x\in B(0,1)$. For any $n \ge 1$, let $\vartheta _{{n}}(x) ={n
}^{d}\vartheta (n x )$. Define $b_n = b * \vartheta _{{n}}$.
We have that $b_{n}$ is a $C_{}^{\infty}$ and Lipschitz vector field such that $b - b_n \in
C^{\theta}_b(\mathbb{R}^d;\mathbb{R}^d)$ and $\| b - b_n\|_{C^{\theta}_b }$ tends to 0 as $n \to \infty$. Let $(\phi^n_t)$ be the associated flow of smooth diffeomorphisms which solves the SDE involving $b_n$ and let $(P_t^n)$ be the corresponding diffusion semigroup. The Bismut-Elworthy-Li formula for $(P_t^n)$ is given by $$D_h P_t^n f (x)= \frac{1}{t} \mathbb{E} [ f(\phi_t^n (x)) \,
\int_0^t \langle (\sigma^{*}a^{-1}) (\phi_u^n(x)) D_h \phi^n_u (x) ,
dW_u \rangle ],\; t>0,\; x \in \mathbb{R}^d, \, n
\in \mathbb N.$$ Note that $
D_h P_t^n f (x)
= \mathbb{E} [\langle D
f (\phi_t^n (x)), D_h\phi_t^n (x) \rangle ]
$. Passing to the limit as $n \to \infty$, using the estimates and , we get $$D_h P_t f (x)
= \mathbb{E} [ \langle D f (\phi_t (x)), D_h\phi_t (x) \rangle ]
= \frac{1}{t} \mathbb{E} [ f(\phi_t(x))
\int_0^t \langle \sigma^{-1} (\phi_u(x)) D_h \phi_u (x) ,
dW_u \rangle ],$$ for any $ t>0,\; x \in \mathbb{R}^d .$
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|
---
abstract: |
In this paper, we obtained the three-dimensional Pauli equation for a spin-1/2 particle in the presence of an electromagnetic field in noncommutative phase-space, as well the corresponding deformed continuity equation, where the cases of a constant and non-constant magnetic field are considered. Due to the absence of the current magnetization term in the deformed continuity equation as expected, we had to extract it from the noncommutative Pauli equation itself without modifying the continuity equation. It is shown that the non-constant magnetic field lifts the order of the noncommutativity parameter in both the Pauli equation and the corresponding continuity equation. However, we successfully examined the effect of the noncommutativity on the current density and the magnetization current. By using a classical treatment, we derived the semi-classical noncommutative partition function of the three-dimensional Pauli system of the one-particle and N-particle systems. Then, we employed it for calculating the corresponding Helmholtz free energy followed by the magnetization and the magnetic susceptibility of electrons in both commutative and noncommutative phase-spaces. Knowing that with both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product, we introduced the phase-space noncommutativity in the problems in question.
$\phantom{}$
**Keywords:** 3-D noncommutative phase-space; Pauli equation; continuity equation; current magnetization; semi-classical partition function; magnetic susceptibility
address: 'Laboratoire de Physique Mathématique et de Physique Subatomique (LPMPS), Université Frères Mentouri, Constantine 25000, Algeria'
author:
- '[Ilyas Haouam]{}'
title: '`ON THE THREE-DIMENSIONAL PAULI EQUATION IN NONCOMMUTATIVE PHASE-SPACE` '
---
Introduction
============
It is known that the relativistic wave equation describing the fermions with spin-1/2 is the Dirac equation; on the other hand, the non-relativistic wave equation describing them, namely the Pauli equation, which is a topic of great interest in physics [@key-1; @key-2; @key-3; @key-4]. It is relative to the explanation of many experimental results, and its probability current density changed to including an additional spin-dependent term recognized as the spin current [@key-5; @key-6; @key-7]. Pauli equation shown [@key-8; @key-9; @key-10; @key-11; @key-12] as the non-relativistic limit of Dirac equation. Knowing that historically at first time Pauli in 1927 [@key-13] presented his known spin matrices in modifying the non-relativistic Schrödinger equation to account for Goudsmit-Uhlenbecks hypothesis (1925) [@key-14; @key-15]. Therefore, he applied an ansatz for adding a phenomenological term to the ordinary non-relativistic Hamiltonian in the presence of an electromagnetic field, the interaction energy of a magnetic field and electronic magnetic moment relative to the intrinsic spin angular momentum of the electron. Describing this spin angular momentum through the spin matrices requires replacing the complex scalar wave function by a two-component spinor wave function in the wave equation. Since then, the study of the Pauli equation became a matter of considerable attention.
In 1928 when Dirac presented his relativistic free wave equation in addition to the minimal coupling replacement to include electromagnetic interactions [@key-16], he showed that his equation contained a term involving the electron magnetic moment interacting with a magnetic field, which was the same one inserted by hand in Paulis equation. After that, it became common to account electron spin as a relativistic phenomenon, and the corresponding spin-1/2 term could be inserted into the spin-0 non-relativistic Schrodinger equation as will be discussed in the following to see how this is possible. However, motivated by attempts to understand string theory and describe quantum gravitation using noncommutative geometry and by trying to have drawn considerable attention to the phenomenological implications, we focus here on studying the problem of a non-relativistic spin-1/2 particle in the presence of an electromagnetic field within 3-dimensional noncommutative phase-space.
As a mathematical theory, noncommutative geometry is by now well established, although at first, its progress has been narrowly restricted to some branches of physics such as quantum mechanics. However, recently, the noncommutative geometry has become a topic of great interest. It has been finding applications in many sectors of physics and rapidly has become involved in them, continued to promote fruitful ideas and the search for a better understanding. Such as in the quantum gravity [@key-17]; the standard model of fundamental interactions [@key-18]; as well in the string theory [@key-19]; and its implication in Hopf algebras [@key-20] gives the ConnesKreimer Hopf algebras [@key-21; @key-22; @key-23] etc. There are many papers devoted to the study such various aspects especially in quantum field theory [@key-24; @key-25; @key-26] and quantum mechanics [@key-27; @key-28; @key-29].
This paper is organized as follows. In section 2, we present an analysis review of noncommutative geometry, in particular both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product. In section 3, we investigate the three-dimensional Pauli equation in the presence of an electromagnetic field and the corresponding continuity equation. Besides, we derived the current magnetization term in the deformed continuity equation. Section 4 is devoted to calculating the semi-classical noncommutative partition function of the Pauli system of the one-particle and N-particle systems. Consequently, we obtain the corresponding magnetization and the magnetic susceptibility through the Helmholtz free energy, all in both commutative and noncommutative phase-spaces and within a classical limit. Therefore, concluding with some remarks.
Review of noncommutative algebra
================================
Firstly, we present the most essential formulas of noncommutative algebra [@key-29]. It is well known that at very tiny scales such as the string scale, the position coordinates do not commute with each other, neither do the momenta.
Let us accept in a d-dimensional noncommutative phase-space the operators of coordinates and momenta $x_{i}^{nc}$ and $p_{j}^{nc}$, respectively. The noncommutative formulation of quantum mechanics corresponds to the following Heisenberg-like commutation relations $$\begin{array}{cccc}
\left[x_{\mu}^{nc},x_{\nu}^{nc}\right]=i\Theta_{\mu\nu}, & \left[p_{\mu}^{nc},p_{\nu}^{nc}\right]=i\eta_{\mu\nu}, & \left[x_{\mu}^{nc},p_{\nu}^{nc}\right]=i\tilde{\hbar}\delta_{\mu\nu} & ,\:(\mu,\nu=1,..d)\end{array},\label{eq:1}$$ the effective Planck constant is the deformed Planck constant, which is given by $$\tilde{\hbar}=\alpha\beta\hbar+\frac{\text{Tr}(\Theta\eta)}{4\alpha\beta\hbar},\label{eq:2}$$ where $\frac{\text{Tr}(\Theta\eta)}{4\alpha\beta\hbar}\ll1$ is the condition of consistency in quantum mechanics. $\Theta_{\mu\nu}$, $\eta_{\mu\nu}$ are constant antisymmetric $d\times d$ matrices and $\delta_{\mu\nu}$ is the identity matrix.
It is shown that $x_{i}^{nc}$ and $p_{j}^{nc}$ can be represented in terms of coordinates $x_{i}$ and momenta $p_{j}$ in usual quantum mechanics through the so-called generalized Bopp-shift as follows [@key-27] $$\begin{array}{cccc}
x_{\mu}^{nc} & = & \alpha x_{\mu}-\frac{1}{2\alpha\hbar}\Theta_{\mu\nu}p_{\nu},\:\text{ and } & p_{\mu}^{nc}=\beta p_{\mu}+\frac{1}{2\beta\hbar}\eta_{\mu\nu}x_{\nu}\end{array},\label{eq:3}$$ with $\alpha=1-\frac{\Theta\eta}{8\hbar^{2}}$ and $\beta=\frac{1}{\alpha}$ are scaling constants.
To the 1rst order of $\Theta$ and $\eta$, in the calculations we take $\alpha=\beta=1$, so the Equations (\[eq:3\], \[eq:2\]) become $$\begin{array}{ccc}
x_{\mu}^{nc} & =x_{\mu}-\frac{1}{2\hbar}\Theta_{\mu\nu}p_{\nu},\; & p_{\mu}^{nc}=p_{\mu}+\frac{1}{2\hbar}\eta_{\mu\nu}x_{\nu}\end{array},\:\text{ and }\tilde{\hbar}=\hbar+\frac{\Theta\eta}{4\hbar}.\label{eq:4}$$
If the system in which we study the effects of noncommutativity is three-dimensional, we limit ourselves to the following noncommutative algebra $$\begin{array}{cccc}
\left[x_{j}^{nc},x_{k}^{nc}\right]=i\frac{1}{2}\epsilon_{jkl}\Theta_{l}, & \left[p_{j}^{nc},p_{k}^{nc}\right]=i\frac{1}{2}\epsilon_{jkl}\eta_{l}, & \left[x_{j}^{nc},p_{k}^{nc}\right]=i\tilde{\hbar}\delta_{jk} & ,\:(j,k,l=1,2,3)\end{array},\label{eq:5}$$ $\Theta_{l}=(0,0,\Theta)$, $\eta_{l}=(0,0,\eta)$ are the real-valued noncommutative parameters with the dimension of $length{}^{2}$, $momentum{}^{2}$ respectively, they are assumed to be extremely small. And $\epsilon_{jkl}$ is the Levi-Civita permutation tensor, with $\epsilon_{123}=\epsilon_{231}=\epsilon_{312}=-\epsilon_{321}=-\epsilon_{132}=-\epsilon_{213}=1$, if $j=k$ or $k=l$, $\epsilon_{jkl}=0$. Therefor, we have $$x_{i}^{nc}=x_{i}-\frac{1}{4\hbar}\epsilon_{ijk}\Theta_{k}p_{j}:\begin{cases}
\begin{array}{ccc}
x^{nc} & = & x-\frac{1}{4\hbar}\Theta p_{y}\\
y^{nc} & = & y+\frac{1}{4\hbar}\Theta p_{x}\\
z^{nc} & = & z
\end{array}\end{cases},\;p_{i}^{nc}=p_{i}+\frac{1}{4\hbar}\epsilon_{ijk}\eta_{k}x_{j}:\begin{cases}
\begin{array}{ccc}
p_{x}^{nc} & = & p_{x}+\frac{1}{4\hbar}\eta y\\
p_{y}^{nc} & = & p_{y}-\frac{1}{4\hbar}\eta x\\
p_{z}^{nc} & = & p_{z}
\end{array}\end{cases}.\label{eq:5-1}$$
$\phantom{}$
In noncommutative quantum mechanics, it is quite possible that we replace the usual product with the Moyal-Weyl ($\star$) product, then the quantum mechanical system will become simply the noncommutative quantum mechanical system. Let $\mathcal{H}\left(x,p\right)$ be the Hamiltonian operator of the usual quantum system, then the corresponding Schrödinger equation on noncommutative quantum mechanics is typically written as $$\mathcal{H}\left(x,p\right)\star\psi\left(x,p\right)=E\psi\left(x,p\right).\label{eq:6}$$
The definition of Moyal-Weyl product between two arbitrary functions $f(x,p)$ and $g(x,p)$ in phase-space is given by [@key-30] $$\begin{array}{c}
(f\star g)(x,p)=\exp[\frac{i}{2}\Theta_{ab}\partial_{x_{a}}\partial_{x_{b}}+\frac{i}{2}\eta_{ab}\partial_{p_{a}}\partial_{p_{b}}]f\left(x_{a},p_{a}\right)g\left(x_{b},p_{b}\right)=f(x,p)g(x,p)\\
+\sum_{n=1}\frac{1}{n!}\left(\frac{i}{2}\right)^{n}\Theta^{a_{1}b_{1}}...\Theta^{a_{n}b_{n}}\partial_{a_{1}}^{x}...\partial_{a_{k}}^{x}f(x,p)\partial_{b_{1}}^{x}...\partial_{b_{k}}^{x}g(x,p)\\
+\sum_{n=1}\frac{1}{n!}\left(\frac{i}{2}\right)^{n}\eta^{a_{1}b_{1}}...\eta^{a_{n}b_{n}}\partial_{a_{1}}^{p}...\partial_{a_{k}}^{p}f(x,p)\partial_{b_{1}}^{p}...\partial_{b_{k}}^{p}g(x,p)
\end{array},\label{eq:7}$$ with $f(x,p)$ and $g(x,p)$, assumed to be infinitely differentiable. If we consider the case of noncommutative space the definition of Moyal-Weyl product will be reduced to [@key-31]
$$(f\star g)(x)=\exp[\frac{i}{2}\Theta_{ab}\partial_{x_{a}}\partial_{x_{b}}]f\left(x_{a}\right)g\left(x_{b}\right)=f(x)g(x)+\sum_{n=1}\frac{1}{n!}\left(\frac{i}{2}\right)^{n}\Theta^{a_{1}b_{1}}...\Theta^{a_{n}b_{n}}\partial_{a_{1}}...\partial_{a_{k}}f(x)\partial_{b_{1}}...\partial_{b_{k}}g(x).\label{eq:8}$$
Due to the nature of the $\star$product, the noncommutative field theories for low-energy fields ([$E^{2}\apprle1/\Theta$]{}) at classical level are completely reduced to their commutative versions. However, this is just the classical result and quantum corrections always reveal the effects of $\Theta$ even at low-energies.
$\phantom{}$
On noncommutative phase-space the $\star$product can be replaced by a Bopp’s shift, i.e. the $\star$product can be changed into the ordinary product by replacing $\mathcal{H}\left(x,p\right)$ with $\mathcal{H}\left(x^{nc},p^{nc}\right)$. Thus the corresponding noncommutative Schrödinger equation can be written as $$\mathcal{H}\left(x,p\right)\star\psi\left(x,p\right)=\mathcal{H}\left(x_{i}-\frac{1}{2\hbar}\Theta_{ij}p_{j},\;p_{\mu}+\frac{1}{2\hbar}\eta_{\mu\nu}x_{\nu}\right)\psi=E\psi.\label{eq:9}$$
Note that $\Theta$ and $\eta$ terms always can be treated as a perturbation in quantum mechanics.
If $\Theta=\eta=0$, the noncommutative algebra reduces to the ordinary commutative one.
Pauli equation in noncommutative phase-space
=============================================
Formulation of noncommutative Pauli equation
--------------------------------------------
The nonrelativistic Schrödinger equation that describes an electron in interaction with an electromagnetic potential $\left(A_{0},\overrightarrow{A}\right)$ ($\hat{\overrightarrow{p}}$ is replaced with $\hat{\overrightarrow{\pi}}=\hat{\overrightarrow{p}}-\frac{e}{c}\overrightarrow{A}$ and $\hat{E}$ with $\hat{\epsilon}=i\hbar\frac{\partial}{\partial t}-e\phi$ ) iswhere $\hat{\overrightarrow{p}}=i\hbar\overrightarrow{\nabla}$ is the momentum operator, $m$, $e$ are the mass and charge of the electron, and $c$ is the speed of light. $\psi\left(r,t\right)$ is Schrödingers scalar wave function. The appearance of real-valued electromagnetic Coulomb and vector potentials, $\phi\left(\overrightarrow{r},t\right)$ and $\overrightarrow{A}\left(\overrightarrow{r},t\right)$, is a consequence of using the gauge-invariant minimal coupling assumption to describe the interaction with the external magnetic and electric fields defined by $$\overrightarrow{E}=-\overrightarrow{\nabla}\phi-\frac{1}{c}\frac{\partial\overrightarrow{A}}{\partial t},\quad\overrightarrow{B}=\overrightarrow{\nabla}\times\overrightarrow{A}.\label{eq:11}$$
However, the electron gains potential energy when the spin interacts with the magnetic field, therefore the Pauli equation of an electron with spin is given by [@key-1; @key-8] $$\frac{1}{2m}\left(\overrightarrow{\sigma}\hat{\overrightarrow{\pi}}\right)^{2}\psi\left(r,t\right)+e\phi\psi\left(r,t\right)=\frac{1}{2m}\left(\hat{\overrightarrow{p}}-\frac{e}{c}\overrightarrow{A}\right)^{2}\psi\left(r,t\right)+e\phi\psi\left(r,t\right)+\mu_{B}\overrightarrow{\sigma}\overrightarrow{B}\psi\left(r,t\right)=i\hbar\frac{\partial}{\partial t}\psi\left(r,t\right),\label{eq:12}$$ where $\psi\left(r,t\right)=\left(\begin{array}{cc}
\psi_{1} & \psi_{2}\end{array}\right)^{T}$ are the spinor wave function, which replaces the scalar wave function. With $\mu_{B}=\frac{\left|e\right|\hbar}{2mc}=9.27\times10^{-24}JT^{-1}$ is Bohr’s magneton, $\overrightarrow{B}$ is the applied magnetic field vector, also $\mu_{B}\overrightarrow{\sigma}$ represents the magnetic moment. $\overrightarrow{\sigma}$’s being the three Pauli matrices ($\text{Tr}\overrightarrow{\sigma}=0$), obey the following algebra $$\left[\sigma_{i},\sigma_{j}\right]=2i\epsilon_{ijk}\sigma_{k},\label{eq:13}$$ $$\sigma_{i}\sigma_{j}=\delta_{ij}I+i\sum_{k}\epsilon_{ijk}\sigma_{k},\label{eq:15-1}$$ $$\left(\overrightarrow{\sigma}\hat{\overrightarrow{a}}\right)\left(\overrightarrow{\sigma}\hat{\overrightarrow{b}}\right)=\hat{\overrightarrow{a}}\hat{\overrightarrow{b}}+i\overrightarrow{\sigma}\left(\hat{\overrightarrow{a}}\times\hat{\overrightarrow{b}}\right),\label{eq:14-1}$$ $\hat{\overrightarrow{a}}$, $\hat{\overrightarrow{b}}$ are any two vector operators that commute with $\overrightarrow{\sigma}$. It must be emphasized that the third term of equation (\[eq:12\]) is the Zeeman term, which is generated automatically by using feature (\[eq:14-1\]) with a correct g-factor of $g=2$ as reduced in the Bohr’s magneton rather than being introduced by hand as a phenomenological term, as is usually done.
The Pauli equation in noncommutative phase-space is $$\mathcal{H}\left(x^{nc},p^{nc}\right)\psi\left(x^{nc},t\right)=\mathcal{H}\left(x,p^{nc}\right)\star\psi\left(x,t\right)=e^{\frac{i}{2}\Theta_{ab}\partial_{x_{a}}\partial_{x_{b}}}\mathcal{H}\left(x_{a},p^{nc}\right)\psi\left(x_{b},t\right)=i\hbar\frac{\partial}{\partial t}\psi\left(x,t\right).\label{eq:14}$$
Here we achieved the noncommutativity in space using Moyal $\star$product then the noncommutativity in phase through Bopp-shift. Using equation (\[eq:8\]), we have $$\mathcal{H}\left(x^{nc},p^{nc}\right)\psi\left(x^{nc},t\right)=\left\{ \mathcal{H}\left(x,p^{nc}\right)+\frac{i}{2}\Theta^{ab}\partial_{a}\mathcal{H}\left(x,p^{nc}\right)\partial_{b}+\sum_{n=2}\frac{1}{n!}\left(\frac{i}{2}\right)^{n}\Theta^{a_{1}b_{1}}...\Theta^{a_{n}b_{n}}\partial_{a_{1}}...\partial_{a_{k}}\mathcal{H}\left(x,p^{nc}\right)\partial_{b_{1}}...\partial_{b_{k}}\right\} \psi.\label{eq:15}$$
In case of a constant real magnetic field $\overrightarrow{B}=\left(0,0,B\right)=B\overrightarrow{e}_{3}$ oriented along the axis (Oz), which is often referred to as the Landau system. We have the following symmetric gauge $$\overrightarrow{A}=\frac{\overrightarrow{B}\times\overrightarrow{r}}{2}=\frac{B}{2}\left(-y,x,0\right),\;\text{with }\:A_{0}\left(x\right)=e\phi=0.\label{eq:16}$$
Therefore, the derivations in the equation (\[eq:15\]) approximately shut down in the first-order of $\Theta$, then the noncommutative Pauli equation in the presence of a uniform magnetic field can be written as follows $$\mathcal{H}\left(x,p^{nc}\right)\star\psi\left(x\right)=\left\{ \frac{1}{2m}\left(\overrightarrow{p}^{nc}-\frac{e}{c}\overrightarrow{A}\left(x\right)\right)^{2}+\mu_{B}\overrightarrow{\sigma}\overrightarrow{B}+\frac{ie}{4mc}\Theta^{ab}\partial_{a}\left(\frac{e}{c}\overrightarrow{A}^{2}-2\overrightarrow{p}^{nc}\overrightarrow{A}\right)\partial_{b}\right\} \psi\left(x\right)+0(\Theta^{2}),\label{eq:17}$$ with $\left[\overrightarrow{p}^{nc},\overrightarrow{A}\right]=0$. We now make use of the Bopp-shift transformation (\[eq:4\]), in the momentum operator to obtain $$\begin{array}{c}
\mathcal{H}\left(x^{nc},p^{nc}\right)\psi\left(x^{nc},t\right)=\left\{ \frac{1}{2m}\left(p_{i}+\frac{1}{2\hbar}\eta_{ij}x_{j}-\frac{e}{c}A_{i}\right)^{2}+\mu_{B}\overrightarrow{\sigma}\overrightarrow{B}\right.\\
\left.-\frac{ie}{4mc}\Theta^{ab}\partial_{a}\left(2\left(p_{i}+\frac{1}{2\hbar}\eta_{ij}x_{j}\right)A_{i}-\frac{e}{c}\overrightarrow{A}^{2}\right)\partial_{b}\right\} \psi(x,t)=i\hbar\frac{\partial}{\partial t}\psi\left(x,t\right),
\end{array}\label{eq:18}$$ we rewrite the above equation in a more compact form $$\begin{array}{c}
\mathcal{H}\left(x,p^{nc}\right)\star\psi\left(x,t\right)=\left\{ \frac{1}{2m}\left(\overrightarrow{p}-\frac{e}{c}\overrightarrow{A}\right)^{2}-\frac{1}{2m}\left(\overrightarrow{x}\times\overrightarrow{p}\right)\overrightarrow{\eta}-\frac{1}{2m}\frac{e}{c\hbar}\left(\overrightarrow{x}\times\overrightarrow{A}\left(x\right)\right)\overrightarrow{\eta}\right.\\
\left.+\frac{1}{8m\hbar^{2}}\eta_{ij}\eta_{\alpha\beta}x_{j}x_{\beta}+\mu_{B}\overrightarrow{\sigma}\overrightarrow{B}+\frac{e}{4\hbar mc}\left(\overrightarrow{\nabla}\left(2\overrightarrow{p}\overrightarrow{A}-\frac{1}{2\hbar}\left(\overrightarrow{x}\times\overrightarrow{A}\left(x\right)\right)\overrightarrow{\eta}-\frac{e}{c}\overrightarrow{A}^{2}\right)\times\overrightarrow{p}\right)\overrightarrow{\Theta}\right\} \psi\left(x,t\right).
\end{array}\label{eq:19}$$
We restrict ourselves only to the first-order of the parameter $\eta$. The only reason behind this consideration is the balance with the noncommutativity in the space considered in the case of constant magnetic field. Thus we have now $$\begin{array}{c}
\mathcal{H}\left(x,p^{nc}\right)\star\psi\left(x,t\right)=\left\{ \frac{1}{2m}\left(\overrightarrow{p}-\frac{e}{c}\overrightarrow{A}\right)^{2}-\frac{1}{2m}\overrightarrow{L}\overrightarrow{\eta}-\frac{e}{2mc\hbar}\left(\overrightarrow{x}\times\overrightarrow{A}\left(x\right)\right)\overrightarrow{\eta}+\mu_{B}\overrightarrow{\sigma}\overrightarrow{B}\right.\\
\left.+\frac{e}{4mc\hbar}\left(\overrightarrow{\nabla}\left(2\overrightarrow{p}\overrightarrow{A}\left(x\right)-\frac{1}{2\hbar}\left(\overrightarrow{x}\times\overrightarrow{A}\left(x\right)\right)\overrightarrow{\eta}-\frac{e}{c}\overrightarrow{A}^{2}\right)\times\overrightarrow{p}\right)\overrightarrow{\Theta}\right\} \psi\left(x,t\right))=i\hbar\frac{\partial}{\partial t}\psi\left(x,t\right).
\end{array}\label{eq:20}$$
The existence of a Pauli equation for all orders of $\Theta$ parameter is explicitly relative to the magnetic field.
In the case of a non-constant magnetic field, we introduce a function depending on $x$ in the Landau gauge as $A_{2}=xBf(x)$ which gives us a non-constant magnetic field. The magnetic field can be calculated easily using the second equation of equation (\[eq:11\]) as follows [@key-26] $$\overrightarrow{B}(x)=Bf(x)\overrightarrow{e}_{3}.\label{eq:21}$$
If we specify $f(x)$ we obtain different classes of the non-constant magnetic field. If take $f(x)=1$ in this case we get a constant magnetic field.
Having the equation (\[eq:20\]) on hand, we calculate the probability density and the current density.
Deformed continuity equation
----------------------------
In the following we calculate the current density, which results from the Pauli equation (\[eq:20\] that describing a system of two coupled differential equations for $\psi_{1}$ and $\psi_{2}$.
By putting $$\begin{array}{cc}
\mathcal{Q}_{\eta}=\mathcal{Q}_{\eta}^{\ast}=\left(\overrightarrow{x}\times\overrightarrow{A}\left(x\right)\right)\overrightarrow{\eta},\; & \mathcal{Q}_{\Theta}=\left(\overrightarrow{\nabla}\left(2\overrightarrow{p}\overrightarrow{A}\left(x\right)-\frac{1}{2\hbar}\mathcal{Q}_{\eta}-\frac{e}{c}\overrightarrow{A}^{2}\left(x\right)\right)\times\overrightarrow{p}\right)\overrightarrow{\Theta}\end{array}=\left(\overrightarrow{\nabla}\mathcal{V}\left(x\right)\times\overrightarrow{p}\right)\overrightarrow{\Theta},\label{eq:22}$$ the noncommutative Pauli equation in the presence of a uniform magnetic field simply reads $$\left\{ \frac{1}{2m}\left(-\hbar^{2}\overrightarrow{\nabla}^{2}+\frac{ie\hbar}{c}\left(\overrightarrow{\nabla}\overrightarrow{A}+\overrightarrow{A}\overrightarrow{\nabla}\right)+\frac{e^{2}}{c^{2}}\overrightarrow{A}^{2}\right)-\frac{\overrightarrow{L}\overrightarrow{\eta}}{2m}-\frac{e\mathcal{Q}_{\eta}}{2mc\hbar}+\mu_{B}\overrightarrow{\sigma}\overrightarrow{B}+\frac{e\mathcal{Q}_{\Theta}}{4mc\hbar}\right\} \psi=i\hbar\frac{\partial}{\partial t}\psi.\label{eq:23}$$
Knowing that $\overrightarrow{\sigma}$, $\overrightarrow{L}$ are Hermitian and the magnetic field is real, and $\mathcal{Q}_{\Theta}^{\ast}$ is the adjoint of $\mathcal{Q}_{\Theta}$.
The adjoint equation of equation (\[eq:23\]) reads $$\frac{1}{2m}\left\{ -\hbar^{2}\overrightarrow{\nabla}^{2}\psi^{\dagger}-\frac{ie\hbar}{c}\left(\overrightarrow{\nabla}\overrightarrow{A}+\overrightarrow{A}\overrightarrow{\nabla}\right)\psi^{\dagger}+\frac{e^{2}}{c^{2}}\overrightarrow{A}^{2}\psi^{\dagger}\right\} -\frac{\overrightarrow{L}\overrightarrow{\eta}}{2m}\psi^{\dagger}-\frac{e\mathcal{Q}_{\eta}}{2mc\hbar}\psi^{\dagger}+\mu_{B}\overrightarrow{\sigma}\overrightarrow{B}\psi^{\dagger}+\frac{e}{4mc\hbar}\psi^{\dagger}\mathcal{Q}_{\Theta}^{\ast}=-i\hbar\frac{\partial\psi^{\dagger}}{\partial t}.\label{eq:24}$$
Here $\ast$, $\text{\dag}$ stand for the complex conjugation of the potentials, operators and for the wave-functions successively.
To find the continuity equation, we multiply equation (\[eq:23\]) from left by $\psi^{\dagger}$ and equation (\[eq:24\]) from the right by $\psi$, then making the subtraction of these equations, yields
$$\begin{array}{c}
\frac{-\hbar^{2}}{2m}\left\{ \psi^{\dagger}\overrightarrow{\nabla}^{2}\psi-\left(\overrightarrow{\nabla}^{2}\psi^{\dagger}\right)\psi\right\} +\frac{ie\hbar}{2mc}\left\{ \psi^{\dagger}\left(\overrightarrow{\nabla}\overrightarrow{A}+\overrightarrow{A}\overrightarrow{\nabla}\right)\psi+\left[\left(\overrightarrow{\nabla}\overrightarrow{A}+\overrightarrow{A}\overrightarrow{\nabla}\right)\psi^{\dagger}\right]\psi\right\} \\
+\frac{e}{4mc\hbar}\left(\psi^{\dagger}\mathcal{Q}_{\Theta}\psi-\psi^{\dagger}\mathcal{Q}_{\Theta}^{\ast}\psi\right)=i\hbar\left(\psi^{\dagger}\frac{\partial}{\partial t}\psi+\psi\frac{\partial}{\partial t}\psi^{\dagger}\right),
\end{array}\label{eq:25}$$
after some minor simplefications, we have
$$\frac{-\hbar}{2m}div\left\{ \psi^{\dagger}\overrightarrow{\nabla}\psi-\psi\overrightarrow{\nabla}\psi^{\dagger}\right\} +\frac{ie}{mc}div\left\{ \overrightarrow{A}\psi^{\dagger}\psi\right\} +\frac{e}{4mc\hbar^{2}}\left(\psi^{\dagger}\mathcal{Q}_{\Theta}\psi-\psi^{\dagger}\mathcal{Q}_{\Theta}^{\ast}\psi\right)=i\frac{\partial}{\partial t}\psi^{\dagger}\psi.\label{eq:26}$$
This will be recognized as the deformed continuity equation. The obtained equation (\[eq:26\]) contains new quantity, which is the deformation due to the effect of the phase-space noncommutativity on the Pauli equation.
The third term on the left-hand side, which is the deformation quantity, can be simplified as follows $$\frac{ie}{4mc\hbar^{2}}\left(\psi^{\dagger}\mathcal{Q}_{\Theta}\psi-\psi^{\dagger}\mathcal{Q}_{\Theta}^{\ast}\psi\right)=\frac{ie}{4mc\hbar^{2}}\left(\psi^{\dagger}\left(\mathcal{V}\left(x\right)\star\psi\right)-\left(\psi^{\dagger}\star\mathcal{V}\left(x\right)\right)\psi\right),\label{eq:27}$$ using the propriety $\left(\overrightarrow{a}\times\overrightarrow{b}\right)\overrightarrow{c}=\overrightarrow{a}\left(\overrightarrow{b}\times\overrightarrow{c}\right)=\overrightarrow{b}\left(\overrightarrow{c}\times\overrightarrow{a}\right)$, also we must pay attention to the order, $\psi^{\dagger}$ is the first and $\psi$ the second factor, we have $$\frac{ie}{4mc\hbar^{2}}\left(\psi^{\dagger}\mathcal{Q}_{\Theta}\psi-\psi^{\dagger}\mathcal{Q}_{\Theta}^{\ast}\psi\right)=\frac{e}{8mc\hbar^{2}}\overrightarrow{\nabla}\mathcal{V}\left(x\right)\left(\overrightarrow{\Theta}\times\overrightarrow{\nabla}\left(\psi^{\dagger}\psi\right)\right)=\overrightarrow{\nabla}\overrightarrow{\xi}^{nc}.\label{eq:28}$$
Using the following identity also gives the same equation above [@key-6] $$\upsilon^{\dagger}\left(\overrightarrow{\pi}\tau\right)-\left(\overrightarrow{\pi}\upsilon\right)^{\dagger}\tau=-i\hbar\overrightarrow{\nabla}\left(\upsilon^{\dagger}\tau\right),\label{eq:29}$$ where $\upsilon$, $\tau$ are arbitrary two-component spinor. Noting that $\overrightarrow{A}$ does not appear on the right-hand side of the identity; and that this identity is related to the fact that $\overrightarrow{\pi}$ is Hermitian.
It is evident that the noncommutativity affects the current density, and the deformation quantity may apear as a correction to it. The deformed current density satisfies the current conservation, which means, we have a conservation of the continuity equation in the noncommutative phase-space. Equation (\[eq:26\]) may be contracted as $$\frac{\partial\rho}{\partial t}+\overrightarrow{\nabla}\overrightarrow{j}^{nc}=0,\label{eq:30}$$ where $$\rho=\psi^{\dagger}\psi=\left|\psi\right|^{2},\label{eq:31}$$ is the probability density and $$\overrightarrow{j}^{nc}=\overrightarrow{j}+\overrightarrow{\xi}^{nc}=\frac{-i\hbar}{2m}\left\{ \psi^{\dagger}\overrightarrow{\nabla}\psi-\psi\overrightarrow{\nabla}\psi^{\dagger}\right\} -\frac{e}{mc}\left\{ \overrightarrow{A}\psi^{\dagger}\psi\right\} +\overrightarrow{\xi}^{nc},\label{eq:32}$$ is the deformed current density of the electrons. The deformation quantity is $$\overrightarrow{\xi}^{nc}=\frac{e}{8mc\hbar^{2}}\mathcal{V}\left(x\right)\left(\overrightarrow{\Theta}\times\overrightarrow{\nabla}\left(\psi^{\dagger}\psi\right)\right)=\frac{e}{8mc\hbar^{2}}\left(2\overrightarrow{p}\overrightarrow{A}-\frac{1}{2\hbar}\mathcal{Q}_{\eta}-\frac{e}{c}\overrightarrow{A}^{2}\right)\left(\overrightarrow{\Theta}\times\overrightarrow{\nabla}\left(\psi^{\dagger}\psi\right)\right).\label{eq:33}$$
The existence of a deformed continuity equation for all orders of $\Theta$ parameter also proportional to the magnetic field. Actually, one can explicitly calculate the conserved current to all orders of $\Theta$. In the case of a non-constant magnetic field, and using equation (\[eq:21\]), we have $$\frac{\partial\rho}{\partial t}+\overrightarrow{\nabla}\overrightarrow{j}+\frac{ie}{4mc\hbar^{2}}\left\{ \psi^{\dagger}\left(\mathcal{V}\left(x\right)\star\psi\right)-\left(\psi^{\dagger}\star\mathcal{V}\left(x\right)\right)\psi\right\} =0,\label{eq:34}$$ we calculate the $n^{th}$ order term in the general deformed continuity equation (\[eq:34\]) as follows $$\begin{array}{c}
\left.\psi^{\dagger}\left(\mathcal{V}\left(x\right)\star\psi\right)-\left(\psi^{\dagger}\star\mathcal{V}\left(x\right)\right)\psi\right|_{n^{th}}=\frac{1}{n!}\left(\frac{i}{2}\right)^{n}\Theta^{a_{1}b_{1}}...\Theta^{a_{n}b_{n}}\\
\times\left(\psi^{\dagger}\partial_{a_{1}}...\partial_{a_{k}}\mathcal{V}\left(x\right)\partial_{b_{1}}...\partial_{b_{k}}\psi-\partial_{a_{1}}...\partial_{a_{k}}\psi^{\dagger}\left(\partial_{b_{1}}...\partial_{b_{k}}\mathcal{V}\left(x\right)\right)\psi+(-1)^{n}cc.\right).
\end{array}\label{eq:35}$$
We note the absence of the magnetization current term in equation (\[eq:32\]), as in commutative case when this was asserted by authors [@key-1; @key-4; @key-8; @key-13; @key-16; @key-20], where at first they attempted to cover this deficiency by explaining how to derive this additional term from the non-relativistic limit of the relativistic Dirac probability current density. Then, Nowakowski and others [@key-6] provided a superb explanation of how to extract this term through the non-relativistic Pauli equation itself.
Knowing that, in commutative background the magnetization current $\overrightarrow{j}_{M}$ from the probability current of Pauli equation is proportional to $\overrightarrow{\nabla}\times\left(\psi^{\dagger}\overrightarrow{\sigma}\psi\right)$. However, the existence of such an additional term is important and it should be discussed when talking about the probability current of spin-1/2 particles. In following, we try to derive the current magnetization in noncommutative background without changing the continuity equation, and seek if such additional term is affected by noncommutativity or not.
Derivation of the magnetization current
---------------------------------------
At first it must be clarified that the authors Nowakowski and others (2011) in [@key-4; @key-6] derived the non-relativistic current density for a spin-1/2 particle using **minimally coupled Pauli equation**. In contrast, Wilkes, J. M (2020) in [@key-32] derived the non-relativistic current density for a free spin-1/2 particle using directly **free Pauli equatio**n. However, we show here that the current density can be derived from the minimally coupled Pauli equation in noncommutative phase-space.
Starting with the **noncommutative minimally coupled Pauli** equation written in the form $$\mathcal{H}_{Pauli}^{nc}\psi=\frac{1}{2m}\left(\overrightarrow{\sigma}\hat{\overrightarrow{\pi}}^{nc}\right)^{2}\psi=i\hbar\frac{\partial}{\partial t}\psi,\label{eq:38-1}$$ we multiply the above equation from left by $\psi^{\dagger}$ and the adjoint equation of equation (\[eq:38-1\]) from the right by $\psi$, the subtraction of these equations yields the following continuity equation
$$2m\left\{ \left(\left(\overrightarrow{\sigma}\hat{\overrightarrow{\pi}}^{nc}\right)^{2}\psi\right)^{\dagger}\psi-\psi^{\dagger}\left(\overrightarrow{\sigma}\hat{\overrightarrow{\pi}}^{nc}\right)^{2}\psi\right\} =i\hbar\left(\psi^{\dagger}\frac{\partial\psi}{\partial t}+\psi\frac{\partial\psi^{\dagger}}{\partial t}\right),\label{eq:40-1}$$
noting that the noncommutativity of $\pi^{nc}$ has led us to express the two terms as follows $$\frac{i}{2m\hbar}\sum_{i,j}\left\{ \left(\hat{\pi_{i}}^{nc}\hat{\pi_{j}}^{nc}\psi\right)^{\dagger}\sigma_{j}\sigma_{i}\psi-\psi^{\dagger}\sigma_{i}\sigma_{j}\left(\hat{\pi_{i}}^{nc}\hat{\pi_{j}}^{nc}\psi\right)\right\} =\frac{\partial\rho}{\partial t}.\label{eq:41-1}$$
While with only $p_{i}$, we would have no reason for preferring $p_{i}p_{j}\psi$ over $p_{j}p_{i}\psi$.
It is easy to verify that the identity (\[eq:29\]) remains valid for $\overrightarrow{\pi}^{nc}$ because the fact that $\overrightarrow{\pi}^{nc}$ is Hermitian. Therefore, through identity (\[eq:29\]), we have $$\frac{-1}{2m}\sum_{i,j}\nabla_{i}\left\{ \left(\hat{\pi_{j}}^{nc}\psi\right)^{\dagger}\sigma_{j}\sigma_{i}\psi+\psi^{\dagger}\sigma_{i}\sigma_{j}\left(\hat{\pi_{j}}^{nc}\psi\right)\right\} +\frac{i}{2m\hbar}\sum_{i,j}\left\{ \left(\hat{\pi_{j}}^{nc}\psi\right)^{\dagger}\sigma_{j}\sigma_{i}\left(\hat{\pi_{i}}^{nc}\psi\right)-\left(\hat{\pi_{i}}^{nc}\psi\right)^{\dagger}\sigma_{i}\sigma_{j}\left(\hat{\pi_{j}}^{nc}\psi\right)\right\} =\frac{\partial\rho}{\partial t},\label{eq:42-1}$$ then $$\frac{-1}{2m}\sum_{i,j}\nabla_{i}\left\{ \left(\hat{\pi_{j}}^{nc}\psi\right)^{\dagger}\sigma_{j}\sigma_{i}\psi+\psi^{\dagger}\sigma_{i}\sigma_{j}\left(\hat{\pi_{j}}^{nc}\psi\right)\right\} =\frac{\partial\rho}{\partial t}.\label{eq:43-1}$$
Knowing that the $2{}^{nd}$ sum in equation (\[eq:42-1\]) gives zero by swapping $i$ and $j$ for one of the sums, then the probability current vector from the above continuity equation is $$j_{i}=\frac{1}{2m}\sum_{j}\left\{ \left(\hat{\pi_{j}}^{nc}\psi\right)^{\dagger}\sigma_{j}\sigma_{i}\psi+\psi^{\dagger}\sigma_{i}\sigma_{j}\left(\hat{\pi_{j}}^{nc}\psi\right)\right\} .\label{eq:44-1}$$
Using the property (\[eq:15-1\]), equation (\[eq:44-1\]) becomes $$j_{i}=\frac{1}{2m}\sum_{j}\left\{ \left(\hat{\pi_{j}}^{nc}\psi\right)^{\dagger}\psi+\psi^{\dagger}\left(\hat{\pi_{j}}^{nc}\psi\right)+i\sum_{k}\left[\epsilon_{jik}\left(\hat{\pi_{j}}^{nc}\psi\right)^{\dagger}\sigma_{k}\psi+\epsilon_{ijk}\psi^{\dagger}\sigma_{k}\left(\hat{\pi_{j}}^{nc}\psi\right)\right]\right\} ,\label{eq:45-1}$$ with $\epsilon_{jik}=-\epsilon_{ijk}$, and using one more time identity (\[eq:29\]), we find (this is similar to investigation by [@key-6] in the case of commutative phase-space) $$j_{i}=\frac{1}{2m}\left[\left(\hat{p_{j}}^{nc}\psi\right)^{\dagger}\psi-\frac{e}{c}\left(A_{j}^{nc}\psi\right)^{\dagger}\psi+\psi^{\dagger}\hat{p_{j}}^{nc}\psi-\frac{e}{c}\psi^{\dagger}A_{j}^{nc}\psi\right]+\frac{\hbar}{2m}\sum_{j,k}\epsilon_{ijk}\nabla_{j}\left(\psi^{\dagger}\sigma_{k}\psi\right).\label{eq:46-1}$$
In the right-hand side of the above equation, the first term will be interpreted as the noncommutative current vector $\overrightarrow{j}^{nc}$ given by equation (\[eq:33\]), and the second term is the requested additional term, namely current magnetization $\overrightarrow{j}_{M}$, where $$\underset{M}{j}_{i}=\frac{\hbar}{2m}\left(\overrightarrow{\nabla}\times\left(\psi^{\dagger}\overrightarrow{\sigma}\psi\right)\right)_{i}.\label{eq:47-1}$$
Besides, $\overrightarrow{j}_{M}$ can also be shown to be a part of the conserved Noether current [@key-33], resulting from the invariance of the Pauli Lagrangian under the global phase transformation U(1).
What can be concluded here is that the magnetization current is not affected by the noncommutativity, perhaps because the spin operator could not be affected by noncommutativity. This is in contrast to what was previously found around the current density, which showed a great influence of noncommutativity.
Noncommutative Semi-classical Partition Function
================================================
In this part of our work, we investigate the magnetization and the magnetic susceptibility quantities of our Pauli system using the partition function in noncommutative phase-space. We concentrate, at first, on the calculation of the semi-classical partition function Z. While our system is not completely classical but contains a quantum interaction concerning the spin, therefore, the noncommutative partition function is separable into two independent parts as follows $$Z^{nc}=Z_{clas}^{nc}Z_{spin}.$$
Based on the proposal that noncommutative observables corresponding to the commutative one [@key-34], and for non-interacting particles, the classical partition function in the canonical ensemble in noncommutative phase-space is given by the following formula [@key-35; @key-36] $$Z_{clas}^{nc}=\frac{1}{N!\left(2\pi\bar{\hbar}\right)^{3N}}\int e^{-\beta\mathcal{H}_{clas}^{nc}\left(x,p\right)}d^{3N}x^{nc}d^{3N}p^{nc},\label{eq:36}$$ which is written for a $N$ particle, $\frac{1}{N!}$ is the Gibbs correction factor, considered due to accounting for indistinguishability, which means that there are $N!$ ways of arranging $N$ particles at $N$ sites. $\bar{h}\sim\triangle x^{nc}\triangle p^{nc}$, with $\frac{1}{\bar{\hbar}^{3}}$ is a factor that makes the volume of the noncommutative phase-space dimensionless.
$\beta$ defined as $\frac{1}{K_{B}T}$, $K_{B}$ is the Boltzmann constant, where $K_{B}=1.38\times10^{-23}JK^{-1}$. The Helmholtz free energy is $$F=-\frac{1}{\beta}\text{ln}Z,\label{eq:37}$$ we may derive the magnetization as follows $$\left\langle M\right\rangle =-\frac{\partial F}{\partial B}.\label{eq:38}$$
For a single particle, The noncommutative classical partition function is then $$Z_{clas,1}^{nc}=\frac{1}{\bar{\hbar}^{3}}\int e^{-\beta\mathcal{H}_{Clas}^{nc}\left(x,p\right)}d^{3}x^{nc}d^{3}p^{nc},\label{eq:39}$$ where $d^{3}$ is a shorthand notation serving as a reminder that the $x$ and $p$ are vectors in three-dimensional phase-space. The relation between equation (\[eq:36\]) and (\[eq:39\]) is given by the following formula $$Z_{clas}^{nc}=\frac{\left(Z_{clas,1}^{nc}\right)^{N}}{N!}.\label{eq:40}$$
Knowing that using equation (\[eq:5-1\]), we have $$d^{3}x^{nc}d^{3}p^{nc}=\left(1-\frac{\Theta\eta}{8\hbar^{2}}\right)d^{3}xd^{3}p,\label{eq:42}$$ and according to uncertainty principle [@key-36] $$\bar{h}^{3}=h^{3}\left(1+\frac{\Theta\eta}{4}\right).\label{eq:43}$$
For an electron with spin in interaction with an electromagnetic potential, once the magnetic field $\overrightarrow{B}$ be in the z-direction, and by equation (\[eq:16\]), bear in mind that $\left[\overrightarrow{p}^{nc},\overrightarrow{A}^{nc}\right]=0$, then for the sake of simplicity, the noncommutative Pauli Hamiltonian from equation (\[eq:20\]) takes the form $$\mathcal{H}_{Pauli}\left(x^{nc},p^{nc}\right)=\frac{1}{2m}\left\{ \overrightarrow{p}^{nc}\overrightarrow{p}^{nc}-2\frac{e}{c}\overrightarrow{p}^{nc}\overrightarrow{A}^{nc}+\left(\frac{e}{c}\right)^{2}\overrightarrow{A}^{nc}\overrightarrow{A}^{nc}\right\} +\mu_{B}\hat{\sigma}_{z}B.\label{eq:44}$$
We split the noncommutative Pauli Hamiltonian as $\mathcal{H}_{Pauli}^{nc}=\mathcal{H}_{cla}^{nc}+\mathcal{H}_{spin}$, with $\mathcal{H}_{spin}=\mu_{B}\hat{\sigma}_{z}B$ .
It is easy to verify that $$\overrightarrow{p}^{nc}\overrightarrow{p}^{nc}=\left(p_{x}^{nc}\right)^{2}+\left(p_{y}^{nc}\right)^{2}+\left(p_{z}^{nc}\right)^{2}=p_{x}^{2}+p_{y}^{2}+p_{z}^{2}-\frac{\eta}{2\hbar}L_{z}+\frac{\eta^{2}}{16\hbar^{2}}\left(x^{2}+y^{2}\right),\label{eq:45}$$ $$\overrightarrow{p}^{nc}\overrightarrow{A}=p_{x}^{nc}A_{x}^{nc}+p_{y}^{nc}A_{y}^{nc}=\frac{B}{2}\left\{ -\frac{\Theta}{4\hbar}\left(p_{x}^{2}+p_{y}^{2}\right)-\frac{\eta}{4\hbar}\left(y^{2}+x^{2}\right)+\left(1+\frac{\Theta\eta}{16\hbar^{2}}\right)L_{z}\right\} ,\label{eq:46}$$ $$\overrightarrow{A}^{nc}\overrightarrow{A}^{nc}=\left(A_{x}^{nc}\right)^{2}+\left(A_{y}^{nc}\right)^{2}=\frac{B^{2}}{4}\left\{ x^{2}+y^{2}-\frac{\Theta}{2\hbar}L_{z}+\frac{\Theta^{2}}{16\hbar^{2}}\left(p_{x}^{2}+p_{y}^{2}\right)\right\} .\label{eq:47}$$
Using the three equations above, our noncommutative classical Hamiltonian becomes $$\mathcal{H}_{cla}=\frac{1}{2\tilde{m}}\left(p_{x}^{2}+p_{y}^{2}\right)+\frac{1}{2m}p_{z}^{2}-\tilde{\omega}L_{z}+\frac{1}{2}\tilde{m}\tilde{\omega}^{2}\left(x^{2}+y^{2}\right),\label{eq:48}$$ where $L_{z}=p_{y}x-p_{x}y=\left(x_{i}\times p_{i}\right)_{z}$, and $$\tilde{m}=\frac{m}{\left(1+\frac{eB\Theta}{8c\hbar}\right)^{2}},\text{ and }\tilde{\omega}=\frac{c\eta+2e\hbar B}{4c\hbar\tilde{m}\left(1+\frac{eB\Theta}{c8\hbar}\right)},\quad\frac{1}{2}\tilde{m}\tilde{\omega}^{2}=\frac{1}{2m}\left(\frac{\eta eB}{4c\hbar}+\frac{\eta^{2}}{16\hbar^{2}}+\frac{e^{2}B^{2}}{c^{2}4}\right).\label{eq:49}$$
Now, following the definition given in equation (\[eq:39\]) we express the single particle noncommutative classical partition function as [ $$Z_{clas,1}^{nc}=\frac{1}{\bar{h}^{3}}\int e^{-\beta\left[\frac{1}{2\tilde{m}}\left(p_{x}^{2}+p_{y}^{2}\right)+\frac{1}{2m}p_{z}^{2}-\tilde{\omega}L_{z}+\frac{1}{2}\tilde{m}\tilde{\omega}^{2}\left(x^{2}+y^{2}\right)\right]}d^{3}x^{nc}d^{3}p^{nc}.\label{eq:50}$$ ]{}
It should be noted that once we want to factorize our Hamiltonian into a momentum and a position term. This is not always possible when there are matrices (or operators) in the exponent. However, within the classical limit, it is possible. Otherwise, to separate the operators in the exponent, we use the Baker-Campbell-Hausdorff (BCH) formula given by (first few terms) [ $$e^{\left[\hat{A}+\hat{B}\right]}=e^{\left[\hat{A}\right]}e^{\left[\hat{B}\right]}e^{\left[-\frac{1}{2}\left[\hat{A},\hat{B}\right]\right]}e^{\frac{1}{6}\left(2\left[\hat{A},\left[\hat{A},\hat{B}\right]\right]+\left[\hat{B},\left[\hat{A},\hat{B}\right]\right]\right)}...\label{eq:50-1}$$ ]{}
We can now start to replace some of the operators in the exponent[ $$Z_{clas,1}^{nc}=\frac{1}{\bar{h}^{3}}\int e^{-\beta\left[\frac{1}{2\tilde{m}}\left(p_{x}^{2}+p_{y}^{2}\right)+\frac{1}{2m}p_{z}^{2}\right]}e^{-\beta\left[\frac{1}{2}\tilde{m}\tilde{\omega}^{2}\left(x^{2}+y^{2}\right)\right]}e^{\beta\tilde{\omega}L_{z}}d^{3}p^{nc}d^{3}x^{nc}.\label{eq:62}$$ ]{}
We should expand exponentials containing $\tilde{\omega}$, and by consider terms up to the second-order of $\tilde{\omega}$, we obtain $$Z_{clas,1}^{nc}=\int e^{-\frac{\beta}{2}\left[\frac{p_{x}^{2}+p_{y}^{2}}{\tilde{m}}+\frac{p_{z}^{2}}{m}\right]}\left(1+\beta\tilde{\omega}L_{z}+\frac{1}{2}\beta^{2}\tilde{\omega}^{2}L_{z}^{2}\right)\left(1-\beta\tilde{\omega}^{2}\frac{\tilde{m}}{2}\left(x^{2}+y^{2}\right)\right)d^{3}p^{nc}d^{3}x^{nc},\label{eq:63}$$ therefore, we have the appropriate expression for $Z_{clas,1}^{nc}$[ $$\begin{array}{c}
Z_{clas,1}^{nc}=\frac{1+\frac{\Theta\eta}{4}}{h^{3}}\int e^{-\frac{\beta}{2}\left[\frac{p_{x}^{2}+p_{y}^{2}}{\tilde{m}}+\frac{p_{z}^{2}}{m}\right]}d^{3}pd^{3}x+\frac{\left(1+\frac{\Theta\eta}{4}\right)\beta\tilde{\omega}}{\bar{h}^{3}}\int e^{-\frac{\beta}{2}\left[\frac{p_{x}^{2}+p_{y}^{2}}{\tilde{m}}+\frac{p_{z}^{2}}{m}\right]}L_{z}d^{3}pd^{3}x\\
+\frac{\left(1+\frac{\Theta\eta}{4}\right)\beta^{2}\tilde{\omega}^{2}}{2\bar{h}^{3}}\int e^{-\frac{\beta}{2}\left[\frac{p_{x}^{2}+p_{y}^{2}}{\tilde{m}}+\frac{p_{z}^{2}}{m}\right]}L_{z}^{2}d^{3}pd^{3}x-\frac{\left(1+\frac{\Theta\eta}{4}\right)\beta\tilde{\omega}^{2}}{2\bar{h}^{3}}\int e^{-\frac{\beta}{2}\left[\frac{p_{x}^{2}+p_{y}^{2}}{\tilde{m}}+\frac{p_{z}^{2}}{m}\right]}\left(x^{2}+y^{2}\right)d^{3}pd^{3}x.
\end{array}\label{eq:64}$$ ]{}
In the right-hand side of above equation, it is easy to check that the second integral goes to zero, the third and last integrals cancel each other, and thus we obtain[ $$Z_{clas,1}^{nc}=\frac{1+\frac{\Theta\eta}{4}}{h^{3}}\int e^{-\frac{\beta}{2}\left[\frac{p_{x}^{2}+p_{y}^{2}}{\tilde{m}}+\frac{p_{z}^{2}}{m}\right]}d^{3}pd^{3}x.\label{eq:65}$$ ]{}
Using the integral of Gaussian function $\int e^{-ax^{2}}dx=\sqrt{\frac{\pi}{a}}$, we have [ $$Z_{clas,1}^{nc}=\frac{1+\frac{\Theta\eta}{4}}{h^{3}}\int d^{3}x^{nc}\int e^{-\frac{\beta}{2}\left[\frac{p_{x}^{2}+p_{y}^{2}}{\tilde{m}}+\frac{p_{z}^{2}}{m}\right]}d^{3}p^{nc}=\frac{V}{\varLambda^{3}}\frac{1+\frac{\Theta\eta}{4}}{\left(1+\frac{eB\Theta}{8c\hbar}\right)^{2}},\label{eq:66}$$ ]{}where $V$, $\varLambda=h\left(2m\pi K_{B}T\right)^{-\frac{1}{2}}$ are respectively the volume and the thermal de Broglie wavelength. The quantum partition function is $$Z_{spin}=Z_{spin,1}^{N}=\left(\sum_{\sigma_{z}=\pm1}e^{\beta\mu_{B}\hat{\sigma}_{z}B}\right)^{N}=2^{N}\text{cosh}{}^{N}\left(\beta\mu_{B}B\right).\label{eq:67}$$
Finally, the Pauli partition function for a system of $N$ particles in a three-dimensional noncommutative phase-space is $$Z^{nc}=\frac{\left(2V\right)^{N}}{\varLambda^{3N}N!}\frac{\left(1+\frac{\Theta\eta}{4}\right)^{N}\text{cosh}{}^{N}\left(\beta\mu_{B}B\right)}{\left(1+\frac{eB\Theta}{8c\hbar}\right)^{2N}}.\label{eq:68}$$
In the limit of the noncommutativity, i.e. $\Theta\rightarrow0$, $\eta\rightarrow0$, the above expression of $Z^{nc}$ tends to the result of $Z$ in the usual commutative phase-space, which is $$Z=\frac{\left(2V\right)^{N}}{\varLambda^{3N}N!}\text{cosh}{}^{N}\left(\beta\mu_{B}B\right).\label{eq:69}$$
Using formulae (\[eq:37\]) and (\[eq:38\]), we find the magnetization in noncommutative and commutative phase-space, thus $$F^{nc}=-\frac{N}{\beta}\text{ln}\frac{\left(2V\right)}{\varLambda^{3}N!}\frac{\left(1+\frac{\Theta\eta}{4}\right)\text{cosh}\left(\beta\mu_{B}B\right)}{\left(1+\frac{eB\Theta}{8c\hbar}\right)^{2}},\label{eq:70}$$ and the noncommutative magnetization be $$\left\langle M^{nc}\right\rangle =-\frac{\partial F^{nc}}{\partial B}=2\frac{N}{\beta}\frac{e\Theta}{\left(8c\hbar+eB\Theta\right)}+N\mu_{B}\text{tanh}\left(\beta\mu_{B}B\right).\label{eq:71}$$
The commutative magnetization is $$\left\langle M\right\rangle =-\frac{\partial F}{\partial B}=N\mu_{B}\text{tanh}\left(\beta\mu_{B}B\right),\label{eq:72}$$ it is obvious that $\left.\left\langle M^{nc}\right\rangle \right|_{\Theta=0}=\left\langle M\right\rangle $. We may derive the magnetic susceptibility of electrons $\chi=\frac{1}{V}\frac{\partial\left\langle M\right\rangle }{\partial B}$ in noncommutative phase-space using the magnetization (\[eq:71\]) by $$\chi^{nc}=-2\frac{N}{V\beta}\frac{\left(e\Theta\right)^{2}}{\left(8c\hbar+eB\Theta\right)^{2}}+\frac{N}{V}\beta\mu_{B}^{2}\left(1-\text{tanh}^{2}\left(\beta\mu_{B}B\right)\right),\label{eq:73}$$ where the commutative magnetic susceptibility $\chi=\chi^{nc}\left(\Theta=0\right)$ is $$\chi=\frac{N}{V}\beta\mu_{B}^{2}\left(1-\text{tanh}^{2}\left(\beta\mu_{B}B\right)\right).\label{eq:74}$$
Finally, we conclude with the following special cases. Let us first consider $B=0$, then we have $$\left\langle M\right\rangle =0;\;\left\langle M^{nc}\right\rangle =2\frac{N}{\beta}\frac{e\Theta}{8c\hbar};\text{ and }\;\chi^{nc}=-2\frac{N}{V\beta}\frac{\left(e\Theta\right)^{2}}{\left(8c\hbar\right)^{2}}+\frac{N}{V}\beta\mu_{B}^{2}.\label{eq:76}$$
For $B\rightarrow\infty$ and $T=C^{st}$, $\text{tanh}\left(\beta\mu_{B}B\right)=1$, we obtain $$\left\langle M^{nc}\right\rangle =\left\langle M\right\rangle =N\mu_{B};\text{ and }\chi^{nc}=\chi\sim0.\label{eq:77}$$
As well when $T\rightarrow\infty$,$\beta\rightarrow0$ (with $B=C^{st}$), $\text{tanh}\left(\beta\mu_{B}B\right)=0$, we obtain $$\left\langle M^{nc}\right\rangle \rightarrow\infty,\left\langle M\right\rangle =0;\text{ and }\chi^{nc}\rightarrow\infty,\chi=0.\label{eq:78}$$
Armed with the partition function $Z$, we can compute other important thermal quantities, such as the average energy $U=-\frac{\partial}{\partial\beta}\text{ln}z$, the entropy $S=\text{ln}z-\beta\frac{\partial}{\partial\beta}\text{ln}z$ and the specific heat $C=\beta^{2}\frac{\partial^{2}}{\partial^{2}\beta}\text{ln}z$.
Conclusion
==========
In this work, we have exactly studied the three-dimensional Pauli equation and the corresponding continuity equation for a spin-1/2 particle in the presence of an electromagnetic field in noncommutative phase-space, considering constant and non-constant magnetic fields. It is shown that the non-constant magnetic field lifts the order of the noncommutativity parameter in both the Pauli equation and the corresponding continuity equation. Given the known absence of the magnetization current term in the continuity equation, even in noncommutative phase-space as confirmed by our calculations, we extracted the magnetization current term from the Pauli equation itself without modifying the continuity equation. Furthermore, we found that the density current is conserved, which means, we have a conservation of the deformed continuity equation.
By using the classical treatment (within the classical limit), the magnetization and the magnetic susceptibility quantities are explicitly determined in both commutative and noncommutative phase-spaces through a semi-classical partition function of the Pauli system of the one-particle and N-particle systems in three dimensions. Besides, to see the behaviour of these deformed quantities, we carried out some special cases in commutative and noncommutative phase-spaces.
Finally, we can say that we successfully examined the influence of the noncommutativity on the problems in question, where the noncommutativity was introduced using both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product. Further, the noncommutative corrections to the nonrelativistic Pauli equation and the continuity equation are also valid up to all orders in the noncommutative parameter. Our results limits are in good agreement with those obtained by other authors as discussed and in the literature.
The author would like to thank Dr. Mojtaba Najafizadeh for his valuable discussion on the classical partition function.
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abstract: |
In this work we present an active Compton scattering polarimeter as a focal plane instrument able to extend the X-ray polarimetry towards hard X-rays.
Other authors have already studied various instrument design by means of Monte Carlo simulations, in this work we will show for the first time the experimental measurements of “tagging efficiency" aimed to evaluate the polarimeter sensitivity as a function of energy. We performed a characterization of different scattering materials by measuring the tagging efficiency that was used as an input to the Monte Carlo simulation. Then we calculated the sensitivity to polarization of a design based on the laboratory set-up. Despite the geometry tested is not optimized for a realistic focal plane instrument, we demonstrated the feasibility of polarimetry with a low energy threshold of 20 keV. Moreover we evaluated a Minimum Detectable Polarization of 10$\%$ for a 10 mCrab source in 100 ks between 20 and 80 keV in the focal plane of one multilayer optics module of NuSTAR. The configuration used consisted of a doped p-terphenyl scatterer 3 cm long and 0.7 cm of diameter coupled with a 0.2 cm thick LaBr$_{3}$ absorber.
address:
- |
Università di Roma “Tor Vergata", Dipartimento di Fisica, Via della Ricerca Scientifica 1, 00133 Rome, Italy\
INAF-IAPS, Via del Fosso del Cavaliere 100, 00133 Rome, Italy
- 'INAF-IAPS, Via del Fosso del Cavaliere 100, 00133 Rome, Italy'
- 'INAF-IAPS, Via del Fosso del Cavaliere 100, 00133 Rome, Italy'
- 'INAF-IAPS, Via del Fosso del Cavaliere 100, 00133 Rome, Italy'
- 'INAF-IAPS, Via del Fosso del Cavaliere 100, 00133 Rome, Italy'
- 'INAF-IAPS, Via del Fosso del Cavaliere 100, 00133 Rome, Italy'
- 'INAF-IAPS, Via del Fosso del Cavaliere 100, 00133 Rome, Italy'
author:
- Sergio Fabiani
- Riccardo Campana
- Enrico Costa
- Ettore Del Monte
- Fabio Muleri
- Alda Rubini
- Paolo Soffitta
bibliography:
- 'References.bib'
title: Characterization of scatterers for an active focal plane Compton polarimeter
---
X-ray ,polarimetry ,Compton scattering ,scintillation detector ,astrophysics ,GEANT4 ,Monte Carlo simulations
Introduction
============
X-ray polarimetry offers the possibility to reach a deep understanding of violent processes taking place around compact objects such as neutron stars, pulsators and black holes which are particularly bright in this energy band. Focal plane polarimeters based on Compton scattering have been already proposed by @Soffitta2010, @Hayashida2010 and @Beilicke2011. Recent works by @Krawczynski2011a and @Chattopadhyay2012 evaluated theoretical sensitivities for possible different designs of hard X-ray polarimeters, including focal plane configurations. We complement such studies by measuring experimentally the sensitivity of a focal plane active Compton polarimeter. We demonstrate a procedure for the instrumental characterization. From the laboratory measurements on the scatterer tagging efficiency, we arrive at the evaluation of the final instrument response by means of Monte Carlo simulations. For such an experiment the realistic sensitivity strongly depends on the lower achievable energy threshold. For example, at 20 keV and 90$^\circ$ of scattering angle, the deposited energy in the scatterer is only 750 eV and the efficiency in detecting the produced faint scintillation signal in organic scatterers needs to be determined.
In Sect. \[sec:Polarimeter\] the design of the focal plane active Compton polarimeter is described. In Sect. \[sec:Measurements\] we present the characterization of different scattering materials in terms of tagging efficiency at 22.6 keV with a $^{109}$Cd radioactive source. The best available scatterer sample was also tested at 59.5 keV with an $^{241}$Am source. In Sect. \[sec:tageffthreshold\] the tagging efficiency results are used to evaluate the charge threshold for the signals detection capability in the experimental set-up. Tagging efficiency is evaluated at different energies by applying this threshold to simulated coincidence spectra. In Sect. \[sec:realpolarimeter\] the sensitivity, expressed as Minimum Detectable Polarization (MDP), of a realistic detector design is calculated. Tagging efficiency results are applied to the simulation of a Compton polarimeter based on the experimental set-up. Tested scatterer geometry are not optimized to reach the maximum polarimetric performance, therefore there is still the possibility to improve the sensitivity estimates presented here.
Compton polarimeter description {#sec:Polarimeter}
===============================
The Compton scattering is effective to measure the polarization in the hard X-ray band. The sensitivity to the polarization of the incident photon is given by the Klein-Nishina cross section [@Heitler1954]: $$\biggl(\frac{d\sigma}{d\Omega}\biggr)_\mathrm{KN}=\frac{{r_0}^2}{2}\frac{{E^\prime}^2}{{E}^2}\Biggr[ \frac{E}{E^\prime}+\frac{E^\prime}{E}-2\sin^2 \theta \cos^2 \phi \Biggl] \label{eq:KN}$$ where $$\frac{E'}{E}=\frac{1}{1+\frac{E}{m_e c^2}(1-\cos \theta)}\label{eq:EsuE}$$ $E$ and $E^\prime$ are the energies of the incident and scattered photons respectively, $\theta$ is the scattering angle measured from the incident direction of the incoming photon and $\phi$ is the azimuthal angle measured from the plane containing both the incoming direction and the electric vector of the incident photon. The probability to have a Compton interaction is higher at $\phi=90^\circ$ and $270^\circ$ for a fixed value of $\theta$ and conversely has a minimum for $\phi=0^\circ$ and $180^\circ$. Thus, linearly polarized photons are preferentially scattered perpendicularly to the direction of the incident photon electric field. The statistical distribution of the $\phi$ emission directions produced by a beam of polarized radiation is then modulated. The theoretical modulation factor achievable by an ideal Compton polarimeter is: $$\mu(\theta)=\frac{N_\mathrm{max}(\theta)-N_\mathrm{min}(\theta)}{N_\mathrm{max}(\theta)+N_\mathrm{min}(\theta)}=\frac{(\frac{d\sigma}{d\Omega})_{\phi=\frac{\pi}{2}}-(\frac{d\sigma}{d\Omega})_{\phi =0}}{(\frac{d\sigma}{d\Omega})_{\phi=\frac{\pi}{2}}+(\frac{d\sigma}{d\Omega})_{\phi =0}}=\frac{\sin^2\theta }{\frac{E}{E^\prime}+\frac{E^\prime}{E}-\sin^2 \theta} \label{eq:Muphi}$$
![Modulation factor dependence on the scattering angle $\theta$ at different energies. Polarized low energy radiation leads to higher modulation factors which maxima occur for scattering angles approaching 90$^\circ$.[]{data-label="fig:DifferentialMu"}](Fig01_Differential_mu.eps){width="10cm"}
The modulation factor, as expressed by Eq. \[eq:Muphi\], is shown for different energies in Fig. \[fig:DifferentialMu\].
A good Compton polarimeter should exploit the property of the Klein-Nishina cross section which allows to reach a larger modulation factor for scattering angles $\theta \simeq 90^\circ$ at low energy. For a real scatterer the angular distribution of scattered photons is the product of the Klein-Nishina formula and the *scattering function* $S(\chi,Z)$ [@Hubbel1975]: $$\frac{d\sigma}{d\Omega}=\biggl(\frac{d\sigma}{d\Omega}\biggr)_\mathrm{KN} \cdot S(\chi,Z)\label{eq:KNSF}$$ where $Z$ is the atomic number and $\chi=\sin(\frac{\theta}{2})/\lambda[\AA]$ where $\lambda$ is the incident photon wavelength expressed in Angstroms. The scattering function takes into account the influence of the atomic electrons distribution and binding energies. A complete review on the scattering function theoretical derivation and numerical calculation is given by @Hubbel1975. As an example we show the simplified case of the hydrogen atom for which: $$S(\chi,H)=1-[F(\chi,H)]^2\label{eq:scat1}$$ where $F(\chi,H)=(1+4\pi^2 r_0^2 \chi^2)^{-2}$ is the hydrogen form factor. Therefore the scattering function is: $$S(\chi,H)=1-\biggl[\frac{1}{1+4\pi^2 r_0^2 \chi^2}\biggr]^4\label{eq:scat2}$$ For large values of the variable $\chi$ the Eq. \[eq:scat2\] reduces to 1 which is the hydrogen atomic number. Substituting the $\chi$ variable in Eq. \[eq:scat2\] we have: $$S(\chi,H)=1-\biggl[\frac{1}{1+4\pi^2 r_0^2 \bigl(\frac{\sin(\theta/2)}{\lambda}\bigr)^2}\biggr]^4\label{eq:scat3}$$ that tells us that the scattering function essentially suppresses forward scattering with respect to the Klein-Nishina formula, since for $\theta=0$ the left term of Eq. \[eq:scat3\] vanishes. These properties are verified also for the other elements (see tabulation in [@Hubbel1975]). Therefore the directions of scattering around $\theta=90^\circ$ are still suitable for an efficient detection of the scattered photon.
The choice of the scattering material should maximize the probability of Compton interaction, while minimizing the photoelectric absorption, which prevails at low energy and sets the low energy threshold of detection. Since the ratio between Compton scattering and photoelectric absorption cross sections is larger for lower $Z$ materials, these ones can be used in the scatterer. Conversely, a small ratio between scattering and absorption cross sections is required for the absorber. Therefore, we consider a polarimeter design in which a low-$Z$ scatterer is coupled with an absorber made of a high-$Z$ material to detect efficiently the scattered photon via the photoelectric effect and thus determining the energy of the incoming photon [@Costa1995]. Moreover the scatterer should be long enough to ensure a high probability of Compton interaction, while maintaining a small width to avoid multiple scattering events.
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![Section of a Compton polarimeter composed by a scattering scintillating rod, surrounded by a cylindrical array of absorber detectors.[]{data-label="fig:GEANT4pol"}](Fig02_GEANT4_pho.eps "fig:")
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Therefore a convenient geometry for a focal plane Compton polarimeter consists of a long and thin scattering scintillating rod surrounded by a cylindrical array of absorber detectors (see Fig. \[fig:GEANT4pol\]). This configuration allows for minimizing systematic effects of spurious modulation, as induced for example by a squared array geometry of absorbers [@Hayashida2010; @Beilicke2011; @Krawczynski2011a]. Taking into account only signals in coincidence between the scatterer and the absorbers, it is possible to reduce the background.
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![Compton energy deposits calculated with Eq. \[eq:DeltaE\] for some scattering angles around 90$^\circ$.[]{data-label="fig:DeltaE"}](Fig03_DeltaE.eps "fig:")
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From Eq. \[eq:EsuE\] the amount of energy released in the scintillator per scattering event is: $$\Delta E(\theta)= E-E^\prime =\frac{E^2(1-\cos \theta)}{m_e c^2+E(1-\cos \theta)}\label{eq:DeltaE}$$ This energy is converted into a scintillation signal. Our study is intended to demonstrate the feasibility of a polarimeter able to perform measurements starting from about 20 keV. Such a low energy threshold corresponds to an energy release within the scattering scintillator of about 750 eV for a 90$^\circ$ scattering which is a faint scintillation signal to read-out. In Fig. \[fig:DeltaE\] are reported Compton energy deposits calculated with Eq. \[eq:DeltaE\] for some scattering angles around 90$^\circ$.
A polarimeter exploiting this design may be placed at the focal plane of a multilayer optics X-ray telescope taking advantage of its large collecting area. It would be possible to perform sensitive polarimetry with a low or negligible background, especially at low energy because of the larger effective area of the optics and the larger intensity of X-ray celestial sources.
Experimental measurements of photon tagging efficiency {#sec:Measurements}
======================================================
The geometry of a polarimeter is the result of a complex trade-off of various design parameters including the maximization of sensitivity, the control of systematics, the energy band of the telescope and the priorities on the astrophysical targets. In our design one of the parameters of highest relevance is the capability to detect weak signals by the scatterer. This determines the efficiency of the system and it is an important driver of the instrument design. In this work we measure the “tagging efficiency", which we define as the fraction of events which, after the scattering in the central rod, give rise to a coincidence between the scatterer and the absorber. To detect a single scattering event in the scintillator we need a scintillation signal at the read-out device higher than a minimum detection threshold. The characterization of the scatterer material in terms of absolute scintillation output and the choice of the best wrapping to minimize the scintillation signal loss is mandatory. Also the optical coupling between the scintillator and the read-out device is important to collect as much scintillation photons as possible.
Scintillators and wrapping materials {#subsec:Scatterers}
------------------------------------
[p[6.5cm]{}ll]{} & BC-404 & p-terphenyl\
Chemical Composition & H$_{11}$C$_{10}$ & C$_{18}$H$_{14}$\
Light Output (10$^4$ photons/MeV) & 1.36 & 2.7\
Decay time (ns) & 1.8 & 3.7\
Wavelength of Max Emission (nm) & 408 & 420\
Refractive index at Wavelength of Max Emission & 1.58 & 1.65\
Density (g cm$^{-3}$) &1.023 & 1.23\
\[tab:scintmat\]
Scintillation materials tested are plastic scintillator BC-404 (Polyvinyltoluene) by Saint-Gobain Crystals [@BC404cristal] and doped p-terphenyl crystal by Cryos-Beta [@PTCryosBeta] (see Fig. \[fig:PTmu\]). Their properties are listed in Tab. \[tab:scintmat\]. Doped p-terphenyl has a better scintillation light yield which is useful to achieve a low detection threshold, conversely the BC-404 is characterized by a shorter decay time and it is usually employed for fast coincidences. Since the p-terphenyl crystal has a higher density of about 20$\%$ with respect to the BC-404, it can be used to make a shorter scatterer rod, while maintaining a comparable Compton scattering efficiency.
The wavelengths of maximum emission, 408 nm and 420 nm respectively for BC-404 and doped p-terphenyl, match very well with the sensitivity spectrum of the PMT we employed, a Hamamatsu H10721-110 PMT-sel. [@HamaH10721110PMT], which peaks at about 400 nm and that has a superbialkali photocathode selected by the vendor to have the highest quantum efficiency of the lot, that in our case is 41$\%$.
![P-terphenyl (solid lines) and BC-404 (dashed lines) interaction coefficients. In red is represented the photoelectric absorption, in blue the incoherent scattering, in orange the coherent scattering and in black the total interaction coefficient. P-terphenyl and BC-404 are suitable to study the energy release for Compton scattering down to about 20 keV. [@XCOM][]{data-label="fig:PTmu"}](Fig04_BC404-PT_mu.eps){width="12cm"}
All scintillators have a cylindrical geometry. Short ones are 1 cm long and have a diameter of 1 cm for both materials. Long ones are 3 cm long and have a diameter of 0.5 cm or 0.7 cm for BC-404 and doped p-terphenyl respectively.
Wrapping materials tested are BC-620 white painting [@BC620paint] provided by the manufacturer on a BC-404 sample, commercial PTFE (Teflon), TETRATEX tape [@TETRATEXref] and a VM2000 double reflecting film.
Measurement procedure
---------------------
![Experimental set-up for tagging efficiency measurements. []{data-label="fig:PhotoSetUp"}](Fig05_PhotoSetUp_3cm_w_arxiV.eps){width="10cm"}
The measurement set-up is shown in Fig. \[fig:PhotoSetUp\]. A radionuclide is held in a lead shielding box. A brass collimator wrapped with lead has a diaphragm on the bottom side to allow the radiation to impinge on the top of the scatterer. Scintillation signal is detected by the Hamamatsu H10721-110 PMT-sel. The sources we used are $^{109}$Cd and $^{241}$Am. $^{109}$Cd emission of interest is given by K$_{\alpha}$ and K$_{\beta}$ superposition of lines of Ag which are not resolved singularly by the scintillator and result in a convoluted line at about 22.6 keV. $^{241}$Am emission line of interest is at 59.5 keV produced as a consequence of the $\alpha$ decay. The energy release within the scintillator for Compton scattering at about $\theta \simeq 90^\circ$ for $^{109}$Cd lines is about 960 eV, and for $^{241}$Am is about 6.2 keV (see Eq. \[eq:DeltaE\] and Fig.\[fig:DeltaE\]). The optical contact between the scintillator and the PMT is guaranteed by an optical grease.
Scattered radiation is absorbed by a Brillance 380 Detector manufactured by Saint Gobain Inc. [@LaBr3PMT], which is a sealed PMT with a cylindrical head-on LaBr$_{3}$ crystal 2.5 mm thick and 2.54 cm of diameter behind a 220 $\mu$m thick Beryllium window. The whole set-up is covered by a black thick cloth and placed inside a darkroom to minimize the contamination by background stray light.
The block diagram of the electronic chain is reported in Fig. \[fig:ElectricChain\]. The signal from H10721-110 PMT-sel. after pre-amplification (SILENA 207) and amplification (SILENA 7611/L) stages is 1 $\mu$s temporally delayed (TENNELEC TC 215) and sent to a 1024 channels Multi Channel Analyser (MCA Amptek 8000A). The latter provides digitization after applying the *veto*, depending on the signal from the LaBr$_{3}$ PMT. The acquisition from the H10721-110 PMT-sel. is performed only if a photon is detected by the LaBr$_{3}$ PMT within an energy window tuned around the scattered line (ORTEC 4890 Pre-amplifier SCA Amplifier). LaBr$_{3}$ PMT signal triggers a 3 $\mu$s time window (CANBERRA Fast/Slow Coincidence 2144A) to allow MCA to apply the *veto* and to perform digitization.
![Block diagram of the electronic chain employed for the experimental set-up.[]{data-label="fig:ElectricChain"}](Fig06_ElectronicChain2.eps){width="15cm"}
To minimize dark current rate, a low threshold of 4 ADC channels has been chosen for the MCA acquisition, using an input dynamics range of 0–5 V. Thanks to this instrumental configuration the dynamic at low energy was maximized while excluding dark current from coincidence measurements. We performed also measurements with an input dynamics range 0–10 V, with the same channel threshold, to extend the energy band, to include the $^{109}$Cd peak of photoelectric absorption. The $^{241}$Am photoelectric peak at 59.5 keV was outside the 0–10 V input dynamics range. ADC spectra obtained with H10721-110 PMT-sel. has been calibrated in term of charge by using a pulse generator for both input dynamics ranges.
The scintillator spectrum
-------------------------
Spectra obtained with a 3 cm long doped p-terphenyl scintillator illuminated by the $^{109}$Cd source without applying any coincidence veto are shown in Fig. \[fig:signalandbackground\]. The red histogram represents the spectrum obtained with the set-up of Fig. \[fig:PhotoSetUp\]. It is evident the peak due to photoelectric absorption of $^{109}$Cd emission. The black histogram represents the spectrum acquired by blocking the $^{109}$Cd emission with a 0.45 mm thick lead tape in front of the external face of the collimator hole. Around such a tape, the brass diaphragm is still partially transparent to the $^{109}$Cd line at 88.04 keV. Therefore the plateau between about 5 and 10 pC, visible in the red and black spectra, is due to Compton energy deposits of $^{109}$Cd 88.04 keV photons passing through the collimator. Moreover such a radiation excites fluorescences at 74.97 keV and 72.08 keV of the lead placed around the Hamamatsu PMT, employed to reduce backscattering. Such fluorescences interact via Compton adding their contribution to the resulting spectrum. For tagging efficiency measurements the *veto* trigger starts when the LaBr$_{3}$ PMT acquires a signal gated around the Compton scattered line, therefore excluding such a plateau. The blue histogram represent the spectrum acquired removing the $^{109}$Cd from the source holder. The Compton plateau disappears, but some low charge signal is still present, probably depending on environmental stray light reaching the PMT window or background radiation interacting with the scintillator. This interpretation is supported by the fact that using a neoprene cap obscuring completely the PMT window, low amplitude signals are strongly suppressed (green histogram spectrum). Blue and green histograms demonstrate that the experimental set-up is characterized by a very low background.
![Red histogram: spectrum obtained in the PT-1-11 scintillator with $^{109}$Cd source without applying any coincidence veto. On the left of the photoelectric peak a plateau produced by $^{109}$Cd Compton energy deposits of 88.04 keV and by fluorescences generated by the same line exciting the PMT lead cladding. Black histogram: spectrum acquired by blocking direct radionuclide low energy radiation with a lead tape in front of the collimator hole. Blue histogram: spectrum acquired taking away the $^{109}$Cd from the source holder. The Compton plateau disappears, but some low charge signal is still present. Green histogram: the spectrum acquired by replacing the scintillator with a neoprene cap obscuring completely the PMT window.[]{data-label="fig:signalandbackground"}](Fig07_signal_and_background.eps){width="10cm"}
![$^{109}$Cd Spectra of scintillation signal within the doped p-terphenyl scintillator 3 cm of length and 0.7 cm of diameter wrapped with TETRATEX layer. The spectrum without applying coincidence *veto* is shown in black. The spectrum obtained by applying coincidence *veto* between the signal from H10721-110 PMT-sel. and LaBr$_{3}$ PMT is shown in red.[]{data-label="fig:coincidenceBC404"}](Fig08_3cmPT-1-11TetratexCalibration_vs_Coincidences_Cd109_ylog.eps){width="10cm"}
Measurements of tagging efficiency {#subsec:Tagging}
----------------------------------
In this section we discuss the results of measurement of tagging efficiency obtained with BC-404 and doped p-terphenyl scintillators of various size and wrapped with different materials.
The tagging efficiency is defined as: $$\epsilon_\mathrm{tag} = \frac{R_\mathrm{coinc \ net}}{R_\mathrm{tot \ net}} \label{eq:TaggingEff}$$ where $R_\mathrm{coinc \ net}$ is the net coincidence rate (without spurious coincidences) between H10721-110 PMT-sel. and LaBr$_{3}$ PMT and $R_\mathrm{tot \ net}$ is the total net rate given only by the scattered events detected in the LaBr$_{3}$ PMT, therefore subtracting the background rate $R_\mathrm{bkg}$. The rate of spurious coincidences due to H10721-110 PMT-sel. electronic noise which accidentally encounters the LaBr$_{3}$ PMT trigger is negligible and it was measured to be (2.3$\pm$0.1)$\cdot$10$^{-3}$ counts s$^{-1}$. This value was obtained by shielding the H10721-110 PMT-sel. entrance window with a black cap made of Neoprene below the scattering element for preventing light illumination and applying coincidence *veto*. Coincidence measurements then allow to fix the problem of dark current (see low charge region of the spectrum represented by the black histogram in Fig. \[fig:coincidenceBC404\]). The background rate $R_\mathrm{bkg}$ detected by the LaBr$_{3}$ PMT is produced by environmental background falling within the energy window of the LaBr$_{3}$ PMT and source photons scattered by the mechanical set-up. $R_\mathrm{bkg}$ was measured by removing the scatterer from the above of the H10721-110 PMT-sel. window. The background rate in the energy window of $^{109}$Cd scattered emission was acquired three times during long measurements lasting for 3 days each one and it were found to be 0.397$\pm$0.003 counts s$^{-1}$, 0.341$\pm$0.001 counts s$^{-1}$ (MCA dynamic range 0–5V) and 0.331$\pm$0.002 counts s$^{-1}$ (MCA dynamic range 0–10 V). For measurements with $^{241}$Am, the background rate was acquired once and it was found to be 0.287$\pm$0.001 counts s$^{-1}$ (MCA dynamic range 0–10 V). Thus, the Eq. \[eq:TaggingEff\], expressing the tagging efficiency, can be rewritten as: $$\epsilon_\mathrm{tag} = \frac{R_\mathrm{coinc}-R_\mathrm{sp \ coinc}}{R_\mathrm{tot}-R_\mathrm{bkg}} \label{eq:TaggingEffExplicit}$$ where $R_\mathrm{coinc}$ is the coincidence rate effectively measured, $R_\mathrm{sp \ coinc}$ is the rate of spurious coincidences (negligible in our case) and $R_\mathrm{tot}$ is the total rate effectively measured with the LaBr$_{3}$ PMT.
In some cases multiple evaluations of tagging efficiency were measured for the same combinations of scintillators and wrapping materials by repeating $R_\mathrm{coinc}$ and $R_\mathrm{tot}$ measurements after adjusting the set-up (i.e. removing and replacing the scatterer) to verify the repeatability of the measurements. Thus, a weighted average of tagging efficiency value $\overline{\epsilon_\mathrm{tag}}$ have been calculated. In other cases no multiple measurements were performed so that a single efficiency value is reported. All results are listed in Tab. \[tab:taggeff\].
[lllllllll]{} &Scintillator &Scintillator Dimension &Wrapping & MCA &number $n$ of & & & Used for\
&Material & $h$ (cm) $\times$ $d$ (cm) & Material&input dynamics (V) &measurements &$\overline{\epsilon_\mathrm{tag}}$ ($\%$)& $\overline{\sigma_{\epsilon_\mathrm{tag}}}$&Sensitivity\
$^{109}$Cd & & & & & & & &\
&BC-404 &1.0 $\times$ 1.0 & Teflon& 0-5 &1 &34.44& 0.76&\
\
&BC-404& 3.0 $\times$ 0.5 & BC-620 paint&0-5 &2 & 14.06& 0.43&\
& & &+ Teflon& & & & &\
&BC-404 & 3.0 $\times$ 0.5 & Teflon&0-5 &1 & 33.74& 0.80&\
&BC-404 &3.0 $\times$ 0.5 &VM2000 &0-5 &1 &36.27& 0.51&\
\
&PT-1-11 &3.0 $\times$ 0.7 &TETRATEX& 0-5 &1 &47.48& 0.23&\
&PT-1-11 &3.0 $\times$ 0.7 &TETRATEX&0-10 &1 & 36.15 &0.26&\*\
\
&PT-2-11 &3.0 $\times$ 0.7 &TETRATEX&0-5 &4 &38.50& 0.16&\
&PT-2-11 &3.0 $\times$ 0.7 &Teflon&0-5 &1&38.47 &0.39&\
&PT-2-11 & 3.0 $\times$ 0.7 & VM2000&0-5 &2 &40.59& 0.16&\
\
&PT-3-11&1.0 $\times$ 1.0 &Teflon&0-5 &1 &45.5& 1.0&\
&PT-3-11 &1.0 $\times$ 1.0 &VM2000&0-5 &1&47.20& 0.24&\
&PT-3-11 &1.0 $\times$ 1.0 & No wrapping&0-5 &1 & 28.3& 1.9&\
$^{241}$Am & & & & & & & &\
&PT-1-11 &3.0 $\times$ 0.7 &TETRATEX&0-10 &1 & 87.1 & 2.7& \*\
\[tab:taggeff\]
Systematic effects were found with respect to the comparison of measurements depending on the rods diameters. The structure holding the PMT glass window has a cavity with a diameter of 1 cm, that encircles the photocathode active region which has a diameter of 0.8 cm. Scintillator rods 1 cm long have also a diameter of 1 cm. Thus, when coupled with the PMT they are blocked by the structure holding the PMT window, but they exceed the photocathode active region. Even though a fraction of scintillation photons reaching the rod bottom side is lost, such fraction is the same for all 1 cm long rods. On the contrary, rods which are 3 cm long have a diameter of 0.7 cm (doped p-terphenyl) and 0.5 cm (BC-404), therefore smaller than the PMT active window. Moreover, they can move partially out from the PMT active window causing some scintillation photons loss. Such effect can affect the reproducibility of measurements, allowing only the evaluation of maximum values for tagging efficiency. Therefore it make sense to consider multiple measurements for different wrapping/scintillator combinations looking for the reproducibility of results. Whenever single measurements values were compatible within the $3\sigma$ error we performed the average of tagging efficiency to obtain results reported in Tab. \[tab:taggeff\]. Therefore, due to the diameter size larger than the active window of the PMT for 1 cm long rod, we cannot compare directly collected scintillation signal from rods of different length. Despite this fact we notice that the length is an important parameter: longer rods increase scattering efficiency while leading to a larger scintillation signal loss, which must be reduced using an efficient wrapping material.
Two rods of doped p-terphenyl 3 cm long with 0.7 cm of diameter were tested. One of them (PT-1-11) shows a higher tagging efficiency with respect to the other (PT-2-11). For preserving the better sample, we decided to leave it wrapped with its original TETRATEX layer as delivered by the vendor. The difference of tagging efficiency arises from a larger light output of the PT-1-11 with respect to PT-2-11 allowing to get more coincidences and therefore to achieve a higher tagging efficiency. At a visual inspection crystals do not show any evident difference, but even small structural defects may contribute to change scintillation photons production and collection properties.
Comparison between doped P-terphenyl and BC-404 {#subsec:materials}
-----------------------------------------------
Doped p-terphenyl is characterized by an expected larger light output with respect to BC-404 (see Tab. \[tab:scintmat\]). This is confirmed by the spectra reported in Fig. \[fig:PT311vsBC404\] showing the comparison of 1 cm long scintillators wrapped with Teflon tape. Therefore, the photoelectric absorption peak of doped p-terphenyl scintillator is shifted towards a larger value of charge collection if compared with BC-404. Since doped p-terphenyl is able to produce more scintillation photons with respect to BC-404, more events will be detectable and therefore doped p-terphenyl tagging efficiency is expected to be larger than BC-404 one.
![Spectra of doped p-terphenyl PT-3-11 (green) and BC-404 (black) 1 cm long rods both wrapped with Teflon. A stronger scintillation signal is achieved by employing PT-3-11 which has a theoretical light output about 2 times larger than BC-404 (see Tab. \[tab:scintmat\]).[]{data-label="fig:PT311vsBC404"}](Fig09_PT-3-11_vs_BC-404.eps){width="10cm"}
This fact is confirmed by results reported in Tab. \[tab:taggeff\], where doped p-terphenyl scintillators are systematically more efficient than BC-404 ones. In particular tagging efficiencies of rods of 1 cm of length, wrapped with Teflon, are $(34.44\pm0.76) \%$ for BC-404 and $(45.5\pm 1.0) \%$ for PT-3-11. Therefore BC-404 tagging efficiency is $75.7\%$ of the p-terphenyl tagging efficiency. If rods of 3 cm of length, wrapped with Teflon, are considered the difference in tagging efficiency decreases. In fact the BC-404 tagging efficiency is $(33.74\pm0.80) \%$ while the PT-2-11 tagging efficiency is $(38.47\pm0.39) \%$. Therefore the BC-404 tagging efficiency is $87.7\%$ of the p-terphenyl one. This demonstrate how the p-terphenyl crystal is less efficient to collect scintillation photons, since if longer rods are considered its tagging efficiency decreases more with respect to the BC-404. The p-terphenyl crystal could present internal inhomogeneities which reduce its scintillating and visible light collection performance. This problem can be more significant for long rods with respect to shorter one.
Comparison between VM2000, TETRATEX and Teflon
----------------------------------------------
A good wrapping plays a crucial role to enhance the tagging efficiency owing to the preservation of scintillation signal.
![Spectra from doped p-terphenyl PT-3-11 sample (1 cm long rod) wrapped with VM2000 (red), Teflon (green) and without any wrapping (black). The VM2000 reflecting foil allows to preserve the larger fraction of scintillation signal, and it is also the one that leads to the highest tagging efficiency.[]{data-label="fig:betterWrappingPT"}](Fig10_PT-3-11_wrapping.eps){width="10cm"}
The fact that a better tagging efficiency is associated to a higher collection of light can be demonstrated by comparing the peaks of photoelectric absorption from doped p-terphenyl 1 cm long rods wrapped with different materials (see Fig. \[fig:betterWrappingPT\]) and tagging efficiency results. VM2000, which allows for collecting the most intense scintillation signal, it is also the one which leads to the highest tagging efficiency. Tab. \[tab:taggeff\] shows clearly as VM2000 reflecting layer allows to obtain a higher tagging efficiency with respect to the other wrapping materials.
Here we omit any discussion on the other aspects such as the uniformity of the wrapping and the impact on the systematics. These considerations may be faced for a real polarimetric experiment, but they are not relevant to the present discussion.
The tagging efficiency as a low amplitude signal threshold {#sec:tageffthreshold}
==========================================================
The energy deposited by a scattered photon produces a scintillation signal which is larger for higher energy release (see Eq. \[eq:DeltaE\]) and for smaller scintillation light loss. Since the tagging efficiency identifies the fraction of such a Compton energy deposit effectively read-out above the detection threshold, which is independent on the incident radiation energy, different values of tagging efficiency are expected. The higher is the energy of incident radiation, the higher will be the tagging efficiency.
To evaluate the sensitivity of a Compton polarimeter across the X-ray energy band, it is necessary to quantify the tagging efficiency energy dependence across the energy spectrum. A way to obtain such an information is to perform direct experimental measurements of tagging efficiency for all the energies of interest. Unfortunately there is a limited number of long lived radioactive sources emitting lines at the energies of our interest, not emitting also higher energy lines interfering with the measurements. An alternative strategy is necessary. We measured the tagging efficiency of $^{109}$Cd and $^{241}$Am (see Sect. \[sec:Measurements\]). Then, by simulating the coincidence Compton spectra of $^{109}$Cd and $^{241}$Am with the experimental set-up shown in Fig. \[fig:PhotoSetUp\], we found the threshold corresponding to the measured tagging efficiency for each radionuclide spectrum. The two values found are two estimates of the same threshold which is independent on the incident energy, therefore we calculated their weighted average. Subsequently, we simulated coincidence Compton spectra for different incident energies with the same set-up, and we calculated the tagging efficiency corresponding to the threshold we obtained before. Finally, by simulating a realistic polarimeter design (see Fig. \[fig:GEANT4pol\]) and multiplying such tagging efficiencies for the Compton polarimeter efficiencies at the same energies, it is possible to obtain the total efficiency and then to evaluate the sensitivity for the polarimeter design considered.
Simulations of laboratory set-up
--------------------------------
A simulator of the the Compton energy deposition within the scintillator in the experimental set-up (see Fig. \[fig:PhotoSetUp\]) was developed using the GEANT 4 toolkit [@Agostinelli2003]. The simulated scintillator chosen is the doped p-terphenyl PT-1-11 wrapped with TETRATEX to reproduce measures performed with the input dynamics range 0–10 V with $^{109}$Cd and $^{241}$Am (see measures labelled with “\*" in Tab. \[tab:taggeff\]). The simulator output file reports the energy deposits occurred within the scintillator at different depths. Coincidences between scatterer and absorber are tagged with a specific flag to be later analysed.
The GEANT 4 simulator only models the energy losses in the scintillator. We choose to model the scintillation light propagation within the scatterer by means of a Monte Carlo simulator specifically developed for this purpose. The input of this simulation stage are the coincident events energy deposits and their interaction depth within the scintillator as obtained by the GEANT 4 simulator. The output are the coincidence spectra of $^{109}$Cd and $^{241}$Am suitable to be compared with the experimental ones as obtained by the H10721-110 PMT-sel.
In this Monte Carlo simulator the X-ray photons are assumed to be Compton scattered along the scintillator vertical axis. Since the X-ray beam is collimated and centred, this approximation is good. The number of scintillation photons produced for each X-ray scattering event is extracted from a Poissonian distribution of expected value: $$m_{0}=\frac{\Delta E}{\delta}\label{eq:m0}$$ where $\Delta E$ is the energy deposit as given by GEANT 4 simulator and $\delta$ is the scintillator light output (see Tab. \[tab:scintmat\]).
The path length $s$ of each scintillation photon is evaluated extracting a random value from an exponential distribution that depends on the mean free path $\lambda$ of visible light within the scintillation material. Therefore $s$ is given by: $$s=-\lambda \ln(R)\label{eq:scintpathlenght}$$ where $R$ is a random number between 0 and 1. The mean free path value that gives the better agreement with measured data is 2.0 cm. Indeed the scintillation light extinction at small distance (few centimetres) from the visible light production site is large [@Nicoll1970] and $\lambda$ is very small. The probability of scintillation light to be reflected or transmitted through a discontinuity layer between two different materials is evaluated by means of Fresnel equations. Also the total internal reflection, if conditions are verified, is assumed. Since the scintillator is wrapped with the TETRATEX diffuser on the top and side faces, visible light passing through its edges encounters air and then it is diffused by TETRATEX. Therefore in this case the discontinuity between p-terphenyl and air is taken into account to evaluate the probability of reflection or diffusion. Light exiting from the scintillator faces enters into the diffuser which has a probability of 95$\%$ [@TERTATEXreflectivity] to diffuse it back. Since the diffuser produces a randomisation of directions the diffusion angle is a random value between $-\pi$ and $\pi$ with respect to the normal to the discontinuity layer. In the Monte Carlo simulator we developed, scintillation photons propagate within vertical planes passing thought the scatterer vertical axis, therefore reflection by the scintillator edges, but also diffusion by the wrapping, does not produce any variation of the azimuthal direction. When a scintillation photon reaches the bottom face of the scatterer the following discontinuities are considered:
1. between the scintillator and the optical grease,
2. between the optical grease and the PMT glass window,
3. between the PMT glass window and the photocathode
The refractive indexes are: for the p-terphenyl scintillator 1.65 (see Tab. \[tab:scintmat\]), for the optical grease 1.5, for the PMT glass window 1.55 and for the superbialkali photocathode 2.15 (the real part) and 1.2 (the imaginary part). This last value was estimated from @Motta2005 that studied the reflection properties of bialkali photocathodes. Then survived photons crossing the PMT glass window are converted into photoelectrons by the photocathode and start the multiplication avalanche.
The average number of photoelectrons generated by the photocathode is $\mu=mq$ where $q$ is the photocathode quantum efficiency. Assuming all photoelectrons are collected by the first dynode, the probability that $n$ photoelectrons will start the multiplication is given by the Poisson relation [@Bellamy1994]: $$P(n; \mu)=\frac{\mu^n e^{-\mu}}{n!}\label{eq:poisson}$$
The multiplication process for $n$ photoelectrons, each starting a mutually independent chain, can be approximated by a Gaussian distribution, if the coefficient of secondary electron emission by the first dynode is large ($>$ 4) and the coefficient of secondary electron collection by the initial dynodes multiplication stages is close to one [@Bellamy1994]: $$G_n(x)=\frac{1}{\sigma_1 \sqrt{2 \pi n}}e^{-\frac{(x-nQ_1)^2}{2 n{\sigma_1}^2}}\label{eq:gauss}$$ where $x$ is the PMT output signal expressed in charge, $Q_1$ is the PMT charge output signal at the end of the multiplication chain for one photoelectron, $\sigma_1$ is the standard deviation of the one photoelectron peak charge distribution. Obviously $Q_1= e g$ where $e$ is the elementary charge and $g$ is the PMT gain. For $n=0$ we have simply zero charge at the output signal. The convolution of Poisson (Eq. \[eq:poisson\]) and Gaussian distributions (Eq. \[eq:gauss\]) gives [@Bellamy1994]: $$S_\mathrm{output}(x)=P(n;\mu) \otimes G_n(x)=\sum_{n=1}^{n_\mathrm{max}}{\frac{\mu^n e^{-\mu}}{n!} \frac{1}{\sigma_1 \sqrt{2 \pi n}}e^{-\frac{(x-nQ_1)^2}{2 n{\sigma_1}^2}} }\label{eq:poisgaussconvolution}$$ The Eq. \[eq:poisgaussconvolution\] describes the PMT response to an event which produces $\mu$ average photoelectrons at the photocathode.
In the case of the $^{109}$Cd source, most of the PMT output is given by low amplitude signals since the energy deposit within the scintillator is small and the scintillation light transport will allow only to a small number of scintillation photons to reach the PMT entrance window. The experimental characterization of small amplitude signals is an important issue in order to model properly the photomultiplier response. A weak signal like the pulse from a single primary electron can show significant fluctuations. In the spectrum of single photoelectron a fraction of 10–20$\%$ could be due to small pulses below 1/3 of peak amplitude position. Those signals are produced by photoelectrons that are inelastically back-scattered by the first dynodes [@ManualPMT; @Knoll2010]. They distort the peak shape producing a wing in the low amplitude part of the spectrum. Therefore the single photoelectron peak ($n=1$ in Eq. \[eq:poisgaussconvolution\]) is an intrinsic characteristic of the specific PMT. The measurement of the single photoelectron spectrum requires a proper experimental set-up. For example a single photon light pulse from a LED can be used to trigger the PMT acquisition [@Barnhill2008]. At the moment we do not have the possibility to perform such a kind of study to measure properly the single photoelectron spectrum. There is an other method in the literature [@Bauleo2004] according to which the single photoelectron spectrum is measured by integrating the dark current after the PMT was left for a long time in the dark. However in this case the signal produced by single electrons emitted for thermal agitation from the photocathode can be distorted by low amplitude signals coming from the emission of electrons from subsequent dynodes. Therefore, we did not perform the characterization of the single photoelectron spectrum and we decided to modify Eq. \[eq:gauss\] and then Eq. \[eq:poisgaussconvolution\] to reproduce low amplitude signals at least by maintaining the total probability of the Gaussian statistics. The low amplitude wing of the Gaussian of Eq. \[eq:poisgaussconvolution\], when $n$ is a small number, forms a not negligible part of the spectrum and therefore of the Gaussian probability. We mirrored the Gaussian lower tail for negative values of the the charge. Following this procedure the total probability of the Gaussian is preserved.
The energy dependent tagging efficiency as derived by the detection threshold in $^{109}$Cd and $^{241}$Am
----------------------------------------------------------------------------------------------------------
Eq. \[eq:poisgaussconvolution\] is used to simulate coincidence spectra of $^{109}$Cd (superposition of lines at about 22.6 keV) and $^{241}$Am (59.5 keV) (see Fig. \[fig:simCd109Am241\]) whose measured tagging efficiency is reported in Tab. \[tab:taggeff\] (measurements labelled with “\*"). Since the measured tagging efficiency is known, the corresponding detection threshold $x_\mathrm{thr}$, expressed in charge, is the one that gives the same simulated tagging efficiency calculated as follows: $$\epsilon_\mathrm{tag}(x_\mathrm{thr})=\frac{\Biggl[ \biggl( N_\mathrm{tot}-\sum_{i=0}^{N_\mathrm{tot}}{P(n=0; \mu_i)} \biggr) \cdot \mathcal{I}(S_\mathrm{output}(x_\mathrm{thr})) \Biggr]}{N_\mathrm{tot}} \label{eq:taggingeffsim}$$ where $$\mathcal{I}(S_\mathrm{output}(x_\mathrm{thr})) =\frac{\int_{x_\mathrm{thr}}^{x_\mathrm{max}}{S_\mathrm{output}(x)}}{ \int_{0}^{x_\mathrm{max}}{S_\mathrm{output}(x)} }dx\label{eq:I}$$ In Eq. \[eq:taggingeffsim\] $N_\mathrm{tot}$ is the total number of simulated X-ray coincidence scattering events simulated in coincidence, the term $\sum_{i=0}^{N_\mathrm{tot}}{P(n=0; \mu_i)}$ gives the total number of events which extract zero photoelectrons from the photocathode. Thus, $\biggl(N_\mathrm{tot}-\sum_{i=0}^{N_\mathrm{tot}}{P(n=0; \mu_i)}\biggr)$ gives the number of events that produce an output charge signal at the end of dynodes chain (which have $n \ge 1$). The term $\mathcal{I}(S_\mathrm{output}(x_\mathrm{thr}))$ represents the fraction of simulated charge signal above the charge threshold. The weighted average of $^{109}$Cd and $^{241}$Am charge threshold is finally evaluated.
![$^{109}$Cd (superposition of lines at about 22.6 keV)and $^{241}$Am (59.5 keV) measured (blue) and simulated (red solid line) coincidence spectra. The orange line is the charge threshold obtained from simulated spectra applying the measured tagging efficiency.[]{data-label="fig:simCd109Am241"}](Fig11_PoissonPlusGaussCd109.eps "fig:"){width="8cm"} ![$^{109}$Cd (superposition of lines at about 22.6 keV)and $^{241}$Am (59.5 keV) measured (blue) and simulated (red solid line) coincidence spectra. The orange line is the charge threshold obtained from simulated spectra applying the measured tagging efficiency.[]{data-label="fig:simCd109Am241"}](Fig12_PoissonPlusGaussAm241.eps "fig:"){width="8cm"}
Measured (blue points) and simulated spectra (red solid line) are reported in Fig. \[fig:simCd109Am241\]. The simulated detection threshold $x_\mathrm{thr}$ (vertical orange solid line) is also shown in Fig. \[fig:simCd109Am241\]. It can be compared with the hardware experimental threshold corresponding to the sharp cut of the spectra on the left.
Tagging efficiencies for 20, 25, 30, 35, 40, 45, 50, 60, 70, 80, 90 and 100 keV are obtained by simulating each coincidence spectrum and substituting into the Eq. \[eq:taggingeffsim\] the threshold value $x_\mathrm{thr}$ previously calculated.
![Comparison of coincidence simulated spectra at different energies.[]{data-label="fig:SimChargeSpectrumConfrontation"}](Fig13_SimChargeSpectrumConfrontation.eps){width="12cm"}
Coincidence spectra at different energies are compared in Fig. \[fig:SimChargeSpectrumConfrontation\].
Tagging efficiency values obtained are reported in Fig. \[fig:taggeffplot\]. The efficiency increases from lower to higher energies and experimental measured values are compatible with simulated ones.
![Simulated tagging efficiency (black) plotted with measured one (red) for $^{109}$Cd and $^{241}$Am.[]{data-label="fig:taggeffplot"}](Fig14_TaggingEfficiency_plot.eps){width="12cm"}
From tagging efficiency to realistic polarimeter sensitivity {#sec:realpolarimeter}
============================================================
In this section we show the simulated sensitivity of a polarimeter design based on the experimental set-up characterized in previous sections.
Polarimeters performance can be evaluated by means of the Minimum Detectable Polarization (MDP), that is the minimum polarization degree detectable at a certain confidence level. @Weisskopf2010 calculated the MDP for the 99$\%$ confidence level as: $$MDP(99\%)=\frac{4.29}{\mu R_\mathrm{source} }\sqrt{\frac{B+R_\mathrm{source}}{ T}}\label{eq:MDP}$$ where $R_\mathrm{source}$ is the coincidence rate from the source, $B$ is the background coincidence rate, $T$ is the observing time and $\mu$ is the modulation factor.
In case of a focal plane polarimeter, for which the background can be neglected because the instrument is largely source dominated (possibly by the addition of an external anticoincidence detector), the MDP formula simplifies in: $$MDP(99\%)\simeq \frac{4.29}{\mu \sqrt{T R_\mathrm{source}}} \ \ \ \ \ \ \mbox{if} \ \ \ \ \ \ B \ll R_\mathrm{source} \label{eq:MDPsimple}$$
The coincidence source rate detected is proportional to the tagging efficiency $\epsilon_\mathrm{tag}$, such that Eq. \[eq:MDPsimple\] gives: $$MDP \propto \frac{1}{\sqrt{R_\mathrm{source}}} \propto \frac{1}{\sqrt{\epsilon_\mathrm{tag}}} \label{eq:MDPpropto}$$
A polarimeter design as described in Sect. \[sec:Polarimeter\] was simulated by means of the GEANT 4 toolkit. The scatterer is a 3 cm long rod which diameter is 0.7 cm, made of doped p-terphenyl (the material which gave the higher light output response) and wrapped with TETRATEX. A beam of monochromatic radiation impinges on axis at the center of the scatterer upper face. A LaBr$_{3}$ absorber is placed all around the scatterer at a distance of 5 cm away from the scatterer axis. The distance between the scatterer and the absorber is an important geometric parameter. If they are placed at a short distance scattering angles quite different from 90$^\circ$ are possible for coincidences and then a low modulation is achieved (see Fig. \[fig:DifferentialMu\]). The absorber is subdivided in 48 pixels, 3 cm long and 0.2 cm thick, laying parallel to the rod extension to measure the azimuthal directions of scattering. We derived from this simulation the efficiency as a function of energy, that we then multiplied by the tagging efficiency (evaluated in Sect. \[sec:tageffthreshold\]), and the modulation factor.
To evaluate the polarimeter performance then we calculated the MDP achievable at the focal plane of one optics module of NuSTAR [@NuSTAROptics; @Harrison2010], that will observe for the first time the sky in the hard X-rays between 5 and 80 keV. NuSTAR telescope is composed by two 10.14 m focal length identical optics modules based on Wolter I geometry employing multilayer technology.
In Fig. \[fig:MDPplot\] is reported the MDP for various sources. An MDP of about 10$\%$ between 20 and 80 keV is obtained for 10 mCrab sources in 100 ks of observation. The MDP achievable with both optics modules can be evaluated taking into account that the coincidence source rate $R_\mathrm{source}$ scales with the effective collecting area $A_\mathrm{eff}$, so that from Eq. \[eq:MDPpropto\] we have: $$MDP \propto \frac{1}{\sqrt{R_\mathrm{source}}} \propto \frac{1}{\sqrt{A_{\mathrm{eff}}}}\label{eq:MDPAeff}$$ Thus, doubling the area (i.e. observing with both optics modules) the MDP reduces by a factor $\sqrt{2}$.
Neutron stars with cyclotron lines could be observed, if the lines fall within the 20 – 80 keV energy band. Among them an MDP lower than 3$\%$ is achievable for Vela X-1 which shows cyclotron lines at about 23 and 50 keV [@Kreykenbohm2002], X0115+63 which presents multiple cyclotron absorption features from 10 to 50 keV [@Santangelo1999; @Heindl2000] and Her X-1 which has a line at about 40 keV [@Truemper1978; @Gruber2001; @Vasco2011].
![MDP for a Compton polarimeter based on the experimental set-up at the focal plane of one module of NuSTAR telescope. For evaluating the MDP achievable with both optics modules the plotted values must be divided by a factor $\sqrt{2}$, because the MDP scales with the root square of the effective collecting area. []{data-label="fig:MDPplot"}](Fig15_MDP_v13_Mu000_NuSTAR_x1_20120306_Scatt20120202ptTETRATEX3cm_Be150um_ER018_20.0-80.0_keV.eps)
From the point of view of the optimization of the polarimeter we can make some considerations. Since a long crystal scattering rod could have internal inhomogeneities that reduce the collection of scintillating light (see Sect. \[subsec:materials\]), the choice to enhance the detector sensitivity by means of enhancing the Compton efficiency with a rod longer then the 3 cm, requires probably to change the scintillating material passing to a plastic one (i.e. BC-404). For what concerns the wrapping material, the VM2000 reflecting film is the one which allows to obtain the higher tagging efficiency. For our experimental measurements we decided to preserve the most efficient crystal sample by avoiding to replace its TETRATEX wrapping with the VM2000 film, but a smaller MDP would be expected in this other a case.
Conclusions
===========
We defined a procedure based on experimental laboratory measurements and Monte Carlo simulations aimed to characterize the physical response of an active Compton polarimeter composed by a central low-$Z$ scatterer rod surrounded by a high-$Z$ absorber. The cylindrical geometry of the absorbers array allows for reducing systematics effect. In this study the experimental characterization is employed to fix simulator parameters needed to evaluate the instrument response. We characterized BC-404 (Polyvinyltoluene) plastic scintillator and doped p-terphenyl crystal. The latter showed a larger light output leading to a higher tagging efficiency with respect to the other one. We verified also that VM2000 reflecting film is the best wrapping material with respect to PTFE (commercial Teflon), TETRATEX and BC-620 white painting, giving a higher tagging efficiency depending on a better preservation of scintillation signal within the scintillator.
We concluded by showing the sensitivity of a polarimeter design based on the laboratory set-up with a 3 cm long doped p-terphenyl rod wrapped with TETRATEX coupled with LaBr$_{3}$ absorber placed at the focal plane of one of the two optics module of NuSTAR telescope. The Minimum Detectable Polarization achievable is 10$\%$ between 20 and 80 keV for a 10 mCrab source in 100 ks of observation. The sensitivity of this kind of polarimeter can be improved. If a longer scatterer is employed, the Compton efficiency is enhanced. However a longer p-terphenyl crystal could be affected by internal inhomogeneities which could reduce the scintillation light collection. Therefore, it might be that BC-404 plastic scintillator is better suited for long scatterers. For what concerns the wrapping material, if the VM2000 reflecting film is employed instead of TETRATEX, a higher tagging efficiency is found. A scatterer characterization aimed to find best polarimetric results must be performed, there is thus the possibility to improve significantly the polarimeter sensitivity with respect to the tested set-up.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was supported by a fellowship of the PhD school in Astronomy of University of Rome “Tor Vergata".
References {#references .unnumbered}
==========
|
---
abstract: 'In this series of papers, we develop a two-fluid model for VLBI jets. The idea is that the jet itself is non- or mildly-relativistic (electrons and protons), while the radiating blobs are relativistic electron-positron ‘clouds’ moving on helical paths wrapped around the jet. In this work, the emphasis is on the physical description of the clouds, and not on the structure or origin of the trajectory. In the simple case where the magnetic field is uniform and homogeneous accross the cloud, and the properties of the cloud are constant, the present paper shows synthetic maps of VLBI jets in different configurations, as well as the variation of different observational parameters along the trajectory.'
author:
- 'V. Despringre'
- 'D. Fraix-Burnet'
date: 'Received June 13, 1996; accepted September 18, 1996'
subtitle: 'I. Homogeneous and stationary synchrotron emission simulations.'
title: 'Two-fluid model for VLBI jets'
---
Introduction
============
Extragalactic jets at the parsec scale are present in numerous Active Galactic Nuclei (AGN; see a review by Zensus 1995). Impressive progress has been made by the Very Large Baseline Interferometry (VLBI), and details still closer to the central engine are expected with the advent of millimeter VLBI. Already, a lot of information can be gained from the structure of these jets with typically 1 parsec resolution. It has been possible to detect motions within a few years in about 100 sources (Vermeulen 1995), a lot of them being superluminal with apparent speeds up to $10 c$. The motions detected are those of blobs moving on curved trajectories. Generally speaking, these paths are wiggling, reminiscent of more or less helical lines seen in projection, and apparently different from one blob to the other, and the blob velocities vary along the trajectory (Zensus 1995; Qian et al. 1996).
Not much information is available on the nature of the blobs themselves. They are very generally believed to be shock fronts, because i) shock waves are expected in these jets and ii) they are an excellent means of accelerating particles through the first-order Fermi acceleration process as has been worked out in the kpc scale jets. Recently, in a series of papers, Gómez et al. (1993, 1994a, 1994b) performed numerical simulations of a VLBI jet where the blobs are shock fronts traveling along a helical relativistic jet. Nevertheless, the reality of these shock fronts is far from established.
The hypothesis of a relativistic jet is also debatable. Firstly, due to the Compton drag close to the black hole, it is very difficult to extract a jet with Lorentz factors higher than 2 or 3 (Phinney 1987, Henri & Pelletier 1991). Secondly, at the kpc scale, jets are probably non- or only mildly-relativistic (e.g. Parma et al. 1987, Fraix-Burnet 1992). Some authors conclude that the jets should decelerate (Bowman et al. 1996) from super- to subluminal speeds, but obviously, the lost energy should be observed in a manner or in another.
An interesting alternative to relativistic shocked jets is the two-fluid concept, in which the bulk of the jet (electrons and protons ejected from the accretion disk in the form of a collimated wind) is non- or mildly-relativistic at all scales, and synchrotron radiation is produced by a beam of relativistic electrons/positrons. This idea has been worked out theoretically by Sol, Pelletier and Asséo (1989) and applied to kpc jets (Pelletier & Roland 1986, 1988; Fraix-Burnet & Pelletier 1991; Fraix-Burnet 1992) for the particle acceleration problem. At small scales, observed relativistic phenomena can be produced by the relativistic electrons/positrons, and Pelletier & Roland (1989) found a very interesting application for cosmology using superluminal radio sources.
In this series of papers, we propose to apply this two-fluid concept to VLBI jets. The idea is based on the correlation between outbursts of AGNs and the subsequent appearance of VLBI blobs. If these bursts are explained by bursts of high-energy particles (as in Marcowith et al. 1996), then it is probable that these particles propagate on a few parsecs away. A relativistic beam propagating within the jet plasma has been shown (Sol et al. 1989; Achatz, Lesch & Schlickeiser 1990; Pelletier & Sol 1992; Achatz & Schlickeiser 1992) to be stable relatively to the excitation of Langmuir, Alfvén and whistler waves, on scales up to several hundreds of parsecs. Hanasz & Sol (1996) recently showed that large scale fluid (Kelvin-Helmoltz) stability is also possible. Hence, we suggest that the blobs seen in VLBI jets are these ‘clouds’ of relativistic electron-positron pairs propagating along helical trajectories wrapped around a non-relativistic jet. The term [*cloud*]{} is defined in this work as an ensemble of relativistic particles occupying a limited region of the jet, but these particles and the jet plasma are fully interpenetrated, making a two-component plasma. [*Cloud*]{} should [*not*]{} be understood in the fluid sense of an isolated component with a well defined boundary. We thus consider that the jet itself does not radiate. Its magnetohydrodynamics determines the structure of the trajectories (magnetic field lines?) that the radiating clouds will follow. The emphasis is on the physics of the clouds, because in a later paper, the properties of these clouds will be taken from high-energy emission models from AGNs (Marcowith et al. 1996). This two-fluid concept will thus build a coherent picture of extragalactic jets from their extraction in the AGN to the largest scale up to the extended lobes.
In this first paper, the basic model is presented in a simple configuration where the magnetic field is supposed to be uniform and oriented along the helix. The characteristics of the cloud are constant in time (stationary case). Synthetic maps are presented as well as the evolution of apparent speed and brightness of the clouds along a period of the helix. In a subsequent paper, a turbulent component of the magnetic field will be added, and polarization maps will be computed. Then, in a third paper, the temporal evolution of the cloud will be considered together with the self-Compton radiation. The model is presented in Sect. 2 while the numerical method is described in Sect. 3. Results are shown in Sect. 4 and a discussion is given in Sect. 5.
The model
=========
The description of the model in this section is divided in three parts. The geometrical aspects deal with the shapes of the jet and the cloud (see Introduction), the description of the helical trajectory and the definition of the different reference frames. The physical aspects of the model include the magnetic field characteristics and properties of the particles within the cloud. The synchrotron radiation is then computed through the Stokes parameters.
Geometry and kinematics
-----------------------
### Geometry
psfig.sty
We consider a cylindrical jet of radius ${R_{\rm jet}}$ making an angle $\alpha$ with the line of sight. The trajectory of the cloud is defined by a helix wrapped around the jet with the same axis (Fig. \[croquis\]). The ratio of the pitch $h$ to the radius is given by: $r_p={h/{R_{\rm jet}}}$. The shape of the cloud is taken to be an ellipsoid because we have in mind the study, in a future paper, of the temporal evolution of a spherical cloud of radius $a$ propagating along a magnetic field line. We intuitively expect a stretch of the cloud in the direction of propagation to a half large axis $b$. In a reference frame [*R’*]{} linked to the cloud in which the $y'$ axis is along the trajectory, the equation of the ellipsoid writes:
$${(x'-x'_c)^2\over a^2}+{(y'-y'_c)^2\over b^2}+
{(z'-z'_c)^2\over a^2}=1$$
The coordinates of the ellipsoid center $x'_c, y'_c, z'_c$ define the helix considered above and are parametrized in a frame [*R”*]{} linked to the jet:
$$\left\{
\begin{array}{ll}
x''_c(t) & = h\omega t/ 2\pi \\
y''_c(t) & = {R_{\rm jet}}\cos(\omega t) \\
z''_c(t) & = {R_{\rm jet}}\sin(\omega t)
\end{array}
\right.$$ The $x''$ axis is parallel to the jet axis, the $y''$ axis lies in the plane of the sky (Fig. \[croquis\]) and $\omega$ is the angular speed if $t$ is interpreted as the time. We make the further asumption: $b<<h$ so that the curvature of the helix along the cloud is negligible, or in other words, the magnetic field is uniform across the cloud.
Finally, the observer frame [*R*]{} has its $x$ axis along the line of sight and its $y$ axis parallel to the $y''$ axis (Fig. \[croquis\]).
### Kinematics
We assume that the jet and the parent AGN are at rest with respect to the observer. Relativistic effects only concern the cloud moving at a speed $\beta$ along the helix. The $y'$ axis of the cloud reference frame [*R’*]{} is defined by this velocity vector which makes an angle $\theta$ with the line of sight. Naturally, $\theta$ varies along the trajectory. The Doppler factor $\delta$ is then: $ \delta = \Gamma^{-1} \left( 1 - \beta \cos \theta \right)^{-1} $ where $\Gamma$ is the Lorentz factor of the cloud.
Physical characteristics
------------------------
The magnetic field is split in two components: $\vec B=\vec B_0 + \vec B_1$, where $\vec B_0$ is uniform throughout the jet and always tangent with the helical trajectory, and $\vec B_1$ is a non-uniform component. In this first paper, we take: $\vec B_1=0$. Since we do not consider here the origin of the magnetic field and its structure, there is no need to precise further the physics of the jet.
The relativistic cloud is made of electron-positron pairs. The energy distribution per unit volume of radiating particles is assumed to be a power law: $ N(E) dE = N_0 E^{-p} dE $, and the velocity distribution is isotropic in the cloud reference frame.
In contrast with Gómez et al. (1993, 1994a, 1994b), we take into account the upper cutoff energy $E_{\rm max}=\gamma_{\rm max}mc^2$ because it plays a role in high energy spectra of AGNs we will consider in a later paper. The global particle density (cm$^{-3}$) is thus:
$$N_{\rm e}= \int_{E_{\rm min}}^{E_{\rm max}}N(E){\rm d}E= N_{0} {1 \over 1 -
p}
\left[ E_{\rm max}^{1-p} - E_{\rm min}^{1-p} \right]$$
where $E_{\rm min}=\gamma_{\rm min}mc^2$ is the lower cutoff energy.
Transfer of synchrotron radiation
---------------------------------
The synchrotron radiation from the relativistic cloud is computed through the Stokes parameters $I, U, Q$ and $V$. All the necessary background and formulae for an uniform density distribution of particles with isotropic velocity distribution can be found in Pacholczyk (1970) and can also be found in Gómez et al. (1993). We neglect the elliptical polarisation ($V = 0$), and focalize only on the intensity $I$ in this paper since polarization will be the subject of a forthcoming paper.
The magnetic field is here assumed to be uniform across the cloud (see Sect. 2.1) which is supposed to be homogeneous, so that we are allowed to use the analytical resolution of the full transfer equations described by Pacholczyk (1970). This of course saves us considerable CPU time for this first stage, but resolution of the transfer equation via numerical techniques will be necessary in the next paper with an additional non-uniform magnetic field .
The observed frequency $\nu$ and the rest frequency $\nu'$ in the cloud frame are related by: $ \nu = \delta * \nu' $. Likewise, the emission and absorption coefficients are computed in the cloud frame [*R’*]{} (respectively $\epsilon'(\nu')$ and $\kappa'(\nu')$) but the transfer equations are solved in the observer frame [*R*]{} with:
$ \epsilon(\nu) = \delta^{2}\ \epsilon'(\nu') \;\; ; \;\;
\kappa(\nu) = \kappa'(\nu') / \delta .$
Cosmological corrections would imply the Doppler factor $\delta$ to be replaced by $\delta / (1 + z)$, with $z$ the redshift of the source. In this work we take $z=0$.
Parameters of the model
-----------------------
Our model of a VLBI jet considered in this first paper requires 11 parameters to be defined:
[*Jet:* ]{}
[1.]{} ${R_{\rm jet}}$, radius of the jet;
[2.]{} $r_p$, ratio of pitch to jet radius;
[3.]{} $\alpha$, angle of the jet to the line of sight;
[4.]{} $B_0$, magnetic field;
[*Cloud:*]{}
[5.]{} $a,b$, half small and large axes of ellipsoidal cloud;
[6.]{} $\beta$, cloud speed;
[7.]{} $N_{\rm e}$, particle density (cm$^{-3}$);
[8.]{} $\gamma_{\rm min},\gamma_{\rm max}$, lower and upper cutoff energy;
[9.]{} $p$, spectral index of the particle energy distribution;
[*Observer:*]{}
[10.]{} $\nu$, frequency of the observations;
[11.]{} $D$, distance to the source.
[ ]{}[ ]{}
Numerical coding of the model
=============================
Definition of the trajectory
----------------------------
As described in Sect. 2.1.1, the helical trajectory is parametrized in the reference frame [*R”*]{} where the $x''$ is the jet axis. Then, a simple rotation by the angle $\alpha$ around the $y$ or $y''$ axis defines the trajectory of the cloud in the observer reference frame, especially the projection onto the plane of the sky. The tangent of the trajectory gives the direction of the cloud velocity vector and $\vec B_0$.
The ellipsoidal cloud
---------------------
The center of the cloud moves along the trajectory defined above. At each position, a cloud reference frame [*R’*]{} is defined where the $y'$ axis is tangent to the helix and makes an angle $\theta$ to the line of sight. The large axis of the ellipsoid is parallel to this $y'$ axis. In this frame, the ellipsoid is given by Eq. 1, and a simple transformation entirely defines the 3-D ellipsoid in the observer reference frame.
At this stage, the sky plane is discretized into 2-D cells (pixels). Each cell is associated with the depth $s$ of the cloud along the line of sight and is given a size of $5~10^{-3}$ pc.
Synchrotron radiation
---------------------
Since the cloud is homogeneous and the magnetic field uniform across the cloud, $s$ is the only quantity varying from a cell to the other. The transfer equation in this simple case is then solved analytically for each cell.
Doppler effects are the same for all cells, but, for a given configuration of the jet (i.e. $\alpha$, $r_p$, $\beta$), vary depending on the position of the cloud on the trajectory (because the angle $\theta$ varies).
Results
=======
Given the relatively important number of parameters of the model, many types of jet can be produced. In this section, only two geometrical configurations are studied. The emphasis is put on observational diagnostics as well as on the understanding of the effect of the different parameters. Some of the parameters listed in Sect. 2.4, are kept constant in all the results presented in this paper: $D=15~$Mpc, ${R_{\rm jet}}=0.25$ pc, $a=0.2{R_{\rm jet}}$, $b=0.5 {R_{\rm jet}}$, $r_p=30$, $p=2$, $\gamma_{\rm min}=10^2$, $\gamma_{\rm max}=10^7$. All distances have been fixed because they are “morphological” and are more or less constrained by the observations. We think the chosen values are typical for close extragalactic VLBI jets (i.e. M87). The value for $p$ is also typical for these objects. The parameter $\gamma_{\rm min}$ is kept constant because it is coupled to $N_e$ through Eq. (2), whereas $\gamma_{\rm max}$ has no influence on the results of this paper since we are not concerned with high-energy radiation. Changing all these parameters would not affect very much the results presented here. The synchrotron intensity would be modified if a different cloud size is chosen, but the particle density or the magnetic field intensity have about the same effect.
The variable parameters considered in the following are thus: $\alpha,
\vec B_0, \beta, N_{\rm e}, \nu$. For clarity, results are shown for one cloud moving over one period of the helix, although real jets have several clouds propagating at the same time, possibly on different trajectories.
Maps
----
The resulting jet from our model with $\vec B_0=10^{-2}$ G, $\beta=0.99$, $N_{\rm e}=10^4$ cm$^{-3}$ is shown in Fig. \[map-large\] ($\alpha=70\degr$ and $\nu=10^{9}$ Hz) and Fig. \[map-small\] ($\alpha=5\degr$ and $\nu=10^{12}$ Hz). In the first case, the cloud is optically thick. Each figure is a set of 6 maps corresponding to 6 positions of the cloud along one period of the helix. A motionless object of constant arbitrary intensity is added to reproduce the core of AGN. This object has no means in our model and is placed on the axis of the jet, hence not on the trajectory. To mimic realistic observations, all maps have been convolved with a gaussian of FWHM=${R_{\rm jet}}$.
The resemblance with some observed VLBI jets is obvious. One interesting point to note here is that the cloud initially appears to move in a direction nearly perpendicular to the axis of the jet. Also, on Fig. \[map-small\], the intensity of the cloud changes dramatically along the trajectory, in contrast with the optically thick case of Fig. \[map-large\]. This flux variation of the cloud is illustrated on Fig. \[figmap2\] and Fig. \[figmap3\]. The apparent speed of the cloud is also plotted in these figures. It is always superluminal here (up to 7$c$ in the case of Fig. \[map-small\]), but more importantly it greatly varies along the trajectory.
Flux of cloud
-------------
The variation of the cloud flux along the helix for different speeds is shown in Fig. \[flux-large\] ($\alpha=70\degr$) and Fig. \[flux-small\] ($\alpha=5\degr$).
Two phenomena are competing in the flux variation along the helix: the Doppler effect and the orientation of the magnetic field. The Doppler factor depends on the cosine of the angle $\theta$ between the velocity vector and the line of sight, while the synchrotron flux depends on the sine of this same angle (because magnetic field and cloud velocity vector are parallel and both tangent to the helix, and the synchrotron intensity depends on the magnetic field component which is perpendicular to the line of sight). Hence, at low speeds, the intrinsic flux is maximum where this angle is the largest (middle of the curves in our examples), whereas the Doppler factor creates the opposite behaviour at very high speeds. At intermediate speeds, two maxima can appear due to the two competing effects.
In the case of a large angle to the line of sight (Fig. \[flux-large\]), the Doppler factor is smaller than 1 (flux dimming) and decreases with increasing cloud speed (for $\beta\ga 0.5$). Here, the effect of the magnetic field is dominated by the Doppler effect at speeds as low as $\beta = 0.5$. This is because the variation of the angle $\theta$ between the magnetic field and the line of sight is small. In the opposite case (Fig. \[flux-small\]), the Doppler factor $\delta$ is always larger than 1 (flux amplification) and increases with the cloud speed. The effect of the magnetic field is dominant at speeds as high as $\beta =0.96$ where two maxima are present. The consequence of the competition between these two phenomena is that the flux does not simply increase with $\beta$. This is true only for a limited range of speeds and at some locations along the trajectory.
Contrast
--------
The ratio $F_{max}$/$F_{min}$ (that we call contrast) of the maximum to the minimum fluxes of the cloud over one period of the helix, is plotted vs frequency in Fig. \[contrast\] for several speeds.
The angle of the jet to the line of sight is set to $30\degr$ for this figure. The contrast depends on the optical thickness of the cloud: it is higher in the optically thin regime at high frequencies. The competition between the two effects discussed in Sect. 4.2 is illustrated by the fact that the difference in this contrast between the two regimes has a minimum for $\beta\simeq 0.7$. The constrast increases with speed in the optically thick regime, but it first decreases and then increases with increasing speed in the optically thin regime.
Spectra
-------
Synchrotron spectra of the cloud at two positions distant by half a period of the helix for $\alpha=70\degr$ (corresponding to the first and fourth frame in Fig. \[map-large\]) are presented in Fig. \[spectra\].
At small frequencies the slope is $+5/2$ (the cloud is optically thick), and at larger frequencies, the slope is $-1/2$, as given by synchrotron radiation theory for a particle energy distribution spectral index of 2 in the optically thin regime.
As the cloud moves along the helix, these two slopes naturally remain the same. But the transition frequency $\nu_{m}$, where the flux of the cloud is maximum, increases with the projected magnetic field and the Doppler factor. The same two competing effects discussed in Sect. 4.2 are again in play here. In the case of Fig. 9, it has been shown in Sect. 4.2 that this is the Doppler effect that dominates. In general, the variation of $\nu_{m}$ along the trajectory could imply an apparent transition between the two regimes of optical thickness if the source is observed at a fixed frequency.
The influence of the particle density $N_e$ and the magnetic field $B_0$ on the synchrotron spectra is shown in Fig. \[spectra-parameters\]. Increasing the particle density or the magnetic field shifts upward the optically thin part of the spectrum. The optically thick part is not sensitive to the particle density, while it is shifted downward with increasing magnetic field.
Discussion and conclusion
=========================
The previous section shows that it is possible to explain observed VLBI jets with the two-fluid concept, even with a jet at rest. The presence of a relativistic ‘cloud’ (see Introduction) propagating inside the jet is the key ingredient in our model. We think that the idea of non- or mildly-relativistic jets in AGN and radiosources is now fully viable at all scales. It reconciles observed relativistic phenomena at scales smaller than the parsec and/or at VLBI scales, with non-relativistic jets both at large scale (observations) and at the central part of AGNs (theories of jet extraction).
The helical trajectory, observationally suggested, relaxes the constraint on the angle between the jet and the line of sight. The consequences of curved paths of VLBI blobs have not been fully appreciated, but AGN “unification” models would certainly benefit from such considerations. The helical trajectory also yields the observed behaviour that the initial direction of propagation of a blob can be nearly perpendicular to the jet axis. This is observed in quite a few sources (e.g. Mrk 501, Conway & Wrobel 1995). The case of a small angle to the line of sight shown in Fig. \[map-small\] is rather reminiscent of the BL Lacertae object 0235+164 (Chu et al 1996).
From the synthesized maps, different observational quantities are presented in Sect. 4. This helps in understanding the origin of flux variation along the trajectory. These are also observational curves that could bring some information on the different parameters of the model. Even if it requires multiepoch and multifrequency data, our model can probably be already applied in some cases. For instance, Qian et al. (1996) used a helical model to interpret the intrinsic evolution of the VLBI blobs in 3C345. As has been seen in Sect. 4.2, the orientation of the magnetic field also yields a variation of the flux along the trajectory. This has not been taken into account by these authors, but it could lead to different results.
Naturally, the present work is very simplistic, but undoubtly justifies sophistication of the simulations. Such simulations are necessarily limited because the reality encompasses so many physical phenomena. The originality of our work is that no ad-hoc assumption is made in the sense that the physics of the radiating cloud can be entirely derived from theories of jet extraction and high-energy radiation. In the same way, the trajectory could also be precised from physical calculations. Our goal here is to build a fully physically coherent picture of AGNs from the accretion disk up to the VLBI jet, under observational constraints from the radio to the high energy radiation. The next step will be the complete simulation of the stationary jet, by including the polarization with the addition of a turbulent magnetic field. In a later stage, the evolution of the cloud along its way from the region where it produces $\gamma$-rays to the VLBI scale will be theoretically studied and implemented in the numerical simulations.
We would like to thank an anonymous referee for very useful comments.
Achatz U., Lesch H., Schlickeiser R., 1990, A&A 233, 391 Achatz U., Schlickeiser R., 1992, in:[*Extragalactic radio sources: from beams to jets*]{}, eds. J. Roland, H. Sol and G. Pelletier (Cambridge University Press), p. 256 Bowman M., Leahy J.P., Komissarov S.S., 1996, MNRAS 279, 899 Chu H.S., Bååth L.B., Rantakyrö F.T., Zhang F.J., Nicholson G., 1996, A& A 307, 15 Conway J.E., Wrobel J.M., 1995, ApJ 439, 98 Fraix-Burnet D., 1992, A& A 259, 445 Fraix-Burnet D., Pelletier G., 1991, ApJ 367, 86 Gómez J.L., Alberdi A., Marcaide J.M., 1993, A& A 274, 55 Gómez J.L., Alberdi A., Marcaide J.M., 1994a, A& A 284, 51 Gómez J.L., Alberdi A., Marcaide J.M., Marscher A.P., Travis J.P., 1994b, A& A 292, 33 Hanasz M., Sol H., 196, MNRAS in press Henri G., Pelletier G., 1991, ApJ 383, L7 Marcowith A., Henri G., Pelletier G., 1995, MNRAS 277, 681 Pelletier G., Roland J., 1986, A& A 163, 9 Pelletier G., Roland J., 1988, A& A 196, 71 Pelletier G., Roland J., 1989, A& A 224, 24 Pelletier G., Sol H., 1992, MNRAS 254, 635 Phinney E.S., 1987, in:[*Superluminal Radio Sources*]{}, eds. J.A. Zensus and T.J. Pearson (Cambridge University Press), p. 301 Qian S.J., Krichbaum T.P., Zensus J.A., Steffen W., Witzel A., 1996, A& A 308, 395 Sol H., Pelletier G., Asséo E., 1989, MNRAS 237, 411 Vermeulen R., 1995, in:[*IAU Symposium 175, Extragalactic Radio Sources*]{}, (Dordrecht: Kluwer) Zensus J.A., 1995, in:[*IAU Symposium 175, Extragalactic Radio Sources*]{}, (Dordrecht: Kluwer)
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abstract: 'We consider several extensions of the Standard Model (SM) which can explain the anomalies observed by the Atomki collaboration in the decay of excited states of Beryllium via a new boson with a mass around 17 MeV yielding $e^+e^-$ pairs. We show how both spin-0 and 1 solutions are possible and describe the Beyond the SM (BSM) scenarios that can accommodate these. They include BSM frameworks with either an enlarged Higgs, or gauge sector, or both.'
author:
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Luigi Delle Rose$^{a,b}$, Shaaban Khalil$^{c}$, Simon J. D. King$^{b,d}$, Stefano Moretti$^{b,e}$\
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title: '**New Physics Suggested by Atomki Anomaly**'
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Introduction {#sec1:intro}
============
[[The quest for New Physics (NP) above and Beyond the Standard Model (BSM) has always seen a twofold approach. On the one hand, the high energy frontier has been pursued, typically through multi-purpose experiments at hadron accelerators, like the $Sp\bar pS$, Tevatron and LHC. On the other hand, the high precision frontier has been exploited, typically at lepton collider experiments, like LEP and SLC. Alongside this time honoured two-prong pursuit, over the years, a transversal dimension, covering both hadron and lepton colliders, centered on flavour physics, has also developed. So that, presently, the attention of the particle physics community in unveiling some NP has mainly been concentrated upon these three research strands. However, surprises may arise in other contexts, notably from (much) lower energy experiments. In this respect, results from $(g-2)$ of the muon are prototypical. Another interesting result which has recently been reported is the one in Ref. [@Krasznahorkay:2015iga] [[(see also [@Krasznahorkay:2017gwn; @Krasznahorkay:2017bwh; @Krasznahorkay:2017qfd; @Krasznahorkay:2018snd])]{}]{}, by the Atomki experiment [@Gulyas:2015mia]. The latter is a pair spectrometer for measuring multi-polarities of nuclear transitions, specifically, using a multi-detector array designed and constructed for the simultaneous measurement of energy and angular correlations of electron-positron pairs, in turn emerging via internal pair creation from a variety of nuclear transitions in various isotopes, such as $^{16}$O, $^{12}$C and $^8$Be. The intriguing result reported in [@Krasznahorkay:2015iga] concerns $e^+e^-$ correlations measured for the isovector magnetic dipole 17.64 MeV state (with spin-parity and isospin, $J^P=1^+$, $T=1$, respectively), and the isoscalar magnetic dipole 18.15 MeV state ($J^P =1^+$, $T=0$) in their transitions to the ground state ($J^P =0^+$, $T=0$) for the Beryllium case. Significant deviations from the internal pair creation rate were observed at large angles in the angular correlation for the isoscalar transition with a confidence level of more than $5\sigma$. This observation may indicate that, in an intermediate step, a (light) neutral boson with a mass of $16.70\pm0.35\,({\rm stat})\pm0.5\,({\rm sys})$ MeV has been created. In fact, also the 17.64 MeV transition eventually appeared to present a similar anomaly, albeit less significant, with a boson mass broadly compatible with the above one, i.e., $17.0\pm 0.5\, ({\rm stat})\pm 0.5\, ({\rm sys})$ MeV[^1]. ]{}]{}
[[The purpose of this review is to discuss possible solutions to these results, assuming that the neutral boson could be either a spin-1 or spin-0 object, belonging to a variety of BSM scenarios. The plan is as follows. In the next section we consider the characteristics of the results reported by the Atomki experiment. Then we describe possible candidate particles for such a light bosonic state. Finally, we illustrate the embedding of such solutions in possible theoretical models, in presence of a variety of experimental constraints emerging from both low and high energy experiments. We finally conclude.]{}]{}
The Atomki experiment and 17 MeV Beryllium anomaly
==================================================
The Atomki pair spectrometer experiment [@Gulyas:2015mia] was set up for searching $e^+ e^-$ internal pair creation in the decay of excited $^8$Be nuclei (henceforth, $^8{{\rm Be}^*}$), the latter being produced with the help of a beam of protons directed on a $^7$Li target. The proton beam was tuned in such a way that the different $^8$Be excitations could be separated in energy with high accuracy.
In the data collection stage, a clear anomaly was observed in the decay of $^8{{\rm Be}^*}$ with $J^P=1^+$ into the ground state $^8{\rm Be}$ with spin-parity $0^+$ (both with $T=0$), where $^8{{\rm Be}^*}$ had an excitation energy of 18.15 MeV [@Krasznahorkay:2015iga]. Upon analysis of the electron-positron properties, the spectra of both their opening angle $\theta$ and invariant mass $M$ presented the characteristics of an excess consistent with an intermediate boson (henceforth, $X$) being produced on-shell in the decay of the $^8{{\rm Be}^*}$ state, with the $X$ object subsequently decaying into $e^+e^-$ pairs. As mentioned, the best fit to the mass $M_X$ of $X$ was given as $
M_X = 16.7 \pm 0.35\ \text{(stat)}\ \pm 0.5\ \text{(sys)\ MeV},
$ [@Krasznahorkay:2015iga] in correspondence of a ratio of Branching Ratios (BRs) obtained as (Xe\^+ e\^-) = 5.8 10\^[-6]{}. \[eq:BeAnomaly\] The signal appeared as a bump over the monotonically decreasing background from pure Quantum Electro-Dynamics (QED) interactions, i.e., internal pair creation via $\gamma^*\to e^+e^-$ splittings. This excess appeared only for symmetric energies of $e^+ e^-$, as expected from an on-shell non-relativistic particle. In addition, the opening angle of electron-positron pair and their invariant mass distributions presented the characteristics of an excess consistent with an intermediate boson. The measurements yielded the mentioned value $M_X$ from the invariant mass $m_{e^+ e^-}$, in correspondence of an angular excess around $\sim 135^\circ$, as shown in Fig. \[invmass\].
![Angular and invariant mass distributions of the internal conversion electron-positron pairs measured by the Atomki spectrometer (from [@Feng:2016ysn]).[]{data-label="invmass"}](invmass "fig:") -0.75cm
The best fit to data was obtained for a new particle interpretation, in which case the statistical significance of the excess is 6.8 sigma. [[The aforementioned result from the 17.64 MeV transition yielded $M_X=17.0\pm 0.5\, ({\rm stat})\pm 0.5\, ({\rm sys})$ as best fit, in correspondence of an angular peak around $155^\circ$ with ${\mathcal B}=4.0\times10^{-6}$. The corresponding significance is nowhere near discovery though.]{}]{}
Candidates for the new boson
============================
An explanation of the nature of the intermediate particle, $X$, decaying to electron-positron pairs, was attempted by considering it as boson either with spin zero (scalar or pseudoscalar) or with spin one (vector or axial-vector). We introduce all possible combinations in turn.
Scalar particle
---------------
If the intermediate particle $X$ is a scalar, $\phi$ ($J^P = 0^+$), then the decay $^8{\rm Be}^*(1^+) \to {}^8{\rm Be}(0^+) + \phi$ implies, due to angular momentum conservation, that $\phi$ should have $L=1$. Also, from parity conservation, it must have a parity equal to $(-1)^L$, which is $-1$ and this contradicts the assumption that $\phi$ is scalar with even parity. Therefore, one can conclude that a scalar intermediate particle is ruled out.
Pseudoscalar particle
---------------------
[[The situation is different if the intermediate particle is a pseudoscalar, $A$ ($J^P = 0^-$) [@Ellwanger:2016wfe]. In this case, given the quantum numbers of the $^8$Be$^*$ and $^8$Be states, the intermediate boson can indeed be a $J^P = 0^-$ pseudoscalar particle if it was emitted with $L = 1$ orbital momentum. It was in fact shown in Ref. [@Ellwanger:2016wfe] that $A$ can account for the Atomki results if its Yukawa couplings with the SM fermions are of order of the Yukawa couplings of the SM Higgs.]{}]{}
Vector particle
---------------
A neutral vector boson is the most common example considered for explaining this signal [@Feng:2016jff; @Feng:2016ysn; @Gu:2016ege; @Chen:2016dhm; @Liang:2016ffe; @Jia:2016uxs; @Kitahara:2016zyb; @Chen:2016tdz; @Seto:2016pks; @Neves:2016ugb; @Chiang:2016cyf]. It was emphasised that it can be a valid candidate if its coupling is constrained as $g' \sim 10^{-3}$.
Axial-Vector particle
---------------------
The pure axial-vector boson is also considered and it was shown that it can be a candidate if its coupling satisfies $g' \sim 10^{-4}$, as done in [@Kozaczuk:2016nma; @Feng:2016ysn; @Kahn:2016vjr]. The case of general spin-one boson, with no definite parity, [i.e.]{}, it is a mix of vector and axial-vector, could be a possible candidate after taking care of stringent constraints from atomic parity violation.
The couplings of these new light bosons with the SM particles remain an open question and subject to severe constraints from several experiments.
Experimental constraints on the pseudoscalar explanation
========================================================
The reduced couplings $\xi_q$ of a pseudoscalar $A$ to quarks is defined as $${\mathcal L}_{Aqq} = \xi_q \frac{m_q}{v} A \bar{q} i \gamma_5 q,$$ with $v\sim 246$ GeV. [[Assuming such fundamental interactions and adopting the nuclear shell model wave functions with definite isospin $T = 0$ of Ref. [@Ellwanger:2016wfe], one finds that $$\
\label{xiud}
\xi_u + \xi_d \approx 0.6$$ or, for $\xi_u=\xi_d\equiv \xi$, $\xi \approx 0.3$. Furthemore, if $A$ has Yukawa couplings to quarks and leptons which are proportional to the Yukawa couplings of the SM Higgs boson rescaled by generation independent factors $\xi_d \approx \xi_u \approx \xi_e$ (or $\xi_u \ll \xi_d$), and the Yukawa couplings to BSM fermions are not much larger than the electric charge $e$, $A$ has a BR of about 99% into $e^+ e^-$ and only about 1% into $\gamma\gamma$. Its total width is then dominated by $A\to e^+ e^-$ and given by $$\Gamma(A) = \xi_e^2 \frac{m_e^2}{8\pi v^2}M_A = \xi_e^2\cdot 2.9\times 10^{-15}\ \text{GeV}$$ for $M_A=17$ MeV. Its decay length is $$l_A=\frac{p_A}{M_A \Gamma(A)}\; .$$ For the decay $^8{{\rm Be}^*}\to{^8{\rm Be}}+A$ with $M(^8{{\rm Be}^*})-M(^8{\rm Be})=18.15$ MeV we obtain $$l_A \sim \frac{1}{\xi_e^2}\cdot 2.5\ \text{cm}.$$ (For $M_A=17.9$ MeV, $2\,\sigma$ above the central value in $M_X$ from the 18.15 MeV transition, we obtain $l_A \sim \frac{1}{\xi_e^2}\times 1.1\ \text{cm}$.) In order to explain the observed anomaly in the Atomki pair spectrometer experiment [@Krasznahorkay:2015iga], $l_A$ should then not be much larger than 1 cm leading to $$\label{xiegt1}
\xi_e {\gsim} 1\; ,$$ depending somewhat on the precise value of $M_A$. ]{}]{}
Light pseudoscalars are subject to constraints from searches for axions or axion-like particles. For recent summaries of constraints relevant for light pseudoscalars decaying dominantly into $e^+ e^-$, see [@Dolan:2014ska; @Andreas:2010ms; @Essig:2010gu; @Hewett:2012ns; @Dobrich:2015jyk]. However, since we allow for different Yukawa type couplings rescaled by $\xi_u$, $\xi_d$ and $\xi_e$ with respect to SM Higgs couplings, at least some experimental constraints studied therein have to be reconsidered. Constraints from $\pi^0\to \gamma + X$ from the NA48/2 experiment, which play a major role for the $Z'$ scenario [@Feng:2016jff; @Feng:2016ysn], do not apply here since the decay $\pi^0\to \gamma + A$ would violate parity.
Constraints also originate from flavour violating meson decays, analysed recently in [@Dolan:2014ska], and are mainly due to the following decays: $K^+ \to \pi^+ + A$ (constrained by the $K_{\mu 2}$ experiment [@Yamazaki:1984vg]), $K^+ \to \pi^+ + {\sl invisible}$ (measured by the experiments E787 [@Adler:2004hp] and BNL-E949 [@Artamonov:2009sz]), $B_s\to \mu^+\mu^-$ (measured by the LHCb collaboration [@Aaij:2013aka] and the CMS collaboration [@Chatrchyan:2013bka], see [@CMS:2014xfa] for a LHCb/CMS combination) and $B^0\to K^0_S + invisible$ (measured by CLEO [@Ammar:2001gi]). It turns out that the most stringent Flavour Changing Neutral Current (FCNC) constraint is due to $K^+ \to \pi^+ + A$ from the $K_{\mu 2}$ experiment [@Yamazaki:1984vg]. This process depends on a loop-induced $Asd$ vertex (with $W$ bosons and up-type quarks in the loop) which depends, in turn, on the couplings of $A$ to $d$- and $u$-type quarks. Constraints from [@Yamazaki:1984vg] can lead to $$\label{boundxid}
\xi_d \lsim 2\times 10^{-2}.$$ A similar constraint can be obtained from the process $B \to K +A$.
Constraints from searches for $K^+ \to \pi^+ + {\sl invisible}$ from E787 and BNL-E949 [@Adler:2004hp; @Artamonov:2009sz] apply only if $A$ decays outside the detectors, i.e., if $\xi_e$ is small enough. According to [@Andreas:2010ms], identifying now $C_{Aff}$ in [@Andreas:2010ms] with $\xi_e$, this is not the case for $\xi_e \gsim 0.3$.
According to [@Dolan:2014ska], the constraints from $B_s\to \mu^+\mu^-$ (through an off-shell $A$) rule out any $\xi \gsim 0.7$ which is weaker than the constraint from $K^+ \to \pi^+ + A$. Again, the loop contributions to the $Asb$ vertex considered in [@Dolan:2014ska] are incomplete within an Ultra-Violet (UV) complete extension of the Higgs sector, and could again be cancelled by additional BSM contributions as in the case of the $Asd$ vertex.
The constraints from $B^0\to K^0_S + invisible$ measured by CLEO [@Ammar:2001gi] apply only if the pseudoscalar $A$ produced in $B^0\to K^0_S + A$ decays outside the detector. Accordingly these constraints depend both on the BR$(B^0\to K^0_S + A)$, hence on the $Asb$ vertex or on $\xi_u,\xi_d$, and on the $A$ decay length which depends on $\xi_e$. These quantities are identified in [@Dolan:2014ska] where a limit $\xi \gsim 3.5$ on all flavours satisfies the constraints, since then the $A$ decay length becomes short enough despite the large production rate. Using this constraint only for $\xi_e$ is conservative, if $\xi_u,\xi_d < \xi_e$ is assumed.
Finally, $\xi_e \gsim 3.5$ satisfies also bounds on $A$ production in radiative $\Upsilon$ decays $\Upsilon \to \gamma + invisible$ interpreted as $\Upsilon \to \gamma + A$ from CLEO [@Balest:1994ch] and BaBar [@Aubert:2008as], which apply only if $A$ decays outside the detectors. For $M_A\sim 17$ MeV, following [@Andreas:2010ms], this is not the case for $\xi_e \gsim 1.5$.
Other important constraints on light pseudoscalars originate from beam dump experiments. From the Orsay experiment of Ref. [@Davier:1989wz], lifetimes $\tau_A$ in the range $5\times 10^{-12}\ \text{s} \lsim \tau_A \lsim 2\times 10^{-9}$ s are ruled out for $M_A\sim 17-18$ MeV. This has already been translated into constraints on a reduced pseudoscalar-fermion Yukawa coupling $C_{Aff}$ in [@Andreas:2010ms], where $C_{Aff} = \xi_e$ in our notation. Following [@Andreas:2010ms], $0.4 \lsim C_{Aff} \lsim 4$ is ruled out by this constraint. Since $\xi_e < 0.4$ is incompatible with , one is left with $$\label{orsay}
\xi_e {\gsim} 4\; .$$ This constraint leads automatically to the satisfaction of the lower bound $\xi_e \gsim 3.5$ from $B^0\to K^0_S +\ invisible$, as well as to a short enough decay length for the Atomki pair spectrometer experiment. It is also compatible with the exclusion from the NA64 experiment [@Banerjee:2018vgk] provided that $\xi_e \lesssim 15$.
Another potentially relevant experiment is the proton beam dump on copper CHARM experiment [@Bergsma:1985qz]. In [@Bergsma:1985qz] constraints were derived assuming that the production cross section and decay length of light pseudoscalars correspond to those of axions, which is not the case here. Relevant is the analysis in [@Dolan:2014ska] which uses the production of light pseudoscalars in $K\to \pi + A$ and $B \to X + A$ decays. For universally rescaled Yukawa couplings the region $\xi \gsim 1$ satisfies the constraints, since then the decay length of $A$ is too short to reach the decay region of the CHARM experiment. This constraint does not supersede the one in eq. .
Explanation of the Beryllium anomaly with a Pseudoscalar
========================================================
One of the less well studied solutions is that of the pseudoscalar, but this has been done in [@Ellwanger:2016wfe]. It was initially dismissed by [@Feng:2016jff; @Feng:2016ysn] and subsequent authors by the argument that for such axion-like pseudoscalars $A$, fermion loops generate couplings of the form $g_{A\gamma\gamma} A F^{\mu\nu}(\gamma)\tilde{F}_{\mu\nu}(\gamma)$ which are strongly constrained by axion searches. However, light pseudoscalars in this mass range with tree level Yukawa couplings to electrons decay dominantly into electron-positron pairs, unless Yukawa couplings to other charged fermions $f$ with mass $m_f$ are much larger than $m_f/m_e$ compensating $g_{A\gamma\gamma}\approx 1/(8\pi m_f)$. For solutions to the Atomki anomaly, we require such couplings to electrons and hence one should dismiss the pseudoscalar solution.
To summarise the previous section investigating the constraints, couplings of the form $\xi _u + \xi _d \sim 0.6$ and $\xi_e > 4$ should satisfy all aforementioned constraints and provide an explanation to the Atomki anomaly, with the caveat that FCNCs must be suppressed by loop contributions at the level of at least $10\%$.
[[Ultimately, it will be the Atomki experiment itself which will be in a position to either confirm or disprove the light pseudoscalar hypothesis. In fact, the experiment is currently planning to study the $\gamma\gamma$ decays of the 17 MeV particle, also in $4He\to\gamma\gamma$ [@Krasznahorkay:2017bwh], in order to distinguish between a vector boson and pseudoscalar boson scenario. According to the Landau-Yang theorem, the (on-shell) decay of a vector boson by double $\gamma$-emission is forbidden, however, the decay of a pseudoscalar one is allowed [@Moretti:2014rka]. The angular correlation of the $\gamma$-rays will be measured by using 15 large ($3"\times3"$) LaBr$_3$ detectors. If the $A$ boson with a mass of 17 MeV is created in the decay of the $J^P=0^-$ state and in turn decays into two $\gamma$-rays, their angular correlation $\theta$ should peak at $$\cos\theta = 1-\frac{M^2_A}{2E_{\gamma}E_{\gamma^\prime}},$$ where $M_A$ is the mass of the $A$ boson (17 MeV) and $E_{\gamma,\gamma^\prime}$ are the energies of the two photons. However, it should be kept in mind that a light pseudoscalar with tree level coupling to electrons would have a loop-induced BR to di-photons of only one percent or so, hence hardly visible with current Atomki data sets. At any rate, results in this respect, are eagerly awaited.]{}]{}
Experimental constraints on the spin-1 explanation {#sec:constraints}
===================================================
Let us assume that the generic coupling of a new vector boson, $Z'$, to the SM fermions is given by the following interaction Lagrangian -[L\_]{} = Z’\_\_f |\_f \^(C\_[f,V]{}+ \_5 C\_[f,A]{}) \_f .
Experimental constraints on the lepton couplings {#experimental-constraints-on-the-lepton-couplings .unnumbered}
------------------------------------------------
We have not seen a $Z'$ in the electron beam dump experiment SLAC E141. Therefore, a $Z'$ has not been produced, hence C\_[e,V]{}\^2 + C\_[e,A]{}\^2 < 10\^[-17]{} or, else, a $Z'$ has been caught in the dump, hence 3.7 10\^[-9]{}. We have not seen a $Z'$ either in the electron beam dump experiment NA64 [@Banerjee:2018vgk]. If a $Z'$ has been caught in the dump, this places the (stronger than E141) condition 1.6 10\^[-8]{} . \[eq:NA64\]
The parity-violating M[ø]{}ller scattering measured at the SLAC E158 experiment [@Anthony:2005pm] imposes a constraint on the product $C_{e,V} C_{e,A}$ of the $Z'$, namely C\_[e,V]{} C\_[e,A]{} 10\^[-8]{}, for $M_{Z'} \simeq 17$ MeV [@Kahn:2016vjr].
Furthermore, there could be contributions of a $Z'$ to the magnetic moments of electron and muon. The one-loop ones $\delta a_{l}$, mediated by a $Z'$, lead to a\_[l]{}= , where $r_{m_l} \equiv (m_l/M_{Z'})^2$ and $g_V,g_A$ are given by$$\begin{aligned}
g_V(r)= \int^1_0 dz\frac{z^2(1-z)}{1-z+r z^2}, \qquad
g_A(r)= \int^1_0 dz\frac{(z-z^2)(4-z) + 2r z^3}{1-z+r z^2} \,.\end{aligned}$$ The light boson contribution to the anomalous magnetic moment of the electron is required to be within the $2\sigma$ uncertainty of the departure of the SM prediction from the experimental result [@Giudice:2012ms]. Concerning the muon anomalous magnetic moment [@Altmannshofer:2016brv], which has been measured at Brookhaven National Laboratory (BNL) to a precision of $0.54$ parts per million, the current average of the experimental results is given by [@Bennett:2006fi; @Blum:2013xva; @Lindner:2016bgg] a\^[exp]{}\_=11 659 208.9(6.3)10\^[-10]{}, which is different from the SM prediction by $3.3\sigma$ to $3.6\sigma$: $
\Delta a_{\mu}=a^{\rm exp}_{\mu}-a^{\rm SM}_{\mu}=(28.3 \pm
8.7\ {\rm to}\ 28.7 \pm
8.0)\times 10^{-10}.
$ We require again that the contribution of a $Z'$ to $(g-2)_\mu$, which is mainly due to its axial-vector component, is less than the $2\sigma$ uncertainty of the discrepancy between the SM result and the experimental measure. For $M_{Z'} \simeq 17$ MeV, one then finds a\_[e]{} &=& 7.6 10\^[-6]{} C\_[e,V]{}\^2 -3.8 10\^[-5]{} C\_[e,A]{}\^2 -10.5(8.1) 10\^[-13]{},\
a\_ &=&0.009 C\_[,V]{}\^2 -C\_[,A]{}\^2 2.9 (90) 10\^[-9]{}.
Electron-positron colliders (like KLOE2) would be sensitive to a new spin-1 gauge boson via the channel $e^+ e^- \to \gamma, Z, Z' \to e^+ e^-$. From this process one finds (C\_[e,V]{}\^2 + C\_[e,A]{}\^2) [BR]{}(Z’ e\^+ e\^-) 3.7 10\^[-7]{}.
Similarly, $Z'$ contributions to neutrino-electron scattering implies a bound on the product of the electron and neutrino couplings to the $Z'$ [@Deniz:2009mu; @Bilmis:2015lja].
Experimental constraints on the quark couplings {#experimental-constraints-on-the-quark-couplings .unnumbered}
-----------------------------------------------
The couplings of a light $Z'$ state with quarks are, in general, strongly constrained from $\pi^0 \to Z' +\gamma$ searches at the NA48/2 experiment [@Raggi:2015noa]. The process is proportional to the anomaly factor $ N_\pi = \frac{1}{2} (2 C_{u,V} + C_{d,V})^2$. Therefore, one gets the following limit: 2 C\_[u,V]{} + C\_[d,V]{} for $M_{Z'} \simeq 17$ MeV. The contribution of the axial components is induced by chiral symmetry breaking effects and is, therefore, suppressed by the light quark masses.
Furthermore, atomic parity violation in Cesium (Cs) must be considered. In fact, very strong constraints on a light $Z'$ can be extracted from the measurement of the effective weak charge of the Cs atom [@Davoudiasl:2012ag; @Bouchiat:2004sp]: Q\_w = C\_[e,A]{} ( ) 0.71 at $2\sigma$ [@Porsev:2009pr].
A $U(1)^\prime$ extension of the SM with a light and weakly interacting $Z'$
============================================================================
We consider a generic extension to the SM described by a new Abelian group $U(1)'$ [@Fayet:1980rr; @Fayet:1980ad; @Fayet:1990wx; @Fayet:2007ua; @Fayet:2008cn; @Fayet:2016nyc; @DelleRose:2017xil]. Due to the presence of two such Abelian symmetries, $U(1)_{Y} \times U(1)'$, the most general kinetic Lagrangian of the corresponding fields, $\hat B_\mu$ and $\hat B'_\mu$, allows for a gauge invariant operator mixing the two field strengths. In particular, the quadratic Lagrangian for the two gauge fields is given by \[eq:lag\_mixing\] L\_ = - F\_ F\^ - F’\_ F\^[’]{} - F’\_ F\^, with $\kappa$ being the kinetic mixing parameter. Since the parameterisation above may be inconvenient for practical computations, it is often useful to recast the kinetic Lagrangian into a diagonal form by removing the mixing operator through a rotation and rescaling of the Abelian fields. This transformation, while diagonalising Eq. (\[eq:lag\_mixing\]), introduces a non-diagonal term in the interactions such that the covariant derivative may be written as D\_ = \_+ …+ i g\_1 Y B\_+ i (g Y + g’ z) B’\_, where $Y$ and $z$ are, respectively, the hypercharge and the $U(1)'$ charge, and $B_\mu,B'_\mu$ are the rotated fields. The parameter $\tilde g$ replaces $\kappa$ and describes the mixing between the two Abelian groups while $g'$ is the usual gauge coupling associated to the extra Abelian symmetry $U(1)'$.
Due to the mixing term in the gauge covariant derivative, after spontaneous symmetry breaking, the EW Vacuum Expectation Value (VEV) contributes to the $U(1)'$ breaking even if the Higgs sector is neutral under the new Abelian symmetry. For instance, in a scenario with only one Higgs doublet, the neutral gauge boson mass matrix can be extracted from the Higgs Lagrangian and reads as - L\_ = (g\_2 W\_\^3 - g\_1 B\_- g\_B’\_)\^2 + B\_\^[’2]{} + …, where $g_\Phi = \tilde g + 2 z_\Phi g'$ with $z_\Phi$ being the $U(1)'$ charge of the SM Higgs or a combination of charges in multi-Higgs doublet scenarios. As stated above, a non-vanishing $g_\Phi$ can be achieved either by the non-zero $U(1)'$ charges of the Higgs sector, $z_\Phi \neq 0$, or by the presence of the kinetic mixing $\tilde g \neq 0$. Both of them contribute to a $Z - Z'$ mass mixing. The mass term $m_{B'}^2$ represents a possible source for the $Z'$ mass from a SM neutral sector. This can be realised, for instance, by the VEV $v'$ of a SM-singlet complex scalar $\chi$, with a $z_\chi$ charge under $U(1)'$. In this case $m_{B'} = g' z_\chi v'$. We remark here that, for our purposes, it is not necessary to specify the origin of the $B'$ mass term and other mechanisms, beside Spontaneous Symmetry Breaking (SSB) with a complex scalar, can be also envisaged. Moreover, the mixing in the neutral gauge sector is only triggered by the $g_\Phi$ parameter and, as such, is unaffected by the details of the scalar sector in which the $B'$ mass term is generated.
The diagonalisation of the mass matrix provides the relation between the interaction and the mass eigenstates and is described by the rotation matrix (
[c]{} B\^\
W\_3\^\
B’\^
) = (
[ccc]{} \_w & - \_w ’ & \_w ’\
\_w & \_w ’ & - \_w ’\
0 & ’ & ’
) (
[c]{} A\^\
Z\^\
Z’\^
) where $\theta_w$ is the usual weak mixing angle and $\theta'$ is a new mixing angle, with $-\pi/4 \le \theta' \le \pi/4$, defined as [@Accomando:2016sge] \[eq:mixing1Higgs\] 2 ’ = , where $g_Z = \sqrt{g_1^2 + g_2^2}$ is the EW coupling, $g_\Phi = \tilde g + 2 z_\Phi g'$ and $g_{\Phi^2} = g_{\Phi}^2$. The masses of the $Z$ and $Z'$ gauge bosons are then given by \[eq:ZZpmass1Higgs\] M\_[Z,Z’]{} = g\_Z \^. For a light and weakly interacting $Z'$, namely $g', \tilde g \ll g_Z$ and $m_{B'}^2 \ll v^2$, the mixing angle and the masses can be expanded at leading order as \[eq:expansion\] M\_Z\^2 g\_Z\^2 v\^2 , M\_[Z’]{}\^2 m\_[B’]{}\^2, 2 ’ - 2 . While the SM $Z$ mass is correctly reproduced by the EW VEV, the mass of the $Z'$ is controlled by the $m_{B'}$ parameter or, equivalently, by the VEV $v'$ of the SM-singlet $\chi$ which is then given by $v' = M_{Z'}/(g' z_\chi)$. The $Z'$ massless limit for $m_{B'} = 0$ is naively expected since if SSB is turned off in the scalar sector, no scalar degrees of freedom can provide the longitudinal component of a massive $Z'$. For a 17 MeV $Z'$ with $g' \sim 10^{-3}$ the VEV of $\chi$ is $v' \sim 10$ GeV.
The expansions in Eq. (\[eq:expansion\]) are applicable if the Higgs sector is populated by only one $SU(2)$ doublet, as in the SM. This assumption can be obviously relaxed and more Higgs doublets can be implemented. We show, indeed, in the following sections that this possibility leads to an interesting phenomenology in the $Z'$ sector and provides alternative solutions to the $\Be$ anomaly.
For instance, in a scenario with two $SU(2)$ doublet scalars, $\Phi_1$ and $\Phi_2$ with the same hypercharge $Y=1/2$ and two different charges $z_{\Phi_1}$ and $z_{\Phi_2}$ under the extra $U(1)'$, the diagonalisation of the neutral gauge mass matrix is obtained through the mixing angle $\theta'$ in Eq. (\[eq:mixing1Higgs\]) with g\_&=& (g + 2 g’ z\_[\_1]{}) \^2 + (g + 2 g’ z\_[\_2]{}) \^2 ,\
g\_[\^2]{} &=& (g + 2 g’ z\_[\_1]{})\^2 \^2 + (g + 2 g’ z\_[\_2]{})\^2 \^2 . The angle $\beta$ is defined as usual as $\tan \beta = v_2/v_1$ with $v^2 = v_1^2 + v_2^2$. In the small coupling limit the $Z'$ mass is given by M\_[Z’]{}\^2 m\_[B’]{}\^2 + \^2 ( z\_[\_1]{} - z\_[\_2]{} )\^2 \^2(2 ), which, differently from the previous case, is non-vanishing even when $m_{B'} \simeq 0$ due to mismatch between $z_{\Phi_1}$ and $z_{\Phi_2}$. In the limit in which there is no contribution from the dark scalar sector, one finds for $M_{Z'} \simeq 17$ MeV and $v \simeq 246$ GeV, $\tilde g \sim g' \sim 10^{-4}$. Interestingly, as we will show below, the same order of magnitude of the gauge couplings is required to explain the $\Be$ anomaly with a $Z'$ gauge boson characterised by axial-vector couplings.
In summary, for the case of one Higgs doublet, we showed that the limit $m_{B'} \ll v$ leads to $M_{Z'} \simeq m_{B'}$ with the SM Higgs sector playing no role in the generation of the $Z'$ mass. In contrast, in a multi-Higgs scenario, like in a 2-Higgs Doublet Model (2HDM), if $z_{\Phi_1} \neq z_{\Phi_2}$, the symmetry breaking of the $U(1)'$ can actually be realised without any contribution from the dark scalar sector, namely with $v' = 0$. In fact, the longitudinal degree of freedom of the $Z'$ is provided by the typical CP-odd state of the 2HDM spectrum which, differently from standard constructions, is characterised by a missing pseudoscalar field among the physical states. Before moving to this 2HDM realisation, though, we ought to discuss the $Z'$ interactions with the SM fermions emerging from the present construct.
The $Z'$ interactions with the SM fermions
------------------------------------------
The interactions between the SM fermions and the $Z'$ gauge boson are described by the Lagrangian $\mathcal L_\textrm{int} = - J^\mu_{Z'} Z'_\mu$ where the gauge current is given by J\^\_[Z’]{} = \_f |\_f \^( C\_[f, L]{} P\_L + C\_[f, R]{} P\_R ) \_f with coefficients C\_[f,L]{} &=& - g\_Z s’ ( T\^3\_f - s\_w\^2 Q\_f ) + (g Y\_[f, L]{} + g’ z\_[f, L]{}) c’ ,\
C\_[f,R]{} &=& g\_Z s\_w\^2 s’ Q\_f + (g Y\_[f, R]{} + g’ z\_[f, R]{}) c’ . In the previous equations we have adopted the shorthand notation $s_w \equiv \sin \theta_w$, $c_w \equiv \cos \theta_w$, $s' \equiv \sin \theta'$ and $c' \equiv \cos \theta'$ and introduced $Y_f$ the hypercharge, $z_f$ the $U(1)'$ charge, $T^3_f$ the third component of the weak isospin and $Q_f$ the electric charge. Analogously, the vector and axial-vector components of the $Z'$ interactions are [@DelleRose:2017xil] C\_[f, V]{} &=& = ,\
C\_[f, A]{} &=& = , The vector and axial-vector coefficients simplify considerably in the limit $g', \tilde g \ll g_Z$. By noticing that $s' \simeq - g_\Phi/g_Z$, we get \[eq:CVA\_expanded\] C\_[f, V]{} && g c\_w\^2 Q\_f + g’ ,\
C\_[f, A]{} && g’ , where we have introduced the vector and axial-vector $U(1)'$ charges $z_{f,V/A} = 1/2(z_{f,R} \pm z_{f,L})$ and $z_\Phi$ can be either the $U(1)'$ charge of the Higgs or $z_{\Phi_1} \cos^2 \beta + z_{\Phi_2} \sin^2 \beta$ in a 2HDM scenario.
The $Z'$ couplings are characterised by the sum of three different contributions. The kinetic mixing $\tilde g$ induces a vector-like term proportional to the Electro-Magnetic (EM) current which is the only source of interactions when all the SM fields are neutral under $U(1)'$. In this case the $Z'$ is commonly dubbed *dark photon*. The second term is induced by the $z_\Phi$, the $U(1)'$ charge in the Higgs sector, and leads to a *dark Z*, namely a gauge boson mixing with the SM $Z$ boson. Finally there is the standard gauge interaction proportional to the fermionic $U(1)'$ charges $z_{f,V/A}$. We can delineate different scenarios depending on the structure of the axial-vector couplings of the $Z'$ boson. In particular, the $C_{f,A}$ coefficients can be suppressed with respect to the vector-like counterparts (see also [@Kahn:2016vjr]). This is realised, for instance, when only one $SU(2)$ doublet is considered and the gauge invariance of the Yukawa Lagrangian under the new Abelian symmetry is enforced. Indeed, the latter requires the $U(1)'$ charge of the Higgs field to satisfy the conditions \[eq:yukawa\_gaugeinv\] z\_= z\_[Q]{} - z\_[d]{} = - z\_[Q]{} + z\_[u]{} = z\_L - z\_e . Inserting the previous relations into Eq. (\[eq:CVA\_expanded\]), we find $C_{f, A} \simeq 0$ which describes a $Z'$ with only vector interactions with charged leptons and quarks. We stress again that the suppression of the axial-vector coupling is only due to the structure of the scalar sector, which envisions only one $SU(2)$ doublet, and the gauge invariance of the Yukawa Lagrangian. This feature is completely unrelated to the $U(1)'$ charge assignment of the fermions, the requirement of anomaly cancellation and the matter content potentially needed to account for it.
In contrast, in the scenario characterised by two Higgs doublets, the axial-vector couplings of the $Z'$ are, in general, of the same order of magnitude of the vector ones and the cancellation between the two terms of $C_{f, A}$ in Eq. (\[eq:CVA\_expanded\]) is not achieved regardless of the details of the Yukawa Lagrangian (such as which type 2HDM). The same result can be achieved if a single Higgs doublet is considered but the conditions in Eq. (\[eq:yukawa\_gaugeinv\]) are not satisfied as in scenarios in which the fermion masses are generated radiatively or through horizontal symmetries.
To summarise, we can identify three different situations that can provide a light $Z'$ with interactions potentially explaining the Beryllium anomaly. In all of them, the SM is extended by an additional Abelian gauge group.
$1.$ The SM scalar sector is unchanged, being characterised by only one Higgs doublet. In this case the mass of the $Z'$ is entirely generated in the dark sector. The Yukawa Lagrangian preserves the SM structure and its gauge invariance under the $U(1)'$ necessary implies that the $Z'$ has only vector interactions with the SM fermions at leading order in the couplings $\tilde g, g'$. $2.$ The SM scalar sector is extended by an additional Higgs doublet. Even though the Yukawa Lagrangian is invariant under the local $U(1)'$ symmetry, the cancellation between the two terms in $C_{f,A}$ in Eq. (\[eq:CVA\_expanded\]) does not occur and both the vector and axial-vector couplings of the $Z'$ are non-vanishing. The mass of the $Z'$ acquires contribution from both the dark and the EW sectors. $3.$ The SM scalar sector is characterised by a single Higgs doublet but the constraints in Eq. (\[eq:yukawa\_gaugeinv\]) are avoided by relying on more complicated Yukawa structures. As such, the cancellation providing $C_{f,A} \simeq 0$ is not realised and the vector and axial-vector interactions of the $Z'$ are of the same order of magnitude.
We will discuss the three scenarios in the following sections focusing on their implications in the interpretation of the $\Be$ anomaly.
Before concluding this section we briefly go through the conditions required by the cancellation of gauge and gravitational anomalies which strongly constrain the charge assignment of the SM spectrum under the extra $U(1)'$ gauge symmetry. These conditions can be eventually combined with the requirement of gauge invariance of the Lagrangian responsible for the generation of the fermion masses which may also involve non-renormalisable operators. We will also allow for extra SM-singlet fermions which can be easily interpreted as right-handed neutrinos. We assign the charges $z_Q$ and $z_L$ for the $SU(2)$ quark and lepton doublets, $z_u, z_d, z_e$ for the corresponding right-handed components and $z_{\nu}$ for the $n_R$ right-handed neutrinos. We obtain the following gauge and gravitational anomaly cancellation conditions: $$\begin{aligned}
\label{eq:anomaly}
& U(1)'SU(3)SU(3): & \sum_i^{3} (2 z_{Q_i} - z_{u_i} - z_{d_i}) = 0 \,, {\tilde{\nu}}\\
& U(1)'SU(3)SU(3): & \sum_i^{3} \, ( 3 z_{Q_i} + z_{L_i}) = 0 \,, {\tilde{\nu}}\\
& U(1)'U(1)_YU(1)_Y: & \sum_i^{3} \left( \frac{z_{Q_i}}{6} - \frac{4}{3} z_{u_i} - \frac{z_{d_i}}{3} + \frac{z_{{L_i}}}{2} - z_{e_i}\right) = 0 \,, {\tilde{\nu}}\\
& U(1)'U(1)'U(1)_Y: & \sum_i^{3} \left( z_{Q_i}^2 - 2 z_{u_i}^2 + z_{d_i}^2 - z_{{L_i}}^2 + z_{e_i}^2 \right) = 0 \,, {\tilde{\nu}}\\
& U(1)'U(1)'U(1)': & \sum_i^{3} \left( 6 z_{Q_i}^3 - 3 z_{u_i}^3 - 3 z_{d_i}^3 + 2 z_{{L_i}}^3 - z_{e_i}^3 \right) + \sum_i^{n_R} z_{\nu _i} = 0 \,, {\tilde{\nu}}\\
& U(1)'GG: & \sum_i^{3} \left( 6 z_{Q_i} - 3 z_{u_i} - 3 z_{d_i} + 2 z_{{L_i}} - z_{e_i} \right) + \sum_i^{n_R} z_{\nu _i} = 0 .
\end{aligned}$$ A simple solution is found for instance in the family universal case with $n_R = 3$ and $z_{\nu_{i}} = z_{\nu}$ and it is defined in terms of only two $U(1)'$ charges, $z_Q$ and $z_u$ as shown in Tab. \[tab:charges\]. As an example, the $U(1)_{B-L}$ is reproduced by $z_Q = z_u = 1/3$ while the sequential $U(1)'$ is obtained for $z_Q = 1/6$ and $z_u = 2/3$.
$SU(3)$ $SU(2)$ $U(1)_Y$ $U(1)'$
--------- --------- --------- ---------- -----------------
$Q_L$ 3 2 1/6 $z_Q$
$u_R$ 3 1 2/3 $z_u$
$d_R$ 3 1 -1/3 $2 z_Q - z_u$
$L$ 1 2 -1/2 $-3 z_Q$
$e_R$ 1 1 -1 $-2 z_Q - z_u$
$\nu_R$ 1 1 0 $- 4 z_Q + z_u$
: Family universal charge assignment in the $U(1)'$ extension of the SM. \[tab:charges\]
$Z'$ with vector couplings {#sec:ZpVector}
--------------------------
The simplest $U(1)'$ extension of the SM, which may account for an extra neutral light gauge boson potentially explaining the $\Be$ anomaly, is characterised by a single Higgs doublet. As already explained above, the gauge invariance of the Yukawa interactions fixes the $U(1)'$ charge of the Higgs to satisfy the restrictions in Eq. (\[eq:yukawa\_gaugeinv\]) thus leading to a suppression of the $Z'$ axial-vector couplings to the quarks and charged leptons with respect to the vector ones.\
In this scenario, the anomalous internal pair creation transition of the excited stated of the Beryllium described by the normalised BR is given by = (C\_[p, V]{} + C\_[n, V]{})\^2 in which any dependence from the nuclear matrix elements factors out in the ratio of BRs. Moreover, the partial decay width of the $Z'$ into SM fermions is \[eq:Zpdecaywidth\] (Z’ f |f) = . Since $M_{Z'} \simeq 17$ MeV, the light $Z'$ can only decay into electrons and active neutrinos (assuming heavier right-handed neutrinos, if any).\
While $C_{f, A} \simeq 0$, the explicit expressions of the vector couplings of the $Z'$ are C\_[p, V]{} &=& g c\_w\^2 - 2 g’ z\_H s\_w\^2 + g’ (z\_H + 3 z\_Q) ,\
C\_[n, V]{} &=& - g’ ( z\_H - 3 z\_Q ) ,\
C\_[e, V]{} &=& - g c\_w\^2 + 2 g’ z\_H s\_w\^2 - g’ (z\_H - z\_L),\
C\_[, V]{} &=& - C\_[, A]{} = (z\_H + z\_L) , where we have introduced the proton and neutron couplings $C_{p,V} = 2 C_{u,V} + C_{d,V}$, $C_{n,V} = C_{u,V} + 2 C_{d,V}$ and we have exploited the gauge invariance of the Yukawa Lagrangian. Moreover, the cancellation of the anomaly in the $U(1)'SU(2)SU(2)$ triangle diagram given in Eq. (\[eq:anomaly\]) leads to $3 z_Q + z_L = 0$, namely $C_{\nu,V} = - 2 C_{n,V}$.\
The acceptable range of couplings is [@Feng:2016jff; @Feng:2016ysn] |C\_[p, V]{}| && 1.2 10\^[-3]{} e ,\
|C\_[n, V]{}| &=& (2 - 10) 10\^[-3]{} e ,\
|C\_[e, V]{}| &=& (0.2 - 1.4) 10\^[-3]{} e ,\
&& 3 10\^[-4]{} e, where $\textrm{BR}(Z' \to e^+ e^-) = 1$ has been assumed. The first two conditions ensure that the Atomki anomaly is correctly reproduced while avoiding, at the same time, the strong constraint from the $\pi^0 \to Z' \gamma$ decay. As the coupling to proton is smaller than the corresponding one to neutron, the $Z'$ realising this particular configuration has been dubbed *protophobic*. The bound on the electron coupling is mainly obtained from KLOE2, $(g-2)_e$ and beam dump experiments, while the neutrino coupling is constrained by neutrino scattering off electrons at the Taiwan EXperiment On Neutrinos (TEXONO) [@Deniz:2009mu]. Reinterpreting the bounds obtained in [@Bilmis:2015lja], where a $B-L$ scenario without mixing has been considered, for a general vector-like $Z'$, one can show that the $C_\nu$ coupling must be much smaller than the typical value of $C_{n,V}$ required to explain the $\Be$ anomaly, thus invalidating the $C_{\nu,V} = - 2 C_{n,V}$ condition required by the consistency of this simple model.
A possible way to suppress the neutrino coupling, without affecting the neutron one, could be to invoke the presence of additional neutral fermionic degrees of freedom, charged under the $U(1)'$ symmetry and mixed to the left-handed neutrinos, so that the effective coupling of the $Z'$ to the physical neutrino mass eigenstate would be significantly reduced. This mixing is commonly realised in the seesaw mechanism, which is naturally envisaged in the Abelian extension considered here since right-handed neutrinos are required to cancel the gauge anomalies, but it can hardly account for the bounds determined by the neutrino-electron scattering experiments. Such a strategy has been discussed in [@Feng:2016ysn], however, here we show two alternative solutions based on the exploitation of the $Z'$ axial-vector interactions.
Explanation of the Beryllium anomaly with a family universal $U(1)'$
====================================================================
In this section we investigate the explanation of the Atomki anomaly in a scenario characterised by an extra $U(1)'$ model and two Higgs doublets.
One possibility studied as a solution to the Atomki anomaly is a well-known realisation of the scalar potential and Yukawa interactions with two scalar doublets is the so-called type-II in which the up-type quarks couple to one Higgs (conventionally chosen to be $\Phi_2$) while the down-type quarks couple to the other ($\Phi_1$). The constraint from anomaly cancellation arising from the $U(1)'SU(3)SU(3)$ triangle diagram together with the gauge invariance of the Yukawa Lagrangian require $2 z_Q - z_d - z_u = z_{\Phi_1} - z_{\Phi_2} = 0$, in the type-II scenario. In order to satisfy this condition with $z_{\Phi_1} \neq z_{\Phi_2}$, extra coloured states are necessarily required which will bring additional terms into the anomaly cancellation conditions and so the equation above will be modified and no longer require equal Higgs charges under the new gauge group. These states must be vector-like under the SM gauge group and chiral under the extra $U(1)'$. This option has been explored in detail in [@Kahn:2016vjr]. In this work, the constraints on new vector bosons with axial vector couplings in a family-universal scenario which includes extra coloured states to cancel anomaly conditions is considered. In this review focus on a different, more minimal scenario than this, which does not require additional states, but modifies the scalar theory to affect the condition of anomaly cancellation.
The gauge invariance condition above is modified when the scalar sector reproduces the structure of the type-I 2HDM in which only one ($\Phi_2$) of the two Higgs doublets participates in the Yukawa interactions. In this theory, the corresponding Lagrangian is the same as the SM one and its gauge invariance simply requires $z_{\Phi_2} = - z_Q + z_u = z_{Q} - z_d = z_{L} - z_e$, without constraining the $U(1)'$ charge of $\Phi_1$, in the type-I scenario. In this way, we allow for gauge invariance even when $z_{\Phi _1} \neq z_{\Phi _2}$. Differently from the type-II scenario in which extra coloured states are required to build an anomaly-free model, in the type-I case the UV consistency of the theory can be easily satisfied introducing only SM-singlet fermions as demanded by the anomaly cancellation conditions of the $U(1)'U(1)'U(1)'$ and $U(1)'GG$ correlators. Nevertheless, the mismatch between $z_\Phi$ and $z_{f,A}=\pm z_{\Phi_2}/2$ (for up-type and down-type quarks, respectively) prevents $C_{f,A}$ to be suppressed and the $Z'$ interactions are given by [@DelleRose:2017xil], $$\begin{aligned}
&C_{u, V} = \frac{2}{3} \tilde g c_w^2 + g' \left[ z_\Phi \left(\frac{1}{2} - \frac{4}{3} s_w^2 \right) + z_{u,V} \right], {\tilde{\nu}}\\
&C_{u, A} = - \frac{g'}{2} \cos^2 \beta (z_{\Phi_1} - z_{\Phi_2}) \,, {\tilde{\nu}}\\
&C_{d, V} = -\frac{1}{3} \tilde g c_w^2 + g' \left[ z_\Phi \left(-\frac{1}{2} + \frac{2}{3} s_w^2 \right) + z_{d,V} \right], {\tilde{\nu}}\\
&C_{d, A} = \frac{g'}{2} \cos^2 \beta (z_{\Phi_1} - z_{\Phi_2}) \,, {\tilde{\nu}}\\
&C_{e, V} = - \tilde g c_w^2 + g' \left[ z_\Phi \left(-\frac{1}{2} + 2 s_w^2 \right) + z_{e,V} \right], {\tilde{\nu}}\\
&C_{e, A} = \frac{g'}{2} \cos^2 \beta (z_{\Phi_1} - z_{\Phi_2}) \,, {\tilde{\nu}}\\
&C_{\nu, V} = - C_{\nu, A} = \frac{g'}{2} \left( z_{\Phi} + z_L \right) .
\label{couplings}\end{aligned}$$
As pointed out in [@Feng:2016ysn], the contribution of the axial-vector couplings to the $\Be^* \rightarrow \Be \, Z'$ decay is proportional to $|\vec{k}_{Z'}|/M_{Z'} \ll 1$, where $\vec{k}_{Z'}$ is the momentum of the $Z'$, while the vector component is suppressed by $|\vec{k}_{Z'}|^3/M_{Z'}^3$. Therefore, in our case, being $C_{f,V} \sim C_{f,A}$, we can neglect the effects of the vector couplings of the $Z'$ and their interference with the axial counterparts. For a $Z'$ with only axial-vector couplings to quarks, the transition $\Be^* \rightarrow \Be \, Z'$ is described by the partial width [@Kozaczuk:2016nma] = ( 2 + ) | a\_n 0 || \^n || 1 + a\_p 0 || \^p || 1 |\^2, where the neutron and proton coefficients $a_n = (a_0-a_1)/2$ and $a_p = (a_0+a_1)/2$ are defined in terms of a\_0 &=& ( C\_[u,A]{} + C\_[d,A]{}) ( u\^[(p)]{} + d\^[(p)]{} ) + 2 C\_[s,A]{} s\^[(p)]{} ,\
a\_1 &=& ( C\_[u,A]{} - C\_[d,A]{}) ( u\^[(p)]{} - d\^[(p)]{} ), with $\Delta u^{(p)} = 0.897(27)$, $\Delta d^{(p)} = -0.367(27)$ and $\Delta s^{(p)} = - 0.026(4)$ [@Bishara:2016hek]. The reduced nuclear matrix elements of the spin operators have been computed in [@Kozaczuk:2016nma] and are given by $\langle 0 || \sigma^n || 1 \rangle = -0.132 \pm 0.033$, $\langle 0 || \sigma^p || 1 \rangle = -0.047 \pm 0.029$ for the isoscalar $\Be^* \rightarrow \Be$ transition and $\langle 0 || \sigma^n || 1 \rangle = -0.073 \pm 0.029$, $\langle 0 || \sigma^p || 1 \rangle = 0.102 \pm 0.028$ for the isovector $\Be^{*'} \rightarrow \Be$ transition.
Notice that the axial couplings of the quarks and, therefore, the width of the $\Be^* \rightarrow \Be \, Z'$ decay are solely controlled by the product $g' \cos^2 \beta$ while the kinetic mixing $\tilde g$ only affects the $\textrm{BR}(Z' \rightarrow e^+e^-)$ since the $Z' \rightarrow \nu \nu$ decay modes are allowed (we assume that the $Z' \rightarrow \nu_R \nu_R$ decays are kinematically closed). For definiteness, we consider a $U(1)_\textrm{B-L}$ charge assignment with $z_{Q_{L}} =z_{u_R}=1/3$, with other charges defined using Tab. \[tab:charges\], and $z_{\Phi_2} = 0$, $z_{\Phi_1} = 1$ and $\tan \beta = 1$. Analogue results may be obtained for different $U(1)'$ charge assignments and values of $\tan \beta$. We show in Fig. \[fig:typeI\] the parameter space explaining the Atomki anomaly together with the most constraining experimental results.
The orange region, where the $Z'$ gauge couplings comply with the best-fit of the $\Be^*$ decay rate in the mass range $M_{Z'} = 16.7 \, {\rm MeV} - 17.6 \, {\rm MeV}$ [@Krasznahorkay:2015iga; @Feng:2016ysn], encompasses the uncertainties on the computation of the nuclear matrix elements [@Kozaczuk:2016nma]. The region above it is excluded by the non-observation of the same transition in the isovector excitation ${\Be^{*}}'$ [@Krasznahorkay:2015iga]. The horizontal grey band selects the values of $g'$ accounting for the $Z'$ mass in the negligible $m_{B'}$ case in which the $U(1)'$ symmetry breaking is driven by the two Higgs doublets. Furthermore, among all other experimental constraints involving a light $Z'$ that may be relevant for this analysis we have shown the most restrictive ones.
The strongest bound comes from the atomic parity violation in Cs and it represents a constraint on the product of $C_{e,A}$ and a combination of $C_{u,V}$ and $C_{d,V}$. This bound can be avoided if the $Z'$ has either only vector or axial-vector couplings but in the general scenario considered here, it imposes severe constraints on the gauge couplings $g',\tilde g$ thus introducing a fine-tuning in the two gauge parameters.
![ Allowed parameter space (orange region) explaining the anomalous $\Be^*$ decay. The white region above is excluded by the non-observation of the same anomaly in the ${\Be^{*}}'$ transition. Also shown (shaded regions) is the allowed parameter space by the $g-2$ of electrons and muons and the M[ø]{}ller scattering at SLAC E158 and pion decay from NA48/2. The beam dump experiment NA64 allows parameter space outside the red shaded region with dashed line. Finally, the blue line selects values of $g'$ and $\tilde g$ compatible with the weak nuclear charge measurement of Cesium. The horizontal grey band delineates values of $g'$ for which the $Z'$ mass is solely generated by the SM vev. \[fig:typeI\]](2HDM_NA64_new){width="9.cm" height="9.cm"}
We finally comment on the constraints imposed by neutrino-electron scattering processes [@Vilain:1994qy; @Deniz:2009mu; @Bellini:2011rx], the strongest one being from $\bar \nu_e e$ scattering at the TEXONO experiment [@Deniz:2009mu], which affect a combination of $C_{e, V/A}$ and $C_{\nu,V}$. As discussed above, in the protophobic scenario, in which the $Z'$ has only vector interactions, the constrained $\nu$ coupling to the $Z'$ boson is in high tension with the measured $\Be^*$ decay rate since $C_{\nu,V} = -2 C_{n,V}$ and a mechanism to suppress the neutrino coupling must be envisaged [@Feng:2016ysn]. This bound is, in general, alleviated if one attempts to explain the Atomki anomaly with a $Z'$ boson with axial-vector interactions since the required gauge couplings $g',\tilde g$ are smaller than the ones needed in the protophobic case. [ Neutrino couplings are also constrained by meson decays, like, for instance $K^\pm \rightarrow \pi^\pm \nu \nu$ which has been studied in [@Davoudiasl:2014kua] and where it has been shown that the corresponding constraint is relaxed by a destructive interference effect induced by the charged Higgs. As the results presented in [@Davoudiasl:2014kua] relies on the Goldstone boson equivalence approximation, we have computed the full one-loop corrections to the $K^\pm \rightarrow \pi^\pm Z'$ process in the $U(1)'$-2HDM scenario. The results are in agreement with the estimates in [@Davoudiasl:2014kua]. In our setup, for $g' \sim 10^{-4}$ and $\tan \beta = 1$, $M_{H^\pm} \sim 600$ GeV can account for the destructive interference quoted above between the $W^\pm$ and $H^\pm$ loops. For instance, we find $\textrm{BR}(K^\pm \rightarrow \pi^\pm Z' \rightarrow \pi^\pm \nu \nu) \simeq 0.1 \, \textrm{BR}(K^\pm \rightarrow \pi^\pm \nu \nu)_\textrm{exp}$ for $M_{H^\pm} \sim 615$ GeV with $\textrm{BR}(Z' \rightarrow \nu\nu) \simeq 30\%$ which is the maximum value for the invisible $Z'$ decay rate in the allowed region (orange and grey shaded area) shown in Fig. \[fig:typeI\]. A similar constraint arises from the $B$ meson decay to invisible but is less severe than the one discussed above [@Patrignani:2016xqp]. The $B^\pm \rightarrow K^\pm Z'$ process is characterised by the same loop corrections appearing in $K^\pm \rightarrow \pi^\pm Z'$, with the main difference being the dependence on the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. Therefore, the suppression effect induced by the charge Higgs mass affects both processes in the same region of the parameter space, thus ensuring that the bound from the invisible $B$ decays is satisfied once the constraint from the analogous $K$ meson decay is taken into account. ]{}\
Explanation of the Beryllium anomaly with a family non-universal $U(1)'$
========================================================================
The final alternative is to consider a single Higgs doublet, as with the SM, but non-standard Yukawa interactions, to allow axial couplings through the violation of Eq. (\[eq:yukawa\_gaugeinv\]), as done in [@DelleRose:2018eic]. This is done for the first two generations of fermions and the third has SM-like gauge-invariance, motivated by $\mathcal{O}(1)$ couplings. We begin by modifying the Yukawa couplings for the first two generations as follows, - \_[Yuk]{} &=& \^[u]{} \_[L,i]{}u\_[R,j]{} + \^[d]{} \_[L,i]{} H d\_[R,j]{}\
&+&\^[e]{} \_[i]{} H e\_[R,j]{}+ h.c., where the exponent, $n_{ij}$, of the non-renormalisable scale, $M$, is defined by the $U(1)'$ charges of the fields, such that these new Yukawa terms are gauge invariant. Subsequently, one may obtain fermion masses either at tree-level or radiatively by the method of Ref. [@Froggatt:1978nt][^2]. There are several models which motivate radiative mass generation for the lighter generations, as done in [@Demir:2005ti], alternatively, there exist mass generation dynamics by horizontal symmetries, as in [@Froggatt:1978nt]. We do not specify these dynamics, and simply leave an effective approach. We finally enforce that the first two generations are flavour universal, differing from the third, $z_{i_1}=z_{i_2}$ for $i=\{Q,u_R,d_R,L,e_R\}$, where the condition (\[eq:yukawa\_gaugeinv\]) is not applied to $z_{i_{1,2}}$. We now consider further constraints on the charge assignment. We also enforce the chiral anomaly cancellation conditions in Eq. (\[eq:anomaly\]), which will be satisfied by solely the fermionic content of the SM, supplemented by two right-handed neutrinos.
Our remaining constraints on the charge assignment are motivated by the non-observation of BSM physics. As discussed above, there are strong constraints on coupling to neutrinos, which would enhance processes such as $K^\pm \rightarrow \pi ^\pm \nu \nu$ [@Davoudiasl:2014kua], as well as electron-neutrino interactions, measured by the TEXONO experiment [@Feng:2016ysn; @Deniz:2009mu; @Bilmis:2015lja; @Khan:2016uon]. To avoid these stringent constraints, we therefore impose no couplings to the neutrinos, i.e., $C_{V,\nu} = C_{A,\nu} = 0$. This subsequently yields a relation between the neutrino and Higgs charges, $$z_{L_1}=z_{L_2}=z_{L_3}=-z_{H}.$$ Another constraint is to require that one indeed does have axial couplings for the up/down quarks to the $Z'$, as required to explain the anomaly, $$\begin{aligned}
&-z_{Q_{1,2}} -z_H +z_{u_{1,2}} \neq 0, \\
&-z_{Q_{1,2}} + z_H +z_{d_{1,2}} \neq 0.\end{aligned}$$ Our final constraint is from the atomic parity violation in Cs. As can be seen from other solutions, this provides a stringent bound on models with axial couplings for electrons. We thus also forbid interactions of this kind, and due to requiring universality for the first two generations, this will also forbid axial couplings for the muon, $$\begin{aligned}
C_{e,A} = C_{\mu,A} = 0. \end{aligned}$$ Preventing the appearance of these axial couplings will also help to avoid bounds from both $(g-2)_e$ and avoid worsening the discrepancy in $(g-2)_\mu$.
Combining all these constraints yields a single, unique charge assignment. We have a normalisation choice, and choose to set $z_H =1$. This unique choice is shown in Tab. \[tab:charges1HDM\].
------------- --- --- ------ --------
$Q_{1}$ 3 2 1/6 $1/3$
$Q_{2}$ 3 2 1/6 $1/3$
$Q_{{3}}$ 3 2 1/6 $1/3$
$u_{R_{1}}$ 3 1 2/3 $-2/3$
$u_{R_{2}}$ 3 1 2/3 $-2/3$
$u_{R_{3}}$ 3 1 2/3 $4/3$
$d_{R_{1}}$ 3 1 -1/3 $4/3$
$d_{R_{2}}$ 3 1 -1/3 $4/3$
$d_{R_{3}}$ 3 1 -1/3 $-2/3$
$L_{1}$ 1 2 -1/2 $-1$
$L_{2}$ 1 2 -1/2 $-1$
$L_{3}$ 1 2 -1/2 $-1$
$e_{R_{1}}$ 1 1 -1 $0$
$e_{R_{2}}$ 1 1 -1 $0$
$e_{R_{3}}$ 1 1 -1 $-2$
$H$ 1 2 1/2 $1$
------------- --- --- ------ --------
: Charge assignment of the SM particles under the family-dependent (non-universal) $U(1)'$. This numerical charge assignment satisfies the discussed anomaly cancellation conditions, enforces a gauge invariant Yukawa sector of the third generation and family universality in the first two fermion generations as well as no coupling of the $Z'$ to the all neutrino generations.[]{data-label="tab:charges1HDM"}
Now, we consider constraints on the new gauge coupling, and gauge-kinetic mixing parameters $(g', \tilde{g})$, given this charge selection. Unlike the previous scenarios considered, since this is family non-universal, one finds tree level FCNCs, which should be analysed. In diagonalising the quarks into the mass basis, off-diagonal couplings are generated, due to different coupling strengths between the first two and third quark generations. We now discuss the consequences of this on experimental observables. We begin with $K \rightarrow \pi e^+ e^-$ through a tree-level exchange of an on-shell $Z'$. There are no contributions to the $\mu ^+ \mu ^-$ decay as $M_{Z'} \sim 17$ MeV $< 2m_{\mu}$. There are stringent limits from LHCb [@Aaij:2015dea], though these are inapplicable in our case due to the small invariant mass of the $e^+ e^-$ pair. There is only sensitivity to energies above 20 MeV, due to photon conversion in the detector, and so energy resolution strongly degrades around these invariant masses. It is possible that future upgrades will lower this threshold and thus act as a discovery tool, or to disprove this scenario.
Another flavour observable is from meson mixing measurements. We begin with $B^0 - \bar{B}^0$, following the procedure as done in [@Becirevic:2016zri], but now assuming a much lighter propagator than their scenario, $P \equiv (m_{B} ^2 - M_{Z'}^2)^{-1} \simeq m_{B}^{-2}$, as opposed to their $P \simeq M_{Z'}^{-2}$. One subsequently finds the requirement $$|g^{L(R)} _{sb}| \lesssim 10^{-6},$$ where (assuming Minimal Flavour Violation (MFV) in the quark sector and using CKM matrix elements), $$\begin{aligned}
g^L _{sb} &= g' \Big( V_{\rm CKM} ^T ~\textrm{Diag}(z_{Q_1} , z_{Q_1} , z_{Q_3} )~ V_{\rm CKM} \Big) _{23} ,\\
g^R _{sb} &= g' \Big( V_{\rm CKM} ^T ~\textrm{Diag}(z_{u_{R_1}} , z_{u_{R_1}} , z_{u_{R_3}} )~ V_{\rm CKM} \Big) _{23} ,\end{aligned}$$ Since our charge assignment is family universal for LH quarks, $g^L _{sb}=0$, see Tab. \[tab:charges1HDM\], only the right-handed sector will contribute to the FCNC. This is suppressed by CKM factors, $g^R _{sb} \propto V_{tb} V_{ts}$, and so one finds a condition on the couplings, $g', \tilde g \lesssim 10^{-4}$.\
Proceeding in a similar faction but for $K - \bar{K}$ oscillations will yield a weaker constraint on the couplings. Although the propagator suppression is less severe, $P \simeq m_K ^{-2} > m_{B} ^{-2}$ , the CKM suppression is much stronger, $g_{sd} ^R \propto V_{td} V_{ts}$, and one finds the constraint $g', \tilde g \lesssim 10^{-3}$. In this review, we do not perform a full flavour analysis, but require these approximate constraints.
Finally, we present the allowed parameter space in Fig. \[fig:Final\_Region\_NA64\] for this scenario with one Higgs doublet extended by a $U(1)'$, with a charge assignment shown in Tab. \[tab:charges1HDM\]. The red, purple and green bands show regions which can explain the Atomki anomaly for 16.7, 17.3 and 17.6 MeV $Z'$ masses, respectively. These overlap in places and are independent of $\tilde{g}$ as the axial coupling depends solely on $g'$ and BR$(Z' \rightarrow e^+ e^-)=1$ everywhere. These bands have upper bounds due to the non-observation of the ${}^8 $Be$^{*'}$ anomaly. Also shown on the plot are the bounds from $(g-2)_\mu$, where the allowed region is inside the dashed line and $(g-2)_e$, where the allowed region is shaded in blue inside the dotted lines. In addition, the allowed region from NA64 is also shown, where one should be outside the red shaded region. The overall allowed region is therefore between the NA64 and $(g-2)_e$ lines, in the overlap shaded in blue. The other experimental constraints (electron positron collider (KLOE2), Moller scattering (E158), pion decay (NA48/2), E141, and atomic parity violation of Cs), similar to $(g-2)_\mu$, do not limit the allowed parameter space in blue, and are not shown on the plot.
$M_{Z'} ~(\textrm{MeV}) $ ${\mathcal B}$
--------------------------- ----------------------
16.7 $5.8 \times 10^{-6}$
17.3 $2.3 \times 10^{-6}$
17.6 $5.0 \times 10^{-7}$
: Solutions to the Atomki anomaly, with best fit mass value (16.7 MeV) from [@Krasznahorkay:2015iga] and subsequent alternative masses (17.3 MeV and 17.6 MeV) from [@Feng:2016ysn] along with the corresponding ratio of BRs, ${\mathcal B}$, as defined in Eq. (\[eq:BeAnomaly\]).[]{data-label="tab:Br"}
![Allowed parameter space mapped on the $(g',\tilde{g})$ plane explaining the anomalous ${}^8 \textrm{Be}^*$ decay for $Z'$ solutions with mass 16.7 (red), 17.3 (purple) and 17.6 (green) MeV. The white regions are excluded by the non-observation of the same anomaly in the ${}^8 \textrm{Be}^{*'}$ transition. Also shown are the constraints from $(g-2)_\mu$, to be within the two dashed lines; $(g-2)_e$, to be inside the two dotted lines (shaded in blue) and the electron beam dump experiment, NA64, to be outside the shaded red region, which lies between the two solid lines. The surviving parameter space lies at small positive and negative $\tilde{g}$ (though not at $\tilde{g}=0$), inside the shaded blue region which overlaps the Atomki anomaly solutions.[]{data-label="fig:Final_Region_NA64"}](Final_Region_NA64_new){width="0.7\linewidth"}
Fig. \[fig:BRPlotDensity\_high\] shows the quantity $\mathcal{B}$, as defined in Eq. (\[eq:BeAnomaly\]), over a range of $Z'$ masses. For each fixed mass value, a scan is performed over $(g',\tilde{g})$, in a range compatible with other experimental constraints, and the Atomki anomaly (i.e., over the dark blue and coloured regions in Fig. \[fig:Final\_Region\_NA64\]). For each scanned point in $\{ M_{Z'},~g'.~\tilde{g} \}$, there is a range of branching ratios, due to uncertainties in the Nuclear Matrix Elements (NMEs). This lower limit for all points is lower than the Atomki branching ratios, so only the upper ${\mathcal B}$ is of importance, and this is plotted. Also drawn, in orange, is the required branching ratio, as published by the Atomki collaboration, see Tab. \[tab:Br\]. A given point is then allowed if the upper ${\mathcal B}$ limit lies above the orange dots. For larger $M_{Z'}$ values, the largest ${\mathcal B}$ decreases, and a larger number of the scanned points lie above the Atomki points. This suggests that at higher masses, there is slightly more parameter space available for the 17.6 MeV solution, in comparison to the 16.7 MeV one. This is reflected in the slightly different widths shown in Fig. \[fig:Final\_Region\_NA64\].
![Values of the upper limit ${\mathcal B}$ (lower limits are smaller than the scale of the plot), as defined in Eq. (\[eq:BeAnomaly\]), versus the mass of the $Z'$ obtained scanning over the allowed parameter space in $(g',\tilde{g})$, obtained from Fig. \[fig:Final\_Region\_NA64\] for each mass step taken (in blue). The Atomki collaboration solutions are also shown (in orange).[]{data-label="fig:BRPlotDensity_high"}](FinalBRPlotDensityScaled){width="0.6\linewidth"}
Conclusions
===========
While there remains the possibility that the Atomki anomaly can be explained as a statistical fluctuation combined with yet unknown nuclear physics properties and/or unforeseen experimental conditions, the fact that presently such an effect has been determined with a $6.8\sigma$ significance, including a near-perfect fit of both the mass and angular excesses to the possibility of a new particle with a mass of about 17 MeV been produced, calls for a thorough investigation of plausible theoretical explanations.
With this in mind, in this review, we have presented particle physics scenarios that extend the SM to include the presence of either a spin-0 (pseudoscalar, $A$) boson or a spin-1 (axial-vector, $Z'$) boson, both of which can be made compliant with a variety of experimental data. Assuming the standard Lagrangian structures describing $A$ and $Z'$ interactions with SM fermionic currents in both the lepton and quark sectors, we have determined the required couplings of such bosons to explain the Beryllium data.
As for the theoretically embeddings of these solutions, we can conclude the following. A light pseudoscalar state can appear in models with extended Higgs sectors in which an approximate ungauged global symmetry is spontaneously broken, examples of which include (type-II) 2HDMs with a SM-singlet near the Peccei-Quinn or $R$-symmetric limit, although in this case isospin breaking effects and non-universality in the Yukawa couplings of the new state to electrons and $d$-quarks must be allowed for. As for light gauge bosons with significant axial-vector couplings, two possible theoretical frameworks have been proven to be viable. Both require an additional $U(1)'$ group mixing with the SM one, $U(1)_Y$. In one case, which retains the SM Higgs sector, a family non-universal set of $Z'$ couplings to the known fermions must be invoked. In the other case, $Z'$ couplings to quarks and fermions of the SM can be retained in their universal form, yet this requires an enlarged Higgs sector, which we have identified as possibly being a type-I 2HDM. Needless to say, these two theoretical frameworks were constructed in presence of gauge invariance and anomaly cancellations plus they do not require isospin breaking.
While the above list of possible theoretical setups is clearly not exhaustive, it at least provides somewhat minimal frameworks (only containing enlarged Higgs and gauge sectors, possibly including heavy neutrinos but no exotic particles) within which further data upcoming from the Atomki experiment can be interpreted to pave the way for more dedicated phenomenological studies, which may in turn lead to refinements on the theoretical side.
Acknowledgements {#acknowledgements .unnumbered}
================
The work of LDR and SM is supported in part by the NExT Institute. SM also acknowledges partial financial contributions from the STFC Consolidated Grant ST/L000296/1. Furthermore, the work of LDR has been supported by the STFC/COFUND Rutherford International Fellowship Programme (RIFP). SJDK and SK have received support under the H2020-MSCA grant agreements InvisiblesPlus (RISE) No. 690575 and Elusives (ITN) No. 674896. In addition SK was partially supported by the STDF project 13858. All authors acknowledge support under the H2020-MSCA grant agreement NonMinimalHiggs (RISE) No. 645722.
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[^1]: It should however be mentioned that this second anomaly was never documented in a published paper, only in proceedings contributions.
[^2]: Lagrangians of this form have been used to motivate solutions to the flavour problem, so it may be of interest to investigate whether this $U(1)'$ may explain the allowed masses and mixings, but we perform no such careful investigation here.
|
---
abstract: 'The ideas of topology have found tremendous success in Hermitian physical systems, but even richer properties exist in the more general non-Hermitian framework. Here, we theoretically propose and experimentally demonstrate a new topologically-protected bulk Fermi arc which—unlike the well-known surface Fermi arcs arising from Weyl points in Hermitian systems—develops from non-Hermitian radiative losses in photonic crystal slabs. Moreover, we discover half-integer topological charges in the polarization of far-field radiation around the Fermi arc. We show that both phenomena are direct consequences of the non-Hermitian topological properties of exceptional points, where resonances coincide in their frequencies and linewidths. Our work connects the fields of topological photonics, non-Hermitian physics and singular optics, and paves the way for future exploration of non-Hermitian topological systems.'
author:
- Hengyun Zhou
- Chao Peng
- Yoseob Yoon
- Chia Wei Hsu
- 'Keith A. Nelson'
- Liang Fu
- 'John D. Joannopoulos'
- Marin Soljačić
- Bo Zhen
title: |
Observation of Bulk Fermi Arc and Polarization\
Half Charge from Paired Exceptional Points
---
[^1]
[^2]
[^3]
In recent years, topological physics has been widely explored in closed and lossless Hermitian systems, revealing novel phenomena such as topologically non-trivial band structures [@CastroNeto2009; @Armitage2017; @Tarruell2012; @Wan2011; @Xu2015; @Lv2015; @Lu2015; @Noh2017; @Yang2017] and promising applications including back-scattering-immune transport [@Hasan2010; @Qi2011; @Lu2014; @Haldane2008; @Wang2009; @Fang2012; @Hafezi2013; @Rechtsman2013a; @Khanikaev2013; @Yang2015; @Susstrunk2015; @Wu2015]. However, most systems, particularly in photonics, are generically non-Hermitian due to radiation into open space or material gain/loss. Non-Hermiticity enables even richer topological properties, often with no counterpart in Hermitian frameworks [@Moiseyev2011; @Cao2015; @Leykam2017; @Shen2017]. One such example is the emergence of a new class of degeneracies, commonly referred to as exceptional points (EPs), where two or more resonances of a system coalesce in both eigenvalues and eigenfunctions [@Berry2004a; @Rotter2009; @Heiss2012]. So far, isolated EPs in parameter space [@Dembowski2001; @Mailybaev2005; @Liertzer2012; @Gao2015; @Doppler2016; @Xu2016; @Hassan2017] and continuous rings of EPs in momentum space [@Zhen2015; @Cerjan2016a; @Xu2017] have been studied across different wave systems due to their intriguing properties, such as unconventional transmission/reflection [@Lin2011a; @Feng2013; @Longhi2014], relations to parity-time symmetry [@Bender1998; @Makris2008; @Guo2009; @Ruter2010; @Chong2011; @Regensburger2012; @Konotop2016], as well as their unique applications in sensing [@Hodaei2017; @Chen2017] and single-mode lasing [@Hodaei2014; @Feng2014; @Peng2014a]. Here, we theoretically design and experimentally realize a new configuration of isolated EP pairs in momentum space, which allows us to reveal the unique topological signatures of EPs in the band structure and far-field polarization, and to extend topological band theory into the realm of non-Hermitian systems. Specifically, we demonstrate that a Dirac point (DP) with nontrivial Berry phase can split into a pair of EPs [@Kirillov2005; @Seyranian2005; @Kozii2017] when radiation loss—a form of non-Hermiticity—is added to a 2D-periodic photonic crystal (PhC) structure. The EP-pair generates a distinct double-Riemann-sheet topology in the complex band structure, which leads to two novel consequences: bulk Fermi arcs and polarization half charges. First, we discover and experimentally demonstrate that this pair of EPs is connected by an open-ended isofrequency contour—we refer to it as a bulk Fermi arc—in direct contrast to the common intuition that isofrequency contours are necessarily closed trajectories. The bulk Fermi arc here is a unique topological signature of non-Hermitian effects in paired EPs, and resides in the bulk dispersion of a 2D system. This is fundamentally different from the previously known surface Fermi arcs that arise from the 2D projection of Weyl points in 3D Hermitian systems. Moreover, we find experimentally that near the Fermi arc frequency, the system exhibits a robust half-integer winding number in the far-field polarization [@Zhan2009; @Berry2003; @Freund2005; @Miyai2006; @Dennis2009; @Bauer2015; @Fosel2017], analogous to the orientation-reversal on a Mobius strip. We show that this is a direct consequence of the topological band-switching properties across the Fermi arc connecting the EP pair, and is direct experimental evidence of the $\nu=\pm1/2$ topological index associated with an EP [@Shen2017]. With comprehensive comparisons between analytical models, numerical simulations, and experimental measurements, our results are a direct validation of non-Hermitian topological band theory, and present its novel application to the field of singular optics.
{width="12cm"}
We start by outlining our scheme to split a single DP into a pair of EPs, which directly leads to the emergence of a bulk Fermi arc. First, consider a 2D-periodic photonic crystal (PhC) with a square lattice of circular air holes introduced into a dielectric material. In this Hermitian system (no material gain/loss or radiation loss), the crystalline symmetry ($C_{4v}$) ensures a quadratic band degeneracy at the center of the Brillouin zone (Supplementary Fig. S1A). As this $C_{4v}$ symmetry is broken, e.g. by shearing the structure into a rhombic lattice with elliptical holes (Fig. 1A), the quadratic degeneracy point splits into a pair of DPs situated at $(\pm k_\text{D}, 0)$ along the $k_{x}$ axis. The same splitting behavior is shown in both analytical models and numerical simulations (Supplementary Information, Section I) [@Chong2008; @Zhou2016] .
Next, we consider a non-Hermitian system of a finite thickness PhC slab (inset of Fig. 1B), where modes near the DP become resonances with finite lifetime due to radiative losses towards the top and bottom. Adopting the even and odd $y$-mirror-symmetric eigenstates at the DP as basis, and taking into account the radiation losses via non-Hermitian perturbations, the effective Hamiltonian in the vicinity of the original DP at $(k_\text{D}, 0)$ can be written as [@Zhou2016; @Shen2017]: $$\label{eqn:NH-Hamiltonian}
H_{\rm eff} = \omega_\text{D} - i\gamma_0 + (v_g \delta k_x- i\gamma) \sigma_z + v_g \delta k_y \sigma_x,$$ with complex eigenvalues of $$\label{eqn:EPring}
\omega_{\pm} = \omega_\text{D} - i\gamma_0 \pm \sqrt{(v_g^2 \delta k^2 -\gamma^2) - 2i\gamma v_g\delta k_x}.$$ Here $\sigma_{x,z}$ are Pauli matrices, $\omega_{\text{D}}$ is the DP frequency, and $(\delta k_{x}, \delta k_{y})$ is the momentum displacement from $(k_\text{D},0)$, $\delta k^2 = \delta k_x^2 + \delta k_y^2$. Meanwhile, $\gamma_{0} \pm \gamma$ are the radiation decay rates of even and odd $y$-mirror-symmetric modes, taking into account the fact that the two modes have different coupling strengths to the continuum; $v_g$ is the group velocity describing the dispersion around the DP, which for simplicity is here chosen to be the same along all directions (see Supplementary Information, Sections I & II, for the general case). The real part of the complex eigenvalues $\omega_{\pm}$ characterizes the resonance frequency, while the imaginary part represents the linewidth of the resonance.
The eigenvalue spectrum exhibits a pair of EPs at $(\delta k_{x},\delta k_{y})= (0, \pm \gamma/v_{g})$ when the square root term in Eq. (2) vanishes and the two eigenvectors coalesce (Fig. 1B). Furthermore, this pair of EPs are connected in momentum space by an open-ended arc—a bulk Fermi arc, along which the real part of the complex eigenvalues are degenerate at $\omega_\text{D}$ (Fig. 1C middle panel). Although sharing features similar to previously-studied Fermi arcs—both are open-ended isofrequency contours—our bulk Fermi arc resides in the bulk dispersion rather than on the surface of a 3D Hermitian Weyl system, and originates from non-Hermiticity rather than the presence of Weyl points. As the frequency $\omega$ decreases from above $\omega_\text{D}$, the closed isofrequency contour at $\omega$ shrinks (Fig. 1C top panel), eventually turning into the open Fermi arc when $\omega = \omega_\text{D}$ (Fig. 1C middle panel), and expands out again into a closed contour at even lower frequencies (Fig. 1C bottom panel). Taken together, the band structure around the EPs forms a double-Riemann-sheet topology (Fig. 1B). This originates from the complex square root term in the dispersion in Eq. (2), which, depending on the sign choice of the square root, results in two sheets. The two eigenvalues continuously evolve on each sheet, and their real parts become degenerate along a curve—the bulk Fermi arc. We further verify the existence of bulk Fermi arcs in realistic PhC slab structures via numerical simulations (Fig. 1C circles), showing a good agreement with analytical results (Fig. 1C solid lines); see Supplementary Information, Sections II & III, for details.
![[**Fabricated PhC slab and measurement setup.**]{} [**A,**]{} SEM images of the PhC samples: side view (top panel) and top view (bottom panel). [**B,**]{} Schematic drawing of the scattering measurement setup. Vertically-polarized light from a tunable continuous-wave Ti-Sapphire laser (TL) scatters off the PhC slab and is collected using a CCD camera placed at the focal plane of the lens (L). The specular reflection is blocked to ensure only scattered light is imaged. POL and QWP are used in polarimetry measurements of the scattered light. TL, tunable laser; BS, beam-splitter; L, convex lens with $10$ cm focal length; POL, polarizer; QWP, quarter wave plate; BB, beam block.](fig2_v8.pdf){width="6cm"}
To experimentally demonstrate the bulk Fermi arc, we use interference lithography to fabricate PhC slabs in Si$_3$N$_4$ (refractive index $n = 2.02$, thickness $t=220$ nm) on top of a silica substrate ($n = 1.46$) [@Lee2012]. The PhC structure consists of rhombic unit cells with side length $a=525$ nm, unit cell angle $\theta=65.5^\circ$, and elliptical air holes with long axis length $w=348$ nm and short axis length $h=257$ nm (see Supplementary Information, Section IV, for details). Scanning electron microscope (SEM) images of the fabricated samples are shown in Fig. 2A. The structure is immersed in an optical liquid with refractive index matched to that of the silica substrate to create an up-down symmetric environment.
We performed angle-resolved scattering measurements (setup shown in Fig. 2B) to image isofrequency contours of the sample. The PhC sample is illuminated with a tunable continuous-wave Ti:Sapphire laser (MSquared) that is vertically polarized, while scattered light—arising from natural fabrication imperfections of the sample—is collected with a CCD camera (Princeton Instruments) placed at the focal plane of a convex lens with $10$ cm focal length. Due to resonant-enhancement, the scattered light will have strongest intensity only along directions where the underlying resonances share the same frequency as the pump laser, and thus the isofrequency contours of the sample are directly imaged onto the CCD [@Regan2016; @Shi2010] (see Supplementary Information, section IV, for details of the setup and the scattering process). To show the full shrinking-and-reexpanding feature of the isofrequency contours around the bulk Fermi arc, the laser wavelength is tuned from $794$ nm down to $788$ nm at steps of approximately $0.2$ nm. Furthermore, the polarization at each point along a given isofrequency contour is determined through polarimetry measurements, by optionally inserting a quarter-wave plate and a polarizer (Thorlabs) in front of the CCD (see Supplementary Information, Section V, for details).
![[**Experimental demonstration of a bulk Fermi arc.**]{} [**A,**]{} Experimentally measured isofrequency contours and [**B,**]{} numerically simulated spectral density of states at five representative wavelengths. The bulk Fermi arc appears at $791.0$ nm (middle row), when the isofrequency contour becomes open-ended. The regions of interest are highlighted in all panels to emphasize the shrinking (top two rows) and re-expanding (bottom two rows) feature of isofrequency contours near the bulk Fermi arc. The numerical results are offset by $0.5$ nm for better comparison. ](fig3_v8.pdf){width="8cm"}
In Fig. 3A, the experimental results of isofrequency contours are shown at a few representative wavelengths around the Fermi arc. These are compared to numerical results of isofrequency contours (Fig. 3B) obtained from simulating fitted structural parameters, showing a good agreement with each other. Here, for better comparison, the numerical results are offset by $0.5$ nm relative to the experiments. See Supplementary Information, section IV, about this wavelength offset, and Supplementary Movie I for the full set of isofrequency contours measured at different wavelengths. To focus on the bulk Fermi arc, we highlight the region of interest in both panels, where the isofrequency contours clearly demonstrate the shrinking and re-expanding behavior. As shown in Fig. 3, as the wavelength decreases from $794.0$ nm, the corresponding isofrequency contour shrinks (top two rows), and eventually becomes an open-ended arc at $791.0$ nm (middle row), consistent with our previous theoretical predictions in Fig. 1C. As the wavelength is further decreased down to $789.5$ nm and $788.7$ nm, the arc expands out into closed contours again (bottom two rows). The bending feature of the contours is a result of higher-order terms in the band dispersion (Supplementary Information section II). The open contour at $791$ nm (middle row) is a clear, direct observation of the bulk Fermi arc.
So far, we have shown one direct consequence of the unique double-Riemann-sheet topology near paired EPs—the bulk Fermi arc. Next, we demonstrate another consequence: half-integer topological charges in the polarization configuration, which also serve as a direct experimental validation of the $\nu = \pm 1/2$ topological index of an EP. These topological charges describe the direction (clockwise or counterclockwise) and number of times the polarization vector winds around a point/line singularity in the optical field, and in our particular system we observe a robust 180-degree winding around the Fermi arc, corresponding to a half-integer charge.
To fully reconstruct the far-field polarization configurations of the resonances, we perform polarimetry measurements by recording the intensity of isofrequency contours after passing through six different configurations of polarizers and/or waveplates (see Supplementary Information, section V, for details). Although the incoming light is vertically polarized, the scattered light at each point along the contour is, in general, elliptically polarized, reflecting the polarization state of its underlying resonance [@Hsu2017]. Taking points X and Z in Fig. 4A as examples: after passing through a vertical polarizer, the scattered light is weak(strong) at point X(Z); while after a horizontal polarizer, the relative intensity of the scattered light switches between points X and Z. This clearly shows that the far field of the underlying resonance at point X(Z) is mostly horizontally(vertically) polarized.
Examples of the fully-reconstructed spatial polarizations (blue ellipses) at representative points along the $794$ nm isofrequency contour (red solid line) are shown in the top panel of Fig. 4B, which agree well with numerical results (Fig. 4B bottom panel). Furthermore, both experimental and numerical results show 180-degree winding of the polarization long axis, as illustrated by the green arrows in Fig. 4B: as the momentum point starts from point X, traverses the full contour in the counterclockwise direction, and returns to point X, the polarization long axis flips direction by rotating $180$ degrees in the clockwise direction—corresponding to a $-1/2$ topological charge being enclosed in the loop. These results thus indicate that the far-field emission from our PhC is a vector-vortex beam with half-integer topological charge, in stark contrast to the integer vector beams realized in photonic crystal surface emitting lasers [@Miyai2006; @Iwahashi2011].
{width="12cm"}
We now elaborate on the fundamental connections between the half-integer topological charges observed in the far-field polarization and the half-integer topological index of an EP [@Shen2017], manifested as its mode-switching property. For more details see Supplementary Information, section VI. Along the $k_x$ axis, the two bands forming the EP pair in our system have orthogonal linear polarizations due to the $y$-mirror symmetry: one is horizontal (e.g. mode X in Fig. 4C), while the other is vertical (e.g. mode Z and W). As we follow a closed path in momentum space X$\rightarrow$Y$\rightarrow$Z$\rightarrow$W that encircles one of the EPs in the counterclockwise direction, the initial eigenstate X (horizontally polarized) on the top sheet adiabatically evolves into state Z (vertically polarized) and eventually into final state W (vertically polarized) on the bottom sheet, due to the mode-switching topological property of the EP [@Dembowski2001; @Doppler2016; @Xu2016; @Hassan2017]. The switching behavior of the eigenmodes—from X to W—directly follows from their eigenvalue swapping behavior on the complex plane (see Supplementary Section II and Fig. S2). Equivalently, one complex eigenvalue winds around the other one by half a circle, thus implying that the topological index of an EP is a half-integer. The orthogonal nature between the polarizations at X and Z, arising from the mode-switching property of the EP, guarantees a $(n+1/2)\pi$-rotation ($n\in \mathbb{Z}$) of the polarization vector along half the contour. Again using the $y$-mirror symmetry, the full isofrequency contour will accumulate twice the rotation angle, to a combined $(n+1/2)\times 2\pi$-rotation, corresponding to a half-integer charge of $n+1/2$. We have thus shown the intimate connection between polarization vector winding in singular optics and the double-Riemann-sheet topology of paired EPs. Our experimental demonstration—generating half-integer vector-vortex beams directly from the topological properties of EPs—not only distinguishes us from the previously known integer topological charges of polarization around bound states in the continuum [@Zhen2014], but also proves the nontrivial topology of EPs.
In this work, we have demonstrated that the topological properties of paired EPs endow the band structure and far-field emission with unique features, manifested as the emergence of bulk Fermi arcs and polarization half charges. Application-wise, our structure provides a simple-to-realize method to create half-integer vector-vortex beams [@Dorn2003; @Zhan2009] at a wide range of frequencies. Future prospects leveraging the topological landscape around paired EPs may enable PhC lasers with exotic emission profiles [@Cai2012; @Hirose2014], such as twisted Mobius strips. The isolated EPs found in our structure also provide a straightforward platform to study the influence of EPs and their topology on light-matter interactions, such as modified Purcell factors for spontaneous emission enhancement and nonlinear optics generation [@Lin2016; @Pick2017; @Pick2017a]. Finally, our study paves the way for the exploration of topological band theory in general non-Hermitian wave systems, ranging from photonic and acoustic to electronic and polaritonic systems.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank Dr. Tim Savas for fabrication of the samples. The authors also thank Steven Johnson, Eugene Mele, Ling Lu, Yichen Shen, Scott Skirlo, Shang Liu, Jong Yeon Lee, Francisco Machado, Nicholas Rivera, Grace Zhang and Sam Moore for helpful discussions. Research supported as part of the Army Research Office through the Institute for Soldier Nanotechnologies under contract no. W911NF-13-D-0001 (photon management for developing nuclear-TPV and fuel-TPV mm-scale-systems). Also supported as part of the S3TEC, an Energy Frontier Research Center funded by the US Department of Energy under grant no. DE-SC0001299 (for fundamental photon transport related to solar TPVs and solar-TEs). H.Z. acknowledges support from the Undergraduate Research Opportunities Program at MIT. C.P. acknowledges support from the National Natural Science Foundation of China under Grants No. 61575002, 61320106001, and the China Scholarship Council. C.W.H. acknowledges support from the National Science Foundation under grant NSF DMR-1307632. Y.Y. and K.A.N. were supported in part by Skoltech as part of the Skoltech Next Generation Program.
Author Contributions {#author-contributions .unnumbered}
====================
H.Z. and B.Z. conceived the idea. H.Z. performed the analytical calculations and numerical simulations. H.Z., B.Z., C.P. and Y.Y. conducted the experiments and analyzed the data. H.Z. and B.Z. wrote the manuscript, with input from all authors. B.Z., M.S. and J.D.J. supervised the research. All authors contributed to the analysis and discussion of the results.
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[^1]: These authors contributed equally to this work.
[^2]: These authors contributed equally to this work.
[^3]: To whom correspondence should be addressed; E-mail: bozhen@sas.upenn.edu.
|
---
abstract: 'We present a brief overview of some recent observations of colliding galaxies and relevant numerical simulations. These are compared, and details of the locations and history of collision induced star formation are explored, with possible application to star formation at earlier epochs.'
author:
- 'Susan A. Lamb'
- 'Nathan C. Hearn'
title: Galaxy Collisions and Star Formation
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} = \#1 1.25in .125in .25in
Introduction
============
Galaxy collisions may be the predominant cause of star formation in galaxies over the course of their lifetimes. In the local universe, strong collisions between comparable mass galaxies are rare, and star formation in gaseous disks usually takes place in relatively quiescent circumstances. However, it is thought that there was an epoch in the past when both strong and glancing collisions between comparable mass galaxies were much more common. These interactions would have produced strong tidal torques and density waves, as observed in the rare, strongly interacting disk galaxies of the local universe. Observations of recently impacted disk galaxies usually show evidence of enhanced rates of recent and current star formation (starbursts). Our three-dimensional numerical simulations of collisions between comparable mass galaxies demonstrate that the gas volume density in the disk can be increased by significant factors, both in the nucleus and in pronounced features that form well away from the galactic center. These collision-induced density enhancements can give rise to ring galaxies, strong, one-armed spirals, and grand-design spirals. The very high density regions produced experience strong shocks in many circumstances, which may play an important role in determining the subsequent amount and exact location of star formation. We present comparisons between multi-wavelength observations of star forming, impacted galaxies and our 3-D numerical simulations of galaxy collisions involving disk galaxies. We show that the timescale for massive star formation can be very short, and that the resulting morphology and velocity structure in the disk can be understood for several well-observed systems.
The collision and subsequent merger of clumps of stars and gas to form the current galaxies is a central tenet of the theory of hierarchical galaxy formation. (This is reviewed elsewhere in this volume.) However, the details of the interactions are somewhat unclear, as yet. Do the “dark matter” halos merge at an early stage, while luminous clumps of material have a large relative velocity within these halos, forming tight, high velocity galaxy groups? Are the currently observed compact groups (see Sulentic 2000) the remnants of such a population? The amount and location of star formation in this interacting material, as in current galaxies, will be effected by, among other things, the relative velocities and morphologies of the colliding galaxies, the amount of molecular gas that exists, and the background gravitational potential. The effects of these parameters can be explored for systems in the nearby universe and the results applied judiciously to studies of star formation at earlier times.
One area of exploration that has been given considerable observational attention is that of the relationship between interactions, star formation, and the presence of molecular gas. Recently, this has been investigated for an optically-selected sample of interacting galaxies (see Bushouse et al. 2000, and references within). The choice of sample is important because many previous studies were of infrared-bright selected galaxies which might give somewhat different results. (There are numerous references on this subject, for example Young et al. 1996.) The Bushouse sample spans a large range of interaction strengths and star formation rates, as well as infrared properties. Several of the conclusions drawn were consistent with those found from the IR-selected samples, namely, that 1) interacting galaxies are, on average, marginally richer in molecular gas than a comparison sample of isolated spiral galaxies (assuming the standard conversion factor of CO to H$_2$), 2) the interacting galaxies also have a mean infrared luminosity to H$_2$ mass ratio that is a factor of 1.3 higher than the isolated galaxies, and 3) there is a strong correlation between relative H$_2$ content and the star formation rate, indicating that interaction-induced star formation activity is highly dependent upon the available gas supply. However, other results could only be found in a study of this type of sample. These included, that some galaxies have moderate amounts of H$_2$ but much lower than normal star formation rates and IR emission. (This implies that molecular gas is a necessary, but not sufficient, prerequisite for star formation in these systems.) An apparent strong correlation between interaction strength and both star formation rate and relative H$_2$ content is found in these studies. However, this correlation could be completely removed by invoking non-standard CO-to-H$_2$ conversion factors. Thus there is no strong evidence that interactions initiate conversion of HI to H$_2$ gas, only that the star formation rate is usually higher when more H$_2$ is present.
Detailed exploration of nearby systems
======================================
Observations of recently impacted objects in the local universe, when compared with numerical simulations of galaxy collision, can be used to investigate some important aspects of the induced star formation. In this section we review the results from two studies involving comparisons between multiwavelength observations of star-forming impacted disk galaxies and numerical simulations of colliding galaxies. The systems Arp 118 and Arp 119 both contain a strongly perturbed disk galaxy and an identifiable intruding companion which collided recently, that is, within the last few tens of millions of years. In the studies of these systems, it has been shown that fully dynamical, 3D numerical simulations of an elliptical galaxy colliding with a disk galaxy can reproduce the morphology of the disturbed disk galaxy and the general shape of the elliptical. The results of the simulations are consistent with the approximate relative positions of the two galaxies within each pair, and with the velocity gradients across the disks. The ability to simulate the gross behaviors of both the stars and gas in specific colliding systems leads to the possibility of investigating the timing and amount of massive star formation following the collision, of identifying the size of some of the large-scale physical parameters of the gas involved in the new star formation, and of following the resulting flows of gas in the impacted disk.
All of the ’best-fit’ models to the colliding pairs mentioned above were chosen from a grid of simulations presented in Gerber (1993), Gerber, Lamb, & Balsara (1996), and Lamb, Hearn, & Gerber (2000). These simulations explore the results of face-on collisions (ones parallel to the spin-axis of the disk) between an elliptical galaxy and a disk galaxy, and are comprised of combined N-body/hydrodynamical simulations. The model of the rotating disk galaxy consists of gas, stars, and dark matter, while that of the spherical galaxy is composed of stars and dark matter only. Mass ratios from 1:1 to 10:1 have been explored, with impact parameters ranging from zero (head-on) to slightly larger than the disk radius. The gas dynamics is treated using the smoothed particle hydrodynamics (SPH) method, and the gravitational forces are calculated using standard particle-mesh (PM) techniques. Both the n-body (stellar and dark matter) and SPH (gas) particles contribute to the gravitational potential.
The numerical simulations can be physically scaled to individual galaxy systems if the masses and sizes of the galaxies involved are known. Thus, an estimate of the distance, as well as observations at a variety of wavelengths, must be available if a model fitting is to be attempted for a particular system. The choice of real systems to compare with simulations is mostly dictated by the existence of observational data sets for comparable mass galaxies in pairs. Further selection is based upon the general morphology of the disk galaxy, where indications that the impact was roughly parallel to the spin axis of the disk is sought. Such an impact produces several unique morphological features, such as full or partial rings. These impact-induced morphologies are explored and discussed in Gerber & Lamb (1994). Once the present-day match is found between a real system and a model taken from a simulation, observed regions of recent, current, and potential star formation can be matched with regions of high gas density and shocks during the course of the simulation. Some understanding of the history of, and the conditions influencing, star formation can then be obtained from the simulation.
Arp 118 is comprised of two galaxies, an elliptical (NGC 1143) and a disturbed disk galaxy (NGC 1144) containing a very extended starburst ring and connected arc. These two galaxies have a separation of 42 kpc on the sky (assuming $H_{0}=75$ ), and a relative velocity between their nuclei of 300 along the line of sight. NGC 1143 is a luminous infrared galaxy with a total luminosity of approximately $3\times10^{11}~$L$_{\odot}$ and L$_{IR}=2.5\times10^{11}~$L$_{\odot}$ (Appleton & Struck-Marcel 1987), a very strong CO(1-0) emission (L$_{CO}=10^{10}~$L$_{\odot}$, Gao et al. 1997), and an extreme velocity gradient across the ring of 1100 (see Hippelein 1989a, b). Recent observations of the neutral and ionized gas in this system are presented in Bransford et al. (1999). The nucleus is displaced from the center of the ring structure and is classified as Seyfert 2 (Hippelein 1989a).
Lamb, Hearn, & Gao (1998) showed that the disturbed morphology, the extreme velocity gradients, and the morphological relationships between the emitting regions as observed in radio, H$\alpha$, and CO, can be explained as resulting from a collision between a rotating disk galaxy and a somewhat less massive elliptical which passed through the disk approximately 42 Myr ago at a slight angle to the perpendicular to the disk. The detailed modeling does not support a prior suggestion (see Gao 1996, and references within) that a merger was needed to explain the morphology and large velocity gradients of NGC 1144. The overall dynamics and morphology of the Arp 118 system are well represented by a numerical model of an off-center collision in which the disk galaxy has a mass four times that of the intruding elliptical. In particular, the most notable large-scale feature of the system, the extensive arm of compressed gas edged with a dark dust lane, is very well matched by the model.
The sequence of the various star formation episodes in NGC 1144 was determined by matching the models at three different stages of the simulation with observations in three wavelength bands mentioned above, each of which corresponds to a well-defined epoch of star formation. The radio and H$\alpha$ emission of a starbursting galaxy are an indication of the star formation activity that occurred at earlier times, the H$\alpha$ being from the more recently formed stars. The CO observations provide information about those regions of the gas that are currently dense and likely to form stars shortly. The dimensions of the simulation were scaled to the physical system using the observed length scales and mass of the spiral galaxy as given by Gao et al. (1997) and Hippelein (1998a). These scaled units were then used to determine the times at which the collision and subsequent star formation episodes took place.
This study demonstrates for the first time the connection between the spatial and temporal progression of star formation, and the changing locations of the very dense regions in the gas of a massive disk galaxy in the aftermath of its collision with a massive elliptical. The collision generated a strong, non-linear density wave in the stars and gas in the disk of NGC 1144, causing the gas to became clumped on a large scale. This wave produced a series of superstarclusters along arcs and rings that emanate from the central point of impact. The locations of these star forming regions match those of the regions of increased gas density predicted in the time sequence of models.
It is thought that stars, particularly massive stars, are formed in the cores of giant molecular clouds in the highest density regions. Both multiwavelength observations and the numerical models indicate that strong shocks in the gas, together with large increases in the gas volume density, are associated with star formation over volumes of 1 kpc$^3$. The observed morphology of the regions of dense gas and the clustering observed in the newly formed, H$\alpha$ emitting stars suggest that the observed clumping of the young stars results from a clumping of the densest gas on the same scale. The work by Marston & Appleton (1995) and Appleton & Marston (1997) also provides evidence that the clumping observed in the optical images of collisionally produced ring galaxies is not due to patchy dust obscuration, because the same clustering is also observed in the near infrared. Gas clumping on this same scale is found in the numerical simulations, suggesting that there is a global explanation for the observed morphology of the dense gas and the resulting giant stellar formations in these systems.
The Arp 119 system (also known as CPG 29) has also been studied. The southern member (Arp 119S) of the pair has a strongly disturbed, asymmetric form, while the northern member, Arp 119N, appears to be a normal elliptical, although very devoid of gas (Marziani, et al, 1994). Arp 119S exhibits some remarkable features, including: a bright, circular ring surrounding the nucleus; a luminous arc north of the nucleus that extends to the west; a long arc of knots along the southern edge; and a ÒveilÓ of low-luminosity material in the east. This galaxy nucleus is classified as a LINER. Observations and model fitting of this system are summarized in another paper in this volume (see Hearn, Lamb, Gruendl, & Gerber 2000). A longer paper describing new infrared observations and a simulation fit to these and other observations is in preparation (Hearn & Lamb 2000).
The displacement of the nucleus in Arp 119S from the center of the ring indicates a face-on collision with a significant impact parameter (about 25 percent of the disk’s radius), and the structure of the luminous arc in the northern part of Arp 119 indicated that the elliptical and disk galaxies, including their dark matter halos, are of comparable mass. The analysis of the simulations and the observations suggests that the collision between Arp 119N and Arp 119S took place about 71 million years ago. Multiwavelength image, see, for example, Marziani et al. 1994, reveal locations of past and present starburst events in Arp 119S. These regions have been correlated in space and time with regions of enhanced density and strong shocks in the simulated gas disk. It was found that at least two distinct star formation episodes occurred as a result of the collision. One is currently ongoing, and the other occurred approximately 24 Myr after the impact. The first burst occurred when the expanding, rotating structure of dense gas produced by the collision had formed a ring around the inner third of the disk. This ring of gas produced the bright ring of stars around the nucleus that is now observed in the B-band image. This high density region in the gas has now spread and opened up, and reaches to the edge of the visible disk. It shows up as a northern arc of molecular gas that leads to a bright arc of B-band and H$\alpha$ emission. The long arc of star-forming knots at the southern edge of Arp 119S is coincident with high-density knots in the gas, forming a southern arc in the simulation. The gas density is relatively low in the eastern half of the simulation, matching the region of low luminosity in the optical image. The study of the colliding pair Arp 119 indicates that the star formation triggered by a collision is not continuous in time or space through the impacted disk, but rather occurs in distinct episodes.
Conclusions
===========
The studies described above demonstrate that a careful comparison between high resolution observations and detailed models can yield insight into the sequence of star formation that takes place in a gas-rich galaxy after a major collision. The relative velocities between the two galaxies can be high, as illustrated by the Arp 119 system and the observations of compact groups (Sulentic 2000), and the first encounter can produce considerable morphological change and trigger star formation. The time for a merger of the members of the pair or group may be very large, but the star formation can be triggered early, and perhaps often, by collisions.
The intensity and location of the starburst at any particular epoch will depend upon the speed with which density waves are propagating through the expanding disk. Such quantities can now be predicted quite accurately from current models of colliding galaxies. Thus star formation on the scale of several hundred parsecs, as it occurs in these systems at the current epoch, can now be investigated more thoroughly. The rate of galaxy collisions in the past was larger than it is today, so a considerable portion of the star formation that took place in disk galaxies at earlier epochs was likely triggered by galaxy collisions. We expect, therefore, that studies like the ones summarized here will help in understanding this earlier star formation and its current consequences.
A longer overview of studies of this type, accompanied by a video illustrating some of the results, is given in Lamb, Hearn, & Gerber (2000).
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---
abstract: 'Let ${\mathrm{R}}$ be a real closed field and ${\mathrm{D}}\subset {\mathrm{R}}$ an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincaré characteristic of real algebraic as well as semi-algebraic subsets of ${\mathrm{R}}^k$, which are defined by symmetric polynomials with coefficients in ${\mathrm{D}}$. We give algorithms for computing the generalized Euler-Poincaré characteristic of such sets, whose complexities measured by the number the number of arithmetic operations in ${\mathrm{D}}$, are polynomially bounded in terms of $k$ and the number of polynomials in the input, assuming that the degrees of the input polynomials are bounded by a constant. This is in contrast to the best complexity of the known algorithms for the same problems in the non-symmetric situation, which are singly exponential. This singly exponential complexity for the latter problem is unlikely to be improved because of hardness result ($\#\mathbf{P}$-hardness) coming from discrete complexity theory.'
address:
- |
Department of Mathematics\
Purdue University, West Lafayette\
USA
- |
Aalto Science Institute\
Aalto University, Espoo\
Finland
author:
- Saugata Basu
- Cordian Riener
bibliography:
- 'master.bib'
title: 'Efficient algorithms for computing the Euler-Poincar[é]{} characteristic of symmetric semi-algebraic sets'
---
[^1]
Introduction
============
Let ${\mathrm{R}}$ be a real closed field which is fixed for the remainder of the paper, and let ${\mathrm{C}}$ denote the algebraic closure of ${\mathrm{R}}$. Given a semi-algebraic set $S\subset {\mathrm{R}}^k$, i.e., a set defined by unions and intersections of polynomial inequalities, it is a fundamental question of computational algebraic geometry to compute topological information about $S$. This problem of designing efficient algorithms for computing topological invariants – such as the Betti numbers as well as the Euler-Poincar[é]{} characteristic – has a long history. The first algorithms [[@SS]]{} used the technique of cylindrical algebraic decomposition and consequently had doubly exponential complexity. Algorithms for computing the zeroth Betti number (i.e. the number of semi-algebraically connected components) of semi-algebraic sets using the critical points method were discovered later [[@Canny87; @GR92; @GV92; @BPR99]]{} and improving this complexity bound remains an active area of research even today. Later, algorithms with singly exponential complexity for computing the first Betti number [[@BPRbettione]]{}, as well as the first few Betti numbers [[@Bas05-first]]{} were discovered. Algorithms with singly exponential complexity for computing the Euler-Poincaré characteristic are also known [[@BPR-euler-poincare]]{}. It remains an open problem to design an algorithm with singly exponential complexity for computing all the Betti numbers of a given semi-algebraic set. Algorithms with polynomially bounded complexity for computing the Betti numbers of semi-algebraic sets are known in a few cases – for example, for sets defined by a few (i.e. any constant number of) quadratic inequalities [@BP'R07joa; @BP'R07jems]. Also note that the problem of expressing the Euler-Poincaré characteristic of real algebraic varieties in terms of certain algebraic invariants of the polynomials defining the variety has been considered by several other authors (see for example [[@Dutertre2003]]{} and [[@Szafraniec94]]{}). But these studies do not take into account the computational complexity aspect of the problem.
In this article we will restrict to the case when $S$ is defined by *symmetric* polynomial inequalities whose degree is at most $d\in{\mathbb{N}}$, which we will think of as a fixed constant. It is known [@Timofte03; @Riener] that in this particular setup one can decide emptiness of $S$ in a time which is polynomial in $k$ - the number of variables. Despite being a rather basic property, it is known to be a $\textbf{NP}$-hard problem (in the Blum-Shub-Smale model) to decide if a given real algebraic variety $V \subset {\mathrm{R}}^{k}$ defined by one polynomial equation of degree at most $4$ is empty or not [[@BSS89]]{}. Following this notable difference it is natural to ask, if in general symmetric semi-algebraic sets are algorithmically more tractable than general semi-algebraic sets and if it is possible to obtain polynomial time (for fixed degree $d$) algorithms for computing topological invariants of such sets. In this article we answer this in the affirmative for the problem of computing the generalized Euler-Poincar[é]{} characteristic (both the ordinary as well as the equivariant versions) of symmetric semi-algebraic sets. The problem of computing the generalized Euler-Poincar[é]{} characteristic is important in several applications both theoretical and practical. For example, such an algorithm is a key ingredient in computing the integral (with respect to the Euler-Poincar[é]{} measure) of constructible functions, and this latter problem has been of recent interest in several applications [@Ghrist2010].
Before proceeding further we first fix some notation.
For $P \in {\mathrm{R}}[X_{1}, \ldots,X_{k} ]$ (respectively $P \in {\mathrm{C}}[ X_{1},
\ldots,X_{k} ]$) we denote by ${\mathrm{Zer}}(P, {\mathrm{R}}^{k})$ (respectively, ${\mathrm{Zer}}(P,{\mathrm{C}}^k)$) the set of zeros of $P$ in ${\mathrm{R}}^{k}$ (respectively, ${\mathrm{C}}^k$). More generally, for any finite set $\mathcal{P} \subset {\mathrm{R}}[ X_{1}, \ldots,X_{k} ]$ (respectively, $\mathcal{P} \subset {\mathrm{C}}[ X_{1}, \ldots,X_{k} ]$), we denote by ${\mathrm{Zer}}(\mathcal{P},{\mathrm{C}}^k)$ ${\mathrm{Zer}}(\mathcal{P}, {\mathrm{R}}^{k})$ (respectively, ${\mathrm{Zer}}(\mathcal{P},{\mathrm{C}}^k)$) the set of common zeros of $\mathcal{P}$ in ${\mathrm{R}}^{k}$ (respectively, ${\mathrm{C}}^k$).
Let $\mathcal{P}\subset {\mathrm{R}}[ X_{1}, \ldots,X_{k} ]$ be a finite family of polynomials.
1. We call any Boolean formula $\Phi$ with atoms, $P \; \sim \; 0, P \in \mathcal{P}$, where $\sim$ is one of $=,>,$ or $<$, to be a *$\mathcal{P}$-formula*. We call the realization of $\Phi$, namely the semi-algebraic set $$\begin{aligned}
{\mathrm{Reali}}(\Phi, {\mathrm{R}}^{k}) & = & \{ {\mathbf{x}}\in {\mathrm{R}}^{k} \mid
\Phi ({\mathbf{x}}) \}
\end{aligned}$$ a *$\mathcal{P}$-semi-algebraic set*.
2. \[not:sign-condition\] We call an element $\sigma \in \{
0,1,-1 \}^{\mathcal{P}}$, a *sign condition* on $\mathcal{P}$. For any semi-algebraic set $Z \subset {\mathrm{R}}^{k}$, and a sign condition $\sigma \in
\{ 0,1,-1 \}^{\mathcal{P}}$, we denote by ${\mathrm{Reali}}(\sigma,Z)$ the semi-algebraic set defined by $$\left\{ {\mathbf{x}}\in Z \mid \operatorname{sign}(P ({\mathbf{x}})) = \sigma (P) ,P \in \mathcal{P} \right\},$$ and call it the *realization* of $\sigma$ on $Z$.
3. We call a Boolean formula without negations, and with atoms $P \;\sim\; 0, P\in \mathcal{P}$, and $\sim$ one of $\{\leq,\geq\}$, to be a *$\mathcal{P}$-closed formula*, and we call the realization, ${\mathrm{Reali}}(\Phi, {\mathrm{R}}^{k})$, a *$\mathcal{P}$-closed semi-algebraic set*.
4. \[not:sign-condition\] For any finite family of polynomials $\mathcal{P}
\subset {\mathrm{R}}[ X_{1}, \ldots,X_{k} ]$, we call an element $\sigma \in \{
0,1,-1 \}^{\mathcal{P}}$, a *sign condition* on $\mathcal{P}$. For any semi-algebraic set $Z \subset {\mathrm{R}}^{k}$, and a sign condition $\sigma \in
\{ 0,1,-1 \}^{\mathcal{P}}$, we denote by ${\mathrm{Reali}}(\sigma,Z)$ the semi-algebraic set defined by $$\left\{ {\mathbf{x}}\in Z \mid \operatorname{sign}(P ({\mathbf{x}})) = \sigma (P) ,P \in \mathcal{P} \right\},$$ and call it the *realization* of $\sigma$ on $Z$.
5. \[7:def:sign\]\[7:not:sign\] We denote by $$\operatorname{SIGN}(\mathcal{P}) \subset {\{0,1, - 1\}^{\mathcal{P}}}$$ the set of all sign conditions $\sigma$ on $\mathcal{P}$ such that ${\mathrm{Reali}}(\sigma,{\mathrm{R}}^k) \neq \emptyset$. We call $\operatorname{SIGN}(\mathcal{P})$ the set of *realizable sign conditions of $\mathcal{P}$*.
For any semi-algebraic set $X$, and a field of coefficients ${\mathbb{F}}$, we will denote by ${\mathrm{H}}_{i} (X,{\mathbb{F}})$ the *$i$-th homology group* of $X$ with coefficients in ${\mathbb{F}}$, by $b_{i} (X,{\mathbb{F}}) = \dim_{{\mathbb{F}}} {\mathrm{H}}_{i} (X,{\mathbb{F}})$.
Note here that we work over any real closed field. Therefore the definition of homology groups is a little bit more delicate, in particular because ${\mathrm{R}}$ might be non-archimedean. In case of a closed and bounded semi-algebraic set, $S$ the homology ${\mathrm{H}}_{i} (S,{\mathbb{F}})$ can be defined as the $i$-th simplicial homology group associated to a semi-algebraic triangulation of $S$. The general case then is taken care of by constructing to a general semi-algebraic set $S$ a semi-algebraic set $S'$, which is closed, bounded, and furthermore semi-algebraically homotopy equivalent to $S$. We refer the reader to [@BPRbook3 Chapter 6] for details of this construction. The topological Euler-Poincar[é]{} characteristic of a semi-algebraic set $S \subset {\mathrm{R}}^{k}$ is the alternating sum of the Betti numbers of $S$. More precisely,
$$\begin{aligned}
\chi^{\mathrm{top}}(S,{\mathbb{F}}) & = & \sum_{i} (-1)^{i} \dim_{{\mathbb{F}}} {\mathrm{H}}_{i}
(S,{\mathbb{F}}).
\end{aligned}$$
For various applications (such as in motivic integration [@Cluckers-Loeser] and other applications of Euler integration [@Schapira91; @Schapira95; @Viro-euler; @Ghrist2010]) the generalized Euler-Poincar[é]{} characteristic has proven to be more useful than the ordinary Euler-Poincaré characteristic. The main reason behind the usefulness of the generalized Euler-Poincar[é]{} characteristic of a semi-algebraic set is its additivity property, which is not satisfied by the topological Euler-Poincar[é]{} characteristic. The generalized Euler-Poincar[é]{} characteristic agrees with the topological Euler-Poincar[é]{} characteristic for compact semi-algebraic sets, but can be different for non-compact ones (see Example \[eg:Euler\]). Nevertheless, the generalized Euler-Poincar[é]{} characteristic is intrinsically important because of the following reason.
The Grothendieck group $K_{0} (\mathbf{s}\mathbf{a}_{\mathrm{R}})$ of semi-algebraic isomorphic classes of semi-algebraic sets (two semi-algebraic sets being isomorphic if there is a continuous semi-algebraic bijection between them) (see for example [@Cluckers-Loeser Proposition 1.2.1]) is isomorphic to ${\mathbb{Z}}$, and the generalized Euler-Poincar[é]{} characteristic of a semi-algebraic set can be identified with its image under the isomorphism that takes the class of a point (or any closed disk) to $1$.
\[def:ep-general\] The generalized Euler-Poincar[é]{} characteristic, $\chi^{{\mathrm{gen}}} (S)$, of a semi-algebraic set $S$ is uniquely defined by the following properties [[@Dries]]{}:
1. $\chi^{{\ensuremath{\operatorname{gen}}}}$ is invariant under semi-algebraic homeomorphisms.
2. $\chi^{{\ensuremath{\operatorname{gen}}}}$ is multiplicative, i.e. $\chi^{{\ensuremath{\operatorname{gen}}}} (A
\times B) = \chi^{{\ensuremath{\operatorname{gen}}}} (A) \cdot \chi^{{\ensuremath{\operatorname{gen}}}} (B)$.
3. $\chi^{{\ensuremath{\operatorname{gen}}}}$ is additive, i.e. $\chi^{{\ensuremath{\operatorname{gen}}}} (A \cup B
) = \chi^{{\ensuremath{\operatorname{gen}}}} (A) + \chi^{{\ensuremath{\operatorname{gen}}}} (B) - \chi^{{\ensuremath{\operatorname{gen}}}}
(A \cap B)$.
4. $\chi^{{\ensuremath{\operatorname{gen}}}} ([ 0,1 ]) =1$.
The following examples are illustrative.
\[eg:Euler\] For every $n \geq 0$, $$\begin{aligned}
\chi^{{\ensuremath{\operatorname{gen}}}}([0,1]^n) &= \chi^{{\ensuremath{\operatorname{gen}}}}([0,1])^n = 1, \\
\chi^{\mathrm{top}}([0,1]^n) &= 1,\\
\chi^{{\ensuremath{\operatorname{gen}}}}((0,1)^n)& = (\chi^{{\ensuremath{\operatorname{gen}}}}(0,1))^n = (\chi^{{\ensuremath{\operatorname{gen}}}}([0,1]) - \chi^{{\ensuremath{\operatorname{gen}}}}({0}) - \chi^{{\ensuremath{\operatorname{gen}}}}({1}))^n = (-1)^n, \\
\chi^{\mathrm{top}}((0,1)^n) &= 1.
\end{aligned}$$
Let $\mathfrak{S}_{{k}}$ denote the symmetric group on $k$-letters. Throughout the article we will consider the more general case of products of symmetric groups and we fix the following notation.
\[not:mult\] For $\omega\in{\mathbb{N}}$ and $\mathbf{k}= (k_{1}, \ldots,k_{\omega}) \in {\mathbb{Z}}_{>0}^{\omega}$, with $k= \sum_{i=1}^{\omega} k_{i}$ denote $\mathfrak{S}_{\mathbf{k}}=\mathfrak{S}_{{k_1}}\times\ldots\mathfrak{S}_{{k_\omega}}$. If $\omega =1$, then $k=k_{1}$, and we will denote $\mathfrak{S}_{\mathbf{k}}$ simply by $\mathfrak{S}_{k}$.
A set $X\subset{\mathrm{R}}^k$ is said to be symmetric, if it is closed under the action of $\mathfrak{S}_{\mathbf{k}}$. For such a set we will denote by $X/\mathfrak{S}_{\mathbf{k}}$ the [[*orbit space*]{}]{} of this action.
\[not:equivariant-betti\] For any $\mathfrak{S}_{\mathbf{k}}$ symmetric semi-algebraic subset $S \subset {\mathrm{R}}^{k}$ with $\mathbf{k}= (k_{1}, \ldots
,k_{\omega}) \in {\mathbb{Z}}_{>0}^{\omega}$, with $k= \sum_{i=1}^{\omega} k_{i}$, and any field ${\mathbb{F}}$, we denote
1. $ \chi^{\mathrm{top}}(S,{\mathbb{F}}) = \sum_{i \geq 0} (-1)^{i} b_{i} (
S,{\mathbb{F}}),$
2. $\chi_{\mathfrak{S}_{\mathbf{k}}} (S,{\mathbb{F}}) = \sum_{i \geq 0} (
-1)^{i} b_{\mathfrak{S}_{\mathbf{k}}}^{i} (S,{\mathbb{F}}),$
3. $\chi_{\mathfrak{S}_{\mathbf{k}}}^{{\mathrm{gen}}} (S)
= \chi^{{\mathrm{gen}}} (S/\mathfrak{S}_{\mathbf{k}}) =
\chi^{{\mathrm{gen}}} (\phi_{\mathbf{k}} (S))$.
Main result
-----------
We describe new algorithms for computing the *generalized Euler-Poincar[é]{} characteristic* (see Definition \[def:ep-general\]) of semi-algebraic sets defined in terms of symmetric polynomials. The algorithms we give here have complexity which is polynomial (for fixed degrees and the number of blocks) in the number of symmetric variables. Since for systems of equations with a finite set of solutions, the generalized Euler-Poincar[é]{} characteristic of the set of solutions coincides with its cardinality, it is easily seen that that computing the generalized Euler-Poincar[é]{} characteristic of the set of solutions of a polynomial system with a fixed degree bound is a $\#\mathbf{P}$-hard problem in general (i.e. in the non-symmetric situation). Thus, this problem is believed to be unlikely to admit a polynomial time solution.
We prove the following theorems.
\[thm:algorithm-algebraic\]Let ${\mathrm{D}}$ be an ordered domain contained in a real closed field ${\mathrm{R}}$. Then, there exists an algorithm that takes as input:
1. a tuple $\mathbf{k}= (k_{1}, \ldots,k_{\omega}) \in
{\mathbb{Z}}_{>0}^{\omega}$, with $k= \sum_{i=1}^{\omega} k_{i}$;
2. a polynomial $P \in {\mathrm{D}}[ {\mathbf{X}}^{(1)}, \ldots
,{\mathbf{X}}^{(\omega)} ]$, where each ${\mathbf{X}}^{(i)}$ is a block of $k_{i}$ variables, and $P$ is symmetric in each block of variables ${\mathbf{X}}^{(i)}$;
and computes the generalized Euler-Poincar[é]{} characteristics $$\chi^{{\mathrm{gen}}} \left({\mathrm{Zer}}\left(P, {\mathrm{R}}^{k} \right)
\right),
\chi_{\mathfrak{S}_{\mathbf{k}}}^{{\mathrm{gen}}} \left(
{\mathrm{Zer}}\left(P, {\mathrm{R}}^{k} \right) \right).$$ The complexity of the algorithm measured by the number of arithmetic operations in the ring ${\mathrm{D}}$ (including comparisons) is bounded by $(\omega k d)^{O (D)}$, where $d= \deg (P)$ and $D= \sum_{i=1}^{\omega} \min (k_{i},2d)$.
Notice that in case, $\omega =1$ and $\mathbf{k}= (k)$, the complexity is polynomial in $k$ for fixed $d$.
We have the following result in the semi-algebraic case.
\[thm:algorithm-sa\]Let ${\mathrm{D}}$ be an ordered domain contained in a real closed field ${\mathrm{R}}$. Then, there exists an algorithm that takes as input:
1. a tuple $\mathbf{k}= (k_{1}, \ldots,k_{\omega}) \in
{\mathbb{Z}}_{>0}^{\omega}$, with $k= \sum_{i=1}^{\omega} k_{i}$;
2. a set of $s$ polynomials $\mathcal{P}= \{ P_{1}, \ldots,P_{s} \}
\subset {\mathrm{D}}[ {\mathbf{X}}^{(1)}, \ldots,{\mathbf{X}}^{(\omega)} ]$, where each ${\mathbf{X}}^{(i)}$ is a block of $k_{i}$ variables, and each polynomial in $\mathcal{P}$ is symmetric in each block of variables ${\mathbf{X}}^{(i)}$ and of degree at most $d$;
3. a $\mathcal{P}$-semi-algebraic set $S$, described by $$\begin{aligned}
S & = & \bigcup_{\sigma \in \Sigma} {\mathrm{Reali}}\left(\sigma, {\mathrm{R}}^{k} \right),
\end{aligned}$$ where $\Sigma \subset \{ 0,1,-1 \}^{\mathcal{P}}$;
and computes the generalized Euler-Poincar[é]{} characteristics $
\chi^{{\mathrm{gen}}} (S),
\chi_{\mathfrak{S}_{\mathbf{k}}}^{{\mathrm{gen}}} (S).
$ The complexity of the algorithm measured by the number of arithmetic operations in the ring ${\mathrm{D}}$ (including comparisons) is bounded by $${\mathrm{card}}(
\Sigma)^{O (1)} + s^{D'} k^{d } d^{O (D'D'')} + s^{D'} d^{O(D'')} ( k \omega D)^{O (D''')},$$ where $D=d (D'' \log d+ D'\log s))$, $D' = \sum_{i=1}^{\omega} \min (k_{i},d)$, $D'' = \sum_{i=1}^{\omega} \min (k_{i},d)$, and $D''' = \sum_{i=1}^\omega \min(k_i, 2D)$.
The algorithm also involves the inversion matrices of size $s^{D'} d^{O (D'')}$ with integer coefficients.
Notice that the complexity in the semi-algebraic case is still polynomial in $k$ for fixed $d$ and $s $ in the special case when $\omega =1$, and $\mathbf{k}= (k)$. Also note that, as a consequence of Proposition \[prop:SIGN\] below, the number of sign conditions with non-empty realizations in $\Sigma$ is bounded by $s^{D'} d^{O (D'')}$.
An important point to note is that we give algorithms for computing both the ordinary as well as the equivariant generalized Euler-Poincar[é]{} characteristics. For varieties or semi-algebraic sets defined by symmetric polynomials with degrees bounded by a constant, the ordinary generalized Euler-Poincar[é]{} characteristic can be exponentially large in the dimension $k$. Nevertheless, our algorithms for computing it have complexities which are bounded polynomially in $k$ for fixed degree.
Outline of the main techniques
------------------------------
Efficient algorithms (with singly exponential complexity) for computing the Euler-Poincaré characteristics of semi-algebraic sets [@BPRbook3 Chapter 13] usually proceed by first making a deformation to a set defined by one inequality with smooth boundary and non-degenerate critical points with respect to some affine function. Furthermore, the new set is homotopy equivalent to the given variety and the Euler-Poincaré characteristic of this new set can be computed from certain local data in the neighborhood of each critical point (see [@BPRbook3 Chapter 13] for more detail). Since the number of critical points is at most singly exponential in number, such algorithms have a singly exponential complexity.
The approach used in this paper for computing the Euler-Poincaré characteristics for symmetric semi-algebraic sets is similar – but differs on two important points. Firstly, unlike in the general case, we are aiming here for an algorithm with polynomial complexity (for fixed $d$). This requires that the perturbation, as well as the Morse function both need to be equivariant. The choices are more restrictive (see Proposition \[prop:non-degenerate\]).
Secondly, the topological changes at the Morse critical points need to be analyzed more carefully (see Lemmas \[lem:equivariant\_morseA\] and \[lem:equivariant\_morseB\]). The main technical tool that makes the good dependence on the degree $d$ of the polynomial possible is the so called “half-degree principle” [[@Riener; @Timofte03]]{} (see Proposition \[prop:half-degree\]), and this is what we use rather than the Bezout bound to bound the number of (orbits of) critical points. The proofs of these results appear in [@BC2013], where they are used to prove bounds on the equivariant Betti numbers of semi-algebraic sets.
Using these results, we prove exact formulas for the ordinary as well as the equivariant Euler-Poincaré characteristic of symmetric varieties (see and in Theorem \[thm:equivariant-ep\]), which form the basis of the algorithms described in this paper.
We adapt several non-equivariant algorithms from [@BPRbook3] to the equivariant setting. The proofs of correctness of the algorithms described for computing the ordinary as well as the equivariant (generalized) Euler-Poincar[é]{} characteristics of algebraic as well as semi-algebraic sets (Algorithms \[alg:ep-general-BM\], \[alg:ep-sign-conditions\] and \[alg:ep-sa\]) follow from the equivariant Morse lemmas (Lemmas \[lem:equivariant\_morseA\] and \[lem:equivariant\_morseB\]). The complexity analysis follows from the complexities of similar algorithms in the non-equivariant case [@BPRbook3], but using the half-degree principle referred to above. In the design of Algorithms \[alg:ep-general-BM\], \[alg:ep-sign-conditions\] and \[alg:ep-sa\] we need to use several subsidiary algorithms which are closely adapted from the corresponding algorithms in the non-equivariant situation described in [@BPRbook3]. In particular, one of them, an algorithm for computing the set of realizable sign conditions of a family of symmetric polynomial (Algorithm \[alg:sampling\]), whose complexity is polynomial in the dimension for fixed degree could be of independent interest.
The rest of the paper is organized as follows. In §\[sec:prelim\], we recall certain facts from real algebraic geometry and topology that are needed in the algorithms described in the paper. These include definitions of certain real closed extensions of the ground field ${\mathrm{R}}$ consisting of algebraic Puiseux series with coefficients in ${\mathrm{R}}$. We also recall some basic additivity properties of the Euler-Poincaré characteristic. In §\[sec:deformation\], we define certain equivariant deformations of symmetric varieties and state some topological properties of these deformations, that mirror similar ones in the non-equivariant case. The proofs of these properties appear in [@BC2013] and we give appropriate pointers where they can be found in that paper. In §\[sec:algorithms\] we describe the algorithms for computing the Euler-Poincar[é]{} characteristics of symmetric semi-algebraic sets proving Theorems \[thm:algorithm-algebraic\] and \[thm:algorithm-sa\].
Mathematical Preliminaries {#sec:prelim}
==========================
In this section we recall some basic facts about real closed fields and real closed extensions.
Real closed extensions and Puiseux series
-----------------------------------------
We will need some properties of Puiseux series with coefficients in a real closed field. We refer the reader to [@BPRbook3] for further details.
For ${\mathrm{R}}$ a real closed field we denote by ${\mathrm{R}}\left\langle {\varepsilon}\right\rangle$ the real closed field of algebraic Puiseux series in ${\varepsilon}$ with coefficients in ${\mathrm{R}}$. We use the notation ${\mathrm{R}}\left\langle {\varepsilon}_{1},
\ldots, {\varepsilon}_{m} \right\rangle$ to denote the real closed field ${\mathrm{R}}\left\langle {\varepsilon}_{1} \right\rangle \left\langle {\varepsilon}_{2} \right\rangle
\cdots \left\langle {\varepsilon}_{m} \right\rangle$. Note that in the unique ordering of the field ${\mathrm{R}}\left\langle {\varepsilon}_{1}, \ldots, {\varepsilon}_{m}
\right\rangle$, $0< {\varepsilon}_{m} \ll {\varepsilon}_{m-1} \ll \cdots \ll {\varepsilon}_{1} \ll 1$.
For elements $x \in {\mathrm{R}}\left\langle {\varepsilon}\right\rangle$ which are bounded over ${\mathrm{R}}$ we denote by $\lim_{{\varepsilon}} x$ to be the image in ${\mathrm{R}}$ under the usual map that sets ${\varepsilon}$ to $0$ in the Puiseux series $x$.
\[not:extension\] If ${\mathrm{R}}'$ is a real closed extension of a real closed field ${\mathrm{R}}$, and $S
\subset {\mathrm{R}}^{k}$ is a semi-algebraic set defined by a first-order formula with coefficients in ${\mathrm{R}}$, then we will denote by $\operatorname{Ext}(S, {\mathrm{R}}') \subset {\mathrm{R}}'^{k}$ the semi-algebraic subset of ${\mathrm{R}}'^{k}$ defined by the same formula. It is well-known that $\operatorname{Ext}(S, {\mathrm{R}}')$ does not depend on the choice of the formula defining $S$ [@BPRbook3].
\[not:ball\] For ${\mathbf{x}}\in {\mathrm{R}}^{k}$ and $r \in {\mathrm{R}}$, $r>0$, we will denote by $B_{k} ({\mathbf{x}},r)$ the open Euclidean ball centered at ${\mathbf{x}}$ of radius $r$, and we denote by $S^{k-1}({\mathbf{x}},r)$ the sphere of radius $r$ centered at ${\mathbf{x}}$. If ${\mathrm{R}}'$ is a real closed extension of the real closed field ${\mathrm{R}}$ and when the context is clear, we will continue to denote by $B_{k} ({\mathbf{x}},r)$ (respectively, $S^{k-1}({\mathbf{x}},r)$) the extension $\operatorname{Ext}(B_{k} ({\mathbf{x}},r), {\mathrm{R}}')$ (respectively, $\operatorname{Ext}(S^{k-1}({\mathbf{x}},r),{\mathrm{R}}')$). This should not cause any confusion.
Tarski-Seidenberg transfer principle
------------------------------------
In some proofs that involve Morse theory (see for example the proof of Lemma \[lem:equivariant\_morseB\]), where integration of gradient flows is used in an essential way, we first restrict to the case ${\mathrm{R}}=\mathbb{R}$. After having proved the result over $\mathbb{R}$, we use the Tarski-Seidenberg transfer theorem to extend the result to all real closed fields. We refer the reader to [@BPRbook3 Chapter 2] for an exposition of the Tarski-Seidenberg transfer principle.
Additivity property of the Euler-Poincaré characteristics
---------------------------------------------------------
We need the following additivity property of the Euler-Poincaré characteristics that follow from the Mayer-Vietoris exact sequence.
\[prop:MV\]If $S_{1},S_{2}$ are closed semi-algebraic sets, then for any field ${\mathbb{F}}$ and every $i \geq 0,$ $$\begin{aligned}
\chi^{\mathrm{top}}(S_{1} \cup S_{2},{\mathbb{F}}) & = & \chi^{\mathrm{top}}(S_{1},{\mathbb{F}}) +
\chi^{\mathrm{top}}(S_{2},{\mathbb{F}}) - \chi^{\mathrm{top}}(S_{1} \cap S_{2},{\mathbb{F}}).
\label{eqn:MV2-ep}
\end{aligned}$$
See for example [@BPRbook3 Proposition 6.36].
We also recall the definition of the Borel-Moore homology groups of locally closed semi-algebraic sets and some of its properties.
Borel-Moore homology groups
---------------------------
\[def:BM\]Let $S \subset {\mathrm{R}}^{k}$ be a locally closed semi-algebraic set and let $S_{r} =S \cap B_{k} (0,r)$. The *$p$-th Borel-Moore homology group* of $S$ with coefficients in a field ${\mathbb{F}}$, denoted by ${\mathrm{H}}_{p}^{ \mathrm{BM}} (S,{\mathbb{F}})$,\[6:not-28\] is defined to be the $p$-th simplicial homology group of the pair $\left(
\overline{S_{r}}, \overline{S_{r}} \setminus S_{r} \right)$ with coefficients in ${\mathbb{F}}$, for large enough $r>0$.
\[not:ep-BM\]For any locally closed semi-algebraic set $S$ we denote $$\begin{aligned}
\chi^{ \mathrm{BM}} (S,{\mathbb{F}}) & = & \sum_{i \geq
0} (-1)^{i} \dim_{{\mathbb{F}}} {\mathrm{H}}_{i}^{ \mathrm{BM}}
(S,{\mathbb{F}}).
\end{aligned}$$
It follows immediately from the exact sequence of the homology of the pair $\left(\overline{S_{r}}, \overline{S_{r}} \setminus S_{r} \right)$ that
\[prop:BM-pair\]If $S$ is a locally closed semi-algebraic set then for all $r>0$ large enough $$\begin{aligned}
\chi^{ \mathrm{BM}} (S,\mathbb{Q}) & = & \chi^{\mathrm{top}} \left(
\overline{S_{r}},\mathbb{Q} \right) - \chi^{\mathrm{top}}(S \cap S^{k-1} (0,r)
,\mathbb{Q}).
\end{aligned}$$
It follows from the fact that $\chi^{\mathrm{BM}}(\cdot,{\mathbb{Q}})$ is additive for locally closed semi-algebraic sets (cf. [@BPRbook3 Proposition 6.60]), and the uniqueness of the valuation $\chi^{\mathrm{gen}}(\cdot)$ that:
\[prop:ep-gen-BM\] If $S$ is a locally closed semi-algebraic set, then $$\begin{aligned}
\chi^{{\mathrm{gen}}} (S) & = &
\chi^{ \mathrm{BM}} (S,\mathbb{Q}).
\end{aligned}$$ Moreover, if $S$ is a closed and bounded semi-algebraic set then, $$\chi^{{\mathrm{gen}}} (S) = \chi^{ \mathrm{BM}} (S,\mathbb{Q}) = \chi^{\mathrm{top}}(S,\mathbb{Q}).$$
The following proposition is an immediate consequence of Definition \[def:BM\], Notation \[not:ep-BM\] and Propositions \[prop:BM-pair\] and \[prop:ep-gen-BM\].
\[prop:ep-unbounded\] Let $S \subset {\mathrm{R}}^{k}$ be a closed semi-algebraic set.Then, $$\begin{aligned}
\chi^{{\mathrm{gen}}} (S) & = &
\chi^{{\mathrm{gen}}} \left(S \cap \overline{B_{k} (0,r
)} \right) - \chi^{{\mathrm{gen}}} (S \cap S^{k-1} (0,r
))
\end{aligned}$$ for all large enough $r>0$.
By the theorem on conic structure of semi-algebraic sets at infinity (see [@BPRbook3 Proposition 5.49]) we have that $S$ is semi-algebraically homeomorphic to $S \cap B_{k} (0,r)$ for all large enough $r>0$. Also, note that $S \cap \overline{B_{k} (0,r)}$ is a disjoint union of $S \cap B_{k} (0,r)$ and $S \cap S^{k-1} (0,r)$. The proposition follows from the additivity of $\chi^{{\ensuremath{\operatorname{gen}}}} (\cdot
)$.
\[cor:additive\]Let $S \subset {\mathrm{R}}^{k}$ be a $\mathcal{P}$-closed semi-algebraic set. Let $\Gamma \subset \{ 0,1,-1 \}^{\mathcal{P}}$ be the set of realizable sign conditions $\gamma$ on $\mathcal{P}$ such that ${\mathrm{Reali}}\left(\gamma, {\mathrm{R}}^{k} \right) \subset S.
$ Then, $$\begin{aligned}
\chi^{{\mathrm{gen}}} (S) & = & \sum_{\gamma \in
\Gamma} \chi^{{\mathrm{gen}}} \left({\mathrm{Reali}}\left(\gamma,
{\mathrm{R}}^{k} \right) \right).
\end{aligned}$$
Clear from the definition of the generalized Euler-Poincar[é]{} characteristic (Definition \[def:ep-general\]).
Equivariant deformation\[sec:deformation\]
==========================================
In this section we recall the definition of certain equivariant deformations of symmetric real algebraic varieties that were introduced in [@BC2013]. These are adapted from the non-equivariant case (see for example [@BPRbook3]), but keeping everything equivariant requires additional effort.
For $i\in{\mathbb{N}}$ let $p_{i}^{(k)}:=\sum_{j=1}^{(k)} X_{j}^i$ denote the $i$-th Newton sum and \[not:def\]for any $P \in {\mathrm{R}}[ X_{1}, \ldots,X_{k} ]$ we denote $$\operatorname{Def}(P, \zeta,d) = P - \zeta \left(1+ p_{d}^{(k)}
\right),$$ where $\zeta$ is a new variable.
Notice that if $P$ is symmetric in $X_{1}, \ldots,X_{k}$, so is $\operatorname{Def}(P,
\zeta,d)$.
Properties of $\operatorname{Def}(P,\zeta,d)$
---------------------------------------------
We now state some key properties of the deformed polynomial $\operatorname{Def}(P,\zeta,d)$ that will be important in proving the correctness, as well as the complexity analysis, of the algorithms presented later in the paper. Most of these properties, with the exception of the key Theorem \[thm:equivariant-ep\], have been proved in [@BC2013] and we refer the reader to that paper for the proofs. We reproduce the statements below for ease of reading and completeness of the current paper.
[@BC2013 Proposition 3] \[prop:alg-to-semialg\] Let $d \geq 0$ be even, $\mathbf{k}= (k_{1}, \ldots,k_{\omega}) \in {\mathbb{Z}}_{>0}^{\omega}$, with $k= \sum_{i=1}^{\omega} k_{i}$, and $P \in {\mathrm{R}}[ {\mathbf{X}}^{(1)}, \ldots,{\mathbf{X}}^{(\omega)} ]_{\leq d}$, where each ${\mathbf{X}}^{(i)}$ is a block of $k_{i}$ variables, such that $P$ is non-negative and symmetric in each block of variable ${\mathbf{X}}^{(i)}$. Also suppose that $V = {\mathrm{Zer}}(P, {\mathrm{R}}^{k})$ is bounded. Then, $\operatorname{Ext}(V, {\mathrm{R}}\langle \zeta \rangle^{k})$ is a semi-algebraic deformation retract of the (symmetric) semi-algebraic subset $S$ of ${\mathrm{R}}\langle \zeta \rangle^{k}$, consisting of the union of the semi-algebraically connected components of the semi-algebraic set defined by the inequality $\operatorname{Def}(P, \zeta,d) \leq 0$, which are bounded over ${\mathrm{R}}$. Hence, $\operatorname{Ext}(V,{\mathrm{R}}{\langle}\zeta{\rangle})$ is semi-algebraically homotopy equivalent to $S$. Moreover, $\phi_{\mathbf{k}} (\operatorname{Ext}(V, {\mathrm{R}}\langle \zeta \rangle^{k}))$ is semi-algebraically homotopy equivalent to $\phi_{\mathbf{k}} (S)$.
[@BC2013 Proposition 4] \[prop:non-degenerate\]Let $P \in {\mathrm{R}}[ X_{1}, \ldots,X_{k} ] $, and $d$ be an even number with $\deg (P) < d=p+1$, with $p$ a prime. Let $F=p_{1}^{(k)}(X_{1}, \ldots,X_{k})$. Let $$V_{\zeta} = {\mathrm{Zer}}\left(\operatorname{Def}(P,\zeta,d), {\mathrm{R}}\langle \zeta \rangle^{k} \right).$$ Suppose also that $\gcd (p,k) =1$. Then, the critical points of $F$ restricted to $V_{\zeta}$ are finite in number, and each critical point is non-degenerate.
For any pair $(\mathbf{k},
\boldsymbol{\ell}
$, where $\mathbf{k}= (k_{1},
\ldots,k_{\omega}) \in {\mathbb{Z}}_{>0}^{\omega}$, $k= \sum_{i=1}^{\omega}
k_{i}$, and $\boldsymbol{\ell}= (\ell_{1}, \ldots, \ell_{\omega})$, with $1
\leq \ell_{i} \leq k_{i}$, we denote by $A_{\mathbf{k}}^{\boldsymbol{\ell}}$ the subset of ${\mathrm{R}}^{k}$ defined by $$\begin{aligned}
A^{\boldsymbol{\ell}}_{\mathbf{k}} & = & \left\{ x= (x^{(1)}, \ldots x^{(
\omega)}) \mid {\mathrm{card}}\left(
\bigcup_{j=1}^{k_{i}} \{ x_{j}^{(i)} \} \right) = \ell_{i} \right\}.
\end{aligned}$$
[@BC2013 Proposition 5] \[prop:half-degree\]Let $\mathbf{k}= (k_{1}, \ldots,k_{\omega})
\in {\mathbb{Z}}_{>0}^{\omega}$, with $k= \sum_{i=1}^{\omega} k_{i}$, and $$P
\in {\mathrm{R}}[ {\mathbf{X}}^{(1)}, \ldots
,{\mathbf{X}}^{(\omega)} ],$$ where each ${\mathbf{X}}^{(i)}$ is a block of $k_{i}$ variables, such that $P$ is non-negative and symmetric in each block of variable ${\mathbf{X}}^{(i)}$ and $\deg (P) \leq d$. Let $(X_{1}
, \ldots,X_{k})$ denote the set of variables $(
{\mathbf{X}}^{(1)}, \ldots,{\mathbf{X}}^{(
\omega)})$ and let $F=p_{1}^{(k)} (X_{1}, \ldots,X_{k})$. Suppose that the critical points of $F$ restricted to $V= {\mathrm{Zer}}\left(P, {\mathrm{R}}^{k}
\right)$ are isolated. Then, each critical point of $F$ restricted to $V$ is contained in $A^{\boldsymbol{\ell}}_{\mathbf{k}}$ for some $\boldsymbol{\ell}= (\ell_{1}
, \ldots, \ell_{\omega})$ with each $\ell_{i} \leq d$.
With the same notation as in Proposition \[prop:half-degree\]:
\[cor:half-degree\] Let $\mathcal{P} \subset {\mathrm{R}}[{\mathbf{X}}^{(1)},\ldots,{\mathbf{X}}^{(k_\omega)}]$ be a finite set of polynomials, such that for each $P \in \mathcal{P}$, $P$ is non-negative and symmetric in each block of variable ${\mathbf{X}}^{(i)}$, and $\deg (P) \leq d$. Let $C$ be a bounded semi-algebraically connected component of ${\mathrm{Zer}}(\mathcal{P}, {\mathrm{R}}^k)$. Then, $C \cap A^{\boldsymbol{\ell}}_{\mathbf{k}} \neq \emptyset$, for some $\boldsymbol{\ell}= (\ell_{1}
, \ldots, \ell_{\omega})$, where for each $i, 1 \leq i \leq \omega$, $1 \leq \ell_{i} \leq 2d$.
Let $d'$ be the least even number such that $d' >d$ and such that $d' -1$ is prime. By Bertrand’s postulate we have that $d' \leq 2d$. Now, if $p$ divides $k$, replace each $P \in \mathcal{P}$ by the polynomial $$P {}{}+X_{k+1}^{2},$$ and let $\omega' = \omega +1$, $k' =k+1$, and $\mathbf{k}' = (\mathbf{k},1
)$. Otherwise, let $\omega' = \omega +1$, $k' =k$, and $\mathbf{k}' = (
\mathbf{k},0)$. In either case, we have that $\gcd (p,k') =1$, and $k'
\leq k+1$.
Let $Q = \sum_{P \in \mathcal{P}} P$, and let $V_\zeta = {\mathrm{Zer}}(\operatorname{Def}(Q,\zeta,d'),{\mathrm{R}}{\langle}\zeta{\rangle}^{k'})$. Then for every bounded semi-algebraically connected component $C$ of ${\mathrm{Zer}}(Q,{\mathrm{R}}^{k'})$, there exists a semi-algebraically connected component of $C_\zeta$ of $V_\zeta$ bounded over ${\mathrm{R}}$, such that $\lim_\zeta C_\zeta \subset C$ (see [@BPRbook3 Proposition 12.51]). Now every bounded semi-algebraically connected component $C_\zeta$ of $V_\zeta$ contains at least two critical points of the polynomial $e^{(k)}_1$ restricted to $V_\zeta$, and they are isolated by Proposition \[prop:non-degenerate\]. The corollary now follows from Proposition \[prop:half-degree\].
The next theorem which gives an exact expression for both ${\chi}(S,{\mathbb{F}})$ as well as ${\chi}(S/\mathfrak{S}_{\mathbf{k}},F)$ (where $S$ is as in Proposition \[prop:alg-to-semialg\]) is the key result needed for the algorithms in the paper. We defer its proof to the appendix.
Before stating the theorem we need to introduce a few more notation.
\[not:partition\](Partitions) We denote by $\Pi_{k}$ the set of partitions of $k$, where each partition $\pi = (\pi_{1}, \pi_{2}, \ldots
, \pi_{\ell}) \in \Pi_{k}$, where $\pi_{1} \geq \pi_{2} \geq \cdots \geq
\pi_{\ell} \geq 1$, and $\pi_{1} + \pi_{2} + \cdots + \pi_{\ell} =k$. We call $\ell$ the length of the partition $\pi$, and denote ${\mathrm{length}}(\pi) = \ell$.
More generally, for any tuple $\mathbf{k}= (k_{1}, \ldots,k_{\omega})
\in {\mathbb{Z}}_{>0}^{\omega}$, we will denote by ${\boldsymbol{\Pi}}_{\mathbf{k}} = \Pi_{k_{1}} \times \cdots
\times \Pi_{k_{\omega}}$, and for each ${\boldsymbol{\pi}}= (
\pi^{(1)}, \ldots, \pi^{(\omega)}) \in
{\boldsymbol{\Pi}}_{\mathbf{k}}$, we denote by ${\mathrm{length}}({\boldsymbol{\pi}}) =
\sum_{i=1}^{\omega} {\mathrm{length}}(\pi^{(i)})$. We also denote for each $\boldsymbol{\ell}= (\ell_{1}, \ldots, \ell_{\omega}) \in
{\mathbb{Z}}_{>0}^{\omega}$, $$\begin{aligned}
| \boldsymbol{\ell} | & = & \ell_{1} + \cdots + \ell_{\omega}.
\end{aligned}$$
\[noy:L-pi\] Let ${\boldsymbol{\pi}}\in
{\boldsymbol{\Pi}}_{\mathbf{k}}$ where $\mathbf{k}= (k_{1},
\ldots,k_{\omega}) \in {\mathbb{Z}}_{>0}^{\omega}$, with $k=
\sum_{i=1}^{\omega} k_{i}$.
For $1 \leq i \leq \omega$, and $1 \leq j \leq
{\mathrm{length}}(\pi^{(i)})$, let $L_{\pi^{(i
)}_{j}} \subset {\mathrm{R}}^{k}$ be defined by the equations $$\begin{aligned}
X^{(i)}_{\pi_{1}^{(i)} + \cdots + \pi_{j-1}^{(i)} +1} & = \cdots = &
X^{(i)}_{\pi_{1}^{(i)} + \cdots + \pi_{j}^{(i)}},
\end{aligned}$$ and let $$\begin{aligned}
L_{{\boldsymbol{\pi}}} & = & \bigcap_{1 \leq i \leq \omega}
\bigcap_{1 \leq j \leq {\mathrm{length}}(\pi^{(i)})}
L_{\pi^{(i)}_{j}}.
\end{aligned}$$
\[not:L-fixed\] Let $L \subset {\mathrm{R}}^{k}$ be the subspace defined by $\sum_{i} X_{i} =0$, and ${\boldsymbol{\pi}}= (\pi^{(1)},
\ldots, \pi^{(\omega)}) \in {\boldsymbol{\Pi}}_{\mathbf{k}}$. Let for each $i$, $1 \leq i \leq \omega$, $\pi^{(i)} = (\pi^{(i)}_{1},
\ldots, \pi^{(i)}_{\ell_{i}})$, and for each $j,1 \leq j \leq \ell_{i}
,$ let $L^{(i)}_{j}$ denote the subspace $L \cap L_{\pi^{(i)}_{j}}$ of $L$, and $M^{(i)}_{j}$ the orthogonal complement of $L^{(i)}_{j}$ in $L$. We denote $$L_{{\mathrm{fixed}}} =L \cap L_{{\boldsymbol{\pi}}}.$$
We have the following theorem which gives an exact expression for the Euler-Poincar[é]{} characteristic of a symmetric semi-algebraic set defined by one polynomial inequality satisfying the same conditions as in Lemmas \[lem:equivariant\_morseA\] and \[lem:equivariant\_morseB\] above. The proof of the theorem which depends on the properties of $\operatorname{Def}(P,\zeta,d)$ stated above is given in §\[sec:appendix\].
\[thm:equivariant-ep\]Let $\mathbf{k}= (k_{1}, \ldots,k_{\omega})
\in {\mathbb{Z}}_{>0}^{\omega}$, with $k= \sum_{i=1}^{\omega} k_{i}$, and let $S \subset {\mathrm{R}}^{k}$ be a bounded symmetric basic semi-algebraic set defined by $P \leq 0$, where $P \in {\mathrm{R}}[ {\mathbf{X}}^{(1)},\ldots,{\mathbf{X}}^{(\omega)} ]^{\mathfrak{S}_{\mathbf{k}}}$. where each ${\mathbf{X}}^{(i)}$ is a block of $k_{i}$ variables.
Let $W= {\mathrm{Zer}}(P, {\mathrm{R}}^{k})$ be non-singular and bounded. Let $(X_{1}, \ldots,X_{k})$ denote the variables $({\mathbf{X}}^{(1)}, \ldots
,{\mathbf{X}}^{(\omega)})$ and suppose that $F=p_{1}^{(k)}(X_{1}, \ldots,X_{k})$ restricted to $W$ has a finite number of critical points, all of which are non-degenerate. Let $C$ be the finite set of critical points ${\mathbf{x}}$ of $F$ restricted to $W$ such that $\sum_{1 \leq i \leq
k} \dfrac{\partial P}{\partial X_{i}} ({\mathbf{x}}) <0$, and let ${\mathrm{Hess}}({\mathbf{x}})$ denote the Hessian of $F$ restricted to W at ${\mathbf{x}}$. Then, for any field of coefficients ${\mathbb{F}}$, $$\begin{aligned}
\label{eqn:equivariant-ep1}
&&\\
\nonumber
\chi^{\mathrm{top}}(S,{\mathbb{F}}) & = & \sum_{{\boldsymbol{\pi}}= (\pi^{(
1)}, \ldots, \pi^{(\omega)}) \in
{\boldsymbol{\Pi}}_{\mathbf{k}}} \sum_{{\mathbf{x}}\in C \cap
L_{{\boldsymbol{\pi}}}} (-1
)^{{\mathrm{ind}}^{-} ({\mathrm{Hess}}({\mathbf{x}}))} \binom{k_{1}}{\pi^{(1)}} \cdots \binom{k_{\omega}}{\pi^{(
\omega)}}, \\
\label{eqn:equivariant-ep2}
&& \\
\nonumber
\chi_{\mathfrak{S}_{k}} (S,{\mathbb{F}}) & = &
\sum_{{\boldsymbol{\pi}}\in
{\boldsymbol{\Pi}}_{\mathbf{k}}} \sum_{{\mathbf{x}}\in C \cap
L_{{\boldsymbol{\pi}}},L^{-} ({\mathbf{x}}) \subset
L_{{\mathrm{fixed}}}} (-1
)^{{\mathrm{ind}}^{-} ({\mathrm{Hess}}({\mathbf{x}}))},
\end{aligned}$$ (where for $\pi = (\pi_{1}, \ldots, \pi_{\ell}) \in \Pi_{k}$, $\binom{k}{\pi}$ denotes the multinomial coefficient $\binom{k}{\pi_{1},
\ldots, \pi_{\ell}}$).
See §\[sec:appendix\] (Appendix).
Theorem \[thm:equivariant-ep\] is illustrated by the following simple example.
![\[fig:figure1\]](example.jpg)
In this example, the number of blocks $\omega =1$, and $k=k_{1} =2$. Consider the polynomial $$\begin{aligned}
P & = & (X_{1}^{2} -1)^{2} + (X_{2}^{2} -1)^{2} - {\varepsilon},
\end{aligned}$$ for some small ${\varepsilon}>0$. The sets ${\mathrm{Zer}}\left(P, {\mathrm{R}}^{2} \right)$, and $S=
\left\{ x \in {\mathrm{R}}\langle \zeta \rangle^{2} \mid \bar{P} \leq 0 \right\}$, where $\bar{P} = \operatorname{Def}(P, \zeta,6)$ is shown in the Figure \[fig:figure1\].
The polynomial $p_{1}^{(2)}(X_{1},X_{2}) =X_{1} +X_{2}$ has $16$ critical points, corresponding to $12$ critical values, $v_{1} < \cdots <v_{12}$, on ${\mathrm{Zer}}\left(\bar{P}, {\mathrm{R}}\langle \zeta \rangle^{2} \right)$ of which $v_{5}$ and $v_{9}$ are indicated in Figure \[fig:figure1\] using dotted lines. The corresponding indices of the critical points, the number of critical points for each critical value, the sign of the polynomial $\dfrac{\partial
\bar{P}}{\partial X_{1}} + \dfrac{\partial \bar{P}}{\partial X_{2}}$ at these critical points, and the partition $\pi \in \Pi_{2}$ such that the corresponding critical points belong to $L_{\pi}$ are shown in Table \[tab:table1\]. The critical points corresponding to the shaded rows are then the critical points where $\left(\dfrac{\partial \bar{P}}{\partial
X_{1}} + \dfrac{\partial \bar{P}}{\partial X_{2}} \right) <0$, and these are the critical points which contribute to the sums in Eqns. and .
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Critical values Index $ \operatorname{SIGN}\left( $\pi$ $L^{-} (p)$ $L_{{\mathrm{fixed}}}$ $L^{-} (p) \subset
\dfrac{\partial \bar{P}}{\partial X_{1}} + \dfrac{\partial L_{{\mathrm{fixed}}}$
\bar{P}}{\partial X_{2}} \right)$
----------------- ------- ------------------------------------------------------------------ --------- ------------- ------------------------ -----------------------------
$v_{1}$ 0 $-1$ $(2)$ $0$ $0$ yes
$v_{2}$ 0 $1$ $(2)$ 0 $0$ yes
$v_{3}$ 1 $-1$ $(2)$ $L$ $0$ no
$v_{4}$ 1 $1$ $(2)$ $L$ $0$ no
$v_{5}$ $0$ $-1$ $(1,1)$ $0$ $L$ yes
$v_{6}$ 0 1 $(1,1)$ $0$ $L$ yes
$v_{7}$ $1$ $-1$ $(1,1)$ $L$ $L$ yes
$v_{8}$ $1$ $1$ $(1,1)$ $L$ $L$ yes
$v_{9}$ $0$ $-1$ $(2)$ $0$ $0$ yes
$v_{10}$ 0 $1$ $(2)$ $0$ $0$ yes
$v_{11}$ $1$ $-1$ $(2)$ $L$ $0$ no
$v_{12}$ $1$ 1 $(2)$ $L$ $0$ no
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: \[tab:table1\]
It is now easy to verify using Eqns. and that, $$\begin{aligned}
\chi^{\mathrm{top}} \left({\mathrm{Zer}}\left(P, {\mathrm{R}}^{k} \right),\mathbb{Q} \right) & = & \chi^{\mathrm{top}}(
S,\mathbb{Q})\\
&= & (-1)^{0} \binom{2}{2} + (-1)^{1} \binom{2}{2} + (-1)^{0}
\binom{2}{1,1} \\
&& \;\; +\;\; (-1)^{1} \binom{2}{1,1} + (-1)^{0} \binom{2}{2} + (
-1)^{1} \binom{2}{2}\\
& = & 1-1+2-2+1-1=0.\\
\chi_{\mathfrak{S}_{2}} \left({\mathrm{Zer}}\left(P, {\mathrm{R}}^{k} \right),\mathbb{Q}
\right) & = & \chi_{\mathfrak{S}_{2}} (S,\mathbb{Q})\\
& = & (-1)^{0} + (-1)^{0} + (-1)^{1} + (-1)^{0}\\
& = & 2.
\end{aligned}$$
Algorithms and the proofs of the main theorems {#sec:algorithms}
==============================================
In this section we describe new algorithms for computing the (generalized) Euler-Poincar[é]{} characteristic of symmetric semi-algebraic subsets of ${\mathrm{R}}^{k}$, prove their correctness and analyze their complexities. As a consequence we prove Theorems \[thm:algorithm-algebraic\] and \[thm:algorithm-sa\].
We first recall some basic algorithms from [@BPRbook3] which we will need as subroutines in our algorithms.
Algorithmic Preliminaries
-------------------------
In this section we recall the input, output and complexities of some basic algorithms and also some notations from the book [@BPRbook3]. These algorithms will be the building blocks of our main algorithms described later.
\[2:def:Thom encoding\]Let $P \in {\mathrm{R}}[X]$ and $\sigma \in \{0,1, -
1\}^{\operatorname{Der}(P)}$, a sign condition on the set $\operatorname{Der}(P)$ of derivatives of $P$. The sign condition $\sigma$ is\[2:def:thom\] a *Thom encoding* of $x \in {\mathrm{R}}$ if $\sigma (P) =0$ and ${\mathrm{Reali}}(
\sigma) = \{x\}$, i.e. $\sigma$ is the sign condition taken by the set $\operatorname{Der}(P)$ at $x$.
\[12:def:rurassociated\]A *$k$-univariate representation* $u$ is a $k+2$-tuple of polynomials in ${\mathrm{R}}[T]$, $$u= (f(T),g(T)), \ensuremath{\operatorname{with}} g= (g_{0} (T),g_{1}
(T), \ldots,g_{k} (T)),$$ such that $f$ and $g_{0}$ are co-prime. Note that $g_{0} (t) \neq 0$ if $t
\in {\mathrm{C}}$ is a root of $f (T)$. The points *associated* to a univariate representation $u$ are the points $$x_{u} (t) = \left(\frac{g_{1} (t)}{g_{0} (t)}, \ldots, \frac{g_{k}
(t)}{g_{0} (t)} \right) \in {\mathrm{C}}^{k} \text{\label{12:eq:assoc}}$$ where $t \in {\mathrm{C}}$ is a root of $f (T)$.
Let $\mathcal{P} \subset {\mathrm{R}}[X_{1}, \ldots,X_{k} ]$ be a finite set of polynomials such that ${\mathrm{Zer}}(\mathcal{P}, {\mathrm{C}}^{k})$ is finite. The $k+2$-tuple $u= (f(T),g(T))$, [*represents*]{} ${\mathrm{Zer}}(\mathcal{P},
{\mathrm{C}}^{k})$ if $u$ is a univariate representation and $${\mathrm{Zer}}(\mathcal{P}, {\mathrm{C}}^{k}) \text{=} \left\{ x \in {\mathrm{C}}^{k} | \exists
t \in {\mathrm{Zer}}(\mathcal{f}, {\mathrm{C}}) x=x_{u} (t) \right\}.$$ A [*real k-univariate representation*]{} is a pair $u, \sigma$ where $u$ is a $k$-univariate representation and $\sigma$ is the Thom encoding of a root of $f$, $t_{\sigma} \in{{\mathrm{R}}}$. The point *associated* to the real univariate representation $u, \sigma$ is the point
$$x_{u} (t_{\sigma}) = \left(\frac{g_{1} (t_{\sigma})}{g_{0} (t_{\sigma}
)}, \ldots, \frac{g_{k} (t_{\sigma})}{g_{0} (t_{\sigma})} \right) \in
{\mathrm{R}}^{k}.$$
For the rest of this section we fix an ordered domain ${\mathrm{D}}$ contained in the real closed field ${\mathrm{R}}$. By complexity of an algorithm whose input consists of polynomials with coefficients in ${\mathrm{D}}$, we will mean (following [@BPRbook3]) the maximum number of arithmetic operations in ${\mathrm{D}}$ (including comparisons) used by the algorithm for an input of a certain size.
We will use four algorithms from the book [@BPRbook3]: namely, Algorithm 10.98 (Univariate Sign Determination), Algorithm 12.64 (Algebraic Sampling), Algorithm 12.46 (Limit of Bounded Points), and Algorithm 10.83 (Adapted Matrix). We refer the reader to [@BPRbook3] for the descriptions of these algorithms and their complexity analysis.
Computing the generalized Euler-Poincar[é]{} characteristic of symmetric real algebraic sets
--------------------------------------------------------------------------------------------
We now describe our algorithm for computing the generalized Euler-Poincar[é]{} characteristic for real varieties, starting as usual with the bounded case. Note that using Proposition \[prop:ep-gen-BM\], for a closed and bounded semi-algebraic set $S$, $$\begin{aligned}
\chi^{{\mathrm{gen}}} (S) & = & \chi^{\mathrm{top}}(S,\mathbb{Q}).\end{aligned}$$
#### Proof of correctness
The correctness of the algorithm follows from Propositions \[prop:alg-to-semialg\], \[prop:half-degree\], \[prop:non-degenerate\], Theorem \[thm:equivariant-ep\], as well as the correctness of Algorithms 12.64 (Algebraic Sampling) and Algorithm 10.98 (Univariate Sign Determination) in [@BPRbook3].
#### Complexity analysis
The complexity of Step \[alg:ep-bounded:step1\] is bounded by $d^{O (
1)}$. The complexities of Steps \[alg:ep-bounded:step2\], \[alg:ep-bounded:step3\], \[alg:ep-bounded:step5.1\] are all bounded by $(\omega
k)^{O (d)}$. Using the complexity analysis of Algorithm 12.64 (Algebraic Sampling) in [@BPRbook3], the complexity of Step \[alg:ep-bounded:step5.3\] is bounded by $({\mathrm{length}}(
{\boldsymbol{\pi}}) d)^{O ({\mathrm{length}}(
{\boldsymbol{\pi}}))}$. The number and the degrees of the real univariate representations output in Step \[alg:ep-bounded:step5.3\] are bounded by $d^{O (
{\mathrm{length}}({\boldsymbol{\pi}}))}$. The complexity of Step \[alg:ep-bounded:step5.5\] is bounded by $d^{O (
{\mathrm{length}}({\boldsymbol{\pi}}))}$ using the complexity analysis of Algorithm 10.98 (Univariate Sign Determination) in [@BPRbook3]. Each arithmetic operation in the Gauss-Jordan elimination in Step \[alg:ep-bounded:step5.6\] occurs in a ring $D [ \zeta ] [ T ] / (f (T))$ (where $u_{{\mathbf{z}}} = (
f,g_{0,} \ldots,g_{{\mathrm{length}}(
{\boldsymbol{\pi}})}), \rho_{{\mathbf{z}}}$) with $\deg_{T, \zeta} (f)
=d^{O ({\mathrm{length}}({\boldsymbol{\pi}})
)}$). The number of such operations in the ring $D [ \zeta ] [ T ] / (f (T)
)$ is bounded by $({\mathrm{length}}(
{\boldsymbol{\pi}}) +k)^{O (1)}$. Thus, the total number of arithmetic operations in the ring ${\mathrm{D}}$ performed in Step \[alg:ep-bounded:step5.6\] is bounded by $({\mathrm{length}}({\boldsymbol{\pi}}) k d
)^{O ({\mathrm{length}}({\boldsymbol{\pi}})
)}$.
The number of iterations of Step \[alg:ep-bounded:step5\] is bounded by the number of partitions ${\boldsymbol{\pi}}\in \Pi_{\mathbf{k},\boldsymbol{\ell}}$ with $\boldsymbol{\ell}= (\ell_{1}, \ldots, \ell_{\omega}),1 \leq \ell_{i} \leq \min
(k_{i},d'),1 \leq i \leq \omega$, which is bounded by $$\begin{aligned}
\sum_{\boldsymbol{\ell}= (\ell_{1}, \ldots, \ell_{\omega}),1 \leq \ell_{i}
\leq \min (k_{i},d')} p (\mathbf{k},\boldsymbol{\ell}) & = & k^{O (D)},\end{aligned}$$ where $D = \sum_{i=1}^{\omega} \min(k_i,2d)$. Thus, the total complexity of the algorithm measured by the number of arithmetic operations (including comparisons) in the ring ${\mathrm{D}}$ is bounded by $(\omega k d)^{O (D)}$.
\[alg:ep-general-BM:step1\]
\[alg:ep-general-BM:step2\]
\[alg:ep-general-BM:step3\]
#### Proof of correctness
Since $V= {\mathrm{Zer}}\left(P,
{\mathrm{R}}^{m+k} \right)$ is closed, by Proposition \[prop:ep-unbounded\] we have that $$\begin{aligned}
\nonumber
\chi^{{\ensuremath{\operatorname{gen}}}} (V) & = & \chi^{{\ensuremath{\operatorname{BM}}}} (V,\mathbb{Q}) \\
\nonumber
& = & \chi^{\mathrm{top}} \left(\operatorname{Ext}\left(V, {\mathrm{R}}\langle 1/ \Omega \rangle \right) \cap
\overline{B_{k} (0, \Omega)} \right) - \\
\nonumber
&&\chi^{\mathrm{top}} \left(\operatorname{Ext}\left(V, {\mathrm{R}}\langle 1/ \Omega \rangle \right) \cap S^{k-1} (0, \Omega) \right)
\nonumber\\
& = & \chi^{\mathrm{top}} \left(\operatorname{Ext}\left(V, {\mathrm{R}}\langle 1/ \Omega \rangle \right) \cap
\overline{B_{k} (0, \Omega)} \right) - \chi^{(2)}. \label{eqn:proof1}\end{aligned}$$ Now ${\mathrm{Zer}}\left(P_{1}, {\mathrm{R}}\langle 1/ \Omega \rangle^{k+1} \right)$ is semi-algebraically homeomorphic to two copies of $$\operatorname{Ext}\left(V, {\mathrm{R}}\langle 1/
\Omega \rangle \right) \cap \overline{B_{k} (0, \Omega)},$$ glued along a semi-algebraically homeomorphic copy of $$\operatorname{Ext}\left(V, {\mathrm{R}}\langle 1/ \Omega
\rangle \right) \cap S^{k-1} (0, \Omega) = {\mathrm{Zer}}\left(P_{2}, {\mathrm{R}}\langle 1/
\Omega \rangle^{k} \right).$$ It follows that, $$\begin{aligned}
\chi^{(1)} &=&\chi^{{\ensuremath{\operatorname{gen}}}} \left({\mathrm{Zer}}\left(P_{1}, {\mathrm{R}}\langle 1/ \Omega
\rangle^{k+1} \right) \right) \\
& = & 2 \chi^{\mathrm{top}} \left(\operatorname{Ext}\left(V, {\mathrm{R}}\langle 1/ \Omega \rangle \right)
\cap \overline{B_{k} (0, \Omega)} \right) - \chi^{{\ensuremath{\operatorname{gen}}}} \left({\mathrm{Zer}}\left(P_{2}, {\mathrm{R}}\langle 1/ \Omega \rangle^{k} \right) \right)\\
& = & 2 \chi^{\mathrm{top}} \left(\operatorname{Ext}\left(V, {\mathrm{R}}\langle 1/ \Omega \rangle \right)
\cap \overline{B_{k} (0, \Omega)} \right) - \chi^{(2)},\end{aligned}$$ and hence $$\begin{aligned}
\chi^{\mathrm{top}} \left(\operatorname{Ext}\left(V, {\mathrm{R}}\langle 1/ \Omega \rangle \right) \cap
\overline{B_{k} (0, \Omega)} \right) & = & \tfrac{1}{2} (\chi^{(1)} +
\chi^{(2)}). \label{eqn:proof2}\end{aligned}$$ It now follows from Eqns. (\[eqn:proof1\]) and (\[eqn:proof2\]) that $$\begin{aligned}
\chi^{{\ensuremath{\operatorname{gen}}}} (V) & = & \tfrac{1}{2} (\chi^{(1)} + \chi^{(2)}) -
\chi^{(2)}\\
& = & \tfrac{1}{2} (\chi^{(1)} - \chi^{(2)}).\end{aligned}$$ The proof for the correctness of the computation of $\chi^{{\ensuremath{\operatorname{gen}}}}_{\mathfrak{S}_{\mathbf{k}}} (V)$ is similar and omitted.
#### Complexity analysis
The complexity of the algorithm measured by the number of arithmetic operations (including comparisons) in the ring ${\mathrm{D}}$ is bounded by $(\omega k d)^{O (D)}$, where $D = \sum_{i=1}^\omega \min(k_i,2d)$. This follows directly from the complexity analysis of Algorithm \[alg:ep-bounded\].
The correctness and the complexity analysis of Algorithm \[alg:ep-general-BM\] prove Theorem \[thm:algorithm-algebraic\].
Computing the generalized Euler-Poincar[é]{} characteristic of symmetric semi-algebraic sets
--------------------------------------------------------------------------------------------
We now consider the problem of computing the (generalized) Euler-Poincar[é]{} characteristic of semi-algebraic sets. We reduce the problem to computing the generalized Euler-Poincar[é]{} characteristic of certain symmetric algebraic sets for which we already have an efficient algorithm described in the last section. This reduction process follows very closely the spirit of a similar reduction that is used in an algorithm for computing the generalized Euler-Poincar[é]{} characteristic of the realizations of all realizable sign conditions of a family of polynomials given in [[@BPR-euler-poincare]]{} (see also [@BPRbook3]).
We first need an efficient algorithm for computing the set of realizable sign conditions of a family of symmetric polynomials which will be used later. The following algorithm can be considered as an equivariant version of a very similar algorithm – namely, Algorithm 13.9 (Computing Realizable Sign Conditions) in [@BPRbook3] – for solving the same problem in the non-equivariant case.
$\mathcal{P}^{\star}_{i} \gets \{ P_{i} \pm \gamma \delta_{i},P_{i} \pm \delta_{i} \}$.
\[alg:sampling:step3\]
#### Proof of correctness
We first need a lemma whose proof can be found in [@BC2013].
For any finite family $\mathcal{P} \subset {\mathrm{R}}[ X_{1}, \ldots,X_{k} ]$ and $\ell \geq 0$, we say that $\mathcal{P}$ is in $\ell$-general position with respect to a semi-algebraic set $V \subset {\mathrm{R}}^{k}$ if for any subset $\mathcal{P}' \subset \mathcal{P}$, with ${\mathrm{card}}(
\mathcal{P}') > \ell$, ${\mathrm{Zer}}(\mathcal{P}',V) = \emptyset$.
Let $\mathbf{k}= (k_{1}, \ldots,k_{\omega})$ with $k=
\sum_{i=1}^{\omega} k_{i}$, and $$\mathcal{P} = \{ P_{1}, \ldots,P_{s} \} \subset {\mathrm{R}}[{\mathbf{X}}^{(1)}, \ldots,{\mathbf{X}}^{(\omega)} ]^{\mathfrak{S}_{\mathbf{k}}}$$ be a fixed finite set of polynomials where ${\mathbf{X}}^{(i)}$ is a block of $k_{i}$ variables. Let $\deg (P_{i}) \leq d$ for $1 \leq
i \leq s$. Let $\overline{{\varepsilon}} = \left({\varepsilon}_{1}, \ldots, {\varepsilon}_{s} \right)$ be a tuple of new variables, and let $\mathcal{P}_{\overline{{\varepsilon}}} =
\bigcup_{1 \leq i \leq s} \left\{ P_{i} \pm {\varepsilon}_{i} \right\}$.
The following lemma appears in [@BC2013].
[@BC2013 Lemma 7] \[lem:gen-pos1-with-parameters\]Let $$\begin{aligned}
D' & = & \sum_{i=1}^{\omega} \min (k_{i},d).
\end{aligned}$$ The set of polynomials $\mathcal{P}_{\overline{{\varepsilon}}} \subset {\mathrm{R}}' [
{\mathbf{X}}^{(1)}, \ldots,{\mathbf{X}}^{(
\omega)} ]$ is in $D'$-general position for any semi-algebraic subset $Z
\subset {\mathrm{R}}^{k}$ stable under the action of $\mathfrak{S}_{\mathbf{k}}$, where ${\mathrm{R}}' = {\mathrm{R}}\langle \overline{{\varepsilon}} \rangle$.
Now observe that Lemma \[lem:gen-pos1-with-parameters\] implies that the set $\bigcup_{1 \leq i \leq s} \mathcal{P}_{i}^{\star}$ is in $D'$-general position. Propositions 13.1 and 13.7 in [@BPRbook3] together imply that the image under the $\lim_{\gamma}$ map of any finite set of points meeting every bounded semi-algebraically connected component of each algebraic set defined by polynomials $$\displaylines{
Q_{i_{1}} \in \mathcal{P}_{i_{1}}^{\star}, \ldots,Q_{i_{j}} \in \mathcal{P}_{i_{j}}^{\star},\cr
Q_{i_{1}} \in \mathcal{P}_{i_{1}}^{\star}, \ldots,Q_{i_{j}} \in \mathcal{P}_{i_{j}}^{\star}, Q_0,
}$$ where $1 \leq i_{1} < \cdots <i_{j}
\leq s$, $1 \leq j \leq D'$, and $Q_0 = |{\varepsilon}|(|{\mathbf{X}}^{(1)}|^2+ \cdots + |{\mathbf{X}}^{(\omega)}|^2)-1$, will intersect every semi-algebraically connected component of ${\mathrm{Reali}}\left(\sigma, {\mathrm{R}}^{k} \right)$ for every $\sigma \in {\ensuremath{\operatorname{SIGN}}} (\mathcal{P})$.
Moreover, noticing that the degrees of the polynomials $Q_{i_j}$ above are bounded by $2d$, it follows from Corollary \[cor:half-degree\] that each semi-algebraically connected component of the algebraic sets listed above has a non-empty intersection with $A^{\boldsymbol{\ell}}_{{\mathbf{k}}}$, for some $\boldsymbol{\ell}= (\ell_1,\ldots,\ell_\omega)$, and $1 \leq \ell_1 \leq \min(k_i,4d), 1\leq i \leq \omega$.
The correctness of the algorithm now follows from the correctness of Algorithm 12.64 (Algebraic Sampling) and Algorithm 10.98 (Univariate Sign Determination) in [@BPRbook3].
#### Complexity analysis
The complexity of Step \[alg:sampling:step2\] measured by the number of arithmetic operations in the ring ${\mathrm{D}}[ \delta_{1}, \ldots, \delta_{s}, \gamma ]$ is bounded by $$O
\left(D'\binom{k+d}{k} \right),$$ where $D' = \sum_{i=1}^{\omega} \min(k_i,d)$.
It follows from the complexity analysis of Algorithm 12.64 (Algebraic Sampling) in [@BPRbook3] that each call to Algorithm 12.64 (Algebraic Sampling) in Step \[alg:sampling:step2.2.2\] requires $d^{O ({\mathrm{length}}({\boldsymbol{\pi}}))}$ arithmetic operations in the ring ${\mathrm{D}}[{\varepsilon}, \delta_{1}, \ldots, \delta_{s},
\gamma ]$. The number and degrees of the real univariate representations $u_{{\boldsymbol{\pi}},i}$ output in Step \[alg:sampling:step2.2.2\] is bounded by $d^{O ({\mathrm{length}}({\boldsymbol{\pi}}))}$. Using the complexity analysis of Algorithm 12.46 (Limit of Bounded Points) in [@BPRbook3], each call to Algorithm 12.46 (Limit of Bounded Points) in Step \[alg:sampling:step2.2.3\] requires $d^{O ({\mathrm{length}}(
{\boldsymbol{\pi}}))}$ arithmetic operations in the ring ${\mathrm{D}}[{\varepsilon}, \delta_1,\ldots,\delta_s,\gamma]$, and thus the total complexity of this step in the whole algorithm across all iterations measured by the number of arithmetic operations in the ring ${\mathrm{D}}[ {\varepsilon},\delta_{1}, \ldots, \delta_{s}]$ is bounded by $$\sum_{j=1}^{D'} 2^{j} \binom{s}{j} \left(d^{O (D'')} +O
\left(D' \binom{k+d}{k} \right) \right),$$ where $D'' = \sum_{i=1}^\omega \min(k_i,4d)$, noting that ${\mathrm{length}}({\boldsymbol{\pi}}) \leq D''$.
Similarly, using the complexity analysis of Algorithm 10.98 (Univariate Sign Determination) in [@BPRbook3], each call to Algorithm 10.98 (Univariate Sign Determination) in Step \[alg:sampling:step2.2.5\] requires $d^{O ({\mathrm{length}}(
{\boldsymbol{\pi}}))}$ arithmetic operations in the ring ${\mathrm{D}}[{\varepsilon}, \delta_1,\ldots,\delta_s]$, and thus the total complexity of this step in the whole algorithm across all iterations measured by the number of arithmetic operations in the ring ${\mathrm{D}}[ {\varepsilon},\delta_{1}, \ldots, \delta_{s}]$ is bounded by
$$\sum_{j=1}^{D'} 2^{j} \binom{s}{j} \left(d^{O (D'')} +O
\left(D' \binom{k+d}{k} \right) \right).$$ However, notice that in each call to Algorithm 12.64 (Algebraic Sampling) from [@BPRbook3] in Step \[alg:sampling:step2.2.2\], to Algorithm 12.46 (Limits of Bounded Points) in [@BPRbook3] in Step \[alg:sampling:step2.2.3\], as well as and also in the calls to Algorithm 10.98 (Univariate Sign Determination) from [@BPRbook3] in Step \[alg:sampling:step2.2.5\], the arithmetic is done in a ring ${\mathrm{D}}$ adjoined with $O (D')$ infinitesimals. Hence, the total number of arithmetic operations in ${\mathrm{D}}$ is bounded by
$$\displaylines{
\sum_{j=1}^{D'} 2^{j} \binom{s}{j} \left(d^{O (
D'D'' )} +O \left(D' \binom{k+d}{k}
\right) \right) =
s^{D'} k^{d} d^{O (
D'D'')}.
}$$
The total number of real univariate representations produced in Step \[alg:sampling:step2.2.2\] is bounded by
$$\sum_{j=1}^{D'} 2^{j} \binom{s}{j} d^{O (D'')} =s^{D'} d^{O (D'')}.$$ Their degrees are bounded by $d^{O (D'')}$. Thus, the total number of real points associated to these univariate representations, and hence also $${\mathrm{card}}({\ensuremath{\operatorname{SIGN}}} (\mathcal{P})) = s^{D'} d^{O (D'')}.$$
The complexity analysis of Algorithm \[alg:sampling\] yields the following purely mathematical result.
\[prop:SIGN\] Let $\mathbf{k}= (k_{1}, \ldots,k_{\omega}) \in
{\mathbb{Z}}_{>0}^{\omega}$, with $k= \sum_{i=1}^{\omega} k_{i}$, and let $\mathcal{P}= \{ P_{1}, \ldots,P_{s} \} \subset {\mathrm{R}}[
{\mathbf{X}}^{(1)}, \ldots,{\mathbf{X}}^{(
\omega)} ]$ be a finite set of polynomials, where each ${\mathbf{X}}^{(i)}$ is a block of $k_{i}$ variables, and each polynomial in $\mathcal{P}$ is symmetric in each block of variables ${\mathbf{X}}^{(i)}$. Let ${\mathrm{card}}(\mathcal{P}) =s$, and $\max_{P \in \mathcal{P}} \deg (P) =d$. Then, $$\begin{aligned}
{\mathrm{card}}( \operatorname{SIGN}(
\mathcal{P})) & = & s^{D'} d^{O (D''
)} ,
\end{aligned}$$ where $D'= \sum_{i=1}^{\omega} \min (k_{i},d)$, and $D''= \sum_{i=1}^{\omega} \min (k_{i},4d)$.
In particular, if for each $i,1 \leq i \leq \omega $, $d \leq k_{i}$, then ${\mathrm{card}}( \operatorname{SIGN}(
\mathcal{P}))$ can be bounded independent of $k$.
Given $P \in {\mathrm{R}}[X_{1}, \ldots,X_{k} ]$, we denote $$\begin{aligned}
{\mathrm{Reali}}(P=0,S) & = & \{x \in S \hspace{0.75em} \mid \hspace{0.75em} P(x)=0\}
,\\
{\mathrm{Reali}}(P>0,S) & = & \{x \in S \hspace{0.75em} \mid \hspace{0.75em} P(x)>0\}
,\\
{\mathrm{Reali}}(P>0,S) & = & \{x \in S \hspace{0.75em} \mid \hspace{0.75em} P(x)<0\}
,
\end{aligned}$$ and ${\chi}^{{\mathrm{gen}}} (P=0,S),
{\chi}^{{\mathrm{gen}}} (P>0,S),
{\chi}^{{\mathrm{gen}}} (P<0,S)$ the Euler-Poincar[é]{} characteristics of the corresponding sets. The Euler-Poincar[é]{}-query of $P$ for $S$ is $$\operatorname{EuQ}(P,S) = {\chi}^{{\mathrm{gen}}} (P>0,S) -
{\chi}^{{\mathrm{gen}}} (P<0,S).$$ If $P$ and $S$ are symmetric we denote by $${\chi}^{{\mathrm{gen}}}_{\mathfrak{S}_{k}} (P=0,S),
{\chi}^{{\mathrm{gen}}}_{\mathfrak{S}_{k}} (P>0,S),
{\chi}^{{\mathrm{gen}}}_{\mathfrak{S}_{k}} (P<0,S)$$ the Euler-Poincar[é]{} characteristics of the corresponding sets. The equivariant Euler-Poincar[é]{}-query of $P$ for $S$ is $$\operatorname{EuQ}_{\mathfrak{S}_{k}} (P,S) =
{\chi}^{{\mathrm{gen}}}_{\mathfrak{S}_{k}} (P>0,S) -
{\chi}^{{\mathrm{gen}}}_{\mathfrak{S}_{k}} (P<0,S).$$ Let $\mathcal{P} =P_{1}, \ldots,P_{s}$ be a finite list of polynomials in ${\mathrm{R}}[X_{1}, \ldots,X_{k} ]$.
\[13:def:realization signcondition2\]Let $\sigma$ be a sign condition on $\mathcal{P}$. The *realization of the sign condition $\sigma$ over $S$* is defined by $${\mathrm{Reali}}(\sigma,S) = \{x \in S \mid
\bigwedge_{P \in \mathcal{P}} \operatorname{sign}(P(x))= \sigma (P)\},$$ and its generalized Euler-Poincar[é]{} characteristic is denoted $${\chi}^{{\mathrm{gen}}} (\sigma,S).$$ Similarly, if $P$ and $S$ are symmetric with respect to $\mathfrak{S}_{\mathbf{k}}$ for some $\mathbf{k}= (k_{1}, \ldots,k_{\omega}) \in {\mathbb{Z}}_{>0}^{\omega}$, the equivariant Euler-Poincar[é]{} characteristic of ${\mathrm{Reali}}(\sigma,S)$ is denoted $${\chi}^{{\mathrm{gen}}}_{\mathfrak{S}_{\mathbf{k}}} (
\sigma,S) :=
\chi^{{\mathrm{gen}}}_{\mathfrak{S}_{\mathbf{k}}} \left(
\phi_{\mathbf{k}} \left({\mathrm{Reali}}(\sigma,S) \right),\mathbb{Q} \right).$$
\[13:not:chichibar\]\[13:not:signdet\]Given a finite family $\mathcal{P} \subset {\mathrm{R}}[ X_{1}, \ldots,X_{k} ]$ we denote by ${\chi}^{{\mathrm{gen}}} (\mathcal{P})$ the list of generalized Euler-Poincar[é]{} characteristics $${\chi}^{{\mathrm{gen}}} (\sigma) =
{\chi}^{{\mathrm{gen}}} ({\mathrm{Reali}}(\sigma, {\mathrm{R}}^{k}))$$ for $\sigma \in \operatorname{SIGN}(\mathcal{P})$.
Given $\alpha \in \{0,1,2\}^{\mathcal{P}}$ and $\sigma \in \{0,1, - 1\}^{\mathcal{P}}$, we denote $$\sigma^{\alpha} = \prod_{P \in \mathcal{P}} \sigma (P)^{\alpha (P)},$$ and $$\mathcal{P}^{\alpha}= \prod_{P \in \mathcal{P}} P^{\alpha (P)}.$$
When ${\mathrm{Reali}}(\sigma,Z) \ne
\emptyset$, the sign of $\mathcal{P}^{\alpha}$ is fixed on ${\mathrm{Reali}}(\sigma,Z)$ and is equal to $\sigma^{\alpha}$ with the understanding that $0^{0} =1$.
We order the elements of $\mathcal{P}$ so that $\mathcal{P} = \{P_{1}, \ldots
,P_{s} \}$. We order $\{0,1,2\}^{\mathcal{P}}$ lexicographically. We also order $\{0,1, - 1\}^{\mathcal{P}}$ lexicographically (with $0 \prec 1 \prec
-1$).
Given $A= \alpha_{1}, \ldots, \alpha_{m}$, a list of elements of $\{0,1,2\}^{\mathcal{P}}$ with $\alpha_{1} <_{\operatorname{lex}} \ldots <_{\operatorname{lex}}
\alpha_{m}$, we define $$\begin{aligned}
\mathcal{P}^{A} & = & \mathcal{P}^{\alpha_{1}}, \ldots,
\mathcal{P}^{\alpha_{m}},\\
{\mathrm{EuQ}}(\mathcal{P}^{A},S) & = &
{ {\mathrm{EuQ}}(\mathcal{P}^{\alpha_{1}},S),
\ldots, {\mathrm{EuQ}}(\mathcal{P}^{\alpha_{m}},S).}\end{aligned}$$ Given $\Sigma = \sigma_{1}, \ldots, \sigma_{n}$, a list of elements of $\{0,1, - 1\}^{\mathcal{P}}$, with $\sigma_{1} <_{\operatorname{lex}} \ldots <_{\operatorname{lex}}
\sigma_{n}$,we define $$\begin{aligned}
{\mathrm{Reali}}(\Sigma,S) & = & {\mathrm{Reali}}(\sigma_{1},Z), \ldots, {\mathrm{Reali}}(\sigma_{n},Z)
,\\
\chi^{{\mathrm{gen}}} (\Sigma,S) & = &
\chi^{{\mathrm{gen}}} (\sigma_{1},Z), \ldots,
\chi^{{\mathrm{gen}}} (\sigma_{n},Z).\end{aligned}$$
We denote by $\ensuremath{\operatorname{Mat}} (A, \Sigma)$ the $m \times s$ matrix of signs of $\mathcal{P}^{A}$ on $\Sigma$ defined by $$\begin{aligned}
\ensuremath{\operatorname{Mat}} (A, \Sigma)_{i,j} & = &
\sigma_{j}^{\alpha_{i}}.\end{aligned}$$
\[13:prop:matrix of signs\]If $\cup_{\sigma \in \Sigma} {\mathrm{Reali}}(\sigma,S)
=S$, then $$\ensuremath{\operatorname{Mat}} (A, \Sigma) \cdot
{\chi}^{{\mathrm{gen}}} (\Sigma,S) = \operatorname{EuQ}(
\mathcal{P}^{A},S).$$
See [@BPRbook3 Proposition 13.44].
We consider a list $A$ of elements in $\{0,1,2\}^{\mathcal{P}}$ *adapted to sign determination* for $\mathcal{P}$ (cf. [@BPRbook3 Definition 10.72]), i.e. such that the matrix of signs of $\mathcal{P}^{A}$ over $\operatorname{SIGN}(
\mathcal{P})$ is invertible. If [$\mathcal{P} =P_{1}, \ldots,P_{s}$]{}, let ${\mathcal{P}_{i} =P_{i}, \ldots,P_{s}}$, for $0 \le i \le s$. A method for determining a list $A (\mathcal{P})$ of elements in $\{0,1,2\}^{\mathcal{P}}$ adapted to sign determination for $\mathcal{P}$ from $\operatorname{SIGN}(\mathcal{P})$ is given in Algorithm 10.83 (Adapted Matrix) in [@BPRbook3].
We are ready for describing the algorithm computing the generalized Euler-Poincar[é]{} characteristic. We start with an algorithm for the Euler-Poincar[é]{}-query.
\[12:alg:speuler:step1\]
\[12:alg:speuler:step2\]
\[12:alg:speuler:step3\]
#### Proof of correctness
The algebraic set ${\mathrm{Zer}}(Q_{+}, {\mathrm{R}}^{k+1})$ is semi-algebraically homeomorphic to the disjoint union of two copies of the semi-algebraic set defined by $(P>0) \wedge (Q=0)$, and the algebraic set defined by $(P=0) \wedge (Q=0)$. Hence, using Corollary \[cor:additive\], we have that $$\begin{aligned}
2 {\chi}^{{\ensuremath{\operatorname{gen}}}} (P>0,Z) &=& {\chi}^{{\ensuremath{\operatorname{gen}}}} ({\mathrm{Zer}}(Q_{+},
{\mathrm{R}}^{k+1})) - {\chi}^{{\ensuremath{\operatorname{gen}}}} ({\mathrm{Zer}}((Q,P), {\mathrm{R}}^{k})),\\
2 {\chi}^{{\ensuremath{\operatorname{gen}}}}_{\mathfrak{S}_{k}} (P>0,Z) &=&
{\chi}^{{\ensuremath{\operatorname{gen}}}}_{\mathfrak{S}_{k}} ({\mathrm{Zer}}(Q_{+}, {\mathrm{R}}^{k+1})) -
{\chi}^{{\ensuremath{\operatorname{gen}}}}_{\mathfrak{S}_{k}} ({\mathrm{Zer}}((Q,P), {\mathrm{R}}^{k})).\end{aligned}$$
Similarly, we have that $$\begin{aligned}
2 {\chi}^{{\ensuremath{\operatorname{gen}}}} (P<0,Z) & = & {\chi}^{{\ensuremath{\operatorname{gen}}}} ({\mathrm{Zer}}(Q_{-},
{\mathrm{R}}^{k+1})) - {\chi}^{{\ensuremath{\operatorname{gen}}}} ({\mathrm{Zer}}((Q,P), {\mathrm{R}}^{k})),\\
2 {\chi}^{{\ensuremath{\operatorname{gen}}}}_{\mathfrak{S}_{k}} (P<0,Z) & = &
{\chi}^{{\ensuremath{\operatorname{gen}}}}_{\mathfrak{S}_{k}} ({\mathrm{Zer}}(Q_{-}, {\mathrm{R}}^{k+1})) -
{\chi}^{{\ensuremath{\operatorname{gen}}}}_{\mathfrak{S}_{k}} ({\mathrm{Zer}}((Q,P), {\mathrm{R}}^{k})).\end{aligned}$$
#### Complexity analysis
The complexity of the algorithm is $(\omega k d)^{O (D'')}$, where $D'' = \sum_{i=1}^\omega \min(k_i,4d)$, using the complexity analysis of Algorithm \[alg:ep-general-BM\].
We are now ready to describe our algorithm for computing the Euler-Poincar[é]{} characteristic of the realizations of sign conditions.
\[alg:ep-sign-conditions:step1\]
\[alg:ep-sign-conditions:step2\]
\[alg:ep-sign-conditions:step3\]
\[alg:ep-sign-conditions:step4\]
\[alg:ep-sign-conditions:step5\]
#### Proof of correctness
The correctness follows from the correctness of Algorithm \[alg:sampling\] and the proof of correctness of the corresponding algorithm (Algorithm 13.12) in [@BPRbook3].
#### Complexity analysis
The complexity analysis is very similar to that of Algorithm 13.12 in [@BPRbook3]. The only difference is the use of the bound on ${\mathrm{card}}(\mathcal{P})$ afforded by Proposition \[prop:SIGN\] in the symmetric situation instead of the usual non-symmetric bound. By Proposition \[prop:SIGN\]
$${\mathrm{card}}(\operatorname{SIGN}(\mathcal{P})) \leq s^{D'} d^{O (
D'')},$$ where $D' = \sum_{i=1}^{\omega} \min (k_{i},d)$, and $D''= \sum_{i=1}^{\omega} \min (k_{i},4d)$. The number of calls to to Algorithm \[12:alg:speuler\] (Euler-Poincar[é]{}-query) is equal to ${\mathrm{card}}(\operatorname{SIGN}(\mathcal{P}
))$. The calls to Algorithm \[12:alg:speuler\] (Euler-Poincar[é]{}-query) are done for polynomials which are products of at most $$\log ({\mathrm{card}}(\operatorname{SIGN}(\mathcal{P}))) =O (D''
\log d+ D' \log s))$$ products of polynomials of the form $P$ or $P^{2}$, $P \in \mathcal{P}$ by Proposition 10.84 in [[@BPRbook3]]{}, hence of degree bounded by $D=O (d
(D'' \log d+ D' \log s))$. Using the complexity analysis of Algorithm \[alg:sampling\] (Sampling) and the complexity analysis of Algorithm \[12:alg:speuler\] (Euler-Poincar[é]{}-query), the number of arithmetic operations is bounded by
$$s^{D'} k^{d } d^{O (D'D'')} + s^{D'} d^{O(D'')} ( k \omega D)^{O (D''')},$$ where $D=d (D'' \log d+ D'\log s))$, $D' = \sum_{i=1}^{\omega} \min (k_{i},d)$, $D'' = \sum_{i=1}^{\omega} \min (k_{i},d)$, and $D''' = \sum_{i=1}^\omega \min(k_i, 2D)$.
The algorithm also involves the inversion matrices of size $s^{D'} d^{O (D'')}$ with integer coefficients.
\[alg:ep-sa:step1\]
\[alg:ep-sa:step2\]
\[alg:ep-sa:step3\]
\[alg:ep-sa:step4\]
#### Proof of correctness
The correctness of Algorithm \[alg:ep-sa\] follows from the correctness of Algorithms \[alg:sampling\] and \[alg:ep-sign-conditions\], and the additive property of the generalized Euler-Poincar[é]{} characteristic (see Definition \[def:ep-general\]).
#### Complexity analysis
The complexity is dominated by Step \[alg:ep-sa:step3\], and is thus bounded by $${\mathrm{card}}(
\Sigma)^{O (1)} + s^{D'} k^{d } d^{O (D'D'')} + s^{D'} d^{O(D'')} ( k \omega D)^{O (D''')},$$ where $D=d (D'' \log d+ D'\log s))$, $D' = \sum_{i=1}^{\omega} \min (k_{i},d)$, $D'' = \sum_{i=1}^{\omega} \min (k_{i},d)$, and $D''' = \sum_{i=1}^\omega \min(k_i, 2D)$.
The algorithm also involves the inversion matrices of size $s^{D'} d^{O (D'')}$ with integer coefficients.
The correctness and the complexity analysis of Algorithm \[alg:ep-sa\] prove Theorem \[thm:algorithm-sa\].
Appendix {#sec:appendix}
========
\[not:pi-of-x\] For ${\mathbf{x}}\in {\mathrm{R}}^k$ or ${\mathrm{C}}^k$, let $G_{\mathbf{x}}$ be the isotropy subgroup of ${\mathbf{x}}$ with respect to the action of $\mathfrak{S}_k$ on ${\mathrm{R}}^k$ or ${\mathrm{C}}^k$ permuting coordinates. Then, it is easy to verify that $$G_{\mathbf{x}}\cong \mathfrak{S}_{\ell_1} \times \cdots \times \mathfrak{S}_{\ell_m},$$ where $k \geq \ell_1 \geq \ell_2 \geq \cdots \geq \ell_m > 0, \sum_i \ell_i = k$, and $\ell_1,\ldots,\ell_m$ are the cardinalities of the sets $$\{i \mid 1 \leq i \leq k, x_i = x\}, x \in \bigcup_{i=1}^k \{x_i \}$$ in non-decreasing order. We denote by $\pi({\mathbf{x}})$ the partition $(\ell_1,\ldots,\ell_m) \in \Pi_k$.
More generally, for $\mathbf{k}= (k_{1}, \ldots,k_{\omega})
\in {\mathbb{Z}}_{>0}^{\omega}$, with $k= \sum_{i=1}^{\omega} k_{i}$, and ${\mathbf{x}}= ({\mathbf{x}}^{(1)},\ldots,{\mathbf{x}}^{(\omega)}) \in {\mathrm{R}}^k$, where each ${\mathbf{x}}^{(i)} \in {\mathrm{R}}^{k_i}$, we denote $${\boldsymbol{\pi}}({\mathbf{x}}) = (\pi({\mathbf{x}}^{(1)}),\ldots,\pi({\mathbf{x}}^{(\omega)})) \in {\boldsymbol{\Pi}}_{\mathbf{k}}.$$
In the following proposition we use Notation \[not:L-fixed\].
[@BC2013 Proposition 7] \[prop:orthogonal\] Let $L'_{{\mathrm{fixed}}}
\subset L_{{\mathrm{fixed}}}$ any subspace of $L_{{\mathrm{fixed}}}$, and $I \subset \{ (i,j) \mid 1
\leq i \leq \omega,1 \leq j \leq \ell_{i} \}$. Then the following hold.
A. \[itemlabel:prop:orthogonal:1\] The dimension of $L_{{\mathrm{fixed}}}$ is equal to $
\sum_{i=1}^{\omega} \ell_{i} -1= {\mathrm{length}}({\boldsymbol{\pi}}) -1$.
B. \[itemlabel:prop:orthogonal:2\] The product over $i \in [ 1, \omega ]$ of the subgroups $\mathfrak{S}_{\pi^{(i)}_{1}} \times \mathfrak{S}_{\pi^{(i)}_{2}}
\times \cdots \times \mathfrak{S}_{\pi^{(i)}_{\ell_{i}}}$ acts trivially on $L_{{\mathrm{fixed}}}$.
C. \[itemlabel:prop:orthogonal:3\] For each $i,j,1 \leq i \leq \omega,1 \leq j \leq \ell_{i}$, $M^{(i
)}_{j}$ is an irreducible representation of $\mathfrak{S}_{\pi^{(i
)}_{j}}$, and the action of $\mathfrak{S}_{\pi^{(i')}_{j'}}$ on $M^{(
i)}_{j}$ is trivial if $(i,j) \neq (i',j')$.
D. \[itemlabel:prop:orthogonal:4\] There is a direct decomposition $L=L_{{\mathrm{fixed}}} \oplus \left(
\bigoplus_{1 \leq i \leq \omega,1 \leq j \leq \ell_{i}} M^{(i)}_{j}
\right)$.
E. \[itemlabel:prop:orthogonal:5\] Let $\mathbf{D}$ denote the unit disc in the subspace $L_{{\mathrm{fixed}}}' \oplus \left(\bigoplus_{(i,j) \in I} M^{(i)}_{j}
\right)$. Then, the space of orbits of the pair $(\mathbf{D}, \partial
\mathbf{D})$ under the action of $\mathfrak{S}_{\mathbf{k}}$ is homotopy equivalent to $(\ast, \ast)$ if $I \neq \emptyset$. Otherwise, the space of orbits of the pair $(\mathbf{D}, \partial \mathbf{D})$ under the action of $\mathfrak{S}_{\mathbf{k}}$ is homeomorphic to $(
\mathbf{D}, \partial \mathbf{D})$.
[@BC2013 Lemma 5] \[lem:equivariant\_morseA\] Then, for $1 \leq i<N$, and for each $c \in
[ c_{i},c_{i+1})$, $\phi_{\mathbf{k}} (S_{\leq c})$ is semi-algebraically homotopy equivalent to $\phi_{\mathbf{k}} (S_{\leq
c_{i}})$.
Let $L^{+} ({\mathbf{x}}) \subset
L$ and $L^{-} ({\mathbf{x}}) \subset L$ denote the positive and negative eigenspaces of the Hessian of the function $p_{1}^{(k)}$ restricted to $W$ at ${\mathbf{x}}$. Let ${\mathrm{ind}}^{-} ({\mathbf{x}}) = \dim L^{-} ({\mathbf{x}})$.
The proof of the following lemma follows closely the proof of a similar result (Lemma 6) in [@BC2013].
\[lem:equivariant\_morseB\] Let $G_c$ denote a set of representatives of orbits of critical points ${\mathbf{x}}$ of $F$ restricted to $W$ with $F({\mathbf{x}}) = c$. Then, for all small enough $t>0$, $$\begin{aligned}
\label{eqn:inequality2}
\chi^{\mathrm{top}}(\phi_{\mathbf{k}} (S_{\leq c}),{\mathbb{F}}) & =& \chi^{\mathrm{top}}(
\phi_{\mathbf{k}} (S_{\leq c-t}),{\mathbb{F}}) +
\sum_{{\mathbf{x}}}
(-1)^{{\mathrm{ind}}^-({\mathbf{x}})},
\end{aligned}$$ where the sum is taken over all ${\mathbf{x}}\in G_c$ with $\sum_{1 \leq i \leq k} \dfrac{\partial P}{\partial X_{i}} ({\mathbf{x}}) <0$.
We first prove the proposition for ${\mathrm{R}}=
\mathbb{R}$. We will also assume that the function $F$ takes distinct values on the distinct orbits of the critical points of $F$ restricted to $W$ for ease of exposition of the proof. Since the topological changes at the critical values are local near the critical points which are assumed to be isolated, the general case follows easily using a standard partition of unity argument. Also, note that the value of ${\mathrm{ind}}^{-1}({\mathbf{x}})$ (respectively, $\operatorname{sign}(\sum_{1 \leq i \leq k} \dfrac{\partial P}{\partial X_{i}} ({\mathbf{x}}) )$) are equal for all critical points ${\mathbf{x}}$ belonging to one orbit.
Suppose that for each critical point ${\mathbf{x}}\in W$, with $F ({\mathbf{x}}) =c$, $$\sum_{1 \leq i \leq k} \dfrac{\partial P}{\partial X_{i}} ({\mathbf{x}}) >0.$$ We prove that in this case, for for all small enough $t>0$, $$\begin{aligned}
\label{eqn:inequality1}
\chi^{\mathrm{top}}(\phi_{\mathbf{k}} (S_{\leq c}),{\mathbb{F}}) & = & \chi^{\mathrm{top}}(
\phi_{\mathbf{k}} (S_{\leq c-t}),{\mathbb{F}}).
\end{aligned}$$
If $$\sum_{1 \leq i \leq k} \dfrac{\partial P}{\partial X_{i}} ({\mathbf{x}})
>0,$$ then $S_{\leq c}$ retracts $\mathfrak{S}_{\mathbf{k}}$-equivariantly to a space $S_{\leq c-t} \cup_{B}
A$ where the pair $(A,B) = \coprod_{{\mathbf{x}}} ( A_{{\mathbf{x}}},B_{{\mathbf{x}}})$, and where the disjoint union is taken over the set critical points ${\mathbf{x}}$ with $F ({\mathbf{x}}) =c$, and each pair $(A_{{\mathbf{x}}},B_{{\mathbf{x}}})$ is homeomorphic to the pair $(
\mathbf{D}^{i} \times [ 0,1 ], \partial \mathbf{D}^{i} \times [ 0,1 ] \cup
\mathbf{D}^{i} \times \{ 1 \})$, where $i$ is the dimension of the negative eigenspace of the Hessian of the function $e_1^{(k)}$ restricted to $W$ at ${\mathbf{x}}$. This follows from the basic Morse theory (see [@BPRbook3 Proposition 7.21]). Since the pair $(
\mathbf{D}^{i} \times [ 0,1 ], \partial \mathbf{D}^{i} \times [ 0,1 ] \cup
\mathbf{D}^{i} \times \{ 1 \})$ is homotopy equivalent to $(\ast, \ast
)$, $S_{\leq c}$ is homotopy equivalent to $S_{\leq c-t}$, and it follows that $\phi_{\mathbf{k}} (S_{\leq c})$ is homotopy equivalent to $\phi_{\mathbf{k}} (S_{\leq c-t})$ as well, because of the fact that retraction of $S_{\leq c}$ to $S_{\leq c-t} \cup_{B} A$ is chosen to be equivariant. The equality then follows immediately.\
We now consider the case when for each critical point ${\mathbf{x}}\in W$, with $F ({\mathbf{x}}) =c$, $\sum_{1 \leq i \leq k} \dfrac{\partial
P}{\partial X_{i}} ({\mathbf{x}}) < 0$. Let $T_{{\mathbf{x}}} W$ be the tangent space of $W$ at ${\mathbf{x}}$. The translation of $T_{{\mathbf{x}}} W$ to the origin is then the linear subspace $L \subset {\mathrm{R}}^{k}$ defined by $\sum_{i} X_{i} =0$. Let ${\mathbf{x}}\in
L_{{\boldsymbol{\pi}}}$ where ${\boldsymbol{\pi}}= (\pi^{(1)}, \ldots, \pi^{(
\omega)}) \in {\boldsymbol{\Pi}}_{\mathbf{k}}$, where for each $i,1 \leq i
\leq \omega$, $\pi^{(i)} = (\pi^{(i)}_{1}, \ldots, \pi^{(i
)}_{\ell_{i}}) \in \Pi_{k_{i}}$. The subspaces $L^{+} ({\mathbf{x}}),L^{-} ({\mathbf{x}})$ are stable under the the natural action of the subgroup $\prod_{1 \leq i
\leq \omega,1 \leq j \leq \ell_{i}} \mathfrak{S}_{\pi^{(i)}_{j}}$ of $\mathfrak{S}_{\mathbf{k}}$. For $1 \leq i \leq \omega,1 \leq j \leq
\ell_{i}$, let $L^{(i)}_{j}$ denote the subspace $L \cap L_{\pi^{(i
)}_{j}}$ of $L$, and $M^{(i)}_{j}$ the orthogonal complement of $L^{(i
)}_{j}$ in $L$. Let $L_{{\mathrm{fixed}}} =L \cap L_{{\boldsymbol{\pi}}}$. It follows from Parts , , and of Proposition \[prop:orthogonal\] that:
1. For each $i,j, \,1 \leq i \leq \omega,1 \leq j \leq
\ell_{i}$, $M^{(i)}_{j}$ is an irreducible representation of $\mathfrak{S}_{\pi_{i}}$, and the action of $\mathfrak{S}_{\pi^{(i'
)}_{j'}}$ on $M^{(i)}_{j}$ is trivial if $(i,j) \neq (i',j')$. Hence, for each $i, j ,1 \leq i \leq \omega,1 \leq j
\leq \ell_{i}$, $L^{-} (p) \cap M^{(i)}_{j} =0 {\ensuremath{\operatorname{or}}} M^{(i
)}_{j}$.
2. The subgroup $\prod_{1 \leq i \leq \omega,1 \leq j \leq \ell_{i}}
\mathfrak{S}_{\pi^{(i)}_{j}}$ of $\mathfrak{S}_{\mathbf{k}}$ acts trivially on $L_{{\mathrm{fixed}}}$.
3. There is an orthogonal decomposition $L=L_{{\mathrm{fixed}}} \oplus
\left(\bigoplus_{1 \leq i \leq \omega,1 \leq j \leq \ell_{i}} M^{(i
)}_{j} \right)$.
It follows that $$\begin{aligned}
L^{-} (p) & = & L'_{{\mathrm{fixed}}} \oplus \left(\bigoplus_{(i,j) \in
I} M^{(i)}_{j} \right),
\end{aligned}$$ where $L'_{{\mathrm{fixed}}}$ is some subspace of $L_{{\mathrm{fixed}}}$ and $I
\subset \{ (i,j) \mid 1 \leq i \leq \omega,1 \leq j \leq \ell_{i} \}$.
It follows from the proof of Proposition 7.21 in [@BPRbook3] that for all sufficiently small $t >0$ then $S_{\leq c}$ is retracts $\mathfrak{S}_{\mathbf{k}}$-equivariantly to a space $S_{\leq c-t} \cup_{B}
A$ where the pair $(A,B) = \coprod_{{\mathbf{x}}} ( A_{{\mathbf{x}}},B_{{\mathbf{x}}})$, and the disjoint union is taken over the set critical points ${\mathbf{x}}$ with $F ({\mathbf{x}}) =c$, and each pair $(A_{{\mathbf{x}}},B_{{\mathbf{x}}})$ is homeomorphic to the pair $(\mathbf{D}^{{\mathrm{ind}}^{-} ({\mathbf{x}})}, \partial \mathbf{D}^{{\mathrm{ind}}^{-} ({\mathbf{x}})})$. It follows from the fact that the retraction mentioned above is equivariant that $\phi_{\mathbf{k}} (S_{\leq c})$ retracts to a space obtained from $\phi_{\mathbf{k}} (S_{\leq c-t})$ by glueing ${\mathrm{orbit}}_{\mathfrak{S}_{\mathbf{k}}} \left(\coprod_{{\mathbf{x}}} A_{{\mathbf{x}}} \right)$ along ${\mathrm{orbit}}_{\mathfrak{S}_{\mathbf{k}}} \left(\coprod_{{\mathbf{x}}} B_{{\mathbf{x}}}
\right)$. Now there are the following cases to consider:
(a) ${\mathrm{ind}}^{-} ({\mathbf{x}}) =0$. In this case $${\mathrm{orbit}}_{\mathfrak{S}_{\mathbf{k}}} (\coprod_{{\mathbf{x}}} A_{{\mathbf{x}}},
\coprod_{{\mathbf{x}}} B_{{\mathbf{x}}})$$ is homotopy equivalent to $(\ast, \emptyset)$.
(b) $L^{-} ({\mathbf{x}}) \subset L_{{\mathrm{fixed}}}$ (i.e. $I= \emptyset$ in this case). In this case $${\mathrm{orbit}}_{\mathfrak{S}_{\mathbf{k}}}
(\coprod_{{\mathbf{x}}} A_{{\mathbf{x}}}, \coprod_{{\mathbf{x}}} B_{{\mathbf{x}}})$$ is homeomorphic to $(\mathbf{D}^{{\mathrm{ind}}^{-} ({\mathbf{x}})}, \partial \mathbf{D}^{{\mathrm{ind}}^{-}
({\mathbf{x}})})$ by Part of Proposition \[prop:orthogonal\].
(c) Otherwise, there is a non-trivial action on $L^{-} ({\mathbf{x}})$ of the group $$\prod_{(i,j) \in I} \mathfrak{S}_{\pi^{(i)}_{j}},$$ and it follows from Part of Proposition \[prop:orthogonal\] that in this case $${\mathrm{orbit}}_{\mathfrak{S}_{\mathbf{k}}}(\coprod_{{\mathbf{x}}} A_{{\mathbf{x}}}, \coprod_{{\mathbf{x}}} B_{{\mathbf{x}}})$$ is homotopy equivalent to $(\ast, \ast)$.
The equality follow immediately from .
This finishes the proof in case ${\mathrm{R}}=\mathbb{R}$. The statement over a general real closed field ${\mathrm{R}}$ now follows by a standard application of the Tarski-Seidenberg transfer principle (see for example the proof of Theorem 7.25 in [@BPRbook3]).
The proof follows immediately from Lemmas \[lem:equivariant\_morseA\] and \[lem:equivariant\_morseB\].
[^1]: The first author was partially supported by NSF grants CCF-1319080 and DMS-1161629.
|
---
abstract: 'Molecular dynamics simulations predict a giant electrocaloric effect at the ferroelectric-antiferroelectric phase boundary in PZT (PbTiO$_3$-PbZrO$_3$). These large-scale simulations also give insights into the atomistic mechanisms of the electrocaloric effect in [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{}. We predict a positive electrocaloric effect in ferroelectric PZT, but antiferroelectric PZT exhibits a negative to positive crossover with increasing temperature or electric field. At the antiferroelectric-to-ferroelectric phase boundary we find complex domain patterns. We demonstrate that the origin of giant electrocaloric change of temperature is related to the easy structural response of the dipolar system to external stimuli in the transition region.'
author:
- 'A. V. Kimmel'
- 'O. T. Gindele'
- 'D. M. Duffy'
- 'R.E. Cohen'
bibliography:
- 'ECE-revision.bib'
title: 'Giant electrocaloric effect at the antiferroelecrtric-to-ferroelectric phase boundary in [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{}'
---
[^1]
The electrocaloric effect is a reversible temperature change ($\Delta T$) in materials under adiabatic conditions in response to applied electric (or magnetic) field. The discovery of a giant 12 K electrocaloric effect (ECE) in thin films of Zr-rich lead titanate compositions fuelled interest into the development of novel ferroelectric-based ECE materials[@Mischenko2006].
Giant and moderate ECE’s have since been reported for classical ferroelectrics like BaTiO$_3$[@Kar-Narayan2010] and for several relaxor materials [@Lu2010]. Pb(Zr${_{1-x}}$Ti${_x}$)O${_3}$ (PZT) is a disordered solid solution ABO$_3$ perovskite, with Pb atoms occupying the A-site, and Ti and Zr cations randomly arranged among the B-sites. PbTiO$_3$ (PTO), the $x$=0.0 end member of [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{}, is a classical ferroelectric (FE), and the other end member PbZrO$_3$ (PZO) ($x$=1.0) is antiferroelectric (AFE). Near $x$=0.95 there is a phase boundary that separates AFE and FE phases[@Woodward2005]. Pb(Zr${_{1-x}}$Ti${_x}$)O${_3}$ (PZT) remains an active area of research for novel ECE materials[@Zhang2016; @Zuo2015]. The response of PZT to the applied electric field in the transition region between its ferroelectric and antiferroelectric phases is of particular interest since a giant electrocaloric response has been found experimentally for compositions close to this region[@Mischenko2006]. Studies of electrocaloric response of AFE Pb$_{0.97}$La$_{0.02}($Zr$_{0.95}$Ti$_{0.05}$)O${_3}$ have provided an insight into a mechanism for the negative electrocaloric response. Authors suggested that misaligning of non-collinear dipoles provides different entropy contribution depending on the direction of the applied electric field [@Geng2015]. Several theoretical works discuss caloric effects in perovskites. Large electrocaloric effects have been observed in the vicinity of ferroelectric-paraelectric phase transition, however, little is known about the ECE near AFE-FE phase boundary. Recent work with effective Hamiltonians reveals a strong potential of electrocalorics for thin PZO films with FE and AFE phase competition[@Ponomareva2018]. Phenomenological modelling for an AFE system predicted the negative electrocaloric effect in PZO ceramics, which agrees well with direct measurements of the EC temperature change in this system[@Pirc2014]. Molecular dynamics (MD) methods, using shell model potentials fit to first principles calculations, are promising models for computing the thermal behaviour of materials, since they do not require assumptions about the important degrees of freedom. Such models have been used to study ECE in LiNbO$_3$, PMN-PT, and BaTiO${_3}$[@Rose2012; @*Erratum14; @WuCohen2017a; @WuCohen2017b]. These simulations provide insight into the universal principles related to optimal operating temperature for the electrocaloric effect.
In this work we studied the effects of composition on electrocaloric properties of PZT using large scale MD simulations with first-principles based shell model potentials[@Gindele2015]. We modelled a wide range of ferroelectric and antiferroelectric compositions of [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{}. We found that the electrocaloric response of PZT correlates with the type of ferroelectric order and that a giant electrocaloric response exists at the phase boundary of PZT, where antiferroelectric and ferroelectric order coexist.
To model the electrocaloric properties of PZT we use a core-shell force field, which includes all degrees of freedom. This $ab$ $initio$ based interatomic potential reproduces a set of temperature and composition induced phase transitions characteristic of [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{}[@Gindele2015]. The potential model underestimates the Curie temperatures with respect to experiment for PbTiO$_3$ (600 K versus 750 K[@Woodward2005]) and PbZrO$_3$ (400 K versus 507 K[@Pirc2014]), which is a reasonable error for this type of force field.
A set of [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{} compositions were modelled using the DL$\textunderscore$POLY code[@Todorov2006]. We study AFE and FE compositions with $x$ equal to 0.5, 0.9, 1 (corresponds to AFE PbZrO$_3$), together with $x$=0 (that corresponds to FE PbTiO$_3$), 0.7, 0.8, 0.85, 0.95 shown in Supplementary Information (SI). The B-site cations, Ti and Zr, were randomly distributed over the B-sites. We use the adiabatic shell model (also known as dynamical model [@Fincham93]) as a method of incorporating polarisability into a molecular dynamics simulation with the shell masses varying as 3.5 %, 8.3 %, 17.12 % and 12.5 % of the atomic mass of Pb, Ti, Zr and O, respectively. We used relatively large 20 $\times$ 20 $\times$ 20 super-cells (80 000 core and shell particles). Each composition was equilibrated at 100 K for 40 ps, followed by application of an electric field along the polar axis. The direction of the polar axis depends on the composition of [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{} and was taken as \[001\] for PZO, \[111\] for the Zr content from 0.95 to 0.50, and as \[001\] for $x>=0.4$ . The strength of the applied electric field was 0, 5, 10, 15, 20, 25, 50, 75, 100 MV/m. We used a 0.2 fs timestep and NST ensemble with the Nosé-Hoover thermostat (0.01 ps) and barostat (0.5 ps) for equilibration of individual systems during 8 ps. The equilibration was followed by a 12 ps production run over which the polarisation value was calculated.
To study the electrocaloric effect we used the indirect method, where the change of temperatures were calculated from Maxwell related expression: $$\label{eq:ECE2}
\Delta T=- \int \limits_{0}^{E} \frac{TV}{C_{p,E}} \left(\frac{\partial P}{\partial T}\right)_E \mathrm{d}E,$$ Here, $E$ is the applied electric field, $T$ is the temperature, $V$ is the volume of the simulation cell and $C_{p,E}$ is the heat capacity per cell under constant electric field and pressure. We calculate the ECE change of temperature ($\Delta T$), by integrating equation (\[eq:ECE2\]) numerically. The values of $C_{p,E}$ were calculated as the derivative of the total energy with temperature ($\frac{\partial E_{tot}}{\partial T}$) at a given value of electric field, $E$ and are in agreement with experiment [@Morimoto2003] (See Supplementary Information).
The temperature and field dependence of the electrocaloric change in temperature, $\Delta T$, were calculated for FE PTO via expression eq. (1) ( see Supplementary Information Fig. 1a.) A characteristic dominant peak at 650 K (the PTO Curie temperature reproduced by our force field) moves towards higher temperatures for larger applied electric fields, typical for ferroelectrics [@Rose2012; @*Erratum14]. The magnitude of electrocaloric effect calculated for PTO is good agreement with similar method computations for LiNbO$_3$ that gives 17 K at applied 50 MV/m field versus 16 K in our computations for PTO[@Rose2012].
The morphotropic phase boundary (MPB) is found in a narrow compositional range around $x$=0.5, where the FE phase with rhombohedral symmetry transforms to the tetragonal phase. It is now known that there is a monoclinic transition region between the rhombohedral and tetragonal phases [@Noheda1999; @*Cohen2018; @*Glazer2014; @*Bogdanov2016; @*FuCohen2000; @*Ahart2008]. We found that the electrocaloric effect in FE [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{} with $x$ = 0.5 and 0.7 exhibits very similar behaviour. The peaks of $\Delta T$ broaden, which reflects the B-site cations disorder and reduction of the correlation length in the material [@Rica2016] (See Fig. 1a, b and Supplementary Information, Fig. 1b, c). The $\Delta T$ curves peak above T$_c$ with increasing electric field (Fig. 1b), similar to what was computed for LiNbO$_3$[@Rose2012].
The transition boundary between AFE and FE phases in [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{} has been shown to exist within a composition region around $x$=0.95-0.9[@Woodward2005]. It is challenging to identify the precise composition of the transition region between AFE and FE phases experimentally, due to purity of the samples, composition variance, especially for solid solution materials, and the presence of surface effects that may stabilise the FE phase. The force field used in this work is able to reproduce the composition induced AFE-FE phase boundary, but the model gives a boundary wider than seen experimentally – we find composites with $x>$0.8 exhibit antiferroelectric properties[@Gindele2015]. Further, we have performed calculations of the electrocaloric properties for several of the AFE PZT compositions with $x$ of 0.9, and 1 (PbZrO$_3$), while the result for 0.85, 0.95 are given in Supplementary Information.
We found that the electrocaloric response of AFE’s is very different to that of the FE systems. In AFE’s the applied electric field causes $T_c$ to decrease (Fig. 1c-f, SI Fig. 2), whereas ferroelectric materials show the opposite tendency. A common feature of all studied AFE PZT is a negative-to-positive crossover that varies with temperature and composition. Positive values of the EC $\Delta T$ are related to the reduction of isothermal entropy. In classical FE’s this is related to the drop of macroscopic polarisation with rising temperature. However, in AFE’s the polarisation may exhibit an opposite behaviour, i.e. increasing with rising temperature under applied field. This occurs simply because the applied field favours net polarisation and dielectric susceptibility then increases with temperature. The latter results in negative change of isothermal entropy, and reverse electrocaloric effect (See SI).
PZO does not exhibit a macroscopic polarisation at zero field, as expected for an AFE. Applied electric field induces a polarisation that increases up to the critical temperature, $T_c$, and then falls with further temperature rise (see Fig. 1 e, f). However, the induced polar state of PZO at an applied field of 100 MV/m is only 18 $\mu$ C/m$^2$, which is 40 % lower than that of PTO. Ferroelectrics can also show negative ECE originated from polarisation rotation, where the polarisation along the field direction increases with temperature due to approaching phase transition [@WuCohen2017a; @WuCohen2017b] Calculated behaviour of $\Delta T$ for PZO exhibits a crossover from negative to positive values in the vicinity of $T_c$ as shown in Fig. 1f. For applied electric fields $<$50 MV/m the EC change in temperature exhibits negative values below $T_c$ (at T=250 K the values of ECE are -0.7 K with applied field of 25 MV/m ).
At zero applied field AFE PZT ($x$=0.95, 0.9, 0.85, 0.8) shows zero macroscopic polarisation, but local dipoles, as will be shown later, form competing AFE and FE domains. Application of an electric field enhances the polarisation, which reaches its maximum at temperatures of 400 K, 350 K and 300 K characteristic for each composition with $x$=0.85, 0.90, 0.95, respectively (Fig. 1c, d, SI Fig. 2b, c, d). The electrocaloric response of studied AFE’s is characterised by the negative-to-positive crossover. In PZO the EC $\Delta T$ changes its sign once, whereas AFE PZT exhibits more complex EC behaviour.
We have found that, in general, the EC effect in FE and AFE [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{} with $x>0$ is smaller compared to the pure FE PTO (22.01 K at 100 MV/m of applied field (See SI Fig. 2a)), but at lower temperatures and, thus, more usable under ordinary conditions. At the AFE-FE boundary an enhanced caloric response comparable to MPB PZT. The smallest EC response is observed in the pure AE PZO of about 5 K at 100 MV/m of applied field. The AFE PZT with $x=0.8$ exhibits the EC effect of 6.1 K at a similar field (See SI, Fig. 1). Meanwhile, PZT with $x$=0.95, 0.9 exhibit values of EC $\Delta T$ of about 10 K (SI Fig. 2), which is comparable with the EC response of MPB PZT at similar stimuli.
To understand the origin of the giant EC effect and negative-to-positive crossover at the AFE-FE phase boundary we analysed the evolution of local dipoles in response to applied fields. We found that an AFE system may adopt complex dipole arrangements with a variety of possible states, such as dipole FE order, dipole disorder, and various AFE dipole arrangements characterised by zero macroscopic polarisation. In particular, at small applied fields and low temperatures the AFE [PbZr${_{0.95}}$Ti${_{0.05}}$O$_3$]{} exhibits a dynamically stable 2$\times$1 pattern (Fig. 2a). Here, the local dipoles are arranged as antiparallel double pairs along $X$ cartesian direction, and single antiparallel arrangement along $Z$ axis (See directing arrows in the inset of Fig. 2a). At higher temperatures the order of local dipoles changes to a 1$\times$1 pattern, where single antiparallel dipoles are alternating with the sites of dipole disorder (Fig. 2b). Increasing the applied field to the critical value of 25 MV/m leads to the rotation of local dipoles, so the system turns into an induced polar FE state.
Increasing the Ti content leads to stabilisation of a zig-zag pattern of AFE local dipoles (Fig. 2c). Here, the local dipoles are arranged into antiparallel pairs. As the field increases the system develops competing AFE and FE domains, with widths which correlate with the strength of applied field. The critical field of 50 MV/m switches the system to an induced polar FE monodomain state.
Higher Ti content in PbZr$_{0.85}$Ti$_{0.15}$O$_3$ increases the correlation length of the material,[@Rica2016] which leads to the formation of stripe ordering, with AFE dipole arrangement alternating with FE stripes (Fig. 2d).
We suggest that the nature of giant EC temperature change, $\Delta T$, in AFE PZT is related to the formation of competing AFE and induced FE orders that respond easily to applied fields and temperature. In ferroelectrics the configurational entropy is related to the order maintained by a dipolar system. In the absence of the applied field the change of polarisation with temperature, $\frac{\delta P}{\delta T}$, is relatively small, except the vicinity of the critical temperature, where this value is large. Thus, the maximum of EC $\Delta T$ in ferroelectrics occurs when the system switches from FE to PE.
An AFE system may adopt complex configurations with a variety of possible dipole states - as dipole FE order, dipole disorder, and various AFE dipole arrangements with zero macroscopic polarisation (as shown in Fig. 3, where the system maintained 2$\times$1, 1$\times$1 patterns). We suggest that in AFE systems the change from AFE to FE order happen via a sequence of local minima with a partially preserved AFE order and the formation of competing AFE and FE domains.
We assume that at applied field the aligned dipole configurations may became more advantageous to anti-polar configurations. This leads to destabilisation of anti-aligned arrangements of local dipoles in an AFE material leading to their partial, or complete alignment, and, consequently formation of competing antiferroelectric and induced ferroelectric domains. The transition process may happen through initial canting as proposed in ref. , and follows complete rotation similar to the mechanism proposed for FE’s in ref. . In bulk PZO the change of polarisation $\frac{\delta P}{\delta T}$, is relatively small (SI Fig. 2), because our system is free of defects, and grain boundaries and electrode contacts. Thus, each local dipole has to overcome a barrier for rotation. However, in AFE [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{} the presence of different types of B-site cations increases the configurational entropy of the system, and supports multiple domain configurations. The Ti sites act as nucleation centres for the FE phase, facilitating fast response of local dipoles to applied electric fields. At the AFE-FE phase boundary the concentration of Ti centres is such that there are no FE ordered regions in the absence of an applied electric field, however, the application of an external electric field gives rise to FE ordering, which competes with the AFE order (FIG. 3d). The maximum of EC $\Delta T$ in AFE composites occurs when the system switches to an induced polar FE monodomain.
We have studied the electrocaloric effect in PZT using molecular dynamics simulations with shell model forces fields. Our results show giant electrocaloric effects for FE PTO, in good agreement with similar calculations performed for FE LiNbO$_3$[@Rose2012]. We found a crossover from negative to positive EC temperature change for all studied AFE PZT. The crossover temperatures correlate with composition, which we believe to be related to the correlation length increase in the material[@Rica2016]. We have found that compositions close to the AFE-FE boundary of PZT exhibits an enhanced caloric response, comparable to that of MPB PZT but with the maximum EC temperature change occurring at temperatures closer to ambient temperatures. Our methodology allows to to investigate the details of the polarisation response at an atomistic level. Close to the AFE-FE boundary we identified complex dipole configurations, with competing FE and AFE domain patterns. We postulate that the small energy barriers associated with growing/ reducing these domains are responsible for the easy response of the polarisation to the applied field and temperature and, hence, for the enhanced calorific response. Despite the high EC response, the critical temperature in many ferroelectric materials is considerably higher than room temperature, which substantially limits the potential for the application in solid-state devices. We have found that AFE PZT exhibits extrema of EC $\Delta T$ close to room temperature, in the range 300-400 K. In addition, solid solution [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{} offers great variability in critical temperatures and in ECE magnitude, which allows for compositional engineering of materials for electrocaloric applications. In summary, our findings suggest pathways for tuning the operating temperatures of ECE devices and find solutions for a broad range of operating conditions.
Supplementary Information {#supplementary-information .unnumbered}
=========================
We show calculated heat capacity, and isothermal change of entropy in AFE and FE [Pb(Zr$_{x}$Ti$_{1-x}$)O$_3$]{}. We also calculated electrocaloric effect in FE PbTiO$_3$, FE [PbZr${_{0.7}}$Ti${_{0.3}}$O$_3$]{}, together with polarisation and electrocaloric temperature change in AFE [PbZr${_{0.95}}$Ti${_{0.05}}$O$_3$]{}, and AFE [PbZr${_{0.85}}$Ti${_{0.15}}$O$_3$]{}.
Authors acknowledge UCL computational facilities LEGION and MYRIAD. AK is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sk$\l$odowska-Curie grant agreement No 796781. REC was supported by the U. S. Office of Naval Research Grants No. N00014-12-1-1038, N00014-14-1-0516, and N00014-17-1-2768, the Carnegie Institution for Science, and the European Research Council Advanced Grant ToMCaT.
[^1]: *Corresponding author* a.kimmel@nanogune.eu
|
---
abstract: |
The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branching theorems that include the multiplicity-free property of some of known results such as the Clebsh–Gordan–Pieri formula for tensor products, the Plancherel theorem for Hermitian symmetric spaces (also for line bundle cases), the Hua–Kostant–Schmid $K$-type formula, and the canonical representations in the sense of Vershik–Gelfand–Graev. Our method works in a uniform manner for both finite and infinite dimensional cases, for both discrete and continuous spectra, and for both classical and exceptional cases.
Primary 22E46, Secondary 32A37, 05E15, 20G05, 53C35.
author:
- Toshiyuki KOBAYASHI
title: 'Multiplicity-free theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs'
---
**Contents**
Introduction and statement of main results {#sec:1}
==========================================
The purpose of this article is to give a quite detailed account of the theory of multiplicity-free representations based on a non-standard method (*visible actions* on complex manifolds) through its application to branching problems. More precisely, we address the question of restricting irreducible highest weight representations $\pi$ of reductive Lie groups $G$ with respect to symmetric pairs $(G,H)$. Then, our main goal is to give a simple and sufficient condition on the triple $(G,H,\pi)$ such that the restriction $\pi|_H$ is multiplicity-free. We shall see that our method works in a uniform way for both infinite and finite dimensional representations, for both classical and exceptional cases, and for both continuous and discrete spectra.
This article is an outgrowth of the manuscript [@xkmf] which I did not publish, but which has been circulated as a preprint. From then onwards, we have extended the theory, in particular, to the following three directions: 1)the generalization of our main machinery (Theorem \[thm:2.2\]) to the vector bundle case ([@mfbdle]), 2)the theory of on complex manifolds ([@visiblesymm; @xkgencar; @xksovisible]), 3)‘multiplicity-free geometry’ for coadjoint orbits ([@xknasrin]).
We refer the reader to our paper [@RIMS] for a precise statement of the general results and an exposition of the related topics that have recently developed.
In this article, we confine ourselves to the line bundle case. On the one hand, this is sufficiently general to produce many interesting consequences, some of which are new and some others may be regarded as prototypes of various multiplicity-free branching theorems (e.g. [@xsaid; @xvdh; @xkleiden; @xko; @xkratten; @xnere; @xokada; @xstemgl; @xyawa; @xzhang]). On the other hand, the line bundle case is sufficiently simple, so that we can illustrate the essence of our main ideas without going into technical details. Thus, keeping the spirit of [@xkmf], we have included here the proof of our method (Theorem \[thm:2.2\]), its applications to multiplicity-free theorems (Theorems \[thm:A\]–\[thm:F\]), and the explicit formulae (Theorems \[thm:gHKS\], \[thm:tensordeco\], and \[thm:upqupq\]), except that we referred to another paper [@visiblesymm] for the proof of some algebraic lemmas on the triple of involutions of Lie algebras (Lemmas \[lem:5.1\] and \[lem:7.5\]).
Definition of multiplicity-free representations {#subsec:1.2}
-----------------------------------------------
Let us begin by recalling the concept of the multiplicity-free decomposition of a unitary representation.
Suppose $H$ is a Lie group of type I in the sense of von Neumann algebras. Any reductive Lie group is of type I as well as any algebraic group. We denote by $\widehat{H}$ the unitary dual of $H$, that is, the set of equivalence classes of irreducible unitary representations of $H$. The unitary dual $\widehat{H}$ is endowed with the Fell topology.
Suppose that $(\pi, \mathcal{H})$ is a unitary representation of $H$ defined on a (second countable) Hilbert space $\mathcal{H}$. By a theorem of Mautner, $\pi$ is decomposed uniquely into irreducible unitary representations of $H$ in terms of the direct integral of Hilbert spaces: $$\label{eqn:1.2.1}
\pi \simeq \int_{\widehat{H}} m_\pi(\mu) \mu \, d \sigma (\mu) \, ,$$ where $d \sigma (\mu)$ is a Borel measure on $\widehat{H}$, and the multiplicity function $m_\pi : \widehat{H} \to \mathbb{N}
\cup \{\infty\}$ is uniquely defined almost everywhere with respect to the measure $d \sigma$. Let ${\operatorname{End}}(\mathcal{H})$ be the ring of continuous operators on $\mathcal{H}$, and ${\operatorname{End}}_H(\mathcal{H})$ the subring of $H$-intertwining operators, that is, the commutant of $\set{\pi(g)}{g \in H}$ in ${\operatorname{End}}(\mathcal{H})$.
\[def:1.2\] We say that the unitary representation $(\pi, \mathcal{H})$ is *multiplicity-free* if the ring ${\operatorname{End}}_H(\mathcal{H})$ is commutative.
It is not difficult to see that this definition is equivalent to the following property: $$\text{$m_\pi (\mu) \le 1$ for almost all $\mu \in \widehat{H}$
with respect to the measure $d \sigma (\mu)$ }$$ by Schur’s lemma for unitary representations. In particular, it implies that any irreducible unitary representation $\mu$ of $H$ occurs at most once as a subrepresentation of $\pi$.
Multiplicities for inductions and restrictions {#subsec:1.3}
----------------------------------------------
With regard to the question of finding irreducible decompositions of unitary representations, there are two fundamental settings: one is the induced representation from smaller groups (e.g. harmonic analysis on homogeneous spaces), and the other is the restriction from larger groups (e.g. tensor product representations).
To be more rigorous, suppose $G$ is a Lie group, and $H$ is a closed subgroup of $G$. The $G$-irreducible decomposition of the induced representation $L^2$-$\operatorname{Ind}_H^G \tau$ ($\tau \in \widehat H$) is called the [*Plancherel formula*]{}, while the $H$-irreducible decomposition of the restriction $\pi |_H$ ($\pi \in \widehat G$) is referred to as the [*branching law*]{}.
This subsection examines multiplicities in the irreducible decomposition of the induction and the restriction for reductive symmetric pairs $(G,H)$ (see Subsection \[subsec:3.1\] for definition).
Let us start with the induced representation. Van den Ban [@xban] proved that the multiplicity in the Plancherel formula for $L^2$-$\operatorname{Ind}_H^G \tau$ is finite as far as $\dim \tau < \infty$. In particular, this is the case if $\tau$ is the trivial representation $\mathbf{1}$. Over the past several decades, the induced representation $L^2$-$\operatorname{Ind}_H^G \mathbf{1}$ has developed its own identity (harmonic analysis on reductive symmetric spaces $G/H$) as a rich and meaningful part of mathematics.
In contrast, the multiplicities of the branching law of the restriction $\pi|_H$ ($\pi \in \widehat G$) are usually infinite. For instance, we saw in [@xkbdlejp] that this is the case if $(G, H)=(GL(p+q,{\mathbb {R}}), GL(p,{\mathbb{R}})
\times GL(q,{\mathbb{R}}))$ where $\min(p,q) \ge 2$, for any tempered representation $\pi$ of $G$. In this article, we illuminate by Example \[ex:finite infinite\] this wild behavior. In light of such a wild phenomenon of branching laws for reductive symmetric pairs $(G,H)$ with $H$ non-compact, we proposed in [@xkdecomp; @xkdecoalg] to seek for a ‘nice’ class of the triple $(G, H, \pi)$ in which a systematic study of the restriction $\pi|_H$ could be launched.
Finiteness of multiplicities is a natural requirement for this program. By also imposing discrete decomposability on the restriction $\pi|_H$, we established the general theory for *admissible restriction* in [@xkdecomp; @xkdecoalg; @xkdecoass] and found that there exist fairly rich triples $(G, H, \pi)$ that enjoy this nice property. It is noteworthy that new interesting directions of research in the framework of admissible restrictions have been recently developed by M. Duflo, D. Gross, J.-S. Huang, J.-S. Li, S.-T. Lee, H.-Y. Loke, T. Oda, P. Pandžić, G. Savin, B. Speh, J. Vargas, D. Vogan, and N. Wallach (see [@xkbeijing; @xkronsdecomp] and references therein).
Multiplicity-freeness is another ideal situation, in which we may expect an especially simple and detailed study of the branching law of $\pi|_H$. Thus, we aim for principles that lead us to abundant family of multiplicity-free cases. Among them, a well-known one is the dual pair correspondence, which has given fruitful examples in infinite dimensional theory in the following setting:
$G$ is the metaplectic group, and $\pi$ is the Weil representation.
$H = H_1 \cdot H_2$ forms a dual pair, that is, $H_1$ is the commutant of $H_2$ in $G$, and vice versa.
This paper uses a new principle that generates multiplicity-free representations. The general theory discussed in Section \[sec:2\] brings us to uniformly bounded multiplicity theorems (Theorems \[thm:B\] and \[thm:D\]) and multiplicity-free theorems (Theorems \[thm:A\], \[thm:C\], \[thm:E\] and \[thm:F\]) in the following setting:
$\pi$ is a unitary highest weight representation of $G$ (see Subsection \[subsec:1.4\]),
$(G,H)$ is a symmetric pair (see Subsection \[subsec:1.5\]).
We note that we allow the case where continuous spectra occur in the branching law, and consequently, irreducible summands are not always highest weight representations.
We remark that our bounded multiplicity theorems for the restriction $\pi|_H$ ($\pi$: highest weight module) may be regarded as the counterpart of the bounded multiplicity theorem for the induction $L^2$-$\operatorname{Ind}_H^G \tau$ ($\tau$: finite dimensional representation) due to van den Ban.
Unitary highest weight modules {#subsec:1.4}
------------------------------
Let us recall the basic notion of highest weight modules.
Let $G$ be a non-compact simple Lie group, $\theta$ a Cartan involution of $G$, and $K := \set{g \in G}{\theta g = g}$. We write $\mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ for the Cartan decomposition of the Lie algebra $\mathfrak{g}$ of $G$, corresponding to the Cartan involution $\theta$.
We assume that $G$ is of [*Hermitian type*]{}, that is, the Riemannian symmetric space $G/K$ carries the structure of a Hermitian symmetric space, or equivalently, the center $\mathfrak{c}(\mathfrak{k})$ of $\mathfrak{k}$ is non-trivial. The classification of simple Lie algebras ${\mathfrak {g}}$ of Hermitian type is given as follows: $${\mathfrak {su}}(p,q)\,,\
{\mathfrak {sp}}(n,{\mathbb{R}})\,,\
{\mathfrak {so}}(m,2) \ (m \ne 2)\,,\
{\mathfrak {e}}_{6(-14)}\,,\
{\mathfrak {e}}_{7(-25)} \, .$$ Such a Lie algebras $\mathfrak{g}$ satisfies the rank condition: $$\label{eqn:rankGK}
\operatorname{rank} G = \operatorname{rank} K \, ,$$ or equivalently, a Cartan subalgebra of $\mathfrak{k}$ becomes a Cartan subalgebra of $\mathfrak{g}$. By a theorem of Harish-Chandra, the rank condition is equivalent to the existence of (relative) discrete series representations of $G$. Here, an irreducible unitary representation $(\pi,\mathcal{H})$ is called a *(relative) discrete series representation* of $G$ if the matrix coefficient $g \mapsto (\pi(g)u,v)$ is square integrable on $G$ (modulo its center) for any $u,v \in \mathcal{H}$.
If $\mathfrak{g}$ is a simple Lie algebra of Hermitian type, then there exists a characteristic element $Z \in \mathfrak{c}(\mathfrak{k})$ such that $$\label{eqn:gkppz}
\mathfrak{g}_\mathbb{C}
:= \mathfrak{g} \otimes \mathbb{C}
= \mathfrak{k}_\mathbb{C} \oplus \mathfrak{p}_+ \oplus \mathfrak{p}_-$$ is the eigenspace decomposition of $\operatorname{ad}(Z)$ with eigenvalues $0$, $\sqrt{-1}$ and $-\sqrt{-1}$, respectively. We note that $\dim \mathfrak{c}(\mathfrak{k}) = 1$ if $\mathfrak{g}$ is a simple Lie algebra of Hermitian type, and therefore $\mathfrak{c}(\mathfrak{k}) = \mathbb{R} Z$.
Suppose $V$ is an irreducible $({\mathfrak {g}}_{\mathbb{C}}, K)$-module. We set $$\label{eqn:Hpk}
V^{\mathfrak{p}_+}
:= \set{v \in V}{Y v = 0
\ \text{ for any } Y \in \mathfrak{p}_+}\, .$$ Since $K$ normalizes $\mathfrak{p}_+$, $V^{\mathfrak{p}_+}$ is a $K$-submodule. Further, $V^{\mathfrak{p}_+}$ is either zero or an irreducible finite dimensional representation of $K$. We say $V$ is a *highest weight module* if $V^{\mathfrak{p}_+} \neq \{0\}$.
\[def:1.4\] Suppose $\pi$ is an irreducible unitary representation of $G$ on a Hilbert space $\mathcal{H}$. We set $\mathcal{H}_K:=\{v\in\mathcal{H}:
\dim_{\mathbb{C}} \mathbb{C}\text{-span}
\{\pi(k)v:k\in K\} < \infty \}
$. Then, $\mathcal{H}_K$ is a dense subspace of $\mathcal{H}$, on which the differential action $d\pi$ of the Lie algebra $\mathfrak{g}$ (and consequently that of its complexified Lie algebra $\mathfrak{g}_{\mathbb{C}}$) and the action of the compact subgroup $K$ is well-defined. We say $\mathcal{H}_K$ is *the underlying $(\mathfrak{g}_{\mathbb{C}},K)$-module of $(\pi,\mathcal{H})$*. We say $(\pi,\mathcal{H})$ is a *unitary highest weight representation* of $G$ if $\mathcal{H}_K^{\mathfrak{p}_+} \ne \{0\}$. Then, $\pi$ is *of scalar type* (or *of scalar minimal $K$-type*) if $\mathcal{H}^{\mathfrak{p}_+}_K$ is one dimensional; $\pi$ is a *(relative) holomorphic discrete series representation* for $G$ if the matrix coefficient $g \mapsto (\pi(g)u,v)$ is square integrable on $G$ modulo its center for any $u, v \in \mathcal{H}$. Lowest weight modules and anti-holomorphic discrete series representations are defined similarly with $\mathfrak{p}_+$ replaced by $\mathfrak{p}_-$.
This definition also applies to $G$ which is not simple (see Subsection \[subsec:8.1\]).
The classification of irreducible unitary highest weight representations was accomplished by Enright–Howe–Wallach [@xhew] and H. Jakobsen [@xjak] independently; see also [@xej]. There always exist infinitely many (relative) holomorphic discrete series representations of scalar type for any non-compact simple Lie group of Hermitian type.
Involutions on Hermitian symmetric spaces {#subsec:1.5}
-----------------------------------------
Suppose $G$ is a non-compact simple Lie group of Hermitian type. Let $\tau$ be an involutive automorphism of $G$ commuting with the Cartan involution $\theta$. We use the same letter $\tau$ to denote its differential. Then $\tau$ stabilizes $\mathfrak{k}$ and also $\mathfrak{c}(\mathfrak{k})$. Because $\tau^2 = \operatorname{id}$ and $\mathfrak{c}(\mathfrak{k}) = \mathbb{R} Z$, we have the following two possibilities: $$\begin{aligned}
\tau Z &= Z \, ,
\label{eqn:1.5.1}
\\
\tau Z &= -Z \, .
\label{eqn:1.5.2}\end{aligned}$$ Geometric meanings of these conditions become clear in the context of the embedding $G^{\tau}/ K^{\tau} \hookrightarrow G/K$, where $G^\tau := \set{g \in G}{\tau g = g}$ and $K^\tau := G^\tau \cap K$ (see [@xfo; @xjafbams; @xjafjdg; @xkobanaga]). The condition implies: $\tau$ acts [**holomorphically**]{} on the Hermitian symmetric space $G/K$, $G^\tau/K^\tau \hookrightarrow G/K$ defines a complex submanifold, whereas the condition implies: $\tau$ acts [**anti-holomorphically**]{} on $G/K$, $G^\tau/K^\tau \hookrightarrow G/K$ defines a totally real submanifold.
\[def:holo-anti\] We say the involutive automorphism $\tau$ is *of holomorphic type* if is satisfied, and is of *anti-holomorphic type* if is satisfied. The same terminology will be applied also to the symmetric pair $(G,H)$ (or its Lie algebras $(\mathfrak{g}, \mathfrak{h})$) corresponding to the involution $\tau$.
Here, we recall that $(G,H)$ is called a *symmetric pair* corresponding to $\tau$ if $H$ is an open subgroup of $G^\tau$ (see Subsections \[subsec:3.1\] and \[subsec:3.2.ex\]). We note that the Lie algebra $\mathfrak{h}$ of $H$ is equal to $\mathfrak{g}^\tau
:= \set{X \in \mathfrak{g}}{\tau X = X}$. The classification of symmetric pairs $(\mathfrak{g}, \mathfrak{g}^\tau)$ for simple Lie algebras $\mathfrak{g}$ was accomplished by M. Berger [@xber]. The classification of symmetric pairs $({\mathfrak {g}}, {\mathfrak {g}}^{\tau})$ of holomorphic type (respectively, of anti-holomorphic type) is regarded as a subset of Berger’s list, and will be presented in Table \[tbl:3.3.1\] (respectively, Table \[tbl:3.3.2\]).
Multiplicity-free restrictions — infinite dimensional case {#subsec:1.6}
----------------------------------------------------------
We are ready to state our main results. Let $G$ be a non-compact simple Lie group of Hermitian type, and $(G,H)$ a symmetric pair.
\[thm:A\] If $\pi$ is an irreducible unitary highest weight representation of scalar type of $G$, then the restriction $\pi|_H$ is multiplicity-free.
The branching law of the restriction $\pi |_H$ may and may not contain discrete spectra in Theorem \[thm:A\]. If $(G,H)$ is of holomorphic type then the restriction $\pi |_H$ is discretely decomposable (i.e. there is no continuous spectrum in the branching law); see Fact \[fact:3.4.1\]. Besides, the following theorem asserts that the multiplicities are still uniformly bounded even if we drop the assumption that $\pi$ is of scalar type.
\[thm:B\] We assume that the symmetric pair $(G,H)$ is of holomorphic type. Let $\pi$ be an irreducible unitary highest weight representation of $G$. The restriction $\pi|_H$ splits into a discrete Hilbert sum of irreducible unitary representations of $H$: $$\pi|_H \simeq \sideset{}{^\oplus}\sum_{\mu \in \widehat{H}}
m_\pi(\mu) \mu \, ,$$ and the multiplicities are uniformly bounded: $$C(\pi) := \sup_{\mu \in \widehat{H}}
m_\pi(\mu) < \infty \, .$$ $C(\pi) = 1$ if $\pi$ is of scalar type.
The second statement is a direct consequence of Theorems \[thm:A\] and \[thm:B\] (1). As we shall see in Section \[sec:6\], such uniform boundedness theorem does not hold in general if $\pi$ is not a highest weight representation (see Examples \[exam:6.3\] and \[ex:finite infinite\]).
Here are multiplicity-free theorems for the decomposition of tensor products, which are parallel to Theorems \[thm:A\] and \[thm:B\]:
\[thm:C\] Let $\pi_1$ and $\pi_2$ be irreducible unitary highest (or lowest) weight representations of scalar type. Then the tensor product $\pi_1 \widehat\otimes \pi_2$ is multiplicity-free as a representation of $G$.
Here, $\pi_1 \widehat{\otimes} \pi_2$ stands for the tensor product representation of two unitary representations $(\pi_1, \mathcal{H}_1)$ and $(\pi_2, \mathcal{H}_2)$ realized on the completion $\mathcal{H}_1 \widehat{\otimes} \mathcal{H}_2$ of the pre-Hilbert space $\mathcal{H}_1 \otimes \mathcal{H}_2$. (We do not need to take the completion if at least one of $\mathcal{H}_1$ or $\mathcal{H}_2$ is finite dimensional.) Theorem \[thm:C\] asserts that multiplicities in the direct integral of the irreducible decomposition are not greater than one in both discrete and continuous spectra. We note that continuous spectra appear in the irreducible decomposition of the tensor product representation $\pi_1 \widehat{\otimes} \pi_2$ only if $$\begin{aligned}
&\begin{cases}
\text{$\pi_1$ is a highest weight representation, and} \\
\text{$\pi_2$ is a lowest weight representation,}
\end{cases}\end{aligned}$$ or in reverse order.
If $\pi_1$ and $\pi_2$ are simultaneously highest weight representations (or simultaneously lowest weight representations), then the tensor product $\pi_1 \widehat\otimes \pi_2$ decomposes discretely. Dropping the assumption of scalar type, we have still a uniform estimate of multiplicities:
\[thm:D\] Let $\pi_1$ and $\pi_2$ be two irreducible unitary highest weight representations of $G$. The tensor product $\pi_1 \widehat\otimes \pi_2$ splits into a discrete Hilbert sum of irreducible unitary representations of $G$: $$\pi_1 \widehat\otimes \pi_2 \simeq
\sideset{}{^\oplus}\sum_{\mu \in \widehat{G}}
m_{\pi_1, \pi_2}(\mu) \mu \, ,$$ and the multiplicities $m_{\pi_1,\pi_2}(\mu)$ are uniformly bounded: $$C(\pi_1,\pi_2) := \sup_{\mu \in \widehat{G}}
m_{\pi_1, \pi_2}(\mu) < \infty \, .$$ $C(\pi_1, \pi_2) = 1$ if both $\pi_1$ and $\pi_2$ are of scalar type.
\[rem:D\] For classical groups, we can relate the constants $C(\pi)$ and $C(\pi_1,\pi_2)$ to the *stable constants* of branching coefficients of finite dimensional representations in the sense of F. Sato [@xsatof] by using the see-saw dual pair correspondence due to R. Howe [@xhoweseesaw].
Our machinery that gives the above multiplicity-free theorems is built on complex geometry, and we shall explicate the general theory for the line bundle case in Section \[sec:2\]. The key idea is to transfer properties on representations (e.g. unitarity, multiplicity-freeness) into the corresponding properties of reproducing kernels, which we analyze by geometric methods.
Multiplicity-free restrictions — finite dimensional case {#subsec:1.7}
--------------------------------------------------------
Our method yields multiplicity-free theorems not only for infinite dimensional representations but also for finite dimensional representations.
This subsection presents multiplicity-free theorems that are regarded as ‘finite dimensional version’ of Theorems \[thm:A\] and \[thm:C\]. They give a unified explanation of the multiplicity-free property of previously known branching formulae obtained by combinatorial methods such as the Littlewood–Richardson rule, Koike–Terada’s Young diagrammatic methods, Littelmann’s path method, minor summation formulae, etc. (see [@xhowemultone; @xkoiterjalg; @xmacd; @xokada; @xproctor; @xstem] and references therein). They also contain some ‘new’ cases, for which there are, to the best of our knowledge, no explicit branching formulae in the literature.
To state the theorems, let $\mathfrak{g}_{\mathbb{C}}$ be a complex simple Lie algebra, and $\mathfrak{j}$ a Cartan subalgebra. We fix a positive root system $\Delta^+ (\mathfrak{g}_{\mathbb{C}}, \mathfrak{j})$, and write $\alpha_1,\dots,\alpha_n$ for the simple roots. Let $\omega_1,\dots,\omega_n$ be the corresponding fundamental weights. We denote by $\pi_\lambda \equiv \F{\mathfrak{g}_{\mathbb{C}}}{\lambda}$ the irreducible finite dimensional representation of $\mathfrak{g}_{\mathbb{C}}$ with highest weight $\lambda$.
We say $\pi_\lambda$ is of **pan type** if $\lambda$ is a scalar multiple of some $\omega_i$ such that the nilradical of the maximal parabolic subalgebra corresponding to $\alpha_i$ is abelian (see Lemma \[lem:7.3.1\] for equivalent definitions).
\[thm:E\] Let $\pi$ be an arbitrary irreducible finite dimensional representation of $\mathfrak{g}_{\mathbb{C}}$ of pan type, and $(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ be any symmetric pair. Then, the restriction $\pi|_{\mathfrak{h}_{\mathbb{C}}}$ is multiplicity-free.
\[thm:F\] The tensor product $\pi_1 \otimes \pi_2$ of any two irreducible finite dimensional representations $\pi_1$ and $\pi_2$ of pan type is multiplicity-free.
Theorems E and F are the counterpart to Theorems \[thm:A\] and \[thm:C\] for finite dimensional representations. The main machinery of the proof is again Theorem \[thm:2.2\].
Alternatively, one could verify Theorems \[thm:E\] and \[thm:F\] by a classical technique: finding an open orbit of a Borel subgroup. For example, Littelmann [@xlittelmann] and Panyushev independently classified the pair of maximal parabolic subalgebras $(\mathfrak{p}_1, \mathfrak{p}_2)$ such that the diagonal action of a Borel subgroup $B$ of a complex simple Lie group $G_{\mathbb{C}}$ on $G_{\mathbb{C}} / P_1 \times G_{\mathbb{C}} / P_2$ has an open orbit. Here, $P_1, P_2$ are the corresponding maximal parabolic subgroups of $G_{\mathbb{C}}$. This gives another proof of Theorem \[thm:F\].
The advantage of our method is that it enables us to understand (or even to discover) the multiplicity-free property simultaneously, for both infinite and finite dimensional representations, for both continuous and discrete spectra, and for both classical and exceptional cases by the single principle. This is because our main machinery (Theorem \[thm:2.2\]) uses only a *local* geometric assumption (see Remark \[rem:2.3.2\] (2)). Thus, we can verify it at the same time for both compact and non-compact complex manifolds, and in turn get finite and infinite dimensional results, respectively.
Once we tell a priori that a representation is multiplicity-free, we may be tempted to find explicitly its irreducible decomposition. Recently, S. Okada [@xokada] found explicit branching laws for some classical cases that arise in Theorems \[thm:E\] and \[thm:F\] by using minor summation formulae, and H. Alikawa [@xalikawa] for $(\mathfrak{g}, \mathfrak{h}) = (\mathfrak{e}_6, \mathfrak{f}_4)$ corresponding to Theorem \[thm:E\]. We note that the concept of pan type representations includes *rectangular-shaped* representations of classical groups (see [@xkratten; @xokada]).
There are also some few cases where $\pi_1 \otimes \pi_2$ is multiplicity-free even though neither $\pi_1$ nor $\pi_2$ is of pan type. See the recent papers [@xkleiden] or [@xstemgl] for the complete list of such pairs $(\pi_1, \pi_2)$ for $\mathfrak{g}_{\mathbb{C}} = \mathfrak{gl}(n, \mathbb{C})$. The method in [@xkleiden] to find all such pairs is geometric and based on the ‘vector bundle version’ of Theorem \[thm:2.2\] proved in [@mfbdle], whereas the method in [@xstemgl] is combinatorial and based on case-by-case argument.
We refer the reader to our papers [@visiblesymm; @xkgencar; @xksovisible] for some further results relevant to Theorems \[thm:E\] and \[thm:F\] along the same line of argument here.
$SL_2$ examples {#subsec:1.8}
---------------
We illustrate the above theorems by $SL_2$ examples.
\[exam:1.8\] 1)We denote by $\pi_n$ the holomorphic discrete series representation of $G=SL(2,\mathbb{R})$ with minimal $K$-type $\chi_n$ ($n \ge 2$), where we write $\chi_n$ for the character of $K=SO(2)$ parametrized by $n \in \mathbb{Z}$. Likewise $\pi_{-n}$ denotes the anti-holomorphic discrete series representation of $SL(2, \mathbb{R})$ with minimal $K$-type $\chi_{-n}$ ($n \ge 2$). We note that any holomorphic discrete series of $SL(2,\mathbb{R})$ is of scalar type.
We write $\pi^\varepsilon_{\sqrt{-1} \nu}$ ($\varepsilon = \pm 1, \nu \in \mathbb{R})$ for the unitary principal series representations of $SL(2, \mathbb{R})$. We have a unitary equivalence $\pi^{\varepsilon}_{\sqrt{-1} \nu} \simeq
\pi^{\varepsilon}_{- \sqrt{-1} \nu}$. We write $\chi_\zeta$ for the unitary character of $SO_0(1,1) \simeq \mathbb{R}$ parametrized by $\zeta \in \mathbb{R}$.
Let $m \ge n \ge 2$. Then, the following branching formulae hold. All of them are multiplicity-free, as is ‘predicted’ by Theorems \[thm:A\] and \[thm:C\]:
$$\begin{aligned}
\pi_n|_{SO_0(1,1)}
&\simeq \int_{-\infty}^\infty \chi_{\zeta} \, d \zeta \, ,
\label{eqn:1.8.1a}
\\
\pi_n|_{SO(2)}
&\simeq \sideset{}{^\oplus}\sum_{k\in \mathbb{N}} \chi_{n + 2 k} \, ,
\label{eqn:1.8.1b}
\\
\pi_m \widehat\otimes \pi_{-n}
&\simeq
\int_{0}^\infty \pi^{(-1)^{m-n}}_{\sqrt{-1} \nu} d \nu \quad
\oplus \sum \Sb k \in \mathbb{N} \\ 0 \le 2k \le m-n-2 \endSb
\pi_{m-n-2k} \, ,
\label{eqn:1.8.1c}
\\
\pi_m \widehat\otimes \pi_n
&\simeq \sideset{}{^\oplus}\sum_{k\in \mathbb{N}} \pi_{m+n + 2 k} \, .
\label{eqn:1.8.1d}\end{aligned}$$
The key assumption of our main machinery (Theorem \[thm:2.2\]) that leads us to Theorems \[thm:A\] and \[thm:C\] is illustrated by the following geometric results in this $SL_2$ case: Given any element $z$ in the Poincaré disk $D$, there exists $\varphi \in \mathbb{R}$ such that $e^{\sqrt{-1}\varphi} z = \overline{z}$. In fact, one can take $\varphi = -2 \arg z$. This is the geometry that explains the multiplicity-free property of . Given any two elements $z, w \in D$, there exists a linear fractional transform $T$ on $D$ such that $T (z) = \overline{z}$ and $T (w) = \overline{w}$. This is the geometry for . These are examples of the geometric view point that we pursued in [@visiblesymm] for symmetric pairs. Here is a of the above example. Let $\pi_n$ be the irreducible $n+1$-dimensional representation of $SU(2)$. Then we have the following branching formulae: For $m, n \in \mathbb{N}$,
$$\begin{aligned}
\addtocounter{equation}{4}
\hphantom{\pi_n|_{SO_0(1,1)}}
\llap{$\pi_n|_{SO(2)}$}
&\simeq
\rlap{$\chi_{n} \oplus \chi_{n-2} \oplus \cdots \oplus \chi_{-n} \, ,$}
{\hphantom{\int_{0}^\infty \pi^{(-1)^{m-n}}_{\sqrt{-1} \nu} d \nu \quad
\oplus \sum \Sb k \in \mathbb{N} \\ 0 \le 2k \le m-n-2 \endSb
\pi_{m-n-2k} \, ,}}
{\vphantom{\bigg|}}
\label{eqn:1.8.1e}
\\
\pi_m \otimes \pi_n
&\simeq
\pi_{n+m} \oplus \pi_{n+m-2} \oplus \cdots \oplus \pi_{|n-m|}
\, .
\label{eqn:1.8.1f}\end{aligned}$$
The formula corresponds to the character formula, whereas is known as the Clebsch–Gordan formula. The multiplicity-free property of these formulae is the simplest example of Theorems \[thm:E\] and \[thm:F\].
Analysis on multiplicity-free representations {#subsec:analysismf}
---------------------------------------------
Multiplicity-free property arouses our interest in developing beautiful analysis on such representations, as we discussed in Subsection \[subsec:1.7\] for finite dimensional cases. This subsection picks up some recent topics about detailed analysis on multiplicity-free representations for infinite dimensional cases.
Let $G$ be a connected, simple non-compact Lie group of Hermitian type. We begin with branching laws without continuous spectra, and then discuss branching laws with continuous spectra.
[1)]{}(Discretely decomposable case)Let $(G,H)$ be a symmetric pair of holomorphic type. Then, any unitary highest weight representation $\pi$ of $G$ decomposes discretely when restricted to $H$ (Fact \[fact:3.4.1\]). Suppose now that $\pi$ is a holomorphic discrete series representation. L.-K. Hua [@xhua], B. Kostant, W. Schmid [@xschmidherm] and K. Johnson [@xjohnson] found an explicit formula of the restriction $\pi|_K$ ($K$-type formula). This turns out to be multiplicity-free. Alternatively, the special case of Theorem \[thm:B\] (2) by setting $H=K$ gives a new proof of this multiplicity-free property. Furthermore, we consider a generalization of the Hua–Kostant–Schmid formula from compact $H$ to noncompact $H$, for which Theorem \[thm:B\] (2) still ensures that the generalization will be multiplicity-free. This generalized formula is stated in Theorem \[thm:gHKS\], which was originally given in [@xkmfjp Theorem C]. In Section \[sec:8\], we give a full account of its proof. W. Bertram and J. Hilgert [@xbehi] obtained some special cases independently, and Ben Saïd [@xsaid] studied a quantative estimate of this multiplicity-free $H$-type formula (see also [@xyawa; @xzhangtensor] for some *singular* cases). The branching formulae of the restriction of *singular* highest weight representations $\pi$ are also interesting. For instance, the restriction of the Segal–Shale–Weil representation $\varpi$ of $Mp(n, \mathbb{R})$ with respect to $U(p, n-p)$ (more precisely, its double covering) decomposes discretely into a multiplicity-free sum of the so called [*[ladder representations]{}*]{} of $U(p, n-p)$ (e.g. [@xkv Introduction]). This multiplicity-free property is a special case of Howe’s correspondence because $(U(p, n-p), U(1))$ forms a dual pair in $Mp(n, \mathbb{R})$, and also is a special case of Theorem \[thm:A\] because $(\mathfrak{sp}(n, \mathbb{R}), {\mathfrak{u}}(p, n-p))$ forms a symmetric pair. Explicit branching laws for most of classical cases corresponding to Theorems \[thm:B\] (2) and \[thm:D\] (2) (see Theorems \[thm:gHKS\], \[thm:tensordeco\], \[thm:upqupq\]) can be obtained by using the , which we hope to report in another paper. (Branching laws with continuous spectra)Suppose $\pi_1$ is a highest weight module and $\pi_2$ is a lowest weight module, and both being of scalar type. If both $\pi_1$ and $\pi_2$ are discrete series representations in addition, then the tensor product $\pi_1 \widehat \otimes \pi_2$ is unitarily equivalent to the regular representation on $L^2(G/K, \chi)$, the Hilbert space of $L^2$-sections of the $G$-equivalent line bundle $G\times_K \mathbb{C}_\chi \to G/K$ associated to some unitary character $\chi$ of $K$ (R. Howe [@xhoweseesaw], J. Repka [@xrep]). In particular, Theorem \[thm:C\] gives a new proof of the multiplicity-free property of the Plancherel formula for $L^2(G/K,\chi)$. Yet another proof of the multiplicity-free property of $L^2(G/K, \chi)$ was given in [@RIMS Theorem 21] by still applying Theorem \[thm:2.2\] to the *crown domain* (equivalently, the Akhiezer–Gindikin domain) of the Riemannian symmetric space $G/K$. The explicit decomposition of $L^2(G/K, \chi)$ was found by J. Heckman [@xhecsch] and N. Shimeno [@xshimeno] that generalizes the work of Harish-Chandra, S. Helgason, and S. Gindikin–F. Karpelevich for the trivial bundle case.
In contrast to Riemannian symmetric spaces, it is known that multiplicity-free property in the Plancherel formula fails for (non-Riemannian) symmetric spaces $G/H$ in general (see [@xbsann; @xde] for the description of the multiplicity of the most continuous series representations for $G/H$ in terms of Weyl groups). Similarly to the case 2-a), the restriction $\pi|_H$ for a symmetric pair $(G,H)$ of non-holomorphic type is multiplicity-free and is decomposed into only continuous spectra if $\pi$ is a holomorphic discrete series of scalar type. This case was studied by G. Ólafsson–B. Ørsted ([@xoo]). Theorem \[thm:C\] applied to non-discrete series representations $\pi_1$ and $\pi_2$ (i.e. tensor products of *singular* unitary highest weight representations) provides new settings of multiplicity-free branching laws. They might be interesting from the view point of representation theory because they construct small representations as discrete summands. (We note that irreducible unitary representations of reductive Lie groups have not been classified even in the spherical case. See [@xbarbasch] for the split case.) They might be interesting also from the view point of spectral theory and harmonic analysis which is relevant to the [*[canonical representation]{}*]{} in the sense of Vershik–Gelfand–Graev. Once we know the branching law is a priori multiplicity-free, it is promising to obtain its explicit formula. Some special cases have been worked on in this direction so far, for $G= SL(2,\mathbb{R})$ by V. F. Molchanov [@xmol]; for $G= SU(2,2)$ by B. Ørsted and G. Zhang [@xoz]; for $G= SU(n,1)$ by G. van Dijk and S. Hille [@xvdh]; for $G = SU(p,q)$ by Y. Neretin and G. Ol’shanskiĭ [@xnere; @xneol]. See also G. van Dijk–M. Pevzner [@xDijkPev], M. Pevzner [@xpevzner] and G. Zhang [@xzhang]. Their results show that a different family of irreducible unitary representations (sometimes, spherical complementary series representations) can occur in the same branching laws and each multiplicity is not greater than one.
Organization of this article {#subsec:1.9}
----------------------------
This paper is organized as follows: In Section \[sec:2\], we give a proof of an abstract multiplicity-free theorem (Theorem \[thm:2.2\]) in the line bundle setting. This is an extension of a theorem of Faraut–Thomas [@xft], whose idea may go back to Gelfand’s proof [@xgelsph] of the commutativity of the Hecke algebra $L^1(K \backslash G/K)$. Theorem \[thm:2.2\] is a main method in this article to find various multiplicity-free theorems. In Section \[sec:3\], we use Theorem \[thm:2.2\] to give a proof of Theorem \[thm:A\]. The key idea is the reduction of the geometric condition (*strongly visible action* in the sense of [@RIMS]) to the existence problem of a nice involutive automorphism $\sigma$ of $G$ satisfying a certain rank condition. Section \[sec:4\] considers the multiplicity-free theorem for the tensor product representations of two irreducible highest (or lowest) weight modules and gives a proof of Theorem \[thm:C\]. Sections \[sec:5\] and \[sec:6\] examine our assumptions in our multiplicity-free theorems (Theorems \[thm:A\] and \[thm:C\]). That is, we drop the assumption of ‘scalar type’ in Section \[sec:5\] and prove that multiplicities are still uniformly bounded (Theorems \[thm:B\] and \[thm:D\]). We note that multiplicities can be greater than one in this generality. In Section \[sec:6\], we leave unchanged the assumption that $(G,H)$ is a symmetric pair, and relax the assumption that $\pi$ is a highest weight module. We illustrate by examples a wild behavior of multiplicities without this assumption. In Section \[sec:7\], analogous results of Theorems \[thm:A\] and \[thm:C\] are proved for finite dimensional representations of compact groups. In Section \[sec:8\], we present explicit branching laws that are assured a priori to be multiplicity-free by Theorems \[thm:A\] and \[thm:C\]. Theorem \[thm:tensordeco\] generalizes the Hua–Kostant–Schmid formula. In Section \[sec:9\] (Appendix) we present some basic results on homogeneous line bundles for the convenience of the reader, which give a sufficient condition for the assumption in Theorem \[thm:2.2\].
Main machinery from complex geometry {#sec:2}
====================================
J. Faraut and E. Thomas [@xft], in the case of trivial twisting parameter, gave a sufficient condition for the commutativity of ${\operatorname{End}}_H(\mathcal{H})$ by using the theory of reproducing kernels, which we extend to the general, twisted case in this preliminary section. The proof parallels to theirs, except that we need just find an additional condition when we formalize Theorem \[thm:2.2\] in the line bundle setting.
Basic operations on holomorphic line bundles {#subsec:2.1}
--------------------------------------------
Let $\mathcal{L} \to D$ be a holomorphic line bundle over a complex manifold $D$. We denote by $\mathcal{O}(\mathcal{L}) \equiv \mathcal{O}(D,\mathcal{L})$ the space of holomorphic sections of $\mathcal{L} \to D$. Then $\mathcal{O}(\mathcal{L})$ carries a Fréchet topology by the uniform convergence on compact sets. If a Lie group $H$ acts holomorphically and equivariantly on the holomorphic line bundle $\mathcal{L} \to D$, then $H$ defines a (continuous) representation on $\mathcal{O}(\mathcal{L})$ by the pull-back of sections.
Let $\{U_\alpha\}$ be trivializing neighborhoods of $D$, and $g_{\alpha \beta} \in \mathcal{O}^\times(U_\alpha \cap U_\beta)$ the transition functions of the holomorphic line bundle $\mathcal{L} \to D$. Then an anti-holomorphic line bundle $\overline{\mathcal{L}} \to D$ is a complex line bundle with the transition functions $\overline{g_{\alpha \beta}}$. We denote by $\overline{\mathcal{O}}(\overline{\mathcal{L}})$ the space of anti-holomorphic sections for $\overline{\mathcal{L}} \to D$.
Suppose $\sigma$ is an anti-holomorphic diffeomorphism of $D$. Then the pull-back $\sigma^* \mathcal{L} \to D$ is an anti-holomorphic line bundle over $D$. In turn, $\overline{\sigma^* \mathcal{L}} \to D$ is a holomorphic line bundle over $D$ (see Appendix for more details).
Abstract multiplicity-free theorem {#subsec:2.2}
----------------------------------
Here is the main machinery to prove various multiplicity-free theorems of branching laws including Theorems \[thm:A\] and \[thm:C\] (infinite dimensional representations) and Theorems \[thm:E\] and \[thm:F\] (finite dimensional representations).
\[thm:2.2\] Let $(\pi,\mathcal{H})$ be a unitary representation of a Lie group $H$. Assume that there exist an $H$-equivariant holomorphic line bundle $\mathcal{L} \to D$ and an anti-holomorphic involutive diffeomorphism $\sigma$ of $D$ with the following three conditions:
\[eqn:2.2.1\] There is an injective (continuous) $H$-intertwining map $\mathcal{H} \to \mathcal{O}(\mathcal{L})$.
\[eqn:2.2.2\] There exists an isomorphism of $H$-equivariant holomorphic line bundles $\Psi: \mathcal{L} \overset{\sim}{\to} \overline{\sigma^* \mathcal{L}}$.
\[eqn:2.2.3\] Given $x \in D$, there exists $g \in H$ such that $\sigma (x) = g \cdot x$.
Then, the ring ${\operatorname{End}}_H(\mathcal{H})$ of continuous $H$-intertwining operators on $\mathcal{H}$ is commutative. Consequently, $(\pi, \mathcal{H})$ is multiplicity-free (see Definition \[def:1.2\]).
Remarks on Theorem \[thm:2.2\]
------------------------------
This subsection gives brief comments on Theorem \[thm:2.2\]. First, we consider a special case, and also a generalization.
1)Suppose $\mathcal{L} \to D$ is the trivial line bundle. Then, the condition is automatically satisfied. In this case, Theorem \[thm:2.2\] was proved in [@xft]. 2)An extension of Theorem \[thm:2.2\] to the equivariant vector bundle $\mathcal{V} \to D$ is the main subject of [@mfbdle], where a more general multiplicity-free theorem is obtained under an additional condition that the isotropy representation of $H_x = \{ h \in H: h \cdot x = x \}$ on the fiber $\mathcal{V}_x$ is multiplicity-free for generic $x \in D$. Obviously, the $H_x$-action on $\mathcal{V}_x$ is multiplicity-free for the case $\dim \mathcal{V}_x = 1$, namely, for the line bundle case.
Next, we examine the conditions and .
\[rem:2.3.2\] 1)In many cases, the condition is naturally satisfied. We shall explicate how to construct the bundle isomorphism $\Psi$ in Lemma \[lem:9.6\] for a Hermitian symmetric space $D$. 2)As the proof below shows, Theorem \[thm:2.2\] still holds if we replace $D$ by an $H$-invariant open subset $D'$. Thus, the condition is **local**. The concept of ‘**visible action**’ (see [@xkleiden; @mfbdle; @xkgencar]) arises from the condition on the base space $D$. 3)The condition is automatically satisfied if $H$ acts transitively on $D$. But we are interested in a more general setting where each $H$-orbit has a positive codimension in $D$. We find in Lemma \[lem:3.2\] a sufficient condition for in terms of rank condition for a symmetric space $D$.
Reproducing kernel {#subsec:2.4}
------------------
This subsection gives a quick summary for the reproducing kernel of a Hilbert space $\mathcal{H}$ realized in the space $\mathcal{O}(\mathcal{L})$ of holomorphic sections for a holomorphic line bundle $\mathcal{L}$ (see [@mfbdle] for a generalization to the vector bundle case). Since the reproducing kernel $K_{\mathcal{H}}$ contains all the information on the Hilbert space $\mathcal{H}$, our strategy is to make use of $K_{\mathcal{H}}$ in order to prove Theorem \[thm:2.2\].
Suppose that there is an injective and continuous map for a Hilbert space $\mathcal{H}$ into the Fréchet space $\mathcal{O}(\mathcal{L})$. Then, the point evaluation map $$\mathcal{O}(\mathcal{L}) \supset \mathcal{H} \to \mathcal{L}_z \simeq \mathbb{C}\, , \quad f \mapsto f(z)$$ is continuous with respect to the Hilbert topology on $\mathcal{H}$.
Let $\{\varphi_\nu\}$ be an orthonormal basis of $\mathcal{H}$. We define $$K_{\mathcal{H}}(x,y) \equiv K(x,y)
:= \sum_{\nu} \varphi_\nu(x) \overline{\varphi_\nu(y)}
\in
\mathcal{O}(\mathcal{L}) \widehat{\otimes}
\overline{\mathcal{O}}(\overline{\mathcal{L}}) \, .$$ Then, $K(x,y)$ is well-defined as a holomorphic section of $\mathcal{L} \to D$ for the first variable, and as an anti-holomorphic section of $\overline{\mathcal{L}} \to D$ for the second variable. The definition is independent of the choice of an orthonormal basis $\{\varphi_\nu\}$. $K(x,y)$ is called the reproducing kernel of $\mathcal{H}$.
\[lem:2.4\] [1)]{}For each $y \in D$, $K(\cdot, y) \in \mathcal{H} \otimes \overline{\mathcal{L}_y} \ (\simeq \mathcal{H})$ and $(f(\cdot), K(\cdot,y))_{\mathcal{H}} = f(y)$ for any $f \in \mathcal{H}$. Let $K_i(x,y)$ be the reproducing kernels of Hilbert spaces $\mathcal{H}_i \ \subset \mathcal{O}(\mathcal{L})$ with inner products $(\ , \ )_{\mathcal{H}_i}$, respectively, for $i = 1, 2$. If $K_1 \equiv K_2$, then $\mathcal{H}_1 = \mathcal{H}_2$ and $(\ , \ )_{\mathcal{H}_1} = (\ , \ )_{\mathcal{H}_2}$. If $K_1(x,x) = K_2(x,x)$ for any $x \in D$, then $K_1 \equiv K_2$.
\(1) and (2) are standard. We review only the way how to recover $\mathcal{H}$ together with its inner product from a given reproducing kernel. For each $y \in D$, we fix an isomorphism $\mathcal{L}_y \simeq \mathbb{C}$. Through this isomorphism, we can regard $K(\cdot, y) \in \mathcal{H} \otimes \overline{\mathcal{L}_{y}}$ as an element of $\mathcal{H}$. The Hilbert space $\mathcal{H}$ is the completion of the $\mathbb{C}$-span of $\set{K(\cdot, y)}{y \in D}$ with pre-Hilbert structure $$(K(\cdot, y_1), K(\cdot, y_2))_{\mathcal{H}} := K(y_2, y_1) \in
\mathcal{L}_{y_2} \otimes \overline{\mathcal{L}_{y_1}}
\ (\simeq \mathbb{C}) \, .
\label{eqn:2.4.1}$$ This procedure is independent of the choice of the isomorphism ${\mathcal{L}}_y \simeq \mathbb{C}$. Hence, the Hilbert space $\mathcal{H}$ together with its inner product is recovered. We denote by $\overline{D}$ the complex manifold endowed with the conjugate complex structure on $D$. Then, $\overline{\mathcal{L}} \to \overline{D}$ is a holomorphic line bundle, and $K(\cdot,\cdot) \equiv K_{\mathcal{H}}(\cdot,\cdot)$ is a holomorphic section of the holomorphic line bundle $\mathcal{L} \boxtimes \overline{\mathcal{L}} \to D \times \overline{D}$. As the diagonal embedding $\iota: D \to D \times \overline{D}, z \mapsto (z, z)$ is totally real, $(K_1-K_2)|_{\iota(D)} \equiv 0$ implies $K_1-K_2\equiv 0$ by the unicity theorem of holomorphic functions.
Construction of $J$ {#subsec:2.5}
-------------------
Suppose we are in the setting of Theorem \[thm:2.2\]. We define an anti-linear map $$J : \mathcal{O}(\mathcal{L}) \to \mathcal{O}(\mathcal{L})\, ,
\quad f\mapsto J f$$ by $J f(z) := \overline{f(\sigma(z))}$ ($z \in D$). $Jf$ is regarded as an element of $\mathcal{O}(\mathcal{L})$ through the isomorphism $
\Psi_* : \mathcal{O}(\mathcal{L}) \simeq
\mathcal{O}(\overline{\sigma^* \mathcal{L}})
$ (see ).
\[lem:2.5\] In the setting of Theorem \[thm:2.2\], we identify $\mathcal{H}$ with a subspace of $\mathcal{O}(\mathcal{L})$. Then, the anti-linear map $J$ is an isometry from $\mathcal{H}$ onto $\mathcal{H}$.
We put $\widetilde{\mathcal{H}} := J(\mathcal{H})$, equipped with the inner product $$(J f_1, J f_2)_{\widetilde{\mathcal{H}}} := (f_2, f_1)_{\mathcal{H}}
\quad \text{for } f_1, f_2 \in \mathcal{H} \, .
\label{eqn:2.5.1}$$ If $\{\varphi_\nu\}$ is an orthonormal basis of $\mathcal{H}$, then $\widetilde{\mathcal{H}}$ is a Hilbert space with orthonormal basis $\{J \varphi_\nu\}$. Hence, the reproducing kernel of $\widetilde{\mathcal{H}}$ is given by $K_{\widetilde{\mathcal{H}}}(x,y) = K_{\mathcal{H}}(\sigma(y), \sigma(x))$ because $$ K_{\widetilde{\mathcal{H}}}(x,y)
=\sum_\nu J \varphi_\nu(x) \overline{J \varphi_\nu(y)}
=\sum_\nu \overline{\varphi_\nu(\sigma(x))} \
\overline{\overline{\varphi_\nu(\sigma(y))}}
= K_{\mathcal{H}}(\sigma(y), \sigma(x)) \, .
\label{eqn:2.5.2}$$ We fix $x \in D$ and take $g \in H$ such that $\sigma (x) = g \cdot x$ (see ). Substituting $x$ for $y$ in , we have $$K_{\widetilde{\mathcal{H}}}(x,x)
= K_{\mathcal{H}}(\sigma (x), \sigma (x))
= K_{\mathcal{H}}(g \cdot x, g \cdot x)
= K_{\mathcal{H}}(x, x) \, .$$ Here, the last equality holds because $\{ \varphi_\nu (g \cdot {}) \}$ is also an orthonormal basis of $\mathcal{H}$ as $(\pi, \mathcal{H})$ is a unitary representation of $H$. Then, by Lemma \[lem:2.4\], the Hilbert space $\widetilde{\mathcal{H}}$ coincides with $\mathcal{H}$ and $$(J f_1, J f_2)_{\mathcal{H}} = (f_2, f_1)_{\mathcal{H}}
\quad \text{for } f_1, f_2 \in \mathcal{H}\, .
\label{eqn:2.5.3}$$ This is what we wanted to prove.
Proof of $A^* = JAJ^{-1}$
-------------------------
\[lem:2.6\] Suppose $A \in {\operatorname{End}}_H(\mathcal{H})$. Then the adjoint operator $A^*$ of $A$ is given by $$A^* = J A J^{-1} \, .
\label{eqn:JAJ}$$
We divide the proof into three steps. $\underline {\text{Step 1}}$ (positive self-adjoint case):Assume $A \in {\operatorname{End}}_H(\mathcal{H})$ is a positive self-adjoint operator. Let $\mathcal{H}_A$ be the Hilbert completion of $\mathcal{H}$ by the pre-Hilbert structure $$(f_1, f_2)_{\mathcal{H}_A} := (A f_1, f_2)_{\mathcal{H}}
\quad \text{for } f_1, f_2 \in \mathcal{H} \, .
\label{eqn:2.6.2}$$ If $f_1, f_2 \in \mathcal{H}$ and $g \in H$, then $$\begin{aligned}
(\pi(g) f_1, \pi(g) f_2)_{{\mathcal{H}}_A}
&= (A \pi(g) f_1, \pi(g) f_2)_{\mathcal{H}}
\\
&= (\pi(g) A f_1, \pi(g) f_2)_{\mathcal{H}}
= (A f_1, f_2)_{\mathcal{H}}
= (f_1, f_2)_{{\mathcal{H}}_A} \, .\end{aligned}$$ Therefore, $(\pi, \mathcal{H})$ extends to a unitary representation on ${\mathcal{H}}_A$. Applying (2.5.3) to both $\mathcal{H}_A$ and $\mathcal{H}$, we have $$\begin{gathered}
(A f_1, f_2)_{\mathcal{H}}
= (f_1, f_2)_{\mathcal{H}_A}
= (J f_2, J f_1)_{\mathcal{H}_A}
= (A J f_2, J f_1)_{\mathcal{H}}
\\
= (J f_2, A^* J f_1)_{\mathcal{H}}
= (J f_2, J J^{-1} A^* J f_1)_{\mathcal{H}}
= (J^{-1} A^* J f_1, f_2)_{\mathcal{H}} \, .\end{gathered}$$ Hence, $A = J^{-1} A^* J$, and follows.
$\underline{\text{Step 2}}$ (self-adjoint case):Assume $A \in {\operatorname{End}}_H(\mathcal{H})$ is a self-adjoint operator. Let $A = \int \lambda d E_\lambda$ be the spectral decomposition of $A$. Then every projection operator $E_\lambda \in {\operatorname{End}}(\mathcal{H})$ also commutes with $\pi(g)$ for all $g \in H$, namely, $E_\lambda \in {\operatorname{End}}_H(\mathcal{H})$. We define $$A_+ := \int_{\lambda \ge 0} \lambda d E_\lambda \, ,
\qquad
A_- := \int_{\lambda< 0} \lambda d E_\lambda \, .$$ Then $A = A_+ + A_-$. Let $I$ be the identity operator on $\mathcal{H}$. As a positive self-adjoint operator $A_+ + I$ is an element of ${\operatorname{End}}_H(\mathcal{H})$, we have $
(A_+ + I)^* = J (A_+ + I) J^{-1}
$ by Step 1, whence $
A_+^* = J A_+ J^{-1}.
$ Applying Step 1 again to $- A_-$, we have $
A_-^* = J A_- J^{-1}.
$ Thus, $$A^* = A_+^* + A_-^* = J A_+ J^{-1} + J A_- J^{-1}
= J (A_+ + A_-) J^{-1}
= J A J^{-1} \, .$$
$\underline{\text{Step 3}}$ (general case):Suppose $A \in {\operatorname{End}}_H(\mathcal{H})$. Then $A^*$ also commutes with $\pi(g)$ ($g \in H$) because $\pi$ is unitary. We put $B := \frac{1}{2}(A+ A^*)$ and $C := \frac{\sqrt{-1}}{2}(A^* - A)$. Then, both $B$ and $C$ are self-adjoint operators commuting with $\pi(g)$ ($g \in H$). It follows from Step 2 that $
B^* = J B J^{-1}
$ and $
C^* = J C J^{-1} .
$ As $J$ is an anti-linear map, we have $$(\sqrt{-1}\, C)^*
= - \sqrt{-1}\, C^*
= - \sqrt{-1}\, J C J^{-1}
= J (\sqrt{-1}\, C) J^{-1} \, .$$ Hence, $A = B + \sqrt{-1}\, C$ also satisfies $
A^* = J A J^{-1}.
$
Proof of Theorem \[thm:2.2\] {#subsec:2.7}
----------------------------
We are now ready to complete the proof of Theorem \[thm:2.2\]. Let $A$, $B \in {\operatorname{End}}_H(\mathcal{H})$. By Lemma \[lem:2.6\], we have $$\begin{aligned}
& AB = J^{-1}(AB)^* J = (J^{-1} B^* J) (J^{-1} A^* J) = BA \, .\end{aligned}$$ Therefore, ${\operatorname{End}}_H(\mathcal{H})$ is commutative.
Proof of Theorem \[thm:A\] {#sec:3}
==========================
This section gives a proof of Theorem \[thm:A\] by using Theorem \[thm:2.2\]. The core of the proof is to reduce the geometric condition to an algebraic condition (the existence of a certain involution of the Lie algebra). This reduction is stated in Lemma \[lem:3.2\]. The reader who is familiar with symmetric pairs can skip Subsections \[subsec:3.1\], \[subsec:3.2.ex\], \[subsec:3.3\] and \[subsec:3.5\].
Reductive symmetric pairs {#subsec:3.1}
-------------------------
Let $G$ be a Lie group. Suppose that $\tau$ is an involutive automorphism of $G$. We write $$G^\tau := \{ g \in G: \tau g = g \}$$ for the fixed point subgroup of $\tau$, and denote by $G_0^\tau$ its connected component containing the unit element. The pair $(G,H)$ (or the pair $(\mathfrak{g}, \mathfrak{h})$ of their Lie algebras) is called a *symmetric pair* if the subgroup $H$ is an open subgroup of $G^\tau$, that is, if $H$ satisfies $$G_0^\tau \subset H \subset G^\tau .$$ It is called a *reductive symmetric pair* if $G$ is a reductive Lie group; a *semisimple symmetric pair* if $G$ is a semisimple Lie group. Obviously, a semisimple symmetric pair is a reductive symmetric pair.
We shall use the same letter $\tau$ to denote the differential of $\tau$. We set $$\mathfrak{g}^{\pm \tau} := \set{Y \in \mathfrak{g}}{\tau Y = \pm Y}\,.$$ Then, it follows from $\tau^2 =\operatorname{id}$ that we have a direct sum decomposition $$\mathfrak{g} = \mathfrak{g}^\tau \oplus \mathfrak{g}^{-\tau}.$$
Suppose now that $G$ is a semisimple Lie group. It is known that there exists a Cartan involution $\theta$ of $G$ commuting with $\tau$. Take such $\theta$, and we write $K := G^\theta =\set{g \in G}{\theta g = g}$. Then, $K$ is compact if $G$ is a linear Lie group. The direct sum decomposition $$\mathfrak{g} = \mathfrak{k}\oplus \mathfrak{p}
\equiv \mathfrak{g}^\theta \oplus \mathfrak{g}^{-\theta}$$ is called a Cartan decomposition. Later, we shall allow $G$ to be non-linear, in particular, $K$ is not necessarily compact. The *real rank* of $\mathfrak{g}$, denoted by ${\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}$, is defined to be the dimension of a maximal abelian subspace of $\mathfrak{g}^{-\theta}$.
As $(\tau\theta)^2 = \operatorname{id}$, the pair $(\mathfrak{g}, \mathfrak{g}^{\tau\theta})$ also forms a symmetric pair. The Lie group $$G^{\tau\theta} = \{ g \in G : (\tau\theta)(g) = g \}$$ is a reductive Lie group with Cartan involution $\theta|_{G^{\tau\theta}}$, and its Lie algebra $\mathfrak{g}^{\tau\theta}$ is reductive with Cartan decomposition $$\label{eqn:gtaut}
\mathfrak{g}^{\tau\theta} = \mathfrak{g}^{\tau\theta,\theta}
\oplus \mathfrak{g}^{\tau\theta,-\theta}
= \mathfrak{g}^{\tau,\theta} \oplus \mathfrak{g}^{-\tau,-\theta} \, .$$ Here, we have used the notation $\mathfrak{g}^{-\tau,-\theta}$ and alike, defined as follows: $$\mathfrak{g}^{-\tau,-\theta}
:= \{ Y \in \mathfrak{g} :
(-\tau) Y = (-\theta) Y = Y \} \, .$$ Then, the dimension of a maximal abelian subspace $\mathfrak{a}$ of $\mathfrak{g}^{-\tau,-\theta}$ is equal to the real rank of $\mathfrak{g}^{\tau\theta}$, which is referred to as the *split rank* of the semisimple symmetric space $G/H$. We shall write ${\mathbb{R}\text{-}\operatorname{rank}}G/H$ or ${\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}/\mathfrak{g}^\tau$ for this dimension. Thus, $${\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}^{\theta\tau} = {\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}/\mathfrak{g}^\tau
\, .
\label{eqn:3.1.1}$$ In particular, we have ${\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g} = {\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}/\mathfrak{k}$ if we take $\tau$ to be $\theta$.
The Killing form on the Lie algebra $\mathfrak{g}$ is non-degenerate on $\mathfrak{g}$, and is also non-degenerate when restricted to ${\mathfrak {h}}$. Then, it induces an ${\operatorname{Ad}}(H)$-invariant non-degenerate bilinear form on $\mathfrak{g}/\mathfrak{h}$, and therefore a $G$-invariant pseudo-Riemannian structure on the homogeneous space $G/H$, so that $G/H$ becomes a symmetric space with respect to the Levi–Civita connection and is called a *semisimple symmetric space*. In this context, the subspace ${\mathfrak {a}}$ has the following geometric meaning: Let $A := \exp(\mathfrak{a})$, the connected abelian subgroup of $G$ with Lie algebra $\mathfrak{a}$. Then, the orbit $A \cdot o$ through $o := eH \in G/H$ becomes a flat, totally geodesic submanifold in $G/H$. Furthermore, we have a (generalized) Cartan decomposition:
\[fact:genCar\] $G = KAH$.
The direct sum decomposition of the Lie algebra $$\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{g}^{-\tau,-\theta}
\oplus \mathfrak{g}^{\tau,-\theta}$$ lifts to a diffeomorphism: $$\mathfrak{g}^{-\tau,-\theta} + \mathfrak{g}^{\tau,-\theta}
\overset{\sim}{\to}
K\backslash G\, ,
\quad
(X,Y) \mapsto K e^X e^Y.$$ Since $\exp(\mathfrak{g}^{\tau,-\theta}) \subset H$, the decomposition $G = KAH$ follows if we show $$\label{eqn:HKa}
{\operatorname{Ad}}(H \cap K) \mathfrak{a}
= \mathfrak{g}^{-\tau,-\theta}.$$ The equation is well-known as the key ingredient of the original Cartan decomposition $G^{\tau\theta} = K^\tau A K^\tau$ in light of .
Furthermore, suppose that $\sigma$ is an involutive automorphism of $G$ such that $\sigma$, $\tau$ and $\theta$ commute with one another. We set $$G^{\sigma, \tau} := G^\sigma \cap G^\tau
=
\set{g \in G}{\sigma g = \tau g = g} \, .$$ Then $(G^\sigma, G^{\sigma, \tau})$ forms a reductive symmetric pair, because $\sigma$ and $\tau$ commute. The commutativity of $\sigma$ and $\theta$ implies that the automorphism $\sigma: G \to G$ stabilizes $K$ and induces a diffeomorphism of $G/K$, for which we use the same letter $\sigma$.
Examples of symmetric pairs {#subsec:3.2.ex}
---------------------------
This subsection presents some basic examples of semisimple (and therefore, reductive) symmetric pairs.
\[ex:gpmfd\] Let $G'$ be a semisimple Lie group, and $G := G' \times G'$. We define an involutive automorphism $\tau$ of $G$ by $\tau(x,y) := (y,x)$. Then, $G^\tau = \{ (g,g): g \in G' \}$ is the diagonal subgroup, denoted by ${\operatorname{diag}}(G')$, which is isomorphic to $G'$. Thus, $(G' \times G', {\operatorname{diag}}(G'))$ forms a semisimple symmetric pair.
We set $$\begin{aligned}
I_{p,q} &:=
\begin{pmatrix}
\;
\begin{matrix}
1 \\ &\ \rotatebox{-5}{$\ddots$} \\ &&\ 1
\end{matrix}
\kern-3em\smash{\raisebox{4.9ex}{\rotatebox{-45}{$\overbrace{~~~~~~~~~~~~~}$}} }
\kern-1.5em\smash{\raisebox{3.2ex}{$p$}}
&\mbox{\Huge$0$}
\\
\mbox{\Huge$0$}
&\begin{matrix}
\kern-.1em -1 \\ &\rotatebox{-5}{$\ddots$} \\ &&\kern-.1em -1
\end{matrix}
\kern-2.8em\smash{\raisebox{4.8ex}{\rotatebox{-45}{$\overbrace{~~~~~~~~~~~~~}$}} }
\kern-1.5em\smash{\raisebox{3.2ex}{$q$}}
\kern1em
\end{pmatrix}
\\
J &:=
\begin{pmatrix}
\parbox[c]{3em}{\hfil\Huge$0$\hfil}
& \parbox[c]{3em}{\hfil\Large$I_n$\hfil}
\\[3 ex]
\parbox[c]{3em}{\hfil\Large$-I_n$\hfil}
&\parbox[c]{3em}{\hfil\Huge$0$\hfil}
\end{pmatrix}\end{aligned}$$
\[ex:SLSU\] Let $G = SL(n,\mathbb{C})$, and fix $p,q$ such that $p+q=n$. Then, $$\tau(g) := I_{p,q} \, g^* I_{p,q}
\quad (g \in G)$$ defines an involutive automorphism of $G$, and $G^\tau = SU(p,q)$ (the indefinite unitary group). Thus, $(SL(n,\mathbb{C}), SU(p,q))$ forms a semisimple symmetric pair.
\[ex:SLSL\] Let $G = SL(n,\mathbb{C})$, and $\sigma(g) := \overline{g}$. Then $\sigma$ is an involutive automorphism of $G$, and $G^\sigma = SL(n,\mathbb{R})$. We note that $\sigma$ commutes with the involution $\tau$ in the previous example, and $$\begin{aligned}
G^{\sigma,\tau}& = \{ g \in SL(n,\mathbb{C}):
\overline{g} = g = I_{p,q} \, \overline{{}^t g} \, I_{p,q} \}
\\
& = SO(p,q) \, .\end{aligned}$$ Thus, $(SL(n,\mathbb{C}), SL(n,\mathbb{R}))$, $(SU(p,q), SO(p,q))$, $(SL(n,\mathbb{R}), SO(p,q))$ are examples of semisimple symmetric pairs.
\[ex:SLSp\] Let $G := SL(2n,\mathbb{R})$, and $\tau(g) := J \, {}^t g^{-1} J^{-1}$. Then, $G^\tau = Sp(n,\mathbb{R})$ (the real symplectic group). Thus, $(SL(2n,\mathbb{R}), Sp(n,\mathbb{R}))$ forms a semisimple symmetric pair.
Reduction of visibility to real rank condition {#subsec:3.2}
----------------------------------------------
The following lemma gives a sufficient condition for . Then, it plays a key role when we apply Theorem \[thm:2.2\] to the branching problem for the restriction from $G$ to $G^\tau$ (with the notation of Theorem \[thm:2.2\], $D = G/K$ and $H = G_0^\tau$). This lemma is also used in reducing ‘visibility’ of an action to an algebraic condition ([@visiblesymm Lemma 2.2]).
\[lem:3.2\] Let $\sigma$ and $\tau$ be involutive automorphisms of $G$. We assume that the pair $(\sigma, \tau)$ satisfies the following two conditions:
$\sigma$, $\tau$ and $\theta$ commute with one another. \[eqn:3.2.1\]
${\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}^{\tau\theta}
= {\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}^{\sigma, \tau\theta}$. \[eqn:3.2.2\]
Then for any $x \in G/K$, there exists $g \in G^\tau_0$ such that $
\sigma (x) = g \cdot x
$.
It follows from the condition that $\theta|_{G^\sigma}$ is a Cartan involution of a reductive Lie group $G^\sigma$ and that $\tau|_{G^\sigma}$ is an involutive automorphism of $G^\sigma$ commuting with $\theta|_{G^\sigma}$. Take a maximal abelian subspace $\mathfrak{a}$ in $$\mathfrak{g}^{-\theta, \sigma, \tau\theta}
:=
\set{Y \in \mathfrak{g}}{(-\theta) Y = \sigma Y = \tau\theta Y = Y} \,
.$$ From definition, we have $\dim \mathfrak{a} = {\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}^{\sigma, \tau\theta}$, which in turn equals ${\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}^{\tau\theta}$ by the condition . This means that $\mathfrak{a}$ is also a maximal abelian subspace in $$\mathfrak{g}^{-\theta, \tau\theta}
=
\set{Y \in \mathfrak{g}}{(-\theta) Y = \tau\theta Y = Y} \, .$$ Let $A = \exp(\mathfrak{a})$. Then it follows from Fact \[fact:genCar\] that we have a generalized Cartan decomposition $$G = G^\tau_0 A K \, .
\label{eqn:3.2.3}$$ Let $o := e K \in G/K$. Fix $x \in G/K$. Then, according to the decomposition , we find $h \in G^\tau_0$ and $a \in A$ such that $$x = h a \cdot o \, .$$ We set $g := \sigma(h)\; h^{-1}$. We claim $g \in G^\tau_0$. In fact, by using $\sigma \tau = \tau \sigma$ and $\tau h = h$, we have $$\tau (g) = \tau \sigma(h)\; \tau(h^{-1})
= \sigma \tau (h)\; \tau(h)^{-1}
= \sigma (h)\; h^{-1} = g \, .$$ Hence, $g \in G^\tau$. Moreover, since the image of the continuous map $$G^\tau_0 \to G\, ,
\quad h \mapsto \sigma(h)\; h^{-1}$$ is connected, we have $g \in G^\tau_0$.
On the other hand, we have $\sigma(a) = a$ because $\mathfrak{a} \subset
\mathfrak{g}^{-\theta, \sigma, -\tau} \subset
\mathfrak{g}^\sigma$. Therefore we have $$\sigma (x) = \sigma(h)\; \sigma(a) \cdot o
= \sigma(h)\; h^{-1} h a \cdot o
= g \cdot x \, ,$$ proving the lemma.
Hermitian Symmetric Space $G/K$ {#subsec:3.3}
-------------------------------
Throughout the rest of this section, we assume that $G$ is a simple, non-compact, Lie group of Hermitian type. We retain the notation of Subsection \[subsec:1.4\].
Let $G_\mathbb{C}$ be a connected complex Lie group with Lie algebra $\mathfrak{g}_\mathbb{C}$, and $Q^-$ the maximal parabolic subgroup of $G_\mathbb{C}$ with Lie algebra $\mathfrak{k}_\mathbb{C} + \mathfrak{p}_-$. Then we have an open embedding $G/K \hookrightarrow G_\mathbb{C}/Q^-$ because $\mathfrak{g}_{\mathbb{C}} = \mathfrak{g} +
(\mathfrak{k}_{\mathbb{C}} + \mathfrak{p}_-)$. Thus, a $G$-invariant complex structure on $G/K$ is induced from $G_{\mathbb{C}} / Q^-$. (We remark that the embedding $G/K \hookrightarrow G_\mathbb{C}/Q^-$ is well-defined, even though $G$ is not necessarily a subgroup of $G_{\mathbb{C}}$.) Suppose $\tau$ is an involutive automorphism of $G$ commuting with $\theta$. We recall from Subsection \[subsec:1.5\] that we have either $$\begin{aligned}
{2}
\tau Z &= Z
&\quad&\mbox{(holomorphic type),}
\tag{\ref{eqn:1.5.1}}
\\
\intertext{or}
\tau Z &= -Z
&\quad&\mbox{(anti-holomorphic type).}
\tag{\ref{eqn:1.5.2}}\end{aligned}$$ Here is the classification of semisimple symmetric pairs $(\mathfrak{g}, \mathfrak{g}^\tau)$ with $\mathfrak{g}$ simple such that the pair $(\mathfrak{g}, \mathfrak{g}^\tau)$ satisfies the condition (respectively, ). Table \[tbl:3.3.2\] is equivalent to the classification of totally real symmetric spaces $G^\tau/K^\tau$ of the Hermitian symmetric space $G/K$ (see [@xfo; @xjafbams; @xjafjdg; @xkobanaga]).
$$\vbox{
\offinterlineskip
\def\tablerule{\noalign{\hrule}}
\def\ct{&\cr\tablerule}
\halign{\strut#&\vrule#&
\;\;\hfil#\hfil\hfil\;\;&\vrule#&
\;\;\hfil#\hfil\hfil\;\;&\vrule#\cr\tablerule
&&\multispan3\hfil $(\mathfrak{g}, \mathfrak{g}^\tau)$ is of holomorphic
type
\hfil \ct
&& $\mathfrak{g}$ && $\mathfrak{g}^\tau$ \ct
&& $\mathfrak{su}(p,q)$ && $\hphantom{mmi}\mathfrak{s}(\mathfrak{u}(i,j) + \mathfrak{u}(p-i,q-j))$ \ct
&& $\mathfrak{su}(n,n)$ && $\hphantom{mmi}\mathfrak{so}^*(2n)$ \ct
&& $\mathfrak{su}(n,n)$ && $\hphantom{mmm}\mathfrak{sp}(n, \mathbb{R})$ \ct
&& $\mathfrak{so}^*(2n)$ && $\hphantom{mm}\mathfrak{so}^*(2p) + \mathfrak{so}^*(2n-2p)$ \ct
&& $\mathfrak{so}^*(2n)$ && $\hphantom{mmm}\mathfrak{u}(p,n-p)$ \ct
&& $\mathfrak{so}(2,n)$ && $\hphantom{m,}\mathfrak{so}(2,p)+\mathfrak{so}(n-p)$ \ct
&& $\mathfrak{so}(2,2n)$ && $\hphantom{mm}\mathfrak{u}(1,n)$ \ct
&& $\mathfrak{sp}(n,\mathbb{R})$&& $\hphantom{mmm}\mathfrak{u}(p,n-p)$ \ct
&& $\mathfrak{sp}(n,\mathbb{R})$&& $\hphantom{mi}\mathfrak{sp}(p,\mathbb{R})+\mathfrak{sp}(n-p,\mathbb{R})$\ct
&& $\mathfrak{e}_{6(-14)}$ && $\hphantom{,}\mathfrak{so}(10)+\mathfrak{so}(2)$\ct
&& $\mathfrak{e}_{6(-14)}$ && $\mathfrak{so}^*(10)+\mathfrak{so}(2)$\ct
&& $\mathfrak{e}_{6(-14)}$ && $\mathfrak{so}(8,2)+\mathfrak{so}(2)$\ct
&& $\mathfrak{e}_{6(-14)}$ && $\hphantom{i}\mathfrak{su}(5,1)+\mathfrak{sl}(2, \mathbb{R})$\ct
&& $\mathfrak{e}_{6(-14)}$ && $\mathfrak{su}(4,2)+\mathfrak{su}(2)$\ct
&& $\mathfrak{e}_{7(-25)}$ && $\hphantom{i}\mathfrak{e}_{6(-78)} +\mathfrak{so}(2)$\ct
&& $\mathfrak{e}_{7(-25)}$ && $\hphantom{i}\mathfrak{e}_{6(-14)} +\mathfrak{so}(2)$\ct
&& $\mathfrak{e}_{7(-25)}$ && $\mathfrak{so}(10,2) +\mathfrak{sl}(2,\mathbb{R})$\ct
&& $\mathfrak{e}_{7(-25)}$ && $\mathfrak{so}^*(12) +\mathfrak{su}(2)$\ct
&& $\mathfrak{e}_{7(-25)}$ && $\hphantom{mi}\mathfrak{su}(6,2)$\ct
\hfil\cr}}$$
$$\vbox{
\offinterlineskip
\def\tablerule{\noalign{\hrule}}
\def\ct{&\cr\tablerule}
\halign{\strut#&\vrule#&
\;\;\hfil#\hfil\hfil\;\;&\vrule#&
\;\;\hfil#\hfil\hfil\;\;&\vrule#\cr\tablerule
&&\multispan3\hfil $(\mathfrak{g}, \mathfrak{g}^\tau)$ is of
anti-holomorphic type
\hfil \ct
&& $\mathfrak{g}$ && \quad $\mathfrak{g}^\tau$ \ct
&& $\mathfrak{su}(p,q)$ && $\hphantom{mm}\mathfrak{so}(p,q)$ \ct
&& $\mathfrak{su}(n,n)$ && $\mathfrak{sl}(n,\mathbb{C}) + \mathbb{R}$ \ct
&& $\mathfrak{su}(2p,2q)$ && \hphantom{mm}$\mathfrak{sp}(p,q)$ \ct
&& $\mathfrak{so}^*(2n)$ && $\hphantom{mm}\mathfrak{so}(n,\mathbb{C})$ \ct
&& $\mathfrak{so}^*(4n)$ && $\mathfrak{su}^*(2n) + \mathbb{R}$ \ct
&& $\mathfrak{so}(2,n)$ && $\hphantom{mm}\,\,\,\mathfrak{so}(1,p)+\mathfrak{so}(1,n-p)$ \ct
&& $\mathfrak{sp}(n,\mathbb{R})$&& $\hphantom{mm}\,\,\mathfrak{gl}(n,\mathbb{R})$ \ct
&& $\mathfrak{sp}(2n,\mathbb{R})$&& $\hphantom{mm}\mathfrak{sp}(n,\mathbb{C})$\ct
&& $\mathfrak{e}_{6(-14)}$ && $\hphantom{mm}\mathfrak{f}_{4(-20)}$\ct
&& $\mathfrak{e}_{6(-14)}$ && $\hphantom{mm}\mathfrak{sp}(2,2)$\ct
&& $\mathfrak{e}_{7(-25)}$ && $\phantom{mi}\;\mathfrak{e}_{6(-26)} +\mathfrak{so}(1,1)$\ct
&& $\mathfrak{e}_{7(-25)}$ && $\hphantom{mmmi}\mathfrak{su}^*(8)$ \ct
\hfil\cr}}$$
Holomorphic realization of highest weight representations {#subsec:3.5}
---------------------------------------------------------
It is well-known that an irreducible highest weight representation $\pi$ of $G$ can be realized as a subrepresentation of the space of global holomorphic sections of an equivariant holomorphic vector bundle over the Hermitian symmetric space $G/K$. We supply a proof here for the convenience of the reader in a way that we shall use later.
\[lem:3.5\] Let $(\pi, \mathcal{H})$ be an irreducible unitary highest weight module. We write $\chi$ for the representation of $K$ on $U := \mathcal{H}_K^{\mathfrak{p}_+}$ (see Definition \[def:1.4\]). Let $\mathcal{L} := G \times_K U \to G/K$ be the $G$-equivariant holomorphic vector bundle associated to $\chi$. Then, there is a natural injective continuous $G$-homomorphism $\mathcal{H} \to \mathcal{O}(\mathcal{L})$.
Let $(\ , \ )_{\mathcal{H}}$ be a $G$-invariant inner product on $\mathcal{H}$. We write $(\ , \ )_U$ for the induced inner product on $U$. Then, $K$ acts unitarily on $\mathcal{H}$, and in particular on $U$. We consider the map $$G \times \mathcal{H} \times U \to \mathbb{C}\, ,
\ \
(g,v,u) \mapsto (\pi(g)^{-1} v, u)_{\mathcal{H}} =
(v, \pi(g) u)_{\mathcal{H}} \, .$$ For each fixed $g \in G$ and $v \in \mathcal{H}$, the map $U \to \mathbb{C}, \ u \mapsto (\pi(g)^{-1} v, u)_{\mathcal{H}}$ is an anti-linear functional on $U$. Then there exists a unique element $F_v(g) \in U$ by the Riesz representation theorem for the finite dimensional Hilbert space $U$ such that $$(F_v(g), u)_U = (\pi(g)^{-1} v, u)_{\mathcal{H}}
\ \
\text{ for any } u \in U\, .$$ Then it is readily seen that $F_v(g k) = \chi(k)^{-1} F_v(g)$ and $F_{\pi(g') v}(g) = F_v({g'}^{-1} g)$ for any $g, g' \in G$, $k \in K$ and $v \in \mathcal{H}$. As $u$ is a smooth vector in $\mathcal{H}$, $(F_v(g), u)_U = (v, \pi (g) u)_{\mathcal{H}}$ is a $C^\infty$-function on $G$. Then $F_v(g)$ is a $C^\infty$-function on $G$ with value in $U$ for each fixed $v \in \mathcal{H}$. Thus, we have a non-zero $G$-intertwining operator given by $$F: {\mathcal{H}} \to C^\infty(G \times_K U)\, ,
\quad
v \mapsto F_v \, .$$ As $U$ is annihilated by $\mathfrak{p}_+$, $F_v$ is a holomorphic section of the holomorphic vector bundle $G \times_K U \to G/K$, that is, $F_v \in \mathcal{O}(G \times_K U)$. Then, the non-zero map $F: \mathcal{H} \to \mathcal{O}(G \times_K U)$ is injective because $\mathcal{H}$ is irreducible. Furthermore, $F$ is continuous by the closed graph theorem. Hence, Lemma \[lem:5.1\] is proved.
Reduction to real rank condition {#subsec:3.6}
--------------------------------
The next Lemma is a stepping-stone to Theorem \[thm:A\]. It becomes also a key lemma to the theorem that the action of a subgroup $H$ on the bounded symmetric domain $G/K$ is ‘strongly visible’ for any symmetric pair $(G,H)$ (see [@visiblesymm]).
\[lem:5.1\] Suppose $\mathfrak{g}$ is a real simple Lie algebra of Hermitian type. Let $\tau$ be an involutive automorphism of $\mathfrak{g}$, commuting with a fixed Cartan involution $\theta$. Then there exists an involutive automorphism $\sigma$ of $\mathfrak{g}$ satisfying the following three conditions:
\[eqn:3.6.one\] $\sigma$, $\tau$ and $\theta$ commute with one another.
\[eqn:3.6.2\] ${\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}^{\tau \theta}
= {\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}^{\sigma, \tau \theta}$.
\[eqn:3.6.3\] $\sigma Z = -Z$.
We shall give a proof in the special case $\tau = \theta$ in Subsection \[subsec:4.1\]. For the general case, see [@visiblesymm Lemma 3.1] or [@xkmf Lemma 5.1].
Proof of Theorem \[thm:A\] {#subsec:proofthmA}
--------------------------
Now, we are ready to complete the proof of Theorem \[thm:A\].
Without loss of generality, we may and do assume that $G$ is simply connected. Let $(\pi,\mathcal{H})$ be an irreducible unitary highest weight representation of scalar type. We define a holomorphic line bundle by $\mathcal{L} := G \times_K \mathcal{H}_K^{\mathfrak{p}_+}$ over the Hermitian symmetric space $D := G/K$. Then, it follows from Lemma \[lem:3.5\] that there is an injective continuous $G$-intertwining map $\mathcal{H} \to \mathcal{O}(\mathcal{L})$.
Suppose $(G,H)$ is a symmetric pair. We first note that for an involutive automorphism $\tau$ of $G$, there exists $g \in G$ such that $\tau^g \theta = \theta \tau^g$ if we set $$\tau^g(x) := g \tau(g^{-1} x g)g^{-1}$$ for $x \in G$. Then, $G^{\tau_g} = gHg^{-1}$ is $\theta$-stable. Since the multiplicity-free property of the restriction $\pi|_H$ is unchanged if we replace $H$ by $gHg^{-1}$, we may and do assume that $\theta H = H$, in other words, $\theta\tau = \tau\theta$.
Now, by applying Lemma \[lem:5.1\], we can take $\sigma$ satisfying , and . We use the same letter $\sigma$ to denote its lift to $G$. It follows from that the induced involutive diffeomorphism $\sigma : G/K \to G/K$ is anti-holomorphic (see Subsection \[subsec:1.5\]). In light of the conditions and , we can apply Lemma \[lem:3.2\] to see that for any $x \in D$ there exists $g \in H$ such that $\sigma(x) = g \cdot x$.
Moreover, by using Lemma \[lem:9.6\] in the Appendix, we have an isomorphism $\overline{\sigma^* \mathcal{L}} \simeq \mathcal{L}$ as $G$-equivariant holomorphic line bundles over $G/K$. Therefore, all the assumptions of Theorem \[thm:2.2\] are satisfied. Thus, we conclude that the restriction $\pi|_H$ is multiplicity-free by Theorem \[thm:2.2\].
Proof of Theorem \[thm:C\] {#sec:4}
==========================
In this section we give a proof of Theorem \[thm:C\].
Throughout this section, we may and do assume that $G$ is simply connected so that any automorphism of $\mathfrak{g}$ lifts to $G$. We divide the proof of Theorem \[thm:C\] into the following cases: Case I.Both $\pi_1$ and $\pi_2$ are highest weight modules. Case I$'$.Both $\pi_1$ and $\pi_2$ are lowest weight modules. Case II. $\pi_1$ is a highest weight module, and $\pi_2$ is a lowest weight module. Case II$'$.$\pi_1$ is a lowest weight module, and $\pi_2$ is a highest weight module.
Reduction to real rank condition {#subsec:4.1}
--------------------------------
The following lemma is a special case of Lemma \[lem:5.1\] with $\tau = \theta$. We shall see that Theorem \[thm:C\] in Case I (likewise, Case I$'$) reduces to this algebraic result.
\[lem:4.1.1\] Suppose $\mathfrak{g}$ is a real simple Lie algebra of Hermitian type. Let $\theta$ be a Cartan involution. Then there exists an involutive automorphism $\sigma$ of $\mathfrak{g}$ satisfying the following three conditions:
\[eqn:4.1.1\] $\sigma$ and $\theta$ commute.
\[eqn:4.1.2\] ${\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}= {\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}^\sigma$.
\[eqn:4.1.3\] $\sigma Z = -Z$.
We give a proof of the Lemma based on the classification of simple Lie algebras $\mathfrak{g}$ of Hermitian type.
We recall that for any involutive automorphism $\sigma$ of $G$, there exists $g \in G$ such that $\sigma^g \theta = \theta \sigma^g$. Thus, is always satisfied after replacing $\sigma$ by some $\sigma^g$. The remaining conditions and (cf. Table \[tbl:3.3.2\]) are satisfied if we choose $\sigma \in \operatorname{Aut}(G)$ in the following Table \[tbl:4.1.2\] for each simple non-compact Lie group $G$ of Hermitian type:
$$\vbox{
\offinterlineskip
\def\tablerule{\noalign{\hrule}}
\halign{\strut#&\vrule#&
\;\;\hfil#\hfil\hfil\;\;&\vrule#&
\;\;\hfil#\hfil\hfil\;\;&\vrule#&
\hfil#\hfil\hfil\,&\vrule#&
\;\;\hfil#\hfil\hfil\;\;&\vrule#\cr\tablerule
&&\multispan7\hfil $(\mathfrak{g}, \mathfrak{g}^\sigma)$ satisfying \eqref{eqn:4.1.2}
and \eqref{eqn:4.1.3}
\hfil &\cr\tablerule
&& ${\mathfrak{g}}$ && $\mathfrak{g}^\sigma$
&&&& ${\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g} = {\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}^\sigma$ &\cr\tablerule
&& ${\mathfrak{su}}(p,q)$ && ${\mathfrak{so}}(p,q)$ &&&& $\min(p,q)$ &\cr\tablerule
&& ${\mathfrak{so}}^*(2n)$ && ${\mathfrak{so}}(n,\mathbb{C})$ &&&& $[\frac{1}{2} n]$ &\cr\tablerule
&& ${\mathfrak{sp}}(n,\mathbb{R})$ && ${\mathfrak{gl}}(n,\mathbb{R})$ &&&& $n$ &\cr\tablerule
&& ${\mathfrak{so}}(2,n)$ && ${\mathfrak{so}}(1,n-1)+{\mathfrak{so}}(1,1)$ &&&& $\min(2,n)$ &\cr\tablerule
&& ${\mathfrak{e}}_{6 (-14)}$ && ${\mathfrak{sp}}(2,2)$&&&& $2$ &\cr\tablerule
&& ${\mathfrak{e}}_{7(-25)}$ && $\mathfrak{su}^*(8)$ &&&& $3$&\cr\tablerule
\hfil\cr}}$$
Here, we have proved Lemma.
\[rem:4.1.3\] The choice of $\sigma$ in Lemma \[lem:4.1.1\] is not unique. For example, we may choose $\mathfrak{g}^\sigma \simeq {\mathfrak{e}}_{6(-26)} \oplus \mathbb{R}$ instead of the above choice $\mathfrak{g}^\sigma \simeq \mathfrak{su}^*(8)$ for $\mathfrak{g} = {\mathfrak{e}}_{7(-25)}$.
Proof of Theorem \[thm:C\] in Case I {#subsec:3.7}
------------------------------------
Let $G$ be a non-compact simply-connected, simple Lie group such that $G/K$ is a Hermitian symmetric space.
Let $(\pi_1,\mathcal{H}_1)$ and $(\pi_2,\mathcal{H}_2)$ be two irreducible unitary highest weight representations of scalar type. By Lemma \[lem:3.5\], we can realize $(\pi_i,\mathcal{H}_i)$ in the space $\mathcal{O}(\mathcal{L}_i)$ of holomorphic sections of the holomorphic line bundle $\mathcal{L}_i := G \times_K (\mathcal{H}_i)_K^{\mathfrak{p}_+}$ $(i = 1,2)$ over the Hermitian symmetric space $G/K$. We now define a holomorphic line bundle $\mathcal{L} := \mathcal{L}_1 \boxtimes \mathcal{L}_2$ over $D := G/K \times G/K$ as the outer tensor product of $\mathcal{L}_1$ and $\mathcal{L}_2$. Then, we have naturally an injective continuous $(G \times G)$-intertwining map $\mathcal{H}_1 \widehat{\otimes} \mathcal{H}_2 \to
\mathcal{O}(\mathcal{L})$.
Let us take an involution $\sigma'$ of $\mathfrak{g}$ as in Lemma \[lem:4.1.1\] (but we use the letter $\sigma'$ instead of $\sigma$), and lift it to $G$. We set $\sigma := \sigma' \times \sigma'$. Then it follows from that $\sigma'$ acts anti-holomorphically on $G/K$, and so does $\sigma$ on $D$. Furthermore, we have isomorphisms of holomorphic line bundles $\overline{(\sigma')^* \mathcal{L}_i} \simeq \mathcal{L}_i$ $(i = 1,2)$ by Lemma \[lem:9.6\] and thus $\overline{\sigma^* \mathcal{L}} \simeq \mathcal{L}$.
We now introduce another involutive automorphism $\tau$ of $G \times G$ by $\tau(g_1, g_2) := (g_2, g_1)$. Then $(G \times G)^\tau = {\operatorname{diag}}(G) :=\set{(g,g)}{g \in G}$. We shall use the same letter $\theta$ to denote the Cartan involution $\theta \times \theta$ on $G \times G$ (and $\theta \oplus \theta$ on $\mathfrak{g} \oplus \mathfrak{g}$). Then, we observe the following isomorphisms: $$\begin{aligned}
{2}
&(\mathfrak{g} \oplus \mathfrak{g})^{\tau\theta}
= \{ (X, \theta X): X \in \mathfrak{g} \}
&&\simeq \mathfrak{g} \, ,
\\
&(\mathfrak{g} \oplus \mathfrak{g})^{\sigma,\tau\theta}
= \{ (X, \theta X) : X \in \mathfrak{g}^{\sigma'} \}
&&\simeq \mathfrak{g}^{\sigma'} .\end{aligned}$$Thus, the condition implies $${\mathbb{R}\text{-}\operatorname{rank}}(\mathfrak{g} \oplus
\mathfrak{g})^{\tau\theta}
={\mathbb{R}\text{-}\operatorname{rank}}(\mathfrak{g} \oplus \mathfrak{g})^{\sigma, \tau\theta}
\, .$$ Therefore, given $(x_1,x_2) \in D \simeq (G \times G)/(K \times K)$, there exists $(g,g) \in (G \times G)^\tau$ satisfying $(g \cdot x_1, g \cdot x_2) = (\sigma'(x_1), \sigma'(x_2))$ $(= \sigma(x_1,x_2))$ by Lemma \[lem:3.2\].
Let us apply Theorem \[thm:2.2\] to the setting $(\mathcal{L} \to D,
\mathcal{H}_1\widehat\otimes\mathcal{H}_2, {\operatorname{diag}}(G),
\sigma)$. Now that all the assumptions of Theorem \[thm:2.2\] are satisfied, we conclude that the tensor product $\pi_1 \widehat\otimes \pi_2$ is multiplicity-free as a $G$-module, that is, Theorem \[thm:C\] holds in the case I.
Proof of Theorem \[thm:C\] in Case II {#subsec:4.2}
-------------------------------------
Let us give a proof of Theorem \[thm:C\] in the case II. We use the same $\tau$ as in Subsection \[subsec:3.7\], that is, $\tau(g_1,g_2) := (g_2,g_1)$ and define a new involution $\sigma$ by $\sigma := \tau \theta$, that is, $\sigma(g_1,g_2) = (\theta g_2,\theta g_1)$ for $g_1,g_2 \in G$. Obviously, $\sigma$, $\tau$ and the Cartan involution $\theta$ of $G \times G$ all commute.
We write $M$ for the Hermitian symmetric space $G/K$, and $\overline M$ for the conjugate complex manifold. Then $\sigma$ acts anti-holomorphically on $D := M \times \overline{M}$ because so does $\tau$ and because $\theta$ acts holomorphically.
By the obvious identity $(\mathfrak{g} \oplus \mathfrak{g})^{\tau\theta}
= (\mathfrak{g} \oplus \mathfrak{g})^{\sigma,\tau\theta}$, we have ${\mathbb{R}\text{-}\operatorname{rank}}(\mathfrak{g} \oplus \mathfrak{g})^{\tau\theta}
= {\mathbb{R}\text{-}\operatorname{rank}}(\mathfrak{g} \oplus \mathfrak{g})^{\sigma,\tau\theta}$ $(= {\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g})$. Therefore, it follows from Lemma \[lem:3.2\] that for any $(x_1, x_2) \in D$ there exists $(g,g) \in (G \times G)^\tau$ such that $\sigma(x_1, x_2) = (g,g) \cdot (x_1, x_2)$.
Suppose $\pi_1$ (respectively, $\pi_2$) is a unitary highest (respectively, lowest) weight representation of scalar type. We set $\mathcal{L}_1 := G \times_K (\mathcal{H}_1)_K^{\mathfrak{p}_+}$ and $\mathcal{L}_2 := G \times_K (\mathcal{H}_2)_K^{\mathfrak{p}_-}$. Then, $\mathcal{L}_1 \to M$ and $\mathcal{L}_2 \to \overline{M}$ are both holomorphic line bundles, and we can realize $\pi_1$ in $\mathcal{O}(M,\mathcal{L}_1)$, and $\pi_2$ in $\mathcal{O}(\overline{M},\mathcal{L}_2)$, respectively. Therefore, the outer tensor product $\pi_1 \boxtimes \pi_2$ is realized in a subspace of holomorphic sections of the holomorphic line bundle $\mathcal{L} := \mathcal{L}_1 \boxtimes \mathcal{L}_2$ over $D = M \times \overline{M}$.
Now, we apply Theorem \[thm:2.2\] to $(\mathcal{L} \to D, \mathcal{H}_1 \widehat{\otimes} \mathcal{H}_2,
{\operatorname{diag}}(G), \sigma)$. The condition holds by Lemma \[lem:9.6\]. Hence, all the assumptions of Theorem \[thm:2.2\] are satisfied, and therefore, Theorem \[thm:C\] holds in the case II.
Hence, Theorem \[thm:C\] has been proved.
Uniformly bounded multiplicities — Proof of Theorems \[thm:B\] and \[thm:D\] {#sec:5}
============================================================================
This section gives the proof of Theorems \[thm:B\] and \[thm:D\]. Since the proof of Theorem \[thm:B\] parallels to that of Theorem \[thm:D\], we deal mostly with Theorem \[thm:D\] here. Without loss of generality, we assume $G$ is a non-compact simple Lie group of Hermitian type.
General theory of restriction {#subsec:3.4}
-----------------------------
A unitary representation $(\pi,\mathcal{H})$ of a group $L$ is *discretely decomposable* if $\pi$ is unitarily equivalent to the discrete Hilbert sum of irreducible unitary representations of $L$: $$\pi \simeq
\sideset{}{^\oplus}\sum_{\mu \in \widehat{L}}
m_\pi (\mu) \mu \, .$$ Furthermore, we say $\pi$ is $L$-*admissible* ([@xkdecomp]) if all the multiplicities $m_\pi (\mu)$ are finite. In this definition, we do not require $m_\pi(\mu)$ to be uniformly bounded with respect to $\mu$.
Suppose $L'$ is a subgroup of $L$. Then, the restriction of $\pi$ to $L'$ is regarded as a unitary representation of $L'$. If $\pi$ is $L'$-admissible, then $\pi$ is $L$-admissible ([@xkdecomp Theorem 1.2]).
We start with recalling from [@xkdisc] a discrete decomposability theorem of branching laws in the following settings:
\[fact:3.4.1\] [1)]{} Suppose $\tau$ is of holomorphic type (see Definition \[def:holo-anti\]) and set $H := G_0^\tau$. If $\pi$ is an irreducible unitary highest weight representation of $G$, then $\pi$ is . In particular, $\pi$ is $H$-admissible. The restriction $\pi |_H$ splits into a discrete Hilbert sum of irreducible unitary highest weight representations of $H$: $$\pi|_{H} \simeq {\sum_{\mu \in \widehat{H}}}^\oplus
m_\pi(\mu) \mu
\qquad
\text{(discrete Hilbert sum)},
\label{eqn:3.4.1}
$$ where the multiplicity $m_{\pi}(\mu)$ is finite for every $\mu$. Let $\pi_1, \pi_2$ be two irreducible unitary highest weight representations of $G$. Then the tensor product $\pi_1 \widehat\otimes \pi_2$ is under the diagonal action. Furthermore, $\pi_1 \widehat{\otimes} \pi_2$ splits into a discrete Hilbert sum of irreducible unitary highest weight representations of $G$, each occurring with finite multiplicity. Furthermore, if at least one of $\pi_1$ or $\pi_2$ is a holomorphic discrete series representation for $G$, then any irreducible summand is a holomorphic discrete series representation.
See [@xkdisc Theorem 7.4] for the proof. The main idea of the proof is taking normal derivatives of holomorphic sections, which goes back to S. Martens [@xmartens]. The same idea was also employed in a number of papers including Lipsman ([@xlipad Theorem 4.2]) and Jakobsen–Vergne ([@xjv Corollary 2.3]).
\[rem:Fact51\] Fact \[fact:3.4.1\] (1) holds more generally for a closed subgroup $H$ satisfying the following two conditions:
1)$H$ is $\theta$-stable.
2)The Lie algebra $\mathfrak{h}$ of $H$ contains $Z$.
Here, we recall that $Z$ is the generator of the center of $\mathfrak{k}$. The proof is essentially the same as that of Fact \[fact:3.4.1\] (1).
Theorem \[thm:B\] (2) follows from Theorem \[thm:A\] and Fact \[fact:3.4.1\] (1). Likewise, Theorem \[thm:D\] (2) follows from Theorem \[thm:C\] and Fact \[fact:3.4.1\] (2). What remains to show for Theorems \[thm:B\] and \[thm:D\] is the uniform boundedness of multiplicities.
Remarks on Fact \[fact:3.4.1\] {#fact3.4.1}
------------------------------
Some remarks on Fact \[fact:3.4.1\] are in order.
\[rem:3.4.2\] A Cartan involution $\theta$ is clearly of holomorphic type because $\theta Z = Z$. If $\theta = \tau$ then $H = K$ and any irreducible summand $\mu$ is finite dimensional. In this case, the finiteness of $m_\pi (\mu)$ in Fact \[fact:3.4.1\] (1) is a special case of Harish-Chandra’s admissibility theorem (this holds for any irreducible unitary representation $\pi$ of $G$).
Fact \[fact:3.4.1\] asserts in particular that there is no continuous spectrum in the irreducible decomposition formula. The crucial assumption for this is that $(G,H)$ is of holomorphic type. In contrast, the restriction $\pi|_H$ is not discretely decomposable if $(G,H)$ is of anti-holomorphic type and if $\pi$ is a holomorphic discrete series representation of $G$ ([@xkdecomp Theorem 5.3]). In this setting, R. Howe, J. Repka, G. Ólafsson, B. Ørsted, van Dijk, S. Hille, M. Pevzner, V. Molchanov, Y. Neretin, G. Zhang and others studied irreducible decompositions of the restriction $\pi|_{H}$ by means of the $L^2$-harmonic analysis on Riemannian symmetric spaces $H/H \cap K$ ([@xvd; @xvdh; @xDijkPev; @xhoweseesaw; @xmol; @xnere; @xoo; @xoz; @xrep]). The key idea in Howe and Repka [@xhoweseesaw; @xrep] is that a holomorphic function on $G/K$ is uniquely determined by its restriction to the totally real submanifold $H/H \cap K$ (essentially, the unicity theorem of holomorphic functions), and that any function on $H/H\cap K$ can be approximated (in a sense) by holomorphic functions on $G/K$ (essentially, the Weierstrass polynomial approximation theorem).
A finite multiplicity theorem of the branching law with respect to semisimple symmetric pairs $(G,H)$ holds for more general $\pi$ (i.e. $\pi$ is not a highest weight module), under the assumption that $\pi$ is discretely decomposable as an $(\mathfrak{h}_{\mathbb{C}}, H \cap K)$-module (see [@xkdecoass Corollary 4.3], [@xkbeijing]). However, the multiplicity of the branching law can be infinite if the restriction is not discretely decomposable (see Example \[ex:finite infinite\]).
Theorems \[thm:B\] and \[thm:D\] assert that multiplicities $m_\pi (\mu)$ in Fact \[fact:3.4.1\] are [**uniformly bounded**]{} when we vary $\mu$. This is a distinguished feature for the restriction of highest weight representations $\pi$. A similar statement may fail if $\pi$ is not a highest weight module (see Example \[exam:6.3\]).
Reduction to the scalar type case {#subsec:4.3}
---------------------------------
In order to deduce Theorem \[thm:D\] (1) from Theorem \[thm:D\] (2), we use the idea of ‘coherent family’ of representations of reductive Lie groups (for example, see [@xvg]). For this, we prepare the following Lemma \[lem:4.3\] and Proposition \[lem:4.4\].
\[lem:4.3\] Suppose that $(\pi, \mathcal{H})$ is an irreducible unitary highest weight representation of $G$. Then there exist an irreducible unitary highest weight representation $\pi'$ of scalar type and a finite dimensional representation $F$ of $G$ such that the underlying $(\mathfrak{g}_{\mathbb{C}},K)$-module $\pi_K$ occurs as a subquotient of the tensor product $\pi_K' \otimes F$.
Without loss of generality, we may and do assume that $G$ is simply connected. Since $G$ is a simple Lie group of Hermitian type, the center $\mathfrak{c}(\mathfrak{k})$ of $\mathfrak{k}$ is one dimensional. We take its generator $Z$ as in Subsection \[subsec:1.5\], and write $C$ for the connected subgroup with Lie algebra $\mathfrak{c}(\mathfrak{k})$. Then, $K$ is isomorphic to the direct product group of $C$ and a semisimple group $K'$.
As $(\pi, \mathcal{H})$ is an irreducible unitary highest weight representation of $G$, $\mathcal{H}_K^{\mathfrak{p}_+}$ is an irreducible (finite dimensional) unitary representation of $K$. The $K$-module $\mathcal{H}_K^{\mathfrak{p}_+}$ has an expression $\sigma\otimes \chi_0$, where $\sigma \in \widehat{K}$ such that $\sigma|_C$ is trivial and $\chi_0$ is a unitary character of $K$.
Let $\chi'$ be a unitary character of $K$ such that $\chi'$ is trivial on the center $Z_G$ of $G$ (namely, $\chi'$ is well-defined as a representation of $\operatorname{Ad}_G(K) \simeq K/Z_G$). For later purposes, we take $\chi'$ such that $-\sqrt{-1}\, d \chi'(Z) \gg 0$. There exists an irreducible finite dimensional representation $F$ of $G$ such that $F^{\mathfrak{p}_+} \simeq \sigma \otimes \chi'$ as $K$-modules because $\sigma \otimes \chi'$ is well-defined as an algebraic representation of ${\operatorname{Ad}}_G(K)$.
We set $\chi := \chi_0 \otimes (\chi')^*$ of $K$. Because $- \sqrt{-1}\, d \chi (Z) \ll 0$, the irreducible highest weight $(\mathfrak{g}_{\mathbb{C}},K)$-module $V'$ such that $(V')^{\mathfrak{p}_+} \simeq \chi$ is unitarizable. Let $(\pi', \mathcal{H}')$ denote the irreducible unitary representation of $G$ whose underlying $(\mathfrak{g}_{\mathbb{C}}, K)$-module $\mathcal{H}_K'$ is isomorphic to $V'$. Since $\mathcal{H}'_K$ is an irreducible $(\mathfrak{g}_{\mathbb{C}},K)$-module, $\mathcal{H}_K' \otimes F$ is a $(\mathfrak{g}_{\mathbb{C}},K)$-module of finite length. Furthermore, as $\mathcal{H}'_K$ is a highest weight module, so are all subquotient modules of $\mathcal{H}'_K \otimes F$. Then, $\mathcal{H}_K$ arises as a subquotient of $\mathcal{H}_K' \otimes F$ because the $K$-module $\mathcal{H}_K^{\mathfrak{p}_+}$ occurs as a subrepresentation of $(\mathcal{H}_K'\otimes F)^{\mathfrak{p}_+}$ in view of $$\mathcal{H}_K^{\mathfrak{p}_+}
\simeq
\sigma \otimes \chi_0
\simeq
\chi \otimes (\sigma \otimes \chi')
\simeq
(\mathcal{H}_K')^{\mathfrak{p}_+} \otimes F^{\mathfrak{p}_+}
\subset (\mathcal{H}_K' \otimes F)^{\mathfrak{p}_+} \, .$$ Hence, we have shown Lemma \[lem:4.3\].
Uniform estimate of multiplicities for tensor products {#subsec:4.4}
------------------------------------------------------
Let $(\pi,X)$ be a [$(\mathfrak{g}_\mathbb{C}, K)$]{}-module of finite length. This means that $\pi$ admits a chain of submodules $$\label{eqn:YiX}
0 = Y_0 \subset Y_1 \subset \cdots \subset Y_N = X$$ such that $Y_i/Y_{i-1}$ is irreducible for $i=1,\dots,N$. The number $N$ is independent of the choice of the chain , and we will write $$m(\pi) := N \, .$$ That is, $m(\pi)$ is the number of irreducible [$(\mathfrak{g}_\mathbb{C}, K)$]{}-modules (counted with multiplicity) occurring as subquotients in $\pi$. Here is a uniform estimate of $m(\pi)$ under the operation of tensor products:
\[lem:4.4\] Let $F$ be a finite dimensional representation of a real reductive connected Lie group $G$. Then there exists a constant $C \equiv C(F)$ such that $$m(\pi \otimes F) \le C$$ for any irreducible [$(\mathfrak{g}_\mathbb{C}, K)$]{}-module $\pi$.
Before entering the proof, we fix some terminologies:
\[def:Grothen\] We write $\mathcal{F}(\mathfrak{g}_{\mathbb{C}},K)$ for the category of $(\mathfrak{g}_{\mathbb{C}},K)$-modules of finite length. The *Grothendieck group* $\mathcal{V}(\mathfrak{g}_{\mathbb{C}},K)$ of $\mathcal{F}(\mathfrak{g}_{\mathbb{C}},K)$ is the abelian group generated by $(\mathfrak{g}_{\mathbb{C}},K)$-modules of finite length, modulo the equivalence relations $$X \sim Y + Z$$ whenever there is a short exact sequence $$0 \to Y \to X \to Z \to 0$$ of $(\mathfrak{g}_{\mathbb{C}},K)$-modules. Then $$m: \mathcal{F}(\mathfrak{g}_{\mathbb{C}},K) \to \mathbb{N}$$ induces a group homomorphism of abelian groups: $$m: \mathcal{V}(\mathfrak{g}_{\mathbb{C}},K) \to \mathbb{Z} \, .$$
The Grothendieck group $\mathcal{V}(\mathfrak{g}_{\mathbb{C}},K)$ is isomorphic to the free abelian group having irreducible $(\mathfrak{g}_{\mathbb{C}},K)$-modules as its set of finite generators.
Suppose $(\pi,X)$ is a $(\mathfrak{g}_{\mathbb{C}},K)$-module of finite length. Then, in the Grothendieck group $\mathcal{V}(\mathfrak{g}_{\mathbb{C}},K)$, we have the relation $$\label{eqn:XmpiY}
X = \bigoplus_Y m_\pi(Y) Y \, ,$$ where the sum is taken over irreducible $(\mathfrak{g}_{\mathbb{C}},K)$-modules. Then we have $$\label{eqn:mpiY}
m(\pi) = \sum_Y m_\pi(Y) \, .$$ Suppose $(\pi',X')$ is also a $(\mathfrak{g}_{\mathbb{C}},K)$-modules of finite length. We set $$\begin{aligned}
[\pi:\pi'] :={}
&\dim{\operatorname{Hom}}_{(\mathfrak{g}_{\mathbb{C}},K)}
(\bigoplus_Y m_\pi(Y)Y, \bigoplus_Y m_{\pi'}(Y)Y)
\label{eqn:numberGro}
\\
={}
&\sum_Y m_\pi(Y) m_{\pi'}(Y) \, .
\label{eqn:numberGro2}\end{aligned}$$ The definition makes sense in a more general setting where one of $X$or $X'$ is not of finite length. To be more precise, we recall from [@xkdecoass Definition 1.1]:
\[def:infdeco\] Let $\mathcal{A}(\mathfrak{g}_{\mathbb{C}},K)$ be the category of $(\mathfrak{g}_{\mathbb{C}},K)$-modules $(\pi,X)$ having the following properties: ($K$-admissibility) $\dim{\operatorname{Hom}}_K(\tau,\pi) < \infty$ for any $\tau\in\widehat{K}$. (discretely decomposability, see [@xkdecoass Definition 1.1]) $X$ admits an increasing filtration $$0 = Y_0 \subset Y_1 \subset Y_2 \subset \cdots$$ of $\mathfrak{g}_{\mathbb{C}}$-modules such that $Y_i/Y_{i-1}$ is of finite length and that $X = \bigcup_{i=1}^\infty Y_i$.
We refer the reader to [@xkdecoass] for algebraic results on discretely decomposable $(\mathfrak{g}_{\mathbb{C}},K)$-modules such as:
\[lemma:infdeco\] Suppose $X \in \mathcal{A}(\mathfrak{g}_{\mathbb{C}},K)$. Any submodule or quotient of $X$ is an object of $\mathcal{A}(\mathfrak{g}_{\mathbb{C}},K)$. The tensor product $X\otimes F$ is also an object of $\mathcal{A}(\mathfrak{g}_{\mathbb{C}},K)$ for any finite dimensional $(\mathfrak{g}_{\mathbb{C}},K)$-module.
For $X \in \mathcal{A}(\mathfrak{g}_{\mathbb{C}},K)$, we can take the filtration $\{Y_i\}$ such that $Y_i/Y_{i-1}$ is irreducible as a $(\mathfrak{g}_{\mathbb{C}},K)$-module for any $i$. Then, for any irreducible $(\mathfrak{g}_{\mathbb{C}},K)$-module, $$\#\{i: \text{$Y_i/Y_{i-1}$ is isomorphic to $Y$}\}$$ is finite and independent of the filtration, which we will denote by $m_\pi(Y)$.
\[def:mfagk\] Suppose $X \in \mathcal{A}(\mathfrak{g}_{\mathbb{C}},K)$. We say the $(\mathfrak{g}_{\mathbb{C}},K)$-module $X$ is *multiplicity-free* if $$m_\pi(Y) \le 1
\quad\text{for any irreducible $(\mathfrak{g}_{\mathbb{C}},K)$-module
$Y$}.$$
This concept coincides with Definition \[def:1.2\] if $X$ is the underlying $(\mathfrak{g}_{\mathbb{C}},K)$-module of a unitary representation of $G$. The point of Definition \[def:mfagk\] is that we allow the case where $X$ is not unitarizable.
Generalizing , we set $$[\pi:\pi'] := \sum_Y m_\pi(Y) m_{\pi'} (Y)$$ for $\pi,\pi' \in \mathcal{A}(\mathfrak{g}_{\mathbb{C}},K)$. Here are immediate results from the definition:
\[lem:pi\] Let $\pi,\pi' \in \mathcal{A}(\mathfrak{g}_{\mathbb{C}},K)$. $[\pi:\pi'] < \infty$ if at least one of $\pi$ and $\pi'$ belongs to $\mathcal{F}(\mathfrak{g}_{\mathbb{C}},K)$. $\dim{\operatorname{Hom}}_{(\mathfrak{g}_{\mathbb{C}},K)} (\pi,\pi')
\le [\pi: \pi']$. $[\pi: \pi'] = [\pi': \pi]$. $m_\pi(Y) = [\pi: Y]$if $Y$ is an irreducible $(\mathfrak{g}_{\mathbb{C}},K)$-module. $[\pi:\pi'] \le m(\pi)$if $\pi'$ is multiplicity-free.
Now, we return to Proposition \[lem:4.4\].
We divide the proof into three steps: $\underline{\text{Step 1}}$ ($\pi$ is a finite dimensional representation): We shall prove $$m(\pi \otimes F) \le \dim F
\label{eqn:4.4.1}$$ for any finite dimensional representation $\pi$ of $G$.
Let $\mathfrak{b} = \mathfrak{t} + \mathfrak{u}$ be a Borel subalgebra of $\mathfrak{g}_\mathbb{C}$ with $\mathfrak{u}$ nilradical. We denote by $H^j(\mathfrak{u}, V)$ the $j$th cohomology group of the Lie algebra $\mathfrak{u}$ with coefficients in a $\mathfrak{u}$-module $V$. Since the Lie algebra $\mathfrak{b}$ is solvable, we can choose a $\mathfrak{b}$-stable filtration $$F = F_k \supset F_{k-1} \supset \dots \supset F_0 = \{0\}$$ such that $\dim F_i/F_{i-1} = 1$.
Let us show by induction on $i$ that $$\dim H^0(\mathfrak{u}, \pi \otimes F_i) \le i \, .
\label{eqn:4.4.2}$$ This will imply $m(\pi\otimes F) =
\dim H^0(\mathfrak{u},\pi \otimes F)\le k = \dim F$.
The inequality is trivial if $i=0$. Suppose holds for $i-1$. The short exact sequence of $\mathfrak{b}$-modules $$0 \to \pi \otimes F_{i-1} \to \pi \otimes F_i \to \pi \otimes (F_i/F_{i-1})
\to 0$$ gives rise to a long exact sequence $$\begin{gathered}
0 \to
H^0(\mathfrak{u}, \pi \otimes F_{i-1})
\to
H^0(\mathfrak{u}, \pi \otimes F_i)
\to H^0(\mathfrak{u}, \pi \otimes (F_i/F_{i-1}))
\\
\to
H^1(\mathfrak{u}, \pi \otimes F_{i-1})
\to \dots\end{gathered}$$ of $\mathfrak{t}$-modules. In particular, we have $$\dim H^0(\mathfrak{u}, \pi \otimes F_i)
\le \dim H^0(\mathfrak{u}, \pi \otimes F_{i-1})
+
\dim H^0(\mathfrak{u}, \pi \otimes (F_i/F_{i-1})) \, .
\label{eqn:4.4.3}$$ Because $F_i/F_{i-1}$ is trivial as a $\mathfrak{u}$-module, we have $$H^0(\mathfrak{u}, \pi \otimes (F_i/F_{i-1}))
=
H^0(\mathfrak{u}, \pi) \otimes (F_i/F_{i-1}) \, .
\label{eqn:4.4.4}$$ By definition $H^0(\mathfrak{u}, \pi)$ is the space of highest weight vectors, and therefore the dimension of the right-hand side of is one. Now, the inductive assumption combined with implies $\dim H^0(\mathfrak{u}, \pi \otimes F_i) \le i$, as desired. $\underline{\text{Step 2}}$ ($\pi$ is a principal series representation): In this step, we consider the case where $\pi$ is a principal series representation. We note that $\pi$ may be reducible here.
Let $P = L N$ be a Levi decomposition of a minimal parabolic subgroup $P$ of $G$, $W$ an irreducible (finite dimensional) representation of $L$, and $\operatorname{Ind}_P^G (W)$ the underlying [$(\mathfrak{g}_\mathbb{C}, K)$]{}-module of a principal series representation induced from the representation $W \boxtimes
\mathbf{1} $ of $P = L N$ (without $\rho$-shift). Then, the socle filtration is unchanged so far as the parameter lies in the equisingular set, and thus, there are only finitely many possibilities of the socle filtration of $\operatorname{Ind}_P^G (W)$ for irreducible representations $W$ of $L$. We denote by $m(G)$ the maximum of $m(\operatorname{Ind}_P^G (W))$ for irreducible representations $W$ of $L$.
Let $F$ be a finite dimensional representation of $G$. Then we have an isomorphism of [$(\mathfrak{g}_\mathbb{C}, K)$]{}-modules $$\operatorname{Ind}_P^G (W) \otimes F \simeq
\operatorname{Ind}_P^G (W \otimes F) \, ,$$ where $F$ is regarded as a $P$-module on the right-hand side. We take a $P$-stable filtration $$W_n := W \otimes F \supset W_{n-1} \supset \dots
\supset W_0 = \{0\}$$ such that each $W_{i}/W_{i-1}$ is irreducible as a $P$-module. We notice that $n \le \dim F$ by applying Step 1 to the $L$-module $F|_L$. As $\operatorname{Ind}_P^G (W \otimes F)$ is isomorphic to $\bigoplus_{i=1}^n \operatorname{Ind}_P^G (W_{i}/W_{i-1})$ in the Grothendieck group $\mathcal{V}(\mathfrak{g}_\mathbb{C}, K)$, we have shown that $$m(\operatorname{Ind}_P^G (W) \otimes F)
\le n \; m(G)
\le (\dim F) \; m(G)$$ for any irreducible finite dimensional representation $W$ of $L$. $\underline{\text{Step 3}}$ (general case): By Casselman’s subrepresentation theorem (see [@xwal Chapter 3]), any irreducible $(\mathfrak{g}_{\mathbb{C}},K)$-module $\pi$ is realized as a subrepresentation of some induced representation $\operatorname{Ind}_P^G (W)$. Then $$m(\pi \otimes F) \le m(\pi \otimes \operatorname{Ind}_P^G (W)) \le C$$ by step 2. Thus, Proposition \[lem:4.4\] is proved.
Proof of Theorem \[thm:D\] {#subsec:4.5}
--------------------------
Now let us complete the proof of Theorem \[thm:D\].
Let $\pi = \pi_1 \boxtimes \pi_2$ be an irreducible unitary highest weight representation of $G':=G \times G$. It follows from Lemma \[lem:4.3\] that there exist an irreducible unitary highest weight representation $\pi'= \pi_1' \boxtimes \pi_2'$ of scalar type and a finite dimensional representation $F$ of $G'$ such that $\pi_K$ occurs as a subquotient of $\pi_K' \otimes F$.
By using the notation , we set $[V_1:V_2] := [(V_1)_K:(V_2)_K]$ for $G$-modules $V_1$ and $V_2$ of finite length. Then, for $\mu\in\widehat{G}$, we have $$\begin{aligned}
m_{\pi_1,\pi_2}(\mu)
&= \dim{\operatorname{Hom}}_G (\mu, \pi|_{{\operatorname{diag}}(G)})
\nonumber
\\
&\leq \left[\mu: \pi |_{{\operatorname{diag}}(G)}\right]
\nonumber
\\
&\leq \left[\mu:(\pi' \otimes F)|_{{\operatorname{diag}}(G)}\right]
\nonumber
\\
&= \left[\mu \otimes (F^* |_{{\operatorname{diag}}(G)}): \pi'|_{{\operatorname{diag}}(G)}\right]
\nonumber
\\
&\leq m(\mu \otimes (F^* |_{{\operatorname{diag}}(G)}))
\label{eqn:4.5.1}\\
&\leq C(F^*) \, .
\nonumber\end{aligned}$$ Here the inequality follows from Lemma \[lem:pi\] (5) because $\pi'|_{{\operatorname{diag}}(G)}\simeq \pi_1' \widehat \otimes \pi_2'$ is multiplicity-free (see Theorem \[thm:D\] (2)). In the last inequality, $C(F^*)$ is the constant in Proposition \[lem:4.4\]. This completes the proof of Theorem \[thm:D\] (1).
The argument in Subsections \[subsec:8.3\] and \[subsec:pf tensordeco\] gives a different and more straightforward proof of Theorem \[thm:D\].
Counter examples {#sec:6}
================
In this section, we analyze the assumptions in Theorems \[thm:A\] and \[thm:B\] by counterexamples, that is, how the conclusions fail if we relax the assumptions on the representation $\pi$.
Let $(G, H)$ be a reductive symmetric pair corresponding to an involutive automorphism $\tau$ of $G$, and $\pi$ an irreducible unitary representation of $G$. We shall see that the multiplicity of an irreducible summand occurring in the restriction $\pi|_H$ can be: **greater than one** if $\pi$ is not of scalar type (but we still assume that $\pi$ is a highest weight module); **finite but not uniformly bounded** if $\pi$ is not a highest weight module (but we still assume that $\pi|_H$ decomposes discretely); **infinite** if $\pi|_H$ contains continuous spectra. Although our concern in this paper is mainly with a non-compact subgroup $H$, we can construct such examples for (1) and (2) even for $H = K$ (a maximal compact subgroup modulo the center of $G$).
Case (1) will be discussed in Subsection \[subsec:6.2\], (2) in Subsection \[subsec:6.3\], and (3) in Subsection \[subsec:6.4\], respectively. To construct an example for (3), we use those for (1) and (2).
Failure of multiplicity-free property {#subsec:6.2}
-------------------------------------
Let $G = Sp(2, \mathbb{R})$. Then, the maximal compact subgroup $K$ is isomorphic to $U(2)$. We take a compact Cartan subalgebra $\mathfrak{t}$. Let $\{f_1, f_2\}$ be the standard basis of $\sqrt{-1}\, \mathfrak{t}^*$ such that $\Delta(\mathfrak{g}, \mathfrak{t}) =
\{\pm f_1 \pm f_2, \pm 2 f_1, \pm 2 f_2\}$, and we fix a positive system $\Delta^+(\mathfrak{k}, \mathfrak{t}) := \{f_1 - f_2\}$. In what follows, we shall use the notation $(\lambda_1, \lambda_2)$ to denote the character $\lambda_1 f_1 + \lambda_2 f_2$ of $\mathfrak{t}$.
Given $(p,q) \in \mathbb{Z}^2$ with $p \ge q$, we denote by $\F{U(2)}{(p,q)}$ the irreducible representation of $U(2)$ with highest weight $(p,q) = p f_1 + q f_2$. Then $\dim \F{U(2)}{(p,q)} = p-q+1$.
The set of holomorphic discrete series representations of $G$ is parametrized by $\lambda := (\lambda_1, \lambda_2) \in \mathbb{N}^2$ with $\lambda_1 > \lambda_2 > 0$. We set $\mu \equiv (\mu_1,\mu_2):=(\lambda_1+1,\lambda_2+2)$ and denote by $\hwm{G}{\mu} \equiv \hwm{Sp(2,\mathbb{R})}{(\mu_1,\mu_2)}$ the holomorphic discrete series representation of $G$ characterized by $$\begin{aligned}
{2}
&\ \text{$Z(\mathfrak{g})$-infinitesimal character }
= (\lambda_1, \lambda_2)
&&\quad\text{(Harish-Chandra parameter)},
\\
&\ \text{minimal $K$-type }
= \hwm{U(2)}{(\mu_1, \mu_2)}
&&\quad\text{(Blattner parameter)}.\end{aligned}$$ We note that $\hwm{G}{\mu}$ is of scalar type if and only if $\mu_1 = \mu_2$.
We know from Theorem \[thm:B\] that multiplicities of $K$-type $\tau$ occurring in $\hwm{G}{\mu}$ are uniformly bounded for fixed $\mu = (\mu_1,\mu_2)$. Here is the formula:
\[exam:6.1\] $$\sup_{\tau \in \widehat{K}} \dim {{\operatorname{Hom}}}_K(\tau, \hwm{G}{\mu}|_K)
= \left[\frac{\mu_1 - \mu_2+2}{2}\right] \, .
\label{eqn:6.2.1}$$ The right side of $=1$ if and only if either of the following two cases holds:
$$\begin{aligned}
{2}
\mu_1 &= \mu_2
&&\text{(i.e. $\hwm{G}{\mu}$ is of scalar type),}
\label{eqn:6.2.2a}\\
\mu_1 &= \mu_2 + 1 \quad
&&\text{(i.e. $\hwm{G}{\mu}$ is of two dimensional minimal $K$-type).}
\label{eqn:6.2.2b}$$
Thus, the branching law of the restriction $\hwm{G}{\mu} |_K$ is multiplicity-free if and only if $\mu_1 = \mu_2$ or $\mu_1 = \mu_2 + 1$. The multiplicity-free property for $\mu_1 = \mu_2$ (i.e. for $\hwm{G}{\mu}$ of scalar type) follows from Theorem \[thm:A\]. The multiplicity-free property for $\mu_1 = \mu_2 + 1$ is outside of the scope of this paper, but can be explained in the general framework of the ‘vector bundle version’ of Theorem \[thm:2.2\] (see [@RIMS Theorem 2], [@mfbdle]).
It follows from the Blattner formula for a holomorphic discrete series representation ([@xjohnson], [@xschmidherm]) that the $K$-type formula of $\hwm{G}{\mu}$ is given by $$\begin{aligned}
\hwm{G}{\mu}|_K
&\simeq
\F{U(2)}{(\mu_1,\mu_2)}
\otimes
S(\mathbb{C}^3)
\nonumber
\\
&=\F{U(2)}{(\mu_1,\mu_2)}
\otimes
\bigoplus \Sb a \ge b \ge 0 \\ (a,b) \in \mathbb{N}^2 \endSb
\F{U(2)}{(2 a, 2 b)} \, ,
\label{eqn:holoU2}\end{aligned}$$ where $K=U(2)$ acts on $\mathbb{C}^3 \simeq S^2(\mathbb{C}^2)$ as the symmetric tensor of the natural representation. We write $n_\mu(p,q)$ for the multiplicity of the $K$-type $\hwm{U(2)}{(p,q)}$ occurring in $\hwm{G}{\mu} \equiv \hwm{Sp(2,\mathbb{R})}{(\mu_1,\mu_2)}$, that is, $$n_\mu(p,q) := \dim {\operatorname{Hom}}_K
(\hwm{K}{(p,q)}, \hwm{G}{\mu} |_K) \, .$$ Then, applying the Clebsch–Gordan formula to , we obtain $$n_\mu(p,q) = \# \{ (a,b) \in \mathbb{N}^2:
\text{$(a,b)$ satisfies $a \ge b \ge 0$,
\eqref{eqn:ab1} and \eqref{eqn:ab2}} \},$$ where $$\begin{aligned}
& \,
p+q = \mu_1 + \mu_2 + 2a + 2b \, ,
\label{eqn:ab1}
\\
&\max (2a+\mu_2, 2b+\mu_1) \le p \le 2a+\mu_1 \, .
\label{eqn:ab2}\end{aligned}$$ In particular, for fixed $(\mu_1, \mu_2)$ and $(p,q)$, the integer $b$ is determined by $a$ from , whereas the integer $a$ satisfies the inequalities $p-\mu_1 \le 2a \le p-\mu_2$. Therefore, $$n_\mu(p,q) \le \left[ \frac{(p-\mu_2)-(p-\mu_1)}{2} \right] +1
= \left[ \frac{\mu_1-\mu_2+2}{2} \right] \, .
\qquad\qed$$
Failure of uniform boundedness {#subsec:6.3}
------------------------------
We continue the setting of Subsection \[subsec:6.2\]. Let $B$ be a Borel subgroup of $G_{\mathbb{C}} \simeq Sp(2,\mathbb{C})$. Then, there exist $4$ closed orbits of $K_\mathbb{C} \simeq GL(2,\mathbb{C})$ on the full flag variety $G_{\mathbb{C}}/B$. (By the Matsuki duality, there exist $4$ open orbits of $G = Sp(2,\mathbb{R})$ on $G_{\mathbb{C}}/B$. This observation will be used in the proof of Example \[ex:finite infinite\].) By the Beilinson–Bernstein correspondence, we see that there are $4$ series of discrete series representations of $G$. Among them, two are holomorphic and anti-holomorphic discrete series representations, that is, $\hwm{G}{\mu}$ and $(\hwm{G}{\mu})^*$ (the contragredient representation) with notation as in Subsection \[subsec:6.2\]. The other two series are non-holomorphic discrete series representations. Let us parametrize them. For $\lambda := (\lambda_1, \lambda_2) \in \mathbb{Z}^2$ ($\lambda_1 > -\lambda_2 >0$), we write $W_\lambda$ for the discrete series representation of $G$ characterized by $$\begin{aligned}
{2}
&\ \text{$Z(\mathfrak{g})$-infinitesimal character }
= (\lambda_1, \lambda_2)
&&\quad\text{(Harish-Chandra parameter)},
\\
&\ \text{minimal $K$-type }
= \F{U(2)}{(\lambda_1 + 1, \lambda_2)}
&&\quad\text{(Blattner parameter)}.\end{aligned}$$ Then, non-holomorphic discrete series representations are either $W_\lambda$ or its contragredient representation $W_\lambda^*$ for some $\lambda \in \mathbb{Z}^2$ with $\lambda_1 > -\lambda_2 > 0$. We define a $\theta$-stable Borel subalgebra $\mathfrak{q} = \mathfrak{t}_{\mathbb{C}} + \mathfrak{u}$ of $\mathfrak{g}_{\mathbb{C}} =
\mathfrak{k}_{\mathbb{C}} + \mathfrak{p}_{\mathbb{C}}$ such that $$\Delta(\mathfrak{u} \cap \mathfrak{p}_{\mathbb{C}}, \mathfrak{t})
:= \{2 f_1, f_1 + f_2, -2 f_2\}\, ,
\quad
\Delta(\mathfrak{u} \cap \mathfrak{k}_{\mathbb{C}}, \mathfrak{t})
:= \{f_1 - f_2\} \, .$$ Then, the Harish-Chandra module $(W_\lambda)_K$ is isomorphic to the cohomological parabolic induction $\mathcal{R}_{\mathfrak{q}}^1 (\mathbb{C}_{(\lambda_1, \lambda_2)})$ of degree $1$ as $(\mathfrak{g}_{\mathbb{C}},K)$-modules with the notation and the normalization as in [@xvr]. We set $\mu_1 := \lambda_1+1$ and $\mu_2 := \lambda_2$.
\[exam:6.3\] We write $m_\lambda(p,q)$ for the multiplicity of the $K$-type $\F{U(2)}{(p,q)}$ occurring in $W_\lambda$, that is, $$m_\lambda(p,q) := \dim {\operatorname{Hom}}_K(\F{U(2)}{(p,q)}, W_\lambda|_K) \, .$$ Then, $m_\lambda(p,q) \neq 0$ only if $(p,q) \in \mathbb{Z}^2$ satisfies $$p \ge \mu_1\, ,
\
p-q \ge \mu_1 - \mu_2
\text{ and }
p-q \in 2 \mathbb{Z} + \mu_1 + \mu_2 \, .
\label{eqn:6.3.1}$$ Then, $$m_\lambda(p,q)
= 1 + \min(\left[\frac{p-\mu_1}{2}\right],
\frac{p-q-\mu_1 +\mu_2}{2}) \, .
\label{eqn:6.3.2}$$ In particular, for each fixed $\lambda$, the $K$-multiplicity in $W_\lambda$ is not uniformly bounded, namely, $$\sup_{\tau \in \widehat{K}} \dim {\operatorname{Hom}}_K(\tau, W_\lambda|_K)
=\sup_{(p,q) \text{ satisfies \eqref{eqn:6.3.1} }} m_{\lambda}(p,q)
= \infty \, .$$
For $p,q\in\mathbb{Z}$, we write $\mathbb{C}_{(p,q)}$ for the one dimensional representation of $\mathfrak{t}_{\mathbb{C}}$ corresponding to the weight $pf_1 + qf_2 \in \mathfrak{t}_{\mathbb{C}}^*$. According to the $\mathfrak{t}_{\mathbb{C}}$-module isomorphism: $$\mathfrak{u} \cap \mathfrak{p}_{\mathbb{C}}
\simeq \mathbb{C}_{(2,0)} \oplus \mathbb{C}_{(1,1)} \oplus
\mathbb{C}_{(0,-2)} \, ,$$ the symmetric algebra $S(\mathfrak{u} \cap \mathfrak{p}_{\mathbb{C}})$ is decomposed into irreducible representations of $\mathfrak{t}_{\mathbb{C}}$ as $$\begin{aligned}
S(\mathfrak{u} \cap \mathfrak{p}_{\mathbb{C}})
&\simeq \bigoplus_{a,b,c \in \mathbb{N}}
S^a(\mathbb{C}_{(2,0)})
\otimes S^b(\mathbb{C}_{(1,1)})
\otimes S^c(\mathbb{C}_{(0,-2)})
\nonumber
\\
&\simeq \bigoplus \Sb a, b, c \in \mathbb{N} \endSb
\mathbb{C}_{(2a + b, b-2c)} \, .
\label{eqn:Supabc}\end{aligned}$$ We denote by $H^j(\mathfrak{u} \cap \mathfrak{k}_\mathbb{C}, \pi)$ the $j$th cohomology group of the Lie algebra $\mathfrak{u} \cap \mathfrak{k}_\mathbb{C}$ with coefficients in the $\mathfrak{u} \cap \mathfrak{k}_\mathbb{C}$-module $\pi$. If $\pi$ is a $\mathfrak{k}_{\mathbb{C}}$-module, then $H^j(\mathfrak{u} \cap \mathfrak{k}_\mathbb{C}, \pi)$ becomes naturally a $\mathfrak{t}_{\mathbb{C}}$-module. Then, Kostant’s version of the Borel–Weil–Bott theorem (e.g. [@xvg Chapter 3]) shows that $$\label{eqn:ukcohom}
H^j(\mathfrak{u} \cap \mathfrak{k}_\mathbb{C}, \F{U(2)}{(p,q)})
= \begin{cases} \mathbb{C}_{(p,q)} & (j=0)\, , \\
\mathbb{C}_{(q-1,p+1)} & (j=1)\, , \\
\{0\} & (j\neq 0,1)\, .
\end{cases}$$ By using the Blattner formula due to Hecht–Schmid (e.g. [@xvg Theorem 6.3.12]), the $K$-type formula of $W_\lambda$ is given by $$\begin{aligned}
m_\lambda(p,q)&=
\dim {\operatorname{Hom}}_K(\F{U(2)}{(p,q)}, W_\lambda|_K)
\\
&=
\sum_{j=0}^1 (-1)^j \dim
{\operatorname{Hom}}_{\mathfrak{t}_{\mathbb{C}}}
(H^j(\mathfrak{u} \cap \mathfrak{k}_\mathbb{C}, \F{U(2)}{(p,q)}),
S(\mathfrak{u} \cap \mathfrak{p}_\mathbb{C}) \otimes
\mathbb{C}_{(\mu_1, \mu_2)}) \, .
\\
\intertext{Now, comparing \eqref{eqn:Supabc} with
the above formula \eqref{eqn:ukcohom}
as $\mathfrak{t}_{\mathbb{C}}$-modules, we see}
m_\lambda(p,q)
&=\#\set{(a,b,c) \in \mathbb{N}^3}{p = 2a + b + \mu_1, q = b - 2c + \mu_2}
\\
&\phantom{={}}
-\#\set{(a,b,c) \in \mathbb{N}^3}{q-1 = 2a + b + \mu_1, p+1 = b - 2c + \mu_2}
\\
&=\#\set{(a,b,c) \in \mathbb{N}^3}{p = 2a + b + \mu_1, q = b - 2c + \mu_2}
\\
&= 1 + \min(\left[\frac{p-\mu_1}{2}\right],
\frac{p-q-\mu_1 +\mu_2}{2}) \, .\end{aligned}$$ Thus, the formula has been verified.
Failure of finiteness of multiplicities {#subsec:6.4}
---------------------------------------
Multiplicities of the branching laws can be infinite in general even for reductive symmetric pairs $(G,H)$. In this subsection, we review from [@xkaspm Example 5.5] a curious example of the branching law, in which the multiplicity of a discrete summand is non-zero and finite and that of another discrete summand is infinite. Such a phenomenon happens only when continuous spectra appear.
\[ex:finite infinite\] Let $(G_{\mathbb{C}},G)$ be a reductive symmetric pair $(Sp(2,\mathbb{C}), Sp(2,\mathbb{R}))$. We note that $(G_{\mathbb{C}},G)$ is locally isomorphic to the symmetric pair $(SO(5,\mathbb{C}), SO(3,2))$. We take a Cartan subgroup $H = TA$ of $G_{\mathbb{C}}$. We note that $T \simeq \mathbb{T}^2$ and $A \simeq \mathbb{R}^2$, and identify $\widehat{T}$ with $\mathbb{Z}^2$.
Let $\varpi \equiv \varpi_{(a,b)}^{Sp(2,\mathbb{C})}$ be the unitary principal series representation of $G_{\mathbb{C}}$ induced unitarily from the character $\chi$ of a Borel subgroup $B$ containing $H=TA$ such that $$\chi |_H \simeq \mathbb{C}_{(a,b)} \boxtimes \mathbf{1} \, .$$ We assume $a,b \ge 0$ and set $$c(\mu_1,\mu_2;a,b) :=
\# \{ (s,t,u) \in \mathbb{N}^3:
a = \mu_1+2s+t, \,
b=\mu_2+t+2u \} \, .$$ Then, the discrete part of the branching law of the restriction $\varpi_{(a,b)}^{Sp(2,\mathbb{C})} |_{Sp(2,\mathbb{R})}$ is given by the following spectra: $$\bigoplus_{\mu_1 \ge \mu_2 \ge 3}
c(\mu_1,\mu_2;a,b)
( \hwm{Sp(2,\mathbb{R})}{(\mu_1,\mu_2)} \oplus
\left( \hwm{Sp(2,\mathbb{R})}{(\mu_1,\mu_2)} \right)^* )
\oplus
\sideset{}{^\oplus}\sum_{\lambda_1>-\lambda_2>0}
\infty (W_\lambda \oplus W^*_\lambda)
\, ,
\label{eqn:discSp}$$ with the notation as in Examples \[exam:6.1\] and \[exam:6.3\].
The first term of is a finite sum because there are at most finitely many $(\mu_1,\mu_2)$ such that $c(\mu_1,\mu_2;a,b) \ne 0$ for each fixed $(a,b)$. For instance, the first term of amounts to $$\bigoplus_{\substack{3\le\mu_1\le a \\ \mu_1\equiv a\bmod 2}}
\hwm{Sp(2,\mathbb{R})}{(\mu_1,3)} \oplus
\bigoplus_{\substack{3\le\mu_1\le a \\ \mu_1\equiv a\bmod 2}}
\left(\hwm{Sp(2,\mathbb{R})}{(\mu_1,3)}\right)^*
\quad\text{(multiplicity-free)}$$ if $b=3$.
The second term of is nothing other than the direct sum of all non-holomorphic discrete series representations of $G=Sp(2,\mathbb{R})$ with infinite multiplicities for any $a$ and $b$.
There exist $4$ open $G$-orbits on $G_{\mathbb{C}}/B$, for which the isotropy subgroups are all isomorphic to $T \simeq \mathbb{T}^2$. By the Mackey theory, the restriction $\varpi^{G_{\mathbb{C}}}_{(a,b)}|_G$ is unitarily equivalent to the direct sum of the regular representations realized on $L^2$-sections of $G$-equivariant line bundles $G \times_T \mathbb{C}_{(\pm a, \pm b)} \to G/T$. That is, $$\varpi^{G_{\mathbb{C}}}_{(a,b)}|_G
\simeq \bigoplus_{\varepsilon_1,\varepsilon_2=\pm1}
L^2 (G/T, \mathbb{C}_{(\varepsilon_1 a,\varepsilon_2 b)} ) \, .$$ Therefore, an irreducible unitary representation $\sigma$ of $G$ occurs as a discrete spectrum in $\varpi_{(a,b)}^{G_{\mathbb{C}}} |_G$ if and only if $\sigma$ occurs as a discrete summand in $L^2(G/T,\mathbb{C}_{(\varepsilon_1 a, \varepsilon_2 b)})$ for some $\varepsilon_1, \varepsilon_2 = \pm 1$. Further, the multiplicity is given by $$\dim{\operatorname{Hom}}_G (\sigma, \varpi_{(a,b)}^{G_{\mathbb{C}}} |_G)
= \sum_{\varepsilon_1,\varepsilon_2 = \pm1}
\dim{\operatorname{Hom}}_{\mathbb{T}^2}
( \mathbb{C}_{(\varepsilon_1 a,\varepsilon_2 b)},
\sigma |_{\mathbb{T}^2} )$$ by the Frobenius reciprocity theorem.
Since $T$ is compact, $\sigma$ must be a discrete series representation of $G = Sp(2,\mathbb{R})$ if $\sigma$ occurs in $L^2(G/T, \mathbb{C}_{(\varepsilon_1 a, \varepsilon_2 b)})$ as a discrete summand. We divide the computation of multiplicities into the following two cases:
Case I. $\sigma$ is a holomorphic series representation or its contragredient representation. Let $\sigma = \hwm{Sp(2,\mathbb{R})}{\mu}$. Combining with the weight formulae $$\begin{aligned}
S (\mathbb{C}^3) |_{\mathbb{T}^2}
& \simeq
\bigoplus_{s,t,u\in\mathbb{N}}
S^s (\mathbb{C}_{(2,0)}) \otimes
S^t (\mathbb{C}_{(1,1)}) \otimes
S^u (\mathbb{C}_{(0,2)})
\simeq
\bigoplus_{s,t,u\in\mathbb{N}}
\mathbb{C}_{(2s+t, t+2u)} \, ,
\\
\hwm{U(2)}{(\mu_1,\mu_2)} |_{\mathbb{T}^2}
& \simeq \bigoplus_{\substack{p+q=\mu_1+\mu_2 \\
\mu_2\le p \le\mu_1}}
\mathbb{C}_{(p,q)} \, ,\end{aligned}$$ we have $$\dim{\operatorname{Hom}}_{\mathbb{T}^2}
(\mathbb{C}_{(a,b)}, \hwm{Sp(2,\mathbb{R})}{\mu} |_{\mathbb{T}^2})
= c(\mu_1,\mu_2;a,b)\, .$$
Case II. $\sigma$ is a non-holomorphic discrete series representation. Let $\sigma = W_\lambda$. It follows from the $K$-type formula of $W_\lambda$ that we have $$\dim{\operatorname{Hom}}_{\mathbb{T}^2}
(\mathbb{C}_{(a,b)}, W_{\lambda} |_{\mathbb{T}^2} )
= \sum_{p\ge q} m_\lambda(p,q)\dim{\operatorname{Hom}}_{\mathbb{T}^2}
(\mathbb{C}_{(a,b)},\hwm{U(2)}{(p,q)})
= \infty \, .$$ Likewise for $\sigma = W_\lambda^*$ (the contragredient representation).
Hence, the discrete part of the branching law is given by .
Finite Dimensional Cases — Proof of Theorems \[thm:E\] and \[thm:F\] {#sec:7}
====================================================================
Infinite v.s. finite dimensional representations {#subsec:7.1}
------------------------------------------------
Our method applied to infinite dimensional representations in Sections \[sec:3\] and \[sec:4\] also applies to [**finite**]{} dimensional representations, leading us to multiplicity-free theorems, as stated in Theorems \[thm:E\] and \[thm:F\] in Section \[sec:1\], for the restriction with respect to symmetric pairs.
The comparison with multiplicity-free theorems in the infinite dimensional case is illustrated by the following correspondence: $$\begin{aligned}
{3}
&\text{a non-compact simple group $G$}
&&\quad \leftrightarrow
&&\quad \text{a compact simple group $G_U$}
\\
&\text{a unitary highest weight module}
&&\quad \leftrightarrow
&&\quad \text{a finite dimensional module}
\\
&\text{{scalar type (Definition~\ref{def:1.4})}}
&&\quad \leftrightarrow
&&\quad \text{\lq\lq pan type\rq\rq\
(Definition~\ref{def:7.3.2})} \\
&\text{Theorems~\ref{thm:A} and \ref{thm:B}}
&&\quad \leftrightarrow
&&\quad \text{Theorems~\ref{thm:E} and \ref{thm:F}}.\end{aligned}$$ The main goal of this section is to give a proof of Theorems \[thm:E\] and \[thm:F\] by using Theorem \[thm:2.2\]. Geometrically, our proof is built on the fact that the $H_U$ action on the Hermitian symmetric space is strongly visible if $(G_U,H_U)$ is a symmetric pair (see [@visiblesymm]).
Representations associated to maximal parabolic subalgebras {#subsec:7.2}
-----------------------------------------------------------
Let $\mathfrak{g}_\mathbb{C}$ be a complex simple Lie algebra. We take a Cartan subalgebra $\mathfrak{j}$ of $\mathfrak{g}_\mathbb{C}$, and fix a positive system $\Delta^+(\mathfrak{g}_\mathbb{C}, \mathfrak{j})$. We denote by $\{\alpha_1, \dots, \alpha_n\}$ the set of simple roots, and by $\{\omega_1, \dots, \omega_n\} \ (\subset \mathfrak{j}^*)$ the set of the fundamental weights.
We denote by $\F{\mathfrak{g}_\mathbb{C}}{\lambda}$ irreducible finite dimensional representation of $\mathfrak{g}_\mathbb{C}$ with highest weight $\lambda = \sum_{i=1}^n m_i \omega_i$ for $m_1, \dots, m_n \in \mathbb{N}$. It is also regarded as a holomorphic representation of $G_\mathbb{C}$, a simply connected complex Lie group with Lie algebra $\mathfrak{g}_\mathbb{C}$.
We fix a simple root $\alpha_i$, and define a maximal parabolic subalgebra $${\mathfrak{p}}^{-}_{i\mathbb{C}} :=
{\mathfrak{l}}_{i\mathbb{C}} + {\mathfrak{n}}^{-}_{i\mathbb{C}}$$ such that the nilradical ${\mathfrak{n}}^{-}_{i\mathbb{C}}$ and the Levi part ${\mathfrak{l}}_{i\mathbb{C}}\ (\ \supset \mathfrak{j})$ are given by $$\begin{aligned}
\Delta({\mathfrak{l}}_{i\mathbb{C}}, \mathfrak{j})
&= \mathbb{Z}\text{-span of }
\{\alpha_1, \dots, \overset{\wedge}{\alpha_i}, \dots, \alpha_n\}
\cap \Delta(\mathfrak{g}_\mathbb{C}, \mathfrak{j}) \, ,
\\
\Delta({\mathfrak{n}}^{-}_{i\mathbb{C}}, \mathfrak{j})
&= \Delta^-(\mathfrak{g}_\mathbb{C}, \mathfrak{j}) \setminus
\Delta({\mathfrak{l}}_{i\mathbb{C}}, \mathfrak{j}) \, .\end{aligned}$$ We shall see that irreducible finite dimensional representations realized on generalized flag varieties $G_{\mathbb{C}}/P_{\mathbb{C}}$ is multiplicity-free with respect to any symmetric pairs if $P_{\mathbb{C}}$ has an abelian unipotent radical.
We write ${P}^{-}_{i\mathbb{C}} = {L}_{i\mathbb{C}} {N}^{-}_{i\mathbb{C}}$ for the corresponding maximal parabolic subgroup of $G_\mathbb{C}$.
Let ${\operatorname{Hom}}(\mathfrak{p}^{-}_{i\mathbb{C}}, \mathbb{C})$ be the set of Lie algebra homomorphisms over $\mathbb{C}$. Since any such homomorphism vanishes on the derived ideal $[\mathfrak{p}^{-}_{i\mathbb{C}}, \mathfrak{p}^{-}_{i\mathbb{C}}]$, ${\operatorname{Hom}}({\mathfrak{p}}^{-}_{i\mathbb{C}}, \mathbb{C})$ is naturally identified with $${\operatorname{Hom}}({\mathfrak{p}}^{-}_{i\mathbb{C}}/[{\mathfrak{p}}^{-}_{i\mathbb{C}},
{\mathfrak{p}}^{-}_{i\mathbb{C}}], \mathbb{C}) \simeq \mathbb{C} \omega_i
\, .$$
Next, let ${\operatorname{Hom}}(P^{-}_{i\mathbb{C}}, \mathbb{C}^\times)$ be the set of complex Lie group homomorphisms. Then, we can regard ${\operatorname{Hom}}({P}^{-}_{i\mathbb{C}}, \mathbb{C}^\times) \subset
{\operatorname{Hom}}({\mathfrak{p}}^{-}_{i\mathbb{C}}, \mathbb{C})$. As its subset, ${\operatorname{Hom}}({P}^{-}_{i\mathbb{C}}, \mathbb{C}^\times)$ is identified with $\mathbb{Z} \omega_i$ since $G_\mathbb{C}$ is simply connected.
For $k \in \mathbb{Z}$, we write $\mathbb{C}_{k \omega_i}$ for the corresponding character of ${P}^{-}_{i\mathbb{C}}$, and denote by $$\mathcal{L}_{k \omega_i}
:= G_{\mathbb{C}} \times_{{P}^{-}_{i\mathbb{C}}} \mathbb{C}_{k \omega_i}
\to G_\mathbb{C}/{P}^{-}_{i\mathbb{C}}
\label{eqn:7.2.1}$$ the associated holomorphic line bundle. We naturally have a representation of $G_\mathbb{C}$ on the space of holomorphic sections $\mathcal{O}\left(\mathcal{L}_{k \omega_i}\right)$. Then, by the Borel–Weil theory, $\mathcal{O}(\mathcal{L}_{k\omega_i})$ is non-zero and irreducible if $k \ge 0$ and we have an isomorphism of representations of $G_\mathbb{C}$ (also of $\mathfrak{g}_\mathbb{C}$): $$\F{\mathfrak{g}_\mathbb{C}}{k \omega_i}
\simeq \mathcal{O}\left(\mathcal{L}_{k \omega_i}\right).
\label{eqn:7.2.2}$$
Parabolic subalgebra with abelian nilradical {#subsec:7.3}
--------------------------------------------
A parabolic subalgebra with abelian nilradical is automatically a maximal parabolic subalgebra. Conversely, the nilradical of a maximal parabolic subalgebra is not necessarily abelian. We recall from Richardson–Röhrle–Steinberg [@xrrs] the following equivalent characterization of such parabolic algebras:
\[lem:7.3.1\] Retain the setting of Subsection \[subsec:7.2\]. Then, the following four conditions on the pair $(\mathfrak{g}_\mathbb{C}, \alpha_i)$ are equivalent: The nilradical ${\mathfrak{n}}^{-}_{i\mathbb{C}}$ is abelian. $(\mathfrak{g}_\mathbb{C}, {\mathfrak{l}}_{i\mathbb{C}})$ is a symmetric pair. The simple root $\alpha_i$ occurs in the highest root with coefficient one. $(\mathfrak{g}_\mathbb{C}, \alpha_i)$ is in the following list if we label the simple roots $ \alpha_1, \dots, \alpha_n$ in the Dynkin diagram as in Table \[tbl:7.3.1\].
[Type]{} $A_n$ $\alpha_1, \alpha_2, \dots, \alpha_n$ \[eqn:7.3.1\]
[Type]{} $B_n$ $\alpha_1$ \[eqn:7.3.2\]
[Type]{} $C_n$ $\alpha_n$ \[eqn:7.3.3\]
[Type]{} $D_n$ $\alpha_1, \alpha_{n-1}, \alpha_n$ \[eqn:7.3.4\]
[Type]{} $E_6$ $\alpha_1, \alpha_{6}$ \[eqn:7.3.5\]
[Type]{} $E_7$ $\alpha_7$ \[eqn:7.3.6\]
For types $G_2$, $F_4$, $E_8$, there are no maximal parabolic subalgebras with abelian nilradicals.
$$\begin{aligned}
&(A_n)
&&{\mathop\circ\limits_{\rlap{$\alpha_1$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_2$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}\qquad\cdots\qquad {\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_{n-1}$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\ \alpha_n$}}}
\\[\medskipamount]
&(B_n)
&&{\mathop\circ\limits_{\rlap{$\alpha_1$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_2$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}\qquad\cdots\qquad {\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_{n-1}$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord= \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord= \mkern-3mu$}\hfill
\mkern-6mu \mathord=$}$\mkern-3mu\Rightarrow$}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\ \alpha_n$}}}
\\[\medskipamount]
&(C_n)
&&{\mathop\circ\limits_{\rlap{$\alpha_1$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_2$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}\qquad\cdots\qquad {\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_{n-1}$}}} {\kern-.35em}{\hbox to 2em{$\Leftarrow\mkern-3mu${$\mathsurround=0pt\mathord= \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord= \mkern-3mu$}\hfill
\mkern-6mu \mathord=$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\ \alpha_n$}}}
\\[\medskipamount]
&(D_n)
&&{\mathop\circ\limits_{\rlap{$\alpha_1$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_2$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}\qquad\cdots\qquad {\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}\rlap{\kern0.07em{\raisebox{\circlength}{\rotatebox{90}{{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}}}}}
\rlap{\kern0.13em\raisebox{2.2em}{$\circ\, \alpha_{n-1}$}}
{\mathop\circ\limits_{\rlap{$\alpha_{n-2}$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\ \alpha_n$}}}
\\[\medskipamount]
&(E_6)
&&{\mathop\circ\limits_{\rlap{$\alpha_1$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_3$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}\rlap{\kern0.07em{\raisebox{\circlength}{\rotatebox{90}{{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}}}}}
\rlap{\kern0.13em\raisebox{2.2em}{$\circ\, \alpha_2$}}
{\mathop\circ\limits_{\rlap{$\alpha_4$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_5$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_6$}}}
\\[\medskipamount]
&(E_7)
&&{\mathop\circ\limits_{\rlap{$\alpha_1$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_3$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}\rlap{\kern0.07em{\raisebox{\circlength}{\rotatebox{90}{{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}}}}}
\rlap{\kern0.13em\raisebox{2.2em}{$\circ\, \alpha_2$}}
{\mathop\circ\limits_{\rlap{$\alpha_4$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_5$}}} {\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_6$}}}{\kern-.35em}{\hbox to 2em{{$\mathsurround=0pt\mathord- \mkern-6mu
\cleaders\hbox{$\mkern-3mu \mathord- \mkern-3mu$}\hfill
\mkern-6mu \mathord-$}}}{\kern-.35em}{\mathop\circ\limits_{\rlap{$\alpha_7$}}}\end{aligned}$$
See [@xrrs] for the equivalence (i) $\Leftrightarrow$ (iii) $\Leftrightarrow$ (iv). The implication (iv) $\Rightarrow$ (ii) is straightforward. For the convenience of the reader, we present a table of the symmetric pairs $(\mathfrak{g}_{\mathbb{C}}, \mathfrak{l}_{i\mathbb{C}})$ corresponding to the index $i$ in (iv).
-------- ---------------------------------- ------------------------------------------------------- -----------------
Type $\mathfrak{g}_{\mathbb{C}}$ $\mathfrak{l}_{i\mathbb{C}}$ $i$
$A_n$ $\mathfrak{sl}(n+1,\mathbb{C})$ $\mathfrak{sl}(i,\mathbb{C}) $i=1,2,\dots,n$
+ \mathfrak{sl}(n+1-i,\mathbb{C}) + \mathbb{C}$
$B_n$ $\mathfrak{so}(2n+1,\mathbb{C})$ $\mathfrak{so}(2n-1,\mathbb{C})+\mathbb{C}$ $ i=1 $
$C_n$ $\mathfrak{sp}(n,\mathbb{C})$ $\mathfrak{gl}(n,\mathbb{C})$ $ i=n $
$D_n$ $\mathfrak{so}(2n,\mathbb{C})$ $\mathfrak{so}(2n-2,\mathbb{C}) $ i=1 $
+ \mathbb{C} $
$ \mathfrak{so}(2n,\mathbb{C})$ $\mathfrak{gl}(n,\mathbb{C}) $ $ i=n-1,n $
$E_6 $ $\mathfrak{e}_6$ $\mathfrak{so}(10,\mathbb{C}) + \mathbb{C}$ $ i=1,6 $
$E_7$ $\mathfrak{e}_7$ $\mathfrak{e}_6+\mathbb{C}$ $ i=1 $
-------- ---------------------------------- ------------------------------------------------------- -----------------
If $(\mathfrak{g}_\mathbb{C}, {\mathfrak{l}}_{i\mathbb{C}})$ is a symmetric pair, then $[{\mathfrak{n}}^{-}_{i\mathbb{C}}, {\mathfrak{n}}^{-}_{i\mathbb{C}}]
\subset {\mathfrak{n}}^{-}_{i\mathbb{C}} \cap {\mathfrak{l}}_{i\mathbb{C}} = \{0\}$, whence (ii) $\Rightarrow$ (i).
\[def:7.3.2\] We say the representation $\F{\mathfrak{g}_\mathbb{C}}{k \omega_i}$ $(k=0, 1,2,\dots)$ is of *pan type*, or a pan representation if $(\mathfrak{g}_\mathbb{C}, \alpha_i)$ satisfies one of (therefore, all of) the equivalent conditions of Lemma \[lem:7.3.1\]. Here, [**pan**]{} stands for a [**p**]{}arabolic subalgebra with [**a**]{}belian [**n**]{}ilradical.
Examples of pan representations {#subsec:7.4}
-------------------------------
\[ex:pangl\] Let $\mathfrak{g}_{\mathbb{C}} = \mathfrak{gl}(n,\mathbb{C})$ and $\lambda = (\lambda_1,\dots,\lambda_n) \in \mathbb{Z}^n$ with $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_n$. (This $\mathfrak{g}_{\mathbb{C}}$ is not a simple Lie algebra, but the above concept is defined similarly.) Then, $\pi_{\lambda}$ is of pan type if and only if $$\lambda_1 = \cdots = \lambda_i \ge \lambda_{i+1} = \cdots = \lambda_n$$ for some $i$ $(1 \le i \le n-1)$. Then, $(\mathfrak{l})_{i\mathbb{C}} \simeq
\mathfrak{gl}(i,\mathbb{C}) + \mathfrak{gl}(n-i,\mathbb{C})$.
In particular, the $k$th symmetric tensor representations $S^k (\mathbb{C}^n)$ $(k \in \mathbb{N})$ and the $k$th exterior representations $\Lambda^k (\mathbb{C}^n)$ $(0 \le k \le n)$ are examples of pan representations since their highest weights are given by $(k, 0, \dots, 0)$ and $(\underbrace{1,\dots,1}_{k}, \, \underbrace{0,\dots,0}_{n-k} )$, respectively.
S. Okada [@xokada] studied branching laws for a specific class of irreducible finite dimensional representations of classical Lie algebras, which he referred to as “rectangular-shaped representations”. The notion of “pan representations” is equivalent to that of rectangular-shaped representations for type $(A_n)$, $(B_n)$, and $(C_n)$. For type $(D_n)$, $\pi_{k\omega_{n-1}}, \pi_{k\omega_n}$ $(k \in \mathbb{N})$ are rectangular-shaped representations, while $\pi_{k\omega_1}$ $(k \in \mathbb{N})$ are not.
Reduction to rank condition {#subsec:7.5}
---------------------------
Suppose $(\mathfrak{g}_{\mathbb{C}}, \alpha_i)$ satisfies the equivalent conditions in Lemma \[lem:7.3.1\]. Let $\theta$ be the complex involutive automorphism of the Lie algebra $\mathfrak{g}_\mathbb{C}$ that defines the symmetric pair $(\mathfrak{g}_{\mathbb{C}},\mathfrak{l}_{i\mathbb{C}})$. We use the same letter $\theta$ to denote the corresponding holomorphic involution of a simply connected $G_\mathbb{C}$. We take a maximal compact subgroup $G_U$ of $G_\mathbb{C}$ such that $\theta G_U = G_U$. Then $K := G_U^\theta = G_U \cap L_{i \mathbb{C}}$ becomes a maximal compact subgroup of ${L}_{i\mathbb{C}}$.
Let $\tau$ be another complex involutive automorphism of $\mathfrak{g}_{\mathbb{C}}$, and $(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\mathbb{C}})$ the symmetric pair defined by $\tau$. We also use the same letter $\tau$ to denote its lift to $G_{\mathbb{C}}$. We recall from Subsection \[subsec:proofthmA\] the ‘twisted’ involution $\tau^g$ for $g \in G_{\mathbb{C}}$ is given by $$\tau^g(x) = g \tau(g^{-1} x g) g^{-1} \quad (x \in G_\mathbb{C}) \,
.$$
\[lem:7.5\] Let $(\theta, \tau)$ be as above. There exist an involutive automorphism $\sigma$ of $G_U$ and $g \in G_\mathbb{C}$ satisfying the following three conditions (by an abuse of notation, we write $\tau$ for $\tau^{g}$):
$\tau \mathfrak{g}_U = \mathfrak{g}_U$, $\sigma \theta = \theta \sigma$, $\sigma \tau = \tau \sigma$. \[eqn:7.5.1\]
The induced action of $\sigma$ on $G_U/K$ is anti-holomorphic. \[eqn:7.5.2\]
$(\mathfrak{g}_U)^{\sigma, -\tau, -\theta}$ contains a maximal abelian subspace in $(\mathfrak{g}_U)^{-\tau, -\theta}$. \[eqn:7.5.3\]
[2)]{}For any $x \in G_U/K$, there exists $h \in (G_U^\tau)_0$ such that $\sigma(x) = h \cdot x$. In particular, each $(G_U^\tau)_0$-orbit on $G_U/K$ is preserved by $\sigma$.
1\) See [@visiblesymm Lemma 4.1] for the proof. 2) The second statement follows from the first statement and a similar argument of Lemma \[lem:3.2\].
Proof of Theorem \[thm:E\] {#subsec:7.7}
--------------------------
We are now ready to complete the proof of Theorem \[thm:E\] in Section \[sec:1\].
Let $\pi = \F{\mathfrak{g}_\mathbb{C}}{k \omega_i}$ be a representation of pan type. As in Subsection \[subsec:7.2\], we consider the holomorphic line bundle $\mathcal{L}_{k\omega_i} \to G_{\mathbb{C}}/P_{i\mathbb{C}}^-$ and realize $\pi$ on the space of holomorphic sections $\mathcal{O}(\mathcal{L}_{k \omega_i})$. We fix a $G_U$-invariant inner product on $\mathcal{O}(\mathcal{L}_{k\omega_i})$. With notation as in Subsection \[subsec:7.5\], we have a diffeomorphism $$G_U/K \simeq G_{\mathbb{C}}/P_{i\mathbb{C}}^- \, ,$$ through which the holomorphic line bundle $\mathcal{L}_{k\omega_i} \to G_{\mathbb{C}}/P_{i\mathbb{C}}^-$ is naturally identified with the $G_U$-equivariant holomorphic line bundle $\mathcal{L} \to D$, where we set $\mathcal{L} := G_U \times_K \mathbb{C}_{k\omega_i}$ and $D := G_U/K$ (a compact Hermitian symmetric space).
Now, applying Lemma \[lem:7.5\], we take $\sigma$ and set $H := (G_U^\tau)_0$. We note that the complexification of the Lie algebra of $H$ is equal to $\mathfrak{h}_{\mathbb{C}}$ up to a conjugation by $G_{\mathbb{C}}$. By Lemma \[lem:7.5\], the condition in Theorem \[thm:2.2\] is satisfied. Furthermore, we see the condition holds by a similar argument of Lemma \[lem:9.6\]. Therefore, the restriction $\pi|_{(G_U)^\tau_0}$ is multiplicity-free by Theorem \[thm:2.2\]. Hence, Theorem \[thm:E\] holds by Weyl’s unitary trick.
Proof of Theorem \[thm:F\] {#subsec:7.8}
--------------------------
Suppose $\pi_1$ and $\pi_2$ are representations of pan type. We realize $\pi_1$ and $\pi_2$ on the space of holomorphic sections of holomorphic line bundles over compact symmetric spaces $G_U/K_1$ and $G_U/K_2$, respectively. We write $\theta_i$ for the corresponding involutive automorphisms of $G_U$ that define $K_i$ $(i=1,2)$. In light of Lemma \[lem:7.3.1\] (iv), we can assume that $\theta_1 \theta_2 = \theta_2 \theta_1$. Then, applying Lemma \[lem:7.5\] to $(\theta_1, \theta_2)$ we find an involution $\sigma' \in {\operatorname{{Aut}}}(G_U)$ satisfying the following three conditions:
$\sigma' \theta_i = \theta_i \sigma'$ $(i =1, 2)$. \[eqn:7.8.1\]
The induced action of $\sigma'$ on $G_U/K_i$ $(i=1,2)$ is anti-holomorphic. \[eqn:7.8.2\]
$(\mathfrak{g}_U)^{\sigma', -\theta_1, -\theta_2}$ contains a maximal abelian subspace of $(\mathfrak{g}_U)^{-\theta_1, -\theta_2}$. \[eqn:7.8.3\]
We remark that the condition for $i=2$ is not included in Lemma \[lem:7.5\], but follows automatically by our choice of $\sigma$.
We define three involutive automorphisms $\tau$, $\theta$ and $\sigma$ on $G_U \times G_U$ by $\tau(g_1, g_2) := (g_2, g_1)$, $\theta:=(\theta_1, \theta_2)$ and $\sigma := (\sigma', \sigma')$, respectively. Then $(G_U \times G_U)^\tau = {\operatorname{diag}}(G_U)$. By using the identification $$(\mathfrak{g}_U \oplus \mathfrak{g}_U)^{-\tau}
= \set{(X, -X)}{X \in \mathfrak{g}_U}
\overset{\sim}{\to} \mathfrak{g}_U \, ,
\quad (X, -X) \mapsto X \, ,$$ we have isomorphisms $$\begin{aligned}
(\mathfrak{g}_U \oplus \mathfrak{g}_U)^{-\tau, -\theta}
&\simeq (\mathfrak{g}_U)^{-\theta_1, -\theta_2} \, ,
\\
(\mathfrak{g}_U \oplus \mathfrak{g}_U)^{\sigma, -\tau, -\theta}
&\simeq (\mathfrak{g}_U)^{\sigma', -\theta_1, -\theta_2} \, .\end{aligned}$$ Thus, the condition implies that $(\mathfrak{g}_U \oplus \mathfrak{g}_U)^{\sigma, -\tau, -\theta}$ contains a maximal abelian subspace of $(\mathfrak{g}_U \oplus \mathfrak{g}_U)^{-\tau, -\theta}$. Then, by Lemma \[lem:7.5\] and by a similar argument of Lemma \[lem:3.2\] again, for any $(x, y) \in G_U/K_1 \times G_U/K_2$ there exists a $g \in G_U$ such that $\sigma'(x) = g\cdot x$ and $\sigma'(y) = g \cdot y$ simultaneously. Now, Theorem \[thm:F\] follows readily from Theorem \[thm:2.2\].
List of multiplicity-free restrictions {#subsec:7.9}
--------------------------------------
For the convenience of the reader, we present the list of the triple $(\mathfrak{g}_\mathbb{C}, \mathfrak{h}_\mathbb{C}, i)$ for which we can conclude from Theorem \[thm:E\] that the irreducible finite dimensional representation $\hwm{\mathfrak{g}_\mathbb{C}}{k \omega_i}$ of a simple Lie algebra $\mathfrak{g}_\mathbb{C}$ is multiplicity-free when restricted to $\mathfrak{h}_\mathbb{C}$ for any $k \in \mathbb{N}$ by Theorem \[thm:E\].
$$\vbox{
\offinterlineskip
\def\tablerule{\noalign{\hrule}}
\halign{\strut#&\vrule#&
\;\;\hfil#\hfil\hfil\,&\vrule#&
\;\;\hfil#\hfil\hfil\,&\vrule#&
\hfil#\hfil\hfil\,&\vrule#&
\;\;\hfil#\hfil\hfil\,&\vrule#\cr\tablerule
&& ${\mathfrak{g}_\mathbb{C}}$ && $\mathfrak{h}_\mathbb{C}$
&&&& $i$
&\cr\tablerule
&& $\mathfrak{sl}(n+1,\mathbb{C})$
&& $\mathfrak{sl}(p,\mathbb{C}) + \mathfrak{sl}(n+1-p, \mathbb{C}) + \mathbb{C}$
&&&& $1, 2, \dots, n$
&\cr\tablerule
&& $\mathfrak{sl}(n+1,\mathbb{C})$ && $\mathfrak{so}(n+1,\mathbb{C})$
&&&& $1, 2, \dots, n$ &\cr\tablerule
&& $\mathfrak{sl}(2m,\mathbb{C})$ && $\mathfrak{sp}(m,\mathbb{C})$
&&&& $1, 2, \dots, 2m-1$ &\cr\tablerule
&& $\mathfrak{so}(2n+1,\mathbb{C})$ && $\mathfrak{so}(p,\mathbb{C})+\mathfrak{so}(2n+1-p, \mathbb{C})$
&&&& $1$ &\cr\tablerule
&& $\mathfrak {sp}(n,\mathbb{C})$ && $\mathfrak{sp}(p,\mathbb{C}) + \mathfrak{sp}(n-p, \mathbb{C})$
&&&& $n$ &\cr\tablerule
&& $\mathfrak {sp}(n,\mathbb{C})$ && $\mathfrak{gl}(n,\mathbb{C})$
&&&& $n$ &\cr\tablerule
&& $\mathfrak {so}(2n,\mathbb{C})$ && $\mathfrak{so}(p,\mathbb{C}) + \mathfrak{so}(2n-p,\mathbb{C})$
&&&& $1, n-1, n$ &\cr\tablerule
&& $\mathfrak {so}(2n,\mathbb{C})$ && $\mathfrak{gl}(n,\mathbb{C})$
&&&& $1, n-1, n$ &\cr\tablerule
&& $\mathfrak {e}_6$ && $\mathfrak{so}(10, \mathbb{C})+\mathfrak{so}(2, \mathbb{C})$
&&&& $1, 6$ &\cr\tablerule
&& $\mathfrak {e}_6$ && $\mathfrak{sl}(6, \mathbb{C})+\mathfrak{sl}(2, \mathbb{C})$
&&&& $1, 6$ &\cr\tablerule
&& $\mathfrak {e}_6$ && $\mathfrak{f}_4$
&&&& $1, 6$ &\cr\tablerule
&& $\mathfrak {e}_6$ && $\mathfrak{sp}(4, \mathbb{C})$
&&&& $1, 6$ &\cr\tablerule
&& $\mathfrak {e}_7$ && $\mathfrak{e}_6+\mathfrak{so}(2, \mathbb{C})$
&&&& $7$ &\cr\tablerule
&& $\mathfrak {e}_7$ && $\mathfrak{so}(12, \mathbb{C})+\mathfrak{sl}(2, \mathbb{C})$
&&&& $7$ &\cr\tablerule
&& $\mathfrak {e}_7$ && $\mathfrak{sl}(8, \mathbb{C})$
&&&& $7$ &\cr\tablerule
\noalign{\smallskip}
\cr}}$$
Some of the above cases were previously known to be multiplicity-free by case-by-case argument, in particular, for the case $\operatorname{rank} \mathfrak{g}_{\mathbb{C}} =
\operatorname{rank} \mathfrak{h}_{\mathbb{C}}$. Among them, the corresponding explicit branching laws have been studied by S. Okada [@xokada] and H. Alikawa [@xalikawa]. There are some few representations $\pi$ that are not of pan type, but are multiplicity-free when restricted to symmetric subgroups $H$. Our method still works to capture such cases, but we do not go into details here (see [@xkleiden; @xkgencar; @xksovisible]).
Generalization of the Hua–Kostant–Schmid Formula {#sec:8}
================================================
This section discusses an explicit irreducible decomposition formula of the restriction $\pi|_H$ where the triple $(\pi,G,H)$ satisfies the following two conditions:
1\) $\pi$ is a holomorphic discrete series representation of scalar type (Definition \[def:1.4\]).
2\) $(G,H)$ is a symmetric pair defined by an involution $\tau$ of holomorphic type (Definition \[def:holo-anti\]).
We know a priori from Theorem \[thm:B\] (1) that the branching law is discrete and multiplicity-free. The main result of this section is Theorem \[thm:gHKS\], which enriches this abstract property with an explicit multiplicity-free formula. The formula for the special case $H = K$ corresponds to the Hua–Kostant–Schmid formula ([@xhua; @xjohnson; @xschmidherm]). We also present explicit formulas for the irreducible decomposition of the tensor product representation (Theorem \[thm:tensordeco\]) and of the restriction $U(p,q) \downarrow U(p-1,q)$ (Theorem \[thm:upqupq\]).
Let us give a few comments on our proof of Theorem \[thm:gHKS\]. Algebraically, our key machinery is Lemma \[lem:HnG\] which assures that the irreducible $G$-decomposition is determined only by its $K$-structure. Geometrically, a well-known method of taking normal derivatives (e.g. S. Martens [@xmartens], Jakobsen–Vergne [@xjv]) gives a general algorithm to obtain branching laws for highest weight modules. This algorithm yields explicit formulae by using the observation that the fiber of the normal bundle for $G^\tau/K^\tau \subset G/K$ is the tangent space of another Hermitian symmetric space $G^{\tau \theta}/K^\tau$. The key ingredient of the geometry here is the following nice properties of the two symmetric pairs $(G, G^\tau)$ and $(G, G^{\tau\theta})$:
a\) $K \cap G^\tau = K \cap G^{\tau\theta}$,
b\) $\mathfrak{p} = (\mathfrak{p} \cap \mathfrak{g}^\tau)
\oplus (\mathfrak{p} \cap \mathfrak{g}^{\tau\theta})$.
Unless otherwise mentioned, we shall assume $H$ is connected, that is, $H=G_0^{\tau}$ throughout this section.
Notation for highest weight modules {#subsec:8.1}
-----------------------------------
We set up the notation and give a parametrization of irreducible highest weight modules for both finite and infinite dimensional cases.
First, we consider finite dimensional representations. Let us take a Cartan subalgebra $\mathfrak{t}$ of a reductive Lie algebra $\mathfrak{k}$ and fix a positive system ${\Delta}^+(\mathfrak{k}, \mathfrak{t})$. We denote by $\hwm{\mathfrak{k}}{\mu}$ the irreducible finite dimensional representation of $\mathfrak{k}$ with highest weight $\mu$, if $\mu$ is a dominant integral weight. A $\mathfrak{k}$-module $\hwm{\mathfrak{k}}{\mu}$ will be written also as $\hwm{K}{\mu}$ if the action lifts to $K$.
Next, let $G$ be a connected reductive Lie group, $\theta$ a Cartan involution, $K = \{ g \in G: \theta g = g \}$, ${\mathfrak {g}}={\mathfrak {k}}+ {\mathfrak {p}}$ the corresponding Cartan decomposition and $\mathfrak{g}_{\mathbb{C}} = \mathfrak{k}_{\mathbb{C}}
+ \mathfrak{p}_{\mathbb{C}}$ its complexification. We assume that there exists a central element $Z$ of ${\mathfrak {k}}$ such that $$\label{eqn:gkpp}
{\mathfrak {g}}_{\mathbb{C}}= {\mathfrak {k}}_{\mathbb{C}}+{\mathfrak {p}}_+ + {\mathfrak {p}}_-$$ is the eigenspace decomposition of $\frac 1{\sqrt{-1}} \operatorname{ad} (Z)$ with eigenvalues 0, 1, and $-1$, respectively. This assumption is satisfied if and only if $G$ is locally isomorphic to a direct product of connected compact Lie groups and non-compact Lie groups of Hermitian type (if $G$ is compact, we can simply take $Z=0$). We set $$\label{eqn:sqrtZ}
\widetilde{Z} := \frac{1}{\sqrt{-1}} Z \, .$$
As in Definition \[def:1.4\], we say an irreducible $(\mathfrak{g}_{\mathbb{C}},K)$-module $V$ is a *highest weight module* if $$V^{\mathfrak{p}_+} = \{ v \in V:
Yv = 0 \quad\mbox{for all $Y \in \mathfrak{p}_+$} \}$$ is non-zero. Then, $V^{\mathfrak{p}_+}$ is irreducible as a $K$-module, and the $(\mathfrak{g}_{\mathbb{C}},K)$-module $V$ is determined uniquely by the $K$-structure on $V^{\mathfrak{p}_+}$. If $\mu$ is the highest weight of $V^{\mathfrak{p_+}}$, we write $V$ as $\hwm{\mathfrak{g}}{\mu}$. That is, the irreducible $(\mathfrak{g}_{\mathbb{C}},K)$-module $\hwm{\mathfrak{g}}{\mu}$ is characterized by the $K$-isomorphism: $$\label{eqn:pigpik}
(\hwm{\mathfrak{g}}{\mu})^{\mathfrak{p}_+}
\simeq \hwm{\mathfrak{k}}{\mu} \, .$$ An irreducible unitary highest weight representation $\pi$ of $G$ will be denoted by $\hwm{G}{\mu}$ if the underlying $(\mathfrak{g}_{\mathbb{C}},K)$-module of $\pi$ is isomorphic to $\hwm{\mathfrak{g}}{\mu}$. Let $\Lambda_G$ be the totality of $\mu$ such that $\hwm{\mathfrak{g}}{\mu}$ lifts to an irreducible unitary representation of $G$. For simply connected $G$, irreducible unitary highest weight representations were classified, that is, the set $\Lambda_G$ $(\subset \sqrt{-1} \mathfrak{t}^*)$ was explicitly found in [@xhew] and [@xjak] (see also [@xej]). In particular, we recall from [@xhew] that $$\lambda(\widetilde{Z}) \in \mathbb{R}
\quad\text{for any $\lambda \in \Lambda_G$}$$ and $$\label{eqn:ubdlmd}
c_G := \sup_{\lambda\in\Lambda_G} \lambda(\widetilde{Z})
< \infty$$ if $G$ is semisimple.
The highest weight module $\hwm{\mathfrak{g}}{\mu}$ is the unique quotient of the generalized Verma module $$\label{eqn:Verma}
N^{\mathfrak{g}} (\mu) := U(\mathfrak{g}_{\mathbb{C}})
\otimes_{U(\mathfrak{k}_{\mathbb{C}} + \mathfrak{p}_+)}
\hwm{\mathfrak{k}}{\mu} \, ,$$ where $\hwm{\mathfrak{k}}{\mu}$ is regarded as a module of the maximal parabolic subalgebra $\mathfrak{k}_{\mathbb{C}} + \mathfrak{p}_+$ by making $\mathfrak{p}_+$ act trivially. Furthermore, $\hwm{\mathfrak{g}}{\mu}$ has a $Z(\mathfrak{g}_\mathbb{C})$-infinitesimal character $\mu + \rho_\mathfrak{g} \in \mathfrak{t}^*_{\mathbb{C}}$ via the Harish-Chandra isomorphism $${\operatorname{Hom}}_{\text{$\mathbb{C}$-algebra}}
(Z(\mathfrak{g}_{\mathbb{C}}),\mathbb{C})
\simeq \mathfrak{t}_{\mathbb{C}}^* / W \, ,$$ where $Z(\mathfrak{g}_{\mathbb{C}})$ is the center of the enveloping algebra $U(\mathfrak{g}_{\mathbb{C}})$, $W$ is the Weyl group of the root system $\Delta(\mathfrak{g},\mathfrak{t})$, and $\rho_{\mathfrak{g}}$ is half the sum of positive roots ${\Delta}^+(\mathfrak{g}, \mathfrak{t})
:= {\Delta}^+(\mathfrak{k}, \mathfrak{t}) \cup {\Delta}(\mathfrak{p}_+,
\mathfrak{t})$.
Strongly orthogonal roots {#subsec:8.2}
-------------------------
Let $G$ be a non-compact simple Lie group of Hermitian type, and $\tau$ an involution of holomorphic type which commutes with the Cartan involution $\theta$.
We take a Cartan subalgebra $\mathfrak{t}^\tau$ of the reductive Lie algebra $$\mathfrak{k}^\tau := \set{X \in \mathfrak{k}}{\tau X = X}$$ and extend it to a Cartan subalgebra $\mathfrak{t}$ of $\mathfrak{k}$. We note that $\mathfrak{t}^\tau = \mathfrak{k}^\tau \cap \mathfrak{t}$. The pair $(\mathfrak{k}, \mathfrak{k}^\tau)$ forms a reductive symmetric pair, and $\mathfrak{t}$ plays an analogous role to the fundamental Cartan subalgebra with respect to this symmetric pair. Thus, using the same argument as in [@xvalg], we see that if $\alpha \in \Delta(\mathfrak{k},\mathfrak{t})$ satisfies $\alpha|_{\mathfrak{t}^\tau} = 0$ then $\alpha=0$. Thus, we can take positive systems ${\Delta}^+(\mathfrak{k}, \mathfrak{t})$ and ${\Delta}^+(\mathfrak{k}^\tau, \mathfrak{t}^\tau)$ in a compatible way such that $$\alpha|_{\mathfrak{t}^\tau}
\in {\Delta}^+(\mathfrak{k}^\tau, \mathfrak{t}^\tau)
\quad\text{if $\alpha \in {\Delta}^+(\mathfrak{k}, \mathfrak{t})$} \, .
\label{eqn:8.2.1}$$ Since $\tau$ is of holomorphic type, we have $\tau Z = Z$, and therefore $\tau \mathfrak{p}_+ = \mathfrak{p}_+$. Hence, we have a direct sum decomposition $\mathfrak{p}_+ = \mathfrak{p}_+^\tau \oplus \mathfrak{p}_+^{-\tau}$, where we set $$\mathfrak{p}_+^{\pm\tau} := \set{X \in \mathfrak{p}_+}{\tau X = \pm X}
\, .$$
Let us consider the reductive subalgebra $\mathfrak{g}^{\tau\theta}$. Its Cartan decomposition is given by $$\mathfrak{g}^{\tau\theta} =
(\mathfrak{g}^{\tau\theta} \cap \mathfrak{g}^\theta)
+ (\mathfrak{g}^{\tau\theta} \cap \mathfrak{g}^{-\theta})
= \mathfrak{k}^\tau + \mathfrak{p}^{-\tau} \, ,$$ and its complexification is given by $$\label{eqn:8.2.2}
\mathfrak{g}_{\mathbb{C}}^{\tau\theta}
= \mathfrak{k}_{\mathbb{C}}^\tau \oplus
\mathfrak{p}_+^{-\tau} \oplus \mathfrak{p}_-^{-\tau} \, .$$ The Cartan subalgebra $\mathfrak{t}^\tau$ of $\mathfrak{k}^\tau$ is also a Cartan subalgebra of $\mathfrak{g}^{\tau\theta}$.
Let $
{\Delta}(\mathfrak{p}_+^{-\tau}, \mathfrak{t}^\tau)
$ be the set of weights of $\mathfrak{p}_+^{-\tau}$ with respect to $\mathfrak{t}^\tau$. The roots $\alpha$ and $\beta$ are said to be [*strongly orthogonal*]{} if neither $\alpha + \beta$ nor $\alpha - \beta$ is a root. We take a maximal set of strongly orthogonal roots $\{ \nu_1, \nu_2, \dots, \nu_l \}$ in ${\Delta}(\mathfrak{p}_+^{-\tau}, \mathfrak{t}^\tau)$ such that i) $\nu_1$ is the lowest root among the elements in $
{\Delta}(\mathfrak{p}_+^{-\tau}, \mathfrak{t}^\tau),
$ ii) $\nu_{j+1}$ is the lowest root among the elements in $
{\Delta}(\mathfrak{p}_+^{-\tau}, \mathfrak{t}^\tau)
$ that are strongly orthogonal to $\nu_1, \dots, \nu_j$.
A special case applied to $\tau = \theta$ shows $\mathfrak{k}^\tau = \mathfrak{k}$, $ \mathfrak{t}^\tau = \mathfrak{t}$, $ \mathfrak{p}^{-\tau} = \mathfrak{p}$, and $\Delta(\mathfrak{p}_+^{-\tau},\mathfrak{t}^\tau)
= \Delta(\mathfrak{p}_+,\mathfrak{t})$. In this case, we shall use the notation $\{ {\bar{\nu}}_1, {\bar{\nu}}_2,\dots,{\bar{\nu}}_{\bar{l}} \}$ for a maximal set of strongly orthogonal roots in $\Delta (\mathfrak{p}_+, \mathfrak{t})$ such that
1. ${\bar{\nu}}_1$ is the lowest root among $\Delta(\mathfrak{p}_+,\mathfrak{t})$,
2. ${\bar{\nu}}_{j+1}$ is the lowest root among the elements in $\Delta(\mathfrak{p}_+,\mathfrak{t})$ that are strongly orthogonal to ${\bar{\nu}}_1,\dots,{\bar{\nu}}_j$ $(1 \le j \le {\bar{l}})$.
Then, $\bar{l} = {\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}$ by [@xkwolf]. Likewise, in light of for the Hermitian symmetric space $G^{\tau\theta} / G^{\tau\theta} \cap K =
G^{\tau\theta} / G^{\tau,\theta}$, we have $l = {\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}^{\tau \theta}$. In general, $l \le \bar{l}$.
Branching laws for semisimple symmetric pairs {#subsec:8.5}
---------------------------------------------
It follows from that the highest weight module $\hwm{\mathfrak{g}}{\mu}$ is of scalar type, namely, $(\hwm{\mathfrak{g}}{\mu})^{\mathfrak{p}_+}$ is one dimensional, if and only if $$\langle \mu, \alpha \rangle = 0
\qquad
\text{ for any }
\alpha \in {\Delta}(\mathfrak{k}, \mathfrak{t}) \, .
\label{eqn:8.5.1}$$ Furthermore, the representation $\hwm{G}{\mu}$ is a (relative) holomorphic discrete series representation of $G$ if and only if $$\langle \mu + \rho_\mathfrak{g}, \alpha \rangle < 0
\qquad
\text{ for any }
\alpha \in {\Delta}(\mathfrak{p}_+, \mathfrak{t}) \, .
\label{eqn:8.5.2}$$
We are now ready to state the branching law of holomorphic discrete series representations $\hwm{G}{\mu}$ of scalar type with respect to semisimple symmetric pairs $(G,H)$:
\[thm:gHKS\] Let $G$ be a non-compact simple Lie group of Hermitian type. Assume that $\mu \in \sqrt{-1}\, \mathfrak{t}^*$ satisfies and . Let $\tau$ be an involutive automorphism of $G$ of holomorphic type, $H=G_0^\tau$ (the identity component of $G^\tau$), and $\{\nu_1,\dots,\nu_l \}$ be the set of strongly orthogonal roots in $\Delta(\mathfrak{p}_+^{-\tau}, \mathfrak{t}^\tau)$ as in Subsection \[subsec:8.2\]. Then, $\hwm{G}{\mu}$ decomposes discretely into a multiplicity-free sum of irreducible $H$-modules: $$\hwm{G}{\mu} |_H
\simeq
\sideset{}{^\oplus}\sum_{
\Sb a_1 \ge \dots \ge a_l \ge 0\\ a_1, \dots, a_l \in \mathbb{N} \endSb}
\hwm{H}{\mu|_{\mathfrak{t}^\tau}- \sum_{j=1}^l a_j \nu_j}
\quad
\text{(discrete Hilbert sum)}.
\label{eqn:8.5.3}$$
The formula for the case $H=K$ (that is, $\tau=\theta$) was previously found by L.-K. Hua (implicit in the classical case), B. Kostant (unpublished) and W. Schmid [@xschmidherm] (see also Johnson [@xjohnson] for an algebraic proof). In this case, each summand in the right side is finite dimensional.
For $\tau \ne \theta$, some special cases have been also studied by H. Jakobsen, M. Vergne, J. Xie, W. Bertram and J. Hilgert [@xbehi; @xjak; @xjv; @xxie]. Further, quantitative results by means of reproducing kernels are obtained in [@xsaid]. The formula in the above generality was first given by the author [@xkmfjp].
We shall give a proof of Theorem \[thm:gHKS\] in Subsection \[subsec:8.3\].
Irreducible decomposition of tensor products {#subsec:8.6}
--------------------------------------------
As we saw in Example \[ex:gpmfd\], the pair $(G \times G, {\operatorname{diag}}(G))$ forms a symmetric pair. Correspondingly, the tensor product representation can be regarded as a special (and easy) case of restrictions of representations with respect to symmetric pairs. This subsection provides a decomposition formula of the tensor product of two holomorphic discrete series representations of scalar type. This is regarded as a counterpart of Theorem \[thm:gHKS\] for tensor product representations.
We recall from Subsection \[subsec:8.2\] that $\{\bar{\nu}_1,\dots,\bar{\nu}_{\bar{l}}\}$ is a maximal set of strongly orthogonal roots in $\Delta(\mathfrak{p}_+,\mathfrak{t})$ and $\bar{l}={\mathbb{R}\text{-}\operatorname{rank}}\mathfrak{g}$.
\[thm:tensordeco\] Let $G$ be a non-compact simple Lie group of Hermitian type. Assume that $\mu_1, \mu_2 \in \sqrt{-1}\, \mathfrak{t}^*$ satisfy the conditions and . Then, the tensor product representation $\hwm{G}{\mu_1} \widehat{\otimes} \hwm{G}{\mu_2}$ decomposes discretely into a multiplicity-free sum of irreducible $G$-modules: $$\hwm{G}{\mu_1} \widehat{\otimes} \hwm{G}{\mu_2}
\simeq \sum_{\substack{a_1 \ge\dots\ge a_{\bar{l}} \ge 0 \\
a_1,\dots,a_{\bar{l}} \,\in\, \mathbb{N}}}
\hwm{G}{\mu_1 + \mu_2 - \sum_{j=1}^{\bar{l}} a_j {\bar{\nu}}_j}.$$
The proof of Theorem \[thm:tensordeco\] will be given in Subsection \[subsec:pf tensordeco\].
Eigenvalues of the central element $Z$ {#subsec:Spec}
--------------------------------------
Our proof of Theorems \[thm:gHKS\] and \[thm:tensordeco\] depends on the algebraic lemma that the $K$-type formula determines the irreducible decomposition of the whole group (see Lemma \[lem:HnG\]). This is a very strong assertion, which fails in general for non-highest weight modules. This subsection collects some nice properties peculiar to highest weight modules that will be used in the proof of Lemma \[lem:HnG\].
For a $K$-module $V$, we define a subset of $\mathbb C$ by $$\operatorname{Spec}_{\widetilde{Z}}(V):=\{\text{eigenvalues of
$\widetilde Z$ on $V$}\} \, ,$$ where we set $$\widetilde Z:=\frac {1}{\sqrt{-1}} Z \, .$$ For instance, $\operatorname{Spec}_{\widetilde{Z}}(V)$ is a singleton if $V$ is an irreducible $K$-module. We also note that $\operatorname{Spec}_{\widetilde Z}
({\mathfrak {g}}_{\mathbb{C}})=\{0, \pm 1\}$ by .
\[lem:spec\] Suppose $V$ is an irreducible [$(\mathfrak{g}_\mathbb{C}, K)$]{}-module. Then, $\operatorname{Spec}_{\widetilde{Z}}(V)\subset a_0 + {\mathbb{Z}}$ for some $a_0 \in {\mathbb{C}}$. If $\sup \operatorname{Re} \operatorname{Spec}_{\widetilde{Z}}(V)< \infty$, then $V$ is a highest weight module. If $V$ is a highest weight module $\hwm {\mathfrak g} {\lambda}$, then $\operatorname{Spec}_{\widetilde{Z}}(V)
\subset -\mathbb{N} + \lambda(\widetilde Z)$ and $\sup\operatorname{Re} \operatorname{Spec}_{\widetilde Z}(V)
= \operatorname{Re}\lambda(\widetilde Z)$. If $V$ is a unitary highest weight module, then $\operatorname{Spec}_{\widetilde Z}(V) \subset (-\infty, c_G]$, where $c_G$ is a constant depending on $G$. If both $V$ and $F$ are highest weight modules of finite length, then any irreducible subquotient $W$ of $V \otimes F$ is also a highest weight module.
1)For $a \in {\mathbb{C}}$, we write the eigenspace of $\widetilde Z$ as $
V_{a}:=\set{v \in V}{\widetilde Z v = a v}
$. Then, it follows from the Leibniz rule that $${\mathfrak {p}}_+ V_{a} \subset V_{a+1} \, , \quad
{\mathfrak {k}}_{\mathbb{C}} V_{a} \subset V_{a} \, ,
\quad\text{and}\quad
{\mathfrak {p}}_- V_{a} \subset V_{a-1} \, .$$ An iteration of this argument shows that $$\operatorname{Spec}_{\widetilde{Z}}(U({\mathfrak {g}}_{\mathbb{C}})V_{a})
\subset a + {\mathbb{Z}} \, .$$ Now we take $a_0$ such that $V_{a_0}\ne \{0\}$. Since $V$ is irreducible, we have $V=U(\mathfrak g_{\mathbb C})V_{a_0}$, and therefore $\operatorname{Spec}_{\widetilde Z}(V) \subset a_0 + \mathbb Z$. 2)Suppose $\sup \operatorname{Re}\operatorname{Spec}_{\widetilde Z}(V) < \infty$. Since $\operatorname{Re}\operatorname{Spec}_{\widetilde Z}(V)$ is discrete by (1), there exists $a \in \operatorname{Spec}_{\widetilde{Z}}(V)$ such that $\operatorname{Re}a$ attains its maximum. Then $${\mathfrak {p}}_+ V_{a} \subset V_{a+1}=\{0\} \, .$$ Thus, $V_{a} \subset V^{{\mathfrak{p}}_+}$. Hence, $V$ is a highest weight module. 3)The highest weight module $\hwm{\mathfrak {g}}{\lambda}$ is isomorphic to the unique irreducible quotient of the generalized Verma module $N^{\mathfrak {g}}(\lambda)
=U(\mathfrak g_{\mathbb C})
\otimes _{U(\mathfrak k_{\mathbb{C}} + {\mathfrak p}_+)}\hwm{\mathfrak k}{\lambda}$. By the Poincaré–Birkhoff–Witt theorem, $N^{\mathfrak{g}}(\lambda)$ is isomorphic to $
S(\mathfrak{p}_-) \otimes \pi_\lambda^{\mathfrak{k}}
$ as a $\mathfrak{k}$-module. Thus, any $\mathfrak k$-type $\hwm{\mathfrak k}{\mu}$ occurring in $\hwm{\mathfrak g}{\lambda}$ is of the form $$\mu = \lambda+ \sum_{\alpha \in \Delta({\mathfrak p}_-, {\mathfrak t})}m_{\alpha} \alpha$$ for some $m_{\alpha} \in {\mathbb{N}}$. As $\alpha(\widetilde Z)=-1$ for any $\alpha \in \Delta(\mathfrak p_-, \mathfrak t)$, we have $$\label{eqn:eigenKG}
\mu(\widetilde Z)=\lambda(\widetilde Z)-
\sum_{\alpha\in\Delta(\mathfrak{p}_-,\mathfrak{t})} m_{\alpha} \, .$$ In particular, we have the following equivalence: $$\label{eqn:Rekg}
\operatorname{Re}\mu(\widetilde Z)=\operatorname{Re}\lambda(\widetilde Z)
\ \Longleftrightarrow\ \mu=\lambda \, ,$$ and we also have $$\label{eqn:Ngl}
\operatorname{Spec}_{\widetilde Z}(\hwm{\mathfrak g}{\lambda})
\subset
\set{\lambda(\widetilde Z)-
\sum_{\alpha\in\Delta(\mathfrak{p}_-,\mathfrak{t})}
m_{\alpha}}{m_{\alpha} \in {\mathbb{N}}}
=-{\mathbb N} + \lambda(\widetilde Z) \, .$$ Furthermore, since the $\mathfrak{k}$-type $\hwm{\mathfrak k} {\lambda}$ occurs in $\hwm{\mathfrak g}{\lambda}$, we have $\lambda(\widetilde{Z}) \in
\operatorname{Spec}_{\widetilde Z}(\hwm{\mathfrak g}{\lambda})$. Here, $
\sup \operatorname{Re}\operatorname{Spec}_{\widetilde Z}(\hwm{\mathfrak g}{\lambda})
= \operatorname{Re} \lambda(\widetilde Z)
$. 4)This statement follows from and from (3). 5)For two subsets $A$ and $B$ in $\mathbb C$, we write $A+B:=\set{a+b \in \mathbb C}{a \in A, b \in B}$. Then, $
\operatorname{Spec}_{\widetilde Z}(V \otimes F)
\subset
\operatorname{Spec}_{\widetilde Z}(V)
+
\operatorname{Spec}_{\widetilde Z}(F)\, .
$ Therefore, $$\begin{aligned}
\sup \operatorname{Re} \operatorname{Spec}_{\widetilde Z}(W)
& \le
\sup \operatorname{Re}\operatorname{Spec}_{\widetilde Z} (V)
\\
& \le
\sup \operatorname{Re} \operatorname{Spec}_{\widetilde Z}(V)
+
\sup \operatorname{Re} \operatorname{Spec}_{\widetilde Z}(F)
< \infty.\end{aligned}$$ Hence, $W$ is also a highest weight module by (2).
Bottom layer map {#subsec:ktog}
----------------
The following lemma finds an irreducible summand (‘bottom layer’) from the $K$-type structure.
\[lem:ktog\] Let $V$ be a [$(\mathfrak{g}_\mathbb{C}, K)$]{}-module. We assume that $V$ decomposes into an algebraic direct sum of (possibly, infinitely many) irreducible highest weight modules. We set $$\operatorname{Supp}_{\mathfrak k}(V)
:=\set{\mu \in \sqrt{-1}\mathfrak t^*}
{{\operatorname{Hom}}_{\mathfrak k}(\hwm{\mathfrak k}{\mu}, V)\ne \{0\}} \, .$$ If the evaluation map $$\operatorname{Supp}_{\mathfrak k}(V) \to \mathbb{R}\, , \quad
\mu \mapsto \operatorname{Re} \mu(\widetilde Z)$$ attains its maximum at $\mu_0$, then $${\operatorname{Hom}}_{(\mathfrak g_{\mathbb C}, K)}(\hwm{\mathfrak g}{\mu_0},
V)\ne\{0\} \, .$$
Take a non-zero map $q \in {\operatorname{Hom}}_{\mathfrak k}(\hwm{\mathfrak k}{\mu_0}, V)$. As $V$ is an algebraic direct sum of irreducible highest weight modules, there exists a projection $p:V \to \hwm{\mathfrak g}{\lambda}$ for some $\lambda$ such that $p \circ q \ne 0$. This means that $\hwm{\mathfrak k}{\mu_0}$ occurs in $\hwm{\mathfrak g}{\lambda}$, and therefore we have $$\operatorname{Re}\mu_0(\widetilde Z)
\le \sup \operatorname{Re}\operatorname{Spec}_{\widetilde Z}(\hwm{\mathfrak g}{\lambda})
=\operatorname{Re}\lambda(\widetilde Z)
\, .$$ Here, the last equality is by Lemma \[lem:spec\] (3).
Conversely, the maximality of $\mu_0$ implies that $\operatorname{Re}\mu_0(\widetilde Z)\ge \operatorname{Re}\lambda(\widetilde Z)$. Hence, $\operatorname{Re} \mu_0 (\widetilde{Z})
= \operatorname{Re} \lambda(\widetilde{Z})$, and we have then $\mu_0 = \lambda$ by . Since $\hwm{\mathfrak{g}}{\lambda}$ is an irreducible summand of $V$, we have ${\operatorname{Hom}}_{(\mathfrak g_{\mathbb C}, K)}(\hwm{\mathfrak g}{\mu_0}, V) \ne
\{0\}$.
Determination of the $\mathfrak{g}_{\mathbb{C}}$-structure by $K$-types {#subsec:8.11}
-----------------------------------------------------------------------
In general, the $K$-type formula is not sufficient to determine the irreducible decomposition of a unitary representation even in the discretely decomposable case. However, this is the case if any irreducible summand is a highest weight module. Here is the statement that we shall use as a main machinery of the proof of Theorems \[thm:gHKS\] and \[thm:tensordeco\].
\[lem:HnG\] Suppose $(\pi, \mathcal H)$ is a $K$-admissible unitary representation of $G$, which splits discretely into a Hilbert direct sum of irreducible unitary highest weight representations of $G$. Let $\mathcal{H}_K$ be the space of $K$-finite vectors of $\mathcal{H}$. Assume that there exists a function $n_\pi: \mathfrak{t}_{\mathbb{C}}^* \to \mathbb{N}$ such that $\mathcal{H}_K$ is isomorphic to the following direct sum as $\mathfrak{k}$-modules: $$\label{eqn:Hnalg}
\mathcal H_K \simeq \bigoplus_{\lambda} n_{\pi}(\lambda) \hwm{\mathfrak g}{\lambda}
\quad
\text{(algebraic direct sum)}.$$ Then, $n_\pi(\lambda)\ne0$ only if $\lambda \in \Lambda_G$, that is, $\hwm{\mathfrak{g}}{\lambda}$ lifts to an irreducible unitary representation $\hwm{G}{\lambda}$ of $G$. Furthermore, the identity holds as a $(\mathfrak{g}_{\mathbb{C}},K)$-module isomorphism, and the unitary representation $\pi$ has the following decomposition into irreducible unitary representations of $G$: $$\label{eqn:HnG}
\pi \simeq {\sum_{\lambda}}^{\oplus} n_{\pi}(\lambda) \hwm G{\lambda}
\quad
\text{(discrete Hilbert sum)}.$$
We write an abstract irreducible decomposition of $\mathcal H$ as $$\mathcal H \simeq {\sum_{\lambda\in\Lambda_G}}^{\oplus} m_{\lambda} \hwm
G{\lambda} \quad\text{(discrete Hilbert sum)}.$$ Since $\mathcal{H}$ is $K$-admissible, the multiplicity $m_{\lambda}< \infty$ for all $\lambda$, and we have an isomorphism of [$(\mathfrak{g}_\mathbb{C}, K)$]{}-modules with the same multiplicity $m_\lambda$ (see [@xkaspm Theorem 2.7]): $$\label{eqn:Hmk}
\mathcal H_K \simeq \bigoplus_{\lambda\in\Lambda_G}
m_{\lambda} \hwm{\mathfrak g}{\lambda}
\quad
(\text{algebraic direct sum}).$$ Let us show $n_\pi(\lambda) = m_\lambda$ for all $\lambda$. For this, we begin with an observation that $$\operatorname{Spec}_{\widetilde{Z}} (\mathcal{H}_K)
= \bigcup_{\substack{\lambda\text{ such that}\\
m_\lambda\ne0}}
\operatorname{Spec}_{\widetilde{Z}} (\hwm{\mathfrak{g}}{\lambda})$$ is a subset in $\mathbb{R}$ and has an upper bound. This follows from Lemma \[lem:spec\] (4) applied to each irreducible summand in .
First, we consider the case where there exists $a \in \mathbb{R}$ such that $$\label{eqn:congr}
\lambda(\widetilde{Z})\equiv a \bmod \mathbb{Z}
\quad\text{for any $\lambda$ satisfying $n_\pi(\lambda)\ne0$.}$$ Then, the set $$\label{eqn:wtH}
\{ \lambda(\widetilde{Z}): \lambda\in\mathfrak{t}^*_{\mathbb{C}},
n_\pi(\lambda)\ne 0 \}$$ is contained in $\operatorname{Spec}_{\widetilde{Z}}(\mathcal{H}_K)$ by , and is discrete by . Hence, it is a discrete subset of $\mathbb{R}$ with an upper bound. Thus, we can find $\mu_0 \in \mathfrak{t}_{\mathbb{C}}^*$ such that $n_\pi(\mu_0) \ne 0$ and that $\mu_0(\widetilde{Z})$ attains its maximum in . In turn, the evaluation map $\operatorname{Supp}_{\mathfrak k}(\mathcal H_K) \to \mathbb R$, $\mu \mapsto \mu(\widetilde Z)$ attains its maximum at $\mu_0 \in \operatorname{Supp}_{\mathfrak
k}(\mathcal H_K)$ by and Lemma \[lem:spec\] (3). Therefore, ${\operatorname{Hom}}_{(\mathfrak g_{\mathbb C}, K)}(\hwm{\mathfrak g}{\mu_0}, \mathcal H_K)\ne \{0\}$ by Lemma \[lem:ktog\]. Thus, we have shown $m_{\mu_0} \ne 0$, that is, $\hwm{G}{\mu_0}$ occurs as a subrepresentation in $\mathcal{H}$.
Next, we consider the unitary representation $\pi'$ on $$\mathcal{H}' :=
\sideset{}{^\oplus} \sum_{\lambda\ne\mu_0}
m_\lambda \hwm{G}{\lambda}
\oplus (m_{\mu_0} -1) \hwm{G}{\mu_0} \, ,$$ the orthogonal complement of a subrepresentation $\hwm{G}{\mu_0}$ in $\mathcal{H}$. Then, the $K$-type formula for $(\pi',\mathcal{H}')$ holds if we set $$n_{\pi'} (\lambda)
:= \begin{cases}
n_\pi (\lambda) - 1 & (\lambda = \mu_0)\, , \\
n_\pi (\lambda) & (\lambda \ne \mu_0)\, .
\end{cases}$$ Hence, by the downward induction on $\sup\operatorname{Spec}_{\widetilde{Z}}
(\mathcal{H}_K)$, we have $n_\pi(\lambda) = m_\lambda$ for all $\lambda$.
For the general case, let $A$ be the set of complete representatives of $\{\lambda(\widetilde{Z}) \in \mathbb{C} \mod \mathbb Z:
n_\pi({\lambda}) \ne 0\}$. For each $a \in A$, we define a subrepresentation $\mathcal{H}_a$ of $\mathcal{H}$ by $$\mathcal H_{a}
:=\sideset{}{^\oplus}
\sum_{\lambda(\widetilde{Z}) \equiv a \operatorname{mod}\mathbb Z}
m_{\lambda}\hwm G{\lambda}
\quad\text{(discrete Hilbert sum)}.$$ Then, we have an isomorphism of unitary representations of $G$: $$\mathcal H \simeq {\sum_{a \in A}}^{\oplus} \mathcal
H_{a} \, .$$ Since $\operatorname{Spec}_{\widetilde{Z}}(\hwm{\mathfrak{g}}{\lambda})
\subset a+\mathbb{Z}
$ if and only if $\lambda(\widetilde{Z})\equiv a \bmod \mathbb{Z}$ by Lemma \[lem:spec\] (3), we get from the following $K$-isomorphism $$\label{eqn:Hnalga}
(\mathcal{H}_a)_K \simeq
\bigoplus_{\lambda(\widetilde{Z})\equiv a\bmod\mathbb{Z}}
n_\pi(\lambda) \hwm{\mathfrak{g}}{\lambda}$$ for each $a \in A$. Therefore, our proof for the first step assures $n_\pi(\lambda)=m_\lambda$ for any $\lambda$ such that $\lambda\equiv a \bmod \mathbb{Z}$. Since $a \in A$ is arbitrary, we obtain Lemma in the general case.
Proof of Theorem \[thm:gHKS\] {#subsec:8.3}
-----------------------------
In this section, we give a proof of Theorem \[thm:gHKS\]. This is done by showing a more general formula in Lemma \[lem:8.3\] without the scalar type assumption . Then, Theorem \[thm:gHKS\] follows readily from Lemma \[lem:8.3\] because the assumption makes $\dim \hwm{\mathfrak{k}}{\mu} = 1$ and $\mathbb{S}_{(a_1,\dots,a_l)} (\mu) =
\{ \mu - \sum_{j=1}^l a_j \nu_j \}$ (see for notation).
For a discussion below, it is convenient to use the concept of a multiset. Intuitively, a multiset is a set counted with multiplicities; for example, $\{a,a,a,b,c,c\}$. More precisely, a multiset $\mathbb{S}$ consists of a set $S$ and a function $m: S \to \{ 0,1,2,\dots,\infty \}$. If $\mathbb{S}' = \{S,m'\}$ is another multiset on $S$ such that $m'(x) \le m(x)$ for all $x \in S$, we say $\mathbb{S}'$ is a [*[submultiset]{}*]{} of $\mathbb{S}$ and write $\mathbb{S}' \subset \mathbb{S}$.
Suppose we are in the setting of Subsection \[subsec:8.2\] and recall $\tau$ is an involution of holomorphic type. For a ${\Delta}^+(\mathfrak{k}, \mathfrak{t})$-dominant weight $\mu$, we introduce a multiset $\mathbb{S}(\mu)$ consisting of ${\Delta}^+(\mathfrak{k}^\tau, \mathfrak{t}^\tau)$-dominant weights: $$\mathbb{S}(\mu):=
\bigcup_{\Sb a_1 \ge \dots \ge a_l \ge 0\\ a_1, \dots, a_l \in \mathbb{N} \endSb}
\mathbb{S}_{(a_1,\dots,a_l)} (\mu) \, ,$$ where we define the multiset $\mathbb{S}_{(a_1,\dots,a_l)} (\mu)$ by $$\label{eqn:Samu}
\parbox{25em}{$\{$highest weight of irreducible $\mathfrak{k}^\tau$-modules
occurring in\newline
\hspace*{1em} $\hwm{\mathfrak{k}^\tau}{-\sum_{j=1}^l a_j \nu_j} \otimes
\hwm{\mathfrak{k}}{\mu}|_{\mathfrak{k}^\tau} $
counted with multiplicities$\}$.}$$ Because the central element $\widetilde{Z} = \frac{1}{\sqrt{-1}} Z$ of $\mathfrak{k}_{\mathbb{C}}$ acts on the irreducible representation $\hwm{\mathfrak{k}}{\mu}$ by the scalar $\mu(\widetilde{Z})$ and because $\nu_j (\widetilde{Z}) = 1$ for all $j$ $(1 \le j \le l)$, any element $\nu$ in $\mathbb{S}_{(a_1,\dots,a_l)}(\mu)$ satisfies $
\nu(\widetilde{Z}) = - \sum_{j=1}^l a_j + \mu(\widetilde{Z})
$. Therefore, the multiplicity of each element of the multiset $\mathbb{S}(\mu)$ is finite.
\[lem:8.3\] Let $\tau$ be an involution of $G$ of holomorphic type, and $H = G_0^\tau$. If $\hwm{G}{\mu}$ is a (relative) holomorphic discrete series representation of $G$, then it decomposes discretely into irreducible $H$-modules as: $$\hwm{G}{\mu} |_H
\simeq
\sideset{}{^\oplus}\sum_{\nu \in \mathbb{S}(\mu)}
\hwm{H}{\nu}
\quad
\text{(discrete Hilbert sum).}
$$
It follows from Fact \[fact:3.4.1\] (1) that $\hwm{G}{\mu}$ is $(H\cap K)$-admissible, and splits discretely into a Hilbert direct sum of irreducible unitary representations of $H$.
Applying Lemma \[lem:HnG\] to $H = G_0^\tau$, we see that Lemma \[lem:8.3\] is deduced from the following $\mathfrak{k}^\tau$-isomorphism: $$\label{eqn:kgeneral}
\hwm{\mathfrak{g}}{\mu}
\simeq
\bigoplus_{\nu \in \mathbb{S}(\mu)}\hwm{\mathfrak{g}^\tau}{\nu}
\quad
\text{(algebraic direct sum).}$$ The rest of the proof is devoted to showing .
Since $\hwm{G}{\mu}$ is a holomorphic discrete series, $\hwm{\mathfrak{g}}{\mu}$ is isomorphic to the generalized Verma module $N^{\mathfrak{g}}(\mu)
= U(\mathfrak{g}_\mathbb{C})
\otimes_{U(\mathfrak{k}_\mathbb{C} + \mathfrak{p}_+)} \hwm{\mathfrak{k}}{\mu}
$ as a $\mathfrak{g}$-module, which in turn is isomorphic to the $\mathfrak{k}$-module $
S(\mathfrak{p}_-) \otimes \hwm{\mathfrak{k}}{\mu}
$.
According to the decomposition $\mathfrak{p}_- = \mathfrak{p}_-^\tau \oplus \mathfrak{p}_-^{-\tau}$ as $\mathfrak{k}^\tau$-modules, we have then the following $\mathfrak{k}^\tau$-isomorphism: $$\label{eqn:pipp}
\hwm{\mathfrak{g}}{\mu}
\simeq S(\mathfrak{p}_-) \otimes \hwm{\mathfrak{k}}{\mu}
\simeq S(\mathfrak{p}_-^\tau) \otimes
S(\mathfrak{p}_-^{-\tau}) \otimes
\hwm{\mathfrak{k}}{\mu} \, .$$ Now, we consider the Hermitian symmetric space $G^{\tau\theta} / G^{\tau,\theta}$, for which the complex structure is given by the decomposition $\mathfrak{g}_{\mathbb{C}}^{\tau\theta} =
\mathfrak{k}_{\mathbb{C}}^\tau \oplus
\mathfrak{p}_+^{-\tau} \oplus
\mathfrak{p}_-^{-\tau}
$ (see ). Then, the Hua–Kostant–Schmid formula ([@xschmidherm Behauptung c]) applied to $G^{\tau\theta}/G^{\tau,\theta}$ decomposes the symmetric algebra $S(\mathfrak{p}_-^{-\tau})$ into irreducible $\mathfrak{k}^\tau$-modules: $$\label{eqn:HKStau}
S(\mathfrak{p}_-^{-\tau})
\simeq
\bigoplus
\Sb a_1 \ge \dots \ge a_l \ge 0\\ a_1, \dots, a_l \in \mathbb{N} \endSb
\hwm{\mathfrak{k}^\tau}{- \sum_{j=1}^l a_j \nu_j} \, .$$ It follows from the definition of $\mathbb{S}(\mu)$ that we have the following irreducible decomposition as $\mathfrak{k}^\tau$-modules: $$S(\mathfrak{p}_-^{-\tau}) \otimes \hwm{\mathfrak{k}}{\mu}
\simeq \bigoplus_{\nu\in\mathbb{S}(\mu)}
\hwm{\mathfrak{k}^\tau}{\nu}.$$ Combining this with , we get a $\mathfrak{k}^\tau$-isomorphism $$\hwm{\mathfrak{g}}{\mu}
\simeq \bigoplus_{\nu\in\mathbb{S}(\mu)}
S(\mathfrak{p}_-^\tau) \otimes
\hwm{\mathfrak{k}^\tau}{\nu} \, .$$ Next, we consider the Verma module $N^{\mathfrak{g}^\tau}(\nu) =
U(\mathfrak{g}^\tau_{\mathbb{C}})
\otimes_{U(\mathfrak{k}^\tau_{\mathbb{C}}+\mathfrak{p}^\tau_+)}
\hwm{\mathfrak{k}^\tau}{\nu}$ of the subalgebra $\mathfrak{g}^\tau$. Then, $\hwm{\mathfrak{g}^\tau}{\nu}$ is the unique irreducible quotient of $N^{\mathfrak{g}^\tau}(\nu)$. We shall show later that $N^{\mathfrak{g}^\tau}(\nu)$ is irreducible as a $\mathfrak{g}^\tau$-module, but at this stage we denote by $\hwm{\mathfrak{g}^\tau}{\nu},
\hwm{\mathfrak{g}^\tau}{\nu'},
\hwm{\mathfrak{g}^\tau}{\nu''},
\ldots$ the totality of irreducible subquotient modules of $N^{\mathfrak{g}^\tau}(\nu)$. (There are at most finitely many subquotients, and all of them are highest weight modules.) Then, as $\mathfrak{k}^\tau$-modules, we have the following isomorphisms: $$\begin{aligned}
S(\mathfrak{p}_-^\tau) \otimes \hwm{\mathfrak{k}^\tau}{\nu}
& \simeq N^{\mathfrak{g}^\tau}(\nu)
\\
& \simeq \hwm{\mathfrak{g}^\tau}{\nu} \oplus
\hwm{\mathfrak{g}^\tau}{\nu'} \oplus
\hwm{\mathfrak{g}^\tau}{\nu''} \oplus \cdots \, .\end{aligned}$$ Therefore, we get a $\mathfrak{k}^\tau$-isomorphism: $$\hwm{\mathfrak{g}}{\mu}
\simeq \bigoplus_{\nu\in\mathbb{S}(\mu)}
(\hwm{\mathfrak{g}^\tau}{\mu} \oplus
\hwm{\mathfrak{g}^\tau}{\nu'} \oplus
\hwm{\mathfrak{g}^\tau}{\nu''} \oplus \cdots) \, .$$ Accordingly, the restriction $\hwm{G}{\mu}|_H$ splits discretely into irreducible unitary representations of $H$ by Lemma \[lem:HnG\]: $$\hwm{G}{\mu} |_H
\simeq \; \sideset{}{^\oplus}\sum_{\nu \in \mathbb{S}(\mu)}
(\hwm{H}{\nu} \oplus
\hwm{H}{\nu'} \oplus
\hwm{H}{\nu''} \oplus \cdots ) \, .$$ Since $\hwm{G}{\mu}$ is a (relative) holomorphic discrete series representation of $G$, all irreducible summands in the right-hand side must be (relative) holomorphic discrete series representations of $H$ by Fact \[fact:3.4.1\] (1). Therefore, $N^{\mathfrak{g}^\tau}(\nu)$ is irreducible, and the other subquotients $\hwm{\mathfrak{g}^\tau}{\nu'}, \hwm{\mathfrak{g}^\tau}{\nu''},
\ldots$ do not appear. Hence, the $\mathfrak{k}^\tau$-structures of the both sides of are the same. Thus, Lemma \[lem:8.3\] is proved.
Proof of Theorem \[thm:tensordeco\] {#subsec:pf tensordeco}
-----------------------------------
For two irreducible representations $\hwm{\mathfrak{k}}{\mu_1}$ and $\hwm{\mathfrak{k}}{\mu_2}$, we define a multiset $\mathbb{S}(\mu_1,\mu_2)$ consisting of $\Delta^+(\mathfrak{k},\mathfrak{t})$-dominant weights by $$\mathbb{S}(\mu_1,\mu_2)
:= \bigcup_{\substack{a_1\ge\dots\ge a_{\bar{l}}\ge0 \\
a_1,\dots,a_{\bar{l}}\in\mathbb{N} }}
\mathbb{S}_{(a_1,\dots,a_{\bar{l}})}
(\mu_1,\mu_2) \, ,$$ where $\mathbb{S}_{(a_1,\dots,a_{\bar{l}})} (\mu_1,\mu_2)$ is the multiset consisting of highest weights of irreducible $\mathfrak{k}$-modules occurring in $\hwm{\mathfrak{k}}{-\sum_{j=1}^{\bar{l}} a_j\bar{\nu}_j}
\otimes \hwm{\mathfrak{k}}{\mu_1}
\otimes \hwm{\mathfrak{k}}{\mu_2}
$ counted with multiplicities.
Theorem \[thm:tensordeco\] is derived from the following more general formula:
\[lem:tensordeco\] The tensor product of two (relative) holomorphic discrete series representations $\hwm{G}{\mu_1}$ and $\hwm{G}{\mu_2}$ decomposes as follows: $$\hwm{G}{\mu_1} \widehat{\otimes} \, \hwm{G}{\mu_2}
\simeq \;
\sideset{}{^\oplus}\sum_{\nu \in \mathbb{S}(\mu_1,\mu_2)}
\hwm{G}{\nu} \, .$$
We define two injective maps by: $$\begin{aligned}
&{\operatorname{diag}}: \mathfrak{p}_+ \to \mathfrak{p}_+ \oplus \mathfrak{p}_+ \, ,
\quad
X \mapsto (X,X) \, ,
\\
&{\operatorname{diag}}': \mathfrak{p}_+ \to \mathfrak{p}_+ \oplus \mathfrak{p}_+ \, ,
\quad
X \mapsto (X,-X) \, .\end{aligned}$$ It then follows that we have $\mathfrak{k}$-isomorphisms: $$\begin{aligned}
S(\mathfrak{p}_-) \otimes S(\mathfrak{p}_-)
& \simeq S(\mathfrak{p}_- \oplus \mathfrak{p}_-)
\\
& \simeq S({\operatorname{diag}}(\mathfrak{p}_-)) \otimes
S({\operatorname{diag}}'(\mathfrak{p}_-) )
\\
& \simeq \bigoplus_{\substack{a_1 \ge\cdots\ge a_{\bar{l}}\ge0\\
a_1,\dots,a_{\bar{l}}\in\mathbb{N}}}
S({\operatorname{diag}}(\mathfrak{p}_-))
\otimes
\hwm{\mathfrak{k}}{-\sum^{\bar{l}}_{j=1}a_j\bar{\nu}_j} \, . \end{aligned}$$ This brings us the following $\mathfrak{k}$-isomorphisms: $$\begin{aligned}
\hwm{\mathfrak{g}}{\mu_1} \otimes \hwm{\mathfrak{g}}{\mu_2}
& \simeq S(\mathfrak{p}_-) \otimes \hwm{\mathfrak{k}}{\mu_1}
\otimes S(\mathfrak{p}_-) \otimes \hwm{\mathfrak{k}}{\mu_2}
\\
& \simeq \bigoplus_{\nu\in\mathbb{S}(\mu_1,\mu_2)}
S({\operatorname{diag}}(\mathfrak{p}_-)) \otimes \hwm{\mathfrak{k}}{\nu}
\\
& \simeq \bigoplus_{\nu\in\mathbb{S}(\mu_1,\mu_2)}
N^{\mathfrak{g}}_\nu \, .\end{aligned}$$ The rest of the proof goes similarly to that of Lemma \[lem:8.3\].
Restriction $U(p,q) \downarrow U(p-1,q)$ and $SO(n,2) \downarrow SO(n-1,2)$ {#subsec:8.4}
---------------------------------------------------------------------------
Suppose $(G,H)$ is a reductive symmetric pair whose complexification $(\mathfrak{g}_\mathbb{C}, \mathfrak{h}_\mathbb{C})$ is one of the following types:
$(\mathfrak{sl}(n, \mathbb{C}), \mathfrak{gl}(n-1, \mathbb{C}))$ (or $(\mathfrak{gl}(n,\mathbb{C}),\mathfrak{gl}(1,\mathbb{C})
+\mathfrak{gl}(n-1,\mathbb{C}))$),
$(\mathfrak{so}(n, \mathbb{C}), \mathfrak{so}(n-1, \mathbb{C}))$.
As is classically known (see [@xvk]), for compact $(G,H)$ such as $(U(n),U(1) \times U(n-1))$ or $(SO(n),SO(n-1))$, any irreducible finite dimensional representation $\pi$ of $G$ is multiplicity-free when restricted to $H$. For non-compact $(G,H)$ such as $(U(p,q), U(1)\times U(p-1,q))$ or $(SO(n,2), SO(n-1,2))$, an analogous theorem still holds for highest weight representations $\pi$:
\[thm:minus1\] If $(\mathfrak{g}, \mathfrak{h}) = ({\mathfrak{u}}(p,q),
\mathfrak{u}(1)+{\mathfrak{u}}(p-1,q))$ or $(\mathfrak{so}(n,2), \mathfrak{so}(n-1,2))$, then any irreducible unitary highest weight representation of $G$ decomposes discretely into a multiplicity-free sum of irreducible unitary highest weight representations of $H$.
In contrast to Theorem \[thm:A\], the distinguishing feature of Theorem \[thm:minus1\] is that $\pi$ is not necessarily of scalar type but an arbitrary unitary highest weight module. The price to pay is that the pair $(G, H)$ is very special. We do not give the proof here that uses the vector bundle version of Theorem \[thm:2.2\] (see [@mfbdle]). Instead, we give an explicit decomposition formula for holomorphic discrete series $\pi$. The proof of Theorem \[thm:minus1\] for the case $(G,H) = (SO_0(n,2), SO_0(n-1,2))$ can be also found in Jakobsen and Vergne [@xjv Corollary 3.1].
Branching law for $U(p,q)\downarrow U(p-1,q)$
---------------------------------------------
This subsection gives an explicit branching law of a holomorphic discrete series representation $\hwm{G}{\mu}$ of $G=U(p,q)$ when restricted to $H=U(1)\times U(p-1,q)$. Owing to , such $\hwm{G}{\mu}$ is parametrized by $\mu=(\mu_1,\dots,\mu_{p+q}) \in \mathbb{Z}^{p+q}$ satisfying $$\mu_1\ge\cdots\ge\mu_p, \mu_{p+1}\ge\cdots\ge\mu_{p+q},
\mu_{p+q}\ge\mu_1 +p+q \, .$$ Here is the formula:
\[thm:upqupq\] Retain the above setting. Then, the branching law of $\hwm{G}{\mu}$ of the restriction to the subgroup $H$ is multiplicity-free for any $\mu$; it is given as follows: $$\label{holoupq}
\hwm{G}{\mu}|_H
\simeq {}
\sideset{}{^\oplus}\sum_{a=0}^\infty
\sideset{}{^\oplus}\sum_{\substack{
{\mu_1\ge\lambda_2\ge\mu_2\ge\cdots\ge\lambda_p\ge\mu_p} \\
{\lambda_{p+1}\ge\mu_{p+1}\ge\cdots\ge\lambda_{p+q}\ge\mu_{p+q}}\\
{\sum_{i=1}^q(\lambda_{p+i}-\mu_{p+i})=a}}}
\mathbb{C}_{\sum_{i=1}^p \mu_i - \sum_{i=1}^p \lambda_i - a}
\boxtimes
\hwm{U(p-1,q)}{(\lambda_2,\dots,\lambda_p,\lambda_{p+1},\dots,\lambda_{p+q})}
\, .$$
For $
(G,H) \equiv (G,G^\tau)
= (U(p,q), U(1) \times U(p-1,q))
$, we have $$\begin{aligned}
{2}
&G^{\tau\theta}
&&\simeq U(1,q) \times U(p-1),
\\
&H \cap K
\ (= K^\tau = K^{\tau\theta})
&&\simeq U(1) \times U(p-1) \times U(q),\end{aligned}$$ $\mathfrak{t}^\tau = \mathfrak{t}$, and $$\Delta^+ (\mathfrak{p}_+^{-\tau}, \mathfrak{t}^\tau)
= \{ e_1 - e_{p+i} : 1 \le i \le q \}$$ by using the standard basis of $\Delta(\mathfrak{g}, \mathfrak{t})
= \{ \pm (e_i - e_j) :
1 \le i < j \le p+q \}$. Thus, $l = {\mathbb{R}\text{-}\operatorname{rank}}G^{\tau\theta} = 1$ and $\nu_1 = e_1 - e_{p+1}$. Hence, the $\mathfrak{k}^\tau$-type formula amounts to $$\begin{aligned}
\label{eqn:Supq}
S (\mathfrak{p}_-^{-\tau})
&\simeq \bigoplus_{a=0}^\infty
\hwm{H\cap K}{-a(e_1 - e_{p+1})}
\nonumber
\\
&\simeq \bigoplus_{a=0}^\infty
\mathbb{C}_{-a} \boxtimes \mathbf{1} \boxtimes
\hwm{U(q)}{(a,0,\dots,0)}\end{aligned}$$ as $H \cap K \simeq U(1) \times U(p-1) \times U(q)$ modules. Here, $\mathbf{1}$ denotes the trivial one dimensional representation of $U(p-1)$.
On the other hand, we recall a classical branching formula $U(p) \downarrow U(p-1)$: $$\hwm{U(p)}{(\mu_1,\dots,\mu_p)} |_{U(1)\times U(p-1)}
\simeq \bigoplus_{\mu_1 \ge \lambda_2 \ge \mu_2 \ge
\cdots \ge \lambda_p \ge
\mu_p}
\mathbb{C}_{\sum_{i=1}^p \mu_i - \sum_{i=2}^p \lambda_i}
\otimes \hwm{U(p-1)}{(\lambda_2,\dots,\lambda_p)} \, ,$$ whereas the classical Pieri rule says $$\hwm{U(q)}{(a,0,\dots,0)} \otimes \hwm{U(q)}{(\mu_{p+1},\dots,\mu_{p+q})}
\simeq
\bigoplus_{\substack{\lambda_{p+1}\ge\mu_{p+1}\ge\cdots
\ge\lambda_{p+q}\ge\mu_{p+q}\\
{\sum_{i=1}^q (\lambda_{p+i}-\mu_{p+i})=a}}}
\hwm{U(q)}{(\lambda_{p+1},\dots,\lambda_{p+q})} \, .$$ These two formulae together with yield the following $\mathfrak{k}^\tau$-isomorphisms: $$\begin{aligned}
&S(\mathfrak{p}_-^{-\tau}) \otimes
\hwm{\mathfrak{k}}{\mu} |_{\mathfrak{k}^\tau}
\\
&{}\simeq
\bigoplus_{a=0}^\infty
(( \mathbb{C}_{-a} \boxtimes \mathbf{1}) \otimes
\hwm{U(p)}{(\mu_1,\dots,\mu_p)}
|_{U(1)\times U(p-1)} )
\boxtimes (\hwm{U(q)}{(a,0,\dots,0)} \otimes
\hwm{U(q)}{(\mu_{p+1},\dots,\mu_{p+q})})
\\
&{}\simeq
\bigoplus_{a=0}^\infty \
\bigoplus_{\substack{
{\mu_1\ge\lambda_2\ge\mu_2\ge\cdots\ge\lambda_p\ge\mu_p} \\
{\lambda_{p+1}\ge\mu_{p+1}\ge\cdots\ge\lambda_{p+q}\ge\mu_{p+q}}\\
{\sum_{i=1}^q(\lambda_{p+i}-\mu_{p+i})=a}}}
\mathbb{C}_{\sum_{i=1}^p\mu_i-\sum_{i=2}^p\lambda_i-a}
\boxtimes
\hwm{U(p-1)}{(\lambda_2,\dots,\lambda_p)}
\boxtimes
\hwm{U(q)}{(\lambda_{p+1},\dots,\lambda_{p+q})} \, .\end{aligned}$$ In view of the $\mathfrak{k}^\tau$-isomorphisms $$\hwm{\mathfrak{g}}{\mu}\simeq S(\mathfrak{p}^\tau_-) \otimes
S(\mathfrak{p}^{-\tau}_-) \otimes \hwm{\mathfrak{k}}{\mu}|_{\mathfrak{k}^\tau}$$ and $N^{\mathfrak{g}^\tau} (\nu) \simeq S(\mathfrak{p}^\tau_-) \otimes
\hwm{\mathfrak{k}^\tau}{\nu}$, we have now shown that the $\mathfrak{k}^\tau$-structure of $\hwm{\mathfrak{g}}{\mu}$ coincides with that of $$\bigoplus_{a=0}^\infty \
\bigoplus_{\substack{
{\mu_1\ge\lambda_2\ge\mu_2\ge\cdots\ge\lambda_p\ge\mu_p} \\
{\lambda_{p+1}\ge\mu_{p+1}\ge\cdots\ge\lambda_{p+q}\ge\mu_{p+q}}\\
{\sum_{i=1}^q(\lambda_{p+i}-\mu_{p+i})=a}}}
N^{\mathfrak{g}^\tau}
(\sum_{i=1}^p \mu_i - \sum_{i=2}^p \lambda_i - a,
\lambda_2, \dots, \lambda_{p+q}) \, .$$ As in the last part of the proof of Theorem \[thm:gHKS\], we see that any generalized Verma module occurring in the right-hand side is irreducible (and is isomorphic to the underlying $(\mathfrak{g}^\tau_{\mathbb{C}}, H\cap K)$-module of a holomorphic discrete series of $H$). Therefore, Theorem follows from Lemma \[lem:HnG\].
Appendix: Associated Bundles on Hermitian Symmetric Spaces {#sec:9}
==========================================================
In this Appendix, we explain standard operations on homogeneous vector bundles. The results are well-known and elementary, but we recall them briefly for the convenience of the reader. The main goal is Lemma \[lem:9.6\] which is used to verify the condition in Theorem \[thm:2.2\].
Homogeneous vector bundles {#subsec:9.1}
--------------------------
Let $M$ be a real manifold, and $V$ a (finite dimensional) vector space over $\mathbb{C}$. Suppose that we are given an open covering $\{ U_\alpha \}$ of $M$ and transition functions $$g_{\alpha \beta} : U_\alpha \cap U_\beta \to GL_\mathbb{C}(V)$$ satisfying the following compatibility conditions: $$g_{\alpha \beta} \; g_{\beta \gamma} \; g_{\gamma \alpha}
\equiv \operatorname{id}
\quad \text{on} \
U_\alpha \cap U_\beta \cap U_\gamma \; ;
\qquad
g_{\alpha \alpha}
\equiv \operatorname{id}
\quad \text{on} \
U_\alpha \, .$$ A complex vector bundle $\mathcal{V}$ over $M$ with typical fiber $V$ is constructed as the equivalence class of $
\coprod_\alpha (U_\alpha \times V),
$ where $
(x,v) \in U_\beta \times V \text{ and } (y,w) \in U_\alpha \times V
$ are defined to be equivalent if $y = x$ and $w = g_{\alpha \beta}(x) v$. Then, the space of sections $\Gamma(M, \mathcal{V})$ is identified with the collection $$\set{(f_\alpha)}{f_\alpha\in C^\infty(U_\alpha, V),
\ f_\alpha(x) = g_{\alpha \beta}(x) f_\beta(x),
\text{ for } x \in U_\alpha \cap U_\beta} \, .
\label{eqn:9.1.1}$$ If $M$ is a complex manifold and if every $g_{\alpha \beta}$ is holomorphic (or anti-holomorphic), then $\mathcal{V} \to M$ is a holomorphic (or anti-holomorphic, respectively) vector bundle. Next, let $G$ be a Lie group, $K$ a closed subgroup of $G$, and $M:= G/K$ the homogeneous manifold. Then, we can take an open covering $\{ U_\alpha \}$ of $M$ such that for each $\alpha$ there is a local section $\varphi_\alpha : U_\alpha \to G$ of the principal bundle $G \to G/K$. Given a representation $\chi : K \to GL_\mathbb{C}(V)$, we define the homogeneous vector bundle, $\mathcal{V} := G \times_K (\chi, V)$. Then $\mathcal{V}$ is associated with the transition functions: $$g_{\alpha \beta}: U_\alpha \cap U_\beta \to GL_{\mathbb{C}}(V),
\quad
g_{\alpha \beta}(x) := \chi(\varphi_\alpha(x)^{-1}
\varphi_\beta(x)) \, .$$ The section space $\Gamma(M, \mathcal{V})$ is identified with the following subspace of $C^\infty(G,V)$: $$\set{f \in C^\infty(G, V)}{f(g k) = \chi^{-1}(k) f(g), \ \text{for }
g \in G, k \in K} \, .
\label{eqn:9.2.1}$$
Pull-back of vector bundles {#subsec:9.3}
---------------------------
Let $G'$ be a Lie group, $K'$ a closed subgroup of $G'$, and $M' := G'/K'$ the homogeneous manifold. Suppose that $\sigma : G' \to G$ is a Lie group homomorphism such that $\sigma(K') \subset K$. We use the same letter $\sigma$ to denote by the induced map $M' \to M$, $g' K' \mapsto \sigma(g') K$. Then the pull-back of the vector bundle $\mathcal{V} \to M$, denoted by $\sigma^* \mathcal{V} \to M'$, is associated to the representation $$\chi \circ \sigma : K' \to GL_{\mathbb{C}}(V) \, .$$ Then we have a commuting diagram of the pull-back of sections (see ): $$\begin{aligned}
{7}
&\sigma^*
&&: {}
&&\;\;\Gamma(M, \mathcal{V})
&&\to \;\;
&&\Gamma(M', \sigma^* \mathcal{V})\, ,
\quad
&&(f_\alpha)_\alpha
&&\mapsto (f_\alpha \circ \sigma)_\alpha \, ,
\\
&
&&
&&{}\qquad \cap
&&
&& \qquad\cap
&&
&&
\\
&\sigma^*
&&: {}
&&C^\infty(G, V)
&&\to \;\;
&&C^\infty(G', V)\, ,
\quad
&&\;\;f
&&\mapsto
\;\;f \circ \sigma \, .\end{aligned}$$
Push-forward of vector bundles {#subsec:9.4}
------------------------------
Suppose that $V$ and $W$ are complex vector spaces and that $\xi: V \to W$ is an anti-linear bijective map. Then, we have an anti-holomorphic group isomorphism $$GL_\mathbb{C}(V) \to GL_\mathbb{C}(W)\, ,
\quad
g \mapsto g^\xi := \xi \circ g \circ \xi^{-1} \, .$$ Let $\mathcal{V} \to M$ be a complex vector bundle with transition functions $g_{\alpha \beta}: U_\alpha \cap U_\beta \to GL_{\mathbb{C}}(V)$. Then, one constructs a complex vector bundle $\xi_* \mathcal{V} \to M$ with the transition functions $g_{\alpha \beta}^\xi : U_\alpha \cap U_\beta \to GL_\mathbb{C}(W)$. We have a natural homomorphism $$\xi_* : \Gamma(M, \mathcal{V}) \to \Gamma(M, \xi_* \mathcal{V})\, ,
\quad
(f_\alpha) \mapsto (\xi \circ f_\alpha) \, ,$$ which is well-defined because the compatibility condition in is satisfied as follows: If $x \in U_\alpha \cap U_\beta$ then $$g_{\alpha \beta}^\xi(x) (\xi\circ f_\beta)(x)
=
(\xi \circ g_{\alpha \beta}(x)\circ \xi^{-1}) (\xi\circ f_\beta)(x)
=
\xi \circ g_{\alpha \beta}(x) f_\beta(x)
=
\xi \circ f_\alpha(x) \, .$$
If $\mathcal{V}$ is the homogeneous vector bundle $G \times_K (\chi, V)$ associated to a representation $\chi : K \to GL_{\mathbb{C}}(V)$, then $\xi_*\mathcal{V}$ is isomorphic to the homogeneous vector bundle $G \times_K (\chi^\xi, W)$ associated to the representation $$\chi^\xi : K \to GL_{\mathbb{C}}(W)\, ,
\quad
k \mapsto \chi^\xi(k) := \xi \circ \chi(k) \circ \xi^{-1} \, .$$
A sufficient condition for the isomorphism $\xi_*\sigma^*\mathcal{V}\simeq\mathcal{V}$ {#subsec:9.5}
--------------------------------------------------------------------------------------
We are particularly interested in the case where $G' = G$, $K' = K$, $V = W = \mathbb{C}$ and $\xi(z) := \bar z$ (the complex conjugate of $z$) in the setting of Subsections \[subsec:9.3\] and \[subsec:9.4\].
By the identification of $GL_\mathbb{C}(\mathbb{C})$ with $\mathbb{C}^\times$, we have $g^\xi = \overline{g}$ for $g \in GL_\mathbb{C}(V)\simeq \mathbb{C}^{\times}$. Then, $\chi^\xi$ coincides with the conjugate representation $$\overline{\chi} : K \to GL_\mathbb{C}(W) \simeq \mathbb{C}^\times\, ,
\quad
k \mapsto \overline{\chi(k)}$$ for $\chi \in {\operatorname{Hom}}(K, \mathbb{C}^\times)$. Thus, we have an isomorphism of $G$-equivariant holomorphic line bundles: $$\xi_* \sigma^* \mathcal{V} \simeq G
\times_K (\overline{\chi\circ \sigma}, \mathbb{C})
\label{eqn:9.5.1}$$ with the following correspondence of sections: $$\xi_* \circ \sigma^* :
\Gamma(M, \mathcal{V}) \to \Gamma(M, \xi_* \sigma^* \mathcal{V})\, ,
\quad
(f_\alpha) \mapsto (\overline{f_\alpha \circ \sigma}) \, .$$
We now apply the formula to the setting where $M=G/K$ is an irreducible Hermitian symmetric space.
\[lem:9.6\] Let $\chi : K \to \mathbb{C}^\times$ be a unitary character. We denote by $\mathcal{V}$ the homogeneous line bundle $G \times_K (\chi, \mathbb{C})$. Suppose $\sigma$ is an involutive automorphism of $G$ of anti-holomorphic type (see Definition \[def:holo-anti\]). Then we have an isomorphism of $G$-equivariant holomorphic line bundles: $
\xi_* \sigma^* \mathcal{V} \simeq \mathcal{V}.
$
In view of , it suffices to show $
\overline{\chi \circ \sigma} = \chi.
$ As the character $\chi$ of $K$ is unitary, we have $\overline{\chi(k)} = \chi(k^{-1})$ for any $k \in K$. Let $Z$ be a generator of the center $\mathfrak{c}(\mathfrak{k})$ of $\mathfrak{k}$. Since $\sigma$ is of anti-holomorphic type, we have $\sigma Z = -Z$, and then $$\overline{\chi \circ \sigma(\exp t Z)}
=
\overline{\chi (\exp (- t Z))}
=
\chi (\exp t Z)
\qquad
(t \in \mathbb{R})\, .$$ On the other hand, if $k \in [K, K]$, then $\overline{\chi \circ \sigma(k)} = 1 = \chi(k)$ because $[K,K]$ is a connected semisimple Lie group. As $K= \exp \mathfrak{c}(\mathfrak{k}) \cdot [K,K]$, we have shown $
\overline{\chi \circ \sigma} = \chi.
$ Hence Lemma.
I owe much to J. Faraut for enlightening discussions, in particular, for clarifying the idea of his joint work [@xft] with E. Thomas at an early stage on the occasion of the CIMPA school on Invariant Theory in Tunisia organized by P. Torasso in 1996, when I formalized Theorem \[thm:A\] and gave its proof in the classical case.
Some of the results of this article were presented in various places including the conferences Representation Theory at Saga in 1997 (see [@xkmfjp]) organized by K. Mimachi, at Summer Solstice Days at Paris University VI and VII in 1999, at Copenhagen organized by B. Ørsted and H. Schlichtkrull in 1999, and at Oberwolfach organized by A. Huckleberry and K.-H. Neeb and J. Wolf in 2000 and 2004, at the MSRI program “Integral Geometry” organized by S. Gindikin, L. Barchini, and R. Zierau in 2001, at the workshop “Representation theory of Lie groups, harmonic analysis on homogeneous spaces and quantization” at the Lorentz center in Leiden in 2002 (see [@xkleiden]) organized by G. van Dijk and V. F. Molchanov, and at the workshop “Representation theory and automorphic forms” in Korea in 2005 coorganized by J. Yang and W. Schmid with the author. I express my deep gratitude to the organizers of these conferences and to the participants for helpful and stimulating comments on various occasions. I also thank an anonymous referee who read the manuscript very carefully. Special thanks are due to Ms. Suenaga for her help in preparing for the final manuscript.
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|
---
author:
- |
Jun H. Lee\
\
\
Donghun Lee\
\
\
Dae-Kyoo Kim\
\
\
- |
Sooyong Park\
\
\
bibliography:
- 'zuna.bib'
title: |
A Semantic-Based Approach for Detecting\
and Decomposing God Classes
---
Bad smell, God class, Large class, reengineering, refactoring, semantic analysis.
Introduction {#sec:introduction}
============
Related Work {#sec:relatedwork}
============
Overview of Approach {#sec:overview}
====================
Semantic Similarity {#sec:csm}
===================
Extracting responsibilities of God Class {#sec:badsmell}
========================================
Case Study: JMeter {#sec:casestudy}
==================
Conclusion {#sec:conclusion}
==========
[**Acknowledgements.**]{}\
This research was supported by the MKE(The Ministry of Knowledge Economy), Korea, under the ITRC(Information Technology Research Center) support program supervised by the NIPA. (National IT Industry Promotion Agency(NIPA-2011-(C1090-1131-0008)))
|
---
abstract: 'In the context of Monte Carlo (MC) simulation of particle transport Uncertainty Quantification (UQ) addresses the issue of predicting non statistical errors affecting the physical results, i.e. errors deriving mainly from uncertainties in the physics data and/or in the model they embed. In the case of a single uncertainty a simple analytical relation exists among its the Probability Density Function (PDF) and the corresponding PDF for the output of the simulation: this allows a complete statistical analysis of the results of the simulation. We examine the extension of this result to the multi-variate case, when more than one of the physical input parameters are affected by uncertainties: a typical scenario is the prediction of the dependence of the simulation on input cross section tabulations.'
address: |
National Institute for Nuclear Physics (I.N.F.N.)\
Via Dodecaneso, 33\
16146 Genova (Italy)
author:
- P Saracco$^1$ and M G Pia$^2$
title: An exact framework for uncertainty quantification in Monte Carlo simulation
---
Introduction and problem definition
===================================
Uncertainty Quantification (UQ) may refer to a wide variety of specific problems pertaining different scientific areas: related studies involve many methodologies or techniques and even terminologies[@g1]-[@g10], so that it is necessary to better delimit the problem: we are interested in studying how uncertainties of the input data needed for physics simulation transfer into the output. Even within this more restricted domain the problem seems quite complex because the meaning of [*input data needed for physics simulation*]{} is rather undetermined: it may refer to those data needed to specify the experimental setup to be simulated - like, e.g., geometrical data or to the composition of the materials - to some conditions externally applied to the system - like temperature, pressure or electromagnetic fields - or to physical data - cross sections or, often, physical models - needed to describe the transport of particles. All these data are generally affected by uncertainties, which necessarily imply uncertainties in the output of the simulation. So from a responsible MC user’s perspective [*input data*]{} should refer both to those that are under his direct experimental control and to (many) other data, whose values and uncertainties come from different experiments: it is remarkable that often the latter are not considered. In this contribution we summarize previous results[@SarPia2012; @SarPia2013] together with some new ones.
To give our analysis a well established mathematically ground we make a fundamental assumption: to be able to disentangle input data uncertainty from the process of simulation itself. In other words we assume that the process of simulation relies on the knowledge of the values of some parameters $x_1,\ldots,x_N$, with their associate uncertainties, and that these values cannot vary during the simulation (or that these values do not depend on the simulation itself): this assumption enables to assume [*a priori*]{} the knowledge of the Probability Distribution Function (PDF) $f\left(x_1,\ldots,x_N\right)$ for the input parameters $x_1,\ldots,x_N$, that can then be assumed as stochastic parameters.
This is our main hypothesis: its validity in the generic case must be assessed by users case by case.
We just mention two typical cases under which this hypothesis can be not valid or questionable: the first concerns the local temperature of some experimental apparatus we want to simulate; to some extent it can be assumed as fixed by external conditions with their associate uncertainties, external temperature and/or cooling system of the apparatus if it exists, but local significant temperature variations can be present depending on the local rate of energy deposition due to the transported particles. This last quantity is really tracked by the simulation itself, but its consequence in terms of local temperature variation is not normally taken into account automatically by the MC code: the relation between energy deposition and local temperature variations can be taken into account by coupling the MC simulation with some thermo-mechanical code or model, but even if this can be done, not all MC codes are able to use different cross sections data (cross sections data at different temperatures) in the same simulation. The second case concerns the density of some isotopes, which can vary due to activation/decay processes.
Under the quoted assumption the probability that MC simulation of some physical observable $Y$ have a result between $y$ and $y+dy$ has density $$g_{MC}(y)\simeq\int_{-\infty}^{+\infty}\,dx_1\cdots dx_N\,f\left(x_1,\ldots,x_N\right)\sqrt{\frac{\displaystyle N_E}{2\pi\sigma_{y_0}^2}}
\exp\left[-\frac{\left(y-y_0\left(x_1,\ldots,x_N\right)\right)^2}{2\sigma_{y_0}^2/N_E}\right]
\label{eqn:PDFSimul}$$ for a simulation encompassing $N_E$ events. Here $y_0\left(x_1,\ldots,x_N\right)$ is the sampled mean for the physical observable $Y$ when the input unknowns assume values $\left(x_1,\ldots,x_N\right)$ and $\sigma_{y_0}^2$ is its sampled variance. This result derives from the Central Limit Theorem (CLT) if $N_E$ is sufficiently large to make $\sigma_{y_0}^2$ independent from $N_E$ itself. In the limit $N_E\to\infty$ $$g(y)=\int_{-\infty}^{+\infty}\,dx_1\cdots dx_N\,f\left(x_1,\ldots,x_N\right)\delta\left(y-y_0\left(x_1,\ldots,x_N\right)\right)\,.
\label{eqn:PDFExact}$$ Equation (\[eqn:PDFExact\]) mathematically states how uncertainties in the input data [*exactly transfer*]{} into our knowledge of the observable $Y$ because this is an assignment of a probability for each possible outcome of this observable.
Even if we derived (\[eqn:PDFExact\]) from (\[eqn:PDFSimul\]) we could as well make the opposite: (\[eqn:PDFExact\]) represents the forward propagation of uncertainty for a deterministic problem having solution $y_0$ which, in turn, depends (parametrically, not dynamically) on some unknown input values whose PDF is given by $f\left(x_1,\ldots,x_N\right)$. Searching the solution of such deterministic problem by MC simulation - that is we go back from (\[eqn:PDFExact\]) to (\[eqn:PDFSimul\]) - results in a (gaussian) statistical blurring of the exact result $y=y_0\left(x_1,\ldots,x_N\right)$ of the underlying deterministic problem. This dual interpretation of (\[eqn:PDFExact\]) and (\[eqn:PDFSimul\]) is crucial to our investigation: it is a clear indication of which is the natural goal for any UQ project - the determination of $g(y)$ - and of its prerequisite - the knowledge of $f\left(x_1,\ldots,x_N\right)$. Unfortunately, but quite obviously, the determination of $g(y)$ requires also the knowledge of the parametric dependence of the exact solution $y=y_0\left(x_1,\ldots,x_N\right)$ on the input uncertainties and on their PDFs: except for very simple examples we are not able to exploit this dependance, but clearly we can use simulation to extract these required informations, as we shall see. It is relevant to stress that our attention has turned to the determination of the exact form of $g(y)$ rather than the one of $g_{MC}(y)$: with this passage we obtain an important result both on the conceptual - as we have seen - and on the practical side. In fact the inherent dependency of $g_{MC}(y)$ from the details of the simulation[^1] makes it impossible to determine it other than by statistical sampling: this implies the necessity of running a very large number of simulations, each time sampling $(x_1,\ldots,x_N)$ out of $f(x_1,\ldots,x_N)$. On the contrary the determination of the dependence $y_0=y_0\left(x_1,\ldots,x_N\right)$ can be accurately obtained with fewer runs, as we shall discuss.
Uncertainty quantification directly derives from (\[eqn:PDFExact\]): in fact if we know $g(y)$ the determination of any required statistical information of the physical output, e.g. its confidence intervals, from the unknown input parameters is straightforward. So the task of any UQ project is not the study of how uncertainties of the input data needed for physics simulation transfer into the output (of the simulation), that is the determination of $g_{MC}(y)$, rather the use of simulation to determine the properties of $g(y)$ as better as possible. We note, [*en passant*]{}, that techniques we are going to develop to study $g(y)$ are then applicable to more general contexts than simulation, for instance also when simulations are coupled to deterministic codes.
Moreover we learn that the necessary prerequisite is a precise, as far as possible, knowledge of the PDF for the input data, because it is directly transferred into the physical output probability distribution.
In most of the cases uncertainties in the data which are under the user’s control are independent from uncertainties in the physical data needed for transport. So our approach enables to clearly distinguish two very different phases of UQ projects: (i) a [*validation phase*]{} and (ii) a [*problem specific analysis phase*]{}.
(i) [**The validation phase**]{} - All physical data needed for transport should be validated: all general purpose MC codes make use of cross section tabulations as well as of physical models. All these data should be carefully analyzed giving to users at least the knowledge about their confidence intervals (better, their PDF). It should be clear that this cannot be a users’s responsibility: without a previous validation phase any UQ project is meaningless. On the other side how much these data are individually relevant in any given problem cannot be a priori established.
(ii) [**The problem specific analysis phase**]{} - Most of realistic situations under simulation may involve [*a priori*]{} hundreds of physical parameters in their full definition: they include parameters whose confidence intervals (or PDF) come from the validation phase, as well as problem specific parameters, like, e.g., geometrical or material composition parameters: for these last is user’s responsibility to establish at least proper confidence intervals, better PDF. It will become clear that a full UQ process is out of human possibilities and largely meaningless. So a very detailed analysis of the problem at hand should be carried on to identify those parameters which are likely to be the most relevant in the given experimental configuration. Methods we shall expose in Section \[sect:ThePath\] can be applied to the selected set of parameters. If necessary successive iterations must be carried on other parameters.
The path to UQ\[sect:ThePath\]
==============================
Once we have selected a set of parameters on which perform a UQ (or sensitivity) analysis we can proceed on the basis of (\[eqn:PDFExact\]) and (\[eqn:PDFSimul\]): in most of the cases two further assumptions hold, namely (i) variations of the input parameters are independent and (ii) $y_0$ is linear in each of the parameters in their range of variability. We will discuss later these assumptions. If this is true (\[eqn:PDFExact\]) simplifies greatly to $$g(y)=\int_{-\infty}^{+\infty}\,dx_1\cdots dx_N f_1(x_1)\cdots f_N(x_N)\delta\left[y-\bar y_0-\sum_{k=1}^N
a_k(x-\bar x_k)\right]
\label{eqn:PDFLinear}$$ where $f_j(x_j)$ and $\bar x_j$ can be assumed as known from validation phase or from problem specific analysis phase. Problem defined by (\[eqn:PDFLinear\]) is a completely defined mathematical problem once we are able to determine $\bar y_0$ and $a_j=\left.\frac{\displaystyle\partial y_0\left(x_1,\ldots,x_N\right)}{\displaystyle\partial x_j}\right\vert_{x_k=\bar x_k}$: it is obvious that we can obtain a statistical estimate of these required values with a predetermined accuracy by means of $N+1$ simulations each run with a different set of input parameters, namely $\bar x_1,\ldots,\bar x_N$ and $\bar x_1,\ldots,\bar x_j+\Delta,\ldots,\bar x_N$, so that indicating with $y_{MC}\left(x_1,\ldots,x_N\right)$ the output of the simulation with a given set of values for the input parameters $$\begin{aligned}
\bar y_0 &=& y_{MC}\left(\bar x_1,\ldots,\bar x_N\right)\nonumber\\
a_j &=& \left[y_{MC}\left(\bar x_1,\ldots,\bar x_j+\Delta,\ldots,\bar x_N\right)-y_{MC}\left(\bar x_1,\ldots,\bar x_N\right)
\right]/\Delta\nonumber\,.\end{aligned}$$ Obviously these values are affected by statistical errors of the simulation that are of the order of $\sigma_{y_0}/\sqrt{N_E}$[^2]: then a rough estimate of the number of events needed in each of the simulations comes from $$\nonumber \sigma_{y_0}/\sqrt{N_E}\ll \left\vert y_{MC}\left(\bar x_1,\ldots,\bar x_j+\Delta,\ldots,\bar x_N\right)-y_{MC}\left(\bar x_1,\ldots,\bar x_N\right)\right\vert
\label{eqn:NECond}$$ It should be realized that the use of (\[eqn:PDFLinear\]) entails a large reduction in the computer time required with respect to any attempt to directly determine $g_{MC}(y)$, a task that would require thousands of MC runs.
From (\[eqn:PDFLinear\]) it is easy to extract $<y>$ and $<y^2>$; if for instance $\bar x_k=<x_k>$, often a convenient choice $$\begin{aligned}
<y>&=&\bar y_0\nonumber\\ \nonumber
<y^2>&=&\bar y_0^2+\sum_{k=1}^N a_k^2<(x_k-\bar x_k)^2>=\bar y_0^2+\sum_{k=1}^N a_k^2\sigma_{k}^2\end{aligned}$$ so that $\sigma^2_y=\sum_{k=1}^N a_k^2\sigma_{k}^2$. This quantity can be assumed as a first estimate of the uncertainty, but this is really correct only if $g(y)$ is - at least approximately - a normal distribution: this is certainly true if the $f_j$ are normal distributions as well, but not in other cases. So we are left with a classical problem in probability theory, namely the determination of the distribution for the weighted sum of $N$ independent stochastic variables obeying (a priori) arbitrary distributions.
In some cases the calculation of $g(y)$ can be carried on exactly. We quoted the case when all the $f_j$s are normal distributions: this is a specific example of a general class of distributions which [*characteristic functions*]{} - the Fourier transform of the given distributions - are closed under product: these are the so-called [**stable distributions**]{}. In probability theory, a random variable is said to be stable (or to have a stable distribution) if it has the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. The stable distribution family is also sometimes referred to as the [*Lévy*]{} [@Levy1925] [*$\alpha$-stable distribution*]{}. Distributions belonging to this family [@Zolo1980; @FoNo1999] have characteristic functions of the form $$\phi(q;\mu,c,\alpha,\beta)=\exp\left[i t\mu-\left\vert c t\right\vert^\alpha\left(1-i\beta{\rm sgn}(t)\right)\Phi\right]$$ where $\Phi=\tan(\pi\alpha/2)$ if $\alpha\not=1$ and $\Phi=-2\log\vert t\vert/2$ if $\alpha=1$. This is a 4 parameters ($-\infty<\mu<\infty, 0<c<\infty,0<\alpha\le 2$ and $-1\le\beta\le 1$) family which is closed under product for fixed $\alpha$: the analytic form of the corresponding PDF is known only for some special values of the parameters. Among these we mention the normal distribution ($\alpha=2$, $c=\sigma/\sqrt 2$), the Cauchy distribution ($\alpha=1$, $\beta=0$), and the Lévy distribution ($\alpha=1/2$, $\beta=1$). All these distributions apart the normal one are [*heavy tailed*]{}, that is their behavior for large $x$ is $\vert x\vert^{-1-\alpha}$. In the limit $\alpha\to 0$ or $c\to 0$ they approach a Dirac delta.
The most common case of applicability of these properties to UQ is when we have to analyze a certain number of input parameters affected by different [*statistical errors*]{}.
Another useful case in practice is when we have a certain number of input parameters with flat distributions: this is the case when input parameters are given in the form $\bar x_j\pm\Delta x_j/2$. In this case $g(x)$ is given by a generalization of the [*Irwin-Hall distribution*]{} [@Irw1927; @Hall1927] recently revisited [@BraGu2020].
Unfortunately all these results apply only to cases when all the input unknowns have the same distributions with different parameters: for instance they can be all normal distributions, with different means and variances, or all flat distributions, with different means and experimental errors, and so on. It should be clear that in a generic UQ problem this is a strong limitation, because we can have the case of input parameters obeying different distributions as well: this problem is clearly not soluble in full generality so we must work on some approximation scheme to the search for $g(y)$.
A more general result
=====================
We recently succeeded in finding an exact analytical expression for the weighted sum of $N$ independent stochastic variables obeying arbitrary polynomial distributions supported on bounded intervals, whose proof can be found in [@SarPia2013a]. Here we simply quote our central result and we show how this can be useful in determining $g(y)$ with some predetermined error: an arbitrary distribution $f(x)$ supported on a bounded interval $a\le x\le b$ can always be approximated with a predetermined accuracy by a sequence of polynomials each defined on different sub-intervals of $[a, b]$: the most simple case of such procedure is to subdivide the interval $[a, b]$ in $k$ sub-intervals $x_0=a,x_1,\ldots x_k=b$ and to approximate the given distribution with sequence of segments joining the values $f\left(x_0\right),\ldots f\left(x_k\right)$. More accurate results can be obtained with a lower number of subintervals by using splines. Accuracy of an approximation is defined with respect to some given norm, measuring the [*distance*]{} between the exact result and its approximation; in our case, to the purpose of studying the propagation of error, it is convenient to make the choice $$\nonumber ||f||={{\rm sup}\atop{[a, b]}}\vert f(x)\vert$$ so that the distance between two function is the maximum of their absolute differences. We assert without proof that if $f$ is continuous (or if it has at most a finite number of discontinuities) it is always possible to find a subdivision of $[a, b]$ such that the distance between $f$ and its polynomial approximation $f_{\rm app}$ is bounded by a predefined $\varepsilon>0$[^3]. Then if we have $N$ such distributions, each approximated by this procedure, the maximum error of the distribution of the sum $x_1+\ldots+x_N$ is clearly $N\varepsilon$: for a weighted sum $a_1 x_1+\ldots+a_N x_N$ the bound on the error turns out to be $\varepsilon\displaystyle \sum_j \left\vert a_j\right\vert$. This means that with this procedure we can obtain the required distribution $g(y)$ with any predetermined error provided we are able to make exactly the convolution[^4] of generic polynomial forms defined over different intervals.
Thanks to the linearity of the convolution this is equivalent to the ability of performing the convolution of $N$ monomial forms with arbitrary exponents supported on different bounded intervals: this is exactly what we proved in [@SarPia2013a]. The convolution of $N$ monomials $x_1^{p_1},\ldots,x_N^{p_N}$ that are different from zero only over the intervals $-c_1\le x_1\le c_1$, $\ldots$, $-c_N\le x_N\le c_N$ is given by $$\begin{aligned}
\label{eqn:GenConv}
I(x;\vec c;\vec p;N)&=&\frac{\displaystyle \prod_{k=1}^N({\rm max}(p_k,1))!}{\displaystyle 2\left(\sum_{k=1}^N(p_k+1)-1\right)!}
\prod_{k=1}^N\hat Q\left(p_k,\lambda_k\right)\\ &&\times\left.
\sum_{m\in{\mathbb M}_N(x,\vec c,\vec\lambda)}(-1)^{C(m)}m^{\sum_{k=1}(p_k+1)-1}
{\rm sgn}\left(m\vert_{\vec\lambda=1}\right)\right\vert_{\vec\lambda=1}
\nonumber\end{aligned}$$ where $\displaystyle \hat O(q,\lambda)=\sum_{s=0}^q\frac{\displaystyle (-1)^s}{\displaystyle s!}
\frac{\displaystyle\partial^s}{\displaystyle\partial\lambda^s}$ and ${\mathbb M}_N(x,\vec c,\vec\lambda)=\left\{m(x)=x\pm c_1\lambda_1\pm c_2\lambda_2\ldots\pm c_N\lambda_N\right\}$. Equation (\[eqn:GenConv\]) appears as a generalization of the result [@BraGu2020]. We remark that calculation of $g(y)$ along the quoted guidelines can be tedious and often long, so we are planning to release a specific dedicated software to automate the process.
Anyway the outlined procedure [**is always able to give a definite answer on the expected PDF**]{} $g(y)$ with a predefined accuracy: it is then possible to extract confidence intervals for the required physical quantity, or any other statistical information we can need.
Two sources of errors must be accurately followed up: the first concerns the (statistical) errors in the values required for the calculation ($\bar y_0$ and the $a_j=\displaystyle\left\vert\frac{\partial y_0}{\partial x_j}\right\vert_{x_k=\bar x_k}$) that, as we told, can be extracted from accurate simulations; the second concerns propagation of errors in the procedure of approximating individual input PDFs with polynomial forms over finite intervals.
In the most generic case some of the input PDF can be defined over infinite intervals - for instance they may be normal distributions: if this is the case we can follow the described procedure provided we [*a priori*]{} define some confidence for the whole calculation; that is to say that a normal distribution can be approximated by a distribution over a finite interval with a given confidence. So the worst case, for which we cannot easily apply the described procedure, is the case when some of the input PDF are heavy tailed.
Conclusions and outlook
=======================
Results presented in the previous section provide a well defined mathematical framework to obtain the probability density function for any observable $Y$ depending on some given set of input physical parameters and on their associate uncertainties.
We made three assumptions about which some comment is useful.
First we assumed that the values of the input parameters, and implicitly their probability distributions, do not vary along the simulation: on the basis of the discussion we made about equations (\[eqn:PDFSimul\]) and (\[eqn:PDFExact\]) this condition should be more properly reformulated as: the input parameters and their probability distributions do not depend on the dynamical evolution of the system. It should be clear that this condition can, at least in principle, be given up: this possibility depends on our ability to build a more complete dynamical description of the system under examination including also the possible evolution of these dynamically variable parameters: to remain within the examples we made, local temperatures can be derived by some thermomechanical or thermohydraulic model which must be coupled to particle transport, or isotope concentrations can be obtained by coupling transport to some solver for Bateman’s equations; even if this is in principle possible, in practice it depends on a major reformulation of the existing simulation codes.
The second major assumption is about the statistical independence of different input uncertainties: we currently do not see any way-out to this assumption; however this is the most common case in practice.
The third assumption is about the linearity of the response $y_0(x_1,\ldots,x_N)$ with respect to each of the $x_k$, see equation (\[eqn:PDFLinear\]): it is clearly possible to give up this assumption by properly subdividing the region of integration in (\[eqn:PDFExact\]), as we noted earlier, for example using methods from [@LuSte2004], but this process introduce additional approximation errors whose propagation must be studied case by case for their effect on the precision on the knowledge of $g(y)$.
So we can conclude that the presented conceptual mathematical framework for UQ is really consistent and it relies in principle only on the assumption of mutual independence of the input unknowns: however its practical usability is not straightforward in its general formulation because of the error tracking process. Then we plan the release of a dedicated software tool to automate this task.
References {#references .unnumbered}
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[^1]: From (\[eqn:PDFSimul\]) the dependence on the number of generated events $N_E$ is explicit, but obviously $g_{MC}(y)$ depends also on the specific way MC simulation is implemented.
[^2]: We can naturally assume $\sigma_{y_0}$ as constant for small variations $\Delta$ of the parameters.
[^3]: This is a consequence of Weierstrass approximation theorem.
[^4]: We remind the reader that the (weighted) sum of $N$ stochastic variables is expressed by the convolution of their (rescaled) distributions.
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abstract: 'The gap between two component debris disks is often taken to be carved by intervening planets scattering away the remnant planetesimals. We employ $N$-body simulations to determine how the time needed to clear the gap depends on the location of the gap and the mass of the planets. We invert this relation, and provide an equation for the minimum planet mass, and another for the expected number of such planets, that must be present to produce an observed gap for a star of a given age. We show how this can be combined with upper limits on the planetary system from direct imaging non-detections (such as with GPI or SPHERE) to produce approximate knowledge of the planetary system.'
author:
- |
Andrew Shannon$^1$, Amy Bonsor$^1$, Quentin Kral$^1$, & Elisabeth Matthews$^2$\
$^1$Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, UK, CB3 0HA\
$^2$Astrophysics Group, University of Exeter, Physics Building, Stocker Road, Exeter, UK, EX4 4QL
bibliography:
- 'clearing.bib'
title: The unseen planets of double belt debris disk systems
---
minor planets, asteroids, general, planet-disc interactions, stars: circumstellar matter, stars: planetary systems, methods: miscellaneous
Introduction
============
Debris disks are circumstellar dust disks, produced by the destructive collisions of planetesimals leftover from the planet formation process . There exist a significant number of debris disks with two temperature components [@2008ApJ...677..630H]. Modelling suggests that in at least a significant fraction of cases, these two temperature disks harbour two concentric debris rings, with a significant gap between them [@2014MNRAS.444.3164K] - somewhat analogous to the asteroid and Kuiper belts of the Solar system.
Also by analogy with the Solar system, the gap is often inferred to have been opened by planets scattering away the remnant planetesimals. @2007MNRAS.382.1823F modelled the gap clearing as caused by multi-planet instabilities [@1996Icar..119..261C] producing ‘Nice model’ like clearing of massive planetesimal belts [@2005Natur.435..466G]. However, attempts to match such instabilities to observed debris disks suggest they must be rare events overall [@2009MNRAS.399..385B], and thus they are unlikely to be the principle mechanism for gap clearing. This rarity should also apply to the formation of a double ring by a single, eccentric, dynamically unstable planet, as modelled by @2015MNRAS.453.3329P.
The time for a single planet to clear its chaotic zone was considered by @2015ApJ...799...41M and @2015ApJ...798...83N. In the case of the observed gaps opened in double debris disk systems, the necessary planet mass is often too large to have escaped detection by direct imaging attempts. This led @2014IAUS..299..318S to suggest that the observed gaps may be opened by several planets scattering away the remnant planetesimals. Despite some attempts [@2007ApJ...666..423Z; @2011MNRAS.418.1043Q; @2011ApJ...735..109W; @2011ApJ...739...31L], a general theory of the stability of many-planet systems has not yet been developed. Great success, however, has been enjoyed by $N$-body simulations [@1996Icar..119..261C; @2009Icar..201..381S; @2014MNRAS.437.3727K; @2015ApJ...807...44P]. Thus, to consider the case where gaps in double debris disks are caused by multiple planets scattering away the planetesimals leftover from the planet formation epoch, we use $N$-body simulations to calculate the clearing time for a given planetary system. By inverting this relation, we recover an equation for the minimal planetary system that must be present in a gap for a system of a given age (figure \[fig:schematic\]).
Simulations
===========
To fill a gap that extends from $a_1$ to $a_2$ with $N$ planets spaced by $K$ mutual hill radii $\left(R_H\right)$, the planets must have mass $$\mu = \left(\frac{m_p}{m_*}\right) = \frac{12}{K^3} \frac{\left[\left(\frac{a_2}{a_1}\right)^{\frac{1}{N-1}}-1\right]^3}{\left[\left(\frac{a_2}{a_1}\right)^{\frac{1}{N-1}}+1\right]^3} .
\label{eq:pmass}$$ At the inner and outer edges of the gap, the chaotic zone will slightly widen the gap [@1980AJ.....85.1122W], but this can be a rather nuanced problem [e.g., @1989Icar...82..402D; @2009ApJ...693..734C; @2012MNRAS.419.3074M; @2015MNRAS.448..684S]. As this zone is small compared to the inter-planet spacing, we neglect it for this simple model. @2013ApJ...767..115F showed that the typical separation between planets in Kepler multi-planet systems is $21.7 \pm 9.5 R_{H}$. We thus adopt $K = 20$ for our typical separation, and space the planets evenly. Planets were given eccentricity distributed randomly and linearly from $e = 0$ to $e = 0.02$ [roughly the Kepler multi-planet value, @2014ApJ...787...80H], and inclinations distributed randomly and linearly from $ i = 0\degree$ to $i = 2\degree$ [again, following Kepler multi-planet systems, @2014ApJ...790..146F]. All planets are assigned a density of $\rho = 4~\rm{g}~\rm{cm}^{-3}$. We place 100 test particles evenly between $a_1$ and $a_2$, with eccentricities from $e = 0$ to $e = 0.1$ and inclinations from $ i = 0\degree$ to $i = 10\degree$.
We define the clearing time $\tau_{\rm{clear}}$ as the time it takes for half of the initial particles to no longer have a star-particle separation of between $a_1$ and $a_2$, whether they collide with a planet, the star, or are scattered or ejected from the belt. In a few cases (which all failed equation \[eq:minnum\]), we cut off simulations after $5 \times 10^8$ or more orbits at $a_1$; those are represented as lower limits in figure \[fig:result\], and not used in the fit for equation \[eq:minmass\]. As particles scattered to eccentric orbits move in and out of the $a_1 \rightarrow a_2$ belt, we take the first and last time 50 particles reside in the belt as our uncertainty. Simulations were performed with MERCURY [@1999MNRAS.304..793C]. The values of $a_1$ and $a_2$ we simulated are listed in table \[tab:sims\].
[0.47]{}[| r || @ r r r r r r |]{} $a_1$ & $a_2$ & $a_2$ & $a_2$ & $a_2$ & $a_2$ & $a_2$\
1 au & 2 au & 3 au & 10 au & 30 au & 100 au &\
3 au & 6 au & & 10 au & 30 au & 100 au & 300 au\
10 au & 20 au & & & 30 au & 100 au & 300 au\
30 au & 60 au & & & & 100 au & 300 au\
100 au & 200 au & & & & & 300 au\
Examining the data (figure \[fig:result\]), we notice that in most cases, as the number of planets increased and their mass decreased for a given belt, the clearing time lengthened. This trend held for systems as long as $$\frac{N}{2}-1 > \log{\frac{a_2}{a_1}} ,
\label{eq:minnum}$$ but for cases with wide belts and few planets, this trend can flatten or reverse (figure \[fig:failure\]). We exclude those cases when fitting the clearing times.
Fitting the simulation results with a power law of the form $$\tau_{\rm{clear}} = \alpha \left(\frac{a_2}{\rm{1 au}}\right)^{\beta} \left(\frac{m_p}{m_\oplus}\right)^{\gamma} ,
\label{eq:formfit}$$ we find $\alpha = 4 \pm 1 \times 10^6~\rm{yrs}$, $\beta = 1.6 \pm 0.05$ and $\gamma = -0.94 \pm 0.04$ (figure \[fig:result\]). Simulations with $K = 16$ but otherwise the same parameters gave $\alpha = 2 \pm 0.2 \times 10^6$ but otherwise the same results, therefore we infer this method is not strongly sensitive to the exact choice of $K$. After fitting the $a_2$ and $m_p$ dependence, $\tau_{\rm{clear}}$ has no further dependence on $a_1$ nor $N$. Equation \[eq:formfit\] has the same scaling as, and is comparable in magnitude to, the secular interaction time for two equal-mass planets on nearby orbits[^1]. Thus we posit secular resonances may be key to clearing the test particles - and correspondingly, with few planets, the resonances are too sparse to cover the gap sufficiently to clear away most particles, resulting in the breakdown for systems that fail equation \[eq:minnum\]. As such, we set the scaling with stellar mass as it is for secular interactions. We verify that this scaling corrects for stellar mass in figure \[fig:stellarmass\].
This result can be applied to observed double debris disk systems to infer the minimum planetary system that should be present in the gap. For a star of age $\tau$, the minimum mass of the planets in the gap is $$m_p = \left(\frac{4 \rm{Myrs}}{\tau}\right) \left(\frac{a_2}{1~\rm{au}}\right)^{\frac{3}{2}}\left(\frac{M_*}{M_{\odot}}\right)^{\frac{1}{2}} m_\oplus ,
\label{eq:minmass}$$ and assuming typical spacing, the number of planets in the gap is $$N = 1 + \frac{\log{\left(\frac{a_2}{a_1}\right)}}{\log{\left(\frac{1 + 0.13\left(\frac{m_p}{m_\oplus}\right)^{\frac{1}{3}}\left(\frac{M_{\odot}}{M_*}\right)^{\frac{1}{3}}}{1 - 0.13\left(\frac{m_p}{m_\oplus}\right)^{\frac{1}{3}}\left(\frac{M_{\odot}}{M_*}\right)^{\frac{1}{3}}}\right)}} \ .
\label{eq:numberofplanets}$$
Equation \[eq:numberofplanets\] can also be applied to the upper mass limit derived from observational non-detection to envision the maximal planetary system and thus produce a complete picture of what planetary systems could lie within the system (figure \[fig:schematic\]).
There is some scatter about the relation but the only systematic trend is the breakdown when the mass of planets is large, corresponding to number of planets being small (i.e., for systems that do not obey equation \[eq:minnum\]). This result is for equal mass planets with spacings typical of extra-solar systems; one might expect unusually compact or sparse systems to clear faster or slower. For unequal masses, the situation is likely to be more complicated, but for the application considered here, it is reasonable to assume that the clearing will proceed no faster than equation \[eq:formfit\] for the most massive planet. For unequal spacings, @2015ApJ...807...44P showed that the [*first*]{} instability occurred sooner than for equal spacings. However, here we consider the case of the average instability time for a very large number ($10^{6} \sim 10^{12}$) of planetesimals, and so expect the difference between equal and unequal spacings to be minimal.
Real systems
============
Validation
----------
### A young system: HR 8799
HR 8799 is orbited by four giant planets [@2008Sci...322.1348M; @2010Natur.468.1080M], nestled snugly between two debris disks [@2009ApJ...705..314S], making it an ideal case for evaluating this model. The outer edge of the cleared zone is $\sim 145~\rm{au}$ [@2016MNRAS.tmpL..24B]. The most common age estimate is $\sim 30~\rm{Myrs}$ [@2010lyot.confE..42D; @2011ApJ...732...61Z; @2012ApJ...761...57B], and the stellar mass is $M_* \sim 1.5~M_{\odot}$ [@1999AJ....118.2993G]. From equation \[eq:minmass\], this requires a minimum planet mass of $\sim~285~m_{\oplus}$ to have cleared the gap between the two belts. With the inner belt extending to $\sim 15~\rm{au}$, this mass requires $\sim 2.3$ planets to fill the gap (equation \[eq:numberofplanets\]). Since we require an integer number of planets, the minimal planetary system is three $285~m_\oplus$ planets, packed more tightly than average. This is about a factor of $\sim 10$ less than the best estimates of the masses of the four planets [@2011ApJ...729..128C], so our inferred lower limit is indeed compatible with the real system, whose unusual compactness is due to its special dynamic state [@2010ApJ...710.1408F; @2014MNRAS.440.3140G].
### An old system: Solar System
The Solar system has a gap between its two debris disks which extends from the asteroid belt at about 3.5 au to the Kuiper belt at around 39 au. With an age of $4.56 \times 10^9$ years, the minimum planet mass needed to clear the gap in our system is $\sim 0.2~m_\oplus$, roughly twice the mass of Mars, about seventeen of which would fit between the two belts. The four planets observed between the asteroid and Kuiper belt are indeed more massive than this minimum. Thus, we show applying our model to both the young system HR 8799, and the old Solar system, the only two systems for which we have good data on multiple planets between two debris disks, we recover a minimum planet mass that is less than the observed planet masses.
Example Future observations
---------------------------
### HD 38206
New planet finding instruments such as GPI [@2014PNAS..11112661M] and SPHERE [@2008SPIE.7014E..18B] are able to image planets down to a few $m_J$. For instance, an early paper on the 20 Myr system PZ Tel suggested a detection limit of $\sim$3$~m_J$ at 0.5" . We consider the double debris disk star HD 38206, a 30 Myr old A0V star with a mass of $2.3~M_{\odot}$ at a distance of 75 parsecs. HD 38206 was identified as a two temperature debris disk likely to contain two debris belts by @2014MNRAS.444.3164K. Assuming blackbody grains, the debris rings are located at 15 au and 180 au. We estimate the best contrast achievable by direct imaging as the 5-sigma contrast limit from unpublished SPHERE data for other systems (Matthews et al., in prep), by measuring the standard deviation in concentric annuli of the reduced image. This contrast is scaled to the distance and host magnitude for HD38206, and the detection limit is then converted to a mass limit using the COND models . Thus, if an observation of HD 38206 results in a non-detection of planets, the most massive planets that may be present would have $m_p \sim 5~m_{J}$, although it depends slightly on the separation from the star. Using equation \[eq:minmass\], we calculate the minimum mass of planets needed to clear the gap in 30 Myrs to be $\sim 1.4~m_{J}$. The mass limits correspond to three planets in the lower case, and two planets in the upper case. We plot this example in figure \[fig:sphereexample\].
Thus, by exploiting our knowledge of the debris disk, we are able to complement the upper mass limit derived from imaging with a lower mass limit derived from dynamics. For this system this leads to an approximate knowledge of the planetary system, with the masses of the unseen planets known to less than an order of magnitude, and their number to $\pm1$.
Discussion
==========
Implicit in this approach is the assumption that planets form surrounded by a sea of planetesimals that still retain a significant fraction of their mass. There is some theoretical basis to believe planet formation may be $\sim 50\%$ efficient . There is some circumstantial observational evidence of this; the estimated mass of the Oort cloud [@2005ApJ...635.1348F; @2015MNRAS.454.3267F] and calculated fraction of small bodies that end up in the Oort cloud [e.g. @2013Icar..225...40B; @2015MNRAS.446.2059S] imply the mass of solids scattered by the planets was comparable to the mass of solids in the planets. Similar mass clouds may be commonly present around other stars [@2014MNRAS.445.4175V]. Modelling of debris disks also suggests their total mass is comparable to the solid mass of planetary systems . The observational evidence does not strongly indicate that the proto-comets were co-spatial with the planets; if future observations fail to find the minimal planetary systems envisioned here, it will be significant evidence that planets do not clear gaps, but rather that planetesimal gaps form because planet formation is $\sim 100\%$ efficient, or that giant planets clear gas gaps that also removes solids [as in @2016ApJ...825...77D].
Very recently, @2016ApJ...823..118M published a study on the maximal planetary system that can fit dynamically between two debris disks. This provides a stronger constraint on older systems, and thus might provide a more stringent upper limit than direct imaging for older systems.
This model necessitates a caveat: we have neglected the mass of the planetesimals in our study. If the mass of planetesimals is comparable to, or in excess of, that of the planets, they may cause migration of the planets [@1984Icar...58..109F]. @2014Icar..232..118M published a set of criteria for when planets in a planetesimal disk may start to migrate. If the minimum planetary system predicted by this study is such that migration might occur during the clearing phase, the model presented here may be inappropriate. For the young systems most favourable to direct imaging, and most likely to host double debris disks, the minimum mass will be higher (equation \[eq:minmass\]), and migration is unlikely to be a concern. For instance, for HD 38206 we inferred at least $1500~m_{\oplus}$ in planets, while a typical A star debris disk is inferred to have a mass of $\sim 10~m_{\oplus}$ [@2007ApJ...663..365W]. Consequently, from @2014Icar..232..118M we expect no migration, which only occurs for $m_p < 3~m_{disk}$. A massive disk would also gravitationally self-excite, spreading the planetesimals [@1996Icar..123..180K], and viscously spreading the small bodies . This could allow them to encounter secular resonances and be cleared on shorter timescales. As the spreading will depend on the mass and size distribution in the debris, there is no good way to estimate the appropriate timescale.
Summary
=======
We present a simple equation for the minimum mass of planet needed to clear the gap in double debris disk systems (equation \[eq:minmass\]), and the number of such planets that would typically be found in the gap (equation \[eq:numberofplanets\]). At least one direct imaging survey [@2015ApJ...800....5M] has begun targetting double debris disks to search for planets. Currently, if no planets are detected, we can only infer that planets with masses less than the detection limit may lie within the gap. By imposing constraints on both the minimum and maximum planetary systems that could be present, the observational non-detection of planet(s) can be recast as more positive knowledge about the planetary system harboured by the star in question. The use of the clearing time provides the strongest constraints on young systems, as does direct imaging , providing a natural synergy.
Acknowledgements
================
Andrew, Amy, and Quentin are supported by the European Union through ERC grant number 279973. We thank Sasha Hinkley for prompting us to consider the problem, Grant Kennedy for useful discussions, and the anonymous referee whose comments led to an improved manuscript.
[^1]: @1999ssd..book.....M exercise question 7.1
|
---
abstract: 'Deep convolutional neural networks (ConvNets) of 3-dimensional kernels allow joint modeling of spatiotemporal features. These networks have improved performance of video and volumetric image analysis, but have been limited in size due to the low memory ceiling of GPU hardware. Existing CPU implementations overcome this constraint but are impractically slow. Here we extend and optimize the faster Winograd-class of convolutional algorithms to the $N$-dimensional case and specifically for CPU hardware. First, we remove the need to manually hand-craft algorithms by exploiting the relaxed constraints and cheap sparse access of CPU memory. Second, we maximize CPU utilization and multicore scalability by transforming data matrices to be cache-aware, integer multiples of AVX vector widths. Treating 2D ConvNets as a special case, we demonstrate a 5 to 25-fold improvement in throughput compared to previous state-of-the-art.'
bibliography:
- 'refs.bib'
---
Introduction
============
Although convolutional neural networks (ConvNets) have been successfully applied to solve non-trivial image processing problems since the 1990s [@lecun1989backpropagation; @lecun2012efficient], their adoption as a de facto standard for image classification [@russakovsky2015imagenet] and segmentation [@long2015fully] is due largely to recent breakthroughs in network architecture. Beginning with AlexNet in 2012 [@krizhevsky2012imagenet], the annual ImageNet classification challenge (ILSVRC) has been dominated by progressively deeper networks with smaller kernels [@szegedy2015going; @simonyan2014very]. Recent solutions to the issues of vanishing and exploding gradients [@glorot2010understanding] have allowed these networks to extend even deeper, with the ILSVRC15 winner (ResNet [@he2016deep]) being 8-fold deeper than VGG.
It is easy to see why the “deeper is better" trend has led to better performing ConvNets. Constructing even a modest 7 x 7 receptive field with stacked $k=3$ kernels requires $45\%$ fewer parameters than a single kernel of size $k=7$. Intuitively it also captures a richer set of features due to additional non-linearity. Recent studies have begun to formalize the expressive power of deep versus shallow networks, finding that classification boundaries acquire local curvature and expressivity as an exponential function of network depth but not breadth [@poole2016exponential]. The only obvious trade-off to this performance is the extra memory necessary to store the intermediate activations in deeper networks.
Motivated by the success of these models in image processing tasks, researchers have begun to investigate ConvNet applications in the video processing domain. Example applications include video classification [@karpathy2014large], segmentation [@couprie2013indoor] and de-noising [@shi2016real]. An important observation that has emerged from these studies is the importance of 3D convolutional primitives for modelling joint spatio-temporal features; the naïve application of traditional 2D ConvNets frame-by-frame does not capture motion continuity or other rich temporal correlations [@ledig2016photo; @tran2015learning]. It is thus unsurprising that simple 3D ConvNets have yielded state-of-the-art performance on video classification benchmarks [@tran2015learning] and volumetric image segmentation, e.g. tracing neurons between electron microscopy samples [@lee2015recursive].
Given the early success and conceptual simplicity of 3D ConvNets, it is interesting to note that many popular deep learning libraries (e.g. Caffe [@jia2014caffe]) do not provide native support. One simple explanation is that these libraries are optimized for execution on GPUs, and higher-order convolutions require prohibitively large volumes of data with respect to the 16 GB ceiling of today’s most advanced GPU hardware. These limitations are clear in previous studies, which either (a) limit the network size [@tran2015learning], (b) down-sample images to lower resolution [@ji20133d], or (c) include 3D primitives for only a subset of network layers [@lee2015recursive].
There are many potential options for circumventing the issue of ConvNet memory usage. The first is to split the network across multiple GPUs, which requires the careful coordination of activation and gradient flow [@dean2012large]. Even in the case of the most successful distributed frameworks for ConvNets [@abadi2016tensorflow], GPU memory management is largely unresolved. The TensorFlow authors propose two partial solutions warranting further investigation: (a) re-computing versus storing large tensors; and (b) transferring long-lived tensors from GPU to host CPU memory. Instead, we propose an alternative to horizontal scalability for overcoming GPU memory constraints – a fast implementation of $N$-dimension convolution optimized for multicore CPU systems, which have access to practically unbounded memory on a single node.
Prior Art
=========
Algorithms for fast convolution have existed in signal processing literature since the 1980s [@winograd1980arithmetic]. The general recipe is to transform both data and kernel into a new space, where expensive sliding window-style convolutions reduce to cheaper element-wise products. The first examples of this approach in ConvNet literature involved Fast Fourier Transforms (FFTs) exploiting the convolution theorem [@mathieu2013fast; @vasilache2014fast]. More recently, Lavin and Gray have pioneered the use of the more general class of Winograd-style algorithms [@lavin2016fast; @winograd1980arithmetic]. Their implementation and its cuDNN derivatives [@chetlur2014cudnn] have produced state-of-the-art GPU performance on deep networks of small kernels. In this Section we provide a brief overview of the theory underlying this approach, focusing on the aspects that are important for exploiting the architecture of multicore CPUs.
Fast Vector Convolution {#ss:vector}
-----------------------
Consider the 1-dimension convolution $\mathbf{s} = \mathbf{g} * \mathbf{d}$, where the kernel and data vectors are of length $G$ and $D$. This problem can be rephrased as one of polynomial multiplication by introducing the associated polynomials $d(x)$, $g(x)$ and $s(x) = g(x)d(x)$, where the coefficients $s_i = \sum_{k}g_{i - k}d_k$ of $x^{i}$ are the solution to the desired convolution. This computation can be distributed across a set of efficient local computations by considering the Chinese Remainder Theorem (CRT) [@ding1996chinese], as summarized in Theorem 1. By observing that $s(x) = \left[g(x)d(x)\right] \,\mathrm{mod}\, m(x)$ for any polynomial $m(x)$ of sufficiently high degree, we can exploit Theorem 1 to efficiently calculate $s(x)$, as shown in Algorithm 1, which can be conveniently rephrased in terms matrix-vector products: $$\label{eq:winograd_1d}
\mathbf{s} = \mathbf{A} \left[ \,(\mathbf{Cg}) \odot (\mathbf{Bd}) \, \right],$$ where $\mathbf{C}$, $\mathbf{B}$ and $\mathbf{A}$ are introduced as the kernel, data and inverse transforms respectively. With respect to Algorithm 1, Step (1) is implemented by the kernel and data transforms, Step (2) by their transformed element-wise product and Step (3) by the final inverse transform.
\
Let $m(x)=\Pi_{k=1}^{r} m^{(k)}(x)$, where $m^{(k)}(x)$ are pairwise coprime. If $b^{(1)}(x),...,b^{(r)}(x)$ are a set of polynomials then there must exist a unique polynomial $s(x)$ which satisfies the set of congruences: $$\begin{aligned}
s(x) &\equiv b^{(1)}(x)\,\mathrm{mod}\,m^{(1)}(x) \\
s(x) &\equiv b^{(2)}(x)\,\mathrm{mod}\,m^{(2)}(x) \\
\vdots \\
s(x) &\equiv b^{(r)}(x)\,\mathrm{mod}\,m^{(r)}(x),
\end{aligned}$$ provided the degree of m(x) is not less than that of s(x).
$g(x)$, $d(x)$, $m(x)$ (1) Compute residual polynomials for $g(x)$ and $d(x)$: $$\begin{aligned}
g^{(k)}(x) \equiv g(x)\,\mathrm{mod}\,m^{(k)}(x)\\
d^{(k)}(x) \equiv d(x)\,\mathrm{mod}\,m^{(k)}(x)
\end{aligned}$$ (2) Compute residual polynomial multiplications: $$s^{(k)}(x) = \left[g^{(k)}(x) d^{(k)}(x)\right]\,\mathrm{mod}\,m^{(k)}(x)$$ (3) Reduce partial convolutions to solve $s(x)$: $$s(x) = \sum_{k=1}^{r} s^{(k)}(x)a^{(k)}(x)$$
Minimal Winograd Algorithms {#ss:minimal}
---------------------------
In the above formulation (\[eq:winograd\_1d\]), the matrices $\mathbf{C}$ and $\mathbf{B}$ are the remainders of the polynomial divisions $g(x)$ and $d(x)$ by $m^{(k)}(x)$ respectively. The derivation of $\mathbf{A}$ is more involved and is omitted for brevity. Importantly, the only parameter required to synthesize these matrices (in addition to the kernel $g(x)$ and data $d(x)$) is the polynomial $m(x)$.
Traditionally, the selection of $m(x)$ has been subject to many constraints. First, it should be chosen such that the transform matrices contain only degree-1 (scalar) values. Lavin has published code that automates this procedure using the Cook-Toom algorithm to produce transformed kernels and data both of length $D$ [@lavin_git]. For an unpadded convolution $\mathbf{s} = \mathbf{g} * \mathbf{d}$ of length $S = D - G + 1$ and ignoring the cost of applying the transforms, this fast algorithm therefore requires $SG/D$ fewer computations to calculate than the standard sliding-window approach. Inappropriate selection of $m(x)$ would yield matrices of polynomials (degree $> 1$) that require considerably more scalar multiplications to compute. In reality, the transformations themselves require expensive matrix multiplications that can outweigh the above saving. Accordingly, existing implementations of fast convolution aim to synthesize matrices enriched for “simple" (e.g. integer) values. There are two motivations for this. First, it improves numeric stability which can have an impact on double-precision convolutions [@lavin2016fast]. More importantly, it supports the hand-crafting of minimal algorithms. These algorithms reduce the cost of applying transform matrices by identifying and eliminating redundant sub-expressions. A famous instance of this approach was documented by Winograd [@winograd1980arithmetic]. Consider the following matrices: $$\begin{aligned}
\mathbf{A} &= \begin{bmatrix}
1 & 1 & 1 & 0 \\
0 & 1 & -1 & -1
\end{bmatrix} \\
\mathbf{B} &= \begin{bmatrix}
1 & 0 & -1 & 0 \\
0 & 1 & 1 & 0 \\
0 & -1 & 1 & 0 \\
0 & 1 & 0 & -1
\end{bmatrix} \quad \mathbf{C} = \begin{bmatrix}
1 & 0 & 0 \\
\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\[0.3em]
\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\[0.3em]
0 & 0 & 1
\end{bmatrix}.\end{aligned}$$ By substituting these matrices into (\[eq:winograd\_1d\]) and factoring out redundant computations, we arrive at the following minimal algorithm for vector convolution: $$\mathbf{d} * \mathbf{g} = \begin{bmatrix}
m_1 + m_2 + m_3 \\[0.3em]
m_2 - m_3 - m_4
\end{bmatrix},$$ where: $$\begin{aligned}
m_1 = (d_0 - d_2)g_0, \quad m_2 &= (d_1 + d_2)\frac{g_0 + g_1 + g_2}{2}, \\
m_4 = (d_1 - d_3)g_2, \quad m_3 &= (d_2 - d_1)\frac{g_0 - g_1 + g_2}{2}.\end{aligned}$$ This is a so-called $F(S,G)$ algorithm for vector convolution, here for $S = 2$ and $G = 3$. Importantly, this algorithm only works for fixed length kernel and data vectors (here $D = 4$). Generating $F(S, G)$ algorithms for different combinations requires both (a) searching over the space of possible $m(x)$ polynomials as input to Lavin’s or similar code [@lavin_git], and (b) reducing the matrix multiplications to a minimal set of addition, multiplication and shifting operations. To our knowledge there are no automated solutions to either step and thus only a small set of hand-crafted Winograd-style algorithms (e.g. $F(2,3)$, $F(3,4)$ and $F(2,5)$) have been released as fast CPU [@nnpack] or GPU primitives [@chetlur2014cudnn].
Deep Tensor Convolution
=======================
Below we present an alternative approach to fast convolution that removes the need to hand-craft minimal algorithms. This new approach is better suited to video and volumetric image processing for two main reasons. First, the number of terms involved in a closed-form solution for 3 and higher-dimensional convolutions makes Winograd-style refactoring impractical. Second, by removing numeric simplicity as a constraint we are instead able to synthesize transforms optimized to CPU architectural constraints, e.g. data that are integer multiples of the AVX register width. This is made possible by the relaxed memory constraints of CPUs and allows us to close the previous CPU-GPU performance gap by a full order-of-magnitude.
We first define $N$-dimensional convolution and describe how existing fast algorithms can be extended to this general case. Instead of crafting a minimal algorithm, we show how relaxed memory constraints and efficient sparse linear algebra of CPU systems can be leveraged to amortize transform costs. Later we show how architecture-aware transform synthesis can lead to further acceleration.
Convolution in $N$-Dimensions
-----------------------------
Mathematically, the standard convolutional layer used in 2D ConvNets extends trivially to higher-dimensional tensors. Consider a network where for each layer $i$, kernel $j$ and channel $m$, the kernel weights $\boldsymbol{\mathcal{G}}^{(i,j,m)} = (g\,_{p,q,r})$ and resulting feature map $\boldsymbol{\mathcal{D}}^{(i,j)} = (d\,_{x,y,z})$ are both 3D tensors. This calculation can be expressed element-wise as: $$\label{eq:3d}
d\,^{(i+1,j)}_{x,y,z} = f\left( b^{(i,j)} + \sum_m \sum_{p, q, r} g\,^{(i,j,m)}_{p,q,r} d\,^{(i,\, j)}_{x+p,\, y+q,\, z+r} \right),$$ [where $ b^{(i,j)}$ is the bias term and $f$ is a ReLU or other non-linear activation function. This extends to higher dimensions by looping over additional subscripts on $g$ and $d$.]{}
The dimensionality of feature maps is clearly preserved in (\[eq:3d\]), e.g. a video at the input produces a video at the output. The triple $(p, q, r)$-loop ranges from 0 to the layer-$i$ kernel size to perform sliding-window convolution, and the $m$-loop is a reduction over the previous layer’s output channels. This differs from previous studies where the temporal axis is encoded as network channels and flattened after the first layer [@karpathy2014large; @simonyan2014two], producing a single 2D image or class label at the output. These methods have been shown to produce less accurate results on a broad range of video processing tasks when compared to true 3D ConvNets [@tran2015learning].
It is also evident from (\[eq:3d\]) why higher-dimensional ConvNets suffer from issues of impractical memory consumption. Each layer of an $N$-dimensional network requires $\boldsymbol{\mathcal{G}}$ and $\boldsymbol{\mathcal{D}}$ to be stored as $N+2$ and $N+1$–dimensional tensors, owing to their operation over multiple kernels and channels. We believe that this multiplicative effect has likely stalled the adoption of the deeper network architectures that dominate image processing tasks, with recent studies instead compromising on network expressiveness to fit within the 16 GB memory constraints of today’s top-end GPUs [@ji20133d; @lee2015recursive; @tran2015learning].
Accelerating Tensor Convolution
-------------------------------
Sidestepping memory constraints by shifting from GPU to CPU hardware is conceptually trivial, as most popular ConvNet frameworks support execution on both CPU and GPU environments. However, the issue preventing the widespread adoption of CPU implementations is not a lack of software support but the large perceived gap between CPU and GPU performance. This is reminiscent of a large ongoing CPU-vs-GPU debate, with various studies claiming that GPUs provide anywhere from 100-to-1000x speed-up across broad problem domains [@lee2010debunking]. A recent review has demonstrated a similar performance gap in the order of 50x across the most popular ConvNet frameworks [@shi2016benchmarking]. Even if distributed GPU solutions like TensorFlow require tensors to be re-computed or swapped between GPU and host CPU memory [@abadi2016tensorflow], this overhead is easy to justify if the alternative is a 50-fold increase in single-node execution time.
Here we describe how fast algorithms for convolution can be extended to the general case of $N$-dimensional tensors, where the theoretical speed-up is a substantial $(SG/D)^N$. Although recent studies have begun to explore extensions of FFT-based convolution to 3-dimensions [@zlateski2016znni], to our knowledge there have been no attempts to extend Lavin and Gray’s Winograd-style approach [@lavin2016fast]. In order to extend the fast vector algorithm to 1 to $N$-dimensions, we consider the $n$-mode product of a tensor, $\boldsymbol{\mathcal{X}} \in \mathbb{R}^{I_1 \times I_2 \times \dots \times I_N}$, with a matrix, $\mathbf{U} \in \mathbb{R}^{J\times I_n}$, herein denoted as $\boldsymbol{\mathcal{X}} \times_n \mathbf{U}$ [@kolda2009tensor]: $$\label{eq:tensorprod}
(\boldsymbol{\mathcal{X}} \times_n \mathbf{U})_{i_1, \dots, i_{n-1}, j, i_{n+1}, \dots, i_N} = \sum_{i_n = 1}^{I_n}x_{i_1, \dots, i_N}u_{j, i_n}.$$ In our case $\mathbf{U}$ is sparse and $\boldsymbol{\mathcal{X}}$ is dense, so we implement (\[eq:tensorprod\]) such that $\mathbf{U}$ is traversed in the outermost two loops. We also introduce the following notation for brevity: $$\boldsymbol{\mathcal{X}} \times_{n=1}^{N} \mathbf{U}_n = \boldsymbol{\mathcal{X}} \times_{1} \mathbf{U}_1 \times_{2} \dots \times_{N} \mathbf{U}_N.$$
The fast algorithm for tensor convolution applies the transforms $\mathbf{C}_n$, $\mathbf{B}_n$ and $\mathbf{A}_n$ separately to each dimension $n$ of the kernel and data tensors, $\boldsymbol{\mathcal{G}}$ and $\boldsymbol{\mathcal{D}}$: $$\label{eq:winond}
\boldsymbol{\mathcal{S}} =\left[\,( \boldsymbol{\mathcal{G}}\times_{n=1}^{N} \mathbf{C}_{n} )\odot ( \boldsymbol{\mathcal{D}}\times_{n=1}^{N} \mathbf{B}_{n} ) \,\right]\times_{n=1}^{N} \mathbf{A}_{n}.$$ It is straightforward to show that (\[eq:winograd\_1d\]) is a special case of (\[eq:winond\]) by considering the following equivalence: $$\label{eq:equiv}
\boldsymbol{\mathcal{Y}} = \boldsymbol{\mathcal{X}} \times_n \mathbf{U} \Leftrightarrow \mathbf{Y}_{(n)} = \mathbf{UX}_{(n)},$$ where the matrix $\mathbf{X}_{(n)}$ is the mode-$n$ major unfolding of tensor $\boldsymbol{\mathcal{X}}$ [@kolda2009tensor]. In the 1-dimensional case, $\mathbf{X}_{(1)}$ is simply $\mathbf{x}$ and thus $\boldsymbol{\mathcal{X}} \times_1 \mathbf{U} = \mathbf{Ux}$. Likewise in 2D, as $\boldsymbol{\mathcal{X}}\times_1 \mathbf{U} = \mathbf{UX}$ and $\boldsymbol{\mathcal{X}}\times_2 \mathbf{U} = \mathbf{UX}^\top$ then (\[eq:winond\]) reduces to the case reported by [@lavin2016fast]: $$\mathbf{S} = \mathbf{A} \left[ \,(\mathbf{CGC^\top}) \odot (\mathbf{BDB^\top}) \, \right]\mathbf{A^\top}.$$
Amortizing Transform Costs
--------------------------
Manually reducing transform costs via Winograd-style minimal algorithms is important for 2-dimensional GPU implementations. However, this is less important for a CPU implementation of higher-dimensional convolution. The reasons are two-fold: (a) the matrix multiplication cost can be amortized across a larger number of kernels and channels due to relaxed memory constraints; and (b) CPUs are able to directly leverage the sparse structure of these matrices for further acceleration. Although efficient sparse linear algebra is possible on GPUs, this typically involves reshuffling sparse matrices into a dense representation (e.g. COO, CSR or ELLPACK [@grewe2011automatically]) and introduces unnecessary computational overhead.
As a simple example, consider Winograd’s minimal F(2,3) algorithm presented in Section \[ss:minimal\]. Computing the output $\mathbf{s}$ of length $S=2$ requires a total of 6 multiplications – 4 between the data and kernel, and 2 by a constant factor of 0.5. The 4 additions are ignored as modern CPUs can compute fused multiply-accumulate operations in a single cycle. By contrast, computing $\mathbf{s}$ explicitly by equation (\[eq:winograd\_1d\]) requires 28 multiplications – 4 for the element-wise product, 16 for the data transform and 8 for the inverse transform (assuming transformed kernels are cached at training time). Even leveraging sparsity in the transform matrices requires 19 multiplications, which is more than triple that required for Winograd’s minimal algorithm.
The game changes when one considers these approaches in the context of a ConvNet layer with multiple channels and kernels. Without loss of generality, assume the numbers of kernels and channels are both equal to $M$. As the inverse transform can be applied once over the reduced output and the data transform once across all kernels, the required number of multiplications is just $4M^2 + 24M$ (versus $6M^2$ for Winograd). This can be reduced further to $4M^2 + 15M$ by exploiting the sparsity of $\mathbf{A}$ and $\mathbf{B}$.
![Reduction in computations achieved by fast tensor convolution (forward pass) for a C3D kernel ($3\times3\times3$) as a function of number of layer channels and kernels. Dashed line indicates direct convolution baseline.[]{data-label="fig:speedup"}](fig1.pdf){width="0.83\columnwidth"}
-0.3in
Although it is also possible to restructure Winograd’s algorithm to exploit the size of the network, for larger networks the $4M^2$ multiplications required by the element-wise product quickly renders the linear transform cost negligible. It is also impractical to construct similar minimal algorithms in higher dimensions. Consider the C3D network of $3\times 3\times 3$ kernels that has yielded state-of-the-art performance across many video processing benchmarks [@tran2015learning]. As an example, we synthesize the following transform matrices such that convolution reduces to a $6\times 6\times 6$ element-wise product: $$\begin{aligned}
\mathbf{A} &= \begin{bmatrix}
1 &1 &1 & 1 & 1 & 0\\
0 &1 & -1 & \frac{1}{3} & -\frac{1}{3} & 0\\
0 & 1& 1 & \frac{1}{9} & \frac{1}{9} & 0\\
0 &1 & -1 & \frac{1}{27}& -\frac{1}{27} & 1
\end{bmatrix}\\
\mathbf{B} &= \begin{bmatrix}
\frac{1}{9} & 0 & -\frac{10}{9}& 0 & 1& 0\\
0 & -\frac{1}{9} & -\frac{1}{9} & 1 & 1 & 0\\
0 &\frac{1}{9} &-\frac{1}{9} & -1 & 1 & 0\\
0 & -\frac{1}{3} & -1 & \frac{1}{3} & 1 &0\\
0& \frac{1}{3} & -1 & -\frac{1}{3} & 1& 0\\
0 & \frac{1}{9} & 0 & -\frac{10}{9} & 0 &1
\end{bmatrix}\\
\mathbf{C} &= \begin{bmatrix}
9 & \frac{9}{16} & \frac{9}{16} & -\frac{81}{16} & -\frac{81}{16} & 0\\
0 & \frac{9}{16} & -\frac{9}{16} & -\frac{27}{16} & \frac{27}{16} & 0\\
0 & \frac{9}{16} & \frac{9}{16} & -\frac{9}{16} & -\frac{9}{16} & 1
\end{bmatrix}^\top.\end{aligned}$$ The theoretical ceiling on speed-up obtainable using these matrices is 8-fold, ignoring the cost of the matrix-tensor products required when applying (\[eq:winond\]). Figure \[fig:speedup\] demonstrates the actual reduction in computations as a function of kernels and channels. For a network of just 100 kernels and 100 channels, it is possible to obtain greater than 6-fold acceleration with respect to direct sliding-window convolution. This is triple the performance margin that could be gained if the network was constrained to 10 kernels and channels due to a lower memory ceiling.
We can further improve this performance margin by exploiting the sparsity of the matrices themselves, as it is comparatively straightforward to implement efficient sparse linear algebra for CPUs. One might worry that the transform matrix sparsity is inversely proportional to the degree of $m(x)$. However, this simply suggests that our fast algorithm is best suited for networks of small kernels, which is fortunately well-aligned with recent trends in deep ConvNet architecture [@he2016deep; @simonyan2014very; @szegedy2015going]. Sparsity and numerical precision also decrease as a function of $D$. In practice, the data matrix $\textbf{D}$ is not the full feature map (e.g. an ImageNet image) but rather one of many small, overlapping input tiles (each of size $D\times D$, stepping by $S$ along both axes) whose $S\times S$ outputs are stitched together to form the final convolution result. In Section \[ss:avxwino\] we discuss how the fully-automated nature of our implementation can leverage this property for further performance improvement.
Optimizing for CPU Architecture
===============================
There are a myriad of algorithmic tricks that can be applied to reduce the number of computations required for convolution. Consider the special case where our transforms are the discrete Fourier transform (DFT) and inverse DFT matrices. As the Fourier transform of a real-valued signal has Hermitian symmetry, the number of unique terms in the element-wise product can be reduced [@mathieu2013fast]. More generally, one could also apply the Strassen algorithm to reduce the number of steps required for matrix multiplication [@cong2014minimizing].
In practice, the merit of any of these approaches depends intimately on whether they can be implemented to effectively leverage hardware. Consider the 50-to-1 performance ratio observed between existing GPU and CPU implementations [@shi2016benchmarking]. For the devices used in this study (Titan X versus Xeon E7-8890), the ratio of theoretical throughput is actually less than to 5-to-1. This seems to suggest that current CPU performance limitations are largely issues of software rather than hardware.
Although some previous studies have discussed CPU-specific performance optimizations for neural networks [@vanhoucke2011improving], these guidelines have not necessarily translated to optimal implementations. For example, the Eigen 3.2 linear algebra library (used until recently by TensorFlow) does not provide native support for AVX (vectorized) instructions, introducing a tight bottleneck on theoretical throughput. Looking beyond a single core, a recent review demonstrates poor multicore scalability across all major ConvNet frameworks [@shi2016benchmarking]. Solving these two issues alone has the potential to close the CPU-GPU gap by a full order-of-magnitude, and this improvement is multiplicative with the algorithmic savings described earlier.
Single-Core Utilization
-----------------------
Although our fast algorithm requires theoretically fewer computations to execute than naïve convolution (e.g. 8-fold for C3D kernels), it is considerably more difficult to implement with high CPU utilization. Consider the element-wise product $\boldsymbol{\mathcal{G}}^\prime \odot \boldsymbol{\mathcal{D}}^\prime$, summed for each channel $m = 1\dots, M$ to produce the $N$-dimensional tensor $\boldsymbol{\mathcal{S}}^\prime$. We can compute the ratio of computations, i.e. 1 multiply and 1 accumulate operation per $(g,\,d)$-pair, to the volume of memory loaded: $$\frac{\mathrm{computations}}{\mathrm{memory\,accesses}} = \frac{2D^NM}{2D^NM} = 1.$$ Little’s Law shows this is problematic for effective CPU utilization, as convolution expressed in this form is bottlenecked by memory bandwidth [@little1961proof]. To solve this problem, recall that $\boldsymbol{\mathcal{D}}$ is one of many small, overlapping tiles that span the full-size feature map. Considering $T$ of these tiles, we introduce the following matrices: $$\label{eq:practical}
\hat{\mathbf{S}}^{(i)} = \hat{\mathbf{D}}^{(i)} \times \hat{\mathbf{G}}^{(i)},$$ where $\hat{\mathbf{D}}^{(i)} \in \mathbb{R}^{T\times M}$ (tiles-by-channels) and $\hat{\mathbf{G}}^{(i)} \in \mathbb{R}^{M\times K}$ (channels-by-kernels). Each matrix $i \in 1,\dots, D^N$ captures a single $(x, y)$ coordinate in the earlier $\boldsymbol{\mathcal{G}}^\prime \odot \boldsymbol{\mathcal{D}}^\prime$ element-wise product, which is fused with the channel-wise reduction into end-to-end matrix multiplications: $$\frac{\mathrm{computations}}{\mathrm{memory\,accesses}} = \frac{2D^NMTK}{D^N(MT + MK)} = \frac{2\,TK}{T+K}.$$
0.05in $\texttt{FMA}\left(\hat{\mathbf{s}}_{\,t,\, k}^{(i\,:\,i+W)},\, \hat{\mathbf{d}}_{\,t,\, m}^{(i\,:\,i+W)},\, \hat{\mathbf{g}}_{\,m,\, k}^{(i\,:\,i+W)} \right)$
As $T$ can be any number of the small $D^N$ input tiles, we can select $T = K$ to demonstrate a compute-to-memory ratio that grows linearly in the number of kernels. The fast convolutional form in (\[eq:practical\]) is also well-suited to a number of other practical CPU performance optimizations [@vanhoucke2011improving]. Foremost among these is the effective use of AVX (vectorized) and FMA (fused multiply-accumulate) floating-point SIMD operations. Consider the function `FMA`($\mathbf{x},\, \mathbf{y},\, \mathbf{z}$), which calculates the sum of vector $\mathbf{x}$ with the element-wise product $\mathbf{y} \odot \mathbf{z}$ and stores the result in $\mathbf{x}$, all in a single CPU cycle. This function can be leveraged for an efficient practical implementation of (\[eq:practical\]), as presented in Algorithm \[alg:efficient\] for a single tile-kernel pair $s_{\,t,\, k}^{(i)} \in \hat{\mathbf{S}}^{(i)}$ and an AVX vector of width $W$. An illustration of the 2-dimensional case is provided in Figure \[fig:avx\]. On our Xeon CPU with 256-bit AVX registers and two dedicated FMA units, this optimization alone can yield a 32-fold speed-up over naïve implementations. This margin is expected to double with the introduction of 512-bit AVX registers for Intel Skylake and Xeon Phi.
![Illustration of Algorithm \[alg:efficient\] using 2-dimensional ConvNets as an example. Both the element-wise product $\mathbf{G}^\prime \odot \mathbf{D}^\prime$ and reduction down $M$ channels are captured within matrix multiplication. Multiple elements in $\hat{\mathbf{s}}_{\,t,\, k}$ can be calculated simultaneously by filling AVX registers into-the-page. This technique generalizes trivially to $N$-dimensions by substituting $D^2$ for $D^N$.[]{data-label="fig:avx"}](fig5c.pdf){width="1.0\columnwidth"}
-0.3in
We benchmarked the performance of our fast convolution algorithm on a 1.44 TFLOP/s Xeon E7-8890 CPU and observe that it executes at $\sim$70% maximum utilization. This includes all steps from input to output, including all necessary data reshuffling. As a point of comparison, Intel’s own MKL convolutional primitive runs at just $20\%$ (excluding reshuffling) on the same processor. The Eigen 3.2. linear algebra library is lower utilization still, capped at just $3.5\%$ due to a lack of AVX and FMA support. Both of these libraries have been widely used by popular ConvNet frameworks including Caffe, CNTK, TensorFlow and Torch.
AVX-Aware Transform Synthesis {#ss:avxwino}
-----------------------------
The fully automated nature of our transform generation allows for the synthesis of transform matrices that optimize for CPU architectural constraints. From Figure \[fig:avx\], it is clear the full utilization can only be achieved if $D^N$ is an integer multiple of the AVX vector width $W$. This is an important optimization, as data volumes are constantly small (invariant of numbers of channels and kernels) and thus there is little opportunity to amortize padding overhead.
Table \[tab:transforms\] summarizes statistics for example transforms that we have generated for square 2 and 3-dimensional kernels, enumerated automatically using [@lavin_git]. In each case, we generate transforms for the smallest possible $\mathbf{D}\in\mathbb{R}^{D\times D}$ such that $SG/D > 1$ and $D^2 \,\mathrm{mod}\, W = 0$. The matrices are provided in the Supplementary Materials.
----- ----- ----- ----- -------------- -------------- -------------- ------ -------
$D$ $G$ $S$ $\mathbf{A}$ $\mathbf{B}$ $\mathbf{D}$ 2D 3D
(a) 4 2 3 0.33 0.50 0.25 2.25 3.38
(b) 4 3 2 0.25 0.50 0.33 2.25 3.38
(c) 8 4 5 0.20 0.31 0.19 6.25 15.63
(d) 8 5 4 0.19 0.31 0.20 6.25 15.63
(e) 8 6 3 0.17 0.31 0.21 5.06 11.39
----- ----- ----- ----- -------------- -------------- -------------- ------ -------
: Size, transform sparsity and algorithmic speed-up statistics for example transforms matrices. Associated matrices are provided in the Supplementary Materials.[]{data-label="tab:transforms"}
Multicore Scalability
---------------------
Single-core utilization is just one dimension of performance optimization. Many modern systems contain both multiple CPU chips, with shared access to host RAM; and multiple cores per chip, with shared access to faster L3 cache. We adopt a relatively simple parallelization scheme where threads simultaneously operate on different subsets of $T$ input tiles. To avoid memory contention and other concurrency issues we adopt the Cilk Plus work-stealing scheduler supported by GCC 4.8 [@blumofe1996cilk; @robison2013composable], simply applying its fork-join primitive to all for-loops with no iteration dependencies. The number of tiles $T$ per thread is empirically tuned to simultaneously maximize L3 cache utilization ($T$ cannot be too large) and compute-to-memory ratio ($T$ cannot be too small).
We observe that even this simple parallelization scheme yields near-optimal linear scalability. In Figure \[fig:multicore\] we present ConvNet throughput as a function of processor cores for both (a) our fast algorithm, and (b) our own multicore implementation of naïve convolution (which is comparatively simple to implement). Scalability is measured across a single convolution layer for a $1024\times 1024$ image with kernels of size $4\times 4$. To avoid NUMA issues relating to expensive inter-chip communication, we spawn independent instances for each CPU in our 4-socket shared-memory server such that all 18 threads in Figure \[fig:multicore\] are bound to a single chip. When using all 18 cores of our Intel Xeon E7-8890 CPU the scalability of (a) is 95% theoretically optimal. As a point of comparison, a recent review examined the scalability of popular ConvNet frameworks Caffe, CNTK, TensorFlow and Torch on a similar 16-core Xeon E5-2630 CPU [@shi2016benchmarking]. They reported multicore scalability ranging from $16\%$ (Caffe) to $42\%$ (TensorFlow), which is equivalent to a 2.3 to 5.9-fold improvement with our implementation.
![Multicore scalability of our cache-aware and Cilk-optimized implementations of (a) fast convolution, and (b) naïve convolution. Dashed line indicates theoretical scalability limit with respect to a single-core implementation. Executed on 18-core Intel Xeon E7-8890 processor with 45 MB L3-cache.[]{data-label="fig:multicore"}](fig4.pdf){width="0.92\columnwidth"}
-0.3in
Performance Benchmarking
------------------------
The most popular ConvNet benchmarks focus exclusively on GPU performance [@chintala2015convnet]. The only study we could find presenting thorough CPU benchmarking is that of Shi *et al.*, comparing the throughput of Caffe, CNTK, Tensorflow and Torch for the AlexNet and ResNet architectures [@shi2016benchmarking]. Although this is a useful study for ball-parking our multicore scalability, it is difficult to extrapolate fair comparisons to our overall system throughput for many reasons. Foremost is that the authors do not select CPU-optimized implementations. They adopt an earlier version of TensorFlow that uses the Eigen 3.2 library (no AVX/FMA support), and otherwise use the default framework-specific implementations of convolution rather than linking to optimized packages such as Intel MKL.
We benchmark 2D ConvNet performance against two popular frameworks: TensorFlow, using the newer Eigen 3.3 library (with AVX support); and Caffe, compiled to use Intel’s optimized MKL library. We consider the propagation time of a $224\times224$ ImageNet image through three convolution layers to capture any necessary inter-layer reshuffling. We choose this simple architecture over a named network because we are not interested in comparing execution times of pooling, fully-connected or other layers. We also select an obscure kernel size ($4\times4$) for which there have been no Winograd-style fast algorithms published, in order to demonstrate the generality of our implementation to arbitrary kernels. Each layer contains a modest 32 channels and 32 kernels for spreading the cost associated with applying transform matrices. Results presented are the fastest across batch sizes of 1, 20 and 200. An important innovation of our approach is that it is batch size-agnostic, making it suitable for single-image autoregressive models common in generative modelling and deep reinforcement learning.
![Measured throughput (megavoxels per second) of (a) our fast 2D convolution implementation (as a special case of our $N$-dimensional algorithm), (b) TensorFlow, using the latest Eigen 3.3, and (c) Caffe, using Intel MKL. Throughput is calculated by propagating $224\times 224$ images through 3 convolutional layers.[]{data-label="fig:tensorflow"}](fig6_new.pdf){width="0.85\columnwidth"}
-0.3in
Our performance benchmarks are presented in Figure \[fig:tensorflow\]. The single-core throughput of (a) our fast algorithm is 0.89 MVox/s, compared to (b) 0.18 for TensorFlow and (c) 0.19 for Caffe. Increasing cores from 1 to 18, our throughput improves to 10.9 MVox/s compared to 1.77 for TensorFlow and 0.41 for Caffe. This is equivalent to an approximate 5 to 25-fold improvement in overall performance. In terms of multicore scalability, this is (a) 68% versus (b) 55% and (c) 12%. We note that our performance here is lower than the $95\%$ presented in Figure \[fig:multicore\] for a larger input size (i.e. $T$ is much larger, yielding a better compute-to-memory ratio), and that the scalability for TensorFlow and Caffe are both similar to those reported in [@shi2016benchmarking].
Discussion
==========
Motivated by the recent success of 3-dimensional ConvNets in video and volumetric image processing [@lee2015recursive; @tran2015learning], we have proposed a transition to CPU hardware to overcome the memory constraints limiting the size and expressivity of these networks. Key to this transition is overcoming the impractical performance gap between existing CPU and GPU implementations. To achieve this, we extended previous algorithms of fast convolution to the $N$-dimensional case, yielding an order-of-magnitude reduction in computations for popular networks such as C3D. Importantly, our implementation diverges from previous studies that focus on the hand-crafting of minimal Winograd-style algorithms. We instead exploit the relaxed memory constraints, efficient sparse access and other architectural considerations of CPU hardware to overcome the cost of applying transform matrices.
The obvious alternative to our approach is to overcome memory constraints by splitting large networks across multiple GPU devices. Distributed frameworks such as TensorFlow are valuable for a broad class of machine learning problems, e.g. many of the data mining tasks faced by large organizations where the data itself is often sharded across different machines. However, it is important to recognize that the horizontal scalability paradigm is not a one-size-fits-all solution. Consider the increasing demand for real-time CPU solutions to image and video processing, particularly on mobile devices. Moving forward, we expect that intensive ConvNet-driven tasks such as video classification and de-noising will continue to migrate from the realm of academic research to practical realization [@shi2016real]. Efficient CPU implementations of ConvNets and other deep learning algorithms will play a fundamental role in this transition.
At the opposite end of the spectrum, some “big data" problems in the image processing domain are, counterintuitively, too big to be solved in a distributed setting. Consider the emerging field of high-throughput connectomics [@meirovitch2016multi]. Multi-beam electron microscopes image cross-sectional slices of neural tissue at nanometer-resolution, which are then segmented by ConvNets to reconstruct the 3-dimensional morphology and interconnectivity of individual neurons [@ronneberger2015u]. The major issue here is simply one of scale – a seemingly modest cubic millimeter volume of neural tissue takes several months to image at the TB/hr pace of modern electron microscopes, which exceeds maximum data transfer rates. To avoid introducing communication bottlenecks to the connectomics pipeline, it is necessary that segmentation can execute in real-time on a server physically co-located in the same room as the microscope [@lichtman2014big; @matveev2016]. Shared-memory CPU systems can support hundreds of cores and terabytes of memory in a single server, and it is critical that systems be implemented to exploit these valuable resources.
Treating 2D ConvNets as a special case of tensor convolution, our implementation yields 5 to 25-fold improved throughput compared to previous state-of-the-art on CPU. This is an important step toward bridging the performance gap between CPU and GPU hardware and is particularly important in the context of emerging hardware trends, e.g. Intel announcing that future generations of CPUs will contain dedicated deep learning accelerators. More importantly, we believe that removing constraints on 3D ConvNet size will herald new opportunities in the machine learning community; particularly in the context of generative models [@denton2015deep; @goodfellow2014generative], where rich temporal correlations are currently ignored when learning latent manifolds [@ledig2016photo].
Acknowledgements {#acknowledgements .unnumbered}
================
Support is gratefully acknowledged from the National Science Foundation (NSF) under grants IIS-1447786 and CCF-1563880, and the Intelligence Advanced Research Projects Activity (IARPA) under grant 138076-5093555.
|
---
abstract: 'Black-hole perturbation theory is a useful tool to investigate issues in astrophysics, high-energy physics, and fundamental problems in gravity. It is often complementary to fully-fledged nonlinear evolutions and instrumental to interpret some results of numerical simulations. Several modern applications require advanced tools to investigate the linear dynamics of generic small perturbations around stationary black holes. Here, we present an overview of these applications and introduce extensions of the standard semianalytical methods to construct and solve the linearized field equations in curved spacetime. Current state-of-the-art techniques are pedagogically explained and exciting open problems are presented.'
address: |
CENTRA, Departamento de Física, Instituto Superior Técnico,\
Universidade Técnica de Lisboa - UTL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal\
&\
Institute for Theory $\&$ Computation, Harvard-Smithsonian CfA, 60 Garden Street, Cambridge, MA, USA\
paolo.pani@ist.utl.pt
author:
- Paolo Pani
bibliography:
- 'slowrot.bib'
title: 'Advanced Methods in Black-Hole Perturbation Theory[^1]'
---
A perturbative approach to BH dynamics {#sec:intro}
======================================
The scope of these notes is to introduce some state-of-the-art tools to investigate the dynamics of small perturbations around stationary and axisymmetric black holes (BHs) at linear level. A perturbative analysis of BH dynamics is crucial in several contexts, ranging from astrophysics to high-energy physics. There exists a number of excellent reviews on the subject [@Nollert:1999ji; @Kokkotas:1999bd; @Ferrari:2007dd; @Berti:2009kk; @Konoplya:2011qq; @Cardoso:2012qm] to which we refer for details and for an exhaustive account of the literature. The stability analysis of BH spacetimes, BH ringdown after binary mergers, gravitational-wave emission in astrophysical processes and even the gravity/gauge correspondence are just the most noteworthy contexts in which BH perturbation theory is relevant. The problem is – directly or indirectly – reduced to solving the linearized dynamics of some fields on a curved background. Hence, we wish to address the following question: *“How do small perturbations propagate on the background of a stationary BH?”*
The dawn of BH linear perturbation theory dates back to 1957 due to the pioneering work by Regge and Wheeler. During the BH Golden Age (1963–1973) the field experienced a tremendous boost thanks to the fundamental contributions by Zerilli, Vishveshwara, Teukolsky and Press (original references can be found in the reviews mentioned above). Already in 1973, the master equations governing the massless perturbations of the Kerr metric were known for scalar, electromagnetic and gravitational perturbations and they have been more recently extended to include fields of spin $1/2$ and $5/2$.
The great advantage of Teukolsky’s equations is that the angular dependence has been completely separated by a suitable choice of the angular basis, written in terms of spheroidal harmonics. Therefore, the remaining equations only contain a radial and a time dependence: the problem is reduced to a $1+1$ evolution in the time domain or, due to the stationarity of the background, to a simple one-dimensional problem in the frequency domain.
Over the years several numerical methods have been implemented in order to solve the master equations subjected to some initial and boundary conditions in the time domain and to physically-motivated boundary conditions in the frequency domain. In the latter case, the equations reduce to a one-dimensional eigenvalue problem. Imposing boundary conditions at the BH horizon and at infinity singles out an infinite [@Kokkotas:1999bd] number of complex frequencies, $\omega=\omega_R+i\omega_I$. The nonzero imaginary part of the modes is due to dissipation, both at infinity (because of the emission of gravitational waves) and at the horizon (which behave as a one-way, viscous membrane). An important class of eigenfrequencies are the so-called BH quasi-normal modes (QNMs) which are thoroughly discussed in the reviews above and that we shall also discuss in some detail.
In the frequency domain, the most popular techniques to compute the BH eigenfrequencies include: WKB approximations, highly-efficient continued-fraction techniques, series solutions for asymptotically Anti de Sitter (AdS) BHs, Breit-Wigner resonance method for long-lived modes and monodromy techniques \[see also [@Fiziev:2012mw] for other approaches\]. The spectrum of spinning BHs is extremely rich and each of these methods is best-suited to explore some specific region [@Berti:2009kk].
Nevertheless Teukolsky’s approach – based on a Newman-Penrose tetrad decomposition in terms of the components of the Weyl tensor [@Chandra] – is limited to cases in which the angular dependence is separable. This is a fairly restrictive requirement, because the Kerr metric in four dimensions represents an exception in this regard. Indeed, separability usually requires that the background spacetime enjoys special symmetries. The underlying property that, at least in some cases [^2] allows for the exceptional separability of the perturbation equations on a Kerr spacetime is the fact that the latter is of Petrov type-D.
In recent times, it has become clear that the standard Teukolsky’s approach is inadequate to deal with more generic classes of background metrics, which naturally emerge in a variety of applications. We list here the most noteworthy ones:
- **Tests of the no-hair theorem.** Uniqueness theorems in general relativity (GR) guarantee that a stationary BH is necessarily axisymmetric and described by the Kerr-Newman metric. This is not generically the case in modified theories of gravity, whose spinning BH solutions deviate parametrically from their GR counterparts. Furthermore, even in those theories which share the same BH solutions as in GR [@Sotiriou:2011dz], the dynamics of linear perturbations is different [@Barausse:2008xv] and encodes information of the underlying theory. Near-future gravitational-wave observations will probe regions of strong gravitational field (e.g. by detecting the signal from a BH-binary merger or from the inspiral of small compact objects around supermassive BHs [@AmaroSeoane:2012km]) and are in principle able to detect deviations from the Kerr solution. This, however, requires (at the very least) to understand the gravitational-wave emission from nonKerr BHs in alternative theories and Teukolsky’s approach does not seem adequate in this case.
- **Spinning BHs in higher-dimensions.** Uniqueness theorems do not extend to dimensions $D$ higher than four and multiple spinning black objects with the same asymptotic charges are known when $D>4$. In addition, several axes of rotation exist and, correspondingly, these objects are characterized by multiple angular momenta. The phase diagram depends on the number of dimensions and on the angular momenta and it generically shows bifurcation points and phase transitions. Correspondingly, several spinning objects in higher dimensions are unstable [@Shibata:2010wz; @Dias:2009iu] and their linear stability analysis is still an open question in a generic setup. Furthermore, in a semiclassical treatment of BH evaporation, the calculation of greybody factors (which may be of direct interest for ongoing experiments [@Cardoso:2012qm]) relies on our ability to understand wave scattering in rotating BH spacetimes. Extensions of Teukolsky’s approach to dimensions higher than four are both challenging and of great relevance.
- **Kerr-Newman BHs in GR.** Despite the 40-year-long effort, gravito-electromagnetic perturbations on a Kerr-Newman metric do not appear to be separable in the standard Teukolsky’s formalism [@Chandra] \[see Ref. [@Pani:2013ija; @Pani:2013wsa] for a recent attempt in the slowly-rotation approximation discussed below\]. This is highly disappointing because, as mentioned above, the Kerr-Newman metric describes the most general stationary BH solution in GR and it is remarkably simple, being defined by three parameters only: the mass, the spin and the electric charge.
- **Massive bosonic perturbations of a Kerr BH.** Interestingly, not all probe-field perturbations are separable even in a Kerr background. Massive perturbations of spin equal or greater than one do not appear to be separable in the standard approach. Besides their theoretical interest *per sé*, light massive bosonic fields around spinning BHs give rise to interesting effects [@Arvanitaki:2009fg; @Cardoso:2011xi] which can be revealed by a linearized analysis. These fields are ubiquitous in extensions of the standard model, for instance in the so-called axiverse scenario [@Arvanitaki:2009fg] or in models describing light vector fields and massive gravitons [@Babichev:2013una; @Brito:2013wya]. Astrophysical signatures of the dynamics of these fields around BHs may open new windows to test particle physics beyond the standard model.
- **Astrophysical BHs.** Realistic BHs that are formed as end-states of sufficiently massive stars are surrounded by matter. The prototypical example are accretion disks, but spinning BHs are also believed to host magnetic fields and give rise to jet emissions. These configurations are typically dynamical and not particularly symmetric but, even when they can be approximately treated as stationary and axisymmetric, their gravitational perturbations are coupled to those of the surrounding matter, requiring some extension of the standard approach.
- **BHs in the gravity/gauge duality.** Last but not least, all previous considerations about the challenge of studying linear perturbations in nonKerr spacetimes apply to the case of asymptotically AdS BHs. These solutions are of great relevance in the so-called gravity/gauge duality and in its phenomenological applications to strongly-coupled condensed-matter systems. In the correspondence, some correlation functions and transport coefficients of the dual holographic theory are related to the lowest order BH QNMs and to the BH linear response in general [@Hartnoll:2009sz; @Berti:2009kk; @Pani:2012zz]. In this context, BHs endowed with nontrivial (scalar, electromagnetic, nonAbelian and fermionic) fields are usually considered, and understanding the thermalization processes in the dual theory relies on the ability of solving the linear dynamics on the hairy background.
Of course, in most cases listed above a linear analysis cannot be conclusive and must be complemented with exact solutions and extended by full-fledged numerical evolutions. The latter however, would greatly benefit by a detailed linearized analysis. The two approaches are often complementary to each other and have their own disjoint domain of validity.
The number of interesting applications that require advanced tools in BH perturbation theory is a good predictor of the relevance of the topics we are going to discuss and of the exciting time lying ahead those who will embrace this field.
### Notation {#notation .unnumbered}
Hereafter Greek indices stand for spacetime coordinates. Capital Latin indices are used to denote nonangular coordinates, whereas lower-case Latin indices at the beginning of the alphabet (e.g. $a,b,...$) denote angular coordinates. Latin indices in the middle of the alphabet (e.g. $i,j,k,n,I,L...$) denote unspecified indices (e.g. matrix indices). According to this notation, the four-dimensional coordinate vector reads $x^\mu=(y^A,z^a)$ with $y^A=(t,r)$ and $z^a=(\vartheta,\varphi)$. Unless otherwise stated, we adopt natural units $\hbar=G=c=1$.
Perturbations of nonspinning BHs {#sec:nonspinning}
================================
We start by discussing the simpler case of nonrotating BHs. We consider static and spherically symmetric spacetimes in four dimensions, although in the nonspinning case most of the discussion can be easily extended to higher dimensions and to other topologies (e.g. to higher-dimensional black branes). The line element reads $$ds^2\equiv g_{\mu\nu}dx^\mu dx^\nu=-F(r)dt^2+B(r)^{-1}dr^2+r^2 d\Omega^2+\delta g_{\mu\nu}dx^\mu dx^\nu\,,\label{metric}$$ where $F(r)$ and $B(r)$ are background quantities that depend on the specific solution and $\delta g_{\mu\nu}$ are first order terms. Our goal in this section is to derive the perturbation equations for $\delta g_{\mu\nu}$ and solve them numerically in a quite generic class of problems. In doing so, we shall present several methods that can be directly adapted to study perturbations of rotating metrics.
Harmonic decomposition
----------------------
In order to derive the perturbation equations, we follow a standard decomposition of the metric and possible other fields in tensor spherical harmonics [@Chandra]. The decomposition of the metric is based on the transformation properties of the ten components of the perturbation tensor $\delta g_{\mu\nu}$ under a rotation of the frame around the origin. When considered as covariant quantities on the sphere, they transform as three $SO(2)$ scalars $\delta g_{AB}$, two $SO(2)$ vectors $\delta g_{Aa}$ and one $SO(2)$ second–order tensor $\delta g_{ab}$ and they can be expanded in the complete basis constituted by the spherical harmonics of different rank.
Furthermore, perturbations naturally divide into two classes, accordingly to their transformation properties under parity, namely $$\delta g_{\mu\nu}(t,r,\vartheta,\varphi)=\delta g_{\mu\nu}^{\rm odd}(t,r,\vartheta,\varphi)+\delta g_{\mu\nu}^{\rm even}(t ,r,\vartheta,\varphi)$$ with $$\label{oddpart}
\delta g_{\mu\nu}^{\rm odd} =
\begin{pmatrix}
0 & 0 & h_0^\ell S_\vartheta^{\ell} & h_0^\ell S_\vartheta^{\ell} \\
* & 0 & h_1^\ell S_\vartheta^{\ell} & h_1^\ell S_\vartheta^{\ell} \\
* & * & -h_2^\ell\frac{X^\ell}{\sin\vartheta} & h_2^\ell\sin\vartheta W^\ell \\
* & * & * & h_2^\ell\sin\vartheta X^\ell
\end{pmatrix}\,,$$ $$\label{evenpart}
\delta g_{\mu\nu}^{\rm even}=
\begin{pmatrix}
g_{tt}^{(0)} H_0^\ell Y^\ell & H_1^\ell Y^\ell & \eta_0^\ell Y_{,\vartheta}^\ell& \eta_0^\ell Y_{,\varphi}^\ell\\
* & g_{rr}^{(0)} H_2^\ell Y^\ell & \eta_1^\ell Y_{,\vartheta}^\ell & \eta_1^\ell Y_{,\varphi}^\ell\\
* & * & r^2\left[K^\ell Y^\ell+G^\ell W^\ell\right] & r^2 G^\ell X^\ell \\
* & * & * & r^2\sin^2\vartheta\left[K^\ell Y^\ell-G^\ell W^\ell\right]
\end{pmatrix}\,.$$ where asterisks represent symmetric components, $Y^{\ell}=Y^{\ell}(\vartheta,\varphi)$ are the scalar spherical harmonics and we have defined $$\begin{aligned}
(S_\vartheta^{\ell},S_\varphi^{\ell})&\equiv&\left(-\frac{Y^{\ell}_{,\varphi}}{\sin\vartheta}
,\sin\vartheta Y^{\ell}_{,\vartheta}\right)\,.\\
(X^{\ell},W^{\ell})&\equiv&\left(2(Y^{\ell}_{,\vartheta\varphi}-\cot\vartheta Y^{\ell}_{,\varphi}),Y^{\ell}_{,\vartheta\vartheta}-\cot\vartheta Y^{\ell}_{,\vartheta}-\frac{Y^{\ell}_{,\varphi\varphi}}{\sin^2\vartheta}\right)\,. \label{XW}\end{aligned}$$ Here and in the following, a sum over the harmonic indices $\ell$ and $m$ (such that $|m|\leq\ell$) is implicit[^3]. Under parity transformations ($\vartheta\rightarrow\pi-\vartheta$, $\varphi\rightarrow\varphi+\pi$): polar and axial perturbations are multiplied by $(-1)^\ell$ and $(-1)^{\ell+1}$, respectively. The odd and even sectors are also referred to as “axial” and “polar” and we shall use the two notations indistinctly. The functions $(H_0,H_1,H_2,K,G,\eta_0,\eta_1)^{\ell}$ and $(h_0,h_1,h_2)^{\ell}$ only depend on $t$ and $r$ and describe the polar parity metric perturbations and the axial parity metric perturbations, respectively.
Depending on the number of polarizations of the graviton, there can be a residual gauge freedom in the metric perturbations that can be used to simplify the equations. For a massless graviton it is convenient to adopt the so-called Regge-Wheeler gauge, in which $\eta_i^\ell\equiv G^\ell\equiv h_2^\ell\equiv0$. In this gauge, we are then left with four polar functions and two axial functions. However, in modified theories of gravity the graviton can propagate more than two polarizations. For example a massive graviton propagates five degrees of freedom and there is no residual gauge freedom in the expansion above.
Finally, in presence of other fundamental fields, we decompose them in spherical harmonics of the corresponding type. Vector fields are decomposed in a basis of vector spherical harmonics, whereas scalar fields are decomposed in scalar spherical harmonics. This procedure is very general and can be performed in any spherically symmetric spacetime. Noteworthy, in this decomposition axial and polar perturbations belong to two separate sets of equations and also perturbations with different harmonic index $\ell$ are separated. For a given $\ell$ we are then left with two systems of equations, one for the axial sector and one for the polar sector, which completely characterize the linear response of the system.
Computing the eigenfrequencies {#sec:computingQNMs}
------------------------------
Typically, for a given $\ell$, the axial and polar sectors can be separately written as a coupled system of the form[^4]: $$\left[-\frac{d^2}{dt}+\frac{d^2}{dr_*^2}\right]\mathbf{Y}-\mathbf{V}(r)\mathbf{Y}=0\,. \label{systemt}$$ where $r_*$ are some suitable coordinate (we assume $r_*\to-\infty$ as $r\to r_+$ and $r_*\to\infty$ as $r\to \infty$), $\mathbf{Y}$ is a $N$-vector and $\mathbf{V}$ is a $N\times N$ matrix, which depends only on $r$ and $\ell$ and not on $t$ and $m$ if the background is stationary and spherically symmetric. It is often convenient to Fourier-transform to the frequency domain. By defining $\mathbf{Y}=\int dt e^{-i\omega t} \mathbf{\tilde{Y}}$, we get $$\left[\frac{d^2}{dr_*^2}+\omega^2-\mathbf{V}(r)\right]\mathbf{\tilde{Y}}=0\,. \label{system}$$ For brevity, in the rest of this section we omit the tilde, but all quantities have to be understood as Fourier transforms. Furthermore, we assume $V_{ij}\to0$ at the BH outer horizon, $r\to r_+$, and $V_{ij}\to \mu^2 \delta_{ij}$ at infinity, $r\to \infty$. The latter is the typical behavior of the potential for massive fields and it reduces to the more common massless case when $\mu=0$. Possible generalizations to different classes of potentials are straightforward and left for exercise. The case of asymptotically (A)dS spacetime, in which $V_{ij}\to\infty$ at infinity is discussed in the next sections.
When physically motivated boundary conditions at the horizon and at infinity are imposed, the system forms an eigenvalue problem for the frequency $\omega$. Our goal in this section is to compute the eigenfrequency spectrum.
Close to the horizon, the solution behaves as a superposition of ingoing and outgoing waves and physical boundary conditions require a purely ingoing-wave condition [@Berti:2009kk]. Therefore, the desired behavior of the solution close to the horizon reads: $$Y_i\sim e^{-i \omega r_*}\sum_n b_n^{(i)}(r-r_+)^n \qquad r\to r_+\,,\label{series_hor}$$ where $n>0$, $Y_i$ is the $i$th-component of $\mathbf{Y}$ and the coefficients $b_n^{(i)}$ can be computed in terms of $b_0^{(i)}$ by solving the near-horizon equations order by order. The general asymptotic behavior at infinity reads: $$Y_i\sim B_{(i)} e^{-k_\infty r_*}
+C_{(i)} e^{k_\infty r_*} \qquad r\to \infty\,,\label{BCinf}$$ where $k_\infty=\sqrt{\mu^2-\omega^2}$ and, without loss of generality, we choose the root such that $\rm{Re}[k_\infty]>0$. The boundary conditions $B_{(i)}=0$ define purely outgoing waves at infinity, i.e. QNMs [@Berti:2009kk]. In the case of massive perturbations the condition $C_{(i)}=0$ is also allowed and physically motivated. The latter defines states which are spatially localized within the vicinity of the BH and decay exponentially at spatial infinity, i.e. bound states [@Dolan:2007mj; @Rosa:2011my]. In fact, if such modes exist in a BH spacetime they are “quasi” bound because, even if they do not propagate energy to infinity, they dissipate energy at the BH event horizon. Dissipation at the horizon allows for interesting effects related to the superradiance of spinning BH spacetimes [@Teukolsky:1974yv; @Cardoso:2012zn; @Cardoso:2011xi] and may also produce instabilities [@Detweiler:1980uk; @Pani:2012vp], whose timescale $1/\omega_I$ can be computed within the linearized approximation.
### Matrix-valued continued-fraction method
Since the seminal work by Leaver [@Leaver:1985ax], it is well-known that many classes of eigenvalue problems in GR can be solved through continued-fraction techniques. This is a highly-efficient method which is well-suited to Schroedinger-like potentials that contain only (fractions of) powers of $1/r$. In this case, the eigenfunction can be written as a series whose coefficients satisfy a finite-term recurrence relation. A robust method to solve three-term recurrence relations is available and any higher-order recurrence relation can be reduced to a three-term one via Gaussian elimination [@Berti:2009kk]. The efficiency of this method makes it one of the optimal tools to solve linear eigenvalues problems. Here, we discuss a generalization of the method, to solve coupled systems of equations in the form [@Rosa:2011my; @PhysRevE.59.5344].
In order to optimize the recurrence relation, it is important to choose a suitable ansatz for the eigenfunctions. Let us consider the case in which the background solution has a single horizon $r_+$, such that $F(r_+)=0$[^5]. Then a convenient ansatz for the solution of the system reads: $$Y_i=e^{-i\omega r_*}r^{-\nu} e^{qr} \sum_n a_n^{(i)}F(r)^n$$ where $\nu$ is a constant that depends on the specific problem and $q=\pm\sqrt{\mu^2-\omega^2}$. In the case of massive fields, $\mu\neq0$, the sign of the real part of $q$ selects the correct boundary condition at infinity: the plus sign refers to QN frequencies, whereas the minus sign refers to quasi-bound states. Inserting the equation above into Eq. , it is possible to obtain a recurrence relation for the vectors $\mathbf{a}_n$. Let us start with a simple case and assume that the system reduces to a three-term matrix-valued recurrence relation: $$\begin{aligned}
&&\boldsymbol{\alpha}_0 \mathbf{a}_{1} + \boldsymbol{\beta}_0 \mathbf{a}_{0} = 0\qquad n=0\,, \label{recurrence0}\\
&&\boldsymbol{\alpha}_n \mathbf{a}_{n+1} + \boldsymbol{\beta}_n \mathbf{a}_{n} + \boldsymbol{\gamma}_n \mathbf{a}_{n-1} = 0\,, \qquad n > 0\,,\label{recurrencen}\end{aligned}$$ The matrices $\boldsymbol{\alpha}_n$, $\boldsymbol{\beta}_n$ and $\boldsymbol{\gamma}_n$ are generically nondiagonal for coupled systems. In this case, a three-term recurrence relation as the one above can be solved in the following way. First, we define the ladder matrix $\mathbf{R}_n^+$ such that $$\mathbf{a}_{n+1}=\mathbf{R}_n^+ \mathbf{a}_n\,.$$ Taking Eq. with $n\to n+1$, solving for $\mathbf{a}_{n+1}$ and using the equation above, we obtain: $$\mathbf{R}_n^+=-\left[\boldsymbol{\beta}_{n+1}+\boldsymbol{\alpha}_{n+1} \mathbf{R}_{n+1}^+\right]^{-1}\boldsymbol{\gamma}_{n+1}\,. \label{Rn}$$ Finally, the recurrence relation is solved by imposing Eq. , i.e. by searching for roots of the equation $\mathbf{M}\mathbf{a}_0=0$, with $$\mathbf{M}\equiv \boldsymbol{\beta_0}-\boldsymbol{\alpha}_0\left[\boldsymbol{\beta}_{1}-\boldsymbol{\alpha}_{1}(\boldsymbol{\beta}_2+\boldsymbol{\alpha}_2\mathbf{R}_{2}^+) \boldsymbol{\gamma}_2\right]^{-1}\boldsymbol{\gamma}_1\,,$$ where $\mathbf{R}_{2}^+$ is obtained recursively from Eq. . Therefore, for nontrivial solutions the eigenfrequencies are the roots of the determinant: $${\rm det}\mathbf{M}=0\,.$$ In practice, one usually fixes a large truncation order $N$ and initializes $\mathbf{R}_N^+$ arbitrarily. Then, Eq. is used to obtain $\mathbf{R}_{N-1}^+$, $\mathbf{R}_{N-2}^+$ and so on, down to $\mathbf{R}_2^+$. After this cascade of matrix-inversions, the matrix $\mathbf{M}$ can be constructed. Clearly, convergence of the results for different values of $N$ must be verified a posteriori.
This procedure might appear a bit abstract at first sight, but it is indeed straightforward to implement. In the notebook [CF\_matrix\_3terms.nb](CF_matrix_3terms.nb) [@webpage], we present a short implementation of the matrix-valued continued-fraction method to compute scalar, electromagnetic and gravitational modes of a Schwarzschild BH in GR. Since in this case the equations are decoupled, the matrices defining the recurrence relation are diagonal. Note that, with a few lines of code, it is possible to compute the modes of perturbations of different spin in a single step.
Let us consider the case in which the recurrence relation involves more than three terms. As an example, we consider a four-term recurrence relation: $$\begin{aligned}
&&\boldsymbol{\alpha}_0 \mathbf{a}_{1} + \boldsymbol{\beta}_0 \mathbf{a}_{0} = 0\,, \quad \hspace{5cm} n=0\,,\nn\\
%%%
&&\boldsymbol{\alpha}_1 \mathbf{a}_{2} + \boldsymbol{\beta}_1 \mathbf{a}_{1} + \boldsymbol{\gamma}_1 \mathbf{a}_{0} = 0\,, \quad \hspace{3.85cm} n =1\,,\nn \\
%%%
&&\boldsymbol{\alpha}_n \mathbf{a}_{n+1} + \boldsymbol{\beta}_n \mathbf{a}_{n} + \boldsymbol{\gamma}_n \mathbf{a}_{n-1} + \boldsymbol{\delta}_n \mathbf{a}_{n-2} = 0\,, \quad \hspace{1.375cm} n > 1\,,\nn \end{aligned}$$ where $\mathbf{a}_n$ is a $N$-dimensional vector and $\boldsymbol{\alpha}_n$, $\boldsymbol{\beta}_n$, $\boldsymbol{\gamma}_n$, $\boldsymbol{\delta}_n$, $\boldsymbol{\rho}_n$ and $\boldsymbol{\sigma}_n$ are $N\times N$ *invertible* matrices. The order of the recurrence relation can be reduced by using a matrix-valued version of the Gaussian elimination [@Leaver:1990zz; @Berti:2009kk]. By defining $$\begin{aligned}
\boldsymbol{\tilde\alpha}_n &=& \boldsymbol{\alpha}_n \,,\\
\boldsymbol{\tilde\beta}_0 &=& \boldsymbol{\beta}_0 \,,\\
\boldsymbol{\tilde\gamma}_0 &=& \boldsymbol{\gamma}_0 \,,\\
\boldsymbol{\tilde\beta}_n &=& \boldsymbol{\beta}_n-\boldsymbol{\delta}_n\left[\boldsymbol{\tilde\gamma}_{n-1} \boldsymbol{\tilde\alpha}_{n-1}\right]^{-1} \quad n>0\,, \\
%%%%
\boldsymbol{\tilde\gamma}_n &=& \boldsymbol{\gamma}_n-\boldsymbol{\delta}_n\left[\boldsymbol{\tilde\gamma}_{n-1} \boldsymbol{\tilde\beta}_{n-1}\right]^{-1} \quad n>0\,,\end{aligned}$$ the tilded matrices satisfy the same three-term recurrence relation as –. This procedure can be extended to reduce any matrix-value recurrence relation (provided some matrices are invertible) to a thee-term one[^6], which can be solved as explained above.
### Matrix-valued direct integration
It is possible to compute the characteristic frequencies of the system also using a direct integration shooting method [@Ferrari:2007rc; @Rosa:2011my; @Pani:2012bp]. The idea is to integrate the system from the horizon with boundary conditions outwards to infinity, where we impose either $B_{(i)}=0$ or $C_{(i)}=0$, depending on the physical problem at hand. The procedure is explained here in general and an example is given in the notebook [DCS\_DI.nb](DCS_DI.nb) [@webpage], where we compute the QNMs of a Schwarzschild BH in Dynamical Chern-Simons (DCS) gravity \[see also Section \[sec:example\] below\] with this method.
Let us start with a system of $N$ second-order ordinary differential equations (ODEs) for $N$ perturbation functions as in Eq. . Starting with a near-horizon solution as $\eqref{series_hor}$, a family of solutions at infinity is then characterized by $N$ parameters, corresponding to the $N$-dimensional vector of the near-horizon coefficients, $\mathbf{b_0}=\{b_0^{(i)}\}$ ($i=1,...,N$). At infinity we look either for exponentially decaying solutions, $C_{(i)}=0$, or for QNMs, $B_{(i)}=0$. In both cases, the spectrum can be obtained as follows. We first choose a suitable orthogonal basis for the $N$-dimensional space of the initial coefficients $b_0^{(i)}$. We perform $N$ integrations from the horizon to infinity and construct the $N\times N$ matrix: $$\begin{aligned}
\label{matrix_coupled}
\mathbf{S}(\omega)=
\begin{pmatrix}
A_1^{(1)} & A_1^{(2)} & ... & A_1^{(N)}\\
A_2^{(1)} & A_2^{(2)} & ... & ... \\
... & ... & ... & ... \\
... & ... & ... & ... \\
A_N^{(1)} & ... & ... & A_N^{(N)} \\
\end{pmatrix}\,,\end{aligned}$$ where $A\equiv B$ if we want to compute QNMs, whereas $A\equiv C$ if we want to compute quasi-bound states, respectively (of course, mixed boundary conditions are possible). The superscripts denote a particular vector of the basis, i.e. $A_{i}^{(1)}$ corresponds to $\mathbf{b_0}=\{1,0,0,...,0\}$, $A_i^{(2)}$ corresponds to $\mathbf{b_0}=\{0,1,0,...,0\}$ and $A_i^{(N)}$ corresponds to $\mathbf{b_0}=\{0,0,0,...,1\}$. Finally, the characteristic frequencies $\omega_0=\omega_R+i\omega_I$ are obtained by imposing $$\det{\mathbf{S}(\omega_0)}=0\,.\label{det_modes}$$ To summarize, by performing $N$ integrations from the horizon to infinity we can construct the single-valued complex function $\det{\mathbf{S}(\omega_0)}$ and the problem of finding the eigenfrequencies is reduced to finding the complex roots of this function. This can be implemented, for instance, by a simple one-parameter shooting method \[cf. notebook [DCS\_DI.nb](DCS_DI.nb) [@webpage] for an example\].
Note that a direct integration performs extremely well to compute quasi-bound state modes, because in this case the condition $C_{i}=0$ can be imposed from the leading behavior of the fields at infinity. In many cases, the accuracy of the results may exceed that achievable by continued-fraction techniques, whose convergence properties deteriorate in some case (e.g. for ultra-slowly-damped modes). On the other hand, the condition $B_{(i)}=0$ requires to extract the subdominant, exponentially suppressed behavior at large distance and this can be contaminated by numerical errors. This makes the direct-integration approach nonoptimally suited to compute QNMs. Nonetheless, if the imaginary part of the mode is sufficiently small compared to the real part, precise results can be obtained by integrating up to moderately large values of $r$ and including higher-order terms in the series expansion at infinity to reduce truncation errors. Typically this allows to compute the fundamental mode and possibly the first few overtones. In spite of this limitation, the direct integration technique is extremely flexible, because it does not rely on any particular property of the matrix-valued potential $\mathbf{V}(r)$ and can be applied to essentially any class of boundary value problems.
### Breit-Wigner resonance method
When dealing with complicated systems of coupled equations, the direct integration discussed above can be time demanding. In cases in which the eigenfrequency spectrum supports slowly damped modes, i.e. those with $\omega_I\ll\omega_R$, we can adopt an approximate procedure known as Breit-Wigner resonance method, or the standing-wave approach [@1969ApJ...158..997T; @Chandrasekhar:1992ey; @Ferrari:2007rc; @Berti:2009wx]. As we now discuss, the great advantage of this method is that the eigenvalue problem can be solved by looking for minima of a real-valued function of a real variable [@Chandrasekhar:1992ey].
By expanding Eq. about $\omega_R$ and assuming $\omega_I\ll
\omega_R$ we get [@Ferrari:2007rc] $$\det{\mathbf{S}(\omega_0)}\simeq\det{\mathbf{S}(\omega_R)}
+i\omega_I\left.\frac{d\left[\det{\mathbf{S}(\omega)}\right]}{d\omega}\right|_{\omega_R}=0\,.$$ We consider the function $\det{\mathbf{S}}$ restricted to real values of $\omega$. Using the relation above, a Taylor expansion for real $\omega$ close to $\omega_R$ yields: $$\det{\mathbf{S}(\omega)}\simeq\det{\mathbf{S}
(\omega_R)}\left[1-\frac{\omega-\omega_R}{i\omega_I}
\right]\propto\omega-\omega_R-i\omega_I\,.$$ Therefore, in the region of the real–$\omega$ axis close to the real part of the mode, we have $$|\det{\mathbf{S}\left(\omega\right)}|^2\propto
\left(\omega-\omega_R\right)^2+\omega_I^2\,,\label{parabola}$$ that is, the function $|\det{\mathbf{S}}|^2$ is simply a parabola when $\omega\approx\omega_R$. To summarize, to find the slowly-damped modes it is sufficient to integrate the system $N$ times for real values of the frequency $\omega$, construct the matrix ${\mathbf{S}\left(\omega\right)}$ and find the minima of the function $|\det{\mathbf{S}}|^2$, which represent the real part of the modes. Then the imaginary part (in modulus) of the mode can be extracted through a quadratic fit, as in Eq. . We postpone an application of the Breit-Wigner method to the case of slowly-rotating BHs discussed below, in which the great efficiency of this method becomes evident.
### Matrix-valued series method
The methods discussed so far can be implemented in asymptotically flat spacetimes (and with some minor modification in asymptotically de Sitter spacetimes). However, computing the eigenfrequencies of BHs in asymptotically AdS spacetime usually requires a separate treatment, due to the different behavior of the fields at the AdS boundary. When the problem is described by a single second-order ODE, a series method [@Horowitz:1999jd] proves to be very efficient. In this method, local solutions near the regular singular points (at the horizon and at spatial infinity) are represented in terms of convergent Frobenius series. In various cases of interest, the radius of convergence of the series is equal to or larger than the interval of interest. This is the case of large spherically symmetric BHs (i.e. $r_+\gg L$, $L$ being the AdS radius) or of black branes. On the other hand, for small BHs ($r_+\ll L$) the convergence properties of the series are very poor. In such cases, if the spectrum supports slowly-damped modes, these can be computed by the Breit-Wigner method discussed above [@Berti:2009wx]. Since large AdS BHs and black branes are relevant in the context of the gauge/gravity duality [@Hartnoll:2009sz], here we discuss an extension of the series method to deal with coupled systems which arise quite naturally in the case of hairy AdS BHs [@Delsate:2011qp].
We consider a coupled system of $N$ equations as in Eq. . The near-horizon behavior of the solution is given by Eq. but, at variance with the asymptotically flat case, the equations present a regular singularity at spatial infinity. Generically, near spatial infinity the solution behaves as $$Y_i\to A_i r^{\alpha_i}+B_i r^{\beta_i} \,, \label{BCinfAdS}$$ where $\alpha_i$ and $\beta_i$ depend on the specific problem at hand. Typically, $\alpha_i\beta_i<0$, so that only one of the two terms above is regular. By imposing that the coefficient of the irregular term vanishes (Dirichlet boundary conditions) the eigenvalue problem is specified. In some cases, for example for gravitational perturbations of a Schwarzschild-AdS BH, both terms in Eq. are regular and several inequivalent choices of the boundary conditions are possible. In the context of the AdS/CFT correspondence, Robin boundary conditions have been also considered [@Moss:2001ga; @Dias:2013sdc]. Here for simplicity we focus on Dirichlet boundary conditions, i.e. $Y_i\to0$ at spatial infinity.
By defining a new variable $x=1/r$, and factorizing the near-horizon behavior $Y_i(x)=e^{-i\omega r_*}Z_i(x)$, the system of equations can be written in the form: $$(x-x_+)s(x)\frac{d^2\mathbf{Z}}{dx^2}+ t(x)\frac{d\mathbf{Z}}{dx}+\frac{\mathbf{u(x)}}{x-x_+}\mathbf{Z}=0\,,$$ where $x_+=1/r_+$ and $\mathbf{u}(x)$ is a matrix related to $\mathbf{V}(r)$. The method consists in finding a local Frobenius solution near the singular point $x=x_+$, $$Z_i=(x-x_+)^{\gamma_{i}}\sum_{n=0}^\infty a_n^{(i)}(\omega) (x-x_+)^n\,, \label{Frobenius}$$ where $\gamma_i$ depend on the specific problem. The series coefficients $a_n^{(i)}$ can be computed iteratively and they only depend on the $N$-dimensional vector $\mathbf{a_0}\equiv\left\{a_0^{(i)}\right\}$.
As discussed in the case of direct integration, we can choose a suitable orthogonal basis for the $N$-dimensional space of the initial coefficients $a_0^{(i)}$. For each element of the basis, we construct the $N\times N$ matrix: $$\begin{aligned}
\label{matrix_coupledZ}
\mathbf{S}(\omega)=\lim_{r\to\infty}
\begin{pmatrix}
Z_1^{(1)} & Z_1^{(2)} & ... & Z_1^{(N)}\\
Z_2^{(1)} & Z_2^{(2)} & ... & ... \\
... & ... & ... & ... \\
... & ... & ... & ... \\
Z_N^{(1)} & ... & ... & Z_N^{(N)} \\
\end{pmatrix}\,,\end{aligned}$$ where again the superscripts denote a particular vector of the basis, i.e. $Z_{i}^{(1)}$ corresponds to $\mathbf{a_0}=\{1,0,0,...,0\}$, $Z_1^{(2)}$ corresponds to $\mathbf{a_0}=\{0,1,0,...,0\}$ and $Z_1^{(N)}$ corresponds to $\mathbf{a_0}=\{0,0,0,...,1\}$. As before, the characteristic frequencies are obtained by imposing $\det{\mathbf{S}(\omega_0)}=0$, i.e. Eq. .
In a region where the radius of convergence of the series is large enough, the series method is extremely efficient. Indeed, it only requires to compute $N$ Frobenius series in the form for a given truncation order and to construct the complex single-valued function $\det{\mathbf{S}(\omega)}$. Then, a standard shooting method can be implemented to compute the root. In the case of AdS BHs, the radius of convergence strongly depends on the size of the BH with respect to the AdS radius. When $r_+\gg L$, the series converges quickly and the method is efficient. Fortunately, this is the case of major interest for holographic applications. On the other hand, the convergence properties of the Frobenius series are poor for small BHs, and the method becomes practically inefficient when $r_+\ll L$. In the latter case, the spectrum supports slowly-damped modes which can be computed through a Breit-Wigner method [@Berti:2009wx]. A pedagogical implementation of the matrix-valued series method is presented in the notebook [series\_method\_DCS.nb](series_method_DCS.nb) [@webpage], where we compute the QNMs of a Schwarzschild-AdS BH in DCS gravity \[see also next section\].
Example: QNMs of Schwarzschild BHs in Dynamical Chern-Simons gravity {#sec:example}
--------------------------------------------------------------------
It is instructive to consider on an example in which the perturbation equations form a coupled system of ODEs. For concreteness, we focus on a prototype theory, in which the Einstein-Hilbert action is modified by adding an extra scalar field coupled to higher-curvature terms. We consider the following Lagrangian density: $${\cal L}_{\rm DCS}=\sqrt{-g}\left( \frac{R}{16\pi} - \frac{1}{2}\nabla_\rho\phi\nabla^\rho\phi +\frac{\alpha}{4} \phi {}^*RR-V(\phi)\right)+{\cal L}_{\rm matter}\,,\label{actionCS}$$ where ${}^*RR\equiv R_{\mu\nu\rho\sigma}{}^*R^{\nu\mu\rho\sigma}=\epsilon_{\sigma\rho\tau\eta}{R_{\mu\nu}}^{\tau\eta}R^{\mu\nu\rho\sigma}/2$. This theory is usually referred to as Dynamical Chern-Simons gravity [@Alexander:2009tp]. Interestingly, it admits all spherically symmetric GR solutions while it deviates from GR in case of rotation. In addition, even though spherically symmetric BHs are described by the Schwarzschild metric as in GR, their linear perturbations obey different equations and, in particular, gravitational and scalar perturbations are coupled to each other in this theory.
The Lagrangian density above is an example of what we generically expect in modified theories of gravity in a Lagrangian formulation: $${\cal L}={\cal L}(g_{\mu\nu},\partial_\sigma g_{\mu\nu},...,\phi,\partial_\sigma\phi,...)\,,$$ where $\phi$ represents an extra fundamental field of generic spin and, in the case of DCS gravity, it is a pseudoscalar field. While it is straightforward to derive the field equations from the Lagrangian above, the procedure can be lengthy and tedious depending on the extra terms in the Lagrangian. It is then particularly useful to obtain the equations with a symbolic manipulation software. This is presented in the notebook [field\_eqs.nb](field_eqs.nb) [@webpage], which makes use of the external package [<span style="font-variant:small-caps;">xTensor</span>]{} [@xTensor] for tensorial calculus[^7].
The field equations read [@Cardoso:2009pk; @Molina:2010fb] $$\begin{aligned}
G_{\mu\nu}&=&8\pi T_{\mu\nu}+8\pi\left[\partial_\mu\phi\partial_\nu\phi-\frac{g_{\mu\nu}}{2}(\partial\phi)^2-g_{\mu\nu}V(\phi)\right]-16\pi\alpha C_{\mu\nu}\,,\\
\square\phi&=&V'(\phi)-\frac{\alpha}{4}{}^*RR\end{aligned}$$ where $$C^{\mu\nu}=\nabla_\rho\phi\epsilon^{\rho\sigma\tau(\mu}\nabla_\tau R^{\nu)}_{~~\sigma}+\nabla_\rho\nabla_\sigma\phi\,^*R^{\sigma(\mu\nu)\rho}\,.\label{Ctensor}$$ Since in any spherically symmetric background $C^{\mu\nu}\equiv0$ and $\epsilon_{\rho\sigma \tau \eta}{R_{\mu\nu}}^{\tau \eta}R^{\mu\nu\rho\sigma}\equiv0$, spherically symmetric GR solutions are also solutions of this theory. In the following we consider the case $V(\phi)=\Lambda/(8\pi)$ in order to allow for a possible cosmological constant. Therefore, a vanishing scalar field and a Schwarzschild-(A)dS metric, $$F(r)=B(r)^{-1}=1-\frac{2M}{r^2}+\frac{\Lambda}{3} r^2\,.\label{backgroundDCS}$$ are a consistent vacuum solution of the field equations[^8].
Let us apply the harmonic decomposition discussed above to the case of gravito-scalar perturbations of a Schwarzschild BH in DCS gravity. We consider the vacuum case ($T_{\mu\nu}\equiv0$). Together with metric perturbations, we also decompose the scalar field in spherical harmonics as: $$\phi(t,r,\vartheta,\varphi)=\frac{\Theta^\ell(r,t)}{r}Y^{\ell}(\vartheta,\varphi)\,.\label{expscal}$$ The perturbation equations are derived in the notebook [DCS\_pert\_eqs.nb](DCS_pert_eqs.nb) [@webpage], where we insert the harmonic decomposition of the metric and of the scalar field into the field equations and solve them at first order in the perturbations. In the frequency-domain the gravitational-axial sector and the scalar sector are described by the following system [@Molina:2010fb] $$\label{systemDCS}
\left\{
\begin{array}{c}
\left[\frac{d^2}{d r_*^2}+\omega^2-V_{RW}(r)\right]\!Q^{\ell}(t,r)=T_{RW}(r)\Theta^{\ell}(t,r)\\
\left[\frac{d^2}{d r_*^2}+\omega^2-V_{S}(r)\right]\!\Theta^{\ell}(t,r)=T_S(r)Q^{\ell}(t,r)
\end{array}
\right.$$ where $r_*$ is the tortoise coordinate defined by $dr/dr_*=F$, $Q^\ell$ is the Regge-Wheeler function in terms of which $h_0^\ell$ and $h_1^\ell$ can be expressed, and the potentials read $$\begin{aligned}
&&V_{RW}(r)=F\left(\frac{\ell(\ell+1)}{r^2}-\frac{6M}{r^3}\right)\,,\label{potentialRW}\\
%
&&T_{RW}(r)=F\frac{96i\pi M\omega \alpha}{r^5}\,,\\
%
&&V_{S}(r)=F\left(\frac{\ell(\ell+1)}{r^2}\left[1+\frac{576\pi
M^2\alpha^2}{r^6\beta}\right]+\frac{2M}{r^3}-\frac{2\Lambda}{3}\right)\,,\label{potentialScalar}\\
%
&&T_S(r)=-F\frac{(\ell+2)!}{(\ell-2)!}\frac{6Mi\alpha}{r^5\beta\omega}\,.
%\end{aligned}$$ On the other hand, the polar sector is described by the same Zerilli equation as in GR, and it can be solved by standard methods and gives the well-known QNMs of a Schwarzschild BH [@Berti:2009kk].
The system is already in the form and can be solved with the methods described above. In the asymptotically flat case, $\Lambda=0$, the modes can be computed via matrix-valued direct integration (cf. [DCS\_DI.nb](DCS_DI.nb)). When $\Lambda<0$, the gravito-scalar modes of the Schwarzschild-AdS background can be computed via a matrix version of the series method (cf. [series\_method\_DCS.nb](series_method_DCS.nb)). In the corresponding notebooks the dependence of the modes from the coupling constant $\alpha$ is shown.
Perturbations of spinning BHs {#sec:spinning}
=============================
In this section we discuss perturbations of *generic* stationary and axisymmetric spacetimes. As discussed in the introduction, this topic is still largely an open problem and, besides the special case of Kerr metric in four-dimensional GR, not much is known on perturbations of other spinning geometries.
For any stationary and axisymmetric spacetime, the linearized field equations reduce to a set of coupled partial differential equations which depend on the radial coordinate $r$, on the angular coordinate $\vartheta$ and on the time $t$. In the time-domain, one then needs to perform a $2+1$ evolution of the coupled system, whereas in the frequency domain one is left with a two-dimensional boundary problem in the $(r,\vartheta)$ variables.
Robust methods to evolve $2+1$ coupled systems are available [@Yoshino:2012kn; @Strafuss:2004qc; @Witek:2012tr] and will be covered elsewhere [@Hiro; @Helvi]. Such simulations are numerically challenging and time consuming. Furthermore, with the current computational power it is possible to follow the evolution on a timescale not larger than $10^4M$ with sufficient precision [@Yoshino:2012kn; @Strafuss:2004qc; @Witek:2012tr]. This is usually sufficient for many purposes, for example for some stability analysis or to extract the characteristic frequencies through a spectral decomposition. However, in other relevant situations one wishes to perform longer and precise simulations, up to $t\sim 10^6 M$ or more. An emblematic example is the evolution of a massive Klein-Gordon field around a Kerr BH [@Dolan:2012yt]. It is well-known that the system develops an exponentially-growing instability which is caused by superradiant amplification of low-frequency waves near the Kerr horizon [@Teukolsky:1974yv; @Press:1972zz; @Detweiler:1980uk; @Cardoso:2004nk; @Dolan:2007mj]. The maximum $e$-fold time of the instability is about $10^6M$ which requires a long and stable evolution to capture the exponential growth. Shorter evolutions may lead to misinterpretion of the instability timescale, especially because the time-domain signal shows beating effects due to interference of several quasi-bound state modes with similar frequencies [@Witek:2012tr].
The aim of the following sections is to introduce some recent tools that can be viewed as alternative and complementary techniques to “hard numerics”[^9]. The latter requires advanced numerical methods that are covered in other work presented at the School [@webpage] and we refer to other lecture’s notes [@Hiro; @Helvi; @Miguel] on this topic.
In Section \[sec:time\] we briefly review an analytical technique to transform some $2+1$ problems into simpler $1+1$ problems at *any* order in the spin parameter and the numerical tools required to achieve stable extra-long ($t\sim 10^6M$) evolutions of the field equations. As an example, we shall focus on the case of massive scalar perturbations of a Kerr BH [@Dolan:2012yt]. In more complicated settings, this dimensional reduction might be impractical and other approximation schemes are required. In Section \[sec:slowrot\] we introduce a slow-rotation expansion of the linearized perturbation equations. Considering an arbitrary power of the dimensionless spin parameter, the dynamics of *any* small perturbation propagating on a generic stationary and axisymmetric spacetime can be reduced to a $1+1$ problem in the time-domain, or to a simple one-dimensional eigenvalue problem in the frequency domain. Finally, in Section \[sec:spectral\] we present spectral methods that have been recently applied to the stability analysis of higher-dimensional spinning BHs [@Dias:2009iu; @Dias:2010eu]. The variety of analytical and numerical tools we present are in some sense complementary to each other and best-suited for different classes of problems.
Reduction to a $1+1$ problem {#sec:time}
----------------------------
In this section, we briefly discuss the time evolution of the linearized equations on a stationary and axisymmetric geometries. It is not our scope to give a detailed presentation, but we simply review some recent developments to perform extra-long evolutions of wavepackets propagating on a spinning BH background [@Dolan:2012yt]. We briefly present the main ideas and refer to other works for a more detailed discussion.
Given a $2+1$ problem where the radial and angular dependences are not manifestly separable, it is nonetheless possible to expand all perturbation variables in a complete basis of spherical harmonics. As we shall discuss in detail, the orthogonality properties of the spherical harmonics can be used to eliminate the angular dependence, at cost of introducing $\ell$-mode-mixing couplings. In principle, this procedure can be performed for any stationary and axisymmetric spacetime, at least when the background is known in closed form. Therefore, the dimensionality of the problem can be lowered to $1+1$ dimensions.
### Example: Massive scalar perturbations of Kerr BHs
A particularly illuminating example is the case of massive Klein-Gordon probe-fields on a Kerr BH [@Dolan:2012yt]. Following Dolan’s original work, here we focus on this case, but the combination of analytical and numerical tools that are presented can find application to a variety of other interesting problems.
We start with the massive Klein-Gordon equation $$\square\phi=\mu^2\phi\,,\label{KG}$$ on the exact Kerr metric in Boyer-Lindquist coordinates: $$\begin{aligned}
&&ds_{\rm Kerr}^2=-\left(1-\frac{2Mr}{\Sigma}\right)dt^2
+\frac{\Sigma}{\Delta}dr^2-\frac{4rM^2}{\Sigma}\tilde{a}\sin^2\vartheta d\varphi dt \nonumber\\
&&+{\Sigma}d\vartheta^2+
\left[(r^2+M^2\tilde{a}^2)\sin^2\vartheta +
\frac{2rM^3}{\Sigma}\tilde{a}^2\sin^4\vartheta \right]d\varphi^2\,,\label{kerrmetric}\end{aligned}$$ where $\Sigma=r^2+M^2\tilde{a}^2\cos^2\vartheta$, $\Delta=(r-r_+)(r-r_-)$ and $r_\pm=M(1\pm\sqrt{1-\tilde a^2})$. In Eq. $\mu$ is related to the mass $m_s$ of the scalar field by $\mu=m_s \hbar c /G$. In natural units, the quantity $M\mu$ is dimensionless.
In an axisymmetric background, the scalar field can be decomposed as $$\phi(t,r,\vartheta,\varphi)=\frac{\Psi(t,r,\vartheta)}{r}e^{im\varphi}\,,$$ and the field equation can be written as: [@Dolan:2012yt] $$\left[{\cal D}_{tr}-\Delta\left({\cal
D}_{\vartheta\tilde\varphi}-V-V_c-\mu^2 r^2 \right) \right](\Psi
e^{i m\tilde\varphi})=0\,,\label{KGtime}$$ with $$\begin{aligned}
{\cal
D}_{tr}&\equiv&(r^2+\tilde{a}^2M^2)^2[\partial_{tt}
-\partial_ { r_*r_* } ]-M^2\tilde{a}^2\Delta\partial_{tt} +4i\tilde{a}
mM^2 r\partial_t\nn\\
&&-\left[2i\tilde{a}Mm(r^2+\tilde{a}^2M^2)-\frac{2\tilde{a}^2M^2\Delta}{r}\right]\partial_{r_*}\,,\\
{\cal D}_{\vartheta\tilde\varphi}&\equiv&\partial_{\vartheta\vartheta}+\cot\vartheta\partial_\vartheta+\frac{\partial_{\tilde\varphi\tilde\varphi}}{\sin^2\vartheta}\,,\\
V&\equiv& \frac{2M}{r}\left(1-\frac{\tilde{a}^2M}{r}\right)+\frac{2i\tilde{a}mM}{r}\,,\\
V_c&\equiv& \tilde{a}^2M^2(\mu^2+\partial_{tt})\cos^2\vartheta\,,\end{aligned}$$ where $r_*$ is the tortoise coordinate defined by $dr/dr_*=\Delta/(r^2+\tilde{a}^2M^2)$ and a new azimuthal coordinate, $d\tilde{\varphi}=d\varphi+\tilde{a}M dr/\Delta$ has been introduced [@Dolan:2012yt]. In the frequency domain, the radial and angular dependence of the equation above can be completely separated using a basis of spheroidal harmonics [@Detweiler:1980uk; @Dolan:2007mj]. However, the spheroidal eigenvalues are frequency-dependent and this prevents a similar separation in the time domain. Without further reduction, Eq. is suitable for standard $2+1$ evolution in the time domain.
The problem can be made computationally less demanding by reducing the dimensionality. This can be achieved by further expanding the field in *spherical* harmonics: $$\Psi(t,r,\vartheta) e^{im\tilde\varphi}=\psi_\ell(t,r)Y^\ell(\vartheta,\tilde\varphi) \label{KGdec}$$ (hereafter a sum over $m$ and $\ell\geq|m|$ is understood) and noting that the spherical harmonics $Y^\ell$ are eigenfunctions of ${\cal
D}_{\vartheta\tilde\varphi}$, which is the Laplace operator on the sphere \[cf. Eq. in \[app:orthogonality\]\]. If $V_c\equiv0$, this would be sufficient to separate the angular part and obtain a single radial equation for each index $\ell$.
However, the term $\cos^2\vartheta$ in $V_c$ prevents this decoupling. Nonetheless, such term can be written as a combination of spherical harmonics with harmonic indices $\ell$ and $\ell\pm2$. To show this, we start by the following property of the scalar spherical harmonics: $$\cos\th Y^{\ell}=\cQ_{\ell+1}Y^{\ell+1}+\cQ_{\ell}Y^{\ell-1}\,,\label{ident1}$$ where $${\cal Q}_\ell=\sqrt{\frac{\ell^2-m^2}{4\ell^2-1}}\,.\label{Qpm}$$ By repeated use of the equation above, we obtain[^10] $$\cos^2\th Y^{\ell}=\left(\cQ_\lp^2 + \cQ_\ell^2\right)Y^\ell+\cQ_\lp \cQ_\lpp Y^\lpp + \cQ_\ell \cQ_\lm Y^\lmm\,.\label{ident3}$$ Other similar identities are listed in Eqs. – below. Using the property above, Eq. can be written as $$\begin{aligned}
&&Y^\ell\left[\frac{{\cal D}_{tr}}{\Delta}-\ell(\ell+1)-V-\mu^2 r^2\right]\psi_\ell\nn\\
&&=\tilde{a}^2M^2(\mu^2+\partial_{tt})\left[\left(\cQ_\lp^2 +
\cQ_\ell^2\right)Y^\ell+\cQ_\lp \cQ_\lpp Y^\lpp + \cQ_\ell \cQ_\lm
Y^\lmm\right]\psi_\ell\,.\end{aligned}$$ Finally, using the orthogonality properties of the spherical harmonics, $$\int Y^{\ell} Y^{*\,\ell'}d\Omega=\delta^{\ell\ell'}\,,\label{orthoscalar}$$ and integrating over the two-sphere \[cf. also \[app:orthogonality\]\], we obtain the final equation: $$\begin{aligned}
&&{\cal D}_{tr}\psi_\ell-\Delta \left\{\ell(\ell+1)+V+\mu^2 r^2
+\tilde{a}^2M^2\left(\cQ_\lp^2 +
\cQ_\ell^2\right)(\mu^2+\partial_{tt})\right\}\psi_\ell\nn\\
&&=\tilde{a}^2M^2\Delta(\mu^2+\partial_{tt})\left[\cQ_\ell
\cQ_{\ell-1}\psi_{\ell-2} + \cQ_{\ell+2} \cQ_{\ell+1}
\psi_{\ell-2}\right]\,. \label{KGfinaltime}\end{aligned}$$ The angular dependence has been completely eliminated and the problem has been reduced to a $1+1$ equation. However, in this equation the field $\psi_\ell$ is coupled to $\psi_{\ell\pm2}$. Because $\ell\geq|m|$, Eq. actually contains an *infinite* number of coupled equations. In practice, the coupled system can be solved by truncating the sum over $\ell$ implicit in Eq. to some order $L$, and checking convergence of the results when $L$ is sufficiently large [@Pani:2012vp; @Dolan:2012yt].
### Time evolution
A standard approach to solve equations in the form of Eq. is to use the so-called method of lines and a finite-difference approximation on spatial slides [@NumericalRecipes]. Defining a one-dimensional grid along the radial direction, spatial derivatives are substituted with finite differences of various order. The system is then reduced to a second-order-in-time set of ODEs. The system can be reduced to first-order form: $$\frac{d\mathbf{y}(t)}{dt}=\mathbf{A}\mathbf{y}(t)\,, \label{MoL}$$ where $\mathbf{y}$ is a vector containing the variables $\psi_\ell$ and their momenta $\partial_t \psi_\ell$ discretized on the grid. We refer to classical books [@NumericalRecipes] and to other lecture notes [@Helvi; @Hiro] for advanced methods to solve this class of problems.
### Spectral analysis
Assuming a stable time evolution of the system is achieved, it is useful to perform a Fourier analysis of the waveform in order to extract the eigenfrequency spectrum and the amplitudes of the single modes. The power spectrum at a given frequency and for a given harmonic index $\ell$ is $$P_\ell(\omega)=|f_\ell(\omega)|^2\,,\label{power}$$ where the Fourier amplitude reads $$f_\ell(\omega_j)=\frac{1}{N}\sum_{k=0}^{N-1}\psi_\ell(t_k) e^{-i\omega_j t_k}\,,\label{fourier}$$ where $\psi_\ell$ is evaluated at a fixed radial position, $t_k=k \Delta t$ with $\Delta t=t_F/(N-1)$ and we assume a time evolution in the domain $[0,t_F]$ discretized in $N$ equidistant points [@Dolan:2012yt]. The resolution in frequency is given by $2\pi/t_F$, so that the longer the simulation the more refined is the frequency spectrum.
Near an eigenfrequency $\omega_R+i\omega_I$, the power spectrum – considered as a function on the real axis – has the typical Breit-Wigner form: $$P(\omega)\approx \frac{1}{(\omega-\omega_R)^2+\omega_I^2}\,,\qquad \omega\approx \omega_R \label{PowerBW}$$ and the real and imaginary parts can be extracted through a quadratic fit around the power peak. This procedure is more precise when $\omega_I\ll\omega_R$, so that $\omega_R$ is approximately a pole of the power spectrum and the width of the resonance is related to the imaginary part. To achieve more precision, filtering techniques may be used. A simple example is explained in Ref. [@Dolan:2012yt], where the modes in the Fourier amplitude are isolated using a filter peaked at $\omega_R$. An inverse Fourier-transform is performed on the filtered signal, and the imaginary part can be precisely extracted from the waveform at intermediate time. The results obtained by this technique are quite impressive, as they allow to extract the fundamental unstable mode and the first overtones with good precision, even when the evolution is followed only up to one tenth of the instability timescale [@Dolan:2012yt].
### Limitations
The $1+1$ reduction discussed above can be applied to other linearized equations and to other metric backgrounds, at least as long as the angular dependence of the perturbation equations can be written in terms of simple trigonometric functions. However, if the field equations are more involved (for example in the case of massive spin-1 fields [@Rosa:2011my; @Pani:2012vp; @Pani:2012bp] or in more general cases) such harmonic decomposition would introduce couplings not only between the nearest-neighbor modes, but also to the next-to-nearest ones and so on, and it would also introduce parity-mixing couplings. In this case a higher truncation order $L$ might be needed and this would result in a large number of $1+1$ coupled differential equations, which may be challenging to evolve for long times. Furthermore, the convergence properties of the solution at a given truncation order might deteriorate. Nonetheless, in some cases this procedure can still be more convenient (or complementary) to a brute-force $2+1$ evolution.
An alternative method to reduce more involved problems is to introduce a series of infinite couplings to higher $\ell$ modes [@Racz:2011qu]. If the coupling terms die away sufficiently fast, the coupled system is still suitable for stable evolution.
To overcome these difficulties, in the next section we introduce a further approximation scheme and we consider a slow-rotation expansion of the linearized equations. As we shall discuss, the slowly-rotating framework simplifies the perturbation equations considerably and it is well suited to attack arbitrarily complicated systems of coupled equations.
Slow-rotation expansion {#sec:slowrot}
-----------------------
In this section we discuss a general method to study linear perturbations of slowly rotating BHs that is particularly useful when the perturbation variables are not separable. The method is an extension of Kojima’s work on perturbations of slowly rotating neutron stars [@Kojima:1992ie; @1993ApJ...414..247K; @1993PThPh..90..977K] and it has been recently generalized and put on firmer basis in the context of BH perturbations [@Pani:2012vp; @Pani:2012bp]. The idea is that slowly-rotating backgrounds are “close enough” to spherical symmetry that an approximate separation of the perturbation equations in radial and angular parts becomes possible. Similarly to the case discussed in the previous section, the perturbation functions are expanded in spherical harmonics and they reduce, in general, to a $1+1$ coupled system of differential equations where various couplings between different multipolar indices $\ell$ and between perturbations with different parity are introduced.
However, at variance with what discussed above, the slow-rotation approximation guarantees that only a certain (typically small) number of couplings to higher multipoles contributes to a given order in $\tilde{a}\ll1$. This makes the method well suited to investigate complicated systems of coupled equations. In the Fourier space, one is left with a simple system of ODEs which can be integrated by standard methods.
### Formalism
Let us start by considering the most general stationary axisymmetric spacetime [@Chandra] $$ds^2_0=-H^2dt^2+Q^2dr^2+r^2K^2\left[d\vartheta^2+\sin^2\vartheta(d\varphi-Ldt)^2\right]\,,\nn$$ where $H$, $Q$, $K$ and $L$ are functions of $r$ and $\vartheta$ only. This metric can describe a stationary, axisymmetric compact object (such as a neutron star or a BH). If the object is slowly rotating, one can define a perturbative expansion in the angular momentum $J$ (or in some other parameter linear in $J$, which characterizes the rotation rate). To second order in rotation, the metric above can be expanded as [@Hartle:1967he] $$\begin{aligned}
ds^2_0=&-F(r)\left[1+F_2\right]dt^2 +B(r)^{-1}\left[1+\frac{2B_2}{r-2M}\right]dr^2\nn\\
&+r^2(1+k_2)\left[d{\vartheta}^2+\sin^2\th(d\varphi-\varpi dt)^2\right] \,,
\label{metric2}\end{aligned}$$ where $M$ is the mass of the spacetime, $\varpi$ is a function of $r$ linear in the rotation parameter, and $F_2$, $B_2$ and $k_2$ are functions of $r$ and $\th$ quadratic in the rotation parameter. The functions $H$, $K$ and $Q$ transform like scalars under rotation, and they can be expanded in scalar spherical harmonics which, due to axisymmetry, reduce to the Legendre polynomials $P_\ell(\th)$. To second order, only $\ell=0$ and $\ell=2$ polynomials contribute [@Hartle:1967he], therefore we obtain: $$\begin{aligned}
F_2(r,\vartheta)&=&F_{r}(r)+F_{\vartheta}(r)P_2(\vartheta)\,,\\
B_2(r,\vartheta)&=&B_{r}(r)+B_{\vartheta}(r)P_2(\vartheta)\,,\\
k_2(r,\vartheta)&=&k_{r}(r)+k_{\vartheta}(r)P_2(\vartheta)\,.\end{aligned}$$ At first order in rotation the metric reduces to a much simpler form $$\begin{aligned}
ds^2_0=-F(r)dt^2 +B(r)^{-1}dr^2-2\varpi(r)\sin^2\th d\varphi dt+r^2d^2\Omega\,.\label{metric1}\end{aligned}$$ The metric and can be computed solving Einstein’s equations to second and first order in the rotation, respectively. For example, to first order, the slowly-rotating Kerr metric corresponds to $$F(r)=B(r)=1-2M/r\,,\quad \varpi=2M^2\tilde a/r \,, \label{Kerr1st}$$ where $M$ and $J=\tilde a M^2$ are the mass and the angular momentum of the BH. More generically, given a nonrotating metric the gyromagnetic function $\varpi(r)$ can be computed using the approach originally developed by Hartle [@Hartle:1967he]. This is the case for some BH solutions in modified gravity theories, which are only known perturbatively [@Yunes:2009hc; @Pani:2009wy; @Pani:2011gy; @Yagi:2012ya]. The metric can also be constructed numerically, for instance in the case of slowly-rotating stars [@Kojima:1992ie; @1993ApJ...414..247K; @1993PThPh..90..977K].
Slowly rotating and oscillating compact objects can be studied as perturbations of the axisymmetric, stationary solutions discussed above. Scalar, vector and tensor field equations in the background metric can be linearized in the field perturbations. Any perturbation function $\delta X$ can be expanded in a complete basis of spherical harmonics, as previously discussed for the nonrotating case. Schematically, in the frequency domain we have $$\delta X_{\mu_1\dots}(t,r,\vartheta,\varphi)=
\delta X^{(i)}_{\ell m}(r){\cal Y}_{\mu_1\dots}^{\ell m\,(i)}e^{-i\omega t}\,,
\label{expa}$$ where ${\cal Y}_{\mu_1\dots}^{\ell m\,(i)}$ is a basis of scalar, vector or tensor harmonics, depending on the tensorial nature of the perturbation $\delta X$. As in the spherically symmetric case, the perturbation variables $\delta X^{(i)}_{\ell m}(r)$ can be classified as “polar” or “axial” depending on their behavior under parity transformations.
The linear response of the system is fully characterized by a coupled system of ODEs in the perturbation functions $\delta X^{(i)}_{\ell m}(r)$. As previously discussed, in the case of a spherically symmetric background, perturbations with different values of $(\ell,\,m)$, as well as perturbations with opposite parity, are decoupled. In a rotating, axially symmetric background, perturbations with different values of $m$ are still decoupled but perturbations with different values of $\ell$ are not. At this stage, we present the general schematic form of the perturbation equations, and postpone the derivation of some particular cases to Section \[sec:applications\]. To second order, the perturbation equations schematically read $$\begin{aligned}
0&=&{\cal A}_{\ell}+\tilde a m \bar{\cal A}_{{\ell}}+\tilde{a}^2 \hat{{\cal A}}_\ell\nn\\
&+&\tilde a ({\cal Q}_{{\ell}}\tilde{\cal P}_{\ell-1}+{\cal Q}_{\ell+1}\tilde{\cal P}_{\ell+1})\nn\\
&+&\tilde{a}^2 \left[\cQ_\lm \cQ_\ell \breve{{\cal A}}_\lmm + \cQ_\lpp \cQ_\lp
\breve{{\cal A}}_\lpp \right]+{\cal O}(\tilde{a}^3)\,,\nn\\\label{epF1c}\\
%%%%%
0&=&{\cal P}_{\ell}+\tilde a m \bar{\cal P}_{{\ell}}+\tilde{a}^2 \hat{{\cal P}}_\ell\nn\\
&+&\tilde a ({\cal Q}_{{\ell}}\tilde{\cal A}_{\ell-1}+{\cal Q}_{\ell+1}\tilde{\cal A}_{\ell+1})\nn\\
&+&\tilde{a}^2 \left[\cQ_\lm \cQ_\ell \breve{{\cal P}}_\lmm + \cQ_\lpp \cQ_\lp
\breve{{\cal P}}_\lpp \right]+{\cal O}(\tilde{a}^3)\,,\nn\\\label{epF2c}\end{aligned}$$ where ${\cal Q}_\ell$ were defined in Eq. , and ${\cal A}_{\ell}$, $\bar {\cal A}_{\ell}$, $\tilde {\cal A}_{\ell}$, $\hat {\cal A}_{\ell}$, $\breve {\cal A}_{\ell}$ are *linear* combinations of the axial perturbations with multipolar index $\ell$; similarly, ${\cal P}_{\ell}$, $\bar {\cal
P}_{\ell}$, $\tilde {\cal P}_{\ell}$, $\hat {\cal P}_{\ell}$, $\breve {\cal P}_{\ell}$ are linear combinations of the polar perturbations with index $\ell$.
The structure of Eqs. – is very interesting. In the limit of slow rotation there is a Laporte-like “selection rule” [@ChandraFerrari91]: at first order in $\tilde{a}$, perturbations with a given value of $\ell$ are only coupled to those with $\ell\pm1$ and *opposite* parity. At second order, perturbations with a given value of $\ell$ are also coupled to those with $\ell\pm2$ and *same* parity, and so on. More precisely, perturbations with a given parity and index $\ell$ are coupled to: (i) perturbations with *opposite* parity and index $\ell\pm1$ at order $\tilde{a}$; (ii) perturbations with *same* parity and *same* index $\ell$ up to order $\tilde{a}^2$; (iii) perturbations with *same* parity and index $\ell\pm2$ at order $\tilde{a}^2$. The symmetries of the harmonic expansion guarantee that this scheme is preserved at any order in $\tilde{a}$.
Furthermore, from Eq. it follows that ${\cal Q}_{\pm m}=0$, and therefore if $|m|=\ell$ the coupling of perturbations with index $\ell$ to perturbations with indices $\ell-1$ and $\ell-2$ is suppressed. This general property is usually called [@ChandraFerrari91] “propensity rule” in atomic theory, and states that transitions $\ell\to\ell+1$ are strongly favored over transitions $\ell\to\ell-1$. Indeed, the slow-rotation technique is well known in quantum mechanics and the coefficients ${\cal Q}_{\ell}$ are related to the usual Clebsch-Gordan coefficients.
### Eigenvalue spectrum in the slow-rotation limit
Due to the coupling between different multipolar indices, the spectrum of the solutions of Eqs. – is extremely rich. However, if we are interested in the characteristic modes of the slowly-rotating background to first or to second order in $\tilde{a}$, the perturbation equations can be considerably simplified.
Let us start by considering the first order corrections. We expand all quantities to first order and we ignore the terms $\hat {\cal A}_{\ell}$, $\breve {\cal A}_{\ell}$, $\hat
{\cal P}_{\ell}$ and $\breve {\cal P}_{\ell}$, which are multiplied by $\tilde{a}^2$ in Eqs. and . Crucially, the terms ($\tilde {\cal P}_{\ell},\,\tilde {\cal A}_{\ell}$) do not contribute to the eigenfrequencies at first order in $\tilde{a}$ [@1993PThPh..90..977K; @Pani:2012bp]. Here, we follow the proof that was presented in Ref. [@Pani:2012bp]. At first order, Eqs. and are invariant under the simultaneous transformations
$$\begin{aligned}
& a_{\ell m}\to\mp a_{\ell -m}\,,\qquad p_{\ell m}\to \pm p_{\ell -m}\,,\\
& \tilde{a}\to-\tilde{a}\,,\qquad \hspace{1cm} m\to -m\,,\end{aligned}$$
where $a_{\ell m}$ ($p_{\ell m}$) schematically denotes any axial (polar) perturbation variables with indices $(\ell,m)$. The invariance follows from the linearity of the terms in Eqs. and and from the fact that the ${\cal Q}_{\ell}$’s are *even* functions of $m$. The boundary conditions that define the characteristic modes of the BH are also invariant under the transformations above. Therefore in the slow-rotation limit the eigenfrequencies can be expanded as $$\omega=\omega_0+m\,\omega_1 \tilde{a}+\omega_2
\tilde{a}^2+{\cal O}(\tilde{a}^3)\,, \label{exp_omega}$$ where $\omega_0$ is the eigenfrequency of the nonrotating spacetime and $\omega_n$ is the $n$-th order correction[^11]. Crucially, only the terms ($\bar {\cal P}_{\ell},\,\bar {\cal
A}_{\ell}$) in Eqs. and can contribute to $\omega_1$. Indeed, due to the factor $\tilde{a}$ in front of all terms ($\bar {\cal P}_{\ell},\,\bar {\cal A}_{\ell}$, $\tilde {\cal
P}_{\ell},\,\tilde {\cal A}_{\ell}$) and to their linearity, at first order in $\tilde{a}$ we can simply take the zeroth order (in rotation) expansion of these terms. That is, to our level of approximation the terms ($\bar {\cal P}_{\ell},\,\bar {\cal
A}_{\ell}$, $\tilde {\cal P}_{\ell},\,\tilde {\cal A}_{\ell}$) in Eqs. and only contain the perturbations of the *nonrotating*, spherically symmetric background. Since the latter do not explicitly depend on $m$, the $m$ dependence in Eq. can only arise from the terms ($\bar {\cal P}_{\ell},\,\bar {\cal A}_{\ell}$) to zeroth order.
Therefore, the eigenvalue problem to first order is equivalent to the following *decoupled* sets of equations: $$\begin{aligned}
{\cal A}_{\ell}+\tilde a m \bar{\cal A}_{{\ell}}&=&0\,,\label{epF1}\\
{\cal P}_{\ell}+\tilde a m \bar{\cal P}_{{\ell}}&=&0\,.\label{epF2}\end{aligned}$$ In the equations above polar and axial perturbations – as well as perturbations with different values of the harmonic indices – are decoupled from each other and can be studied independently. In practice, the final eigenvalue problem is very similar to the nonspinning case, the only difference being the introduction of the Zeeman-splitting term proportional to $\tilde{a}m$ that breaks the azimuthal degeneracy.
Let us now move to the the eigenfrequency spectrum at second order in $\tilde{a}$. First, we expand any axial and polar perturbation function (respectively denoted as $a_{\ell m}$ and $p_{\ell m}$) that appears in Eqs. and : $$\begin{aligned}
a_{\ell m}&=&a^{(0)}_{\ell m}+\tilde a\,a^{(1)}_{\ell m}+\tilde a^2a^{(2)}_{\ell m}+{\cal O}(\tilde{a}^3)\nn\\
p_{\ell m}&=&p^{(0)}_{\ell m}+\tilde a\,p^{(1)}_{\ell m}+\tilde a^2p^{(2)}_{\ell m}+{\cal O}(\tilde{a}^3)\,.\end{aligned}$$ The terms $\breve{{\cal A}}_{\ell\pm2}$ and $\breve{{\cal
P}}_{\ell\pm2}$ are multiplied by factors $\tilde{a}^2$, so they only depend on the zeroth-order perturbation functions, $a^{(0)}_{\ell
m}$, $p^{(0)}_{\ell m}$. The terms $\tilde{{\cal A}}_{\ell\pm1}$ and $\tilde{{\cal P}}_{\ell\pm1}$ are multiplied by factors $\tilde{a}$, so they only depend on zeroth- and first-order perturbation functions $a^{(0)}_{\ell m}$, $p^{(0)}_{\ell m}$, $a^{(1)}_{\ell m}$, $p^{(1)}_{\ell m}$.
Since in the nonrotating limit axial and polar perturbations are decoupled, a possible consistent set of solutions of the system (\[epF1c\])–(\[epF2c\]) has $a^{(0)}_{\ell\pm2 m}\equiv0$, which leads to the “axial-led” [@Lockitch:1998nq] subset of Eqs. (\[epF1c\])–(\[epF2c\]): $$\label{epF1bis}
\left\{ \begin{array}{l}
{\cal A}_{\ell}+\tilde a m \bar{\cal A}_{{\ell}}+\tilde{a}^2 \hat{{\cal A}}_\ell+\tilde a ({\cal Q}_{{\ell}}\tilde{\cal P}_{\ell-1}+
{\cal Q}_{\ell+1}\tilde{\cal P}_{\ell+1})=0\,,\\
%%
{\cal P}_{\ell+1}+\tilde a m \bar{\cal P}_{{\ell+1}}+
\tilde a {\cal Q}_{{\ell+1}}\tilde{\cal A}_{\ell}=0\,,\\
%%
{\cal P}_{\ell-1}+\tilde a m \bar{\cal P}_{{\ell-1}}+
\tilde a {\cal Q}_{{\ell}}\tilde{\cal A}_{\ell}=0\,.
\end{array}\right.$$ Similarly, another consistent set of solutions of the same system has $p^{(0)}_{\ell\pm2 m}\equiv0$. The corresponding “polar-led” system reads $$\label{epF2bis}
\left\{ \begin{array}{l}
{\cal P}_{\ell}+\tilde a m \bar{\cal P}_{{\ell}}+\tilde{a}^2 \hat{{\cal P}}_\ell+
\tilde a ({\cal Q}_{{\ell}}\tilde{\cal A}_{\ell-1}+{\cal Q}_{\ell+1}
\tilde{\cal A}_{\ell+1})=0\,,\\
%%
{\cal A}_{\ell+1}+\tilde a m \bar{\cal A}_{{\ell+1}}
+\tilde a {\cal Q}_{{\ell+1}}\tilde{\cal P}_{\ell}=0\,,\\
%%
{\cal A}_{\ell-1}+\tilde a m \bar{\cal A}_{{\ell-1}}+
\tilde a {\cal Q}_{{\ell}}\tilde{\cal P}_{\ell}=0\,,
\end{array}\right.$$ In the second and third equations of the two systems above we have dropped the $\tilde{{\cal A}}_{\ell\pm2}$ and $\tilde{{\cal P}}_{\ell\pm2}$ terms, because they only enter at zeroth order, and we have set $a^{(0)}_{\ell\pm2 m}\equiv0$ and $p^{(0)}_{\ell\pm2 m}\equiv0$[^12]. Interestingly, within this perturbative scheme a notion of “conserved quantum number” $\ell$ is still meaningful, even though, for any given $\ell$, rotation couples terms with opposite parity and different multipolar index.
To summarize, the eigenfrequencies (or at least a subset of the eigenfrequencies) of the general system , can be found, at first order in $\tilde{a}$, by solving the two decoupled sets and for axial and polar perturbations, respectively. At second order in $\tilde{a}$ we must solve either the set or the set for “axial-led” and “polar-led” modes, respectively. The power of this procedure stands in its generality. It can be applied to any slowly-rotating spacetime and to any kind of perturbation. In Section \[sec:applications\], we explicitly derive some particular cases.
Stability of higher-dimensional BHs: Spectral methods {#sec:spectral}
-----------------------------------------------------
Although a detailed overview is beyond the scope of this work, we wish to conclude this section by mentioning another method that was recently developed to study the instability of highly-spinning BHs in higher dimensions [@Dias:2009iu; @Dias:2010eu]. In $D>4$, thermodynamical and perturbative arguments suggest that quasi-extremal or highly-spinning BH geometries should be linearly unstable [@Emparan:2008eg]. A slow-rotation approximation is not promising to study the instability if the latter is a high-spin effect. Furthermore, the parameter space of the perturbations in higher dimensions can be extremely large and a complete characterization of the linear dynamics is not feasible. Finally, as previously discussed the angular part of the perturbation equations does not appear to be separable in the general case, which makes the linearized problem particularly challenging.
In the approaches taken so far, one restricts the analysis to some special subclass of perturbations that preserve some rotational symmetry of the background. In the frequency domain, the idea is to reduce the linearized dynamics to a two-dimensional boundary problem involving coupled partial differential equations or coupled ODEs, depending on the background geometry. Such problem can be efficiently solved by a Chebyshev spectral method [@Monteiro:2009ke]. In a nutshell, the method proceeds as follows. First, the spinning BH metric (for instance a Myers-Perry BH with a single spin [@Dias:2009iu] or a cohomogeneity-one Myers-Perry geometry [@Dias:2010eu]) is embedded into a black string with one extra spatial dimension $z$. A subclass of the stationary perturbations of the black string is considered in the form $\sim e^{-i\omega t}e^{i k z}h_{\mu\nu}$, where $h_{\mu\nu}$ does not depend on $t$ and $z$. In the transverse and traceless gauge, the linearized Einstein equations have the form [@Dias:2009iu; @Dias:2010eu] $$(\Delta_L h)_{\mu\nu}\equiv -\nabla_\rho\nabla^\rho h_{\mu\nu}-2R_{\mu\nu\rho\sigma} h^{\rho\sigma}=-k^2 h_{\mu\nu}\,,$$ where $\Delta_L$ is the Lichnerowicz operator on the corresponding Myers-Perry background. Perturbations of the original spinning BH metric are obtained when $k=0$. As usual, after suitable boundary conditions are imposed at the horizon and at infinity, the equations above define an eigenvalue problem for the complex frequency $\omega$. Depending on the background metric, the problem is reduced to a set of coupled partial differential equations [@Dias:2009iu] or a system of coupled ODEs [@Dias:2010eu]. In the latter case the eigenvalue problem can be solved with the methods described in the previous sections. Nonetheless, if one is interested in finding *purely imaginary* modes, $\omega=i\omega_I$, the task can be simplified by reversing the eigenvalue problem. For a given $\omega_I$ and given spin parameter(s) one seeks for the (real) value of $k$ that solves the boundary problem. Modes with $\omega_I>0$ and $k\neq0$ correspond to black-string metrics that are unstable under the Gregory-Laflamme instability [@Emparan:2008eg]. If such modes exist, by increasing the angular momentum it is possible to track the eigenvalues $k$. The critical value of the spin corresponding to $k=0$ (if it exists) signals the onset of an instability of the associated Myers-Perry background. In order to require the existence of purely imaginary modes in first instance, a particular subclass of perturbations must be considered. [@Dias:2009iu; @Dias:2010eu]
This method is clearly not optimally suited to explore the whole parameter space, but it can be efficiently adopted to prove the existence of a subclass of unstable modes and to construct marginally stable solutions at the bifurcation point (which are new BH solutions with pinched horizons). Adopting this method, singly-spinning Myers-Perry BHs in $D\geq6$ [@Dias:2009iu; @Dias:2010maa], Myers-Perry BHs in $D=9$ with equal angular momenta [@Dias:2010eu] and in $D=7$ with two of the three angular momenta set to be equal [@Dias:2011jg], were all found to be unstable above a critical value of the spin. In fact, arguments have been provided for a generic ultraspinning instability of Myers-Perry BHs in $D>5$.
Let us briefly discuss the spectral method that can be adopted to solve the boundary value problem [@Monteiro:2009ke; @Dias:2010eu; @SpectralMethods]. In order to illustrate how the method works, let us consider a generalized boundary problem defined by $N$ coupled ODEs: $$\boldsymbol{{\cal D}}\mathbf{Y}=-k^2\mathbf{V}\mathbf{Y}\,,\label{boundarySM}$$ where $\boldsymbol{{\cal D}}$ is a $N\times N$ differential operator, $\mathbf{V}$ is a $N\times N$ matrix and $\mathbf{Y}$ is the $N$-dimensional eigenfunction that we want to compute in a finite domain $y\in[y_i,y_f]$. The basic idea of spectral methods is to approximate the eigenfunction by a finite sum of polynomials: $$Y_i(y)=\sum_{j=0}^n a_j^{(i)} y^j\,,$$ where $Y_i$ is the $i$th component of $\mathbf{Y}$ ($i=1,...,N$). In practice, the coefficients $a_j^{(i)}$ are obtained by a polynomial interpolation. In order for the method to be accurate and stable, it is crucial to interpolate the functions in a suitably-chosen set of points. The best repartition depends on the problem at hand, here we consider the so-called Chebyshev nodes, which are the roots of the Chebyshev polynomials of the first kind [@Monteiro:2009ke; @Dias:2010eu]: $$y_l=\frac{y_f+y_i}{2}+\frac{y_f-y_i}{2}\cos\left(\frac{(2l-1)\pi}{2n}\right)\,, \label{Chebyshev}$$ where $l=0,...,n$. This repartition is particularly useful because the corresponding polynomial interpolation minimizes the Runge’s phenomenon [@NumericalRecipes] i.e., roughly speaking, they minimize the errors of higher-order interpolations at the boundary. The interpolation effectively maps $Y_i(y)\to a_j^{(i)}$, where the latter are just constants. Equation can be transformed into an *algebraic* system for the $(n+1)$-dimensional vectors containing $Y_i(y_l)$ as entries and where differential operators are transformed to nondiagonal matrices that mix the various components of the vectors [@SpectralMethods]. Once boundary conditions are imposed on $Y_i(y_0)$ and on $Y_i(y_N)$, i.e. on the first and on the last entries of each vector, the problem effectively reduces to an algebraic eigenvalue problem for $k$ in $N(n-1)$ dimensions [@Dias:2010eu]., which can be solved by standard methods [@NumericalRecipes; @SpectralMethods].
As mentioned above, this method is efficient to find purely imaginary unstable modes, but it is not well-suited to explore the full parameter space. This is because one seeks for purely real values of $k$, and this would require a multivariate search in the complex $\omega$-plane. It would be very interesting to complement this method with a slowly-rotating approximation of the Myers-Perry family. Even though the results of the slow-rotation expansion cannot be extrapolated to the ultra-spinning regime, this method would allow to treat perturbations of higher-dimensional BHs generically and to explore the full parameter space at a given order in $\tilde{a}$. Furthermore, it is not impossible that ultra-spinning instabilities corresponding to purely imaginary modes will have a counterpart in complex unstable modes at lower rotation rate. Possibly, such instabilities can be captured by a slow-rotation analysis. Another interesting application of this approach could be to provide semianalytical arguments in favor of the so-called bar-mode instability of higher-dimensional BHs [@Shibata:2010wz].
Applications of the slow-rotation formalism {#sec:applications}
===========================================
In order to illustrate how the slow-rotation framework presented in Section \[sec:slowrot\] works, in this section we work out some simple application.
Massive scalar perturbations of slowly rotating Kerr BHs {#sec:scalar}
--------------------------------------------------------
Let us again consider the massive Klein-Gordon equation around a Kerr BH in the frequency domain. Even though this problem is separable in the standard Teukolsky approach, one may apply the slow-rotation formalism to this case and check the errors introduced by the slow-rotation approximation by comparing with exact results. Thus, our goal is to derive the perturbation equations up to second order in rotation. The entire derivation reported in this section is also available in the selfconsistent [<span style="font-variant:small-caps;">Mathematica</span>]{} notebook [slow\_rot\_scalar.nb](slow_rot_scalar.nb)[@webpage]. The exact Kerr metric in Boyer-Lindquist coordinates reads as in Eq. . To second order in $\tilde{a}$, the metric can be written in the form and the event horizon $r_+$, the Cauchy horizon $r_-$ and the outer ergosphere $r_{S^+}$ respectively read: $$r_+=2M\left(1-\frac{\tilde{a}^2}{4}\right)\,,\quad r_-=\frac{M\tilde{a}^2}{2}\,, \quad r_{S^+}=2M\left(1-\cos^2\th\frac{\tilde{a}^2}{4}\right)\,.
\label{cauchyhor}$$ Corrections to the horizon location are of second-order and up to first order the ergosphere coincides with the horizon. Again, we decompose the scalar field in spherical harmonics: $$\phi=\sum_{\ell m}\frac{\Psi_\ell(r)}{\sqrt{r^2
+\tilde{a}^2 M^2}}e^{-i\omega t}Y^\ell(\vartheta,\varphi)\,,$$ and expand the square root above to second order in $\tilde{a}$. Schematically, we obtain the following equation: $$A_{\ell} Y^\ell+ D_\ell \cos^2\vartheta Y^\ell=0\,,\label{eq_expY}$$ where a sum over $(\ell,m)$ is implicit, and the explicit form of the *radial* coefficients $A_\ell$ and $D_\ell$ (which depend on $\phi$ and its radial derivatives) is given in the notebook [@webpage; @Pani:2012bp]. Note that the equation above can be seen as an expansion of Eq. to ${\cal O}(\tilde a^2)$ and in the frequency domain. The coefficient $D_\ell$ is proportional to $\tilde{a}^2$, so the second term in the equation above is zero to first order in rotation. Indeed, the angular dependence is already separated to first order, and the linearized Klein-Gordon equation can be cast in the form[^13] $$\hat{\cal D}_2\Psi_\ell-\left[\frac{4m M^2
\tilde{a}\omega}{r^3}+F \frac{2M}{r^3}\right]\Psi_\ell=0\,,\label{scalar1st}$$ where $F=1-2M/r$ and we defined the operator [@Rosa:2011my] $$\hat{\cal D}_2=\frac{d^2}{d r_*^2}+\omega^2
-F\left[\frac{\ell(\ell+1)}{r^2}+\mu^2\right]\,,\label{D2}$$ with $dr/dr_*=F$. Equation coincides with Teukolsky’s master equation [@Teukolsky:1973ha] for spin $s=0$ perturbations expanded to first order in $\tilde{a}$.
Let us separate the angular part of Eq. . This can be achieved by using the identity as well as the orthogonality properties of scalar spherical harmonics \[cf. also \[app:orthogonality\]\]. We obtain: $$A_\ell+(\cQ_{\ell+1}^2+\cQ_{\ell}^2) D_\ell+\cQ_{\ell-1}\cQ_{\ell} D_{\ell-2}+\cQ_{\ell+2}\cQ_{\ell+1} D_{\ell+2 }=0\label{final_schematic}\,.$$ Note that the coupling to perturbations with indices $\ell\pm1$ is absent. This is due to the fact that Klein-Gordon perturbations are polar quantities, and at first order the Laporte-like selection rule implies that polar perturbations with index $\ell$ should couple to axial perturbations with $\ell\pm1$, but the latter are absent in the spin-0 case. At second order, perturbations with harmonic index $\ell$ are coupled to perturbations with the *same* parity and $\ell\pm2$, but this coupling does not contribute to the eigenfrequencies for the reasons discussed in the previous section. We verify this statement below.
As discussed in Section \[sec:time\], by repeated use of the identity we can separate the perturbation equations at *any order* in $\tilde{a}$. In the case of spin-1 or spin-2 perturbations, combinations of vector and tensor spherical harmonics also appear, and these introduce terms such as $(\sin\vartheta)^n Y^\ell_{,\vartheta}$. The latter, can be decoupled in a similar fashion. For example, for computations to first and to second order, two useful relations are[^14] $$\begin{aligned}
\sin\th Y^{\ell}_{,\vartheta}&=&
{\cal Q}_{\ell+1}\ell Y^{\ell+1}-{\cal Q}_{\ell}(\ell+1)Y^{\ell-1}\,,\label{ident2}\\
%%%%
\cos\th\sin\th Y^{\ell}_{,\vartheta}&=&\left(\ell\cQ_\lp^2 -(\ell+1)\cQ_\ell^2\right)Y^\ell+\cQ_\lp \cQ_\lpp \ell Y^\lpp \nn\\
&&- \cQ_\ell \cQ_\lm (\ell+1) Y^\lmm\,.\label{ident4}\end{aligned}$$ Using the explicit form of the coefficients given in the notebook, the field equations schematically read $$\begin{aligned}
\frac{d^2\Psi_\ell}{dr_*^2}+V_\ell \Psi_\ell+\tilde{a}^2&&
\Bigg[U_{\ell+2} \Psi_{\ell+2}+U_{\ell-2} \Psi_{\ell-2}+W_{\ell+2} \frac{d^2\Psi_{\ell+2}}{dr_*^2}+
W_{\ell-2} \frac{d^2\Psi_{\ell-2}}{dr_*^2}\Bigg]=0\,,\nn\\\label{final_tortoise}\end{aligned}$$ where we have defined the tortoise coordinate via $dr/dr_*=
\Delta/(r^2+\tilde{a}^2M^2)$ (expanded at second order) and $V$, $U$ and $W$ are some potentials explicitly given in the notebook [@webpage].
As we expected, the coupling to the $\ell\pm2$ terms is proportional to $\tilde{a}^2$. For a calculation accurate to second order in $\tilde{a}$ the terms in parenthesis can be evaluated at zeroth order, and the functions $\Psi_{\ell\pm2}^{(0)}$ must be solutions of $$\frac{d^2\Psi_{\ell\pm2}^{(0)}}{dr_*^2}+V_{\ell\pm2}^{(0)} \Psi_{\ell\pm2}^{(0)}=0\,.$$ By substituting these relations in Eq. we get $$\begin{aligned}
\frac{d^2\Psi_\ell}{dr_*^2}+V_\ell \Psi_\ell&&+\tilde{a}^2
\left(U_{\ell+2}^{(0)}-V_{\ell+2}^{(0)}W_{\ell+2}^{(0)}\right) \Psi_{\ell+2}^{(0)}+\tilde{a}^2\left(U_{\ell-2}^{(0)}-
V_{\ell-2}^{(0)}W_{\ell-2}^{(0)}\right) \Psi_{\ell-2}^{(0)}=0\,.\nn\end{aligned}$$ Finally, making use of the expressions for $V$, $U$ and $W$, the field equations reduce to $$\begin{aligned}
&&\frac{d^2\Psi_\ell}{dr_*^2}+V_\ell \Psi_\ell=\frac{\tilde{a}^2
M^2(r-2M)\left(\mu ^2-\omega ^2\right)}{r^3} \left[\cQ_{\ell+1} \cQ_{\ell+2} \Psi_{\ell+2}^{(0)}
+\cQ_{\ell-1} \cQ_{\ell} \Psi_{\ell-2}^{(0)}\right]\,,\nn\\\label{final}\end{aligned}$$ where the potential is given by $$\begin{aligned}
V_\ell&&=\omega^2-F\left[
\frac{\ell(\ell+1)}{r^2}+\frac{2M}{r^3}+\mu^2\right]-\frac{4\tilde{a}m\omega M^2}{r^3}\nn\\
&&-\frac{\tilde{a}^2 M^2}{r^6}\left[24 M^2+4 M r
\left(\ell(\ell+1)-3+r^2 \mu ^2\right)+r^4 F (\mu^2 -\omega^2 ) \left(\cQ_{\ell}^2
+\cQ_{\ell+1}^2\right)\right.\nn\\
&&\left.-2 M r^3 \omega ^2-r^2 \left(\ell(\ell+1)+m^2
+r^2 (\mu^2 -\omega^2 ) -1\right)\right]\,.\label{scalpot}\end{aligned}$$ Note the similar structure of Eq. (which is valid to order $\tilde{a}^2$) and Eq. (which is valid to *any* order in $\tilde{a})$. Even if Eq. is approximate, one advantage of the slow-rotation approximation is that the couplings to terms with indices $\ell\pm2$ can be neglected in the calculation of the modes to second order. In the scalar case this can be shown explicitly as follows. If we define $$Z_\ell=\psi_\ell-\tilde{a}^2\left[c_{\ell+2}\psi_{\ell+2}-c_{\ell}\psi_{\ell-2}\right]\,, \label{Zell}$$ where $$c_{\ell}=\frac{M^2\left(\mu ^2-\omega ^2\right) \cQ_\lm \cQ_\ell}{2 (2\ell-1) }\,,$$ then, at second order in rotation, Eq. can be written as a single, Schroedinger-like equation for $Z_\ell$: $$\frac{d^2 Z_\ell}{dr_*^2}+V_\ell Z_\ell=0\,,\label{final2}$$ which can be solved by the methods discussed in Section \[sec:time\]. This equation coincides with Teukolsky’s master equation [@Teukolsky:1973ha] for spin $s=0$ perturbations expanded at second order in $\tilde{a}$. In particular, the coefficients ${\cal Q}_{\ell}$ in Eq. agree with an expansion of Teukolsky’s spheroidal eigenvalues to second order in the BH spin [@Berti:2005gp][^15].
Gravitational perturbations of a Kerr BH in the slow-rotation limit {#sec:Kerr}
-------------------------------------------------------------------
As another relevant application of the slow-rotation expansion, here we derive the gravitational perturbations of a Kerr BH to first order in the angular momentum. Also in this case the linearized equations are known exactly in the Teukolsky formalism. Nonetheless, this exercise is propaedeutic to more involved cases that cannot be treated in the Teukolsky formalism. The entire procedure discussed in this section is presented in the notebook [slow\_rot\_grav\_Kerr.nb](slow_rot_grav_Kerr.nb)[@webpage].
As a background, we consider the Kerr metric to first order in $\tilde{a}$, which is given in Eq. . On this background we consider a harmonic decomposition of the metric perturbations as in Eqs. and with $\eta_i^\ell\equiv G^\ell\equiv h_2^\ell\equiv0$.
Using this decomposition, we can solve Einstein’s equations at linear order in the perturbations and to first order in $\tilde a$. Because of the transformation properties of the perturbation functions, the linearized Einstein equations naturally separate into three groups [@Kojima:1992ie]. By denoting the linearized Einstein equations as $\delta {\cal E}_{\mu\nu}=0$, the first group reads $$\delta{\cal E}_{(I)}\equiv (A^{(I)}_{{\ell}}+{\tilde A}^{(I)}_{{\ell}}\cos\th)Y^{{\ell}}
+B^{(I)}_{{\ell}}\sin\th Y^{{\ell}}_{,\vartheta}+C^{(I)}_{{\ell}} Y^{{\ell}}_{,\varphi}=0,\label{eqG1}$$ where $I=0,1,2,3$ corresponds to $\delta {\cal E}_{tt}=0$, $\delta
{\cal E}_{tr}=0$, $\delta {\cal E}_{rr}=0$ and $\delta {\cal
E}_{\vartheta\vartheta} +{\delta {\cal
E}_{\varphi\varphi}}/{\sin\vartheta^2}=0$, respectively. The second group reads $$\begin{aligned}
&\delta{\cal E}_{(L\vartheta)}&\equiv(\alpha^{(L)}_{{\ell}}+{\tilde \alpha}^{(L)}_{{\ell}}\cos\th) Y^{{\ell}}_{,\vartheta}-
(\beta^{(L)}_{{\ell}}+{\tilde \beta}^{(L)}_{{\ell}}\cos\th)\frac{ Y^{{\ell}}_{,\varphi}}{\sin\th}\nn\\
&&+\eta^{(L)}_{{\ell}}\sin\th Y^{{\ell}}+\xi^{(L)}_{{\ell}}X^{{\ell}}+
\chi^{(L)}_{{\ell}}\sin\th W^{{\ell}}=0,\label{eqG2a}\\
%%%%
&\delta{\cal E}_{(L\varphi)}&\equiv(\beta^{(L)}_{{\ell}}+{\tilde \beta}^{(L)}_{{\ell}}\cos\th) Y^{{\ell}}_{,\vartheta}+
(\alpha^{(L)}_{{\ell}}+{\tilde \alpha}^{(L)}_{{\ell}}\cos\th)\frac{ Y^{{\ell}}_{,\varphi}}{\sin\th}\nn\\
&&+\zeta^{(L)}_{{\ell}}\sin\th Y^{{\ell}}+\chi^{(L)}_{{\ell}}X^{{\ell}}-
\xi^{(L)}_{{\ell}}\sin\th W^{{\ell}}=0,\label{eqG2b}\end{aligned}$$ where $L=0,1$ and the first equation corresponds to $\delta {\cal
E}_{t\vartheta}=0$ and $\delta {\cal E}_{r\vartheta}=0$, whereas the last equation corresponds to $\delta {\cal E}_{t\varphi}=0$ and $\delta {\cal E}_{r\varphi}=0$. Finally the third group is $$\begin{aligned}
\delta {\cal E}_{(\vartheta\varphi)}&\equiv&
f_{{\ell}}\sin\th Y^{{\ell}}_{,\th}+g_{{\ell}} Y^{{\ell}}_{,\varphi}+s_{{\ell}}
\frac{X^{{\ell}}}{\sin\th}+t_{{\ell}}W^{{\ell}}=0, \label{eqG3a}\\
%%%%
\delta {\cal E}_{(-)}&\equiv&
g_{{\ell}}\sin\th Y^{\ell}_{,\th}-f_{{\ell}} Y^{{\ell}}_{,\varphi}-t_{{\ell}}
\frac{X^{{\ell}}}{\sin\th}+s_{{\ell}}W^{{\ell}}=0\,, \label{eqG3b}\end{aligned}$$ corresponding to $\delta {\cal E}_{\vartheta\varphi}=0$ and $\delta {\cal E}_{\vartheta\vartheta} -{\delta {\cal E}_{\varphi\varphi}}/{\sin\vartheta^2}=0$, respectively. In the equations above, $X^\ell$ and $W^\ell$ are the the tensor spherical harmonics defined as in Eq. .
The coefficients appearing in these equations are *linear* and *purely radial* functions of the perturbation variables. Furthermore, they naturally divide into two sets accordingly to their parity: $$\begin{aligned}
\text{Polar:}\qquad && A^{(I)}_{{\ell}},\quad C^{(I)}_{{\ell}},\quad \alpha^{(L)}_{{\ell}},\quad \tilde\beta^{(L)}_{{\ell}},\quad \zeta^{(L)}_{{\ell}},\quad \xi^{(L)}_{{\ell}},\quad s_{{\ell}},\quad f_{{\ell}},\nn\\
%%%
\text{Axial:}\qquad &&\tilde A^{(I)}_{{\ell}},\quad B^{(I)}_{{\ell}},\quad \beta^{(L)}_{{\ell}},\quad \tilde\alpha^{(L)}_{{\ell}},\quad \eta^{(L)}_{{\ell}},\quad \chi^{(L)}_{{\ell}},\quad t_{{\ell}},\quad g_{{\ell}}.\nn\end{aligned}$$ The explicit form of the coefficients is given in the online notebook [@webpage]. The crucial point is to recognize that the coefficients above are purely radial functions, that is, the entire angular dependence has been completely factored out in the linearized Einstein equations.
### Separation of the angular dependence
The decoupling of the angular dependence of the Einstein equations for a slowly-rotating star was performed by Kojima [@Kojima:1992ie] (see also [@ChandraFerrari91]) by using the orthogonality properties of the spherical harmonics, which are derived in \[app:orthogonality\] for completeness. The procedure has been extended to the case of slowly-rotating BHs in Refs. [@Pani:2012vp; @Pani:2012bp]. Here we adopt the same technique.
Multiplying Eq. by $Y^{*\,\ell'}$ and integrating over the sphere, we get $$0=\int d\Omega {Y^*}^\ell \delta{\cal E}_{(I)}=A_\ell^{(I)}+i m C_\ell^{(I)}+{\cal L}_0^{\pm1} \tilde A_{\ell}^{(I)}+{\cal L}_1^{\pm1} B_{\ell}^{(I)}\,,$$ where the operators ${\cal L}_i^{\pm1}$ are defined in \[app:orthogonality\]. Using the explicit forms in the appendix, we obtain[^16] $$\begin{aligned}
&&A^{(I)}_{{\ell}}+i mC^{(I)}_{{\ell}}+\cQ_{{\ell}}\left[{\tilde A}^{(I)}_{{\ell-1}}
+(\ell-1){B}^{(I)}_{{\ell-1}}\right]+\cQ_{{\ell+1}}\left[{\tilde A}^{(I)}_{{\ell+1}}
-(\ell+2){ B}^{(I)}_{{\ell+1}}\right]=0,\nn\\\label{decG1}\end{aligned}$$ where ${\cal Q}_{{\ell}}$ is defined as in Eq. . Similarly, Eqs. - can be decoupled as follows $$\begin{aligned}
0&=&\int d\Omega\left[ {Y^*}^{\ell'}_{,\vartheta}\delta{\cal E}_{(L\vartheta)}+\frac{{Y^*}^{\ell'}_{,\varphi}}{\sin\vartheta}\delta{\cal E}_{(L\varphi)}\right]\\
&=& {\ell(\ell+1)}\alpha_{\ell}^{(L)}+{i} m\left[(\ell-1)(\ell+2)\xi^{(L)}_{{\ell}}
-{\tilde\beta}^{(L)}_{{\ell}}-\zeta^{(L)}_{{\ell}}\right]+{\cal L}_2^{\pm1}\eta_{\ell}^{(L)}+{\cal L}_3^{\pm1}\tilde\alpha_{\ell}^{(L)}+{\cal L}_4^{\pm1}\chi_{\ell}^{(L)}\,,\nn\\
%%%
0&=&\int d\Omega\left[ {Y^*}^{\ell'}_{,\vartheta}\delta{\cal E}_{(L\varphi)}-\frac{{Y^*}^{\ell'}_{,\varphi}}{\sin\vartheta}\delta{\cal E}_{(L\vartheta)}\right]\\
&=&{\ell(\ell+1)} \beta^{(L)}_{{\ell}}+{i} m\left[(\ell-1)(\ell+2)\chi^{(L)}_{{\ell}}
+{\tilde\alpha}^{(L)}_{{\ell}}+\eta^{(L)}_{{\ell}}\right]+{\cal L}_2^{\pm1}\zeta_{\ell}^{(L)}+{\cal L}_3^{\pm1}\tilde\beta_{\ell}^{(L)}-{\cal L}_4^{\pm1}\xi_{\ell}^{(L)}\,.\nn\end{aligned}$$ Using the explicit form of the operators ${\cal L}_i^{\pm1}$ given in \[app:orthogonality\], the equations above reduce to $$\begin{aligned}
&&\ell(\ell+1) \alpha^{(L)}_{{\ell}}+i m\left[(\ell-1)(\ell+2)\xi^{(L)}_{{\ell}}
-{\tilde\beta}^{(L)}_{{\ell}}-\zeta^{(L)}_{{\ell}}\right]+\nn\\
&&\cQ_{{\ell}}(\ell+1)\left[(\ell-2)(\ell-1)\chi^{(L)}_{{\ell-1}}
+(\ell-1){\tilde\alpha}^{(L)}_{{\ell-1}}-\eta^{(L)}_{{\ell-1}}\right]-\nn\\
&&\cQ_{{\ell+1}}\ell\left[(\ell+2)(\ell+3)\chi^{(L)}_{{\ell+1}}
-(\ell+2){\tilde\alpha}^{(L)}_{{\ell+1}}-\eta^{(L)}_{{\ell+1}}\right]=0,
\label{decG2a}\\
%%%
&&\ell(\ell+1) \beta^{(L)}_{{\ell}}+i m\left[(\ell-1)(\ell+2)\chi^{(L)}_{{\ell}}
+{\tilde\alpha}^{(L)}_{{\ell}}+\eta^{(L)}_{{\ell}}\right]-\nn\\
&&\cQ_{{\ell}}(\ell+1)\left[(\ell-2)(\ell-1)\xi^{(L)}_{{\ell-1}}-
(\ell-1){\tilde\beta}^{(L)}_{{\ell-1}}+\zeta^{(L)}_{{\ell-1}}\right]+\nn\\
&&\cQ_{{\ell+1}}\ell\left[(\ell+2)(\ell+3)\xi^{(L)}_{{\ell+1}}
+(\ell+2){\tilde\beta}^{(L)}_{{\ell+1}}+\zeta^{(L)}_{{\ell+1}}\right]=0\,.
\label{decG2b}\end{aligned}$$ Notice that Eqs. - have exactly the same form as Eqs. (14)-(16) as in Kojima’s original work [@Kojima:1992ie]. Therefore, the radial equations have also the same schematic form. Finally, the last two equations and have the same form as Eqs. (18)-(19) in Kojima’s work [@Kojima:1992ie] and their angular dependence can be eliminate by constructing the following relations $$\begin{aligned}
0&=&\int d\Omega \frac{1}{{\ell(\ell+1)}-2}\left[{W^*}^{\ell'}\delta{\cal E}_{(-)}+\frac{{X^*}^{\ell'}}{\sin\vartheta}\delta{\cal E}_{(\vartheta\varphi)}\right]={\ell(\ell+1)} s_{\ell}-im f_{\ell}+{\cal L}_2^{\pm1} g_{\ell} \,, \nn\\
%%%%
0&=&\int d\Omega \frac{1}{{\ell(\ell+1)}-2}\left[{W^*}^{\ell'}\delta{\cal E}_{(\vartheta\varphi)}-\frac{{X^*}^{\ell'}}{\sin\vartheta}\delta{\cal E}_{(-)}\right]={\ell(\ell+1)} t_{\ell}+im g_{\ell}+{\cal L}_2^{\pm1} f_{\ell} \,, \nn\end{aligned}$$ which reduce to $$\begin{aligned}
&&0=\ell(\ell+1) s_{{\ell}}-i m f_{{\ell}}-\cQ_{{\ell}}(\ell+1)g_{{\ell-1}}+\cQ_{{\ell+1}}\ell g_{{\ell+1}} \label{decG3a}\,,\\
&&0=\ell(\ell+1) t_{{\ell}}+i m g_{{\ell}}-\cQ_{{\ell}}(\ell+1)f_{{\ell-1}}+\cQ_{{\ell+1}}\ell f_{{\ell+1}}\,. \label{decG3b}\end{aligned}$$ To summarize, our decoupling procedure in the slow-rotation limit allows to obtain a system of $10$ coupled, ordinary differential equations. To first order, the mixing of different angular functions in Eqs. - results in a mixing of perturbation functions with multipolar indices $\ell$, $\ell+1$ and $\ell-1$ in the linearized radial equations. The final radial equations are ,- and -. Their explicit form is available online [@webpage]. Note that these equations have the general structure given in Eqs. – to first order. Clearly, not all ten linearized Einstein equations are independent and the coupled system can be simplified further.
### First-order corrections to the eigenvalue equations
As previously discussed, the couplings to the $\ell\pm1$ terms do not contribute to the QNM spectrum at first order in $\tilde a$. For this reason we shall neglect these terms in the following. This allows us to treat the axial and polar sector separately.
Let us start with the axial sector. Neglecting the couplings to $\ell\pm1$ terms, the axial sector is fully described by three equations $$\begin{aligned}
0&=&\ell(\ell+1) \beta^{(L)}_{{\ell}}+{i} m\left[(\ell-1)(\ell+2)\chi^{(L)}_{{\ell}}
+{\tilde\alpha}^{(L)}_{{\ell}}+\eta^{(L)}_{{\ell}}\right] ,\nn\\ \label{axial_dec}
0&=&\ell(\ell+1) t_{{\ell}}+{i} m g_{{\ell}} , \label{dec_t}\end{aligned}$$ where $L=0,1$. Only two equations are independent and they can be solved for the axial perturbations $h_0^\ell$ and $h_1^\ell$. We define the Regge-Wheeler function $\Psi^\ell$ as $$h_1^\ell=\frac{r\Psi^\ell}{F},$$ Then, from Eq. with $L=1$, we get $$\begin{aligned}
{h_0^\ell}'&=&\frac{2 (r-2M) \omega h_{00}^\ell-i \left(\left(\ell(\ell+1)-2\right) (2 M-r)+r^3 \omega ^2\right) \Psi^\ell}{r (r-2M) \omega }\nn\\
&&+\frac{2 \tilde{a}M^2 m \left(6 (r-2M) \omega h_{00}^\ell-i \ell (\ell+1) \left(\left(\ell(\ell+1)-2\right) (2 M-r)-r^3 \omega ^2\right) \Psi^\ell\right)}{\ell(\ell+1) r^4 (r-2M) \omega ^2}\,.\nn\end{aligned}$$ Substituting this equation in the remaining Eqs. with $L=0$ and $L=2$ we get a decoupled equation for $\Psi^\ell$ only. By defining $$\Psi^\ell=\psi^\ell\left(1-\frac{2m \tilde{a}M^2}{\omega r^3}\right)\,,$$ the final modified Regge-Wheeler equation describing axial perturbations reads $$\frac{d^2\psi^\ell}{dr_*^2}+\left[\omega^2-\frac{4m \tilde{a}M^2\omega}{r^3}-V_{\rm axial}\right]\psi^\ell=0\,.\label{RWaxial}$$ with $$V_{\rm axial}=F(r)\left(\frac{\ell(\ell+1)}{r^2}-\frac{6M}{r^3}+\frac{24m\tilde{a}M^2(3r-7M)}{\ell(\ell+1)\omega r^6}\right)\,.$$ The Schroedinger-like equation above can then be solved with standard methods, e.g. continued fractions or direct integration.
Let us now turn to the polar sector, which is slightly more involved. Neglecting the coupling to axial perturbations with $\ell\pm1$, the polar sector is described by the following equations:
$$\begin{aligned}
0&=&A^{(I)}_{{\ell}}+i mC^{(I)}_{{\ell}}\,,\\
0&=&\ell(\ell+1) \alpha^{(L)}_{{\ell}}+i m\left[(\ell-1)(\ell+2)\xi^{(L)}_{{\ell}}
-{\tilde\beta}^{(L)}_{{\ell}}-\zeta^{(L)}_{{\ell}}\right]\,,\\
0&=&\ell(\ell+1) s_{{\ell}}-i m f_{{\ell}} \,.\end{aligned}$$
The last equation can be solved for $H_2^\ell$, whereas the first equation with $I=2$ can be solved for $H_0^\ell$. The remaining equations can be reduced to a system of first order equations for $K^\ell$ and $H_1^\ell$. Finally, the system can be reduced to a single second-order equation for a new function $Z^\ell$ in term of which $K^\ell$ and $H_1^\ell$ are defined. The detailed procedure is derived in the notebook [slow\_rot\_grav\_Kerr.nb](slow_rot_grav_Kerr.nb) [@webpage]. The final result is a modified Zerilli equation describing polar perturbations: $$\frac{d^2Z^\ell}{dr_*^2}+\left[\omega^2-\frac{4m \tilde{a}M^2\omega}{r^3}-V_{\rm polar}\right]Z^\ell=0\,.\label{Zerillipolar}$$ with $$\begin{aligned}
V_{\rm polar}&&=F(r)\left[\frac{2 M}{r^3}+\frac{(\ell-1) (\ell+2)}{3} \left(\frac{1}{r^2}+\frac{2 (\ell-1) (\ell+2) \left(\ell(\ell+1)+1\right)}{\left(6 M+r \left(\ell(\ell+1)-2\right)\right)^2}\right)\right.\nn\\
&&\left.+\frac{4 \tilde{a} m M^2 }{r^7 \ell (\ell+1 ) \left(6 M+r \left(\ell(\ell+1)-2\right)\right)^4 \omega } \left(27648 M^6+2592 M^5 r (6 \ell (\ell+1 )-19)\right.\right.\nn\\
&&\left.\left.+144 M^4 r^2 \left(230+\ell (\ell+1 ) (21 \ell (\ell+1 )-148)+6 r^2 \omega ^2\right)\right.\right.\nn\\
&&\left.\left.+12 M^2 r^4 \left(\ell(\ell+1)-2\right)^2\left(\ell (\ell+1 ) (-12+5 \ell (\ell+1 ))+28 r^2 \omega ^2-4\right)\right.\right.\nn\\
&&\left.\left.+12 M^3 r^3 \left(\ell(\ell+1)-2\right) \left(374+\ell (\ell+1 ) (29 \ell (\ell+1 )-200)+72 r^2 \omega ^2\right)\right.\right.\nn\\
&&\left.\left.+r^6 \left(\ell(\ell+1)-2\right)^3 \left(-3 \left(\ell(\ell+1)-2\right) \left(\ell(\ell+1)+2\right)+2 r^2 \left(\ell(\ell+1)-4\right) \omega ^2\right)\right.\right.\nn\\
&&\left.\left.+M r^5 \left(\ell(\ell+1)-2\right)^2 \left(\left(\ell(\ell+1)-2\right) \left(\ell(\ell+1)+2\right) (7 \ell (\ell+1 )-38)\right.\right.\right.\nn\\
&&\left.\left.\left.+24 r^2 (2 \ell (\ell+1 )-5) \omega ^2\right)\right)\right]\,.\end{aligned}$$ Although the polar potential is more involved than the axial one, we are still left with a single second-order ODE that can be solved by standard methods[^17].
Computing the QNMs in slow-rotation limit
-----------------------------------------
One of the key advantages of the slow-rotation approximation is that, for basically any stationary and axisymmetric background, the linearized field equations have a form which is very similar to the nonrotating case. Thus, all existing methods to solve the linear dynamics around spherically symmetric BHs can be directly applied to the slow-rotation case with only minor modifications. The two principal modifications are:
- The coupling between modes with different parity and differnt harmonic indices $\ell$,
- The behavior of the fields close to the horizon.
The first correction automatically implies that, in the Fourier space, we are dealing with coupled systems of ODEs in the form: $${\cal D}\mathbf{Y}^\ell +\mathbf{V}\mathbf{Y}^\ell=\mathbf{S}_{+1}\mathbf{Y}^{\ell+1}+\mathbf{S}_{-1}\mathbf{Y}^{\ell-1}+\mathbf{S}_{+2}\mathbf{Y}^{\ell+2}+\mathbf{S}_{-2}\mathbf{Y}^{\ell-2}+\dots\,. \label{systemC}$$ where ${\cal D}$ is some differential radial operator, $\mathbf{V}$ and $\mathbf{S}_{\pm n}$ are radial $N\times N$ matrices and $\mathbf{Y}^{\ell}$ is a $N$-dimensional vector which contains all relevant perturbation functions with harmonic index $\ell$.
Since $\ell=0,1,2,..$, the full system formally contains an *infinite* number of equations. In practice, we can truncate it at some given value of $\ell=L$, compute the modes as explained below, and finally check convergence by increasing the truncation order. Let us suppose we truncate the couplings at order $\ell+p$, i.e. for a given $m$ we assume $$\mathbf{Y}^{L}\equiv0\qquad {\rm when}\qquad L>\ell+p\,.$$ Therefore, we can recast the system into a system of $(2p+1)N$ coupled second-order ODEs or, equivalently, to a system of $(2p+1)2N$ first-order ODEs in the schematic form $$\frac{d \mathbf{Z}}{dr_*}+\mathbf{W}\mathbf{Z}=0\,, \label{system1st}$$ where $r_*$ is some suitable coordinate and the $(2p+1)2N$-dimensional vector $\mathbf{Z}$ contains $\mathbf{Y}^{\ell}$, $\mathbf{Y}^{\ell\pm 1}$,..., $\mathbf{Y}^{\ell\pm p}$ and their first derivatives. A consistency check of the slow-rotation approximation is that the couplings to perturbations with $\ell-p$ are automatically vanishing when $p>\ell$. Although the system can contain several coupled equations, nonetheless the latter are ODEs and the corresponding eigenvalue problem (or the corresponding time evolution in the time-domain) can be analyzed with the methods discussed in the previous sections. The second modification listed above arises due to the frame-dragging effect. Indeed, it is generically possible to recast the perturbation functions and to choose a suitable radial coordinate $r_*$ such that $$\begin{aligned}
Y_i^\ell(r)&\sim& e^{i\omega r_*},\qquad \hspace{0.4cm}r\to\infty, \label{asym_inf}\\
Y_i^\ell(r)&\sim& e^{-i k_H r_*},\qquad r\to r_+, \label{asymp_hor}\end{aligned}$$ where $r$ is the original radial coordinate as defined in Eq. , $$k_H=\omega-m\Omega_H,\label{kH}$$ and $$\Omega_H=-\lim_{r\to r_+}\frac{g^{(0)}_{t\varphi}}{g^{(0)}_{\varphi\varphi}}\,,$$ is the angular velocity at the horizon of locally non-rotating observers to some given order in $\tilde{a}$. The near-horizon behavior shows that, if $\omega<m\Omega_H$, an observer at infinity would see waves outgoing from the horizon. By computing the energy and angular momentum fluxes carried by these waves, it is possible to show that superradiant amplification occurs in spinning BH spacetimes when the superradiance condition $k_H<0$ is met. This analysis was originally performed for perturbations of a Kerr BH within the Teukolsky formalism [@Teukolsky:1974yv]. We have here shown how the same result can be obtained for any perturbation of generic rotating, axisymmetric BH spacetimes within the slow-rotation approximation. An important point to bear in mind is that, in order to consistently discuss superradiance in a slow-rotation approximation, one needs to include at least *second-order* terms in the expansion [@Pani:2012bp]. Indeed, at superradiance $\omega<\Omega_H\sim{\cal O}(\tilde a)$ and the energy of the wave scales as $\omega^2\sim{\cal O}(\tilde a^2)$, so that a first-order analysis is in principle not sufficient. Nonetheless, at least in some specific case [@Pani:2012bp], the first and second order results are in qualitatively (and sometimes in remarkably good quantitative) agreement even in the superradiant regime where, in principle, deviations of order unity might be expected. One naive reason for this agreement is that, by symmetry arguments, the superradiant condition itself only contains odd powers of $\tilde{a}$, so that the first-order corrections is valid up to third order and the onset of superradiance can be captured already at first order.
### Eigenvalue problems with couplings to different $\ell$: Breit-Wigner method
When considering the couplings to higher multipolar indices, the number of coupled equations that have to be solved may be quite large, depending on the order of the slow-rotation expansion and on the number of perturbation variables for a given $\ell$. In principle, any of the methods previously discussed for coupled systems can be applied to this case. In practice, if the number of equations is large, some of the methods become inefficient.
If the eigenvalue problem admits slowly-damped modes (i.e. modes with $\omega_I\ll\omega_R$) then the Breit-Wigner method discussed above is very convenient because its simplicity makes it well-suited to deal with large systems of ODEs.
Roughly speaking, slowly-damped modes exist if the potential has a sufficiently deep minimum. In a single-ODE problem this happens, for example, if the perturbation field is massive [@Detweiler:1980uk; @Dolan:2007mj] or for small AdS BHs [@Berti:2009wx]. In a coupled-ODE problem the situation is less clear, but slowly-rotating modes are expected in the same settings and indeed they have been found recently [@Rosa:2011my; @Pani:2012vp; @Pani:2012bp].
To illustrate how the matrix-valued Breit-Wigner method works in case of couplings with different harmonic indices $\ell$, let us consider one simple case: massive spin-1 (Proca) perturbations of a Kerr BH to second order in the rotation [@Pani:2012bp]. The method has been implemented in the notebook [BW\_Proca\_2nd\_order.nb](BW_Proca_2nd_order.nb) [@webpage].
The coupled system contains three ODEs for two polar functions $u_{(2)}^\ell$ and $u_{(3)}^\ell$ and an axial function $u_{(4)}^\ell$. As explained before, at first order in $\tilde{a}$ the polar functions are coupled to $u_{(4)}^{\ell\pm1}$ and, at second order in $\tilde{a}$, they are also coupled to $u_{(2)}^{\ell\pm2}$ and $u_{(3)}^{\ell\pm2}$. Let us suppose we truncate the axial sector at $\ell=L$ and the polar sector at $\ell=L+1$. When $m=0$, the system reduces to $N=3L$ coupled second-order ODEs for $L-1$ axial functions and $2L+1$ polar functions, including the monopole [@Pani:2012bp]. When $|m|>0$ the truncated system contains $N=3L-3|m|+2$ second-order ODEs (for $L-|m|$ axial functions and $2L-2|m|+2$ polar functions). In all cases we are left with a system of $N$ second-order ODEs for $N$ perturbation functions[^18].
In this case the Breit-Wigner method proves to be very instructive, because it makes manifest an interesting physical property of the Proca system [@Pani:2012bp]. When the mass of the spin-1 field is small, $M\mu\ll1$, the spectrum has a hydrogen-like behavior: $$\omega_R^2=\mu^2\left[1-\left(\frac{M\mu}{\ell+n+S +1}\right)^2\right]+{\cal O}\left(\mu^4\right)\,,\label{wRProca}$$ where $n\geq0$ is the overtone number and $S$ is the so-called polarization index \[in the Proca case, $S=0$ for transverse axial modes and $S=\pm1$ for longitudinal polar modes\]. The equation above predicts an approximate degeneracy for modes with the same value of $\ell+n+S$ in the small $\mu$ limit. In the Breit-Wigner method, the mode frequencies can be identified as minima of the real-valued function $|\det{\mathbf{S}}|^2$. The approximate degeneracy translates into a series of minima which are very close to each other in the real axis (in fact, their separation scales with $\mu^4$ in the small $\mu$ limit).
In the notebook [BW\_Proca\_2nd\_order.nb](BW_Proca_2nd_order.nb) we show $|\det{\mathbf{S}}|^2$ in a given range of $\omega_R$ for $m=1$ and arbitrary truncation order [@webpage]. In the selected range, the function shows three minima, which correspond to a three-fold degeneracy, $\ell+n+S=1$ in Eq. . The latter can be achieved by $(\ell,n,S)=(1,0,0)$, $(2,0,-1)$, $(1,1,-1)$. Therefore, one minimum corresponds to the axial fundamental mode, whereas the other two minima correspond to the fundamental and to the first overtone of the polar modes. This example shows the advantage of the resonance method: in a single numerical implementation one is able to obtain the *full* quasi-bound spectrum, irrespectively of the parity of the modes and even for different values of $\ell$, up to a given truncation order.
Conclusions and open problems {#sec:conclusions}
=============================
We have presented self-consistent tools to derive, separate and solve numerically the perturbation equations of stationary and axisymmetric BHs within some approximate scheme. In general, the linearized field equations on a stationary and axisymmetric spacetime form a coupled $2+1$ dimensional system which can be evolved using the techniques discussed elsewhere [@Witek:2012tr; @Helvi; @Hiro]. Such evolution requires advanced numerical methods and it is usually time- and resource-consuming. Here, we have introduced complementary techniques to reduce the problem to a $1+1$ time evolution [@Dolan:2012yt] or to a simple one dimensional problem in the frequency domain.
Working in some perturbative scheme, it is possible to solve the linear dynamics on *generic* stationary and axisymmetric BHs. This requires a combination of analytical and numerical tools, which we have discussed in some detail.
Most of the numerical methods presented in this work have been implemented in simple [<span style="font-variant:small-caps;">Mathematica</span>]{} notebooks which are publicly available [@webpage]. We hope this will help students and researchers to adapt and extend them to a multitude of problems that are still open in this field.
Spinning BHs play a crucial role in gravity and in astrophysics, and they are indeed ubiquitous in modern applications. There are several venues in which the techniques we discussed can be useful. The linear response of a BH to gravitational perturbations is mostly unknown if the background metric is not Kerr or if the underlying theory of gravity is not GR. These extensions are important to discuss the BH ringdown and the gravitational-wave emission in modified theories of gravity. On a more theoretical side, the linear dynamics of hairy BHs is relevant in the context of the gauge/gravity duality. If one wishes to describe an axisymmetric theory in the holographic space, understanding the linear dynamics of spinning AdS BHs is crucial. Probably one of the most important applications concerns the study of BH perturbations in higher-dimensions. Besides some particular cases of enhanced symmetries, a generic treatment of perturbed spinning BHs in higher dimensions is still lacking. This prevents a complete stability analysis and a full understanding of the greybody factors of spinning higher-dimensional BHs. Here we have focused on vacuum solutions, but astrophysical BHs are surrounded by various type of matter and magnetic fields. Extending the techniques discussed in this work to the case of nonvacuum solutions is an interesting problem. Finally, some direct applications of the slow-rotation approach include the study of massive spin-2 fields around spinning BHs [@Brito:2013wya] and the analysis of gravito-electromagnetic perturbations of the Kerr-Newman metric in GR [@Pani:2013ija; @Pani:2013wsa]. We hope to report on these interesting topics in the near future.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Leonardo Gualtieri for suggestions and for a careful reading of the manuscript. It is also a pleasure to thank Emanuele Berti, Vitor Cardoso and Leonardo Gualtieri, who have been involved in various projects related to this work, and the Organizer and the Editors of the NR/HEP2 Spring School [@webpage]. This work was supported by the Intra-European Marie Curie contract aStronGR-2011-298297 and by FCT - Portugal through PTDC projects FIS/098025/2008, FIS/098032/2008, CERN/FP/123593/2011. Computations were performed on the “Baltasar Sete-Sois” cluster at IST, the cane cluster in Poland through PRACE DECI-7 “Black hole dynamics in metric theories of gravity”, on Altamira in Cantabria through BSC grant AECT-2012-3-0012, on Caesaraugusta in Zaragoza through BSC grants AECT-2012-2-0014 and AECT-2012-3-0011, XSEDE clusters SDSC Trestles and NICS Kraken through NSF Grant No. PHY-090003, Finis Terrae through Grant CESGA-ICTS-234.
List of publicly available codes [@webpage]
===========================================
Most of the numerical and analytical methods discussed in the main text have been directly implemented in ready-to-be-used [<span style="font-variant:small-caps;">Mathematica</span>]{}^^ notebooks, which are publicly available [@webpage]. Here we give a short description of them:
- [BW\_Proca\_2nd\_order.nb](BW_Proca_2nd_order.nb): Proca quasi-bound states of a Kerr BHs in GR to second order in the spin (including couplings with different harmonic indices) via the Breit-Wigner resonance method.
- [CF\_matrix\_3terms.nb](CF_matrix_3terms.nb): scalar, electromagnetic and gravitational QNMs of a Schwarzschild BH computed with a matrix-valued continued fraction method.
- [DCS\_DI.nb](DCS_DI.nb): Gravito-scalar QNMs of Schwarzschild BHs in Dynamical Chern-Simons gravity computed with a matrix-valued direct integration.
- [DCS\_pert\_eqs.nb](DCS_pert_eqs.nb): Derivation of the gravito-scalar perturbation equations of Schwarzschild BHs in Dynamical Chern-Simons gravity.
- [field\_eqs.nb](field_eqs.nb): Derivation of the field equations starting from a Lagrangian in a modified gravity theory.
- [series\_method\_DCS.nb](series_method_DCS.nb): Gravito-scalar QNMs of Schwarzschild-AdS BH in Dynamical Chern-Simons gravity computed with a matrix-valued series method.
- [slow\_rot\_grav\_Kerr.nb](slow_rot_grav_Kerr.nb): Derivation of the gravitational perturbation equations (axial and polar) of a Kerr background to first order in the angular momentum.
- [slow\_rot\_scalar.nb](slow_rot_scalar.nb): Derivation of the perturbation equations for a massive Klein-Gordon equation on a Kerr background to second order in the angular momentum.
Orthogonality properties of the spherical harmonics {#app:orthogonality}
===================================================
In this appendix we give some useful orthogonality properties of scalar, vector and tensor spherical harmonics. For clarity, here we explicitly append both multipolar indices $\ell$ and $m$. We define the scalar product on the two–sphere as $$\begin{aligned}
<f,g>&\equiv&\int d\Omega f^*g=\int d\vartheta d\varphi\sin\vartheta f^*g\,, \label{scalarproduct}\\
<f_a,g_a>&\equiv&\int d\Omega f_a^*g_b\gamma^{ab}\,, \\
<f_{ab},g_{ab}>&\equiv&\int d\Omega f_{ab}^*g_{cd}\gamma^{ca}\gamma^{db}\,,\end{aligned}$$ where $\gamma_{ab}={\rm diag}(1,\sin^2\vartheta)$ is the induced metric on the two-sphere. By definition, scalar spherical harmonics satisfy the fundamental identity $$Y^{\ell m}_{,\vartheta\vartheta}+\cot\vartheta Y^{\ell m}_{,\vartheta}+
\frac{1}{\sin^2\vartheta}Y^{\ell m}_{,\varphi\varphi}=-\ell(\ell+1) Y^{\ell m}\,. \label{propY}$$ From this equation and from the orthogonality property for the scalar spherical harmonic \[cf. also Eq. \]: $$<Y^{\ell m},Y^{\ell'm'}>=\delta^{\ell\ell'}\delta^{mm'}\,,
\label{orthoY}$$ we obtain the following relations: $$\begin{aligned}
<Y^{\ell m}_a,Y^{\ell m}_a>&=&<S^{\ell m}_a,S^{\ell m}_a>=\int d\vartheta d\varphi\sin\vartheta\left(Y^{*\ell\,m}_{,\vartheta}Y^{\ell m}_{,\vartheta}
+\frac{1}{\sin^2\vartheta}Y^{*\ell\,m}_{,\varphi}Y^{\ell m}_{,\varphi}\right)\nn\\
&&=-\int d\vartheta d\varphi\sin\vartheta Y^{*\ell\,m}\left(Y^{\ell m}_{,\vartheta\vartheta}+
\cot\vartheta Y^{\ell m}_{,\vartheta}+\frac{1}{\sin^2\vartheta}Y^{\ell m}_{,\varphi\varphi}\right)\nn\\
&&=\ell(\ell+1) \int d\vartheta d\varphi\sin\vartheta Y^{*\ell\,m}Y^{\ell m}\nn\\
&&=2(n+1)\,,
\label{orthoYS}\end{aligned}$$ where $2n\equiv(\ell-1)(\ell+2)$ and we have defined the polar and axial vector harmonics, which respectively read: $$\begin{aligned}
Y_{a}^{\ell m}&=&(Y^{\ell m}_{,\vartheta},Y^{\ell m}_{,\varphi})\,,\\
S_{a}^{\ell m}&=&(- Y^{\ell m}_{,\varphi}/\sin\vartheta,\sin\vartheta Y^{\ell m}_{,\vartheta})\,.\end{aligned}$$
The spin–two harmonics are defined as $$_{-2}S^{\ell m}(\vartheta,\varphi)\equiv\frac{W^{\ell m}(\vartheta,\varphi)-{i}X^{\ell m}(\vartheta,\varphi)/\sin\vartheta}{\ell(\ell+1) (\ell(\ell+1)-2)}\,,$$ where $W^{\ell m}$ and $X^{\ell m}$ are defined as in Eqs. . The spin–two harmonics satisfy the orthogonality property: $$<\,_{-2}S^{\ell m},\,_{-2}S^{\ell'm'}>=\delta^{\ell\ell'}\delta^{mm'}\,.$$ Using this relation, one can obtain the following: $$\begin{aligned}
&&\frac{1}{2}<Z^{\ell m}_{ab},Z^{\ell'm'}_{ab}>=
\frac{1}{2}<S^{\ell m}_{ab},S^{\ell'm'}_{ab}>\nn\\
&&=\int d\vartheta d\varphi\sin\vartheta\left(W^{*\ell\,m}_{ab}W^{\ell'm'}_{cd}
+\frac{X^{*\ell\,m}_{ab}X^{\ell'm'}_{cd}}{\sin^2\vartheta}\right)\gamma^{ca}\gamma^{db}\nn\\
&&=4n(n+1)\delta^{\ell\ell'}\delta^{mm'}\,,\label{orthoZS}\end{aligned}$$ where we have defined the polar and axial tensor harmonics, which respectively read: $$\begin{aligned}
Z_{ab}^{\ell m}&=&\left(\begin{array}{cc}
W^{\ell m}&\quad X^{\ell m}\\
X^{\ell m}&\quad -\sin^2\vartheta W^{\ell m}
\end{array}\right)\,,\\
%%%%%%
S_{ab}^{\ell m}&=&\left(\begin{array}{cc}
-X^{\ell m}/\sin\vartheta&\quad \sin\vartheta W^{\ell m}\\
\sin\vartheta W^{\ell m}&\quad \sin\vartheta X^{\ell m}
\end{array}\right)\,,\,.\end{aligned}$$ Similarly, we have $$\begin{aligned}
\int d\Omega \left[{W^*}^{\ell' m'} Y^{\ell m}_{,\varphi}-{X^*}^{\ell'm'} Y^{\ell \,m}_{,\vartheta}\right]&=&i m(\ell(\ell+1)-2)\delta_{mm'}\delta_{\ell\ell'}\,,\nn\\
\int d\Omega \cos\vartheta\left[{W^*}^{\ell' m'}W^{\ell\,m}+\frac{{X^*}^{\ell'm'}X^{\ell\,m}}{\sin\vartheta^2}\right]&=&2im(\ell(\ell+1)-2)\delta_{mm'}\delta_{\ell\ell'}\,,\nn\\
\int d\Omega \left[\frac{{{W^*}^{\ell'm'}X^{\ell\,m}}-{{X^*}^{\ell'm'}W^{\ell\,m}}}{\sin\vartheta}\right]&=&0\,.\nn\end{aligned}$$ Two other useful identities involving the spherical harmonics are given in Eqs. and . Using those identities, we can evaluate the following operators, acting on a generic function $A_{\ell m}$ $$\begin{aligned}
{\cal L}_0^{\pm1} A_{\ell m} &\equiv& A_{\ell'm'}\int d\Omega {Y^*}^{\ell m}\cos\vartheta Y^{\ell' m'}={\cal Q}_{\ell m} A_{\ell-1 m}+{\cal Q}_{\ell+1 m}A_{\ell+1 m} \,,\nn\\
%%%
{\cal L}_1^{\pm1} A_{\ell m} &\equiv& A_{\ell'm'} \int d\Omega {Y^*}^{\ell m}\sin\vartheta Y^{\ell' m'}_{,\vartheta}=(\ell-1){\cal Q}_{\ell m} A_{\ell-1 m}-(\ell+2){\cal Q}_{\ell+1 m} A_{\ell+1 m} \,,\nn\\
%%%%%
{\cal L}_2^{\pm1} A_{\ell m} &\equiv& \left[-2{\cal L}_0^{\pm1}-{\cal L}_1^{\pm1}\right] A_{\ell m} =-(\ell+1){\cal Q}_{\ell m} A_{\ell-1 m}+\ell Q_{\ell+1 m} A_{\ell+1 m} \,.\nn\\
%%%%
%%%
{\cal L}_3^{\pm1} A_{\ell m} &\equiv& \left[{\ell(\ell+1)}{\cal L}_0^{\pm1}+{\cal L}_1^{\pm1}\right] A_{\ell m} \nn\\
&=&(\ell-1)(\ell+1){\cal Q}_{\ell m} A_{\ell-1 m}+\ell(\ell+2){\cal Q}_{\ell+1 m} A_{\ell+1 m} \,,\nn\\
%%%%%
{\cal L}_4^{\pm1} A_{\ell m} &\equiv&\left[-2({\ell(\ell+1)}-2){\cal L}_0^{\pm1}+({\ell(\ell+1)}+2){\cal L}_1^{\pm1}\right]A_{\ell m} \nn\\
&=&(\ell^2-1)(\ell-2){\cal Q}_{\ell m} A_{\ell-1 m}-\ell(\ell+2)(\ell+3)Q_{\ell+1 m} A_{\ell+1 m}\,,\nn\end{aligned}$$ where ${\cal Q}_{\ell m}$ is defined as in Eq. (omitting the subscript $m$). The operators above are used in the main text to separate the angular dependence of the linearized field equations within the slow-rotation expansion.
[^1]: Based on a series of lectures given at the NR/HEP2: Spring School \[11-14 March, 2013 (Lisbon, Portugal)\]. [<span style="font-variant:small-caps;">Mathematica</span>]{}^^ notebooks are publicly available on the School webpage [@webpage].
[^2]: The Myers-Perry metric is the generalization of the Kerr solution to higher dimensions. Even if this spacetime is Type D in the Petrov classification, separability in the Teukolsky formalism is still an open problem in the general case, see the main text. Another notable example of nonseparability in Type-D background is the case of massive spin-1 perturbations on a Kerr metric. This is discussed later on.
[^3]: Furthermore, from now on we will append the relevant multipolar index $\ell$ to any perturbation variable but we will omit the index $m$, because in an axisymmetric background it is possible to decouple the perturbation equations so that all quantities have the same value of $m$.
[^4]: Equation can be replaced by a generic system of ordinary differential equations which is of second-order in time and of second-order in the radial coordinate. The rest of the discussion would be very similar to that given in the text. We find it convenient to use Eq. , though, mostly to simplify the notation.
[^5]: In case of multiple horizons a slightly different ansatz is more convenient. See e.g. [@Leaver:1990zz; @Berti:2005eb].
[^6]: **Exercise:** Extend the code in [CF\_matrix\_3terms.nb](CF_matrix_3terms.nb) in order to reduce a generic $N$-term recurrence relation into a three-term one.
[^7]: **Exercise:** adapt the code to derive the field equations of so-called quadratic gravity [@Yunes:2009hc; @Pani:2011gy], ${\cal L}=\sqrt{-g}(R+\alpha_1 R^2+\alpha_2 R_{\mu\nu}^2+\alpha_3 R_{\mu\nu\rho\sigma}^2+\alpha_4 {}^*RR)$. In the small-$\alpha_i$ limit, what is the differential order are the field equations?
[^8]: **Exercise:** They are also the *only* static vacuum solution. Why?
[^9]: The distinction between “soft numerics” and “hard numerics” is an interesting concept that emerged during the first NR/HEP Workshop [@Cardoso:2012qm].
[^10]: **Exercise:** derive a similar relation for $\cos^n\th Y^{\ell}$.
[^11]: $\omega_1$ and $\omega_2$ are generically polynomials in $m$ but, due to the above symmetry, $\omega_1$ is an *even* polynomial.
[^12]: Even though in principle there may be modes which do not belong to the classes of “axial-led” or “polar-led” perturbations, all solutions belonging to one of these classes which fulfill the appropriate boundary conditions defining QNMs or bound states are also solutions of the full system – and belong to the eigenspectrum (up to second order in $\tilde{a}$).
[^13]: **Exercise**: the formalism applies to a *generic* stationary and axisymmetric background. Derive the field equation in the general background . To first order, it is easy to show that massive Klein-Gordon perturbations are described by:
$$F B\Psi_{\ell}''+\frac{1}{2}\left[B'F+F'B\right]\Psi_{\ell}'
+\left[\omega^2-\frac{2 m \varpi(r) \omega}{r^2}-F\left(\frac{\ell(\ell+1)}{r^2}+\mu^2+
\frac{B'}{2r}+\frac{B F'}{2r F}\right)\right]\Psi_{\ell}=0\,.$$
[^14]: **Exercise:** using the properties of the spherical harmonics, derive these identities.
[^15]: **Exercise:** by extending our procedure to higher order, reconstruct the Teukolsky scalar potential order by order in $\tilde{a}$.
[^16]: **Exercise:** derive the radial equations ,- and - explicitly.
[^17]: **Exercise:** compute the polar and axial QNMs integrating Eqs. and . The axial eigenvalue problem can be reduced to a six-term recurrence relation and solved by continued fractions. The polar sector can be solved by direct integration. Interestingly, the QNM spectrum is the same for the two sectors. This extends to first order in the rotation the well-known fact that axial and polar perturbations of a Schwarzschild BH are isospectral [@Chandra] and it is consistent with a mode analysis of the Kerr metric in the Teukolsky formalism.
[^18]: **Exercise:** as we have discussed, to second order in $\tilde{a}$, only the couplings to $\ell\pm1$ are important. This property can be verified numerically by truncating the coupled system to some high order and checking that the modes do not change if the truncation order is greater than $\ell+1$.
|
---
abstract: |
We propose a new joint image reconstruction method by recovering edge directly from observed data. More specifically, we reformulate joint image reconstruction with vectorial total-variation regularization as an $l_1$ minimization problem of the Jacobian of the underlying multi-modality or multi-contrast images. Derivation of data fidelity for Jacobian and transformation of noise distribution are also detailed. The new minimization problem yields an optimal $O(1/k^2)$ convergence rate, where $k$ is the iteration number, and the per-iteration cost is low thanks to the close-form matrix-valued shrinkage. We conducted numerical tests on a number multi-contrast magnetic resonance image (MRI) datasets, which show that the proposed method significantly improves reconstruction efficiency and accuracy compared to the state-of-the-arts.
**Keywords.** Joint image reconstruction, matrix norm, optimal gradient method, multi-contrast.
author:
- 'Yunmei Chen [^1]'
- 'Ruogu Fang [^2]'
- 'Xiaojing Ye [^3]'
bibliography:
- 'library.bib'
title: 'Joint image edge reconstruction and its application in multi-contrast MRI'
---
Introduction {#sec:intro}
============
The advances of medical imaging technology have allowed simultaneous data acquisition for multi-modality images such as PET-CT, PET-MRI and multi-contrast images such as T1/T2/PD MRI. Multi-modality/contrast imaging integrates two or more imaging modalities (or contrasts) into one system to produce more comprehensive observations of the subject. Such technology combines the strengths of different imaging modalities/contrasts in clinical diagnostic imaging, and hence can be much more precise and effective than conventional imaging. The images from different modalities or contrasts complement each other and generate high spatial resolution and better tissue contrast. However, effectively utilizing sharable information from different modalities and reconstructing multi-modality images remain as a challenging task especially when only limited data are available. Therefore, the main approach of joint image reconstruction is to incorporate similarities, such as anatomical structure, between all modalities/contrasts into the process to improve reconstruction accuracy [@Bilgic:2011a; @Ehrhardt:2016b; @Huang:2014a]. In this paper, we consider a new approach by jointly reconstructing edges of images, rather than the images themselves, directly from multi-modality/contrast imaging data. The final images can be obtained very easily given the reconstructed edges. We show that this new approach results in an $l_1$-type minimization that can be effectively solved by accelerated prox-gradient methods with an optimal $O(1/k^2)$ convergence rate, where $k$ is the iteration number. Moreover, the subproblems reduce complex vectorial/joint total variation regularizations to simple matrix-valued shrinkages, which often have cheap closed-form solutions. This is in sharp contrast to primal-dual based image reconstruction algorithms, such as primal-dual hybrid gradient method (PDHG) and alternating direction method of multipliers (ADMM), which only yield $O(1/k)$ convergence rate. Therefore, the proposed method can produce high quality reconstructions with much improved efficiency over the state-of-the-arts methods in multi-modality/contrast image reconstruction problems. The contributions of this paper can be summarized as follows. (a) We develop a novel two-step joint image reconstruction method that transforms the vectorial TV regularized minimization of image into an $l_1$ minimization of Jacobian. This enables a numerical scheme with optimal $O(1/k^2)$ convergence rate by employing accelerated gradient descent method (Section \[subsec:edgerecon\]). (b) In the resulting $l_1$ minimization problems, the main subproblem involves matrix norm (instead of vectorial TV) and can be solved easily using generalized shrinkage for matrices. We provide close-form solutions for several cases originated from commonly used vectorial TVs (Section \[subsec:shrinkage\] and Appendix \[apd:shrinkage\]). (c) We analyze the noise distribution after transformation, and incorporate it into the data fidelity of Jacobian in the algorithm which significantly improves reconstruction accuracy and efficiency (Section \[subsec:data\_reform\]). (d) We conduct a series to numerical tests on several multi-contrast MRI datasets and show the very promising performance of the proposed methods (Section \[sec:results\]). Although we only focus on multi-contrast MRI reconstruction where all data is acquired in Fourier space, it is worth noting that the proposed method can be readily extended to other cases, such as those with Radon data, hence is applicable to reconstructions involving other types of imaging modalities.
The remainder of the paper is organized as follows. We first provide an overview of recent literatures in joint image reconstruction in Section \[sec:related\]. Then we present the proposed method and address a number of details in Section \[sec:proposed\]. In Section \[sec:results\], we conduct a number of numerical tests on a variety of multi-contrast MR image reconstruction datasets. Section \[sec:conclusion\] concludes our findings.
Related Work {#sec:related}
============
There have been a significant amount effort devoted to develop models and algorithms that can effectively take the anatomy structure similarities across modalities/contrasts into account during joint reconstructions. As widely accepted, these structure similarities can be exploited using the locations and directions of the edges [@Ehrhardt:2015b; @Ehrhardt:2016a; @Ehrhardt:2015a] characterized by the magnitude and direction of the gradient of an image. The active researches on multi-modal/contrast image reconstructions have focused mainly on how to effectively utilize the complimentary information on these structure similarities to improve the accuracy and robustness of the reconstructions.
Inspired by the success of total variation (TV) based image reconstructions for scalar-valued images, many algorithms for joint reconstruction of multi-modal images extend the classical TV to vectorial TV for joint multi-modal image reconstructions. This extension aims at capturing the sharable edge information and performing smoothing along the common edges across the modalities. There have been several different ways to define the TV regularization for multi-modal images (represented by vector-valued functions). For instance, $\sum_{j=1}^m TV(u_j)$, where $u_j$ is the (scalar-valued) image of modality/channel/contrast $j$ in an $m$-modality imaging problem, is proposed in [@Blomgren:1998a]. In [@Duran:2016a], a number of variations along this direction, called collaborative TV (CTV), are studied and summarized comprehensively. A specific TV takes a particular form to integrate partial derivatives of the image across the modalities. For example, $\sum_{j=1}^m TV(u_j)$ takes $l_2$ norm of image gradient at every point (pixel) for each modality, then $l_1$ norm over all pixels, and finally $l_1$ norm (direct sum) across all modalities. Another commonly used CTV variation, called joint TV (JTV), is formulated as $\int_\Omega (\sum_{j=1}^m \sum_{l=1}^d |\partial_l u_j|^2)^{1/2} \dif x$ and has been successfully applied to color image reconstruction [@Bresson:2008a; @Sapiro:1996b], multi-contrast MR image reconstruction [@Huang:2014a; @Majumdar:2011a], joint PET-MRI reconstruction [@Ehrhardt:2015a], and in geophysics [@Haber:2013a]. As one can see, JTV is taking $l_2$ norm in terms of gradients, $l_2$ norm across modalities, and then finally $l_1$ for pixels. In general, the gradient of a vector-valued image consisting multiple modalities/contrasts is a tensor at each pixel, and the TV takes specific combination of norms regarding partial derivatives (gradients), modalities (channels, or contrasts), and pixels, respectively. Another joint regularization approach is to extend anisotropic TV regularization from scalar to vector-valued case. In this approach, anisotropic TV is employed to replace isotropic TV by an anisotropic term that incorporates directional structures exhibited by either the original data or the underlying image. Anisotropic TV been applied in standard single-modal image reconstruction with successes in [@Estellers:2015a; @Grasmair:2010a; @Lenzen:2015a], where structure tensor is used to provide information of both size and orientation of image gradients. In particular, the model proposed in [@Grasmair:2010a] employs an anisotropic TV regularization of form $\int_\Omega (\dif u^T A(u)\dif u)^{1/2} \dif x$ for image reconstruction. Here, $\dif u=(\partial_1 u,\partial_2 u)^T$ is the gradient of the single-modal, scalar-valued 2D image $u$, the diffusion tensor $A(u)$ is determined by the eigenvalues $(\lambda_1, \lambda_2)$ and corresponding eigenvectors $(v_1,v_2)$ of the structure tensor $J_\rho =k_\rho \dif u \dif u^T$. The model developed in [@Estellers:2015a] alternately solves two minimizations: one that estimates structure tensor using the image from previous iteration, and the other one improves the image with an adaptive regularizer defined from this tensor. The idea of anisotropic TV has been extended to vector-valued images with the anisotropy adopted from the gradients of multi-modal/contrast images, which incorporates similarities of directional structures in joint reconstruction. The method developed in [@Ehrhardt:2016b] projects the gradient in the total variation functional onto a predefined vector field given by the other contrast for joint reconstruction of multi-contrast MR images. In [@Ehrhardt:2016b], a directional TV (DTV) is proposed with the form $DTV_{u_2}(u_1)= \|P_{\zeta} (\dif u_1)\|_1$, where $P_{\zeta} (\dif u_1)$ is the residue of projection of $\dif u_1$ to $\zeta$, and $u_1$ and $u_2$ are scalar-valued functions each representing the image of one modality and $\zeta(x)=\dif u_2(x)/\|\dif u_2(x)\|$ for all $x\in \Omega$. In other words, this is the anisotropic diffusion $\|(I- \zeta\zeta^T) \dif u_1\|_1$ by using the structure tensor $\zeta \zeta^T$ of given reference image $u_2$.
With geometric interpretation of gradient tensor above, authors in [@Di-Zenzo:1986a] suggest to consider a vector-valued image as a parameterized 2-dimensional Riemann manifold with metric $G=D u^T D u$, where $u$ is a vector-valued image and $Du=[\partial_j u_k]_{j,k}$ is the $2\times 2$ Jacobian of $u$. Then the eigenvector corresponding to the larger eigenvalue gives the direction of the vectorial edge. Based on this framework, several forms of vectorial TV (VTV) have been developed. In [@Sapiro:1996a], a family of VTV formulations are suggested as the integral of $f(\lambda_+,\lambda_-)$ over the manifold, where $\lambda_+$ and $\lambda_-$ denote the larger and smaller eigenvalues of the metric $G$, respectively, and $f$ is a suitable scalar-valued function. A special choice of $f(\lambda_+, \lambda_-) = \sqrt{\lambda_+ + \lambda_-}$, i.e., the Frobenius norm of the Jacobian $Du$, reduce to JTV mentioned above. For another special choice of $f(\lambda_+,\lambda_-) = \sqrt{\lambda_+}$ , there is $VTV(u)=\int_\Omega \sigma_1(Du) \dif x$, where $\sigma_1(Du)$ is the largest singular value of the Jacobian $Du$ of $u$. This can be computed by using the dual formulation $\sup_{(\xi,\eta) \in K} \sum_{k=1}^d \int_\Omega u_k \mathrm{div}(\eta_k\xi)\dif x$ with $K=C_0^1(\Omega; S^m \times S^d)$ where $S^d$ is the standard simplex in ${\mathbb{R}}^d$.
Besides joint TV or tensor based regularization, the parallelism of level sets across multi-modality/contrast images are also proposed as joint image regularization in [@Ehrhardt:2014b; @Ehrhardt:2015a; @Ehrhardt:2014a; @Haber:2013a]. The main idea is to exploit the structural similarity of two images $u_1$ and $u_2$ measured by the parallelism of their gradients $\dif u_1$ and $\dif u_2$ at each point using $\delta(\dif u_1,\dif u_2):= f (g(|\dif u_1||\dif u_2|)-g(\langle \dif u_1,\dif u_2 \rangle ))$, for some functions $f$ and $g$. Then the regularization in joint reconstruction takes form $\int_\Omega \delta(\dif u_1, \dif u_2) \dif x$ Several different choices of $f$ and $g$ have been studied in these works. For instance, in [@Ehrhardt:2014a], $f$ and $g$ are taken as identities, or $f(s)=\sqrt{1+s}$ and $g(s)=s^2$. In [@Haber:2013a], $f$ is the identity and $g(s)=s^2$. In [@Ehrhardt:2015a] the side information on the level set, namely, the location and direction of the gradients of a reference image, is available to assist the reconstruction.
Another joint reconstruction approach different from aforementioned methods is to recover gradient information for each of the underlying multi-modal images from the measurements in Fourier domain, then use the recovered gradients to reconstruct images. This approach is motivated by the idea of Bayesian compressed sensing and applied to joint multi-contrast MR image reconstruction in [@Bilgic:2011a]. In [@Bilgic:2011a], gradients of the multi-contrasts images are reconstructed jointly from their measurements in Fourier space under a hierarchical Bayesian framework, where the joint sparsity on gradients across multi-contrast MRI is exploited by sharing the hyper-parameter in the their maximum a posteriori (MAP) estimations. Their experiments show the advantage of using joint sparsity on gradients over conventional sparsity. However, their method requires extensive computational cost. A two-step gradient reconstruction of MR images is also proposed in [@Patel:2012a], however, only single-modality/contrast image is considered. In [@Patel:2012a], the authors showed that this two-step gradient reconstruction approach allows to reconstruct image with fewer number of measurements than required by standard TV minimization method.
Proposed Method {#sec:proposed}
===============
In this section, we propose a new joint image reconstruction method that first restores image edges (gradients/Jacobian) from observed data, and then assembles the final image using these edges. Without loss of generality, we assume all images are $2$-dimensional, i.e., the image domain $\Omega\subset {\mathbb{R}}^2$ (all derivations below can be easily extended to higher dimensional images). For simplicity, we further assume $\Omega$ is rectangular. For single channel/modality/contrast case, we use function $u:\Omega \to {\mathbb{R}}$ represents the image, such that $u(x)$ stands for the intensity of image at $x\in\Omega$. In multi-modality case, $u:\Omega \to {\mathbb{R}}^m$ where $m$ is the number of modalities.
It is also convenient to treat an image $u$ as a matrix in discretized setting which we mainly work on in practice, and further stack the columns of $u$ to form a single column vector in ${\mathbb{R}}^n$, where $n$ is the total number of pixels in the image. For multi-modality case, we have $u_j\in{\mathbb{R}}^n$ to represent the (discretized) image of modality $j$ for $j=1,\dots,m$.
Vectorial total-variation regularization {#subsec:VTV}
----------------------------------------
Standard total-variation (TV) regularized image reconstruction can be formulated as a minimization problem as follows: $$\label{eq:TVbased}
\min_u \alpha TV(u) + h(u)$$ where $h$ represents the data fidelity function, e.g., $h(u)=\frac{1}{2}\|Au-b\|^2$, for some given data sensing matrix $A$ and observed partial/noisy/blurry data $b$. By solving , we obtain a solution $u$ which minimizes the sum of TV regularization term and data fidelity term with a weighting parameter $\alpha>0$ that balances the two terms. It is shown that TV regularization can effectively recover high quality images with well preserved object boundaries from limited and/or noisy data.
In joint multi-modality image reconstruction, the edges of images from different modalities are highly correlated. To take such factor into consideration, the standard TV regularized image reconstruction can be simply replaced by vectorial TV (VTV) regularized counterpart: $$\label{eq:VTVbased}
\min_u \alpha VTV(u) + h(u)$$ where $VTV(u)$ is the vectorial TV of $u$. In the case that $u$ is continuously differentiable, VTV is a direct extension of standard TV as an “$l_1$ of gradient” as $$\label{eq:VTV}
VTV(u)=\int_\Omega \|Du(x)\|_\star \dif x$$ where $Du(x)\in{\mathbb{R}}^{2\times m}$ is the Jacobian matrix at point $x$, and $\star$ indicates some specific matrix norm. For example, let $Q=[q_{ij}]$ be an $d$-by-$m$ matrix, then we may use one of the following matrix norms as $\|\cdot\|_\star$:
- Frobenius norm: $\|Q\|_F=(\sum_{i,j}|q_{ij}|^2)^{1/2}$. This essentially treats $Q$ as an $(dm)$-vector.
- Induced 2-norm: $\|Q\|_2=\sigma_1$ where $\sigma_1$ is the largest singular value of $Q$. This norm is advocated in [@Goldluecke:2010a] with a geometric interpretation when used in VTV.
- Nuclear norm: $\|Q\|_*=\sum_{i=1}^{\min\{d,m\}}\sigma_i$ where $\sigma_1\geq \sigma_2\geq\dots\geq0$ are singular values of $Q$. This is a convex relaxation of matrix rank.
Obviously, there are a number of other choices for VTV due to the many variations of matrix norms. However, in this paper, we only focus on these three norms as they are the mostly used ones in VTV regularized image reconstructions.
It is also worth noting that, in general, the VTV norm with matrix norm $\|\cdot\|_\star$ is defined for any function $u\in L_\text{loc}^1({\mathbb{R}}^2; {\mathbb{R}}^{m})$ (not necessarily differentiable) as $$\label{eq:VTVdef}
VTV(u) = \sup \cbr[3]{ \int_\Omega u(x) \mathrm{div}(\xi(x)) \dif x: \|\xi(x)\|_\bullet\leq 1,\ \xi(x)\in{\mathbb{R}}^{2\times m},\forall x\in \Omega}$$ where $\|\cdot\|_\bullet$ is the dual norm of $\|\cdot\|_\star$. Although we would not always make use of the original VTV definition in discrete setting, we show that they can help to derive closed-form soft-shrinkage with respect to the corresponding matrix $\star$-norm as in Appendix \[apd:shrinkage\].
Joint edge reconstruction {#subsec:edgerecon}
-------------------------
The **main idea** of this paper is to **reconstruct edges (gradients/Jacobian) of multi-modality images jointly**, and then assemble the final image from these edges. To that end, we let $v$ denote the “gradients” (or Jacobian) of $u$. If $u$ is differentiable, then $v(x)=(v_{kj}(x)):=Du(x) \in{\mathbb{R}}^{2\times m}$ is the Jacobian matrix of $u$ at $x\in \Omega$. For example, assuming there are three modalities and $u(x)=(u_1(x),u_2(x),u_3(x))\in{\mathbb{R}}^3$ at every point $x\in\Omega$, then $v(x)$ is a matrix $$v(x)=\del{\dif u_1(x),\dif u_2(x), \dif u_3(x)}=\begin{pmatrix}
v_{11}(x) & v_{21}(x) & v_{31}(x) \\
v_{12}(x) & v_{22}(x) & v_{32}(x)
\end{pmatrix}$$ where $v_{lj}(x):=\partial u_j(x)/\partial x_l$ at every $x=(x_1,x_2)$ for $l=1,2$ and $j=1,\dots.m$ ($m=3$ here). As a result, the VTV of $u$ in simplifies to $\int_\Omega \|v(x)\|_\star \dif x$. If $u$ is not differentiable, then $v$ may become the weak gradients of $u$ (when $u\in W^{1,1}(\Omega;{\mathbb{R}}^m)$) or even a Radon-Nikodym measure (when $u\in BV(\Omega;{\mathbb{R}}^m)$). However, in common practice using finite differences for numerical implementation to approximate partial derivatives of functions, such subtlety does not make much differences. Therefore, we treat $v$ as the Jacobian of $u$ throughout the rest of the paper and in numerical experiments.
Assume that we can derive the relation of the Jacobian $v$ and the original observed data $b$ and form a data fidelity $H(v)$ of $v$, as an analogue of data fidelity $h(u)$ of image $u$ (justification of this assumption will be presented in Section \[subsec:data\_reform\]). Then we can reformulate the VTV regularized image reconstruction problem about $u$ to a matrix $\star$-norm regularized inverse problem about $v$ as follows: $$\label{eq:L1based}
\min_v \alpha \|v\|_\star + H(v).$$ Now we seek for reconstructing $v$ instead of $u$. This reformulation has two significant advantages compared to the original formulation :
- It reduces to an $l_1$ type minimization and can be solved effectively by accelerated gradient method. For example, using Algorithm \[alg:edgeFISTA\] based on FISTA [@Beck:2009b], we can attain an optimal convergence rate of $O(1/k^2)$ to solve , where $k$ is the iteration number. This is in sharp contrast to the best known $O(1/k)$ rate of primal-dual based methods (including the recent, successful PDHG and ADMM) for solving .
- The per-iteration complexity of the $l_1$ type minimization , e.g., Algorithm \[alg:edgeFISTA\], is the same or even lower compared to VTV regularized minimization . In particular, closed-form solution of gradients in , a matrix $\star$-norm variant of soft-shrinkage, is widely available and cheap to compute.
**Input:** Initial $u_0$ and its Jacobian $v_0=D u_0$. Step size $\tau\leq 1/\|\nabla H\|$. Set $k=0$, $t_0=1$, $w_0=v_0$, and iterate – below until stopping criterion is met $$\begin{aligned}
v_{k+1} & = \operatorname*{arg\,min}_{v} \del[3]{\alpha \|v\|_\star + \frac{1}{2}\|v-w_k+\tau \nabla H(w_k)\|_F^2} \label{eq:vsubp}\\
t_{k+1} & = \frac{1+\sqrt{1+4t_k^2}}{2} \label{eq:tsubp}\\
w_{k+1} & = v_{k+1} + \frac{t_k-1}{t_{k+1}} (v_{k+1}-v_k) \label{eq:wsubp}\end{aligned}$$ **Output:** Reconstructed Jacobian (edge) $v \leftarrow v_{k+1}$.
In the remainder of this section, we will answer the following three questions that are critical in the proposed joint image reconstruction method based on :
- In what situations/applications the fidelity $h(u)$ of image $u$ can be converted to fidelity $H(v)$ of Jacobian $v$? (Section \[subsec:data\_reform\])
- How to obtain closed form solution of $v$-subproblem for matrix $\star$-norms, particularly for the three matrix norms in Section \[subsec:VTV\]? (Section \[subsec:shrinkage\])
- How to reconstruct image $u$ using Jacobian $v$ obtained from Algorithm \[alg:edgeFISTA\]? (Section \[subsec:recon\_u\])
Formulating fidelity of Jacobian $v$ {#subsec:data_reform}
------------------------------------
In an extensive variety of imaging technologies, especially medical imaging, the image data are acquired in transform domains. Among those common transforms, Fourier transform and Radon transform are the two most widely used ones in medical imaging. In what follows, we show that the relation between an image $u$ and its (undersampled) data $f$ can be easily converted to that between the Jacobian matrix $v$ and $f$ in the Fourier domains. This allows us to derive the data fidelity of Jacobian $v$ and reconstruct it properly for MRI reconstruction problems. Similar idea can be applied to the case with Radon transforms, subject to modification of data fidelity and noise distribution in a straightforward manner. In this paper, we focus on the Fourier case only as our data are multi-contrast MRIs.
**Data transformation.** Imaging technologies, such as magnetic resonance imaging (MRI) and radar imaging, are based on Fourier transform of images. The image data are the Fourier coefficients of image acquired in the Fourier domain (also known as the $k$-space in MRI community). The inverse problem of image reconstruction in such technologies often refers to recovering image from partial (i.e., undersampled) Fourier data.
In discrete setting, let $u\in{\mathbb{R}}^n$ denote the image to be reconstructed, $F\in{\mathbb{R}}^{n\times n}$ the (discrete) Fourier transform matrix (hence a unitary matrix), and $P\in{\mathbb{R}}^{n\times n}$ the diagonal matrix with binary values ($0$ or $1$) as diagonal entries to represent the undersampling pattern (also called mask in $k$-space), and $f\in{\mathbb{R}}^n$ be the observed partial data with $0$ at unsampled locations. Therefore, the relation between the underlying image $u$ and observed partial data $f$ is given by $PFu=f$.
The gradient (partial derivatives) of an image can be regarded as a convolution. The Fourier transform, on the other hand, is well known to convert a convolution to simple point-wise multiplication in the transform domain. In discrete settings, this simply means that $\hat{D}_i:=F D_i F^T$ is a diagonal matrix where $D_i\in\mathbb{R}^{n\times n}$ is the discrete partial derivative (e.g., forward finite difference) operator along the $x_i$ direction ($i=1,2$). This amounts to a straightforward formulation for the data fidelity of $v$: Let $v_i$ be the partial derivative of $u$ in $x_i$ direction, i.e., $v_i=D_i u$ then we have $$\label{eq:data_trans}
PFv_i=PFD_iu=PFD_i F^T F u = P\hat{D}_iFu=\hat{D}_iPFu=\hat{D}_if$$ where we used the facts that both $P$ and $\hat{D}_i$ are diagonal matrices so they can commute. Therefore, the data fidelity term $H(v)$ in can be, for example, formulated as $$H(v) = \frac{1}{2} \del[2]{ \| PFv_1 - \hat{D}_1 f\|^2 + \| PFv_2 - \hat{D}_2 f\|^2 }$$ As long as the Fourier data is concerned, the data fidelity of the corresponding gradient can be formulated in a similar way as above. **Noise transformation.** We now consider the noise distribution in image reconstruction from Fourier data. Suppose the noise is due to acquisition in Fourier transform domain such that $$\label{eq:data_trans_noise}
f(\omega) = P {\mathcal{F}}[u](\omega) + e(\omega)$$ for each frequency value $\omega$ in Fourier domain. Here ${\mathcal{F}}[u]$ is the Fourier transform of image $u$. By multiplying ${\hat{D}}_i(\omega)$ on both sides of , we see that this fidelity of $v_i$ can be obtained by $${\hat{D}}_i(\omega)f(\omega) = P {\mathcal{F}}[v_i] (\omega) + {\hat{D}}_i(\omega)e(\omega)\ .$$ Suppose that ${\hat{D}}_i(\omega)=A_i(\omega)+ {\mathrm{i}}B_i(\omega)$ where $A_i(\omega)\in{\mathbb{R}}$ and $B_i(\omega)\in{\mathbb{R}}$ are real and imaginary parts of ${\hat{D}}_i(\omega)$ respectively, and $e(\omega)=a(\omega)+ {\mathrm{i}}b(\omega)$ where $a(\omega)\sim N(0,\sigma_a^2)$ and $b(\omega)\sim N(0,\sigma_b^2)$ are independent for all $\omega$. Then we know that the transformed noise ${\hat{D}}_i(\omega)e(\omega)$ are independent for different $\omega$ and distributed as bivariate Gaussian $$\begin{pmatrix}
\text{real}({\hat{D}}_i(\omega)e(\omega))\\
\text{imag}({\hat{D}}_i(\omega)e(\omega))
\end{pmatrix}\sim
N\del{0,C_i(\omega) \Sigma C_i^T(\omega)}$$ where $C_i(\omega)=[A_i(\omega),-B_i(\omega); B_i(\omega), A_i(\omega)]\in{\mathbb{R}}^{2\times2}$ and $\Sigma = \text{diag}(\sigma_a^2,\sigma_b^2)\in{\mathbb{R}}^{2\times2}$. Therefore, we can readily obtain the maximum likelihood of ${\hat{D}}_i(\omega)e(\omega)$ and hence the fidelity of $v_i$. In particular, if $\sigma_a=\sigma_b=\sigma$, then $\text{real}({\hat{D}}_i(\omega)e(\omega))$ and $\text{imag}({\hat{D}}_i(\omega)e(\omega))$ are two i.i.d. $N(0,\sigma^2(A_i^2(\omega)+B_i^2(\omega)))$. i.e., $N(0,\sigma^2|{\hat{D}}_i(\omega)|^2)$. In this case, we denote $\Psi_i=\text{diag}(1/|{\hat{D}}_i(\omega)|^2)$, then data fidelity (i.e., negative log-likelihood) of $v_i$ simply becomes $(1/2)\|P{\mathcal{F}}v_i - {\hat{D}}_i f\|_{\Psi_i}^2$, as in contrast to the fidelity term $(1/2)\|P{\mathcal{F}}u - f\|^2$ of $u$. In the remainder of this paper, we assume that the standard deviation of real and imaginary parts are both $\sigma$. Furthermore, in numerical experiments, we can pre-compute $|{\hat{D}}_i(\omega)|^2$ and only perform an additional point-wise division of $|{\hat{D}}_i(\omega)|^2$ after computing $P{\mathcal{F}}v_i-{\hat{D}}_i f$ in each iteration.
Closed form solution of matrix-valued shrinkage {#subsec:shrinkage}
-----------------------------------------------
As we can see, the algorithm step calls for solution of type $$\label{eq:mtxshrink}
\min_{X\in {\mathbb{R}}^{2\times m}} \alpha \|X\|_\star + \frac{1}{2}\|X-B\|_F^2$$ for specific matrix norm $\|\cdot\|_\star$ and given matrix $B\in{\mathbb{R}}^{2\times m}$. In what follows, we provide the close-form solutions of when the matrix $\star$-norm is Frobenius, induced 2-norm or nuclear norm, as mentioned in Section \[subsec:VTV\]. The derivations are provided in Appendix \[apd:shrinkage\].
- **Frobenius norm.** This is the simplest case since $\|X\|_F$ treats $X$ as a vector in ${\mathbb{R}}^{2m}$ in , for which the shrinkage has close-form solution. More specifically, the solution of is $$\label{eq:Fro_sol}
X^*=\max(\|B\|_F-\alpha,0)\frac{B}{\|B\|_F}\quad .$$
- **Induced 2-norm.** This is advocated the vectorial TV [@Goldluecke:2010a], but now we can provide a close-form solution of as $$\label{eq:2norm_sol}
X^*=B-\alpha \xi\eta^T$$ where $\xi$ and $\eta$ are the left and right singular vectors corresponding to the largest singular value of $B$.
- **Nuclear norm.** This norm promotes low rank and yields a close-form solution of as $$\label{eq:nuclear_sol}
X^*=U \max(\Sigma-\alpha,0) V^T$$ where $(U,\Sigma,V)$ is the singular value decomposition (SVD) of $B$.
The computation of is essential shrinkage of vector and hence very cheap. The computations of and involve (reduced) SVD, however, explicit formula also exists as the matrices have tiny size of $2$-by-$m$ ($3$-by-$m$ if images are 3D), where $m$ is the number of image channels/modalities.
**Remark.** It is worth noting that the computation of is carried out at every pixel independently of others in each iterations. This allows straightforward parallel computing which can further reduce real-world computation time.
Reconstruct image from gradients {#subsec:recon_u}
--------------------------------
Once the Jacobian $v$ is reconstructed from data, the final step is to resemble the image $u$ from $v$. Since this step is performed for each modality, the problem reduces to reconstruction of a scalar-valued image $u$ from its gradient $v=(v_1,v_2)$. In [@Patel:2012a; @Sakhaee:2015a], the image $u$ is reconstructed by solving the Poisson equation $\Delta u=\mathrm{div}(v)=-(D_1^Tv_1+D_2^Tv_2)$ since $v_1=D_1u$ and $v_2=D_2u$ are the partial derivatives of $u$. The boundary condition of this Poisson equation can be either Dirichlet or Neumann depending on the property of imaging modality. In medical imaging applications, such as MRI, CT, and PET, the boundary condition is simply $0$ since it often is just background near image boundary $\partial\Omega$.
Numerically, it is more straightforward to recover $u$ by solving the following minimization $$\label{eq:u_recon}
\min_{u} \|D_1u-v_1\|^2 + \|D_2u-v_2\|^2 + \beta h(u)$$ with some parameter $\beta>0$ to weight the data fidelity $h(u)$ of $u$. The solution is easy to compute since the objective is smooth, and often times closed-form solution may exist. For example, in the MRI case where $h(u)=\frac{1}{2}\|PFu-f\|^2$, the solution of is given by $$\label{eq:usol}
u = {\mathcal{F}}^T\sbr[3]{ \del[2]{{\hat{D}}_1^T{\hat{D}}_1+{\hat{D}}_2^T{\hat{D}}_2+\beta P^TP}^{-1}
\del[2]{{\hat{D}}_1^T{\mathcal{F}}v_1+{\hat{D}}_2^T{\mathcal{F}}v_2+\beta P^Tf} }$$ where ${\hat{D}}_1^T{\hat{D}}_1+{\hat{D}}_2^T{\hat{D}}_2+\beta P^TP$ is diagonal and hence trivial to invert. The main computations are just few Fourier transforms.
In fact, it seems often sufficient to retrieve the base intensity of $u$ from $h(u)$ to reconstruct $u$ from $v$, and hence the result is not sensitive to $\beta$ when solving . It is also worth pointing out that, by setting $\beta=0$, the minimization is just least squares whose normal equation is the Poisson equation mentioned above. To summarize, we propose the two-step Algorithm \[alg:twostep\] which restores image edge $v$ and resembles image $u$ for multi-contrast MRI reconstruction. The two steps are each executed only once (no iteration). Step 1 itself requires iterations which converge very quickly with rate $O(1/k^2)$ where $k$ is iteration number. Step 2 has closed form solution for multi-contrast MRI reconstruction.
**Input:** Multi-contrast partial Fourier data $f=(f_1,\dots,f_m)$, mask $P$. Initial $u_0$. **Step 1:** Jointly reconstruct Jacobian $v=(v_1,\dots,v_m)$ using Algorithm \[alg:edgeFISTA\]. **Step 2:** Reconstruct image $u_j$ of contrast $j$ using $v_j$ and $f_j$ by for $j=1,\dots,m$. **Output:** Multi-contrast image $u=(u_1,\dots,u_m)$.
Numerical Results {#sec:results}
=================
In this section, we conduct a series of numerical experiments on synthetic and real multi-contrast MRI datasets using Algorithm \[alg:twostep\] (for short, we call it **ER**, standing for Edge-based Reconstruction, without adaptive weighting, and **ER-weighted** for the one with adaptive weighting $\Psi$ in Section \[subsec:data\_reform\]). For comparison, we also obtained implementation of two state-of-the-arts method for joint image reconstruction: multi-contrast MRI method Bayesian CS (**BCS**)[^4] [@Bilgic:2011a], and Fast Composite Splitting Algorithm for multi-contrast MRI (**FCSA-MT**)[^5] [@Huang:2014a], which are both publicly available online. We tune the parameters of each methods so they can perform nearly optimally. In particular, FCSA-MT seems to be sensitive to noise level and we need to tune a very different parameter for different $\sigma$. Specifically, we set the $l_1$ weight in BCS as $10^{-8}$ for both $\sigma=4$ and $10$ cases; the TV term and wavelet term weights in FCSA-MT are set to $0.010$ and $0.035$ (resp.) for $\sigma=4$ case, and $2.50$ and $2.50$ (resp.) for $\sigma=10$ case; the $l_1$ weight of ER/ER-weighted is set to $10^{-3}$, and the fidelity weight for to obtain image from edges is set to $\beta=0.001$ for both $\sigma=4$ and $10$ cases.
The image datasets we used are a 2D multi-contrast Shepp-Logan phantom (size $256\times256$), a simulated multi-contrast brain image (size $256\times256$) obtained from BrainWeb[^6], and a 2D in-vivo brain image (size $217\times181$) included in the FCSA-MT code. All images contain three contrasts. In particular, the three contrasts in BrainWeb and in-vivo brain datasets represent the T1, T2, and PD images. Undersampling pattern and ratio are presented below with results. The experiments are performed in Matlab computing environment on a Mac OS with Intel i7 CPU and 16GB of memory. Gaussian white noise with standard deviation $\sigma=4$ and $10$ were added to the k-space for the simulated data. We full Fourier data to obtain reference (ground truth) image $u^*$. The relative (L2) error of reconstruction $u_j$ for modality $j$ is defined by $\|u_j-u_j^*\|/\|u_j^*\|$ and used to measure error of $u_j$ quantitatively for $j=1,\dots,m$.
Comparison of different matrix norms
------------------------------------
Our first test is on the performance of three matrix norms, namely Frobenius, Induced 2-norm, and Nuclear norm, when used in Algorithm \[alg:twostep\]. These norms correspond to three commonly used vectorial TV regularization in multi-modal/channel/contrast joint image reconstruction. All three norms yield closed-form solutions for subproblem as we showed in Section \[subsec:shrinkage\]. We apply the proposed Algorithm \[alg:twostep\] with these three norms to different image data and noise level combinations, and observe very similar performance. For demonstration, we show a typical result using BrainWeb data with no noise in left panel of Figure \[fig:norms\]. As we can see, all norms yield very similar accuracy in terms of reconstruction error.
In terms of real-world computational cost, however, Frobenius norm treats matrix as vector and hence the shrinkage is very cheap to compute, while both induced 2-norm and nuclear norm require (reduced) SVD in and that can be much slower despite that the matrices all have small size ($2\times m$ for 2D images with $m$ modalities/contrasts). We show the same trajectory of relative errors but versus CPU time in the right panel of Figure \[fig:norms\], and it appears that Frobenius norm is the most cost-effective choice for this test. In the remainder of this section, we only use Frobenius norm (corresponding to the specific vectorial TV norm of form $\int_\Omega \|Du(x)\|_F\dif x$) for the proposed Algorithm \[alg:twostep\], but not the induced 2-norm (corresponding to $\int_\Omega \|Du(x)\|_2\dif x$) and nuclear norm (corresponding to $\int_\Omega \|Du(x)\|_*\dif x$).
![Comparisons of Frobenius norm, induced 2-norm, and nuclear norm for Algorithm \[alg:twostep\] (ER) on the BrainWeb image without noise. (a) Relative error vs number of iterations; (b) Relative error vs CPU time.[]{data-label="fig:norms"}](fig/norm_ER_vs_iteration.pdf){width="\textwidth"}
![Comparisons of Frobenius norm, induced 2-norm, and nuclear norm for Algorithm \[alg:twostep\] (ER) on the BrainWeb image without noise. (a) Relative error vs number of iterations; (b) Relative error vs CPU time.[]{data-label="fig:norms"}](fig/norm_ER_vs_time.pdf){width="\textwidth"}
Comparisons on multi-contrast MRI datasets
------------------------------------------
Now we conduct comparison of the proposed algorithm with BSC and FCSA-MT on multi-contrast MRI datasets. We first test the comparison algorithms on a multi-contrast Shepp-Logan phantom image. To demonstrate robustness of the proposed algorithm, we use a radial mask of undersampling ratio $13.5\%$ in Fourier space, but add white complex-valued Gaussian noise with standard deviation $\sigma=4,10$ (for both real and imaginary parts of the noise) to the undersampled Fourier data. Then we apply all comparison algorithm, namely BCS, FSCA-MT, and proposed ER and ER-weighted to the undersampled noisy Fourier data to reconstruct the multi-contrast image. For each method, we record the progress of relative error and show the relative error vs CPU time in Figure \[fig:phantom\_sigma\_error\_time\]. From these plots, we can see that the proposed Algorithm \[alg:twostep\] (ER/ER-weighted) achieves higher efficiency as they reduce reconstruction error faster than the comparison algorithms BCS and FCSA-MT. Moreover, the proposed algorithm with the weight $\Psi$ (ER-weighted) can incorporate the transformed noise distribution and further improve reconstruction accuracy. We observe that ER-weighted sometimes performs slightly slower than ER in terms of CPU time (although always faster in terms of iteration number which is not shown here), since the ER-weighted requires an additional point-wise division using $\Psi$. We believe that this is an issue that can be resolved by further optimizing code, because complexity wise the additional computation required by ER-weighted is rather low compared to other operations (such as Fourier transforms) in ER/ER-weighted.
Besides the plot of relative error vs CPU time, we also show the final reconstruction images using these methods in Figure \[fig:phantom\_sigma4\_visual\] for the noise level $\sigma=4$ case. In Figure \[fig:phantom\_sigma4\_visual\], the radial mask of sampling ratio (13.5%) is show on upper left corner (white pixels indicate sampled location). The final reconstructed multi-contrast images $u=(u_1,u_2,u_3)$ by the four comparison algorithm are shown in the middle column with algorithm name on left. On the right column, we also show their corresponding error image $u_j-u_j^*$ where $u^*=(u_1^*,u_2^*,u_3^*)$ is the ground truth multi-contrast image obtained using full Fourier data. As we can see, the reconstruction error of proposed algorithm is very small compared other others, especially showing less obvious error on edges. This demonstrates the improve reconstruction accuracy using our method.
![Relative error vs CPU time of the comparison algorithms on Shepp-Logan phantom image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:phantom_sigma_error_time"}](fig/phantom_sigma4_error_time_f1 "fig:"){width=".3\textwidth"} ![Relative error vs CPU time of the comparison algorithms on Shepp-Logan phantom image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:phantom_sigma_error_time"}](fig/phantom_sigma4_error_time_f2 "fig:"){width=".3\textwidth"} ![Relative error vs CPU time of the comparison algorithms on Shepp-Logan phantom image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:phantom_sigma_error_time"}](fig/phantom_sigma4_error_time_f3 "fig:"){width=".3\textwidth"} ![Relative error vs CPU time of the comparison algorithms on Shepp-Logan phantom image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:phantom_sigma_error_time"}](fig/phantom_sigma10_error_time_f1 "fig:"){width=".3\textwidth"} ![Relative error vs CPU time of the comparison algorithms on Shepp-Logan phantom image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:phantom_sigma_error_time"}](fig/phantom_sigma10_error_time_f2 "fig:"){width=".3\textwidth"} ![Relative error vs CPU time of the comparison algorithms on Shepp-Logan phantom image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:phantom_sigma_error_time"}](fig/phantom_sigma10_error_time_f3 "fig:"){width=".3\textwidth"}
![Undersampling mask, reconstructed image, and error image for the Shepp-Logan image with noise level $\sigma=4$.[]{data-label="fig:phantom_sigma4_visual"}](fig/phantom_sigma4_visual){width=".8\textwidth"}
We also conduct the same test using the BrainWeb image with a Poisson mask of sampling ratio 25%, also with noise level $\sigma=4$ and $10$. The results are shown in Figure \[fig:brainweb\_sigma\_error\_time\]. For this “near real” image, our method again shows significant improvement of efficiency compared to other methods. In particular, ER-weighted outperforms all methods in both cases, suggesting its superior robustness in multi-contrast image reconstruction. The final reconstruction images and error images, similar to those for Shepp-Logan image, are also plotted in Figure \[fig:brainweb\_sigma4\_visual\] for the $\sigma=4$ case.
![Relative error vs CPU time of the comparison algorithms on BrainWeb image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:brainweb_sigma_error_time"}](fig/brainweb_sigma4_error_time_f1 "fig:"){width=".32\textwidth"} ![Relative error vs CPU time of the comparison algorithms on BrainWeb image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:brainweb_sigma_error_time"}](fig/brainweb_sigma4_error_time_f2 "fig:"){width=".32\textwidth"} ![Relative error vs CPU time of the comparison algorithms on BrainWeb image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:brainweb_sigma_error_time"}](fig/brainweb_sigma4_error_time_f3 "fig:"){width=".32\textwidth"} ![Relative error vs CPU time of the comparison algorithms on BrainWeb image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:brainweb_sigma_error_time"}](fig/brainweb_sigma10_error_time_f1 "fig:"){width=".32\textwidth"} ![Relative error vs CPU time of the comparison algorithms on BrainWeb image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:brainweb_sigma_error_time"}](fig/brainweb_sigma10_error_time_f2 "fig:"){width=".32\textwidth"} ![Relative error vs CPU time of the comparison algorithms on BrainWeb image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:brainweb_sigma_error_time"}](fig/brainweb_sigma10_error_time_f3 "fig:"){width=".32\textwidth"}
![Undersampling mask, reconstructed image, and error image for the BrainWeb image with noise level $\sigma=4$.[]{data-label="fig:brainweb_sigma4_visual"}](fig/brainweb_sigma4_visual){width=".9\textwidth"}
Finally, we conduct the same test on the in-vivo brain image with radial mask of sampling ratio 25%. The relative error vs CPU time plots are shown in Figure \[fig:brain\_sigma\_error\_time\] ($\sigma=4$ and $10$). For this image, our method further shows its promising performance with significant improvement when compared to existing methods. The final reconstruction images and error images are also shown in Figure \[fig:brain\_sigma4\_visual\] for the $\sigma=4$ case.
![Relative error vs CPU time of the comparison algorithms on in-vivo brain image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:brain_sigma_error_time"}](fig/brain_sigma4_error_time_f1 "fig:"){width=".32\textwidth"} ![Relative error vs CPU time of the comparison algorithms on in-vivo brain image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:brain_sigma_error_time"}](fig/brain_sigma4_error_time_f2 "fig:"){width=".32\textwidth"} ![Relative error vs CPU time of the comparison algorithms on in-vivo brain image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:brain_sigma_error_time"}](fig/brain_sigma4_error_time_f3 "fig:"){width=".32\textwidth"} ![Relative error vs CPU time of the comparison algorithms on in-vivo brain image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:brain_sigma_error_time"}](fig/brain_sigma10_error_time_f1 "fig:"){width=".32\textwidth"} ![Relative error vs CPU time of the comparison algorithms on in-vivo brain image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:brain_sigma_error_time"}](fig/brain_sigma10_error_time_f2 "fig:"){width=".32\textwidth"} ![Relative error vs CPU time of the comparison algorithms on in-vivo brain image with noise standard deviation $\sigma=4$ (top row) and $\sigma=10$ (bottom row).[]{data-label="fig:brain_sigma_error_time"}](fig/brain_sigma10_error_time_f3 "fig:"){width=".32\textwidth"}
![Undersampling mask, reconstructed image, and error image for the in-vivo brain image with noise level $\sigma=4$.[]{data-label="fig:brain_sigma4_visual"}](fig/brain_sigma4_visual){width=".9\textwidth"}
Conclusion {#sec:conclusion}
==========
We proposed to reconstruct Jacobian of mutli-modal/contrast/channel image from which we can resemble the underlying image. We showed the relation between Jacobian and the observed data when the underlying transform is Fourier, and formulate the reconstruction problem of Jacobian as an $l_1$ minimization. Our new method then exhibits an optimal $O(1/k^2)$ convergence rate which outperforms the $O(1/k)$ rate of primal-dual based algorithms. We also derived closed-form solutions for the minimization subproblem as matrix-valued shrinkage. The per-iteration complexity is thus very low. Numerical results demonstrated the promising performance of the proposed method when compared to the state-of-the-arts joint image reconstruction methods.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank Yao Xiao and Yun Liang for assisting Fang to run the MATLAB code and collect numerical results. This research is supported in part by National Science Foundation grant DMS-1719932 (Chen), IIS-1564892 (Fang), DMS-1620342 (Ye) and CMMI-1745382 (Ye), and National Key Research and Development Program of China No: 2016YFC1300302 (Fang) and National Natural Science Foundation of China No: 61525106 (Fang).
Appendix {#appendix .unnumbered}
========
Computation of matrix-valued shrinkage {#apd:shrinkage}
--------------------------------------
We show that the minimization problem has closed-form solutions when the matrix $\star$-norm is Frobenius, induced 2-norm, or nuclear norm.
- **Frobenius norm.** In the case, all matrices can be considered as vectors and hence the vector-valued shrinkage formula can be directly applied. We omit the derivations here.
- **Induced 2-norm.** In the case, becomes $$\label{eq:mtx2shrink}
\min_X \|X\|_2+\frac{1}{2\alpha}\|X-B\|_F^2 = \min_X\max_{\|\xi\|=\|\eta\|=1}\xi^T X \eta + \frac{1}{2\alpha}\|X-B\|_F^2$$ where $\xi=(\xi_1,\xi_2)^T\in{\mathbb{R}}^2$ and $\eta=(\eta_1,\dots,\eta_m)^T\in{\mathbb{R}}^m$.
Switching min and max, and solving for $X$ with fixed $\xi$ and $\eta$, we obtain $X=B-\alpha \xi \eta^T$ and the dual problem of becomes $$\max_{\|\xi\|=\|\eta\|=1}\xi^T B \eta-\alpha \xi^T (\xi \eta^T) \eta + \frac{\alpha}{2}\|\xi\eta^T\|_F^2
=\max_{\|\xi\|=\|\eta\|=1}\xi^T B \eta- \frac{\alpha}{2}$$ where we used the facts that $\xi^T (\xi \eta^T) \eta=\|\xi\|^2\|\eta\|^2=1$ and $\|\xi\eta^T\|_F^2=\sum_{i}\sum_{j}(\xi_i\eta_j)^2=\sum_{i}(\xi_i)^2\sum_{j}(\eta_j)^2=1$. Therefore, $\xi$ and $\eta$ are the left and right singular vectors of $B$, and hence the optimal solution of is $X^*=B-\alpha \xi\eta^T$.
To obtain $X^*$, one needs to compute the largest singular vectors of $B$, or the SVD of $B$. Note that $B$ has size $2\times m$, and hence a reduced SVD for is sufficient. In particular, for $m=2$, there is a closed form expression to compute SVD of $X$. In general, SVD of a $2\times m$ matrix is easy to compute and has at most two singular values.
- **Nuclear norm** This corresponds to standard nuclear norm shrinkage: compute SVD of $B=U\Sigma V^T$, and set $X^*=U\max(\Sigma-\alpha,0)V^T$. Computation complexity is comparable to the 2-norm case above.
[^1]: Department of Mathematics, University of Florida, Gainesville, FL 32611, USA. Email: `yun@math.ufl.edu`
[^2]: Department of Biomedical Engineering, University of Florida, Gainesville, FL 32611, USA. Email: `ruogu.fang@bme.ufl.edu`
[^3]: Corresponding author. Department of Mathematics & Statistics, Georgia State University, Atlanta, GA 30303, USA. Email: `xye@gsu.edu`
[^4]: BCS code: <http://martinos.org/~berkin/Bayesian_CS_Prior.zip>
[^5]: FCSA-MT code: <http://ranger.uta.edu/~huang/codes/Code_MRI_MT.zip>
[^6]: BrainWeb: <http://brainweb.bic.mni.mcgill.ca/brainweb/>
|
---
abstract: 'Inspired by recent measurements on individual metallic nanospheres that can not be explained with traditional classical electrodynamics, we theoretically investigate the effects of nonlocal response by metallic nanospheres in three distinct settings: atomic spontaneous emission, electron energy loss spectroscopy, and light scattering. These constitute two near-field and one far-field measurements, with [zero-,]{} [one-,]{} and two-dimensional excitation sources, respectively. We search for the clearest signatures of hydrodynamic pressure waves in nanospheres. We employ a linearized hydrodynamic model and Mie–Lorenz theory is applied for each case. Nonlocal response shows its mark in all three configurations, but for the two near-field measurements we predict especially pronounced nonlocal effects that are not exhibited in far-field measurements. Associated with every multipole order is not only a single blueshifted surface plasmon, but also an infinite series of bulk plasmons that has no counterpart in a local-response approximation. We show that these increasingly blueshifted multipole plasmons become spectrally more prominent at shorter probe-to-surface separations and for decreasing nanosphere radii. For selected metals we predict hydrodynamic multipolar plasmons to be measurable on single nanospheres.'
author:
- Thomas Christensen
- Wei Yan
- Søren Raza
- 'Antti-Pekka Jauho'
- 'N. Asger Mortensen'
- Martijn Wubs
title: 'Nonlocal Response of Metallic Nanospheres Probed by Light, Electrons, and Atoms'
---
**Keywords:** Nonlocal response, nanoplasmonics, EELS, extinction, LDOS, spontaneous emission, multipole plasmons. 1em
------------------------------------------------------------------------
1em A plethora of effects arises in structured metals due to collective excitations of conduction electrons and their interaction with the electromagnetic field. This constitutes plasmonics, a research field with mature roots[@Ritchie:1957; @Pines:1952] that is continuing to develop strongly.[@Stockman:2011] Notably, applications for plasmonics are found in the biochemistry and biomedical fields, *e.g.*, in surface-enhanced Raman spectroscopy (SERS),[@Campion:1998] biosensing[@VazquezMena:2011] and biomedical imaging,[@Khlebtsov:2010] drug delivery,[@Kyrsting:2010] and phototherapy of cancer-cells.[@Lal:2008] Purely photonic applications are also emerging, *e.g.*, in plasmonic waveguiding,[@Bozhevolnyi:2006b] optical nanoantennas[@Novotny:2011; @Muskens:2007], and photovoltaics.[@Wu:2011]
Recent years’ advances in fabrication, synthesis, and characterization techniques have allowed well-controlled experimental investigations of plasmonics even at the nanoscale. Yet in this growing field of nanoplasmonics,[@Stockman:2011; @Stockman:2011_OptExpress] the commonly employed theory for light-matter interaction is still traditional classical electrodynamics, where the response of the material constituents to light is described collectively in terms of local, bulk-material response-functions. Indeed, this approach usually remains very accurate, even for sub-wavelength phenomena.
Interestingly, recent measurements on individual few-nanometer plasmonic particles have shown phenomena that are clearly beyond classical electrodynamics. Electron energy-loss spectroscopy (EELS) of Ag spheres resting on dielectric substrates showed surface plasmon resonance blueshifts up to $0.5\,\mathrm{eV}$ as compared to classical theory[@Scholl:2012; @Raza:2013_Nanophotonics]. Earlier similar measurements were performed on ensembles of nanoparticles.[@vomFelde:1988] Classical electrodynamics was also shown to fail in experiments involving (sub-)nanometer-sized gaps between dimers,[@Kern:2012; @Savage:2012; @Scholl:2013a] or between nanoparticles and a substrate.[@Ciraci:2012]
To explain these features arising beyond the validity of classical electrodynamics, various physical mechanisms are invoked. Firstly, classical electrodynamics assumes a step-function profile of the free-electron density at a metal-dielectric interface. The finite quantum mechanical spill-out[@Lang:1970a] of the electron density redshifts the surface plasmon resonance,[@Liebsch:1993a; @Teperik:2013] may give rise to nonresonant field enhancement,[@Ozturk:2011] and may enable charge transfer between non-touching plasmonic dimers.[@Esteban:2012; @Savage:2012; @Scholl:2013a] Secondly, a stronger confinement of the free electrons gives rise to blueshifts. In cluster physics, it is single-particle excitations that are blueshifted due to quantum confinement,[@DeHeer:1993] while confinement in nanoplasmonics blueshifts collective resonances and gives rise to Friedel oscillations in the electron density.[@Keller:1993; @Townsend:2011a] A third, semi-classical physical mechanism beyond classical electrodynamics is nonlocal response, discussed in more detail below, which becomes important when reducing the particle size or gap size of a dimer down to the range of the nonlocality[@Ginzburg:2013] ($\xi_{\textsc{nl}}$, denoting the spatial extent of significant nonlocal interaction, to be introduced shortly), and blueshifts surface plasmon resonance frequencies.
Large experimental blueshifts of the localized surface plasmon (LSP) dipole resonance seem to indicate that several physical mechanisms add up.[@Raza:2013_Nanophotonics; @Raza:2013_OE] Certainly, in experiments all these physical mechanisms beyond traditional classical electrodynamics are at work simultaneously, thus emphasizing the importance of microscopic theories[@Stella:2013] (*e.g.*, density-functional theory, DFT) or effective models[@CarminaMonreal:2013a] that incorporate multiple mechanisms. Yet at the same time it is important to ascertain the relative strength and compatibility of the various mechanisms. Indeed, it is paramount to know - and to measure - the unique characteristics of each mechanism, that is to say, find their individual “smoking guns”, in order to appreciate the dominant physical mechanisms under different nonstandard circumstances. We foresee an increasing number of such decisive experiments on individual nanoparticles in the near future.
The boundary between cluster physics and nanoplasmonics is an interesting one. Metal clusters require a quantum description of interacting electron states, often studied with DFT. In contrast, nanoplasmonics could be defined to start for nanoparticle sizes that allow an effective quantum description in terms of non-interacting plasmons[@Townsend:2011a]. A current interesting issue is where to place the origin of the observed blueshift of the surface plasmon resonance of individual nanospheres: is it primarily due to quantum confinement of single-particle states,[@vomFelde:1988; @Scholl:2012] or due to confinement of collective modes?[@Keller:1993; @Townsend:2011a; @Raza:2013_Nanophotonics; @CarminaMonreal:2013a] In this article we assume the latter and identify new observable consequences. We focus on nanoparticles that are considered large enough ($2R \geq 3$ nm) that so-called core plasmons, although collective in nature, can be neglected according to DFT calculations.[@Townsend:2011a]
Nonlocal response is a semi-classical effect which emerges in nanoplasmonics at few-nanometer length scales. The general nonlocal relation between the displacement and electric fields, $\mathbf{D}(\mathbf{r},\omega) = \varepsilon_0 \int\varepsilon(\mathbf{r},\mathbf{r}';\omega) \mathbf{E}(\mathbf{r}',\omega) \,\mathrm{d}\mathbf{r}'$ becomes simpler and more familiar in the local-response approximation (LRA), *i.e.*, $\varepsilon(\mathbf{r},\mathbf{r}';\omega) \simeq \varepsilon_{\textsc{lra}}(\mathbf{r},\omega)\delta(\mathbf{r}-\mathbf{r}')$. In many cases this approximation provides an excellent effective description due to the short-range nature of the nonlocal interaction. However, the LRA is not justifiable when the nonlocal interaction length, $\xi_{\textsc{nl}}$, becomes comparable with characteristic feature sizes of structural or optical kind.[@Ginzburg:2013]
Here we consider inclusion of the classically neglected Fermi–Dirac pressure of the electron gas. Its associated pressure waves give rise to a nonlocal optical response. The simplest way to study the effects of Fermi pressure in nanoplasmonics is by assuming a hydrodynamic model,[@Ruppin:1973a; @David:2012; @Raza:2013_Nanophotonics; @Fuchs:1987; @Baltz:1995; @TranThoai:1986; @TranThoai:1988; @Mortensen:2013] which neglects the aforementioned spill-out and confinement effects on the static electron density. In hydrodynamics the nonlocal interaction length becomes $\xi_{\textsc{nl}} = v_{\textsc{f}}/\omega$, with $v_{\textsc{f}} = \hbar\sqrt[3]{3\pi^2n_0}/m$ denoting the Fermi velocity, defined through the effective mass $m$ and free-electron density $n_0$. This corresponds to $\xi_{\textsc{nl}}$-values in the range $2\,\textrm{-}\,5\,\textrm{\AA}$ for typical plasmonic metals at optical frequencies, see Table S1 in the Supporting Information (SI). We will focus on the linearized hydrodynamic model here, but would like to emphasize that the full hydrodynamic model involves both nonlocality and nonlinearity, predicting nonlinear effects such as second-harmonic generation at the surface of metal nanoparticles for larger field-strengths[@Sipe:1980; @Ginzburg:2012_PRB; @Ciraci:2012_PRB].
The strongest evidence of hydrodynamic behavior in metals originate from experiments on thin metal films, where resonances due to standing waves of confined bulk plasmons have been identified, in silver by Lindau and Nilsson[@Lindau:1971], in potassium by Anderegg *et al.*[@Anderegg:1971], in magnesium by Chen[@Chen:1976], and very recently by [" O]{}zer *et al.*[@Oezer:2011] Rather surprisingly, [" O]{}zer *et al.*[@Oezer:2011] could measure confined bulk plasmon resonances (*i.e.*, standing Fermi pressure waves) even for ultrathin magnesium films of only three atomic monolayers, and found qualitative agreement with theory even when neglecting electronic spill-out. For nanospheres on the other hand, the observations of blueshifted dipole-resonances of localized surface plasmons (LSPs) in individual nanospheres[@Scholl:2012; @Raza:2013_Nanophotonics; @Raza:2013_OE] and of broad resonance-features above the plasma frequency in ensembles,[@Duthler:1971] tentatively suggested as associated with confined bulk plasmons,[@Ruppin:1973a] are perhaps less conclusive evidence of hydrodynamic behavior. This may in part be due to a line of reasoning which addresses just a single resonance, namely the dipole.
Our aim in this article is then to examine theoretically which phenomena constitute the clearest evidence of hydrodynamic pressure waves in plasmonic nanospheres, and how best to observe them. Powerful measurement techniques include scattering measurements, as realized *e.g.*, in the infrared regime by Fourier transform infrared spectroscopy (FTIR), scanning near-field optical microscopy (SNOM)[@Greffet:1997], EELS[@Abajo_PhysRevMod2010; @Egerton:2009], and fluorescence microscopy techniques, utilizing decay enhancement of emitters near plasmonic resonances[@Schmelzeisen:2010; @Willets:2013]. In this theoretical article, we systematically explore three prominent measurement techniques, each with different excitation sources, namely the extinction cross-section, the EELS probability, and the electric local density of states (LDOS). The excitation sources are, respectively, a linearly polarized plane wave, a traveling electron with kinetic energy in the $\mathrm{keV}$-range, and an electric dipole emitter, corresponding to a two-, one-, and zero-dimensional source. The three measurement principles represent both far- and near-field types, and we show their spectra to be qualitatively different.
We investigate not only the strongest (dipolar) LSP resonance of nanospheres, but also higher-order multipole LSPs, as well as bulk plasmons, for all three measurements considered. We show that hydrodynamic response leads to a significant spectral separation of the sphere’s multipole plasmons at small radii, allowing them to extend above the LRA asymptotic limit at $\omega_{\mathrm{p}}/\sqrt{2}$. Resonance-features above this limit have already been observed in polydisperse ensembles of nanospheres, and previously been interpreted instead in terms of single-particle confinement.[@vomFelde:1988] We find significant qualitative disparity between properties measurable in the far-field, *i.e.*, *via* extinction, and in the near-field, *i.e.*, *via* EELS or LDOS. Our findings result in concrete suggestions to experimentally observe hydrodynamic nonlocal phenomena in the near-field, by identifying the multipolar plasmon resonances of individual nanospheres of selected metals.
Results and Discussion
======================
#### Theoretical framework.
In a linearized hydrodynamic description, the current density $\mathbf{J}(\mathbf{r},\omega)$ and the electric field $\mathbf{E}(\mathbf{r},\omega)$ are interrelated by the nonlocal relation[@Boardman:1982a; @Raza:2011]:
\[eqs:governing\] $$\label{eq:hydromain}
\frac{\beta_{\textsc{f}}^2}{\omega(\omega+i\eta)}\nabla[\nabla\cdot \mathbf{J}(\mathbf{r},\omega)] + \mathbf{J}(\mathbf{r},\omega) = \sigma(\omega)\mathbf{E}(\mathbf{r},\omega),$$ where $\sigma(\omega) = i\varepsilon_0\omega_\mathrm{p}^2/(\omega+i\eta)$ is the usual Drude conductivity of a free-electron gas with plasma frequency $\omega_{\mathrm{p}}$, including a phenomenological loss-rate $\eta$, and $\beta_{\textsc{f}}^2 = (3/5)v_{\textsc{f}}^2$ is the hydrodynamical velocity of plasma pressure waves in the metal. The hydrodynamic model can be classified as ‘semi-classical’ because Eq. (\[eq:hydromain\]) relates the classical fields $\mathbf{J}$ and $\mathbf{E}$ *via* the parameter $\beta_{\textsc{f}}\propto v_{\textsc{f}}$ which is proportional to $\hbar$. Hydrodynamic response appears as a lowest order spatially nonlocal correction to the local Ohm’s law, with a strength proportional to $\xi_{\textsc{nl}}^{-2}k^2$ in momentum $k$-space.
In addition to Eq. , the electric field must satisfy the Maxwell wave equation $$\label{eq:vectorwave}
\nabla\times\nabla\times\mathbf{E}(\mathbf{r},\omega) - k_0^2\varepsilon_{\infty}(\omega)\mathbf{E}(\mathbf{r},\omega) = i\omega\mu_0 \mathbf{J}(\mathbf{r},\omega),$$
with $k_0 = \omega/c$ denoting the usual free-space wavenumber, and $\varepsilon_{\infty}(\omega)$ the dielectric response of the bound charges, *i.e.*, the response not due to the free-electron plasma. The sum of the bound- and free-electron response gives the transverse response of the metal $\varepsilon_{\textsc{m}}(\omega) = \varepsilon_{\infty}(\omega)+\sigma(\omega)/i\varepsilon_0\omega$, familiar from the LRA. For calculations involving a measured transverse metal response $\varepsilon_{\textsc{m}}(\omega)$, the bound response $\varepsilon_{\infty}(\omega)$ is determined by fixing $\omega_{\mathrm{p}} = \sqrt{n_0 e^2/\varepsilon_0 m}$, *i.e.*, through the free-electron density $n_0$ and effective mass $m$, thus determining the free response $\sigma(\omega)$ and allowing $\varepsilon_{\infty}(\omega)$ to be determined by subtraction.[@David:2012].
The practical solution of Eqs. in structures with curvilinear symmetries can be aided significantly by expansion in the so-called vector wave functions. Concretely, a monochromatic electromagnetic field in a region of uniform dielectric function, can be expanded in the basis composed of the solenoidal, $\mathbf{M}_{\nu}(\boldsymbol{\mathrm{r}})$ and $\mathbf{N}_{\nu}(\boldsymbol{\mathrm{r}})$, and irrotational, $\mathbf{L}_{\nu}(\boldsymbol{\mathrm{r}})$, vector wave functions: [@Chew:book; @Stratton] $$\boldsymbol{\mathrm{E}}(\boldsymbol{\mathrm{r}}) =
\sum_{\nu} a_{\nu}\mathbf{M}_{\nu}(\boldsymbol{\mathrm{r}})
+b_{\nu}\mathbf{N}_{\nu}(\boldsymbol{\mathrm{r}})
+c_{\nu}\mathbf{L}_{\nu}(\boldsymbol{\mathrm{r}}),$$ where $\nu$ denotes a composite expansion index with $a_{\nu}$, $b_{\nu}$, and $c_{\nu}$ being associated expansion coefficients. The functions $\mathbf{M}_{\nu}(\boldsymbol{\mathrm{r}})$ and $\mathbf{N}_{\nu}(\boldsymbol{\mathrm{r}})$ describe the TE and TM parts, respectively, of the electric field, and describe the propagation of transverse, or divergence-free, modes.[@Stratton] The functions $\mathbf{L}_{\nu}(\boldsymbol{\mathrm{r}})$ are irrotational, and as such are irrelevant in media described by the LRA. However, their inclusion is indispensable for the treatment of plasmonic nanoparticles by hydrodynamic response, in order to account for the inclusion of longitudinal modes.
![Sketch of an exciting wave $\boldsymbol{\mathrm{E}}^{\mathrm{ex}}$ interacting with a metallic sphere embedded in a dielectric background, giving rise to scattered and transmitted fields $\boldsymbol{\mathrm{E}}^{\mathrm{sc}}$ and $\boldsymbol{\mathrm{E}}^{\mathrm{tr}}$, respectively.[]{data-label="fig:schematic"}](Figure1)
Next, we consider the case of an arbitrary external exciting field $\boldsymbol{\mathrm{E}}^{\mathrm{ex}}$ that originates in an outer dielectric region and scatters upon a spherical metallic particle of radius $R$ that is centered at the origin. This induces scattered fields $\boldsymbol{\mathrm{E}}^{\mathrm{sc}}$ outside the particle and transmitted fields $\boldsymbol{\mathrm{E}}^{\mathrm{tr}}$ inside, see Figure \[fig:schematic\]. For spherical nanoparticles, the choice of multipolar vector wave functions separates the composite expansion index $\nu$ into the angular-momentum quantum numbers $l$ and $m$, for details see the Methods section.
Outside the nanosphere ($r>R$), the fields $\boldsymbol{\mathrm{E}}^{\mathrm{ex}}$ and $\boldsymbol{\mathrm{E}}^{\mathrm{sc}}$ can be expanded solely in terms of the in- and outgoing transverse multipoles $\{\mathbf{M}_{lm}^{\rm ex},\mathbf{N}_{lm}^{\rm ex}\}$ and $\{\mathbf{M}_{lm}^{\rm sc},\mathbf{N}_{lm}^{\rm sc}\}$, respectively, since the dielectric region does not support longitudinal waves. The corresponding expansion coefficients are $\{a_{lm}^{\mathrm{ex}},b_{lm}^{\mathrm{ex}}\}$ and $\{a_{lm}^{\mathrm{sc}},b_{lm}^{\mathrm{sc}}\}$. The transmitted field $\boldsymbol{\mathrm{E}}^{\mathrm{tr}}$ inside the nanosphere ($r<R$) requires besides ingoing transverse multipoles, $\{\mathbf{M}_{lm}^{\rm tr},\mathbf{N}_{lm}^{\rm tr}\}$, also ingoing longitudinal modes $\mathbf{L}_{lm}^{\mathrm{tr}}$, which correspondingly necessitates three sets of expansion coefficients $\{a_{lm}^{\mathrm{tr}},b_{lm}^{\mathrm{tr}},c_{lm}^{\mathrm{tr}}\}$.
The fields inside and outside the nanosphere are related by boundary conditions (BCs), see the Methods section. This translates into linear relations between the expansion coefficients of the exciting and scattered fields[@Stratton; @Quinten_2011] $$\label{eq:exc_vs_sca}
a_{lm}^{\mathrm{sc}} = t_{l'}^{\textsc{te}} a_{l'm'}^{\mathrm{ex}}\delta_{ll'}\delta_{mm'}, \qquad
b_{lm}^{\mathrm{sc}} = t_{l'}^{\textsc{tm}} b_{l'm'}^{\mathrm{ex}}\delta_{ll'}\delta_{mm'},$$ where $\delta_{jk}$ is the Kronecker delta. The proportionality constants $t_l^{\textsc{te}}$ and $t_l^{\textsc{tm}}$ are known as the Mie–Lorenz coefficients[@Mie:1908]. For nanospheres with nonlocal response they are given by[@Ruppin:1973a; @David:2012]
\[eqs:Miecoefs\] $$\begin{aligned}
t_l^{\textsc{te}} &= \frac
{-j_l(x_{\textsc{m}})[x_{\textsc{d}}j_l(x_{\textsc{d}})]'+j_l(x_{\textsc{d}})[x_{\textsc{m}}j_l(x_{\textsc{m}})]'}
{j_l(x_{\textsc{m}})[x_{\textsc{d}}h_l^{\scriptscriptstyle (1)}(x_{\textsc{d}})]'-h_l^{\scriptscriptstyle (1)}(x_{\textsc{d}})[x_{\textsc{m}}j_l(x_{\textsc{m}})]'}, \label{eq:Miecoefs_TE}\\
t_l^{\textsc{tm}} &= \frac
{-\varepsilon_{\textsc{m}} j_l(x_{\textsc{m}})[x_{\textsc{d}}j_l(x_{\textsc{d}})]'+\varepsilon_{\textsc{d}}j_l(x_{\textsc{d}})\big\lbrace [x_{\textsc{m}} j_l(x_{\textsc{m}})]'+\Delta_l\big\rbrace}
{\varepsilon_{\textsc{m}} j_l(x_{\textsc{m}})[x_{\textsc{d}}h_l^{\scriptscriptstyle (1)}(x_{\textsc{d}})]'-\varepsilon_{\textsc{d}}h_l^{\scriptscriptstyle (1)}(x_{\textsc{d}})\big\lbrace [x_{\textsc{m}} j_l(x_{\textsc{m}})]'+\Delta_l\big\rbrace },\label{eq:Miecoefs_TM}\end{aligned}$$ where $x_{\textsc{d}} = k_{\textsc{d}}R$ and $x_{\textsc{m}} = k_{\textsc{m}}R$ are dimensionless parameters in terms of the dielectric and transverse metal wavenumbers (see Methods), and the radius $R$ of the nanosphere. The primes denote the derivatives with respect to $x_{\textsc{d,m}}$. As for the usual Mie–Lorenz coefficients in the LRA, these hydrodynamic Mie–Lorenz coefficients are independent of the multipole label $m$, due to the spherical geometry of the scatterer. Spatial nonlocality influences the Mie–Lorenz coefficients through the hydrodynamic term[@Ruppin:1973a; @David:2012] $$\label{eq:Mie_HydroCorrec}
\Delta_l = l(l+1) j_l(x_{\textsc{m}})\frac{\varepsilon_{\textsc{m}}-\varepsilon_{\infty}}{\varepsilon_{\infty}}\frac{j_l(x_{\textsc{nl}})}{x_{\textsc{nl}}j_l'(x_{\textsc{nl}})},$$
with $x_{\textsc{nl}} = k_{\textsc{nl}}R$ introducing the longitudinal metal wavenumber (see Methods). As expected, the correction $\Delta_l$ vanishes in the LRA limit, since $|x_{\textsc{nl}}|\rightarrow \infty$ as $\beta_{\textsc{f}}\rightarrow 0$. Note that only the scattering of TM waves is affected by the inclusion of spatial nonlocality. There are no contributions to the magnetic field from the longitudinal multipoles $\mathbf{L}_{lm}^{\rm tr}$, *cf.* the Maxwell–Faraday equation, thus leaving the TE waves, sometimes called the magnetic waves, unaffected.
The significance of the Mie–Lorenz coefficients is that they specify the scattering laws outside the sphere, *i.e.*, they determine the outcome of external measurements. In particular, a general linear measurement $\mathcal{O}$ on a nanosphere can be expressed as a linear combination of them. As discussed in more detail below, all three measurements that we consider can be expressed in the general form $$\label{eq:O_measurement_general}
\mathcal{O} = \sum_{lm} \mathcal{O}_{lm}^{\textsc{te}}\mathrm{Re}(t_l^{\textsc{te}}) + \mathcal{O}_{lm}^{\textsc{tm}}\mathrm{Re}(t_l^{\textsc{tm}}),$$ where the coefficients $\mathcal{O}_{lm}^{\textsc{te},\textsc{tm}}$ contain all information regarding the measurement, *e.g.*, type and position, while $t_l^{\textsc{te},\textsc{tm}}$ contain all information regarding the scattering geometry, *e.g.*, dielectric composition and size. Crucially, the inclusion of hydrodynamic nonlocality modifies only the Mie–Lorenz coefficients $t_l^{\textsc{tm}}$ – but not the measurement coefficients $\mathcal{O}_{lm}^{\textsc{te},\textsc{tm}}$.
For this reason we can first focus on the Mie–Lorenz coefficients and look for the local and nonlocal plasmonic resonances that in principle affect all measurements. After that, we will identify the measurements in which these resonances make a prominent appearance and where the impact of hydrodynamic dispersion is especially strong.
#### Multipole plasmon resonances.
Figure \[fig:MieCoefs\] depicts the frequency dependence of the first few Mie–Lorenz coefficients $t_l^{\textsc{te},\textsc{tm}}$ of a free-electron $R = 2.5\,\mathrm{nm}$ nanosphere. Clearly, large-$l$ multipoles in general scatter significantly weaker than small-$l$ multipoles (notice the log scale). In addition, the $t_l^{\textsc{tm}}$ coefficients exhibit a series of resonances, corresponding to poles of the coefficient, associated with excitation of LSPs of dipole, quadrupole, hexapole (and so on) character, for $l=1,2,3,\ldots$, respectively. By contrast, the $t_l^{\textsc{te}}$ coefficients exhibit no such resonances. Moreover they are several orders of magnitude smaller than their equal-momenta TM correspondents. As a result, the TM-interaction dominates the response of plasmonic nanospheres. It is this dominant TM-interaction which is modified by nonlocal response.
![Absolute value of the Mie–Lorenz coefficients $t_l^{\textsc{te}}$ and $t_l^{\textsc{tm}}$ in (a) and (b), respectively, on a logarithmic scale, as a function of frequency, for the first few values of $l$. Considered is a $R=2.5\, \mathrm{nm}$ sphere with Drude-metal parameters $\omega_{\mathrm{p}} = 10\, \mathrm{eV}$, $\eta = 0.1\,\mathrm{eV}$ and $\varepsilon_{\infty} = 1$ embedded in vacuum, $\varepsilon_{\textsc{d}} = 1$. For comparison, the LRA TM Mie–Lorenz coefficients are illustrated in gray dashed lines. Approximate resonance predictions for LRA and hydrodynamics, as predicted by Eqs. and , are given in dashed and full red lines, respectively.[]{data-label="fig:MieCoefs"}](Figure2)
*Surface plasmon resonance conditions.* A trademark of hydrodynamic response is its blueshift of resonances as compared to local response. Figure \[fig:MieCoefs\] illustrates that for nanospheres these blueshifts show up in the TM Mie–Lorenz coefficients, and are increasingly shifted for larger $l$.[@Yan:2013_GSIM] We study this quantitatively and find the multipole plasmon resonances of order $l$ from the pole of the $t_l^{\textsc{tm}}$ coefficient. The nonretarded limit can be applied to the small spheres under consideration, leading to the plasmon condition[@BoardmanParanjape:1977] $$\label{eq:multipolecondition}
l\varepsilon_{\textsc{m}} + (l+1)(1+\delta_l)\varepsilon_{\textsc{d}} = 0,$$ where $\delta_l=\Delta_l/[j_l(x_{\textsc{m}})(l+1)]$ accounts for the hydrodynamic correction, see SI for additional details (a similar multipole plasmon condition was derived in Ref. for metallic spheres in vacuum, but with a missing factor of $i/x_{\textsc{nl}}$ in their equivalent definition of $\delta_l$). Evidently, nonlocality can be interpreted as modifying the dielectric surrounding, by introducing an effective $l$-dependent dielectric constant $\varepsilon_{l,\textsc{d}}^{\mathrm{eff}} = (1+\delta_l)\varepsilon_{\textsc{d}}$. Since $\delta_l$ itself is a function of frequency and angular momentum, Eq. defines plasmon resonances only implicitly. Nevertheless, their spectral location can be determined by approximation while retaining the essential physics, as we shall show below.
In the LRA limit $\delta_l \rightarrow 0$ and upon neglecting dispersion of the bound response and damping, *i.e.*, taking $\varepsilon_{\textsc{m}}(\omega) = \varepsilon_{\infty}-\omega_{\mathrm{p}}^2/\omega^2$, the well-known local electrostatic plasmon resonances are immediately recovered from Eq. as $$\label{eq:localresonances}
\omega_l^{\textsc{l}} = \frac{\omega_{\mathrm{p}}}{\sqrt{\varepsilon_{\infty}+\tfrac{l+1}{l}\varepsilon_{\textsc{d}}}},$$ Thus, in local theory, for $l=1$ we find the well-known (dipolar) LSP resonance $\omega_l^{\textsc{l}} = \omega_{\mathrm{p}}/\sqrt{\varepsilon_{\infty}+2\varepsilon_{\textsc{d}}}$, which reduces to $\omega_{\mathrm{p}}/\sqrt{3}$ for a free Drude-metal sphere in vacuum. The high-order multipole plasmons tend asymptotically from below towards the local planar-interface surface plasmon $\omega_{\mathrm{p}}/\sqrt{\varepsilon_{\infty}+\varepsilon_{\textsc{d}}}$ for $l\rightarrow \infty$, reducing to $\omega_{\mathrm{p}}/\sqrt{2}$ for a free Drude-metal sphere in vacuum. The $l$-dependence of $\omega_l^{\textsc{l}}$ as described by Eq. is depicted by the red-dashed line in Figure \[fig:MieCoefs\], clearly showing the asymptotic behavior for large $l$.
Turning now from local to nonlocal response, let us assume that $\delta_l$ in Eq. is a small perturbation, which is valid for small $l$ and for $R\gg\beta_{\textsc{f}}/\omega_{\mathrm{p}}$. We circumvent the implicitness of the resonance condition by making a pole approximation, replacing the dispersive function $\delta_l(\omega)$ by its value $\delta_l^{\textsc{l}} = \delta_l(\omega_l^{\textsc{l}})$ in the local resonance frequency $\omega_l^{\textsc{l}}$, the latter given by Eq. . The hydrodynamically corrected resonances $\omega_l^{\textsc{nl}}$ then occur at approximately[@Yan:2013_GSIM] $$\label{eq:approxmultipoleres}
\omega_l^{\textsc{nl}}
\simeq \frac{\omega_{\mathrm{p}}} {\sqrt{ \varepsilon_{\infty} + \tfrac{l+1}{l}(1+\delta_l^{\textsc{L}})\varepsilon_{\textsc{d}}}}
\simeq \omega_l^{\textsc{l}} + \frac{\beta_{\textsc{f}}}{R}\sqrt{\frac{l(l+1)\varepsilon_{\textsc{d}}}{4\varepsilon_{\infty}}},$$ where, at the last step, in addition to a Taylor expansion of the square-root term, we have utilized the large imaginary $x_{\textsc{nl}}$ limit of the hydrodynamic correction, $\delta_l \simeq l\tfrac{\varepsilon_{\textsc{m}}-\varepsilon_{\infty}}{\varepsilon_{\infty}}\tfrac{i}{x_{\textsc{nl}}}$, which is applicable at frequencies below the screened plasma frequency $\omega_{\mathrm{p}}^{\scriptscriptstyle \infty} \equiv \omega_{\mathrm{p}}/\varepsilon_{\infty}$. These approximate nonlocal surface plasmon resonance frequencies are illustrated by the solid red line in Figure \[fig:MieCoefs\]. The approximation captures the exact nonlocal blueshift well but is less accurate for larger $l$, as expected. By implication of these nonlocal blueshifts, excitations appear between the LRA $l= \infty$ mode (the planar surface plasmon) and the volume plasmon at $\omega_{\mathrm{p}}$, classically a resonance-free frequency interval.[@vomFelde:1988]
*Bulk plasmon resonance condition.* Besides blueshifting the multipolar LSP resonances that already exist in the LRA, hydrodynamical theory also predicts the appearance of additional resonances due to confined bulk plasmons for which no LRA counterparts exist.[@Ruppin:1973a; @Raza:2011] More microscopic theories have also predicted the emergence of such bulk plasmons.[@Townsend:2011a; @Stella:2013] These bulk plasmons emerge due to the presence of propagating, longitudinal pressure waves above the plasma frequency. In hydrodynamics, the confined bulk plasmons are then easily interpreted as the standing-wave resonances of longitudinal waves. Table \[tab:multipoles\] depicts isosurfaces of the induced charge density for LSPs and bulk plasmons for comparison.
An approximation for these bulk resonances can be found by neglecting the coupling of the pressure waves to light, *i.e.*, by searching for standing wave solutions of $\mathbf{L}_{lm}^{\mathrm{tr}}$, thus neglecting the transverse components. For nanospheres, this gives radially quantized confined bulk plasmons resonating at the frequencies $\omega_{ln}^{\mathrm{bulk}}$ (see SI for details): $$\label{eq:bulkplasmonapproximation}
\omega_{ln}^{\mathrm{bulk}}(\omega_{ln}^{\mathrm{bulk}} + i \eta) = \frac{\omega_{\mathrm{p}}^{2}}{\varepsilon_{\infty}} + w_{ln}^{2}\bigg(\frac{\beta_{\textsc{f}}}{R}\bigg)^{2},$$ where $w_{ln}$ is the $n^{th}$ positive root of $j_{l}'(w)$, the derivative of the $l^{\mathrm{th}}$-order spherical Bessel function (see Refs. and for lengthier, more accurate approximations). Modes associated with the first root at $n=0$ are in fact not resonant, but are artifacts of the approximation that arise due to having neglected the transverse field-components. Regardless, for every multipole order $l$ there is an infinite number of confined bulk plasmons associated with $n=1,2,\ldots$.
As for the LSP resonances, we first illustrate the signature of these bulk plasmons in the Mie–Lorenz coefficients, before considering the experiments in which their presence is most pronounced. In Figure \[fig:MieCoefs\_BulkFocus\_Insets\] we depict the frequency dependence of the first few Mie–Lorenz transmission coefficients $q_{l}^{\textsc{l}}$ near and above $\omega_{\mathrm{p}}$. These coefficients give the transmission amplitude to a longitudinal mode due to excitation by an incident TM mode, and are defined analogously to the scattering coefficients $t_{l}^{\textsc{te},\textsc{tm}}$ of Eq. through $c_{lm}^{\mathrm{tr}} = q_{l'}^{\textsc{l}} b_{l'm'}^{\mathrm{ex}}\delta_{ll'}\delta_{mm'}$, see SI for their explicit form.
![Absolute value of the Mie–Lorenz transmission coefficients $q_{l}^{\textsc{l}}$ on a logarithmic scale, as a function of frequency. The coefficients give the coupling amplitude between transmitted longitudinal multipoles, and incident TM multipoles. Setup-parameters are identical to those in Figure \[fig:MieCoefs\]. Shown are the dipolar, $q_{1}^{\textsc{l}}$ and the quadrupolar, $q_{2}^{\textsc{l}}$, coefficients in blue. Both exhibit peaks above $\omega_{\mathrm{p}}$, corresponding to a series of confined bulk plasmons labeled by $n=0,1,2,\ldots$. Green curves show approximate resonance positions, see Eq. . The absence of an $n=0$ resonance is apparent. Insets depict logarithmic scale contour plots, with contours separated by factors of 2, of the absolute value of the induced charge density of the bulk resonances, with $[l,n]$ indices labeled, in the $xz$-plane.[]{data-label="fig:MieCoefs_BulkFocus_Insets"}](Figure3)
The first dipolar and quadrupolar bulk plasmon resonances of a nanosphere clearly show up as Lorentzian resonances, and the bulk plasmon approximation Eq. is quite accurate. The resonant charge distributions in the insets illustrate the radial quantization of the confined bulk plasmons. To the best of our knowledge, only the dipole ($l=1$) confined bulk plasmons have been considered previously, *e.g.*, in relation with extinction-features above the plasma frequency in nanospheres.[@Ruppin:1973a; @Raza:2011] In our investigation of EELS and LDOS below, we consider additionally if these higher-$l$ bulk plasmons may influence the spectral response in the near field. First, however, we discuss the properties of higher-order LSP multipoles.
*Large-$l$ plasmonic resonances.* We have seen in Figure \[fig:MieCoefs\] that multipolar hydrodynamic LSP modes blueshift away from the classical limit, the LRA planar surface plasmon at $\omega_{\rm p}/\sqrt{2}$. What is more, Figure \[fig:MieCoefs\_bulkvsvolume\]
![Absolute value of the TM Mie–Lorenz coefficients, $t_l^{\textsc{tm}}$, on a logarithmic scale, as a function of frequency for high-angular momenta. Setup-parameters are identical to those in Figure \[fig:MieCoefs\]. Hydrodynamic results are illustrated in blue solid lines, while LRA results are illustrated by gray dashed lines for comparison. The transition across the plasma frequency is marked by the black dashed line. The red line depicts the approximate LSP resonance of Eq. ; the green lines show Eq. and approximate the first few confined bulk plasmon resonances. The bulk plasmons show up as Fano-like resonances in $|t_l^{\textsc{tm}}|$.[@Tribelsky:2012][]{data-label="fig:MieCoefs_bulkvsvolume"}](Figure4)
illustrates that high-multipole nonlocal LSP resonances can even appear above the plasma frequency $\omega_{\rm p}$. There is no indication that the plasma frequency would mark a qualitative transition. This is despite the change from predominantly imaginary metal wavenumbers ($k_{\textsc{m}}$ and $k_{\textsc{nl}}$) for frequencies $\omega<\omega_{\mathrm{p}}^{\infty}$, to predominantly real metal wavenumbers for $\omega > \omega_{\mathrm{p}}^{\infty}$. In particular, the transition from predominantly imaginary to real wavenumbers does not carry with it a transition from predominantly bound surface modes to volume-like modes as assumed in the past.[@Baltz:1995] \[Such a transition does not emerge since $|x_{\textsc{nl}}|$ remains comparative with $\sqrt{l+1}$, which, *cf.* Eq. and the small-argument asymptotic form $j_l(x)\simeq x^l/(2l+1)!!$ valid for $|x|\ll\sqrt{l+1}$, implies that $|j_l(x)|\sim |j_l(ix)|$ for $|x|<|x_{\textsc{nl}}|$, whereby the charge density is left qualitatively unchanged and surface-bound.\] Hydrodynamic surface plasmons above the plasma frequency have also been found theoretically for a planar metal-dielectric interface, for a thin metal slab, and for planar metamaterials.[@Wei_PRB2012; @Raza:2013_PhysRevB]
It is fruitful to pursue further the analogy between the LSPs of our nanospheres and of planar structures. The analogy is well-known for local response, but the hydrodynamic version holds a surprise. The large-$l$ LSP resonances below and above the plasma frequency can both be characterized by wave propagation along the surface of the nanosphere. The $l^{\mathrm{th}}$ surface mode accommodates exactly $l$ oscillation periods along the periphery of the sphere. One can therefore ascribe an effective surface wavelength $\lambda^{\mathrm{s}}_l = 2\pi R/l$ and an effective surface wavenumber $k^{\mathrm{s}}_l = l/R$ to the $l^{\mathrm{th}}$ mode. For larger $l$, the effective wavelength becomes shorter and the modes perceive the curving surface of the sphere as increasingly flat. For that reason the dispersion would mimic that of a planar metal-dielectric interface for large $l$.
To test this prediction from the analogy, we compute the exact plasmon resonances from Eq. and show them in a pseudo-dispersion plot in Figure \[fig:pseudodispersion\].
![Dispersion of the nonretarded surface plasmon resonances of nanospheres. Material-parameters as in Figure \[fig:MieCoefs\] but with $\eta = 0$. Wavenumbers are normalized to the plasma wavenumber $k_{\mathrm{p}} = \omega_{\mathrm{p}}/\beta_{\textsc{f}}$. The hydrodynamic model is shown in blue and the LRA in gray. The $l=1,4,7,\ldots,34$ multipole LSP resonances are indicated by squares and circles; nonretarded dispersion relations[@Raza:2013_PhysRevB] for a planar interface are shown as solid lines. Insets in panels (a) and (b) show the real parts of the electric field of selected LSP modes in the $xz$-plane along $\theta$-polarization (on separate color scales). Panel (c) depicts contour plots of the absolute value of the hydrodynamic charge density of selected LSP modes in the same nanosphere and in the same plane (contours separated by factors of 10 with separate, logarithmic color scales). []{data-label="fig:pseudodispersion"}](Figure5)
For local response, Figure \[fig:pseudodispersion\] indeed shows the well-known result that for larger $l$ the dispersion of the nanosphere LSPs approaches more and more that of a flat interface. For nonlocal response, also shown in Figure \[fig:pseudodispersion\], we first note that the LSP dispersion indeed does not show a transition at the plasma frequency, as we already guessed from Figure \[fig:MieCoefs\_bulkvsvolume\]. Secondly, there is satisfactory agreement of the hydrodynamic dispersion of LSPs for a nanosphere and for the flat interface, so the analogy is also meaningful for hydrodynamic response. However, and this is the surprising third point, unlike for local response, the agreement does [*not*]{} converge towards a complete agreement as $l$ increases: a discrepancy develops for large $l$. The discrepancy is larger in Figure \[fig:pseudodispersion\](a) for $R=2.5\,{\rm nm}$ spheres than for the twice larger spheres in Figure \[fig:pseudodispersion\](b).
This can be explained by noting that in the LRA all the induced free charge resides only *on* the surface of the sphere, whereas it is distributed *close to* this surface in the hydrodynamic description. The latter is illustrated in Figure \[fig:pseudodispersion\](c). Note the surficial standing-wave quantization of the LSPs in Figure \[fig:pseudodispersion\], and also the absence of radial quantization, being associated only with the bulk plasmons as shown in Figure \[fig:MieCoefs\_BulkFocus\_Insets\]. For large $l$, the neighboring hydrodynamic charge patterns in Figure \[fig:pseudodispersion\] get squeezed into each other due to the finite curvature, producing the discrepancy with the planar interface. An alternative explanation of the discrepancy as due to interaction across antipodal surface points can be ruled out, since the insets of Figures \[fig:pseudodispersion\](a,b) show that the electric fields corresponding to high-$l$ modes are well localized near the surface of the nanosphere, even those above the plasma frequency (in contrast to predictions of Ref. ), so that fields on opposite angular regions of the sphere are spatially well separated. This agrees with recent findings for hydrodynamic LSP modes in a planar thin metal slab, which do not show finite-size effects either for sufficiently large wavevectors. Rather, since the slab has no curvature, the large-$k$ dispersion of its LSP modes does indeed agree with that of the single interface.[@Wei_PRB2012]
#### Extinction, EELS, and LDOS.
Having discussed the characteristics of the multipole plasmons, and in particular the modifications due to hydrodynamic response, we will now consider three distinct measurements, each with a different sensitivity to the various surface and bulk plasmons:
1. **Light scattering.** This measurement gives the extinction cross-section $\sigma_{\mathrm{ext}}(\omega)$, yielding the ratio of power dissipated due to scattering and absorption of a plane-wave relative to incident intensity.
2. **Electron energy-loss spectroscopy.** EELS gives information on the electron loss function $\Gamma(\omega)$, that expresses the probability that a relativistic electron will lose an energy $\hbar\omega$ due to interaction with the particle. We consider electrons traveling with velocity $v\hat{\mathbf{z}}$ and impact parameter $\mathbf{b}$ in the $xy$-plane outside the sphere ($|\mathbf{b}|=b>R$).
3. **Atomic spontaneous emission.** A dipole orientation-averaged measurement of local spontaneous emission rates relates linearly to the electric local optical density of states (or LDOS) $\rho^{\textsc{e}}(\omega)$. We consider emitter positions $\mathbf{b}$ outside the nanosphere ($b > R$).
These three measurements constitute examples of illumination of the sphere by plane-, cylinder-, and spherical-like waves. Extinction is measured in the archetypical far-field scattering setup, while the EELS probability and LDOS can be measured locally in the near-field. Sub-nanometer control of the probe-surface separation is routinely achieved in EELS[@Abajo_PhysRevMod2010] and also demonstrated in fluorescence measurements[@Anger:2006; @Dulkeith:2005], permitting experimental investigation of the various calculated spectra that we will show below.
Let us briefly discuss the computation of these measurements in the multipole basis. The arbitrary exciting field can be decomposed into the multipole basis, *i.e.*, the coefficients $\{a_{lm}^{\mathrm{ex}},b_{lm}^{\mathrm{ex}}\}$ can be determined. The scattered field is then obtained through the Mie–Lorenz coefficients using Eq. . A general linear measurement $\mathcal{O}$ may involve components of the scattered field at a single location, as for the LDOS, or a continuous weighting of different spatial components of the field, as for the extinction cross-section or the EELS probability. In any case, the measurements can be expressed through a weighted $lm$-summation of the scattering amplitudes $t_l^{\textsc{te}} a_{lm}^{\mathrm{ex}}$ and $t_l^{\textsc{tm}} b_{lm}^{\mathrm{ex}}$. As stated above, for the extinction cross-section[@BohrenHuffman:1983], EELS probability[@Abajo:1999; @Abajo_PhysRevMod2010], and LDOS[@Kerker:1980; @Ruppin:1982; @Chew:1987; @Dung:2001pra; @Vos:2009], the measurements $\mathcal{O}$ can all be expressed in terms of the Mie–Lorenz coefficients in the general form of Eq. . For the specific forms that Eq. takes for each of the three measurements, we refer the reader to Eqs. (S3), (S6) and (S9) of the SI.
In the following we normalize the extinction cross-section to the geometric cross-section, $\pi R^2$, yielding the extinction efficiency $Q_{\mathrm{ext}}(\omega) \equiv \sigma_{\mathrm{ext}}(\omega)/\pi R^2$, and similarly normalize the LDOS to the free-space LDOS $\rho^{\textsc{e}}_0(\omega)$, yielding the LDOS enhancement $[\rho^{\textsc{e}}/\rho^{\textsc{e}}_0](\omega)$.
*Near-field versus far-field.* Figure \[fig:EELSLDOSExt\](a)
depicts the probe-to-surface separation dependence of the LDOS and EELS spectra in a Drude-metal nanosphere of $R=1.5\,\mathrm{nm}$, and for comparison also depicts the extinction resonances. Hydrodynamic and LRA calculations are shown to be distinctly different. Most conspicuous in Figure \[fig:EELSLDOSExt\](a) is perhaps that many new resonances appear in the nonlocal EELS and LDOS spectra, many more than in extinction, and that drastic changes occur when we vary $b/R$ from the contact scenario $b/R=1$ to $b/R=4.5$. When fixing $b/R=2$, we obtain the spectra of Figure \[fig:ExtEELSLDOS\_1Dplot\](a).
Below we discuss both figures in more detail, but before that, Figure \[fig:EELSLDOSExt\](a) already makes clear that only a rudimentary understanding of EELS measurements can be obtained by comparing them with calculated extinction or absorption spectra. Such comparisons have nevertheless been quite common.[@Scholl:2012; @Raza:2013_Nanophotonics]
Let us interpret Figures \[fig:EELSLDOSExt\](a) and \[fig:ExtEELSLDOS\_1Dplot\](a) in more detail by first discussing the region below the plasma frequency, where both in local and nonlocal response, the extinction efficiency exhibits just the single dipolar ($l=1$) surface plasmon resonance. Higher-order multipole plasmons do not contribute since the sphere size is much smaller than the wavelength of the incident plane wave.[@Ruppin:1973a]
In stark contrast to these known extinction spectra, several additional multipole LSP resonances are observable in the EELS and LDOS spectra, and better so for smaller probe-to-surface separations. Notice that higher-order LSP modes do exist in the LRA, as we have seen in the analysis of the Mie–Lorenz coefficients, but these additional LSP resonances converge towards the $l=\infty$ limit at $\omega_{\mathrm{p}}/\sqrt{2}$ and rapidly become indistinguishable due to losses. By contrast, the higher-order LSP resonances are much more clearly visible in the hydrodynamic spectra because of the $l$-dependent nonlocal blueshift of Eq. , which pushes the multipole resonances in the EELS and LDOS spectra beyond the LRA $l=\infty$ limit and moreover separates them despite the loss-induced broadening.[@TranThoai:1986; @TranThoai:1988]
Observation of a multipolar resonance above the $l=\infty$ limit was reported by vom Felde *et al.*[@vomFelde:1988] in EELS measurements on ensembles of potassium clusters of radius $1-2\,\mathrm{nm}$ embedded in magnesium oxide. Vom Felde *et al.* attributed this blueshift into the classically quiet region to quantum size effects. Here we show that there is a good alternative explanation, namely collective hydrodynamic multipolar LSP resonances. Thus the ongoing discussion how to interpret the blueshift of the main dipolar LSP resonance as seen in EELS[@Scholl:2012; @Raza:2013_Nanophotonics; @CarminaMonreal:2013a] can now be extended to higher-order LSP resonances, observable both in EELS and LDOS measurements. This improves the outlook of obtaining conclusive evidence for hydrodynamic behavior in plasmonic nanospheres.
Importantly, our calculations performed for aluminum ($\omega_{\rm p} = 14.94,\mathrm{eV}$) in Figures \[fig:EELSLDOSExt\](b) and \[fig:ExtEELSLDOS\_1Dplot\](b), using measured data from Ref. , confirm the feasibility of measuring multipole resonances beyond the $l=\infty$ limit for realistic (*i.e.*, non-Drude) metals: at least four orders of surface plasmons besides both dipole and quadrupole bulk plasmons are discernible. The nanosphere radius considered in Figures \[fig:EELSLDOSExt\] and \[fig:ExtEELSLDOS\_1Dplot\] is, however, relatively small at $R=1.5\,\mathrm{nm}$. While consideration of such small nanospheres eases interpretation and labeling, it also approaches the emergence of the realm of cluster physics. Nevertheless, similar spectral features persist for larger spheres, upholding the pertinence of the analysis. Supporting calculations for $R=3\,\mathrm{nm}$ nanospheres are presented in the SI.
We emphasize that one should not view the results in Figures \[fig:EELSLDOSExt\](b) and \[fig:ExtEELSLDOS\_1Dplot\](b) as being fully representative of experiments: the semi-classical plasma-in-a-box hydrodynamic model necessarily cannot contain all relevant physics. In particular, it is known that the nonlocal blueshift of the *dipolar* SPP for aluminum spheres in vacuum will be more than fully compensated by a redshift due to electronic spill-out.[@Mandal:2013]
However, for higher-order multipoles we expect that the centroid of the induced charge will be pushed inwards at larger multipole orders, and that nonlocality will come to dominate the effects of spill-out. These considerations are supported by calculations in Ref. on planar simple-metal surfaces, which show that the induced charge recedes to the interior of the metal at large momentum transfers, equivalent to high multipole order. This indicates that spill-out does not undo our prediction that higher-order SPP resonances will be well-separated due to nonlocal response, and thus suggests a novel direction for identification of hydrodynamic behavior in nanospheres. The key features of our theoretical near-field spectra for aluminum are encouraging in this respect. Accordingly, experimental investigation and further theoretical study with more microscopic models is highly desirable.
Additionally, we note that electronic spill-out is not a property of the metal nanoparticle alone but also of its surrounding dielectric, in a similar way that the atomic spontaneous-emission rate is not a property of the atom alone but also of its electromagnetic environment. This gives additional experimental freedom: by embedding metal spheres into a solid matrix, electronic spill-out can be controlled and the associated redshift suppressed.[@vomFelde:1988] A high-index dielectric surrounding can significantly reduce the electronic spill-out, even in simple metals. Thus with high-index background dielectrics, our plasma-in-a-box model is expected to be more accurate. The key effects of a non-unity background dielectric function on the SPP and bulk-plasmon resonances of Figures \[fig:EELSLDOSExt\] and \[fig:ExtEELSLDOS\_1Dplot\] can be readily discerned from Eqs. (\[eq:localresonances\])–(\[eq:bulkplasmonapproximation\]).
As further promising experiments, we propose to use the same materials as in Ref. , namely potassium (or Na or Rb) nanospheres in an MgO matrix, but now for doing EELS on an individual nanosphere, so that inhomogeneous broadening would no longer obscure individual multipolar peaks. Similarly, rather than utilizing a continuous embedding matrix, it may be feasible to suppress the electronic spill-out just by coating the nanospheres with a suitable dielectric, thereby also providing protection from oxidization.
The higher-order LSPs that we propose to observe were not seen in the recent EELS measurements on silver nanospheres of Refs. and . This agrees with calculations performed by us for silver, which are detailed in the SI: due to strong interband effects, higher-order multipole LSP resonances are obscured even in individual Ag nanospheres.
Above the plasma frequency, two hydrodynamic peaks can be seen in the (identical) extinction spectra of Figures \[fig:EELSLDOSExt\](a) and \[fig:ExtEELSLDOS\_1Dplot\](a). They clearly have no analogue in the LRA, and correspond to the first two dipolar confined bulk plasmon resonances, with labels $[l,n]=[1,1]$ and $[1,2]$, that we also identified in the hydrodynamic Mie–Lorenz coefficients in Figure \[fig:MieCoefs\_BulkFocus\_Insets\]. They have first been predicted by Ruppin to exist in the extinction spectrum.[@Ruppin:1973a] Interestingly, in the EELS and LDOS spectra of Figures \[fig:EELSLDOSExt\] and \[fig:ExtEELSLDOS\_1Dplot\], we see more resonances above the plasma frequency than the two dipolar bulk plasmons of the extinction spectrum. According to our investigations of the Mie–Lorenz coefficients in Figures \[fig:MieCoefs\_BulkFocus\_Insets\] and \[fig:pseudodispersion\], these additional resonances in principle could be either high-$l$ LSP resonances or quadrupolar and higher-order bulk plasmon resonances. They all turn out to be bulk plasmons, and are therefore labeled accordingly; the high-$l$ LSP resonances are much weaker and absent in the spectrum.
Better than observing shifts in LSP peaks, observing the confined bulk plasmon peaks would constitute a unique identification of hydrodynamic pressure waves in nanospheres. However, since we find them to be three orders of magnitude weaker than the dipolar LSP resonance, actually the same order of magnitude weaker as found in recent density-functional calculations,[@Townsend:2011a] they are difficult to measure in nanospheres. To our knowledge they have not yet been observed (unlike their counterparts in thin films), so to date bulk plasmons are “non-smoking guns” of hydrodynamic pressure waves in nanospheres.
Overall, Figures \[fig:EELSLDOSExt\] and \[fig:ExtEELSLDOS\_1Dplot\] illustrate the importance of the dimensionality of the excitation source. As is well known, the plane wave used in extinction measurements only excites dipole resonances in deeply subwavelength spheres. As to the EELS spectra, the one-dimensional source of a travelling electron excites a cylinder-like field, which for short probe-to-surface separations is sufficiently inhomogeneous to excite higher-order ($l>1$) plasmons as well. Lastly, the LDOS spectra illustrate the largest sensitivity to the multipole modes, with all LSPs discernible and significant response from several bulk plasmon orders. The spherical-like field of the zero-dimensional dipole induces locally a more inhomogeneous excitation field than the traveling electron, thus accounting for the increased multipole-sensitivity in LDOS compared to EELS. At large probe-surface separations shown in Figure \[fig:EELSLDOSExt\], the exciting fields in both EELS and LDOS are almost homogeneous near the sphere, and the response due to higher-order multipoles is diminished. As a consequence, for large probe-to-surface separations the spectral response in extinction, EELS, and LDOS is qualitatively the same. See SI for analytical considerations of this latter point, regarding the asymptotics of the LDOS and EELS spectra.
*Distance-dependence of LDOS.* In the preceding sections we established that the response of high-order plasmons is significantly enhanced with probes of low-dimensionality when examined in the near-field, where the observability of multipolar LSPs is enhanced by hydrodynamics itself. Let us therefore finally focus solely on the LDOS spectra, where the response of these high-order multipoles is most pronounced. In Figure \[fig:LDOS\_bvar\]
, for a $R= 2.5\, \mathrm{nm}$ sphere.[]{data-label="fig:LDOS_bvar"}](Figure10)
we display the variation of the LDOS spectra as a function of the probe-to-surface separation, varying from $b/R=1$ (*i.e.*, source on surface) to $b/R=5$ ($10\,\mathrm{nm}$ separation). For the panels with $b/R < 2$, contributions from high-order multipoles are increasingly important, as the excitation of multiple LSP orders contribute to the spectrum[@Moroz:2010]. Consequently, in the LRA the largest LDOS occurs at $\omega_{\rm p}/\sqrt{2}$, the limiting frequency of the high-order LSPs, coinciding with the pile-up of LRA multipoles. By contrast, the hydrodynamically blueshifted LSPs do not have a finite limiting frequency or an associated similar pile-up of modes, but instead exhibit distinguishable peaks associated with excitation of different multipoles. The qualitative discrepancy between local and nonlocal spectra is even substantial. For larger spheres, the multipole peaks merge and instead give rise to a broadband enhancement above $\omega_{\mathrm{p}}/\sqrt{2}$, even extending beyond the plasma frequency, see SI for supporting calculations on an $R=10\,\mathrm{nm}$ sphere. This suggests an hitherto largely unexplored regime of studying nonlocal response in comparatively large nanostructures but at short surface-to-probe separations.
As is well known, in the extreme limit $b=R$ the LRA LDOS diverges (hence not shown) due to the $1/(b-R)^3$ scaling of the nonradiative decay rate. For the $b/R=1$ panel of Figure \[fig:LDOS\_bvar\] we obtain convergent results for the hydrodynamic response, and associated finite LDOS spectra. The convergence, however, hinges upon the choice of a simple Drude-metal with real-valued $\varepsilon_{\infty}$, as discussed in Ref. . As such, hydrodynamic response does not fully regularize the divergence of the LDOS for real metals with dissipative bound response. Complete regularization in real metals would likely necessitate an appropriate nonlocal treatment of not only the free response, but also the bound response. In addition, for these very close proximities between source and nanosphere, the effect of high-order moments - beyond the dipole – of the source itself, due to the finite size of the source, would certainly modify the decay rates as well.[@Andersen:2011a] For emitters at the larger separations, *e.g.*, in the panel with $b/R = 5$, the dipole mode of the nanosphere is again the primary feature, but with the quadrupolar LSP still imposing a significant spectral feature.
Conclusions
===========
In this paper we have aimed to identify indisputable signatures of hydrodynamic response in plasmonic nanospheres. The corresponding evidence for layered systems is the observation, found both with light[@Lindau:1971; @Anderegg:1971] and with electrons,[@Oezer:2011] of confined bulk plasmons in thin films. Employing the hydrodynamic Drude model we predict the existence of confined bulk plasmons also in nanospheres. An important question then, is whether such excitations would be observable. A series of confined bulk plasmons of dipolar character has been predicted before to show up in extinction spectra.[@Ruppin:1973a] Here we additionally found that besides the dipole series, also series of quadrupolar and higher-order bulk plasmons emerge in near-field EELS and LDOS spectra. However, we find the strength of these bulk plasmon resonances in nanospheres to be about three orders of magnitude weaker than the dominant LSP peak. Their experimental observation in nanospheres, for example with EELS or LDOS, remains an open challenge. Another promising technique is core-level photoemission.[@Oezer:2011]
Of a more immediate, accessible nature experimentally, is our prediction that in the near-field EELS and LDOS spectra also quadrupolar and higher-order LSPs appear, besides the well-known dominant dipolar LSPs. In itself it is no surprise that higher-order LSPs show up in near-field spectra, since already the LRA predicts them[@Moroz:2010]. The salient point here is that LRA LSPs exhibit the surface plasmon $\omega_{\mathrm{p}}/\sqrt{2}$ of a planar interface as a limiting upper frequency, while we predict hydrodynamic LSPs to be observable also above $\omega_{\mathrm{p}}/\sqrt{2}$. This follows from our prediction that higher-$l$ plasmons exhibit a larger nonlocal blueshift. Indeed, we found that high-$l$ LSPs in principle can occur above the plasma frequency in few-nanometer spheres, with their mode profiles still well bound to the surface. An upper limiting frequency for LSPs actually does not exist in the hydrodynamic model.
Not all multipolar LSPs will be observable, though. For silver, we predict all LSPs besides the dipolar one to be suppressed due to interband effects. On the other hand, we predict that for aluminum nanospheres several higher-order LSPs should be observable in near-field EELS and LDOS spectra. In ensembles of alkali metal (Na, K, Rb) nanospheres in an MgO matrix, resonances above the LRA limit $\omega_{\mathrm{p}}/\sqrt{2}$ have actually already been observed, but individual resonance peaks could not be resolved due to ensemble averaging.[@vomFelde:1988] We propose to do these measurements on individual alkali metal nanospheres, something that has already been achieved with silver nanospheres.[@Scholl:2012; @Raza:2013_Nanophotonics]
Would such measurements constitute the unequivocal evidence, the “smoking gun”, of hydrodynamic nonlocal response in nanospheres that we set out to identify? We can only suggest ‘perhaps’ at this stage, because alternative explanations for resonances above $\omega_{\mathrm{p}}/\sqrt{2}$ do exist. In particular, vom Felde *et al.* invoke quantum confinement (cluster physics) rather than hydrodynamics (nanoplasmonics) to explain their intriguing observation of resonances above the LRA limit.[@vomFelde:1988] It is safe to assume, however, that fitting the two distinct models to a measured series of LSP resonances will be more conclusive than fitting only the dominant dipolar LSP, which remains state of the art.[@Scholl:2012; @Raza:2013_Nanophotonics; @CarminaMonreal:2013a] We therefore suggest to measure near-field EELS and LDOS spectra of nanospheres of aluminum and alkali metals embedded in a solid dielectric environment.
The plasmonic resonances emerge with strikingly different weights in the three types of spectra that we calculated, so that for example the state-of-the-art comparison of EELS experiments with theoretical absorption cross sections[@Scholl:2012] or extinction cross sections[@Raza:2013_Nanophotonics] can be of limited use. The comparison happened to be useful for silver nanospheres,[@Scholl:2012; @Raza:2013_Nanophotonics] where interband effects suppress the beyond-dipole LSP resonances that otherwise would show up in near-field EELS and LDOS experiments.
Even for the relatively simple hydrodynamic theory that we used here, the near-field spectra of nanospheres become rather elaborate and rich – but they can be understood rigorously. We therefore expect that our results could also assist in the interpretation of near-field spectra calculated with more microscopic calculations, with some features attributable to hydrodynamic nonlocal response.
Methods
=======
#### Hydrodynamics and multipole basis.
By eliminating the current density in Eqs. , the hydrodynamic equations can be recast solely in terms of the electric field:
\[eqs:hydroJgrp\] $$\begin{aligned}
\left(\nabla^2 + k_{\textsc{m}}^2 \right) \nabla \times \boldsymbol{\mathrm{E}}(\boldsymbol{\mathrm{r}},\omega) &= 0, \label{eq:hydroET} \\
\left(\nabla^2 + k_{\textsc{nl}}^2 \right) \nabla \cdot \boldsymbol{\mathrm{E}}(\boldsymbol{\mathrm{r}},\omega) &= 0, \label{eq:hydroEL}\end{aligned}$$
where $k_\textsc{m}^2 = k_0^2 \varepsilon_{\textsc{m}}$ and $k_{\textsc{nl}}^2 = (\omega_{\mathrm{p}}/\beta_{\textsc{f}})^2 \varepsilon_{\textsc{m}}/[\varepsilon_{\infty}(\varepsilon_{\infty}-\varepsilon_{\textsc{m}})]$ denote the transverse and longitudinal wavenumbers in the metal, respectively. The transverse response of the metal is governed by $\varepsilon_{\textsc{m}}(\omega) = \varepsilon_\infty(\omega) - \sigma(\omega)/i\varepsilon_0\omega$.
The vector wave functions, $\mathbf{M}_{\nu}(\boldsymbol{\mathrm{r}})$, $\mathbf{N}_{\nu}(\boldsymbol{\mathrm{r}})$, and $\mathbf{L}_{\nu}(\boldsymbol{\mathrm{r}})$, are defined in terms of a pilot vector $\mathbf{c}$, and a generating scalar function $\psi_{\nu}(\boldsymbol{\mathrm{r}})$, satisfying the Helmholtz equation $\nabla^2\psi_{\nu}(\boldsymbol{\mathrm{r}}) + k^2\psi_{\nu}(\boldsymbol{\mathrm{r}}) = 0$. In spherically symmetric structures it is natural to express the generating functions in spherical coordinates $\boldsymbol{\mathrm{r}} = (r,\theta,\phi)$ and to choose the pilot vector as the (non-constant) outward radial vector $\mathbf{c} = \boldsymbol{\mathrm{r}}$. In this case, the degeneracy label $\nu$ separates into the angular momentum quantum numbers $l$ and $m$, and the vector wave functions read as
\[eqs:SphericalVectorWaves\] $$\begin{aligned}
\mathbf{M}_{lm}(\boldsymbol{\mathrm{r}}) &= \nabla \times \mathbf{r}\psi_{lm}(\boldsymbol{\mathrm{r}}),\\
\mathbf{N}_{lm}(\boldsymbol{\mathrm{r}}) &= \frac{1}{k}\nabla \times \nabla \times \mathbf{r}\psi_{lm}(\boldsymbol{\mathrm{r}}),\\
\mathbf{L}_{lm}(\boldsymbol{\mathrm{r}}) &= \frac{1}{k}\nabla \psi_{lm}(\boldsymbol{\mathrm{r}}),\end{aligned}$$
with $\psi_{lm}(r,\theta,\phi) = z_l(kr)P_l^m(\cos\theta)\mathrm{e}^{im\phi}$, where $z_l$ denotes spherical Bessel or Hankel functions of the first kind, $j_l$ or $h_l^{\scriptscriptstyle (1)}$, for in- and outgoing waves, respectively. Finally, $P_l^m$ denote the associated Legendre polynomials. In addition, by requirements of continuity along $\phi$ and boundedness at the polar extremes, the angular momentum quantum numbers are restricted to integer values in the ranges $l\in [1,\infty[$ and $m\in[-l,l]$. This particular basis is usually referred to as the multipole basis.
The $k$-dependence of the vector wave functions used in the field expansions varies inside and outside the sphere. By insertion of the external field into the vector Helmholtz equation, $\nabla^2\boldsymbol{\mathrm{E}} + k_{\textsc{d}}^2\boldsymbol{\mathrm{E}} = 0$, which is valid outside the sphere, it is clear that the appropriate choice of wavenumber is $k_{\textsc{d}} = \sqrt{ \varepsilon_{\textsc{d}} }k_0$ outside the sphere. Similarly, by insertion of the internal field into Eqs. , it is clear that the solenoidal vector waves $\mathbf{M}_{lm}^{\mathrm{tr}}$ and $\mathbf{N}_{lm}^{\mathrm{tr}}$ inside the sphere are associated with the transverse wavenumber $k_{\textsc{m}}$, while the irrotational vector wave $\mathbf{L}_{lm}^{\mathrm{tr}}$ is associated with the longitudinal wavenumber $k_{\textsc{nl}}$.
Finally, the matching of internal and external expansions is facilitated by application of BCs. The usual BCs for the electromagnetic field requires the continuity of the tangential components of the electric and magnetic field at $r=R$, *i.e.*, $\boldsymbol{\mathrm{E}}^{\mathrm{ex}}_{\scriptscriptstyle \parallel}+ \boldsymbol{\mathrm{E}}^{\mathrm{sc}}_{\scriptscriptstyle \parallel} = \boldsymbol{\mathrm{E}}^{\mathrm{tr}}_{\scriptscriptstyle \parallel}$ and $\mathbf{H}^{\mathrm{ex}}_{\scriptscriptstyle \parallel}+ \mathbf{H}^{\mathrm{sc}}_{\scriptscriptstyle \parallel} = \mathbf{H}^{\mathrm{tr}}_{\scriptscriptstyle \parallel}$. Furthermore, an additional BC is required to account for the presence of the longitudinal waves inside the metal, which, in the case of an abrupt dielectric boundary, is unambiguously chosen as the continuity of the normal component of the induced current, equivalent to the continuity of the normal component of the bound-charge depolarization at $r=R$, corresponding to $\varepsilon_\textsc{d}\boldsymbol{\mathrm{E}}^{\mathrm{ex}}_{\scriptscriptstyle \perp}+ \varepsilon_\textsc{d}\boldsymbol{\mathrm{E}}^{\mathrm{sc}}_{\scriptscriptstyle \perp} = \varepsilon_{\infty}\boldsymbol{\mathrm{E}}^{\mathrm{tr}}_{\scriptscriptstyle \perp}$ [@Boardman:1982a; @Wei_PRB2012].
The Center for Nanostructured Graphene is sponsored by the Danish National Research Foundation, Project DNRF58. This work was also supported by the Danish Council for Independent Research - Natural Sciences, Project 1323-00087.
Additional information regarding computation of extinction cross-section, EELS probability, and LDOS in the multipole basis along with asymptotics of EELS and LDOS in near-extinction limits. Also provides derivation of the quasistatic multipolar polarizability, associated LSP resonance conditions, Mie–Lorenz transmission coefficients, approximate longitudinal resonance conditions for bulk plasmons, quasistatic LDOS expressions and associated calculations for $R=10\,\mathrm{nm}$ Drude-metal sphere, and supplementary calculations for silver and larger nanospheres.
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abstract: 'The effect of A site disorder on the magnetic field induced melting of charge order (CO) in half doped manganites has been studied using a Monte-Carlo technique. Strong A-site disorder destroys CO even without an applied field. At moderate disorder, the zero field CO state survives but has several intriguing features in its field response. Our spatially resolved results track the broadening of the first order field melting transition due to disorder, explain the unusual dependence of the melting scales on bandwidth and disorder, and allow an unified understanding of CO melting across all manganites. We also present some results on disorder assisted trapping of metastable phases in the low temperature state as seen in recent experiments.'
author:
- Anamitra Mukherjee and Pinaki Majumdar
date: 23 Nov 2008
title: 'A Real Space Description of Field Induced Melting in the Charge Ordered Manganites: II. the Disordered Case'
---
Introduction
============
The half-doped state in the manganites[@mang-book; @tok-rev; @dagotto1; @other-theory] has been thoroughly explored in the recent past. In particular, the magnetic field induced melting of the charge order (CO) has been probed experimentally to map out the hysteretic response, and the bandwidth dependence of the ‘melting’ field in systems with low[@respaud], intermediate[@kuwahara; @tokura-2; @other-exp] and strong[@bicr-prl] disorder. While these results were primarily concerned with the bulk signature of melting, there have also been attempts to probe the spatial nature of field melting, directly[@melt-exp1; @trokiner; @cheong] and indirectly[@melt-exp2; @melt-exp3; @freitas; @parisi].
The main attempts at theoretical understanding have been in the ‘clean’ problem[@kp-am-pm1; @satpathy; @fratini; @cep-hrk1] with only an initial attempt [@short] probing the effect of disorder on field melting. The effort to understand the ‘clean’ problem is understandable because even without disorder there were many issues which have been clarified only recently[@cep-hrk1; @short; @clean-long]. These studies, alongwith an experimental understanding of the systematics of disorder [@atfld1; @tomioka], set the stage for an attempt on the disordered field melting problem.
To put the experimentally observed disorder effects in context, let us recapitulate the physics of the ‘clean’ system. The bandwidth of a manganite is controlled by the mean ionic radius $r_A$. At low $r_A$, the half-doped manganites are insulators with in-plane checkerboard CO, $d_{x^2-r^2}/d_{y^2-r^2}$ orbital order (OO), and CE type magnetic order. We will simply call this the CE-CO-I phase. The large BW materials, with large $r_A$, have a ferromagnetic metallic (FM-M) ground state. These two states, and an ‘A type’ antiferromagnetic state, compete with each other at $x=0.5$.
For CE-CO-I manganites close to the FM-M boundary, the application of a modest field converts the material to a FM-M through a first order phase transition (FOPT). The critical CO melting field were found to be much smaller than the corresponding CO melting temperatures and is understood to be due to closeness of the energies of the competing CE-CO-I and FM-M. The melting field increases with decreasing $r_A$ as one goes further from the FM-M phase. Finally, for very small $r_A$ systems, the effective single electron bandwidth (BW) is so low that the Jahn-Teller (JT) coupling, $\lambda$ say, is able to localize electrons by itself and create a CO state without requiring CE magnetic order. Such charge order is naturally impossible to melt with an applied magnetic field. The applied field at best would convert the CE-CO-I to a FM-CO-I, but not ‘melt’ the charge order. In low disorder systems *e.g.* Ln$_{0.5}$Ca$_{0.5}$MnO$_3$, this broad trend of increasing melting fields with decreasing $r_A$ has been seen[@respaud]. In our calculations[@clean-long], we have shown that at intermediate magnetic fields, the ground state ceases to be a homogeneous FM-M with density $n=0.5$ and instead phase separates (PS). The PS is between FM-M and AFM-M at larger $r_A$, and between FM-M and FM-CO at smaller $r_A$. Further, these coexisting phases are all off $n=0.5$. Thus, the field melted state is at best a percolative metal.
In real life, while low cation mismatch is possible, as in Ln$_{0.5}$Ca$_{0.5}$MnO$_3$, eliminating disorder altogether is difficult. To this end, there is only one attempt to create a disorder free half doped system[@bicr-prl], but it unfortunately doesnot probe the nature of the field melted state. In all other conventional experiments, the disorder, although small is enough to mask the effect above.
This brings us to a point where we need to carefully analyse the effects of weak and strong disorder. First, the origin of A-site disorder. In the process of varying the $r_A$, one inevitably introduces disorder due to cation size mismatch. Disorder makes its appearance in two forms, (i) the randomness introduced in the hopping due to random deviations of the Mn-O-Mn bond angles and, (ii) if the valence state of the dopant is different from the parent material, the dopant locations act as scattering centers. The extent of structural disorder is quantified by the variance, $\sigma_A$, of the distribution of the cation radii in A$_{1-x}$A’$_{x}$BO$_{3}$ and has been measured for the Ca, Sr, and Ba families [@atfld1; @tomioka]. From these measurements it is known that while in the Ca family (Ln$_{0.5}$Ca$_{0.5}$MnO$_3$, where Ln=La, Pr, Nd, Sm, Eu, etc.) $\sigma_A
\sim10^{-3} \AA^2$, for the Sr family (Ln$_{0.5}$Sr$_{0.5}$MnO$_3$) greater size mismatch leads to $\sigma_A \sim 10^{-2}\AA^2 $. For the Ba family disorder is large enough to kill the CE-CO-I state completely even without a magnetic field [@bicr-prl].
In our earlier work[@short; @clean-long] we classified the Ln$_{0.5}$Ca$_{0.5}$MnO$_3$ family as ‘clean’ with a disorder variance $\sigma_A \sim
10^{-3}~A^2$, while the Ln$_{0.5}$Sr$_{0.5}$MnO$_3$ family was classified as disordered with the disorder variance ($\sigma_A \sim 10^{-2}~A^2$). We describe our modelling of disorder in Section III. In our earlier work we compared our results for ideally clean, $\Delta=0$, results with the Ca family and that of moderate disorder with the Sr family. While the $\Delta=0$ results do broadly match with the trends seen in the Ca family (there is a monotonic growth in the critical melting fields $h_c^{\pm}$ with reducing $r_A$) here we demonstate that $\Delta=0$ is not crucial and the same trend can be obtained for a window of low $\Delta$. We also track the disorder induced smearing of the first order transition. Finally, we show that, starting in from $\Delta=0$ and going up to strong disorder, the essential physics can be rationalized by studying the zero field ’bandwidth-disorder’ phase diagram.
One is also interested in the spatial nature of the field melted state. This line of work has been picked up only recently. Experiments on weak and moderately disordered system both at half doping[@melt-exp1; @melt-exp2; @melt-exp3; @trokiner; @freitas] and away, [e.g.]{} LPCMO [@cheong; @parisi], show that disorder inhibits the approach to equilibrium, leading to *non-equilibrium* coexistent states with the phase fractions of the coexistent states being strongly dependent on the path followed to bring the system to its final state[@melt-exp1; @melt-exp2; @melt-exp3]. In the present work we show results on disorder induced trapping of metastable phase fractions. The detailed nature of the nonequilibrium coexistence and its $h-T$ protocol dependence will be reported elsewhere.
The paper is organised as follows. In section II we summarize the key experimental results and in the following section III define the model and describe the method for solving it. Section IV A. discusses the zero field reference state and the effects of disorder on that. Section IV B and C discusses the effects of disorder on field melting.and in sSection IV D, we present our results on disorder assisted trapping of FM-M phase in the low temperature CE-CO-I state. Section V concludes the paper.
Experimental Results
====================
Typically, disorder in manganites ABO$_{3}$ arise because of A or B site substitutions. In this work we consider the effects of A-site disordering on the melting of the CO-state. We have studied the effects of B-site disorder on the CO state at half doping is studied in [@kp-am-pm1].
![Colour online: The (a) clean $ r_A$ variation of CO melting temperatures ($T_{CO}$) and magnetic fields ($h_{CO}^{\pm}$) for the Ca family and (b) disordered for the Sr family. Here, the $\pm$ on $h_{CO}$ imply forward(+) and backward(-) melting fields in a low temperature field sweep experiment. The disorder variance $\Delta$ in (a) is $10^-3$ and that is (b) is $\sim 10^-2$. These plots are reconstructed from experimental data[@respaud; @tokura-2].](Fig1_d.png){width="8.0cm" height="5.0cm"}
Fig.1.(a)-(b) show the key differences between the clean and the disordered results. These plots, reconstructed from experimental data[@respaud; @tokura-2] present the evolution of the critical switching fields and CO melting temperatures with decreasing $r_A$ (increasing BW) for (a) clean ($\Delta=0$) and (b) disordered cases. The $\pm$ on the critical melting fields $h_{CO}$ stand for the critical melting field as obtained in low temperature magnetic field sweeps in the direction of increasing (+) and decreasing (-) field. The switching fields are defined as the magnetic field value where the CO volume fraction switches. This is typically concomitant with a sharp change in the resistivity ($\rho$). The labels on top show the A site dopant and (a) is for the Ln$_{0.5}$Ca$_{0.5}$MnO$_3$ family and (b) is for the Ln$_{0.5}$Sr$_{0.5}$MnO$_3$ family. Increase in $\lambda$ (equivalently decrease in $r_A$ or reduction in BW) for the ’clean’ system Fig1 (a) increases the CO melting temperatures and the switching fields $h_{CO}^{\pm}$.
For a range of $r_A$ 1.36-1.32, while there is a increase in $T_{CO}$ and $h_{CO}^{\pm}$, the corresponding quantities in the disordered case initially rise a bit and then is very strongly suppressed and drops to zero at $r_A\sim 1.32$, which corresponds to Sm$_{0.5}$Sr$_{0.5}$MnO$_3$[@tokura-2]. For materials with larger disorder such as the Ba family, the CO state does not exist even at $h=0$[@bicr-prl].
While these measurements probe the bulk effects of disorder, recent experiments have also probed [@cheong; @trokiner; @melt-exp1; @melt-exp2; @melt-exp3] the effects of disorder on the spatial nature of the melting phenomenon. In particular (Chaddah *et.al.* in their work[@melt-exp2]), Fig.2(a) show that even in the weak disorder samples there is marked non-equilibrium phase coexistence. Here samples of La$_{0.5}$Ca$_{0.5}$MnO$_3$ are field cooled from 320K to 5K at different fields. The sample cooled in higher field has greater fraction of FM-M regions coexisting with AF-I regions. However, as shown, for the 6T field cooled sample, a small increase in temperature to about 80K, after isothermally reducing the applied field to 1T (at 5K), makes the system lose this FM volume fraction. Infact, the magnetization drops to that of the 1 Tesla field cooled sample at the same temperature. While magnetization by itself is not a good probe of amount of FM regions (which domains can be misaligned), the corresponding resistivity shown in Fig.2.(b), too show an increase in the resistivity for the (6 Tesla) field cooled sample at about 80K and essentially coinciding with that of the 1 Tesla field cooled sample at this temperature. These results imply the non-equilibrium nature of the low $T$ state and the tunability of phase volumes depending on $h-T$ protocol.
![Colour online: (a) The variation of magnetization[@melt-exp2] with temperature for samples cooled in 1T and for sample cooled in 6T field. (b) The corresponding resistivities. The red line corresponds to the warming path in 1T field. The multivalued nature of the magnetization and the resistivity is seen in the varied values of these quantities when the systems are cooled in different fields and then isothermally brought to the same $(h,T)$ point. The non-equilibrium nature of the (6T) field cooled system is seen in the fact that the magnetization and the resistivity both approach the 1T field cooled results on increasing temperature to 80K, after decreasing the field(on the 6T sample) to 1T isothermally at low temperature(5K). The thermal hysteresis in the 1T case (in b) is due to the accompanying metallic to insulator first order transition with decreasing temperature.](Fig2_d.png){width="8.0cm" height="4.7cm"}
We will discuss our results in context of the trapping of the FM-M phase in the (CE-CO-I) ground state and its systematics with changing disorder, while we leave the issues of the nonequilibrium nature and the protocol dependence for a later work.
Model and method
================
Model
-----
We consider a two band model for $e_g$ electrons, Hunds coupled to $t_{2g}$ derived core spins, in a two dimensional square lattice. The electrons are also coupled to Jahn-Teller phonons, while the core spins have an AF superexchange coupling between them. These ingredients are all necessary to obtain a CE-CO-OO phase. We include the effect of disorder through an on site potential. $$\begin{aligned}
H &=& \sum_{\langle ij \rangle \sigma}^{\alpha \beta}
t_{\alpha \beta}^{ij}
c^{\dagger}_{i \alpha \sigma} c^{~}_{j \beta \sigma}
+ \sum_i (\epsilon_i -\mu)n_i
~ - J_H\sum_i {\bf S}_i.{\mbox {\boldmath $\sigma$}}_i \cr
&&
+ J_{AF}\sum_{\langle ij \rangle}
{\bf S}_i.{\bf S}_j
- \lambda \sum_i {\bf Q}_i.{\mbox {\boldmath $\tau$}}_i
+ {K \over 2} \sum_i {\bf Q}_i^2
-h\sum_{i}{\bf S}_{i.z}
\nonumber\end{aligned}$$ Here, $c$ and $c^{\dagger}$ are annihilation and creation operators for $e_g$ electrons and $\alpha$, $\beta $ are the two Mn-$e_g$ orbitals $d_{x^2-y^2}$ and $d_{3z^2-r^2}$, labelled $(a)$ and $(b)$ in what follows. $t_{\alpha \beta}^{ij}$ are hopping amplitudes between nearest-neighbor sites with the symmetry dictated form: $t_{a a}^x= t_{a a}^y \equiv t$, $t_{b b}^x= t_{b b}^y
\equiv t/3 $, $t_{a b}^x= t_{b a}^x \equiv -t/\sqrt{3} $, $t_{a b}^y= t_{b a}^y
\equiv t/\sqrt{3} $, where $x$ and $y$ are spatial directions We consider effectively a lattice of Mn ions and treat the alloy disorder due to cationic substitution as a random potential $\epsilon_i$ at the Mn site picked from the distribution $P_A(\epsilon_i)
= {1 \over 2}(\delta(\epsilon_i - \Delta) +
\delta(\epsilon_i + \Delta))$. The $e_g$ electron spin is ${\sigma}^{\mu}_i=
\sum_{\sigma \sigma'}^{\alpha} c^{\dagger}_{i\alpha \sigma}
\Gamma^{\mu}_{\sigma \sigma'}
c_{i \alpha \sigma'}$, where the $\Gamma$’s are Pauli matrices. It is coupled to the $t_{2g}$ spin ${\bf S}_i$ via the Hund’s coupling $J_H$, and we assume $J_H/t \gg 1$. $\lambda$ is the coupling between the JT distortion ${\bf Q}_i = (Q_{ix}, Q_{iz})$ and the orbital pseudospin ${\tau}^{\mu}_i = \sum^{\alpha \beta}_{\sigma}
c^{\dagger}_{i\alpha \sigma}
\Gamma^{\mu}_{\alpha \beta} c_{i\beta \sigma}$, and $K$ is the lattice stiffness. We set $t=1$, $K=1$, and treat the ${\bf Q}_i$ and ${\bf S}_i$ as classical variables. The chemical potential $\mu$ is adjusted so that the electron density remains $n=1/2$ which is also $x= 1-n =1/2$. Note that this modelling of disorder as random fluctuating potentials is based on the fact the while hopping disorder is crucial to the formation of spin-glass phases, it is onsite disorder rather than hopping disorder that plays the dominant role in the kind of physics we are interested in[@sanjeev-kampf].
Method
------
We employ a real space exact diagonalization (ED) based Monte Carlo (MC) technique[@tca] that allows us to perfrom calculations on system sizes of up to $40^{2}$. Details are given in [@clean-long]. Typically results are averaged over 20-30 disorder realizations.
Physical quantities
-------------------
In order to study the evolution of the system with applied magnetic field we track various physical quantities in real space and momentum space. We compute the ‘one point’ distribution of lattice distortions, $P({Q})=\sum_i\delta({Q}-{Q_i} )$, where $Q_i = \vert {\bf Q}_i \vert$, spatial ${Q}_i,{Q}_j$ correlations, $D_Q(\textbf{q})=\sum_{ij} \langle
{\bf Q}_i {\bf Q}_j \rangle
e^{i \textbf{q}.({ \bf r}_i- {\bf r}_j)}$, and spin-spin correlations, $S(\textbf{q})=\sum_{ij} \langle {\bf S}_i.{\bf S}_j \rangle
e^{i \textbf{q}.( {\bf r} _i- {\bf r}_j)}$. Angular brackets represent a thermal average. We also compute the volume fraction of the charge ordered region in the lattice from direct spatial snapshots of the charge distribution. To measure the volume fraction, we tag a site with a particular color if the site has $n > 0.5$ and is surrounded by the four nearest neighbor sites with $\textbf{n} < 0.5$ and vice verse (i.e. a site with local anti-ferro-charge correlation is marked with a particular color). Similarly, if the difference between the charge density at a site with its nearest neighbours is less than a threshold, that site is tagged by a different color, *i.e*., the charge uniform regions are marked by this color. For intermediate cases, we use an interpolative colour scheme. The volume fraction is necessary for studying inhomogeneous melting where the momentum space structure factors are not a good measure of the amount of CO in the system. Further, the spatial snapshots of the real space charge density also directly provide visual information on the melting process. While the indicators above measure the spatial correlations and spatial evolution, the metallic or insulating character is tracked via (low frequency) conductivity, $\sigma_{dc}$, and the density of states (DOS), $N(\omega)= \langle {1 \over N} \sum_n
\delta(\omega-\epsilon_n) \rangle $, where $\epsilon_n$ are the electronic eigenvalues in some MC background and the angular brackets indicate thermal average. We track all the above quantities as a function of temperature and applied magnetic fields for studying the CO melting phenomenon. Results are disorder averaged over 15-20 distinct disorder realizations.
Results
=======
We begin by looking at the effects of disorder without any applied field. We will then look at the combined effects of both disorder and magnetic fields and discuss the fate of the charge-orbital-spin ordered state.
Zero field
----------
We first look at the zero field disorder effects on the charge ordered state by cooling the system in zero field at different disorder strengths for various values of electron-phonon coupling($\lambda$). Our goal here is to probe the effects of disorder on the charge ordered state using bulk indicators like the volume of the CO state($V_{CO}$) indicating the extent of the CO in the ground state, the charge order structure factor, giving the coherence of the CO regions and analysis of spatial snapshots of the charge density fields ($n_i$).
{width="8.0cm" height="4.0cm"}
Figure 3 (a) and (b) show the disorder averaged volume fraction of the charge ordered regions in the ground state as a function of disorder strengths for various $\lambda$ values indicated in (a) and the corresponding $\pi,\pi$ charge order structure factors in (b). Both for small $\lambda \sim 1.45$ and for large $\lambda \sim 2.0$, the volume fractions of the charge ordered regions decrease rapidly with $\Delta$, the disorder strength, while for intermediate $\lambda
\sim 1.6-1.7$, the volume fraction of the CO regions remains robust till $\Delta
\sim 0.12 $, with the $\lambda=1.7$ case having its CO volume fraction suppressed slightly more rapidly than that for $\lambda\sim 1.6$ case. The CO structure factor provides a clearer picture of the way disorder destabilizes the CO state. For weak and strong $\lambda$ cases till about $\Delta=0.1$, while the volume fraction of the CO regions is about $80\%$, the corresponding structure factors diminish rapidly, indicating formation of domain walls between different CO regions. For the intermediate $\lambda$ case, the system presents a single phase CO state till $\lambda\sim 0.12$, beyond which the $\lambda=1.7$ case is more prone to losing the CO state than the $\lambda=1.6$ case. This fact is seen exemplified in the spatial snapshot analysis to which we turn next. Figure 4 shows the snapshots of the CO state ($n_i$) at three values of disorder, from weak, through intermediate, to strong from top to bottom, for the four $\lambda$ values increasing from left to right (see the corresponding caption for the values). Figure 5 shows the snapshots of the corresponding magnetic states.
(i) *Small and large $\lambda$ regimes:* At small disorder, the low $\lambda$ case (the extreme left column) begins developing metallic regions, which are concomitant with local disruption of the CE chains. With increase in disorder, these metallic regions grow along with appearance of line-like AF states. Further increase of disorder to $\sim
0.15$ leads to formation of distinct FM-M regions. The strong coupling case, which essentially consists of almost site localized electrons with weak overlap with neighbouring sites, starts forming clear domain walls at low and intermediate disorder and finally at large disorder, forms a badly disordered polaronic state. The corresponding magnetic state is intially disrupted by formation of local CE regions of opposite handedness, which at large disorder develops patches with local $(\pi,\pi)$ magnetic correlations. The small and larger $\lambda$ states under the effect of disorder can be understood in terms of disorder hindering the growth of the ideal CE-CO-I state from the intermediate temperature states. For low $\lambda$, at intermediate temperature and $\Delta=0$, the system is in a FM-M phase which in absence of disorder gives way to a CE-CO-I state at low temperature. However in the presence of disorder, a fraction of the FM-M phase remains trapped at low temperatures. Similarly, for large $\lambda$ case, since the effective charge order stiffness is small at large $\lambda$ values[@short], the intermediate state is a disordered polaronic phase and it is this state which remains partially trapped due to disorder when the system is cooled to low temperatures.
Since the metallic regions at low T can gain kinetic energy by delocalizing electrons within the regions, double exchange promotes FM order and the system develops FM-M patches, for the low $\lambda$ case. For large $\lambda$ case however, the since the on site localization tendency (which grows as $\lambda^{2}$) is larger than the gain from kinetic energy, the double exchange is suppressed and antiferromagnetic-superexchange dictates a local $(\pi,\pi)$ antiferromagnetic regions existing in a matrix of small CE regions(which exists in regions with local CO).
{width="8.0cm" height="6.0cm"}
(ii) *Intermediate $\lambda$ regime:* Increasing $\lambda$ from weak to intermediate values lead to a deepening of the CE-OO-CO minima in the energy landscape small, where, disorder is unable to trapthe system partly in the FM-M phase (which in a Ginzburg-Landau scenario would no longer remain metastable at low temperatures). Similarly decreasing $\lambda$ from strong couplings increases the effective CO stiffness, thus pushing up, in energy, the disordered polaronic phase, which again for weak and intermediate disorder would be unstable at low temperature.
{width="8.0cm" height="6.0cm"}
The trend in stability of the CO state with $\lambda$ suggests a possible intermediate $\lambda$ where the CE-OO-CO state is most robust to disorder. This corresponds to $\lambda\sim 1.6$, for the parameters used in the present work. Results at $\lambda=1.7$ are specifically shown to demonstrate the weakening of the CO state compared to $\lambda=1.6$.
(iii) *$\lambda-\Delta$ phase diagram:* Fig.6.(a) shows the variation of $T_{CO}$ with $\Delta$ at various $\lambda$. 6.(b) shows the $\lambda-\Delta$ phase diagram. In Fig.6.(a), the low disorder $T_{CO}$ increases with increasing $\lambda$, but for $\Delta\sim 0.1$, the $T_{CO}$ at larger $\lambda$ starts falling and by $\Delta\sim0.12 $, goes below those for weaker $\lambda$ values.
To understand this behaviour, we mapped out the the $\lambda-\Delta$ phase diagram shown in Fig.6.(b). Since at very low $\lambda$ and $\Delta=0$, the system is in a FM-M state, at small disorder, this state will be a disordered FM-M. Thus, the CO region in the $\Delta-\lambda$ parameter space, on the lower $\lambda$ side is bounded by a FM-M at zero disorder and by a disordered FM-M at intermediate disorder. On the other hand, at large $\lambda$ and zero disorder, the system, at T=0 will remain in the CO state at all finite $\lambda$ values (ignoring quantum fluctuations). However, at any finite T, small thermal fluctuations will destroy the long range nature of the charge order, although the electrons will be almost site localized. This will happen due to the essential on site localization of the electrons which leads to suppression of the effective charge order stiffness. By the same token, at any finite $\Delta$, there is always a critical $\lambda$, beyond which the system, even at T=0, turns into a disordered polaronic state.
The asymptotic limit where $\lambda$ is large enough to allow perturbation in kinetic energy can be analytically handled. Infact this very general disorder induced suppression of long range order is well understood in classical models. For the random field Ising model (RFIM), this has been studied in detail[@rfim-ref]. The real regime where the materials exist, $\lambda\sim1.5-1.7$, is non-perturbative and has to be understood as the regime interpolating between the large $\lambda$ asymptote and the small $\lambda<1.4$ FM-M. This naturally causes, as seen in Fig. 6(b), the region for stable CO to be finite at any finite value of $\Delta$ and explains why with increasing $\lambda$, the $T_{CO}$ falls more rapidly at finite disorder. Further, the two kinds of CO regions, CD-I and CD-II, in the Fig. 6(b), refer to disordered FM-M and disordered polaronic regime with AF magnetic background. The dashed lines on Fig.6 (b) represents parameter points along which calculations were done. We shall later use the $\lambda-\Delta$ phase diagram to explain experimental observations on the material systematics of the critical melting fields.
{width="8.0cm" height="4.0cm"}
Disorder induced rounding of first order transition
---------------------------------------------------
When a first order phase transition (FOPT) occurs in a disordered background, the transition ceases to be discontinuous and acquires a smooth character. Here we track the magnetic field induced FOPT between the CE-CO-I and the FM-M as it takes place in the disordered background and compare to what we found in the clean case[@clean-long].
{width="8.0cm" height="8.0cm"}
Fig. 7 and 8 show this evolution of the CO state as we sweep the magnetic field after cooling the system to low temperature. While Fig. 7 tracks measures of bulk properties $V_{CO}$, $S_{fm}$ and $\rho$, as a function of applied field in the clean (a), (c) and disordered (b) ,(d) cases, Fig. 8 looks at the actual spatial profile of the CO melting. The spatial profile in Fig. 8, for the clean system (top panel) is seen to resist the field up to $h/t=0.06$, beyond which it abruptly goes to a percolative FM-M state. The disordered case (bottom panel) shows a more gradual trend in the melting, it starts by creating small metallic regions which grow over a window of field values to reach the final percolative metallic state. This gradual loss of CO volume fraction is corroborated by Fig. 7(b) and the abruptness of the transition on the clean case is correlated with Fig. 7(a). The fact the the systems are metallic is seen from the resistivities shown in Fig. 7(c) and (d) which follow the abruptness of the change in $V_{CO}$ in the clean case and smoothened change in disordered case.
{width="18.0cm" height="6.0cm"}
Field melting $\&$ the relation to the $\lambda-\Delta$ phase diagram
---------------------------------------------------------------------
Here we look at the detailed behaviour of the thermal and magnetic melting scales for the CO state with $\lambda$ and disorder. Since it would also help making comparison with experiments, we restate the broad experimental results. While for the systems with $\sigma_A
\sim 10^{-3}A^2$ (e.g. the Ca family), the magnetic melting scales increase with decreasing $r_A$, for systems with $\sigma_A \sim 10^{-2}A^2$ (e.g. the Sr family), the melting scales initially increases with decreasing $r_A$ and then are strongly suppressed and eventually go to zero with further decrease in $r_A$. These are shown in Fig.1(a) and 1.(b).
We follow the same $h,T$ protocols as in the experiments, [*i.e*]{}, we sweep up and down in $h$ at low $T$ after cooling at $h=0$. In terms of the numerics we start from zero field and then first increase the magnetic field up to $h/t\sim 0.2$ and then reduce to zero in steps of 0.01. As discussed earlier [@clean-long], the forward melting field is defined as $h^{+}_{CO}$ and that in the downward sweep is $h^{-}_{CO}$. Fig.9(a)-(d) show the results of our calculation starting with $\Delta/t=0$ in (a) to strong disorder $\Delta
\sim 0.12$ in (d). We clearly see that apart from the expected overall suppression in melting scales with disorder, there is a gradual downturn with increasing $\lambda$ (or decreasing $r_A$). In (a) the melting field diverges, i.e., the CO state is stable without CE order. In (b) the melting scales become finite in the same $\lambda$ regime and begin to get strongly suppressed in (c) and (d).
To explain the results we refer back to Fig.6(b), the $\lambda-\Delta$ phase diagram. Here we have marked by the red dashed lines the ($\lambda$,$\Delta$) combinations for which we have shown results in Fig.9(a)-(d)(with progressive increase in the values of disorder from a to d). From Fig 6(b) it is clear that while at $\Delta=0$, the system has only one boundary with the CD-1 region, so the melting scales are the smallest at $\lambda\sim 1.45$ and increase with increasing $\lambda$. For all other dashed lines the systems encounter boundaries with CD-1 and CD-2, so there is a generic suppression in the magnetic and thermal melting scales. Thus the presence of disorder causes a competition between the long range order of the CO state and random pinning effects of disorder which weakens the effecting CO stiffness and hence the melting temperatures and melting fields[@short]. This at large $\lambda$ and presence of the FM-M at low $\lambda$ places the CE-CO-I in a small $\lambda$window. While the Ca family wold fall in the low $\Delta$ category, the Sr family would, in the same $\lambda$ regime, be placed at $\Delta\sim0.12$. The Ba family would be beyond $\Delta\sim0.15$. The $\lambda-\Delta$ phase diagram gives a natural framework to rationalize both the clean and disordered field melting systematics in an unified manner.
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Disorder induced trapping of metastable states
----------------------------------------------
We now briefly turn to certain aspects of disorder induced coexistence in the half doped manganites The experimental results as discussed in Section II have three major aspects, one the disorder induced trapping of metastable phase in the low temperature state of the system, two, the non-equilibrium nature of such coexistent states and three, the strong protocol dependence of the phase fractions. Here we present our results only on the first aspect, while we will report the remaining two in a later work.
The results shown in Fig. 10 are obtained by cooling the systems in presence of various fixed magnetic fields at different disorder strengths. All results are on $16^2$ systems and the results are averaged over many disorder realizations. We use two indicators for the study, one, is the volume faction of the CO regions($V_{CO}$) and the other is the measure of excess number of FM bonds in the system over that which exists in the CE phase ($\Delta V_{FM}$). In the CE phase, each site in the plane has four bonds two of which are ferromagnetic and two antiferromagnetic. Thus if we define $\Delta
V_{FM}=\frac{N^{FM}_{Tot}-N^{FM}_{CE}}{N^{FM}_{CE}}$, then in the CE phase $\Delta V_{FM}=0$, in the FM phase it is 1 and is negative if the number of bonds antiferromagnetic bonds are more than the number of ferromagnetic bonds. In the case of G-type AF phase it is -1. Although $\Delta
V_{FM}$ cannot distinguish between two spatial patterns with the same number of AF and FM bonds, it is a useful measure of net FM volume on the system and in the present definition gives the net increase or decrease in the FM volume fraction from that which exists in the CE phase.
{width="8.0cm" height="7.0cm"}
(i) *Effects of magnetic field in a clean systems:* We begin by looking at the effect of magnetic field on $V_{CO}$ when there is no disorder in the system. Fig. 10(a) shows the variation of $V_{CO}$ with applied magnetic fields, at three $\lambda$ values. It was shown in [@clean-long], that the system at half doping phase separates, at finite magnetic fields, into ’FM-M AF-M’ at low $\lambda$ and between ’FM-CO AF-M’ at $ \lambda\geq 1.6$. Here we see that for $\lambda<1.6$ the system at low fields ($h\sim 0.02-0.04$), actually lose almost all the CO regions to a FM-M regions and then regain a certain volume fraction of the CO state at intermediate field values. For $\lambda\geq 1.6$, the system directly goes into a phase separated state, without the low field FM-M phase. Fig. 10(c) shows $\Delta V_{FM}$, for low fields, for $\lambda=1.5$. For $h\sim 0-0.01$, the system resists any significant change in the CE- phase. At $h=0.012$, it converts about $10\%$ of the AF bonds to FM bonds. Further increase of field makes the system lose more AF bonds. However, we will restrict our attention to $h\leq 0.01$ and look at the effects of disorder on the CE phase.
(ii) *Interplay of magnetic field and disorder:* Fig. 10(b) shows $V_{CO}$ as a function of magnetic field at two disorder values for a typical low $\lambda(=1.5)$ case. While beyond $h=0.05$, the phase separation tendency masks the disorder effects, we focus our attention to low field. For $h\leq0.05$, we see that while for those field values where the clean system is in the CO state, increasing $\Delta$ reduces the volume fraction of the CO regions and for those values of $h$ where the clean system would be a FM-M, disorder traps some fraction of the system in the CO state. This clearly shows the generic effect of disorder assisted trapping of the nearby (in energy) metastable phases in the low temperature state of the system. Fig. 10(d) shows the variation of $\Delta
V_{FM}$ with disorder at various magnetic field values. At very low fields $h=0.002$, the system presents no change to the CE phase when there is no disorder, however, in the presence of disorder, the ground state has a certain volume fraction of FM region and this fraction grow to about $5\%$ as the disorder strength $\Delta$ grows from 0 to 0.2. At larger $\Delta \sim 0.3$ the system doesnot have any clear order due to strong random pinning effects of disorder. Further, increasing magnetic field and disorder allows one to achieve a continuum of values for $\Delta V_{FM}$, which together with the results in Fig. 10(b), shows the tunability of the CE-CO and FM-M volume fractions by cooling in different applied fields[@melt-exp2].
Conclusions
===========
Using a real space Monte-Carlo scheme we have shown how the charge ordered state in the half-doped manganites is affected by the presence of disorder and an applied magnetic field. We illustrated the disorder induced broadening of the melting transition and mapped out the full $h-T$ phase diagram for various combinations of bandwidth and disorder. We could explain the counterintuitive bandwidth dependence seen in the Sr based manganites. We motivated the trends seen in the detailed numerical results by analysing the stability of the $h=0$ charge ordered state in response to disorder. Finally, we observe that disorder leads to trapping of metastable phases in the low temperature state. Different field cooling pathways will trap different amounts of FM-M in the nominal ‘CE-CO-I’ ground state.
We acknowledge use of the Beowulf cluster at HRI.
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---
abstract: 'The internal motions of DNA immersed in bio-fluid are investigated. The interactions between the fragments of DNA and the surrounding bio-fluid are modeled using the gauge fluid lagrangian. In the model, the bio-fluid is coupled to the standard gauge invariant bosonic lagrangian describing the DNA. It is shown that at non-relativistic limit various equation of motions, from the well-known Sine-Gordon equation to the simultaneous nonlinear equations, can be constructed within a single framework. The effects of bio-fluid are investigated for two cases : single and double stranded DNA. It is argued that the small and large amplitudes of a single stranded DNA motion immersed in bio-fluid can be explained in a natural way within the model as a solitonic wave regardless with the fluid velocity. In contrary the double stranded DNA behaves as regular or damped harmonic oscillator and is highly depending on the fluid velocity.'
author:
- 'A. Sulaiman$^{a,b}$[^1] and L.T. Handoko$^{c,d}$[^2]'
title: 'The effects of bio-fluid on the internal motion of DNA'
---
$^{a)}$Department of Physics, Bandung Institute of Technology[^3], Jl. Ganesha 10, Bandung 40132, Indonesia\
$^{b)}$P3 TISDA BPPT[^4], BPPT Bld. II (19$^{\rm th}$ floor), Jl. M.H. Thamrin 8, Jakarta 10340, Indonesia\
$^{c)}$Group for Theoretical and Computational Physics, Research Center for Physics, Indonesian Institute of Sciences[^5], Kompleks Puspiptek Serpong, Tangerang 15310, Indonesia\
$^{d)}$Department of Physics, University of Indonesia[^6], Kampus UI Depok, Depok 16424, Indonesia\
Keywords : elementary biomatter; biomatter structure; biomatter interaction; DNA; modeling
Introduction
============
Both deoxyribo- and ribo-nucleic acid (DNA and RNA) have been recognized as the most important biomolecules. Especially DNA helical structures undergo a very complex dynamics which plays several important roles in various biological phenomena such as storage of information, inheritance (replication, etc) and the usage of genetic information (transcription, etc). The importance of biopolymers like DNA/RNA is motivated by established observations that the homologous recombination is preceded by recognition and local pairing of intact double stranded DNA fragments, rather than involving known recombination proteins. Therefore, it should be attributed to direct DNA-DNA interactions whose physical origin has not been understood [@burgess; @weiner]. Experimentally, the physical properties of DNA/RNA have been measured in many works, for example : the DNA single-molecule [@smith; @lavery; @strick], double stranded DNA forming bubbles [@altan], the DNA/RNA nucleoside and nucleotides [@peon], the structural transitions of DNA through torques measurements [@bryant], the thermodynamic fluctuations of DNA in a reacting system [@magde], the stretching DNA with a receding meniscus [@bensimon], the electrical transport through single DNA molecules [@porath] and so forth.
From physical point of view, a biopolymer like DNA molecule is considered as a system consisting of many interacting matters in a particular configuration of space-time. Some models treats this kind of DNA dynamics as the phenomena of nonlinear excitations like soliton. This type of models has been pioneered by Englander et.al. using nonlinear dynamics relevant to the transcription process in terms of coupled pendulum chain which generates the sine-Gordon equation and its classical solitons [@englander]. Further, Davydov described the alpha helices in quantum solitons [@davydov]. Following these suggestions, a number of models for the nonlinear DNA have been elaborated in the last decades, in both classical or quantum approaches [@yakushevich; @peyrard; @cardoni]. A typical classical approach is the so-called PDB model which takes into account twisted DNA molecules [@peyrard2; @dauxois; @dauxois2]. On the other hand, there are several models based on the particle interactions [@lee; @knotts; @sulaiman]. Also, the polyelectrolyte model which treats DNA molecule as a cylinder with a net charge homogeneously distributed along its surface, and has further been modified to be the electrostatic zipper motif for DNA aggregation [@kornyshev], to solve high dependency of the electrostatic interaction between DNA duplexer on surface charge patterns [@kornyshev2].
It has also been shown that under particular external conditions the DNA molecules form a double helix, and its (transverse, longitudinal and torsional) motions can be divided into two main regions : the small and large amplitude of internal motions [@yakushevich2]. The small amplitude of motion can be described by the hamiltonian of harmonic oscillator. On the other hand, the large amplitude is described by a non-harmonic one [@mingalev]. Recently, many works have discussed and arrived at the conclusion that the large amplitude of internal motion can be considered as a nonlinear dynamical system where solitary conformational waves can be excited [@yakushevich]. Then nonlinear interaction between molecules in DNA gives rise to a very stable excitation as soliton [@mingalev; @cardoni2].
As mentioned above, DNA is not motionless. It is in a constantly wriggling dynamics state in a medium of bio-organic fluid in the nucleus cell [@julia]. However, the motion of DNA surrounded by fluid is rarely studied. Previous studies are usually done by solving the fluid equations and its wave equations simultaneously using appropriate boundary conditions. On the other hand, in the Hamiltonian formulation the viscous force is considered to be comparable with other forces arising from Hamiltonian [@kovici; @aslin]. The solution is then obtained by expansion and performing order-by-order calculation. In these approaches, anyway the picture of interaction between DNA and its surrounding fluid is not clear. Also, in most models the over-damped DNA dynamics are treated by putting some additional terms by hand in the differential equation to obtain the non-homogenous ones [@irwin]. The stochastic simulations of DNA in flow has been done for a fully parametrized bead–spring chain model by taking into account the fluctuating hydrodynamic interactions [@richard].
In this paper, a new model to describe various internal motions of DNA inspired by gauge fluid theory is proposed. The DNA dynamics is modeled as the result of interactions among matters in a fluid medium using the relativistic and gauge invariant fluid lagrangian. Although the theory is a relativistic one, we take its non-relativistic limit at the final stage to deal with problems in DNA as done in some previous works, for instance in some models using the ideal gas approximation [@fedyanin]. Moreover, the lagrangian is intended for physics at scale of order transport mean free paths, that is the transition region where neither a hydrodynamics nor kinetic theory is valid. Therefore it fits the current interest of modeling “elementary” biomatters like DNA. Just to mention, the lagrangian is originally devoted to model the quark gluon plasma (QGP) as a relativistic fluid system [@mahajan1; @handoko2; @mahajan2; @handoko], inspired by the similarity between the dynamical properties of fluid and electromagnetic field [@marmanis; @marmanis2]. The DNA is treated as strongly coupled system like non-Abelian plasmas where neither a hydrodynamics nor kinetic theory is really valid. Within the model, a single and double stranded biopolymers are described in a general way as the results of interactions among the fluid and matter fields.
We show in two specific cases how to derive the equation of motion (EOM) and investigate the internal motions through its solutions and behaviors. From the EOM of a DNA as a single bulk, we argue that small and large amplitude regions of the internal motion of DNA are determined by its internal dynamics and interactions with surrounding fluid. On the other hand, in the case of double stranded DNA the EOM is solved analytically to investigate the effects of fluid velocity to its internal motion.
The paper is organized as follows. First we briefly introduce the theory of gauge invariant fluid lagrangian, and then provide the allowed interactions within the model. After explaining how to model DNA using the interactions in the lagrangian, we provide two typical examples : 1) the Abelian U($1$) case to model the dynamics of a single bulk of DNA, and 2) the non-Abelian SU($2$) case to describe the internal motion of double stranded DNA. Finally, the paper is ended with summary and discussion.
Theoretical background
======================
Here, a new approach to investigate the interaction between biopolymer and its surrounding bio-fluid is discussed using the lagrangian method. Rather putting it by hand, the interaction is described in a more natural way from first principle, by introducing some symmetries in the lagrangian under consideration.
The lagrangian
--------------
The model is an extension of the original model based on the U($1$) gauge theory devoted for QGP as a magnetofluid system [@mahajan1; @handoko2; @mahajan2]. Thereafter it has been extended to the non-Abelian case to accommodate a system with many matters, either bosonic or fermionic ones [@handoko]. Concerning the fact that an (elementary) matter has no intrinsic degree of freedom like spin, it is considerable to represent its elementary constituents as scalar (boson) fields governed by the bosonic lagrangian, ł\_ = ( \_)\^( \^) + V() , \[eq:lphi\] where $V(\Phi)$ is the potential. For example in the typical $\Phi^4-$theory, V() = - m\_\^2 \^- (\^)\^2 , \[eq:v\] where $m_\Phi$ and $\lambda$ are the mass of matter and the dimensionless coupling constant of matter self-interaction. The hermite conjugate is $\Phi^\dagger \equiv {(\Phi^\ast)}^T$ for a general complex field $\Phi$.
We impose the above bosonic lagrangian to be gauge invariant under local (in general non-Abelian) gauge transformation [@yang; @mills], $U \equiv \exp[-i T^a \theta^a(x)] \approx 1 - i T^a \theta^a(x)$ with $\theta^a \ll 1$. $T^a$’s are generators belong to a particular Lie group and satisfy certain commutation relation $[T^a,T^b] = i f^{abc} T^c$ with $f^{abc}$ is the anti-symmetric structure constant [@chengli]. The matter field is then transformed as $\Phi \stackrel{U}{\longrightarrow} \Phi^\prime \equiv \exp[-i T^a \theta^a(x)] \, \Phi$, with $T^a$ are $n \times n$ matrices while $\Phi$ is an $n \times 1$ multiplet containing $n$ elements, = (
[c]{} \_1\
\_2\
\
\_n\
) \^T = (\_1 \_2 \_n) , \[eq:multiplet\] for $n$ dimension Lie groups as SU($n$), O($n+1$), etc. It is well-known that the symmetry in Eq. (\[eq:lphi\]) is revealed by introducing gauge fields $A_\mu^a$ which are transformed as $U^a_\mu \stackrel{U}{\longrightarrow} {U^a_\mu}^\prime \equiv U^a_\mu - \frac{1}{g} (\pd_\mu \theta^a) + f^{abc} \theta^b U^c_\mu$, and replacing the derivative with the covariant one, $\cd_\mu \equiv \pd_\mu + i g \, T^a U^a_\mu$. Anyway, the number of generators, and also gauge bosons, is determined by the dimension of group under consideration. For an SU($n$) group one has $n^2 - 1$ generators and the index $a$ runs over $1, 2, \cdots, n^2 - 1$. For example the SU(2) group is realized by $2 \times 2$ matrices $T^a \equiv \frac{1}{2} \sigma^a$ with $\sigma^a$ are the Pauli matrices [@chengli], \^1 = (
[cc]{} 0 & 1\
1 & 0\
) , \^2 = (
[cc]{} 0 & -i\
i & 0\
) , \^3 = (
[cc]{} 1 & 0\
0 & -1\
) , \[eq:pm1\] In particular, the Abelian U($1$) case is revealed by putting $T^a \theta^a(x) \rightarrow \theta(x)$, the phase transformation, respectively.
Finally, the gauge invariance leads to the total lagrangian with some additional terms in the lagrangian to keep its gauge invariance, ł= ł\_ + ł\_ + ł\_ , \[eq:l\] where, ł\_ & = & - S\^a\_ [S\^a]{}\^ , \[eq:la\]\
ł\_ & = & -g J\^a\_\^+ g\^2 ( \^T\^a T\^b ) U\_\^a [U\^b]{}\^ . \[eq:li\] The strength tensor is $S^a_{\mu\nu} \equiv \pd_\mu U^a_\nu - \pd_\nu U^a_\mu + g f^{abc} U^b_\mu U^c_\nu$, while the 4-vector current is, J\^a\_= -i . \[eq:j\] The coupling constant $g$ then represents the interaction strength between gauge field and matter. We should note that, the current conservation is realized by the covariant current $\pd^\mu \cj^a_\mu = 0$ with $\cj^a_\mu \equiv -i \left[ (\cd_\mu \Phi)^\dagger T^a \Phi - \Phi^\dagger T^a (\cd_\mu \Phi) \right]$ [@handoko].
The gauge boson $U_\mu$ is interpreted as a “fluid field” with velocity $u_\mu$, and takes the form [@mahajan1; @handoko2; @mahajan2; @handoko], U\^a\_= ( U\_0\^a, u\^a ) u\^a\_ , \[eq:a\] with, u\_\^a (1, -v\^a) , \[eq:ae\] where $\phi$ is an auxiliary boson field, while $\gamma^a \equiv \left( 1 -
|\v^a|^2 \right)^{-1/2}$. Here we adopt the natural unit, the light speed $c
= 1$. Eq. (\[eq:ae\]) is nothing else than rewriting a gauge field in terms of its polarization vector and wave function which represents the fluid distribution in a system. It has further been shown that the non-relativistic fluid equation can be reproduced using Eq. (\[eq:a\]) [@handoko2; @handoko]. This fact actually justifies us to model the DNA dynamics in a fluid medium using the total lagrangian in Eq. (\[eq:l\]).
Now we are ready to model the DNA using the above lagrangian. First, we should investigate the allowed interactions in the present theory.
The interactions
----------------
In order to be specific, let us consider the $\Phi^4-$potential in Eq. (\[eq:v\]) for matter lagrangian in Eq. (\[eq:lphi\]). With a complete lagrangian at hand, we can extract $m-$point interactions for fluid and matter with $m$ is the number of relevant legs involved in an interaction. We list all allowed interactions below for each element in the matter multiplet denoted by the indices $i,j$.
- $2-$point interactions :\
The interactions arise through the kinetic and mass terms of matter in Eqs. (\[eq:lphi\]) and (\[eq:v\]), and the fluid kinetic term in Eq. (\[eq:la\]), & : & ( \_\_i\^)( \^\_i) - m\_\^2 \^\_i \_i . \[eq:2p\]\
UU & : & - ( \_U\^a\_- \_U\^a\_)( \^\^- \^\^) . \[eq:2a\]
- $3-$point interactions :\
These interactions are induced by the fluid self-interaction in Eq. (\[eq:la\]) and the fluid-matter interaction in Eq. (\[eq:li\]), U & : & i g T\^a\_[ij]{} \^ . \[eq:3pa\]\
UUU & : & g f\^[abc]{} [U\^b]{}\^\^( \_U\^a\_- \_U\^a\_) , \[eq:3a\]
- $4-$point interactions :\
These interactions are induced through the matter self-interaction in Eq. (\[eq:lphi\]), the fluid kinetic term in Eq. (\[eq:la\]) and the fluid-matter interaction in Eq. (\[eq:li\]), & : & - ( \_i\^\_i )\^2 , \[eq:4p\]\
UUUU & : & - g\^2 f\^[abc]{} f\^[ade]{} [U\^b]{}\^\^U\^d\_U\^e\_ , \[eq:4a\]\
UU & : & g\^2 \^\_i ( T\^a T\^b )\_[ij]{} \_j U\_\^a [U\^b]{}\^ . \[eq:4pa\]
All of these constitute the so-called Feynman diagrams and its order of magnitudes that will be used soon in the subsequent section. Now we are ready to construct the models relevant for biopolymers.
Modeling the DNA
================
Here, we consider two typical examples on how to describe various dynamics of DNA within the present model. First we present a model for a single bulk of DNA or a fragment of DNA molecule like nucleotide or nucleoside. Further we construct a more complicated picture for the double stranded DNA. The model is a new type of the mesoscale model of DNA that reduces the complexity of a nucleotide to three interactions sites [@knotts].
Single bulk of DNA : the Abelian U($1$) model {#subsec:u1}
---------------------------------------------
The Abelian U($1$) lagrangian involves only a single matter and a fluid field. In this case, the $3-$point interaction in Eq. (\[eq:3a\]) and the $4-$point interaction in Eq. (\[eq:4a\]) vanish. It is also clear that we are not able to construct a realistic model for a biopolymer composed by several different matters in this case [@sulaiman]. However, we can model the dynamics of a single bulk of DNA or its fragment like nucleoside which could be considered as a composite field of sugar and base. This means we investigate the internal dynamics of namely DNA molecules through the EOM of its fragments and study the basic behaviors.
The total lagrangian in this case becomes, ł& = & ( \_\^) ( \^) - m\_\^2 \^- (\^)\^2 + g\^2 U\_U\^ \^\
& & - ( \_U\_- \_U\_)( \^\^- \^\^) + i g U\^ , \[eq:lu1\] using Eqs. (\[eq:v\]) and (\[eq:l\])$\sim$(\[eq:j\]). Imposing the variational principle of action and the Euler-Lagrange equation in term of $\Phi$ [@chengli], - \_ = 0 , \[eq:ele\] we find the EOM for a single matter as follow, ( \^2 + m\_\^2 + 2 g\^2 U\^2 ) + \^3 = 0 . \[eq:eomu1\] for a real $\Phi$ field.
This result leads to a solitonic wave equation for $\lambda \ne 0$ described by the well-known nonlinear Klein-Gordon equation, ( \^2 + \_\^2 ) + \^3 = 0 , \[eq:nlkge\] with $\mb_\Phi^2 \equiv m_\Phi^2 + 2 g^2 \, U^2$, and $U^2 = \phi^2$ from Eqs. (\[eq:a\]) and (\[eq:ae\]). Here $\lambda$ determines the ’level of non-linearity’ for the Klein-Gordon equation. If one puts $\lambda \approx \mb_\Phi^2$, we arrive at the sine-Gordon equation in $4-$dimensional space-time $(t,\x)$, $\partial_t^2 \Phi - \partial^2_\x\Phi - \mb_\Phi^2 \sin \Phi = 0$ using $\sin \Phi \approx \Phi - \frac{1}{3!} \Phi^3 + \cdots$. This kind of equation often appears in the models based on the coupled pendulum chains pioneered by Englander et.al. [@englander]. However, we should note that the equality $\lambda \approx \mb_\Phi^2$ in this model doesn’t make sense since $\lambda$ and $\mb_\Phi^2$ have different dimensions. In this paper, rather than considering that special case, let us solve Eq. (\[eq:nlkge\]) in a general way.
For the sake of simplicity, we consider a traveling wave in $2-$dimensional space-time $(t,x)$, $\Phi(\xp) \equiv \Phi(x-Ct)$, where $C$ is a phase velocity. Since $\partial^2_t\Phi = C^2 \partial^2_\xp \Phi$ and $\partial^2_x\Phi = \partial^2_\xp \Phi$, Eq. (\[eq:nlkge\]) can be rewritten as, \^2\_+ \_\^2 + \^3 = 0 , \[eq:nlkge2\] with $\mt_\Phi^2 \equiv {\mb_\Phi^2}/{(C^2 - 1)}$ and $\lt \equiv \lambda/{[3!(C^2 - 1)]}$. Assuming that $v_x = v$ is a constant makes $\mt_\Phi$ to also be a constant. Hence we can multiply both sides of Eq. (\[eq:nlkge2\]) with $\partial_\xp \Phi$ to obtain, \_= 0 . \[eq:nlkge3\] Concerning that the quantum wave function $\Phi$ has the Gaussian distribution, it is integrable and then leads to the following differential equation, ( \_)\^2 + \_\^2 \^2 + \^4 = 0 . \[eq:nlkge4\] Through standard mathematical procedures, we can straightforwardly get the solution, (x\^) = |\_| ( |\_| x\^) , \[eq:soliton\] for $\lt > 0$, or $|C| > 1$.
The non-relativistic limit can be obtained by performing a transformation $t \rightarrow \tau \equiv i t$ in Eq. (\[eq:nlkge\]), and putting $\gamma \rightarrow 1$ respectively. This leads to the same result as Eq. (\[eq:soliton\]), but $t$ is replaced with $-i \tau$. The behavior of this solitonic wave function is depicted in Fig. \[fig:soliton\] as a function of $x^\prime$ with (solid line) and without (dashed line) surrounding fluid for a fixed parameter set. Anyway, the fluid contribution is independent on its velocity $v$, since the effective mass $m_\Phi^2$ is shifted by $U^2 = \phi^2$. From the figure, we can conclude that the large and small amplitudes can be considered as the effects of fluid surrounding the DNA.
![The solitonic wave function for a 2-dimensional DNA as a function of $x^\prime$ with the coupling constants $g = 0.1$ (solid line) and $g = 0$ (dashed line) for a fixed parameter set $(m_\Phi,\phi,C,\lambda) = (1,1,2,4)$.[]{data-label="fig:soliton"}](nucleotide.eps){width="11cm"}
Double stranded DNA : the non-Abelian SU($2$) model {#subsec:su2}
---------------------------------------------------
Now let us apply the present lagrangian in a more realistic case of double stranded DNA. Concerning the smallest group beyond U($1$), we take the SU($2$) group to construct the model. In this group, we have 2 sub-matters in a doublet of matter field as Eq. (\[eq:multiplet\]) with $n = 2$.
Since we have only 2 different states of matter, $\Phi_1$ and $\Phi_2$, it is convincing to split the nucleotide to be a phosphate and a nucleoside consisting of sugar and base. So, the interaction between two nucleotides, which further form the backbone of DNA molecule, is attributed to the interaction of two different matters, phosphate and nucleoside. On the other hand, the base pair is revealed as the interaction between two identical matters, two neighboring nucleosides belonging to different strands. The model is schematically illustrated in Fig. \[fig:model\] where we have assigned $\Phi_1$ for the nucleosides and $\Phi_2$ for the phosphates. Following the allowed interactions in Eqs. (\[eq:2p\])$\sim$(\[eq:4pa\]), we can easily estimate the order of magnitudes for each interaction relevant to Fig. \[fig:model\] as listed in Fig. \[fig:fd\].
From Fig. \[fig:fd\], it is straightforward to deduce that $I_1$ bound is materialized by vertex $A$, while vertex $B$ is responsible for $I_2$ and $I_3$ bounds. Anyway, we should note that there are another diagrams with spring loops in the vertices $A$ and $B$, however they would be vanishing due to the anti-symmetric $f^{abc}$. Now, we unfortunately face a problem on distinguishing $I_2$ with $I_3$ in Fig. \[fig:model\]. It is quite natural to consider $I_3$ must be larger than $I_2$, since the backbone should be rather strongly tied and rigid than the nucleotide. Therefore in order to resolve this problem we propose an additional contribution to $I_3$ coming from interacting fluid (either fluid absorption or emission) with matters depicted in vertex $D$ of Fig. \[fig:fd\]. Of course, so $I_1$ could receive additional contribution from vertex $C$ too. This scenario could be understood in the following way. Since the backbone is more open to surrounding fluid than the phosphate$-$nucleoside encaged in the nucleotide bound-state, its interaction with surrounding fluid would contributes more significantly, and then should be taken into account.
![The double stranded DNA in the non-Abelian SU($2$) model with nucleosides and phosphates are represented by $\Phi_1$ and $\Phi_2$. The vertices $I_1$, $I_2$ and $I_3$ denote different types of interactions connecting nucleosides ($\Phi_1-\Phi_1$) manifesting base pairs in neighboring strands, nucleoside$-$phosphate ($\Phi_1-\Phi_2$) within a nucleotide, and nucleoside$-$phosphate ($\Phi_1-\Phi_2$) between nucleotides in a strand.[]{data-label="fig:model"}](biopolymer.eps){width="12cm"}
![The $2-$point interactions and its first order contents relevant for double stranded DNA in Fig. \[fig:model\] and its order of magnitudes. The plain and spring lines indicate matter and fluid fields.[]{data-label="fig:fd"}](interaction.eps){width="15cm"}
We might remark that in the present case the nucleoside, consisting of sugar and base, should be considered as a well-confined bound-state. So we are not going into insight to investigate its structure. In consequence of this, we can not distinguish the A$-$T (adenine$-$thymine) with the G$-$C (guanine$-$cytosine) base pairs. Although in principle, these might be explained using multi-loops gauge boson exchanges inside nucleosides, and two different base pairs are attributed to the fluid velocities in the fluid loops (the second diagram of vertex $A$ in Fig. \[fig:fd\]) with opposite sign. However we postpone this point in this paper since it would require larger group like SU($3$) containing more matter fields. Anyway, the opposite torsional motions of neighboring strands forming a DNA molecule can be explained, at time being, qualitatively by considering the surrounding fluid in both strands have the same velocities ($\v$) but with opposite sign each other.
Now, we investigate the EOM in SU($2$) as done in Sec. \[subsec:u1\]. Substituting the full lagrangian, Eqs. (\[eq:l\])$\sim$(\[eq:li\]), into Eq. (\[eq:ele\]), we obtain for each element of matter, \^2 \_1 + m\_\^2 \_1 - 2 g ( \_U\_2\^) \_2 + ( \_1\^2 + \_2\^2 ) \_1 - 4 g U\_2\^( \_\_2 ) & = & 0 , \[eq:eom1\]\
\^2 \_2 + m\_\^2 \_2 + 2 g ( \_U\_2\^) \_1 + ( \_1\^2 + \_2\^2 ) \_2 + 4 g U\_2\^( \_\_1 ) & = & 0 , \[eq:eom2\] for real fields $\Phi_i$ ($i : 1,2$). Eqs. (\[eq:pm1\]), (\[eq:eom1\]) and (\[eq:eom2\]) immediately yield the following EOM, ( \^2 + m\_\^2 -4 i g \_2 U\_2\^\_) + ( \^T ) = 0 , \[eq:eomsu2\] for constant fluid velocity and $\phi$. Comparing this result with Eq. (\[eq:nlkge\]), contribution from the interacting fluid medium also appears in the third term but it contributes differently. Using Eqs. (\[eq:a\]) and (\[eq:ae\]) we arrive at non-relativistic limit, \_\^2 + \_\^2 - m\_\^2 + 4 g \_2 ( \_i v\_) - ( \^T ) = 0 , \[eq:nreomsu2\] for $\v_2 = \v$. This is the nonlinear EOM governing the double stranded DNA dynamics with surrounding fluid medium in the present theory. The plus and minus signs show the dynamics of a strand and its counterpart surrounded by the fluids with opposite velocities.
Obviously, in contrast with the U(1) case it is hard to solve Eq. (\[eq:nreomsu2\]) exactly. For the sake of simplification, let us consider 2-dimensional $(t,x)$ case of Eq. (\[eq:eomsu2\]), - + \_t - \_x + m\_\^2 + \^3 =0 , \[eq:twostrain1\] where $\alpha_t \equiv -4 i g \sigma_2 \gamma \, \phi$ and $\alpha_x = 4 i g \sigma_2 \gamma \, v_x \, \phi$. Borrowing the traveling wave $\Phi(\xp) \equiv \Phi(x-Ct)$ as before we obtain, - + \_\^2 + \^3 =0 , \[eq:twostrain4\] with $\at \equiv {(C \alpha_t + \alpha_x)}/{(C^2 - 1)}$, $\mt_\Phi^2 \equiv {m_\Phi^2}/{(C^2 - 1)}$ and $\lt \equiv {\lambda}/{(3!(C^2 - 1))}$. For $\lt = 0$ it coincides with the equation of inharmonic oscillator, Eq. (\[eq:soliton\]).
For further simplification, we assume that $\lt$ is small enough such that the last term in Eq. (\[eq:twostrain4\]) can be treated perturbatively. Then, we can expand the mass $\mt_\Phi$ in term of $\lt$, $\mt_\Phi \simeq \mt_{\Phi_0} + \lt \mt_{\Phi_1}$, and $\Phi \simeq \Phi_0 + \lt \Phi_1$ up to $O(\lt)$ accuracy. Now we are ready to solve Eq. (\[eq:twostrain4\]) order by order.
The lowest order with respect to $\lt$, $O(1)$, satisfies the following equation, - + \_[\_0]{}\^2 \_0 = 0 . \[eq:twostrain7\] Following the standard mathematical procedures the solution can in general be expressed in the form of $\Phi_0 = N_0^+ \mathrm{e}^{k_+ \xp} + N_0^- \mathrm{e}^{k_- \xp}$ with $k_\pm = \frac{1}{2} \left( \at \pm \sqrt{\at^2 - 4 \mt_{\Phi_0}^2}\right)$. Therefore, the solution of Eq. (\[eq:twostrain7\]) is simply, \_0 = [2 N\_0 \^[ ]{} ]{} {
[lcl]{} 1 & & \^2 = 4 \_[\_0]{}\^2\
[( )]{} & & \^2 > 4 \_[\_0]{}\^2\
[( )]{} & & \^2 < 4 \_[\_0]{}\^2\
. , \[eq:solusi2\] by putting the normalization factor to be $N_0^+ = N_0^- \equiv N_0$. Each solution is corresponding to the over-damped, damped and regular harmonic oscillators respectively.
Subsequently, the next order, $O(\lt)$, is governed by the equation, - + \_[\_0]{}\^2 \_1 + 2 \_[\_0]{} \_[\_1]{} \_0 + \_0\^3 = 0 . \[eq:twostrain8\] The over-damped $\Phi_0$ in Eq. (\[eq:solusi2\]) yields the general solution for $\Phi_1$ should be, \_1 = N\_1\^+ \^[ ]{} + N\_1\^- \^[ ]{} . \[eq:solusis2od1\] Substituting the over-damped $\Phi_0$ and Eq. (\[eq:solusis2od1\]) into Eq. (\[eq:twostrain8\]), we obtain $N_1^- = - {\left( 32 N_0^3 \right)}/{\left( 3 \at^2 + 4 \mt_{\Phi_0}^2\right)}$. In the present case the first term in Eq. (\[eq:solusis2od1\]) is vanishing for any $N_1^+$ since $\at^2 = 4 \mt_{\Phi_0}^2$. This also leads to the result $\mt_{\Phi_1} = 0$ since $\mt_{\Phi_0}, N_0 \neq 0$. Finally, \_1 = - \^[ ]{} , and $\mt_\Phi = \mt_{\Phi_0}$.
In the second case of damped $\Phi_0$, we make use of the equality $\cosh^3 x = 1/2 \cosh(3x) + \cosh x$ to obtain, & &\
+ [4 N\_0\^3 \^[ ]{} ( 2 + \^[-]{} ) ( )]{} & = & 0 . \[eq:solusis22\] Since $\cosh x < \cosh(3x)$ and the last term is enhanced only by a factor of as small as 2, Eq. (\[eq:solusis22\]) can be approximately reduced to be, 0 . \[eq:solusis23\] The solution is given by, \_1 = \^[ ]{} . \[eq:solusis24\] Again substituting it into Eq. (\[eq:solusis23\]) yields, N\_1\^+ & = & - N\_0\^3 ,\
N\_1\^- & = & - N\_0\^3 . From these results, for $\xp > 0$ we can safely omit the sub-dominant $\sinh$ term in Eq. (\[eq:solusis24\]), also because $N_1^+ > 2 N_1^-$ since $\at^2 > 4 \mt_{\Phi_0}^2$. Hence, \_1 = - N\_0\^3 \^[ ]{} (3 ) . \[eq:solusis25\]
![The wave function for 2-dimensional double stranded DNA as functions of $x^\prime$ with $v_x = 1.9$ (solid line) and $v_x = 1.7$ (dashed line) for a fixed parameter sets $(m_{\Phi_0},\phi,C,\lambda,g,N_0) = (1,1,2,4,1,1)$.[]{data-label="fig:su2"}](biopolimer.eps){width="11cm"}
The last case of regular harmonic oscillator is governed by the following equation, & &\
+ [2 N\_0\^3 \^[ ]{} ( 3 + \^[-]{} ) ( )]{} & = & 0 , \[eq:solusiho1\] using the relation $4 \sin^3 x = 3 \sin x - \sin (3x)$. In contrary with previous cases the general solution for Eq. (\[eq:solusiho1\]) is complicated. So, let us assume here that the more rapid oscillation term, the fourth term, is dominant than the last one which reduces the equation to be, = 0 , \[eq:solusiho2\] Hence the general solution is simply, \_1 = \^[ ]{} . \[eq:solusiho3\] Following similar procedures as before, N\_1\^+ & = & N\_0\^3 ,\
N\_1\^- & = & N\_0\^3 . In non-relativistic case, by definition the condition $m_{\Phi_0} > \left| 2 g (C - v_x) \phi \right|$ should be fulfilled. Obviously, for large enough $\mt_{\Phi_0}$ ($\at^2 \ll 4 \mt_{\Phi_0}^2$ or $m_{\Phi_0} \gg \left| 2 g (C - v_x) \phi \right|$) the solution is dominated by $N_1^-$ term, while both terms are comparable for $\at^2 \rightarrow 4 \mt_{\Phi_0}^2$ or $m_{\Phi_0} \rightarrow \left| 2 g (C - v_x) \phi \right|$. These arguments lead to the result, \_1 = N\_0\^3 \^[ ]{} (3 ) . \[eq:solusiho4\]
We should remark that up to the current accuracy there is no need in all cases to calculate the leading order of mass, $\mt_{\Phi_1}$. As a typical example, the wave function for the harmonically oscillating, the sum of Eqs. (\[eq:solusi2\]) and (\[eq:solusiho4\]), double stranded DNA is depicted in Fig. \[fig:su2\] as a function of $x^\prime$ for certain velocities. It can also be seen that the oscillation is sensitive to the fluid velocity.
Summary and discussion
======================
We have introduced a new type of model to describe DNA using the gauge invariant fluid lagrangian. The lagrangian is able to accomodate various internal motions of DNA, from the single bulk to the double stranded of DNA as done in the preceeding section. The EOM’s and its solutions for two typical cases using the Abelian U($1$) and non-Abelian SU($2$) lagrangians have been derived and investigated.
In the case of Abelian U($1$) lagrangian, we have seen from Eq. (\[eq:soliton\]) that the interacting fluid medium characterized by the coupling constant $g$ influences the magnitude and also the width (associated to the dispersion or steppening rate) of solitonic wave equation as well, but regardless with the fluid velocity. On the other hand, obviously the matter self-interaction represented by its coupling constant $\lambda$ could change only the magnitude and not the dispersion or steppening rate of soliton. Actually, this provides a natural explanation for small and large amplitude regions of the internal motion of a single bulk of DNA immersed in bio-fluid without adding any new terms by hand as done in some previous works [@cwlim]. Furthermore, that contribution shifts the matter mass $m_\Phi$ to be $\mb_\Phi$. This is the so-called running mass induced by the dynamical fluctuation of internal kinematics in the system as a result of interaction between matter and fluid. However, the result is again independent on the fluid velocity.
In the second case, using the non-Abelian SU($2$) lagrangian we have constructed a model for double stranded DNA in detail up to the level of its constituents, except for sugar and base composing the nucleoside. It has been shown that the EOM follows a similar form as in the U($1$) model, but the interacting fluid medium contributes in different way. The model requires that the DNA polymer would exist if and only if it resides in a fluid medium, represented by $I_2$ and $I_3$ bounds realized by fluid-matter interactions. Otherwise, the binding interactions $I_2$ and $I_3$ would vanish and the strands are broken. These results could be used to explain the deformation of DNA molecules associated with vanishing interactions in $I_{1,2,3}$. In contrast with the previous case, the fluid velocity plays an important role and changes the dynamics drastically, namely the highly damped, damped and regular harmonic oscillators. This supports a conclusion obtained in [@aslin], that is the effect of hydrodynamic interactions on the dynamics of DNA translocation depends on the fluid velocity.
As mentioned earlier, both strands in a double stranded molecule are considered to follow the same EOM as Eq. (\[eq:nreomsu2\]) with opposite sign of fluid velocities. In contrary, the single fragments of DNA belong to those strands are governed by Eq. (\[eq:nlkge\]) independent on the fluid velocity, and should behave identically no matter with the directions of its surrounding fluid velocities.
Further studies can be done using the lattice gauge simulation to calculate numerically, for instance the finite temperature partition function density $\z = \mathrm{exp}(1/T \int \mathrm{d}^3x \l)$. This kind of macroscopic ensemble provides direct relation between the internal dynamics of DNA and some physical observables like temperature and so on. Actually this is the main advantage of deploying the gauge invariant lagrangian like the present one. Such numerical calculations would be able to simulate quantitatively some phenomena in DNA like critical temperature or pressure related to the deformation of DNA molecules, etc. For example, one can investigate the critical temperature as a double stranded DNA is splitted into single strands [@gerland], $I_1 \rightarrow 0$ in the present model. Such studies are in the progress.
Acknowledgment {#acknowledgment .unnumbered}
==============
We greatly appreciate fruitful discussion with T.P. Djun throughout the work. AS thanks the Group for Theoretical and Computational Physics LIPI for warm hospitality during the work. This work is partially funded by the Indonesia Ministry of Research and Technology and the Riset Kompetitif LIPI in fiscal years 2009 and 2010 (Contract no. 11.04/SK/KPPI/II/2009 and 11.04/SK/KPPI/II/2010).
[99]{} S.M. Burges, N. Kleckner and B.M. Weiner, . B.M. Weiner and N. Kleckner, . S.B. Smith, L. Finzi and C. Bustamante, . R. Lavery, A. Lebrun, J.F. Allemand, D. Bensimon and V. Croquette, . T.R. Strick, M.N. Dessinges, G. Charvin, N.H. Dekker, J.F. Allemand, D. Bensimon and V. Croquette, . G. Altan-Bonnet, A. Libchaber and O. Krichevsky, . J. Peon and A.H. Zewall, . Z. Bryant, M.D. Stone, S.B. Smith. N.R. Cozzarelli and C. Bustamante, . D. Magde, E. Elson and W.W. Webb, . D. Bensimon, A.J. Simon, V. Croquette and A. Bensimon, . D. Porath, A. Bezryadin, S. de Vries and C. Dekker, . S.W. Englander, N.R. Kallenbach, A.J. Heeger, J.A. Krumhansl and A. Litwin, . A.S. Davydov, . L.V. Yakushevich, . M. Peyrard, . M. Cardoni, R. de Leo and G. Gaeta, . M. Peyrard and A.R. Bishop, . T. Dauxois, . T. Dauxois and M. Peyrard, . D.J. Lee, A. Wynveen and A.A. Kornyshev, . T. A. Knotts, N. Rathore, D. C. Schwartz and J. J. de Pablo, . A. Sulaiman, . A.A. Kornyshev and S. Leikin, . A.A. Kornyshev and S. Leikin, ;\
. L.V. Yakushevich, . S.F. Mingalev, P.L. Christiansen, Y.B. Gaididei, M. Johansson and K.O. Rasmussen, . M. Cardoni, R. de Leo and G. Gaeta, . J. A. Berashevich and T. Chakraborty, . S. Zdrakovici, J.A. Tuszynski and M.V. Sataric, . A. Izmitli, D. C. Schwartz, M. D. Graham and J. J. de Pablo, . T. P. Westcott, I. Tobias and W. K. Olson, . R. M. Jendrejack, J. J. de Pablo and M. D. Graham, . V.K. Fedyanin and L.V. Yakushevich, . S.M. Mahajan, . A. Sulaiman and L.T. Handoko, . B.A. Bambah, S.M. Mahajan and C. Mukku, . A. Sulaiman, A. Fajarudin, T.P. Djun and L.T. Handoko, . Marmanis, . Marmanis, . C. N. Yang, . R. Mills, . T.P. Cheng and L.F. Li, C.W. Lim and J-J. Shu, . R. Bundschuh and U. Gerland, .
[^1]: Email : asulaiman@webmail.bppt.go.id, sulaiman@teori.fisika.lipi.go.id
[^2]: Email : handoko@teori.fisika.lipi.go.id, laksana.tri.handoko@lipi.go.id
[^3]: http://www.fi.itb.ac.id
[^4]: http://tisda.bppt.go.id
[^5]: http://teori.fisika.lipi.go.id
[^6]: http://www.fisika.ui.ac.id
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abstract: 'This note is to bring to the reader’s attention the fact that general relativity and quantum mechanics differ from each other in one main aspect. General relativity is based on the diffeomorphism covariant formulation of the laws of physics while quantum mechanics is constructed such that its fundamental laws remain invariant to a change of topology. It is the goal of this paper to show that in order to obtain a complete description of quantum gravity one has to extend the principle of diffeomorphism invariance from general relativity in the sense of quantum mechanics i.e. the laws of physics must be covariant to a change in the topology of spacetime.'
address: 'University College London, Department of Physics and Astronomy, London, WC1E 6BT, UK'
author:
- 'Andrei T. Patrascu'
title: On quantization and general relativity
---
Introduction
============
The prescriptions of general relativity and quantum mechanics are taking away most of the absoluteness associated to choices of coordinates, trajectories followed by particles and states of physical systems in the absence of any accessible information about them. It is my observation that there still remains an epistemological defect associated to these ideas. Not to all arbitrary conventions has been taken their absolute status away. In fact the connectivity of space is probably the last convention that still is considered absolute by many physicists. It is my observation that one cannot assign a specific absolute topology to spacetime itself in the absence of a method of detecting such a topology. It is also my observation that in the absence of specific information related to the topology of space no such property can be rigorously associated to it without arriving at paradoxes and inconsistencies. Because of this, it appears to be necessary for the laws of nature to be specified in a topology-covariant way. This attempt is made in this article.
Covariance principles in physics
================================
The main developments of the past century in physics (special relativity, general relativity and quantum mechanics) have brought to our attention the fact that abstract mathematical conventions should not stand at the fundaments of physics. In general, the role of conventions is to facilitate the comprehension of physical reality and not to assign physical reality to conventional constructions. This idea was noted probably for the first time by Einstein and incorporated in his theory of special relativity as the weak equivalence principle: “the laws of nature should not depend on the arbitrary choice of an inertial reference frame”. Of course, this law was further generalized to the statement that “the general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatsoever (generally co-variant)”\[1\].
It was the goal of general relativity to eliminate any absolute character associated to spacetime coordinates and to refer to them as to arbitrary choices. Geometrical curvature of space-time was to be associated to what was before known as “force of gravity”. Diffeomorphism invariance (i.e. invariance of the laws of physics under the smooth local shift of a path in space-time followed by a reparametrization to the initial numerical coordinate values) was formulated as a fundamental invariance of nature. Indeed, following this way of thinking there is no difference between an accelerated motion and a motion in a gravitational field as seen from a local standpoint. The result was the general theory of relativity which is certainly among the best tested constructions of theoretical physics. There is however a situation in which general relativity fails. The associated phenomenon is known as a “black hole” and is probably one of the most intensely studied phenomena today \[2\],\[3\]. A black hole is in general characterized, among other, by the area of its horizon. Physically, the horizon of a black hole is a classical surface of no return. It is assumed that it also gives non-trivial topology to the space-time that contains it. One can define a black hole as the region encapsulated by such a horizon. General relativity is predicted to fail at some point behind a horizon. However, generally, the connectivity of spacetime invariantly changes when a horizon forms. As probably I do not have to remind here, there is no possible causal connection between the interior of a black hole and the exterior. It results that a consequence of the formation of a black hole is the alteration of the topology (connectivity) of space-time. Let me now consider general diffeomorphism invariance in the presence of a black hole. Diffeomorphism invariance is assumed to be a symmetry of general relativity. That means, if one performs a push-forward of a path and then a coordinate transformation to the original numerical values of the coordinates one should obtain no change in the formal expression of the theory. However, one will encounter problems when dealing with topologically non-trivial spacetimes. One may observe that a hole in space will make some transformations impossible from purely geometrical reasons. In general relativity, there exist points that are associated to divergences of diffeomorphism invariant objects (scalars). These are the points where the prescriptions of general relativity fail.
One aspect has however been overlooked when dealing with this type of problems. In flat or curved spacetime without horizons the laws of physics have to be formulated in a covariant way (due to the strong equivalence principle) and it is essentially impossible to tell the difference between a curved space and an accelerating frame using only local measurements. What about the choice of a topology? Should one be able in principle to infer the existence of a non-trivial topology in the absence of an experimental setup constructed such that the connectedness of spacetime becomes manifest? Otherwise stated is the topology of spacetime a fundamental absolute property of space? The universal coefficient theorems show that in principle there is nothing absolute related to connectedness and classes of homotopy or (co)homology and that one cannot expect to have a universal absolute topological setup over spacetime. Is there a physical reason for this lack of absoluteness? In fact, there are several paradoxes that appear due to the assumption that spacetime connectivity should have a special character.
The problem one is faced with is: should the laws of physics change dramatically in the presence of non-trivial topologies? Is there a special feature associated to a topologically non-trivial space that makes it special among all the other possible choices? On the other side, even in the absence of a macroscopic black hole, measurements of intervals in space-time are possible only if enough energy is concentrated in a small region. The more accurate measurements we want the more energy has to be added to the specific region. Finally one may add sufficient energy such that the spacetime will become topologically non-trivial. By this, the mere procedure of measuring distances may entail non-trivial topologies and consequently horizons and Hawking radiation that may be detected locally from far away. One may ask then, how is it possible to have singled out a whole category of spacetimes that depend only on our choices of performing measurements of some nature? Do the laws of nature change when working with a topologically non-trivial spacetime?
Indeed it is the goal of this article to show that there are no special requirements for topologically non-trivial spacetimes and that the laws of nature are fundamentally independent of the connectivity assigned to a given space. In essence topology is just another word for connectedness. The scope of topology is to identify and measure connected structures in an abstract space. However, it was already clear for Einstein that “all our space-time verifications invariably amount to a determination of space-time coincidences. If for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meetings of the material points of our measuring instruments with other material points (...)”\[1\]. It is at this point where the connection to quantum mechanics must be introduced. There, besides the immaterial character of an absolute spacetime reference, one adds the fact that specific intermediate states of particles (even intermediate positions) are not to be considered physical unless they are practically observed. As a result, following Feynman’s path integral prescription \[4\], one is capable of formulating quantum mechanics in the form of functional integrals over field spaces. All possible paths that connect two events in space-time must be considered as possible alternatives in calculating the amplitudes. Born rules \[5\] will provide us with the probabilities for various events such that we will be able to construct the statistics. The calculation of amplitudes via Feynman path integrals and the construction of probabilities via Born rules represent methods of probing the topology of the given experimental setup. In fact the context independence of the prescriptions of quantum mechanics (i.e. Born rule and the rules for constructing amplitudes via summation over histories are independent of the specific experimental setup) represent a first expression of what I will call a “topologically co-variant” formulation of the laws of nature. If one decides to perform a quantum mechanical experiment, it appears that one is not allowed to assume a pre-existence of states, positions or paths between two measurable events. Moreover, one should not be allowed to assume the existence of one topological structure instead of another. The particular way in which space-time is connected is not observable a-priori. It becomes material only when an experiment that allows the inference of the topology is performed. So, at this moment we have general relativity, a theory that fails when dealing with “holes” in space-time (i.e. non-trivial topology) and quantum mechanics (i.e. a theory that is capable of probing a given topology). However, reasoning in the context of general relativity it appears plausible that to the topology of space-time should not be given an absolute character. As Einstein noted, only the “coincidences” \[1\] matter. In principle “connectedness” is whatever is in the context of quantum mechanics “the non-measurable in between-ness” so one is not supposed to know a-priori how this connectedness is realized. This is a common point in the reasoning of quantum mechanics and general relativity. It appears that in order to construct a quantum theory of gravity one has to pay attention to the way in which the laws of nature are expressed such that they do not depend on a particular topological choice. In my previous notes \[6\] I showed that several properties are to be considered relative if one makes use of several freedoms in defining the topology. It is my goal here to show that it is possible to construct a general theory of quantum gravity that is formulated explicitly independent of a particular topology.
Independence of topology and the Universal Coefficient Theorem
===============================================================
As argued in the previous chapter, the laws of physics should not depend on unobservable properties of spacetime. Specifically the choice of a particular coordinate system or a particular topology should not be relevant for the formulation of the laws of physics. I showed in a previous article \[6\] that specific choices of coefficient groups in cohomology may affect the observable connectedness of space-time (or generally of an abstract space) as measured by topological techniques. Here I focus on an aspect that maybe remained unspecified, namely what changes should be made in a theory in order for it to be independent on the way one choses to regard the topology? It appears to me that not all of the absolute character of the conventions related to spacetime has been removed while imposing diffeomorphism invariance. This appears to be what escaped Einstein in his formulation of general relativity and was unexpectedly introduced in the formulation of quantum mechanics, hence the apparent incompatibility of the two. In order to show what I mean let me start with the prescription of path integral quantization \[4\]. R. Feynman observed that different functionals may give identical results when taken between any two states and argued that this equivalence between functionals is the statement of operator equations in the language of path integrals. I assume that the standard prescription of computing quantum probabilities using quantum amplitudes is well known. If $P_{ac}$ is the quantum probability of measuring event $c$ when it follows the measurement of event $a$ then the probability must be calculated as $P_{ac}=|\varphi_{ac}|^{2}$ where $\varphi_{ac}=\sum_{b}\varphi_{ab}\varphi_{bc}$ where the sum is over the possible intermediate states $b$ which, I emphasize, following Feynman (ref. \[4\], page 3 in manuscript) have no meaningful independent value. In a 1-space and 1-time dimensional context (the generalization to arbitrary dimensions should be straightforward) a succession of measurements may represent a succession of the space-coordinate $x$ at successive times $t_{1},t_{2},...$, where $t_{i+1}=t_{i}+\epsilon$. Let the observed value at $t_{i}$ be $x_{i}$. Classically the successive values of $x_{1},x_{2},...$ define a path $x(t)$ when $\epsilon\rightarrow 0$. If the intermediate positions are actually measured one may talk about such a path with a well defined set of observed positions $x_{1},x_{2},...$ and the probability that the specified path $P(...x_{i},x_{i+1},...)$ lies in a region $R$ is given by the classical formula $$P=\int_{R}P(...x_{i},x_{i+1},...)...dx_{i}dx_{i+1}...$$ where the integral is taken over the ranges of the variables which lie within the region $R$. If the intermediate positions are not measured then one cannot assign a value to them. In this case the probability of finding the outcome of a measurement in $R$ is $|\varphi(R)|^{2}$ and $\varphi(R)$, i.e. the probability amplitude is calculated as $$\varphi(R)=\lim_{\epsilon\rightarrow 0}\int_{R}\Phi(...x_{i},x_{i+1},...)$$ where $\Phi(...x_{i},x_{i+1},...)$ defines the path. In the given limit this object becomes a path functional. There should be no mystery nowadays that the probability amplitude should be calculated as $$\varphi(R)=\lim_{\epsilon\rightarrow\ 0}\int_{R}exp[\frac{i}{\hbar}\sum_{i}S(x_{i+1},x_{i})]...\frac{dx_{i+1}}{A}\frac{dx_{i}}{A}...$$ where $S$ is the action functional for the given path segment. In order to go a step further and define the wavefunction in this context I will continue to follow Feynman’s paper \[4\]. The region $R$ considered above can be divided into future and past with respect to a choice of a time position $t$. One can define the region $R'$ as the past and the region $R''$ as the future. The probability amplitude connecting these regions will be $$\varphi(R',R'')=\int \chi^{*}(x,t)\psi(x,t)dx$$ where $$\psi(x_{k},t)=\lim_{\epsilon\rightarrow 0} \int_{R'}exp[\frac{i}{\hbar}\sum_{i=-\infty}^{k-1}S(x_{i+1},x_{i})]\frac{dx_{k-1}}{A}\frac{dx_{k-2}}{A}...$$ and $$\chi^{*}(x_{k},t)=\lim_{\epsilon\rightarrow 0} \int_{R''}exp[\frac{i}{\hbar}\sum_{i=k}^{\infty}S(x_{i+1},x_{i})]\frac{1}{A}\frac{dx_{k+1}}{A}\frac{dx_{k+2}}{A}...$$ In this way one can separate the “past” and the “future” via the functions $\psi$ and $\chi$. One may also construct a closer equivalence to the matrix representation of quantum mechanics by introducing matrix elements of the form
$$<\chi_{t''}|F|\psi_{t'}>_{S}=\lim_{\epsilon\rightarrow\ 0}\int ... \int \chi^{*}(x'',t'')F(x_{0},...x_{j})exp[\frac{i}{\hbar}\sum_{i=0}^{j-1}S(x_{i+1},x_{i})]\psi(x',t')\frac{dx_{0}}{A}...\frac{dx_{j-1}}{A}dx_{j}$$
In the limit $\epsilon\rightarrow 0$, $F$ is a functional of the path $x(t)$. At this moment one can define various equivalences between functionals. These are to be associated to operator equations in the matrix formulation. One can of course define $\frac{\partial F}{\partial x_{k}}$ and one can calculate the associated matrix element using an action functional $S$. Using the fact that the action functional appears as $exp(\frac{i}{\hbar} S)$ one obtains matrix equations as, say $$<\chi_{t''}|\frac{\partial F }{\partial x_{k}}|\psi_{t'}>_{S}=-\frac{i}{\hbar}<\chi_{t''}|F\frac{\partial S}{\partial x_{k}}|\psi_{t'}>_{S}$$ which can be stated as a functional relation defined for an action $S$ as $$\frac{\partial F }{\partial x_{k}} \leftrightarrow -\frac{i}{\hbar}F\frac{\partial S}{\partial x_{k}}$$ Using the fact that $S=\sum_{i=0}^{j-1}S(x_{i+1},x_{i})$ one can rewrite $$\frac{\partial F }{\partial x_{k}} \leftrightarrow -\frac{i}{\hbar}F[\frac{\partial S(x_{k+1},x_{k})}{\partial x_{k}}+\frac{\partial S(x_{k},x_{k-1})}{\partial x_{k}}]$$ In the case of a simple 1-dimensional problem one can write $$\frac{\partial S(x_{k+1},x_{k})}{\partial x_{k}}=-m(x_{k+1}-x_{k})/\epsilon$$ and $$\frac{\partial S(x_{k},x_{k-1})}{\partial x_{k}}=+m(x_{k}-x_{k-1})/\epsilon-\epsilon V'(x_{k})$$ Neglecting terms of order $\epsilon$ one obtains $$m\frac{(x_{k+1}-x_{k})}{\epsilon}x_{k}-m\frac{(x_{k}-x_{k-1})}{\epsilon}x_{k} \leftrightarrow \frac{\hbar}{i}$$ The important aspect here is that the order of terms in a matrix operator product corresponds to the order in “time” of the corresponding factors in a functional. The order of the factors in the functional is of no importance as long as the indexation of these factors is reflected in the ordering of the operators in the matrix representation. This means the left-most term in the above equation must change order so that one obtains the well known commutation relation $$px-xp=\frac{\hbar}{i}$$
These results should be nowadays generally known. One may observe that the indexation of the measurement outcomes, according to a time index (i.e. $\mathbb{Z}$-group), leads to the well known commutation relations. The ideas behind path integral quantization are kept intact when going to the relativistic context. However, when we have to go to a gravitational context the sum over geometries becomes non-trivial. In this sense one has to construct the (co)homology structure of the space and one has to deal with the universal coefficient theorem. This theorem states that a specific framework, constructed by the choice of a coefficient group in (co)homology is (up to (extension) torsion in (co)homology) equivalent with the choice of an integer coefficient group. However, some choices of coefficient groups may make some observables manifest while others may hide them. Moreover, simple order relations as the ones used in the proof above are no longer uniquely defined. As I showed in my previous notes \[6\], what was identified by Feynman as a natural choice (time ordering) may in fact be just the result of a given coefficient group. In order to make the discussion more practical let me consider the formulation of Feynman’s idea in the context of a space which is topologically non-trivial if looked upon via a coefficient group $G$. Let the probed space be $X$. In this case one cannot perform a trivial path-integral quantization due to the fact that the “hole in space” will make the path integral ill-defined. However, the universal coefficient theorem assures us that we can in principle chose another group structure for probing the space. Let this new structure be $G'$ such that one simply “overlooks” the “hole” in space under this group. I showed that this is possible in my previous notes \[6\]. In the context of Feynman’s path integrals the paths will be indexed considering $G'=\mathbb{Z}_{n}$ case in which the succession of measurements will have the form $(x_{1},x_{2},...,x_{n},x_{1},...)$ but one will not exit a specific domain of the usual integers. One obtains a different topology when probing the space that contains a black hole. In fact, if one continues to use this group the black hole will essentially be invisible for an observer behind the horizon although he will never be able to exit it. The concepts must of course generalize for higher dimensional spaces but the idea is valid anyway. The construction of Feynman path integrals must be altered accordingly: $$\varphi(R)=\lim_{\epsilon\rightarrow\ 0}\int_{R}[exp[\frac{i}{\hbar}\sum_{i}S(x_{i+1},x_{i})]...\frac{dx_{i+1}}{A}\frac{dx_{i}}{A}...]_{\mathbb{Z}_{n}}$$ where the index shows that the sampling must be taken according to the new group structure. We face two distinct situations: one in which a massive black hole alters the topology of space-time and one in which all the diffeomorphism invariant properties are finite and there is no apparent problem. The topology-covariant formulations of the laws of physics are encoded in the statement of the universal coefficient theorem. Indeed the two situations are equivalent if one considers the $Ext$ and/or $Tor$ groups in (co)homology. In this sense homological algebra and the universal coefficient theorem are the mathematical tools required for the topological covariant formulation of the laws of physics somehow in the same way in which tensor calculus was the mathematical tool required for the general covariant formulation of the laws of physics.
Now consider the exterior space also to be taken into account. In general there is no unique formulation of the universal coefficient theorem. One formulation that is particularly important is the following: given the tensor product of modules $H_{i}(X;\mathbb{Z})\otimes A$ one has the short exact sequence $$0\rightarrow H_{i}(X;\mathbb{Z})\otimes A \xrightarrow{\mu} H_{i}(X;A)\rightarrow Tor(H_{i-1}(X;\mathbb{Z}),A)\rightarrow 0$$ Here $A$ is the alternative coefficient group, X is the analyzed space and $Tor$ is the torsion. More practically let for example $(C_{*},\partial)$ be a chain complex over a ring $R$. The chain groups are $C_{*}$. Then there is a map $$Hom_{R}(C_{q},M)\times C_{q} \rightarrow M$$ that evaluates like $$(f,z)\rightarrow f(z)$$ This is a general formulation of a structure that has analogues in the covariant and contravariant structures in general relativity but also in the bra-ket notation of standard quantum mechanics. In quantum mechanics the amplitudes are characterized by complex numbers. The adjoint is defined naturally via hermitian conjugation giving rise to the bra-ket formalism and allowing the construction of theories preserving overall unitarity. In general relativity adjoints are constructed as dual 1-forms that appear as “covariant” indices and together with their contravariant counterparts assure that the theory can be formulated in a diffeomorphism invariant form despite the possible intrinsic curvature of spacetime. In principle the 1-forms take the value of a vector and produce a scalar. If $\tilde{P}$ is a 1-form and $\vec{V}$ is a vector then $<\tilde{P},\vec{V}>=\tilde{P}(\vec{V})=\vec{V}(\tilde{P})$. In the case of black holes there are well known and over-discussed issues related to the unitarity of the description also known under the generic name of “information paradoxes”. These appear because the standard bra-ket construction does not map isomorphically into the (co)homology of the analyzed space and hence cannot be used in the same way. There are several ways in which possible pairings as the ones discussed above can be mapped into the realm of universal coefficient theorems. One possible pairing defined in the way described above is $$< , >:H^{q}(C_{*};M)\times H_{q}(C_{*})\rightarrow M$$ which relates homology with cohomology. This pairing is bilinear and its adjoint is a homomorphism $$H^{q}(C_{*},M)\rightarrow Hom(H_{q}(C_{*});M)$$ Universal coefficient theorems, among other things, provide a measure of how this adjoint fails to be an isomorphism in terms of $Ext^{q}$ and $Tor_{q}$ \[7\]. Essentially what one has to observe is that in order to obtain the situation consisting of a black hole and an exterior asymptotically flat space-time one has to combine a structure with coefficients in $\mathbb{Z}_{n}$ with a structure with coefficients in $\mathbb{Z}$. The universal coefficient theorem puts restrictions on this combination. The map fails to be isomorphic and as a consequence not all observables that have a meaning under one choice of coefficients will have a well defined meaning in the other choice. Essentially it is the calculation of the $Ext$ and $Tor$ structures that tells us where the difference lies. It is only after one uses the universal coefficient theorem that one can talk meaningfully about observations made by an in-falling observer with respect to observations made by an observer standing at a large distance from the horizon. As the sequence is exact the observations of an in-falling observer and the observations of an observer standing far away will not share the same set of observables. Many “questions” asked by the far away observer will have no meaning for the in-falling one. This is the basic generalization of the invariance principles such that they include also the invariance with respect to a different choice of topology. The laws of nature must remain invariant under various choices of coefficient groups in (co)homology and this can be assured by the proper consideration of the $Ext$ and $Tor$ structures. There appears the question if there is some fundamental reason why I am using $\mathbb{Z}_{n}$ as a group structure instead of, say $\mathbb{Q}_{n}$ or $\mathbb{R}_{n}$. The simple answer is that nothing should stop me in using those other groups. As stated before there is no fundamental significance given to a choice of a group. One cannot associate an absolute topology to spacetime (be it discrete, continuous, etc.). Any such choice would be assimilable to the “ether” with respect to special relativity and would be immaterial.
conclusion
==========
The main discussion of this article revolves around the epistemological meaning of connectedness associated to a given space(time) and its fundamentally arbitrary nature. It is argued that in order to be able to formulate a theory that encompasses quantum mechanics and general relativity one must abandon the idea of absolute topological structure and rely on the universal coefficient theorem as on a “topology-covariant” formulation of the laws of physics.
[9]{} A. Einstein, Ann. d. Physik 354 (7), 769-822 (1916) Schwarzschild, K. Sitzungsberichte der Kšniglich Preussischen Akademie der Wissenschaften 7: 189Ð196 (1916) Penrose, R. Gen. Rel. and Grav. 34 (7), 1141 (2002) R. P. Feynman, Rev. Mod. Phys. 20 (2), (1948) M. Born, Zeitschrift fŸr Physik, 37, (12), pp. 863Ð867 (1926) A. Patrascu, arXiv:1404.1800 (submitted to Phys. Rev. D) J. F. Davis, P. Kirk, Lecture Notes in Algebraic Topology (see page 43 and 47 in notes)
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abstract: 'We study the X-ray diffraction spectrum produced by a collectively pinned charge density wave (CDW), for which one can expect a Bragg glass phase. The spectrum consists of two asymmetric divergent peaks. We compute the shape of the peaks, and discuss the experimental consequences.'
author:
- Alberto Rosso
- Thierry Giamarchi
title: 'X-ray diffraction of a disordered charge density wave'
---
The statics and dynamics of disordered elastic objects govern the physics of a wide range of systems, either periodic, such as vortex flux lines [@blatter_vortex_review] and charge density waves (CDW) [@gruner_revue_cdw], or involving propagating interfaces, such as domain walls in magnetic [@lemerle_domainwall_creep] or ferroelectric [@tybell_ferro_creep] systems, contact lines of liquid menisci on rough substrates [@moulinet_contact_line] and propagation of cracks in solids [@gao_crack_elasticity]. It was recently shown that periodic systems have unique properties, quite different from the ones of the interfaces. If topological defects (i.e. dislocations etc.) in the crystal are excluded, displacements grow only logarithmically [@nattermann_pinning; @korshunov_variational_short; @giamarchi_vortex_short], instead of the power-law growth as for interfaces. The positional order is only algebraically destroyed [@giamarchi_vortex_short; @giamarchi_vortex_long] leading to divergent Bragg peaks and a nearly perfect crystal state. Quite remarkably, it was shown that for weak disorder this solution is *stable* to the proliferation of topological defects, and thus that a thermodynamically stable phase having both glassy properties and quasi-long range positional order exists [@giamarchi_vortex_long]. This phase, nicknamed Bragg glass, has prompted many further analytical and experimental studies (see e.g. [@nattermann_vortex_review; @giamarchi_vortex_review] for reviews and further references). Although its existence can be tested indirectly by the consequences on the phase diagram of vortex flux lines, the most direct proof is to measure the predicted algebraic decay of the positional order. Such a measurement can be done by means of diffraction experiments, using either neutrons or X-rays on the crystal. Neutron diffraction experiments have recently provided unambiguous evidence [@klein_brglass_nature] of the existence of the Bragg glass phase for vortex lattices.
Another periodic system in which one can expect a Bragg glass to occur are charge density waves [@gruner_revue_cdw], where the electronic density is spatially modulated. Disorder leads to the pinning of the CDW [@fukuyama_pinning]. In such systems very high resolution X-rays experiments can be performed [@rouzieres_structue_cdw]. The resolution is in principle much higher than the one that can be achieved by neutrons for vortex lattices, consequently CDW systems should be prime candidates to check for the existence of a Bragg glass state. However, compared to the case of vortex lattices the interpretation of the spectrum is much more complicated for two main reasons: (i) the phase of the CDW is the object described by an elastic energy, whereas the X-rays probe the displacements of the atoms in the crystal lattice (essentially a cosine of the phase); (ii) since the impurities substitute some atoms of the crystal, the very presence of the impurities changes the X-ray spectrum. This generates non-trivial terms of interference between disorder and atomic displacements [@ravy_x-ray_peakasymmetry; @rouzieres_structue_cdw]. It is thus necessary to make a detailed theoretical analysis for the diffraction due to a pinned CDW. The study of the spectrum has been carried out so far either for strong pinning or at high temperatures [@rouzieres_structue_cdw; @ravy_x-ray_whiteline; @ravy_x-ray_peakasymmetry; @brazovskii_x-ray_cdwT].
In this paper we focus on the low temperature limit where a well formed CDW exists and on weak disorder, for which one expects to be in the Bragg glass regime. We show that the diffraction spectrum consists in two asymmetric peaks. In contrast to previous assumptions [@ravy_x-ray_peakasymmetry], we show that the asymmetry is present also in the weak pinning limit. The peaks are power-law divergent, with an anisotropy in shape. This form is consistent with the Bragg glass behavior [@giamarchi_vortex_long]. The asymmetry is a subdominant power-law too, with an exponent that we determine. We also briefly discuss the role of unscreened Coulomb interaction for the CDW on the diffraction spectrum.
The general expression [@guiner_xray] for the total diffraction intensity in a crystal is given by $$\begin{aligned}
\label{eq:Sdef}
I(q) = \frac{1}{V} \sum_{i,j} e^{-iq(R_i-R_j)} \left\langle
\overline{f_if_j e^{-iq(u_i-u_j)}} \right\rangle,\end{aligned}$$ where $u_i$ is the atom displacement from the equilibrium position $R_j = ja$, with $a$ indicating the lattice constant, $f_i$ the atomic scattering factor and $\overline{\langle\ldots \rangle}$ denotes the double average over the disorder and over the thermal fluctuations. As an example let us first consider the case of fixed atoms ($u_i=0$). We obtain: $$\begin{aligned}
\label{Laue}
I(q) = \overline{f}^2 \sum_K \delta(q-K)
+ {\Delta f}^2 N_I,\end{aligned}$$ where $\Delta f= f_I -f $ is the difference between the impurity $I$ and the host atom scattering factors, $N_I=n_I(1-n_I) $, with $n_I$ is the impurity concentration and $\overline{f}$ is the average scattering factor. The usual Bragg peaks, in correspondence to the reciprocal lattice vectors $K$, arise from the first term in (\[Laue\]), the second term is responsible for a background intensity, called Laue scattering, due to the disorder.
In a second stage we take into account displacements of the atoms related to the presence of a CDW. To this purpose, we consider an electron density characterized by a sinusoidal modulation: $$\rho(x) = \rho_0 \cos(Qx+\phi(x)).
\label{rho}$$ $\phi$ is the phase of the charge density wave and $Q=2k_F$, where $k_F$ is the Fermi wave vector. The associated Hamiltonian writes: $$\begin{aligned}
\label{eq:Hamiltonian}
H= \int d^dx \frac{c}{2} \left( \nabla \phi(x) \right)^2
\pm V_0\int d^dx \Sigma(x) \rho(x),\end{aligned}$$ where $d$ is the dimension of the space. The first term in the Hamiltonian (\[eq:Hamiltonian\]) represents the elasticity. The elasticity is in fact anisotropic [@feinberg_cdw] along the $Q$-direction: $$\begin{aligned}
\label{eq:anisotropy}
H_{\text{el.}}= \int d x d^{d-1}y
\frac{c_1}{2}
\left( \partial_x \phi \right)^2 +
\frac{c_2}{2} \left( \partial_y \phi \right)^2 ,\end{aligned}$$ where $x \parallel Q$, and $c_1 \gg c_2$. The compression along $x$ corresponds to an increase of electric charge density and thus pays the price of Coulomb repulsion, while distortions along the remaining $d-1$ directions are much easier. We are led back to (\[eq:Hamiltonian\]) by redefining the spatial variables $x' =
x/\sqrt{c_1}$ and $y' = y/\sqrt{c_2}$, with $c=(c_1c^{d-1}_2)^{\frac{1}{2}}$. The main effect in the diffraction spectrum is thus to make the *shape* of the peaks anisotropic, but this will not change the overall divergence. The local, but anisotropic, elasticity (\[eq:anisotropy\]) is valid beyond the distance at which the Coulomb interaction between various parts of the CDW is screened. If this length is very large, or if one want to examine short range regime one should keep the $q$-dependence in the elastic constants. This leads to a more complicated behavior that we will only briefly discuss here and will be examined in details elsewhere [@rosso_cdw_long]. The second term in the Hamiltonian (\[eq:Hamiltonian\]) reflects the effect of the disorder on the electron density. The Gaussian random function $\Sigma(x)$ describes the impurity distribution and is characterized by the correlator $\overline{\Sigma(x)\Sigma(y)} = N_I \delta(x-y) $, $V_0$ is a positive constant which measures the impurity potential and finally the sign $+$ ($-$) is related to the repulsive (attractive) interaction between the electrons and the local impurity. In the following, we restrict our analysis to the repulsive case, $\rho_0$ is absorbed in $V_0$ and we define the disorder strength $D= {V_0}^2 N_I$.
A density modulation is accompanied by a lattice distortion $u$ given at low temperature by $$u(x) = \frac{u_0}{Q} \nabla \cos(Qx+\phi(x)).$$ We are interested in the behavior of the scattering intensity $I(q)$ near a Bragg peak ($q \sim K$). Since $|\delta q| =|q-K|
\ll K$, we can take the continuum limit $i \rightarrow x$ and we obtain from (\[eq:Sdef\]): $$\begin{aligned}
\label{eq:Sdef2}
I(q) = \int_r \left\langle
\overline{f_{\frac{r}{2}}f_{-\frac{r}{2}}
e^{-i \delta q (u(\frac{r}{2})-u(-\frac{r}{2}))}} \right\rangle.\end{aligned}$$ where $\int_r = \frac{1}{a^d} \int d^dr e^{-i \delta q r}$ and $f_{\frac{r}{2}} = \overline{f}+ \Delta f a^{d/2}
\Sigma(\frac{r}{2})$. In (\[eq:Sdef2\]) we have applied the standard decomposition in center of mass $R$ and relative $r$ coordinates ($x = R+\frac{r}{2}$ and $y = R-\frac{r}{2}$). The integration over $R$ has already been performed because $u$ vary slowly at the scale of the lattice spacing. Assuming that in the elastic approximation displacements remain small ($u_i \ll R_i$), one can expand (\[eq:Sdef2\]) as powers of $Ku_0$. Developing up to the second order we get [@ravy_x-ray_whiteline] : $$\begin{aligned}
\label{eq:Intensities}
& & I(q) = I_{\text{d}} + I_{\text{a}}+ I_{\text{tripl.}}, \mbox{with} \\
& & I_{\text{d}} =\overline{f}^2 q^2 \int_r
\langle \overline{u(\frac{r}{2}) u(-\frac{r}{2})} \rangle, \nonumber \\
& & I_{\text{a}} = -iq {\Delta f} a^{d/2}\overline{f} \int_r \left\langle
\overline{\Sigma(-\frac{r}{2}) u(\frac{r}{2}) - \Sigma(\frac{r}{2})
u(-\frac{r}{2}) } \right \rangle, \nonumber \\
& & I_{\text{tripl.}}= -
iq {\Delta f}^2 a^{d} \int_r
\left\langle
\overline{\Sigma(-\frac{r}{2})\Sigma(\frac{r}{2}) (u(\frac{r}{2})
- u(-\frac{r}{2})) } \right \rangle . \nonumber\end{aligned}$$ While the contribution $I_{\text{d}}$ represents the intensity due to the atomic displacements alone, the contributions $I_{\text{a}}$ and $I_{\text{tripl.}}$ are generated by the coupling between the disorder and the displacement. The presence of a CDW is signaled by the formation, around each Bragg peak, of two satellites at reciprocal vectors $K \pm Q$. In absence of disorder ($D=0$ and $\phi \sim \mbox{const.}$) the displacement term has the form $I_{\text{d}}=f^2 q^2 u_0^2 \sum_K \delta(q+K\pm Q)$ and the other terms are vanishing: in this case the two satellites have the same intensity and the broadening is absent. To interpret the experimental findings [@ravy_x-ray_peakasymmetry; @brazovskii_x-ray_cdwT; @rouzieres_structue_cdw], in particular to explain the measured strong asymmetry [^1] between the peaks at $K+Q$ and at $K-Q$, we need to account for the effect of impurities.
In the literature the term $I_{\text{a}}$ was evaluated by means of models [@ravy_x-ray_whiteline; @ravy_x-ray_peakasymmetry; @brazovskii_x-ray_cdwT] which describe the pinning by imposing a constant value $\phi_0$ to the phase in (\[rho\]) in proximity of each impurity, and $I_{\text{tripl.}}$ was conjectured to be negligible [@ravy_x-ray_whiteline; @cowley_x-ray_cdw]. In that approach, the observed satellite asymmetry is seen as a clear sign of the strong disorder; in fact, $\phi_0$ is not constant and, for sufficiently large domains, one should have [@ravy_x-ray_peakasymmetry] $I_{\text{a}} \propto
\overline{\cos(\phi_0)} \sim 0$. To go beyond this phenomenological approach and also deal with the weak disorder limit, in which one expects the Bragg glass, we use a Gaussian variational approach [@mezard_variational_replica; @giamarchi_vortex_long]. We first perform the average over the disorder using the standard replica techniques. The replicated Hamiltonian corresponding to (\[eq:Hamiltonian\]) is $$\label{eq:ham}
H_{\text{\text{eff.}}}= \int d^dr\sum_a \frac{c}{2}
(\nabla \phi_a)^2 - \frac{D}{T} \sum_{a,b}
\cos\left(\phi_a(r) -\phi_b(r)\right) ,$$ where $T$ is the temperature and the sum over the $n$ replica has to be considered in the limit $n \rightarrow 0$. We stress that, moving from the Hamiltonian (\[eq:Hamiltonian\]) to its replicated version we also need to change the correlation functions containing explicitly the disorder: we have, for example, $\left\langle \overline{\Sigma(-\frac{r}{2})
u(\frac{r}{2}) } \right \rangle \rightarrow -\frac{D}{TV_0}
\sum_{a} \left\langle \rho_a(-\frac{r}{2}) u_1(\frac{r}{2})
\right \rangle_{\text{eff.}}$. After some manipulations and using (\[rho\]), we obtain: $$\begin{aligned}
\label{eq:Correlations}
&& I_{\text{d}} = \overline{f}^2 q^2 {u^2}_0 \int_r\left[ e^{-i Q
r} +c.c.\right] C_{\text{d}}(r) \,,
\\
&&I_{\text{a}} = - \overline{f} {\Delta f} q u_0 \sqrt{N_I a^d D }
\int_r \left[ e^{-i Q r} - c.c.\right]
C_{\text{a}}(r) \nonumber \,,\end{aligned}$$ where $C_{\text{d}}(r)=
\left\langle e^{i(\phi_1(\frac{r}{2})
-\phi_1(-\frac{r}{2}))}\right\rangle_{\text{eff.}}$ and $C_{\text{a}}(r) =\frac{1}{Tn} \sum_{a,b}^n
\left\langle e^{i(\phi_a(\frac{r}{2}) -\phi_b(-\frac{r}{2}))}
\right \rangle_{\text{eff.}}$ are the positional correlation functions controlling the behavior of each contribution. We notice that the intensity of the peaks at $q=Q+K$ and $q=K-Q$ is symmetric, as in the case of a pure system, for the displacement term $I_{\text{d}}$, but it is antisymmetric for $I_{\text{a}}$. The sum of these two terms leads to an asymmetry of the peaks. Fig. \[model\] show the behavior of the different contributions.
![Intensities of the different contributions to satellite peaks. The more divergent term, $I_{\text{d}}$, is symmetric. $I_{\text{a}}$ and $I_{\text{tripl.}}$ are antisymmetric, with $I_{\text{a}} \gg I_{\text{tripl.}}$.[]{data-label="model"}](modelb1.eps){width="\figwidth"}
Following the method used in [@giamarchi_vortex_long] for flux lines in presence of weak disorder, we can calculate the various terms in (\[eq:Correlations\]). We look for the best trial Gaussian Hamiltonian $H_0= \int_q G_{ab}(q) \phi_a(q) \phi_b(-q)$ in replica space, which approximates (\[eq:ham\]). Defining $$\begin{aligned}
B_{ab}(r)&=& \langle (\phi_a(r) -\phi_b(0))^2 \rangle_0 \\
&=&2 T\int_q \left[\tilde{G}(q) -G_{ab}(q)
\cos qr \right], \nonumber\end{aligned}$$ where $\tilde{G}$ is the diagonal element of $G_{ab}$, and using the Gaussian approximation, the positional correlation functions become $C_{\text{a}}(r)
= \frac{1}{nT} \sum_{a,b}^n e^{- \frac{ B_{ab}(r)}{2}}
$ and $ C_{\text{d}}(r) = e^{-\frac{\tilde{B}(r)}{2}}$, where $\tilde{B}$ is the diagonal element of $B_{ab}$. Two general classes of solutions exist for this problem: while the first class preserves the permutation symmetry of the replica (RS), the second class (RSB) breaks the replica symmetry. It has been shown [@giamarchi_vortex_long] that the stable solution for $d>2$ corresponds to the RSB class, while the RS solution remains valid at short distance. $C_{\text{d}}$ is similar to the correlation calculated for flux lines [@giamarchi_vortex_long] and will be discussed later. To evaluate the contribution of the interference between disorder and displacement we factorize the antisymmetric term $C_{\text{a}}(r)=\chi(r)C_{\text{d}}(r)$. We first consider the RS approximation: $$\begin{aligned}
\chi(r)=\frac{1}{T}\left[1 - e^{-T \int_q G_c(q)\cos qr} \right],\end{aligned}$$ where $G_{c}=\frac{1}{cq^2}$ is the connected part of $G_{ab}$. In $d=3$ we estimate $$\begin{aligned}
C^{RS}_{a}(r)
&\sim& \frac{2 \pi^2 }{ c r}
e^{- \frac{ \tilde{B}(r)}{2}}.\end{aligned}$$ The triplet term can be evaluated in an analogous way, but it gives non-zero contributions only considering higher order harmonic terms in the electron density. Equation (\[rho\]) becomes: $\rho(x) = \rho_0
\cos(n(Qx+\phi(x)))$, with $n=1,2$. As we have already found for $I_{\text{a}}$, we get an antisymmetric term with a prefactor $\propto {\Delta f}^2 q u_0 N_i a^d D
$ and a correlation $C_{\text{tripl.}}=\frac{1}{2nT^2} \sum_{a,b,c}^n \langle(e^{- i (\phi_c(\frac{r}{2}) -2
\phi_a(\frac{r}{2})+\phi_b(-\frac{r}{2}))} \rangle_{\text{eff.}}$. In $d=3$ and at low temperature we finally obtain $$\begin{aligned}
C^{RS}_{\text{tripl.}}(r)
\sim \frac{2 \pi }{c^2 a r } e^{- \frac{\tilde{B}}{2}}.\end{aligned}$$ It is interesting to evaluate, at this stage, the relative weight of the two antisymmetric terms in a satellite peak. We introduce the Fukuyama-Lee length (or Larkin-Ovchinikov length) [@fukuyama_pinning; @larkin_ovchinnikov_pinning] $R_a= (c^2 / D)^{1/4-d}$ (for $d=3$ $R_a =c^2 / D$) such that $\phi$ varies on scale given by the length $R_a$. The ratio of the two intensity peaks is: $$\begin{aligned}
\frac{I_{\text{tripl.}}}{I_{\text{a}}} = - \frac{\Delta f}{\pi \overline{f}}
\sqrt{N_I} \sqrt{\frac{a}{R_a }} .
\label{stima}\end{aligned}$$ For weak disorder $R_a \gg a$ it follows that $ I_{\text{a}} \gg
I_{\text{tripl.}}$ and we thus need only to consider $I_{\text{a}}$ and $I_{\text{d}}$.
Since for $d=3$ the RS solution is unstable, to obtain the correct physics one has to look for the RSB method. Within this scheme [@giamarchi_vortex_long], the off diagonal elements of $G_{ab}(q)$ are parameterized by $G(q,v)$ where $0<v<1$ and the solution is characterized by a variational breakpoint $v_c$. The form of the symmetric part is given in [@giamarchi_vortex_long]: $$\begin{aligned}
\label{displRSB}
C_{\text{d}}(r)\sim e^{-\frac{\phi^{2}_T}{2}}
\left(\frac{l}{r}\right)^{\eta}\end{aligned}$$ where $\phi^{2}_T \simeq \frac{2 T}{\pi c a}$ measures the strength of thermal fluctuations, and $\eta \sim 1$ is the Bragg glass exponent in $d=3$. At low temperature one has $l \sim R_a$. The algebraic behavior of (\[displRSB\]) is controlled by small $v$ ($v<v_c$). Values of $v$ above the breaking point ($v>v_c$) give the small distance contribution. Finally one finds $v_c=\frac{1}{8} \phi^{2}_T \frac{a}{l}$.
To fully characterize the spectrum it still remains to evaluate $\chi(r)$ in the RSB scenario : $$\begin{aligned}
\label{RSB}
\chi(r) = \frac{1}{T}[1 - \int_0^1 dv e^{-T \int_q
(\tilde{G}(q) - G(q,v)) \cos qr}].\end{aligned}$$ Restricting to the case $d=3$, we write: $$\label{RSB3d}
\tilde{G}(q)-G(q,v) =\frac{1}{c} \left[ \frac{1}{q^2+l^{-2}} +
\frac{2}{l^2}\int_{v/v_c}^1 dt \frac{1}{\left(q^2 +(\frac{t}{l})^2 \right)^2}\right].$$ By integrating (\[RSB3d\]) over q and with some manipulations, (\[RSB\]) becomes: $$\begin{aligned}
\chi(r) &=& \frac{v_c}{T} \left[ 1- \int_0^{1} dz \exp \left(-8\pi^3 \int^{1}_z \frac{dt}{t}
e^{-rt/l} \right) \right]\end{aligned}$$
![Ratio between the RSB and RS solutions for $\chi(r)$. At large distance this ratio tends towards a constant value $b$, where $b \sim 0.018$. This means that the RSB solution affects $\chi(r)$ only by a multiplicative factor.[]{data-label="integral"}](integral.eps){width="\figwidth"}
The low temperature behavior ($l \sim R_a$) of this term is sketched in Fig. \[integral\]. As for the Replica Symmetry case, we have $C_{\text{a}}(r) \propto \frac{1}{r}
e^{- \frac{ \tilde{B}(r)}{2}}$. We can now compare the two terms: $$\begin{aligned}
\label{stima2}
I_{\text{d}}(K+Q) & = & \overline{f}^2 K^2 {u_0}^2 \int_r (\frac{R_a}{r})^{\eta} \\
I_{\text{a}}(K+Q) & = & - 2\pi^2 \overline{f} {\Delta f} \sqrt{N_I} \sqrt{\frac{a}{R_a }}
K u_0 \int_r
(\frac{R_a}{r})^{\eta} \frac{ba}{r}\nonumber .\end{aligned}$$ After executing the $d$-dimensional Fourier Transforms, we conclude that both terms are divergent: in particular, $I_{\text{d}} \propto \frac{1}{q^{d-\eta}}$ and $I_{\text{a}}
\propto \frac{1}{q^{d-\eta-1}}$. This effect, shown in Fig. \[model\], is a clear sign of a quasi-long range positional ordered phase. We have found that the peak at $K+Q$ is smaller than the $K-Q$ one, as the potential between the impurity and CDW is repulsive (we would have the opposite asymmetry in case of an attractive potential). We observe that for an ideal infinite resolution experiment, the symmetric term would be dominant, since $C_{\text{d}} (r)$ decays to zero less rapidly than $C_{\text{a}}(r)$. However, if the divergence in (\[stima2\]) is cut by the finite resolution of the experiment both terms should be taken into account because $I_d$ is quadratic in the small parameter $K u_0$ whereas $I_a$ is only linear.
The powerlaw lineshape is obtained for a short range elasticity. If the Coulomb interaction is unscreened, as might be the case in fully gapped systems such as the blue bronzes, the dispersion of $c_1$ should be kept in (\[eq:anisotropy\]). In that case $c_1(q) \sim q_x^2/q^2$, which leads to peaks diverging even faster than (\[stima2\]) [@rosso_cdw_long].
On the experimental side few detailed diffraction spectra are available at the moment. One case is doped blue bronzes where the lineshape corresponding to the CDW has been obtained after substraction of a Friedel oscillation contribution [@rouzieres_structue_cdw]. The observed asymmetry of the peaks would be compatible with both strong and weak pinning. However given the short correlation length extracted from the data, this particular experiment is most likely still in the strong pinning regime. It would thus be highly desirable to have more detailed analysis of the lineshapes either in this compound, for different impurity concentrations, or in less disordered systems, where one can expect a Bragg glass behavior.
We thank J.-P. Poujet and S. Ravy for stimulating discussions.
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[^1]: The asymmetry of the two peaks can be very strong: it has been observed that the intensity of the lowest peak can even be smaller than the Laue scattering intensity. In this case one talks of white line in the spectrum.
|
---
abstract: 'In this article we review the theory of anafunctors introduced by Makkai and Bartels, and show that given a subcanonical site $S$, one can form a bicategorical localisation of various 2-categories of internal categories or groupoids at weak equivalences using anafunctors as 1-arrows. This unifies a number of proofs throughout the literature, using the fewest assumptions possible on $S$.'
address: |
School of Mathematical Sciences,\
University of Adelaide\
Adelaide, SA 5005\
Australia
author:
- David Michael Roberts
title: 'Internal categories, anafunctors and localisations'
---
Introduction {#section_1}
============
It is a well-known classical result of category theory that a functor is an equivalence (that is, in the 2-category of categories) if and only if it is fully faithful and essentially surjective. This fact is equivalent to the axiom of choice. It is therefore *not* true if one is working with categories internal to a category $S$ which doesn’t satisfy the (external) axiom of choice. This is may fail even in a category very much like the category of sets, such as a well-pointed boolean topos, or even the category of sets in constructive foundations. As internal categories are the objects of a 2-category $\Cat(S)$ we can talk about internal equivalences, and even fully faithful functors. In the case $S$ has a singleton pretopology $J$ (i.e. covering families consist of single maps) we can define an analogue of essentially surjective functors. Internal functors which are fully faithful and essentially surjective are called *weak equivalences* in the literature, going back to [@Bunge-Pare_79]. We shall call them $J$-equivalences for clarity. We can recover the classical result mentioned above if we localise the 2-category $\Cat(S)$ at the class $W_J$ of $J$-equivalences.
We are not just interested in localising $\Cat(S)$, but various full sub-2-categories $C {\hookrightarrow}\Cat(S)$ which arise in the study of presentable stacks, for example algebraic, topological, differentiable, etc. stacks. As such it is necessary to ask for a compatibility condition between the pretopology on $S$ and the sub-2-category we are interested in. We call this condition existence of *base change* for covers of the pretoplogy, and demand that for any cover $p\colon U\to X_0$ (in $S$) of the object of objects of $X\in C$, there is a fully faithful functor in $C$ with object component $p$.
Let $S$ be a category with singleton pretopology $J$ and let $C$ be a full sub-2-category of $\Cat(S)$ which admits base change along arrows in $J$. Then $C$ admits a calculus of fractions for the $J$-equivalences.
Pronk gives us the appropriate notion of a calculus of fractions for a 2-category in [@Pronk_96] as a generalisation of the usual construction for categories [@Gabriel-Zisman]. In her construction, 1-arrows are spans and 2-arrows are equivalence classes of bicategorical spans of spans. This construction, while canonical, can be a little unwieldy so we look for a simpler construction of the localisation.
We find this in the notion of *anafunctor*, introduced by Makkai for plain small categories [@Makkai] (Kelly described them briefly in [@Kelly_64] but did not develop the concept further). In his setting an anafunctor is a span of functors such that the left (or source) leg is a surjective-on-objects, fully faithful functor.[^1] For a general category $S$ with a *subcanonical* singleton pretopology $J$ [@Bartels], the analogon is a span with left leg a fully faithful functor with object component a cover. Composition of anafunctors is given by composition of spans in the usual way, and there are 2-arrows between anafunctors (a certain sort of span of spans) that give us a bicategory $\Cat_\ana(S,J)$ with objects internal categories and 1-arrows anafunctors. We can also define the full sub-bicategory $C_\ana(J) {\hookrightarrow}\Cat_\ana(S,J)$ analogous to $C$, and there is a strict inclusion 2-functor $C {\hookrightarrow}C_\ana(J)$. This gives us our second main theorem.
Let $S$ be a category with subcanonical singleton pretopology $J$ and let $C$ be a full sub-2-category of $\Cat(S)$ which admits base change along arrows in $J$, Then $C {\hookrightarrow}C_\ana(J)$ is a localisation of $C$ at the class of $J$-equivalences.
So far we haven’t mentioned the issue of size, which usually is important when constructing localisations. If the site $(S,J)$ is locally small, then $C$ is locally small, in the sense that the hom-categories are small. This also implies that $C_\ana(J)$ and hence any $C[W_J^{-1}]$ has *locally* small hom-categories i.e. has only a set of 2-arrows between any pair of 1-arrows. To prove that the localisation is locally essentially small (that is, hom-categories are equivalent to small categories), we need to assume a size restriction axiom on the pretopology $J$, called WISC (Weakly Initial Sets of Covers).
WISC can be seen as an extremely weak choice principle, weaker than the existence of enough projectives, and states that for every object $A$ of $S$, there is a set of $J$-covers of $A$ which is cofinal in all $J$-covers of $A$. It is automatically satisfied if the pretopology is specified as an assignment of a *set* of covers to each object.
Let $S$ be a category with subcanonical singleton pretopology $J$ satisfying WISC, and let $C$ be a full sub-2-category of $\Cat(S)$ which admits base change along arrows in $J$. Then any localisation of $C$ at the class of $J$-equivalences is locally essentially small.
Since a singleton pretopology can be conveniently defined as a certain wide subcategory, this is not a vacuous statement for large sites, such as $\Top$ or $\Grp(E)$ (group objects in a topos $E$). In fact WISC is independent of the Zermelo-Fraenkel axioms (without Choice) [@vdBerg_12; @Roberts_13]. It is thus possible to have the theorem fail for the topos $S = \Set_{\neg AC}$ with surjections as covers.
Since there have been many very closely related approaches to localisation of 2-categories of internal categories and groupoids, we give a brief sketch in the following section. Sections 3 to 6 of this article then give necessary background and notation on sites, internal categories, anafunctors and bicategories of fractions respectively. Section 7 contains our main results, while section 8 shows examples from the literature that are covered by the theorems from section 7. A short appendix detailing superextensive sites is included, as this material does not appear to be well-known (they were discussed in the recent [@Shulman_12], Example 11.12).
This article started out based on the first chapter of the author’s PhD thesis, which only dealt with groupoids in the site of topological spaces and open covers. Many thanks are due to Michael Murray, Mathai Varghese and Jim Stasheff, supervisors to the author. The patrons of the $n$-Category Café and $n$Lab, especially Mike Shulman and Toby Bartels, provided helpful input and feedback. Steve Lack suggested a number of improvements, and the referee asked for a complete rewrite of this article, which has greatly improved the theorems, proofs, and hopefully also the exposition. Any delays in publication are due entirely to the author.
Anafunctors in context
======================
The theme of giving 2-categories of internal categories or groupoids more equivalences has been approached in several different ways over the decades. We sketch a few of them, without necessarily finding the original references, to give an idea of how widely the results of this paper apply. We give some more detailed examples of this applicability in section 8.
Perhaps the oldest related construction is the distributors of Bénabou, also known as modules or profunctors [@Benabou_73] (see [@Elephant] for a detailed treatment of internal profunctors, as the original article is difficult to source). Bénabou pointed out [@Benabou_email], after a preprint of this article was released, that in the case of the category $\Set$ (and more generally in a finitely complete site with reflexive coequalisers that are stable under pullback, see [@MMV2012]), the bicategory of small (resp. internal) categories with representable profunctors as 1-arrows is equivalent to the bicategory of small categories with anafunctors as 1-arrows. In fact this was discussed by Baez and Makkai [@Baez-Makkai_emails], where the latter pointed out that representable profunctors correspond to *saturated* anafunctors in his setting. The author’s preference for anafunctors lies in the fact they can be defined with weaker assumptions on the site $(S,J)$, and in fact in the sequel [@Roberts2], do not require the 2-category to have objects which are internal categories. In a sense this is analogous to [@Street_80], where the formal bicategorical approach to profunctors between objects of a bicategory is given, albeit still requiring more colimits to exist than anafunctors do.
Bénabou has pointed out in private communication that he has an unpublished distributor-like construction that does not rely on existence of reflexive coequalisers; the author has not seen any details of this and is curious to see how it compares to anafunctors.
Related to this is the original work of Bunge and Paré [@Bunge-Pare_79], where they consider functors between indexed categories associated to internal categories, that is, the *externalisation* of an internal category and stack completions thereof. This was one motivation for considering weak equivalences in the first place, in that a pair of internal categories have equivalent stack completed externalisations if and only if they are connected by a span of internal functors which are weak equivalences.
Another approach is constructing bicategories of fractions à la Pronk [@Pronk_96]. This has been followed by a number of authors, usually followed up by an explicit construction of a localisation simplifying the canonical one. Our work here sits at the more general end of this spectrum, as others have tailored their constructions to take advantage of the structure of the site they are interested in. For example, *butterflies* (originally called papillons) have been used for the category of groups [@Noohi_05b; @Aldrovandi-Noohi_09; @Aldrovandi-Noohi_10], abelian categories [@Breckes_09] and semiabelian categories [@AMMV_10; @MMV2012]. These are similar to the meromorphisms of [@Pradines_89], introduced in the context of the site of smooth manifolds; though these only use a 1-categorical approach to localisation.
Vitale [@Vitale_10], after first showing that the 2-category of groupoids in a regular category has a bicategory of fractions, then shows that for protomodular regular categories one can generalise the pullback congruences of Bénabou in [@Benabou_89] to discuss bicategorical localisation. This approach can be applied to internal categories, as long as one restricts to invertible 2-arrows. Similarly, [@MMV2012] give a construction of what they call *fractors* between internal groupoids in a Mal’tsev category, and show that in an efficiently regular category (e.g. a Barr-exact category) fractors are 1-arrows in a localisation of the 2-category of internal groupoids. The proof also works for internal categories if one considers only invertible 2-arrows.
Other authors, in dealing with internal groupoids, have adopted the approach pioneered by Hilsum and Skandalis [@Hilsum-Skandalis_87], which has gone by various names including Hilsum-Skandalis morphisms, Morita morphisms, bimodules, bibundles, right principal bibundles and so on. All of these are very closely related to saturated anafunctors, but in fact no published definition of a saturated anafunctor in a site other than $\Set$ ([@Makkai]) has appeared, except in the guise of internal profunctors (e.g. [@Elephant], section B2.7). Note also that this approach has only been applied to internal groupoids. The review [@Lerman_10] covers the case of Lie groupoids, and in particular orbifolds, while [@Mrcun_01] treats bimodules between groupoids in the category of affine schemes, but from the point of view of Hopf algebroids.
The link between localisation at weak equivalences and presentable stacks is considered in (of course) [@Pronk_96], as well as more recently in [@Carchedi_12], [@Schappi_12], in the cases of topological and algebraic stacks respectively, and for example [@Tu-Xu-LaurentGengoux_04] in the case of differentiable stacks.
A third approach is by considering a model category structure on the 1-category of internal categories. This is considered in [@Joyal-Tierney_91] for categories in a topos, and in [@Everaert_et_al_05] for categories in a finitely complete subcanonical site $(S,J)$. In the latter case the authors show when it is possible to construct a Quillen model category structure on $\Cat(S)$ where the weak equivalences are the weak equivalences from this paper. Sufficient conditions on $S$ include being a topos with nno, being locally finitely presentable or being finitely complete regular Mal’tsev – and additionally having enough $J$-projective objects. If one is willing to consider other model-category-like structures, then these assumptions can be dropped. The proof from [@Everaert_et_al_05] can be adapted to show that for a finitely complete site $(S,J)$, the category of groupoids with source and target maps restricted to be $J$-covers has the structure of a category of fibrant objects, with the same weak equivalences. We note that [@Colman-Costoya_09] gives a Quillen model structure for the category of orbifolds, which are there defined to be proper topological groupoids with discrete hom-spaces.
In a similar vein, one could consider a localisation using *hammock* localisation [@Dwyer-Kan_80a] of a category of internal categories, which puts one squarely in the realm of $(\infty,1)$-categories. Alternatively, one could work with the $(\infty,1)$-category arising from a 2-category of internal categories, functors and natural *isomorphisms* and consider a localisation of this as given in, say [@Lurie_HTT]. However, to deal with general 2-categories of internal categories in this way, one needs to pass to $(\infty,2)$-categories to handle the non-invertible 2-arrows. The theory here is not so well-developed, however, and one could see the results of the current paper as giving toy examples with which one could work. This is one motivation for making sure the results shown in this paper apply to not just 2-categories of groupoids. Another is extending the theory of presentable stacks from stacks of groupoids to stacks of categories [@Roberts1].
Sites {#sites_categories}
=====
The idea of *surjectivity* is a necessary ingredient when talking about equivalences of categories—in the guise of just essential surjectivity—but it doesn’t generalise in a straightforward way from the category $\Set$. The necessary properties of the class of surjective maps are encoded in the definition of a Grothendieck pretopology, in particular a singleton pretopology. This section gathers definitions and notations for later use.
A *Grothendieck pretopology* (or simply *pretopology*) on a category $S
$ is a collection $J$ of families $$\{ (U_i \to A)_{i\in I} \}_{A\in \Obj(S)}$$ of morphisms for each object $A \in S$ satisfying the following properties
1. $(A' \stackrel{\sim}{\to} A)$ is in $J$ for every isomorphism $A'\simeq A$.
2. Given a map $B \to A$, for every $(U_i \to A)_{i\in I}$ in $J$ the pullbacks $B \times_A A_i$ exist and $(B \times_A A_i \to B)_{i\in I}$ is in $J$.
3. For every $(U_i \to A)_{i\in I}$ in $J$ and for a collection $(V_k^i \to
U_i)_{k\in K_i}$ from $J$ for each $i \in I$, the family of composites $$(V_k^i \to A)_{k\in K_i,i\in I}$$ are in $J$.
Families in $J$ are called *covering families*. We call a category $S$ equipped with a pretopology $J$ a *site*, denoted $(S,J)$ (note that often one sees a site defined as a category equipped with a Grothendieck *topology*).
The pretopology $J$ is called a *singleton* pretopology if every covering family consists of a single arrow $(U \to A)$. In this case a covering family is called a *cover* and we call $(S,J)$ a *unary* site.
Very often, one sees the definition of a pretopology as being an assignment of a *set* covering families to each object. We do not require this, as one can define a singleton pretopology as a subcategory with certain properties, and there is not necessarily then a set of covers for each object. One example is the category of groups with surjective homomorphisms as covers. This distinction will be important later.
One thing we will require is that sites come with *specified* pullbacks of covering families. If one does not mind applying the axiom of choice (resp. axiom of choice for classes) then any small site (resp. large site) can be so equipped. But often sites that arise in practice have more or less canonical choices for pullbacks, such as the category of ZF-sets.
The prototypical example is the pretopology $\mathcal{O}$ on $\Top$, where a covering family is an open cover. The class of numerable open covers (i.e. those that admit a subordinate partition of unity [@Dold_63]) also forms a pretopology on $\Top$. Much of traditional bundle theory is carried out using this site; for example the Milnor classifying space classifies bundles which are locally trivial over numerable covers.
\[defn:effective\_cov\_fam\] A covering family $(U_i \to A)_{i\in I} $ is called *effective* if $A$ is the colimit of the following diagram: the objects are the $U_i$ and the pullbacks $U_i \times_A
U_j$, and the arrows are the projections $$U_i \leftarrow U_i \times_A U_j \to U_j.$$ If the covering family consists of a single arrow $(U \to A)$, this is the same as saying $U \to A$ is a regular epimorphism.
A site is called *subcanonical* if every covering family is effective.
On $\Top$, the usual pretopology $\mathcal{O}$ of opens, the pretopology of numerable covers and that of open surjections are subcanonical.
In a regular category, the class of regular epimorphisms forms a subcanonical singleton pretopology.
In fact we can make the following definition.
For a category $S$, the largest class of regular epimorphisms of which all pullbacks exist, and which is stable under pullback, is called the *canonical singleton pretopology* and denoted ${\underline{c}}$.
This is a to be contrasted to the canonical *topology* on a category, which consists of covering sieves rather than covers. The canonical singleton pretopology is the largest subcanonical singleton pretopology on a category.
\[defn:saturation\] Let $(S,J)$ be a site. An arrow $P \to A$ in $S$ is called a *$J$-epimorphism* if there is a covering family $(U_i \to A)_{i\in I}$ and a lift $$\xymatrix{
& P \ar[d] \\
U_i \ar@{-->}[ur] \ar[r] & A
}$$ for every $i \in I$. A $J$-epimorphism is called *universal* if its pullback along an arbitrary map exists. We denote the singleton pretopology of universal $J$-epimorphisms by $J_{un}$.
This definition of $J$-epimorphism is equivalent to the definition in III.7.5 in [@MacLane-Moerdijk]. The dotted maps in the above definition are called local sections, after the case of the usual open cover pretopology on $\Top$. If $J$ is a singleton pretopology, it is clear that $J \subset J_{un}$.
The universal $\mathcal{O}$-epimorphisms for the pretopology $\mathcal{O}$ of open covers on $\Diff$ form $Subm$, the pretopology of surjective submersions.
\[eg:split\_epis\] In a finitely complete category the universal $triv$-epimorphisms are the split epimorphisms, where $triv$ is the *trivial pretopology* where all covering families consist of a single isomorphism. In $\Set$ with the axiom of choice there are all the epimorphisms.
Note that for a finitely complete site $(S,J)$, $J_{un}$ contains $triv_{un}$, hence all the split epimirphisms.
Although we will not assume that all sites we consider are finitely complete, results similar to ours have, and so in that case we can say a little more, given stronger properties on the pretopology.
A singleton pretopology $J$ is called *saturated* if whenever the composite $A \stackrel{h}{\to} B \stackrel{g}{\to} C$ is in $J$, then $g\in J$.
The concept of a saturated pretopology was introduced by Bénabou under the name *calibration* [@Benabou_75a]. It follows from the definition that a saturated singleton pretopology contains the split epimorphisms (take $h$ to be a section of the epimorphism $g$).
The canonical singleton pretopology ${\underline{c}}$ in a regular category (e.g. a topos) is saturated.
Given a pretopology $J$ on a finitely complete category, $J_{un}$ is saturated.
Sometimes a pretopology $J$ contains a smaller pretopology that still has enough covers to compute the same $J$-epimorphisms.
If $J$ and $K$ are two singleton pretopologies with $J \subset K$, such that $K \subset J_{un}$, then $J$ is said to be *cofinal* in $K$.
Clearly $J$ is cofinal in $J_{un}$ for any singleton pretopology $J$.
If $J$ is cofinal in $K$, then $J_{un} = K_{un}$.
We have the following lemma, which is essentially proved in [@Elephant], C2.1.6.
\[subcanonical\_goes\_up\_cofinal\] If a pretopology $J$ is subcanonical, then so any pretopology in which it is cofinal. In particular, $J$ subcanonical implies $J_{un}$ subcanonical.
As mentioned earlier, one may be given a singleton pretopology such that each object has more than a set’s worth of covers. If such a pretopology contains a cofinal pretopology with set-many covers for each object, then we can pass to the smaller pretopology and recover the same results (in a way that will be made precise later). In fact, we can get away with something weaker: one could ask only that the category of all covers of an object (see definition \[cover\_slice\] below) has a set of weakly initial objects, and such set may not form a pretopology. This is the content of the axiom WISC below. We first give some more precise definitions.
A category $C$ has a *weakly initial set* $\mathcal{I}$ of objects if for every object $A$ of $C$ there is an arrow $O\to A$ from some object $O\in \mathcal{I}$.
For example the large category $\Fields$ of fields has a weakly initial set, consisting of the prime fields $\{\mathbb{Q},\mathbb{F}_p|p\textrm{ prime}\}$. To contrast, the category of sets with surjections for arrows doesn’t have a weakly initial set of objects. Every small category has a weakly initial set, namely its set of objects.
We pause only to remark that the statement of the adjoint functor theorem can be expressed in terms of weakly initial sets.
\[cover\_slice\] Let $(S,J)$ be a site. For any object $A$, the *category of covers of $A$*, denoted $J/A$ has as objects the covering families $(U_i \to A)_{i\in I}$ and as morphisms $(U_i \to A)_{i\in I} \to (V_j \to A)_{j\in J}$ tuples consisting of a function $r\colon I\to J$ and arrows $U_i \to V_{r(i)}$ in $S/A$.
When $J$ is a singleton pretopology this is simply a full subcategory of $S/A$. We now define the axiom WISC (Weakly Initial Set of Covers), due independently to Mike Shulman and Thomas Streicher.
A site $(S,J)$ is said to *satisfy WISC* if for every object $A$ of $S$, the category $J/A$ has a weakly initial set of objects.
A site satisfying WISC is in some sense constrained by a small amount of data for each object. Any small site satisfies WISC, for example, the usual site of finite-dimensional smooth manifolds and open covers. Any pretopology $J$ containing a cofinal pretopology $K$ such that $K/A$ is small for every object $A$ satisfies WISC.
Any regular category (for example a topos) with enough projectives, equipped with the canonical singleton pretopology, satisfies WISC. In the case of $\Set$ ‘enough projectives’ is the Presentation Axiom (PAx), studied, for instance, by Aczel [@Aczel] in the context of constructive set theory.
$(\Top,\mathcal{O})$ satisfies WISC, using AC in $\Set$.
Choice may be more than is necessary here; it would be interesting to see if weaker choice principles in the site $(\Set,surjections)$ are enough to prove WISC for $(\Top,\mathcal{O})$ or other concrete sites.
If $(S,J)$ satisfies WISC, then so does $(S,J_{un})$.
It is instructive to consider an example where WISC fails in a non-artificial way. The category of sets and surjections with all arrows covers clearly doesn’t satisfy WISC, but is contrived and not a ‘useful’ sort of category. For the moment, assume the existence of a Grothendieck universe $\mathbb{U}$ with cardinality $\lambda$, and let $\mathrm{Set}_\mathbb{U}$ refer to the category of $\mathbb{U}$-small sets. Clearly we can define WISC relative to $\mathbb{U}$, call it WISC${}_\mathbb{U}$. Let $G$ be a $\mathbb{U}$-large group and $\mathbf{B}G$ the $\mathbb{U}$-large groupoid with one object associated to $G$. The boolean topos $\mathrm{Set}_\mathbb{U}^{\mathbf{B}G}$ of $\mathbb{U}$-small $G$-sets is a unary site with the class $epi$ of epimorphisms for covers. One could consider this topos as being an exotic sort of forcing construction.
\[U-small\_G-sets\] If $G$ has at least $\lambda$-many conjugacy classes of subgroups, then $(\mathrm{Set}_\mathbb{U}^{\mathbf{B}G},epi)$ does not satisfy WISC${}_\mathbb{U}$.
Alternatively, one could work in foundations where it is legitimate to discuss a proper class-sized group, and then consider the topos of sets with an action by this group. If there is a proper class of conjugacy classes of subgroups, then this topos with its canonical singleton pretopology will fail to satisfy WISC. Simple examples of such groups are $\mathbb{Z}^\mathbb{U}$ (given a universe $\mathbb{U}$) and $\mathbb{Z}^K$ (for some proper class $K$).
Recently, [@vdBerg_12] (relative to a large cardinal axiom) and [@Roberts_13] (with no large cardinals) have shown that the category of sets may fail to satisfy WISC. The models constructed in [@Karaglia_12] are also conjectured to not satisfy WISC.
Perhaps of independent interest is a form of WISC with a bound: the weakly initial set for each category $J/A$ has cardinality less than some cardinal $\kappa$ (call this WISC${}_\kappa$). Then one could consider, for example, sites where each object has a weakly initial finite or countable set of covers. Note that the condition ‘enough projectives’ is the case $\kappa = 2$.
Internal categories {#internal_cats}
===================
Internal categories were introduced in [@Ehresmann_63], starting with differentiable and topological categories (i.e. internal to $\Diff$ and $\Top$ respectively). We collect here the necessary definitions, terminology and notation. For a thorough recent account, see [@HDA5] or the encyclopedic [@Elephant].
Fix a category $S$, referred to as the *ambient category*.
\[def:cat\] An *internal category* $X$ in a category $S$ is a diagram $$X_1 \times_{X_0} X_1 \times_{X_0} X_1\rightrightarrows X_1 \times_{X_0}
X_1 \xrightarrow{m} X_1 \stackrel{s,t}{{\rightrightarrows}} X_0 \xrightarrow{e} X_1$$ in $S$ such that the *multiplication* $m$ is associative (we demand the limits in the diagram exist), the *unit map* $e$ is a two-sided unit for $m$ and $s$ and $t$ are the usual *source* and *target*. An *internal groupoid* is an internal category with an involution $$(-)^{-1}\colon X_1 \to X_1$$ satisfying the usual diagrams for an inverse.
Since multiplication is associative, there is a well-defined map $X_1 \times_{X_0} X_1 \times_{X_0} X_1 \to X_1$, which will also be denoted by $m$. The pullback in the diagram in definition \[def:cat\] is $$\xymatrix{
X_1 \times_{X_0} X_1 \ar[r] \ar[d] & X_1 \ar[d]^-{s}\\
X_1 \ar[r]_-{t} & X_0\;.
}$$ and the double pullback is the limit of $X_1 \stackrel{t}{\rightarrow} X_0 \stackrel{s}{\leftarrow} X_1
\stackrel{t}{\rightarrow} X_0 \stackrel{s}{\leftarrow}X_0$. These, and pullbacks like these (where source is pulled back along target), will occur often. If confusion can arise, the maps in question will be explicity written, as in $X_1 \times_{s,X_0,t} X_1$. One usually sees the requirement that $S$ is finitely complete in order to define internal categories. This is not strictly necessary, and not true in the well-studied case of $S = \Diff$, the category of smooth manifolds.
Often an internal category will be denoted $X_1 {\rightrightarrows}X_0$, the arrows $m,s,t,e$ (and $(-)^{-1}$) will be referred to as *structure maps* and $X_1$ and $X_0$ called the object of arrows and the object of objects respectively. For example, if $S = \Top$, we have the space of arrows and the space of objects, for $S = \Grp$ we have the group of arrows and so on.
\[eg:cech\_gpd\] If $X \to Y$ is an arrow in $S$ admitting iterated kernel pairs, there is an internal groupoid $\check{C}(X)$ with $\check{C}(X)_0 = X$, $\check{C}(X)_1 = X \times_Y X$, source and target are projection on first and second factor, and the multiplication is projecting out the middle factor in $X \times_Y X \times_Y X$. This groupoid is called the *Čech groupoid* of the map $X \to Y$. The origin of the name is that in $\Top$, for maps of the form $\coprod_I U_i \to Y$ (arising from an open cover), the Čech groupoid $\check{C}(\coprod_I U_i)$ appears in the definition of Čech cohomology.
\[eg:disc-codisc\_gpd\] Let $S$ be a category with binary products. For each object $A \in S$ there is an internal groupoid $\disc(A)$ which has $\disc(A)_1 = \disc(A)_0 = A$ and all structure maps equal to $id_A$. Such a category is called *discrete*. There is also an internal groupoid $\codisc(A)$ with $$\codisc(A)_0 = A,\
\codisc(A)_1 = A \times A$$ and where source and target are projections on the first and second factor respectively. Such a groupoid is called *codiscrete*.
\[def:functor\] Given internal categories $X$ and $Y$ in $S$, an *internal functor* is a pair of maps $$f_0\colon X_0 \to Y_0 \quad\textrm{and}\quad f_1\colon X_1 \to Y_1$$ called the object and arrow component respectively. Both components are required to commute with all the structure maps.
If $A\to C$ and $B\to C$ are maps admitting iterated kernel pairs, and $A \to B$ is a map over $C$, there is a functor $\check{C}(A) \to \check{C}(B)$.
\[functors2discrete\] If $(S,J)$ is a subcanonical unary site, and $U \to A$ is a cover, a functor $\check{C}(U) \to \disc(B)$ gives a unique arrow $A\to B$. This follows immediately from the fact $A$ is the colimit of the diagram underlying $\check{C}(U)$.
\[def:nat\_iso\] Given internal categories $X,Y$ and internal functors $f,g\colon X \to Y$, an *internal natural transformation* (or simply *transformation*) $$a\colon f \Rightarrow g$$ is a map $a\colon X_0 \to Y_1$ such that $s \circ a = f_0,\ t\circ a = g_0$ and the following diagram commutes $$\label{diag:naturality}
\xymatrix{
X_1 \ar[r]^-{(g_1,a\circ s)} \ar[d]_{(a \circ t,f_1)} &
Y_1 \times_{Y_0} Y_1 \ar[d]^{m} \\
Y_1 \times_{Y_0} Y_1 \ar[r]^-{m} & Y_1
}$$ expressing the naturality of $a$.
Internal categories (resp. groupoids), functors and transformations in a locally small category $S$ form a locally small 2-category $\Cat(S)$ (resp. $\Gpd(S)$) [@Ehresmann_63]. There is clearly an inclusion 2-functor $\Gpd(S) \to \Cat(S)$. Also, $\disc$ and $\codisc$, described in example \[eg:disc-codisc\_gpd\], are 2-functors $S \to \Gpd(S)$, whose underlying functors are left and right adjoint to the functor $$\Obj\colon\Cat(S)_{\leq 1} \to S,\qquad (X_1{\rightrightarrows}X_0)\mapsto X_0.$$ Here $\Cat(S)_{\leq 1}$ is the 1-category underlying the 2-category $\Cat(S)$. Hence for an internal category $X$ in $S$, there are functors $\disc(X_0) \to X$ and $X \to \codisc(X_0)$, the arrow component of the latter being $(s,t):X_1\to X_0^2$.
We say a natural transformation is a *natural isomorphism* if it has an inverse with respect to vertical composition. Clearly there is no distinction between natural transformations and natural isomorphisms when the codomain of the functors is an internal groupoid. We can reformulate the naturality diagram (\[diag:naturality\]) in the case that $a$ is a natural isomorphism. Denote by $-a$ the inverse of $a$. Then the diagram (\[diag:naturality\]) commutes if and only if the diagram $$\label{naturality}
\xymatrix{
X_0 \times_{X_0} X_1 \times_{X_0} X_0
\ar[rr]^{-a\times f_1 \times a}
\ar[d]_{\simeq} &&Y_1 \times_{Y_0} Y_1 \times_{Y_0} Y_1 \ar[d]^m \\
X_1 \ar[rr]_{g_1} && Y_1
}$$ commutes, a fact we will use several times.
If $X$ is a category in $S$, $A$ is an object of $S$ and $f,g:X \to \codisc(A)$ are functors, there is a unique natural isomorphism $f\stackrel{\sim}{\Rightarrow} g$.
An *internal* or *strong equivalence* of internal categories is an equivalence in the 2-category of internal categories. That is, an internal functor $f
\colon X\to Y$ such that there is a functor $f'\colon Y\to X$ and natural isomorphisms $f\circ f' \Rightarrow \id_Y$, $f'\circ f \Rightarrow \id_X$.
\[induced\_cat\] For an internal category $X$ and a map $p:M\to X_0$ in $S$ the *base change of $X$ along $p$* is any category $X[M]$ with object of objects $M$ and object of arrows given by the pullback $$\xymatrix{
M^2 \times_{X_0^2} X_1 \ar[r] \ar[d] & X_1 \ar[d]^{(s,t)} \\
M^2 \ar[r]_{p^2} & X_0^2
}$$
If $C\subset \Cat(S)$ denotes a full sub-2-category and if the base change along any map in a given class $K$ of maps exists in $C$ for all objects of $C$, then we say $C$ *admits base change along maps in $K$*, or simply *admits base change for $K$*.
In all that follows, ‘category’ will mean object of $C$ and similarly for ‘functor’ and ‘natural transformation/isomorphism’.
The strict pullback of internal categories $$\xymatrix{
X \times_Y Z \ar[r] \ar[d] & Z \ar[d] \\
X \ar[r] & Y
}$$ when it exists, is the internal category with objects $X_0 \times_{Y_0} Z_0$, arrows $X_1 \times_{Y_1} Z_1$, and all structure maps given componentwise by those of $X$ and $Z$. Often we will be able to prove that certain pullbacks exist because of conditions on various component maps in $S$. We do not assume that all strict pullbacks of internal categories exists in our chosen $C$.
It follows immediately from definition \[induced\_cat\] that given maps $N\to M$ and $M\to X_0$, there is a canonical isomorphism $$\label{induced_cat_1}
X[M][N] \simeq X[N].$$ with object component the identity map, when these base changes exist.
\[remark:pullback\_of\_x\_along\_x\] If we agree to follow the convention that $M \times_N N = M$ is the pullback along the identity arrow $\id_N$, then $X[X_0] = X$. This also simplifies other results of this paper, so will be adopted from now on.
One consequence of this assumption is that the iterated fibre product $$M\times_M M \times_M \ldots \times_M M,$$ bracketed in any order, is *equal* to $M$. We cannot, however, equate two bracketings of a general iterated fibred product; they are only canonically isomorphic.
\[strict\_pullbacks\_triv\_J\_fibrations\] Let $Y\to X$ be a functor in $S$ and $j_0\colon U \to X_0$ a map. If the base change along $j_0$ exists, the following square is a strict pullback $$\xymatrix{
Y[Y_0\times_{X_0}U] \ar[r] \ar[d] & X[U] \ar[d]^j \\
Y \ar[r] & X
}$$ assuming it exists.
Since base change along $j_0$ exists, we know that we have the functor $Y[Y_0\times_{X_0}U] \to Y$, we just need to show it is a strict pullback of $j$. On the level of objects this is clear, and on the level of arrows, we have $$\begin{aligned}
(Y_0\times_{X_0}U)^2 \times_{Y_0^2}Y_1 &\simeq U^2\times_{X_0^2} Y_1\\
&\simeq (U^2\times_{X_0^2}X_1) \times_{X_1}Y_1 \\
&\simeq X[U]_1\times_{X_1}Y_1\end{aligned}$$ so the square is a pullback.
We are interested in 2-categories $C$ which admits base change for a given pretopology $J$ on $S$, which we shall cover in more detail in section \[examples\].
Equivalences in $\Cat$—assuming the axiom of choice—are precisely the fully faithful, essentially surjective functors. For internal categories, however, this is not the case. In addition, we need to make use of a pretopology to make the ‘surjective’ part of ‘essentially surjective’ meaningful.
\[def:weak\_equiv\] Let $(S,J)$ be a unary site. An internal functor $f:X \to Y$ in $S$ is called
1. *fully faithful* if $$\xymatrix{
X_1 \ar[r]^{f_1} \ar[d]_{(s,t)} & Y_1 \ar[d]^{(s,t)}\\
X_0 \times X_0 \ar[r]_{f_0 \times f_0} & Y_0 \times Y_0
}$$ is a pullback diagram;
2. *$J$-locally split* if there is a $J$-cover $U\to Y_0$ and a diagram $$\xymatrix{
Y[U] \ar[d]_{\bar f} \ar@/^.5pc/[dr]_{\ }="s1"^{u}& \\
X\ar[r]_{f}^(.33){\ }="t1"&Y
\ar@{=>}"s1";"t1"
}$$ commuting up to a natural isomorphism;
3. a *$J$-equivalence* if it is fully faithful and $J$-locally split.
The class of $J$-equivalences will be denoted $W_J$. If mention of $J$ is suppressed, they will be called *weak equivalences*.
There is another defintion of full faithfulness for internal categories, namely that of a functor $f\colon Z\to Y$ being *representably fully faithful*. This means that for all categories $Z$, the functor $$f_\ast\colon \Cat(S)(Z,X) \to \Cat(S)(Z,Y)$$ is fully faithful. It is a well-known result that these two notions coincide, so we shall use either characterisation as needed.
If $f:X \to Y$ is a fully faithful functor such that $f_0$ is in $J$, then $f$ is $J$-locally split.
That is, the canonical functor $X[U] \to X$ is a $J$-equivalence whenever the base change exists. Also, we do not require that $J$ is subcanonical. We record here a useful lemma.
\[rep\_ff\_functors\_closed\_under\_iso\] Given a fully faithful functor $f\colon X \to Y$ in $C$ and a natural isomorphism $f \Rightarrow g$, the functor $g$ is also fully faithful. In particular, an internal equivalence is fully faithful.
This is a simple application of the definition of representable full faithfulness and the fact that the result is true in $\Cat$.
The first definition of weak equivalence of internal categories along the lines we are considering appeared in [@Bunge-Pare_79] for $S$ a regular category, and $J$ the class of regular epimorphisms (i.e. ${\underline{c}}$), in the context of stacks and indexed categories. This was later generalised in [@Everaert_et_al_05] to more general finitely complete sites to discuss model structures on the category of internal categories. Both work only with saturated singleton pretopologies.
Note that when $S$ is finitely complete, the object $X_1^{iso} {\hookrightarrow}X_1$ of isomorphisms of a category $X$ can be constructed as a finite limit [@Bunge-Pare_79], and in the case when $X$ is a groupoid we have $X_1^{iso} \simeq X_1$.
[[@Bunge-Pare_79; @Everaert_et_al_05]]{}\[def:PB\_weak\_equiv\] For a finitely complete unary site $(S,J)$ with $J$ saturated, a functor $f$ is called *essentially $J$-surjective* if the arrow labelled $\circledast$ below is in $J$. $$\xymatrix{
&\ar[dl] X_0 \times_{Y_0} Y_1^{iso} \ar@/^1pc/[ddr]^\circledast \ar[d]&\\
X_0 \ar[d]_{f_0} & \ar[dl]^s Y_1^{iso} \ar[dr]_t &\\
Y_0 && Y_0
}$$ A functor is called a *Bunge-Paré $J$-equivalence* if it is fully faithful and essentially $J$-surjective. Denote the class of such maps by $W_J^{BP}$.
Definition \[def:weak\_equiv\] is equivalent to the one in [@Bunge-Pare_79; @Everaert_et_al_05] in the sites they consider but seems more appropriate for sites without all finite limits. Also, definition \[def:weak\_equiv\] makes sense in 2-categories other than $\Cat(S)$ or sub-2-categories thereof.
\[BP\_equiv\_iff\_weak\_equiv\] Let $(S,J)$ be a finitely complete unary site with $J$ saturated. Then a functor is a $J$-equivalence if and only if it is a Bunge-Paré $J$-equivalence.
Let $f\colon X \to Y$ be a Bunge-Paré $J$-equivalence, and consider the $J$-cover given by the map $U := X_0 \times_{Y_0} Y_1^{iso} \to Y_0$. Denote by $\iota\colon U\to Y_1^{iso}$ the projection on the second factor, by $-\iota$ the composite of $\iota$ with the inversion map $(-)^{-1}$ and by $s_0\colon U\to X_0$ the projection on the first factor. The arrow $s_0$ will be the object component of a functor $s\colon Y[U] \to X$, we need to define the arrow component $s_1$. Consider the composite $$\begin{aligned}
Y[U]_1 \simeq U\times_{Y_0} Y_1 \times_{Y_0} U \xrightarrow{(s,\iota)\times\id\times(-\iota,s)}
(X_0 \times_{Y_0} Y_1^{iso}) \times_{Y_0} Y_1 \times_{Y_0} ( Y_1^{iso} \times_{Y_0} X_0) \\
\hookrightarrow X_0 \times_{Y_0} Y_3 \times_{Y_0} X_0 \xrightarrow{\id\times m\times\id}
X_0 \times_{Y_0} Y_1 \times_{Y_0} X_0 \simeq X_1\end{aligned}$$ where the last isomorphism arises from $f$ being fully faithful. It is clear that this commutes with source and target, because these are given by projection on the first and last factor at each step. To see that it respects identities and composition, one can use generalised elements and the fact that the $\iota$ component will cancel with the $-\iota = (-)^{-1}\circ \iota$ component.
We define the natural isomorphism $f\circ s \Rightarrow j$ (here $j\colon Y[U] \to Y$ is the canonical functor) to have component $\iota$ as denoted above. Notice that the composite $f_1\circ s_1$ is just $$Y[U]_1 \simeq U \times_{Y_0} Y_1 \times_{Y_0} U \xrightarrow{\iota\times\id
\times -\iota} Y_1^{iso} \times_{Y_0} Y_1 \times_{Y_0} Y_1^{iso} \hookrightarrow
Y_3 \xrightarrow{m} Y_1.$$ Since the arrow component of $Y[U] \to Y$ is $U \times_{Y_0} Y_1 \times_{Y_0} U
\xrightarrow{\pr_2} Y_1$, $\iota$ is indeed a natural isomorphism using the diagram (\[naturality\]). Thus a Bunge-Paré $J$-equivalence is a $J$-equivalence.
In the other direction, given a $J$-equivalence $f\colon X\to Y$, we have a $J$-cover $j\colon U\to Y_0$ and a map $(\overline{f},a)\colon U \to X_0 \times Y_1^{iso}$ such that $j = (t\circ pr_2)\circ(\overline{f},a)$. Since $J$ is saturated, $(t\circ pr_2)\in J$ and hence $f$ is a Buge-Paré $J$-equivalence.
We can thus use definition \[def:weak\_equiv\] as we like, and it will still refer to the same sorts of weak equivalences that appear in the literature.
Anafunctors
===========
We now let $J$ be a *subcanonical* singleton pretopology on the ambient category $S$. In this section we assume that $C{\hookrightarrow}\Cat(S)$ admits base change along arrows in the given pretopology $J$. This is a slight generalisation of what is considered in [@Bartels], where only $C = \Cat(S)$ is considered.
[[@Makkai; @Bartels]]{}\[def:anafunctor\] An *anafunctor* in $(S,J)$ from a category $X$ to a category $Y$ consists of a $J$-cover $(U \to X_0)$ and an internal functor $$f\colon X[U] \to Y.$$ Since $X[U]$ is an object of $C$, an anafunctor is a span in $C$, and can be denoted $$(U,f)\colon X {-\!\!\!\mapsto}Y.$$
\[eg:ordinary\_functor\] For an internal functor $f\colon X \to Y$ in $S$, define the anafunctor $(X_0,f)\colon
X {-\!\!\!\mapsto}Y$ as the following span $$X \xleftarrow{=} X[X_0] \xrightarrow{f} Y.$$ We will blur the distinction between these two descriptions. If $f=id\colon X \to X$, then $(X_0,id)$ will be denoted simply by $id_X$.
If $U \to A$ is a cover in $(S,J)$ and $\mathbf{B}G$ is a groupoid with one object in $S$ (i.e. a group in $S$), an anafunctor $(U,g)\colon\disc(A) {-\!\!\!\mapsto}\mathbf{B}G$ is the same thing as a Čech cocycle.
[[@Makkai; @Bartels]]{} Let $(S,J)$ be a site and let $$(U,f),(V,g)\colon X {-\!\!\!\mapsto}Y$$ be anafunctors in $S$. A *transformation* $$\alpha\colon (U,f) \Rightarrow (V,g)$$ from $(U,f)$ to $(V,g)$ is a natural transformation $$\xymatrix{
& \ar[dl] X[U\times_{X_0}V] \ar[dr] & \\
X[U] \ar[dr]_f & \stackrel{\alpha}{\Rightarrow} & X[V] \ar[dl]^g\\
& Y &
}$$ If $\alpha$ is a natural isomorphism, then $\alpha$ will be called an *isotransformation*. In that case we say $(U,f)$ is isomorphic to $(V,g)$. Clearly all transformations between anafunctors between internal groupoids are isotransformations.
\[eg:ordinary\_transf\] Given functors $f,g\colon X \to Y$ between categories in $S$, and a natural transformation $a\colon f \Rightarrow g$, there is a transformation $a\colon (X_0,f)
\Rightarrow (X_0,g)$ of anafunctors, given by the component $X_0\times_{X_0}X_0
= X_0 \xrightarrow{a} Y_1$.
If $(U,g),(V,h)\colon \disc(A) {-\!\!\!\mapsto}\mathbf{B}G$ are two Čech cocycles, a transformation between them is a coboundary on the cover $U\times_A V\to A$.
Let $(U,f)\colon X {-\!\!\!\mapsto}Y$ be an anafunctor in $S$. There is an isotransformation $1_{(U,f)}\colon (U,f) \Rightarrow (U,f)$ called the *identity transformation*, given by the natural transformation with component $$\label{id_transf_component}
U \times_{X_0} U \simeq (U \times U) \times_{X_0^2} X_0 \xrightarrow{id_U^2
\times e} X[U]_1 \xrightarrow{f_1} Y_1$$
[[@Makkai]]{}\[renaming\_transf\] Given anafunctors $(U,f)\colon X\to Y$ and $(V,f\circ k)\colon X \to Y$ where $k
\colon V\to U$ is a cover (over $X_0$), a *renaming transformation* $$(U,f)\Rightarrow(V,f\circ k)$$ is an isotransformation with component $$1_{(U,f)}\circ (k\times \id):V\times_{X_0} U \to U\times_{X_0} U \to Y_1.$$ (We also call its inverse for vertical composition a renaming transformation.) If $k$ is an isomorphism, then it will itself be referred to as a *renaming isomorphism*.
We define (following [@Bartels]) the composition of anafunctors as follows. Let $$(U,f)\colon X {-\!\!\!\mapsto}Y \quad \textrm{and} \quad (V,g)\colon Y {-\!\!\!\mapsto}Z$$ be anafunctors in the site $(S,J)$. Their composite $(V,g)\circ(U,f)$ is the composite span defined in the usual way. It is again a span in $C$: $$\xymatrix{
&& \ar[dl] X[U\times_{Y_0}V] \ar[dr]^{f^V} & \\
&\ar[dl]X[U] \ar[dr]_f & & Y[V] \ar[dl] \ar[dr]^g\\
X&& Y &&Z
}$$ The square is a pullback by lemma \[strict\_pullbacks\_triv\_J\_fibrations\] (which exists because $V\to
Y_0$ is a cover), and the resulting span is an anafunctor because $V \to Y_0$, hence $U\times_{Y_0}V\to X_0$, are covers, and using the isomorphism (\[induced\_cat\_1\]). We will sometimes denote the composite by $(U\times_{Y_0}V,g\circ f^V)$.
Here we are using the fact we have specified pullbacks of covers in $S$. Without this we would not end up with a bicategory (see theorem \[anafunctors\_are\_a\_bicat\]), but what [@Makkai] calls an *anabicategory*. This is similar to a bicategory, but composition and other structural maps are only anafunctors, not functors.
Consider the special case when $V = Y_0$, so that $(Y_0,g)$ is just an ordinary functor. Then there is a renaming transformation (the identity transformation!) $(Y_0,g)\circ(U,f) \Rightarrow (U,g\circ f)$, using the equality $U \times_{Y_0} Y_0= U$ (by remark \[remark:pullback\_of\_x\_along\_x\]). If we let $g=\id_Y$, then we see that $(Y_0,\id_Y)$ is a strict unit on the left for anafunctor composition. Similarly, considering $(V,g)\circ(Y_0,\id)$, we see that $(Y_0,\id_Y)$ is a two-sided strict unit for anafunctor composition. In fact, we have also proved
\[coherent\_composition\] Given two functors $f\colon X\to Y$, $g\colon Y \to Z$ in $S$, their composition as anafunctors is equal to their composition as functors: $$(Y_0,g)\circ(X_0,f) = (X_0,g\circ f).$$
As a concrete and relevant example of a renaming transformation we can consider the triple composition of anafunctors $$\begin{aligned}
(U,f)\colon & X {-\!\!\!\mapsto}Y,\\
(V,g)\colon & Y {-\!\!\!\mapsto}Z,\\
(W,h)\colon & Z {-\!\!\!\mapsto}A.\end{aligned}$$ The two possibilities of composing these are $$\left((U\times_{Y_0} V)\times_{Z_0}W,h\circ(gf^V)^W\right)\quad \text{and}\quad \left(U
\times_{Y_0} (V\times_{Z_0} W),h\circ g^W\circ f^{V\times_{Z_0}W}\right).$$
\[lemma:associator\] The unique isomorphism $(U\times_{Y_0} V)\times_{Z_0}W \simeq U\times_{Y_0} (V
\times_{Z_0} W)$ commuting with the various projections is a renaming isomorphism. The isotransformation arising from this renaming transformation is called the *associator*.
A simple but useful criterion for describing isotransformations where one of the anafunctors involved is a functor is as follows.
\[anafun\_iso2\_fun\] An anafunctor $(V,g)\colon X {-\!\!\!\mapsto}Y$ is isomorphic to a functor $(X_0,f)\colon X {-\!\!\!\mapsto}Y$ if and only if there is a natural isomorphism $$\xymatrix{
& \ar[dl] X[V] \ar[dr]^g \\
X \ar@/_1.5pc/[rr]_(.6){f}& \stackrel{\sim}{\Rightarrow} & Y
}$$
Just as there is a vertical composition of natural transformations between internal functors, there is a vertical composition of transformations between internal anafunctors [@Bartels]. This is where the subcanonicity of $J$ will be used in order to construct a map locally over some cover. Consider the following diagram $$\xymatrix{
&& \ar[dl] X[U\times_{X_0} V\times_{X_0} W] \ar[dr]\\
& \ar[dl] X[U\times_{X_0} V] \ar[dr] &
& \ar[dl] X[V\times_{X_0} W] \ar[dr]\\
X[U] \ar[drr]_f & \stackrel{a}{\Rightarrow} & X[V] \ar[d]^g& \stackrel{b}
{\Rightarrow} & X[W] \ar[dll]^h \\
&&Y&&
}$$ We can form a natural transformation between the leftmost and the rightmost composites as functors in $S$. This will have as its component the arrow $$\widetilde{ba}\colon U\times_{X_0} V\times_{X_0} W \xrightarrow{\id\times \Delta
\times \id} U\times_{X_0}V\times_{X_0}V\times_{X_0} W \xrightarrow{a\times b}
Y_1\times_{Y_0} Y_1 \xrightarrow{m} Y_1$$ in $S$. Notice that the Čech groupoid of the cover $$\label{iterated_cover}
U\times_{X_0} V\times_{X_0} W \to U \times_{X_0} W$$ is $$U\times_{X_0} V\times_{X_0} V\times_{X_0} W {\rightrightarrows}U\times_{X_0} V\times_{X_0} W,$$ with source and target arising from the two projections $V\times_{X_0} V \to V$. Denote this pair of parallel arrows by $s,t\colon UV^2W {\rightrightarrows}UVW$ for brevity. In [@Bartels], section 2.2.3, we find the commuting diagram $$\label{tobys_diag}
\xymatrix{
UV^2W \ar[r]^t \ar[d]_s & UVW \ar[d]^{\widetilde{ba}}\\
UVW \ar[r]_{\widetilde{ba}} & Y_1
}$$ (this can be checked by using generalised elements) and so we have a functor $$\check{C}(U\times_{X_0} V\times_{X_0} W) \to \disc(Y_1).$$ Our pretopology $J$ is assumed to be subcanonical, so example \[functors2discrete\] gives us a unique arrow $ba\colon U\times_{X_0} W \to Y_1$, which is the data for the composite of $a$ and $b$.
In the special case that $U\times_{X_0} V\times_{X_0} W \to U \times_{X_0} W$ is split (e.g. is an isomorphism), the composite transformation has $$U \times_{X_0} W\to U\times_{X_0} V\times_{X_0} W \xrightarrow{\widetilde{ba}} Y_1$$ as its component arrow. In particular, this is the case if one of $a$ or $b$ is a renaming transformation.
\[eg:transf\_compose2\] Let $(U,f):X{-\!\!\!\mapsto}Y$ be an anafunctor and $U'' \xrightarrow{j'} U' \xrightarrow{j} U$ successive refinements of $U \to X_0$ (e.g isomorphisms). Let $(U',f_{U'})$ and $(U'',f_{U''})$ denote the composites of $f$ with $X[U'] \to X[U]$ and $X[U''] \to X[U]$ respectively. The arrow $$U \times_{X_0} U'' \xrightarrow{\id_U\times j\circ j'} U \times_{X_0} U \to Y_1$$ is the component for the composition of the isotransformations $(U,f)
\Rightarrow(U',f_{U'}),\Rightarrow(U'',f_{U''})$ described in example \[renaming\_transf\]. Thus we can see that the composite of renaming transformations associated to isomorphisms $\phi_1,\phi_2$ is simply the renaming transformation associated to their composite $\phi_1\circ \phi_2$.
This can be used to show that the associator satisfies the necessary coherence conditions.
\[eg:transf\_compose1\] If $a\colon f\Rightarrow g,\ b\colon g\Rightarrow h$ are natural transformations between functors $f,g,h\colon X\to Y$ in $S$, their composite as transformations between anafunctors $$(X_0,f),(X_0,g),(X_0,h)\colon X{-\!\!\!\mapsto}Y.$$ is just their composite as natural transformations. This uses the equality $$X_0\times_{X_0} X_0\times_{X_0} X_0= X_0\times_{X_0} X_0 = X_0,$$ which is due to our choice in remark \[remark:pullback\_of\_x\_along\_x\] of canonical pullbacks.
Even though we don’t have pseudoinverses for weak equivalences of internal categories, one might guess that the local splitting guaranteed to exist by definition is actually more than just a splitting of sorts. This is in fact the case, if we use anafunctors.
\[anafunctors-r-inverses\] Let $f\colon X \to Y$ be a $J$-equivalence in $S$. There is an anafunctor $$(U,\bar{f})\colon Y {-\!\!\!\mapsto}X$$ and isotransformations $$\begin{aligned}
\iota\colon (X_0,f)\circ (U,\bar{f}) & \Rightarrow id_Y\\
\epsilon\colon(U,\bar{f})\circ (X_0,f) & \Rightarrow id_X\end{aligned}$$
We have the anafunctor $(U,\bar{f})$ by definition as $f$ is $J$-locally split. Since the anafunctors $\id_X,\ \id_Y$ are actually functors, we can use lemma \[anafun\_iso2\_fun\]. Using the special case of anafunctor composition when the second is a functor, this tells us that $\iota$ will be given by a natural isomorphism $$\xymatrix{
& X \ar[dr]^{f}_(0.2){\ }="s" & \\
Y[U] \ar[rr]^{\ }="t" \ar[ur]^{\bar{f}} && Y
\ar@{=>}"s";"t"
}$$ with component $\iota\colon U \to Y_1$. Notice that the composite $f_1\circ \bar{f}_1$ is just $$Y[U]_1 \simeq U \times_{Y_0} Y_1 \times_{Y_0} U \xrightarrow{\iota\times\id
\times -\iota} Y_1 \times_{Y_0} Y_1 \times_{Y_0} Y_1 \hookrightarrow
Y_3 \xrightarrow{m} Y_1.$$ Since the arrow component of $Y[U] \to Y$ is $U \times_{Y_0} Y_1 \times_{Y_0} U
\xrightarrow{\pr_2} Y_1$, $\iota$ is indeed a natural isomorphism using the diagram (\[naturality\]).
The other isotransformation $\epsilon$ is between $(X_0\times_{Y_0} U,\bar{f}\circ \pr_2)$ and $(X_0,\id_X)$, and is given by the component $$\epsilon\colon X_0 \times_{X_0} X_0\times_{Y_0} U = X_0\times_{Y_0} U
\xrightarrow{\id\times (\bar{f}_0,\iota)} X_0\times_{Y_0} (X_0\times_{Y_0} Y_1) \simeq X_0^2
\times_{Y_0^2} Y_1 \simeq X_1$$ The diagram $$\xymatrix{
(X_0\times_{Y_0^2} U)^2 \times_{X_0^2} X_1 \ar[d]_\simeq \ar[rr]^{\pr_2}
& &X_1 \ar[dd]^\simeq\\
U \times_{Y_0} X_1 \times_{Y_0}U \ar[d]_{-\iota\times f\times\iota} & \\
(X_0 \times_{Y_0} Y_1) \times_{Y_0} Y_1 \times_{Y_0} (Y_1 \times_{Y_0} X_0)
\ar[rr]_(.6){\id\times m \times \id}
&& X_0\times_{Y_0} Y_1 \times_{Y_0} X_0
}$$ commutes (a fact which can be checked using generalised elements), and using (\[naturality\]) we see that $\epsilon$ is natural.
The first half of the following theorem is proposition 12 in [@Bartels], and the second half follows because all the constructions of categories involved in dealing with anafunctors outlined above are still objects of $C$.
[[@Bartels]]{}\[anafunctors\_are\_a\_bicat\] For a site $(S,J)$ where $J$ is a subcanonical singleton pretopology, internal categories, anafunctors and transformations form a bicategory $\Cat_\ana(S,J)$. If we restrict attention to a full sub-2-category $C$ which admits base change for arrows in $J$, we have an analogous full sub-bicategory $C_\ana(J)$.
In fact the bicategory $C_{ana}(J)$ fails to be a strict 2-category only in the sense that the associator is given by the non-identity isotransformation from lemma \[lemma:associator\]. All the other structure is strict.
There is a strict 2-functor $C_\ana(J) \to \Cat_\ana(S,J)$ which is an inclusion on objects and fully faithful in the strictest sense, namely being the identity functor on hom-categories. The following is the main result of this section, and allows us to relate anafunctors to the localisations considered in the next section.
\[W-inverting\_alpha\] There is a strict, identity-on-objects 2-functor $$\alpha_J\colon C \to C_\ana(J)$$ sending $J$-equivalences to equivalences, and commuting with the respective inclusions into $\Cat(S)$ and $\Cat_\ana(S,J)$.
We define $\alpha_J$ to be the identity on objects, and as described in examples \[eg:ordinary\_functor\], \[eg:ordinary\_transf\] on 1-arrows and 2-arrows (i.e. functors and transformations). We need first to show that this gives a functor $C(X,Y)
\to C_\ana(J)(X,Y)$. This is precisely the content of example \[eg:transf\_compose1\]. Since the identity 1-cell on a category $X$ in $C_\ana(J)$ is the image of the identity functor on $S$ in $C$, $\alpha_J$ respects identity 1-cells. Also, lemma \[coherent\_composition\] tells us that $
\alpha_J$ respects composition. That $\alpha_J$ sends $J$-equivalences to equivalences is the content of lemma \[anafunctors-r-inverses\].
The 2-category $C$ is locally small (i.e. enriched in small categories) if $S$ itself is locally small (i.e. enriched in sets), but *a priori* the collection of anafunctors $X{-\!\!\!\mapsto}Y$ do not constitute a set for $S$ a large category.
Let $(S,J)$ be a locally small, subcanonical unary site satisfying WISC and let $C$ admit base change along arrows in $J$. Then $C_\ana(J)$ is locally essentially small.
Given an object $A$ of $S$, let $I(A)$ be a weakly initial set for $J/A$. Consider the locally full sub-2-category of $C_\ana(J)$ with the same objects, and arrows those anafunctors $(U,f):X {-\!\!\!\mapsto}Y$ such that $U \to X_0$ is in $I(X_0)$. Every anafunctor is then isomorphic, by example \[renaming\_transf\], to one in this sub-2-category. The collection of anafunctors $(U,f):X {-\!\!\!\mapsto}Y$ for a fixed $U$ forms a set, by local smallness of $C$, and similarly the collection of transformations between a pair of anafunctors forms a set by local smallness of $S$.
Examples of locally small sites $(S,J)$ where $C_\ana(J)$ is not known to be locally essentially small are the category of sets from the model of ZF used in [@vdBerg_12], the model of ZF constructed in [@Roberts_13] and the topos from proposition \[U-small\_G-sets\]. We note that local essential smallness of $C_\ana(J)$ seems to be a condition just slightly weaker than WISC.
Localising bicategories at a class of 1-cells {#localisation}
=============================================
Ultimately we are interesting in inverting all $J$-equivalences in $C$ and so need to discuss what it means to add the formal pseudoinverses to a class of 1-cells in a 2-category – a process known as *localisation*. This was done in [@Pronk_96] for the more general case of a class of 1-cells in a bicategory, where the resulting bicategory is constructed and its universal properties examined. The application in *loc. cit.* is to show the equivalence of various bicategories of stacks to localisations of 2-categories of smooth, topological and algebraic groupoids. The results of this article can be seen as one-half of a generalisation of these results to more general sites.
[[@Pronk_96]]{} Let $B$ be a bicategory and $W \subset B_1$ a class of 1-cells. A *localisation of $B$ with respect to $W$* is a bicategory $B[W^{-1}]$ and a weak 2-functor $$U \colon B \to B[W^{-1}]$$ such that $U$ sends elements of $W$ to equivalences, and is universal with this property i.e. precomposition with $U$ gives an equivalence of bicategories $$U^* \colon Hom(B[W^{-1}],D) \to Hom_W(B,D),$$ where $Hom_W$ denotes the sub-bicategory of weak 2-functors that send elements of $W$ to equivalences (call these *$W$-inverting*, abusing notation slightly).
The universal property means that $W$-inverting weak 2-functors $F\colon B \to D$ factor, up to an equivalence, through $B[W^{-1}]$, inducing an essentially unique weak 2-functor $\widetilde{F}\colon B[W^{-1}] \to D$.
[[@Pronk_96]]{}\[bicat\_fracs\] Let $B$ be a bicategory with a class $W$ of 1-cells. $W$ is said to *admit a right calculus of fractions* if it satisfies the following conditions
1. $W$ contains all equivalences
2. a\) $W$ is closed under composition\
b) If $a\in W$ and there is an isomorphism $a \stackrel{\sim}{\Rightarrow} b$ then $b\in W$
3. For all $w\colon A' \to A,\ f\colon C \to A$ with $w\in W$ there exists a 2-commutative square $$\xymatrix{
P \ar[dd]^v \ar[rr]^g && A'\ar[dd]^w_{\ }="s" \\
\\
C \ar[rr]^{f}="t" & & A
\ar@{=>}_{\simeq} "s"; "t"
}$$ with $v\in W$.
4. If $\alpha\colon w \circ f \Rightarrow w \circ g$ is a 2-arrow and $w\in W$ there is a 1-cell $v \in W$ and a 2-arrow $\beta\colon f\circ v \Rightarrow g \circ v$ such that $\alpha\circ v = w \circ \beta$. Moreover: when $\alpha$ is an isomorphism, we require $\beta$ to be an isomorphism too; when $v'$ and $\beta'$ form another such pair, there exist 1-cells $u,\,u'$ such that $v\circ u$ and $v'\circ u'$ are in $W$, and an isomorphism $\epsilon\colon v\circ u \Rightarrow v' \circ u'$ such that the following diagram commutes: $$\label{2cf4.diag}
\xymatrix{
f \circ v \circ u \ar@{=>}[rr]^{\beta\circ u}
\ar@{=>}[dd]_{f\circ \epsilon}^\simeq &&
g\circ v \circ u \ar@{=>}[dd]^{g\circ \epsilon}_\simeq \\
\\
f\circ v' \circ u' \ar@{=>}[rr]_{\beta'\circ u'} && g\circ v' \circ u'
}$$
For a bicategory $B$ with a calculus of right fractions, [@Pronk_96] constructs a localisation of $B$ as a bicategory of fractions; the 1-arrows are spans and the 2-arrows are equivalence classes of bicategorical spans-of-spans diagrams.
From now on we shall refer to a calculus of right fractions as simply a calculus of fractions, and the resulting localisation constructed by Pronk as a bicategory of fractions. Since $B[W^{-1}]$ is defined only up to equivalence, it is of great interest to know when a bicategory $D$, in which elements of $W$ are sent to equivalences by a 2-functor $B \to D$, is equivalent to $B[W^{-1}]$. In particular, one might be interested in finding such an equivalent bicategory with a simpler description than that which appears in [@Pronk_96].
[[@Pronk_96]]{}\[comparison\_thm\] A weak 2-functor $F:B \to D$ which sends elements of $W$ to equivalences induces an equivalence of bicategories $$\widetilde{F} \colon B[W^{-1}] \xrightarrow{\sim} D$$ if the following conditions hold
1. $F$ is essentially surjective,
2. For every 1-cell $f \in D_1$ there are 1-cells $w \in W$ and $g\in B_1$ such that $Fg \stackrel{\sim}{\Rightarrow} f \circ Fw$,
3. $F$ is locally fully faithful.
Thanks are due to Matthieu Dupont for pointing out (in personal communication) that proposition \[comparison\_thm\] actually only holds in the one direction, not in both, as claimed in *loc. cit.*
The following is useful in showing a weak 2-functor sends weak equivalences to equivalences, because this condition only needs to be checked on a class that is in some sense cofinal in the weak equivalences.
\[inverting\_special\_we\] Let $V \subset W$ be two classes of 1-cells in a bicategory $B$ such that for all $w\in W$, there exists $v\in V$ and $s\in W$ and an invertible 2-cell $$\xymatrix{
&& a \ar[dd]^w \\
& & \\
b \ar[rr]_v^{\ }="t1" \ar[uurr]^s_{\ }="s1" && c\; .
\ar@{=>}"s1";"t1"^{\simeq}
}$$ Then a weak 2-functor $F\colon B \to D$ that sends elements of $V$ to equivalences also sends elements of $W$ to equivalences.
In the following the coherence arrows will be present, but unlabelled. It is enough to prove that if in a bicategory $D$ with a class of maps $M$ (in our case $M=F(W)$) such that for all $w\in M$ there is an equivalence $v$ and an isomorphism $\alpha$, $$\xymatrix{
&& a \ar[dd]^w \\
& & \\
b \ar[rr]_v^{\ }="t1" \ar[uurr]^s_{\ }="s1" && c
\ar@{=>}"s1";"t1"^{\simeq}_\alpha
}$$ where $s\in M$, then all elements of $M$ are also equivalences.
Let $\bar v$ be a pseudoinverse for $v$ and let $j = s \circ \bar v$. Then there is sequence of isomorphisms $$w\circ j \Rightarrow (w\circ s)\circ \bar v \Rightarrow v \circ \bar v
\Rightarrow I.$$
Since $s\in M$, there is an equivalence $u$, $t\in M$ and an isomorphism $\beta$ giving the following diagram $$\xymatrix{
d \ar[dd]_{t} \ar[rr]^{u}_{\ }="s2" && a \ar[dd]^w \\
& & \\
b \ar[rr]_v^{\ }="t1" \ar[uurr]^s="t2"_{\ }="s1" && c \; .
\ar@{=>}"s1";"t1"^\alpha
\ar@{=>}"s2";"t2"_\beta
}$$ Let $\bar u$ be a pseudoinverse of $u$. We know from the first part of the proof that we have a pseudosection $k = t\circ \bar u$ of $s$, with an isomorphism $s \circ k \Rightarrow I$. We then have the following sequence of isomorphisms: $$j\circ w
= (s\circ \bar v) \circ w
\Rightarrow ((s\circ \bar v) \circ w) \circ (s \circ k)
\Rightarrow s \circ ((\bar v \circ v) \circ (t\circ \bar u))
\Rightarrow (s\circ t) \circ u
\Rightarrow \bar u \circ u
\Rightarrow I.$$ Thus all elements of $M$ are equivalences.
2-categories of internal categories admit bicategories of fractions {#main}
===================================================================
In this section we prove the result that $C{\hookrightarrow}\Cat(S)$ admits a calculus of fractions for the $J$-equivalences, where $J$ is a singleton pretopology on $S$.
The following is the first main theorem of the paper, and subsumes a number of other, similar theorems throughout the literature (see section \[examples\] for details).
\[bicat\_frac\_exists\] Let $S$ be a category with a singleton pretopology $J$. Assume the full sub-2-category $C {\hookrightarrow}\Cat(S)$ admits base change along maps in $J$. Then $C$ admits a right calculus of fractions for the class $W_J$ of $J$-equivalences.
We show the conditions of definition \[bicat\_fracs\] hold.
- An internal equivalence is clearly $J$-locally split. Lemma \[rep\_ff\_functors\_closed\_under\_iso\] gives us the rest.
- - That the composition of fully faithful functors is again fully faithful is trivial. Consider the composition $g\circ f$ of two $J$-locally split functors, $$\xymatrix{
Y[U] \ar[d] \ar@/^.5pc/[dr]_{\ }="s1"^{u}&Z[V] \ar[d]\ar@/^.5pc/[dr]_(.5){\ }="s2"^{v}& \\
X\ar[r]_{f}^(.33){\ }="t1"&Y \ar[r]_{g}^(.33){\ }="t2" & Z
\ar@{=>}"s1";"t1"
\ar@{=>}"s2";"t2"
}$$ By lemma \[strict\_pullbacks\_triv\_J\_fibrations\] the functor $u$ pulls back to a functor $Z[U\times_{Y_0}V] \to Z[V]$. The composite $Z[U\times_{Y_0}V] \to Z$ is fully faithful with object component in $J$, hence $g\circ f$ is $J$-locally split.
- Lemma \[rep\_ff\_functors\_closed\_under\_iso\] tells us that fully faithful functors are closed under isomorphism, so we just need to show $J$-locally split functors are closed under isomorphism.
Let $w,f\colon X\to Y$ be functors and $a\colon w \Rightarrow f$ be a natural isomorphism. First, let $w$ be $J$-locally split. It is immediate from the diagram $$\xymatrix{
Y[U] \ar[dd] \ar@/^.7pc/[ddrr]_{\ }="s1"^{u} \\
\\
X\ar@/^1pc/[rr]^{w}="t1"_{\ }="s2" \ar@/_1pc/[rr]_{f}^{\ }="t2"
&&Y
\ar@{=>}"s1";"t1"
\ar@{=>}"s2";"t2"^{a}
}$$ that $f$ is also $J$-locally split.
- Let $w\colon X\to Y$ be a $J$-equivalence, and let $f\colon Z\to Y$ be a functor. From the definition of $J$-locally split, we have the diagram $$\xymatrix{
Y[U] \ar[d] \ar@/^.5pc/[dr]_{\ }="s1"^{u}& \\
X\ar[r]_{w}^(.33){\ }="t1"&Y
\ar@{=>}"s1";"t1"
}$$ We can use lemma \[strict\_pullbacks\_triv\_J\_fibrations\] to pull $u$ back along $f$ to get a 2-commuting diagram $$\xymatrix{
& Z[U\times_{Y_0} Z_0] \ar[dr]^{v} \ar[dl] \\
Y[U] \ar[d] \ar@/^.5pc/[dr]_{\ }="s1"^{u}& &Z \ar[dl]^f\\
X\ar[r]_{w}^(.33){\ }="t1"&Y
\ar@{=>}"s1";"t1"
}$$ with $v\in W_J$ as required.
- Since $J$-equivalences are representably fully faithful, given $$\xymatrix{
&Y \ar[dr]^w \\
X \ar[ur]^f \ar[dr]_g & \Downarrow a & Z\\
& Y \ar[ur]_w
}$$ where $w\in W_J$, there is a unique $a'\colon f
\Rightarrow g$ such that $$\raisebox{36pt}{
\xymatrix{
&Y \ar[dr]^w \\
X \ar[ur]^f \ar[dr]_g & \Downarrow a & Z\\
& Y \ar[ur]_w
}
}
{\hspace{10pt}=\hspace{10pt}}\raisebox{36pt}{
\xymatrix{
&&\\
X \ar@/^1.5pc/[rr]^f \ar@/_1.5pc/[rr]_g&\Downarrow a'& Y \ar[r]^w & Z
\,.
}
}$$ The existence of $a'$ is the first half of 2CF4, where $v=\id_X$. Note that if $a$ is an isomorphism, so if $a'$, since $w$ is representably fully faithful. Given $v'\colon W\to X \in W_J$ such that there is a transformation $$\xymatrix{
&X \ar[dr]^f \\
W \ar[ur]^{v'} \ar[dr]_{v'} & \Downarrow b & Y\\
& X \ar[ur]_g
}$$ satisfying $$\begin{aligned}
\label{antiwhisker_eqn}
\raisebox{36pt}{
\xymatrix{
&X \ar[dr]^f \\
W \ar[ur]^{v'} \ar[dr]_{v'} & \Downarrow b & Y \ar[r]^w & Z\\
& X \ar[ur]_g
}
}
{\hspace{10pt}=\hspace{10pt}}&
\raisebox{36pt}{
\xymatrix{
&&Y \ar[dr]^w \\
W \ar[r]^{v'} &X \ar[ur]^f \ar[dr]_g & \Downarrow a & Z\\
&& Y \ar[ur]_w
}
} \nonumber \\
{\hspace{10pt}=\hspace{10pt}}&
\raisebox{36pt}{
\xymatrix{
&&\\
W\ar[r]^{v'}&X \ar@/^1.5pc/[rr]^f \ar@/_1.5pc/[rr]_g &\Downarrow a'&
Y \ar[r]^w & Z
}
}\, ,\end{aligned}$$ then uniqueness of $a'$, together with equation (\[antiwhisker\_eqn\]) gives us $$\raisebox{36pt}{
\xymatrix{
&X \ar[dr]^f \\
W \ar[ur]^{v'} \ar[dr]_{v'} & \Downarrow b & Y \\
& X \ar[ur]_g
}
}
{\hspace{10pt}=\hspace{10pt}}\raisebox{36pt}{
\xymatrix{
&&\\
W\ar[r]^{v'}&X \ar@/^1.5pc/[rr]^f \ar@/_1.5pc/[rr]_g
&\Downarrow a'
& Y
}
}\, .$$ This is precisely the diagram (\[2cf4.diag\]) with $v=\id_X$, $u=v'$, $u'=\id_W$ and $\epsilon$ the identity 2-arrow. Hence 2CF4 holds.
The proof of theorem \[bicat\_frac\_exists\] is written using only the language of 2-categories, so can be generalised from $C$ to other 2-categories. This approach will be taken up in [@Roberts2].
The second main result of the paper is that we want to know when this bicategory of fractions is equivalent to a bicategory of anafunctors, as the latter bicategory has a much simpler construction.
\[anafunctors\_localise\] Let $(S,J)$ be a subcanonical unary site and let the full sub-2-category $C{\hookrightarrow}\Cat(S)$ admit base change along arrows in $J$. Then there is an equivalence of bicategories $$C_\ana(J) \simeq C[W_J^{-1}]$$ under $C$.
Let us show the conditions in proposition \[comparison\_thm\] hold. To begin with, the 2-functor $\alpha_J\colon C \to C_{ana}(J)$ sends $J$-equivalences to equivalences by proposition \[W-inverting\_alpha\].
- $\alpha_J$ is the identity on 0-cells, and hence surjective on objects.
- This is equivalent to showing that for any anafunctor $(U,f)\colon X{-\!\!\!\mapsto}Y$ there are functors $w,g$ such that $w$ is in $W_J$ and $$(U,f) \stackrel{\sim}{\Rightarrow} \alpha_J(g)\circ\alpha_J(w)^{-1}$$ where $\alpha_J(w)^{-1}$ is some pseudoinverse for $\alpha_J(w)$.
Let $w$ be the functor $X[U] \to X$ and let $g=f\colon X[U] \to Y$. First, note that $$\xymatrix{
& \ar[dl] X[U] \ar[dr]^= &\\
X && X[U]
}$$ is a pseudoinverse for $$\alpha_J(w)
{\hspace{10pt}=\hspace{10pt}}\left(\raisebox{24pt}{
\xymatrix{
& \ar[dl]_{=} X[U][U] \ar[dr] &\\
X[U] && X
}
}\right)\,.$$ Then the composition $ \alpha_J(f)\circ\alpha_J(w)^{-1}$ is $$\xymatrix{
& \ar[dl] X[U\times_U U \times_U U]\ar[dr]\\
X && Y\; ,
}$$ which is just $(U,f)$ (recall we have the equality $U\times_U U \times_U U = U$ by remark \[remark:pullback\_of\_x\_along\_x\]).
- If $a\colon(X_0,f)\Rightarrow(X_0,g)$ is a transformation of anafunctors for functors $f,g\colon X\to Y$, it is given by a natural transformation $$f \Rightarrow g\colon X = X[X_0 \times_{X_0} X_0] \to Y.$$ Hence we get a unique natural transformation $a\colon f\Rightarrow g$ such that $a$ is the image of $a'$ under $\alpha_J$.
We now give a series of results following from this theorem, using basic properties of pretopologies from section \[sites\_categories\].
\[equivalent\_anafunctors\] When $J$ and $K$ are two subcanonical singleton pretopologies on $S$ such that $J_{un}=K_{un}$, for example $J$ cofinal in $K$, there is an equivalence of bicategories $$C_\ana(J) \simeq C_\ana(K).$$
The class of maps in $\Top$ of the form $\coprod U_i \to X$ for an open cover $\{U_i\}$ of $X$ form a singleton pretopology. This is because $\mathcal{O}$ is a *superextensive* pretopology (see the appendix). Given a site with a superextensive pretopology $J$, we have the following result which is useful when $J$ is not a singleton pretopology (the *singleton* pretopology $\amalg J$ is defined analogously to the case of $\Top$, details are in the appendix).
Let $(S,J)$ be a superextensive site where $J$ is a subcanonical pretopology. Then $$C[W_{J_{un}}^{-1}] \simeq C_\ana(\amalg J).$$
This essentially follows by lemma \[J-coprodJ-epis\].
Obviously this can be combined with previous results, for example if $K$ is cofinal in $\amalg J$, for $J$ a non-singleton pretopology, $K$-anafunctors localise $C$ at the class of $J_{un}$-equivalences.
Finally, given WISC we have a bound on the size of the hom-categories, up to equivalence.
Let $(S,J)$ be a subcanonical unary site satisfying WISC with $S$ locally small and let $C{\hookrightarrow}\Cat(S)$ admit base change along arrows in $J$. Then any localisation $C[W_J^{-1}]$ is locally essentially small.
Recall that this localisation can be chosen such that the class of objects is the same as the class of objects of $C$, and so it is not necessary to consider additional set-theoretic mechanisms for dealing with large (2-)categories here.
We note that the issue of size of localisations is not touched on in [@Pronk_96]. even though such issues are commonly addressed in localisation of 1-categories. If we have a specified bound on the hom-sets of $S$ and also know that some WISC${}_\kappa$ holds, then we can put specific bounds on the size of the hom-categories of the localisation. This is important if examining fine size requirements or implications for localisation theorems such as these, for example higher versions of locally presentable categories.
Examples
========
The simplest example is when we take the trivial singleton pretopology $triv$, where covering families are just single isomorphisms: $triv$-equivalences are internal equivalences and, up to equivalence, localisation at $W_{triv}$ does nothing. It is worth pointing out that if we localise at $W_{triv_{un}}$, which is equivalent to considering anafunctors with source leg having a split epimorphism for its object component, then by corollary \[equivalent\_anafunctors\] this is equivalent to localising at $W_{triv}$, so $C_{ana}(triv_{un}) \simeq C_{ana}(triv)\simeq C$.
The first non-trivial case is that of a regular category with the canonical singleton pretopology ${\underline{c}}$. This is the setting of [@Bunge-Pare_79]. Recall that $W_J^{BP}$ is the class of Bunge-Paré $J$-equivalences (definition \[def:PB\_weak\_equiv\]). For now, let $C$ denote either $\Cat(S)$ or $\Gpd(S)$.
Let $(S,J)$ be a finitely complete unary site with $J$ saturated. Then we have $$C[(W_J^{BP})^{-1}] \simeq C[W_J^{-1}]$$
This is merely a restatement of the fact Bunge-Paré $J$-equivalences and ordinary $J$-equivalences coincide in this case.
\[PB\_are\_the\_same\_weak\_equivs\] The canonical singleton pretopology ${\underline{c}}$ on a finitely complete category $S$ is saturated. Hence $W_{{\underline{c}}}^{BP} = W_{{\underline{c}}}$ for this site, and $$C[(W_{{\underline{c}}}^{BP})^{-1}] \simeq C[W_{{\underline{c}}}^{-1}]\simeq C_\ana({\underline{c}})$$
We can combine this corollary with corollary \[equivalent\_anafunctors\] so that the localisation of either $\Cat(S)$ or $\Gpd(S)$ at the Bunge-Paré weak equivalences can be calculated using $J$-anafunctors for $J$ cofinal in ${\underline{c}}$. We note that ${\underline{c}}$ does not satisfy WISC in general (see proposition \[U-small\_G-sets\] and the comments following), so the localisation might not be locally essentially small.
The previous corollaries deal with the case when we are interested in the 2-categories consisting of all of the internal categories or groupoids in a site. However, for many applications of internal categories/groupoids it is not sufficient to take all of $\Cat(S)$ or $\Gpd(S)$. One widely used example is that of Lie groupoids, which are groupoids internal to the category of (finite-dimensional) smooth manifolds such that source and target maps are submersions (more on these below). Other examples are used in the theory of algebraic stacks, namely groupoids internal to schemes or algebraic spaces. Other types of such *presentable* stacks use groupoids internal to some site with specified conditions on the source and target maps. Although it is not covered explicitly in the literature, it is possible to consider presentable stacks of categories, and this will be taken up in future work [@Roberts1].
We thus need to furnish examples of sub-2-categories $C$, specified by restricting the sort of maps that are allowed for source and target, that admit base change along some class of arrows. The following lemma gives a sufficiency condition for this to be so.
\[lemma:existence\_of\_base\_change\] Let $\Cat^\mathcal{M}(S)$ be defined as the full sub-2-category of $\Cat(S)$ with objects those categories such that the source and target maps belong to a singleton pretopology $\mathcal{M}$. Then $\Cat^\mathcal{M}(S)$ admits base change along arrows in $\mathcal{M}$, as does the corresponding 2-category $\Gpd^\mathcal{M}(S)$ of groupoids.
Let $X$ be an object of $\Cat^\mathcal{M}(S)$ and $f\colon M\to X_0 \in \mathcal{M}$. In the following diagram, all the squares are pullbacks and all arrows are in $\mathcal{M}$. $$\SelectTips {cm}{}\xymatrix{
X[M]_1 \ar[d] \ar[r] \ar @/_2.4pc/ [dd]_{s'} \ar @/^1pc/[rr]^{t'} & X_1\times_{X_0} M \ar[r] \ar[d] & M \ar[d] \\
M\times_{X_0} X_1 \ar[d] \ar[r] & X_1 \ar[r] \ar[d] & X_0 \\
M \ar[r] & X_0
}$$ The maps marked $s',t'$ are the source and target maps for the base change along $f$, so $X[M]$ is in $\Cat^\mathcal{M}(S)$. The same argument holds for groupoids verbatim.
In practice one often only wants base change along a subclass of $\mathcal{M}$, such as the class of open covers sitting inside the class of open maps in $\Top$. We can then apply theoerems \[bicat\_frac\_exists\] and \[anafunctors\_localise\] to the 2-categories $\Cat^\mathcal{M}(S)$ and $\Gpd^\mathcal{M}(S)$ with the classes of $\mathcal{M}$-equivalences, and indeed to sub-2-categories of these, as we shall in the examples below.
We shall focus of a few concrete cases to show how the results of this paper subsume similar results in the literature proved for specific sites. The category of smooth manifolds is not finitely complete so the localisation results in this section so far do not apply to it. There are two ways around this. The first is to expand the category of manifolds to a category of smooth spaces which *is* finitely complete (or even cartesian closed). In that case all the results one has for finitely complete sites can be applied. The other is to take careful note of which finite limits are actually needed, and show that all constructions work in the original category of manifolds. There is then a hybrid approach, which is to work in the expanded category, but point out which results/constructions actually fall inside the original category of manifolds. Here we shall take the second approach. First, let us pin down some definitions.
Let $\Diff$ be the category of smooth, finite-dimensional manifolds. A *Lie category* is a category internal to $\Diff$ where the source and target maps are submersions (and hence the required pullbacks exist). A *Lie groupoid* is a Lie category which is a groupoid. A *proper* Lie groupoid is one where the map $(s,t)\colon X_1 \to X_0 \times X_0$ is proper. An *étale* Lie groupoid is one where the source and target maps are local diffeomorphisms.
By lemma \[lemma:existence\_of\_base\_change\] the 2-categories of Lie categories, Lie groupoids and proper Lie groupoids admit base change along any of the following classes of maps: open covers ($\amalg\mathcal{O}$), surjective local diffeomorphisms ($\acute{e}t$), surjective submersions ($Subm$). The 2-categories of étale Lie groupoids and proper étale Lie groupoids admit base change along arrows in $\acute{e}t$ and $Subm$. We should note that we have $\amalg\mathcal{O}$ cofinal in $\acute{e}t$, which is cofinal in $Subm$.
We can thus apply the main results of this paper to the sites $(\Diff,\mathcal{O})$, $(\Diff,\amalg\mathcal{O})$, $(\Diff,\acute{e}t)$ and $(\Diff,Subm)$ and the 2-categories of Lie categories, Lie groupoids, proper Lie goupoids and so on. However, the definition of weak equivalence we have here, involving $J$-locally split functors, is not one that apppears in the Lie groupoid literature, which is actually Bunge-Paré $Subm$-equivalence. However, we have the following result:
\[Subm-equiv\_are\_BP-equiv\] A functor $f\colon X\to Y$ between Lie categories is a $Subm$-equivalence if and only if it is a Bunge-Paré $Subm$-equivalence.
Before we prove this, we need a lemma proved by Ehresmann.
[[@Ehresmann_59]]{} For any Lie category $X$, the subset of invertible arrows, $X_1^{iso} {\hookrightarrow}X_1$ is an open submanifold.
Hence there is a Lie groupoid $X^{iso}$ and an identity-on-objects functor $X^{iso} \to X$ which is universal for functors from Lie groupoids. In particular, a natural isomorphism between functors with codomain $X$ is given by a component map that factors through $X_1^{iso}$, and the induced source and target maps $X_1^{iso} \to X_0$ are submersions.
(proposition \[Subm-equiv\_are\_BP-equiv\]) Full faithfulness is the same for both definitions, so we just need to show that $f$ is $Subm$-locally split if and only if it is essentially $Subm$-surjective. We first show the forward implication.
The special case of a $\amalg\mathcal{O}$-equivalence between Lie groupoids is a small generalisation of the proof of proposition 5.5 in [@Moerdijk-Mrcun_03], which states than an internal equivalence of Lie groupoids is a Bunge-Paré $Subm$-equivalence. Since $\amalg\mathcal{O}$ is cofinal in $Subm$, a $Subm$-equivalence is a $\amalg\mathcal{O}$-equivalence, hence a Bunge-Paré $Subm$-equivalence.
For the case when $X$ and $Y$ are Lie categories, we use the fact that we can define $X_0\times_{Y_0}Y_1^{iso}$ and that the local sections constructed in Moerdijk-Mrčun’s proof factor through this manifold to set up the proof as in the groupoid case.
For the reverse implication, the construction in the first half of the proof of proposition \[BP\_equiv\_iff\_weak\_equiv\] goes through verbatim, as all the pullbacks used involve submersions.
The need to localise the category of Lie groupoids at $W_{Subm}$ was perhaps first noted in [@Pradines_89], where it was noted that something other than the standard construction of a category of fractions was needed. However Pradines lacked the necessary 2-categorical localisation results. Pronk considered the sub-2-category of étale Lie groupoids, also localised at $W_{Subm}$, in order to relate these groupoids to differentiable étendues [@Pronk_96]. Lerman discusses the 2-category of orbifolds *qua* stacks [@Lerman_10] and argues that it should be a localisation of the 2-category of proper étale Lie groupoids (again at $W_{Subm}$). These three cases use different constructions of the 2-categorical localisation: Pradines used what he called *meromorphisms*, which are equivalence classes of butterfly-like diagrams and are related to Hilsum-Skandalis morphisms, Pronk introduces the techniques outlined in this paper, and Lerman uses Hilsum-Skandalis morphisms, also known as right principal bibundles.
Interestingly, [@Colman_10] considers this localisation of the 2-category of Lie groupoids then considers a further localisation, not given by the results of this paper.[^2] Colman in essence shows that the full sub-2-category of topologically discrete groupoids, i.e. ordinary small groupoids, is a localisation at those internal functors which induce an equivalence on fundamental groupoids.
Our next example is that of topological groupoids, which correspond to various flavours of stacks on the category $\Top$. The idea of weak equivalences of topological groupoids predates the case of Lie groupoids, and [@Pradines_89] credits it to Haefliger, van Est, and [@Hilsum-Skandalis_87]. In particular the first two were ultimately interested in defining the fundamental group of a foliation, that is to say, of the topological groupoid associated to a foliation, considered up to weak eqivalence.
However more recent examples have focussed on topological stacks, or variants thereon. In particular, in parallel with the algebraic and differentiable cases, the topological stacks for which there is a good theory correspond to those topological groupoids with conditions on their source and target maps. Aside from étale topological groupoids (which were considered by [@Pronk_96] in relation to étendues), the real advances here have come from work of Noohi, starting with [@Noohi_05a], who axiomatised the concept of *local fibration* and asked that the source and target maps of topological groupoids are local fibrations.
A singleton pretopology $LF$ in $\Top$ is called a class of *local fibrations* if the following conditions hold:[^3]
1. $LF$ contains the open embeddings
2. $LF$ is stable under coproducts, in the sense that $\coprod_{i\in I} X_i \to Y$ is in $LF$ if each $X_i\to Y$ is in $LF$
3. $LF$ is local on the target for the open cover pretopology. That is, if the pullback of a map $f\colon X\to Y$ along an open cover of $Y$ is in $LF$, then $f$ is in $LF$.
Conditions 1. and 2. tell us that $\amalg\mathcal{O} \subset LF$, and that $LF$ is $\amalg J$ for some superextensive pretopology $J$ containing the open embeddings as singleton ‘covering’ families (beware the misleading terminology here: covering families are not assumed to be jointly surjective). Note that $LF$ will not be subcanonical, by condition 1. As an example, given any of the following pretopologies $K$:
- Serre fibrations,
- Hurewicz fibrations,
- open maps,
- split maps,
- projections out of a cartesian product,
- isomorphisms;
one can define a class of local fibrations by choosing those maps which are in $K$ on pulling back to an open cover of the codomain. Such maps are then called *local $K$*. As an example of the usefulness of this concept, the topological stacks corresponding to topological groupoids with local Hurewicz fibrations as source and target have a nicely behaved homotopy theory. The case of étale groupoids corresponds to the last named class of maps, which give us local isomorphisms, i.e. étale maps. We can then apply lemma \[lemma:existence\_of\_base\_change\] and theorem \[bicat\_frac\_exists\] to the 2-category $\Grp^{LF}(\Top)$ to localise at the class $W_{\amalg \mathcal{O}}$ (as $\amalg \mathcal{O} \subset LF$), or any other singleton pretopology contained in $LF$, using anafunctors whenever this pretopology is subcanonical. Note that if $C$ satisfies WISC, so will the corresponding $LF$, although this is probably not necessary to consider in the presence of full AC.
A slightly different approach is taken in [@Carchedi_12], where the author introduces a new pretopology on the category $CGH$ of compactly generated Hausdorff spaces. We give a definition equivalent to the one in *loc cit*.
A (not necessarily open) cover $\{V_i{\hookrightarrow}X\}_{i\in I}$ is called a $\mathcal{CG}$-cover if for any map $K\to X$ from a compact space $K$, there is a finite open cover $\{U_j {\hookrightarrow}K\}$ which refines the cover $\{V_i\times_X K\to K\}_{i\in I}$. $\mathcal{CG}$-covers form a pretopology $\mathcal{CG}$ on $CGH$.
Compactly generated stacks then correspond to groupoids in $CGH$ such that source and target maps are in the pretopology $\mathcal{CG}_{un}$. Again, we can localise $\Gpd^{\mathcal{CG}}(CGH)$ at $W_{\mathcal{CG}_{un}}$ using lemma \[lemma:existence\_of\_base\_change\] and theorem \[bicat\_frac\_exists\], and anafunctors can be again pressed into service.
We now arrive at the more involved case of algebraic stacks (cf. the continually growing [@StacksProj] for the extent of the theory of algebraic stacks), which were the first presentable stacks to be defined. There are some subtleties about the site of definition for algebraic stacks, and powerful representability theorems, but we can restrict to three main cases: groupoids in the category of affine schemes $\Aff = \Ring^{op}$; groupoids in the category $\Sch$ of schemes; and groupoids in the category $\AlgSp$ of algebraic spaces. Algebraic spaces reduce to algebraic stacks on $\Sch$ represented by groupoids with trivial automorphism groups, and the category of schemes is a subcategory of $Sh(\Aff)$, so we shall just consider the case when our ambient category is $\Aff$. In any case, all the special properties of classes of maps in all three sites are ultimately defined in terms of properties of ring homomorphisms. Note that groupoids in $\Aff$ are exactly the same thing as cogroupoid objects in $\Ring$, which are more commonly known as *Hopf algebroids*.
Despite the possibly unfamiliar language used by algebraic geometry, algebraic stacks reduce to the following semiformal definition. We fix three singleton pretopologies on our site $\Aff$: $J$, $E$ and $D$ such that $E$ and $D$ are local on the target for the pretopology $J$. An algebraic stack then is a stack on $\Aff$ for the pretopology $J$ which ‘corresponds’ to a groupoid $X$ in $\Aff$ such that source and target maps belong to $E$ and $(s,t)\colon X_1 \to X_0^2$ belongs to $D$. We recover the algebraic stacks by localising the 2-category of such groupoids at $W_E$ (this claim of course needs substantiating, something we will not do here for reasons of space, referring rather to [@Pronk_96; @Schappi_12] and the forthcoming [@Roberts1]).
In practice, $D$ can be something like closed maps (to recover Hausdorff-like conditions) or all maps, and $E$ consists of either smooth or étale maps, corresponding to Artin and Deligne-Mumford stacks respectively. $J$ is then something like the étale topology (or rather, the singleton pretopology associated to it, as the étale topology is superextensive), and we can apply lemma \[lemma:existence\_of\_base\_change\] to see that base change exists along $J$, along with the fact that asking for $(s,t) \in D$ is automatically stable under forming the base change. In practice, a variety of combinations of $J,E$ and $D$ are used, as well as passing from $\Aff$ to $\Sch$ and $\AlgSp$, so there are various compatibilities to check in order to know one can apply theorem \[bicat\_frac\_exists\].
A final application we shall consider is when our ambient category consists of algebraic objects. As mentioned in section 2, a number of authors have considered localising groupoids in Mal’tsev, or Barr-exact, or protomodular, or semi-abelian categories, which are hallmarks of categories of algebraic objects rather than spatial ones, as we have been considering so far.
In the case of groupoids in $\Grp$ (which, as in any Mal’tsev category, coincide with the internal categories) it is a well-known result that they can be described using *crossed modules*.
A *crossed module* (in $\Grp$) is a homomorphism $t\colon G\to H$ together with a homomorphism $\alpha\colon H\to \Aut(G)$ such that $t$ is $H$-equivariant (using the conjugation action of $H$ on itself), and such that the composition $\alpha\circ t\colon G\to\Aut(G)$ is the action of $G$ on itself by conjugation. A crossed module is often denoted, when no confusion will arise, by $(G\to H)$. A morphism $(G \to H) \to (K\to L)$ of crossed modules is a pair of maps $G\to K$ and $H\to L$ making the obvious square commute, and commuting with all the action maps.
Similar definitions hold for groups internal to cartesian closed categories, and even just finite-product categories if one replaces $H\to \Aut(G)$ with its transpose $H\times G\to G$. Ultimately of course there is a definition for crossed modules in semiabelian categories (e.g. [@AMMV_10]), but we shall consider just groups. There is a natural definition of 2-arrow between maps of crossed modules, but the specifics are not important for the present purposes, so we refer to [@Noohi_05c definition 8.5] for details. The 2-categories of groupoids internal to $\Grp$ and crossed modules are equivalent, so we shall just work with the terminology of the latter.
Given the result that crossed modules correspond to pointed, connected homotopy 2-types, it is natural to ask if all maps of such arise from maps between crossed modules. The answer is, perhaps unsurprisingly, no, as one needs maps which only *weakly* preserve the group structure. One can either write down the definition of some generalised form of map ([@Noohi_05c definition 8.4]), or localise the 2-category of crossed modules ([@Noohi_05c] considers a model structure on the category of crossed modules). To localise the 2-category of crossed modules we can consider the singleton pretopology $epi$ on $\Grp$ consisting of the epimorphisms, and localise $\Gpd(\Grp)$ at $W_{epi}$.
There are potentially interesting sub-2-categories of crossed modules that one might want to consider, for example, the one corresponding to *nilpotent* pointed connected 2-types. These are crossed modules $t\colon G \to H$ where the cokernel of $t$ is a nilpotent group and the (canonical) action of $\coker t$ on $\ker t$ is nilpotent. The correspondence between such crossed modules and the corresponding internal groupoids is a nice exercise, as well as seeing that this 2-category admits base change for the pretopology $epi$.
Superextensive sites
====================
The usual sites of topological spaces, manifolds and schemes all share a common property: one can (generally) take coproducts of covering families and end up with a cover. In this appendix we gather some results that generalise this fact, none of which are especially deep, but help provide examples of bicategories of anafunctors. Another reference for superextensive sites is [@Shulman_12].
[[@Carboni-Lack-Walters_93]]{} A *finitary* (resp. *infinitary*) *extensive* category is a category with finite (resp. small) coproducts such that the following condition holds: let $I$ be a a finite set (resp. any set), then, given a collection of commuting diagrams $$\xymatrix{
X_i \ar[r] \ar[d] &Z \ar[d] \\
A_i \ar[r] & \coprod_{i\in I} A_i\;,
}$$ one for each $i\in I$, the squares are all pullbacks if and only if the collection $\{X_i
\to Z\}_{i\in I}$ forms a coproduct diagram.
In such a category there is a strict initial object: given a map $A \to 0$ with $0$ initial, we have $A \simeq 0$.
$\Top$ is infinitary extensive. $\Ring^{op}$, the category of affine schemes, is finitary extensive.
In $\Top$ we can take an open cover $\{U_i\}_I$ of a space $X$ and replace it with the single map $\coprod_I U_i \to X$, and work just as before using this new sort of cover, using the fact $\Top$ is extensive. The sort of sites that mimic this behaviour are called *superextensive*.
[(Bartels-Shulman)]{} A *superextensive site* is an extensive category $S$ equipped with a pretopology $J$ containing the families $$(U_i \to \coprod_I U_i)_{i\in I}$$ and such that all covering families are bounded; this means that for a finitely extensive site, the families are finite, and for an infinitary site, the families are small. The pretopology in this instance will also be called superextensive.
Given an extensive category $S$, the *extensive pretopology* has as covering families the bounded collections $(U_i \to \coprod_I U_i)_{i\in I}$. The pretopology on any superextensive site contains the extensive pretopology.
The category $\Top$ with its usual pretopology of open covers is a superextensive site.
An elementary topos with the coherent pretopology is finitary superextensive, and a Grothendieck topos with the canonical pretopology is infinitary superextensive.
Given a superextensive site $(S,J)$, one can form the class $\amalg J$ of arrows of the form $\coprod_I U_i \to A$ for covering families $\{U_i \to A\}_{i\in I}$ in $J$ (more precisely, all arrows isomorphic in $S/A$ to such arrows).
The class $\amalg J$ is a singleton pretopology, and is subcanonical if and only if $J$ is.
Since isomorphisms are covers for $J$ they are covers for $\amalg J$. The pullback of a $\amalg J$-cover $\coprod_I U_i \to A$ along $B \to A$ is a $\amalg J$-cover as coproducts and pullbacks commute by definition of an extensive category. Now for the third condition we use the fact that in an extensive category a map $$f\colon B \to \coprod_I A_i$$ implies that $B\simeq \coprod_I B_i$ and $f=\coprod_i f_i$. Given $\amalg J$-covers $\coprod_I U_i \to A$ and $\coprod_J V_j \to (\coprod_I U_i)$, we see that $\coprod_J V_j \simeq \coprod_I W_i$ for some objects $W_i$. By the previous point, the pullback $$\coprod_I U_k \times_{\coprod_I U_{i'}} W_i$$ is a $\amalg J$-cover of $U_i$, and hence $(U_k \times_{\coprod_I U_{i'}} W_i \to U_k)_{i\in I}$ is a $J$-covering family for each $k\in I$. Thus $$(U_k \times_{\coprod_I U_{i'}} W_i \to A)_{i,k\in I}$$ is a $J$-covering family, and so $$\coprod_J V_j \simeq \coprod_{k\in I} \left( \coprod_{i\in I} U_k \times_{\coprod_I U_{i'}}
W_i\right) \to A$$ is a $\amalg J$-cover.\
The map $\coprod_I U_i \to A$ is the coequaliser of $\coprod_{I\times I} U_i \times_A
U_j {\rightrightarrows}\coprod_I U_i$ if and only if $A$ is the colimit of the diagram in definition \[defn:effective\_cov\_fam\]. Hence $(\coprod_I U_i \to A)$ is effective if and only if $
(U_i \to A)_{i\in I}$ is effective
Notice that the original superextensive pretopology $J$ is generated by the union of $\amalg J$ and the extensive pretopology.
One reason we are interested in superextensive sites is the following.
\[J-coprodJ-epis\] In a superextensive site $(S,J)$, we have $J_{un} = (\amalg J)_{un}$.
This means we can replace the singleton pretopology $J_{un}$ (e.g. local-section-admitting maps of topological spaces) with the singleton pretopology $\amalg J$ (e.g. disjoint unions of open covers) when defining anafunctors. This makes for much smaller pretopologies in practice.
One class of extensive categories which are of particular interest is those that also have finite/small limits. These are called *lextensive*. For example, $\Top$ is infinitary lextensive, as is a Grothendieck topos. In contrast, an elementary topos is in general only finitary lextensive. We end with a lemma about WISC.
If $(S,J)$ is a superextensive site, $(S,J)$ satisfies WISC if and only if $(S,\amalg J)$ does.
One reason for why superextensive sites are so useful is the following result from [@Schappi_12].
Let $(S,J)$ be a superextensive site, and $F$ a stack for the extensive topology on $S$. Then the associated stack $\widetilde{F}$ on the site $(S,\amalg J)$ is also the associated stack for the site $(S,J)$.
As a corollary, since every weak 2-functor $F\colon S\to \Gpd$ for extensive $S$ represented by an internal groupoid is automatically a stack for the extensive topology, we see that we only need to stackify $F$ with respect to a singleton pretopology on $S$. This will be applied in [@Roberts1].
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[^1]: Anafunctors were so named by Makkai, on the suggestion of Pavlovic, after profunctors, in analogy with the pair of terms anaphase/prophase from biology. For more on the relationship between anafunctors and profunctors, see below.
[^2]: In fact this is the only 2-categorical localisation result involving internal categories or groupoids known to the author to *not* be covered by theorem \[bicat\_frac\_exists\] or its sequel [@Roberts2].
[^3]: We have packaged the conditions in a way slightly different to [@Noohi_05a], but the definition is in fact identical.
|
---
abstract: 'We present a simple and effective instrument for simultaneous real-time imaging and hysteresis of the anisotropic magnetic domain dynamics in thin films using the Magneto-Optical Kerr Effect (MOKE). We furthermore illustrate that magnetic imaging allows a more accurate interpretation of the magnetization reversal processes than the conventional hysteresis characterization. In particular, we present a case where the onset of a double-step reversal observed in imaging remains invisible in the spatially integrated hysteresis loops. When complemented by precise tuning of the external magnetic field orientation, our system reveals the singular anisotropic variations of the domain dynamics near the hard-axis in epitaxial thin films, thus shedding light on the reported, but as yet unexplained, hard axis coercivity behavior.'
author:
- Pavel Chvykov
- Vladimir Stoica
- Roy Clarke
bibliography:
- 'instr\_ref.bib'
title: 'Real-time MOKE microscopy made simple'
---
Introduction
============
The study of magnetism in thin-films and related nanostructures has seen an increasing popularity over the past decade, as these topics form the core of fundamental and applied research in spintronics [@spintronics]. Nonetheless, achieving high spatio-temporal resolution in the imaging of the magnetic domain structures is an intricate matter because magnetic domain imaging instruments are usually complex and expensive. For example, prominent magnetic imaging techniques offering nanoscale spatial resolution include the Magnetic Force Microscopy (MFM)[@MFM; @MFM_crossover], Scanning Electron Microscopy with Polarization Analysis (SEMPA)[@SEMPA; @SEMPA2010], and Magnetic Transmission X-ray Microscopy (MTXM)[@MTXM; @MTXM2010]. Despite their distinct advantage of high spatial resolution, these techniques require intricate setups and offer limited flexibility in, e.g., quick adjustment of field of view and resolution, while the former two techniques cannot be easily be implemented for time-resolved studies [@domains_book].
Magneto-Optic Kerr Effect (MOKE) microscopy using visible light sources is another common imaging technique known for its simplicity and flexibility, while also being capable of femtosecond time and diffraction-limited spatial resolutions. [@old_MOKE; @MOKE_rever_anis; @fs_MOKE] However, this method generally suffers from a weak signal (defined as the imaging contrast between spin orientations), especially if time-resolved imaging is needed, which limits the use time-averaging or scanning-type setups for signal-to-noise ratio improvement purposes. The most common MOKE microscopy setups either employ MOKE in the polar geometry, giving larger sensitivity compared to the longitudinal or transverse geometries, or use the help of Magneto-Optic Imaging Films (MOIF) for signal enhancement [@MOIF1]. However, the polar MOKE geometry can only record the magnetization component orthogonal to the film surface[@PolarMOKE], while imaging the more common in-plane magnetization with MOIF [@MOIF2; @MOIF_new] requires a magneto-optical transducer placed directly on top of the film surface, thus causing undesirable interference with the sample. Further, as the MOIF materials are not commercially available, they can be difficult to obtain.
In this paper, we show that an effective MOKE microscope can be easily assembled using basic components available in most optics laboratories. Moreover, with our experimental setup we could perform simultaneous studies of vector-resolved magnetic hysteresis and real-time-resolved longitudinal MOKE imaging of magnetic domain dynamics in thin films. The imaging of the transverse magnetization component can also be achieved through a straightforward system reconfiguration, as can the more common polar geometry. We demonstrate that when complemented by high angular resolution in manipulating the sample orientation with respect to the external magnetic field, we can observe the fine details of the anisotropic behavior of magnetic domains during the magnetization reversal. In particular, we study the reversal dynamics on the hard-axis coercivity spike – a sadden increase in the coercive field when the magnetic field direction is oriented along the hard axis, which was observed in from hysteresis, but the domain structure could not be previously imaged with time-resolution. Finally, we show that key physical information about the reversal can be missed by the conventional hysteresis measurements, but is picked up in imaging, and thus simultaneous imaging and hysteresis data acquisition are essential for interpreting the magnetization dynamics.
The Experimental Setup
======================
Our setup is schematically presented in fig. \[fig:setup\]. A 15 mW HeNe laser beam passes through a polarizer and is directed onto the sample by lens 1 for illumination. We observe that the size and quality of the illumination spot here are not crucial, as long the field of view is illuminated uniformly and with sufficient brightness. A lens is used for imaging the sample surface onto a CCD camera with 12 bit dynamic range, acquiring 30 frames per second with 0.1 to 1 ms variable exposure time. After the collection lens, the beam passes through an analyzer. Although the laser beam divergence does affect the analyzer performance, as the incident angle deviates from normal, this decline is insignificant and does not pose problems for the imaging contrast. A beam splitter is then used for directing the beam to both the CCD and a commercial photodiode (Thorlabs PDA10A), for which the data acquisition is performed in real time using a USB digitizer (National Instruments USB-6221). The photodiode simply provides the averaged intensity from the imaged spot, but with a much higher temporal resolution for faster and more precise hysteresis measurements than those possible with the CCD. The sample is mounted on a motorized rotation stage with $ 0.01{^{\circ}}$ angular resolution and with the plane of rotation being parallel to the film surface and to the external field $ H $ applied by an electromagnet (GMW 3470). The magnet is driven by a sinusoidal current from its power supply (Kepco BOP 50-8M) seeded by a function generator or, with more flexibility, by a PC. The system provides a maximum field of 10 kOe with a field uniformity across the sample of 2%. The magnetic field is measured by a Gauss probe next to the sample and all data processing occurs in real time following the acquisition on a PC using a USB data acquisition board. A LabView software routine was programmed for experiment control, which uses the Gauss-probe signal to trigger the hysteresis loop measurements, and appropriately averages typically between 1 and 30 hysteresis loops, while further plotting the loops in real-time to correlate the photodiode MOKE signal relative to the Gauss-probe-measured field. Additionally, the software routine numerically analyses the measured loops to find the parameters of interest, such as coercivity, remanence and the magnetic field at saturation, and plots these parameters live during the measurement.
In our system, for simplicity and flexibility, the sample was mounted using adhesive tape to the end of the rotation stage mount (the drift of the sample orientation settled within a few minutes after taping and did not cause problems). Due to the small spacing between the poles of the electromagnet, which was necessary to apply larger magnetic fields, the sample could not be mounted directly on a tilt-stage, and so the tilting of the sample was restricted, and thus alignment of the plane of the film precisely with the plane of rotation had some deviations of up to 5${^{\circ}}$. This caused the reflected beam to precess slightly as the sample rotated. However, this could easily be compensated by replacing the collection lens with two confocal lenses. This scheme allowed placing the first of these lenses sufficiently close to the sample, ensuring a high enough numerical aperture for capturing the precessing reflected beam from the sample. We further stress that in our setup, and hence for all the images presented in this paper, the imaging was done by a regular convex lens and not by a microscope objective. Based on single element lenses, with different focal lengths, and by adjusting the object and image distances, we could easily control the resolution and the field of view. For the case when diffraction-limited resolution is needed, which was not necessary in our studies, where 5 - 10 [$ \mathrm{\mu} $m]{} resolution was sufficient, the lens can be replaced by a high-magnification objective. Such high numerical aperture objective usually contains multiple optical elements that will have to be very close to the sample and, thus to the electromagnet, thus possibly causing a Faraday polarization rotation in the glass of the objective. Such an effect will then mix with the polarization rotation of the longitudinal MOKE signal. This problem was also observed with the short focal-length lenses that we have used. Nonetheless, this Faraday rotation induced in the lenses is precisely linear with the applied magnetic field, and thus could be filtered out digitally by looking at the slope of the measured intensity vs. $ H $ in the hysteresis loop at saturation. The calibration of the digital filter needed to be repeated only once per setup configuration.
![Experimental arrangement for observation of anisotropic reversal dynamics, including the simultaneous detection of spatially integrated hysteresis with a photodiode and with MOKE microscopy with a CCD[]{data-label="fig:setup"}](setup){width="35.00000%"}
Measurements
============
This setup allows for three different types of measurements. In the first configuration, the polarizer is set such that the beam arrives with s-polarization on the sample – a geometry that can be used to measure the surface magnetization component parallel to the field based on the laser polarization rotation on reflection from the sample (longitudinal MOKE [@domains_book]). To optimally detect this polarization rotation, we set the analyzer a few degrees from extinction, such that the derivative of the transmission curve (namely $ \sin^2(\theta) $ – $ \theta $ being the analyzer angle) is significant, while the transmitted laser intensity is still low, giving the largest percent intensity variation. Further, to re-check the alignment of the polarizer and to minimize the residual effects of birefringence from the lenses used in the setup, we can set the analyzer exactly at extinction to verify that the resulting measured intensity remains independent of the magnetization. Because the slope of the transmission curve is zero at extinction, any observed dependence will indicate effects other than longitudinal MOKE such as transverse and higher-order MOKE signals, which can mostly be suppressed by adjusting the polarizer and analyzer alignment as well as their relative polarization axis orientation. Further, for the case of samples used in the present study, the resulting MOKE signal is strong enough to be clearly seen on the CCD in real-time during magnetic field reversal without any additional averaging or filters, other than background subtraction (see fig. \[fig:100rev\]). We note here that the spatial and temporal resolutions in magnetic domain imaging are diffraction and frame recording rate limited, respectively. Note also that the time-resolution requirements can be somewhat relaxed by decreasing the field sweep rate, but not completely removed, as the observed reversal processes are dynamic and non-equilibrium. Finally, in this configuration, the photodiode simultaneously provides longitudinal MOKE hysteresis measurements with high temporal resolution of around 50 000 points/second.
In the second setup configuration, we choose a p-polarized incident beam while removing the analyzer altogether, in order to measure the transverse MOKE magnetization component (orthogonal to the magnetic field), based on the reflectivity variations (transverse MOKE [@domains_book]). However, in such measurements the magneto-optical signal is much weaker, and imaging on the CCD requires additional data processing including temporal averaging, which precluded the use of real-time imaging. On the other hand, the photodiode detection is more sensitive and permits averaging, thus readily providing the spatially integrated transverse MOKE signal in real time. Nonetheless, if transverse MOKE imaging is needed, it can be achieved by redirecting the beam such that the plane of incidence is orthogonal to the horizontal scattering plane in fig. \[fig:setup\] and to the sample. This geometry can then take advantage of the stronger signal of longitudinal MOKE for detecting the magnetization component perpendicular to the magnetic field direction and parallel to the sample plane. Finally, for the case of the samples studied here, the dominant contribution of the dipolar field confined the magnetization along the thin film plane and thus no significant contributions from the polar MOKE were detected.
The possibility of switching between three different measurement configurations with our setup allows performing complementary measurements that can be used to simultaneously characterize the magnetic reversal in a sample. These setup configurations include the measurement of hysteresis loops of both components of the magnetization, as well as time-resolved longitudinal MOKE imaging of magnetic domains during the magnetization reversal. Such imaging is particularly interesting for studies of epitaxial ferromagnetic films that exhibit anisotropic magnetic hysteresis loops. Moreover, simultaneous probing of the component-resolved hysteresis and magnetic domain structure during the reversal have allowed for unique identification of the magnetization orientation across the imaged region, ensuring complete visual information about the anisotropy of domain dynamics.
{width="\textwidth"}
Example: two-step reversal from 4-fold anisotropy {#example-two-step-reversal-from-4-fold-anisotropy .unnumbered}
-------------------------------------------------
In figure \[fig:100rev\] we show the combined measurement capability of our setup, where $\alpha = 7{^{\circ}}$ for a $ {\left <}100{\right >}$ Fe$ _{0.85} $Ga$ _{0.15} $ film – the red and black loops illustrate respectively the longitudinal and transverse MOKE hysteresis, while the two horizontal image strips show two separate runs of the real-time domain dynamics, taken under identical conditions.[^1] The polar plot then shows the reversal of the vector magnetization, with the colormap indicating the external field $ H $, computed from the hysteresis. The points whose magnitude is less than 1 indicate multi-domain configurations corresponding to the shown images.
FeGa films have recently become particularly important due to their enhanced magnetostrictive properties [@FeGa_magnetostriction], which can allow sensitive control of the film’s magnetization using, e.g., thermal or electrical influences. We know from previous work in hysteresis[@dbl_step_hyst] and imaging , that because the $ {\left <}100{\right >}$ films have two easy axes, the reversal usually proceeds via two $ 90{^{\circ}}$ domain-mediated jumps. Our measurements have confirmed this behavior for this particular film for $ \alpha > 7{^{\circ}}$. However, for $ \alpha<5{^{\circ}}$, both hysteresis and imaging have shown a single $ 180{^{\circ}}$-step reversal between initial and final states. Then, within the narrow transition region from one mode to the other, we have observed a curious mixture of the two pathways, which has not been reported previously, and is presented in fig. \[fig:100rev\].
In these images, we clearly see the presence of grey 90$ {^{\circ}}$ domains, where the magnetization orientation is as indicated in the frames. These domains form either as the thin stripes emanating from the main black 180$ {^{\circ}}$ domain, or as the characteristic grey and black bands over the triangular features emphasized in the images (fig. c,d). This behavior is, however, completely missed by the hysteresis measurements, which still indicate only a single $ 180{^{\circ}}$-step jump, as seen from the straight, uninterrupted magnetization jumps measured (fig. a,b). This illustrates the necessity of imaging in the analysis of reversal pathways. Additionally, we stress that the observation of this mixture was only possible with the high contrast, spatio-temporal and angular resolutions provided by our MOKE imaging setup.
Procedures {#sec: meas}
==========
Next, we move on to elaborate on the measurement procedure. As illustrated by the two movie-strips in fig. \[fig:100rev\], the precise reversal dynamics and domain wall motion are different between the cycles of the external field, although after comparing the images vertically, we see that the overall structure and features of the reversal remain (this is also shown in [@real-time]). Thus, in order to sharply resolve the individual domain wall dynamics, real-time single-exposure imaging is necessary. Averaging over images obtained at the same applied field value in multiple field cycles (by triggering the CCD relative to the driving field) can still be useful to identify repeated reversal patterns, such as trap sites, but will smear out the important details of the wall motion seen in the figures here. Due to the practical restriction on the frame rate of the CCD (to 30 fr/s in our case), the real-time imaging requires a slowly varying external field near the coercivity value so as to decrease the domain wall speed and allow several frames to be recorded during the remagnetization process. Depending on the particular reversal pathway, the field sweep rates that we found practical were around 2 to 20 Oe/s. Although this can be achieved by simply decreasing the frequency of the sinusoidal seed from the signal generator, the resulting period can be up to one minute, with only about 10 frames containing non-trivial contrast information, resulting in extensive recording deadtime. This can be vastly improved (down to an acquisition period of around 5 s) by generating the seed on a computer such that the sweep rate is low only around the coercivity value, where it is needed.
The CCD is also connected to a PC, allowing to perform the background subtraction, while recording the resulting images in real-time, exactly as they are seen in the figures in this paper. The background image is chosen at the single-domain state at saturation, once at each new orientation of the magnetic field, and then subtracted from every measured frame in real time. We have seen that the system has enough stability that the background measurement need not be updated other than after system readjustments, while this method provides for easier interpretation of images than the common differential imaging technique (subtraction of subsequent frames).
On the other hand, for hysteresis studies, particularly in the transverse configuration, averaging (over 5 to 30 loops) and large number of data points (such as anisotropy measurements of $ 360{^{\circ}}$ at $ 0.1{^{\circ}}$ resolution) are often required. Since the temporal resolution of the photodiode is much higher than that of the CCD, it is convenient to increase the frequency of the driving field (to about 10 to 100 Hz), and hence the field sweep rate (to around 20 kOe/s). Although a correlation between the sweep rate and the reversal process has been reported in [@sweep_rate], it is only manifested at frequencies above 100 Hz, and we have checked that in our system there were no significant differences between the hysteresis loops obtained at the two sweep rates. The signal from the photodiode is then fed into a computer, where and the data processing is performed, while the result is displayed live by the LabView software routine. Furthermore, our setup could then be automated to allow for fast and high angular resolution anisotropy measurements, thus allowing the detection of singular anisotropy features (e.g. spikes – see below) that might otherwise be missed. To realize this at any given sample angle $ \alpha $, the field was cycled a pre-set number of times and the hysteresis averaged to obtain a clean measurement. The parameters of interest (such as coercivity and remanence) are then computed, and the rotation stage is subsequently rotated to the next specified angle, where the process is repeated. Automating this process typically allows completing an entire $ 360{^{\circ}}$ anisotropy measurement at $ 0.1{^{\circ}}$ resolution in 3 to 15 minutes depending on the precision requirements.[^2]
{width="\textwidth"}
Example: hard axis coercivity spike from four-fold anisotropy {#example-hard-axis-coercivity-spike-from-four-fold-anisotropy .unnumbered}
--------------------------------------------------------------
Figure \[fig:spike\] shows one interesting scenario where high angular resolution is necessary to detect a distinct feature of the anisotropy – the hard axis coercivity spike, which, despite notable interest over more than a decade (for example, see the hysteresis studies [@spike2003; @spike2011]), is still not well understood. To study this, we have recorded the magnetic domain dynamics along the hard axis and slightly away from this direction in order to look at the mechanism behind the jump in the coercivity characteristic for the hard axis spike. In contrast to a prior report on the static magnetic domain imaging of domain structures near the hard axis direction , our measurements could probe distinct differences in the magnetic domain dynamics, which could not be achieved previously.
Fig. \[fig:spike\]a shows the coercivity anisotropy plot measured with the automated system as described above , where the coercivity of the hysteresis obtained at each angle is plotted. The aforementioned spikes, measuring $ <1{^{\circ}}$ wide, are clearly observed at the hard-axes. The three film-strips then show how the domain behaviour changes going from off-spike (b) to on-spike on either side of the peak (c,d). The most dramatic change is in the domain wall speed – as seen from the field step-size between the frames, the wall sweeps through the field of view much faster slightly away from the hard-axis spike (in fact, there, it is difficult to capture more than one frame in the multi-domain configuration).The average wall speed under otherwise identical conditions is about 15 times slower on the spike – an observation which can only be detected with real-time imaging, again emphasizing its importance. Additionally, while away from the hard-axis spike the reversal proceeds through the usual double-step transition (with only the first step shown, as the coercivity of the second step diverges here), which is characterized by straight or zigzag domain walls (as in fig. \[fig:100rev\]), along the hard-axis where the spike is observed, the reversal process is entirely different, resulting in amorphous walls whose shape is mostly governed by trapping behavior. Also note that the reversal process along the spike direction has a second stage, as seen by the remaining vertical triangular domains after the wall sweeps through. These domains then uniformly fade out, rather than moving through, thus marking another distinction from the usual double-step transition. Finally, we can also see that along the spike direction, the reversal dynamics are especially sensitive to the sample angle – a rotation of only $ 0.2{^{\circ}}$ results in the flip of the direction of domain wall propagation.
Conclusion
==========
In conclusion, we have presented a cost-effective and easy to implement MOKE setup that is capable of directly imaging the time-resolved domain dynamics during the magnetization reversal in thin films. This was tested in particular for [Fe$ _{1-x} $Ga$ _{x} $]{}films, and can easily be applied to other epitaxial ferromagnetic films. Our setup simultaneously allows for fast high-resolution anisotropy measurements based on the longitudinal and transverse MOKE hysteresis. Furthermore, we have illustrated the necessity of imaging, as well as the need for fine-tuning the anisotropy angle, in studies of the magnetic properties of epitaxial thin films. This was emphasized on several specific examples, where hysteresis measurements alone can miss crucial aspects of the reversal behavior, or even give misleading interpretations. We were also able to carry out real-time reversal imaging on the hard-axis coercivity spike, which should improve the understanding of such behavior in further studies.
Our real-time MOKE microscopy approach allows to easily conduct anisotropy studies of surface domain dynamics, little of which has thus far been explored. As we have seen in a preliminary survey (which includes the above images), such studies can often lead to unexpected and interesting observations that can help deepen our understanding of magnetic domain dynamics in thin films.
[^1]: We note that all images presented here are taken with a single exposure of the CCD (no averaging) and do not use any specialized filters besides the background subtraction.
[^2]: The LabView automation routine with a user guide is available from the authors upon request.
|
---
abstract: 'The latest CMS jet measurements in p-p collisions at $\sqrt{s}$ = 7 TeV, sensitive to small-x QCD physics, are discussed. These include inclusive forward jet and simultaneous forward-central jet production, as well as production ratios and azimuthal angle decorrelations of jets widely separated in rapidity.'
author:
- |
Pedro Cipriano[^1], on behalf of the CMS Collaboration\
[Deutsches Elektronen-Synchrotron, Notkestrasse 85, 22607 Hamburg, Germany ]{}\
title: 'Small-x QCD physics probed with jets in CMS [^2]'
---
Introduction
============
The measurement of forward jets provides an important testing ground for QCD predictions of the Standard Model in the low-x region. The LHC (Large Hadron Collider) can reach $Q^2$ and $x$ values previously inaccessible to Hera as displayed in figure \[fig:intro\]. To access the low-x region one must look at high rapidity. For such task the rapidity coverage of up to $|\eta| $ = 5.2 in CMS [@cms] has been used.
![Kinematic phase–space accessible to Hera and LHC [@parton_fig].[]{data-label="fig:intro"}](partonkinematics.eps){height="0.4\textheight"}
The jet–rapidity and transverse–momenta is well described by the calculations at next-to- leading-order (NLO) in perturbative quantum chromodynamics (QCD) using the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) [@dglap1; @dglap2; @dglap3; @dglap4; @dglap5] approach and collinear factorization. The dijet cross-section is also well described [@dijets_2011]. When the collision energy $\sqrt{s}$ is considerably larger than the hard scattering scale given by the jet transverse momentum, $p_{T}$, calculations in perturbative QCD require a resummation of large $\log(1/x)$ terms. This leads to the prediction of new dynamic effects, expected to be described by Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution [@bfkl1; @bfkl2; @bfkl3] and $k_{T}$ factorization [@ktfact1; @ktfact2; @ktfact3]. An effective theory has been developed which describes strong interactions in this kinematic domain [@smallx]. This description is particularly useful in events with several jets with large rapidity separation, which are not well described by DGLAP predictions.
To extend the study of the parton evolution equations, the azimuthal angle differences were also measured. This observable has a sensitivity to BFKL effects when both jets are widely separated in rapidity (eg: Mueller-Navelet jets).
Inclusive forward jet production
================================
The inclusive forward jet cross-section was measured from an integrated luminosity of 3.14 $pb^{-1}$ [@jets1]. Jets were reconstructed with the anti-$k_{T}$ clustering algorithm [@antikt; @fastjet] with a distance parameter R = $\sqrt{(\Delta\eta)^2 + (\Delta\phi)^2}$ = 0.5. The energy depositions in the calorimeter cells were used as input for the clustering. Assuming massless jets, a four–momentum is associated with them by summing the energy of the cells above a given threshold.
![Feynman diagram for inclusive forward jet production[]{data-label="fig:feynman_inclusive_forward"}](pp-fwdjet.eps){width="60.00000%"}
The forward region is defined as 3.2 $ < |\eta| < $ 4.7. The jets are required to have a transverse momentum above $p_{T}$ = 35 GeV. If more than one jet is present, the one with with highest $p_{T}$ is considered, as is illustrated in figure \[fig:feynman\_inclusive\_forward\]. The jets are corrected for the following systematic effects: $p_T$ and $\eta$–dependent response of the calorimeters, overlap with other proton–proton interactions and the migration of events across the $p_{T}$ bins due to jet energy resolution.
![Inclusive forward jet production uncertainty [@jets1].[]{data-label="fig:inclusive_forward_unc"}](inclusive_forward_unc.eps){width="36.00000%"}
In figure \[fig:inclusive\_forward\_unc\] the experimental systematic uncertainties are shown for the leading forward jet as function of $p_T$. The jet energy scale is the dominant systematic uncertainty and the total uncertainty is around -25+30%.
![Inclusive forward jet cross-section compared with different Monte Carlo predictions [@jets1].[]{data-label="fig:inclusive_forward"}](inclusive_forward.eps){width="85.00000%"}
The inclusive forward jet production cross–section corrected to hadron level is presented in figure \[fig:inclusive\_forward\]. Although all predictions describe the data within the uncertainty band, some of them do better. [Powheg]{} [@powheg] + [Pythia 6]{} [@pythia6] gives the best description. [Pythia 6]{} and [Pythia 8]{} [@pythia8] describe the data reasonably well. [Cascade]{} [@cascade] underestimates the cross-section while [Herwig 6]{} [@herwig] + [Jimmy]{} [@jimmy] tends to overestimate. NLOJET++ overestimates the data but is still within the large theoretical and experimental uncertainties.
Forward-central dijet production
================================
The selection procedure for the simultaneous forward–central dijet production is similar to the one for for the inclusive forward jet production. In addition, a central jet within $|\eta| <$ 2.8 with a transverse momentum above $p_T$ = 35 GeV is required. A Feynman diagram of the process is shown in figure \[fig:feymamn\_forwardcentral\].
![Feynmann diagram for forward–central dijet production[]{data-label="fig:feymamn_forwardcentral"}](pp-fwd-central-jet.eps){width="60.00000%"}
Several MC predictions compared to the data cross-section is presented in figures \[fig:forwardcentral1\] and \[fig:forwardcentral2\] [@jets1]. Forward jet cross-section is steeper than the central jet. The shape of the forward jet is poorly described when compared with the central jet. HEJ [@hej] provides the best description being followed closely by [Herwig 6]{} and [Herwig ++]{} [@herwigpp]. Both [Pythia 6]{}, [Pythia 8]{} and the CCFM [Cascade]{} have troubles describing the data for the central jets and for low $p_T$ forward jets. [Powheg + Pythia 6]{}, which was the best prediction for inclusive forward jet production, yelds similar result as [Pythia 6]{} alone.
![Forward–central dijet production compared with different Monte Carlo predictions [@jets1].[]{data-label="fig:forwardcentral1"}](central_forward1.eps){width="80.00000%"}
![Forward–central dijet production compared with different Monte Carlo predictions [@jets1].[]{data-label="fig:forwardcentral2"}](central_forward2.eps){width="80.00000%"}
Azimuthal–angle decorrelations of jets widely separated in rapidity
===================================================================
The reconstruction and correction procedure is similar as for the inclusive forward jet production [@mn1]. Mueller-Navelet jets are the dijet pair with the highest rapidity separation. In this analysis only jets with $p_T$ above 35 GeV and $|\eta| < $ 4.7 were considered. The azimuthal angle decorrelations of jets widely separated in rapidity is presented in figures \[fig:azimuthal\_decorrelations1\] and \[fig:azimuthal\_decorrelations2\] as function of rapidity separation.
![Azimuthal–angle decorrelations of jets widely separated in rapidity compared with different Monte Carlo predictions [@mn1].[]{data-label="fig:azimuthal_decorrelations1"}](azimutal_correlations1.eps "fig:"){width="45.00000%"} ![Azimuthal–angle decorrelations of jets widely separated in rapidity compared with different Monte Carlo predictions [@mn1].[]{data-label="fig:azimuthal_decorrelations1"}](azimutal_correlations2.eps "fig:"){width="45.00000%"}
![Azimuthal–angle decorrelations of jets widely separated in rapidity compared with different Monte Carlo predictions [@mn1].[]{data-label="fig:azimuthal_decorrelations1"}](azimutal_correlations3.eps "fig:"){width="45.00000%"} ![Azimuthal–angle decorrelations of jets widely separated in rapidity compared with different Monte Carlo predictions [@mn1].[]{data-label="fig:azimuthal_decorrelations1"}](azimutal_correlations4.eps "fig:"){width="45.00000%"}
![Azimuthal–angle decorrelations of jets widely separated in rapidity compared with different Monte Carlo predictions [@mn1].[]{data-label="fig:azimuthal_decorrelations1"}](azimutal_correlations5.eps "fig:"){width="45.00000%"} ![Azimuthal–angle decorrelations of jets widely separated in rapidity compared with different Monte Carlo predictions [@mn1].[]{data-label="fig:azimuthal_decorrelations1"}](azimutal_correlations6.eps "fig:"){width="45.00000%"}
The first row of figure \[fig:azimuthal\_decorrelations1\] displays the azimuthal angle difference $\Delta\phi$ for jets with a rapidity separation $\Delta y$ less than 3. [Pythia 6]{} and [Herwig ++]{} describe the data within uncertainties, while [Pythia 8]{} and [Sherpa 1.4]{} [@sherpa] with parton matrix elements matched show deviations at small and intermediate $\Delta\phi$. The second row shows $\Delta\phi$ for a rapidity separation between 3 and 6. [Herwig ++]{} provides the best description, but all predictions show deviation beyond the experimental uncertainties. The last row shows the azimuthal–angle difference for $\Delta y$ between 6 and 9. The dijets are strongly decorrelated. [Herwig ++]{} provides the best description while [Pythia 6]{} and [Pythia 8]{} fail for the lower $\Delta\phi$ region.
The figure \[fig:azimuthal\_decorrelations2\] shows $\Delta\phi$ for Mueller-Navelet jets with different rapidity separations compared with with different [Pythia 6]{} predictions. The contributions of the angular ordering (AO) and multi–parton interactions (MPI) are very similar. The intermediate $\Delta y$ region is better described without MPI. Overall the data is better described with AO and MPI.
![Azimuthal angle decorrelations of jets widely separated in rapidity compared with different PYTHIA6 predictions [@mn1].[]{data-label="fig:azimuthal_decorrelations2"}](delta_phi_low_deta2.eps "fig:"){width="32.00000%"} ![Azimuthal angle decorrelations of jets widely separated in rapidity compared with different PYTHIA6 predictions [@mn1].[]{data-label="fig:azimuthal_decorrelations2"}](delta_phi_medium_deta2.eps "fig:"){width="32.00000%"} ![Azimuthal angle decorrelations of jets widely separated in rapidity compared with different PYTHIA6 predictions [@mn1].[]{data-label="fig:azimuthal_decorrelations2"}](delta_phi_high_deta2.eps "fig:"){width="32.00000%"}
Fourier coefficients ratio of the average azimuthal cosines
===========================================================
Using the same selection as in the previous section, the Fourier coefficients of the average cosines have been measured [@mn1] and is presented in the figure \[fig:average\_cosines\].
$$C_{n} : d\sigma/d(\Delta\phi) \sim \sum C_{n};
\hspace{5mm}
C_{n} = <cos(n(\pi - \Delta\phi))>$$
![Fourier coefficients ratio of the average azimuthal cosines compared with different Monte Carlo predictions [@mn1].[]{data-label="fig:average_cosines"}](ratio_fourier_cosines1.eps "fig:"){width="45.00000%"} ![Fourier coefficients ratio of the average azimuthal cosines compared with different Monte Carlo predictions [@mn1].[]{data-label="fig:average_cosines"}](ratio_fourier_cosines2.eps "fig:"){width="45.00000%"}
DGLAP contributions are expected to partly cancel in the $C_{n+1} / C_n$ ratio, which are described the by LL DGLAP–based generators towards low $\Delta y$. [Sherpa]{}, [Pythia 8]{} and [Pythia 6]{} overestimate $C_2 / C_1$ while [Herwig]{} underestimate it. The CCFM–based [Cascade]{} predicts too small $C_{n+1} / C_n$. At $\Delta y > 4$, a BFKL NLL calculation describe $C_2 / C_1$ within uncertainties.
Ratios of dijets production
===========================
Using jets with $p_T > $ 35 GeV and $|\eta| < $ 4.7 the ratio of the inclusive to exclusive dijet production was measured as a function of $\Delta y$ [@kfactor]. With increasing $\Delta y$ a larger phase–space for radiation is opened. The inclusive dijet sample consists of events with at least 2 jets over the threshold and exclusive requires exactly two jets. The ratio of inclusive to exclusive dijet production is shown in the figure \[fig:ratio\_dijets1\]. [Pythia 6]{} and [Pythia 8]{} agree well with the data while [Herwig ++]{} and [Hej + Ariadne]{} [@ariadne] overestimate the data at higher $\Delta y$. [Cascade]{} is completly off. MPI gives only a small contribution.
![Ratios of inclusive/exclusive dijets production compared with different Monte Carlo predictions [@kfactor].[]{data-label="fig:ratio_dijets1"}](k-factor_inclusive1.eps "fig:"){width="45.00000%"} ![Ratios of inclusive/exclusive dijets production compared with different Monte Carlo predictions [@kfactor].[]{data-label="fig:ratio_dijets1"}](k-factor_inclusive2.eps "fig:"){width="45.00000%"}
The ratio of inclusive to exclusive Mueller-Navelet dijets is presented in \[fig:ratio\_dijets2\]. At low $\Delta y$ the ratio of Muller-Navelet over exclusive is, by definition, smaller than inclusive over exclusive and at higher $\Delta y$ it is the same. The conclusions of the comparison between data and MC are the same as for the ratio inclusive over exclusive.
![Ratios of Mueller-Navelet/exclusive dijets production compared with different Monte Carlo predictions [@kfactor].[]{data-label="fig:ratio_dijets2"}](k-factor_mn1.eps "fig:"){width="45.00000%"} ![Ratios of Mueller-Navelet/exclusive dijets production compared with different Monte Carlo predictions [@kfactor].[]{data-label="fig:ratio_dijets2"}](k-factor_mn2.eps "fig:"){width="45.00000%"}
Summary
=======
Inclusive measurements of forward and central–forward jets, are reasonably well described by the MC predictions while more exclusive measurements are poorly described. A summary of the MC description is presented in table \[summary\]. The DGLAP–based generators, [Pythia]{} and [Herwig]{}, seem to do a better job than the BFKL–inspired [Cascade]{}. The effort of description of the underlying events, development of the parton showers and tuning of [Pythia]{} and [Herwig]{} play an huge role into this result.
Observable [Pythia]{} [Herwig]{} [Cascade]{} [HEJ]{}
----------------------------- ------------ ------------ ------------- --------- --
Forward jet $p_{T}$ Good Acceptable Acceptable Good
Central-forward jet $p_{T}$ Bad Acceptable Bad Good
Azimuthal correlations Acceptable Good Bad –
Fourier coefficients ratio Acceptable Bad Bad –
Dijet ratios Good Acceptable Bad Bad
: Monte Carlo description of the measurements[]{data-label="summary"}
Acknowlegements {#acknowlegements .unnumbered}
===============
To the CMS collaboration for the oportunity to join this conference and to Hannes Jung for supervision in writing this proceeding.
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[^1]: pedro.cipriano@desy.de
[^2]: Presented at the Low x workshop, May 30 - June 4 2013, Rehovot and Eilat, Israel
|
---
abstract: 'Time–delayed feedback is exploited for controlling noise–induced motion in coherence resonance oscillators. Namely, under the proper choice of time delay, one can either increase or decrease the regularity of motion. It is shown that in an excitable system, delayed feedback can stabilize the frequency of oscillations against variation of noise strength. Also, for fixed noise intensity, the phenomenon of entrainment of the basic oscillation period by the delayed feedback occurs. This allows one to steer the timescales of noise-induced motion by changing the time delay.'
author:
- 'N.B. Janson$^{1,2}$, A.G. Balanov$^{1}$, E. Schöll$^1$'
title: 'Delayed feedback as a means of control of noise-induced motion'
---
Very often in practical application the need arises to control the properties of oscillations. Usually control assumes an enhancement in regularity of motion [@arrhythmia; @BAB02; @Balanov_CSF03]. However, in some cases, for instance in medical applications, one aims to disorder oscillations, since too strong coherence might be undesirable and even lead to damaging consequences, e.g. epilepsy or Parkinson’s disease, [@epilepsy; @Tass]. During the last decade new methods for control of irregular self-sustained oscillations in deterministic systems have been developed, including suppression of chaos by an external (periodic) signal [@chaos_supp], stabilization of unstable periodic orbits embedded in a chaotic attractor by time-discrete control [@OGY], or the use of a time-delayed feedback loop [@Pyragas_PLA92; @SOC94] for the same purpose.
Whereas the existing methods are designed to control deterministic oscillations or, most recently, noise-induced enhancement of deterministic oscillations [@Gammaitoni99; @Lindner01] and self-oscillations affected by noise [@Goldobin], there is a large class of systems that do not oscillate autonomously; but if they are forced even by a purely random process featuring no specific timescales, they demonstrate motion resembling a self-oscillatory process [@CR_Gang; @CR_Strogatz]. Prominent representatives of this class are excitable systems like neurons [@CR_neuron], chemical reaction systems [@Chem], and semiconductor nanostructures [@UNK03]. The degree of closeness of their oscillations to ideally periodic ones, i.e. coherence, can depend resonantly on noise intensity [@CR_Gang], which is why it was called [*coherence resonance*]{} (CR) [@Pikovsky_PRL97]. CR has been shown to occur in systems close to bifurcations [@CR_Neiman], in excitable systems [@Pikovsky_PRL97], and in bistable systems [@Lindner00]. Remarkably, CR oscillators possess the fundamental property of self-oscillators, namely, the ability to synchronize [@CR_Postnov].
Frequently the timescale of oscillations in a CR system varies substantially depending on noise intensity. Since the latter is not easily controllable in practice, there is need to make a CR device robust against variation in the properties of noise. Another important task is to find a reliable way to deliberately change the timescales of noise-induced oscillations in a universal way without affecting intrinsic system parameters. Finally, the obvious need is to control the regularity of noise-induced motion. At present, all three problems remain a challenge. In the present Letter we propose to exploit time-delayed feedback control to tackle all three issues.
As a first example of a CR oscillator, we consider the noisy Van der Pol system closely before the Hopf bifurcation, extended by a delayed feedback loop $$\begin{aligned}
\label{VDP}
\frac{dx}{dt}&=&y, \\
\frac{dy}{dt}&=&(\nu -x^2)y-\omega_0^2x +K(y_{\tau}-y) +D \xi(t).
\nonumber\end{aligned}$$ Here, $x$ and $y$ denote phase variables at time $t$, while $y_{\tau}$ denotes the delayed variable $y(t-\tau)$; $K$ is the strength of delayed feedback. $\xi(t)$ is a random variable with Gaussian distribution, zero mean and unity variance, $D$ is the noise intensity. We set the parameters $\nu=-0.01$ and $\omega_0=1$ at which a stable focus exists.
First consider $K=0$, i.e. no delayed feedback in Eq. (\[VDP\]). While no limit cycle occurs at $D=0$, application of noise induces oscillatory motion as illustrated by the phase portrait in Fig. 1 (a). The coherence of oscillations may be quantified by the correlation time $t_{cor}$, estimated from the normalized autocorrelation function $\Psi(s)$ of $y$ as $t_{cor}=\int_0^{\infty} |\Psi(s)|ds$. In Fig. 2(a) the grey (green on-line) line shows $t_{cor}$ vs $D$ for $K=0$. At small noise intensity D the oscillations are more coherent.
In deterministic self-oscillatory systems, application of a delayed feedback in the form above acts as follows. If there exists an unstable periodic orbit of period $T$ in the phase space, a feedback with delay time $\tau=T$ can stabilize this orbit in some range of the control strength $K$. A CR system may have no periodic orbits, but noise may induce oscillations with a well-defined timescale that is associated with the spectral peak. We suppose that application of a delayed feedback can act by analogy with a system containing a periodic orbit, provided that $\tau$ is equal or close enough to the basic period $T_0$ of the noise-induced motion [@comment1] without feedback. Namely, it should suppress deviations from a reference state and thus enhance the regularity of oscillations.
To test this expectation, we switch on the control force in Eq. (\[VDP\]). We set $\tau=T_0$ with $T_0=6.17283951 \approx
2\pi/\omega_0$. The phase portrait with control at $K=0.2$ is shown in Fig. 1(b) for the same $D$ as in (a). Fig. 1(b) reveals a remarkable ordering of the oscillation as compared with Fig. 1(a).
To quantify the ordering due to the feedback, we estimate $t_{cor}$ in dependence on $D$ as above. It is given for $K=0.2$ by the black line in Fig. 2(a). One can see that for any $D$ the coherence of noise-induced oscillations becomes larger when the delayed feedback loop is switched on and $\tau$ is close to $T_0$. On the other hand, it was found that if $\tau$ is far from an integer multiple of $T_0$, the coherence of oscillations, on the contrary, decreases. Also, for $\tau=T_0$ the coherence of noise-induced oscillations in Eq. (\[VDP\]) was found to increase monotonically with increasing $K$. In the following we fix $K=0.2$.
{width="45.00000%"}
{width="48.00000%"}
Next, we study how the feedback can affect the system’s timescales. For this purpose, we set the noise at the intensity $D=0.003$ as for Figs. 1(a), (b), and follow the evolution of Fourier power spectra with $\tau$, which is illustrated by Fig. 3(a). Without feedback ($\tau=0$), system (\[VDP\]) has one pronounced peak $f_0$ in the spectrum. As $\tau$ increases from zero, the peak frequency, height and width change. At $\tau>8$, new peaks that change their positions, heights and widths with $\tau$ become clearly visible.
Since the control parameter of delayed feedback is the time interval $\tau$, we propose to describe the response of the system in terms of periods rather than frequencies. With feedback, we select all spectral peaks, and for each peak introduce the period $T$ as the inverse of the peak frequency. Denote the period of highest peak by $T_1$. The dependences of $T$ on $\tau$ are given in Fig. 4 (a): $T_1$ by circles (yellow on-line), other $T$ by crosses (blue on-line). One can see that variation of $\tau$ changes $T_1$ in a certain range, doing so most effectively as $\tau < T_0/2$. As $\tau$ increases beyond $T_0/2$, $T_1$ drops quickly, and again increases with $\tau$ with a similar slope as before. After $\tau$ has increased by about $T_0$, $T_1$ again drops abruptly to a lower branch, and again follows $\tau$, although with a smaller slope. These abrupt transitions to successive lower branches occur roughly every $T_0$ time units, and each subsequent entrainment happens at a lower slope. The plot of $T_1$ [*vs*]{} $\tau$ exhibits a piecewise approximately linear dependence, the larger the $\tau$, the closer each segment is to a straight line.
The numerical results obtained above for the Van der Pol system can be understood in terms of a general theory of a canonical nonlinear oscillator with time-delayed feedback $$\begin{aligned}
\label{can_delay}
\ddot{x}+f(x,\dot{x})-K(\dot{x}_{\tau}-\dot{x})=0.\end{aligned}$$ Note that Eq. (\[VDP\]) fits the form Eq. (\[can\_delay\]) if rewritten as a single second-order differential equation with $f(x,\dot{x})=-(\nu -x^2)\dot{x}+\omega_0^2x$ and $y=\dot{x}$. Without feedback ($K=0$ or $\tau=0$) the fixed point $(x_0,0)$ is a stabe focus if $$\label{ders}
0<\frac{\partial f}{\partial \dot{x}}<2\sqrt{\frac{\partial
f}{\partial x}},$$ where partial derivatives are taken at the fixed point. In Eq. (\[VDP\]) $x_0=0$, and Eq. (\[ders\]) is true for the given parameters. Setting $K>0$ does not change $x_0$. Either with, or without feedback the noise-induced oscillations take place in the close vicinity of the fixed point. It is to be expected that the motion is influenced by the local properties of this point. At $\tau=0$ the stable focus has a pair of complex conjugate eigenvalues $\lambda_0=p_0 \pm iq_0$, $p_0<0$, $q_0
\ne 0$, and the value of $q_0$ should give an estimate of the angular frequency. Indeed, the only peak of the power spectrum (Fig. 3(a)) has frequency $f_0 \approx |q_0|/2\pi$. With $\tau>0$, the system becomes infinite-dimensional, and possesses a countable set of eigenvalues $\lambda$. In order to exclude that the delayed feedback might induce the birth of a stable limit cycle via a Hopf bifurcation, thus providing a trivial explanation for the remarkable ordering of oscillations, we perform a linear stability analysis of the fixed point of Eq. (\[can\_delay\]). Following the standard routine of linearizing Eq. (\[can\_delay\]) around the fixed point [@Hale], the characteristic equation for $\lambda$ is derived: $$\begin{aligned}
\lambda^2+\lambda\frac{\partial f}{\partial \dot{x}}+
\frac{\partial f}{\partial x}-K\lambda(e^{-\lambda \tau}-1)=0,
\label{pq-f}\end{aligned}$$ Substituting $\lambda=p + iq$, real and imaginary parts can be separated. The condition for a Hopf bifurcation is $p=0$, $q \ne 0$. Substituting it into the imaginary part of Eq. (\[pq-f\]) we obtain: $$\begin{aligned}
\label{nozerop}
\cos q\tau =\frac{K+\partial f/\partial \dot{x}}{K}.\end{aligned}$$ Since the right-hand side is larger than unity due to Eq. (\[ders\]), the Hopf bifurcation condition is not satisfied for any $K$ and $\tau$. Thus, the delayed feedback in the given form [*cannot*]{} induce a Hopf bifurcation.
The numerical solution of Eq. (\[pq-f\]) with $f(x,\dot{x})$ from Eq. (\[VDP\]) yields the eigenvalue spectrum $\lambda=p + iq$ as a function of $\tau$. The eigenperiods defined as $T^e=2\pi/|q|$ (dots in Fig. 4(a)) coincide remarkably with the inverse peak frequencies of the power spectrum of the noise-induced oscillations as a function of $\tau$. The corresponding real parts $p$ are given by dots in Fig. 4(b) (the seven largest $p$ are shown). All $p$ remain negative, but, as seen from Fig. 4(b), nonmonotonically change with $\tau$. As $\tau$ increases, separate branches of $p$ cross, thus providing a striking explanation of the strongly nonmonotonic, discontinuous evolution of the dominant spectral peak of the noise-induced motion under delayed feedback: The period $T_1$ of the highest spectral peak (circles (yellow on-line) in Fig. 4(a)) always coincides with the period $T^e$ of the least stable eigenmode, i.e. the one with the largest real part which we denote as $p_1$ (circles (yellow on-line) in Fig. 4(b)). The more stable eigenmodes result in the side peaks of the frequency spectrum. The more stable the modes are, the lower the peaks are.
As $p_1$ oscillates with $\tau$, the degree of stability of the fixed point of the deterministic system is modulated, thus leading to modulation of the coherence of the stochastic motion, quantified by the correlation time $t_{cor}$ (solid line (green on-line) in the upper part of Fig. 4(b)). The local maxima of coherence occur when $p_1$ is close to zero, and $T_1$ is close to $T_0$.
The entrainment of $T_1$ by $\tau$, which manifests itself in the almost piecewise linear dependence of $T_1$ on $\tau$ for large $\tau$, can be understood as follows. As shown above, it is related to the eigenvalue whose real part $p_1$ is closest to zero. Assuming $p_1 \approx 0$ in Eq. (\[pq-f\]) we obtain Eq. (\[nozerop\]). With $\partial f/\partial \dot{x}=-\nu \ll K$, and $(K-\nu)/K\approx 1$, this gives $\cos(q \tau) \approx 1$ and $|q| \tau \approx 2\pi n$, where $n$ is integer. Then the eigenperiod $T^e$ is $$\begin{aligned}
\label{linear}
T^e=2\pi/|q| \approx \frac{\tau}{n}.\end{aligned}$$ As illustrated by Fig. 4(b), $p_1$ is close enough to zero only for large $\tau$, for which the relation (\[linear\]) holds most accurately. To obtain the location of the maxima of $p_1$, i.e. the maxima of coherence, substitute $p_1 \approx 0$, $q = 2\pi n/\tau$ into the real part of Eq. (\[pq-f\]), which yields $\tau = 2\pi n/\omega_0=n T_0$.
To summarize, delayed feedback applied to an oscillatory system of the form (\[can\_delay\]), gives rise to a countable set of eigenmodes of the fixed point, whose eigenperiods and stability are controlled by $\tau$. The highest peak in the spectrum of the noise-induced motion is due to excitation of the least stable eigenmode. The coherence of oscillations is the higher, the less stable the mode is. The range of modulation of the peak frequency is largest for small tau and large K.
{width="48.00000%"}
{width="48.00000%"}
Next, we consider another example of a CR oscillator, the FitzHugh-Nagumo system, which serves as a prototype of an excitable system. Extending it again by a delayed feedback loop, we obtain $$\begin{aligned}
\label{FHN}
\epsilon \frac{dx}{dt}&=&x-\frac{x^3}{3}-y, \\
\frac{dy}{dt}&=&x+a+K(y_{\tau}-y)+D \xi(t). \nonumber\end{aligned}$$ We set the parameters $\epsilon=0.01$ and $a=1.1$ such that a stable node is the only attractor of the system in the absence of feedback.
Without feedback ($K=0$), the mechanism for inducing oscillations by noise is different from the Van der Pol oscillator (\[VDP\]). In Fig. 1(c), (d) dashed lines show the null-clines defined by $dy/dt=0$ (vertical) and by $dx/dt=0$ (cubic parabola). They intersect at the fixed point, which is slightly displaced to the left of the minimum of the parabola for the parameters chosen. The null-clines divide the phase plane into four regions with different directions of phase velocity. Dots in Fig. 1(c) show the phase portrait with noise $D=0.09$.
In Fig. 2(b) the grey (green on-line) line shows $t_{cor}$ for Eq. (\[FHN\]) versus noise intensity, exhibiting a distinct maximum at $D=0.09$. Also, grey (green on-line) circles show the basic period $T_0$ of oscillations. Unlike Eq. (\[VDP\]), here $T_0$ changes substantially with noise, as was earlier shown in [@Lindner00].
Now, switch on the feedback with $K=0.2$ and set $\tau$ equal to the value of $T_0=4.12694$ at optimum noise. The black line in Fig. 2(b) denotes $t_{cor}$ vs $D$, and shows that for any $D$ the coherence of oscillations is higher with the feedback. However, this feature is not visible in the phase portrait (Fig. 1(d)). Black circles in Fig. 2(b) show the basic period $T_1$ with feedback. It is evident that delayed feedback substantially reduces the variation of the noise-induced basic timescale. Note, however, that this may not be so if $\tau$ is very different from $T_0$.
Next, we study how the feedback can control the timescales and the regularity of noise-induced motion. Fix $D$ at an optimum value $0.09$, $K$ at $0.2$ and change $\tau$. The spectrum in dependence on $\tau$ is given in Fig. 3(b). With increasing $\tau$, the spectral peaks move towards zero, and the spectrum is gradually enriched by new peaks. As with the Van der Pol oscillator, select all visible peaks with periods $T$, and denote the period of the highest peak as $T_1$. In Fig. 4(c) $T$ of several peaks are given by crosses (blue on-line), and $T_1$ by circles (yellow on-line), depending on $\tau$. These dependencies are qualitatively very much like those for Eq. (\[VDP\]) in Fig. 4(a). In Fig. 4(d) $t_{cor}$ is given depending on $\tau$, exhibiting oscillatory features. Local maxima of coherence occur when $T_1$ is equal to $T_0$. Unlike in case of the Van der Pol oscillator, the mechanism of delayed feedback control can not be explained by a local analysis of the fixed point since the oscillations are characterized by large excursions in phase space. Rather, a global analysis would be needed which is clearly beyond the scope of the present Letter.
In conclusion, time-delayed feedback in the form of the difference between the current and a delayed state of the system can be used to control oscillations that are induced merely by noise. The most crucial parameter of such a control is the time delay $\tau$, depending on which the coherence of noise-induced oscillations increases or decreases. With this, a phenomenon of entrainment of the basic period of noise-induced motion by the time delayed feedback is discovered. The latter ability is somehow reminiscent of classical synchronization phenomena, in that the externally imposed timescale $\tau$ tunes the basic period of oscillations in the system, although it involves quite different mechanisms.
This work was supported by DFG in the framework of Sfb 555. The authors gratefully acknowledge discussions with A. Nikitin and A. Amann.
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|
---
abstract: |
We propose [*PanopticFusion*]{}, a novel online volumetric semantic mapping system at the level of [*stuff*]{} and [*things*]{}. In contrast to previous semantic mapping systems, PanopticFusion is able to densely predict class labels of a background region ([*stuff*]{}) and individually segment arbitrary foreground objects ([*things*]{}). In addition, our system has the capability to reconstruct a large-scale scene and extract a labeled mesh thanks to its use of a spatially hashed volumetric map representation. Our system first predicts pixel-wise panoptic labels (class labels for [*stuff*]{} regions and instance IDs for [*thing*]{} regions) for incoming RGB frames by fusing 2D semantic and instance segmentation outputs. The predicted panoptic labels are integrated into the volumetric map together with depth measurements while keeping the consistency of the instance IDs, which could vary frame to frame, by referring to the 3D map at that moment. In addition, we construct a fully connected conditional random field (CRF) model with respect to panoptic labels for map regularization. For online CRF inference, we propose a novel unary potential approximation and a map division strategy.
We evaluated the performance of our system on the ScanNet (v2) dataset. PanopticFusion outperformed or compared with state-of-the-art offline 3D DNN methods in both semantic and instance segmentation benchmarks. Also, we demonstrate a promising augmented reality application using a 3D panoptic map generated by the proposed system.
author:
- 'Gaku Narita, Takashi Seno, Tomoya Ishikawa, Yohsuke Kaji$^{1}$ [^1]'
bibliography:
- 'paper.bib'
title: |
**PanopticFusion: Online Volumetric Semantic Mapping\
at the Level of Stuff and Things**
---
INTRODUCTION
============
Geometric and semantic scene understanding in 3D environments has an important role in autonomous robotics and context-aware augmented reality (AR) applications. Geometric scene understanding such as visual simultaneous localization and mapping (SLAM) and 3D reconstruction has been widely discussed since the early days of both the robotics and computer vision communities. In recent years, semantic mapping, which not only reconstructs the 3D structure of a scene but also recognizes what exists in the environment, has attracted much attention because of the great progress of deep neural networks.
Semantic mapping systems could take a variety of approaches in terms of geometry and semantics. When we think about robotic and AR applications that deeply interact with the real world, what kind of properties are required for the ideal semantic mapping system? In terms of geometry, it needs to be able to reconstruct a large-scale scene, not sparsely but densely. Additionally, the 3D reconstruction desirably needs to be represented as a volumetric map, not just point clouds or surfels, because it is difficult to directly utilize point clouds and surfels for robot–object collision detection or robot navigation. In terms of semantics, which we mainly focus on in this paper, we believe that it is important for the mapping system to have a [*holistic*]{} scene understanding capability, that is to say, dense semantic labeling as well as individual object discrimination. This is because densely labeled semantics is a crucial cue for intelligent robot navigation, and also, discriminating individual objects is essential for robot–object interaction.
![PanopticFusion realizes an online volumetric semantic mapping at the level of [*stuff*]{} and [*things*]{}. The system performs large-scale 3D reconstruction, as well as dense semantic labeling on [*stuff*]{} regions and segmentation of individual [*things*]{} in an online manner, as shown in the top figure. It is also able to restore the class labels of [*things*]{} and yield a colored mesh, as shown in the bottom figures. The results obtained with scene0645\_01 of ScanNet v2 are shown.[]{data-label="fig_panopticfusion"}](./img/panopticfusion_v2_LQ.png){height="6.5cm"}
Turning our eyes to the field of 2D image recognition, an image understanding task called [*panoptic*]{} segmentation has been proposed recently [@kirillov2018panoptic]. In the panoptic segmentation task, semantic classes are defined as a set of [*stuff*]{} classes (amorphous regions, such as floors, walls, the sky and roads) and [*thing*]{} classes (countable objects, such as chairs, tables, people and vehicles) and one needs to predict class labels on [*stuff*]{} regions and both class labels and instance IDs on [*thing*]{} regions, where the predictions should be performed for each pixel. Extending this point of view to 3D mapping, in this paper we propose the [*PanopticFusion*]{} system. To the best of our knowledge, it is the first semantic mapping system that realizes scene understanding at the level of [*stuff*]{} and [*things*]{}. Our system incrementally performs large-scale 3D surface reconstruction online, as well as dense class label prediction on the background region and segmentation and recognition of individual foreground objects, as shown in Fig. \[fig\_panopticfusion\].
Our approach first passes the incoming RGB frame to 2D semantic and instance segmentation networks and obtains a panoptic label image in which class labels are assigned to [*stuff*]{} pixels and instance IDs to [*thing*]{} pixels. The predicted panoptic labels and depth measurements are integrated into the volumetric map. Before integration, we keep the consistency of instance IDs, which possibly change from frame to frame, by referring to the volumetric map at that moment. In addition, we regularize the map using a fully connected CRF model with respect to panoptic labels. For CRF inference, we propose a unary potential approximation using limited information stored in the map. We also present a map division strategy that achieves a significant reduction in computational time without a drop in accuracy.
We evaluated the performance of our system on the ScanNet v2 dataset [@dai2017scannet], a richly annotated large-scale dataset for indoor scene understanding. The results revealed that PanopticFusion is superior or comparable to the state-of-the-art offline 3D DNN methods in the both 3D semantic and instance segmentation tasks. Note that our system is not limited to indoor scenes. Finally, we demonstrated a promising AR application using the 3D panoptic map generated by our system.
The main contributions of this paper are the following:
- The first reported semantic mapping system that realizes scene understanding at the level of [*stuff*]{} and [*things*]{}.
- Large-scale 3D reconstruction and labeled mesh extraction thanks to the use of a spatially hashed volumetric map representation.
- Map regularization using a fully connected CRF with a novel unary potential approximation and map division strategy.
- Superior or comparable results in both 3D semantic and instance segmentation tasks, in comparison with the state-of-the-art offline 3D DNN methods.
RELATED WORK
============
Previously proposed representative semantic mapping systems related to our PanopticFusion system are shown in Table \[table\_related\_work\]. These systems can be divided into two categories from the perspective of semantics: the dense labeling approach and the object-oriented approach.
The dense labeling approach builds a single 3D map and assigns a class label or a probability distribution of class labels to each surfel or voxel to realize a dense 3D semantic segmentation. Hermans [*et al.*]{} [@hermans2014dense] utilize random decision forests for 2D semantic segmentation and transfer the inferred probability distributions to point clouds with a Bayesian update scheme. Extending the approach of Hermans [*et al.*]{} [@hermans2014dense], SemanticFusion [@mccormac2017semanticfusion] improves the recognition performance by using CNNs for 2D semantic segmentation and makes use of ElasticFusion [@Whelan2015ElasticFusionDS] for a SLAM system to generate a globally consistent map. Xiang [*et al.*]{} [@xiang2017rnn] presented KinectFusion[@newcombe2011kinectfusion]-based volumetric mapping with novel data associated RNNs for improving the segmentation accuracy. While these methods realize dense scene understanding, they suffer from the drawback that they are not able to distinguish individual objects in the scene.
Methods adopted in the early days of the object-oriented approach leverage 3D model databases. SLAM++ [@salas2013slam++] performs point pair feature-based object detection and feeds the detected objects into a pose graph. Tateno [*et al.*]{} [@tateno20162] proposed a 3D object detection and pose estimation system that combines unsupervised geometric segmentation and global 3D descriptor matching. These methods, however, require the shapes of objects in the scene to be exactly the same as the 3D models in the database. Recently, several studies on the object-oriented approach using a CNN-based 2D object detector have been reported. S[ü]{}nderhauf [*et al.*]{} [@sunderhauf2017meaningful] and Nakajima [*et al.*]{} [@nakajima2019efficient] combine a 2D object detector and unsupervised geometric segmentation in order to detect objects in point clouds or a surfel map. MaskFusion [@runz2018maskfusion], Fusion++ [@mccormac2018fusion++] and MID-Fusion [@xu2018mid] introduced an object-oriented map representation that individually builds 3D maps for each object based on 2D object detection. The object-oriented map representation enables tracking of individual objects [@runz2018maskfusion; @xu2018mid] and an object-level pose graph optimization [@mccormac2018fusion++]. However, the quantitative recognition performance of these methods is not clear because they mainly evaluate the camera trajectory accuracy. Furthermore, they focus on foreground objects, resulting in a lack of semantics and/or geometry of background regions.
{height="2.38cm"}
In contrast to these related studies, PanopticFusion realizes holistic scene reconstruction and dense semantic labeling with the ability to discriminate individual objects. Our system builds a single volumetric map, similar to dense labeling approaches, yet each voxel stores neither class labels nor class probability distributions but DNN-predicted panoptic labels in order to seamlessly manage both [*stuff*]{} and [*things*]{} semantics. The class labels of foreground objects can be restored by a probability integration process. In addition, our 3D reconstruction leverages the truncated signed distance field (TSDF) volumetric map with the voxel hashing data structure [@niessner2013real], which allows us to reconstruct a large-scale scene as well as extract labeled meshes by using marching cubes [@lorensen1987marching], in contrast to the 3D maps of previous methods, which are based on point clouds [@hermans2014dense; @sunderhauf2017meaningful], surfels [@mccormac2017semanticfusion; @nakajima2019efficient; @runz2018maskfusion] and a fixed-sized voxel grid [@xiang2017rnn; @mccormac2018fusion++]. It should be noted that, with 3D DNN methods that directly apply deep networks to 3D data such as point clouds or voxel grids, high recognition performance has been reported [@qi2017pointnet++; @dai20183dmv; @yi2018gspn; @hou20183d]. Nevertheless, with those methods, it is basically necessary to reconstruct the whole scene in advance, requiring offline processing, which could limit their application to robotics and AR. On the contrary, PanopticFusion is an online and incremental framework.
METHOD
======
Fig. \[fig\_system\_overview\] shows the system overview of PanopticFusion. Our system first feeds an incoming RGB frame into 2D semantic and instance segmentation networks and obtains pixel-wise panoptic labels by fusing the two outputs (Section \[sec\_pano\_seg\]). The panoptic labels are carefully tracked by referring to the volumetric map at that moment (Section \[sec\_label\_tracking\]) and are integrated into the map with depth measurements (Section \[sec\_volumetric\_integration\]). Probability distributions of class labels for foreground objects are also incrementally integrated (Section \[sec\_label\_prob\_integration\]). In addition, online map regularization with a fully-connected CRF model is performed for a further improvement of the recognition accuracy. Note that camera poses with respect to the volumetric map are given by an external vSLAM, and labeled meshes are extracted by using marching cubes [@lorensen1987marching].
Notations
---------
We denote all class labels by $\mathcal{L}$, and they are divided into [*stuff*]{} labels $\mathcal{L}^\mathrm{St}$ and [*thing*]{} labels $\mathcal{L}^\mathrm{Th}$: such that $\mathcal{L} = \mathcal{L}^\mathrm{St} \cup \mathcal{L}^\mathrm{Th}$ and $\mathcal{L}^\mathrm{St} \cap \mathcal{L}^\mathrm{Th} = \emptyset$. A set of instance IDs for discriminating individual [*things*]{} is denoted by $\mathcal{Z}$. Here we define a set of panoptic labels $\mathcal{L}^\mathrm{Pa} = \mathcal{L}^\mathrm{St} \cup \mathcal{Z} \cup l_\mathrm{unk}$ in order to seamlessly manage [*stuff*]{} and [*things*]{} level semantics in the 3D map. $l_\mathrm{unk}$ denotes the [*unknown*]{} label.
Volumetric Map
--------------
We use the TSDF-based volumetric map representation with a voxel hashing approach [@niessner2013real], which manages spatially hashed small regular voxel grids called voxel blocks. This approach is memory efficient compared with a single voxel grid approach like the original KinectFusion [@newcombe2011kinectfusion] and enables us to reconstruct large-scale scenes. Our implementation is based on voxblox [@oleynikova2017voxblox], which is a CPU-based TSDF mapping system, but we extend it to integrate the semantics.
Our volumetric map stores the truncated signed distance $\mathtt{D}_t(\mathbf{v}) \in \mathbb{R}$, the RGB color $\mathtt{C}_t(\mathbf{v}) \in \mathbb{R}^3$ and the associated weight $\mathtt{W}_t^\mathrm{D}(\mathbf{v}) \in \mathbb{R}_{\geq 0}$ at each voxel location $\mathbf{v} \in \mathbb{R}^3$, as with [@newcombe2011kinectfusion]. Our system additionally stores the panoptic label $\mathtt{L}_t^\mathrm{Pa}(\mathbf{v}) \in \mathcal{L}^\mathrm{Pa}$ and its weight $\mathtt{W}_t^\mathrm{L}(\mathbf{v}) \in \mathbb{R}_{\geq 0}$. Here $t$ denotes the time index.
2D Panoptic Label Prediction\[sec\_pano\_seg\]
----------------------------------------------
For the incoming RGB frame, we predict pixel-wise panoptic labels by fusing both 2D semantic and instance segmentation outputs. We utilize the state-of-the-art CNN architectures of PSPNet [@zhao2017pyramid] and Mask R-CNN [@he2017mask] for 2D semantic and instance segmentation, respectively. PSPNet infers pixel-wise class labels $L_t(\mathbf{u}) \in \mathcal{L}$, where $\mathbf{u} \in \mathbb{R}^2$ denotes the image coordinates. Mask R-CNN outputs instance IDs for each pixel $Z_t(\mathbf{u}) \in \mathcal{Z} \cup l_\mathrm{unk}$, where the regions without any foreground objects are filled with $l_\mathrm{unk}$. The foreground object probability $p_t(z, \mathcal{O})$ and conditional probability distribution of [*thing*]{} labels $p_t(z, l^\mathrm{Th} | \mathcal{O})$ with respect to instance $z$ are utilized in the probability integration step described in Section \[sec\_label\_prob\_integration\]. We obtain pixel-wise panoptic labels $L_t^\mathrm{Pa}(\mathbf{u})$ from $L_t(\mathbf{u})$ and $Z_t(\mathbf{u})$ preceding the instance IDs: $$\begin{aligned}
L_t^\mathrm{Pa}(\mathbf{u}) = \begin{cases}
Z_t(\mathbf{u}) & Z_t(\mathbf{u}) \neq l_\mathrm{unk} \\
L_t(\mathbf{u}) & Z_t(\mathbf{u}) = l_\mathrm{unk} \land L_t(\mathbf{u}) \in \mathcal{L}^\mathrm{St} \\
l_\mathrm{unk} & \mathrm{otherwise.}
\end{cases}\end{aligned}$$
Panoptic Label Tracking\[sec\_label\_tracking\]
-----------------------------------------------
Direct integration of raw panoptic labels $L_t^\mathrm{Pa}(\mathbf{u})$ into the volumetric map induces label inconsistency because Mask R-CNN does not necessarily output a consistent instance ID for the same object through multiple frames. To avoid this problem, we need to estimate consistency-resolved panoptic labels $\hat{L}_t^\mathrm{Pa}(\mathbf{u})$ before the integration. The simplest way is to track the foreground objects in the 2D image sequence using a visual object tracker. This approach unfortunately is not able to re-identify an object in the case of a loopy camera trajectory. Therefore, we take a map reference approach similar to [@runz2018maskfusion; @mccormac2018fusion++].
We first prepare the reference panoptic labels $\tilde{L}_{t-1}^\mathrm{Pa}(\mathbf{u})$ by accessing the map. Here, $\mathbf{T}_{t}$ denotes the live camera pose, $\mathbf{K}$ the camera intrinsic parameters, and $D_t(\mathbf{u})$ the live depth map: $$\begin{aligned}
\tilde{L}_{t-1}^\mathrm{Pa}(\mathbf{u}) = \mathtt{L}_{t-1}^\mathrm{Pa}(\mathbf{T}_{t} \mathbf{K}^{-1} D_t(\mathbf{u}) [\mathbf{u}, 1]^\mathrm{T}).\end{aligned}$$ To track labels, we compute the intersection over union (IoU) $U(\tilde{z}, z)$ of instance ID $z$ of raw panoptic labels $L_t^\mathrm{Pa}(\mathbf{u})$ and instance ID $\tilde{z}$ of reference panoptic labels $\tilde{L}_{t-1}^\mathrm{Pa}(\mathbf{u})$: $$\begin{aligned}
U(\tilde{z}, z) = \mathrm{IoU}\bigr(\{\mathbf{u} | \tilde{L}_{t-1}^\mathrm{Pa}(\mathbf{u}) = \tilde{z}\}, \{\mathbf{u} | L_t^\mathrm{Pa}(\mathbf{u}) = z\}\bigl)\end{aligned}$$ Here, IoU is defined as $\mathrm{IoU}(A,B) = |A \cap B| / |A \cup B|$.
When the maximum value of IoU exceeds a threshold $\theta_U$, $\tilde{z}$ giving the maximum value is associated with $z$. Otherwise a new instance ID is assigned to $z$: $$\begin{aligned}
\hat{z} = \begin{cases}
\mathrm{arg}\max_{\tilde{z}} U(\tilde{z}, z) & \max_{\tilde{z}} U(\tilde{z}, z) > \theta_U \\
z_\mathrm{new} & \mathrm{otherwise.}
\end{cases} \end{aligned}$$ The association is processed in descending order in the mask area $|\{\mathbf{u} | L_t^\mathrm{Pa}(\mathbf{u}) = z\}|$. Once a reference instance ID $\tilde{z}$ is associated with $z$, that instance ID is not associated with any other $z$. The utilization of IoU instead of an overlap ratio, as used in [@runz2018maskfusion; @mccormac2018fusion++], and the exclusive label association is for avoiding under-segmentation of foreground objects in the map. From the associated instance IDs and raw [*stuff*]{} labels, we obtain the consistency-resolved panoptic labels $\hat{L}_t^\mathrm{Pa}(\mathbf{u})$ as follows, which are used in the integration step: $$\begin{aligned}
\hat{L}_t^\mathrm{Pa}(\mathbf{u}) = \begin{cases}
L_t^\mathrm{Pa}(\mathbf{u}) & L_t^\mathrm{Pa}(\mathbf{u}) \in \mathcal{L}^\mathrm{St} \\
\hat{z} & L_t^\mathrm{Pa}(\mathbf{u}) \in \mathcal{Z} \\
l_\mathrm{unk} & \mathrm{otherwise.}
\end{cases} \end{aligned}$$
Volumetric Integration\[sec\_volumetric\_integration\]
------------------------------------------------------
For integration, we take the raycasting approach, as with [@oleynikova2017voxblox]. For each pixel $\mathbf{u}$, we cast a ray from the sensor origin $\mathbf{s}$ to the back-projected 3D point $\mathbf{p}_\mathbf{u} = \mathbf{T}_{t} \mathbf{K}^{-1} D_t(\mathbf{u}) [\mathbf{u}, 1]^\mathrm{T}$ and update the voxels along the ray within a truncated distance. Regarding TSDF values, we update them by weighted averaging, similar to [@newcombe2011kinectfusion]: $$\begin{aligned}
\mathtt{D}_t(\mathbf{v}) &= \frac{\mathtt{W}_{t-1}^\mathrm{D}(\mathbf{v}) \mathtt{D}_{t-1}(\mathbf{v}) + w_t(\mathbf{v}, \mathbf{p}_\mathbf{u}) d_t(\mathbf{v}, \mathbf{p}_\mathbf{u}, \mathbf{s})}{\mathtt{W}_{t-1}^\mathrm{D}(\mathbf{v}) + w_t(\mathbf{v}, \mathbf{p}_\mathbf{u})}, \\
\mathtt{W}_t^\mathrm{D}(\mathbf{v}) &= \mathtt{W}_{t-1}^\mathrm{D}(\mathbf{v}) + w_t(\mathbf{v}, \mathbf{p}_\mathbf{u}). \label{eq_tsdf_integ}\end{aligned}$$ Here, $d_t$ denotes the distance between the voxel and the surface boundary, and $w_t$ a quadric weight [@oleynikova2017voxblox] that takes the reliability of depth measurements into account. Similar updating is applied to the voxel color $\mathtt{C}_t(\mathbf{v})$.
In contrast to TSDF and colors of continuous values, weighted averaging cannot be applied to panoptic labels of discrete values. The most reliable and simplest way to manage panoptic labels is to record all integrated labels. This, unfortunately, results in a significant increase in memory usage and frequent memory allocation. Instead we store a single label at each voxel and update its weight by the increment/decrement strategy. If the pixel-wise panoptic label $\hat{L}_t^\mathrm{Pa}(\mathbf{u})$ estimated in the previous section is the same as the current voxel panoptic label $\mathtt{L}_{t-1}^\mathrm{Pa}(\mathbf{v})$, we increment the weight $\mathtt{W}_t^\mathrm{L}(\mathbf{v})$ with the quadric weight: $$\begin{aligned}
\mathtt{L}_t^\mathrm{Pa}(\mathbf{v}) = \mathtt{L}_{t-1}^\mathrm{Pa}(\mathbf{v}), \
\mathtt{W}_t^\mathrm{L}(\mathbf{v}) = \mathtt{W}_{t-1}^\mathrm{L}(\mathbf{v}) + w_t(\mathbf{v}, \mathbf{p}_\mathbf{u}). \label{eq_label_integ_case1}\end{aligned}$$ In contrast, if those panoptic labels do not coincide, we decrement the weight: $$\begin{aligned}
\mathtt{L}_t^\mathrm{Pa}(\mathbf{v}) = \mathtt{L}_{t-1}^\mathrm{Pa}(\mathbf{v}), \
\mathtt{W}_t^\mathrm{L}(\mathbf{v}) = \mathtt{W}_{t-1}^\mathrm{L}(\mathbf{v}) - w_t(\mathbf{v}, \mathbf{p}_\mathbf{u}). \label{eq_label_integ_case2}\end{aligned}$$ Note that in the case where $w_t > \mathtt{W}_{t-1}^\mathrm{L}$, that is, when the weight considerably falls, we replace the voxel label with the newly estimated label: $$\begin{aligned}
\mathtt{L}_t^\mathrm{Pa}(\mathbf{v}) = \hat{L}_t^\mathrm{Pa}(\mathbf{u}), \
\mathtt{W}_t^\mathrm{L}(\mathbf{v}) = w_t(\mathbf{v}, \mathbf{p}_\mathbf{u}) - \mathtt{W}_{t-1}^\mathrm{L}(\mathbf{v}). \label{eq_label_integ_case3}\end{aligned}$$
Thing Label Probability Integration\[sec\_label\_prob\_integration\]
--------------------------------------------------------------------
The [*thing*]{} label predicted by Mask R-CNN is frequently uncertain even while the segmentation mask is accurate, especially in the case where a small part of the object is visible. Hence we probabilistically integrate [*thing*]{} labels instead of assigning a single label to each foreground object: $$\begin{aligned}
p_{1 \cdots t}(z, l^\mathrm{Th}) = \frac{\sum_t p_t(z, \mathcal{O}) p_t(z, l^\mathrm{Th}|\mathcal{O})}{\sum_t p_t(z, \mathcal{O})}.\end{aligned}$$ Weighting the probability distributions with the detection confidence $p_t(z, \mathcal{O})$ allows the final distribution to preferentially reflect reliable detections.
Online Map Regularization\[sec\_map\_regularization\]
-----------------------------------------------------
While the integration scheme described above yields a reliable 3D panoptic map, it is possible to further improve the recognition accuracy by using a map regularization with a fully connected CRF model. A fully connected CRF with Gaussian edge potentials has been widely used in 2D image segmentation since an efficient inference method was proposed [@krahenbuhl2011efficient]. Recently, several studies that apply it to a 3D map, such as surfels or occupancy grids, have been reported [@hermans2014dense; @mccormac2017semanticfusion; @yang2017]. In those approaches, CRF models are constructed with respect to class labels whose number is fixed, whereas we consider the CRF with respect to panoptic labels whose number depends on the scene and is theoretically not limited. Here we are faced with two problems: how to properly compute unary potentials for panoptic labels, and how to infer a CRF whose number of labels is potentially large within a practical time.
### Problem Setting
We construct a fully connected graph whose nodes are individual voxels. We assign a label variable $x_v \in \mathcal{L}^\mathrm{Pa}$ to each node and infer the optimal labels $\mathbf{x} = \{x_v\}$ that minimize the Gibbs energy $E$ by the mean-field approximation and a message passing scheme: $$\begin{aligned}
E(\mathbf{x}) = \sum_v \psi_u(x_v) + \sum_{v<v^\prime} \psi_p(x_v, x_{v^\prime}).\end{aligned}$$ While it is non-trivial which unary potentials should be used for a panoptic label CRF, we use a negative logarithm of a probability distribution following a standard class label CRF: $$\begin{aligned}
\psi_u(x_v) = -\log p(x_v). \label{eq_unary_potential}\end{aligned}$$ We utilize a linear combination of Gaussian kernels for pairwise potentials because the efficient inference method [@krahenbuhl2011efficient] can be applied: $$\begin{aligned}
\psi_p(x_v, x_{v^\prime}) = \mu(x_v, x_{v^\prime}) \sum_m w^{(m)} k^{(m)}(\mathbf{f}_v, \mathbf{f}_{v^\prime}).\end{aligned}$$ Here, $\mu(x_s, x_s^\prime) = 1_{[x_s \neq x_s^\prime]}$ is a simple Potts model. As in [@krahenbuhl2011efficient], we chose the following two kernels which regularize the map with respect to voxel colors and locations, respectively: $$\begin{aligned}
k^{(1)}(\mathbf{f}_v, \mathbf{f}_{v^\prime}) &= \exp\biggr(-\frac{|\mathbf{v} - \mathbf{v}^{\prime}|^2}{2 \theta_\alpha^2} - \frac{|\mathtt{C}(\mathbf{v}) - \mathtt{C}(\mathbf{v}^{\prime})|^2}{2 \theta_\beta^2}\biggl), \\
k^{(2)}(\mathbf{f}_v, \mathbf{f}_{v^\prime}) &= \exp\biggr(-\frac{|\mathbf{v} - \mathbf{v}^{\prime}|^2}{2 \theta_\alpha^2}\biggl).\end{aligned}$$
### Unary Potential Approximation
Previous approaches [@hermans2014dense; @mccormac2017semanticfusion; @yang2017] assigned a probability distribution to each surfel or voxel, which can be used directly to compute unary potentials; in contrast, from the viewpoint of memory efficiency, we store only a single label in each voxel. Therefore, we approximate the unary potentials using only a single label, and weights stored in a voxel, based on a certain assumption described as follows.
Here let us focus on the integration scheme of panoptic labels shown in Eq. -. We denote the set of times when the predicted panoptic label is the same as, and not the same as, the current voxel label by $\mathcal{T}_{+} = \{\tau \ | \ \hat{L}_\tau^\mathrm{Pa}(\mathbf{u}) = \mathtt{L}_t^\mathrm{Pa}(\mathbf{v})\}$ and $\mathcal{T}_{-} = \{\tau \ | \ \hat{L}_\tau^\mathrm{Pa}(\mathbf{u}) \neq \mathtt{L}_t^\mathrm{Pa}(\mathbf{v})\}$, respectively. If $\mathtt{L}_\tau^\mathrm{Pa}(\mathbf{v}) = \mathtt{L}_t^\mathrm{Pa}(\mathbf{v})$ for all $\tau = 1, \cdots, {t-1}$, that is to say, the voxel label has not changed, Eq. holds strictly. If $p(x_v = \mathtt{L}_t^\mathrm{Pa}(\mathbf{v})) > 0.5$ and the number of integrations is sufficiently large, Eq. holds asymptotically: $$\begin{aligned}
\sum_{t \in \mathcal{T}_{+}} w_t(\mathbf{v}, \mathbf{p}_\mathbf{u}) - \sum_{t \in \mathcal{T}_{-}} w_t(\mathbf{v}, \mathbf{p}_\mathbf{u}) \simeq \mathtt{W}_t^\mathrm{L}(\mathbf{v}). \label{eq_label_weight_approx}\end{aligned}$$ In addition, from the TSDF update scheme in Eq. we have, $$\begin{aligned}
\sum_{t \in \mathcal{T}_{+}} w_t(\mathbf{v}, \mathbf{p}_\mathbf{u}) + \sum_{t \in \mathcal{T}_{-}} w_t(\mathbf{v}, \mathbf{p}_\mathbf{u}) = \mathtt{W}_t^\mathrm{D}(\mathbf{v}).\end{aligned}$$ Consequently, the probability that the current panoptic label in the voxel is actually correct can be calculated as, $$\begin{aligned}
p(x_v = \mathtt{L}_t^\mathrm{Pa}(\mathbf{v})) &= \frac{\sum_{t \in \mathcal{T}_{+}} w_t(\mathbf{v}, \mathbf{p}_\mathbf{u})}{\sum_{t \in \mathcal{T}_{+}} w_t(\mathbf{v}, \mathbf{p}_\mathbf{u}) + \sum_{t \in \mathcal{T}_{-}} w_t(\mathbf{v}, \mathbf{p}_\mathbf{u})} \nonumber \\
&\simeq \frac{1}{2} \biggl(1 + \frac{\mathtt{W}_t^\mathrm{L}(\mathbf{v})}{\mathtt{W}_t^\mathrm{D}(\mathbf{v})}\biggr). \label{eq_approx_positive_prob}\end{aligned}$$ It is unfortunately not possible to calculate the exact probability that the voxel takes a label other than the current label because the map does not record all the information about previously integrated labels. Therefore, we approximate the probability as follows, where $M$ denotes the number of panoptic labels in the map: $$\begin{aligned}
p(x_v) = \frac{1}{M-1} \bigl(1 - p(x_v = \mathtt{L}_t^\mathrm{Pa}(\mathbf{v}))\bigr) \ \ (x_v \neq \mathtt{L}_t^\mathrm{Pa}(\mathbf{v})). \label{eq_approx_negative_prob}\end{aligned}$$
Finally, we obtain the unary potential from Eq. , and . In spite of the approximated approach, it realizes quantitative and qualitative improvements in recognition accuracy, as shown in Section \[sec\_eval\_crf\].
### Map Division for Online Inference
The computational complexity of the inference algorithm proposed by Kr[ä]{}henb[ü]{}hl [*et al.*]{} [@krahenbuhl2011efficient] is $\mathcal{O}(NM)$, where $N$ and $M$ are the numbers of voxels and panoptic labels, respectively. In our problem setting, however, $M$ is theoretically limitless and could in practice be large, e.g. several hundreds, which would make online inference impracticable. To solve this problem, we present a map division strategy. When we divide the volumetric map into $S$ spatially contiguous submaps, the number of panoptic labels in each submap can be expected to be $\mathcal{O}(M/S)$. Hence, the total computational complexity could be reduced to $S \times \mathcal{O}(N/S \times M/S) = \mathcal{O}(NM) / S$. The map is divided by the block-wise region growing approach based on the predefined maximum number of voxel blocks. The division process has little effect on computational time.
EVALUATION
==========
Experimental Setup
------------------
{height="7cm"}
For evaluating the performance of our system, we used the ScanNet v2 dataset [@dai2017scannet], a large-scale dataset for indoor scene understanding. It provides RGB-D images captured by hand-held consumer-grade depth sensors, camera trajectories, reconstructed 3D models, and 2D/3D semantic annotations. In the following experiments, we used RGB-D images of size 640$\times$480 pixels and the provided camera trajectories for fair comparison. The dataset was composed of 1201 training scenes and 312 [*open*]{} test scenes. In addition, 100 [*hidden*]{} test scenes without publicly available semantic annotations are provided for the ScanNet Benchmark Challenge [@scannet_benchmark_challenge]. For quantitative evaluations, 20 class annotations are generally used. In this paper, we define the wall and floor as the [*stuff*]{} class $\mathcal{L}^\mathrm{St}$ and the other 18 classes, such as chairs and sofas, as the [*thing*]{} class $\mathcal{L}^\mathrm{Th}$. Note that our system is not limited to indoor scenes, and the numbers of [*stuff*]{} and [*thing*]{} classes can be arbitrarily defined.
We employed ResNet-50 for the backbone of PSPNet. The network was initialized with the ADE20K [@zhou2016semantic] pre-trained weights, and was then fine-tuned using a SGD optimizer for 30 epochs with a learning rate of 0.01 and a batch size of 2. We leveraged ResNet-101-FPN for the Mask R-CNN’s backbone. After initialization with MS COCO [@lin2014microsoft] pre-trained weights, the network was fine-tuned by 4-step alternating learning [@ren2015faster] using an ADAM optimizer for 25 epochs with a learning rate of 0.001 and a batch size of 1[^2].
We used the following parameters for the integration process: voxel size of 0.024 m, a truncation distance of 4$\times$0.024 m, 16$\times$16$\times$16 voxels per voxel block, IoU threshold $\theta_U = 0.25$. In the map regularization, $w^{(1)} = 10$, $w^{(2)} = 15$, $\theta_\alpha = 0.05 \ \mathrm{m}$, $\theta_\beta = 20$ were used with 5 iterations of CRF inference. The following experiments were performed on a computer equipped with an Intel Core i7-7800X CPU at 3.50 GHz and two NVIDIA GeForce GTX 1080Ti GPUs.
Quantitative and Qualitative Results
------------------------------------
Fig. \[fig\_qualitative\_results\] shows examples of 3D panoptic maps generated by our system. Unfortunately, there are no semantic mapping systems or 3D DNNs that can recognize a 3D scene at the level of [*stuff*]{} and [*things*]{}. Therefore, we evaluated the performance on two sub-tasks, 3D semantic segmentation and instance segmentation, for a quantitative comparison. In this evaluation, we used the hidden test set of ScanNet v2. We show the results in Tables \[table\_scannet\_sema\_seg\_benchmark\] and \[table\_scannet\_inst\_seg\_benchmark\]. In the tables, the state-of-the-art methods that apply 3D DNNs to points or volumetric grids are listed. Note that the methods of [@huang2018texturenet; @dai20183dmv; @hou20183d] leverage RGB images with associated camera poses as well. Our system that uses only 2D-based recognition modules surprisingly achieves comparable or superior performance compared with those methods, thanks to the careful integration of multi-view predictions. In terms of the class-wise accuracy, the results revealed that our system has advantages especially in the case of small objects such as sinks and pictures, and objects that are confusing to recognize only by their geometry, such as beds, bookshelves, and curtains. In Table \[table\_scannet\_sema\_seg\_benchmark\], several semantic segmentation methods outperform our system because of their large receptive fields in 3D space. However, these methods basically need to reconstruct the entire scene in advance, assuming offline process, while our system is an online and incremental framework. How to apply 3D DNNs to partial observations and how to integrate them into an online mapping system are left for future work.
Additionally, we evaluated 3D panoptic quality on the open test set of ScanNet v2, although there are no quantitatively comparable methods. We employed the evaluation criteria originally proposed in [@kirillov2018panoptic]. Note that the quality was evaluated with respect to each vertex instead of each pixel, and, as with the ScanNet 3D semantic instance benchmark, we ignored the predicted [*things*]{} with less than 100 vertices. We show the panoptic quality (PQ) as well as the segmentation quality (SQ) and recognition quality (RQ) in Table \[table\_panoptic\_quality\]. We hope these results will invigorate research in this field.
\[0.8\]
avg. wall floor cab bed chair sofa tabl door wind bkshf pic cntr desk curt fridg showr toil sink bath ofurn
----------------------------------------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
ScanNet [@dai2017scannet] 30.6 43.7 78.6 31.1 36.6 52.4 34.8 30.0 18.9 18.2 50.1 10.2 21.1 34.2 0.2 24.5 15.2 46.0 31.8 20.3 14.5
PointNet++ [@qi2017pointnet++] 33.9 52.3 67.7 25.6 47.8 36.0 34.6 23.2 26.1 25.2 45.8 11.7 25.0 27.8 24.7 21.2 14.5 54.8 36.4 58.4 18.3
SPLATNet [@su2018splatnet] 39.3 [69.9]{} 92.7 31.1 51.1 65.6 51.0 38.3 19.7 26.7 60.6 0.0 24.5 32.8 40.5 0.1 24.9 59.3 27.1 47.2 22.7
Tangent Conv. [@tatarchenko2018tangent] 43.8 63.3 91.8 36.9 64.6 64.5 56.2 42.7 27.9 35.2 47.4 14.7 35.3 28.2 25.8 28.3 29.4 61.9 48.7 43.7 29.8
3DMV [@dai20183dmv] 48.4 60.2 79.6 42.4 53.8 60.6 50.7 41.3 37.8 53.9 64.3 21.4 31.0 43.3 57.4 [53.7]{} 20.8 69.3 47.2 48.4 30.1
TextureNet [@huang2018texturenet] [56.6]{} 68.0 [93.5]{} [49.4]{} 66.4 [71.9]{} 63.6 [46.4]{} [39.6]{} [56.8]{} [67.1]{} 22.5 [44.5]{} 41.1 67.8 41.2 53.5 79.4 [56.5]{} **67.2** [35.6]{}
SparseConvNet [@graham20183d] **72.5** **86.5** **95.5** **72.1** **82.1** **86.9** **82.3** **62.8** **61.4** **68.3** **84.6** **32.5** **53.3** **60.3** **75.4** **71.0** **87.0** **93.4** **72.4** [64.7]{} **57.2**
**PanopticFusion (Ours)** 52.9 60.2 81.5 38.6 [68.8]{} 63.2 [64.9]{} 44.2 29.3 56.1 60.4 [24.1]{} 22.5 [43.4]{} [70.5]{} 49.9 [66.9]{} [79.6]{} 50.7 49.1 34.8
\[0.8\]
avg. cab bed chair sofa tabl door wind bkshf pic cntr desk curt fridg showr toil sink bath ofurn
--------------------------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- --------- ---------- ---------- ---------- ---------- ---------- ---------- ----------- ----------
SGPN [@wang2018sgpn] 14.3 6.5 39.0 27.5 35.1 16.8 8.7 13.8 16.9 1.4 [2.9]{} 0.0 6.9 2.7 0.0 43.8 11.2 20.8 4.3
GSPN [@yi2018gspn] 30.6 [34.8]{} 40.5 [58.9]{} 39.6 [27.5]{} 28.3 24.5 31.1 2.8 **5.4** 12.6 6.8 21.9 21.4 82.1 33.1 50.0 29.0
3D-SIS [@hou20183d] 38.2 19.0 43.2 57.7 **69.9** 27.1 [32.0]{} 23.5 24.5 7.5 1.3 3.3 26.3 **42.2** **85.7** 88.3 11.7 **100.0** 24.0
MASC [@liu2019masc] [44.7]{} **38.2** [55.5]{} **63.3** [63.9]{} **38.6** **36.1** [27.6]{} [38.1]{} [32.7]{} 0.2 **26.0** [50.9]{} [45.1]{} 57.1 **98.0** [36.7]{} 52.8 [43.2]{}
**PanopticFusion (Ours)** **47.8** 25.9 **71.2** 55.0 59.1 26.7 25.0 **35.9** **59.5** **43.7** 0.0 [17.5]{} **61.3** 41.1 **85.7** [94.4]{} **48.5** [66.7]{} **43.4**
\[0.75\]
method metric all *things* *stuff* wall floor cab bed chair sofa tabl door wind bkshf pic cntr desk curt fridg showr toil sink bath ofurn
-------- -------- ------ ---------- --------- ------ ------- ------ ------ ------- ------ ------ ------ ------ ------- ------ ------ ------ ------ ------- ------- ------ ------ ------ -------
PQ 29.7 26.7 56.7 37.5 76.0 18.6 29.1 37.8 38.2 29.5 13.8 14.1 13.0 26.5 8.3 14.9 11.6 38.0 28.8 72.4 33.3 28.0 24.3
SQ 71.2 71.4 69.5 62.3 76.7 69.4 68.5 69.3 72.3 70.1 74.6 69.9 70.7 72.9 65.0 60.6 70.5 75.3 75.8 79.2 71.9 74.0 75.3
RQ 41.1 36.8 79.6 60.2 99.0 26.8 42.5 54.6 52.8 42.1 18.5 20.1 18.4 36.3 12.8 24.6 16.4 50.4 37.9 91.3 46.4 37.8 32.2
PQ 33.5 30.8 58.4 40.4 76.4 23.8 35.8 46.7 42.1 34.8 18.0 19.3 16.4 26.4 10.4 16.1 16.6 39.5 36.3 76.1 36.7 31.0 27.7
SQ 73.0 73.3 70.7 64.0 77.4 71.1 70.1 74.3 74.6 74.3 76.0 72.5 73.9 71.2 65.1 61.7 72.3 77.7 79.5 81.4 72.7 75.3 75.8
RQ 45.3 41.3 80.9 63.1 98.7 33.5 51.1 62.8 56.3 46.9 23.6 26.7 22.2 37.1 16.0 26.0 23.0 50.8 45.7 93.5 50.5 41.2 36.5
![Results of the map regularization with the map division strategy. The relationship between the maximum number of voxel blocks and (a) recognition accuracy and (b) computational time. Note the computational time is shown in a logarithmic scale.[]{data-label="fig_crf_with_map_division"}](./img/offline_crf_with_map_division.png){width="8.5cm"}
Evaluation of Map Regularization\[sec\_eval\_crf\]
--------------------------------------------------
In this section, we evaluate the map regularization proposed in Section \[sec\_map\_regularization\]. First, we evaluated the effects of the map division on the recognition accuracy and computational time. We used the open test set for the recognition accuracy and typical scenes in ScanNet v2 for the computational time. The result is shown in Fig. \[fig\_crf\_with\_map\_division\]. Note that, in this experiment, we applied regularization to the pre-generated map as a post process to evaluate solely the effects of CRF.
As can been seen, the recognition performance was improved by the map regularization with the proposed unary potential approximation regardless of whether or not map division was used. The results also show that the map division strategy drastically reduced the computational time without a decrease in recognition performance, compared with the case of building a CRF model for a whole map.
Based on the above results, our online system employed map regularization with the map division strategy. We chose a maximum number of voxel blocks of 25 because of the better recognition accuracy and acceptable computational time. Table \[table\_panoptic\_quality\] shows the difference in recognition performance due to whether or not map regularization was used in online processing. This result shows that the map regularization improved the recognition performance even when the system ran online. Note that the scores of almost all the classes were boosted by the proposed regularization. See Fig. \[fig\_crf\_qualitative\_reuslts\] for qualitative effects of the map regularization.
![Qualitative results of map regularization. The noisy predictions within red circles are appropriately regularized, taking a spatial context into account.[]{data-label="fig_crf_qualitative_reuslts"}](./img/crf_qualitative_results_LQ.png){width="8.5cm"}
Run-time Analysis
-----------------
Table \[table\_runtime\_analysis\] shows computational times for each component of our system, which are measured on scene0645\_01, a typical large-scale scene in ScanNet v2 (shown in Fig. \[fig\_panopticfusion\]). PSPNet and Mask R-CNN each run on GPUs, and the other components are processed on a CPU. All components are basically processed in parallel. The throughput of our system is around 4.3 Hz, which is determined by Mask R-CNN, the bottleneck process of our system. Although our current implementation is not highly optimized, our system is able to run at a rate allowing interaction. Note that the computational time except for the map regularization does not depend on the scale of scenes nor the number of [*things*]{} because we utilize the raycasting approach for the integrations. The processing time of the map regularization increases to about 10 seconds at the end of the sequence, but it could be reduced by processing only the voxel blocks near the camera frustum.
\[0.95\]
**Frequency** **Component** **time**
------------------------- ------------------------------- -------------
Every Mask R-CNN frames PSPNet 80 ms
Mask R-CNN 235 ms
Panoptic label fusion 2 ms
Reference panoptic label gen. 19 ms
Panoptic label tracking 9 ms
Volumetric integration 139 ms
Probability integration $\sim 1$ ms
Every 10 sec. Map regularization 4.5 s
Every 1 sec. Mesh extraction 14 ms
**Throughput** 4.3 Hz
: Run-time analysis.[]{data-label="table_runtime_analysis"}
APPLICATIONS
============
In this section, we demonstrate a promising AR application utilizing a 3D panoptic map generated online by the proposed system. A 3D panoptic map reconstructed as 3D meshes allows us to realize the following visualizations according to the context of the scene:
- Path planning on [*stuff*]{} regions such as floors and walls.
- Interaction with individual objects, or the [*thing*]{} regions.
- Interaction appropriate for the semantics of each region.
- Natural occlusion and collision visualization.
We show an example of an AR application utilizing the above visualizations in Fig. \[fig\_ar\_application\]. Humanoids and insect-type robots are able to locomote on the floor and wall meshes, respectively, according to the automatic path planning. Additionally, the semantics of each object realizes context-aware interactions such that humanoids sit and lie on chairs and sofas, respectively, and CG objects appear on tables. Moreover, we can naturally visualize the occlusion effects, which are important for AR, because the 3D meshes of the scene are extracted. Note that, taking advantage of the accurately recognized 3D panoptic map, we can easily estimate the poses of seats of chairs and sofas, and top panels of tables by using simple normal- and curvature-based segmentation and plane detection.
We believe that our system is useful not only for AR scenarios but also for autonomous robots that explore scenes and manipulate objects.
![An example of AR application using a 3D panoptic map generated by PanopticFusion system.[]{data-label="fig_ar_application"}](./img/ar_application_v2_LQ.png){width="8.5cm"}
CONCLUSIONS
===========
In this paper, we have introduced a novel online volumetric semantic mapping system at the level of [*stuff*]{} and [*things*]{}. It performs dense semantic labeling while discriminating individual objects, as well as large-scale 3D reconstruction and labeled mesh extraction thanks to the use of a spatially hashed volumetric map representation. This was realized by pixel-wise panoptic label prediction and its volumetric integration with careful label tracking. In addition, we constructed a fully connected CRF model with respect to panoptic labels and inferred it online with a novel unary potential approximation and a map division strategy, which further improved the recognition performance. The experimental results showed that our system outperformed or compared well with state-of-the-art offline 3D DNN methods in terms of both 3D semantic and instance segmentation. In future work, we plan to extend our system to ensure global consistency against long-term pose drift, to perform high-throughput mapping by network reduction, and to support dynamic environments.
We believe that the [*stuff*]{} and [*things*]{}-level semantic mapping will open the way to new applications of intelligent autonomous robotics and context-aware augmented reality that deeply interact with the real world.
[^1]: $^{1}$The authors are with R&D Center, Sony Corporation. [{gaku.narita, takashi.seno, tomoya.ishikawa, yohsuke.kaji}@sony.com]{}
[^2]: We used a publicly available implementation of [@pspnet_keras_tensorflow] and [@matterport_maskrcnn_2017] for PSPNet and Mask R-CNN, respectively.
|
---
abstract: 'A transient in the Local Group dwarf irregular galaxy NGC6822 (Barnard’s Galaxy) was discovered on 2017 August 2 and is only the second classical nova discovered in that galaxy. We conducted optical, near-ultraviolet, and X-ray follow-up observations of the eruption, the results of which we present here. This ‘very fast’ nova had a peak $V$-band magnitude in the range $-7.41>M_V>-8.33$mag, with decline times of $t_{2,V} = 8.1 \pm 0.2$d and $t_{3,V} = 15.2 \pm 0.3$d. The early- and late-time spectra are consistent with an spectral class. The H$\alpha$ emission line initially has a full width at half-maximum intensity of $\sim 2400$kms$^{-1}$ – a moderately fast ejecta velocity for the class. The H$\alpha$ line then narrows monotonically to $\sim1800$kms$^{-1}$ by 70d post-eruption. The lack of a pre-eruption coincident source in archival *Hubble Space Telescope* imaging implies that the donor is a main sequence, or possibly subgiant, star. The relatively low peak luminosity and rapid decline hint that AT2017fvz may be a ‘faint and fast’ nova.'
author:
- 'M. W. Healy,$^{1}$[^1] M. J. Darnley,$^{1}$[^2] C. M. Copperwheat,$^{1}$ A. V. Filippenko,$^{2,3}$'
- 'M. Henze,$^{4}$ J. C. Hestenes,$^{2}$ P. A. James,$^{1}$ K. L. Page,$^{5}$'
- |
S. C. Williams,$^{6}$ and W. Zheng$^{2}$\
$^{1}$Astrophysics Research Institute, Liverpool John Moores University, Liverpool, L3 5RF, UK\
$^{2}$Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA\
$^{3}$Miller Senior Fellow, Miller Institute for Basic Research in Science, University of California, Berkeley, CA 94720, USA\
$^{4}$Department of Astronomy, San Diego State University, San Diego, CA 92182, USA\
$^{5}$X-Ray and Observational Astronomy Group, Department of Physics & Astronomy, University of Leicester, LE1 7RH, UK\
$^{6}$Physics Department, Lancaster University, Lancaster, LA1 4YB, UK\
bibliography:
- 'ResearchPapers.bib'
date: 'Accepted 2019 April 15. Received 2019 April 15; in original form 2018 November 13'
title: 'AT2017fvz: a nova in the dwarf irregular galaxy NGC6822'
---
\[firstpage\]
novae, cataclysmic variables – stars: individual (AT 2017fvz)
Introduction {#Introduction}
============
Classical novae (CNe) belong to the class of accreting binaries known as cataclysmic variables. As first proposed by @1954PASP...66..230W, these are closely interacting binaries consisting of a white dwarf (WD) accreting material from a donor – either a main sequence, subgiant, or red giant star [see @2012ApJ...746...61D]. Through Roche-lobe overflow or the stellar wind of an evolved donor, hydrogen-rich material from the donor streams, usually via an accretion disk [@1995CAS....28.....W], onto the WD where severe heating and compression take place. Given favourable conditions, this results in a thermonuclear runaway (TNR) within the accreted envelope on the WD with a proportion of that envelope subsequently ejected — the nova eruption [@1976IAUS...73..155S]. The luminosity of these systems typically increases to a few $\times10^{4}\, L_{\odot}$ [see, e.g., @2010AN....331..160B] with peak absolute magnitudes reaching $M_V \approx -10.5$ in extreme cases [@2009ApJ...690.1148S; @2018MNRAS.474.2679A].
Following the TNR, stable H-burning continues within any material remaining on the WD surface. This results in the emission of a large amount of X-rays typically peaking in the range $30-50$eV — the so-called super-soft X-ray source . The SSS is initially obscured by optically thick ejecta surrounding the nova; however, once the optical depth has decreased sufficiently, the SSS is unveiled.
All novae are predicted to recur [@2005ApJ...623..398Y], but the broad range of times between consecutive eruptions has led to segregation based on recurrence period ($P_\mathrm{rec}$). CNe have been observed to erupt just once. Recurrent novae (RNe) are systems with a high-mass WD and high accretion rates that have been recorded erupting multiple times.
One can categorise novae into different speed classes based on their decline times, $t_2$ and $t_3$ [@1957gano.book.....G]. These denote the time taken to decay by 2 or 3 mag (respectively) from maximum light. @1936PASP...48..191Z first proposed a relationship between the decline time and the maximum absolute visual magnitude of a nova. Subsequently, @1945PASP...57...69M and @1956AJ.....61...15A developed the ‘maximum magnitude – rate of decline’ relation [MMRD; see, e.g., @2000AJ....120.2007D]. However, the MMRD suffers from a large scatter, and the relation has been diluted by the discovery of ‘faint and fast’ novae [@2011ApJ...735...94K; @2016ApJS..227....1S] and short-cycle RNe ($P_\mathrm{rec}<10$yr; Darnley & Henze, in prep.).
Independent studies of Galactic novae using [*Gaia*]{} data release 2 parallaxes appear to show contradictory results. @2018MNRAS.481.3033S proposes that the MMRD is an unusable distance determination method and that it should no longer be employed. However, show that the MMRD relationship is strengthened once [*Gaia*]{} distances are assumed. Although there is some overlap, these two studies use different samples of novae. @2017ApJ...839..109S used a large sample of M87 novae to clearly demonstrate (see their Figure 1) the impact of ‘faint and fast’ novae on the MMRD distribution and therefore that the concept is inherently flawed.
Novae may be divided into two spectroscopic classes based on the prominent non-Balmer emission lines in their early-post-maximum spectra: either He/N or [@1992AJ....104..725W; @1994ApJ...426..279W]. The contribution of novae from each class varies between different galaxies, possibly due to variations in the dominant stellar population and metallicity of a given host [@2013AJ....145..117S]. Novae from younger (disk) populations have higher mean WD masses than those from older (bulge) populations. High-mass WDs create lower mass but higher velocity ejecta than their low-mass counterparts, and are believed to produce the He/N dominant spectra, with the lower mass WDs creating the class [@2012AJ....144...98W; @2013AJ....145..117S].
The study of novae in extragalactic environments provides the only way to explore how the local environment (e.g., the metallicity and star-formation rate) can affect the nova rate and characteristics of nova eruptions [@2016ApJS..227....1S]. The M31 nova population is dominated by the class [82%; @2011ApJ...734...12S]. Yet ‘bulgeless’ galaxies show similar numbers of each class. For example, five of the ten spectroscopically classified novae in M33 are ; the M33 spectral type distribution differs from that of M31 at the 98% confidence level [@2012ApJ...752..156S; @2014ASPC..490...77S]. The fraction of novae in the LMC is also $\sim 50\%$ [@2013AJ....145..117S].
NGC6822 is a dark matter dominated [@2003MNRAS.340...12W] dwarf irregular galaxy in the Local Group at a distance $476\pm44$kpc [@2014ApJ...794..107R]. It provides a low-metallicity environment compared to the majority of Local Group novae: \[Fe/H\] $\approx -0.5$ .
AT2017fvz[^3] (aka KAIT-17bm) is only the second nova to be discovered within NGC6822 (see Section \[PreviousNova\] for a discussion of the first nova). It was discovered on 2017 Aug. 2.384 UT with an unfiltered magnitude of 17.6 at $\alpha=19^\mathrm{h}45^\mathrm{m}1^\mathrm{s}\!.03$, $\delta=-14\degr46^\prime50^{\prime\prime}\!\!.74$ [J2000; @2017TNSTR.831....1H] by the Katzman Automatic Imaging Telescope [KAIT; @2001ASPC..246..121F] of the Lick Observatory Supernova Search (LOSS). This nova was also observed by the All-Sky Automated Survey for Supernovae [ASAS-SN; see @2014ApJ...788...48S] on Aug. 3.190 and then with the Asteroid Terrestrial-impact Last Alert System [ATLAS; @2018PASP..130f4505T; @2018AJ....156..241H] on Aug. 3.386.
Here we report optical, near-ultraviolet (NUV), and X-ray observations of the eruption of AT2017fvz. In Section \[Observations and Data Analysis\] we describe the observations and data analysis. In Section \[Results\] we present the results of the photometry, spectroscopy, and X-ray analysis, and we discuss these in Section \[Discussion\]. We summarise our findings in Section \[Summary and Conclusions\]. Throughout, all times are quoted in coordinated universal time (UT), all uncertainties are quoted to 1$\sigma$, and all upper limits to 3$\sigma$.
Observations and Analysis {#Observations and Data Analysis}
=========================
Ground-based photometry
-----------------------
The field containing the nova had been monitored by KAIT using its clear filter since 2017 July 15.404 without any associated detections until the discovery on Aug. 2.384, after which the nova was followed until Aug. 31.284. ATLAS monitored a similar field from July 5.477 using its ‘orange’ filter, approximately covering $r'$ and $i'$ (5600–8200Å)[^4], without any associated detections until the first detection on Aug. 3.386. Like KAIT, the nova was monitored after discovery by ATLAS for the next 47d until Sep. 19.317 using the orange filter and also a ‘cyan’ filter which approximately covers $V$ and $r'$ (4200–6500Å). A few hours before the ATLAS detection, the nova was detected by ASAS-SN on 2017 Aug. 3.190 with a $V$-band filter. A Liverpool Telescope [LT; @2004SPIE.5489..679S] follow-up campaign began 7.53d post-discovery; observations were taken with IO:O[^5] through $u'BVr'i'$ filters.
Debiasing and flatfielding of the LT data were performed by the automatic LT reduction pipeline. Aperture photometry was calculated from these data using standard tools within PyRAF and calibrated against stars from the Local Group Galaxies Survey [LGGS; @2007AJ....133.2393M]. The $u'r'i'$ magnitudes of the LGGS stars were calculated using transformations from @2005AJ....130..873J [their Table 1]. Each time spectra were obtained with the SPectrograph for the Rapid Acquisition of Transients [SPRAT; @2014SPIE.9147E..8HP see Section \[Spectroscopy\]] by the LT, acquisition images were also taken using the SPRAT detector. These images were reduced in the same manner as the IO:O data. The acquisition images were unfiltered, but the photometry was calibrated relative to the $r'$ filter.
The KAIT data were reduced using a custom pipeline [@2010ApJS..190..418G]. Point-spread-function (PSF) photometry was obtained using the IDL implementation of DAOPHOT [@1987PASP...99..191S; @1993ASPC...52..246L]. Several nearby stars from the APASS catalog [@2009AAS...21440702H] were used to calibrate the KAIT clear-band data, with their magnitudes converted to the Landolt $R$-band system using the empirical prescription presented by R. Lupton[^6].
ATLAS carries out difference imaging of every frame with respect to a reference sky and the photometry reported here is from those images. The photometry was carried out as described by [@2018PASP..130f4505T] and [@2017ApJ...850..149S].
Spectroscopy {#Spectroscopy}
------------
The optical spectra of AT2017fvz were taken using SPRAT on the LT. SPRAT is a spectrograph with a slit $95^{\prime\prime}$ long and $1^{\prime\prime}\!\!.8$ wide giving a resolution of 18Å per pixel, corresponding to $R\approx350$ at the centre of the spectrum. It covers visible wavelengths in the range 4000–8000Å. The details of the spectra, which were obtained using the blue-optimised mode, are summarised in Table \[SpectroscopyTable\]. All spectra were extracted, wavelength calibrated, and flux calibrated using the SPRAT pipeline [@2014SPIE.9147E..8HP], except for the Aug. 25 spectrum which was not flux calibrated owing to poor sky transparency (clouds). The spectra were then analysed using routines with PyRAF.
\[SpectroscopyTable\]
UT Date$^\mathrm{a}$ MJD (d) $t-t_0$ (d) Exposure time (s)
---------------------- ----------- ------------- -------------------
2017-08-09.900 57974.900 8.016 $3 \times 600$
2017-08-15.924 57980.924 14.040 $3 \times 600$
2017-08-19.894 57984.894 18.010 $3 \times 600$
2017-08-25.885 57990.885 24.001 $3 \times 600$
2017-09-12.879 58008.879 41.995 $3 \times 900$
2017-10-10.848 58036.848 69.964 $3 \times 900$
: LT SPRAT spectroscopy of AT2017fvz.
$^\mathrm{a}$The date refers to the midpoint of each observation.
*Swift* NUV and X-ray observations {#UV and X-ray observations}
----------------------------------
Five target-of-opportunity (ToO) observations with the Neil Gehrels [*Swift*]{} Observatory [@2004ApJ...611.1005G], totalling 20.0ks, were utilised to follow the NUV and X-ray evolution of the AT2017fvz (Target ID: 10268). We summarise all of the *Swift* data in Table \[Swift\].
------------------ ------------------- ------- -------------------- ------------------- ---------- ----------- ---------- -----------
Exp$^\mathrm{a}$ Date$^\mathrm{b}$ MJD $t-t_0^\mathrm{c}$ $uvw1^\mathrm{d}$
(ks) (UT) (d) (d) (mag) 0.3–1keV 0.3–10keV 0.3–1keV 0.3–10keV
3.9 2017-09-09 58005 38.12 $18.6\pm0.1$ $<1.9$ $<1.9$ $<0.8$ $<0.8$
3.7 2017-10-08 58034 67.12 $19.4\pm0.2$ $<3.5$ $<3.3$ $<1.4$ $<1.3$
3.4 2017-11-07 58064 97.12 $19.6\pm0.3$ $<2.5$ $<3.2$ $<1.0$ $<1.3$
3.7 2018-04-27 58235 268.12 $20.2\pm0.3$ $<3.4$ $<3.2$ $<1.3$ $<1.3$
4.0 2018-08-25 58355 388.12 $19.9\pm0.2$ $<2.5$ $<3.0$ $<1.0$ $<1.2$
------------------ ------------------- ------- -------------------- ------------------- ---------- ----------- ---------- -----------
$^\mathrm{a}$Dead-time corrected XRT exposure time.
$^\mathrm{b}$Start date of the observation.
$^\mathrm{c}$Time since day of eruption on 2017 Aug. 1.884.
$^\mathrm{d}$Vega magnitudes for the $uvw1$ filter (central wavelength: 2600Å).
$^\mathrm{e}$X-ray luminosity upper limits (unabsorbed, blackbody fit, 0.3–1keV or 0.3–10keV, as indicated).
NUV data were obtained with the UV/Optical Telescope [UVOT; @2005SSRv..120...95R] through the $uvw1$ filter. X-ray data were collected by the X-ray Telescope [XRT; @2005SSRv..120..165B] in photon-counting mode. The NUV data were processed with HEASoft tools [v6.24; @1995ASPC...77..367B] and using the most recent calibration files. We extracted the count-rate upper limits from the X-ray data using the online [*Swift*]{} XRT tool[^7] [@2009MNRAS.397.1177E].
Results {#Results}
=======
Reddening {#Reddening}
---------
NGC6822 has a Galactic longitude and latitude of $\ell= 25.4\degr$ and $b = -18.4\degr$, respectively . This results in that galaxy being affected by a modest amount of foreground Milky Way extinction. @1967AJ.....72..134K found the Galactic reddening toward the outer regions of NGC6822 to be $E_{B-V}=0.27 \pm 0.03$mag, as did @1995AJ....110.2715M with $E_{B-V}=0.26$mag. These are consistent with @1996AJ....112.1928G and @2007AJ....133.2393M who found $E_{B-V}=0.24 \pm 0.03$mag and $E_{B-V}=0.25$mag, respectively. The online dust-mapping tool[^8] [@2018MNRAS.478..651G] returns a Galactic reddening toward NGC6822 of $E_{B-V}=0.22 \pm 0.02$mag.
Cepheid variables within NGC6822 have been employed to estimate the internal reddening. @1983ApJ...273..539M found $E_{B-V}=0.36$mag, @2006ApJ...647.1056G reported a similar average reddening of $E_{B-V}=0.36 \pm 0.01$mag. @2014ApJ...794..107R used optical and infrared data for Cepheids to determine that the foreground reddening along the line of sight to NGC6822 is $E_{B-V}=0.35\pm 0.04$mag.
We have no knowledge of the radial displacement of AT2017fvz within NGC6822 so we adopt the two most extreme values of reddening. The foreground reddening toward NGC6822 gives the lower limit, the addition of reddening internal to NGC6822 gives the upper limit.
Photometric evolution
---------------------
The AT2017fvz photometry from ASAS-SN, ATLAS, KAIT, LT, and *Swift* are presented in Figure \[Lightcurve\] and in Tables \[Photometry1\]–\[Photometry3\]. The light curves illustrate that the nova was discovered prior to peak optical magnitude. We calculate the time of eruption to be 2017 Aug. $1.9 \pm 0.5$, the midpoint between the last nondetection (KAIT) with $m_\mathrm{clear}
> 18.1$mag on 2017 Aug. 1.384 and the discovery on Aug. 2.384. Throughout, we refer to the time of eruption as $t_{0}$.
{width="94.00000%"} {width="94.00000%"}
\[6Lightcurve\]
The $u'$, $B$, and $V$ bands all fade at approximately the same rate from peak until around 40d, while $i'$ fades more slowly and $r'$ even slower owing to the strong influence of the H$\alpha$ emission line on the broad-band photometry. We estimated the decline times ($t_2$ and $t_3$) of AT2017fvz by taking the brightest data point as the peak of the eruption and assuming a power-law decline (in luminosity) [see, e.g., @2006ApJS..167...59H]. The decline times for each filter are recorded in Table \[Decline Times\]. If we utilise the decline times with the MMRD relation [@2000AJ....120.2007D], then we would expect peak absolute magnitudes of $M_{V} = -9.0 \pm 0.5$ and $M_{V} = -9.0 \pm 0.7$ for $t_2$ and $t_3$, respectively. It should be noted that this MMRD was derived from Galactic novae and, in addition to other limitations, may not be reliable within the differing environment of NGC6822 (see Section \[PossibleRN\]).
Filter $t_2$ (d) $t_3$ (d)
-------- ---------------- ------------------------------------
$u'$ $7.1 \pm 0.2$ $13.7 \pm 0.3$
$B$ $7.0 \pm 0.2$ $13.3 \pm 0.3$
$V$ $8.1 \pm 0.2$ $15.2 \pm 0.3$
$r'$ $15.5 \pm 0.4$ $33\phantom{.0} \pm 1\phantom{.0}$
$i'$ $13.0 \pm 0.3$ $25.3 \pm 0.6$
: Decline times of AT2017fvz in each filter.[]{data-label="Decline Times"}
Taking the distance modulus of NGC6822 as $\mu_{0} = 23.38 \pm 0.02$mag [@2014ApJ...794..107R], correcting for foreground reddening using $E_{B-V}=0.22 \pm 0.02$mag, we derive a lower limit for the peak absolute magnitude of $M_{V} = -7.41 \pm 0.07$. Here, we assumed that the peak observed magnitude (ASAS-SN) corresponded to the peak of the eruption. By extrapolating the $V$-band light curve power-law fit back to the final pre-eruption nondetection, we can estimate an upper limit on the peak eruption magnitude. Combining this upper limit with the estimate of the total (foreground and internal) reddening ($E_{B-V}=0.36 \pm 0.01$mag) yields an upper limit for the peak absolute magnitude of $M_{V} = -8.33 \pm 0.05$.
There is evidence for a ‘plateau’ in the $u'$, $B$, $V$, and $i'$-band light curves around $t = 25$d. As such, this nova would belong to the ‘plateau’ class [P-class; @2010AJ....140...34S], where an otherwise smoothly declining light curve is interrupted by a short period when the optical magnitude remains approximately constant.
Spectroscopic evolution
-----------------------
To aid the analysis of the AT2017fvz spectra, we made extensive use of spectral line data from @1945CoPri..21....1M and @2012AJ....144...98W. All of the spectra of AT2017fvz are plotted in Figure \[AllSpectra\]. These spectra can be split into three groups: the first contains the first four spectra that are within a $\sim16$-day-long period during early decline; the fifth spectrum was taken at $t \approx 42$d during the plateau, and the sixth was taken at $t \approx 70$d during the nebular phase. The spectra are presented in the rest frame of the observer. The average radial velocity of NGC6822 is $-57$kms$^{-1}$ [@2004AJ....128...16K]. The flux and full width at half-maximum intensity (FWHM) velocity were calculated by fitting Gaussian profiles to the emission lines using the SPLAT package in STARLINK; the fluxes and velocities are reported in Tables \[SpecFlux\] and \[SpecVel\], respectively.
{width="97.00000%"}
------------------------- --------------- --------------- --------------- --------------- ---------------
Line identification
(rest wavelength \[Å\]) $t = 8.016$d $t = 14.040$d $t = 18.010$d $t = 41.995$d $t = 69.964$d
H$\delta$ (4102) 6 $\pm$ 3 8 $\pm$ 2 7 $\pm$ 3 2 $\pm$ 1 1.7 $\pm$ 0.4
H$\gamma$ (4341) 12 $\pm$ 1 9 $\pm$ 1 7 $\pm$ 1 2.4 $\pm$ 0.5 2.0 $\pm$ 0.6
H$\beta$ (4861) 30 $\pm$ 3 31 $\pm$ 4 27 $\pm$ 2 6.8 $\pm$ 0.5 2.4 $\pm$ 0.5
[\[\]]{} (5007) [–]{} [–]{} [–]{} [–]{} 2.6 $\pm$ 0.4
(5018) 4.6 $\pm$ 0.7 5 $\pm$ 1 [–]{} [–]{} [–]{}
H$\alpha$ (6563) 139 $\pm$ 6 210 $\pm$ 10 230 $\pm$ 10 61 $\pm$ 3 24 $\pm$ 2
(7773) 28 $\pm$ 4 22 $\pm$ 3 16 $\pm$ 2 [–]{} [–]{}
------------------------- --------------- --------------- --------------- --------------- ---------------
We have not included the spectrum from 24.001d after eruption because the fluxes are not reliable.
------------------------- ---------------- ---------------- ---------------- --------------- ---------------- -------------------------------------------
Line identification
(rest wavelength \[Å\]) $t = 8.016$d $t = 14.040$d $t = 18.010$d $t = 24.001$d $t = 41.995$d $t = 69.964$d
H$\delta$ (4102) 2600 $\pm$ 900 2100 $\pm$ 300 2100 $\pm$ 600 [–]{} 900 $\pm$ 50 1500 $\pm$ 300
H$\gamma$ (4341) 2500 $\pm$ 200 2000 $\pm$ 200 2200 $\pm$ 300 [–]{} 1400 $\pm$ 200 [*3500*]{} $\pm$ [*800*]{}$\ ^\mathrm{a}$
H$\beta$ (4861) 2300 $\pm$ 200 2100 $\pm$ 200 1900 $\pm$ 100 [–]{} 1800 $\pm$ 100 1600 $\pm$ 200
[\[\]]{} (5007) [–]{} [–]{} [–]{} [–]{} [–]{} 1900 $\pm$ 200
(5018) 1900 $\pm$ 200 2200 $\pm$ 200 [–]{} [–]{} [–]{} [–]{}
H$\alpha$ (6563) 2430 $\pm$ 70 2300 $\pm$ 100 2070 $\pm$ 70 2000 $\pm$ 90 1840 $\pm$ 60 1900 $\pm$ 100
(7773) 2800 $\pm$ 300 2200 $\pm$ 200 2000 $\pm$ 200 [–]{} [–]{} [–]{}
------------------------- ---------------- ---------------- ---------------- --------------- ---------------- -------------------------------------------
$^\mathrm{a}$ This velocity is an upper limit as the H$\gamma$ line is blended with other lines around this wavelength.
### Early Decline {#Early Decline}
The first spectrum was taken at $t = 8.016$d when the nova was in the early decline phase [@2017ATel10630....1W]. By this time, we have missed the optically thick ‘fireball’ stage which occurs on the rise until around peak brightness. We may have caught the very end of this transition with some of the lines and the H$\delta$ emission line still showing tentative signs of PCygni profiles. The H$\delta$ line has a small blueshifted absorption component with a midpoint of $4052 \pm 8$Å and an equivalent width of $29 \pm 6$Å. Also, the emission component may have a different profile than the other Balmer lines with a FWHM of $\sim 2400$kms$^{-1}$. In Figure \[PCygni\] we show the Balmer lines from the first three spectra to illustrate the tentative evidence for an H$\delta$ P Cygni profile.
{width="95.00000%"}
As the predominant non-Balmer emission lines are those of iron, AT2017fvz is consistent with the class. The broad Balmer lines lie close to the border value of $2500$kms$^{-1}$ which defines the broad-lined novae class [b; @2009ApJ...690.1148S].
The other prominent features of this first spectrum are the H$\alpha$, H$\beta$, and H$\gamma$ emission lines and the double-peaked (1) emission line at 7773Å, all of which have FWHM velocities of $\sim2400$kms$^{-1}$. There is an (42) triplet redward of H$\beta$ at 4924, 5018, and 5169Å, as well as a fairly strong D emission line at $\sim 5892$Å. There may be a weak (74) multiplet blueward of H$\alpha$ with 6148Å and 6456Å lines visible. However, with the 6248Å and 6417Å lines clearly absent, the feature at 6456Å is more likely to be associated with nitrogen. Another explanation for this line at 6148Å could be $\lambda$6158. Between these lines, there is a feature at around 6300Å, which is almost certainly \[\], that persists until the ‘plateau’ phase. We also see tentative evidence for (37) lines at 4556Å and 4629Å.
The second AT2017fvz spectrum, taken 14.040d post-eruption, maintains all aforementioned emission lines including the lines, which show slightly lower FWHM velocities of $\sim 2100$kms$^{-1}$, and many lines. In addition, a prominent feature has developed at around 4640Å which may be a blend of and emission lines at 4638Å and 4676Å, respectively (see Section \[‘Plateau’ Phase\]). Other emission lines could be present at this location including $\lambda$4658, \[\] $\lambda$4658, or (18) at 4655Å. The \[\] $\lambda$5577 line can be seen alongside the (3) line at 5679Å, similar to V1494Aquilæ (Nova Aql 1999) $\sim14$d post-maximum and to SN2010U[^9], 15.3d post-maximum [@2013ApJ...765...57C see their Figure 11].
The third spectrum ($t = 18.010$d) is similar to the previous two, with all lines except H$\alpha$ (see Section \[HaEvolution\]), H$\beta$, and the blend at $\sim 4640$Å having weakened. Unfortunately, the fourth spectrum ($t=24.001$d) has low signal-to-noise ratio owing to poor observing conditions, and only Balmer and \[\] $\lambda$6300 lines are apparent.
### ‘Plateau’ Phase {#`Plateau' Phase}
The fifth spectrum was taken 41.995d post-eruption during the apparent plateau phase. The emission lines still dominate, but these are joined by nitrogen emission lines such as (24) at 5001Å, (3) at 5679Å, and \[\] $\lambda$5755. During this evolutionary phase, we might expect to see a considerable enhancement of nitrogen lines — the so-called ‘nitrogen flaring’ — caused by the Bowen fluorescence mechanism whereby is ‘pumped’ by the UV resonance lines of [@1934PASP...46..146B; @1935ApJ....81....1B]. suggested that this ‘Bowen Blend’ ($\sim4640$Å) may be more naturally explained by ‘oxygen flaring,’ whereby there is flaring of the multiplet (V1) in the range 4638–4696Å.
Such N- or O-flaring may manifest in the spectrum of AT2017fvz through a broad amalgamation of lines at approximately 4640Å, where it is difficult to distinguish the individual lines owing to the low spectral resolution. We assume that they are the (5) multiplet at 4614Å, 4621Å, and 4630Å, and the (1) multiplet at 4647Å, 4650Å, and 4651Å, as well as other nitrogen species.
### Nebular Phase
The final spectrum was taken 69.964d post-eruption. Here, there is evidence for the \[\] nebular lines at 4959 and 5007Å. Only a handful of novae beyond the Magellanic Clouds have been observed spectroscopically during this phase [@Williams_2017]. The appearance of \[\] often roughly coincides with the beginning of the SSS phase when the ejecta from the nova are becoming optically thin to UV radiation and collisions are still occurring owing to the sufficiently high density providing a cooling mechanism [@2018ApJ...853...27M]. Additionally, the ‘Bowen Blend’ is still visible but has broadened and taken on a ‘dome-like’ appearance.
At this time, the density of the ejecta must be less than the critical density ($n_e^\mathrm{crit} = 6.8 \times 10^5$cm$^{-3}$) for the collisional de-excitation of \[\]. One might also expect the \[\] auroral line at 4363Å; however, as this is a relatively weak line, it is most likely blended with H$\gamma$ or hidden by the night-sky line at 4358Å. Even so, we can use the ratio of these three emission lines to estimate an upper limit for the electron temperature within this part of the ejecta of 5000K [see Figure 5.1 in @2006agna.book.....O].
### H$\alpha$ evolution {#HaEvolution}
![H$\alpha$ emission-line evolution from $t = 8$d to $t = 70$d in terms of normalised flux and velocity. The $t = 24$d spectrum is not included owing to lack of flux calibration.[]{data-label="AllVelocityHaBoth"}](VelocityHa){width="0.95\columnwidth"}
The evolution of the H$\alpha$ emission-line profile is shown in Figure \[AllVelocityHaBoth\]. After the first spectrum at $t \approx 8$d, when the line has a FWHM of $2430 \pm 70$kms$^{-1}$, the line progressively narrows from $2300 \pm 100$kms$^{-1}$ to $2070 \pm 70$kms$^{-1}$ and then $2000 \pm 90$kms$^{-1}$ at $t \approx 14$d, $t \approx 18$d, and $t \approx 24$d, respectively. The line width then remains constant between the fifth and sixth spectra with the FWHM being $1840 \pm 60$kms$^{-1}$ at $t \approx 42$d and $1900 \pm 100$kms$^{-1}$ at $t \approx 70$d. With no evidence for substantial circumbinary material, such line narrowing is probably due to decreasing emissivity as the ejecta expand, rather than a deceleration.
X-rays {#X-rays}
------
Utilising the $r'$-band decline time ($t_{2} \approx 15$d; see Table \[Decline Times\]), we used the correlations presented by to predict the expected SSS properties of AT2017fvz. These indicate that a SSS with blackbody temperature $kT \approx 50$eV should have appeared at $t_\mathrm{on}\approx72$d and turned off at $t_\mathrm{off}\approx243$d.
A Galactic foreground column density of $N_\mathrm{H} = 10^{21}$cm$^{-2}$ toward AT2017fvz was derived from the HEASARC $N_\mathrm{H}$ tool based on the Galactic neutral hydrogen map by . We used the PIMMS software (v4.8f) with this column and the estimated SSS temperature to convert from counts to unabsorbed flux. We then derived X-ray luminosities by assuming a distance of 476kpc to NGC6822; these are presented in Table \[Swift\].
We do not detect any X-ray emission from AT2017fvz in any of the five visits between 38d and 388d post-eruption. The luminosity upper limits, calculated from the X-ray count limits (0.3–1keV), assuming $kT \approx 50$eV in Table \[Swift\], are all below $1.4 \times 10^{37}$ergs$^{-1}$. The assumed temperature is low compared in particular to fast RNe such as M31N2008-12a [$\sim 120$eV; @2016ApJ...833..149D] and RSOph [$\sim 90$eV; @2011ApJ...727..124O]; therefore, AT2017fvz must not have had a bright SSS phase during our observational window.
The nova progenitor {#ProgenitorSection}
-------------------
A nova system may harbour either a main sequence, subgiant, or red giant donor. If AT2017fvz hosted a red giant or a luminous accretion disk then it could have been detectable with *Hubble Space Telescope* (*HST*) owing to the proximity of NGC6822 [@2014ApJS..213...10W]. AT2017fvz is located within archival *HST* Wide-Field Planetary Camera 2 (WFPC2) images (GO-11079) taken through the F170W, F255W, F336W, F439W, F555W, and F814W filters.
As described by @2009ApJ...705.1056B, , and in detail by @2014ApJS..213...10W, we used reference stars in the LT images and an F814W *HST* image to compute a precise astrometric transformation between the datasets. We extended the technique by employing all 18 of the $i'$-band and $r'$-band LT images of AT2017fvz to calculate the average nova position (and subsequent scatter) to more precisely and accurately constrain the nova position in the *HST* data, as shown in Figure \[Progenitor\].
{width="32.33330%"} \[NovaLT\] {width="32.33330%"} \[Progenitor1\] {width="32.33330%"} \[Progenitor814\]
We performed crowded-field PSF fitting photometry with DOLPHOT [v2.0; @2000PASP..112.1383D using standard WFPC2 parameters] on all detected objects in the *HST* image, recovering a source that is within $5.14\sigma$ (2.05 WFPC2/PC pixels) of AT2017fvz, an angular separation of $0^{\prime\prime}\!\!.0931$, or a projected distance of 0.21pc (see Figure \[Progenitor\] for the position and Table \[HSTPhotometry\] for the photometry of the source). While seemingly close, we have no knowledge of the line-of-sight separation of the two objects. A colour-magnitude diagram based on these [*HST*]{} data was used to determine a limiting magnitude of $m_{\mathrm{F814W}} \approx 23.5$. Using the method described by @2016ApJ...817..143W, the probability of a coincidental alignment between AT2017fvz and this source is 18%. This does not meet the criterion ($\leq5\%$) employed by [@2016ApJ...817..143W] to confirm a likely nova candidate. The astrometric separation indicates with high confidence that this is indeed a chance alignment. The absence of a detected progenitor within the *HST* data indicates that the system is highly likely to harbour a main sequence or subgiant donor, and that the mass accretion rate is modest at best.
Filter Photometry (mag)
-------- --------------------
F170W $18.481 \pm 0.480$
F255W [–]{}
F336W $20.982 \pm 0.392$
F439W [–]{}
F555W $23.372 \pm 0.201$
F814W $22.259 \pm 0.137$
: [*HST*]{}/WFPC2 photometry of the nearby source.$^\mathrm{a}$[]{data-label="HSTPhotometry"}
$^\mathrm{a}$No source was detected in the F255W or F439W data.
The proximity of this bright source to the nova ($\sim0^{\prime\prime}\!\!.1$) may have contaminated the ground-based and [*Swift*]{} photometry. Therefore, we determined this source’s luminosity in the F814W, F555W, F336W, and F170W filters. Its spectral energy distribution (SED) is shown in Figure \[SED\] and compared to the SED evolution of AT2017fvz. The source is extremely bright in the NUV, indicating that it is most likely to be an O or B star. The AT2017fvz SEDs clearly illustrate the influence that the H$\alpha$ emission of the nova has on the $r'$-band photometry. The final AT2017fvz $u'$-band observation ($\sim 103$d post-eruption) is consistent with the *HST* F336W photometry (similar wavelengths), indicating that the late-time $u'$ photometry is contaminated by this nearby source. The [*Swift*]{} photometry is similarly adversely affected.
A fit to the SED ([*HST*]{} plus [*Swift*]{} data) of the nearby source is consistent with the Rayleigh-Jeans tail of a blackbody with $T_\mathrm{eff}=40000\pm8000$K and $M\approx-10.1$mag ($\chi_\mathrm{red}^{2}=1.68$), at the distance of NGC6822 and assuming $E_{B-V}=0.22$mag. Such a temperature and luminosity are consistent with an O-star. However, the F814W photometry is significantly brighter than would be expected for such a star.
![SED of AT2017fvz (from 8d to 103d post-eruption) and the source within $0^{\prime\prime}\!\!.1$. Black points are optical photometry of AT2017fvz, grey points are from the $uvw1$ photometry. The light grey points are photometry associated with the nearby source, with one of the points being the apparently discrepant F814W photometry. The grey bars at the base are the combined systematic uncertainties from the distance and extinction toward NGC6822.[]{data-label="SED"}](ProgenitorSED_Extinction_Grey){width="0.99\columnwidth"}
Discussion {#Discussion}
==========
The previous nova in NGC 6822 {#PreviousNova}
-----------------------------
There has only been one previous observed nova in NGC6822, which was discovered independently by @1999IAUC.7208....3K and @1999IAUC.7209....2W. That nova, located at $\alpha=19^\mathrm{h}45^\mathrm{m}0^\mathrm{s}\!.31$, $\delta=-14\degr50^\prime10^{\prime\prime}\!\!.3$ (J2000), was discovered by KAIT in unfiltered images taken on 1999 June 23.40 and 23.44 with $m\approx17.3$mag, and by the Beijing Astronomical Observatory Supernova Survey on June 23.69 and 24.72 with an unfiltered magnitude of 18. The nova was then imaged on June 24.38 by LOSS with an unfiltered apparent magnitude of $\sim 17.0$ and by the 1m telescope at Sutherland Observatory on June 26.08 and 28.07 with $V$-band apparent magnitudes of $19.0\pm0.1$ and $19.6\pm0.1$, respectively [@1999IAUC.7211....3B].
This nova was spectroscopically confirmed on 1999 July 9 using the Kast spectrograph on the 3m Shane telescope at Lick Observatory [@1999IAUC.7220....2F]. If we assume that the optical peak occurred at discovery, then this spectrum was taken on $t \approx 16$d, roughly comparable to the $t \approx 14$d and $t \approx 18$d spectra of AT2017fvz. This spectrum is published for the first time in Figure \[1999Nova\] alongside a stacked spectrum of AT2017fvz from $t \approx 8$d, $t \approx 14$d, and $t \approx 18$d for comparative purposes.
{width="95.00000%"}
Just as we see in the spectra of AT2017fvz, there are prominent Balmer lines and many of the same lines. Blueward of H$\beta$ there is (37) at 4629Å and 4666Å and redward there is (42) at 4924Å, 5018Å, and 5169Å. The (74) multiplet is located to the blue of H$\alpha$ at 6148Å, 6248Å, 6417Å, and 6456Å, and again $\lambda$6158 may contribute to the emission line at 6156Å. The D and \[\] emission lines at 5892Å and 6300Å (respectively) are also present and much more apparent. As well as the large number of lines between H$\beta$ and H$\gamma$ that were not clearly visible in the AT2017fvz spectrum, we see the (49) multiplet at 5235Å, 5276Å, and 5326Å to the red of the (42) multiplet. There is a feature in the spectrum at approximately 5533Å which may also be and a feature at 5573Å which is likely to be \[\] $\lambda$5577 given the prominence of the \[\] $\lambda$6300 emission line.
Many of the lines, such as D, (42), and the Balmer lines have PCygni profiles indicating that this spectrum was taken as the nova transitioned from the fireball stage. Comparing directly to the evolution of AT2017fvz implies that the 1999 nova evolved more slowly, which is consistent with the much narrower emission lines. See Table \[1999SpecVel\] for the emission-line velocities of many of the prominent emission lines with the corresponding velocities for AT2017fvz.
---------------- ------------ ---------------- ----------------
Line Wavelength 1999 nova AT2017fvz
identification (Å) (NGC6822) ($t=14.040$d)
H$\gamma$ 4341 900 $\pm$ 120 2000 $\pm$ 200
(37) 4491 910 $\pm$ 60 [–]{}
H$\beta$ 4861 970 $\pm$ 50 2100 $\pm$ 210
(42) 4924 840 $\pm$ 50 [–]{}
(42) 5018 840 $\pm$ 80 2200 $\pm$ 230
(42) 5169 1500 $\pm$ 110 [–]{}
(49) 5235 860 $\pm$ 30 [–]{}
(49) 5276 1120 $\pm$ 50 [–]{}
(49) 5326 1600 $\pm$ 150 [–]{}
5533 840 $\pm$ 90 [–]{}
[\[\]]{} 5577 850 $\pm$ 60 [–]{}
D 5892 800 $\pm$ 130 [–]{}
[\[\]]{} 6300 650 $\pm$ 40 [–]{}
H$\alpha$ 6563 830 $\pm$ 20 2300 $\pm$ 100
---------------- ------------ ---------------- ----------------
: Comparison of emission-line FWHM velocities (kms$^{-1}$).[]{data-label="1999SpecVel"}
The lack of X-rays
------------------
We do not detect X-ray emission from AT2017fvz in any of the five [*Swift*]{} observations. This presents two scenarios: either the emission was not detectable, or it was detectable but we did not observe the system at the correct time.
There are two reasons why the X-rays emanating from the WD surface may not have been detectable. One option is that the X-ray emission may have ceased before the ejecta surrounding the nova were sufficiently diffuse to permit observation — that is, $t_{\mathrm{off}} < t_{\mathrm{on}}$.
The alternative is that the SSS may have been too faint to be detected, below the X-ray luminosity upper limit of $\sim10^{37}$ergs$^{-1}$. A number of SSSs in M31 have been particularly faint, but these have been limited to (suspected) slow novae. M31N2003-08c had a luminosity of $3.5 \times 10^{36}$ergs$^{-1}$ when it was first detected $\sim1540$d post-eruption and M31N2006-09c had a luminosity $\leq 4.0 \times 10^{36}$ergs$^{-1}$ $\sim 426$d post-eruption . Both lacked photometric data to compute decline times, but we can reasonably assume that they are slow novae owing to their low ejecta velocities. The FWHM of the H$\alpha$ emission line in M31N2003-08c is 900kms$^{-1}$ [@2003IAUC.8231....4D] and M31N2006-09c has an expansion velocity of $570 \pm 45$kms$^{-1}$ . Given their low ejection velocities, the observed turn-on times for these novae are fairly consistent with estimates determined from for $t_\mathrm{on}$. As such, we would not expect such a late $t_\mathrm{on}$ for AT2017fvz. As AT2017fvz does not belong to the slow speed class, a faint X-ray luminosity is potentially explained by the low-metallicity environment of NGC6822. Depending upon the amount of mixing between the accreted envelope and the underlying WD, the metallicity of the accreted shell will either only weakly (strong mixing) or strongly (little mixing) depend upon the metallicity of the donor. As the TNR operates via the hot-CNO cycle, a lower metallicity shell might therefore be expected to produce a lower luminosity, but a longer lived SSS phase. In such a scenario, low metallicity alone might explain the lack of any X-ray detection. @2013AstRv...8a..71O provides further discussion about SSS populations within the SMC, a possibly similar environment to that of NGC6822.
Alternatively, if the X-ray emission was in principle detectable, then the reasons for not observing this SSS phase revolve around the timing of the observations. It may also indicate that the correlations used to predict $t_{\mathrm{on}}$ and $t_{\mathrm{off}}$ (derived from CNe in M31) are not valid in the lower metallicity environment of NGC6822 [see, e.g., @Williams_2017]. Firstly, the supersoft X-ray source may occur after 388d (our last [*Swift*]{} observation), so we have simply observed too early, indicative of high-mass ejecta and also a low-mass WD. Secondly, the whole SSS phase may have taken place within one of the observing gaps, either between 38d and 67d, between 67d and 97d, between 97d and 268d, or between 243d and 388d. Though unlikely, there are examples of very short SSS phases in fast novae such as M31N2007-12d, which had an extremely short SSS phase of $< 20$d . Finally, the most tantalising option is that the entire SSS phase took place before our first [*Swift*]{} observation, 38d post-eruption. This would imply low-mass ejecta and a high-mass WD, and potentially a recurrent nova.
A possibly ‘faint and fast’ or recurrent nova? {#PossibleRN}
----------------------------------------------
With $t_{2,V} = 8.1 \pm 0.2$d, AT2017fvz is a ‘very-fast’ fading nova. We calculated from the MMRD relations of @2000AJ....120.2007D an expected peak $V$-band absolute magnitude of $M_V \approx -9$, but with a peak absolute magnitude in the range $-7.41>M_V>-8.33$mag AT2017fvz may be substantially fainter than ‘expected.’ Given this range, and after accounting for expected differences between the $V$-band and $g$ filters [see @2009ApJ...690.1148S], AT2017fvz would lie below the MMRD (broadly consistent with the position of M31N2008-11a) as presented by @2011ApJ...735...94K [see their Figure 12], which plots six ‘faint and fast’ novae by their $t_2$ and their peak absolute $g$-band magnitude. Here we suggest caution, as the upper end of this range (high internal extinction contribution and missed light-curve peak) is marginally consistent with the MMRD. We also note that @2011ApJ...735...94K employed the Balmer decrement to correct for extinction toward many of their M31 novae; however, Case B recombination is not valid in the early stages of nova evolution [see @Williams_2017 for a discussion].
The ‘faint and fast’ region of the MMRD phase space is populated by a number of Galactic [@2011ApJ...735...94K see their Figure 13] and M31 RNe. @2014ApJ...788..164P defined a number of key indicators for a RN masquerading as a CN (i.e., only one observed eruption). AT2017fvz satisfies some of these; for example the short $t_2$ implies the presence of a high-mass WD. The high ejecta velocities (for an nova) inferred from the H$\alpha$ emission line ($2430 \pm 70$kms$^{-1}$) further reinforce this suggestion.
Additionally, there is a plateau in the optical light curve from around day 25 to day 45. It has been proposed that such plateaus are produced by the SSS irradiating a reformed, or surviving, accretion disk and the donor. The subsequent reprocessed optical light then dominates the light emitted by the nova ejecta, temporarily halting the decline of the light curve [@2000ApJ...528L..97H; @2008ASPC..401.....E; @2016ApJ...833..149D]. This could indicate that the accretion disk survived the eruption, pointing to a high accretion rate and/or low ejected mass — a reasonable indicator of a RN. However, it does not provide strong evidence in isolation. Additionally, the spectrum obtained during the plateau shows no evidence for narrow (or any) lines, a key signature of a hot disk [as in seen during the plateau phase of known recurrent novae; e.g., @2018ApJ...857...68H].
The other criteria suggested by @2014ApJ...788..164P require either far superior spectroscopy or identification of the quiescent system. AT2017fvz matches all of their RN indicators that we can reasonably test. The lack of a detected progenitor also indicates the absence of a luminous accretion disk, therefore at most only a modest accretion rate. Even if this system were a RN, it certainly would not be a short-cycle recurrent system.
Summary and Conclusions {#Summary and Conclusions}
=======================
In this paper we present observations and analysis of AT2017fvz, the second nova observed in the Local Group dwarf irregular galaxy NGC6822. We carried out detailed photometric and spectroscopic observations of the nova from its initial rise through to the nebular phase. We summarise as follows.
1. AT2017fvz was spectroscopically confirmed as an nova, but exhibited broader than typical emission lines.
2. The light-curve evolution indicates that AT2017fvz may belong to the P-class (plateau) novae, a proposed indication of a surviving or reformed accretion disk.
3. As a ‘very fast’ nova with a decline time $t_{2 (V)} \approx 8$d, the MMRD predicted peak magnitude is $M_V\approx-9$. Yet we estimate the observed peak is in the range $-7.41>M_V>-8.33$.
4. The rapid decline and possible low luminosity suggest that AT2017fvz may be a ‘faint and fast’ nova.
5. No X-rays were detected between 38 and 388 days post-eruption, therefore the SSS must have occurred within the first $\sim40$d, been fully obscured by the ejecta, or simply been too faint to be detectable — a possible metallicity effect.
6. The progenitor system was not recoverable from [*HST*]{} data, indicating a main sequence or subgiant donor.
We have also included, for the first time, the sparse available data for the other confirmed nova in NGC6822. Although currently limited in number, the study of novae across a range of galaxy types will permit systematic studies of how environment — particularly metallicity — can affect the properties of novae.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Mike Shara for his role in refereeing the manuscript, and all those that have contributed to the discussion about this object. M.W.H. acknowledges a PhD studentship from the UK Science and Technology Facilities Council (STFC). M.J.D. received funding from STFC. K.L.P. received funding from the UK Space Agency. The work of A.V.F.’s group at UC Berkeley has been generously supported by the TABASGO Foundation, the Christopher R. Redlich Fund, and the Miller Institute for Basic Research in Science (UC Berkeley); additional funding was provided by NASA/[*HST*]{} grant AR-14295 from the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under NASA contract NAS5-26555. We thank the staff of the various observatories at which data were obtained. This work made extensive use of the Liverpool Telescope, which is operated by LJMU on the island of La Palma in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial support from STFC. This work has made use of data from the Asteroid Terrestrial-impact Last Alert System (ATLAS) project. ATLAS is primarily funded to search for new near-Earth asteroids, through NASA grant NN12AR55G issued under the guidance of Lindley Johnson and Kelly Fast. A byproduct of this search is a collection of images and catalogs of the survey area. The ATLAS science products have been made possible through the contributions of the Institute for Astronomy, the University of Hawaii, the Queen’s University Belfast, STScI, and Harvard University. KAIT and its ongoing operation were made possible by donations from Sun Microsystems, Inc., the Hewlett-Packard Company, AutoScope Corporation, Lick Observatory, the NSF, the University of California, the Sylvia & Jim Katzman Foundation, and the TABASGO Foundation. Research at Lick Observatory is partially supported by a generous gift from Google. This research has made use of the APASS database, located at the AAVSO web site. Funding for APASS has been provided by the Robert Martin Ayers Sciences Fund. PyRAF is a product of STScI, which is operated by AURA for NASA.
Photometry of AT 2017fvz
========================
In Table \[Photometry1\] we provide the optical photometry of AT2017fvz from the Liverpool Telescope and ASAS-SN. Table \[Photometry2\] lists the photometry of AT2017fvz from KAIT. Table \[Photometry3\] presents the ATLAS photometry of AT2017fvz.
\[Photometry1\]
UT Date MJD (d) $t-t_0$ (d) Telescope & instrument Exposure time (s) Filter Photometry (mag)
---------------- ----------- ------------- ------------------------ ------------------- -------- --------------------
2017-08-09.916 57974.916 8.032 LT IO:O 60 $u'$ 18.522 $\pm$ 0.118
2017-08-15.908 57980.908 14.024 LT IO:O 60 $u'$ 18.737 $\pm$ 0.063
2017-08-17.894 57982.894 16.010 LT IO:O 60 $u'$ 19.248 $\pm$ 0.131
2017-08-19.910 57984.910 18.026 LT IO:O 60 $u'$ 19.337 $\pm$ 0.111
2017-08-23.889 57988.889 22.005 LT IO:O 60 $u'$ 19.633 $\pm$ 0.127
2017-08-30.883 57995.883 28.999 LT IO:O 120 $u'$ 20.265 $\pm$ 0.191
2017-09-04.939 58000.939 34.055 LT IO:O 120 $u'$ 20.722 $\pm$ 0.335
2017-09-19.936 58015.936 49.052 LT IO:O 120 $u'$ 20.378 $\pm$ 0.158
2017-09-24.896 58020.896 54.012 LT IO:O 120 $u'$ 20.571 $\pm$ 0.274
2017-10-17.842 58043.842 76.958 LT IO:O 120 $u'$ 21.228 $\pm$ 0.131
2017-11-12.822 58098.822 102.938 LT IO:O 120 $u'$ 21.895 $\pm$ 0.270
2017-08-09.917 57974.917 8.033 LT IO:O 60 $B$ 18.224 $\pm$ 0.019
2017-08-15.910 57980.910 14.026 LT IO:O 60 $B$ 19.003 $\pm$ 0.025
2017-08-17.895 57982.895 16.011 LT IO:O 60 $B$ 19.198 $\pm$ 0.032
2017-08-19.911 57984.911 18.027 LT IO:O 60 $B$ 19.380 $\pm$ 0.038
2017-08-23.890 57988.890 22.006 LT IO:O 60 $B$ 19.903 $\pm$ 0.049
2017-08-30.885 57995.885 29.001 LT IO:O 120 $B$ 20.204 $\pm$ 0.086
2017-09-04.941 58000.941 34.057 LT IO:O 120 $B$ 19.987 $\pm$ 0.081
2017-09-19.941 58015.941 49.057 LT IO:O 120 $B$ 20.591 $\pm$ 0.072
2017-09-24.901 58020.901 54.017 LT IO:O 120 $B$ 20.875 $\pm$ 0.106
2017-10-17.847 58043.847 76.963 LT IO:O 120 $B$ 21.066 $\pm$ 0.059
2017-11-12.827 58098.827 102.943 LT IO:O 120 $B$ 21.321 $\pm$ 0.071
2017-08-03.190 57968.190 1.306 ASAS-SN 270 $V$ 16.654
2017-08-09.918 57974.918 8.034 LT IO:O 60 $V$ 17.905 $\pm$ 0.017
2017-08-15.911 57980.911 14.027 LT IO:O 60 $V$ 18.722 $\pm$ 0.017
2017-08-17.896 57982.896 16.012 LT IO:O 60 $V$ 19.035 $\pm$ 0.029
2017-08-19.912 57984.912 18.028 LT IO:O 60 $V$ 19.428 $\pm$ 0.045
2017-08-23.891 57988.891 22.007 LT IO:O 60 $V$ 19.691 $\pm$ 0.048
2017-08-30.887 57995.887 29.003 LT IO:O 120 $V$ 20.128 $\pm$ 0.073
2017-09-04.943 58000.943 34.059 LT IO:O 120 $V$ 20.148 $\pm$ 0.097
2017-09-19.946 58015.946 49.062 LT IO:O 120 $V$ 20.341 $\pm$ 0.061
2017-09-24.906 58020.906 54.022 LT IO:O 120 $V$ 20.565 $\pm$ 0.092
2017-10-17.852 58043.852 76.968 LT IO:O 120 $V$ 20.758 $\pm$ 0.056
2017-11-12.832 58098.832 102.948 LT IO:O 120 $V$ 21.079 $\pm$ 0.077
2017-08-09.919 57974.919 8.035 LT IO:O 60 $r'$ 17.186 $\pm$ 0.011
2017-08-15.912 57980.912 14.028 LT IO:O 60 $r'$ 17.589 $\pm$ 0.009
2017-08-17.897 57982.897 16.013 LT IO:O 60 $r'$ 17.729 $\pm$ 0.010
2017-08-19.913 57984.913 18.029 LT IO:O 60 $r'$ 17.886 $\pm$ 0.011
2017-08-23.892 57988.892 22.008 LT IO:O 60 $r'$ 18.212 $\pm$ 0.013
2017-08-30.889 57995.889 29.005 LT IO:O 120 $r'$ 18.681 $\pm$ 0.023
2017-09-04.944 58000.944 34.060 LT IO:O 120 $r'$ 18.750 $\pm$ 0.024
2017-09-19.951 58015.951 49.067 LT IO:O 60 $r'$ 19.153 $\pm$ 0.028
2017-09-24.911 58020.911 54.027 LT IO:O 60 $r'$ 19.255 $\pm$ 0.038
2017-10-17.857 58043.857 76.973 LT IO:O 60 $r'$ 19.729 $\pm$ 0.030
2017-11-12.837 58098.837 102.953 LT IO:O 60 $r'$ 20.143 $\pm$ 0.046
2017-08-09.890 57974.890 8.006 LT SPRAT 10 $r'$ 17.391 $\pm$ 0.027
2017-08-15.915 57980.915 14.031 LT SPRAT 10 $r'$ 17.983 $\pm$ 0.031
2017-08-19.886 57984.886 18.002 LT SPRAT 10 $r'$ 18.268 $\pm$ 0.035
2017-08-25.874 57990.874 23.990 LT SPRAT 10 $r'$ 17.951 $\pm$ 0.174
2017-09-11.884 58007.884 41.000 LT SPRAT 10 $r'$ 19.286 $\pm$ 0.043
2017-09-12.865 58008.865 41.981 LT SPRAT 10 $r'$ 19.368 $\pm$ 0.050
2017-10-10.834 58036.834 69.950 LT SPRAT 10 $r'$ 20.040 $\pm$ 0.067
2017-08-09.920 57974.920 8.036 LT IO:O 60 $i'$ 17.246 $\pm$ 0.021
2017-08-15.913 57980.913 14.029 LT IO:O 60 $i'$ 18.044 $\pm$ 0.017
2017-08-17.898 57982.898 16.014 LT IO:O 60 $i'$ 18.230 $\pm$ 0.023
2017-08-19.914 57984.914 18.030 LT IO:O 60 $i'$ 18.496 $\pm$ 0.026
2017-08-23.893 57988.893 22.009 LT IO:O 60 $i'$ 18.913 $\pm$ 0.030
2017-08-30.890 57995.890 29.006 LT IO:O 120 $i'$ 19.449 $\pm$ 0.068
2017-09-04.946 58000.946 34.062 LT IO:O 120 $i'$ 19.569 $\pm$ 0.050
2017-09-19.954 58015.954 49.070 LT IO:O 60 $i'$ 19.957 $\pm$ 0.065
2017-09-24.914 58020.914 54.030 LT IO:O 60 $i'$ 19.649 $\pm$ 0.082
2017-10-17.860 58043.860 76.976 LT IO:O 60 $i'$ 20.378 $\pm$ 0.076
2017-11-12.840 58098.840 102.956 LT IO:O 60 $i'$ 20.547 $\pm$ 0.095
\[Photometry2\]
UT Date MJD (d) $t-t_0$ (d) Telescope & instrument Filter Photometry (mag)
---------------- ----------- ------------- ------------------------ -------- ------------------
2017-08-02.384 57967.384 0.500 KAIT Clear 17.61 $\pm$ 0.09
2017-08-03.289 57968.289 1.405 KAIT Clear 16.44 $\pm$ 0.11
2017-08-08.211 57973.211 6.327 KAIT Clear 16.83 $\pm$ 0.04
2017-08-09.293 57974.293 7.409 KAIT Clear 16.87 $\pm$ 0.06
2017-08-10.370 57975.370 8.486 KAIT Clear 17.04 $\pm$ 0.06
2017-08-11.366 57976.366 9.482 KAIT Clear 17.22 $\pm$ 0.05
2017-08-12.369 57977.369 10.485 KAIT Clear 17.22 $\pm$ 0.04
2017-08-13.362 57978.362 11.478 KAIT Clear 17.41 $\pm$ 0.05
2017-08-14.362 57979.362 12.478 KAIT Clear 17.48 $\pm$ 0.05
2017-08-15.360 57980.360 13.476 KAIT Clear 17.57 $\pm$ 0.04
2017-08-16.355 57981.355 14.471 KAIT Clear 17.66 $\pm$ 0.04
2017-08-17.337 57982.337 15.453 KAIT Clear 17.68 $\pm$ 0.09
2017-08-18.347 57983.347 16.463 KAIT Clear 17.82 $\pm$ 0.10
2017-08-19.339 57984.339 17.455 KAIT Clear 17.88 $\pm$ 0.05
2017-08-20.334 57985.334 18.450 KAIT Clear 17.95 $\pm$ 0.06
2017-08-22.333 57987.333 20.449 KAIT Clear 18.13 $\pm$ 0.07
2017-08-23.335 57988.335 21.451 KAIT Clear 18.19 $\pm$ 0.13
2017-08-24.316 57989.316 22.432 KAIT Clear 18.31 $\pm$ 0.07
2017-08-25.336 57990.336 23.452 KAIT Clear 18.33 $\pm$ 0.06
2017-08-26.339 57991.339 24.455 KAIT Clear 18.41 $\pm$ 0.09
2017-08-27.307 57992.307 25.423 KAIT Clear 18.42 $\pm$ 0.06
2017-08-28.320 57993.320 26.436 KAIT Clear 18.61 $\pm$ 0.06
2017-08-29.291 57994.291 27.407 KAIT Clear 18.74 $\pm$ 0.15
2017-08-30.278 57995.278 28.394 KAIT Clear 18.47 $\pm$ 0.22
2017-08-31.284 57996.284 29.400 KAIT Clear 18.62 $\pm$ 0.17
\[Photometry3\]
UT Date MJD (d) $t-t_0$ (d) Telescope & instrument Filter Photometry (mag)
---------------- ----------- -------------------- ------------------------ -------- --------------------
2017-08-03.389 57968.389 1.505 $\pm$ 0.005 ATLAS Orange 16.511 $\pm$ 0.095
2017-08-04.440 57969.440 2.556 $\pm$ 0.006 ATLAS Orange 16.219 $\pm$ 0.036
2017-08-09.455 57974.455 7.571 $\pm$ 0.005 ATLAS Orange 16.955 $\pm$ 0.068
2017-08-11.436 57976.436 9.552 $\pm$ 0.005 ATLAS Orange 17.315 $\pm$ 0.032
2017-08-12.412 57977.412 10.528 $\pm$ 0.005 ATLAS Orange 17.447 $\pm$ 0.023
2017-08-13.446 57978.446 11.562 $\pm$ 0.005 ATLAS Orange 17.502 $\pm$ 0.015
2017-08-15.424 57980.424 13.540 $\pm$ 0.005 ATLAS Orange 17.520 $\pm$ 0.031
2017-08-18.399 57983.399 16.515 $\pm$ 0.005 ATLAS Orange 17.958 $\pm$ 0.098
2017-08-22.384 57987.384 20.500 $\pm$ 0.005 ATLAS Orange 18.209 $\pm$ 0.241
2017-08-23.405 57988.405 21.521 $\pm$ 0.005 ATLAS Orange 18.544 $\pm$ 0.319
2017-08-26.394 57991.394 24.510 $\pm$ 0.004 ATLAS Orange 20.143 $\pm$ 0.239
2017-08-28.370 57993.370 26.486 $\pm$ 0.004 ATLAS Orange 19.877 $\pm$ 0.118
2017-08-16.394 57981.394 14.510 $\pm$ 0.005 ATLAS Cyan 18.750 $\pm$ 0.169
2017-08-17.417 57982.417 15.533 $\pm$ 0.005 ATLAS Cyan 18.519 $\pm$ 0.442
2017-08-21.411 57986.411 19.527 $\pm$ 0.010 ATLAS Cyan 18.134 $\pm$ 0.410
2017-09-17.329 58013.329 46.445 $\pm$ 0.004 ATLAS Cyan 20.376 $\pm$ 0.403
$^\mathrm{a}$The ‘orange’ filter covers the $r'$ and $i'$ bands. The ‘cyan’ filter covers the $V$ and $r'$ bands.
$^\mathrm{b}$The date listed here is the mean time of multiple observations taken on this date.
$^\mathrm{c}$The magnitude listed here is the mean magnitude calculated from multiple observations taken on this date with the associated standard uncertainty.
\[lastpage\]
[^1]: E-mail: M.W.Healy@2017.ljmu.ac.uk
[^2]: E-mail: M.J.Darnley@ljmu.ac.uk
[^3]: <https://wis-tns.weizmann.ac.il/object/2017fvz>
[^4]: <http://www.fallingstar.com/specifications.php>
[^5]: <http://telescope.livjm.ac.uk/TelInst/Inst/IOO>
[^6]: <http://sdss.org/dr7/algorithms/sdssUBVRITransform.html>
[^7]: <http://www.swift.ac.uk/user_objects/>
[^8]: <http://argonaut.skymaps.info>
[^9]: Not a supernova [@2013ApJ...765...57C]!
|
---
abstract: 'A formalism is introduced for the non-perturbative, purely numerical, solution of the reduced Rayleigh equation for the scattering of light from two-dimensional penetrable rough surfaces. As an example, we apply this formalism to study the scattering of p- or s-polarized light from two-dimensional dielectric or metallic randomly rough surfaces by calculating the full angular distribution of the co- and cross-polarized intensity of the scattered light. In particular, we present calculations of the mean differential reflection coefficient for glass and silver surfaces characterized by (isotropic or anisotropic) Gaussian and cylindrical power spectra. The proposed method is found, within the validity of the Rayleigh hypothesis, to give reliable results. For a non-absorbing metal surface the conservation of energy was explicitly checked, and found to be satisfied to within 0.03%, or better, for the parameters assumed. This testifies to the accuracy of the approach and a satisfactory discretization.'
author:
- 'T. Nordam'
- 'P. A. Letnes'
- 'I. Simonsen'
bibliography:
- 'references.bib'
title: 'Numerical Simulations of Scattering of Light from Two-Dimensional Surfaces Using the Reduced Rayleigh Equation'
---
=1
Introduction {#sec:Introduction}
============
Wave scattering from rough surfaces is an old discipline which keeps attracting a great deal of attention from the scientific and technological community. Several important technologies in our society rely on such knowledge, with radar being a prime example. In the past, the interaction of light with rough surfaces was often considered an extra complication that had to be taken into account in order to properly interpret or invert scattering data. However, with the advent of nanotechnology, rough structures can be used to design novel materials with tailored optical properties. Examples include: metamaterials [@MetaMaterials; @AgrGar06], photonic crystals [@PhotonicCrystals], spoof plasmons [@SpoofPlasmons], optical cloaking [@Cloaking-1; @Cloaking-2; @Cloaking-3], and designer surfaces [@DesignerSurfaces; @Simonsen2001-6]. These developments have made it even more important to have available efficient and accurate simulation tools to calculate both the far- and near-field behavior of the scattered and transmitted fields for any frequency of the incident radiation, including potential resonance frequencies of the structure.
Lord Rayleigh was the first to perform systematic studies of wave scattering from rough surfaces when, in the late 1800s, he studied the intensity distribution of a wave scattered from a sinusoidal surface [@Rayleigh1907; @Book:Rayleigh]. More than three decades later, Mandel’shtam studied light scattering from *randomly rough* surfaces [@Madelstam-1913] thereby initiating the field of wave scattering from surface disordered systems. Since the initial publication of these seminal works, numerous studies on wave scattering from randomly rough surfaces have appeared in the literature [@Book:Bass-1979; @Book:Ogilvy-1991; @Book:Voronovich-1999; @Book:Nieto-Vesperinas-2006; @Book:Maradudin-2007; @Zayats-2005; @Simonsen-2010], and several new multiple scattering phenomena have been predicted and confirmed experimentally. These phenomena include the enhanced backscattering and enhanced transmission phenomena, the satellite peak phenomenon, and coherent effects in the intensity-intensity correlation functions [@Backscattering-1; @Backscattering-2; @Backscattering-3; @EnhancedTransmission; @Satellitepeaks; @Simonsen-2010].
These studies, and the methods they use, can be categorized as either perturbative or purely numerical (and non-pertubative). While the former group of methods is mainly limited to weakly rough surfaces, and therefore have limited applicability, the latter group of methods can be applied to a wider class of surface roughnesses. Rigorous numerical methods can in principle be used to study the wave scattering from surfaces of any degree of surface roughness. Such simulations are routinely performed for systems where the interface has a one-dimensional roughness, i.e., where the surface structure is constant along one of the two directions of the mean plane [@Maradudin1990255; @Simonsen-2010]. However, for the practically more relevant situation of a two-dimensional rough surface, the purely numerical and rigorous methods are presently less used due to their computationally intensive nature. The reason for this complexity is the fact that for a randomly rough surface there is no symmetry or periodicity in the surface structure that can be used to effectively reduce the simulation domain. For a periodic surface, it is sufficient to simulate a single unit cell, while for a random surface the unit cell is in principle infinite.
A wide range of simulation methods are currently available for simulating the interaction of light with matter, including the finite-difference time-domain (FDTD) method [@FDTD], the finite-element method (FEM) [@Book:FEM-1; @Book:FEM-2], the related surface integral equation techniques also known as the boundary element method (BEM) or the method of moments (MoM) [@Book:BEM-1; @Book:BEM-2; @Book:MoM; @Book:SpectralMethods; @Simonsen2010-04], the reduced Rayleigh equation (RRE) technique [@brown1984381; @Madrazo:1997uq; @Simonsen_OptComm; @Simonsen2009-5; @Mcgurn:1996fk; @PhysRevB.63.245411; @Soubret:01; @Zayats-2005], and spectral methods [@Book:SpectralMethods].
The FDTD and FEM methods discretize the whole volume of the simulation domain. Due to the complex and irregular shape of a (randomly) rough surface, it is often more convenient, and may give more accurate results (for the same level of numerical complexity) [@Kern:09], to base numerical simulations on methods where only the surface itself needs to be discretized. This is the case, for example, for the surface integral technique and the reduced Rayleigh equation methods.
The reduced Rayleigh equation is an integral equation where the unknown is either the scattering amplitude or the transmission amplitude. In the former (latter) case, one talks about the reduced Rayleigh equation for reflection (transmission). For reflection this equation was originally derived by Brown *et al.* [@brown1984381], and subsequently by Soubret *et al.* [@PhysRevB.63.245411; @Soubret:01]. Later it has also been derived for transmission [@RRE_Transmission] and film geometries [@PhysRevB.63.245411; @Lekova_RRE; @OUR_Satellite_Paper].
In the past, the surface integral technique has been used to study light scattering from two-dimensional randomly rough, perfectly conducting or penetrable surfaces [@Simonsen2009-1; @Simonsen2009-9; @Simonsen2010-04]. However, to date, a direct numerical and non-perturbative solution of the two-dimensional reduced Rayleigh equation has not appeared in the literature, even if its one-dimensional analog has been solved numerically and has been used to study the scattering from, and transmission through, one-dimensional rough surfaces [@Madrazo:1997uq; @Simonsen_OptComm; @Simonsen2009-5]. The lesson learned from the one-dimensional scattering studies reported in Refs. [@Madrazo:1997uq; @Simonsen_OptComm; @Simonsen2009-5] is that simulations based on a direct numerical solution of the reduced Rayleigh equation may give accurate non-perturbative results for systems where alternative methods struggle to give the same level of accuracy. Moreover, the reduced Rayleigh method also requires less memory for the same surface dimensions when compared to, e.g., the rigorous surface integral technique.
The main aim of this paper is to present a numerical method and formalism for the solution of the two-dimensional reduced Rayleigh equation for reflection. While we exclusively consider reflection, the formalism for transmission will be almost identical, and the resulting equation will have a similar form as for reflection. Additionally, the equation for transmission or reflection for a film geometry, i.e., for a film of finite thickness on top of a substrate, where only one interface is rough, will also have a similar form. The method presented will be illustrated by applying it to the study of the scattering of p- or s-polarized light from two-dimensional metallic or dielectric media separated from vacuum by an isotropic or anisotropic randomly rough surface.
This paper is organized as follows: First, in Sec. \[sec:geometry\] we present the scattering geometry to be considered. We will then present some relevant scattering theory, including the reduced Rayleigh equation for the geometry under study (Sec. \[sec:scatteringtheory\]), followed by a detailed description of how the equation can be solved numerically (Sec. \[sec:solving\]). Next, we will present some simulation results obtained by the introduced method (Sec. \[sec:results\]). We then discuss some of the computational challenges of this method (Sec. \[sec:discussion\]), and, finally, in Sec. \[sec:Conclusion\] we draw some conclusions.
Scattering Geometry {#sec:geometry}
===================
![(Color online) A sketch of the scattering geometry assumed in this work. The figure also shows the coordinate system used, angles of incidence $(\theta_0,\phi_0)$ and scattering $(\theta_s,\phi_s)$, and the corresponding lateral wavevectors ${{{\mathbf{k}}_{\parallel}}}$ and ${{{\mathbf{q}}_{\parallel}}}$, respectively.[]{data-label="fig:geometry"}](scatteringgeometry)
We consider a system where a rough surface separates two regions. Region 1 is assumed to be vacuum (${{{\varepsilon_{\mathrm{1}}}}}= 1$), and region 2 is filled with a metal or dielectric characterized by a complex dielectric function ${{{\varepsilon_{\mathrm{2}}}}}(\omega)$, where the angular frequency is $\omega=2\pi c/\lambda$, with $\lambda$ being the wavelength of the incident light in vacuum and $c$ the speed of light in vacuum. The height of the surface measured in the positive $x_3$ direction from the $x_1 x_2$-plane is given by the single-valued function $x_3=\zeta({{{{\mathbf{x}}_{\parallel}}}})$, where ${{{{\mathbf{x}}_{\parallel}}}}=(x_1, x_2, 0)$, which is assumed to be at least once differentiable with respect to $x_1$ and $x_2$. Angles of incidence $(\theta_0,\phi_0)$ and scattering $(\theta_s,\phi_s)$ are defined positive according to the convention given in Fig. \[fig:geometry\].
In principle, the theory to be presented in Sec. \[sec:scatteringtheory\] can be used to calculate the scattering of light from any surface, provided it is not too rough. However, in this paper, we will consider randomly rough surfaces where $\zeta({{{{\mathbf{x}}_{\parallel}}}})$ constitutes a stationary random process defined by $$\begin{aligned}
\begin{aligned}
\label{eq:surface_definition}
\left\langle\zeta({{{{\mathbf{x}}_{\parallel}}}})\right\rangle
&=
0,
\\
\left\langle
\zeta({{{{\mathbf{x}}_{\parallel}}}})
\zeta({{{{\mathbf{x}}_{\parallel}}}}')
\right\rangle
&=
\delta^2
W({{{{\mathbf{x}}_{\parallel}}}}-{{{{\mathbf{x}}_{\parallel}}}}'),
\end{aligned}\end{aligned}$$ where the angle brackets denote an average over an ensamble of surface realizations. In writing Eqs. (\[eq:surface\_definition\]) we have defined the root-mean-square height of the surface, $\delta=\left<\zeta^2({{{\mathbf{x}}_{\parallel}}})\right>^{1/2}$, and $W({{{{\mathbf{x}}_{\parallel}}}}-{{{{\mathbf{x}}_{\parallel}}}}')$ denotes the height-height auto-correlation function of the surface, normalized so that $W({\mathbf{{0}}})=1$ [@Simonsen-2010]. According to the Wiener-Khinchin theorem [@Book:VanKampen-2007], the power spectrum of the surface profile function is given by $$\begin{aligned}
\label{eq:powerspectrum}
g({{{\mathbf{k}}_{\parallel}}} )
&=
\int\! \mathrm{d}^2x_\parallel\;
W({{{\mathbf{x}}_{\parallel}}} ) \exp\left(-\mathrm{i}{{{\mathbf{k}}_{\parallel}}}\cdot{{{\mathbf{x}}_{\parallel}}} \right).\end{aligned}$$ The power spectra that will be considered in this work are of either the Gaussian form [@Simonsen2010-04] $$\begin{aligned}
\label{eq:gaussian}
g({{{{\mathbf{k}}_{\parallel}}}}) &= \sqrt{\pi}a_1a_2
\exp\left( -\frac{k_1^2 a_1^2}{4}
-\frac{k_2^2 a_2^2}{4}
\right),\end{aligned}$$ where $a_i$ ($i=1,2$) denotes the lateral correlation length for direction $i$, or the cylindrical form [@Mcgurn:1996fk] $$\begin{aligned}
\label{eq:cylindrical}
\begin{aligned}
g(k_\parallel) =& \frac{4\pi}{k_+^2 - k_-^2}\left[\theta(k_\parallel-k_-)\theta(k_+ - k_\parallel)\right],
\end{aligned}\end{aligned}$$ where $k_\parallel=|{{{{\mathbf{k}}_{\parallel}}}}|$, $\theta$ denotes the Heaviside unit step function, and $k_\pm$ are wavenumber cutoff parameters, with $k_-<k_+$. The cylindrical form in Eq. (\[eq:cylindrical\]) is a two-dimensional generalization of the power spectrum used in the experiments where West and O’Donnell confirmed the existence of the enhanced backscattering phenomenon for weakly rough surfaces [@Backscattering-2].
Scattering Theory {#sec:scatteringtheory}
=================
We consider a linearly p- or s-polarized plane wave which is incident on the surface from region 1, with the electric field given by ${{\mathbf{E}^{\mathrm{inc}}}}({{\mathbf{x}}};t)={{\mathbf{E}^{\mathrm{inc}}}}({{\mathbf{x}}}|\omega) \exp(-{\mathrm{i}}\omega t)$ where
\[eq:E\_inc\_total\] $$\begin{aligned}
\label{eq:E_inc}
{{\mathbf{E}^{\mathrm{inc}}}}({{\mathbf{x}}}| \omega)
={}&
{\bm{\mathcal{E}}^\mathrm{inc}}({{{{\mathbf{k}}_{\parallel}}}})\exp\left[{\mathrm{i}}{{{{\mathbf{k}}_{\parallel}}}}\cdot{{{{\mathbf{x}}_{\parallel}}}}-{\mathrm{i}}{{{\alpha_{\mathrm{1}}}}}(k_\parallel)x_3\right],\end{aligned}$$ with $$\begin{aligned}
\label{eq:E_inc_amplitudes}
\begin{aligned}
{\bm{\mathcal{E}}^\mathrm{inc}}({{{{\mathbf{k}}_{\parallel}}}})
={}&
-\frac{c}{\omega}
\left[
{{{{{\mathbf{\hat{k}}}_{\parallel}}}}}{{{\alpha_{\mathrm{1}}}}}(k_\parallel)+{{\mathbf{\hat{x}}}}_3 k_\parallel
\right]
{{\mathcal{E}^\mathrm{inc}_{\mathrm{p}}}}({{{{\mathbf{k}}_{\parallel}}}})
\\
&+
\left({{\mathbf{\hat{x}}}}_3 \times {{{{{\mathbf{\hat{k}}}_{\parallel}}}}}\right){{\mathcal{E}^\mathrm{inc}_{\mathrm{s}}}}({{{{\mathbf{k}}_{\parallel}}}}),
\end{aligned}\end{aligned}$$ and $$\begin{aligned}
\label{eq:alpha_1}
{{{\alpha_{\mathrm{1}}}}}(k_\parallel)
&=
\sqrt{\frac{\omega^2}{c^2}-k_\parallel^2}
,\;\; \mathrm{Re}\,{{{\alpha_{\mathrm{1}}}}}\geq 0,\;\mathrm{Im}\,{{{\alpha_{\mathrm{1}}}}}\geq 0.\end{aligned}$$
Here, and in the rest of the paper, a caret over a vector indicates a unit vector. The expressions in front of the amplitudes ${{\mathcal{E}^\mathrm{inc}_{\mathrm{\alpha}}}}({{{{\mathbf{k}}_{\parallel}}}})$ ($\alpha=\mathrm{p,s}$) in Eq. (\[eq:E\_inc\_amplitudes\]) correspond to unit polarization vectors for incident light of linear polarization $\alpha$. Moreover, ${{{{\mathbf{k}}_{\parallel}}}}= (k_1, k_2, 0)$ denotes the lateral component of the wave vector ${\mathbf{{k}}}={{{\mathbf{k}}_{\parallel}}}-\alpha(k_\parallel){\mathbf{{\hat{x}}}}_3$. When the lateral wavenumber satisfies $k_\parallel \leq \omega/c$, as will be assumed here, ${{{\mathbf{k}}_{\parallel}}}$ is related to the angles of incidence according to $$\begin{aligned}
\label{eq:k-parallel}
{{{\mathbf{k}}_{\parallel}}} &=
\frac{\omega}{c} \sin\theta_0 \left(\cos\phi_0, \sin\phi_0, 0 \right),\end{aligned}$$ where $c$ denotes the speed of light in vacuum and $\theta_0$ and $\phi_0$ are the polar and azimuthal angles of incidence, respectively (Fig. \[fig:geometry\]). When writing the field of incidence, ${{\mathbf{E}^{\mathrm{inc}}}}({{\mathbf{x}}};t)$, a time harmonic dependence of the form $\exp(-{\mathrm{i}}\omega t)$ was assumed. A similar time dependence will be assumed for all field expressions, but not indicted explicitly.
Above the surface roughness region, i.e., for $x_3 > \max \zeta({{{{\mathbf{x}}_{\parallel}}}})$, the scattered field can be written as a superposition of [*upwards*]{} propagating reflected plane waves:
\[eq:E\_sc\_total\] $$\begin{aligned}
\begin{aligned}
\label{eq:E_sc}
{{\mathbf{E}^{\mathrm{sc}}}}({{\mathbf{x}}}| \omega)
={}& \int \frac{\mathrm{d}^2q_\parallel}{(2\pi)^2}
{\bm{\mathcal{E}}^\mathrm{sc}}({{{{\mathbf{q}}_{\parallel}}}})
\\
&\times\exp\left[{\mathrm{i}}{{{{\mathbf{q}}_{\parallel}}}}\cdot{{{{\mathbf{x}}_{\parallel}}}}+{\mathrm{i}}{{{\alpha_{\mathrm{1}}}}}(q_\parallel)x_3\right],
\end{aligned}\end{aligned}$$ where $$\begin{aligned}
\begin{aligned}
{\bm{\mathcal{E}}^\mathrm{sc}}({{{{\mathbf{q}}_{\parallel}}}})
={}&
\frac{c}{\omega}
\left[
{{{{{\mathbf{\hat{q}}}_{\parallel}}}}}{{{\alpha_{\mathrm{1}}}}}(q_\parallel)-{{\mathbf{\hat{x}}}}_3 q_\parallel
\right]
{{\mathcal{E}^\mathrm{sc}_{\mathrm{p}}}}({{{\mathbf{q}}_{\parallel}}})
\\
&+
\left({{\mathbf{\hat{x}}}}_3 \times {{{{{\mathbf{\hat{q}}}_{\parallel}}}}}\right){{\mathcal{E}^\mathrm{sc}_{\mathrm{s}}}}({{{\mathbf{q}}_{\parallel}}}).
\end{aligned}\end{aligned}$$
The integration in Eq. (\[eq:E\_sc\]) is over the entire plane, including the evanescent region . Therefore, both propagating and evanescent modes are included in ${{\mathbf{E}^{\mathrm{sc}}}}({{\mathbf{x}}}| \omega)$.
We will assume that a linear relationship exists between the amplitudes of the incident and the scattered fields, and we write (for $\alpha=\mathrm{p,s}$) $$\begin{aligned}
\label{eq:scattering_amplitude}
{{\mathcal{E}^\mathrm{sc}_{\mathrm{\alpha}}}}({{{{\mathbf{q}}_{\parallel}}}})
={}&
\sum_{\beta={\mathrm{p}},{\mathrm{s}}}
R_{\alpha\beta}({{{{\mathbf{q}}_{\parallel}}}}|{{{{\mathbf{k}}_{\parallel}}}})
{{\mathcal{E}^\mathrm{inc}_{\mathrm{\beta}}}}({{{{\mathbf{k}}_{\parallel}}}}).\end{aligned}$$ Here we have introduced the so-called *scattering amplitude* $R_{\alpha\beta}({{{{\mathbf{q}}_{\parallel}}}}|{{{{\mathbf{k}}_{\parallel}}}})$, which describes how incident $\beta$-polarized light characterized by a lateral wave vector ${{{\mathbf{k}}_{\parallel}}}$ is converted by the surface roughness into scattered light of polarization $\alpha$ and lateral wave vector ${{{\mathbf{q}}_{\parallel}}}$. When $q_\parallel \leq \omega/c$, the wave vector ${{{\mathbf{q}}_{\parallel}}}$ is related to the angles of scattering $(\theta_s,\phi_s)$ by $$\begin{aligned}
\label{eq:q-parallel}
{{{\mathbf{q}}_{\parallel}}} &= \frac{\omega}{c} \sin\theta_s
\left(\cos\phi_s, \sin\phi_s, 0 \right).\end{aligned}$$
Below the surface region, i.e., for $x_3< \min\,\zeta({{{\mathbf{x}}_{\parallel}}})$, the transmitted electric field can be written as
\[eq:E\_tr\_total\] $$\begin{aligned}
\label{eq:E_tr}
\begin{aligned}
{{\mathbf{E}^{\mathrm{tr}}}}({{{\mathbf{x}}_{\parallel}}}|\omega)
={}&
\int \frac{{\mathrm{d}}^2 p_\parallel}{(2\pi)^2}
{\bm{\mathcal{E}}^\mathrm{tr}}({{{\mathbf{p}}_{\parallel}}})
\\
&\times\exp \left[ {\mathrm{i}}{{{\mathbf{p}}_{\parallel}}}\cdot {{{{\mathbf{x}}_{\parallel}}}}- {\mathrm{i}}{{{\alpha_{\mathrm{2}}}}}(p_\parallel) x_3 \right]
\end{aligned}\end{aligned}$$ with $$\begin{aligned}
\label{eq:E_tr_amplitudes}
\begin{aligned}
{\bm{\mathcal{E}}^\mathrm{tr}}({{{\mathbf{p}}_{\parallel}}})
={}&
-\frac{1}{\sqrt{\varepsilon_2(\omega)}}
\frac{c}{\omega}
\left[
{\mathbf{{\hat{p}}}}_\parallel {{{\alpha_{\mathrm{2}}}}}(p_\parallel)+{{\mathbf{\hat{x}}}}_3 p_\parallel
\right]
{{\mathcal{E}^\mathrm{tr}_{\mathrm{p}}}}({{{\mathbf{p}}_{\parallel}}})
\\
&+
\left({{\mathbf{\hat{x}}}}_3 \times {\mathbf{{\hat{p}}}}_\parallel \right){{\mathcal{E}^\mathrm{tr}_{\mathrm{s}}}}({{{\mathbf{p}}_{\parallel}}}).
\end{aligned}\end{aligned}$$
In writing Eqs. (\[eq:E\_tr\_total\]) we have introduced wave vectors of the transmitted field ${\mathbf{{p}}}={{{\mathbf{p}}_{\parallel}}}-{{{\alpha_{\mathrm{2}}}}}(p_\parallel){\mathbf{{\hat{x}}}}_3$, where $$\begin{aligned}
\label{eq:alpha_2}
\begin{aligned}
{{{\alpha_{\mathrm{2}}}}}(p_\parallel)
&{}=
\sqrt{\varepsilon_2(\omega)\frac{\omega^2}{c^2}-p_\parallel^2},\\
&\mathrm{Re}\,{{{\alpha_{\mathrm{2}}}}}\geq 0,\;\mathrm{Im}\,{{{\alpha_{\mathrm{2}}}}}\geq 0.
\end{aligned}\end{aligned}$$ In complete analogy to what was done for reflection, a transmission amplitude $T_{\alpha\beta}({{{\mathbf{p}}_{\parallel}}}|{{{\mathbf{k}}_{\parallel}}})$ may be defined via the following linear relation between the amplitudes of the incident and transmitted fields ($\alpha=\mathrm{p,s}$) $$\begin{aligned}
\label{eq:transmission_amplitude}
{{\mathcal{E}^\mathrm{tr}_{\mathrm{\alpha}}}}({{{\mathbf{p}}_{\parallel}}})
={}&
\sum_{\beta={\mathrm{p}},{\mathrm{s}}}
T_{\alpha\beta}({{{\mathbf{p}}_{\parallel}}}|{{{\mathbf{k}}_{\parallel}}})
{{\mathcal{E}^\mathrm{inc}_{\mathrm{\beta}}}}({{{{\mathbf{k}}_{\parallel}}}}).\end{aligned}$$
Since the form of the electric fields given by Eqs. , , and apply far away from the surface region, they are referred to as the [*asymptotic forms*]{} of the electric field. These equations automatically satisfy the boundary conditions at infinity.
In passing we note that once the incident field has been specified, the scattered and transmitted fields are fully specified outside the surface roughness region if the reflection \[$R_{\alpha\beta}({{{\mathbf{q}}_{\parallel}}}|{{{\mathbf{k}}_{\parallel}}})$\] and transmission \[$T_{\alpha\beta}({{{\mathbf{p}}_{\parallel}}}|{{{\mathbf{k}}_{\parallel}}})$\] amplitudes are known. We will now address how the reflection amplitude can be calculated.
The Rayleigh Hypothesis
-----------------------
Above the surface, i.e., in the region $x_3 > \max \zeta({{{{\mathbf{x}}_{\parallel}}}})$, the total electric field is equal to the sum of the incident and the scattered field, ${{\mathbf{E}^{\mathrm{inc}}}}({{\mathbf{x}}}|\omega) + {{\mathbf{E}^{\mathrm{sc}}}}({{\mathbf{x}}}| \omega)$. Below the surface, in the region $x_3 < \min\zeta({{{{\mathbf{x}}_{\parallel}}}})$, it equals the transmitted field, ${{\mathbf{E}^{\mathrm{tr}}}}({{\mathbf{x}}}| \omega)$. In the surface roughness region, $\min \zeta({{{{\mathbf{x}}_{\parallel}}}}) \leq x_3 \leq \max \zeta({{{{\mathbf{x}}_{\parallel}}}})$, these forms of the total field will not generally be valid. In particular, when we are above the surface but still below its maximum point, i.e., $\zeta({{{{\mathbf{x}}_{\parallel}}}}) \leq x_3< \max\zeta({{{{\mathbf{x}}_{\parallel}}}})$, the expression for the scattered field will also have terms containing $\exp\left[{\mathrm{i}}{{{{\mathbf{q}}_{\parallel}}}}\cdot{{{{\mathbf{x}}_{\parallel}}}}-{\mathrm{i}}{{{\alpha_{\mathrm{1}}}}}(q_\parallel)x_3\right]$. Similarly, the transmitted field in the surface region has to contain an additional term similar to Eq. but with the exponential function replaced by $\exp\left[{\mathrm{i}}{{{{\mathbf{q}}_{\parallel}}}}\cdot{{{{\mathbf{x}}_{\parallel}}}}+{\mathrm{i}}{{{\alpha_{\mathrm{2}}}}}(q_\parallel)x_3\right]$ (and a different amplitude).
If the surface roughness is sufficiently weak, however, the asymptotic form of the fields, Eqs. , , and , can be assumed to be a good approximation to the total electric field in the surface roughness region. This assumption is known as the *Rayleigh hypothesis* [@Book:Rayleigh; @Rayleigh1907; @Book:Maradudin-2007], in honor of Lord Rayleigh, who first used it in his seminal studies of wave scattering from sinusoidal surfaces [@Book:Rayleigh; @Rayleigh1907]. For a (one-dimensional) sinusoidal surface, $x_3=\zeta_0\sin(\Lambda x_1)$, the criterion for the validity of the Rayleigh hypothesis, and thus equations that can be derived from it (like the reduced Rayleigh equation to be introduced below), is known to be $\zeta_0\Lambda<0.448$, independent of the wavelength of the incident light [@Millar-1969; @Millar-1971]. For a randomly rough surface, however, the absolute limit of validity of this hypothesis is not generally known, though some numerical studies have been devoted to finding the region of validity for random surfaces [@Tishchenko-2009]. Even if no absolute criterion for the validity of the Rayleigh hypothesis for randomly rough surfaces is known, it remains true that it is a small-slope hypothesis. In particular, if the randomly rough surface is characterized by an rms height $\delta$, and a correlation length $a$ (see Sec. \[sec:geometry\] and Ref. [@Simonsen-2010] for details), there seems to be a consensus in the literature on the Rayleigh hypothesis being valid if $\delta/a\ll 1$ [@Tishchenko-2009; @Book:Maradudin-2007]. We stress that the validity of the Rayleigh hypothesis does not require the amplitude of the surface roughness to be small, only its slope.
The Reduced Rayleigh Equations
------------------------------
Under the assumption that the Rayleigh hypothesis is valid, the total electric field in the surface region, $\min \zeta({{{{\mathbf{x}}_{\parallel}}}}) < x_3 < \max \zeta({{{{\mathbf{x}}_{\parallel}}}})$, can be written in the form given by Eqs. , and \[with Eqs. and \]. Hence, these asymptotic fields can be used to satisfy the usual boundary conditions on the electromagnetic field at the rough surface $x_3=\zeta({{{\mathbf{x}}_{\parallel}}})$ [@Book:Jackson-2007; @Book:Stratton-2007]. In this way, one obtains the so-called Rayleigh equations, a set of coupled inhomogeneous integral equations, which the reflection and transmission amplitudes should satisfy.
In the mid-1980s, it was demonstrated by Brown *et al.* [@brown1984381] that either the reflection or transmission amplitude could be eliminated from the Rayleigh equations, resulting in an integral equation for the remaining amplitude only. Since this latter integral equation contains only the field above (below) the rough surface, it has been termed the *reduced Rayleigh equation* for reflection (transmission). Subsequently, reduced Rayleigh equations for two-dimensional film geometries, i.e., a film of finite thickness on top of an infinitely thick substrate, where only one interface is rough, was derived by Soubret *et al.* [@PhysRevB.63.245411; @Soubret:01] and Leskova [@Lekova_RRE; @OUR_Satellite_Paper]. Moreover, reduced Rayleigh equations for reflection from clean, perfectly conducting, two-dimensional randomly rough surfaces [@RRE_PEC] and reduced Rayleigh equations for transmission through clean, penetrable two-dimensional surfaces [@RRE_Transmission] have been derived.
For the purposes of the present study, we limit ourselves to a scattering system consisting of a clean, penetrable, two-dimensional rough surface $x_3=\zeta({{{\mathbf{x}}_{\parallel}}})$ (Sec. \[sec:geometry\]). If the scattering amplitudes are organized as the $2\times2$ matrix $$\begin{aligned}
{\mathbf{R}}({{{{\mathbf{q}}_{\parallel}}}}|{{{{\mathbf{k}}_{\parallel}}}})
&=
\left(\begin{array}{cc}
R_{pp}({{{{\mathbf{q}}_{\parallel}}}}|{{{{\mathbf{k}}_{\parallel}}}}) & R_{ps}({{{{\mathbf{q}}_{\parallel}}}}|{{{{\mathbf{k}}_{\parallel}}}}) \\
R_{sp}({{{{\mathbf{q}}_{\parallel}}}}|{{{{\mathbf{k}}_{\parallel}}}}) & R_{ss}({{{{\mathbf{q}}_{\parallel}}}}|{{{{\mathbf{k}}_{\parallel}}}})
\end{array}\right),\end{aligned}$$ the reduced Rayleigh equation (for reflection) for this geometry can be written in the form [@Mcgurn:1996fk; @PhysRevB.63.245411; @Soubret:01]
\[eq:RRE-total\]
$$\begin{aligned}
\label{eq:RRE}
\int\frac{{\mathrm{d}}^2q_\parallel}{(2\pi)^2}
\frac
{I\left({{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}})-{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{q}}_{\parallel}}}})|{{{{\mathbf{p}}_{\parallel}}}}-{{{{\mathbf{q}}_{\parallel}}}}\right)}
{{{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}})-{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{q}}_{\parallel}}}})}
{\mathbf{M}}^{+}({{{{\mathbf{p}}_{\parallel}}}}|{{{{\mathbf{q}}_{\parallel}}}})
{\mathbf{R}}({{{{\mathbf{q}}_{\parallel}}}}|{{{{\mathbf{k}}_{\parallel}}}})
=
-\frac
{I\left({{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}})+{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{k}}_{\parallel}}}})|{{{{\mathbf{p}}_{\parallel}}}}-{{{{\mathbf{k}}_{\parallel}}}}\right)}
{{{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}})+{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{k}}_{\parallel}}}})}
{\mathbf{M}}^{-}({{{{\mathbf{p}}_{\parallel}}}}|{{{{\mathbf{k}}_{\parallel}}}}),\end{aligned}$$
where $$\begin{aligned}
\label{eq:I_integral}
I(\gamma|{{{{\mathbf{Q}}_{\parallel}}}})
=&
\int{\mathrm{d}}^2x_\parallel
\exp\left[-{\mathrm{i}}\gamma\zeta({{{{\mathbf{x}}_{\parallel}}}})\right]
\exp\left(-{\mathrm{i}}{{{{\mathbf{Q}}_{\parallel}}}}\cdot{{{{\mathbf{x}}_{\parallel}}}}\right), \end{aligned}$$ and $$\begin{aligned}
{\mathbf{M}}^\pm({{{{\mathbf{p}}_{\parallel}}}}|{{{{\mathbf{q}}_{\parallel}}}})
=
\left(
\begin{array}{cc}
p_\parallel q_\parallel \pm {{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{q}}_{\parallel}}}}){{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}}){{{{{\mathbf{\hat{p}}}_{\parallel}}}}}\cdot {{{{{\mathbf{\hat{q}}}_{\parallel}}}}}&
-\frac{\omega}{c}{{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}})\left[{{{{{\mathbf{\hat{p}}}_{\parallel}}}}}\times {{{{{\mathbf{\hat{q}}}_{\parallel}}}}}\right]_3 \\
\pm\frac{\omega}{c}{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{q}}_{\parallel}}}})\left[{{{{{\mathbf{\hat{p}}}_{\parallel}}}}}\times{{{{{\mathbf{\hat{q}}}_{\parallel}}}}}\right]_3 &
\frac{\omega^2}{c^2}{{{{{\mathbf{\hat{p}}}_{\parallel}}}}}\cdot{{{{{\mathbf{\hat{q}}}_{\parallel}}}}}\end{array}
\right),\end{aligned}$$
where the integrals in Eqs. (\[eq:RRE\]) and (\[eq:I\_integral\]) are over the entire ${{{{\mathbf{q}}_{\parallel}}}}$-plane and ${{{{\mathbf{x}}_{\parallel}}}}$-plane, respectively. Reduced Rayleigh equations for transmission, or film geometries with only one rough interface, will have a similar structure to Eq. [@PhysRevB.63.245411; @Soubret:01], and can be solved in a completely analogous fashion.
It should be mentioned that the reduced Rayleigh equation can serve as a starting point for most, if not all, perturbation theoretical approaches to the study of scattering from rough surfaces [@Simonsen-2010]. For example, McGurn and Maradudin studied the scattering of light from two-dimensional rough surfaces based on the reduced Rayleigh equation, going to fourth order in the expansion in the surface profile function, and demonstrating the presence of enhanced backscattering [@Mcgurn:1996fk].
Mean Differential Reflection Coefficient {#sub:mdrc}
----------------------------------------
The solution of the reduced Rayleigh equation determines the scattering amplitudes $R_{\alpha\beta}({{{{\mathbf{q}}_{\parallel}}}}| {{{{\mathbf{k}}_{\parallel}}}})$. While this quantity completely specifies the total field in the region above the surface, it is not directly measurable in experiments. A more useful quantity is the mean differential reflection coefficient (DRC), which is defined as the time-averaged fraction of the incident power scattered into the solid angle ${\mathrm{d}}\Omega_s$ about the scattering direction ${{\mathbf{q}}}$. The mean DRC is defined as [@Mcgurn:1996fk] $$\begin{aligned}
\label{eq:drc}
\left\langle
\frac{\partial R_{\alpha\beta}}{\partial \Omega_s}
\right\rangle
=
\frac{1}{L^2}
\frac{\omega^2}{4 \pi^2 c^2}
\frac{\cos^2 \theta_s}{\cos \theta_0}
\left<
\left| R_{\alpha\beta}({{{{\mathbf{q}}_{\parallel}}}}| {{{{\mathbf{k}}_{\parallel}}}}) \right|^2
\right>,\end{aligned}$$ where $L^2$ is the area of the plane $x_3=0$ covered by the rough surface. In this work, we are mainly interested in diffuse (incoherent) scattering. Since we consider weakly rough surfaces, the specular (coherent) scattering will dominate, and it will be convenient to separate the mean DRC into its coherent and incoherent parts. By coherent scattering, we mean the part of the scattered light which does not cancel when the ensemble average of $R_{\alpha\beta}$ is taken, i.e., the part where the scattered field is in phase between surface realizations. Conversely, the incoherent part is the part which cancels in the ensemble average. The component of the mean DRC from incoherent scattering is [@Mcgurn:1996fk] $$\begin{aligned}
\begin{aligned}
\label{eq:drc_incoh}
&\left<
\frac{\partial R_{\alpha\beta}}{\partial \Omega_s}
\right>_{\text{incoh}}
=
\frac{1}{L^2}
\frac{\omega^2}{4 \pi^2 c^2}
\frac{\cos^2 \theta_s}{\cos \theta_0}
\\
&\qquad \times
\left[\left<
\left| R_{\alpha\beta}({{{{\mathbf{q}}_{\parallel}}}}| {{{{\mathbf{k}}_{\parallel}}}}) \right|^2
\right>
-\left|\left<
R_{\alpha\beta}({{{{\mathbf{q}}_{\parallel}}}}| {{{{\mathbf{k}}_{\parallel}}}})
\right> \right|^2
\right].
\end{aligned}\end{aligned}$$ The contribution to the mean DRC from the coherently scattered light is given by the difference between Eqs. and .
Conservation of Energy {#sub:energy}
----------------------
As a way to check the accuracy of our results, it is useful to investigate energy conservation. If we consider a metallic substrate with no absorption, the reflected power should be equal to the incident power. The fraction of the incident light of polarization $\beta$ which is scattered into polarization $\alpha$ is given by the integral of the corresponding mean DRC over the upper hemisphere: $$\begin{aligned}
\mathcal{U}_{\alpha\beta}
=&
\int {\mathrm{d}}\Omega_s\;
\left\langle
\frac{\partial R_{\alpha\beta}}{\partial \Omega_s}
\right\rangle.\end{aligned}$$ For a non-absorbing metal, if we send in light of polarization $\beta$, we should have $$\begin{aligned}
\sum_{\alpha} \mathcal{U}_{\alpha\beta} = 1,\end{aligned}$$ if energy is conserved. While the conservation of energy is useful as a relatively simple test, it is important to note that it is a necessary, but not sufficient, condition for correct results.
Numerical Solution of the Reduced Rayleigh Equation {#sec:solving}
===================================================
The starting point for the numerical solution of the reduced Rayleigh equation is a discretely sampled surface, from which we wish to calculate the reflection. We will limit our discussion to quadratic surfaces of size $L \times L$, sampled on a quadratic grid of $N_x \times N_x$ points with a grid constant $$\begin{aligned}
\label{eq:deltax}
\Delta x
=
\frac{L}{N_x}.\end{aligned}$$
In this paper, we will present results for numerically generated random surfaces. These are generated by what is known as the Fourier filtering method. Briefly, it consists of generating uncorrelated random numbers with a Gaussian distribution, transforming them to Fourier space, filtering them with the square root of the surface power spectrum $g({{{{\mathbf{k}}_{\parallel}}}})$, and transforming them back to real space. The interested reader is referred to, e.g., Refs. [@Simonsen2010-04; @Maradudin1990255].
The next step towards the numerical solution of the reduced Rayleigh equation is the evaluation of the integrals $I(\gamma|{{{\mathbf{Q}}_{\parallel}}})$ defined in Eq. . These integrals are so-called Fourier integrals and care should be taken when evaluating them due to the oscillating integrands [@Book:NR-1992]. Using direct numerical integration routines for their evaluation will typically result in inaccurate results. Instead, a (fast) Fourier transform technique with end point corrections may be adapted for their evaluation, and the details of the method is outlined in Ref. [@Book:NR-1992]. However, these calculations are time consuming, since $I(\gamma|{{{\mathbf{Q}}_{\parallel}}})$ must be evaluated for all values of the arguments $\gamma={{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{p}}_{\parallel}}}})-{{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{q}}_{\parallel}}}})$ and $\gamma={{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{p}}_{\parallel}}}})-{{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{k}}_{\parallel}}}})$ [^1].
Instead, a computationally more efficient way of evaluating $I(\gamma|{{{{\mathbf{Q}}_{\parallel}}}})$ is to assume that the exponential function $\exp\left[-{\mathrm{i}}\gamma\zeta({{{{\mathbf{x}}_{\parallel}}}})\right]$, present in the definition of $I(\gamma|{{{\mathbf{Q}}_{\parallel}}})$, can be expanded in powers of the surface profile function, and then evaluating the resulting expression term-by-term by Fourier transform. This gives
\[eq:I\_fourier\_expansion\] $$\begin{aligned}
\label{eq:I_fourier_expansion_A}
I(\gamma|{{{{\mathbf{Q}}_{\parallel}}}})
=& \sum_{n=0}^{\infty}
\frac{(-{\mathrm{i}}\gamma)^{n}}{n!}
\hat{\zeta}^{(n)}({{{\mathbf{Q}}_{\parallel}}}),\end{aligned}$$ where $\hat{\zeta}^{(n)}({{{\mathbf{Q}}_{\parallel}}})$ denotes the Fourier transform of the $n$th power of the profile function, [[i.e.]{}]{}, $$\begin{aligned}
\label{eq:I_fourier_expansion_B}
\hat{\zeta}^{(n)}({{{\mathbf{Q}}_{\parallel}}})
=&
\int{\mathrm{d}}^2 x_\parallel
\zeta^{n}({{{{\mathbf{x}}_{\parallel}}}})
\exp\left(-{\mathrm{i}}{{{{\mathbf{Q}}_{\parallel}}}}\cdot{{{{\mathbf{x}}_{\parallel}}}}\right). \end{aligned}$$
In practice, the sum in Eq. will be truncated at a finite value $n=J$, and the Fourier transforms are calculated using a fast Fourier transform (FFT) algorithm.
The advantage of using Eqs. for calculating $I(\gamma|{{{{\mathbf{Q}}_{\parallel}}}})$, rather than the method of Ref. [@Book:NR-1992], is that the Fourier transform of each power of $\zeta({{{\mathbf{x}}_{\parallel}}})$ can be performed once, and changing the argument $\gamma$ in $I(\gamma|{{{\mathbf{Q}}_{\parallel}}})$ will not require additional Fourier transforms to be evaluated. This results in a significant reduction in computational time. The same method has previously been applied successfully to the numerical solution of the one-dimensional reduced Rayleigh equation [@Madrazo:1997uq; @Simonsen_OptComm; @Simonsen2009-5].
It should be noted that the Taylor expansion used to arrive at Eq. requires that $\left|\gamma\zeta({{{\mathbf{x}}_{\parallel}}})\right|\ll 1$ to converge reasonably fast, putting additional constraints on the amplitude of the surface roughness which may be more restrictive than those introduced by the Rayleigh hypothesis. Hence, surfaces exist for which the Rayleigh hypothesis is satisfied, but the above expansion method will not converge, and the more time-consuming approach of Ref. [@Book:NR-1992] will have to be applied.
Next, we need to truncate and discretize the integral over ${{{{\mathbf{q}}_{\parallel}}}}$ in Eq. (\[eq:RRE\]). We discretize ${{{{\mathbf{q}}_{\parallel}}}}$ on a grid of equidistant points, with spacing $\Delta q$, such that $$\begin{aligned}
\label{eq:qgrid}
{{{{\mathbf{q}}_{\parallel}}}}_{ij}=\left(-\frac{{\mathcal{Q}}}{2}+i\Delta{}q, -\frac{{\mathcal{Q}}}{2}+j\Delta{}q, 0\right),\end{aligned}$$ where $i, j = 0,1,2,\ldots, N_q-1$, and ${\mathcal{Q}}=\Delta q (N_q-1)$. Here, $N_q$ denotes the number of points along each axis of the grid. Additionally, we limit the integration over ${{{{\mathbf{q}}_{\parallel}}}}$ to the region $q_\parallel \leq {\mathcal{Q}}/2$. The choice of a circular integration domain reduces the computational cost, and will be discussed in more detail in Sec. \[sec:discussion\]. Converting the integral into a sum by using a two-dimensional version of the standard mid-point quadrature scheme, we get the equation:
$$\begin{aligned}
\begin{aligned}
\label{eq:RRE2}
\left(\frac{\Delta{}q}{2\pi} \right)^2
\sum_{{q_\parallel}_{ij} \leq \mathcal{Q}/2}
& \frac
{I\left({{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}})-{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{q}}_{\parallel}}}}_{ij})|{{{{\mathbf{p}}_{\parallel}}}}-{{{{\mathbf{q}}_{\parallel}}}}_{ij}\right)}
{{{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}})-{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{q}}_{\parallel}}}}_{ij})}
{\mathbf{M}}^{+}({{{{\mathbf{p}}_{\parallel}}}}|{{{{\mathbf{q}}_{\parallel}}}}_{ij})
{\mathbf{R}}({{{{\mathbf{q}}_{\parallel}}}}_{ij}|{{{{\mathbf{k}}_{\parallel}}}})
=\\
&-\frac
{I\left({{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}})+{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{k}}_{\parallel}}}})|{{{{\mathbf{p}}_{\parallel}}}}-{{{{\mathbf{k}}_{\parallel}}}}\right)}
{{{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}})+{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{k}}_{\parallel}}}})}
{\mathbf{M}}^{-}({{{{\mathbf{p}}_{\parallel}}}}|{{{{\mathbf{k}}_{\parallel}}}}).
\end{aligned} \end{aligned}$$
Here, the sum is to be taken over all ${{{{\mathbf{q}}_{\parallel}}}}_{ij}$ such that ${q_\parallel}_{ij} \leq {\mathcal{Q}}/2$, where ${q_\parallel}_{ij} = \left|{{{{\mathbf{q}}_{\parallel}}}}_{ij}\right|$. This sum yields a matrix equation where the unknowns are the four components of ${\mathbf{R}}({{{{\mathbf{q}}_{\parallel}}}}_{ij}|{{{{\mathbf{k}}_{\parallel}}}})$. It is evident from Eq. (\[eq:scattering\_amplitude\]) that if we consider incident light of either p or s polarization, we need only calculate two of the components of the scattering amplitude to fully specify the reflected field. Hence, we solve separately for either p-polarized incident light, i.e., $R_{pp}$ and $R_{sp}$, or s-polarized incident light, i.e., $R_{ss}$ and $R_{ps}$. In either case, we have twice as many unknowns as the number of values of ${{{{\mathbf{q}}_{\parallel}}}}_{ij}$ included in the sum in Eq. (\[eq:RRE2\]). Note that the left hand side of the equation system is the same for both incident polarizations, and will also remain the same for all angles of incidence, as ${{{{\mathbf{k}}_{\parallel}}}}$ only enters at the right hand side of Eq. (\[eq:RRE2\]).
In order to solve for all unknowns, we need to discretize ${{{{\mathbf{p}}_{\parallel}}}}$ as well, to obtain a closed set of linear equations. Using the same grid for ${{{{\mathbf{p}}_{\parallel}}}}$ as for ${{{{\mathbf{q}}_{\parallel}}}}$ will give us the necessary number of equations, as Eq. yields two equations for each value of ${{{{\mathbf{p}}_{\parallel}}}}$. Since we integrate over a circular ${{{{\mathbf{q}}_{\parallel}}}}$ domain, with ${{{{\mathbf{q}}_{\parallel}}}}$ discretized on a quadratic grid, the exact number of values of ${{{{\mathbf{q}}_{\parallel}}}}_{ij}$ will depend on the particular values of ${\mathcal{Q}}$ and $N_q$, but will be approximately $(\pi/4) N_q^2$.
In order to take advantage of the method for calculating $I(\gamma|{{{{\mathbf{Q}}_{\parallel}}}})$ described by Eq. (\[eq:I\_fourier\_expansion\]), it is essential that all possible values of ${{{{\mathbf{p}}_{\parallel}}}}-{{{{\mathbf{q}}_{\parallel}}}}$ and ${{{{\mathbf{p}}_{\parallel}}}}-{{{{\mathbf{k}}_{\parallel}}}}$ \[see Eq. \] fall on the grid of wave vectors ${{{{\mathbf{Q}}_{\parallel}}}}$ resolved by the Fourier transform of the surface profile we used in that calculation. First, we note that when ${{{{\mathbf{p}}_{\parallel}}}}$ and ${{{{\mathbf{q}}_{\parallel}}}}$ are discretized on the same quadratic grid, the number of possible values for each component of ${{{{\mathbf{p}}_{\parallel}}}}- {{{{\mathbf{q}}_{\parallel}}}}$ will always be an odd number, $2N_q - 1$, where $N_q$ is the number of possible values for each component of ${{{{\mathbf{p}}_{\parallel}}}}$ and ${{{{\mathbf{q}}_{\parallel}}}}$. Thus, by choosing $N_q$ such that $2N_q-1$ equals the number of elements along each axis of the FFT of the surface profile we used to calculate the integrals in Eq. , we ensure that the required number of points is resolved by the FFT [^2]. Hence, we choose $$\begin{aligned}
\label{eq:Nq}
N_q = \left\lfloor \frac{N_x+2}{2} \right\rfloor,\end{aligned}$$ where $\lfloor x \rfloor$ is the floor function of $x$, which is equal to the largest integer less than or equal to $x$.
Next, we let $\Delta q$ equal the resolution of the FFT [@Book:NR-1992], i.e., $$\begin{aligned}
\label{eq:deltaq}
\Delta q = \frac{2\pi}{L}\end{aligned}$$ and we let ${\mathcal{Q}}$ be equal to the highest wavenumber resolved by the FFT [@Book:NR-1992], $$\begin{aligned}
\label{eq:Q}
{\mathcal{Q}}= \Delta q \lfloor N_x/2 \rfloor.\end{aligned}$$
In the end, we get the equation
$$\begin{aligned}
\begin{aligned}
\label{eq:RRE3}
\left(\frac{\Delta{}q}{2\pi} \right)^2
\sum_{\left|{{{{\mathbf{q}}_{\parallel}}}}_{ij}\right| \leq \mathcal{Q}/2}
& \frac
{I\left({{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}}_{kl})-{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{q}}_{\parallel}}}}_{ij})|{{{{\mathbf{p}}_{\parallel}}}}_{kl}-{{{{\mathbf{q}}_{\parallel}}}}_{ij}\right)}
{{{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}}_{kl})-{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{q}}_{\parallel}}}}_{ij})}
{\mathbf{M}}^{+}({{{{\mathbf{p}}_{\parallel}}}}_{kl}|{{{{\mathbf{q}}_{\parallel}}}}_{ij})
{\mathbf{R}}({{{{\mathbf{q}}_{\parallel}}}}_{ij}|{{{{\mathbf{k}}_{\parallel}}}}_{mn})
=\\
&-\frac
{I\left({{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}}_{kl})+{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{k}}_{\parallel}}}}_{mn})|{{{{\mathbf{p}}_{\parallel}}}}_{kl}-{{{{\mathbf{k}}_{\parallel}}}}_{mn}\right)}
{{{{\alpha_{\mathrm{2}}}}}({{{{\mathbf{p}}_{\parallel}}}}_{kl})+{{{\alpha_{\mathrm{1}}}}}({{{{\mathbf{k}}_{\parallel}}}}_{mn})}
{\mathbf{M}}^{-}({{{{\mathbf{p}}_{\parallel}}}}_{kl}|{{{{\mathbf{k}}_{\parallel}}}}_{mn}),
\end{aligned} \end{aligned}$$
where ${{{{\mathbf{q}}_{\parallel}}}}_{ij}$, as well as ${{{{\mathbf{p}}_{\parallel}}}}_{kl}$ and ${{{{\mathbf{k}}_{\parallel}}}}_{mn}$, are defined on the grid given by Eq. (\[eq:qgrid\]), with $i,j,k,l,m,n=0,1,2,\ldots, N_q-1$, and where $N_q$, $\Delta q$ and ${\mathcal{Q}}$ are given by Eqs. (\[eq:Nq\]), (\[eq:deltaq\]) and (\[eq:Q\]), respectively.
Evaluating Eq. (\[eq:RRE3\]) for all values of ${{{{\mathbf{p}}_{\parallel}}}}_{kl}$ satisfying ${p_\parallel}_{kl} \leq \mathcal{Q}/2$, and assuming one value of ${{{{\mathbf{k}}_{\parallel}}}}_{mn}$, such that ${k_\parallel}_{mn} < \omega/c$, and one incident polarization $\beta$, results in a *closed* system of linear equations in $R_{\alpha\beta}({{{{\mathbf{q}}_{\parallel}}}}_{ij}|{{{{\mathbf{k}}_{\parallel}}}}_{mn})$ where $\alpha={\mathrm{p}},{\mathrm{s}}$. Repeating the procedure for both incident polarizations allows us to obtain all four components of ${\mathbf{R}}({{{{\mathbf{q}}_{\parallel}}}}_{ij}|{{{{\mathbf{k}}_{\parallel}}}}_{mn})$.
With the reflection amplitudes $R_{\alpha\beta}({{{\mathbf{q}}_{\parallel}}}_{ij}|{{{\mathbf{k}}_{\parallel}}}_{mn})$ available, the contribution to the mean differential reflection coefficient from the light that has been scattered incoherently is obtained from Eq. (\[eq:drc\_incoh\]) after averaging over an ensemble of surface realizations.
In passing we note that to avoid loss of numerical precision by operating on numbers with widely different orders of magnitude, we have rescaled all quantities in our problem to dimensionless numbers. When considering an incoming wave of wavelength $\lambda$, angular frequency $\omega$, and wave vector ${{\mathbf{k}}}$, we have chosen to rescale all lengths in our problem by multiplying with $\omega/c$, and all wavenumbers by multiplying with $c/\omega$, effectively measuring all lengths in units of $\lambda/2\pi$, and the magnitude of wave vectors in units of $\omega/c$.
Results {#sec:results}
=======
![(Color online) Incoherent part of the mean differential reflection coefficient \[Eq. (\[eq:drc\_incoh\])\], showing only the in-plane scattering as a function of outgoing lateral wave vector, averaged over 14,200 randomly rough silver surface realizations. The wavelength (in vacuum) of the incident light was $\lambda=\unit{457.9}{\nano\meter}$, and the dielectric function of silver at this wavelength is ${{{\varepsilon_{\mathrm{2}}}}}=-7.5+0.24{\mathrm{i}}$. The surface power spectrum was Gaussian \[Eq. (\[eq:gaussian\])\], with correlation lengths $a_1=a_2=0.25\lambda$ and rms height $\delta=0.025\lambda$. The angle of incidence was $\theta_0=18.24\degree$, the surface covered an area $L \times L$, where $L=25\lambda$, and the surface was discretized on a grid of $319 \times 319$ points. The position of the specular peak (not present in the incoherent part) and the enhanced backscattering peak are indicated by the vertical dashed lines.[]{data-label="fig:ag_1D_drc"}](AgSingleInterface-isotropic-circular-code_drc_in-plane_ik1-85_k2-85){width="\columnwidth"}
![(Color online) Incoherent part of the mean differential reflection coefficient \[Eq. (\[eq:drc\_incoh\])\], showing the full angular distribution as a function of outgoing lateral wave vector. All parameters are the same as in Fig. \[fig:ag\_1D\_drc\]. The specular position is indicated by the white dots.[]{data-label="fig:ag_2D_drc"}](AgSingleInterface-isotropic-circular-code_drc_2D_k1-085_k2-085){width="\columnwidth"}
![(Color online) Ratio of reflected power to incident power, $\mathcal{U}$, as a function of ratio between rms roughness and correlation length, $\delta/a$. Surface size and resolution were the same as for Fig. \[fig:ag\_1D\_drc\], and the surface was randomly rough with a Gaussian power spectrum, correlation length was kept constant at $a=a_1=a_2=0.25\lambda$, while the rms roughness $\delta$ was varied from $0.0$ to $0.045\lambda$. The Fresnel coefficients have been included for comparison.[]{data-label="fig:reflection"}](reflection){width="\columnwidth"}
![(Color online) The same as in Fig. \[fig:ag\_2D\_drc\], except that ${{{\varepsilon_{\mathrm{2}}}}}=2.64$, and the results are averaged over 21,800 randomly rough surfaces.[]{data-label="fig:di_2D_drc"}](Photoresist_drc_2D_k1-085_k2-085){width="\columnwidth"}
![(Color online) The same as in Fig. \[fig:ag\_2D\_drc\], except the correlation length of the Gaussian roughness, which is $a_1=0.25\lambda$ in the $x_1$ direction and $a_2=0.75\lambda$ in the $x_2$ direction, and the results are the average of an ensemble of 6,800 surface realizations.[]{data-label="fig:ag_anisotropic"}](AgSingleInterface-anisotropic-30_drc_2D_k1-085_k2-085){width="\columnwidth"}
![(Color online) Incoherent part of the mean differential reflection coefficient \[Eq. (\[eq:drc\_incoh\])\], showing only the in-plane scattering as function of outgoing lateral wave vector, averaged over 7,000 surface realizations with dielectric constant ${{{\varepsilon_{\mathrm{2}}}}}=-16+1.088{\mathrm{i}}$, which corresponds to silver at $\lambda=\unit{632.8}{\nano\meter}$. The surface power spectrum was of the cylindrical type \[Eq. (\[eq:cylindrical\])\], with $k_- = 0.82\omega/c$, $k_+ = 1.97\omega/c$, and rms roughness $\delta=0.025\lambda$. The angles of incidence were $\theta_0=1.6\degree$ and $\phi_0=45\degree$.[]{data-label="fig:ag_WoD_1D_drc"}](Ag_WoD_drc_in-plane_ik1-80_k2-80){width="\columnwidth"}
To demonstrate the use of the formalism for solving the reduced Rayleigh equation, the first set of calculations we carried out was for two-dimensional randomly rough silver surfaces. The surface roughness was characterized by an rms height of $\delta=0.025\lambda$ and an isotropic Gaussian power spectrum \[Eq. (\[eq:gaussian\])\] of correlation lengths $a_1=a_2=0.25\lambda$. In Figs. \[fig:ag\_1D\_drc\] and \[fig:ag\_2D\_drc\] we present simulation results for the contribution to the mean differential reflection coefficients from light of wavelength (in vacuum) $\lambda=\unit{457.9}{\nano\meter}$ that was scattered incoherently from a rough silver surface of size $25\lambda \times 25\lambda$, discretized into $319\times319$ points. The dielectric function of silver at this wavelength is ${{{\varepsilon_{\mathrm{2}}}}}=-7.5+0.24{\mathrm{i}}$, and the angles of incidence were $\theta_0=18.24\degree$ and $\phi_0=45\degree$.
Figure \[fig:ag\_1D\_drc\] shows the in-plane scattering for this system. The enhanced backscattering peak, a multiple scattering phenomenon, is clearly visible, and is as expected strongest in $\mathrm{p} \to \mathrm{p}$ scattering, since p-polarized light has a stronger coupling to surface plasmon polaritons [@Simonsen-2010]. Figure \[fig:ag\_2D\_drc\] shows the full angular distribution of the mean DRC for the same system. In Figs. \[fig:ag\_2D\_drc\](a)–(c) and Figs. \[fig:ag\_2D\_drc\](d)–(e) the incident light was p- and s-polarized, respectively. Figures \[fig:ag\_2D\_drc\](c) and \[fig:ag\_2D\_drc\](f) show scattering into s-polarization, Figs. \[fig:ag\_2D\_drc\](b) and \[fig:ag\_2D\_drc\](e) show scattering into p-polarization and in Figs. \[fig:ag\_2D\_drc\](a) and \[fig:ag\_2D\_drc\](d) the polarization of the scatted light was not recorded. In particular from Fig. \[fig:ag\_2D\_drc\](b), we observe that the enhancement features seen in Fig. \[fig:ag\_1D\_drc\] at angular position $\theta_s=-\theta_0$, are indeed enhancements in a well-defined direction corresponding to that of retro-reflection, and not some intensity ridge structure about this direction (as has been seen for other scattering systems [@Simonsen2009-1]). Moreover, the structures of the angular distribution of the intensity of the scattered light depicted in Fig. \[fig:ag\_2D\_drc\] are consistent with what was found by recent studies by using other numerical methods [@Simonsen2009-1; @Simonsen2009-9]. The results presented in Figs. \[fig:ag\_1D\_drc\] and \[fig:ag\_2D\_drc\] were obtained by averaging the DRC over an ensemble consisting of 14,200 surface realizations.
A test of energy conservation was performed by simulating the scattering of light from a non-absorbing silver surface ($\mathrm{Im}~{{{\varepsilon_{\mathrm{2}}}}}= 0$) with otherwise the same parameters as those used to obtain the results of Figs. \[fig:ag\_1D\_drc\] and \[fig:ag\_2D\_drc\]. For this scattering system we found $|\mathcal{U}-1| \leq 0.0003$, i.e., energy is conserved to within 0.03%, something that testifies to the accuracy of the approach and a satisfactory discretization.
As a further test, we studied the scattering from a set of (absorbing) silver surfaces with the same parameters used to obtain Figs. \[fig:ag\_1D\_drc\] and \[fig:ag\_2D\_drc\], except that the rms roughness $\delta$ was varied between $0$ and $0.045\lambda$, while the correlation lengths were held constant at $a_1=a_2=0.25\lambda\equiv a$. For the purpose of comparison, we also performed simulations for a similar set of surfaces but assuming no absorption, i.e., we used ${{{\varepsilon_{\mathrm{2}}}}}=-7.5$. The results of these tests are presented in Fig. \[fig:reflection\].
The reduced Rayleigh equation is only valid for surfaces of small slopes [@Book:Maradudin-2007]. We have found that at least for the parameters used in obtaining Fig. \[fig:reflection\], our code gives good results for an rms roughness to correlation-length ratio $\delta/a \lesssim 0.12$, as judged by energy conservation. For larger values of $\delta/a$, the results look qualitatively much the same, but the ratio of reflected to incident power starts to become nonphysical (increasing past $1$), as seen in Fig. \[fig:reflection\]. It is noted that decreasing the sampling interval $\Delta q$, with $\mathcal{Q}$ unchanged, did not change this conclusion in any significant way, indicating that the observed lack of energy conservation was not caused by poor resolution in discretizing the integral over ${{{{\mathbf{q}}_{\parallel}}}}$.
The next set of calculations we performed was for a dielectric substrate characterized by ${{{\varepsilon_{\mathrm{2}}}}}=2.64$. Otherwise, all roughness parameters were the same as for the silver surface used to produce Figs. \[fig:ag\_1D\_drc\] and \[fig:ag\_2D\_drc\]. The mean differential reflection coefficient for light scattered incoherently by the rough dielectric surface is presented in Fig. \[fig:di\_2D\_drc\]. By comparing these results to those presented in Fig. \[fig:ag\_2D\_drc\], we notice that the dielectric reflects less than the silver (the figures show only the incoherent scattering, but the same holds for the coherent part), which is as expected. The ratio of reflected to incident power for these data was $\mathcal{U}=0.0467$ for p-polarized light at an angle of incidence of $\theta_0=18.24\degree$. Moreover, from Fig. \[fig:di\_2D\_drc\] we also notice the absence of the enhanced backscattering peak, which is also to be expected since this phenomenon (for a weakly rough surface) requires the excitation of surface guided modes [@Simonsen-2010]. Note that for a transparent substrate, it is not possible to verify the conservation of energy without also calculating the transmitted field. Therefore, energy conservation has not been tested for the dielectric substrate geometry.
So far, we have exclusively considered surfaces with statistically isotropic roughness. For the results presented in Fig. \[fig:ag\_anisotropic\], we simulated the light scattering from a silver surface of the same parameters as those assumed in producing the results of Figs. \[fig:ag\_1D\_drc\] and \[fig:ag\_2D\_drc\], except that now the surface power spectrum was anisotropic, with correlation lengths $a_1=0.25\lambda$ in the $x_1$ direction and $a_2=0.75\lambda$ in the $x_2$ direction and an rms roughness of $\delta=0.025\lambda$. Figure \[fig:ag\_anisotropic\] shows the incoherent part of the mean DRC averaged over 6,800 surface realizations. In this case, there is more diffuse scattering along the $x_1$ direction than the $x_2$ direction, which is to be expected, since a shorter correlation length means the height of the surface changes more rapidly when moving along the surface in this direction. The interested reader is encouraged to consult Ref. [@Simonsen2010-04] for a more detailed study of light scattering from anisotropic surfaces.
Finally, for the results presented in Fig. \[fig:ag\_WoD\_1D\_drc\], we have simulated the scattering of light from a surface of size $25\lambda \times 25\lambda$, discretized into $319\times319$ points, with ${{{\varepsilon_{\mathrm{2}}}}}=-16+1.088{\mathrm{i}}$, corresponding to silver at a wavelength $\lambda=\unit{632.8}{\nano\meter}$. The surface power spectrum was cylindrical \[see Eq. \], with $k_- = 0.82\omega/c$, $k_+~=~1.97\omega/c$ and rms roughness $\delta=0.025\lambda$, and the angles of incidence were $(\theta_0,\phi_0)=(1.6\degree, 45\degree)$. Figure \[fig:ag\_WoD\_1D\_drc\] shows the in-plane, incoherent part of the mean differential reflection coefficient averaged over 7,000 surface realizations.
From perturbation theory [@Simonsen-2010; @Book:Maradudin-2007], we know that for an incident wave of lateral wave vector ${{{{\mathbf{k}}_{\parallel}}}}$ to be scattered *via single scattering* into a reflected wave of lateral wave vector ${{{{\mathbf{q}}_{\parallel}}}}$, we must have $g({{{{\mathbf{q}}_{\parallel}}}}-{{{{\mathbf{k}}_{\parallel}}}})>0$, where $g({{{{\mathbf{k}}_{\parallel}}}})$ is the surface power spectrum \[Eq. (\[eq:powerspectrum\])\]. Since the power spectrum in this case is zero for , we have no contribution from single scattering in the angular interval from $\theta_s=-53.5\degree$ to $\theta_s=56.7\degree$ (for the angles of incidence assumed here). The enhanced backscattering peak, which is due to multiple scattering processes, is clearly visible in Fig. \[fig:ag\_WoD\_1D\_drc\] (at $\theta_s=-\theta_0$) partly because it is not masked by a strong single scattering contribution.
Discussion {#sec:discussion}
==========
A challenge faced when performing a direct numerical solution of the reduced Rayleigh equation for the scattering of light from two-dimensional rough surfaces is the numerical complexity. In this section, we discuss some of these issues in detail.
Memory Requirements {#sub:memory}
-------------------
Part of the challenge of a purely numerical solution of the reduced Rayleigh equation by the formalism introduced by this study, is that it requires a relatively large amount of memory. With approximately $\mathcal{N}=(\pi/4) N_q^2$ possible values for ${{{{\mathbf{q}}_{\parallel}}}}$, the coefficient matrix of the linear equation system will contain approximately $(2\mathcal{N})^2$ elements, where the factor $2$ comes from the two outgoing polarizations. Hence, the memory required to hold the left hand side of the equation system will be approximately $4\mathcal{N}^2\eta$, where $\eta$ is the number of bytes used to store one complex number.
If each element is a single precision complex number, which is what was used to obtain the results presented in this paper, then $\eta=8\;\mathrm{bytes}$, and the matrix will require approximately $2 \pi^2 N_q^4$ bytes of memory for storage. For instance, with the choice $N_x=319$, which was used in all the simulations presented in this paper, and that corresponds to $N_q=160$ \[Eq. \], the coefficient matrix will take up approximately 12 GB of memory.
Note that if we instead performed the ${{{{\mathbf{q}}_{\parallel}}}}$ integration in Eq. over a square domain of edge ${\mathcal Q}$, the number of elements in the resulting coefficient matrix would be $(2N_q^2)^2=(16/\pi^2)(2{\mathcal N})^2$. Hence, by restricting the ${{{{\mathbf{q}}_{\parallel}}}}$ integration present in the reduced Rayleigh equation to a circular domain of radius ${\mathcal Q}/2$, the memory footprint of the simulation is approximately $\pi^2/16\approx0.62$ of what it would have been if a square integration domain of edge ${\mathcal Q}$ was used. For this reason, a circular integration domain has been used in obtaining the results presented in this paper. However, we have checked and found that using a square ${{{\mathbf{q}}_{\parallel}}}$ integration domain of a similar size will not affect the results in any noticeable way. Indeed, if this was not the case, it would be a sign that $\mathcal{Q}$ was too small.
When determining the system size, we can freely choose the length of the edge of the square surface, $L$, and the number of sampling points along each direction, $N_x$. These parameters will then fix the resolution of the surface, $\Delta x$, the resolution in wave vector space, $\Delta q$, the number of resolved wave vectors, $N_q$, and the cutoff in the ${{{{\mathbf{q}}_{\parallel}}}}$ integral, $\mathcal{Q}$ \[see Eqs. , , , and \]. The combination of $\Delta q$ and $\mathcal{Q}$ then determines the number of resolved wave vectors that actually fall inside the propagating region, $|{{{{\mathbf{q}}_{\parallel}}}}| < \omega / c$, which is identical to the number of data points used to produce, e.g., Fig. \[fig:ag\_2D\_drc\].
As we are not free to choose all the parameters of the system, it is clear that some kind of compromise is necessary. The number of sampling points of the surface along each direction, $N_x$, and how it determines $N_q$ via Eq. , determines the amount of memory needed to hold the coefficient matrix, as well as the time required to solve the corresponding linear set of equations. Hence, the parameter $N_x$ is likely limited by practical considerations, typically by available computer hardware or time. For a given value of $N_x$, it is possible to choose the edge of the square surface, $L$, to get good surface resolution, at the cost of poor resolution in wave vector space, or vice versa. Note also that changing $L$ will change $\mathcal{Q}$ via $\Delta q$ \[see Eqs. and \]. If $\mathcal{Q}$ is not large enough to include evanescent surface modes, like surface plasmon polaritons, multiple scattering will not be correctly included in the simulations, and the results can therefore not be trusted. The optimal compromise between values of $N_x$ and $L$ depends on the system under study.
Calculation Time {#sub:time}
----------------
The simulations presented in this paper were performed on shared-memory machines with 24 GB of memory and two six-core 2.4 GHz AMD Opteron processors, running version 2.6.18 of the Linux operating system. The code was parallelized using the MPI library, and the setup of the linear set of equations ran on all 12 cores in the timing examples given. The linear equation solver used was a parallel, dense solver based on LU-decomposition [@Book:NR-1992] (PCGESV from ScaLAPACK), which runs efficiently on all 12 cores. Setting up the equation system scaled almost perfectly to a large number of cores, while the solver scaled less well, due to the need for communication. Numerically solving the reduced Rayleigh equation for the scattering amplitudes associated with one realization of a rough surface, discretized onto a grid of $319\times319$ points, took approximately 17 minutes on the architecture described above, and required about 12 GB of memory. Out of this time, approximately 100 seconds was spent setting up the equation system, 950 seconds was spent solving it by LU decomposition, and typically around 1 second was spent on other tasks, including writing data to disk. Table \[tab:time\] shows timing and memory details of the calculations, including other system sizes.
Based on the discussion in Sec. \[sub:memory\], we note that the use of a circular ${{{{\mathbf{q}}_{\parallel}}}}$ integration domain also significantly reduces the time required to solve the resulting linear system of equations. When using a dense solver, the time to solve the systems scales as the cube of the number of unknowns. Thus we expect the CPU time to solve the matrix system for a circular integration domain of radius $\mathcal{Q}/2$ to be about half ($\pi^3/2^6$) the time to solve the corresponding system using a square domain of edge $\mathcal{Q}$.
The ratio of the time spent solving one equation system to the total simulation time per surface realization increases with increasing system size, as the time to set up the equation system is $\mathcal{O}(N_x^4)$, while the time to solve the linear system by LU decomposition scales as $\mathcal{O}(N_x^6)$. It is clear from Table \[tab:time\] that for any surface of useful size the runtime is completely dominated by the time spent in solving the linear set of equations.
Since the time solving the equation system dominates the overall simulation time, we investigated if one could improve the performance of the simulations by using an iterative solver instead of a direct solver based on LU decomposition. For example, Simonsen et al. [@PhysRevA.81.013806] recently reported good performance using BiCGStab [@vorst:631] on a dense matrix system of a similar size. In our preliminary investigations into using iterative solvers, we found that convergence with BiCGStab was slow and unreliable for our linear equation systems. However, it should be stressed that we did not use a preconditioning scheme, which could potentially yield significantly improved convergence.
From Eq. (\[eq:RRE\]) it follows that changing the angles of incidence and/or the polarization of the incident light changes [*only*]{} the right hand side of the equation system to be solved. Hence, an advantage of using LU decomposition (over iterative solvers) is that the additional time required to solve the system for several right hand sides is negligible, since the overall majority of time is spent factorizing the matrix. Conversely, the time spent using an iterative solver (like BiCGStab) will scale linearly with the number of right hand sides. For these reasons, we have chosen to use an LU-based solver.
[rdddd]{} & & & &\
199 & 10 & 58 & 69 & 1.8\
239 & 28 & 171 & 200 & 3.8\
279 & 56 & 429 & 486 & 7.0\
319 & 97 & 946 & 1,045 & 12.0\
369 & 154 & 1,916 & 2,074 & 19.2\
399 & 266 & 3,625 & 3,895 & 29.4\
GPU implementation {#sub:GPU implementation}
------------------
Currently, performing simulations like those presented in this paper on a single desktop computer is prohibitively time consuming due to inadequate floating point performance. However, the increasing availability of powerful graphics processing units (GPUs) has the potential to provide computing power comparable to that of a powerful parallel machine, but at a fraction of the cost. As the most time-consuming step in our simulations is the LU decomposition of the system matrix (see Table \[tab:time\]), this is where efforts should be made to optimize the code. With this in mind, the simulation code was adapted to (optionally) employ version 1.0 of the MAGMA library [@agullo2011lu] for GPU-based LU decomposition. Performance was compared between a regular supercomputing service and a GPGPU (General Purpose GPU) testbed. On the regular service, the code was running on a single compute node containing two AMD 2.3 GHz 16-core processors and 32 GB of main memory. On the GPGPU testbed, the hardware consisted of a single Nvidia Fermi C2050 processor with 3 GB of dedicated memory and 32 GB of main system memory. For these two computer systems, the initial performance tests indicated that the LU decomposition took comparable time on the two architectures for a system of size $N_q = 100$ (the difference was less than 10%). The time using the GPGPU testbed included the transfer of the system matrix to and from the Fermi card and the decomposition of the matrix. Even though these results are preliminary, it demonstrates that there is a possibility of performing simulations like those reported in this study without having to resort to costly supercomputing resources. Instead, even with limited financial means, they may be performed on single desktop computers with a state-of-the-art GPU.
Conclusion {#sec:Conclusion}
==========
We have introduced a formalism for performing non-perturbative, purely numerical, solutions of the reduced Rayleigh equation for the reflection of light from two-dimensional penetrable rough surfaces, characterized by a complex dielectric function $\varepsilon(\omega)$.
As an example, we have used this formalism to carry out simulations of the scattering of p- or s-polarized light from two-dimensional randomly rough dielectric and metallic surfaces characterized by isotropic or anisotropic Gaussian and cylindrical power spectra. From the scattering amplitudes, obtained by solving the reduced Rayleigh equation, we calculate the mean differential reflection coefficients, and we calculate the full angular distribution of the scattered light, with polarization information. For the scattering of light from weakly rough metal surfaces, the mean differential reflection coefficient shows a well-defined peak in the retro-reflection direction (the enhanced backscattering phenomenon). From previous experimental and theoretical work, this is to be expected for such scattering systems. Moreover, the obtained angular distributions of the intensity of the scattered light show the symmetry properties found for strongly rough surfaces in recent studies using other simulation methods.
For the purpose of evaluating the accuracy of our simulation results, we used the conservation of energy for a corresponding non-absorbing scattering system. This is a required, but not sufficient, condition for the correctness of the numerical simulations. By this method, we found that within the validity of the reduced Rayleigh equation our code produces reliable results, at least for the parameters assumed in this study. In particular, for a rough non-absorbing metal surface of the parameters used in this study, energy was conserved to within $0.03\%$, or better. This testifies to the accuracy of the approach and a satisfactory discretization. Moreover, we also performed simulations of the scattered intensity for systems where the rms roughness of the surface was systematically increased from zero with the other parameters kept unchanged. It was found that energy conservation was well satisfied (for the parameters assumed) when the ratio of rms roughness ($\delta$) to correlation length ($a$), satisfied $\delta/a \lesssim 0.12$.
We believe that the results of this paper provide an important addition to the collection of available methods for the numerical simulation of the scattering of light from rough surfaces. The developed approach can be applied to a wide range of scattering systems, including clean and multilayered scattering systems, that are relevant for numerous applications.
We would like to acknowledge the help of Dr. Chris Johnson at the EPCC, University of Edinburgh, for help in parallelizing the code. We are also indebted to Dr. A. A. Maradudin for discussions on the topic of this paper. The work of T.N. and P.A.L. was partially carried out under the HPC-EUROPA2 project (project number: 228398) with support of the European Commission – Capacities Area – Research Infrastructures. The work was also supported by NTNU by the allocation of computer time.
[^1]: For the calculations used to generate the results presented in this paper, this would amount to evaluating $I(\gamma|{{{{\mathbf{Q}}_{\parallel}}}})$ on the order of $10^{10}$ times.
[^2]: Note that since the FFT always resolves the zero frequency, and the FFT of a purely real signal is symmetric about the zero frequency under complex conjugation, it is always possible to calculate an odd number of elements along each axis of the FFT
|
---
abstract: 'We report infrared photometry of the extrasolar planet HD209458b during the time of secondary eclipse (planet passing behind the star). Observations were acquired during two secondary eclipses at the NASA Infrared Telescope Facility (IRTF) in September 2003. We used a circular variable filter (1.5% bandpass) centered at 3.8 $\mu$m to isolate the predicted flux peak of the planet at this wavelength. Residual telluric absorption and instrument variations were removed by offsetting the telescope to nearby bright comparison stars at a high temporal cadence. Our results give a secondary eclipse depth of $0.0013\pm0.0011$, not yet sufficient precision to detect the eclipse, whose expected depth is $\sim 0.002 - 0.003$. We here elucidate the current observational limitations to this technique, and discuss the approach needed to achieve detections of hot Jupiter secondary eclipses at 3.8 $\mu$m from the ground.'
author:
- |
Drake Deming$^{1,4}$[^1], L. Jeremy Richardson$^{2}$[^2], & Joseph Harrington$^{3}$ [^3]\
$^{1}$Planetary Systems Laboratory, Code 693, Goddard Space Flight Center, Greenbelt MD 20771 USA\
$^{2}$Exoplanet and Stellar Astrophysics Laboratory, Code 667, Goddard Space Flight Center, Greenbelt MD 20771 USA\
$^{3}$Department of Physics, University of Central Florida, Orlando FL 32816-2385 USA\
$^{4}$Visiting Astronomer at the Infrared Telescope Facility, which is operated by the University of Hawaii\
under Cooperative Agreement no. NCC 5-538 with the National Aeronautics and Space Administration,\
Science Mission Directorate, Planetary Astronomy Program
date:
title: '3.8-Micron Photometry During the Secondary Eclipse of the Extrasolar Planet HD209458b'
---
\[firstpage\]
infrared: techniques: photometric - eclipses - stars:individual: HD 209458 - planetary systems - infrared
Introduction
============
The passage of a transiting extrasolar planet behind its star during secondary eclipse has emerged as a valuable technique for measuring thermal radiation emitted by hot Jupiters. Secondary eclipse detections of hot Jupiters have been accomplished using the Spitzer Space Telescope [@charb05; @deming05a; @deming06a]. The spectra of hot Jupiters are predicted to exhibit prominent flux peaks near 2.2 & 3.8 $\mu$m, where absorption by water vapor is minimal [@sudarsky; @seager; @fortney]. Unfortunately, the bandpasses available from Spitzer do not sample these flux peaks in an optimal way. Spitzer has no capability at 2 $\mu$m, and the IRAC filter bandpass at 3.5 and 4.5 $\mu$m [@fazio] overlap the 3.8 $\mu$m peak minimally.
Observing hot Jupiter extrasolar planets at 2.2 and 3.8 $\mu$m in the near future must depend on ground-based observations. Ground-based observers have attempted both photometry [@snellen; @snellen2] and spectroscopy [@rich03] to detect the 2.2 $\mu$m peak using secondary eclipses. Recently, a tentative detection of OGLE-TR-113 at 2.2 $\mu$m was reported by @snellen2, with eclipse amplitude $0.0017\pm0.0005$. @knutson_a attempted to detect the 3.8 $\mu$m peak (L-band) of TrES-1 spectroscopically, but no investigations have yet reported photometry of hot Jupiters at 3.8 $\mu$m. At this longer wavelength, thermal background emission from the telescope and terrestrial atmosphere is a major impediment. Moreover, observations in the L band are significantly affected by telluric water vapor absorption, which is notoriously variable. In addition, photometric transit searches at visible wavelengths have been significantly affected by ‘red noise’ [@pont], and this problem might become more severe in the infrared (IR). It is therefore of interest to explore the limits of ground-based L-band photometry for secondary eclipse detection.
In this paper we report L-band photometry during the secondary eclipse of HD 209458b. A preliminary description of these measurements was reported by @deming06b; here we report the final results. We do not achieve a detection of the planet’s thermal emission, but we elucidate the observational limitations, and we describe improvements that will allow an L-band detection from the ground.
Observations
============
We observed HD 209458 during two secondary eclipses on 2003 September 9 & 16 UT, using NASA’s 3-metre Infrared Telescope Facility (IRTF) on Mauna Kea. We also observed (and detected) a primary eclipse (transit) on 2003 August 17 UT. Since the HD 209458b transit is well observed at both visible [@brown] and IR wavelengths [@rich06], we concentrate here on the secondary eclipse observations. However, observations on the transit night were very useful in analyzing the properties of our data (see below).
We imaged HD 209458 using the 256x256-pixel IR camera [@shure]. To avoid detector saturation from thermal background radiation, we used a circular-variable-filter (CVF) with a 1.5% bandpass tuned to 3.8 $\mu$m. This wavelength not only corresponds to a predicted peak in the planet’s thermal emission, but also has optimal transmission through the terrestrial atmosphere. Figure 1 shows the CVF bandpass in comparison to the telluric transmittance at Kitt Peak, obtained by convolving the atmospheric transmission atlas of @atlas to lower spectral resolution ($\lambda/\delta\lambda =
100$, comparable to CVF resolution). Since Mauna Kea is higher and drier than Kitt Peak, Figure 1 is a worst case representation of the telluric transmission for our observations.
It is generally not feasible to monitor comparison stars simultaneously with the target star in ground-based thermal IR photometry. Attaining the requisite wide field of view would require an increase in the per-pixel solid angle seen by the detector, to the point where the observations would be unavoidably saturated by thermal background. Instead, our observations alternated between HD 209458 and a comparison star, moving the telescope between them at the most rapid possible cadence (17 sec for telescope motion and settling). Our imaging sequence consisted of a continuing series of exposures in the order $T_a, C_a, C_b, T_b, T_a, ...$, where $T$ indicates the target star (HD 209458), and $C$ indicates a comparison star (see below). The subscripts $a$ and $b$ indicate two distinct nod positions on the detector array, separated by 6 arcsec (110 pixels). The nod is used for subtraction of thermal background (see Infrared Photometry section). The observations commenced each night before the time of secondary eclipse, and continued for as long as possible after eclipse.
The ideal comparison star would have an IR brightness comparable to the target, and would lie at a close angular distance. Unfortunately, no ideal star exists near HD 209458. We must choose between fainter stars in close proximity, or brighter stars at greater angular distances. Since these are exploratory observations, we elected to try both approaches. On 9 Sept we monitored HD 209346 (A2, $V=8.3$, $0.2$ degrees distant). To compensate for this star being 0.7 magnitudes fainter than HD 209458, we added additional comparison images to the observing sequence, viz.: $T_a, C_a, C_b, C_a, C_b, T_b, T_a, ...$. On 16 Sept we monitored HD 210483 (G1,V=7.6, 1.7 degrees distant), following @rich03. For all stars, images consisted of two co-adds of 5 sec exposures. A single 5 sec exposure produced about $2\times 10^8$ electrons of background radiation within our synthetic aperture for photometry (see below), and about $5\times 10^6$ electrons due to HD 209458.
Infrared Photometry
===================
Because the thermal background is intense at this wavelength, weighting the stellar image by the average point-spread-function (PSF) could in principle achieve an optimum signal-to-noise ratio (SNR) for photometry. The stellar PSF at this wavelength exhibits a significant component due to diffraction, as shown in Figure 2. Unfortunately, the PSF also contains a large contribution from seeing, and did not prove to be sufficiently stable to utilize PSF-weighted photometry. We therefore extracted photometric intensitites for the star using simple aperture photometry, with a circular synthetic aperture of radius 20 pixels (1.1 arcsec). The best radius for the synthetic aperture was determined by minimizing the scatter in the photometric intensities. With radii significantly larger than 20 pixels, the greater background noise within the aperture degrades the SNR, and with significantly smaller radii the photometry becomes sensitive to seeing fluctuations.
A significant factor in the quality of our photometry is telescope focus, which changes with temperature and has to be monitored and corrected by the IRTF observer. We found that both the background level and the stellar intensity varied with telescope focus. Figure 3 shows the background level for the HD 209458 transit observations (best night to illustrate this effect). Improvements in telescope focus are accompanied by a decrease in the measured background. We attribute this to less off-axis acceptance of warm radiation from the telescope structure as focus improves. The dependence of background on observed airmass allows us to deduce that $\sim 80\%$ of the background originates from the telescope and warm optics, the remaining $\sim 20\%$ being contributed by the terrestrial atmosphere.
In principle, the stellar photometry should be independent of telescope focus, as long as the synthetic aperture is of sufficient radius to encompass slightly defocused stellar images. In practice, we find that the stellar photometric values change by $\sim 1\%$ when updating telescope focus. We found a weak correlation between intensity and the width of the stellar PSF (correlation coefficient $\sim 0.4$). Broader stellar images - from poorer seeing - tended to yield less intensity from aperture photometry. Although this correlation was dominated by seeing fluctuations, we expect that imperfections in focus have a similar effect. The sense of the correlation - less photometric intensity from broader PSFs - is consistent with not recovering all of the stellar photons within a given synthetic aperture as the image broadens. Using extremely large synthetic apertures is impractical due to background noise. We therefore analyzed our data in blocks between focus updates, measuring HD 209458 relative to the comparison star.
Although thermal IR photometry is daunting in many respects, there are some compensating advantages. These include insensitivity to intra-pixel variations, and the ability to flat-field the detector array nearly simultaneously with the data acquisition. Since the thermal background will be spatially uniform over our $14$ arcsec field of view, we use the thermal background for flat fielding. Within each block defined by focus updates, we compute the median image. Since telescope nods and pointing jitter move the star by significantly more than its FWHM, the median image does not contain the star, and defines an accurate flat field calibration for that data block.
We measure the intensity ratio between HD 209458 and the comparison star, and we do this separately for the ’a’ and ’b’ nod positions in each data block. To increase the precision of the comparison star photometry, we smooth the individual comparison measurements using a 10-point moving average. This increases the effective time scale for our HD 209458 to comparison ratio to $\sim$ 20 minutes. We spline-interpolate the comparison star moving average to the times of the HD 209458 observations, and compute the ratio. Results from this process, for two representative data blocks, are shown in Figure 4. We explored the possibility of improving the precision by decorrelating the stellar intensities against the widths of their PSFs, but we found that the correlation was not sufficiently strong to significantly improve our results.
We compute errors for each photometric measurement of HD 209458. Both the HD 209458 photometry and the comparison star photometry contribute to the total error. The largest source of error is background shot noise, and we compute its magnitude by measuring the per-pixel fluctuations in the flat-fielded background for each frame, using the region immediately adjacent to the photometry aperture. This noise was about two times larger than the theoretical value from the square root of the electron number. Scaling the measured fluctuation as the square root of the number of pixels, we calculate the shot noise in the synthetic aperture due to background. Stellar photon noise and detector read noise are negligible by comparison. We propagate these errors through the relative photometry, and compute the error in the 10-point moving average of the comparison stars. Comparison star and HD 209458 errors are added in quadrature, yielding an average per-point fractional precision of 0.009 per single HD 209458 measurement. (In this paper we quote eclipse depths and errors in units of the stellar intensity, [*not*]{} in magnitudes.) This does not include errors due to the terrestrial atmosphere, which may not cancel perfectly between HD 209458 and the comparison star.
Results and Discussion
======================
Although the photometry exhibits a general decrease in intensity with increasing airmass, there are significant fluctuations on shorter time scales. Measurements of comparison stars are necessary for this reason. Examination of the photometry for the nights of 9 & 16 Sept shows much better results using the closer, albeit fainter, comparison star (9 Sept). This is obvious on Figure 4, where the lower panel (16 Sept) exhibits fluctuations on time scales of $\sim$ 20 minutes that are not similar between HD 209458 and the comparison star. We find that this holds on 16 Sept even when fewer points are used in the moving average of this bright comparison star. Therefore, the data on this night are not useful for eclipse detection. We conclude that the relatively large angular separation between HD 209458 and HD 210483 (1.7 degrees) is the principal factor contributing to this significant degree of atmospheric noise. Physically, this probably results from incoherence of water vapor absorption over the distances separating the lines of sight to these two stars. Note that this conclusion will not necessarily apply to the results of @rich03, since those authors used HD 210483 to perform [*spectroscopy*]{} relative to HD 209458.
In contrast to the 16 Sept results, the data for 9 Sept (e.g., upper panel of Figure 4) show that fluctuations in HD 209458 consistently track those of HD 209346, even though the effective time scale of these data is $\sim$ 20 minutes (10-point moving average of the comparison star). This is also true for the other data blocks on this night. The increase in thermal background per unit airmass was approximately two times lower on 9 Sept than 16 Sept (10% vs. 20%). Nevertheless, we judge that the primary difference in our results is due to the comparison star selection: a nearby, albeit faint, comparison star is more useful than a brighter but more distant one for photometry in the L-band. Certainly this is true in conditions that are less than optimum.
Figure 5 (upper panel) plots the 331 individual observations from 9 Sept versus the planet’s orbital phase, using the ephemeris from @knutson_b. These data are normalized to an average of unity. The lower panel averages the data in bins of width 0.002 in phase, and adds error bars calculated as discussed above. The best-fit eclipse depth ($0.0013\pm0.0011$) does not have sufficient precision for detection, but our results are similar to a result recently reported for TrES-1 by @knutson_a, who used a spectroscopic technique at 3.8 $\mu$m.
The reduced chi-squared of the fit in the lower panel of Figure 5 is 1.96, consistent with a per-point scatter of 0.014 (upper panel), about 50% larger than the errors from background shot noise. Additional error can arise from atmospheric noise that is not common to both stars on the spatial and temporal scales of these observations. This ’red noise’ (amplitude $\sim$ 0.01) can be reduced by using a broader optical bandwidth. New IR cameras such as at the IRTF can utilize the full L’ band without saturating. This will increase the SNR by a factor of $\sim 3.5$ per image (SNR proportional to square root of optical bandwidth). Having greater SNR per frame should increase the precision of fainter comparison stars; for example, HD 209346 should improve from 0.021 to 0.006 precision in 10 seconds of integration. A more significant effect of a broader optical bandwidth is that it will decrease the temporal bandwidth of the red noise, by an order of magnitude. This follows because the time to define the target-to-comparison-star ratio to a given precision shortens as the square of the SNR per image, and we would not need to utilize a 10-point moving average for the comparison star.
Atmospheric noise power often increases as the inverse of the sampling frequency (‘1/f’ noise). We have calculated the improvement expected in both the background shot noise and the atmospheric 1/f noise using the full L’ optical bandwidth. This calculation assumes that any high frequency contribution from the terrestrial atmosphere - similar to scintillation - does not dominate on $\sim 10$ to $30$ second time scales at this wavelength. The result indicates that our 9 Sept final errors will decrease by a factor of two. A secondary eclipse of depth 0.002 in HD 209458 would be detected to $3.6\sigma$ significance. Moreover, the greater planet-to-star contrast of the recently discovered HD 189733 system [@bouchy], coupled with its brighter apparent magnitude, will lead to even more favorable detectability. We conclude that ground-based detection of photons from extrasolar planets is possible at 3.8 $\mu$m.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank our IRTF support scientist, Bobby Bus, for his expert help in making the observations, and we are grateful to John Rayner for assistance with NSFCam and to the IRTF telescope operators for their assistance.
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\[lastpage\]
[^1]: E-mail: ddeming@pop600.gsfc.nasa.gov
[^2]: E-mail: richardsonlj@stars.gsfc.nasa.gov
[^3]: E-mail:jharring@physics.ucf.edu
|
MIT-CTP 4019\
Imperial/TP/2009/JS/01
[A Rotating Holographic Superconductor]{}\
[<span style="font-variant:small-caps;">Julian Sonner</span>]{}\
\
[*and*]{}\
[*Trinity College, University of Cambridge, Cambridge CB2 1TQ, UK*]{}\
[js499@cam.ac.uk]{}\
[**Abstract**]{}
In this paper we initiate the study of SSB in $3+1$ dimensional rotating, charged, asymptotically AdS black holes. The theory living on their boundary, $\mathbb{R}\times S^2$, has the interpretation of a $2+1$ dimensional rotating holographic superconductor. We study the appearance of a marginal mode of the condensate as the temperature is decreased. We find that the transition temperature depends on the rotation. At temperatures just below $T_c$, the transition temperature at zero rotation, there exists a critical value of the rotation, which destroys the superconducting order. This behaviour is analogous to the emergence of a critical applied magnetic field and we show that the superconductor in fact produces the expected London field in the planar limit.
March 2009
Introduction
============
Over the past year or so, there has been considerable activity in the field of AdS/CMT, which attempts to bring a phenomenological approach to the AdS/CFT correspondence to bear on problems in condensed-matter physics [@Hartnoll:2007ip; @Hartnoll:2007ih; @Balasubramanian:2008dm; @Son:2008ye; @Adams:2008wt; @KeskiVakkuri:2008eb; @Gubser:2008wz; @Gubser:2008wv]. The scope of this program is broad, dealing with such diverse problems as the Nernst effect at a superfluid-insulator transition, the quantum Hall effect and cold atoms in the unitarity limit.
In [@Gubser:2008px], Gubser suggested that spontaneous $U(1)$ symmetry breaking by bulk black holes - specifically Reissner-Nordström-AdS black holes - can be used to construct gravitational duals of the transition from normal state to superconducting state in the (not further specificed) boundary theory. This can be done by studying the dynamics of a minimally coupled complex scalar field with Lagrangian density = -F\^2- |[D]{}|\^2 - m\_\^2 ||\^2 , where ${\cal D}_\mu\Psi = (\partial_\mu - ie A_\mu)\Psi$ and the scalar field mass respects the Breitenlohner-Freedman (BF) bound [@Breitenlohner:1982jf] m\_\^2 L\^2 -9/4. In the remainder of this paper, we only consider scalar masses at or above the conformal value of $m^2_\Psi=-2$. In the first instance one works in the approximation where the scalar does not backreact on the geometry. This approximation to the full system is however sufficient for an [*exact*]{} determination of the transition temperature of the holographic superconductor.
Gubser showed numerically that for sufficiently large charge $e$ of the condensate the scalar field on the black hole background allows a marginal mode. This is indicative of an instability towards a ‘hairy’ black hole solution with a finite charged condensate outside the horizon. The scalar hair breaks the $U(1)$ gauge symmetry in the bulk space time. Hartnoll, Herzog and Horowitz [@Hartnoll:2008vx] brought this idea to fruition by numerically constructing the state with broken symmetry. They were able to show that the electrical resistance in the dual field theory indeed drops to zero in the broken phase by studying the fluctuations of the Maxwell field around the state of broken symmetry.
In subsequent works, these authors, and others [@Hartnoll:2008vx; @Albash:2008eh], verified a number of other physical properties of these backgrounds and accumulated evidence that these are gravitationals dual of a type II superconductor. This is despite the fact that a [*local*]{} bulk $U(1)$ symmetry is a priori associated to a [*global*]{} symmetry of the boundary theory, and hence one would expect to find the physics of a charged superfluid [@Herzog:2008he]. However one may weakly gauge the boundary $U(1)$ and in this regime the material is described by the London theory [@Hartnoll:2008vx] of superconductivity.
In this paper we aim to put to explore another feature of superconductors, the [*London moment*]{}, and by doing so, extend the present framework of holographic superconductors to the case of rotating black holes in the bulk. At first sight, this also seems to be a hopeless task, since the holographic superconductor does not have a dynamical photon. However, the work of [@Hartnoll:2008vx] showed that it nevertheless produces the screening currents necessary [*e.g.*]{} for the Meissner effect. Thus we may also be optimistic about being able to observe the London moment.
At zero angular momentum, the system studied here reduces to a subset of those considered in [@Gubser:2008px], with spherical horizon, where the transition to a superconducting state is already known to occur. We thus expect the marginal mode to exist for small values of the rotation parameter. An interesting question is whether this mode always persists for larger values. Physical intuition tells us that it should not: we know from previous studies [@Albash:2008eh] that a holographic superconductor exhibits a critical magnetic field, above which it is energetically favourable to be in the normal phase. We also know that a rotating superconductor dynamically generates a magnetic field, the so-called [*London field*]{} [@London]. Intuitively this arises because the rotation induces a lag between charge carriers close to the surface and the charged superfluid in the bulk of the superconductor. Thus we should expect a given marginal mode at zero rotation to disappear at some critical rotation. Or, in other words, we expect the critical temperature to drop as the angular momentum of the dual black hole is raised. We shall demonstrate this behaviour.
The paper is organised as follows. In section 2 we introduce the dual gravity background, the Kerr-Newman-AdS black hole and explain the influence of the bulk rotation on the boundary theory. In section 3 we study the angular and radial behaviour of marginal modes of a charged complex scalar field in this background and construct the phase diagram of the holographic superconductor. For a tachyonic scalar in the planar limit, [*i.e.*]{} close to the poles, we compute the London field explicitly. Section 4 is devoted to a discussion and appendices A and B contain details on the separation of the wave equation as well as the numerical methods employed in this paper.
Rotation and AdS/CMT
====================
In this section we introduce the gravitational background, which we argue is dual to a two-dimensional material living on the surface of a sphere, exhibiting superconductivity at sufficiently low temperature. The study of rotation in the AdS/CFT context was initiated in [@Hawking:1998kw] (see also [@Berman:1999mh]).
AdS$_4$ backgrounds with Angular Momentum
-----------------------------------------
The desire to introduce rotation into the AdS/CMT story leads us to consider the four-dimensional Kerr-Newman-AdS solution, orginally found by Carter [@Carter:1968ks], who was interested in finding space times in which the Hamilton-Jacobi equation would separate. In Boyer-Lindquist coordinates, the metric is $$\dd s^2= - \frac{\Delta_r}{\rho^2}\left[ \dd t - \frac{a}{\Xi}\sin^2\theta \dd \phi \right]^2 + \frac{\rho^2\dd r^2}{\Delta_r} + \frac{\rho^2 \dd \theta^2}{\Delta_\theta} + \frac{\Delta_\theta\sin^2\theta}{\rho^2}\left[ a\dd t - \frac{r^2 + a^2}{\Xi}\dd \phi \right]^2$$ with gauge field field A=-(t-), and \_r := &(r\^2 + a\^2)(1 + r\^2 L\^[-2]{} ) - 2 M r + q\_e\^2 ,\_&:= 1 - a\^2 L\^[-2]{}\^2\
\^2 := &r\^2 + a\^2\^2&:=1 - a\^2 L\^[-2]{} There is some confusion in the literature as to how exactly the parameters $a$ and $M$ are related to the physical angular momentum and mass-energy of the black hole, respectively. This confusion is cleared up in the work [@Gibbons:2004ai] (see also [@Papadimitriou:2005ii]), by careful thermodynamic considerations, and we adopt their definitions E=,J= for the energy $E$ and angular momentum $J$ of the background. Note that both $E$ and $J$ diverge as $aL^{-1}$ approaches unity. However, $J$ is always strictly bounded above by $EL$, as expected for a rotating black hole. Finally, we quote the result for the Hawking temperature of the black hole [@Caldarelli:1999xj], \[eq:thawking\] T\_H = , which, in the usual way, is identified with the temperature of the dual field theory living on the conformal boundary of the AdS black hole spacetime. The boundary has topology $\mathbb{R}\times S^2$, so we are considering a two-dimensional superconductor living on the surface of a sphere. The quantity $r_+$ is the horizon radius, defined to be the largest real root of the equation $\Delta_r=0$.
Rotation and the Boundary Theory
--------------------------------
Massive rotating bodies exhibit the [*frame-dragging effect*]{}: an inertial reference frame outside the body is set into rotational motion. This effect diminishes as the distance to the rotating body increases. For the original Kerr black hole, which is asymptotically flat, this effect vanishes at infinity. However, for the Kerr-AdS family, it does not [@Gibbons:2004ai]. Its boundary metric is in the conformal class of the Einstein Static Universe s\^2 = -t\^2 + L\^2 ( \^2 + \^2\^2 ) where the new angular coordinates satisfy $$\phi = \hat\phi - \frac{a}{L^2}t\,, \qquad\tan\theta = \sqrt{\Xi}\tan\hat\theta\,.$$ From the first of these see that the angular momentum of the boundary theory is \_= . We learn that the local speed of rotation at the [*equator*]{} of the boundary $S^2$ of radius $L$ reaches the speed of light when $a=L$. In fact the gravitational background becomes singular at that point. We therefore restrict to $a<L$ in this paper.
The meaning of rotation in the field theory becomes apparent if we compute for example the free partition function, [*i.e.*]{} the free energy of the theory at zero coupling. It is given by the expression F=-=T\_[bdry]{} \_i \_[=0]{}\^\_[m=-]{}\^\_i (1-\_i e\^[-( - m \_)]{} ), where $i$ labels the different (particle) species, $\eta_i=\pm 1$ for bosons (fermions) and $\beta=1/T_{\rm bdry}$. The allowed frequencies $\omega$ follow from the corresponding free wave equations. The summation is over the angular momentum quantum numbers of the fields $(\ell,m)$ as indicated. The quantity $\Omega_\infty$ enters as a chemical potential constraining the angular momentum.
Superconducting Instablity
==========================
We now turn to the main objective of this work, which is to identify an instability under perturbations of a charged scalar field propagating on the background. We rely on the separation properties of this equation, but relegate the details of the separation procedure to appendix A. It is not necessary to follow these in order to understand this paper, but we have included them for the interested reader.
Radial Equation
---------------
As is well known from a classical result due to Carter [@Carter:1968ks], the Klein-Gordon equation is separable on the background of the Kerr-AdS black hole in $3+1$ dimensions. This result can be extended to the case of a complex (charged) scalar field on the Kerr-Newman-AdS background above. We assume that $\Psi(t,r,\theta,\phi)\sim e^{-i\omega t - i m\phi}R(r) S(\theta)$, and look for a marginal mode, for which $m=\omega=0$. As in [@Gubser:2008px; @Hartnoll:2008vx], it is consistent to seek real solutions $R(r)$ and $S(\theta)$. However, since the complex scalar $\Psi$ is not a gauge-invariant quantity, this is essentially a choice of gauge, the details of which are explained in appendix A. For later convenience, let us define the [*horizon function*]{} r\^2h(r)=\_r(r). Then we obtain the radial equation \[eq:radialI\] (\_r r\^2h(r) \_r + ( 1-r )\^2 - (a\^2+r\^2) m\_\^2 - )R(r) &=& 0 , where we have set the horizon radius to unity[^1], and $\lambda$ is the separation constant between $S(\theta)$ and $R(r)$. In the Schwarzschild-AdS limit ($a\rightarrow 0$) it reduces to the familiar value $\ell(\ell+1)$, $\ell\in\mathbb{Z}$ being the principal eigenvalue of the spherical harmonics. Sometimes we shall find it useful to label the values of $\lambda$ by the corresponding integer $\ell$ to which they reduce. While there exists no analytical expression for $\lambda$, it is possible to determine the numerical value of this separation constant by treating the $\theta$-equation as an eigenvalue problem for $\lambda$.
To gain a more intuitive understanding of the physics involved in finding the marginal mode, let us manipulate expression [(\[eq:radialI\])]{} into the form of a Schrödinger equation. Then the question of finding the gravitational mode is re-expressed in terms of finding a marginal bound state of the corresponding quantum-mechanical problem. To this end, define the [*tortoise coordinate*]{} \[eq:tortoise\] r\_\* = . Outside the horizon at $r=1$, $r_* $ is real and takes values[^2] $r_* \in (-\infty,0)$. Then the function $Z(r) = r R(r)$ satisfies the Schrödinger equation of a zero-energy particle in a potential: \[eq:schrod\] - V Z(r\_\*)=0. The potential is given implicitly by \[eq:potential\] V(r,) = h (r), where the prime indicates differentiation with respect to the original radial coordinate $r$. Since we cannot invert $r_*(r)$ analytically to get the potential as a function of $r$, we cannot give an explicit functional form of the potential as a function of the tortoise coordinate $r_*$. It is however easy to obtain plots of the potential, numerically or otherwise, which are sufficient to develop the intuition we wish to achieve. Before proceeding to obtain these plots, we must first return to the problem of the separation constant and the angular equation. In the process we will gain insight into the nature of the condensate.
Angular Equation and Localization
---------------------------------
Without further ado, here is the angular equation for our marginal mode: \[eq:angularI\] S() =0 Again, a detailed derviation is given in appendix A. We can gain some understanding of this equation by considering two limits. Firstly, in the non-rotating limit, this equation reduces to the associated Legendre equation, defining the $\theta$-dependent part of the spherical harmonics $Y_{\ell m} (\theta,\phi)$. In this limit, we learn that $\lambda = \ell(\ell + 1)$ and it is a useful check on our computations to note that our results reduce to these integer values in the appropriate limit.
![[ Lowest angular AdS spheroidal harmonics. Regions with high density of the condensate are shown lighter and regions of low density are shown darker. Both panels correspond to a rotation parameter of $\alpha=0.9$. On the left panel, we have chosen a positive mass-squared term, while on the right the mass-squared term is negative.]{}[]{data-label="fig:angular"}](angular1 "fig:"){width="40.00000%"}![[ Lowest angular AdS spheroidal harmonics. Regions with high density of the condensate are shown lighter and regions of low density are shown darker. Both panels correspond to a rotation parameter of $\alpha=0.9$. On the left panel, we have chosen a positive mass-squared term, while on the right the mass-squared term is negative.]{}[]{data-label="fig:angular"}](angular2 "fig:"){width="40.00000%"}
In the flat-space limit, $L\rightarrow \infty$, the angular equation reduces to the [*spheroidal*]{} equation, first obtained in the black-hole context by Teukolsky [@Teukolsky:1972my]. In this limit, no analytical expressions are known for $\lambda$, but its values are tabulated. Again, it is a confidence-inspiring check on our procedures that our values for $\lambda$ agree with those of the spheroidal harmonics in the appropriate limit. For the sake of terminology, let us refer to the solutions of the angular equation [(\[eq:angularI\])]{} as [*AdS spheroidal harmonics*]{}. The deviation of the AdS spheroidal harmonics from being spherical harmonics is measured by the ratio of specific angular momentum of the black hole to the AdS length, so we introduce the deformation parameter $\alpha=\frac{a}{L}$.
The eigenvalues of the deformed spheroidal equation [(\[eq:angularI\])]{} are then labelled $\lambda(\ell,\alpha, a^2 m_\Psi^2)$. Solutions to equation [(\[eq:angularI\])]{}, much like the associated Legendre functions, have definite parity, which allows us to develop a simple, iterative, shooting technique to solve both for the eigenfunction and the eigenvalue to high accuracy. The details are relegated to appendix B. There we also list a range of eigenvalues $\lambda$, corresponding to various values of the deformation parameter $\alpha$.
An interesting feature of the AdS spheroidal harmonics is that, unlike their French cousins, the Legendre functions, the lowest mode is not constant. Rather it has a non-trivial profile over the sphere, controlled by the deformation parameter $\alpha$ causing even the lowest mode to be localized either near the poles or near the equator. They are localized near the poles if $m^2_\Psi<0$ and near the equator if $m^2_\Psi>0$. This is illustrated in figure \[fig:angular\], which shows density plots of the lowest mode on the unit $S^2$. For the special case of $m_\Psi=0$ they are constant.
We now have all the ingredients needed to obtain the critical behaviour of the rotating superconductor. We start by exhibiting marginal modes of the charged scalar field.
Marginal Modes
--------------
### Qualitative Considerations
Evidently, equation [(\[eq:schrod\])]{} cannot have bound state solutions, unless the potential $V(r_*)$ develops a negative well in a certain range of parameters. Therefore we will expect the lowest marginal mode to lie in the sector with $\ell=0$. Figure \[fig:figure1\]a illustrates this point. The occurrence of a negative potential well is closely related to the fact, pointed out by Gubser [@Gubser:2008px], that the effective mass of the scalar field m\_[eff]{}\^2:=m\_\^2 - ( -1 )\^2 gets a contribution due to the coupling to the background gauge potential that can make this sufficiently negative for a sufficiently long interval outside the horizon.
The criterion for instabilty can be illustrated by considering the toy model of the semi-infinite square well of depth $U$ and width $w$. The system will be stable if the potential admits negative-energy bound-state solutions and will become unstable once the last of these boundstates (i.e. the ground state) acquires positive energy and becomes non-normalizable. Elementary quantum mechanics tells us that the condition that there exists at least one bound state is $$w\frac{\sqrt{2mU}}{\hbar} \geq \frac{\pi}{2}\,,$$ where we have briefly reinstated units of $\hbar$. Thus if the potential well is too shallow or too narrow, no bound-state can exist. In particular, for the toy model, a bound state exists for $\nu\leq 4 w$, where $\nu$ is the de-Broglie wavelength of a particle of energy $U$. In searching for the zero-energy marginal bound state, we precisely capture the moment where the ground state exits the Hilbert space of the theory. We interpret this as the transition point for the condensation of the order parameter.
a)![a) Plot of effective potential barrier around Kerr-Newman-AdS black hole for charged scalar waves with $m^2_\Psi=4$. In the chosen coordinates, the horizon is at $r_*=-\infty$ and asymptotic infinity is on the right at $r_*=0$. b) Square well toy model. The depth of the well is $U$ and its width is $w$. AdS boundary conditions are equivalent to putting a reflecting barrier at the origin. []{data-label="fig:figure1"}](a2pot "fig:"){width="50.00000%"}b)![a) Plot of effective potential barrier around Kerr-Newman-AdS black hole for charged scalar waves with $m^2_\Psi=4$. In the chosen coordinates, the horizon is at $r_*=-\infty$ and asymptotic infinity is on the right at $r_*=0$. b) Square well toy model. The depth of the well is $U$ and its width is $w$. AdS boundary conditions are equivalent to putting a reflecting barrier at the origin. []{data-label="fig:figure1"}](SquareWell "fig:"){width="30.00000%"}
With this simple picture in mind, let us return to the physics of the holographic superconductor. Keeping the system at at a [*fixed*]{} temperature $T$ below its transition temperature for zero rotation, $T_0$, the potential well becomes more and more shallow as the specific angular momentum $a$ is increased. In figure \[fig:figure2\] this can be seen as the decrease in depth of the potential well, as one increases the rotation $a$ in units of the AdS length. We see that the well becomes both less deep and more narrow as the rotation is increased. Hence the expectation is that the zero-energy mode will cease to exist at some point.
This section has illustrated the physical mechanism behind the suppression of superconductivity with rotation, but because of the complicated shape of the potential [(\[eq:potential\])]{}, it is necessary to analyze the marginal mode in detail, in order to gain a quantitative understanding of the phenomenon. The issue of stability of AdS black holes under scalar field perturbations is an interesting subject in its own right. For recent mathematical results on (real) scalar fields in AdS black hole backgrounds, see [@Holzegel:2009ye].
![[Plot of effective potential barrier for the $\lambda$ value corresponding to $\ell=0$ and $eL=10, m^2_\Psi=4$. The potential is given as a function of the deformation parameter $\alpha=aL^{-1}$ at [*constant*]{} temperature $T_0$. We clearly see that the potential well disappears as one increases the rotation.]{}[]{data-label="fig:figure2"}](critical1){width="50.00000%"}
### Numerical Solutions
We will now embark on a study of the bevhaviour of the marginal mode as a function of the parameter $\alpha$. It should come as no surprise to the reader that the only successful quantitative approach approach to solving the radial equation is the numerical one. Already in the much simpler case of RNAdS black holes, no analytical solutions are known for the marginal mode, much less for the broken phase.
In order to facilitate the numerical analysis, let us transform to the variable $z=r^{-1}$. Then the question of the marginal mode is transformed into a boundary value problem on the interval $(0,1)$. We wish to solve the equation \[eq:zradial\] ( h(z) ) + R(z)=0 subject to certain boundary conditions. We see that both the horizon at $z=1$ and asymptotic infinity at $z=0$ are regular singular points of this equation. Near the AdS boundary at $z=0$, the solutions take the familiar form R(z)\~\_1 z\^[\_+]{} +\_2 z\^[\_-]{}, where $\Delta_\pm$ is the conformal weight of the dual operator, given by the root of the indicial equation (-3)=m\_\^2 L\^2. Unitarity requires that $\Delta\geq\frac{1}{2}$. Depending on the bulk scalar mass, either or both of the operators \_1 &=& \_1\
\_2&=& \_2 may condense. We have chosen the same normalization for the bulk-boundary coupling as [@Hartnoll:2008vx]. The indicial equation at the horizon tells us that the solution there behaves as R(z) \~R\_[(0)]{} + R\_[(1)]{}(z-1) and regularity demands that $R_{(1)}=0.$
In order to solve [(\[eq:zradial\])]{}, we set either of $\Psi_i=0$, corresponding to either $\langle {\cal O}_1\rangle$ or $\langle {\cal O}_2 \rangle$ condensing. Thus we develop the solution in a Frobenius series around infinity R(z)= z\^( \_[i]{}+\_[n]{}a\_nz\^n )i=12 This can be done to very high order using a symbolic algebra package to solve recursively for the coefficients. Similarly the expansion around the horizon takes the form R(z) = 1 + \_[n]{}b\_n(z-1)\^n, [*i.e.*]{} we have chosen to normalize $R_{(0)}=1$. As remarked in [@Gubser:2008px] it would be more proper to treat $R_{(0)}$ as a small parameter that allows us to justify neglecting the backreaction, but in the end this is a mere rescaling of the mode.
We use numerical integration to match the two series solutions at some intermediate point by adjusting the coefficient $\Psi_i$. This shooting technique can be implemented[^3] very efficiently using a numerical root finder and looping over initial conditions.
![Examples of lowest ([*i.e.*]{} no nodes) marginal modes for condensation of ${\cal O}_1$. We take $eL=8$, $m^2L^2=4$ and $\alpha=0,0.7,0.9$ for the solid, dashed and dot-dashed modes respectively. In the Schrödinger representation these are the radial wavefunctions of the zero-energy bound states.[]{data-label="fig:marginalmodes"}](MarginalModes1){width="40.00000%"}
However, after matching the value of the function, there is no guarantee that the first derivatives will also match at this point. This is because there is only a marginal mode at certain specific values of the background parameters. Evaluating $T_H$, given in equation [(\[eq:thawking\])]{}, at the critical values for which the mode first appears gives us the critical temperature. An illustration of several marginal modes is given in figure \[fig:marginalmodes\].
The family of backgrounds we are interested in depends on the parameters $L,a,q_e, r_+$ and the scalar field has mass $m_\Psi$ and charge $e$. We have chosen units such that $r_+=1$ and analyse the equation for fixed values of $e$ and $m_\Psi$. This results in a curve of critical points in the $(\alpha,q_e)$-plane. For concreteness we fix $q_e=1$ to extract a value for $T_c$. Thus we are looking at projections of the full phase diagram. It would be interesting to extend this analysis to explore the entire phase diagram.
Phase Diagram
-------------
In order to find the critical temperature at a given value of background parameters, we follow the example of [@Denef:2009tp] and focus on the condensation of ${\cal O}_1$ for different parameter values. This is convenient because it always corresponds to a normalizable solution, as long as $m^2_\Psi \geq -\frac{9}{4}$. Quantitatively, this only gives a lower bound on the critical temperature, since in the mass range where both ${\cal O}_1$ and $ {\cal O}_2$ are normalizable, ${\cal O}_1$ may in fact condense first. However, in this paper we are not so much concerned with the actual value of $T_c$, but rather its behaviour as a function of $aL^{-1}$.
a\) ![[Critical temperature for the condensation of ${\cal O}_1 $ at $q_e=1$ as a function of $\alpha=a/L$. Left panel for $m_\Psi^2 L^2=-2$ and right panel for $m_\Psi^2 L^2 = 4$. Superconductivity is increasingly suppressed as the rotation is increased.]{}[]{data-label="fig:phasediagram"}](Figure3a "fig:"){width="45.00000%"}b)![[Critical temperature for the condensation of ${\cal O}_1 $ at $q_e=1$ as a function of $\alpha=a/L$. Left panel for $m_\Psi^2 L^2=-2$ and right panel for $m_\Psi^2 L^2 = 4$. Superconductivity is increasingly suppressed as the rotation is increased.]{}[]{data-label="fig:phasediagram"}](Tcm4 "fig:"){width="45.00000%"}
We find that in all cases $T_c$ decreases as the angular momentum of the superconductor is increased. This means that at a temperature slightly below $T_0$, the critical temperature at zero rotation, there always exists a critical value of $aL^{-1}$ above which the material is forced back into its normal phase.
However, for some choices of parameters, such as in figure \[fig:phasediagram\]a, one can choose the temperature low enough, that no amount of rotation will force the material back into its normal state. Recall that for $aL^{-1}=1$ the rotation speed at the equator equals the speed of light. However, as we have seen in figure \[fig:angular\], the condensate may be localised away from the equator, close to the poles, where the local speed of rotation is arbitrarily small. Thus, it is plausible, that in these cases, enough of the condensate recedes far enough away from the areas of the $S^2$, where the rotation would destroy the superconducting order.
Planar Limit and London Moment
------------------------------
Let us consider what happens if we concentrate on a small region around the north pole of the boundary $S^2$. To this end, we introduce the coordinate u = L, and consider the regime $u\ll L$, with $\alpha=\frac{a}{L}$ held constant. In other words we zoom in on the physics close to the north pole on scales much smaller than the AdS length. In this regime the curvature of the sphere is negligible and we are effectively dealing with a flat horizon rotating with angular velocity $\Omega = \frac{a}{L^2}$. Note that the local speed of rotation is of order ${\cal O}\left( uL^{-1} \right)$, small compared to the speed of light. The angular equation [(\[eq:angularI\])]{} simplifies to give S”(u) + S’(u) + S(u) = -S(u). Compare this with the equation of a planar holographic superconductor immersed in a magnetic field, viz. S”(u) + S’(u) - ()\^2S(u) = -S(u), for a different separation constant of mass dimension two, as defined in [@Hartnoll:2008vx]. This equation was studied in [@Hartnoll:2008vx; @Albash:2008eh], where it was shown to lead to superconducting droplets that are exponentially confined near the origin (in our case the north pole). This behaviour is consistent with our findings for operators that are dual to tachyonic bulk scalars. We conclude that in these cases, the rotating superconductor is equivalent to a static superconductor, immersed in a magnetic field of magnitude B\_L =\_, where $\Omega_\infty$ is the angular momentum of the boundary theory and $m=|m_\Psi|$. Again, this field is weak, of order ${\cal O}\left( L^{-1} \right)$, in the limit we are considering. Furthermore, in the nonrelativistic regime $a^2 L^{-2}\ll 1$, it reduces to the relation between angular momentum and magnetic field first obtained by F. London [@London], by considering his phenomenlogical theory for a rotating superconductor. The work of [@Basu:2008st] studied the case where one adds linear momentum to the system. It would be interesting to see if their results can be seen in the present framework by zooming in on the equator of the solutions.
Discussion
==========
In this paper we have demonstrated that finite-temperature gravitational backgrounds with non-zero angular momentum exhibit SSB much like their static limits. Furthermore we have seen that the additional dependence on rotation causes phenomena akin to those found in studies of superconductors in magnetic fields.
An important difference to this case is that there always exists a critical magnetic field $B_{c}$, even at very low temperature, above which the superconducting order is destroyed. We have seen that this is not always the case for the rotating superconductor on a sphere. This can be explained by noting that the local speed of rotation decreases as one moves away from the equator, and thus there can be regions of material, where the superconducting order is present, even when the equator moves at the speed of light. Near the poles we have seen that the field produced is precisely that predicted by F. London on the basis of his phenomenological theory of superconductors. The field is parametrically weak, of order ${\cal O}\left( L^{-1} \right)$, so at low temperatures, it is not strong enough to destroy the superconducting order. This fact is also borne out in the full numerics, as exemplified in figure 5a.
The London field is often written in terms of the dressed mass $m^*$ of the condensate, rather than the bare mass $m$. In our case, we find that $m^*=\frac{m}{\sqrt{\Xi}}$. Thus the dressed mass corresponds to the relativistic mass of a particle of mass $m$ at the equator of the boundary sphere.
We have seen that even the lowest angular eigenfunction localizes the condensate in a droplet close to the equator for positive mass squared of the dual scalar field, and in a ring around the equator for negative mass squared. It is interesting that the localization behaviour of the condensate is qualitatively different for the case of positive bulk mass compared to negative mass (squared) and only in the latter case have we offered an explanation in terms of the material’s London moment. It would clearly be an interesting question to understand what the different localization properties mean from the point of view of the boundary superconductor. It would appear that in one case the instablitity is towards forming a vortex anti-vortex pair localized on the antipodes, while in the other case we have a pair of lumps of superconducting material on the two poles.
Our analysis was done to first order in the small condensate. The localization behaviour observed in this paper would be expected in the early stages of vortex formation. At the level of approximation of this paper, the model does not allow for dynamically generated screening currents. However, [@Hartnoll:2008vx; @Albash:2008eh] argue that going to higher order in the condensate will generate the screening currents necessary to confine the normal phase in vortex cores whose centres will allow magnetic flux to penetrate. Thus our results confirm the expectation that the rotating holographic superconductor has a London moment generating a London field.
It would be very interesting to extend the computations presented in this work to include the effects of backreaction of the scalar, both on the Maxwell field and on the gravitational background in order to understand the formation of vortices from the localized droplets in our system and to directly see the London field.
Independently of any holographic interpretation, the results of this paper indicate the existence of a new branch of stationary black hole solutions. Here we have found a marginal mode of the stationary asymptotically AdS black hole, which preserves axial symmetry. Just like in the static case, where the existence of a static marginal mode preserving the full spherical symmetry indicates the existence of a new branch of charged static solutions, here we expect the existence of a new branch of charged stationary solutions. Clearly these last two issues are numerically more involved and will require more sophisticated methods than the ones discussed in this paper.
[**Acknowledgements**]{}
It is a pleasure to thank Eduard Antonyan, John McGreevy, Sean Hartnoll, Gustav Holzegel, David Tong, Arttu Rajantie and especially Toby Wiseman for helpful discussions and comments. I gratefully acknowledge generous support by EPSRC and Trinity College, Cambridge. The work presented here was initiated at MIT while I was supported by the NSF under grant PHY-0600465 and the DOE under collaborative research agreement DEFG02-05ER41360.
Separation of Variables
=======================
While it can be shown by direct computation that the charged scalar field allows separable solutions, this procedure is as tedious as it is inelegant. We shall instead use the dyadic index formulation of Newman and Penrose [@Newman:1961qr] which is well suited to the present problem. A suitable Newman-Penrose tetrad is via the differential operators \[eq:npframe\] D=\^\_&=& ( (r\^2 + a\^2)\_t + \_r\_r + a \_)\
=n\^\_&=& ( (r\^2 + a\^2)\_t - \_r\_r + a\_)\
=m\^\_&=& ( \_t + \_+ \_)\
|=|[m]{}\^\_&=& (- \_t + \_- \_), where all quantities are as defined in the main paper and in addition |=r + ia As ever, the metric takes the form s\^2 = 2 ( m |[m]{} - n ) in terms of the one-forms dual to [(\[eq:npframe\])]{}. A simple representation of the non-vanishing spin coefficients is given by $$\begin{aligned}
\pi &=-\bar\delta \ln\bar\rho^* \qquad\qquad\qquad\\
\rho &= D \ln\bar\rho^*\qquad\qquad\qquad\qquad~\,\tau = \delta\ln\bar\rho^*\,,\qquad\qquad\qquad\qquad\nonumber\\
\mu &=-\Delta \ln\bar\rho^*\qquad \qquad\qquad\quad~~\beta = -\frac{1}{4}\delta\ln\left(a^2\sin^2\theta\Delta_\theta \right) \nonumber\\
\alpha &= \frac{1}{2}\bar\delta \ln\left(\frac{a\sin\theta\sqrt{\Delta_\theta}\Delta_r }{(\bar\rho^*)^2} \right)\,\quad \,\gamma= \frac{1}{2}\Delta\ln\left(\frac{a\sin\theta\sqrt{\Delta_\theta}\Delta_r }{(\bar\rho^*)^2} \right)\,,\end{aligned}$$ where, in keeping with the notation introduced by Newman and Penrose, we also use the letter $\rho$ for one of the complex spin coefficients. Finally, the charged scalar field obeys the equation \^[(a)(b)]{}(\_[(a)]{}+ieA\_[(a)]{})(\_[(b)]{}+ieA\_[(b)]{})=m\_\^2where we use Chandrasekhar’s notation of putting indices that refer to the NP tetrad in brackets. Let us introduce the family of differential operators $$\begin{aligned}
{\cal D}_n &= \partial_r - \frac{iK}{\Delta_r} + n \partial_r\ln\Delta_r\nonumber\\
{\cal D}^\dagger_n &=\partial_r + \frac{iK}{\Delta_r} + n \partial_r\ln\Delta_r\nonumber\\
{\cal L}^\dag_n &= \partial_\theta + \frac{H}{\Delta_\theta} + n\partial_\theta\ln\sin\theta\sqrt{\Delta_\theta}\nonumber\\
{\cal L}_n &= \partial_\theta - \frac{H}{\Delta_\theta} + n\partial_\theta\ln\sin\theta\sqrt{\Delta_\theta}\end{aligned}$$ where acting on functions of the form (t,r,,)=e\^[-i(t + m )]{}(r,), the constants $K$ and $H$ are K=(r\^2 + a\^2)+ a m,=(a + ). It is then straightforward to show in a few steps that the charged scalar equation simplifies to the expression $$\begin{aligned}
\left\{\frac{\Delta_r}{2\rho^2}\left[\left({\cal D}_1 - \frac{ieq_er}{\Delta_r} \right)\left({\cal D}_0^\dag + \frac{eiq_er}{\Delta_r} \right) + \left( {\cal D}_1^\dag + \frac{ieq_e r}{\Delta_r} \right) \left( {\cal D}_0 - \frac{eiq_e r}{\Delta_r} \right) \right]\right.\nonumber\\
\left.+\frac{\sqrt{\Delta_\theta}}{2\rho^2}\left( {\cal L}_1^\dag\sqrt{\Delta_\theta}{\cal L}_0 + {\cal L}_1\sqrt{\Delta_\theta}{\cal L}_0^\dag \right)\right\}\Phi=m_\Psi^2\Phi\end{aligned}$$ Now, notice that the operators ${\cal D}_{0,1}$ and their hermitian conjugates are purely radial, while the operators ${\cal L}_{0,1}$ and their hermitian conjugates are purely angular. Thus the equation is separable upon multiplying by $\rho^2$, resulting in the two equations R(r) = 0 and ()=0. Substituting the definitions of the differential operators ${\cal L}_{0,1}$ and ${\cal D}_{0,1}$ results in the radial and angular equations quoted in the main text modulo a gauge choice, on which we now elaborate.
Details on Numerical Methods
============================
This appendix contains details on the gauge choice used in equation [(\[eq:radialI\])]{} and gives details on the numerical algorithms used to compute the separation constants used in the bulk of the paper.
Gauge Choice and Regular Solutions
----------------------------------
The potential contribution to the radial equation contains an effective mass-squared term proportional to the square of the gauge field. This term seemingly blows up at the horizon unless the gauge field $A$ goes to zero there. Notice however, that the function $R(r)$ that we are solving for is itself not a gauge invariant quantity. In particular, it has a gauge-dependent phase. Thus if we choose the radial function to be e\^[ie(r)]{}R(r), we can remove the apparent singular behaviour of the potential at the horizon with a particular gauge choice for $\varphi(r)$. A computation shows that the function (r) = = , which we recognize as the tortoise coordinate [(\[eq:tortoise\])]{}, removes the apparent singular behaviour at the horizon. With this choice of phase, the gauge-invariant part of the radial function satisfies the equation [(\[eq:radialI\])]{}.
Angular Eigenvalues and Eigenfunctions
--------------------------------------
The separated equations are coupled through the separation constant $\lambda$. It is thus important to have precise numerical methods to determine it. With $x=\cos\theta$ equation [(\[eq:angularI\])]{} reads \[eq:kerradsangular\] - \^2 x (1-x\^2) & &\
+S\_[m]{}\^(x)&=&0, where we have briefly reinstated the azimuthal quantum number $m$. In the limit $L\rightarrow\infty$, equation [(\[eq:kerradsangular\])]{} goes over into the spheroidal equation and the eigenfunctions $S_{\ell m}^\gamma(x)$ are the spheroidal harmonics. In many references on spheroidal harmonics the parameter $a^2 m^2_\Psi$ is called $\gamma^2$. In the limit $a\rightarrow 0$ it becomes the associated Legendre equation.
In this paper, we only consider $S_{\ell m}^{\gamma \pm}(x,\lambda)$ for $m=0$, which is all we need in determining the critical temperature. Note, however, that the angular eigenvalue of the (AdS) spheroidal equation depends on $m$, unlike the more well-known case of the spherical harmonics (i.e. the Legendre polynomials). However, our procedures work equally well for non-zero values of $m$.
### Numerics
We now describe the numerical procedure used to extract the eigenvalues $\lambda$ from this equation. It is evident that solutions of equation [(\[eq:kerradsangular\])]{} are either even or odd functions. We will make use of this crucial fact, which makes it easier to use a simple shooting method:
1. Obtain series expansions of $S^\gamma_{\ell m}$ at $x=\pm 1$. There is a regular branch and a logarithmic branch of solutions. As usual we select the regular branch. We set the following boundary conditions S\_[m]{}\^(-1) = S\_[m]{}\^(1)=1 S\_[m]{}\^(x)[even]{}\
S\_[m]{}\^(-1) = -S\_[m]{}\^(1)=1 S\_[m]{}\^(x)[odd]{} The choice of the magnitude of the eigenfunctions at either end of the interval is clearly just a choice of normalization.
2. Numerically integrate inwards to the origin from both ends for a fine grid of values of the eigenvalue $\lambda$ in order to match the Frobenius expansions at either end of the intervals. This gives rise to a series of functions $S_{\ell m}^{\gamma \pm}(x,\lambda)$, where the $\pm$ refers to the numerical integration from $\pm 1$.
3. For odd functions the first derivatives of $S_{\ell m}^{\gamma \pm}(x,\lambda)$ are guaranteed to coincide at the origin. Thus we define the function $$O(\lambda) = S_{\ell m}^{\gamma, +}(0,\lambda) - S_{\ell m}^{\gamma, -}(0,\lambda)$$ by interpolating the values of the discrete grid of values of $\lambda$. For even eigenfunctions, we are guaranteed that the value of $S_{\ell m}^{\gamma \pm}(x,\lambda)$ matches at the origin. Thus we define $$E(\lambda) = \left(S_{\ell m}^{\gamma, +}\right)'(0,\lambda) - \left(S_{\ell m}^{\gamma, -}\right)'(0,\lambda)$$
4. Find the zeroes of $E(\lambda)$ and $O(\lambda)$. Every root corresponds to an eigenvalue of the angular equation.
We now tabulate (a subset of) the $\lambda$ values that were used to find the critical temperatures displayed in figure \[fig:phasediagram\]. Table 1 lists a representative sample of eigenvalues to five significant digits that were used in obtaining the phase curves of critical temperature versus rotation. Our numerical algorithms in fact allow a far higher accuracy, but it would be impractical and of little value to list more significant digits here.
$\alpha$ 0 0.1 0.2 0.4 0.6 0.8 0.9
------------------- --- --------- --------- --------- --------- --------- ---------
$m^2_\Psi L^2=-2$ 0 0.0133 0.0532 0.2115 0.4690 0.8086 0.9921
$m^2_\Psi L^2=4$ 0 -0.0266 -0.1071 -0.4340 -0.9997 -1.8393 -2.3764
: [Angular eigenvalues of AdS spheroidal harmonics. The first line gives the deformation parameter $\alpha=a L^{-1}$ and the following two lines supply the angular eigenvalues for the two mass-squared values used to compute the phase diagram.]{}
[99]{}
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[^1]: This can always be done because of scaling symmetry present in the background. However, physically, this is nothing but a choice of units in which we measure lengths in multiples of the horizon radius.
[^2]: The upper bound can be seen from the neat identity $\sum_{\rm roots}\frac{1}{{\cal Q}'(x_i)}=0$ for any polynomial ${\cal Q}(x)$ at least quadratic in $x$. A simple proof [@zsuzsa08] can be given via contour integration of the function $f(z)=\frac{1}{{\cal Q}(z)}$ defined by analytically continuing ${\cal Q}$ into the complex plane.
[^3]: As with all numerical procedures described in this paper, I shall make Mathematicanotebooks available upon request.
|
---
abstract: 'A compression algorithm is introduced for multi-determinant wave functions which can greatly reduce the number of determinants that need to be evaluated in quantum Monte Carlo calculations. We have devised an algorithm with three levels of compression, the least costly of which yields excellent results in polynomial time. We demonstrate the usefulness of the compression algorithm for evaluating multi-determinant wave functions in quantum Monte Carlo calculations, whose computational cost is reduced by factors of between 1.885(3) and 25.23(4) for the examples studied. We have found evidence of sub-linear scaling of quantum Monte Carlo calculations with the number of determinants when the compression algorithm is used.'
author:
- 'Gihan L. Weerasinghe'
- Pablo López Ríos
- 'Richard J. Needs'
title: 'Compression algorithm for multi-determinant wave functions'
---
Introduction {#sec:intro}
============
The variational and diffusion [@ceperley-1980; @qmcrmp] quantum Monte Carlo (VMC and DMC) methods are the most accurate known for computing the energies of large numbers of interacting quantum particles. The crucial ingredient is an approximate trial wave function which should be easy to evaluate, while giving a good approximation to the true many-body wave function. A standard approach is to use a Slater-Jastrow trial wave function which consists of the product of determinants for the up and down-spin electrons, multiplied by a Jastrow factor that describes dynamical correlation [@DTN_jastrow; @jastrow]. We omit the Jastrow factor in the rest of this paper for conciseness. Static correlation can be included by replacing the determinant product by a multi-determinant expansion, $$\label{eq:sj_mdet}
\Psi_{\rm MD}({\bf R}) =
\sum_{k=1}^{N_s} c_k
\Phi^\uparrow_k({\bf R}_{\uparrow})
\Phi^\downarrow_k({\bf R}_{\downarrow}) \;,$$ where $\Phi^\uparrow_k({\bf R}_{\uparrow}) =
\det\left[\phi_{a_{i,k}^\uparrow}({\bf r}_j^\uparrow)\right]$ and $\Phi^\downarrow_k({\bf R}_{\downarrow}) =
\det\left[\phi_{a_{i,k}^\downarrow} ({\bf r}_j^\downarrow)\right]$ are determinants of up- and down-spin single-particle orbitals, $a_{i,k}^\sigma$ is an index that selects the orbital which occurs in the $i$th row of the $\sigma$-spin determinant in the $k$th term of the expansion, ${\bf R}_\sigma$ denotes the set of $\sigma$-spin electron coordinates, ${\bf R}=\{{\bf R}_{\uparrow},
{\bf R}_{\downarrow}\}$, and $c_k$ is the coefficient of the $k$th term in the expansion. The accuracy of $\Psi$ can be further improved by, for example, increasing the number of terms in the expansion, $N_s$, or by introducing a backflow transformation [@bf-ne; @backflow].
VMC and DMC calculations are normally performed by displacing electrons one at a time, because this has been shown to be the most efficient way to decorrelate consecutive electronic configurations [@backflow; @strategies]. The displacement of a single electron requires the calculation of the ratio of the wave functions at the new and old coordinates, $\Psi({\bf R}^{\prime})/\Psi({\bf R})$. In a standard single-determinant calculation this requires the replacement of a single row of the Slater matrix by the vector of the orbitals at the new position ${\bf R}^{\prime}$, and the required calculation is performed using the Sherman-Morrison formula [@qmcrmp]. In a multi-determinant calculation $N_s$ such calculations must be performed. In a backflow calculation each electronic coordinate in the Slater part of the wave function is replaced by a “quasiparticle coordinate”, which depends on all of the electronic coordinates, so that each entry within each Slater matrix must be recalculated when an electron is displaced, and its determinant must be reevaluated, which is achieved using standard LU decomposition. In a multi-determinant backflow calculation, $N_s$ Slater determinants for each spin must be constructed and evaluated.
The repeated evaluation of the trial wave function and its first two derivatives for different electronic coordinates ${\bf R}$ is the main contribution to the computational cost of a QMC calculation, which is, as discussed above, approximately proportional to $N_s$. Methods to reduce the cost of evaluating multi-determinant wave functions during QMC calculations have been developed in previous studies [@clark_2011; @nukala_2009]. In this paper we introduce a determinant compression algorithm which can significantly reduce the cost of evaluating a multi-determinant trial wave function by reducing the number of determinants in the expansion, by as much as a factor of 26.57 in the examples presented here. We have used the <span style="font-variant:small-caps;">casino</span> code [@casino] for the QMC calculations reported here.
Methodology {#sec:method}
===========
Quantum chemistry methods are often used to provide appropriate multi-determinant wave functions for electronic systems. In practice, the determinants contain $M_s$ distinct orbitals, and different determinants often differ by only a single orbital. Moreover, quantum chemistry methods group determinants into configuration state functions (CSFs), and different CSFs may contain the same determinant product $\Phi^\uparrow_k \Phi^\downarrow_k$. The compression method we present here exploits these two facts.
Basic determinant operations {#sec:combine}
----------------------------
### Identical determinants {#sec:dedup}
To achieve greater efficiency we combine repeated determinant products so that each of them need only be evaluated once at each ${\bf R}$, which is equivalent to simply adding together the terms from identical determinant products, *i.e.*, $$c_1 \Phi^\uparrow \Phi^\downarrow +
c_2 \Phi^\uparrow \Phi^\downarrow =
(c_1+c_2) \Phi^\uparrow \Phi^\downarrow =
c_1^\prime \Phi^\uparrow \Phi^\downarrow \;,$$ where $c_1^\prime=c_1+c_2$ is the coefficient of the term arising from the combination of two repeated determinant products.
We refer to this procedure as “de-duplication”, which is the first operation in our compression algorithm, and its computational cost scales as $\mathcal{O}({N_s}^2)$, since all pairs of determinants need to be compared to determine if they are equal. The number of determinants in the resulting expansion is $N_d\leq
N_s$, while the number of distinct orbitals $M_d$ is equal to $M_s$. Later stages of the compression algorithm can be simplified based on the assumption that the expansion has been de-duplicated.
### Determinants differing by a single orbital {#sec:compress_core}
It is convenient to express each Slater determinant in Eq. (\[eq:sj\_mdet\]) using a compact vector notation consisting of the list of orbitals that the determinant contains, $$\label{eq:detmap}
\left[ \phi_{a_1}, \phi_{a_2}, \ldots, \phi_{a_n} \right] \equiv
\left|
\begin{array}{cccc}
\phi_{a_1}({\bf r}_1) & \phi_{a_2}({\bf r}_1) & \cdots &
\phi_{a_n}({\bf r}_1) \\
\phi_{a_1}({\bf r}_2) & \phi_{a_2}({\bf r}_2) & \cdots &
\phi_{a_n}({\bf r}_2) \\
\vdots & \vdots & \ddots &
\vdots \\
\phi_{a_1}({\bf r}_n) & \phi_{a_2}({\bf r}_n) & \cdots &
\phi_{a_n}({\bf r}_n)
\end{array}
\right| \;.$$ Central to the algorithm is an elementary identity from linear algebra which allows two determinants to be combined if they differ by a single row or column. This is applicable to a multi-determinant expansion for terms where the determinants of one spin type are equal and can be factored out, and the determinants of the other spin type differ by a single column, *e.g.*, $$\begin{aligned}
\label{eq:example_detcombine1}
c^\prime_1 [\phi_{a_1}, \phi_{a_3}, \ldots, \phi_{a_4}] \Phi^\downarrow
+ c^\prime_2 [\phi_{a_2}, \phi_{a_3}, \ldots, \phi_{a_4}] \Phi^\downarrow
\nonumber \\
= [c^\prime_1 \phi_{a_1} + c^\prime_2 \phi_{a_2}, \phi_{a_3}, \ldots,
\phi_{a_4}] \Phi^\downarrow
\nonumber \\
= {\tilde c}_1 [{\tilde \phi}_{{\tilde a}_1}, \phi_{a_3}, \ldots, \phi_{a_4}]
\Phi^\downarrow
\;,\end{aligned}$$ where ${\tilde \phi}_{{\tilde a}_1}$ is a new orbital resulting from the linear combination of two of the original orbitals and ${\tilde
c}_1$ is a new coefficient, satisfying $$\label{eq:example_detcombine2}
{\tilde c}_1 {\tilde \phi}_{{\tilde a}_1} =
c^\prime_1 \phi_{a_1} + c^\prime_2 \phi_{a_2} \;.$$ This operation can be applied to sets of more than two determinant products, provided they all differ in the same column of the same-spin determinant.
The compression algorithm we have developed applies this basic operation repeatedly to all possible sets of determinants. However, there may be multiple mutually-exclusive ways of combining the determinants, and the size of the resulting expansion depends on the choice of operations. We discuss this in Sec. \[sec:group\_terms\].
Representation of compressed expansions {#sec:represent}
---------------------------------------
A compressed multi-determinant expansion is of the form $$\label{eq:compressed_exp}
\Psi_{\rm MD}({\bf R}) =
\sum_{k=1}^{N_c} {\tilde c}_k
\det\left[ {\tilde \phi}_{{\tilde a}^\uparrow_{ik}}
({\bf r}^\uparrow_j) \right]
\det\left[ {\tilde \phi}_{{\tilde a}^\downarrow_{ik}}
({\bf r}^\downarrow_j) \right] \;,$$ where $N_c$ is the number of terms in the compressed expansion, ${\tilde c}$ are the compressed expansion coefficients, ${\tilde \phi}$ are the compressed orbitals, and ${\tilde a}^\sigma_{ik}$ is an index that selects which compressed orbital occurs in the $i$th row of the $\sigma$-spin determinant in the $k$th term of the compressed expansion. The expansion coefficients are $$\label{eq:compressed_coeff}
{\tilde c}_k =
\pm c^\prime_{\nu_{k1}}
\prod_{p=2}^{P_k}
\frac{c^\prime_{\nu_{kp}}}
{c^\prime_{\delta_{kp}}} \;,$$ and the compressed orbitals are $$\label{eq:compressed_orb}
{\tilde \phi}_a ({\bf r}) =
\sum_{x=1}^{X_a}
\pm
\prod_{q=1}^{Q_{ax}}
\frac{c^\prime_{n_{axq}}}
{c^\prime_{d_{axq}}}
\phi_{\mu_{ax}} ({\bf r}) \;,$$ where $P$, $X$, and $Q$ are sum and product lengths, and $\nu$, $\delta$, $n$, $d$, and $\mu$ are indices, all of which arise from the application of the compression operations described above. The $\pm$ signs account for any required row exchanges in the determinants. In this notation, the compression operation exemplified in Eq. (\[eq:example\_detcombine1\]) is such that $${\tilde c}_1 = + c^\prime_1 \;,$$ and $${\tilde \phi}_{{\tilde a}_1} ({\bf r}) =
+ \phi_{a_1}({\bf r})
+ \frac{c^\prime_2}{c^\prime_1} \phi_{a_2}({\bf r}) \;,$$ which satisfies Eq. (\[eq:example\_detcombine2\]), as required. Therefore in this case $P_1=1$, $\nu_{1,1}=1$, $X_{{\tilde a}_1}=2$, $Q_{{\tilde a}_1,1}=0$, $\mu_{{\tilde a}_1,1}=a_1$, $Q_{{\tilde a}_1,2}=1$, $n_{{\tilde a}_1,2,1}=2$, $d_{{\tilde a}_1,2,1}=1$, and $\mu_{{\tilde a}_1,2}=a_2$.
A compressed expansion is fully determined by specifying ${\tilde a}$, $P$, $X$, $Q$, $\nu$, $\delta$, $n$, $d$, $\mu$, and the signs in Eqs. (\[eq:compressed\_coeff\]) and (\[eq:compressed\_orb\]). Expressing the compressed expansion in this manner is useful because the orbitals and coefficients can be quickly reconstructed using Eqs. (\[eq:compressed\_coeff\]) and (\[eq:compressed\_orb\]) when the original expansion coefficients, the $\{c_k\}$ of Eq. (\[eq:sj\_mdet\]), change, as is the case during wave function optimization within QMC.
Choosing the optimal set of operations {#sec:group_terms}
--------------------------------------
It is convenient to express the principles of the compression algorithm using set theory notation. Let $P$ be a set whose elements are the terms in the de-duplicated multi-determinant expansion, $P = \{p_k \equiv
c^\prime_k \Phi_k^\uparrow \Phi_k^\downarrow\}$, of size $|P|=N_d$. Let $u_i$ be a subset of $P$ such that its elements can be combined via the compression operation of Sec. \[sec:compress\_core\], and $U$ the set of all possible such sets, $U=\{u_i\}$. Note that $u_i$ is allowed to contain only one term, and that any two elements $u_i$ and $u_j$ may contain the same term $p_k$.
A valid compressed expansion can be obtained by finding a subset $V=\{v_i\}$ of $U$ that satisfies the conditions that (a) $V$ contains all terms in $P$, $\bigcup_i v_i = P$, and (b) each term in $P$ is contained in only one element of $V$, $v_i \cap v_j = \emptyset
~~ \forall~ i \neq j$. The resulting compressed expansion will contain one term for each element of $V$, and therefore the optimal compression is that for which $V$ has the fewest elements.
Finding the minimal set of sets $V$ that covers a set $P$ is otherwise known as the set-covering problem. This can be expressed as a binary linear program [@Beasley_set_cover], that is, an optimization problem where a linear objective function $f$ of the binary unknowns $\{x_i\}$ is to be optimized subject to a set of linear equalities and/or inequalities involving the unknowns. In the binary linear program associated with a set-covering problem there are $|U|$ unknowns $\{x_i\}$ that determine whether a subset $u_i$ is present in $V$ ($x_i=1$) or not ($x_i=0$). The objective function that must be minimized is the number of subsets in $V$, $$\label{eq:LPObjective}
f(\{x_i\}) = \sum_{i=1}^{|U|} x_i \;,$$ constrained so that each element in $P$ appears exactly once in $V$, $$\label{eq:LPConstraints}
\sum_{j=1}^{|U|} a_{ij} x_j = 1 \;,$$ where the binary element $a_{ij}$ of the constraint matrix indicates whether the term $p_i$ is contained within subset $u_j$.
The size of the enumeration set $U$ can become very large if large sets of combinable determinants are present in the original expansion, since all possible combinations of those determinants are required to be individual elements of $U$. In practice we construct a set $W$ which only contains the sets in $U$ that are either of size one or not wholly contained in another set. The maximum size of $W$ is linear with the original expansion size. The process of constructing $V$ from $W$ differs slightly from that described earlier, in that when an element $w_i$ is added to $V$ we now require that the terms contained in $w_i$ be removed from all other $\{w_j\}_{j \neq i}$. Note that by construction the order in which elements of $W$ are added to $V$ can affect which compression operations are in $V$, but not their number, and therefore the compressed expansion obtained by this procedure is of the same size as that obtained directly from $U$.
The linear program to be solved in this simplified variation of the method has $|W|$ unknowns $\{y_i\}$ that determine whether a subset $w_i$ should be added to $V$. The objective function that is to be minimized is $$\label{eq:LPObjective_W}
g(\{y_i\}) = \sum_{i=1}^{|W|} y_i \;,$$ and the constraints which guarantee that $V$ covers $P$ are $$\label{eq:LPConstraints_W}
\sum_{j=1}^{|W|} b_{ij} y_j \geq 1 \;,$$ where the binary element $b_{ij}$ of the constraint matrix indicates whether the term $p_i$ is contained within subset $w_j$.
We further reduce the size of the linear program by partitioning $W$ into subsets such that each term $p_i$ appears in only one of the subsets. Solving the linear programs for each of the partitions independently is equivalent to solving the linear program for $W$.
There exist efficient methods to solve binary linear programs, such as the iterative simplex method implemented in the <span style="font-variant:small-caps;">lpsolve</span> library [@lpsolve]. However, solving a binary linear program, or otherwise solving the set-covering problem, is in general NP-hard [@Karp_combinatorial], and therefore a good determinant compression algorithm should implement an approximate fall-back method for cases where it is infeasible to obtain the exact solution in a reasonable amount of time. A good approximate solution to the set-covering problem can be found in polynomial time using a “greedy” algorithm [@greedy_setcover], in which $V$ is constructed by adding to it the largest element of $W$, removing all the terms contained in this element from the other elements of $W$, and repeating this process until no non-empty elements remain in $W$. In the examples we have studied in the present work we have not found any cases where we had to resort to the greedy algorithm.
Multiple Iterations {#sec:multi_iter}
-------------------
In some cases it is possible for a set of compressed determinants to be combined in order to yield an even shorter expansion. For example the sum $$\label{eq:multigroupexample_1}
[ \phi_{a_1}, \phi_{a_3} ] + [ \phi_{a_2}, \phi_{a_3} ] +
[ \phi_{a_1}, \phi_{a_4} ] + [ \phi_{a_2}, \phi_{a_4} ] \;,$$ can be compressed into $$\label{eq:multigroupexample_2}
[ \phi_{a_1}+\phi_{a_2}, \phi_{a_3} ] +
[ \phi_{a_1}+\phi_{a_2}, \phi_{a_4} ] \;,$$ which can be further compressed into $$\label{eq:multigroupexample_3}
[ \phi_{a_1}+\phi_{a_2}, \phi_{a_3}+\phi_{a_4} ] \;.$$ The presence of multiply compressible sets of terms is a property of the original expansion. Note that the result of applying a compression operation to already-compressed determinants continues to be of the form given by Eqs. (\[eq:compressed\_exp\]), (\[eq:compressed\_coeff\]), and (\[eq:compressed\_orb\]).
The most straightforward method of dealing with multiply compressible terms is to apply the procedure described in the previous section iteratively until no further decrease in the length of the expansion occurs, which we refer to as the “simple iterative method”. By construction this method will operate on a different row of the determinants at each iteration, and therefore the maximum number of iterations is the number of electrons in the system.
However, the simple iterative method is not guaranteed to give the optimal solution for two reasons. Firstly, the choice of which terms are grouped in earlier iterations affects the size of the final expansion, in such a way that making sub-optimal choices at individual iterations (e.g., using the greedy algorithm) may yield a better overall compression than solving the set-covering problem exactly at all iterations. And secondly, terms in the original expansion should be allowed to contribute to more than one term of the compressed expansion. For example, consider the following compression of a six term expansion into two terms, $$\begin{aligned}
\nonumber
& [ \phi_{a_1}, \phi_{a_2} ] + 2 [ \phi_{a_1}, \phi_{a_3} ] +
[ \phi_{a_2}, \phi_{a_3} ] + & \\ \nonumber
& [ \phi_{a_1}, \phi_{a_4} ] + 2 [ \phi_{a_2}, \phi_{a_4} ] +
[ \phi_{a_3}, \phi_{a_4} ] = & \\ \nonumber
& [ \phi_{a_1}, \phi_{a_2}+\phi_{a_3} ] +
[ \phi_{a_1}+\phi_{a_2}, \phi_{a_3} ] + & \\ \nonumber
& [ \phi_{a_1}+\phi_{a_2}, \phi_{a_4} ] +
[ \phi_{a_2}+\phi_{a_3}, \phi_{a_4} ] = & \\
\label{eq:example_complex_multi_1}
& [ \phi_{a_1}-\phi_{a_4}, \phi_{a_2}+\phi_{a_3} ] +
[ \phi_{a_1}+\phi_{a_2}, \phi_{a_3}+\phi_{a_4} ] \;. &\end{aligned}$$ This compression operation is possible only if the second and fifth terms of the original expansion are used twice; otherwise the result would be a three-term compressed expansion at best. Note that, in the absence of multiply compressible terms, the resulting compressed expansion will contain the same number of terms regardless of whether a term can be used more than once or not.
We have developed a multiple iteration algorithm that solves the first of these issues, although not the second which would require an entirely different methodology, and in our opinion the resulting method would not give significantly better compression ratios. This method, which we refer to as the “unified iteration method”, is similar to that outlined in Sec. \[sec:group\_terms\]. First the enumeration set $U$, which we refer to as $U^{(0)}$ in this context, is constructed. We define $u_i^{(1)}$ as a subset of $U^{(0)}$ such that its elements can be combined, and $U^{(1)}$ is the set of all possible such sets, $U^{(1)}=\{u_i^{(1)}\}$. A similar set can be defined for each recursion level $n>1$, so that $U^{(n)}$ is formed by all possible sets of elements of $U^{(n-1)}$ that can be combined together. Recursion stops at $n=n_{\rm max}$ if $U^{(n_{\rm max})}$ does not contain any terms that can be combined together.
The unknowns of the linear program for the unified iteration method are $\{x_i^{(n)}\}$, where $x_i^{(n)}$ indicates whether set $u_i^{(n)}$ is in $V$ or not. The objective function that is minimized is the number of sets in $V$, $$\label{eq:LPObjective_multi}
f(\{x_i^{(n)}\}) = \sum_{n=0}^{n_{\rm max}}
\sum_{i=1}^{|U^{(n)}|} x_i^{(n)} \;,$$ constrained by $$\label{eq:LPConstraints_multi}
\sum_{n=0}^{n_{\rm max}} \sum_{j=1}^{|U^{(n)}|}
a_{ij}^{(n)} x_j^{(n)} = 1 \;,$$ where the binary element $a_{ij}^{(n)}$ of the constraint matrix indicates whether or not the term $p_i$ is present in $u_j^{(n)}$.
As in the case of the simple iterative method, it is possible to avoid constructing the enumeration set $U^{(0)}$ and instead construct a set $W^{(0)}$ that contains all elements of $U^{(0)}$ that are not contained in other elements. However this simplification cannot be applied to higher recursion levels, and one must construct $U^{(n)}$ explicitly for $n>0$. The reason for this is that eliminating a single term $p_i$ from all $w_j^{(n)}$ during the construction of $V$ may cause the compression operation represented by $w_j^{(n)}$ to become invalid in the absence of $p_i$, an event which is not taken into account by the linear program. Therefore the simplified linear program has the unknowns $\{y_i\}$ and $\{x_i^{(n)}\}_{n=1}^{n_{\rm max}}$, and the objective function $$\label{eq:LPObjective_multi_W}
f(\{y_i\}, \{x_i^{(n)}\}) =
\sum_{i=1}^{|W^{(0)}|} y_i +
\sum_{n=1}^{n_{\rm max}} \sum_{i=1}^{|U^{(n)}|} x_i^{(n)} \;,$$ constrained so that each term of the original expansion appears at least once in the selected operations, $$\label{eq:LPConstraints_multi_W_1}
\sum_{j=1}^{|W^{(0)}|} b_{ij} y_j +
\sum_{n=1}^{n_{\rm max}} \sum_{j=1}^{|U^{(n)}|}
a_{ij}^{(n)} x_j^{(n)} \geq 1 \;,$$ and each term of the original expansion appears at most once in operations of recursion level $n>0$, $$\label{eq:LPConstraints_multi_W_2}
\sum_{n=1}^{n_{\rm max}} \sum_{j=1}^{|U^{(n)}|}
a_{ij}^{(n)} x_j^{(n)} \leq 1 \;.$$ Operations of recursion level $n>0$ must be added to $V$ before those with order $n=0$ to prevent the application of the latter from invalidating the former, as mentioned earlier.
Partitioning can be also applied at recursion level $n=0$ to reduce the potential cost of solving the linear program.
Implementation {#sec:implement}
==============
-------- --------------- ------- ------- ------- ------- ---- -------- ------- ------ ------- -------- ------ ------- ------- -------- ------ ------ ---- --------
$N_{\rm CSF}$ $N_s$ $M_s$ $N_d$ $M_d$ $N_q$ $M_q$ $N_g$ $M_g$ $N_b$ $M_b$
Be$_2$ 61 200 12 200 12 0 001(1) 100 60 0 011(1) 97 72 0 022(1) 97 73 0 023(1)
N 50 1271 51 764 51 0 012(1) 324 191 0 038(2) 324 188 0 044(1) 324 188 0 054(1)
O 100 3386 53 1271 53 0 058(1) 535 365 0 125(1) 534 361 0 144(1) 534 361 0 143(1)
Li 500 8140 105 5824 105 0 332(6) 1226 1023 1 270(2) 1210 1036 1 430(3) 1210 1046 3 144(3)
B 500 14057 105 5703 105 0 79(1) 530 629 1 802(2) 529 629 1 884(7) 529 631 3 73(1)
Be 500 14212 105 10600 105 1 18(1) 2218 1174 4 63(1) 2177 1188 5 17(2) 2163 1198 7 82(3)
Ne 400 22827 105 16260 105 4 56(3) 1844 1182 14 79(8) 1805 1191 15 04(6) 1805 1191 18 77(4)
F 600 57456 105 17174 105 11 43(1) 2801 2553 22 3(2) 2749 2622 22 5(2) 2747 2627 24 18(4)
-------- --------------- ------- ------- ------- ------- ---- -------- ------- ------ ------- -------- ------ ------- ------- -------- ------ ------ ---- --------
We have implemented the multi-determinant expansion compressor as a stand-alone utility [@compress_code] which can be used with any suitably modified quantum Monte Carlo code, and we have modified the <span style="font-variant:small-caps;">casino</span> code [@casino] to be able to use compressed multi-determinant expansions produced by the utility.
The compression utility uses the <span style="font-variant:small-caps;">lpsolve</span> library [@lpsolve] to solve the linear programs described in Secs. \[sec:group\_terms\] and \[sec:multi\_iter\]. This utility implements four operational levels: (a) “de-duplicate”, which performs the de-duplication stage of the compression only (polynomial time); (b) “quick”, which performs de-duplication and uses the greedy algorithm to find an approximate solution to the set-covering problems posed by the simple iterative method (polynomial time); (c) “good”, which performs de-duplication and solves the set-covering problems posed by the simple iterative method exactly using a linear program (potentially NP-hard), and (d) “best”, which performs de-duplication and solves the set-covering problem posed by the unified iteration method exactly using a linear program (potentially NP-hard).
The “best” operational level should be used whenever feasible. The “good” mode is provided for cases where the construction of the enumeration sets $U^{(n)}$ makes the “best” mode too expensive, and the “quick” mode is useful for expansions which make the “good” algorithm exhibit its potential NP-hardness. The utility specifies the compressed expansion in terms of the variables introduced in Eqs. (\[eq:compressed\_exp\]), (\[eq:compressed\_coeff\]), and (\[eq:compressed\_orb\]).
We have modified the <span style="font-variant:small-caps;">casino</span> code to enable it to compute the compressed orbitals as appropriate linear combinations of the original orbitals, as per Eq. (\[eq:compressed\_orb\]). During optimization, the original expansion coefficients are exposed to the optimizer, and the linear coefficients of the compressed determinants and orbitals are re-evaluated when the parameters change.
Results {#sec:results}
=======
We have tested our compression algorithm on multi-determinant expansions for the N, O, Li, B, Be, Ne, and F atoms generated using the <span style="font-variant:small-caps;">atsp2k</span> multi-configurational Hartree-Fock (MCHF) package [@atsp2k], and with a multi-determinant expansion for the Be$_2$ molecule generated using the <span style="font-variant:small-caps;">gamess</span> code [@gamess]. The test cases are intended to represent realistic calculations, such as the multi-determinant calculations of Seth *et al.* [@Seth_2011_atoms].
The wave functions produced by <span style="font-variant:small-caps;">atsp2k</span> are arranged in CSFs, each of which comprises a set of determinants with a certain symmetry. Since the same determinant product may be contained in different CSFs, the atomic wave functions in our tests benefit from de-duplication. In contrast, the <span style="font-variant:small-caps;">gamess</span> code performs de-duplication internally, and as a result the de-duplication stage of our compression utility does not yield any gains for the Be$_2$ wave function.
The results for the compression algorithm are presented in Table \[table:results\], which gives the number of determinants and distinct orbitals for the original and compressed expansions at the different operational levels of the compression utility. Also given is the CPU time taken by the compression utility, averaged over 20 trials, on a modest CPU. (The compression tests were performed on a single core of a 2007 Intel Core 2 Quad CPU.)
De-duplication results in significant reductions in the size of the multi-determinant expansions for the atoms, with $N_s/N_d$ ranging between 1.0 and 3.3. The compression stage provides an even greater reduction, with values of $N_d/N_b$ ranging between 2.1 and 10.8. In total, the compression utility yields compression factors of up to $N_s/N_b = 26.57$.
The different compression levels “quick”, “good”, and “best” yield very similar compression sizes, with “good” giving a small improvement over “quick” of up to $N_q/N_g = 1.024$, and “best” yielding a smaller change over “good” of $N_g/N_b = 1.005$ at most. The CPU times required by the three compression levels are also very similar, which indicates that the linear programs do not exhibit their potential NP-hardness in any of the cases studied here.
The number of distinct orbitals in the compressed expansions appears to be roughly of the order of the number of terms in the compressed expansion, $M \sim N$. For a given system, $M$ can be expected to increase as $N$ decreases. However, this is not always true in our tests, e.g., for Be $M_g < M_q$ even though $N_g < N_q$, and for Ne $M_b < M_g$ even though $N_b = N_g$. These cases are allowed by construction, since our algorithms do not attempt to minimize $M$, and the different operational levels might pick different orbital groupings that yield the same value of $N$ but different values of $M$.
$N_s/N_b$ $N_d/N_b$
-------- ----------- ---- ----------- --------- ---- --------
Be$_2$ 2.06 1 885(3) 2.06 1 905(6)
N 3.92 3 718(9) 2.36 2 276(8)
O 6.34 6 09(1) 2.38 2 309(5)
Li 6.73 6 50(2) 4.81 4 64(1)
B 26.57 25 23(4) 10.78 10 05(2)
Be 6.57 6 48(1) 4.90 4 83(2)
Ne 12.65 13 17(2) 9.01 9 34(1)
F 20.92 21 77(5) 6.25 6 48(1)
: Speed-up provided by the “best” compression algorithm over the uncompressed ($T_s/T_b$) and de-duplicated ($T_d/T_b$) expansions for a fixed number of moves in a multi-determinant VMC calculation. The relative expansion sizes $N_s/N_b$ and $N_d/N_b$ are also shown for comparison. \[table:CASINOcputimes\]
To test the benefits of using compressed multi-determinant expansions in QMC calculations, we have run multi-determinant VMC calculations using the original, de-duplicated and “best” expansions. The CPU time $T$ taken by these runs, averaged over 10 trials, is compared in Table \[table:CASINOcputimes\].
In principle, the cost $T$ of the QMC calculation is at most proportional to the expansion size $N$, and thus $T_s/T_b \leq N_s/N_b$. The results for both the F and Ne atoms are anomalous since this inequality does not hold. Our interpretation is that this is an effect of the reduced memory footprint of the compressed expansion, for which the CPU caches can hold the numerical data for the entire expansion and carry out the operations more rapidly.
The data for the other systems show that the value of $T_s/T_b$ is between 91% and 99% that of $N_s/N_b$, implying that handling the determinants in the expansion is by far the leading contribution to the CPU time of the QMC calculation, and that reducing $N$ produces an almost equal reduction in $T$. The additional CPU time required to compute the orbitals for the compressed expansion as linear combinations of the original orbitals has an insignificant impact on the benefits of using the compression scheme for the examples in Table \[table:CASINOcputimes\].
We have also run VMC and DMC calculations using Jastrow factors and backflow transformations with identical conclusions, but we have omitted the results from Table \[table:CASINOcputimes\] for the sake of conciseness.
![(Color online) Compressed expansion size $N_b$ as a function of the original expansion size $N_s$ for the Be, B, and F atoms. The active space used in the generation of the MCHF expansion includes up to double excitations (D) for B and F, and up to triple excitations (DT) for Be. The results, ignoring the plateau in the case of B, were fitted to $N_b = a N_s^\alpha$. \[fig:detscale\_overview\]](figures/detscale_overview){width="40.00000%"}
![(Color online) Compressed expansion size $N_b$ as a function of the original expansion size $N_s$ for the B atom when the active space used in the generation of the MCHF expansion includes up to double excitations (D) and up to triple excitations (DT). The results, ignoring the plateau in the case of B (D), were fitted to $N_b = a N_s^\alpha$. \[fig:detscale\_boron\]](figures/detscale_boron){width="40.00000%"}
We have applied our “best” compression algorithm to expansions of different sizes to investigate how $N_b$ varies with $N_s$. Results for the Be, B, and F atoms with up to 600 CSFs are plotted in Fig. \[fig:detscale\_overview\], where we also show fits to $N_b = a N_s^\alpha$. The active space used in the generation of the MCHF wave function included up to double excitations for B and F, and up to triple excitations for Be. We find that $\alpha$ is about $1/2$ for B and F, and about $2/3$ for Be.
In the case of B we detect a plateau in $N_b$ as a function of $N_s$. We interpret this as a sign of the exhaustion of the finite active space used in the generation of the multi-determinant wave function. To test this hypothesis we have repeated the scaling tests for B with an active space that includes up to triple excitations, as shown in Fig. \[fig:detscale\_boron\]. As we expected, the plateau disappears when the larger active space is used.
We conclude that the use of the compression algorithm can make multi-determinant QMC calculations scale as $N_s^\alpha$, with $1/2 \leq \alpha < 1$.
Conclusion {#sec:conclude}
==========
In this paper we have presented a compression algorithm for multi-determinant expansions based on a simple identity for combining determinants, which can provide computational cost savings in QMC calculations of about the compression factor $N_s/N_b$, which in our tests ranges between 2.06 and 26.57.
In addition to the full compression algorithm we have implemented a polynomial-scaling fall-back algorithm which has been shown to yield nearly the same compression ratios as the full method. This algorithm avoids the NP-hardness associated with solving the set-covering problem, but in none of our tests did the full algorithm incur costs excessive enough to require the use of the fall-back algorithm.
We find that the compression algorithm makes QMC calculations scale sub-linearly with the number of determinants in the expansion. The cost savings provided by using compressed determinant expansions are expected to permit QMC calculations using much larger multi-determinant expansions than would otherwise be possible. Our compression algorithm can be used in combination with methods for the optimized evaluation of multi-determinant wave functions [@clark_2011; @nukala_2009] for additional efficiency.
We thank John Trail for producing the multi-determinant wave function for Be$_2$. The authors acknowledge financial support from the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom. Computational resources were provided by the Cambridge High Performance Computing Service.
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The compression utility is available from the authors at `http://www.tcm.phy.cam.ac.uk/~pl275/DetCompress`.
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---
abstract: 'We study in detail two families of $q$-Fibonacci polynomials and $q$-Lucas polynomials, which are defined by non-conventional three-term recurrences. They were recently introduced by Cigler and have been then employed by Cigler and Zeng to construct novel $q$-extensions of classical Hermite polynomials. We show that both of these $q$-polynomial families exhibit simple transformation properties with respect to the classical Fourier integral transform.'
---
[**[On Fourier integral transforms ]{}\
[for $q$-Fibonacci and $q$-Lucas polynomials]{}**]{}
[Natig Atakishiyev and Pedro Franco]{}
[Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional\
Autónoma de México, C.P. 62251 Cuernavaca, Morelos, México]{} [E-mail: natig@matcuer.unam.mx, pedro@matcuer.unam.mx]{}
[Decio Levi]{}
[Dipartimento di Ingegneria Elettronica\
Università degli Studi Roma Tre and INFN Sezione di Roma Tre\
Via della Vasca Navale 84, 00146 Roma, Italy]{}
[E-mail: levi@Roma3.infn.it]{}
[Orlando Ragnisco]{}
[Dipartimento di Fisica\
Università degli Studi Roma Tre and INFN Sezione di Roma Tre\
Via della Vasca Navale 84, 00146 Roma, Italy]{}
[E-mail: Ragnisco@Roma3.infn.it]{}
PACS numbers: 02.30.Gp, 02.30.Tb, 02.30.Vv
Mathematics Subject Classification: 33D45, 39A70, 47B39
Introduction
============
The Askey scheme of hypergeometric orthogonal polynomials and their $q$-analogues [@KSL] accumulates current knowledge about a large number of these special functions. Depending on a number of parameters, associated with each polynomial family, they occupy different levels within the Askey hierarchy: for instance, the Hermite polynomials $H_n(x)$ are on the ground level, the Laguerre and Charlier polynomials $L_n^{(\alpha)}(x)$ and $C_n(x;a)$ are one level higher, and so on. All polynomial families in this scheme are characterized by a “canonical" set of properties: they are solutions of differential or difference equations of the second order, they can be generated by three-term recurrence relations, they are orthogonal with respect to weight functions with finite or infinite supports, they obey Rodrigues-type formulas, and so on. Of course, many other polynomial families of interest arise both in pure and applied mathematics, which do not belong to the Askey $q$-scheme only because they lack some of the above mentioned characteristics properties. So this paper is aimed at exploring in detail two particular $q$-polynomial families of this type, namely, $q$-Fibonacci and $q$-Lucas polynomials, which are defined by non-conventional three-term recurrences. They were introduced in [@Cigl-I]–[@Cigl-III] and have been studied in detail in [@Cigl-Zeng]. The Fibonacci and Lucas sequences and polynomials have many physical applications; for example they appear in the study of diatomic chains [@lang], in dynamical systems and chaos theory [@sch], in Ising models [@ising], etc.
Our main result is to show that both of these $q$-polynomials exhibit simple transformation properties with respect to the classical Fourier integral transform.
Throughout this exposition we employ standard notations of the theory of special functions (see, for example, [@GR]–[@KSL]). In sections 2 and 3 we present some basic background facts about Fibonacci and Lucas polynomials and their $q$-extensions, respectively, which are then used in section 4 in order to find explicit forms of Fourier integral transforms for the $q$-Fibonacci and $q$-Lucas polynomials. Section 5 contains the conclusions and a brief discussion of some further research directions of interest. Finally, Appendix concludes this work with the derivation of two transformation formulas for hypergeometric ${}_2F_1$-polynomials, associated with the Chebyshev polynomials $T_n(x)$ and $U_n(x)$.
Fibonacci and Lucas polynomials
===============================
In this Section we review the basic well known facts about the Fibonacci and Lucas polynomials.
The Fibonacci polynomials
--------------------------
$F_{n}(x,s)$ are defined by the three-term recurrence relation F\_[n+1]{}(x,s) = xF\_[n]{}(x,s)+sF\_[n-1]{}(x,s),n1, with initial values $F_0(x,s)=0$ and $F_1(x,s)=1$ [@Cigl-I; @Cigl-II]. They are also given by the explicit sum formula F\_[n+1]{}(x,s)&=&\_[k=0]{}\^[n/2]{}[n-k ()k]{}s\^[k]{}x\^[n-2k]{}\
&=&x\^[n]{}\_2F\_1(-, ;-n|-4s/x\^2),n0, where ${\,n\,\atopwithdelims()\,k\,}= n!/k!(n-k)!$ is a binomial coefficient and $\lfloor x \rfloor$ denotes the greatest integer in $x$. The Fibonacci polynomials $F_{n}(x,s)$ have the following generating function f\_F(x,s;t):=\_[n=0]{}\^F\_[n]{}(x,s)t\^[n]{}=,|t|<1, easily derived through the three-term recurrence relation (2.1).
The Fibonacci polynomials $F_{n}(x,s)$ are normalized so that $F_n(x,1)=f_{n}(x)$ (where $f_{n}(x)$ are the Fibonacci polynomials introduced by Catalan ( see [@Koshy], formula (37.1) on p.443) and for the particular values of $x=s=1$ the recursion (2.1) generates a classical sequence of the Fibonacci numbers $\{F_{n}\}_{n=1}^{\infty}
\equiv\{1,1,2,3,5,8,13,...\}$ and the relation (2.3) reduces to the well-known generating function $f_F(1,1;t)$ for the Fibonacci numbers $\{F_n\}$, which “have been a source of delight to professional and amateur mathematicians for seven centuries” [@Bers] (see also [@Dunlap; @NalliHauk; @StakhAran]).
We call attention to the fact that the Fibonacci polynomials (2.2) (of degree $n$ in $x$ and of $\lfloor\,n/2\,\rfloor$ in $s$) can also be represented as F\_[n+1]{}(x,s)&=&(2)\^[n]{}p\_[n]{} \^[(F)]{}(),\
p\_n\^[(F)]{}(x)&:=&x\^[n]{}\_2F\_1(- ,;-n|-1/x\^2),so that the fundamental properties of $F_{n}(x,s)$ are basically defined by the [*monic*]{} polynomials $p_{\,n}^{\,(F)}(x)$.[^1] Moreover, it turns out that the polynomials $p_{\,n}^{\,(F)}(x)$ [*are essentially*]{} the Chebyshev polynomials of the second kind $U_n(z)$ in an imaginary argument $z\in\CC$ (see p.449 in [@Koshy]). Indeed, recall that the Chebyshev polynomials $U_n(z)$ have an explicit representation (see formula (23) on p.185 in [@HTF]) U\_[n]{}(z)=\_[k=0]{}\^[n/2]{}(-1) \^k[n-k()k]{}(2z)\^[n-2k]{}.Therefore from (2.2), (2.4) and (2.5) one readily deduces that F\_[n+1]{}(x,s)=(-[i]{}) \^[n]{}U\_n([i]{}x/2) and, consequently, $p_{\,n}^{\,(F)}(x)=({-\rm i}/2)^n\,U_n\Big
({\rm i}\,x\Big)$. Observe that from the three-term recurrence relation \[pippo1\] 2zU\_[n]{}(z)= U\_[n+1]{}(z)+U\_[n-1]{}(z) for the Chebyshev polynomials of the second kind $U_n(z)$ it follows at once that p\_[n+1]{}\^[(F)]{}(x) = xp\_[n]{}\^[(F)]{}(x)+14p\_[n-1]{} \^[(F)]{}(x),n1, with initial values $p_{\,0}^{\,(F)}(x)=1$ and $p_{\,1}^{\,(F)}(x)=x$. The coefficients in the three-term recurrence relations (2.8) are $A_n=1$ and $C_n=-1/4$; so that they do not satisfy the conditions $A_n\,C_{n+1}>0$ of Favard’s characterization theorem (see, for example, (7.1.5) on p.175 in [@GR]). This means that there is not a unique positive orthogonality measure for the polynomials $p_{\,n}^{\,(F)}(x)$.
Let us remark that:
1. Since the Chebyshev polynomials of the second kind $U_n(z)$ can be expressed in terms of the hypergeometric ${}_2F_1$ polynomials as (see, for example, (9.8.36) in [@KSL]) U\_n(z)=(n+1)\_2F\_1(-n,n+2;3/2|), then (2.6) is consistent with the second line in (2.2) only if a transformation formula (n+1)\_2F\_1(-n,n+2;3/2| )=(2 z)\^[n]{}\_2F\_1(-, ;-n|1/z\^[2]{}) is valid. A direct proof of (2.10) is given in the Appendix.
2. Since the Chebyshev polynomials of the second kind $U_n(x)$ are known to satisfy the second order differential equation (see (9.8.44) in [@KSL]) U\_n(x)=0, one readily deduces that p\_[n]{}\^[(F)]{}(x)=n(n+2)p\_[n]{}\^[(F)]{}(x).
3. The generating function for the Chebyshev polynomials of the second kind $U_n(x)$ is known to be of the form (see (9.8.56) in [@KSL]) \_[n=0]{}\^t\^[n]{}U\_[n]{}(x)=,|t|<1, so that combining (2.6) with (2.13) we get (2.3).
The Lucas polynomials
----------------------
$L_{n}(x,s)$ for $n\geq3$ ($L_0(x,s)=1$) are defined by the same three-term recurrence relation as in (2.1), but with initial values $L_1(x,s)=x$ and $L_2(x,s)=x^2+2s$ [@Cigl-Zeng]. They have the explicit sum formula L\_[n]{}(x,s)&=&\_[k=0]{}\^[n/2]{} [n-k()k]{}s\^[k]{}x\^[n-2k]{}\
&=& x\^[n]{}\_2F\_1(-,;1-n ;-),n0. The Lucas polynomials $L_{n}(x,s)$ have a generating function of the form f\_L(x,s;t):=\_[n=0]{}\^L\_[n]{}(x,s)t\^[n]{}=,|t|<1.To derive (2.15), multiply both sides of the three-term recurrence relation for $L_{n}(x,s)$ by the factor $t^{\,n+1}$ and sum all three terms with respect to the index $n$ from $n=2$ to infinity by taking into account initial values of $L_0(x,s)$, $L_1(x,s)$ and $L_2(x,s)$.
The Lucas polynomials $L_{n}(x,s)$ are normalized in such a way that $L_{n}(x,1)=l_{n}(x)$ (where $l_{n}(x)$ are the Lucas polynomials studied by Bicknell, see p.459 in [@Koshy]) and for the particular values of $x=s=1$ the sequence $\{L_n(1,1)\}_{n=1}^{\infty}$ reproduces Lucas numbers $\{L_n\}\equiv\{1,3,4,7,11,18,...\}$ and the relation (2.15) reduces to the generating function $f_L(1,1;t)$ for these numbers $\{L_n\}$ (see [@StakhAran] for applications of Lucas numbers and [@NalliHauk] for some generalizations of Lucas polynomials).
We call attention to the fact that the Lucas polynomials (2.14) (of degree $n$ in $x$ and of degree $\lfloor\,n/2\,\rfloor$ in $s$) can also be represented as L\_[n]{}(x,s)&=&s\^[n/2]{}p\_[n]{}\^[(L)]{}(),\
p\_n\^[(L)]{}(x)&:=&x\^[n]{}\_2F\_1(-,;1-n|-), so that the fundamental properties of $L_{n}(x,s)$ are basically defined by the monic polynomials $p_n^{\,(L)}(x)$. Moreover, it turns out that the polynomials $p_n^{\,(L)}(x)$ are in fact the Chebyshev polynomials of the first kind $T_n(z)$ in an imaginary argument $z$. Indeed, recall that the Chebyshev polynomials $T_n(z)$ have an explicit representation (see (23) on p.185 in [@HTF]) T\_[n]{}(z)&=&\_[k=0]{}\^[n/2]{}(2z)\^[n-2k]{}\
&& 2\^[n-1]{}z\^[n]{}\_2F\_1(-,;1-n|1/z\^[2]{}), n1. Therefore from (2.14), (2.16) and (2.17) it follows that L\_[0]{}(x,s)=1,L\_[n]{}(x,s)=2(-[i]{})\^[n]{} T\_n(),n1, and, consequently, $p_{\,0}^{\,(L)}(x)=1\,,\,p_{\,n}^{\,(L)}(x)=2
({-\rm i})^n\,T_n\Big ({\rm i}\,x/2\Big)\,,\,n\geq1$. Observe that from (2.16), (2.18) and the three-term recurrence relation (\[pippo1\]) for the Chebyshev polynomials of the first kind $T_n(z)$ it follows at once that p\_[n+1]{}\^[(L)]{}(x) = xp\_[n]{}\^[(L)]{}(x)+p\_[n-1]{} \^[(L)]{}(x),n1, with initial values $p_{\,0}^{\,(L)}(x)=1$ and $p_{\,1}^{\,(L)}
(x)=x$. The coefficients in the three-term recurrence relations (2.20) are $A_n=-\,C_n=1$; so that they do not satisfy the conditions $A_n\,C_{n+1}>0$ of Favard’s characterization theorem (see, for example, (7.1.5) on p.175 in [@GR]).
It is to be stressed that there are at least [*three*]{} direct consequences of the connection (2.18) between the Lucas polynomials $L_{n}(x,s)$ and the Chebyshev polynomials $T_n\Big({\rm i}\,x/2
\sqrt{s}\Big)$ of the first kind.
1. Since the Chebyshev polynomials of the first kind $T_n(z)$ can be written in terms of the hypergeometric ${}_2F_1$ polynomials as (see, for example, (9.8.35) in [@KSL]) T\_n(z)=\_2F\_1(-n,n;1/2|), then (2.18) is consistent with the second line in (2.14) only if a transformation formula \_2F\_1(-n,n;1/2|) =2\^[n-1]{}z\^[n]{}\_2F\_1(-,;1-n|1/z\^[2]{}),n1, is valid. A proof of this identity is given in Appendix.
2. Since the Chebyshev polynomials of the first kind $T_n(x)$ are known to satisfy the second order differential equation (see (9.8.43) in [@KSL]) T\_n(x)=0, from the relation $p_{\,n}^{\,(L)}(x)=2({-\rm i})^n\,T_n\Big ({\rm i}
\,x/2\Big)$ it follows at once that p\_[n]{}\^[(L)]{}(x)=n\^[2]{}p\_[n]{}\^[(L)]{}(x).
3. The generating function for the Chebyshev polynomials of the first kind $T_n(x)$ is known to be of the form (see (9.8.50) in [@KSL]) \_[n=0]{}\^t\^[n]{}T\_[n]{}(x)=,|t|<1, so that if one combines (2.18) with (2.25), this leads to (2.15).
$q$-Fibonacci and $q$-Lucas polynomials
=======================================
In this Section we review mainly tfacts about the $q$-Fibonacci and $q$-Lucas polynomials.
$q$-Fibonacci polynomials.
---------------------------
Cigler defined in [@Cigl-I; @Cigl-II] a novel $q$-analogue of Fibonacci polynomials $F_{n}(x,s)$, which satisfy the rather non-standard three-term recursion F\_[n+1]{}(x,s|q) = F\_[n]{} (x,s|q)+ sF\_[n-1]{}(x,s|q),n1, where the initial values are $F_{0}(x,s|\,q)=0$ and $F_{1}(x,s|\,q)=1$, and the Hahn $q$-difference operator ${\cal D}_q$ is defined as \_qf(x):=. The $q$-Fibonacci polynomials $F_n(x,s|\,q)$ are explicitly given in the form \[pippo2\] F\_[n+1]{}(x,s|q)= \_[k=0]{}\^[n/2]{}q\^[k(k+1) /2]{}[n-kk ]{}\_q s\^[k]{}x\^[n-2k]{} ,\
\[pippo3\] =x\^[n]{}\_4\_1(q\^[-n/2]{},q\^[(1-n)/2]{},-q\^[-n/2]{} ,-q\^[(1-n)/2]{};q\^[-n]{}|q;-),n0, where ${\,n\,\atopwithdelims []\,k\,}_q$ stands for the $q$-binomial coefficient, \_q :=, \[pippo3a\] and $(z;q)_n$ is the $q$-shifted factorial, that is, $(z;q)_0=1$, $\,(z;q)_n=\prod_{k=0}^{n-1}(1-zq^k)$ for $n\geq 1$. A $q$-extension of the generating function $f_F(x,s;t)$, associated with the $q$-Fibonacci polynomials $F_n(x,s|\,q)$, has the form ([*cf.*]{} (2.3)) f\_F(x,s;t|q)&:=& \_[n=0]{}\^F\_[n]{}(x,s|q)t\^[n]{} =\_1\_1(q;qxt|q;-qst\^2)\
&=& t\_2\_1(-,q;0|q;xt) ,|t|<1. \[pippo4\] The $q$-Fibonacci polynomials $F_n(x,s|\,q)$ are defined in such a way that in the limit as $q\to1$ they reduce to the polynomials $F_n(x,s)$, \[pippo5\] F\_n(x,s|1)\_[q1]{}F\_n(x,s|q)=F\_n(x,s),and the generating function (\[pippo4\]) for $F_n(x,s|\,q)$ coincides therefore in this limit with (2.3), associated with the polynomials $F_n(x,s)$. The appearance in (\[pippo4\]) of the two equivalent expressions in terms of the either ${}_1\phi_1$, or ${}_2\phi_1$ basic hypergeometric functions is fully consistent with the limit case of Heine’s transformation formula (see [@KSL], formula (1.13.13) on p.20) \[pippo6\] \_2\_1(a,b;0|q;z)=\_1\_1(b;bz|q;az), where $a=-\,qst/x$, $b=q$ and $z=xt$ (which means that the quotient $(bz;q)_{\infty}/(z;q)_{\infty}$ simply reduces in this case to the factor $1/(1-z)=1/(1-xt)$).
$q$-Lucas polynomials.
-----------------------
A $q$-extension of the Lucas polynomials $L_{n}(x,s)$ was introduced by Cigler in [@Cigl-II; @Cigl-III] as \[pippo7\] L\_[n]{}(x,s|q)=L\_n(x +(q-1)s[D]{} \_q,s)1, where the Hahn $q$-difference operator ${\cal D}_q$ is defined in (3.2). From (\[pippo7\]) and the three-term recurrence relation for $L_{n}(x,s)$ it then follows that the so introduced $q$-Lucas polynomials satisfy a non-standard three-term recurrence relation of the form of (3.1) for $L_{n}
(x,s\,|\,q)$ for $ n\geq2\,$. The initial values in this case are $L_{0}(x,s|\,q)=1$, $L_{1}(x,s\,|\,q)=x$ and $L_{2}(x,s\,|\,q)= x^2+(1+q)s$. The $q$-Lucas polynomials $L_n(x,s\,|\,q)$ have an explicit sum formula\
([*cf.*]{} (2.14) and (\[pippo2\])) \[pippo8\] L\_[n]{}(x,s|q)= \_[k=0]{}\^[n/2]{} q\^[k(k-1)/2]{}[n-kk]{} \_q s\^[k]{}x\^[n-2k]{}\
\[pippo9\] = x\^[n]{}\_4\_1(q\^[-n/2]{},q\^[(1-n)/2]{},-q\^[-n/2]{} ,-q\^[(1-n)/2]{};q\^[1-n]{}|q;- ), where $[n]_q:=(1-q^n)/(1-q)$. A $q$-extension of the generating function $f_L(x,s;t)$, associated with the $q$-Lucas polynomials $L_n(x,s\,|\,q)$, has the form ([*cf.*]{} (2.15)) f\_L(x,s;t|q)&:=&\_[n=0]{}\^L\_[n]{}(x,s|q)t\^[n]{} =\_1\_1(q;qxt|q; -qst\^2)\
&=&(1+st\^2)\_2\_1(-,q;0|q ;xt),|t|<1. \[pippo10\] In the limit as $q\to1$ the $q$-Lucas polynomials $L_n(x,s\,|\,q)$ reduce to the polynomials $L_n(x,s)$, given in (2.14), and the generating function (\[pippo10\]) for $L_n(x,s\,|\,q)$ coincides in this limit with (2.15). The appearance of the two equivalent expressions in (\[pippo10\]) in terms of the either ${}_1\phi_1$, or ${}_2\phi_1$ basic hypergeometric functions is, as before, fully consistent with a limit case of Heine’s transformation formula (\[pippo6\]).
Since the polynomials $F_{n}(x,s)$ and $L_{n}(x,s)$ satisfy the same recurrence relation (2.1) but with different initial conditions, they are known to be interconnected by the relation $L_{n}(x,s)=F_{n+1}
(x,s)+s\,F_{n-1}(x,s)$ ([*cf.*]{} the classical relation $2T_n(x)=U_n(x)
-U_{n-2}(x)$ for the Chebyshev polynomials). Similarly, a $q$-extension of this relation, \[pippo11\] L\_[n]{}(x,s|q)=F\_[n+1]{}(x,s|q)+sF\_[n-1]{}(x,s|q), interconnects two $q$-polynomial families $F_{n}(x,s|\,q)$ and $L_{n}(x,s|\,q)$. The relation (\[pippo11\]) is readily verified by using the explicit forms (\[pippo3\]) and (\[pippo9\]) of the polynomials $F_{n}(x,s|\,q)$ and $L_{n}(x,s|\,q)$ and an identity $${n-k\atopwithdelims[]k }_q\,=\,\frac{1-q^{n-k}}{1-q^{k}}
\,{n-k-1\atopwithdelims[]k -1}_q$$ for the $q$-binomial coefficient (\[pippo3a\]).
Fourier transforms of $F_n(x,s|\,q)$ and $L_n(x,s|\,q)$
=======================================================
In this section we derive explicit formulas of the classical Fourier integral transform for the $q$-Fibonacci and $q$-Lucas polynomials $F_n(x,s|\,q)$ and $L_n(x,s|\,q)$.
$q$-Fibonacci
-------------
Let us consider the $F_n(x,s|\,q)$ polynomials. To that end let us first define how this $q$-polynomial family changes under the transformation $q\to 1/q$. If one rewrites the defining sum formulas (\[pippo2\]) for $F_n(x,s|\,q)$ as \[pippo4.1\] F\_[n+1]{}(x,s|q)= \_[k=0]{}\^[n/2]{}c\_[n,k]{} \^[(F)]{}(q)s\^[k]{}x\^[n-2k]{}, then the coefficients in (\[pippo4.1\]) are \[pippo4.2\] c\_[n,k]{}\^[(F)]{}(q):= q\^[k(k+1)/2]{}[n-kk ]{}\_q.From definition of the $q$-binomial coefficient ${\,n\,\atopwithdelims[]
\,k\,}_q$ in (\[pippo3a\]) it is not hard to derive an inversion formula \_[1/q]{}=q\^[k(k-n)]{} [nk]{}\_q with respect to the change $q\to 1/q$. Consequently, from (4.2) and (4.3) it follows at once that c\_[n,k]{}\^[(F)]{}(q\^[-1]{})=q\^[k(k-n-1)]{}c\_[n,k]{}\^[(F)]{}(q). This means that ([*cf.*]{} (\[pippo3\])) $$F_{n+1}\Big(x,s\Big|\,q^{-1}\Big)\,\equiv\,\sum_{k=0}^{\lfloor\,n/2\,
\rfloor}\,c_{n,\,k}^{(F)}(q^{-1})\,s^{\,k}\,x^{\,n-\,2\,k}=\sum_{k=0}
^{\lfloor\,n/2\,\rfloor}\,q^{k(k-n-1)}\,c_{n,\,k}^{(F)}(q)\,s^{\,k}
\,x^{\,n-\,2\,k}$$ =x\^[n]{}\_4\_3(q\^[-n/2]{},q\^[(1-n)/2]{},-q\^[-n/2]{},- q\^[(1-n)/2]{};q\^[-n]{},0,0|q;-). Take into account the well-known Fourier transform $$\int_{\RR}\,e^{\,{\rm i}xy -\,x^2/2}\,dx\,=\,\sqrt{2\pi}\,e^{-\,y^2/2}$$ for the Gauss exponential function $e^{-\,x^2/2}$ and computing the Fourier integral transform of the exponential function $\exp\,[\,{\rm i}(n-2k)\,
\kappa\,x - x^2/2\,]$, we get \_e\^[[i]{}xy+ [i]{}(n-2k)x-x\^2/2]{} dx&=&e\^[-\[y+(n-2k)\]\^2/2]{}\
&=&q\^[n\^2/4]{}q\^[k(k-n)]{}e\^[-(n-2k) y-y\^2/2]{}, where $q=e^{-\,2{\kappa}^2}$.
We are now in a position to formulate and prove the following theorem.
The classical Fourier integral transform of the $q$-Fibonacci polynomials $F_{n+1}(a\,e^{\,{\rm i}\,\kappa\,x},s|\,q)$ times the Gauss exponential function $e^{-\,x^2/2}$ has the form: \_F\_[n+1]{}(ae\^[[i]{}x]{},s|q)e \^[[i]{}xy-x\^2/2]{}dx=q\^[n\^2/4]{}F\_[n+1]{} (ae\^[-y]{},qs|q\^[-1]{})e\^[-y\^2/2]{}, where $a$ is an arbitrary constant factor.
Using (\[pippo2\]), the Fourier integral transform (4.6) and the interrelation (4.4) between the coefficients $c_{n,\,k}^{(F)}(q)$ and $c_{n,\,k}^{(F)}(q^{-1})$, we get & & \_F\_[n+1]{}(ae\^[[i]{}x]{},s|q) e\^[[i]{}xy-x\^2/2]{}dx\
& &=\_[k=0]{}\^[n/2]{}c\_[n,k]{}\^[(F)]{}(q)s\^[k]{} a\^[n-2k]{}\_e\^[[i]{}xy+ [i]{}(n-2k) x-x\^2/2]{}dx\
& & =q\^[n\^2/4]{}e\^[-y\^2/2]{}\_[k=0]{}\^[n/2]{}q\^[k(k-n)]{}c\_[n,k]{}\^[(F)]{}(q)s\^[k]{}(ae\^[-y]{})\^[n-2k]{}\
& & =q\^[n\^2/4]{}e\^[-y\^2/2]{}\_[k=0]{}\^[n/2]{}c\_[n,k]{}\^[(F)]{}(q\^[-1]{})(qs)\^[k]{}(ae\^ [-y]{})\^[n-2k]{}\
& &=q\^[n\^2/4]{}F\_[n+1]{}(ae\^[-y]{} ,qs|q\^[-1]{})e\^[-y\^2/2]{}.
$q$-Lucas
---------
We turn now to determine an explicit form of classical Fourier integral transform for the $q$-Lucas polynomials $L_n(x,s\,|\,q)$. If one rewrites the defining sum formula (\[pippo8\]) for $L_n(x,s\,|\,q)$ as L\_[n]{}(x,s|q)= \_[k=0]{}\^[n/2]{} c\_[n,k]{}\^[(L)]{}(q)s\^[k]{}x\^[n-2k]{},then the coefficients in (4.8) are c\_[n,k]{}\^[(L)]{}(q):= q\^[k(k-1)/2]{} [n-kk ]{}\_q. From the starting definition of the symbol $[n]_q$ in (\[pippo8\]) it is not hard to show that \_[1/q]{}=q\^[1-n]{}\[n\]\_q. Consequently, from (4.3) and (4.10) it follows at once that c\_[n,k]{}\^[(L)]{}(q\^[-1]{})=q\^[k(k-n)]{} c\_[n,k]{}\^[(L)]{}(q). This means that ([*cf.*]{} (\[pippo8\], \[pippo9\])) $$L_{n}\Big(x,s\,\Big|\,q^{-1}\Big)\,\equiv\,\sum_{k=0}^{\lfloor\,n/2\,
\rfloor}\,c_{n,\,k}^{(L)}(q^{-1})\,s^{\,k}\,x^{\,n-\,2\,k}= \sum_{k=0}
^{\lfloor\,n/2\,\rfloor}\,q^{k(k-n)}\,c_{n,\,k}^{(L)}(q)\,s^{\,k}\,
x^{\,n-\,2\,k}$$ =x\^[n]{}\_4\_3(q\^[-n/2]{},q\^[(1-n)/2]{},-q\^[-n/2]{},-q\^ [(1-n)/2]{};q\^[1-n]{},0,0|q;-). The next step is to take into account the Fourier integral transform (4.6) in order to prove the following theorem.
The Fourier integral transform of the $q$-Lucas polynomials $L_{n}(b\,
e^{\,{\rm i}\,\kappa\,x},s\,|\,q)$ times the Gauss exponential function $e^{-\,x^2/2}$ has the form: \_L\_[n]{}(be\^[[i]{}x]{},s|q) e\^[[i]{}xy-x\^2/2]{}dx=q\^[n\^2/4]{} L\_[n]{}(be\^[-y]{},s|q\^[-1]{})e\^[-y\^2/2]{} , where $b$ is an arbitrary constant factor.
Starting from (\[pippo8\]) for the $q$-Lucas polynomials, using the Fourier integral transform (4.6) and employing the interrelation (4.11) between the coefficients $c_{n,\,k}^{(L)}(q)$ and $c_{n,\,k}^{(L)}(q^{-1})$, we get
& & \_L\_[n]{}(be\^[[i]{}x]{},s|q) e\^[[i]{}xy-x\^2/2]{}dx\
& &=\_[k=0]{}\^[n/2]{}c\_[n,k]{}\^[(L)]{}(q)s\^[k]{} b\^[n-2k]{}\_e\^[[i]{}xy+ [i]{}(n-2k) x-x\^2/2]{}dx\
& & =q\^[n\^2/4]{}e\^[-y\^2/2]{}\_[k=0]{}\^[n/2]{}q\^[k(k-n)]{}c\_[n,k]{}\^[(L)]{}(q)s\^[k]{}(be\^[-y]{})\^[n-2k]{}\
& & =q\^[n\^2/4]{}e\^[-y\^2/2]{}\_[k=0]{}\^[n/2]{}c\_[n,k]{}\^[(L)]{}(q\^[-1]{})s\^[k]{}(b e\^[-y]{})\^[n-2k]{}\
& &=q\^[n\^2/4]{}L\_[n]{}(be\^[-y]{} ,s|q\^[-1]{})e\^[-y\^2/2]{}.
Concluding remarks and outlook
==============================
We have studied in detail the transformation properties with respect to Fourier transform of the $q$-Fibonacci and $q$-Lucas polynomials, which are governed by the non-conventional three-term recurrences (3.1). In particular, we have proved that these families of $q$-polynomials exhibit a simple transformation behavior (4.7) and (4.13) under the classical Fourier integral transform.
Let us emphasize that one actually may use the Mehta–Dahlquist–Matveev techniques (see [@Mehta; @Dahl; @Matv; @Ata; @NMARueW07]) in order to show that the Fourier integral transformation formulas (4.7) and (4.13) in fact entail a similar behavior of the $q$-Fibonacci and $q$-Lucas polynomials under the discrete (finite) Fourier transform as well.
It is worthwhile to mention here that there are other possibilities (than Cigler’s $F_{n}(x,s|\,q)$ and $L_{n}(x,s|\,q)$) for constructing $q$-extensions of Fibonacci and Lucas polynomials of interest. For instance, two monic $q$-polynomial families r\_[n]{}\^[(F)]{}(x|q)=x\^[n]{}\_2\_1(q\^[-n]{}, q\^[1-n]{};q\^[-2n]{}|q\^[2]{};-), r\_[n]{}\^[(L)]{}(x|q)=x\^[n]{}\_2\_1(q\^[-n]{}, q\^[1-n]{};q\^[2(1-n)]{}|q\^2;-), represent very natural extensions of the Fibonacci and Lucas polynomials $p_{\,n}^{(F)}(x)$ and $p_{\,n}^{(L)}(x)$, defined in (2.4) and (2.16) respectively. Contrary to $F_{n}(x,s|\,q)$ and $L_{n}(x,s|\,q)$, these $q$-polynomial families do satisfy standard three-term recurrence relations of the form r\_[n+1]{}(x|q)=xr\_[n]{}(x|q)+ r\_[n-1]{}(x|q),where by $r_{n}$ we mean either $r_{n}^{(F)}$ or $r_{n}^{(L)}$. Moreover, the $q$-polynomials $r_{n}^{(F)}(x|\,q)$ and $r_{n}^{(L)}(x|\,q)$ are associated with the monic $q$-polynomial families \[5.5\] s\_[n]{}\^[(U)]{}(x|q)=[i]{}\^[-n]{}r\_[n]{}\^[(F)]{}([i]{}x|q),s\_[n]{}\^[(T)]{}(x|q)=[i]{}\^[-n]{}r\_[n]{}\^[(L)]{}([i]{}x|q), which do satisfy the conditions of Favard’s characterization theorem; thus they can be viewed as natural $q$-extensions of the Chebyshev polynomials $U_n(x)$ and $T_n(x)$. It should be noted that the polynomials $s_{n}^{(U)}(x|\,q)$ and $s_{n}^{(T)}(x|\,q)$, explicitly given by \[5.6\] s\_[n]{}\^[(U)]{}(x|q)=x\^[n]{}\_2\_1(q\^[-n]{}, q\^[1-n]{};q\^[-2n]{}|q\^[2]{};), s\_[n]{}\^[(T)]{}(x|q)=x\^[n]{}\_2\_1(q\^[-n]{}, q\^[1-n]{};q\^[2(1-n)]{}|q\^2;), \[5.6\] are of interest on their own.
It is well known that the Chebyshev polynomials $U_n(x)$ and $T_n(x)$ are the special cases of the Jacobi polynomials $P_{n}^{(\alpha,\beta)}(x)$ with the parameters $\alpha=\beta=1/2$ and $\alpha=\beta=-1/2$ respectively. Therefore it seems natural to expect that the continuous $q$-Jacobi polynomials $P_{n}^{(\alpha,\beta)}(x|\,q)$ (which evidently represent $q$-extensions of the Jacobi polynomials $P_{n}^{(\alpha,\beta)}(x)$) with the particular values the parameters $\alpha=\beta=1/2$ and $\alpha=\beta=-1/2$, could provide appropriate $q$-extensions of the Chebyshev polynomials $U_n(x)$ and $T_n(x)$ respectively. Under closer examination however, it turns out that the continuous $q$-Jacobi polynomials $P_{n}^{(1/2,1/2)}(x|\,q)$ and $P_{n}^{(-1/2,-1/2)}(x|\,q)$ [*are only constant (but $q$-dependent) multiples*]{} of the Chebyshev polynomials $U_n(x)$ and $T_n(x)$. In other words, the continuous $q$-Jacobi polynomials $P_{n}^{(1/2,1/2)}(x|\,q)$ and $P_{n}^{(-1/2,-1/2)}(x|\,q)$ differ from the Chebyshev polynomials $U_n(x)$ and $T_n(x)$ only for the choice of the normalization constants; therefore the former two polynomial families are just trivial $q$-extensions of the latter ones.[^2]
The polynomials $ s_{n}^{(U)}(x|\,q)$ and $ s_{n}^{(T)}(x|\,q)$ can be thus viewed as non-trivial compact $q$-extensions of the Chebyshev polynomials $U_n(x)$ and $T_n(x)$, which do not match with the continuous $q$-Jacobi polynomials $P_{n}^{(1/2,1/2)}(x|\,q)$ and $P_{n}^{(-1/2,-1/2)}(x|\,q)$. Observe that both of them can be expressed in terms of the little $q$-Jacobi polynomials $p_{n}(x;a,b|\,q)$ as $$s_{2n}^{(U)}(x|\,q)\,=\,(-1)^n\,q^{n(n-1)}\frac{(q;q^2)_n}{(q^{2(n+1)};q^2)_{n}}
\,p_{n}\Big(x^2;q^{-1},q\Big|\,q^2\Big)\,,$$ \[5.8\] s\_[2n+1]{}\^[(U)]{}(x|q)=(-1)\^nq\^[n(n-1)]{} x p\_[n]{}(x\^2;q,q|q\^2), and $$s_{2n}^{(T)}(x|\,q)\,=\,(-1)^n\,q^{n(n-1)}\frac{(q;q^2)_n}{(q^{2n};q^2)_{n}}
\,p_{n}\Big(x^2;q^{-1},q^{-1}\Big|\,q^2\Big)\,,$$ \[5.9\] s\_[2n+1]{}\^[(T)]{}(x|q)=(-1)\^nq\^[n(n-1)]{} x p\_[n]{}(x\^2;q,q\^[-1]{}|q\^2), where $p_{n}(x;a,b\,|\,q):={}_2\phi_1(q^{- n}, ab\,q^{n+1}; aq\,|\,q\,; qx)$ (see, for example, (14.12.1), p.482 in [@KSL]). It is of considerable interest to examine the properties of the polynomials $ s_{n}^{(U)}(x|\,q)$ and $ s_{n}^{(T)}(x|\,q)$ in more detail, including their transformation properties with respect to the Fourier integral transform. This project is beyond the subject of this paper and will be dealt with elsewhere.
Acknowledgements {#acknowledgements .unnumbered}
================
Discussions with T.H.Koornwinder and K.B.Wolf are gratefully acknowledged. One of us (NMA) would like to thank the Physics Department and Department of Electronic Engineering, University Roma Tre, Italy, for the hospitality extended to him during his visit in September–October 2011, when the final part of this work was carried out. The participation of NMA in this work has been supported by the DGAPA-UNAM IN105008-3 and SEP-CONACYT 79899 projects “Óptica Matemática". DL and OR have been partly supported by the Italian Ministry of Education and Research, 2010 PRIN “Continuous and discrete nonlinear integrable evolutions: from water waves to symplectic maps".
Proof of (2.10)
===============
In order to give a direct proof of the transformation formula (2.10) we start by the defining relation for the hypergeometric ${}_2F_1$-polynomial on the left side of (2.10) and rewrite it \_2F\_1(-n,n+2;3/2|) &:=& \_[k=0]{}\^n\
&=&\_[k=0]{}\^n [n()k]{}\_[l=0]{}\^k[k()l]{}(-z)\^l, by employing the relation $(-n)_k=(-1)^k\,n!/(n-k)!$. The next step is to reverse the order of summation in (A.1) with respect to the indices $k$ and $l$, which leads to the relation
$${}_2F_1\Big(-n\,,n+2\,;\,3/2\,\Big|\frac{1-z}{2}\,\Big)$$ =\_[l=0]{}\^n \_[k=0]{}\^l . The sum over index $k$ in (A.2) represents the hypergeometric polynomial $${}_2F_1\Big(-l\,,2n+2-l\,;\,n-l+3/2\,\Big|\,x\,\Big)$$ for a special value of the variable $x=1/2$, which can be evaluated by Gauss’s second summation theorem (see, for example, (1.7.1.9) on p.32 in [@Slat]) \_2F\_1(2a,2b;a+b+1/2|1/2)&=&,\
a+b+1/2&&-m,m0, with $a=-\,l/2$ and $b=n+1-l/2$ (so that $a+b+1/2=n-l+3/2\geq3/2$ for all $0\leq l\leq n$). The sum over index $k$ in (A.2) thus reduces to \_2F\_1(-l,2n+2-l;n-l+3/2|1/2) =. Since the gamma function $\Gamma(z)$ has poles at the points $z=-\,n$, $\,n\geq 0$, the right-hand side of (6.5) vanishes for all odd values of the index $l$ due to the presence of the factor $\Gamma\Big((1-l)/2\Big)$ in its denominator. This means that only terms with even $l$’s give non-zero contribution into the sum over $l$ in (A.2), that is, & & \_2F\_1(-n,n+2;3/2|)\
& &=\_[m=0]{}\^[n/2]{}\
& &=\_[m=0]{}\^[n/2]{} , where at the last step we employed duplication formula (2z)=(z) (z+1/2) for the gamma function $\Gamma(z)$ and the relation $\Gamma(m+1/2)
\,\Gamma(1/2-m)=\pi/\cos{m\pi}=(-1)^m\,\pi$. Finally, it remains only to use the identity $\Gamma(z+1-n)=(-1)^n\,\Gamma(z+1)/\,(-z)_n$ and again the duplication formula (A.6) in order to show that & & \_2F\_1(-n,n+2;3/2| )\
& &= \_[m=0]{}\^[n/2]{}\
& &= \_[m=0]{}\^[n/2]{}\
& &=\_2F\_1(-, ;-n|z\^[-2]{}). This completes the proof of transformation formula (2.10).
Proof of (2.21)
===============
We prove here the transformation formula (2.21) which was stated in section 2. One starts with the defining relation for the hypergeometric ${}_2F_1$ polynomial on the left side of (2.21) and evaluates first (for $n\geq1$) that \_2F\_1(-n,n;1/2| )&:=& \_[k=0]{}\^n\
&=&\_[k=0]{}\^n [n()k]{}\_[l=0]{}\^k[k()l]{}(-z)\^l, by employing the relation $(-n)_k=(-1)^k\,n!/(n-k)!$. The next step is to reverse the order of summation in (B.1) with respect to the indices $k$ and $l$, which leads to the relation $${}_2F_1\Big(-n\,,n\,;\,1/2\,\Big|\frac{1-z}{2}\,\Big)$$ =n(1/2)\_[l=0]{}\^n \_[k=0]{}\^l. The sum over index $k$ in (B.2) represents the hypergeometric polynomial $${}_2F_1\Big(-l\,,2n-l\,;\,n-l+1/2\,\Big|\,x\,\Big)$$ for a special value of the variable $x=1/2$, which can be evaluated by Gauss’s second summation theorem (see, for example, (1.7.1.9) on p.32 in [@Slat]) \_2F\_1(2a,2b;a+b+1/2|1/2)&=&,\
a+b+1/2&&-m,m0, with parameters $a=-\,l/2$ and $b=n-l/2$ (so that $a+b+1/2=n-l+1/2
\geq1/2$ for all $0\leq l\leq n$). The sum over index $k$ in (B.2) thus reduces to \_2F\_1(-l,2n-l;n-l+1/2|1/2) =. Since the gamma function $\Gamma(z)$ has poles at the points $z=-\,n$, $\,n\geq 0$, the right-hand side of (B.4) vanishes for all odd values of the index $l$ due to the presence of the factor $\Gamma\Big((1-l)/2\Big)$ in its denominator. This means that only terms with even $l$’s give non-zero contribution into the sum over $l$ in (B.2), that is, & & \_2F\_1(-n,n;1/2|)\
& &=n\_[m=0]{}\^[n/2]{}\
& &=\_[m=0]{}\^[n/2]{} , where at the last step we employed duplication formula (A.6) for the gamma function $\Gamma(z)$ and the relation $\Gamma(m+1/2)
\,\Gamma(1/2-m)=\pi/\cos{m\pi}=(-1)^m\,\pi$. Finally, it remains only to use the identity $\Gamma(z+1-n)=(-1)^n\,\Gamma(z+1)/
\,(-z)_n$ and again the duplication formula (A.6) in order to show that
& & \_2F\_1(-n,n;1/2| )\
& &= \_[m=0]{}\^[n/2]{}\
& &= 2\^[n-1]{}\_[m=0]{}\^[n/2]{}\
& &=2\^[n-1]{}z\^n\_2F\_1(-,;1-n|1/z\^[2]{}). This completes the proof of the transformation formula (2.21).
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[^1]: We recall that an arbitrary polynomial $p_n(x)=\sum_{k=0}^n c_{n,\,k}\,x^k$ of degree $n$ can be written in the [*monic form*]{} $p_n^{(M)}(x)=c_{n,\,n}^{-1}\,
p_n(x)=x^n + c_{n,\,n}^{-1}\sum_{k=0}^{n-1}c_{n,\,k}\,x^k$ just by changing its normalization.
[^2]: This curious “$q$-degeneracy" of the continuous $q$-Jacobi polynomials $P_{n}^{(\alpha,\beta)}(x|\,q)$ for the values of the parameters $\alpha=\beta=1/2$ and $\alpha=\beta=-1/2$ was first noticed by R.Askey and J.A.Wilson in their seminal work [@AW]. We are grateful to Tom Koornwinder for reminding us of this fact.
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\[theorem\][Acknowledgement]{} \[theorem\][Algorithm]{} \[theorem\][Axiom]{} \[theorem\][Claim]{} \[theorem\][Conclusion]{} \[theorem\][Condition]{} \[theorem\][Conjecture]{} \[theorem\][Corollary]{} \[theorem\][Criterion]{} \[theorem\][Definition]{} \[theorem\][Example]{} \[theorem\][Exercise]{} \[theorem\][Lemma]{} \[theorem\][Notation]{} \[theorem\][Problem]{} \[theorem\][Proposition]{} \[theorem\][Remark]{} \[theorem\][Solution]{} \[theorem\][Summary]{}
Introduction
============
The optical properties of coherently prepared atoms have been much studied in recent years. The simplest case is when one transition is resonantly pumped with a strong coherent laser beam and the resulting dressed system probed from a third level in a $\Lambda $, V or cascade configuration. These systems can exhibit electromagnetically-induced transparency (EIT) where quantum interference effects allow a resonant probe beam to propagate without absorption and with greatly-enhanced dispersion [@bib:EIT]. The EIT effect has been exploited in a variety of applications: gain without inversion [@bib:GWI], ground state cooling of trapped atoms[@bib:Cooling], ultra-slow light-speed propagation [@bib:UltraSlow], quantum non-demolition measurements [@bib:QND] and efficient non-linear optical processes [@bib:NLO]. This has lead to interest in more complex systems involving multiple electromagnetic fields interacting with either a single transition [@bib:PolyChro] or with several transitions [@bib:Sandhya1997][@bib:Sadeghi1997][@bib:Wei1998] [@bib:Q4WM]. In particular there is now huge interest in exploiting non-linearities in four-level systems with applications in quantum optics [@bib:QNLO] and four-wave mixing [@bib:Q4WM].
Four-level systems of various configurations can be excited with different pumping schemes. Several theoretical studies have considered three-level ladder configurations coherently prepared by two strong laser beams [@bib:Sandhya1997][@bib:Sadeghi1997][@bib:Wei1998]. They predict a three-peak spectrum when the lowest or highest of the three levels is probed via a fourth level or monitored by fluorescence. When the two dressing beams are on resonance this spectrum has the form of a Doppler-free three-photon absorption peak situated centrally within an EIT window [@bib:Sandhya1997]. More generally, the positions and intensities of the three peaks are functions of the Rabi frequencies and detunings of the two strong beams. The trajectories of the peaks as a function of Rabi frequencies and detunings have been described in doubly dressed analyses [@bib:Sadeghi1997][@bib:Wei1998]. None of these three studies was accompanied by experimental verifications.
In this paper we consider a similar scheme to those of references [@bib:Sandhya1997][@bib:Sadeghi1997][@bib:Wei1998] but with the two strong pump beams, or coupling beams as we shall call them, connecting a ground or metastable level to two excited levels. The resulting dressed V system can then be probed from another ground or metastable level, as illustrated in Figure 1(a). An experimental realisation in $^{87}$Rb cooled in a magneto-optic trap (MOT) is indicated in Figure 1(b). Our interest in this system stems from our current programme of work on the strong cross- and self-phase modulation expected in the N-configuration of Figure 1, and the main aim of this paper is to characterise the weak-probe spectrum of this system. For this task we have to be concerned not only with the behaviour of the ideal four-level model (Figure 1(a)) but also with light shifts due to off-resonant interactions with other nearby hyperfine levels and the effects of level degeneracies in the $^{87}$Rb system. Because our experiments are carried out in a Rb MOT, we are free to propagate our beams in any desired directions, and to choose coupling beams on different optical transitions that are well-separated in frequency thus avoiding potentially complex mutual light shifts induced by the two beams, although some simple light shifts and the effects of the trapping beams do have to be taken into account in the system of Figure 1(a).
\
Many manifestations of Zeeman degeneracy have been reported in EIT experiments: optical pumping among Zeeman levels and to other levels [@bib:ZeemanPumping], coherent population trapping by the coupling beam [@bib:CPT], the inversion of EIT dips (i.e. electromagnetically induced absorption) [@bib:EIA], and absorption of the probe beam on Zeeman transitions that are not coupled by the pump beam[@bib:UncoupledAbsorption]. These latter uncoupled absorptions are a constant feature of our probe spectra, superimposed on the V system spectra we are interested in.
This paper is organised as follows. Section II presents a dressed state analysis of the strongly-pumped model V system of Figure 1(a), showing the expected dependance of the probe absorption spectrum on Rabi frequencies and detunings. Section III describes the experimental arrangement and presents EIT spectra with the first coupling field $C_{1}$ applied and the second coupling field $C_{2}$ switched off, thus showing the effects of light shifts and uncoupled absorptions in a relatively simple system. Our main results are presented in Section IV where both coupling fields are applied and probe spectra obtained for various coupling field intensities and detunings. The spectra are interpreted using the 4-state theory of Section II modified phenomenologically by the light shift and uncoupled absorption effects described in Section III. We justify the use of the four-state model and we describe the origin of the uncoupled absorptions relevant to our experiments in the Appendix.
Theory
======
Figure 1(a) shows a four state N configuration, although for the purposes of this discussion we shall consider it as a three state V scheme formed by the states $\left| a\right\rangle $, $\left| c\right\rangle $ and $\left|
d\right\rangle $ and strong coupling fields $C_{1}$ and $C_{2}$ of frequencies $\omega _{1}$ and $\omega _{2}$, probed weakly on the $\left|
b\right\rangle -\left| d\right\rangle $ transition. The Hamiltonian describing the system consisting of the atom and the two coupling fields can be written down in the semiclassical and rotating wave approximations as $${\cal H}={\cal A}+{\cal I}$$ where $$\begin{aligned}
{\cal A} &=&\Delta _{1}\left| a\right\rangle \left\langle a\right| +\left(
\Delta _{1}-\Delta _{2}\right) \left| c\right\rangle \left\langle c\right| \\
{\cal I} &=&\frac{\Omega _{1}}{2}\left( \left| d\right\rangle \left\langle
a\right| +\left| a\right\rangle \left\langle d\right| \right) +\frac{\Omega
_{2}}{2}\left( \left| c\right\rangle \left\langle a\right| +\left|
a\right\rangle \left\langle c\right| \right) .\end{aligned}$$ ${\cal A}$ and ${\cal I}$ represent the atomic and interaction parts of the Hamiltonian ${\cal H}$. $\Delta _{1}=\omega _{1}-\omega _{da}$ ($\Delta
_{2}=\omega _{2}-\omega _{ca}$) is the detuning of coupling field $C_{1}$ ($%
C_{2}$) from the $\left| a\right\rangle -\left| d\right\rangle $ ($\left|
a\right\rangle -\left| c\right\rangle $) transition and $\omega _{\beta
\alpha }$ is the transition frequency of the $\left| \alpha \right\rangle
-\left| \beta \right\rangle $ transition for $\alpha ,\beta =a,b,c,d$. $%
\Omega _{j}={\bf d}_{j}\cdot {\bf E}_{j}$ is the Rabi frequency for field $%
j=1,2$ where the field ${\bf E}_{j}$ interacts only with its quasi-resonant transition with electric dipole moment ${\bf d}_{j}$. We have chosen units such that $\hbar =1$ so energies are measured in units of frequency.
Because there are three basis states which describe the system, in the most general case, we expect a characteristic three line spectrum when performing probe absorption experiments. Writing out the Hamiltonian in matrix form gives $${\cal H}=\left[
\begin{array}{lll}
\Delta _{1} & \Omega _{2}/2 & \Omega _{1}/2 \\
\Omega _{2}/2 & \Delta _{1}-\Delta _{2} & 0 \\
\Omega _{1}/2 & 0 & 0
\end{array}
\right] . \label{eq:HamiltonianMatrix}$$
Following the method in Shore [@bib:Shore1990] we derive the doubly dressed state energies $$\begin{aligned}
{\cal E}_{1} &=&-\frac{1}{3}\alpha +\frac{2}{3}p\cos \left( \frac{\Theta }{3}%
\right) \\
{\cal E}_{2} &=&-\frac{1}{3}\alpha -\frac{2}{3}p\cos \left( \frac{\Theta }{3}%
+\frac{\pi }{3}\right) \\
{\cal E}_{3} &=&-\frac{1}{3}\alpha -\frac{2}{3}p\cos \left( \frac{\Theta }{3}%
-\frac{\pi }{3}\right)\end{aligned}$$ with $$\begin{aligned}
\alpha &=&-2\Delta _{1}+\Delta _{2} \\
\beta &=&\Delta _{1}(\Delta _{1}-\Delta _{2})-\frac{1}{4}\left( \Omega
_{1}^{2}+\Omega _{2}^{2}\right) \\
\gamma &=&\frac{1}{4}\left( \Delta _{1}\Omega _{1}^{2}-\Delta _{2}\Omega
_{1}^{2}\right) \\
p &=&\sqrt{\alpha ^{2}-3\beta } \\
\cos \Theta &=&-\frac{27\gamma +2\alpha ^{3}-9\alpha \beta }{2p^{3}}.\end{aligned}$$ The corresponding dressed state vector for energy ${\cal E}_{\nu }$ is $$\left| {\cal D}_{\nu }\right\rangle =\frac{\left( {\cal E}_{\nu }\frac{%
\Omega _{2}}{2}\left| a\right\rangle +\left( {\cal E}_{\nu }\left( {\cal E}%
_{\nu }-\Delta _{1}\right) -\frac{\Omega _{1}^{2}}{4}\right) \left|
c\right\rangle +\frac{\Omega _{1}\Omega _{2}}{4}\left| d\right\rangle
\right) }{{\cal N}_{\nu }}$$ with $${\cal N}_{\nu }=\sqrt{{\cal E}_{\nu }^{2}\frac{\Omega _{2}^{2}}{4}+\left(
{\cal E}_{\nu }\left( {\cal E}_{\nu }-\Delta _{1}\right) -\frac{\Omega
_{1}^{2}}{4}\right) ^{2}+\frac{\Omega _{1}^{2}\Omega _{2}^{2}}{16}}$$ being the normalisation constant and $\nu =1,2,3$ indexing the dressed state.
We now consider the special case of mutual resonance of the two coupling fields, i.e. $\Delta _{1}=\Delta _{2}=0$. In this case the dressed state energies and vectors are $$\begin{aligned}
{\cal E}_{1} &=&\frac{1}{2}\sqrt{\Omega _{1}^{2}+\Omega _{2}^{2}} \\
{\cal E}_{2} &=&0 \\
{\cal E}_{3} &=&-\frac{1}{2}\sqrt{\Omega _{1}^{2}+\Omega _{2}^{2}}\end{aligned}$$ and $$\begin{aligned}
\left| {\cal D}_{1}\right\rangle &=&\frac{1}{\sqrt{2}}\left( \left|
a\right\rangle +\frac{\Omega _{2}}{\sqrt{\Omega _{1}^{2}+\Omega _{2}^{2}}}%
\left| c\right\rangle +\frac{\Omega _{1}}{\sqrt{\Omega _{1}^{2}+\Omega
_{2}^{2}}}\left| d\right\rangle \right) \\
\left| {\cal D}_{2}\right\rangle &=&0\left| a\right\rangle -\frac{\Omega
_{1}}{\sqrt{\Omega _{1}^{2}+\Omega _{2}^{2}}}\left| c\right\rangle +\frac{%
\Omega _{2}}{\sqrt{\Omega _{1}^{2}+\Omega _{2}^{2}}}\left| d\right\rangle \\
\left| {\cal D}_{3}\right\rangle &=&\frac{1}{\sqrt{2}}\left( -\left|
a\right\rangle +\frac{\Omega _{2}}{\sqrt{\Omega _{1}^{2}+\Omega _{2}^{2}}}%
\left| c\right\rangle +\frac{\Omega _{1}}{\sqrt{\Omega _{1}^{2}+\Omega
_{2}^{2}}}\left| d\right\rangle \right) .\end{aligned}$$ These results can be used to predict the spectrum obtained when probing the doubly driven $\left| c\right\rangle -\left| d\right\rangle $ transition via the $\left| b\right\rangle -\left| d\right\rangle $ transition with a weak field of frequency $\omega _{p}$ and detuning $\Delta _{p}=\omega
_{p}-\omega _{db}$. The resulting spectrum can be thought of as being made up of a Rabi split doublet, with effective Rabi frequency $\Omega _{\text{eff%
}}=\sqrt{\Omega _{1}^{2}+\Omega _{2}^{2}},$ and a three-photon resonance absorption peak at $\Delta _{p}=0$. The central peak is strictly only on the three-photon resonance (i.e. satisfies $\Delta _{p}-\Delta _{1}+\Delta
_{2}=0$) when $\Delta _{1}=\Delta _{2}$, or in appropriate limits. However, for simplicity we shall refer to the central peak as a three-photon absorption peak when it closely approximates this resonance condition. We note that the probe coupling will be dominated by the $\left| b\right\rangle
-\left| d\right\rangle $ transition as the other transitions are significantly off resonance in the bare atomic basis. Also, in the strong coupling regime ($\Omega _{1},\Omega _{2}>>\Omega _{p}$), which applies in our experiments, almost all the population is optically pumped into the state $\left| b\right\rangle $ with negligible population in the dressed states. Under these conditions the probe absorption will be proportional to the coefficient $A_{\nu }=\left| \left\langle d|{\cal D}_{\nu }\right\rangle
\right| ^{2}$. This implies that the peaks corresponding to the absorption from state $\left| b\right\rangle $ to the outer states $\left| {\cal D}%
_{1}\right\rangle $ and $\left| {\cal D}_{3}\right\rangle $ will have equal heights and dominate the spectrum in the limit $\Omega _{1}/\Omega _{2}>>1$, whilst the three-photon absorption peak, corresponding to absorption to $%
\left| {\cal D}_{2}\right\rangle $, will dominate in the limit $\Omega
_{2}/\Omega _{1}>>1$. Graphs showing the energy levels ${\cal E}_{\nu }$ and coefficients $A_{\nu }$ as a function of $\Omega _{2}/\Omega _{1}$ are presented in Figures 2(a) and (b) respectively. These results are similar to those presented for a ladder system by Wei [*et al*]{}. [@bib:Wei1998]
\
Experimental arrangement, light shifts and EIT effects
======================================================
The experimental setup is shown schematically in Figure 3. The cold $^{87}$Rb sample contains between $10^{7}$ and $10^{8}$ atoms in a region of diameter approximately $3%
%TCIMACRO{\unit{mm}}%
%BeginExpansion
\mathop{\rm mm}%
%EndExpansion
$ in a standard MOT similar to the one used in our previous work on EIT[@bib:EITExp].
\
The laser fields are obtained from external-cavity grating-controlled diode lasers (ECDL) in master and master-slave arrangements. The frequency of each master laser is monitored by saturated absorption in a room temperature Rb cell and can be locked via electronic feedback. In all master-slave arrangements, there is an acousto-optic modulator which shifts the frequency of the slave relative to the frequency of the master. All laser fields, except for the second coupling field, were derived from $780%
%TCIMACRO{\unit{nm}}%
%BeginExpansion
\mathop{\rm nm}%
%EndExpansion
$ laser diodes, whilst the second coupling field was derived from a $795%
%TCIMACRO{\unit{nm}}%
%BeginExpansion
\mathop{\rm nm}%
%EndExpansion
$ laser diode.
The trapping lasers $T$ (not shown in Figure 3) are derived from a master-slave system. They are locked and detuned by $-13%
%TCIMACRO{\unit{MHz}}%
%BeginExpansion
\mathop{\rm MHz}%
%EndExpansion
$ from the $5S_{1/2}F=2$ to $5P_{3/2}F=3$ transition. The trapping beam diameters are $\approx 1%
%TCIMACRO{\unit{cm}}%
%BeginExpansion
\mathop{\rm cm}%
%EndExpansion
$ and the total average intensity in the cold atom sample is $\approx 60%
%TCIMACRO{\unit{mW}}%
%BeginExpansion
\mathop{\rm mW}%
%EndExpansion
/%
%TCIMACRO{\unit{cm}}%
%BeginExpansion
\mathop{\rm cm}%
%EndExpansion
^{2}$.
The probe beam $P$ is derived from an ECDL and is scanned across the $%
5S_{1/2}F=2$ to $5P_{3/2}F=2$ transition by piezo-control of the external cavity. $P$ has an average intensity $\approx 2%
%TCIMACRO{\unit{mW}}%
%BeginExpansion
\mathop{\rm mW}%
%EndExpansion
/%
%TCIMACRO{\unit{cm}}%
%BeginExpansion
\mathop{\rm cm}%
%EndExpansion
^{2}$ in a diameter $\approx 1%
%TCIMACRO{\unit{mm}}%
%BeginExpansion
\mathop{\rm mm}%
%EndExpansion
$.
The coupling beam $C_{1}$ and the trap repumping field $R$ are derived from the same laser which is the slave of a locked ECDL. They are resonant with the $5S_{1/2}F=1$ to $5P_{3/2}F=2$ transition. The average intensity of $%
C_{1}$ is $\approx 400%
%TCIMACRO{\unit{mW}}%
%BeginExpansion
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%EndExpansion
/%
%TCIMACRO{\unit{cm}}%
%BeginExpansion
\mathop{\rm cm}%
%EndExpansion
^{2}$ in a roughly elliptical profile $2%
%TCIMACRO{\unit{mm}}%
%BeginExpansion
\mathop{\rm mm}%
%EndExpansion
\times 4%
%TCIMACRO{\unit{mm}}%
%BeginExpansion
\mathop{\rm mm}%
%EndExpansion
$.
A second coupling beam $C_{2}$ is derived from an ECDL and is quasi-resonant with the $5S_{1/2}F=1$ to $5P_{1/2}F=2$ transition at $795%
%TCIMACRO{\unit{nm}}%
%BeginExpansion
\mathop{\rm nm}%
%EndExpansion
$ (the D$_{1}$ line). For resonant experiments, $C_{2}$ was locked, but for detuned experiments it was stepped using the external cavity piezo with the frequency determined by an optical spectrum analyser. The average intensity of $C_{2}$ can be varied up to a maximum of $\approx 300%
%TCIMACRO{\unit{mW}}%
%BeginExpansion
\mathop{\rm mW}%
%EndExpansion
/%
%TCIMACRO{\unit{cm}}%
%BeginExpansion
\mathop{\rm cm}%
%EndExpansion
^{2}$, in a beam diameter $\approx 1.3%
%TCIMACRO{\unit{mm}}%
%BeginExpansion
\mathop{\rm mm}%
%EndExpansion
$.
The probe $P$ and the second coupling field $C_{2}$ are linearly polarised in the horizontal plane whilst the coupling field $C_{1}$ is linearly polarised in the vertical direction. The angle between the coupling fields $%
C_{1}$and $C_{2}$ is about $175%
%TCIMACRO{\UNICODE[m]{0xb0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$ while the probe propagates at an angle of about $20%
%TCIMACRO{\UNICODE[m]{0xb0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$ with respect to $C_{1}$. These angles were found to give a good overlap of the probe with the coupling fields in the MOT. The paths of all three beams are coplanar.
We show in the Appendix how our arrangement of polarisations gives a good approximation to three separate sets of N configurations of the type depicted in Figure 1(a), together with uncoupled absorption. Although we can qualitatively account for the uncoupled absorptions, the relative populations of the various subsystems will be effected by the trap dynamics and optical pumping between the schemes. Complete modelling of these effects is beyond the scope of this work and we have therefore not presented a quantitative theoretical estimate of the uncoupled absorption strength.
While $C_{2}$ is turned off, $C_{1}$ and $P$ form a $\Lambda $-type EIT system. Figure 4(a) shows the probe absorption versus probe detuning with $%
C_{1}$ locked to the $5S_{1/2}F=1$ to $5P_{1/2}F=2$ transition in the saturated absorption cell, and with $C_{2}$ turned off. The probe detuning is taken to be zero when the probe frequency is equal to the $5S_{1/2}F=2$ to $5P_{3/2}F=2$ transition frequency in the saturated absorption cell. It is seen that the spectrum consists of a central peak situated between the two Autler-Townes peaks of a standard asymmetric EIT profile expected with detuned coupling field. The detuning of $C_{1}$ is due to the fact that the $5S_{1/2}F=1$ level of the sample is light-shifted with respect to the same level in the saturated absorption cell because of the interaction of $%
C_{1}$ with neighbouring transitions (mainly the $5S_{1/2}F=1$ to $%
5P_{3/2}F=1$). We have estimated this light shift to be $\Delta _{1}\approx 7%
%TCIMACRO{\unit{MHz}}%
%BeginExpansion
\mathop{\rm MHz}%
%EndExpansion
$. The central peak in the spectrum is caused by $P$ probing Zeeman sublevels that are not coupled by $C_{1}$ [**(**]{}as discussed in the Appendix). We call this peak the uncoupled absorption peak; in our previous work [@bib:EITExp] we were unable to resolve this peak since the coupling field was too weak. It is to be noted that all three peaks in Figure 4(a) are displaced by approximately $9%
%TCIMACRO{\unit{MHz}}%
%BeginExpansion
\mathop{\rm MHz}%
%EndExpansion
$ with respect to the saturated absorption transition. This is because the trapping beams $T$ act as a detuned coupling field with $P$ in a V-type EIT configuration, as can be seen from Figure 1(b). This splits each of the three peaks into two, one of which is very much larger than the other because $T$ is detuned by $-13%
%TCIMACRO{\unit{MHz}}%
%BeginExpansion
\mathop{\rm MHz}%
%EndExpansion
$. Of the smaller peaks, only the one corresponding to the red detuned Autler-Townes peak is just visible in this trace. We note that the linewidths of our spectra, which are seen to be up to three times the natural linewidth of our Doppler-free sample, are broadened by beam profile inhomogeneities, variations of Clebsch-Gordan coefficients between different Zeeman transitions and the spread in the intensities of the six interfering trapping beams in the MOT.
\
Three peak spectra of the V system
==================================
\
\
We now describe the spectra obtained when probing the V system in the N configuration of Figure 1. The probe absorption spectrum shown in Figure 4(b) was obtained with $C_{1}$ and $C_{2}$ tuned to their respective transitions in saturated absorption cells; these are the same conditions that applied in Figure 4(a) except that now $C_{2}$ is switched on and at its maximum intensity. A comparison of the two figures shows that the height $h_{c}$ of the central peak is larger in Figure 4(b) due to the appearance of the three-photon absorption peak on top of the uncoupled absorption peak, as predicted in the Section II. There is also an increase in the splitting, $\delta _{13}={\cal E}_{1}-{\cal E}_{3}$, of the two Autler-Townes peaks. We have taken a series of probe spectra for different powers $P_{2}$ of $C_{2}$. The results are shown in Figures 5(a)&(b), where the height $h_{c}$ and the splitting $\delta _{13}$ are plotted against the power $P_{2}$. The theoretical fits in these figures were obtained from the theory in Section II, as follows. We assume that the contribution of the three-photon absorption peak is proportional to $A_{2}=$ $\left|
\left\langle d|{\cal D}_{2}\right\rangle \right| ^{2},$ and that both coupling fields have a common detuning of $\Delta =7%
%TCIMACRO{\unit{MHz}}%
%BeginExpansion
\mathop{\rm MHz}%
%EndExpansion
$ from their respective transitions in the sample due to the $C_{1}$ - induced light shift of the $5S_{1/2}F=1$ level. We find, to first order in $\Delta $, $$\begin{aligned}
\delta _{13} &=&\sqrt{\Omega _{1}^{2}+\Omega _{2}^{2}}+{\cal O}\left( \Delta
\right) ^{2} \\
h_{c} &=&h_{uc}+B\frac{\Omega _{2}^{2}}{\Omega _{1}^{2}+\Omega _{2}^{2}}+%
{\cal O}\left( \Delta \right) ^{3}\end{aligned}$$ where $h_{uc}$ is the height of the uncoupled absorption peak, i.e. the height of the central peak when $C_{2}$ is turned off, and $B$ is a constant. These equations have been fitted to the data points and the resulting curves shown in Figure 5(a)&(b). The fit in Figure 5(a) gives $%
\Omega _{1}=62\pm 2%
%TCIMACRO{\unit{MHz}}%
%BeginExpansion
\mathop{\rm MHz}%
%EndExpansion
$ and $\Omega _{2}=(22\pm 2%
%TCIMACRO{\unit{MHz}}%
%BeginExpansion
\mathop{\rm MHz}%
%EndExpansion
/%
%TCIMACRO{\unit{mW}}%
%BeginExpansion
\mathop{\rm mW}%
%EndExpansion
^{1/2})P_{2}^{1/2}$ with a maximum of $\Omega _{2}=44\pm 5%
%TCIMACRO{\unit{MHz}}%
%BeginExpansion
\mathop{\rm MHz}%
%EndExpansion
$. These Rabi frequencies are consistent with the values estimated from the parameters of beams $C_{1}$ and $C_{2}$ and the Clebsch-Gordan coefficients of the transitions. The fit in Figure 5(b) gives $\Omega
_{2}/\Omega _{1}=0.8\pm 0.3$ for maximum $\Omega _{2}$, which is consistent with the previous fit; it also yields the constant $B=3\pm 1$ that determines the relative heights of the uncoupled absorption peak and the three-photon absorption peak. We note that for maximum power of $C_{2}$ the three-photon absorption peak accounts for approximately $0.4$ of the total central peak height.
We now consider the case where $C_{2}$ has its maximum intensity and its detuning is stepped across the $5S_{1/2}F=1$ to $5P_{1/2}F=2$ transition, with $C_{1}$ tuned to the saturated absorption line. The detuning of $C_{2}$ is measured by a calibrated spectrum analyser with respect to the saturated absorption line. The traces obtained are plotted in Figure 6. It is seen that as $\Delta _{2}$ is stepped from the red towards the blue, the uncoupled absorption peak, labelled $U$, remains fixed in position as expected, but the peaks labelled 1, 2 an 3 migrate towards the red, with the central peak 2 moving across the uncoupled absorption. The positions of peaks 1, 2 and 3 are shown as points in Figure 7 with the curves showing the corresponding theoretical expectations based on a detuning $\Delta _{1}=$ $7%
%TCIMACRO{\unit{MHz}}%
%BeginExpansion
\mathop{\rm MHz}%
%EndExpansion
$ of $C_{1}$. We note that this behaviour is qualitatively similar to that predicted for a ladder system in [@bib:Wei1998]. We note also that the anticrossings in Figure 7 are similar to subharmonic resonances described in [@bib:PolyChro], although because the two coupling fields are applied to different transitions, only the first subharmonic resonances (at $\Delta
_{2}=$ $\pm \Omega _{1}/2$ in the limit of small $\Omega _{2})$ are observed in the present study.
\
Conclusions
===========
We have presented a dressed state analysis and an experimental study of a doubly-driven V-system probed from a fourth level in an N configuration. The experiments show the growth of the three-photon absorption peak and the increasing separation of the Autler-Towns peaks as the second coupling field intensity is increased. The migration of the three-peak probe absorption spectrum as the detuning of one of the coupling fields is changed is also observed. Our experiments have been carried out in a laser-cooled $^{87}$Rb sample using the levels $5P_{3/2}F=2$, $5S_{1/2}F=1$ and $5P_{1/2}F=2$, probed on the $5S_{1/2}F=2$ to $5P_{3/2}F=2$ transition. After taking account of light shifts, the effects of the trapping beams and the uncoupled absorptions in this real system, the measured spectra are in good agreement with the analytical predictions. This investigation is important for the understanding of a physically realisable N-system that might be used in cross-phase modulation, photon blockade and other related studies.
Acknowledgments
===============
We would like to thank the EPSRC for financial support on this project and Dr T. B. Smith (Open University) for useful discussions. We would also like to thank Roger Bence, Fraser Robertson and Robert Seaton (Open University) for technical assistance.
Appendix
========
This Appendix justifies our use in Section II of a 4-state model to describe our experiment and also shows the origin of the uncoupled absorptions. Figure 8(a) illustrates the individual Zeeman states of the hyperfine levels coupled by the probe $P$ (fine lines) and two driving fields $C_{1}$ (thick lines) and $C_{2}$ (double lines) [@note:SmallAngle]. All three fields are linearly polarised with the polarisation of $C_{1}$ orthogonal to the polarisations of $P$ and $C_{2}$ as shown in Figure 3. This particular choice of orthogonal linear polarisations was chosen because it reduced the uncoupled absorption of the probe field at the frequency of the $5S_{1/2}F=2$ to $5P_{3/2}F=2$ transition. Also shown are the corresponding Clebsch-Gordan coefficients. The states $\left| m\right\rangle _{X}$ are labeled in terms of the magnetic quantum number $m$ and $X$, where $X$ is one of $a$, $b$, $c$ or $d$ which correspond to the $5S_{1/2}F=1$, $%
5S_{1/2}F=2$, $5P_{1/2}F=2$ and $5P_{3/2}F=2$ levels respectively in Figure 1(b).
\
The figure shows that the coupling field $C_{1}$ (thick lines) alone provides a doubly-driven V configuration as well as a separate quadruply-driven ‘W’ configuration. Taking account of the additional states coupled by the fields $C_{2}$ and $P$ it appears that a six state and, separately, a nine state model are necessary to describe the dynamics of our experiment. The three states, $\left| 0\right\rangle _{b}$, $\left|
-2\right\rangle _{c}$ and $\left| 2\right\rangle _{c}$ do not interact with any fields. Surprisingly, however, this complex structure reduces to sets of simple coupled systems with the following change of basis [@note:FurtherDetails]: $$\begin{aligned}
\left| m,\pm \right\rangle _{X} &\equiv &\frac{1}{\sqrt{2}}\left( \left|
m\right\rangle _{X}\pm \left| -m\right\rangle _{X}\right) \\
\left| \alpha \right\rangle _{d} &\equiv &\frac{1}{2}\left| 0\right\rangle
_{d}+\frac{\sqrt{3}}{2}\left| 2,+\right\rangle _{d} \\
\left| \beta \right\rangle _{d} &\equiv &\frac{\sqrt{3}}{2}\left|
0\right\rangle _{d}-\frac{1}{2}\left| 2,+\right\rangle _{d}.\end{aligned}$$ Figure 8(b) shows the transitions in this new basis. Three doubly-driven V configurations, of the type shown in Figure 1(a), are clearly evident in the new basis. Also shown are the effective Clebsch-Gordan coefficients for the new transformed transitions. The dashed lines in Figure 8(b) represent transitions probed by field $P$ but not directly coupled by either of the coupling fields $C_{1}$ and $C_{2}$. They correspond to what we call uncoupled absorptions of the probe.
The five coupled states on the left of Figure 8(b) shows that the (relatively weak) field $P$ simultaneously probes a doubly-driven V configuration as well as the $|2,-\rangle _{b}$ to $|\beta \rangle _{d}$ transition. In the steady state, the $|2,-\rangle _{b}$ to $|\beta \rangle
_{d}$ transition simply causes additional scattering of field $P$ and so essentially this five-state system can be treated as a four-state N configuration with an additional uncoupled absorption. The underlying probe absorption profile is due essentially to three separate N configurations (solid lines) with two uncoupled absorptions (dashed lines).
Each N configuration has a different set of (effective) Clebsch-Gordan coefficients, and so the Rabi frequencies for the fields $P$, $C_{1}$ and $%
C_{2}$ differ from one N configuration to the next. Since the position of the peaks in the probe absorption spectrum depend on the Rabi frequencies $%
C_{1}$ and $C_{2}$, the absorption peaks due to each N configuration occur at different probe detunings. However, these differences are relatively small, corresponding to about a 10% shift in the separation of Autler-Townes peaks. A [*single*]{} four-state model, as given in Section II, is therefore sufficient to model our experiment, provided we allow for broadened absorption peaks.
The square of Clebsch-Gordan coefficients of the dashed transitions is $1/6$. We can compare this with the different scheme in which all fields are linear polarised along the same axis. In this case there are two uncoupled absorption transitions from $\left| \pm 2\right\rangle _{b}$ to $\left| \pm
2\right\rangle _{d}$ for which the square of the corresponding Clebsch-Gordan coefficients is $2/3$. That is, the absorption of the uncoupled transitions in the parallel-linear polarisation scheme is [*four times*]{} that of the orthogonal-linear polarisation scheme for the same degree of occupation. This clearly shows the advantage of our choice of polarisation scheme.
In our experiment, new atoms are continuously moving into and out of the interaction region, the trapping magnetic fields produce Zeeman mixing amongst ground states and the trapping field weakly couples the $5S_{1/2}F=2$ level to the upper $5P_{3/2}F=3$ level. All these effects tend to redistribute population amongst the states of the $5S_{1/2}F=2$ level. The treatment of these effects to calculate the relative occupations of the $%
5S_{1/2}F=2$ states is, however, beyond the scope of this work. Thus, while we can identify the transitions responsible for the uncoupled absorptions and justify our polarisation scheme, we have no quantitative estimates of the strength of the absorptions.
Finally we note that state $\left| 0\right\rangle _{b}$, uncoupled by any field, is a dark state. Away from resonance conditions that produce other dark states, all atoms would eventually decay to this state and so the steady state would be one of complete transparency of the probe field. However, the above mechanisms which tend to redistribute the population of the $5S_{1/2}F=2$ level will also tend to depopulate the state $\left|
0\right\rangle _{b}$.
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We have taken the quantization axis $z$ in Fig. 8(a) to be along the polarisation of $C_{2}$. We have also ignored a small ($\approx 7\%$ of the intensity) contribution of sigma polarisation from the probe field due to the small ($\approx 15%
%TCIMACRO{\UNICODE[m]{0xb0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$) angle between the polarisation of $P$ and $C_{2}$.
Further details of the form of the transformation are beyond the scope of this paper and will be explored elsewhere.
|
---
abstract: 'We have argued that a high-purity Yb-doped silica glass can potentially be cooled via anti-Stokes fluorescence optical refrigeration. This conclusion is reached by showing, using reasonable assumptions for the host material properties, that the non-radiative decay rate of Yb ions can be made substantially smaller than the radiative decay rate. Therefore, an internal quantum efficiency of near unity can be obtained. Using spectral measurements of the fluorescence emission from a Yb-doped silica optical fiber at different temperatures, we estimate the minimum achievable temperature in Yb-doped silica glass for different values of internal quantum efficiency.'
author:
- Esmaeil Mobini
- Mostafa Peysokhan
- Behnam Abaie
- 'Markus P. Hehlen'
- Arash Mafi
title: 'Spectroscopic Investigation of Yb-doped Silica Glass for Solid-State Optical Refrigeration'
---
Introduction
============
In solid-state optical refrigeration, anti-Stokes fluorescence removes thermal energy from the material, resulting in net cooling. Solid-state optical cooling was first proposed by Pringsheim in 1929 [@Pringsheim1929] and was put on a solid thermodynamic foundation by Landau in 1946 [@landau1946thermodynamics]. Solid-state optical cooling was first experimentally observed in 1995 by Epstein’s group at Los Alamos National Laboratory in Yb-doped ZBLANP glass [@epstein1995observation]. Much attention has since been devoted to solid-state optical refrigeration in different materials and geometries due to its interesting basic science properties and potential applications [@epstein2010optical]. The quest for solid-state optical cooling in new configurations and materials is on-going [@seletskiy2016laser].
In particular, solid-state optical refrigeration of Yb-doped silica glass, which is extensively used in high-power fiber lasers, is highly desirable. New generations of high power fiber amplifiers and lasers now operate at few kiloWatt levels [@Richardson]. However, the significant heat-load in high-power operation has hindered the efforts to further scale up the power in fiber lasers and amplifiers [@Richardson; @Smith:11; @Dawson:08; @Jauregui:12]. Different methods have been developed to manage the heat-load in high-power fiber lasers or amplifiers; in particular, solid-state optical refrigeration via anti-Stokes fluorescence has been suggested as a viable path for heat mitigation [@bowman1999lasers; @bowman2010minimizing; @Esmaeil2018josabRBL]. So far, there is no report of solid-state optical refrigeration in Yb-doped silica; this manuscript is intended to highlight its possibility.
In this context, Radiation-Balanced Lasers (RBL) were first introduced by Bowman in 1999 [@bowman1999lasers]. In radiation balancing, the heat that originates from the quantum defect of the laser as well as parasitic absorption can be removed by anti-Stokes fluorescence under a very subtle balance condition between different parameters of a laser (or an amplifier) [@bowman1999lasers; @bowman2010minimizing; @nemova2009athermal]. In other words, the anti-Stokes fluorescence removes the excess heat generated in the medium. Therefore, heat mitigation by radiation-balancing via anti-Stokes fluorescence is highly desirable and will have great practical implications if it can be achieved in Yb-doped silica glass, which is the material of choice for most high-power fiber lasers and amplifiers [@Richardson; @pask1995ytterbium; @paschotta1997ytterbium].
The investigation of solid-state optical refrigeration can be done either directly or indirectly. In a direct investigation, the material is exposed to a laser in a thermally isolated setup, often in a sophisticated vacuum environment [@seletskiy2010laser], and its temperature is measured directly by a thermal camera or similar methods. In an indirect method, the spectroscopic properties of materials at different temperatures are measured to evaluate the possibility of solid-state optical refrigeration [@epstein1995observation; @seletskiy2010laser; @lei1998spectroscopic; @melgaard2010spectroscopy]. In this manuscript, we use the indirect method to argue for the potential of high-quality Yb-doped silica glass for solid-state optical refrigeration and radiation-balancing in lasers and amplifiers.
In order to characterize the cooling potential of Yb-doped silica glass, we use the cooling efficiency $\eta_{c}$ defined as [@seletskiy2010laser; @melgaard2010spectroscopy] $$\begin{aligned}
\label{Eq:cooleff}
\eta_{c}(\lambda_{p},T)=\eta_{q}\,\eta_{abs}(\lambda_{p},T) \frac{\lambda_{p}}{\lambda_{f}(T)}-1.\end{aligned}$$ In Eq. \[Eq:cooleff\], $\lambda_{f}$ is the mean fluorescence wavelength and $\lambda_{p}$ is the pump wavelength. $\eta_{q}$ is the internal quantum efficiency and $\eta_{abs}$ is the absorption efficiency; they are defined as $$\begin{aligned}
\label{Eq:etaq}
&\eta_{q}=\frac{W_{r}}{W_{tot}},\quad W_{tot}=W_{r}+W_{nr},\\
\label{Eq:etaabs}
&\eta_{abs}(\lambda_{p},T)=\frac{\alpha_{r}(\lambda_{p},T)}{\alpha_{r}(\lambda_{p},T)+\alpha_{b}},\end{aligned}$$ where $W_{r}$, $W_{nr}$, and $W_{tot}$ are radiative, non-radiative, and total decay rates of the excited state, respectively. $\alpha_{b}$ is the background absorption coefficient, and $\alpha_{r}$ is the resonant absorption coefficient. Note that $\alpha_{b}$ does not contain the attenuation due to scattering as this process does not lead to heating of the material. We have assumed that due to the small cross sectional area of optical fibers, the fluorescence escape efficiency to be unity [@epstein1995observation; @ruan2006enhanced]. The mean fluorescence wavelength is defined by $$\begin{aligned}
\lambda_{f}(T)=\frac{\int_{\Delta} \lambda~S(\lambda,T) d\lambda}{\int_{\Delta} S(\lambda,T) d\lambda},
\label{Eq:meanwave}\end{aligned}$$ where $S(\lambda,T)$ is the fluorescence power spectral density, which is a function of the glass temperature $T$, and $\Delta$ is the spectral domain encompassing the relevant emission spectral range [@seletskiy2010laser; @melgaard2010spectroscopy].
In order to achieve net solid-state optical refrigeration, it is necessary for the cooling efficiency to be positive. Therefore, we must show that $\eta_c>0$ is attainable over a range of $\lambda_p$ and $T$ values. It can be seen from Eq. \[Eq:cooleff\] that because $\lambda_p$ and $\lambda_f$ are often very close to each other in solid-state optical refrigeration schemes, the internal quantum efficiency $\eta_q$ has to be close to unity ($\eta_{q}\approx 1$) [@epstein1995observation; @seletskiy2010laser]. There are two main processes that lower the internal quantum efficiency: the multi-phonon non-radiative relaxation and the concentration quenching effect [@hehlen2007model; @digonnet2001rare; @hoyt2003advances; @barua2008influences; @van1983nonradiative; @auzel2003radiation; @boulon2008so; @nguyen2012all]. We will argue that the multi-phonon non-radiative relaxation is negligible in Yb-doped silica glass and the concentration quenching process can be prevented if the Yb ion density is kept lower than the characteristic Yb ion quenching concentration.
In order to evaluate the absorption efficiency $\eta_{abs}$, we need to know the background absorption ($\alpha_b$) and resonant absorption ($\alpha_r$) coefficients. For the background absorption coefficient in Yb-doped silica glass, we will use typical values reported in the literature [@jetschke2008efficient; @leich2011highly; @Sidharthan:18]. By capturing the power spectral density of a Yb-doped silica optical fiber from its side at different temperatures, $S(\lambda,T)$, we can also obtain the resonant absorption coefficient ($\alpha_{r}$) as well as the mean fluorescence wavelength ($\lambda_{f}$). Therefore, we will have all the necessary parameters in Eq. \[Eq:cooleff\]; using this information we will estimate the cooling efficiency and show that solid-state optical refrigeration is feasible in Yb-doped silica glass.
Internal quantum efficiency
===========================
The internal quantum efficiency is the fraction of the radiative decay versus the total decay of an excited state in a medium; therefore, the presence of non-radiative decay channels characterized by the non-radiative decay rate $W_{\rm nr}$ in Eq. \[Eq:etaq\] are responsible for decreasing $\eta_q$ below unity. The non-radiative decay channels in a typical Yb-doped silica glass can be broken down according to the following equation: $$\begin{aligned}
W_{\rm nr}=W_{\rm mp}+W_{\rm OH^{-}}+W_{\rm Yb}+\sum_{\rm TM} W_{\rm TM}+\sum_{\rm RE} W_{\rm RE}.
\label{Eq:total-nonradiative}\end{aligned}$$ The partial non-radiative decay channels are as follows: $W_{\rm mp}$ represents the multi-phonon decay of the Yb excited state, $W_{\rm OH^{-}}$ accounts for non-radiative decay of the Yb excited state via the high-energy vibrational modes of ${\rm OH^{-}}$ impurities, $W_{\rm Yb}$ accounts for non-radiative decay in Yb ion clusters, and $W_{\rm TM}$ and $W_{\rm RE}$ represent non-radiative decay due to interactions of the excited state with various transition-metal and rare-earth ion impurities, respectively.
In the following, we will discuss the various non-radiative decay channels in Eq. \[Eq:total-nonradiative\] and show that they can be made sufficiently small to allow for a near-unity internal quantum efficiency value ($\eta_q\approx 1$). We first begin with the multi-phonon relaxation that originates from the coupling of the excited state with the vibrational wavefunctions of the ground state. Using the energy-gap law [@van1983nonradiative; @faure2007improvement; @hehlen2007model; @hoyt2003advances], we can calculate the decay rate from $$\begin{aligned}
\label{Eq:multi-phonon decay}
W_{\rm mp}=W_{0}~e^{-\alpha_{h} ( E_{g}-2E_{p})},\end{aligned}$$ where $E_{p}$ is the maximum phonon energy of the host material, and $E_{g}$ is the energy gap of the dopant ion (Yb). $W_{0}$ and $\alpha_{h}$ are phenomenological parameters, whose values strongly depend on the host-material [@van1983nonradiative; @faure2007improvement; @hehlen2007model; @hoyt2003advances]. Figure \[Fig:omeganr\] shows the multi-phonon non-radiative decay rates of silica and ZBLAN glasses versus the energy gaps of the doped ions at $T\,=\,300$K, using the parameters shown in Table \[silicazblan\].
\[silicazblan\]
Host $W_{0}\,({\rm s}^{-1})$ $\alpha_h\,({\rm cm})$ $E_p\,({\rm cm}^{-1})$
----------- ------------------------- ------------------------ ------------------------
silica $7.8 \times 10^{7}$ $4.7 \times 10^{-3}$ $1.10 \times 10^{3}$
[ZBLAN]{} $1.7 \times 10^{4}$ $2.1 \times 10^{-3}$ $0.58 \times 10^{3}$
: Parameters related to Eq. \[Eq:multi-phonon decay\] and Fig. \[Fig:omeganr\] for silica and ZBLAN glasses [@faure2007improvement; @hehlen2007model; @hoyt2003advances].
The vertical solid line in Fig. \[Fig:omeganr\] marks the energy gap of a ${\rm Yb^{3+}}$ ion. It is evident that for Yb-doped silica glass, the non-radiative decay rate is around $W_{\rm mp}^{\rm silica} \approx 10^{-8}~s^{-1}$, which is much smaller than the Yb-doped ZBLAN glass multi-phonon decay rate $W_{\rm mp}^{\rm ZBLAN} \approx 10^{-4}~s^{-1}$. This comparison suggests that with respect to Yb multi-phonon relaxation, silica glass is a more suitable choice for solid-state optical refrigeration than ZBLAN glass.
![Multi-phonon non-radiative decay rate ($W_{mp}$) of Yb-doped ZBLAN and silica glasses versus energy gap ($E_{g}$) calculated from Eq. \[Eq:multi-phonon decay\] and the parameters listed in Table \[silicazblan\].[]{data-label="Fig:omeganr"}](omeganr.png){width="3.4"}
Considering the advances in materials synthesis of fiber preforms, the term $W_{\rm OH^{-}}$ in Eq. \[Eq:total-nonradiative\] can be made very small (see e.g. dry fiber technology [@thomas2000towards]); therefore, it can be neglected [@barua2008influences]. It has also been shown by Auzel et al. [@auzel2003radiation] that the total effect of the last three terms in Eq. \[Eq:total-nonradiative\], $W_{\rm Yb}+\Sigma W_{\rm TM}+\Sigma W_{\rm RE}$, can be described by a phenomenological equation based on a limited diffusion process, modeled as a non-radiative dipole-dipole interaction between the ions and impurities [@auzel2003radiation; @boulon2008so]. This concentration quenching process can be prevented if the Yb ion density is lower than the critical quenching concentration of the Yb-doped silica glass, which exists because there are impurities. Therefore, the critical quenching concentration is generally a sample specific quantity. That is, it would be higher for lower impurity concentrations. For a Yb ion density smaller than the critical quenching concentration, the internal quantum efficiency can approach $\eta_{q}\approx 1$ [@auzel2003radiation; @boulon2008so; @nguyen2012all]. It must be noted that an internal quantum efficiency of $\eta_{q}$=0.95 is reported in [@jeong2004ytterbium] for Yb-doped silica, which is consistent with our claim that $W_{\rm nr}$ can be made quite small in Yb-doped silica.
Absorption efficiency and mean fluorescence wavelength
======================================================
In order to calculate the cooling efficiency, we still need to obtain the resonant absorption coefficient and the mean fluorescence wavelength, both of which can be obtained from a spectroscopic investigation. The resonant absorption coefficient is used in conjunction with Eq. \[Eq:etaabs\] to determine the absorption efficiency. The setup implemented in our experiment consists of a single-mode Yb-doped silica fiber (DF-1100, from Newport Corporation) that is pumped by a Ti:Sapphire laser at $\lambda$=900nm. The fiber is mounted on a plate whose temperature is changed from nearly 180K up to 360K. The fluorescence of the Yb-doped silica fiber is captured by a multimode fiber from the side of the Yb-doped silica fiber and is sent to an Optical Spectrum Analyzer. Figure. \[Fig:NormEmission\] shows the measured fluorescence spectra (power spectral density $S(\lambda,T)$), normalized to their peak values at $\lambda_{\rm peak}\,\approx$976nm, at different temperatures.
![Measured peak normalized emission spectra of DF-1100 Yb-doped silica fiber at different temperatures.[]{data-label="Fig:NormEmission"}](NormEmission.png){width="3.4"}
By inserting the measured fluorescence spectra into Eq. \[Eq:meanwave\] and considering $\Delta\in\{905{\rm nm},1150{\rm nm}\}$, the dependence of the mean fluorescence wavelength on temperature is obtained. The mean fluorescence wavelength follows approximately the following function: $$\begin{aligned}
\label{Eq:lambdafT}
\lambda_{f}(T) \approx 999~({\rm nm})\,+\,b\times T^{-1}, \quad b\,=\,2735\pm\, 31 {\rm nm/K}.\end{aligned}$$ This behavior at temperatures above 245K to 360K is nearly linear, which is similar to that reported in other host materials, such as ZBLAN [@hehlen2007model; @lei1998spectroscopic].
In order to calculate the the resonant absorption coefficient $\alpha_{r}$, we first calculate the emission cross section $\sigma_{e}$, and then use the McCumber relation to obtain the absorption cross section $\sigma_{a}$ and then the resonant absorption coefficient $\alpha_{r}$ [@fan1989end; @mccumber1964theory; @newell2007temperature]. The emission cross section is obtained from the measured fluorescence power spectral density $S(\lambda,T)$ via the Füchtbauer-Ladenburg equation [@aull1982vibronic; @newell2007temperature]: $$\begin{aligned}
\sigma_{e}(\lambda,T)=\dfrac{\lambda^5}{8\,\pi\,n^2\,c\,\tau_{r}(T)}\times \dfrac{S(\lambda,T)}{\int_{\Delta} \lambda~S(\lambda,T) d\lambda },
\label{Eq:FuchtbauerLadenburg}\end{aligned}$$ where $n$ is the refractive index of the fiber core, $c$ is the speed of light in vacuum, and $\tau_{r}=W_{r}^{-1}$ is the radiative lifetime.
In order to apply Eq. \[Eq:FuchtbauerLadenburg\], the radiative lifetime at each temperature needs to be measured. In high-quality samples for which the non-radiative decay rates are negligible compared to the radiative decay rates, the fluorescence lifetimes are comparable to the radiative lifetimes ($\tau_{f}\approx \tau_{r}$); therefore, we measured the fluorescence lifetimes at different temperatures from the side of the fiber [@Mobini:18]. Using this assumption, the emission cross sections at different temperatures were calculated and are shown in Fig. \[Fig:emission-cross\]. The absorption cross sections can be readily obtained using the McCumber relation: $$\begin{aligned}
\label{Eq:McCumber}
&\sigma_{a}(\lambda,T)=\sigma_{e}(\lambda,T)\times \mathcal{Z}(\lambda,T),\\
\nonumber
&\mathcal{Z}(\lambda,T)\,=\,\exp\left[\frac{hc}{k_{b}T}(\frac{1}{\lambda}-\frac{1}{\lambda_{0}})\right],\end{aligned}$$ where $k_{b}$ is the Boltzmann constant, $h$ is the Planck constant and $\lambda_{0}\,=\,976$nm is the wavelength corresponding to the zero-line phonon energy [@mccumber1964theory; @melgaard2010spectroscopy; @pask1995ytterbium].
![Emission cross section versus wavelength for DF-1000 Yb-doped silica fiber at different temperatures. The spectra were calculated from Eq. \[Eq:FuchtbauerLadenburg\] using the emission spectra shown in Fig. \[Fig:NormEmission\] and the measured radiative lifetimes from Ref. [@Mobini:18].[]{data-label="Fig:emission-cross"}](emission.png){width="3.4"}
The resonant absorption coefficient can be calculated from $\sigma_{a}(\lambda,T)$ in Eq. \[Eq:McCumber\] (and Fig. \[Fig:emission-cross\]) using $$\begin{aligned}
\label{Eq:alphardef}
\alpha_{r}(\lambda,T)\,=\,\sigma_{a}(\lambda,T)\times N.\end{aligned}$$ Here, we will assume a typical Yb ion density of $N\,=5\times 10^{25}\,{\rm m}^{-3}$. We now have all the necessary ingredients to calculate the cooling efficiency $\eta_{c}$ in Eq. \[Eq:cooleff\]. We only need to provide a value for the background absorption coefficient in Eq. \[Eq:etaabs\] to determine the absorption efficiency $\eta_{abs}$. Here, we assume a background absorption coefficient of $\alpha_{b}\,=\,10$dB/km$\approx 2.3\times~10^{-3}$/m, which is a typical value for commercial grade Yb-doped silica fibers. Using this information, we present a contour plot of the cooling efficiency $\eta_{c}$ in Fig. \[Fig:2PD\] as a function of the pump wavelength and temperature, assuming that $\eta_{q}$=1. Note that we only know the values of $\alpha_{r}(\lambda,T)$ at discrete values of temperature $T$ for which our measurements were performed in Fig. \[Fig:NormEmission\]; the density plot in Fig. \[Fig:2PD\] is an interpolation of the measured values. It is seen in Fig. \[Fig:2PD\] that with a decrease in the temperature, the cooling efficiency decreases; this behavior is due to the red-shift of the mean fluorescence wavelength and the decrease in the resonant absorption coefficient with decreasing temperature [@seletskiy2010laser; @hehlen2007model].
In practice, it is impossible to achieve an internal quantum efficiency of unity; therefore, in Fig. \[Fig:cool-eff\] we investigate the effect of a non-ideal internal quantum efficiency on the cooling efficiency, for $\lambda_{p}\,=\,1030$nm, as a function of the temperature. The discrete points in Fig. \[Fig:cool-eff\] signify the values of $\eta_{c}$ obtained for the assumed $\eta_{q}$ at the particular measured temperatures reported in Fig. \[Fig:NormEmission\]. The apparent difference between the cooling efficiency obtained for $\eta_{q}$=1 versus $\eta_{q}$=0.98 highlights the importance of having a high-quality glass for radiative cooling. While the discrete points in Fig. \[Fig:cool-eff\] reveal the main expected behavior of $\eta_{c}$ versus the temperature, it is helpful to estimate the minimum achievable temperature for solid-state optical refrigeration in Yb-doped silica glass, subject to the assumptions made about $\eta_q$, $N$, and $\alpha_b$. In order to do so, we next present an analytical fitting to the discrete points in Fig. \[Fig:cool-eff\] that can be used to estimate the minimum achievable temperature. The analytical fitting, which is described in the next paragraph, is used in conjunction with Eq. \[Eq:cooleff\] to plot the colored lines for each value of $\eta_q$ in Fig. \[Fig:cool-eff\] and is in reasonable agreement with the experimentally measured discrete data.
![Cooling efficiency for different values of quantum efficiency versus temperature with $\alpha_{b}\,=\,10$dB/km for DF-1100 Yb-doped silica fiber calculated from Eq. \[Eq:cooleff\] for different internal quantum efficiencies, $\eta_q$. The colored lines are plotted using Eq. \[Eq:cooleff\] and the fitting presented in Eq. \[Eq:Fit3\].[]{data-label="Fig:cool-eff"}](cooleff.png){width="3.4"}
From the discussions above and Eqs. \[Eq:FuchtbauerLadenburg\], \[Eq:McCumber\], and \[Eq:alphardef\], we note that $\alpha_{r}(\lambda_p,T)$ (at the pump wavelength) can be expressed as: $$\begin{aligned}
\alpha_{r}(\lambda_p,T) \propto \frac{1}{c} \frac{\lambda_p^5}{\tau_r(T)} \times \frac{S(\lambda_p,T)}{\int_{\Delta} \lambda~S(\lambda,T) d\lambda}\times \mathcal{Z}(\lambda_p,T).
\label{Eq:Fit1}\end{aligned}$$ In Ref. [@Mobini:18], we performed fluorescence lifetime measurements in Yb-silica. Here, we present a fitting of $\tau_r(T)$ to an analytical form that is based on a two-level excited state: $$\begin{aligned}
\tau_r(T)=\frac{1+\exp(-\delta E/k_b T)}{\tau^{-1}_1+\tau^{-1}_2\exp(-\delta E/k_b T)}.
\label{Eq:radiative-func}\end{aligned}$$ $\tau_1\,=\,798\pm 2$s, and $\tau_2\,=\,576\pm 27$s are the lifetimes of the first and second energy levels of the excited state, respectively, and $\delta E\,=\,506\pm 56\,{\rm cm^{-1}}$ is the energy difference between these two levels [@zhang1993thermal; @newell2007temperature]. We also present the following approximation: $$\begin{aligned}
\frac{\lambda_p^2\ S(\lambda_p,T)}{\int_{\Delta} \lambda~S(\lambda,T) d\lambda} \approx 7.4\,+\,\left(\frac{d}{T}\right)^5,
\quad d\,=\,205.9 \pm 2.4\,{\rm K}.
\label{Eq:Fit2}\end{aligned}$$ Using Eqs. \[Eq:radiative-func\] and \[Eq:Fit2\], we can approximate $\alpha_{r}(\lambda_p,T)$ \[Eq. \[Eq:Fit1\]\] with the following mathematical form: $$\begin{aligned}
\alpha_{r}(\lambda_p,T) \approx \frac{\alpha_{r,0}}{c~\tau_r(T)} \times \Big(7.4\,+\,(d/T)^5\Big) \times \mathcal{Z}(\lambda_p,T).
\label{Eq:Fit3}\end{aligned}$$ Fitting the analytical in Eq. \[Eq:Fit3\] to the discrete points in Fig. \[Fig:cool-eff\], we find the dimensionless coefficient $\alpha_{r,0}=\,(0.95\pm\,0.01)\times 10^{6}$. The fitted lines in Fig. \[Fig:cool-eff\] show that the minimum achievable temperature can reach down to $T_{\rm min}\,=\,138$K for $\eta_q\,=\,1$, $T_{\rm min}\,=\,175$K for $\eta_q\,=\,0.99$, and $T_{\rm min}\,=\,290$K for $\eta_q\,=\,0.98$. Figure \[Fig:cool-eff\] also shows that the maximum cooling efficiency for Yb-silica glass is around $\eta^{\rm max}_{c}\approx$2% at room temperature for $\lambda_p$=1030nm. Setting the background absorption to zero ($\alpha_b$=0) increases this value to $\eta^{\rm max}_{c}\approx$2.2%. In order to increase the cooling efficiency, the background absorption must be minimized, the internal quantum efficiency has to be close to unity, and the ion dopant density $N$ must be increased to enhance the resonant absorption coefficient. We note that these requirements are not necessarily compatible with each other; e.g. increasing $N$ can potentially decrease $\eta_q$ due to quenching. Therefore, a compromise determined by careful measurements must be obtained.
Discussion and Conclusion
=========================
It must be noted that by taking $N\,=5\times 10^{25}\,{\rm m}^{-3}$ in this manuscript, we have implicitly assumed that the silica glass host is co-doped with some modifiers like ${\rm Al_2O_3}$, in order to shift the quenching concentration to higher values to reduce clustering and ensure an adequate cooling efficiency [@arai1986aluminum; @laegsgaard2002dissolution]. For pure silica, applying the model developed by Auzel et al. [@auzel2003radiation] to the experimental data from Ref. [@barua2008influences], it can be shown that $N\,=0.7\times 10^{25}\,{\rm m}^{-3}$ can guarantee a near unity internal quantum efficiency [@arai1986aluminum; @laegsgaard2002dissolution]. Using $N=0.7 \times 10^{25} m^{-3}$ in pure silica, we have calculated the minimum achievable temperature to be $T_{\rm min}\,=\,216$K for $\eta_q\,=\,1$, and $T_{\rm min}\,=\,262$K for $\eta_q\,=\,0.99$. For $\eta_q\,=\,0.98$, $T_{\rm min}$ is higher than the room temperature. As expected, a decrease in ion density results in a lower cooling efficiency.
In conclusion, we have argued that a high-purity Yb-doped silica glass can potentially be cooled via anti-Stokes fluorescence optical refrigeration. We show that, in principle, the non-radiative decay rate $W_{nr}$ can be made substantially smaller than the radiative decay rate $W_r$. Therefore, an internal quantum efficiency of near unity can be obtained, making Yb-doped silica glass suitable for solid-state optical refrigeration. Our assessment is based on reasonable assumptions for material properties, e.g. we have assumed a typical background absorption coefficient of $\alpha_{b}$=10dB/km and an internal quantum efficiency of larger than $\eta_{q}$=0.98. We have made spectral measurements of the fluorescence from a Yb-doped silica optical fiber at different temperatures. Using these measurements, we have reported the temperature dependence of the mean fluorescence wavelength, and have estimated the minimum achievable temperature in Yb-doped silica glass. Our analysis highlights the potential for Yb-doped silica glass to be used as the gain medium for radiation-balanced high-power fiber lasers and amplifiers.
Acknowledgment {#acknowledgment .unnumbered}
==============
This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-16-1-0362 titled Multidisciplinary Approaches to Radiation Balanced Lasers (MARBLE).
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---
abstract: 'We use optical data on 10 Kuiper Belt objects (KBOs) to investigate their rotational properties. Of the 10, three (30%) exhibit light variations with amplitude $\Delta m \ge 0.15\,$mag, and 1 out of 10 (10%) has $\Delta m \ge 0.40\,$mag, which is in good agreement with previous surveys. These data, in combination with the existing database, are used to discuss the rotational periods, shapes, and densities of Kuiper Belt objects. We find that, in the sampled size range, Kuiper Belt objects have a higher fraction of low amplitude lightcurves and rotate slower than main belt asteroids. The data also show that the rotational properties and the shapes of KBOs depend on size. If we split the database of KBO rotational properties into two size ranges with diameter [*larger*]{} and [ *smaller*]{} than $400\,$km, we find that: (1) the mean lightcurve amplitudes of the two groups are different with 98.5% confidence, (2) the corresponding power-law shape distributions seem to be different, although the existing data are too sparse to render this difference significant, and (3) the two groups occupy different regions on a [*spin period*]{} vs. [*lightcurve amplitude*]{} diagram. These differences are interpreted in the context of KBO collisional evolution.'
author:
- Pedro Lacerda
- Jane Luu
title: Analysis of the Rotational Properties of Kuiper Belt Objects
---
Introduction
============
The Kuiper Belt (KB) is an assembly of mostly small icy objects, orbiting the Sun beyond Neptune. Kuiper Belt objects (KBOs) are likely to be remnants of outer solar system planetesimals [@1993Natur.362..730J]. Their physical, chemical, and dynamical properties should therefore provide valuable information regarding both the environment and the physical processes responsible for planet formation.
At the time of writing, roughly 1000 KBOs are known, half of which have been followed for more than one opposition. A total of $\approx
10^{5}$ objects larger than 50 km are thought to orbit the Sun beyond Neptune [@2000prpl.conf.1201J]. Studies of KB orbits have revealed an intricate dynamical structure, with signatures of interactions with Neptune (Malhotra 1995). The size distribution follows a differential power-law of index $q=4$ for bodies $\gtrsim
50\,$km [@2001AJ....122..457T], becoming slightly shallower at smaller sizes [@2004AJ....128.1364B].
KBO colours show a large diversity, from slightly blue to very red [@1996AJ....112.2310L; @2000Natur.407..979T; @2001AJ....122.2099J], and seem to correlate with inclination and/or perihelion distance [e.g., @2001AJ....122.2099J; @2002AJ....124.2279D; @2002ApJ...566L.125T]. The few low-resolution optical and near-IR KBO spectra are mostly featureless, with the exception of a weak 2$\,\mu$m water ice absorption line present in some of them [@1999ApJ...519L.101B; @2001AJ....122.2099J], and strong methane absorption on 2003$\,$UB$_{313}$ [@2003UB313].
About 4% of known KBOs are binaries with separations larger than 015 [@2002AJ....124.3424N]. All the observed binaries have primary-to-secondary mass ratios $\approx$ 1. Two binary creation models have been proposed. @2002Icar..160..212W favours the idea that binaries form in three-body encounters. This model requires a 100 times denser Kuiper Belt at the epoch of binary formation, and predicts a higher abundance of large separation binaries. An alternative scenario [@2002Natur.420..643G], in which the energy needed to bind the orbits of two approaching bodies is drawn from the surrounding swarm of smaller objects, also requires a much higher density of KBOs than the present, but it predicts a larger fraction of close binaries. Recently, @2004AJ....127.3023S have shown evidence that $2001\,\rm{QG}_{298}$ could be a close or contact binary KBO, and estimated the fraction of similar objects in the Belt to be $\sim10\%$–$20\%$.
Other physical properties of KBOs, such as their shapes, densities, and albedos, are still poorly constrained. This is mainly because KBOs are extremely faint, with mean apparent red magnitude $m_R\sim$23 [@2001AJ....122.2740T].
The study of KBO rotational properties through time-series broadband optical photometry has proved to be the most successful technique to date to investigate some of these physical properties. Light variations of KBOs are believed to be caused mainly by their aspherical shape: as KBOs rotate in space, their projected cross-sections change, resulting in periodic brightness variations.
One of the best examples to date of a KBO lightcurve – and what can be learned from it – is that of (20000)$\,$Varuna [@2002AJ....123.2110J]. The authors explain the lightcurve of (20000)$\,$Varuna as a consequence of its elongated shape (axes ratio, $a/b\sim 1.5$). They further argue that the object is centripetally deformed by rotation because of its low density, “rubble pile” structure. The term “rubble pile” is generally used to refer to gravitationally bound aggregates of smaller fragments. The existence of rubble piles is thought to be due to continuing mutual collisions throughout the age of the solar system, which gradually fracture the interiors of objects. Rotating rubble piles can adjust their shapes to balance centripetal acceleration and self-gravity. The resulting equilibrium shapes have been studied in the extreme case of fluid bodies, and depend on the body’s density and spin rate [@1969efe..book.....C].
@2003Icar..161..174L ([-@2003Icar..161..174L], hereafter ) showed that under reasonable assumptions the fraction of KBOs with detectable lightcurves can be used to constrain the shape distribution of these objects. A follow-up (, hereafter ) on this work, using a database of lightcurve properties of 33 KBOs , shows that although most Kuiper Belt objects ($\sim85\%$) have shapes that are close to spherical ($a/b\leq1.5$) there is a significant fraction ($\sim12\%$) with highly aspherical shapes ($a/b\geq1.7$).
In this paper we use optical data on 10 KBOs to investigate the amplitudes and periods of their lightcurves. These data are used in combination with the existing database to investigate the distributions of KBO spin periods and shapes. We discuss their implications for the inner structure and collisional evolution of objects in the Kuiper Belt.
Observations and Photometry {#Capitulo4Sec.Observations}
===========================
We collected time-series optical data on 10 KBOs at the Isaac Newton 2.5m (INT) and William Herschel 4m (WHT) telescopes. The INT Wide Field Camera (WFC) is a mosaic of 4 EEV 2048$\times$4096 CCDs, each with a pixel scale of 033/pixel and spanning approximately 113$\times$225 in the plane of the sky. The targets are imaged through a Johnson R filter. The WHT prime focus camera consists of 2 EEV 2048$\times$4096 CCDs with a pixel scale of 024/pixel, and covers a sky-projected area of 2$\times$82$\times$164. With this camera we used a Harris R filter. The seeing for the whole set of observations ranged from 1.0 to 1.9FWHM. We tracked both telescopes at sidereal rate and kept integration times for each object sufficiently short to avoid errors in the photometry due to trailing effects (see Table \[Table.ObsCond\]). No light travel time corrections have been made.
We reduced the data using standard techniques. The sky background in the flat-fielded images shows variations of less than 1% across the chip. Background variations between consecutive nights were less than 5% for most of the data. Cosmic rays were removed with the package LA-Cosmic [@2001PASP..113.1420V].
We performed aperture photometry on all objects in the field using the SExtractor software package . This software performs circular aperture measurements on each object in a frame, and puts out a catalog of both the magnitudes and the associated errors. Below we describe how we obtained a better estimate of the errors. We used apertures ranging from 1.5 to 2.0 times the FWHM for each frame and selected the aperture that maximized signal-to-noise. An extra aperture of 5 FWHMs was used to look for possible seeing dependent trends in our photometry. The catalogs were matched by selecting only the sources that are present in all frames. The slow movement of KBOs from night to night allows us to successfully match a large number of sources in consecutive nights. We discarded all saturated sources as well as those identified to be galaxies.
The KBO lightcurves were obtained from differential photometry with respect to the brightest non-variable field stars. An average of the magnitudes of the brightest stars (the “reference” stars) provides a reference for differential photometry in each frame. This method allows for small amplitude brightness variations to be detected even under non-photometric conditions.
The uncertainty in the relative photometry was calculated from the scatter in the photometry of field stars that are similar to the KBOs in brightness (the “comparison” stars, see Fig.\[Fig.VarvsMag\]). This error estimate is more robust than the errors provided by SExtractor (see below), and was used to verify the accuracy of the latter. This procedure resulted in consistent time series brightness data for $\sim100$ objects (KBO + field stars) in a time span of 2–3 consecutive nights.
We observed Landolt standard stars whenever conditions were photometric, and used them to calibrate the zero point of the magnitude scale. The extinction coefficient was obtained from the reference stars.
Since not all nights were photometric the lightcurves are presented as variations with respect to the mean brightness. These yield the correct amplitudes and periods of the lightcurves but do not provide their absolute magnitudes.
The orbital parameters and other properties of the observed KBOs are given in Table \[ParsObsKBO\]. Tables \[Table.1996TO66AbsPhot\], \[Table.1996TS66AbsPhot\], \[Table.1998SN165AbsPhot\], and \[Table.1998WH24AbsPhot\] list the absolute $R$-magnitude photometric measurements obtained for ${(19308)\,1996\,\rm{TO}_{66}}$, ${1996\,\rm{TS}_{66}}$, ${(35671)\,1998\,\rm{SN}_{165}}$, and ${(19521)\,\rm{Chaos}}$, respectively. Tables \[Table.1999DF9RelPhot\] and \[Table.2001CZ31RelPhot\] list the mean-subtracted $R$-band data for ${(79983)\,1999\,\rm{DF}_{9}}$ and ${2001\,\rm{CZ}_{31}}$.
Lightcurve Analysis
===================
The results in this paper depend solely on the amplitude and period of the KBO lightcurves. It is therefore important to accurately determine these parameters and the associated uncertainties.
Can we detect the KBO brightness variation? {#Capitulo4KBOVariability}
-------------------------------------------
We begin by investigating if the observed brightness variations are intrinsic to the KBO, i.e., if the KBO’s intrinsic brightness variations are detectable given our uncertainties. This was done by comparing the frame-to-frame scatter in the KBO optical data with that of ($\sim10-20$) comparison stars.
To visually compare the scatter in the magnitudes of the reference stars (see Section \[Capitulo4Sec.Observations\]), comparison stars, and KBOs, we plot a histogram of their frame-to-frame variances (see Fig. \[Fig.VarHistograms\]). In general such a histogram should show the reference stars clustered at the lowest variances, followed by the comparison stars spread over larger variances. If the KBO appears isolated at much higher variances than both groups of stars (e.g., Fig. \[Fig.VarHistograms\]j), then its brightness modulations are significant. Conversely, if the KBO is clustered with the stars (e.g. Fig. \[Fig.VarHistograms\]f), any periodic brightness variations would be below the detection threshold.
Figure \[Fig.VarvsMag\] shows the dependence of the uncertainties on magnitude. Objects that do not fall on the rising curve traced out by the stars must have intrinsic brightness variations. By calculating the mean and spread of the variance for the comparison stars (shown as crosses) we can calculate our photometric uncertainties and thus determine whether the KBO brightness variations are significant ($\ge$3$\sigma$).
Period determination {#Capitulo4Period.Determination}
--------------------
In the cases where significant brightness variations (see Section \[Capitulo4KBOVariability\]) were detected in the lightcurves, the phase dispersion minimization method was used [PDM, @1978ApJ...224..953S] to look for periodicities in the data. For each test period, PDM computes a statistical parameter $\theta$ that compares the spread of data points in phase bins with the overall spread of the data. The period that best fits the data is the one that minimizes $\theta$. The advantages of PDM are that it is non-parametric, i.e., it does not assume a shape for the lightcurve, and it can handle unevenly spaced data. Each data set was tested for periods ranging from 2 to 24 hours, in steps of 0.01$\,$hr. To assess the uniqueness of the PDM solution, we generated 100 Monte Carlo realizations of each lightcurve, keeping the original data times and randomizing the magnitudes within the error bars. We ran PDM on each generated dataset to obtain a distribution of best-fit periods.
Amplitude determination {#Capitulo4Amplitude.Determination}
-----------------------
We used a Monte Carlo experiment to determine the amplitude of the lightcurves for which a period was found. We generated several artificial data sets by randomizing each point within the error bar. Each artificial data set was fitted with a Fourier series, using the best-fit period, and the mode and central 68% of the distribution of amplitudes were taken as the lightcurve amplitude and $1\sigma$ uncertainty, respectively.
For the null lightcurves, i.e. those where no significant variation was detected, we subtracted the typical error bar size from the total amplitude of the data to obtain an upper limit to the amplitude of the KBO photometric variation.
Results
=======
In this section we present the results of the lightcurve analysis for each of the observed KBOs. We found significant brightness variations ($\Delta m>0.15\,$mag) for 3 out of 10 KBOs (30%), and $\Delta m \ge
0.40\,$mag for 1 out of 10 (10%). This is consistent with previously published results: @2002AJ....124.1757S ([-@2002AJ....124.1757S], hereafter ) found a fraction of 31% with $\Delta m>0.15\,$mag and 23% with $\Delta m>0.40\,$mag, both consistent with our results. The other 7 KBOs do not show detectable variations. The results are summarized in Table \[Table.LCProperties\].
1998$\,$SN$_\mathbf{165}$
-------------------------
The brightness of ${(35671)\,1998\,\rm{SN}_{165}}$ varies significantly ($>5\sigma$) more than that of the comparison stars (see Figs. \[Fig.VarvsMag\] and \[Fig.VarHistograms\]c). The periodogram for this KBO shows a very broad minimum around $P=9\,$hr (Fig. \[Fig.1998SN165Periodogram\]a). The degeneracy implied by the broad minimum would only be resolved with additional data. A slight weaker minimum is seen at $P=6.5\,$hr, which is close to a 24$\,$hr alias of $P=9\,$hr.
@2002NewA....7..359P ([-@2002NewA....7..359P], hereafter ) observed this object in September 2000, but having only one night’s worth of data, they did not succeed in determining this object’s rotational period unambiguously. We used their data to solve the degeneracy in our PDM result. The data have not been absolutely calibrated, and the magnitudes are given relative to a bright field star. To be able to combine it with our own data we had to subtract the mean magnitudes. Our periodogram of ${(35671)\,1998\,\rm{SN}_{165}}$ (centered on the broad minimum) is shown in Fig. \[Fig.1998SN165Periodogram\]b and can be compared with the revised periodogram obtained with our data combined with the data (Fig. \[Fig.1998SN165Periodogram\]c). The minima become much clearer with the additional data, but because of the 1-year time difference between the two observational campaigns, many close aliases appear in the periodogram. The absolute minimum, at $P=8.84\,$hr, corresponds to a double peaked lightcurve (see Fig. \[Fig.1998SN1650884fit\]). The second best fit, $P=8.7\,$hr, produces a more scattered phase plot, in which the peak in the data coincides with our night 2. Period $P=8.84\,$hr was also favored by the Monte Carlo method described in Section \[Capitulo4Period.Determination\], being identified as the best fit in 55% of the cases versus 35% for $P=8.7\,$hr. The large size of the error bars compared to the amplitude is responsible for the ambiguity in the result. We will use $P=8.84\,$hr in the rest of the paper because it was consistently selected as the best fit.
The amplitude, obtained using the Monte Carlo method described in Section \[Capitulo4Amplitude.Determination\], is $\Delta
m=0.16\pm0.01\,$mag. This value was calculated using only our data, but it did not change when recalculated adding the data.
1999$\,$DF$_\mathbf{9}$
-----------------------
${(79983)\,1999\,\rm{DF}_{9}}$ shows large amplitude photometric variations ($\Delta
m_R\sim0.4\,$mag). The PDM periodogram for ${(79983)\,1999\,\rm{DF}_{9}}$ is shown in Fig. \[Fig.1999DF9.periodogram\]. The best-fit period is $P=6.65\,$hr, which corresponds to a double-peak lightcurve (Fig. \[Fig.1999DF90665fit\]). Other PDM minima are found close to $P/2\approx3.3\,$hr and $9.2\,$hr, a $24\,$hr alias of the best period. Phasing the data with $P/2$ results in a worse fit because the two minima of the double peaked lightcurve exhibit significantly different morphologies (Fig. \[Fig.1999DF90665fit\]); the peculiar sharp feature superimposed on the brighter minimum, which is reproduced on two different nights, may be caused by a non-convex feature on the surface of the KBO [@2003Icar..164..346T]. Period $P=6.65\,$hr was selected in 65 of the 100 Monte Carlo replications of the dataset (see Section \[Capitulo4Period.Determination\]). The second most selected solution (15%) was at $P=9\,$hr. We will use $P=6.65\,$hr for the rest of the paper.
The amplitude of the lightcurve, estimated as described in Section \[Capitulo4Amplitude.Determination\], is $\Delta
m_R=0.40\pm0.02\,$mag.
2001$\,$CZ$_\mathbf{31}$
------------------------
This object shows substantial brightness variations ($4.5\sigma$ above the comparison stars) in a systematic manner. The first night of data seems to sample nearly one complete rotational phase. As for ${(35671)\,1998\,\rm{SN}_{165}}$, the ${2001\,\rm{CZ}_{31}}$ data span only two nights of observations. In this case, however, the PDM minima (see Figs. \[Fig.2001CZ31.periodogram\]a and \[Fig.2001CZ31.periodogram\]b) are very narrow and only two correspond to independent periods, $P=4.69\,$hr (the minimum at $5.82\,$ is a $24\,$hr alias of $4.69\,$hr), and $P=5.23\,$hr.
${2001\,\rm{CZ}_{31}}$ has also been observed by in February and April 2001, with inconclusive results. We used their data to try to rule out one (or both) of the two periods we found. We mean-subtracted the data in order to combine it with our uncalibrated observations. Figure \[Fig.2001CZ31.periodogram\]c shows the section of the periodogram around $P=5\,$hr, recalculated using ’s first night plus our own data. The aliases are due to the 1 month time difference between the two observing runs. The new PDM minimum is at $P=4.71\,$hr – very close to the $P=4.69\,$hr determined from our data alone.
Visual inspection of the combined data set phased with $P=4.71\,$hr shows a very good match between ’s first night (2001 Feb 20) and our own data (see Fig. \[Fig.2001CZ31.0471fit\]). ’s second and third nights show very large scatter and were not included in our analysis. Phasing the data with $P=5.23\,$hr yields a more scattered lightcurve, which confirms the PDM result. The Monte Carlo test for uniqueness yielded $P=4.71\,$hr as the best-fit period in 57% of the cases, followed by $P=5.23\,$hr in 21%, and a few other solutions, all below 10%, between $P=5\,$hr and $P=6\,$hr. We will use $P=4.71\,$hr throughout the rest of the paper.
We measured a lightcurve amplitude of $\Delta m=0.21 \pm
0.02\,$mag. If we use both ours and ’s first night data, $\Delta m$ rises to 0.22$\,$mag.
Flat Lightcurves
----------------
The fluctuations detected in the optical data on KBOs ${(19308)\,1996\,\rm{TO}_{66}}$, ${1996\,\rm{TS}_{66}}$, ${(47171)\,1999\,\rm{TC}_{36}}$, ${(66652)\,1999\,\rm{RZ}_{253}}$, ${(80806)\,2000\,\rm{CM}_{105}}$, and ${(38628)\,\rm{Huya}}$, are well within the uncertainties. ${(19521)\,\rm{Chaos}}$ shows some variations with respect to the comparison stars but no period was found to fit all the data. See Table \[Table.LCProperties\] and Fig. \[Fig.FlatLCurves\] for a summary of the results.
Other lightcurve measurements
-----------------------------
The KBO lightcurve database has increased considerably in the last few years, largely due to the observational campaign of , with recent updates in and @2004AJ....127.3023S. These authors have published observations and rotational data for a total of 30 KBOs (their paper includes data for 3 other previously published lightcurves in the analysis). Other recently published KBO lightcurves include those for $(50000)\,$Quaoar and the scattered KBO $(29981)\,1999\,\rm{TD}_{10}$ . Of the 10 KBO lightcurves presented in this paper, 6 are new to the database, bringing the total to 41.
Table \[Table.OtherLC\] lists the rotational data on the 41 KBOs that will be analyzed in the rest of the paper.
Analysis
========
In this section we examine the lightcurve properties of KBOs and compare them with those of main-belt asteroids (MBAs). The lightcurve data for these two families of objects cover different size ranges. MBAs, being closer to Earth, can be observed down to much smaller sizes than KBOs; in general it is very difficult to obtain good quality lightcurves for KBOs with diameters $D<50\,$km. Furthermore, some KBOs surpass the $1000\,$km barrier whereas the largest asteroid, Ceres, does not reach $900\,$km. This will be taken into account in the analysis.
The lightcurve data for asteroids were taken from the Harris Lightcurve Catalog[^1], Version 5, while the diameter data were obtained from the Lowell Observatory database of asteroids orbital elements[^2]. The sizes of most KBOs were calculated from their absolute magnitude assuming an albedo of 0.04. The exceptions are (47171)$\,$1999$\,$TC$_{36}$, (38638)$\,$Huya, (28978)$\,$Ixion, (55636)$\,$2002$\,$TX$_{36}$, (66652)$\,$1999$\,$RZ$_{36}$, (26308)$\,$1998$\,$SM$_{165}$, and (20000)$\,$Varuna for which the albedo has been shown to be inconsistent with the value 0.04 [@2005Icar..176..184G]. For example, in the case of (20000)$\,$Varuna simultaneous thermal and optical observations have yielded a red geometric albedo of 0.070$_{-0.017}^{+0.030}$ [@2001Natur.411..446J].
Spin period statistics
----------------------
As Fig. \[Fig.PeriodDistrib\] shows, the spin period distributions of KBOs and MBAs are significantly different. Because the sample of KBOs of small size or large periods is poor, to avoid bias in our comparison we consider only KBOs and MBAs with diameter larger than $200\,$km and with periods below $20\,$hr. In this range the mean rotational periods of KBOs and MBAs are $9.23\,$hr and $6.48\,$hr, respectively, and the two are different with a 98.5% confidence according to Student’s $t$-test. However, the different means do not rule that the underlying distributions are the same, and could simply mean that the two sets of data sample the same distribution differently. This is not the case, however, according to the Kolmogorov-Smirnov (K-S) test, which gives a probability that the periods of KBOs and MBAs are drawn from the same parent distribution of 0.7%.
Although it is clear that KBOs spin slower than asteroids, it is not clear why this is so. If collisions have contributed as significantly to the angular momentum of KBOs as they did for MBAs , then the observed difference should be related to how these two families react to collision events. We will address the question of the collisional evolution of KBO spin rates in a future paper.
Lightcurve amplitudes and the shapes of KBOs {#Capitulo4Shape.Distribution}
--------------------------------------------
The cumulative distribution of KBO lightcurve amplitudes is shown in Fig. \[Fig.CumulAmpl\]. It rises very steeply in the low amplitude range ($\Delta m < 0.15\,$mag), and then becomes shallower reaching large amplitudes. In quantitative terms, $\sim 70\%$ of the KBOs possess $\Delta m < 0.15\,$mag, while $\sim 12\%$ possess $\Delta m \ge 0.40\,$mag, with the maximum value being $\Delta m=0.68\,$mag. \[Note: Fig. \[Fig.CumulAmpl\] does not include the KBO 2001$\,$QG$_{298}$ which has a lightcurve amplitude $\Delta
m=1.14\pm0.04\,$mag, and would further extend the range of amplitudes. We do not include 2001$\,$QG$_{298}$ in our analysis because it is thought to be a contact binary [@2004AJ....127.3023S]\]. Figure \[Fig.CumulAmpl\] also compares the KBO distribution with that of MBAs. The distributions of the two populations are clearly distinct: there is a larger fraction of KBOs in the low amplitude range ($\Delta m < 0.15\,$mag) than in the case of MBAs, and the KBO distribution extends to larger values of $\Delta m$.
Figure \[Fig.AmplvsSize\] shows the lightcurve amplitude of KBOs and MBAs plotted against size. KBOs with diameters larger than $D=400\,$km seem to have lower lightcurve amplitudes than KBOs with diameters smaller than $D=400\,$km. Student’s $t$-test confirms that the mean amplitudes in each of these two size ranges are different at the 98.5% confidence level. For MBAs the transition is less sharp and seems to occur at a smaller size ($D\sim 200\,$km). In the case of asteroids, the accepted explanation is that small bodies ($D\lesssim100\,$km) are fragments of high-velocity impacts, whereas of their larger counterparts ($D>200\,$km) generally are not . The lightcurve data on small KBOs are still too sparse to permit a similar analysis. In order to reduce the effects of bias related to body size, we can consider only those KBOs and MBAs with diameters larger than 200$\,$km. In this size range, 25 of 37 KBOs (69%) and 10 of 27 MBAs (37%) have lightcurve amplitudes below 0.15$\,$mag. We used the Fisher exact test to calculate the probability that such a contingency table would arise if the lightcurve amplitude distributions of KBOs and MBAs were the same: the resulting probability is 0.8%.
The distribution of lightcurve amplitudes can be used to infer the shapes of KBOs, if certain reasonable assumptions are made (see, e.g., ). Generally, objects with elongated shapes produce large brightness variations due to their changing projected cross-section as they rotate. Conversely, round objects, or those with the spin axis aligned with the line of sight, produce little or no brightness variations, resulting in “flat” lightcurves. Figure \[Fig.AmplvsSize\] shows that the lightcurve amplitudes of KBOs with diameters smaller and larger than $D=400\,$km are significantly different. Does this mean that the shapes of KBOs are also different in these two size ranges? To investigate this possibility of a size dependence among KBO shapes we will consider KBOs with diameter smaller and larger than $400\,$km separately. We shall loosely refer to objects with diameter $D>400\,$km and $D \le 400\,$km as [*larger*]{} and [*smaller*]{} KBOs, respectively.
We approximate the shapes of KBOs by triaxial ellipsoids with semi-axes $a>b>c$. For simplicity we consider the case where $b=c$ and use the axis ratio $\tilde{a}=a/b$ to characterize the shape of an object. The orientation of the spin axis is parameterized by the aspect angle $\theta$, defined as the smallest angular distance between the line of sight and the spin vector. On this basis the lightcurve amplitude $\Delta m$ is related to $\tilde{a}$ and $\theta$ via the relation (Eq. (2) of with $\bar{c}=1$) $$\begin{aligned}
\Delta m=2.5\log \sqrt{ \frac{2\,\tilde{a}^2} {1 + \tilde{a}^2 +
\left( \tilde{a}^2 - 1 \right) \,\cos (2\,\theta )}}\;.
\label{EqnDeltaMagCap4}
\end{aligned}$$
Following we model the shape distribution by a power-law of the form $$\begin{aligned}
f(\tilde{a})\,{\rm d}\tilde{a}\propto\tilde{a}^{-q}\,{\rm
d}\tilde{a}
\label{EqnShapeDist}
\end{aligned}$$ where $f(\tilde{a})\,{\rm d}\tilde{a}$ represents the fraction of objects with shapes between $\tilde{a}$ and $\tilde{a}+{\rm d}\tilde{a}$. We use the measured lightcurve amplitudes to estimate the value of $q$ by employing both the method described in , and by Monte Carlo fitting the observed amplitude distribution . The latter consists of generating artificial distributions of $\Delta m$ (Eq. \[EqnDeltaMagCap4\]) with values of $\tilde{a}$ drawn from distributions characterized by different $q$’s (Eq. \[EqnShapeDist\]), and selecting the one that best fits the observed cumulative amplitude distribution (Fig. \[Fig.CumulAmpl\]). The values of $\theta$ are generated assuming random spin axis orientations. We use the K-S test to compare the different fits. The errors are derived by bootstrap resampling the original data set [@Efron79], and measuring the dispersion in the distribution of best-fit power-law indexes, $q_i$, found for each bootstrap replication.
Following the method we calculate the probability of finding a KBO with $\Delta m \ge 0.15\,$mag: $$\begin{aligned}
p(\Delta m \ge 0.15) \approx \int^{\tilde{a}_{\rm
max}}_{\sqrt{K}} f(\tilde{a})\,\sqrt{\frac{\tilde{a}^2 -
K}{(\tilde{a}^2 - 1) K}} \,\mathrm{d}\tilde{a}.
\label{EqnProb2}
\end{aligned}$$ where $K=10^{0.8\times0.15}$, $f(\tilde{a})=C\,\tilde{a}^{-q}$, and $C$ is a normalization constant. This probability is calculated for a range of $q$’s to determine the one that best matches the observed fraction of lightcurves with amplitude larger than 0.15$\,$mag. These fractions are $f(\Delta m\ge0.15\,{\rm mag};\,D\le400\,{\rm km})=8/19$, and $f(\Delta
m\ge0.15\,{\rm mag};\,D>400\,{\rm km})=5/21$, and $f(\Delta m\ge0.15\,{\rm
mag})=13/40$ for the complete set of data. The results are summarized in Table \[Table.BestFitShapeDist\] and shown in Fig. \[Fig.CumulDistrFit\].
The uncertainties in the values of $q$ obtained using the method ($q=4.3^{+2.0}_{-1.6}$ for KBOs with $D\le400\,$km and $q=7.4^{+3.1}_{-2.4}$ for KBOs with $D>400\,$km ; see Table \[Table.BestFitShapeDist\]) do not rule out similar shape distributions for smaller and larger KBOs. This is not the case for the Monte Carlo method. The reason for this is that the method relies on a single, more robust parameter: the fraction of lightcurves with detectable variations. The sizeable error bar is indicative that a larger dataset is needed to better constrain the values of $q$. In any case, it is reassuring that both methods yield steeper shape distributions for larger KBOs, implying more spherical shapes in this size range. A distribution with $q\sim8$ predicts that $\sim$75% of the large KBOs have $a/b<1.2$. For the smaller objects we find a shallower distribution, $q\sim4$, which implies a significant fraction of very elongated objects: $\sim$20% have $a/b>1.7$. Although based on small numbers, the shape distribution of large KBOs is well fit by a simple power-law (the K-S rejection probability is 0.6%). This is not the case for smaller KBOs for which the fit is poorer (K-S rejection probability is 20%, see Fig. \[Fig.CumulDistrFit\]). Our results are in agreement with previous studies of the overall KBO shape distribution, which had already shown that a simple power-law does not explain the shapes of KBOs as a whole .
The results presented in this section suggest that the shape distributions of smaller and larger KBOs are different. However, the existing number of lightcurves is not enough to make this difference statistically significant. When compared to asteroids, KBOs show a preponderance of low amplitude lightcurves, possibly a consequence of their possessing a larger fraction of nearly spherical objects. It should be noted that most of our analysis assumes that the lightcurve sample used is homogeneous and unbiased; this is probably not true. Different observing conditions, instrumentation, and data analysis methods introduce systematic uncertainties in the dataset. However, the most likely source of bias in the sample is that some flat lightcurves may not have been published. If this is the case, our conclusion that the amplitude distributions of KBOs and MBAs are different would be strengthened. On the other hand, if most unreported non-detections correspond to smaller KBOs then the inferred contrast in the shape distributions of different-sized KBOs would be less significant. Clearly, better observational contraints, particularly of smaller KBOs, are necessary to constrain the KBO shape distribution and understand its origin.
The inner structure of KBOs
---------------------------
In this section we wish to investigate if the rotational properties of KBOs show any evidence that they have a rubble pile structure; a possible dependence on object size is also investigated. As in the case of asteroids, collisional evolution may have played an important role in modifying the inner structure of KBOs. Large asteroids ($D\gtrsim200\,$km) have in principle survived collisional destruction for the age of the solar system, but may nonetheless have been converted to rubble piles by repeated impacts. As a result of multiple collisions, the “loose” pieces of the larger asteroids may have reassembled into shapes close to triaxial equilibrium ellipsoids [@1981Icar...46..114F]. Instead, the shapes of smaller asteroids ($D\le100\,$km) are consistent with collisional fragments , indicating that they are most likely by-products of disruptive collisions.
Figure \[Fig.PvsAmpl\] plots the lightcurve amplitudes versus spin periods for the 15 KBOs whose lightcurve amplitudes and spin period are known. Open and filled symbols indicate the KBOs with diameter smaller and larger than $D=400\,$km, respectively. Clearly, the smaller and larger KBOs occupy different regions of the diagram. For the larger KBOs (black filled circles) the (small) lightcurve amplitudes are almost independent of the objects’ spin periods. In contrast, smaller KBOs span a much broader range of lightcurve amplitudes. Two objects have very low amplitudes: ${(35671)\,1998\,\rm{SN}_{165}}$ and 1999$\,$KR$_{16}$, which have diameters $D\sim400\,$km and fall precisely on the boundary of the two size ranges. The remaining objects hint at a trend of increasing $\Delta m$ with lower spin rates. The one exception is 1999$\,$TD$_{10}$, a Scattered Disk Object ($e=0.872, a=95.703\,$AU) that spends most of its orbit in rather empty regions of space and most likely has a different collisional history.
For comparison, Fig. \[Fig.PvsAmpl\] also shows results of N-body simulations of collisions between “ideal” rubble piles [gray filled circles; @2000Icar..146..133L], and the lightcurve amplitude-spin period relation predicted by ellipsoidal figures of hydrostatic equilibrium [dashed and dotted lines; @1969efe..book.....C; @2001Icar..154..432H]. The latter is calculated from the equilibrium shapes that rotating uniform fluid bodies assume by balancing gravitational and centrifugal acceleration. The spin rate-shape relation in the case of uniform fluids depends solely on the density of the body. Although fluid bodies behave in many respects differently from rubble piles, they may, as an extreme case, provide insight on the equilibrium shapes of gravitationally bound agglomerates. The lightcurve amplitudes of both theoretical expectations are plotted assuming an equator-on observing geometry. They should therefore be taken as upper limits when compared to the observed KBO amplitudes, the lower limit being zero amplitude.
The simulations of @2000Icar..146..133L ([-@2000Icar..146..133L], hereafter ) consist of collisions between agglomerates of small spheres meant to simulate collisions between rubble piles. Each agglomerate consists of $\sim 1000$ spheres, held together by their mutual gravity, and has no initial spin. The spheres are indestructible, have no sliding friction, and have coefficients of restitution of $\sim0.8$. The bulk density of the agglomerates is 2000$\,$kg$\,$m$^{-3}$. The impact velocities range from $\sim$ zero at infinity to a few times the critical velocity for which the impact energy would exceed the mutual gravitational binding energy of the two rubble piles. The impact geometries range from head-on collisions to grazing impacts. The mass, final spin period, and shape of the [*largest remnant*]{} of each collision are registered (see Table 1 of ). From their results, we selected the outcomes for which the mass of the largest remnant is equal to or larger than the mass of one of the colliding rubble piles, i.e., we selected only accreting outcomes. The spin periods and lightcurve amplitudes that would be generated by such remnants (assuming they are observed equator-on) are plotted in Fig. \[Fig.PvsAmpl\] as gray circles. Note that, although the simulated rubble piles have radii of 1$\,$km, since the effects of the collision scale with the ratio of impact energy to gravitational binding energy of the colliding bodies [@1999Icar..142....5B], the model results should apply to other sizes. Clearly, the model makes several specific assumptions, and represents one possible idealization of what is usually referred to as “rubble pile”. Nevertheless, the results are illustrative of how collisions may affect this type of structure, and are useful for comparison with the KBO data.
The lightcurve amplitudes resulting from the experiment are relatively small ($\Delta m<0.25\,$mag) for spin periods larger than $P\sim5.5\,$hr (see Fig. \[Fig.PvsAmpl\]). Objects spinning faster than $P=5.5\,$hr have more elongated shapes, resulting in larger lightcurve amplitudes, up to 0.65 magnitudes. The latter are the result of collisions with higher angular momentum transfer than the former (see Table 1 of ). The maximum spin rate attained by the rubble piles, as a result of the collision, is $\sim 4.5\,$hr. This is consistent with the maximum spin expected for bodies in hydrostatic equilibrium with the same density as the rubble piles ($\rho = 2000\,$kg$\,$m$^{-3}$; see long-dashed line in Fig. \[Fig.PvsAmpl\]). The results of show that collisions between ideal rubble piles can produce elongated remnants (when the projectile brings significant angular momentum into the target), and that the spin rates of the collisional remnants do not extend much beyond the maximum spin permitted to fluid uniform bodies with the same bulk density.
The distribution of KBOs in Fig. \[Fig.PvsAmpl\] is less clear. Indirect estimates of KBO bulk densities indicate values $\rho\sim1000\,$kg$\,$m$^{-3}$ . If KBOs are strengthless rubble piles with such low densities we would not expect to find objects with spin periods lower than $P\sim6\,$hr (dashed line in Fig. \[Fig.PvsAmpl\]). However, one object (${2001\,\rm{CZ}_{31}}$) is found to have a spin period below 5$\,$hr. If this object has a rubble pile structure, its density must be at least $\sim2000\,$kg$\,$m$^{-3}$. The remaining 14 objects have spin periods below the expected upper limit, given their estimated density. Of the 14, 4 objects lie close to the line corresponding to equilibrium ellipsoids of density $\rho=1000\,$kg$\,$m$^{-3}$. One of these objects, (20000)$\,$Varuna, has been studied in detail by @2002AJ....124.1757S. The authors conclude that (20000)$\,$Varuna is best interpreted as a rotationally deformed rubble pile with $\rho \le
1000\,$kg$\,$m$^{-3}$. One object, 2001$\,$QG$_{298}$, has an exceptionally large lightcurve amplitude ($\Delta m=1.14\,$mag), indicative of a very elongated shape (axes ratio $a/b>2.85$), but given its modest spin rate ($P=13.8\,$hr) and approximate size ($D\sim 240\,$km) it is unlikely that it would be able to keep such an elongated shape against the crush of gravity. Analysis of the lightcurve of this object [@2004AJ....127.3023S] suggests it is a close/contact binary KBO. The same applies to two other KBOs, 2000$\,$GN$_{171}$ and (33128) 1998$\,$BU$_{48}$, also very likely to be contact binaries.
To summarize, it is not clear that KBOs have a rubble pile structure, based on their available rotational properties. A comparison with computer simulations of rubble pile collisions shows that larger KBOs ($D>400\,$km) occupy the same region of the period-amplitude diagram as the results. This is not the case for most of the smaller KBOs ($D\le 400\,$km), which tend to have larger lightcurve amplitudes for similar spin periods. If most KBOs are rubble piles then their spin rates set a lower limit to their bulk density: one object (${2001\,\rm{CZ}_{31}}$) spins fast enough that its density must be at least $\rho\sim2000\,$kg$\,$m$^{-3}$, while 4 other KBOs (including (20000)$\,$Varuna) must have densities larger than $\rho\sim1000\,$kg$\,$m$^{-3}$. A better assessment of the inner structure of KBOs will require more observations, and detailed modelling of the collisional evolution of rubble-piles.
Conclusions
===========
We have collected and analyzed R-band photometric data for 10 Kuiper Belt objects, 5 of which have not been studied before. No significant brightness variations were detected from KBOs ${(80806)\,2000\,\rm{CM}_{105}}$, ${(66652)\,1999\,\rm{RZ}_{253}}$, ${1996\,\rm{TS}_{66}}$. Previously observed KBOs ${(19521)\,\rm{Chaos}}$, ${(47171)\,1999\,\rm{TC}_{36}}$, and ${(38628)\,\rm{Huya}}$ were confirmed to have very low amplitude lightcurves ($\Delta m \le 0.1\,$mag). ${(35671)\,1998\,\rm{SN}_{165}}$, ${(79983)\,1999\,\rm{DF}_{9}}$, and ${2001\,\rm{CZ}_{31}}$ were shown to have periodic brightness variations. Our lightcurve amplitude statistics are thus: 3 out of 10 (30%) observed KBOs have $\Delta m \ge 0.15\,$mag, and 1 out of 10 (10%) has $\Delta m \ge 0.40\,$mag. This is consistent with previously published results.
The rotational properties that we obtained were combined with existing data in the literature and the total data set was used to investigate the distribution of spin period and shapes of KBOs. Our conclusions can be summarized as follows:
1. KBOs with diameters $D>200\,$km have a mean spin period of $9.23\,$hr, and thus rotate slower on average than main belt asteroids of similar size ($\langle P \rangle_{\rm MBAs}=6.48\,$hr). The probability that the two distributions are drawn from the same parent distribution is 0.7%, as judged by the KS test.
2. 26 of 37 KBOs (70%, $D>200\,$km) have lightcurve amplitudes below $0.15\,$mag. In the asteroid belt only 10 of the 27 (37%) asteroids in the same size range have such low amplitude lightcurves. This difference is significant at the 99.2% level according to the Fisher exact test.
3. KBOs with diameters $D>400\,$km have lightcurves with significantly (98.5% confidence) smaller amplitudes ($\langle \Delta m \rangle=0.13\,$mag, $D>400\,$km) than KBOs with diameters $D\le 400\,$km ($\langle \Delta m
\rangle=0.25\,$mag, $D\le 400\,$km).
4. These two size ranges seem to have different shape distributions, but the few existing data do not render the difference statistically significant. Even though the shape distributions in the two size ranges are not inconsistent, the best-fit power-law solutions predict a larger fraction of round objects in the $D>400\,$km size range ($f(a/b<1.2)\sim70^{+12}_{-19}\%$) than in the group of smaller objects ($f(a/b<1.2)\sim42^{+20}_{-15}\%$).
5. The current KBO lightcurve data are too sparse to allow a conclusive assessment of the inner structure of KBOs.
6. KBO ${2001\,\rm{CZ}_{31}}$ has a spin period of $P=4.71\,$hr. If this object has a rubble pile structure then its density must be $\rho \gtrsim 2000\,$kg$\,$m$^{-3}$. If the object has a lower density then it must have internal strength.
The analysis presented in this paper rests on the assumption that the available sample of KBO rotational properties is homogeneous. However, in all likelihood the database is biased. The most likely bias in the sample comes from unpublished flat lightcurves. If a significant fraction of flat lightcurves remains unreported then points 1 and 2 above could be strengthened, depending on the cause of the lack of brightness variation (slow spin or round shape). On the other hand, points 3 and 4 could be weakened if most unreported cases correspond to smaller KBOs. Better interpretation of the rotational properties of KBOs will greatly benefit from a larger and more homogeneous dataset.
This work was supported by grants from the Netherlands Foundation for Research (NWO), the Leids Kerkhoven-Bosscha Fonds (LKBF), and a NASA grant to D. Jewitt. We are grateful to Scott Kenyon, Ivo Labbé, and D. J. for helpful discussion and comments.
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[^1]: <http://pdssbn.astro.umd.edu/sbnhtml/asteroids/colors_lc.html>
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|
---
abstract: 'Metal ions play numerous important roles in biological systems being central to the function of biomolecules. In this letter we show that the absorption of X-rays by these ions leads to a complicated chain of ultrafast relaxation steps resulting in the complete degradation of their nearest environment. We conducted high quality [*ab initio*]{} studies on microsolvated Mg$^{2+}$ clusters demonstrating that ionisation of an 1s-electron of Mg leads to a complicated electronic cascade comprising both intra- and intermolecular steps and lasting only a few hundreds femtoseconds. The metal cation reverts to its original charge state at the end of the cascade, while the nearest solvation shell becomes multiply ionised and large concentrations of radical and slow electron species build up in the metal’s vicinity. We conclude that such cascades involving metal ions are essential for understanding the radiation chemistry of solutions and radiation damage to metal containing biomolecules.'
author:
- 'V. Stumpf'
- 'K. Gokhberg'
- 'L. S. Cederbaum'
title: 'The role of metal ions in X-ray induced photochemistry'
---
Metals like Mg, Ca, Fe, Zn or Cu are essential for living organisms [@Bertini07]. They fulfil a number of important roles in biological systems, such as being instrumental for the catalytic activity of enzymes [@Bertini07; @Andreini08] or ensuring structural stability of chromosomes [@Strick01; @Wu10]. In this report we investigate what is the role of these naturally occurring metal species in radiation damage or, more broadly, X-ray induced photochemistry of biological systems. The X-ray absorption spectroscopy of transition metal complexes in solutions offers a glimpse at the transformations accompanying the interaction of X-rays with the metal. The metal ion may become oxidised, reduced or even remain unaltered [@George08; @Mesu06]; the particular outcome depends on the initial metal charge and the nature of ligands. Ample evidence from a different field, that of the X-ray crystallography, demonstrates that the radiation damage inflicted on a sample is highly non-random [@Carugo05]. Importantly, in metallo-proteins the site containing the metal is one of the weak spots; it is often damaged at a dose much lower than the one at which the rest of the protein molecule is damaged [@Yano05]. It is common to consider the photoreduction of the metal ion as a manifestation of the damage [@George12]. This change in the metal’s charge is thought to change the active site’s geometry and correspondingly the diffraction image.
In spite of accumulated knowledge mentioned above, details about the damage produced by the photoabsorption of an X-ray photon by the metal are sparse. Moreover, little is known about a sequence of events leading to this damage. In what follows we present a new mechanism based on high-quality [*ab initio*]{} calculations, which shows that the physico-chemical events started by the photoabsorption are concentrated on the metal species and its immediate vicinity causing thereby much damage to the molecules close to the metal. We find that the damage caused following the absorption of an X-ray photon by the metal can also be substantial in case the metal carries at the end of the process the same charge as before the photoabsorption.
Absorption of an X-ray photon by a metal predominantly removes a core electron and initiates a cascade of relaxation processes. For typical metals the first step is Auger decay during which electrons with characteristic energies are emitted by the metal species which in turn accumulates a positive charge. The emitted Auger electrons propagate through the system causing secondary ionisations and resulting in damaging lesions [@Howell08]. It is clear by now that in a biological medium further damaging relaxation processes are present. We expect the Auger decay to be accompanied and followed by ultrafast electronic decay processes involving the neighbouring molecules [@Gokhberg14].
![[]{data-label="Fig1"}](fig1.eps)
These non-local processes come in two varieties (see Fig.\[Fig1\]) - interatomic Coulombic decay (ICD) driven by energy transfer, and electron transfer mediated decay (ETMD) driven by charge transfer [@Cederbaum97; @Zobeley01; @Jahnke15]. In ICD electronically excited metal species (M) de-excites by ionising a neighbouring molecule (L) $$\nonumber
M^{q+\ast}\cdot L_{n}\rightarrow M^{q+}\cdot L_{n-1} + L^{+} + e_{ICD}.
\label{eq1}$$ In ETMD a neighbour donates an electron to the metal species, while the excess energy is used to ionise the donor (ETMD(2)) or another molecule (ETMD(3))
$$\nonumber
M^{q+}\cdot L_{n}\rightarrow M^{(q-1)+}\cdot L_{n-1} + L^{2+} + e_{ETMD(2)},
\label{eq2a}$$
$$\nonumber
M^{q+}\cdot L_{n}\rightarrow M^{(q-1)+}\cdot L_{n-2} + 2L^{+} + e_{ETMD(3)}.
\label{eq2b}$$
Both ICD and ETMD lead to the ionisation of the nearest molecules and, in addition, to emitting a slow electron. However, in ICD the medium is only singly ionised, while the charge on the metal remains unaltered. In ETMD the medium is doubly ionised and the charge on the metal is reduced by one. The rate of energy transfer processes falls off as an inverse power of the distance, while for the charge transfer processes the rate behaviour is exponential [@Averbukh04; @Zobeley01]. Therefore, previous work on the interatomic decay in rare-gas clusters showed that ICD proceeds on a timescale of tens of femtoseconds compared to the picosecond timescale of ETMD [@Kolorenc08]. As a result, ETMD is usually observed if the decaying state does not have enough excess energy to decay by ICD [@Forstel11; @Sakai11]. This was specifically demonstrated to be the case for multiply charged ions produced in clusters by the Auger decay process [@Stumpf13].
To study the complex chain of physico-chemical events following the X-ray photoabsorption by metal atoms we considered a microsolvated cluster Mg$^{2+}$(H$_{2}$O)$_{6}$. It may serve as a model of Mg$^{2+}$ in solution and previous experience showed that such cluster models work well [@Pokapanich11]. Indeed, the Mg$^{2+}$ cation usually accommodates six water molecules in its first solvation shell. The equilibrium Mg$^{2+}$-O distance in the cluster is 2.078 Å and lies close to the corresponding value of 2.00-2.15 Å in solution [@Ohtaki93]. Therefore, since the interatomic decay processes involve overwhelmingly the nearest neighbours, the decay rates and branching ratios should be also similar. More generally, the physico-chemical processes in the Mg$^{2+}$(H$_{2}$O)$_{6}$ cluster may serve as a paradigm applicable to other metal complexes.
We found (see Supplementary information for the computational details) that the binding energy of the magnesium 1s-electron (K-edge) in Mg$^{2+}$(H$_{2}$O)$_{6}$ is 1317 eV. Removing this electron by an X-ray photon creates a highly energetic, electronically unstable trication. In the following we discuss the fate of this unstable trication. It predominantly decays by emitting an Auger electron and populating a number of Mg$^{4+}$(H$_{2}$O)$_{6}$ states. The states of Mg$^{4+}$ populated in the Auger decay are Mg$^{4+}(2p^{-2} [^1D,^1S])$ (63%), Mg$^{4+}(2s^{-1}2p^{-1} [^1P])$ (21%), Mg$^{4+}(2s^{-1}2p^{-1} [^3P])$ (9%), and Mg$^{4+}(2s^{-2}[^1S])$ (7%). All of them are unstable with respect to intermolecular electronic decay in the presence of water molecules. Since the excess energy in microsolvated Mg$^{4+}(2p^{-2} [^1D,^1S])$ relative to the Mg$^{4+}(2p^{-2} [^3P])$ ground state of the cation is about 4-9 eV, these states cannot decay by ICD. However, they efficiently decay by both ETMD(2) and ETMD(3) pathways. The computed ETMD lifetime is 16.0 fs, which is much shorter than the picosecond lifetimes found in rare-gas clusters [@Kolorenc08]. Shorter ion - neighbour distances lead to larger orbital overlap greatly facilitating this electron transfer driven decay [@Zobeley01]. The ETMD(2) to ETMD(3) branching ratio is 1.0/1.6, therefore, both H$_{2}$O$^{+}$ and H$_{2}$O$^{2+}$ are efficiently produced in the ionisation of the medium. The final products are either Mg$^{3+}(2p^{-1} [^2P])$(H$_{2}$O$^{2+}$)(H$_{2}$O)$_{5}$ or Mg$^{3+}(2p^{-1} [^2P])$(H$_{2}$O$^{+}$)$_{2}$(H$_{2}$O)$_{4}$; the charge on Mg decreases by one as the result of electron transfer from water (see Fig.\[Fig1\]). The ETMD electrons have energies between 11 and 26 eV for ETMD(2) and between 23 and 40 eV for ETMD(3). The greater delocalisation of the positive charge in the final state of ETMD(3) results in faster emitted electrons.
Unlike the states considered above, microsolvated Mg$^{4+}(2s^{-1}2p^{-1} [^1P])$ possesses enough excess energy for the ionisation of the water molecules and, therefore, can decay by ICD. Its computed ICD lifetime is extremely short, 0.7 fs, showing that this interatomic decay occurs even faster than the already very fast local Auger decay on the metal (1.9 fs). We mention that the experimentally determined ICD lifetime of Mg$^{3+}(2s^{-1}) [^2S]$) in liquid water is 1.0-1.5 fs [@Ohrwall10], indicating that such extremely fast interatomic decay in hydrated metal ions is the rule rather than exception, due to the short metal-water distances and a large number of open ICD channels. In this state ICD will also dominate ETMD by about an order of magnitude due to the higher efficiency of energy transfer. The energies of the ICD electrons lie below 7 eV. Since ICD does not change the charge on Mg it produces Mg$^{4+}(2p^{-2})$(H$_{2}$O$^{+}$)(H$_{2}$O)$_{5}$ states. These are the same states of Mg$^{4+}$ we considered above, but now with H$_{2}$O$^{+}$ ion in its vicinity. Can they continue decaying by ETMD?
Model calculations with the water neighbour ionised via ICD replaced by a point charge (for details see Supplementary information) reveal that most ETMD(3) and ETMD(2) channels remain open. The ETMD lifetime grows from 16.0 to 21.8 fs since only five neutral water neighbours are now available for the electronic decay. We also expect an interesting interplay between ETMD and fragmentation of the Mg$^{4+}(2p^{-2})$(H$_{2}$O$^{+}$)(H$_{2}$O)$_{5}$ cluster in a Coulomb explosion which typically proceeds on the femtosecond timescale [@Vendrell10].
The Mg$^{4+}$ states Mg$^{4+}(2s^{-1}2p^{-1} [^3P])$ and Mg$^{4+}(2s^{-2}[^1S])$ which are weakly populated in the Auger decay similarly exhibit complicated de-excitation pathways. They decay through a cascade of ICD and ETMD steps ultimately producing the Mg$^{3+}(2p^{-1} [^2P])$ cation. The lifetimes at all steps involved lie below 25 fs.
We saw that the electronic decay after the core ionisation of magnesium leads to the formation of Mg$^{3+}(2p^{-1} [^2P])$ species. We found that both ETMD(2) and ETMD(3) channels are open for Mg$^{3+}(2p^{-1} [^2P])$(H$_{2}$O)$_{6}$. The corresponding lifetime is 15.5 fs, while the emitted electrons have energies of 0-6 and 5-19 eV, respectively. Following this decay step the Mg$^{2+}$(H$_{2}$O$^{2+}$)(H$_{2}$O)$_{5}$ or Mg$^{2+}$(H$_{2}$O$^{+}$)$_2$(H$_{2}$O)$_{4}$ species are produced with the ratio of 1.0 to 1.2.
We wish to remark at this point the Mg$^{3+}(2p^{-1} [^2P])$ species can also be obtained directly from the initial core ionised Mg ion in the presence of water ligands. Thus, a 2p-electron of Mg may fill the 1s-vacancy and transfer its energy not to another 2p-electron, which would result in the Auger process on the metal, but to a valence electron on water ionising it in a core ICD-like process [@Pokapanich09; @Pokapanich11; @Slavicek14]. This decay leads to population of Mg$^{3+}$(H$_{2}$O$^{+}$)(H$_{2}$O)$_{5}$ states. Since the positive charge in the latter is more delocalised than in the final states of the Auger decay, the electrons emitted in the core ICD-like process have larger energies (1205-1225 eV) compared to the Auger electrons (1111-1157 eV). What process will be more important? The Auger lifetime of the core ionised state was found to be 1.9 fs, while its ICD lifetime was 57 fs. Therefore, the Auger decay takes place in 97 % of Mg$^{4+}$(H$_{2}$O)$_{6}$ systems and core ICD-like only in 3%. This branching ratio, however, can be tilted much more in favour of the interatomic process for other metal ions. Thus in the case of Ca$^{2+}$ the core ICD-like probability was found experimentally to be 10% [@Ottosson12].
From the previous description of the decay processes involving different Mg cations it is clear that these disparate steps can be glued together in one continuous cascade. Its schematic representation is shown in Fig.\[Fig2\], where all steps discussed above can be found. Taking into account the decreasing number of neutral water neighbours available for interatomic decay at each consecutive step of the cascade we find that 90 percent of the core ionised states would cascade through to the final state within only 220 fs. Within this time window a large amount of Coulombic repulsion energy will be accumulated, due to the high metal charge and short distances to the ionised water neighbours. Its release is expected to result in a complicated fragmentation pattern, involving Coulomb explosions and molecular dissociation [@Pedersen13]. Note, that after only 10 fs already 28 % of the core ionised states will undergo the Auger and the interatomic Coulombic decays. This sets off a Coulomb explosion leading to a modification of molecular geometry in the vicinity of the metal ion already at that short timescale.
All the individual decay steps we discussed above for the microsolvated clusters will occur also in solution. The presence of additional solvation shells will further increase the efficiency of ICD, but their overall impact will be moderate, mainly because of their larger distance to the metal ion [@Averbukh04; @Fasshauer14]. The effect of the polarisable medium will be manifested in stabilising of the positively charged ions at different stages of the cascade. Therefore, the energies of the emitted electrons will be somewhat different from the cluster’s case.
![[]{data-label="Fig2"}](fig2.eps)
This cascade may be viewed from two different angles: the one of the metal cation, and the other one of the nearest neighbours or ligands. As we mentioned above the core ionisation of Mg$^{2+}$ creates a highly metastable ion which first decays mostly in the Auger process. Even after the local Auger processes are over, the interatomic decay keeps going on. If for a specific electronic state of the metal both ICD and ETMD channels are available, ICD will be predominant. It will lead to the electronic relaxation of the metal ion without changing its charge. An ETMD step will follow reducing the charge by one. This is illustrated by the decay of Mg$^{4+}(2s^{-1}2p^{-1} [^1P])$ first by ICD to Mg$^{4+}(2p^{-2})$ states and further by ETMD to Mg$^{3+}(2p^{-1} [^2P])$. Even if the ion does not have excess energy and ICD is not possible there can still be available ETMD channels. Therefore, the cascade of the interatomic processes goes on and the ion is being reduced until no interatomic electronic decay is allowed. For our system it leads to the surprising result that Mg ends up in the same electronic state it was in initially.
From the point of view of the nearest neighbours the probability of their damage by the $\approx$1150 eV Auger electrons is small. However, they are directly damaged in the interatomic processes becoming either singly or doubly ionised. In the case of water solutions the single ionisation will lead to proton transfer reaction producing HO$\cdot$ radical, while doubly ionised water is expected to produce atomic oxygen [@Tavernelli08]. The hydroxyl radical is highly reactive and causes oxidative damage to other molecules present in the solution (see e.g. [@Oneill01]). Atomic oxygen is reactive as well and is admittedly a source of further damaging species such as H$_2$O$_2$ [@Gaigeot07]. In addition to such direct damage to the nearest neighbours and the production of radicals, the interatomic processes result in the emission of electrons having energies $<$40 eV. Such electrons can be resonantly captured by the molecules in the near environment initiating efficient bond breaking reactions [@Alizadeh15]. In total, the interatomic decay processes massively degrade the molecules in the immediate vicinity of the metal species through multiple ionisations releasing in their course both reactive electrons and radicals. If we are to neglect the interatomic processes, the decay of the 1s vacancy on Mg$^{2+}$ would result in one (Auger) electron. Taking them into account and counting the damaging particles released in the complete cascade one would obtain on average 2.4 slow electrons in addition to a fast Auger electron and 4.3 radicals per each 1s vacancy.
In the decay cascade presented above the Mg ion which absorbs the X-ray photon reverts back to its initial electronic state within few hundred femtoseconds. At the same time the nearest environment of this ion is multiply ionised implying the production of a large number of reactive particles (radicals and slow electrons) at the metal’s location. This shows that no change of the metal ion’s charge on X-ray irradiation is by no means equal to no damage done. On the ground of our detailed findings one may anticipate that the similar cascades generally hold also for other metal ions independently whether the ion reverts back to its initial charge or photoreduction is observed. Clearly, these cascades will be accompanied by extensive damage to the surrounding molecules. In addition, strategic positioning of metal ions in biomolecules makes such damage particularly disruptive for the functions these molecules fulfil. Experimental evidence is available that irradiation of DNA molecules complexed with Ca$^{2+}$ by X-rays at energies below and above the Ca K-edge showed about 30% enhancement in the induction of double strand breaks when the photon energy was increased across the K-edge [@Takakura96]. Similarly, it was demonstrated that iron containing metallo-enzymes were more efficiently deactivated when irradiated with the X-rays above the iron’s K-edge [@Jawad86]. We hope that the interatomic electronic cascades elucidated in this report will be useful in understanding the X-ray induced photochemistry and radiation damage of metal containing biomolecules.
Methods
=======
All computations were done by *ab initio* electronic structure methods, using triple-$\zeta$ level Dunning basis sets [@Dunning89; @Woon94; @Woon95; @EMSL]. The only exception was the calculation of Auger decay rates and product populations, where quintuple-$\zeta$ level basis sets were required. For the basis set details see the Supplementary information.
The geometry of the Mg$^{2+}$(H$_2$O)$_6$ cluster was obtained by symmetry constrained optimisation, relying on the M[ø]{}ller-Plesset second order perturbation theory (MP2) implemented in the MOLPRO 2010.1 quantum chemistry package [@MOLPRO2010]. The T$_h$ symmetry of the cluster was chosen according to Feller et al. and represents the global minimum geometry [@Glendening96]. All electronic decay processes were studied at this cluster geometry.
To identify open decay channels, the energies of the involved tricationic and tetracationic electronic states were calculated by the Algebraic Diagrammatic Construction (ADC(2)) scheme for the one-particle and two-particle propagator, respectively [@Trofimov05; @Tarantelli06; @Velkov11]. The latter allow the calculation of single and double ionisation potentials relatively to the Mg$^{2+}$(H$_2$O)$_n$ electronic ground state introducing a perturbational expansion scheme of the propagator complete up to second order. The electron integrals and molecular orbitals serving as input for the propagator based computations were calculated by MOLCAS 7.4 software package [@MOLCAS74]. The energies of pentacationic states were obtained by multi-reference configuration interaction method including single and double excitations (DIRECT-CI)[@Saunders83] implemented in the GAMESS-UK 8.0 quantum chemistry package [@gamess_uk]. The reference space was built up starting with the Mg$^{2+}$(H$_2$O)$_n$ ground state Hartree-Fock determinant. The final states of ETMD were constructed by selecting all configurations having a hole in a 2s- or 2p-orbital of Mg plus two additional holes in the valence orbitals localised on water molecules. The final states of ICD were constructed by selecting all configurations having two holes in 2s- 2p-orbitals of Mg plus an additional hole in the valence orbitals localised on water molecules.
The total and partial electronic decay widths were calculated by means of Fano-ADC-Stieltjes method [@Averbukh05; @Kolorenc08]. Due to numerical limitations, the decay widths for the Mg$^{4+}$(H$_2$O)$_6$ states were determined in an additive approximation [@Mueller05; @Kryzhevoi07]. For details of this approximation and calculation of the partial decay widths see the Supplementary information.
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abstract: 'Near-IR imaging of the AGB star IRC+10011 (= CIT3) reveals the presence of a bipolar structure within the central 0.1 arcsec of a spherical dusty wind. We show that the image asymmetries originate from of swept-up wind material in an elongated cocoon whose expansion is driven by bipolar jets. We perform detailed 2D radiative transfer calculations with the cocoon modeled as two cones extending to 1,100 AU within an opening angle of 30, imbedded in a wind with the standard $r^{-2}$ density profile. The cocoon expansion started $\la$ 200 years ago, while the total lifetime of the circumstellar shell is 5,500 years. Similar bipolar expansion, at various stages of evolution, has been recently observed in a number of other AGB stars, culminating in jet breakout from the confining spherical wind. The bipolar outflow is triggered at a late stage in the evolution of AGB winds, and IRC+10011 provides its earliest example thus far. These new developments enable us to identify the first instance of symmetry breaking in the evolution from AGB to planetary nebula.'
author:
- |
Dejan Vinković,$^{1,2}$ Thomas Blöcker,$^3$ Karl-Heinz Hofmann,$^3$ Moshe Elitzur$^2$ and Gerd Weigelt$^3$\
$^1$School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA; dejan@ias.edu\
$^2$Department of Physics & Astronomy, University of Kentucky, Lexington, KY 40506-0055, USA; moshe@uky.edu\
$^3$Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany; khh@mpifr-bonn.mpg.de, weigelt@mpifr-bonn.mpg.de\
title: 'Bipolar outflow on the Asymptotic Giant Branch—the case of IRC+10011'
---
\[firstpage\]
circumstellar matter — dust — infrared: stars — radiative transfer — stars: imaging — stars: individual: IRC+10011 — stars: AGB and post-AGB
Introduction
============
The transition from spherically symmetric Asymptotic Giant Branch (AGB) winds to non-spherical Planetary Nebulae (PNe) represents one of the most intriguing problems of stellar astrophysics. While most PNe show distinct deviations from spherical symmetry, their progenitors, the AGB stars, are conspicuous for the sphericity of their winds (see, e.g., review by Balick & Frank 2002). There have been suggestions, though, that deviations from sphericity may exist in some AGB winds, and perhaps could be even prevalent (Plez & Lambert 1994, Kahane at al. 1997). Thanks to progress in high resolution imaging, evidence of asymmetry has become more conclusive for several objects in recent years (V Hya: Plez & Lambert 1994, Sahai et al. 2003a; X Her: Kahane & Jura 1996; IRC+10216: Weigelt et al. 1998 & 2002, Haniff & Buscher 1998, Skinner et al. 1998, Osterbart et al. 2000; RV Boo: Bergman et al. 2000, Biller et al. 2003; CIT6: Schmidt et al. 2002).
The star IRC+10011 (= IRAS 01037+1219, also known as CIT3 and WXPsc), an oxygen-rich long-period variable with a mean infrared variability period of 660 days (Le Bertre 1993), is one of the most extreme infrared AGB objects. This source served as the prototype for the first detailed models of AGB winds by Goldreich & Scoville (1976) and of the OH maser emission from OH/IR stars by Elitzur, Goldreich, & Scoville (1976). The optically thick dusty shell surrounding the star was formed by a large mass loss rate of yr$^{-1}$. The shell expansion velocity of 20 has been measured in OH maser and CO lines. Various methods and measurements suggest a distance to IRC+10011 in the range of 500 to 800 pc
For an archetype of spherically symmetric AGB winds, the recent discovery by Hofmann et al. (2001; H01 hereafter) of distinct asymmetries in the IRC+10011 envelope came as a surprise. They obtained the first near infrared bispectrum speckle-interferometry observations of IRC+10011 in the J-, H- and K’-band with respective resolutions of 48 mas, 56 mas and 73 mas. While the H- and K’-band images appear almost spherically symmetric, the J-band shows a clear asymmetry. Two structures can be identified: a compact elliptical core and a fainter fan-like structure. H01 also performed extensive one-dimensional radiative transfer modelling to explain the overall spectral energy distribution (SED) and angle-averaged visibility curves. Their model required a dust shell with optical depth $\tau(0.55 \mu m)=30$ around a 2250 K star, with a dust condensation temperature of . This one-dimensional model successfully captured the essence of the circumstellar dusty environment of IRC+10011 but could not address the observed image asymmetry and its variation with wavelength. In addition, the model had difficulty explaining the far-IR flux, requiring an unusual transition from a $1/r^2$ density profile to the flatter $1/r^{1.5}$ for $r$ larger than 20.5 dust condensation radii. Finally, the model produced scattered near-IR flux in excess of observations.
We report here the results of 2D radiative transfer modelling of IRC+10011 that successfully explain the observed asymmetries. After analyzing in §2 general observational implications we describe in §3 our model for a bipolar outflow in IRC+10011. In §4 we present detailed comparison of the model results with the data and resolution of the problems encountered by the 1D modelling. The discussion in §5 advances arguments for the role of bipolar jets in shaping the circumstellar envelope of IRC+10011 and other AGB stars. We conclude with a summary in §6.
Observational Implications
==========================
The near-IR images, especially the J-band, place strong constraints on the dust density distribution in the inner regions. Emission at the shortest wavelengths comes from the hottest dust regions. For condensation temperature 1,000 K the peak emission is at 4, declining rapidly toward shorter wavelengths. At 1.24 , the J-band is dominated by dust scattering. It is easy to show that scattering by a $1/r^p$ dust density distribution produces a $1/r^{p+1}$ brightness profile (Vinković et al, 2003). The J-band image from H01 is elongated and axially symmetric. The decline of brightness from its central peak along this axis of symmetry is different in the opposite directions. In one direction it declines as $1/r^3$, corresponding to the $1/r^2$ density profile typical of stellar winds. But in the other direction the brightness falls off only as $1/r^{1.5}$, corresponding to the flat, unusual $1/r^{0.5}$ density profile.
The large scale structure is not as well constrained by imaging. However, all observations are consistent with the following simple picture: An optically thick spherical wind has the standard $1/r^2$ density profile. Since the buildup of optical depth is concentrated in the innermost regions for this density law, the near-IR imaging penetrates close to the dust condensation region. The wind contains an imbedded bipolar structure of limited radial extent and density profile $1/r^{0.5}$. The system is observed at an inclination from the axis so that the wind obscures the receding part of the bipolar structure, creating the observed asymmetry of the scattering image, which traces directly the density distribution. The inclination angle must be $\la$ 45 since a larger value starts to expose the receding part. But the inclination cannot be too small because the approaching part would get in front of the wind hot dust, leading to a strong 10 absorption feature, contrary to observations. Because of its shallow density profile, the column density of the bipolar structure [*increases*]{} as $r^{0.5}$ away from the condensation cavity, and the size of J-band image corresponds to the distance where the scattering optical depth reaches unity. Regions further out do not show up because of self-absorption. Dust emission is affected also by the temperature distribution, and the central heating by the star tends to produce spherical isotherms. Images taken at longer wavelengths, such as the K-band, can thus appear more symmetric.
Some qualitative estimates of the gas density follow immediately. The wind optical depth at the J-band must be $\ga$ 1. This optical depth is accumulated close to the dust condensation radius, roughly 3 cm for a distance of 650 pc. Assuming a standard dust-to-gas mass ratio of 1:100, the gas density at the condensation radius is 37 . For the bipolar structure, the J-band optical depth is 1 across the size of the observed image, which is 2 cm. This leads to a density estimate of 76 at the condensation radius within the bipolar structure. These rough estimates are within a factor 10 of the results of the detailed modelling described below.
The density at the base of the outflow is about an order of magnitude lower in the bipolar structure than in the wind region. An outflow can bore its way through another denser one only if its velocity is higher so that it plows its way thanks to its ram pressure. The propagation of such high-velocity bipolar outflows has been studied extensively in many contexts, beginning with jets in extragalactic radio sources (Scheuer 1974). The jet terminates in a shock, resulting in an expanding, elongated cocoon similar to the observed bipolar structure. With a $1/r^{0.5}$ density law, most of the bipolar structure mass is concentrated at its outer edge with the largest $r$, consistent with the structure of the expanding cocoon.
2D Modeling of IRC+10011
========================
Detailed 2D radiative transfer modeling was conducted with our newly developed code LELUYA (see Appendix). Starting with the minimal configuration that explains all available observations, we show on physical grounds that this minimal model must be modified and propose a suitable modification.
A Minimal Model {#sec:minimal}
---------------
For the first working model we adopt the geometry shown in figure \[Model\], which requires the [*minimal number of free parameters*]{}. Each polar cone is described by its half-opening angle $\theta\sub{cone}$ and radial extent . Apart from discontinuities across the cone boundaries, the density depends only on $r$. It varies as $1/r^{0.5}$ inside the cones and $1/r^2$ outside, out to some final radius . To complete the description of the geometry we need to specify its inner boundary, and it is important to note that this cannot be done a-priori. Dust exists only where its temperature is below the condensation temperature . Following H01 we select = 900 K. The dust inner boundary, corresponding to the radial distance of dust condensation, , is determined from The equilibrium dust temperature, $T$, is set by balancing its emission with the radiative heating. But the latter includes also the diffuse radiation, which is not known beforehand when the dust is optically thick; it can only be determined from the overall solution. Furthermore, because the spherical symmetry is broken by the cones, the shape of the dust condensation surface can be expected to deviate from spherical and is not known a-priori. Therefore equation \[eq:Rc\] completes the description of the geometry with an implicit definition of the inner boundary $\Rc(\theta)$.
The radiative transfer problem for radiatively heated dust possesses general scaling properties (Ivezić & Elitzur 1997). As a result, is the only dimensional quantity that need be specified. All other properties can be expressed in dimensionless terms. Luminosity is irrelevant, the only relevant property of the stellar radiation is its spectral shape, which we take as black-body at = 2,250 K. For individual dust grains, the only relevant properties are the spectral shapes of the absorption and scattering coefficients. For these we adopt spectral profiles corresponding to the silicate grains of Ossenkopf, Henning & Mathis (1992) with the standard size distribution described by Mathis, Rumpl & Nordsieck (1977; MRN). Our calculations employ isotropic scattering. These properties are the same everywhere.
Density and distance scales do not enter individually, only indirectly through overall optical depth. With two independent density regions, the problem definition requires two independent optical depths. For this purpose we choose and , the overall optical depths at visual wavelengths along the axis and the equator, respectively. Spatial dimensions can be scaled with an arbitrary pre-defined distance, which we choose as the dust condensation radius in the equatorial plane, (90). Radial distance $r$ is thus replaced with $\rho = r/\Rc(90\deg)$ so that, e.g., $\rho_{out} = \Rout/\Rc(90\deg)$. Equation \[eq:Rc\] becomes an equation for the scaled boundary of the condensation cavity. The relation between angular displacement from the star $\vartheta$ and the distance $\rho$ is where $\vartheta_\star$ is the stellar angular size and $\rho_\star = \R/\Rc(90\deg)$ is the scaled stellar radius. Physical dimensions can be set if one specifies a stellar luminosity Ł, which determines the condensation radius (90).
To summarize, in all of our model calculations the following quantities were held fixed: grain properties, = 900 K, = 2,250 K and the outer boundary $\rho_{out} = 1000$. We varied , , $\theta_{cone}$ and $\rho_{cone}$. Once a model is computed, comparison with observations introduces one more free parameter, the viewing angle $i$. A detailed discussion of the data is available in H01. Figure \[SEDfit\] shows the SED for the best fit model which has = 40, = 20, $\rho_{cone}=700$, and $\theta_{cone}=15\deg$. The 10 region is difficult to fit in full detail. Any further improvement would probably require more complicated geometry and/or modified dust properties. The fit yields a bolometric flux of $F_{\rm bol} =
10^{-9}\,\rm W/m^2$, corresponding to $\vartheta_\star$ = 10.82 mas for the stellar angular size, similar to the 10.9 mas derived in H01. The IR-flux measurements determine the total amount of emitting dust. Assuming a standard $n_d\sigma_d/n$ = cm$^{2}$, the overall mass of the IRC+10011 circumstellar shell is 0.13 .
### Shortcomings of the Minimal Model {#subsection:shortcomings}
The minimal-model success in fitting the SED implies that it contains the proper amount of dust. In addition to the SED, this model reproduces adequately all the imaging observations at near-IR. However, these observations probe only the innermost regions of the bipolar structure and do not provide adequate constraints on its full extent.
We now show that the cones cannot extend all the way to $\rho\sub{cone}$ = 700, as required from the fit to the SED by the minimal model. If the cones were that large, the ratio of mass contained in them and in the wind region would be $M\sub{cone}/M\sub{wind}$ = 1.7, that is, most of the circumstellar mass would be in the cones. But such large mass cannot be swept-up wind material because the fractional volume occupied by the cones is only 0.034. And building up this mass with enhanced outflow through the polar regions can also be ruled out, as follows: Mass conservation along stream lines yields $v_1 t = R_1\int\eta\rho^2
d\rho$, where $\eta(\rho) = n(\rho)/n_1$ is the dimensionless density profile, $v_1$ and $n_1$ are the velocity and density at the streamline base $R_1$, and $t$ is the duration of the outflow. Applying this relation to streamlines in the cone ($\eta = \rho^{-1/2}$) and wind ($\eta = \rho^{-2}$) regions yields With $\rho\sub{cone}$ = 700, the product $v_1t/R_1$ is 5.26 in the cone regions while in the wind it is only 1000. Since the wind starts with a sonic velocity $v\sub{1w}$ 1 , the conical outflow would have to start with velocity $v\sub{1c} \simeq 5.2\x\E3\, t\sub{w}/t\sub{c}$ , where $t\sub{w}$ and $t\sub{c}$ are the wind and cones lifetimes. This is impossible since the bipolar structure would extend much further than the wind even for $t\sub{c} = t\sub{w}$; taking a physical $t\sub{c} \ll t\sub{w}$ only makes things worse. This argument can be easily extended to show that, irrespective of the magnitude of $\rho\sub{cone}$, the mass in the cones could not be deposited purely by recent enhancement of polar mass loss rates. A substantial fraction, perhaps even all, of this mass must be swept-up wind material.
A Physical Model {#sec:physical}
----------------
While our minimal model provides successful fits for all observations, $\rho\sub{cone}$ cannot be as large as this model requires. Unfortunately, observations do not yet meaningfully constrain $\rho\sub{cone}$ because at $\rho$ 10 the near-IR brightness drops below current detection capabilities. A physical model for the origin of the bipolar structure (§\[sec:jet\]) suggests that the cones extend to $\rho\sub{cone}$ = 47 (equation \[rho=47\]). Adopting this value, the minimal model must be modified to account for the far-IR flux produced by the large mass removed from the rest of the cones. This mass can be placed elsewhere as long as its temperature distribution corresponds to far-IR wavelengths (equation A7 in Vinković et al. 2003). With the total far-IR flux as the only observational constrain, the only firm limit on this cold dust component is that it starts at $\rho\ \ga$ 100 so that its temperature is $\la$ 100 K; in all other respects, the geometry is arbitrary.
To account for the far-IR excess, the H01 wind model, which did not incorporate the bipolar component, placed additional cold dust in a spherical component with $\rho^{-1.5}$ density profile at $\rho\ \ga\ 20$. Here we propose a simpler alternative: a detached spherical shell of increased dust density due to an earlier phase of higher mass loss rate. The radial wind dust density profile jumps by factor 3 at $\rho = 100$ so that With the cold dust displaced from the cones to the wind, the equatorial optical depth increases from = 40 in the minimal model to 40.72. Along the axis, drops from 20 to 5.1, of which 4.3 comes from the cones. As is evident from figure \[SEDfit\], the SED produced by this physical model is almost identical to that of the minimal model. The contributions of different components to the total flux are shown in the left panel of figure \[Flux\_components\], with the fractional contributions shown in the right panel.
Table 1 summarizes the input parameters, and various properties derived below, of our model. The proposal of a two-shell model is motivated mostly by physical plausibility since other than the SED, current observations do not place meaningful constraints on the cold dust configuration. We have verified that a disk geometry for the cold dust would also successfully reproduce the SED by modeling with disk structures of $\rho=100$ inner radius and radial density profiles $\rho^{-1/2}$ and $\rho^{-2}$. However, a disk geometry for the cold dust component suffers from the same shortcomings as the extended cones of the minimal model (§\[subsection:shortcomings\]).
The actual dust configuration is probably more complicated than our simple description. Imaging observations at 8.55 with spatial resolution of $\rho$ 50 by Marengo et al. (1999) suggest an extension along an axis almost perpendicular to the symmetry axis of the bipolar structure. This possible additional asymmetry is not accounted by our physical model.
-------------------------------- ---------------------
stellar temperature 2250 K
stellar size 10.8 mas (7 AU)
luminosity 1.3$\cdot$4
bolometric flux W m$^{-2}$
dust condensation temperature 900 K
condensation radius (90) 35.4 mas (23 AU)
viewing angle $i$ 25
Wind — Inner shell:
density profile $r^{-2}$
inner boundary ($\theta$) (eq. 1)
outer boundary 100(90)
radial $\tau_{\rm V}$ 39.6
mass$^*$ 0.005
age 550 years
mass loss rate$^*$ 9$\cdot$ yr$^{-1}$
Wind — Outer shell:
density profile $r^{-2}$
inner boundary 100(90)
outer boundary 1000(90)
radial $\tau_{\rm V}$ 1.1
mass$^*$ 0.13
age 5500 years
mass loss rate$^*$ 2$\cdot$ yr$^{-1}$
Cone Properties:
opening angle 2$\theta_{cone}$ 30
density profile $r^{-0.5}$
outer boundary 47(90)
axial $\tau_{\rm V}$ 4.3
mass$^*$
age$^*$ $\la$ 200 years
-------------------------------- ---------------------
: Model parameters and derived properties. Quantities marked by assume a distance of 650 pc, subject to an uncertainty of $\pm$150 pc. Quantities marked by additionally assume a wind velocity of 20. Quantities marked by $^*$ assume $n_d\sigma_d/n$ = cm$^{2}$. Uncertainties and the acceptable range of the various properties are discussed in the text.
Visibility Functions and Images {#sec:Imaging}
===============================
With the model parameters set from the SED, the surface brightness distribution is fully determined, and the visibility functions are calculated from the brightness. For comparison with observations, the visibility must be normalized with the flux collected within the field of view $\Theta\sub{FV}$. If the image is divided into $N\x N$ pixels then the spatial frequency is $q_i =
i/\Theta\sub{FV}$, where $i=1...N$. Adopting $N = 300$, sufficiently large to resolve the highest measured spatial frequencies, and $\Theta\sub{FV}$ from the instrumental system, the results shown in figure \[visibilities\] contain no additional free parameters. In contrast with the SED, the visibility displays a strong sensitivity to the grain size. A change of only 0.05 in the maximum grain size $a\sub{max}$ has a significant effect on the visibility curves. Our physical model has $a_{max}$ = 0.20 , resulting in good fits for both the SED (figure \[SEDfit\]) and the four different visibility curves (figure \[visibilities\]).
The J-band visibility is the most difficult to model because it is dominated by the scattered light and thus very sensitive to fine details of the density distribution and grain size. Since the agreement between data and theory is better for small scales (higher spatial frequency), the quality of the fit to the J-band image can be expected to deteriorate with distance from the star. The model does not explain the puzzling drop in the H-band visibility at $q
\gtrsim 14$ cycles per arcsec, corresponding to structure smaller than the condensation cavity. Since a similar drop is not present in the J-band, it must correspond to material that emits but does not scatter light significantly. Hot gas might be a possible explanation.
Our model images and their convolution with the instrumental PSF of H01 are shown in figure \[images\]. The comparison between the model and observed images is satisfactory, indicating that the overall geometry is properly captured by our simple model. The “halo” around the star in J-band model image is brighter than observed. Possible explanations are dust accumulation close to the equatorial region as well as asymmetric dust scattering. The overall image asymmetry is much more prominent in the J-band, where dust scattering dominates the radiative transfer (see figure \[Flux\_components\]). As the wavelength shifts toward dominance of dust thermal emission, the image becomes more symmetric. The reason is that scattered light traces directly the density distribution while the dust emission is affected also by the temperature distribution. Figure \[2Dtemperature\] shows the dust temperature distribution around the condensation cavity. Because of the central heating, the temperature decreases with radial distance and tends to create circularly symmetric isotherms. The asymmetric diffuse radiation distorts the isotherms, but the deviations from circularity are small, especially at the high dust temperatures traced by the K-band image. As a result, the image becomes more symmetric, especially after convolution with the PSF as shown in the lower panel of figure \[images\].
As evident from figure \[images\], the PSF convolution smears out the star and the nearby fan-shaped structure into one broad elongated peak whose center is shifted from the stellar position. This shift is more clearly noticeable in the brightness profiles, shown in figure \[Image\_profile\]. The shift is 8.3 mas along the major axis in the J-band and 2.8 mas for the H- and K’-bands. The images provide tight constraints on the inclination angle. Neither $i =
20^{\circ}$ nor $i = 30^{\circ}$ produce acceptable fits, so that $i=25^{\circ}\pm 3^{\circ}$.
Discussion
==========
Thanks to the scaling properties of dust radiative transfer, neither luminosity, distance or density absolute scales were specified. The distance to the source of 650$\pm$150 pc fixes those scales, so that the luminosity is 1.34 and the dust condensation radius is $R_c(90\deg) = 23\pm 5$ AU. The wind inner shell extends to 2,300 AU, thus containing all the masers, including the OH 1612 MHz (Elitzur, Goldreich & Scoville 1976), and its mass is 5 . With a wind velocity of 20 , the duration of this phase is 550 years and the corresponding mass loss rate is 9 yr$^{-1}$. The wind outer shell extends to 23,000 AU and its mass is 0.13 . Assuming the same wind velocity, the duration of this phase is 5,500 years and the corresponding mass loss rate is 2 yr$^{-1}$.
Modelling uncertainties allow a few times larger size of the inner shell, resulting in its larger mass and duration. The outer shell has much smaller overall uncertainties since its total mass is constrained by the far-IR flux. We can still derive general conclusions that:
1. the duration of the inner shell is $\lesssim$1,000 years, while the outer shell is 5 to 10 times older;
2. the overall dust opacity comes mostly from the inner shell;
3. the mass is contained almost exclusively in the outer shell, with about 10 to 30 times more mass than the inner shell;
4. the mass loss rate is of the order of yr$^{-1}$, with a few times larger rate in the other shell;
5. the overall circumstellar mass of 0.13 indicates that IRC+10011 is close to the end of its AGB evolution.
Dust Properties
---------------
Our models employ silicate grains from Ossenkopf et al. (1992) with the standard MRN size distribution. We found that the upper limit on the grain sizes had to be reduced to $a\sub{max}$ = 0.20 from the standard 0.25 . While this change made little difference in the SED analysis, it was necessary for proper fits of the visibility curves. The most important effect of $a\sub{max}$ is control of the crossover from scattering to emission dominance, crucial for explanation of the observed change from elongated to circular images between the J- and K-bands (see §\[sec:Imaging\]). Although we cannot claim to have determined the precise magnitude of $a\sub{max}$, the fact that it is smaller than the standard seems certain.
The dust properties in our model were the same everywhere to minimize the number of free parameters. In a detailed study of the proto-planetary nebula IRAS 16342-3814, Dijkstra et al. (2003) find that the maximum grain size varies from 1.3 in a torus around the star to 0.09 in the bipolar lobes. If such a variation in dust properties can occur already on the AGB, the $a\sub{max}$ we find would represent an average over the cones and wind regions.
Jet Model for the Bipolar Structure {#sec:jet}
-----------------------------------
The near-IR brightness observations map a region of the cones that extends to $\rho \sim 8$ and has optical depth $\tau_V \sim 1.4$, corresponding to a gas density at the base of each cone of $n\sub{1c}$ = 1.36 cm$^{-3}$. In contrast, the gas density at the base of the wind region (obtained from = 41) is $n\sub{1w}$ = 1.78 cm$^{-3}$. The large density disparity amplifies our earlier conclusion that the bipolar cones are sustained by high-velocity ram pressure.
The small density at the base of the cones shows that their material has been evacuated and deposited at larger distances by a recent event. We propose the following simple scenario for the bipolar structure: High-velocity low-density jets were recently turned on at the polar regions. The jets cleared out polar cavities but are trapped by the material pushed ahead by their ram pressure, resulting in an expanding cocoon as described first by Scheuer (1974). Our model cones are a description of the current density distribution of the cocoon, a snapshot of an inherently dynamic structure. In this picture, the mass in the cones is swept-up ambient wind material and the cone boundary is then The swept-up mass is only . The leading edge of the cocoon moves at velocity $v\sub{c} = \beta v\sub{w}$, where $v\sub{w}$ is the local wind velocity and $\beta > 1$. From pressure balance during jet confinement, where $n\sub{c}$ and $n\sub{w}$ are densities across the cocoon leading edge and $v\sub{t}$ is the local speed of sound in the wind. This condition requires that the density of the cones be smaller than the ambient density into which they are expanding, i.e., $n\sub{c} < n\sub{w}$, restricting the cone radial extension to $\rho \le$ 26 which is slightly smaller than the derived $\rho\sub{cone}$. We attribute this discrepancy to the approximate nature of our model, in which the complex structure of the cocoon–wind boundary is replaced with the sharp-cutoff of the simple power-law density distribution of the cones. Taking $n\sub{c} \simeq n\sub{w}$ at the cone boundary, pressure balance implies $v\sub{c} \simeq v\sub{w}$, consistent with a recent start of the jet confinement. Assuming that the cocoon radial boundary moves according to $\rho\sub{cone} \propto t^{\alpha}$, with $\alpha$ 1 to ensure acceleration, its velocity is $v\sub{c} = \alpha\rho\sub{cone}R_1/t$. This yields an estimate for the jet lifetime for $\alpha/\beta\ \la$ 1.
Because of the steep decline of the wind density, the expansion accelerates rapidly as the cocoon boundary reaches lower density regions. Eventually it will break out of the wind, exposing the underlying jets. Indeed, a striking example of such a configuration comes from the recent observations, including proper motion measurements, of water masers in W43A by Imai et al.(2002). The observations reveal tightly collimated velocities of 150 at distances up to 0.3 pc at the two ends of an axis through the star. These masers are created by the impact of the jets on clumps in the surrounding medium. In addition to these far-away high-velocity masers, the source displays the usual configuration typical of OH/IR stars – OH and water masers in shells expanding with velocities 9 with radii of 500 AU. Therefore this source displays both the spherical AGB wind and the jets that broke through it. A similar, and probably more evolved, example is IRAS16342-3814. Similar to W43A, this is a “water-fountain” jet PPN source but its bipolar cavities are also seen with HST (Sahai et al. 1999).
Asymmetry Evolution in AGB Stars {#sec:sequence}
--------------------------------
IRC+10011 and W43A can be considered, respectively, the youngest and most evolved examples of sources displaying the evolution of bipolar jets working their way through AGB winds. The proposal that jets, operating at the late AGB or early post-AGB phase, are the primary mechanisms for shaping PNe has been made already by Sahai & Trauger (1998) and since supported by numerous observations. The prototype C-rich star IRC+10216 shows circular shape on the 20 arcsec scale both in V-band (de Laverney 2003) and molecular line images (e.g., Dayal & Bieging 1995). But high-resolution IR imaging at the 0.1 arcsec scale reveal elongated structure similar to that in IRC+10011 (Osterbart et al. 2000, Weigelt et al. 2002). Unlike IRC+10011, though, where only the J-band image gives clear indication of asymmetry, in IRC+10216 it is evident even in the K-band. This strongly suggests that IRC+10216 represents a more advanced stage than IRC+10011 of the evolution of a jet-driven cocoon confined by the ambient spherical wind.
The C-rich star V Hya provides an example that is further along in evolution. Recent CO observations by Sahai et al. (2003a) show that the bulk of the emission comes from an elongated structure centered on the star. In addition, an emission blob is approaching at a projected line-of-sight velocity of 250 along an axis perpendicular to this elongation. This is the expected morphology of a bipolar outflow breaking from the confinement of the high-density region of the AGB wind if the receding blob is obscured by the central torus. A similar structure has been found in the O-rich star X Her. Partially resolved CO observations by Kahane & Jura 1996 reveal a spherical component expanding with only 2.5 and two symmetrically displaced 10 components, likely to be the red and blue shifted cones of a weakly collimated bipolar flow. The bipolar lobes are 1.5 times bigger than the spherical component. Finally, the C-rich star CIT6 presents an even more evolved system. A bipolar asymmetry dominates the image in molecular line mapping by Lindqvist et al. (2000), Keck imaging by Monnier et al. (2000) and HST-NICMOS imaging by Schmidt et al. (2002).
Figure \[fig:Jet\] shows a schematic sketch of our proposed evolutionary sequence. The evolutionary stage of each indicated object is our rough estimate based on current observations. This figure is only meant as an illustration of the time-line suggested by our proposed scenario. The placing of different objects is based on different kinds of data. For example, for X Her we only have single-dish mm-wave observations with their attendant large beams, whereas IRC+10216 and V Hya were observed with the much higher resolution of HST. Also, whereas there is direct kinematic evidence for the high-velocity jets in V Hya and W43A, the same is lacking for the other objects. We can also expect that in objects with different jet properties (e.g., mass flux, speed, opening angle) and different AGB mass-loss rates in the inner region, the jets will follow different time histories of when they “break out” and the opening angle of the bipolar cone which they dig in the AGB envelope will be different. Also, if jets are episodic than they can change their direction (for which there is observational evidence). The actual picture can be expected to be quite more complex than the simple sketch in Fig \[fig:Jet\]. Nevertheless, we expect the displayed sequence to provide useful guidance for future studies.
Summary and Conclusions
=======================
We find that the circumstellar shell of IRC+10011 contains 0.13 , extends to a radial distance of 23,000 AU ( 35) from the star and is 5,500 years old. Most of the mass ( 96%) is contained in the outer shell from 2,300 AU ( 3.5), corresponding to an earlier phase when the mass-loss rate was about factor 3 higher than now. The near-IR image asymmetries discovered within the central 0.1of this system originate from of swept-up wind material in a cocoon elongated along the axis, extending to a radial distance of 1,100 AU. The cocoon expansion is driven by bipolar jets that it confines, and that were switched on $\la$ 200 years ago. The axial symmetry of the J-band image eliminates the possibility of a companion star, unless closer than 5 stellar radii. Higher sensitivity and/or better angular resolution would uncover image asymmetry in the K-band too.
Jet-driven cocoon expansion at various stages of development has now been observed in a number of AGB stars, culminating in breakout from the confining spherical wind (§\[sec:sequence\]). The immediate post-AGB stage is believed to be the proto-planetary-nebula (PPN) phase. A morphological study of a large sample of PPNs suggested the presence of jets that broke through the massive AGB wind, but it was not known if a similar morphology extends back to the AGB phase (Meixner et al. 1999; Ueta, Meixner & Bobrowsky 2000). Indeed, jets are found to be quite common in PPN as shown by the recent observations of IRAS16342-3814 (Sahai et al. 1999) K3-35 (Miranda et al. 2001), Hen 3-1475 (Riera et al. 2003) and IRAS22036+5306 (Sahai et al 2003b), for example. The case of K3-35 is particularly striking because of its great similarity to the AGB star W43A: water masers at the tips of bipolar jets at a large distance from the systemic center, which is surrounded by masers in the standard spherical shell configuration. This strongly suggests that W43A provides a glimpse of the immediate precursor of K3-35.
These new developments enable us to identify the first instance of symmetry breaking in the evolution from AGB to planetary nebula. Bipolar asymmetry appears during the final stages of AGB mass outflow. Mounting evidence suggests that this asymmetry is driven by collimated outflow in the polar regions. More complex geometries emerge in the post-AGB phase from a mixture of various processes that could involve multiple jets, fast winds, etc. These processes operate in the environment shaped by the AGB phase, leading to the myriad of complex structures found in PPN sources (e.g. Su, Hrivnak & Kwok 2001).
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank R. Sahai for most useful comments. Support by NSF grants PHY-0070928 (D.V.) and AST-0206149 (M.E.) is gratefully acknowledged. This work was also supported by National Computational Science Alliance under AST020014 and utilized the HP Superdome cluster at the University of Kentucky. We thank the University of Kentucky’s KAOS group at the Electrical Engineering department for time on their 65-processor Linux cluster KLAT2.
=
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Details of Radiative Transfer Modeling {#appendix}
======================================
LELUYA is our newly developed 2D radiative transfer code that works with axially symmetric dust configurations. It solves the integral equation of the formal solution of radiative transfer including dust scattering, absorption and thermal emission. The solution is based on a long-characteristics approach to the direct method of solving the matrix version of the integral equation (Kurucz 1969). The equations are solved on a highly unstructured triangular self-adaptive grid that enables LELUYA to resolve simultaneously many orders of magnitude in both spatial and optical depth space. It also enables automatic reshaping of the dust-free cavity around the central source according to asymmetries in the diffuse radiation (eq. 1). All grid points are coupled with each other through a correlation matrix based on the dust scattering. A simple matrix inversion determines the solution of radiative transfer for a given dust temperature distribution without any iterations. The temperature is then updated and the procedure repeated. Luminosity conservation within 5% is achieved in only three steps.
Figure \[grid\] shows LELUYA’s computational grid for the best fit minimal model described in §\[sec:minimal\]. Three grids of different resolutions were created for three sets of wavelengths, based on the density and optical depth variation. The first grid has 2982 points and starts with $\tau^{\rm e} =
120$ at 0.2, the shortest wavelength considered; this is the grid shown in the figure. The second grid has 2836 points and starts at wavelengths with $\tau^{\rm e}=1.2$. The third has 2177 grid points for wavelengths with $\tau^{\rm e} \le 0.1$. Angular integration around a grid point is performed over a highly non-uniform self-adaptive angular grid (with about 550 rays on average).
Luminosity Conservation
-----------------------
Luminosity conservation is the test determining convergence to the correct physical solution. A decrease in computed luminosity indicates energy sink due to insufficient spatial grid resolution, while an increase reflects energy excess due to a coarse angular grid. It is important to note that because of the lack of spherical symmetry, [*the bolometric flux does vary over spherical surfaces*]{}. The conserved quantity is luminosity, the energy transmitted per unit time across any surface enclosing the star. For a sphere of radius $\rho$, the luminosity is computed from the radial component of the bolometric flux vector $F\sub{bol,r}$ via and the luminosity conservation relation is at every $\rho$, where Ł is the stellar luminosity. In spherical symmetry $F\sub{bol,r}$ is $\theta$-independent and $4\pi\rho^2F\sub{bol,r}/\L\ = 1$. When the spherical symmetry is broken $F\sub{bol,r}$ becomes $\theta$-[*dependent*]{} and $4\pi\rho^2F\sub{bol,r}(\theta)/\L$ can [*exceed unity*]{} in certain directions, corresponding to locally enhanced energy outflow.
Our model calculations conserve luminosity within 5% at all radii. Figure \[Fbol\] shows the angular variation of $F\sub{bol,r}(\theta)$ and its following five contributions: stellar, inward and outward emission, and inward and outward scattered flux. These angular variations are shown at $\rho$ = 1.1, 500 and 1000. The small spikes in $F\sub{bol,r}$ close to $\theta\sub{cone}$ are real, reflecting the irregular shape of the dust condensation surface. Even though these irregularities are spatially small, their effect on optical depth variations magnifies their importance. At small radii, energy outflow through the cones is enhanced in comparison with the wind and is the main reason for their higher temperature. This region is dominated by stellar contribution. At large radii these roles are reversed, the diffuse radiation (mostly dust emission) takes over and the temperatures inside and outside the cones become equal.
\[lastpage\]
|
---
author:
- 'Li-Fang Zhu'
- 'Bang-Gui Liu'
title: Striped antiferromagnetism and electronic structures of SrFeAsF and their implications
---
Introduction
============
Fe-based superconductors attract more and more attention since superconductivity was found in doped LaFePO[@LaOFeP]. The advent of superconducting F-doped LaFeAsO stimulates a world-wide campaign for more and better Fe-based superconductors [@SC-LaOFeAs]. Replacing La by other lanthanides or doping with F has yielded more superconductors, and higher phase-transition temperatures ($T_c$) have been achieved in some of them [@SC-LaOFeAs-n; @SC-SmOFeAs-n; @SC-SmOFeAs-zhao-55K]. Furthermore, much more superconducting materials were found by using other dopants and other parent compounds [@Peter; @BaFe2As2-ES; @SrFe2As2; @EuFe2As2]. Even $\alpha$ FeSe can be made superconducting by applying high pressure [@FeSe-27K; @FeSe-pressure]. Now there are three series of FeAs-based superconductors: $R$FeAsO ($R$: lanthanide elements), $A$Fe2As2 ($A$: alkaline-earth elements), and LiFeAs. So far, the highest $T_c$ is 55-56 K in the case of doped SmFeAsO [@SC-SmOFeAs-zhao-55K]. Their structural, magnetic, electronic properties are intensively investigated and the microscopic mechanism for the superconductivity in these materials has been explored [@SC-LaOFeAs-n-MO; @SC-LaOFeAs-n-MO1; @add; @LaOFeAs-ES; @LaOFeAs-AFM; @LaOFeAs-150K; @LaOFeAs-ES-SDW; @LaOFeAs-OP; @LaOFeAs-EC; @moment; @mazinprb; @caoprb]. Very recently, superconductivity was found in Co and La doped SrFeAsF materials [@hosono; @wen2; @SrFeAsF-56K; @SrFeAsF-7]. SrFeAsF has the same crystal structure as $R$FeAsO and similar magnetic instability, but does not include any lanthanide [@epl; @wen1; @SrFeAsF-muon]. It has been found that Sm-doped SrFeAsF can become superconducting at 56 K, and higher transition temperatures should be reached with appropriate dopants [@wen2; @SrFeAsF-56K; @SrFeAsF-7]. Because some magnetic fluctuations are believed to mediate the superconductivity in FaAs-based materials, it is highly desirable to investigate the magnetic orders, electronic structures, and magnetic properties of the parent compound SrFeAsF.
Here, we use a state-of-the-art density-functional theory (DFT) method to investigate the structural, electronic, and magnetic properties of the SrFeAsF. Our total energy results show that the striped antiferromagnetic (AF) order, the same as that of LaFeAsO, is the magnetic ground state in the Fe layer, and the interlayer magnetic interaction is tiny. Our calculated position parameters of As and Sr are in good agreement with experiment. The electronic band result shows that there are only two quasi-two-dimensional (quasi-2D) bands near the Fermi level. Our charge and magnetization density analysis shows that the valence charge is mainly distributed in the Fe and F layers, and the magnetic moment is confined to the Fe layer. The spin couplings within the Fe layer are AF due to superexchange through the nearest As atoms. The real moment should be much smaller than the DFT value because of quantum spin fluctuations. More detailed results will be presented in the following.
Computational detail
====================
SrFeAsF usually assumes the tetragonal AsCrSiZr (tP8) structure with space group P4/nmm (No. 129) at high temperature. It transits to an orthorhombic distorted phase at 185 K, and furthermore, there is a magnetic instability at 175 K [@epl; @wen1]. A similar behavior has been observed in LaFeAsO [@SC-LaOFeAs; @SC-LaOFeAs-n-MO; @SC-LaOFeAs-n-MO1]. We shall investigate by self-consistent calculations the possible magnetic structures of the two SrFeAsF phases shown in fig. \[fig.1\]. Figure \[fig.1\]a shows the structure with the familiar checkerboard AF (AF1) order and fig. \[fig.1\]b that with a striped AF (AF2) order. The interlayer magnetic interaction can be ferromagnetic (FM) or AF.
![(color online). The checkerboard structure \[AF1, (a)\], the stripe one \[AF2, (b)\], and the first Brillouin zone of the AF2 structure (c). The largest ball (cyan or white) denotes Sr, the medium (yellow or white) As, the smallest (magenta or gray) F, and the ball with arrow Fe (red or gray). The arrow denotes the spin orientation.[]{data-label="fig.1"}](fig1.eps){width="8.8cm"}
All the calculations are done using a full-potential linearized-augmented-plane-wave (FLAPW) method within the density-functional theory [@dft1; @dft2], as implemented in the package WIEN2K [@wien2k; @wien2ka]. The generalized gradient approximation (GGA) to the exchange-correlation potential is used for the presented results [@pbe96], and local-density-approximation (LDA) calculations are done for comparison [@lda]. Full relativistic effects are calculated for core states, and the scalar relativistic approximation is used for valence states. The spin-orbit coupling [@relsa] is neglected because it has little effect on the system. We use 400 k points in the first Brillouin zone for the two AF structures. We make the harmonic expansion up to $l_{\rm max}$=10 inside the atomic spheres. The radii of the muffin-tin spheres of Sr, Fe, As, and F are 2.3, 2.1, 2.2 and 2.0 atomic unit (a.u.), respectively. $R_{\rm mt}$$\times$$K_{\rm max}$ is set to 8.0. The self-consistent calculations are considered to be converged only when the integrated charge difference per formula unit between input and output charge density is less than 0.0001.
![(color online). Total density of states (black thick solid, in states/eV per formula unit) and those projected in atomic spheres of F (pink or gray dash), Fe1 (blue or gray dot), Fe2 (blue or gray short dash), and As (magenta or gray dash dot) and the interstitial region (black thin sold) of the AF2 structure.[]{data-label="fig.2"}](fig2.eps){width="8.8cm"}
Results and discussion
======================
We use the experimental lattice constants ($a$,$c$) = (3.9930Å,8.9546Å) for the tetragonal structure, and ($a$,$b$,$c$) = (5.6155Å,5.6602Å,8.9173Å) for the orthorhombic structure [@epl; @wen1]. As for the internal position parameters of As and Sr atoms, we use ($z_{\rm
As}$,$z_{\rm Sr}$) = (0.6527,0.1598) and (0.6494,0.1635) as input and optimize them in terms of forces standard (2 mRy/a.u.) for both of the structures [@epl; @wen1]. We obtain ($z_{\rm
As}$,$z_{\rm Sr}$) = (0.6444,0.1634) and (0.6475,0.1637) for the AF1 and AF2 structures, respectively. For the AF2 structure, our GGA result, (0.6475,0.1637), is in good agreement with experiment [@epl; @wen1], in contrast to that from non-magnetic calculations [@nekrasov; @shein] and (0.6357,0.1624) of an LDA calculation of ours. The magnetic moment in the Fe sphere is 1.65 and 1.97$\mu_B$ for the AF1 and AF2 phases. The total Fe moment in the AF2 phase is approximately 2$\mu_B$, twice as large as an LDA result of ours. The parameters $z_{\rm As}$ and $z_{\rm Sr}$ and the magnetization density distributions do not change when we switch the interlayer magnetic interaction from FM to AF. The AF2 phase is lower by 80 meV in total energy per formula unit than the AF1 phase. This means that the striped AF2 phase is the ground-state phase, in agreement with experiment [@epl; @wen1]. The Fe spins align parallel along the $a$ (shorter) axis and antiparallel along the $b$ axis, which is the same as LaFeAsO [@Peter; @LaOFeAs-150K; @mazin]. SrFeAsF has the striped AF order as its ground state, the same as LaFeAsO [@LaOFeAs-AFM; @LaOFeAs-150K; @mazinprb; @caoprb]. The energy difference of 80 meV of the AF1 phase is also comparable with the 75 meV for the checkerboard AF phase of LaFeAsO [@LaOFeAs-AFM]. The moment of the magnetic ground state phase is approximately the same as that of LaFeAsO [@LaOFeAs-AFM; @mazinprb; @caoprb].
![Spin-dependent band structure of the AF2 structure along representative high-symmetry lines in the first Brillouin zone. The left part is that with Fe1 spin-up character shown and the right part with Fe1 spin-down character shown. The band consists of dots, where the bigger the dot, the stronger Fe1 character is at that point.[]{data-label="fig.3"}](fig3.eps){width="8.8cm"}
The total spin-dependent density of states (DOS) and those projected in the muffin-tin spheres of F, Fe1, Fe2, and As atom and in the interstitial region of the AF2 structure are presented in fig. \[fig.2\]. The filled states between -6.9 and 0 eV are the valence states from Fe-3d$^6$4s$^2$, As-4p$^3$, Sr-5s$^2$, and F-2p$^5$ orbitals. The semi-core states such as As-4s$^2$ are lower than -10.7 eV. There is no energy gap in the energy window presented, but two pseudo-gaps are visible, at the Fermi level and -3.85 eV. The DOS between -6.9 and -5.2 eV, having symmetry between the two spin channels, mainly comes from F-2p states, and the Sr states almost disappear from the energy window presented. The As DOS is also almost symmetrical. Concerning spin, it is distributed between -5.2 and 2.5 eV, but its largest part is between -3.8 and -2.2 eV. It is clear that the Fe states are spin-split. Actually, we have two different Fe atoms: Fe1 and Fe2. Their spins are antiparallel. The spin-down part of the Fe1 DOS is mainly distributed between -1 and 2.5 eV and the spin-up part between -3.8 and -1 eV. The Fe2 DOS is equivalent to the Fe1 DOS with the two spin channels interchanged. One feature is a substantial DOS contribution from the interstitial region, which reflects substantial covalence between Fe and As. At the Fermi level, the DOS calculated without spin polarization is substantially larger than that of our spin-polarized calculation [@nekrasov; @shein]. Our DOS results are similar to spin-polarized GGA results of LaFeAsO [@LaOFeAs-AFM].
![(color online). The charge density distributions in the two Fe-As-Fe planes with the Fe atoms along the $x$ axis (a) and the $y$ axis (e), the two different Sr-As-Sr planes (c) and (g), and the two F-Sr-F planes with the F atoms along the $x$ axis (d) and the $y$ axis (h); and the magnetization density distributions in the same two Fe-As-Fe planes (b) and (f). The charge density increment is 0.003$e$/a.u.$^3$ and the magnetization one $\pm$0.003$\mu_B$/a.u.$^3$. The red or gray dot in the charge plots labels the smallest density. In the magnetization plots, the red or gray letters ‘z’, ‘p’, and ‘n’ mean that the magnetization density therein is zero, positive, and negative, respectively.[]{data-label="fig.4"}](fig4.eps){width="8.8cm"}
In fig. \[fig.3\] we show the spin-dependent band structure with Fe1 character. The bands have very little dispersion along the $z$ axis. It is surprising that the band structure near the Fermi level is very simple, consisting of only two bands at the Fermi level. The band that has the maximum along $\Gamma$-Z originates from the Fe1 d$_{xy}$ state, and the other the Fe1 d$_{x^2-y^2}$ one. These two bands near the Fermi level have a much larger weight in the spin-down than in the spin-up channel. This implies that the electronic structure near the Fermi level is quasi-2D and spin-split. This special property is reasonable because the Sr atom is strongly ionic and the Fe moments align antiferromagneticaly along the $y$ axis. The band structure near the Fermi level is similar to that of LaFeAsO [@LaOFeAs-AFM]. It is interesting that the band feature near the Fermi level along $\Gamma$-$Y$ is consistent with that obtained with a phenomenological two-band model [@leedh]. Actually, the detailed band structure near the Fermi level depends sensitively on the structural parameters ($z_{\rm As}$,$z_{\rm Sr}$) and the magnetic moment [@LaOFeAs-AFM; @mazinprb; @caoprb]. When $z_{\rm
As}$ becomes smaller, the moment is correspondingly smaller and the band structure near the Fermi level looks more like that of non-magnetic calculations [@nekrasov; @shein; @LaOFeAs-ES-SDW; @mazinprb].
![(color online). The charge density distributions in the Fe plane (a), the Sr plane (c), the As plane (d), and the F plane (f); and the magnetization density distributions in the Fe plane (b) and the As plane (e). The red or gray dot labels the smallest density. The red or gray letters ‘z’, ‘p’, and ‘n’ and the density increments are the same as in Fig. 4.[]{data-label="fig.5"}](fig5.eps){width="8.8cm"}
In fig. \[fig.4\] we present the charge and magnetization density distributions in the planes defined by the two symmetrical bonds with the same vertex atom. The panels (a) and (e) show the charge density distributions in the two planes defined by the two different Fe-As-Fe chains, respectively, and the (b) and (f) the corresponding magnetization density distributions. One of the planes includes the $x$ axis and the other the $y$ axis. The panels (c) and (g) show the charge density distributions in the two planes defined by the two different Sr-As-Sr chains. The panels (d) and (h) show the charge density distributions in the two planes defined by the two different F-Sr-F chains, respectively. The red or gray dot in the charge plots labels the smallest charge density in $e$/a.u.$^3$ ($e$ is the electron charge): 0.007 in (a), 0.003 in (c), 0.003 in (d), 0.006 in (e), 0.003 in (g), and 0.003 in (h). The lowest contours represent the charge density values in $e$/a.u.$^3$: 0.004, 0.007, 0.010, and so on. In the magnetization plots in (b) and (f), the red or gray letters ‘z’, ‘p’, and ‘n’ mean that the magnetization density in the region is zero, positive, and negative, respectively. The magnetization density values (in $\mu_B$/a.u.$^3$) of the contours near the zero are $\pm 0.003$, $\pm 0.006$, and so on. The magnetization density vanishes in these planes at the same places as in the panels (c), (d), (g), and (h), and thus this is not shown again.
In order to investigate the charge and magnetization density distributions as a whole, we present them by atomic layer in fig. \[fig.5\]. The panels (a) and (d) show the charge density distributions in the Fe and As planes, and the panels (b) and (e) the magnetization density distributions in the same planes. The panels (c) and (f) show the charge density distributions in the Sr and F planes, and the magnetization density in these two planes is near zero and thus not presented. The red or gray dots denote the smallest charge density in $e$/a.u.$^3$: 0.024 in (a), 0.0025 in (c), 0.006 in (d), and 0.0025 in (f). The letters ‘z’, ‘p’, and ‘n’ imply the same as in fig. \[fig.4\]. It is clear that the spin is mainly confined in the Fe layer and the magnetization density is nearly zero in the F layer and the adjacent two Sr layers. Therefore, the interlayer spin interaction must be tiny, which in return supports our computational model.
In addition, we have done the same DFT calculations assuming an AF order along the $z$ axis. The calculated total energies and position parameters of As and Sr are the same as those presented above. This is caused by the strong ionicity of the Sr layers and the zero magnetization density in the Sr and F layers. Actually, one cannot determine the magnetic order along the $z$ axis by using DFT calculations because the interlayer magnetic interaction is too weak. As for the spin order in the Fe layer, the spins align antiferromagnetically along the $y$ axis but ferromagnetically along the $x$ axis. The nearest spin coupling constants along the $x$ and $y$ axes, $J_x$ and $J_y$, are AF. There is an AF coupling $J^\prime$ ($>J_x/2$) between the next nearest Fe spins. Other inter-spin couplings are substantially smaller [@zlf]. These three coupling constants are determined by the super-exchange through the bridging As atoms and other factors [@mazin]. The small difference $\delta=J_y-J_x$ is caused by the structural distortion. As for the moment per Fe atom, our GGA and LDA results are nearly 2$\mu_B$ and 1$\mu_B$, much larger than the experimental value $\sim 0.3\mu_B$ [@SrFeAsF-muon]. This situation is the same as that of LaFeAsO whose experimental moment per Fe atom is $0.25\sim 0.36$ $\mu_B$ [@SC-LaOFeAs-n-MO; @SC-LaOFeAs-n-MO1], much smaller than $1.5\sim 2.3$ $\mu_B$ from GGA calculations [@mazinprb; @caoprb]. For LaFeAsO, the large discrepancies can be reduced by using LDA rather than GGA in optimizing the crystal structure and calculating the moment, but this scheme leads to a large discrepancy of the calculated As position parameter from the experimental result [@LaOFeAs-150K; @mazinprb; @caoprb]. On the other hand, spin-orbit coupling, monoclinic distortion, and p-d hybridization are used to obtain smaller value for the moment [@moment]. In fact, this situation, however, has not been settled even in the case of LaFeAsO. We believe that quantum many-body effects may play some important roles in determining the actual moment. For SrFeAsF, the actual magnetic moment is also much smaller than the GGA value and we also attribute this discrepancy to quantum many-body effects of the spin fluctuations. More investigations are in need to solve this moment problem without leading to other discrepancies.
Conclusion
==========
In summary, we have investigated the structural, magnetic, and electronic properties of SrFeAsF using the state-of-the-art DFT method. Our total energy results show that the striped AF order is the magnetic ground state in the Fe layer and the interlayer magnetic interaction is tiny. The calculated As and Sr positions are in good agreement with experiment. There are only two quasi-2D bands near the Fermi level. The valence charge is mainly distributed in the Fe and F layers, and the magnetic moment is confined to the Fe atoms. All the intra-layer spin couplings are AF due to the superexchange through As atoms. SrFeAsF shares the main features of the structural, electronic, and magnetic properties with LaFeAsO. The actual moment is smaller than our value, which is the same as the situation for LaFeAsO and should be solved by considering quantum many-body effects. Because the Sm-doped SrFeAsF has equaled the $T_c$ record (56 K) of the FeAs-based superconductors, it is believed that higher $T_c$ will be realized in the SrFeAsF series. These results are useful for understanding the structural, electronic, and magnetic properties of SrFeAsF and should have implications achieving better superconductors by appropriate doping.
BGL is grateful to H. H. Wen for informing him of the SrFeAsF series[@hosono; @wen2; @wen1; @epl]. This work is supported by Nature Science Foundation of China (Grant Nos. 10774180 and 10874232), by Chinese Department of Science and Technology (Grant No. 2005CB623602), and by the Chinese Academy of Sciences (Grant No. KJCX2.YW.W09-5).
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*WIEN2k (Karlheinz Schwarz, Techn. Universit[ä]{}t Wien, Austria)* (2001), ISBN 3-9501031-1-2.
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to be published elsewhere.
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abstract: |
The air entrainment due to the turbulence in a free surface boundary layer shear flow created by a horizontally moving vertical surface-piercing wall is studied through experiments and direct numerical simulations.
In the experiments, a laboratory-scale device was built that utilizes a surface-piercing stainless steel belt that travels in a loop around two vertical rollers, with one length of the belt between the rollers acting as a horizontally-moving flat wall. The belt is accelerated suddenly from rest until reaching constant speed in order to create a temporally-evolving boundary layer analogous to the spatially-evolving boundary layer that would exist along a surface-piercing towed flat plate. To complement the experiments, Direct Numerical Simulations (DNS) of the two-phase boundary layer problem were carried out with the domain including a streamwise belt section simulated with periodic boundary conditions. Cinematic Laser-Induced Fluorescence (LIF) measurements of water surface profiles in two vertical planes oriented parallel to the belt surface (wall-parallel profiles) are presented and compared to previous measurements of profiles in a vertical plane oriented normal to the belt surface (wall-normal profiles). Additionally, photographic observations of air entrainment and measurements of air bubble size distributions and motions are reported herein. The bubble entrainment mechanisms are studied in detail through the results obtained by the DNS simulations. Free surface features resembling breaking waves and traveling parallel to the belt are observed in the wall-parallel LIF movies. These free surface features travel up to 3 times faster than the free surface features moving away from the belt in the wall-normal LIF movies. These breaking events are thought to be one of the mechanisms by which the air is entrained into the underlying flow. The bubble size distribution is found to have a characteristic break in slope, similar to the Hinze scale previously observed in breaking waves [@Deane2002]. The number of bubbles, their velocity, and size are reported versus depth from the experimental data. These results are qualitatively similar to results obtained by the simulations. Finally, several entrainment mechanisms are found in the simulations and their prevalence in the free surface boundary layer is assessed.
author:
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Naeem Masnadi$^{1a}$, Martin A. Erinin$^1$, Nathan Washuta$^{1b}$, Farshad Nasiri$^2$,\
Elias Balaras$^2$ and James H. Duncan$^1$,
bibliography:
- 'Library.bib'
title: Air Entrainment and Surface Fluctuations in a Turbulent Ship Hull Boundary Layer
---
INTRODUCTION
============
Turbulent boundary layers near the free surface along ship hulls and surface-piercing flat plates have been explored by a number of authors, see for example [@Stern1989], [@Longo1998], [@Sreedhar1998], and [@Stern1993]. Air entrainment and bubble distributions in these free surface flows have been explored by [@CarricaEtAl1999], [@MoragaEtAl2008], [@PerretCarrica2015], [@CastroEtAl2016] and [@LiEtAl2016]. One obvious region of two-phase flow in the vicinity of a ship is the layer of white water next to the hull, see for example the photograph in Figure \[fig:ship\]. The mechanisms by which air enters this region of the flow is poorly understood. In particular, it is not known to what degree this white water is the result of active spray generation and air entrainment due to turbulence in the boundary layer along the ship hull and to what degree the white water is the result of spray and air bubbles that are generated upstream in the breaking bow wave and then swept downstream with the flow. In the free surface boundary layer, the air entrainment process is controlled by the ratios of the turbulent kinetic energy to the gravitational potential energy and the turbulent kinetic energy to the surface tension energy. The ratio of the turbulent kinetic energy to the gravitational potential energy is given by the square of the turbulent Froude number ($Fr^2 = q^2 / (g L)$) and the ratio of turbulent kinetic energy to surface tension energy is given by the Weber number ($We = \rho q^2 L/ \sigma$), where $g$ is the acceleration of gravity, $\rho$ is the density of water, $\sigma$ is the surface tension of water, $q$ is the characteristic magnitude of the turbulent velocity fluctuations and $L$ is the length scale of this turbulence.
Several authors have applied theory and numerical methods to explore the interaction of turbulence and a free surface, see for example [@Shen2001], [@Guo2009], [@kim:2013] and [@broc:2001]. [@broc:2001] have used scaling arguments to predict the critical Froude and Weber numbers above which air entrainment and spray generation will occur due to strong free-surface turbulence. Figure \[fig:brocchiniperegrine\], which is from their paper, shows the boundaries of various types of surface undulations on a plot of $q$ versus $L$. The upper region of the plot is the region of air entrainment and droplet generation. We have used classical boundary layer correlations to make estimates of $q$ (taken as the root-mean-square vertical component of the turbulent velocity fluctuations) and $L$ (taken as the boundary layer thickness) at three streamwise positions in a ship boundary layer and plotted these points on the $q$-$L$ map in Figure \[fig:brocchiniperegrine\]. As can be seen from the figure, the points are clearly in the air entrainment region of the plot, especially the points near the bow. Thus, air entrainment due to strong turbulent fluid motions in the hull boundary layer at the free surface is a likely cause of the layer of white water.
![Regions of various types of surface motions for free surface turbulence with velocity fluctuation magnitude $q$ (vertical axis) and length scale $L$ (horizontal axis), from [@broc:2001]. Air entrainment and spray production occur in the upper region, above the uppermost curved line. The three data points are values obtained for the turbulent boundary layer on a flat plate with $q$ taken as the rms of the spanwise (which is vertical for the boundary layer along a ship hull) velocity fluctuation ($w'$) and L taken as the boundary layer thickness ($\delta$).[]{data-label="fig:brocchiniperegrine"}](figure2.pdf){width="3in"}
The difficulty with laboratory experiments on bubble entrainment and spray stems from the fact that the experiments are performed in the same gravitational field as found in ship flows and that the only practical liquid available is water, as is also found in the ocean. Thus, with $g$, $\rho$ , and $\sigma$ the same in the field and in the laboratory, one must attempt to achieve full-scale flow speeds in order to obtain Froude, Weber, and Reynolds similarity with field conditions. Also, even if full scale-values of $q$ and $L$ were obtained by towing a surface piercing flat plate with the length of the ship at high speed in a ship model basin, the free surface flow would include a bow wave which would obfuscate the source of the bubbles and spray. Another problem is that in order to obtain realistic entrainment/spray conditions and bubble/droplet size distributions, these experiments should be performed in salt water which is not typically used in ship model basins (note that though the experiments presented in this paper were performed in fresh water, we hope to repeat the experiments in salt water in the furture).
In view of the above difficulties in simulating air entrainment due to the turbulent boundary layer, we have built a novel device that produces an approximation of a full scale ship boundary layer in the laboratory. This device, called the Ship Boundary Layer (SBL) simulator generates a temporally evolving boundary layer on a vertical, surface-piercing flat wall. This vertical wall consists of a stainless steel belt loop that is 1.0 m wide and about 15 m long. The belt is mounted on two vertically oriented rollers as shown in Figure \[fig:TankSchem\]. The rollers are driven by hydraulic motors and the entire device is placed in a large open-surface water tank as shown in the figure. Before each experimental run, the belt and the water in the tank are stationary. The water level is set below the top edge of the belt and the flow outside the belt loop on one of the long lengths between the rollers is studied. The belt is accelerated from rest using a hydraulic control system, which is able to create a highly repeatable belt motion.
In the experiments discussed herein, the belt is accelerated suddenly from rest until it reaches a pre-defined speed which is held steady for a short time. The flow on the surface of the belt in this case is a simulation of the flow seen by a stationary observer in the ocean as a ship, that makes no waves, passes by at constant speed. The temporally-evolving boundary layer created along the belt can be considered equivalent to the spatially-developing boundary layer along a flat ship hull, with the distance along the ship hull corresponding to the distance traveled by the belt at any time $t$. The primary objective of the experimental study is to gain insight into the different entrainment mechanisms via quantitative and qualitative observations of the water free surface as the belt is launched from rest and to quantify the statistics of the entrained bubbles including their diameters, positions and velocities.
In addition to the experiments, a Direct Numerical Simulation (DNS) that reproduces the main features of the above-described experiments is performed. The computations consider a small streamwise section of the flow along the belt in the experiments, and apply periodic boundary conditions along the direction of motion of the belt. The Navier-Stokes equations for incompressible flow are solved in both the air and water portions of the flow, allowing us to examine the entire three-dimensional velocity field. The primary objective of the computational study is to identify and understand the behavior of turbulent structures immediately below the free surface and their impact on air entrainment.
The remainder of the paper is divided into four sections. The experimental setup for the ship boundary layer, surface profile measurements, and bubble measurements, are described in the second section of the paper and the numerical setup for the DNS is reported in the third section. This is followed by the presentation and discussion of the results in the forth section of the paper. Finally, the conclusions of this study are presented in the fifth section.
EXPERIMENTAL DETAILS
====================
The experiments were performed in the same tank and with similar measurement techniques as those described in [@WashutaThesis] and [@Washuta2016]. A brief overview of these facilities and techniques are given below; the interested reader is referred to the original references for further details.
The experiments were performed in an open-surface water tank that is 13.34 m long, 2.37 m wide and 1.32 m deep, see Figure \[fig:TankSchem\]. The tank walls and bottom are made of clear plastic panels for optical access. The top of the tank is open, offering an unobstructed view of the water surface. The main functional component of the Ship Boundary Layer (SBL) simulator is a one-meter-wide 0.8-mm-thick stainless steel belt loop that is driven by two 0.46-meter-diameter, 1.1-meter-long rollers whose rotation axes are vertically oriented and separated by a horizontal distance of approximately 7.5 meters. The rollers are each driven by two bent-axis hydraulic motors via toothed-belt-and-pulley systems. Each roller along with the motors and drive systems form single drive units that are attached to a welded steel frame that maintains the separation between, and relative parallel orientation of the rollers.
The assembled SBL device is placed in a stainless steel sheet metal box (called the dry box). The dry box keeps the assembly essentially dry, while one of the two straight sections of the belt exits the dry box through a set of seals near the roller on the left and travels through the water to the second set of seals near the opposite roller where the belt re-enters the dry box. The lone straight section exposed to water is approximately 6 meters long and pierces the free surface with approximately 0.33 meters of freeboard for the water level used in the experiments presented in this paper. At the location where the belt leaves the dry box and enters the water, a sheet metal fairing is installed to reduce the flow separation caused by the backwards-facing step associated with the shape of the dry box at this location. When performing experiments, the belt is launched from rest and accelerates until reaching constant speed. Throughout these transient experiments, the belt travel is analogous to the passage of a flat-sided ship that makes no bow waves; the length along the hull is equivalent to the total distance traveled by the belt. Belt speeds ranging from 3 to 5 m/s were used and measurements were continued until a specified belt length had passed by the measurement site. The time to accelerate varies depending on the final belt speeds and independent measurements of the belt travel show that during launch the belt travels 0.85, 1.45, and 2.29 m at belt speeds of 3, 4, and 5 m/s, respectively.
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A cinematic Laser Induced Fluorescence (LIF) technique, see Figure \[fig:LIFSchem\], was used to measure the temporally evolving water surface deformation pattern. In this technique, a continuous-wave Argon Ion laser beam is converted to a thin sheet using a system of spherical and cylindrical lenses. This sheet is projected vertically down onto the water surface in two orientations; one with the plane of the light sheet parallel to the plane of the belt and one with the light sheet perpendicular to the belt. The laser emits light primarily at wavelengths of 488 nm and 512 nm. The water in the tank is mixed with fluorescein dye at a concentration of about 5 ppm and dye within the light sheet fluoresces. High-speed cameras viewed the intersection of the light sheet and the water surface from the side with viewing angles of approximately 20 degrees from horizontal. The images seen by the cameras show a sharp line at the intersection of the light sheet with the free surface. Using image processing, instantaneous surface profiles are extracted from these images.
The present research was focused on preliminary measurements of bubbles under the above-described experimental conditions. In these measurements, a single camera viewed the boundary layer region of the flow from underwater as shown in Figure \[fig:bubble\_setup\_planar\]. A sample image from this setup is shown in Figure \[fig:BubblePhotos\]. Analysis of the images allows for quantitative measurement of the bubble diameters, their two-dimensional positions, and trajectories for radii ranging down to 0.5 mm. In future experiments, more accurate measurements of bubbles are planned using cinematic stereo photography and cinematic inline holography. In the stereo measurements, the cameras and lights are mounted in underwater boxes close to the water free surface and the surface of the belt. Each box contains a camera, that is mounted on a Scheimpflug mount and oriented so that the camera looks down at a mirror that turns the camera’s line of sight to horizontal. The lines of sight of the two cameras are oriented to view the belt at $\pm45$ degrees from the normal to the belt surface. Both cameras are calibrated and focused to look at the same portion of the belt. The system is calibrated through images of a known 3D target and yields the 3D positions and equivalent diameters of the bubbles in any image pair. Illumination is provided by LED light sources that are placed in each underwater box. In future experiments we are also planning to use a digital inline holography system to measure the size, velocity, and position of bubbles down to a radius of 20 $\mu$m. The experimental setup consists of a camera fitted with a long distance microfocus lens, oriented vertically next to the belt, looking down into a dry box, as seen in Figure \[fig:bubble\_holography\]. The dry box is always partially submerged so as to negate the light distorting properties of the rough water free surface. A collimated laser beam from a pulsed ND:YLF laser is directed upwards from the bottom of the tank into the camera lens and sensor. When a bubble is in the path of the collimated laser beam a hologram is recorded by the camera sensor. This hologram can then be reconstructed digitally and the size and three-dimensional position of the bubble can be measured. Once the size and location of the bubbles is obtained, the bubbles are tracked in time and their velocities can be obtained from the resulting trajectories. This system has been successfully implemented in our laboratory to measure droplets generated by breaking waves [@Erinin2017]. An example of a hologram from these droplet measurements is shown in Figure \[fig:droplet\_hologram\_example\].
COMPUTATIONAL SETUP
===================
The computational simulations were designed to mimic the conditions in the experiments. The main challenges for the computations are to properly resolve the boundary layer along the moving belt and, at the same time, capture the complex free-surface deformations and air-entrainment phenomena. We only simulate a small part of the moving belt and apply periodic boundary conditions along the direction of motion of the belt. A schematic of a typical computational box is shown in Figure \[fig:domain\], where a portion of the air above the free-surface is also considered. We define the Cartesian domain $(x,y,z)$ in such way that $x$ is the streamwise direction, making $y-z$ the cross-stream plane. The moving wall is located at $y=0$, the undisturbed free surface is at $z=0$ (parallel to $x-y$ plane) and gravity is imposed in the $-z$ direction. The Navier-Stokes equations for incompressible flow are solved in both the air and water portions of the domain and the interface is implicitly advected and tracked using a geometric reconstruction approach [@QIN2015219]. The governing equations are solved on a block-structured Cartesian grid with Adaptive Mesh Refinement (AMR) [@Vanella2010JCP; @Vanella2014]. AMR allows us to cluster grid points at the dynamically evolving interface, as well as the boundary layer in a cost-efficient manner. The equations are advanced in time using an exact projection method. All spatial derivatives are discretized using second-order, central finite-differences. The jump conditions at the interface are imposed in a sharp manner using a variant of the ghost-fluid method [@FEDKIW1999457]. Details on the overall formulation together with a detailed validation in a series of problems of increasing complexity can be found in @DelaneyPhD2014.
In the experiments, the belt starts from rest and quickly reaches its terminal speed. The boundary layer, undergoes transition and gradually thickens as a function of time. The critical Froude number based on the local momentum thickness, $\theta$, and the belt velocity when air entrainment is initiated is $Fr \sim O(10)$. The corresponding Reynolds number at this time instant is $Re_\theta\sim O(10^4)$. Simulating this process starting from the belt at rest and arriving to post-entrainment Reynolds and Froude number has the advantage of well defined initial and boundary conditions but it is prohibitively expensive even on leadership parallel computing platforms. Due to this limitation, and in order to keep the computational cost at reasonable levels, we considered significantly lower Reynolds numbers, but kept the Froude number in the same regime as in the experiments, where high deformations of the free-surface are observed, leading to air-entrainment. This is a significant advantage of the simulations where we can independently change the surface tension and gravity to replicate the conditions in the air entrainment regime in the experiment at lower Reynolds numbers. In particular we consider the boundary layer to evolve from $Re_\theta=900$ to $1400$ and will discuss two Froude numbers: $Fr=4$ and $Fr=12$, for the same Reynolds number, $Re_\theta=1400$. As an additional cost reduction, we started from the fully turbulent regime, bypassing the transition phase. Despite these approximations, which only enable qualitative comparisons to the experimental results, the computations are well positioned to quantify the effects of the Froude number on the flow physics.
[figure9.pdf]{} (80,6) (41,90) (8,25)
In addition to these considerations the grid was dynamically refined to capture the dynamics of the triple contact point, while the resolution at any location in the domain containing the interface was kept at the highest refinement level. We use periodic boundary conditions in the streamwise direction. At the moving wall the impermeability condition is enforced and the velocity in the streamwise direction is set to the reference value. We impose a Sommerfeld radiation condition on the boundary opposite the moving wall in order to convect the surface waves out of the domain. The convective velocity is calculated using the average water phase wall-normal velocity at the boundary. Details can be found in @DelaneyPhD2014. Slip-wall conditions were used at the two remaining boundaries. The domain dimensions were driven by the maximum Reynolds number we wanted to achieve, and were selected based on prior computations of turbulent boundary layers in the literature.
In all computations, we define the *midsection* as the depth range where the effects of the free-surface are not felt and the velocity statistics are identical to the ones in a zero pressure gradient boundary layer. Also, unless otherwise stated, the midsection quantities used to normalize the results are taken from the flow field at $Re_\theta=1400$.
[figure10a.pdf]{} (20,70) (48,3)
[figure10b.pdf]{} (80,70) (48,2)
Quantitative comparisons at the midsection to reference data in the literature are shown in Figure \[fig:valid\]. The velocity statistics at $Re_\theta=1400$ are compared to the DNS by @Spalart1988. The agreement for both the mean velocity and the turbulent intensities is excellent.
The numerical investigation is primarily deployed to study the mechanisms of air entrainment due to the turbulence field beneath the surface. In the remainder of this study, we report on the numerical findings where comparison to experiments is possible, giving us greater confidence in the turbulent entrainment analysis.
RESULTS AND DISCUSSION
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Surface Profiles
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Surface profile measurements were performed at belt speeds of $U=3$, $4$, and $5$ m/s using the cinematic Laser Induced Fluorescence (LIF) technique. Through initial trials, it was determined that a frame rate of 1000 fps was necessary to provide a sufficient temporal resolution so that surface features perpendicular to the belt could be identified and tracked smoothly in successive frames. LIF images of the water free surface with the light sheet oriented perpendicular to the belt in an experimental run with $U$ = 5 m/s are shown in Figure \[fig:overall\]. The five images in the figure are spaced out equally by distance of belt travel, with the first image (a) taken at 0.0 s, the time when the belt first starts to move. The instantaneous belt speed from the beginning of belt motion through the acceleration portion until the belt reaches constant speed has been measured separately and is used to correlate the time of each frame to the belt travel distance. Here and in the following, rather than refer to images and data by the time after the belt has started moving, we refer to them by the distance, $x$, from the leading edge of an equivalent flat plate, which is also the distance that the belt has traveled
$$x = \int_0^t U(t^\prime )dt^\prime,$$ where $t=0$ is the instant that the belt starts to move. This integral is performed numerically with the measured function $U(t)$, which includes an initial phase of nearly constant acceleration, i.e., $dU/dt\approx$ a constant, followed by a longer period of constant speed, i.e., $U = $ a constant. Thus, the images in Figure \[fig:overall\] depict a portion of a run, with images (a), (b), (c), (d) and (e) captured at 0 s, 1.35 s, 2.35 s, 3.35 s, and 4.35 s, respectively, corresponding to $x=$ 0.0, 5.0, 10.0, 15.0 and 20.0 m.
In this subsection, we will quantitatively examine the free surface profiles parallel and perpendicular to the belt surface for $U= 5$ m/s and report some observed entrainment events from the parallel free surface profiles. Then we will look at processed free surface profiles perpendicular and parallel to the belt at belt speed $3$ m/s and compare them to the computational results qualitatively. Finally, the free surface height from the experimental results is analyzed qualitatively for $3$, $4$, and $5$ m/s.
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As discussed in the experimental details section, in our previously reported measurements [@Washuta2016] the plane of the vertical light sheet was oriented normal to the belt surface and two cameras, from upstream and downstream, looked down at an angle of approximately 20 degrees at the intersection of the light sheet and the free surface. The images in Figure \[fig:overall\] are from the downstream camera in one of these wall-normal water surface profile measurements, and these images have been flipped horizontally for convenience in order to match the coordinate system of later plots, so that the belt is near the left side of each image and is moving out of the page. The position of the belt is marked on the left side of image (a) and the intensity pattern to the left of this location is a reflection of the light pattern on the right. This line of symmetry gives a good indication of the position of the belt in each image. The sharp boundary between the upper dark and lower bright region of each image is the intersection of the light sheet and the water surface. The upper regions of the later images contain light scattered from roughness features on the water surface behind the light sheet. These roughness features include bubbles that appear to be floating on the water surface and moving primarily in the direction of the belt motion. The bright area below the boundary is created by the glowing fluorescent in the underwater portion of the light sheet. The complex light intensity pattern here is created by a combination of the refraction of the laser light sheet as it passes down through the water surface and the refraction of the light from the glowing underwater dye as the light passes up through the water surface between the light sheet and the camera, on its way to the lens. It can be seen from these images that surface height fluctuations (ripples) are created close to the belt surface, at the left side of each image, and propagate away from the belt (to the right). As time passes, the surface height fluctuations grow dramatically and eventually surface breaking and air entrainment events begin to occur, resulting in bubble and droplet production.
From these images, it is observed that the free surface remains nearly quiescent during a period of belt travel at the beginning of each run; during this time period, the LIF images appear similar to what is seen in Figure \[fig:overall\] (a). After a short time, the surface suddenly bursts with activity near the belt surface, creating free surface ripples. After this point, see Figure \[fig:overall\] (b), the free surface fluctuations are continually generated close to the belt and this generation region grows in time. As the belt travel length continues to increase, free surface ripples begin to appear to the right side of the image, away from the belt, see Figure \[fig:overall\] (c-d). Qualitatively, from looking at Figure \[fig:overall\] (c-d) it is evident that the free surface ripples are most intense closest to the belt and decay in intensity as they move away from the belt.
In more recent experiments with the light sheet perpendicular to the belt, the two cameras were both placed downstream in a side-by-side configuration with an overlap in their fields of view. By combining the profiles from the two cameras, a higher resolution was achieved (approximately 15 pixels/mm) while viewing a similar horizontal distance away from the belt (approximately 30 cm). Results shown in Figures \[fig:surface\_profiles\] (a) and Figure \[fig:rms\_height\] are obtained using this new configuration.
A corresponding set of images with the plane of the light sheet parallel to the belt are shown in Figure \[fig:LighSheetParallel\]. In these images the laser light sheet was located at a distance of approximately $y = 1.25$ cm away from the belt surface and the camera is looking toward the belt and down at the free surface at a small angle from horizontal. The belt is in the black background traveling from left to right in the series of images shown in the figure. As in the images of the surface profiles perpendicular to the belt, shown in Figure \[fig:overall\], the sharp boundary in the images in Figure \[fig:LighSheetParallel\] is the intersection of the light sheet with the water surface. The images in Figure \[fig:LighSheetParallel\] (a-e) correspond the the same lengths of belt travel as the images in Figure \[fig:overall\].
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![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13a.jpg "fig:"){height=".43in"} ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13f.jpg "fig:"){height=".43in"}
(a) (f)
![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13b.jpg "fig:"){height=".43in"} ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13g.jpg "fig:"){height=".43in"}
(b) (g)
![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13c.jpg "fig:"){height=".43in"} ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13h.jpg "fig:"){height=".43in"}
(c) (h)
![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13d.jpg "fig:"){height=".43in"} ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13i.jpg "fig:"){height=".43in"}
(d) (i)
![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13e.jpg "fig:"){height=".43in"} ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13j.jpg "fig:"){height=".43in"}
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From watching the movies of the free surface profiles parallel to the belt, it is clear that the free surface experiences a sudden burst of activity somewhere between images (a) and (b) in Figure \[fig:LighSheetParallel\]. This burst of activity on the free surface is associated with the growing size of the turbulent boundary layer in the water. Once the free surface become rough, wave-like breaking events can be observed moving from left to right (in the same direction as the motion of the belt) in the midst of other surface features moving parallel and perpendicular to the light sheet. These wave-like breaking events, presented in Figure \[fig:LighSheetParallel\] (b - d), are persistent and can be observed frequently once the free surface becomes rough.
Sometimes, the above-mentioned breaking events appear to entrap pockets of air into the water below. Two of these breaking events are shown in the images in Figure \[fig:parallel\_entrainment\_events\], which were taken from a wall parallel LIF movie for a belt speed of 5 m/s and with the light sheet 1.25 cm from the belt surface. The non-uniformity of the light intensity at the free surface is partially due to the curvature of the free surface, which reflects laser light and focuses it underneath the surface, and partly because water has obstructed the laser light from reaching the areas below. In Figure \[fig:parallel\_entrainment\_events\] (a) to (e) we see a a sequence of images taken at $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. To the left in Figure \[fig:parallel\_entrainment\_events\] (a) we see a jet forming with an air cavity directly below. The jet then proceeds to plunge forward and above the air cavity, from left to right in the direction of the belt motion, in images (b-c). The jet then splashes on the free surface and closes off the pocket of air in (d) . Finally, air is presumably entrained in the flow by image (e) and the jet is no longer present on the free surface. The time span between images (a) through (e) is 30 milliseconds. Images (f-j) in Figure \[fig:parallel\_entrainment\_events\] tell a similar story. A jet of water is moving from left to right over a cavity in (f). The jet proceeds to overtake the air cavity in (g-h) and the jet splashes on the free surface in (i-j), again, presumably entraining air. The sequence of images in (f-j) takes place over a time span of 18 milliseconds. There are many similar wave-like breaking events in the movies of the free surface profiles parallel to the belt surface.
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In addition to qualitative observations of free surface motions, quantitative surface profiles can be extracted from each frame of the LIF movies through the use of gradient-based image processing techniques. Figure \[fig:surface\_profiles\] shows an example of the surface profiles extracted from LIF images using image processing in MATLAB for a belt speed of $3$ m/s. In these plots the horizontal axis is the horizontal distance in each set of movies. The surface profiles in Figure \[fig:surface\_profiles\] (a) come from movies of the laser light sheet perpendicular to the belt, similar to the images show in Figure \[fig:overall\], hence the horizontal axis is in the y direction with the belt located at $y = 0$ mm. The surface profiles in Figure \[fig:surface\_profiles\] (b) and (c) come from movies of the surface profile parallel to the belt at two different distances from the belt (b, $y = 1.25$ cm and c, $y = 2.5$ cm), similar to the surface profile images shown in Figure \[fig:LighSheetParallel\]; hence the horizontal axis is in the $x$ direction. The profiles are spaced in time by $4$ ms in (a) and by $1$ ms in (b) and (c), Each new surface profile is shifted 1 mm up from the previous profile to reduce overlap and show the propagation of surface features through time. The earliest profile in time is shown at the bottom.
Using this plotting technique surface features like ripple crests can be tracked over a number of successive profiles and the slopes of imaginary lines connecting these features indicates their horizontal speed. Analyzing \[fig:surface\_profiles\] (a) we can estimate the speed of surface features moving away from the belt at a speed of $0.34$ m/s, which is much lower than the belt speed of $3$ m/s. It should be noted that there is a constant train of surface features propagating outwards in plot (a). Plot (b) shows the parallel surface profiles at $y = 1.25$ cm away from the belt. The location of the light sheet is shown as a blue dashed line in plot (a). Surface features propagating along the direction of the belt can be seen and their speed is estimated to be around $1$ m/s in the $x$ direction. Similarly, plot (c) shows parallel surface profiles at $y = 2.5$ cm away from the belt (its location shown in a red dashed line on plot (a)), with surface features speed estimated to be about $0.75$ m/s. Theses surface feature speed estimates from plots (b) and (c) are taken fairly close to the surface of the belt, yet their speeds are significantly less than the belt speed of 3 m/s. It’s interesting to note that surface features traveling parallel to the belt ($1$ m/s), measured at a distance of $y = 1.25$ cm away from the belt, travel about three times faster than features moving away from the belt ($0.34$ m/s). It should be kept in mind these velocity estimates are the $y$ or $x$ components of the phase speed.
Comparison of computational results to experiments can be made by considering a succession of surface profiles at the mid-streamwise location of the numerical domain for $Fr=12$ (Figure \[fig:instant\_DNS\_profiles\]), analogous to the experimental data in figure \[fig:surface\_profiles\]. The profiles are plotted in the same manner in both figures. In Figure \[fig:instant\_DNS\_profiles\], the lowermost profile corresponds to $Re_\theta=900$ and the uppermost profile corresponds to $Re_\theta=1400$. The surface disturbances have greater amplitude closer to the moving wall as compared to the outer regions, in agreement with the experiments. In the immediate vicinity of the moving wall ($0<y/\delta<0.25$) the disturbances appear to be uncorrelated and persist for only a few profiles, suggesting that in this region waves are heavily influenced by the underlying turbulent boundary layer flow. In the regions away from the wall however, the waves persist for much longer periods and maintain their shape, similar to the results in the experiments as discussed in the previous paragraph. The straight black lines track the crests of a few of the outer region waves. This shows that the propagation speed of these waves is very nearly constant and that the waves exhibit the behavior of freely moving waves. On average the propagation speed normalized by wall speed is 0.08 which is on the same range as the experiments (the corresponding experimental value for a belt speed of 3 m/s is 0.11). Overall from a qualitative point of view, the computations are in agreement with the experimental results and capture the surface dynamics of the two-phase turbulent boundary layer.
[figure15.png]{} (30,1)
The profile data from the experiments discussed above was used to obtain distributions of the root-mean-square (RMS) water surface height fluctuation as a function of time and space dimensions. The RMS height at any $x$ or $y$ location is obtained as the square root of the average of the squares of the differences between the height and the average height, where the average is taken over the run time and over all experimental runs with the same belt speed. Figure \[fig:rms\_height\] (a) is a plot of the RMS free surface height versus $y$ at belt speeds $U = $5, 4, and 3 m/s averaged over 20 runs for each speed. The belt is located at $y = 0$ mm. The RMS height reaches a maximum near the belt region where the free surface fluctuations were visibly the most violent as seen in Figure \[fig:surface\_profiles\] (a). Further away from the belt the free surface RMS fluctuations decay for all three belt speeds. Figure \[fig:rms\_height\] (b) shows the RMS height, averaged over all $y$, versus time for all three belt speeds. The belt starts to move at $t = 0$ s, however the RMS height for all three speeds does not change until a little bit before $t = 1$ s. The RMS height then increases at similar constant rate for all three belt speeds until about $t = 1.5$ s when the three curves start to diverge. After approximately $t = 5$ s, all three height RMS curves reach a constant value.
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![RMS of surface height fluctuations for light sheet perpendicular to the belt. (a) Height RMS in time versus distance from the belt for speeds of $U$ = 3, 4, and 5 m/s, and (b) RMS surface height averaged over $0\leq y \leq 30$ cm versus time for the same three speeds. Each curve is an ensemble average over 20 identical runs for each belt velocity.[]{data-label="fig:rms_height"}](figure16a.pdf "fig:"){width="3in"}
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![RMS of surface height fluctuations for light sheet perpendicular to the belt. (a) Height RMS in time versus distance from the belt for speeds of $U$ = 3, 4, and 5 m/s, and (b) RMS surface height averaged over $0\leq y \leq 30$ cm versus time for the same three speeds. Each curve is an ensemble average over 20 identical runs for each belt velocity.[]{data-label="fig:rms_height"}](figure16b.pdf "fig:"){width="3in"}
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A similar analysis was carried out on the surface profiles parallel to the belt. Figure \[fig:rms\_height\_parallel\] shows the RMS height versus $x$ for belt speeds of $U =$ 3, 4, and 5 m/s at two different distances from the belt for each speed. The RMS height varies with $x$ in a random manner with a relatively small amplitude; it is thought that this variation would decrease with increasing numbers of runs. In agreement with Figure \[fig:rms\_height\] (a), the data in Figure \[fig:rms\_height\_parallel\] indicates a strong increase in the RMS height with belt speed and little change (except for the $U=$ 3 m/s case) between the two measurement locations, $y = $ 12.5 and 25 mm.
![RMS of surface height fluctuations in time versus x for light sheet parallel to the belt at $y = 12.5$ and $25.0$ mm away from the belt surface.[]{data-label="fig:rms_height_parallel"}](figure17.pdf "fig:"){width="3in"}\
Bubble Statistics
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In this section, preliminary estimates of bubble statistics are reported for a belt speed of 5 m/s. A single camera set up, as shown in Figure \[fig:bubble\_setup\_planar\], was used to record images of entrained bubbles as the belt starts from rest and travels a distance of $24.88$ m. The camera has a field of view of 47 by 47 mm at the plane of the belt surface and a resolution of 67 $\mu$m per pixel. The belt is illuminated with diffuse white light and the movies are recorded at 1,000 fps with a total of 5,000 images taken for each run. The resulting series of images were processed using a MATLAB code which identifies all the bubbles in each image. Bubbles are measured down to a diameter of 0.5 mm. With the lens f-number used in the measurements, bubbles are in focus over a horizontal distance of about 35 mm from the belt surface along the line of sight of the camera lens and this region contains all the bubbles present in the imaged region of the flow. Since most of the larger bubbles are not spherical, an equivalent radius is calculated based on the two-dimensional projected shape imaged by the camera. Each bubble is then subsequently tracked based on the series of frames in which it appears and its two-dimensional position and average equivalent radius are recorded.
Figure \[fig:prob\_dist\_bubbles\_5mps\] is a log-log plot of the experimentally measured number of bubbles per radius bin width versus bubble radius for a belt speed of 5 m/s. The sample population is all of the bubbles that passed through the upstream side of the measurement region during the belt travel of 24.88 m. A uniform bin spacing of $dr = $ 0.118 mm is used and the centers of the bins range from $ r = $ 0.369 to 3.669 mm. Separate linear regions are observed for small-diameter and large-diameter bubbles. The two linear regions are fitted separately using linear regression to a function of the form $Ar^{\alpha}$ for the region of smaller bubbles and $Br^{\beta}$ for the region of larger bubbles. These functions plot as straight lines in Figure \[fig:prob\_dist\_bubbles\_5mps\] and the optimum position for the break in slope between the two regions was determined by an iterative bisection-like routine outlined as follows: First, an initial guess, $r_0$, is estimated as the radius where the break in slope is to occur, the data is split into two distinct sets and a power law, of the form described above, is fitted to each set. The intersection, $r_i$, of the two fitted lines is then found. If the difference between $r_0$ and $r_i$ does not fall within a specific tolerance, a new guess for $r_0$ between the previous values of $r_0$ and $r_i$ is assigned and the processes is repeated until the tolerance is reached. Using this method with an initial guess of $r_0 = $ 1.3 mm, the break in slope is estimated to be approximately $r_i = $ 1.265 mm. The break in slope in the bubble size distribution has long been observed and identified as the Hinze scale, see for example the work of [@Deane2002] on bubble size distributions in breaking waves. Generally speaking, the Hinze scale implies that different physical mechanisms influence the two different sides of the bubble size spectrum. Dean and Stokes suggest that bubbles that were larger than the Hinze scale were fragmented by turbulent flow with a -10/3 power-law scaling, while bubbles smaller than the Hinze scale are stabilized by surface tension and show a -3/2 power-law scaling with the radius [@Deane2002]. The Hinze scale is defined as
$$r_H=2^{-8/5}\epsilon^{-2/5}(\sigma We_c/\rho)^{3/5}$$
where $\epsilon$ is the turbulent dissipation rate and $We_c$ is the critical Weber number and typically takes on a value of 4.7 (see for example [@Deane2002]).
\
In our numerical simulations we found that $r_H /\delta \approx3.5\times10^{-3} $. The largest bubbles in our simulations have a radius of about $\delta/20 \sim 0.16$ with 25-30 computational points across the bubble whereas the smallest observed bubbles have a radius of about $\delta/160\sim 0.02$ with 3-4 points across. The latter is an order of magnitude greater than the Hinze scale. Further refinement in the numerical resolution to capture smaller bubbles is out of the scope of the present DNS, which focused on the primary entrainment events as a result of the turbulent boundary layer interacting with the free surface. Figure \[fig:NvsRad\] shows the number of observed bubbles against bubble radius from the DNS simulation. The radius of a bubble is calculated by considering the equivalent spherical bubble with the same volume. The vertical axis has been normalized with the total number of observations and the radius has been normalized by the largest radius in the data set. A line with a slope of $-10/3$ is plotted to the top right of the data for reference. The scaling agrees fairly well with the $-10/3$ law and shows further qualitative agreement to the experiments (Figure \[fig:prob\_dist\_bubbles\_5mps\]).
![Relative bubble population versus bubble radius from a DNS simulation of the problem. Horizontal axis has been normalized by the maximum bubble radius and the vertical axis by the total number of bubbles observed. The bubble radius is the radius of a spherical bubble with the same volume as the irregularly shaped bubble in the calculations.[]{data-label="fig:NvsRad"}](figure19.pdf){width="\linewidth"}
Figure \[fig:num\_of\_bubbles\_vs\_z\_5mps\] shows the mean number of unique bubbles vs. depth measured in the experiments for a belt travel of $24.88$ m at a belt speed of 5 m/s. Before the launch of the belt, the calm free surface is positioned at $z = $ 0 mm. Once the belt is launched, the free surface fluctuates dramatically in the $z$ direction, making it difficult to measure bubbles close to $z = $ 0 mm. Generally, the free surface does not fluctuate more than 15 mm below it’s original depth, which is the reason why the measurements presented start at $z \approx 14$ mm. Bubbles are tracked in the series of images in which they appear, the average depth of the bubble is obtained by averaging the $z$ position over all the tracked particle trajectories. The $z$ direction is divided from $z = -14.2$ mm to $z = -45.3$ mm by increments of $dz = 1.072$ mm. Each bubble’s mean $z$ position is then placed into each appropriate bin. From Figure \[fig:num\_of\_bubbles\_vs\_z\_5mps\], it can be seen that the number of bubbles slowly decreases from a few hundred bubbles in the area around $z = 15$ mm to tens of bubbles near $z = 50$ mm indicating that the majority of these large bubbles in the boundary layer tend to stay near the surface.
\
Figure \[fig:numBubbleVsDepth\] shows depth of observed bubbles in DNS against their relative population and is qualitatively analogous to figure experimental data in \[fig:num\_of\_bubbles\_vs\_z\_5mps\]. The depth of the bubbles has been normalized by the average bubble radius and has been broken up into twenty equally sized bins. The overall trend is similar to that of the experiment where the majority of the bubbles are found closer to the free surface. It must be noted that the two plots are not directly comparable given the limitations explained earlier
![Relative bubble population with respect to depth (DNS). Depth has been normalized by average bubble radius.[]{data-label="fig:numBubbleVsDepth"}](figure21.pdf){width="\linewidth"}
Given that most of the large bubbles reside near the free surface, it may be of interest to look at the average equivalent bubble radius versus depth. Figure \[fig:mean\_bubble\_radius\_vs\_z\_5mps\] shows the experimentally measured average bubble radius versus depth from $z = -13.68$ mm to $z = -45.84$ mm in increments of $dz = 2.14$ mm. The average depth of each unique bubble is calculated, in a similar way as described in the previous paragraph, and the bubble is placed into the appropriate $z$ bin. Once all bubbles are assigned the the proper bin, the average radius of the bubbles in each bin is calculated. It should be noted that data points close to the free surface are averaged over significantly more bubbles than ones farther away, perhaps accounting for the noisier data at greater depths. The average bubble radius increases by about 0.4 mm (about 30%) from the deepest and shallowest measurement positions.
\
Finally, from the tracked bubble trajectories, we can fit a second order polynomial to estimate the speed of the bubbles as they enter the camera’s field of view. Second order polynomials were fitted to the $x$ and $z$ positions versus time data for each unique bubble. Then the $u$ and $w$ components of velocity were computed for each bubble as it entered the upstream side of the camera’s field of view. The speed was calculated as $|\vec{u}| = \sqrt{u^2+w^2}$. The results of these calculations, including the bubble speeds and the values of the $u$ and $w$ velocity components, are shown in Figure \[fig:mean\_bubble\_speed\_vs\_z\_5mps\]. It is interesting to note that the mean bubble speed does not seem to change dramatically over the range of depths in which the measurements were taken. The $u$ component of velocity is 3 to 4 times larger then the $w$ component.
\
Analysis of turbulent structures
--------------------------------
In this section we will discuss the mechanisms of air entrainment in the context of the numerical simulations. In general, there are three different entrainment mechanism. Water droplets, for example, can break off ligaments and entrain air upon impact with the free-surface (see Figure \[fig:ent\_mech\]a). This type of air entrainment has been studied extensively primarily in simplified configurations [@Esmailizadeh1986; @Oguz1990; @Hasan1990; @TOMITA2007; @Ray2015; @Hendrix2016]. In such case, a droplet falling towards the free surface traps air between it and the water surface. A crater forms on the surface and upon impact of the droplet, the air inside the crater is entrapped. In the numerical study, about 12% of the air entrainment incidents are from surface impact. Alternatively, entrainment is also caused by turbulent motions underneath the surface. There are two types of vortices that result in entrainment and are distinguished by their orientation with respect to the surface. The first are the vortices that are mainly oriented parallel to the free surface (Figure \[fig:ent\_mech\]b). The second are those that are perpendicular to the free surface (Figure \[fig:ent\_mech\]c). In our numerical study, we found that the latter type of vortices are rare ($< 1\%$) and most of the turbulent entrainment comes from the former type ($\sim 88\%$). A small portion of the entraining vortices lie between the two where the orientation with respect to the free surface is not clear.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Proposed air entrainment mechanism schematics, a breaking wave (a), vortices oriented parallel to the free surface (b), and vortices oriented perpendicular to the free surface (c). (d-f) shows an example sequence of a single bubble being entrained from the DNS results.[]{data-label="fig:ent_mech"}](figure24a.png "fig:"){height=".65in"} ![Proposed air entrainment mechanism schematics, a breaking wave (a), vortices oriented parallel to the free surface (b), and vortices oriented perpendicular to the free surface (c). (d-f) shows an example sequence of a single bubble being entrained from the DNS results.[]{data-label="fig:ent_mech"}](figure24d.pdf "fig:"){height=".65in"}
(a) (d)
![Proposed air entrainment mechanism schematics, a breaking wave (a), vortices oriented parallel to the free surface (b), and vortices oriented perpendicular to the free surface (c). (d-f) shows an example sequence of a single bubble being entrained from the DNS results.[]{data-label="fig:ent_mech"}](figure24b.png "fig:"){height=".65in"} ![Proposed air entrainment mechanism schematics, a breaking wave (a), vortices oriented parallel to the free surface (b), and vortices oriented perpendicular to the free surface (c). (d-f) shows an example sequence of a single bubble being entrained from the DNS results.[]{data-label="fig:ent_mech"}](figure24e.pdf "fig:"){height=".65in"}
(b) (e)
![Proposed air entrainment mechanism schematics, a breaking wave (a), vortices oriented parallel to the free surface (b), and vortices oriented perpendicular to the free surface (c). (d-f) shows an example sequence of a single bubble being entrained from the DNS results.[]{data-label="fig:ent_mech"}](figure24c.png "fig:"){height=".65in"} ![Proposed air entrainment mechanism schematics, a breaking wave (a), vortices oriented parallel to the free surface (b), and vortices oriented perpendicular to the free surface (c). (d-f) shows an example sequence of a single bubble being entrained from the DNS results.[]{data-label="fig:ent_mech"}](figure24f.pdf "fig:"){height=".65in"}
(c) (f)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The typical process of this entrainment regime begins with the interface being pulled into the flow beneath, creating a “neck” where the ligament attaches to the free surface (Figure \[fig:ent\_mech\] d). The neck continues to narrow until it breaks and an air bubble is released into the flow. In the numerical studies, we focus our attention on bubble generation and will not discuss the fate of the bubbles later on. We can examine turbulent air entrainment by considering the local turbulent eddies using the vortex identification scheme introduced by @Hunt1988 known as the Q-criterion. Any spatial point where the Eulerian norm of the vorticity tensor dominates that of the rate of strain is designated as being part of the vortical structure. Figure \[fig:all\_vorts\] shows the vortical structures in the vicinity of an entrainment event. The vortex core closest to the neck of the air ligament is identified in the figure. The low pressure center of the vortex has pulled in the interface and is narrowing the neck.
[figure25.pdf]{} (75,48)[(-1,1)[20]{}]{} (50,40)
The Q-criterion can be useful in identifying the vortex cores but is not a reliable method for establishing the size of the vortices and hence their local length scale. We employ the following 8 step procedure to identify and quantify the local scale of vortices that entrain air.\
**Step 1:** Identify an entrainment event.\
**Step 2:** Visually identify the vortex responsible for entrainment by setting the appropriate value for $Q_c$ (Figure \[fig:vortexQuantMethod\]a).\
**Step 3:** Define the centerline of the vortex by the line of minimum pressure along the length of the vortex (Figure \[fig:vortexQuantMethod\]b).\
**Step 4:** Create planes perpendicular to and along the spine (Figure \[fig:vortexQuantMethod\]c) and calculate the vorticity magnitude on the planes.\
**Step 5:** For each plane, identify the isoline for $\omega_c$.\
**Step 6:** For each plane calculate the circulation within the area designated by the $\omega_c$ isoline. Define $\Gamma$ as the average of circulation across all planes.\
**Step 7:** Find the maximum distance from the vortex center to points of the $\omega_c$ isoline for each plane ($r$). The average of $r$ across all planes as the average radius of the vortex $R$ which is also the local length scale of the vortex.\
**Step 8:** Define local Reynolds, Froude and Weber numbers as $Re_l=\frac{\Gamma}{\nu}$, $Fr_l=\frac{\Gamma}{\sqrt{gR^3}}$ and $We_l=\frac{\rho\Gamma^3}{\sigma R}$, respectively.
\
These three dimensionless groups along with the distance of the vortex from the free surface are the independent variables of turbulent entrainment. Owing to the small value of surface tension, the typical local Weber numbers are very large (O($10^4$)) compared to the other three variables and will not be discussed further. We also non-dimensionalized the distance of the vortex from the surface $d$ using the radius of the vortex. As a result we have a three-dimensional parametric space $(d/r,Re_l,Fr_l)$ for all the entraining and non-entraining vortices. By placing these vortices on the parametric space, we seek to find the dominant factors separating the two types of vortices. We gathered data for 447 entraining and 450 non-entraining vortices for all computations at the two aforementioned Froude numbers. Figure \[fig:parameterSpace\]a shows the scatter plot of normalized distance against local Froude number for both entraining and non-entraining vortices. Most of the entraining vortices are clustered in the corner with $Fr_l<50$ and their population thins out as Froude number is increased. Non-entraining vortices however are spread much more uniformly with respect to the Froude number. For $d/r>6$ only a few entraining vortices exist whereas non-entraining vortices are numerous. The opposite is true for $d/r<6$ indicating that a vortex is not able to entrain air if it sits more than six times its own radius below the surface. This is irrespective of Froude number or the Reynolds number as can be seen in Figure \[fig:parameterSpace\]b. The scatter plot of Reynolds number against Froude number is depicted in \[fig:parameterSpace\]c. There is a significant overlap of entraining and non-entraining vortices and therefore the $(Re_l,Fr_l)$ pair cannot predict entrainment. It is worth noting that most of the entraining vortices collocate in the corner of the plot and tend to assume smaller values of $Re_l$.
[figure27a.pdf]{} (15,70)
[figure27b.pdf]{} (15,70)
[figure27c.pdf]{} (20,70)
Figure \[fig:pdf\]a depicts the probability density function of the normalized distance $d/r$. The entraining vortices are closer to the surface with a mean of 3.01 and standard deviation of 1.55. The non-entraining vortices have a mean equal to 9.85 with a standard deviation of 3.25. The two density functions cross at $(5.45,0.05)$ and do not show significant overlap. The value $d/r=5.45$ seems reasonable as the critical distance for entrainment.
[figure28a.pdf]{} (78,70)
[figure28b.pdf]{} (78,70)
[figure28c.pdf]{} (78,70)
Figure \[fig:pdf\]b shows the probability density function of the local Reynolds number $Re_l$. The two profiles cross at $Re_l\approx 200$ and there is significant overlap of the two distributions for Reynolds numbers smaller than this value. However, for Reynolds numbers larger than 200, the overlap is very small in comparison and the probability of finding entraining vortices drops dramatically, much faster than the non-entraining vortices. Figure \[fig:pdf\]c shows the probability density function of $Fr_l$. The curves coincide at $Fr_l\approx 37$ and there is significant overlap below and above this value. Both entraining and non-entraining vortices cluster towards the smaller Froude numbers however this is more pronounced in the case of the entraining vortices.
Figure \[fig:zplusPDF\] depicts the probability distribution function for the wall-normal location of the entraining vortices. The deep water velocity profile has also been included for comparison. The mean is at $z^+\approx100$ and the standard deviation is about 83 meaning that the majority of the entraining vortices reside in buffer layer and log-law regions of the boundary layer.
![Wall-normal distance PDF for entraining vortices. Deep water mean velocity is given for comparison.[]{data-label="fig:zplusPDF"}](figure29.pdf){width="\linewidth"}
CONCLUSIONS
===========
In this research, the problem of the interaction of a turbulent boundary layer with a free surface is studied experimentally using a laboratory-scale device and a DNS simulation of a similar problem. The experimental device utilizes a stainless steel belt, driven by two powered vertically oriented rollers, as a surface piercing vertical wall of infinite length. This belt accelerates in under 0.7 seconds to constant speed $U$ in an effort to mimic the sudden passage of a flat-sided ship. Utilizing the full length and velocity scales of large naval ships, this device creates a temporally-evolving boundary layer analogous to the spatially-evolving boundary layer along the length of a ship, using the transformation $x=Ut$, where $x$ is distance from the leading edge and $t$ is time. Water surface profiles measured along lines perpendicular and parallel to the belt surface were recorded with a cinematic LIF system to study the generation of surface height fluctuations by the sub-surface turbulence. Entrained bubbles were measured using a high-speed camera setup which was able to measure and track bubbles down to a radius of $r = 0.5$ mm. To complement the experiments, DNS of the two-phase boundary layer problem were carried out where a section of the belt was considered. The boundary conditions of the computational domain are similar to that of the experiments. The DNS results allow access to the entire flow field that is otherwise inaccessible through experiments.
From qualitative observation of the free surface profiles in the LIF movies with the light sheet parallel to the belt, it was found that surface features that resemble breaking water waves travel downstream, parallel to the belt surface. Two of these surface features are observed in detail qualitatively and it is hypothesized that they are one of the mechanisms through which air is entrained into the free surface boundary layer. The speed of these free surface features is measured parallel to the belt and it is found that they move about three times faster than similar features moving away from the belt. Also, it is found that the downstream speed of these features decays quickly when they are measured further away from the belt. The experimental free surface profiles are compared to the computational results and are found to agree qualitatively.
Experimental measurements of bubbles are also reported. The bubble size spectrum is found to have a break in slope at around $r = 2.46$ mm with two characteristic linear regions. The break in slope suggests a Hinze-like scale that dominates the bubble radius spectrum. The experimental results are compared to the bubble radius spectrum from the DNS and they seem to agree qualitatively. The number of bubbles per depth increment is found to decrease with depth, in agreement with the DNS results. The mean bubble radius and mean bubble speed versus depth are found to t decrease slowly with depth.
Three entrainment mechanisms, breaking waves, vertically oriented vortices, and horizontally oriented vortices, are studied in detail from the DNS results. It is found that horizontally oriented vortices account for $\approx 88 \%$ of the entrainment events in the DNS, while entrainment events by breaking waves and vertically oriented vortices account for $\approx 11 \%$ and $< 1 \%$ of the total entrainment events, respectively.
The support of the Office of Naval Research under grant number N000141712081 (Program Managers: Ki-Han Kim and Thomas Fu) is gratefully acknowledged.
Discussion
----------
**Discusser I:**
**Discusser:** This is an excellent paper that presents an experimental and numerical study of air entrainment and surface fluctuations in a developing boundary layer. Besides the interest on a fundamental fluid mechanics problem, the work is significant in that provides further insight into entrainment through the boundary layer/free surface contact line and may help develop appropriate models for air entrainment.
**Reply**: We thank for the discusser for his careful reading of our paper and his insightful questions and comments.
**Discusser:** Detailed questions and comments:
1. **Comment/Question**: It would be desirable to add to the introduction some discussion on how the work in this paper and other contributions by the authors could help development of air entrainment models for bubbly wake applications.
**Reply**: Thank you for pointing this out. We have added some references to the bibliography and made a few modifications along these lines to the beginning of the introduction.
2. **Comment/Question**: The discussion on air entrainment mechanisms is interesting. Can the authors really resolve Mesler entrainment, to the extent of 12 % of the bubbles? Bubbles resulting from impact are usually very small.
**Reply**: This portion of the bubbles is mainly due to the larger cavities collapsing and entrapping large blobs of air which subsequently break up. We rarely see entrainment due to narrow columns of splashing water.
**Discusser II:**
We thank for the discusser for his careful reading of our paper and his insightful questions and comments.
**Detailed comments and questions:**
1. **Comment/Question**: The authors mention these experiments should be carried out in salt water. Were these experiments done in fresh or salt water, unclear.
**Reply**: All the experiments presented in this paper were carried out in filtered fresh water. The paper was modified to clarify this point. Once a complete set of bubble measurements is performed in fresh water, we hope to perform an identical set of measurements in salt water.
2. **Comment/Question**: The authors state: It is worth noting that most of the entraining vortices collocate in the corner of the plot and tend to assume smaller values of $Re_l$. However, from Fig 27b, it looks like at higher $Re$ ($ > 600$) almost no vortices have $d/r<6$. It may be worth commenting/discussing this. Is this due to the nature of higher $Re$ or could your grid resolution be playing into it?
**Reply**: The absence of vortices in that area of the plot ($d/r < $ 6 ; $Re > $ 600) is an issue still under investigation. We are at the moment acquiring additional data to rule out the the possibility that this is due to a small dataset. The resolution of the grid is unlikely to be the issue since we keep the near-interface region at the finest refinement level at all times.
|
---
abstract: 'Motivated by the searching for $bb\bar{b}\bar{b}$ states at LHC recently, we calculate the ground-state energies of $bb\bar{b}\bar{b}$ states with quantum numbers $IJ^P=00^+,01^+,02^+$ in a nonrelativistic chiral quark model using the Gaussian expansion method. In our calculations, two structures, meson-meson and diquark-antidiquark, and their coupling, along with all possible color configurations are considered. It is expected that the studies shall be helpful for the experimental searching of fully-heavy exotic tetraquark states.'
author:
- Xiaoyun Chen
title: 'Analysis of hidden-bottom $bb\bar{b}\bar{b}$ states'
---
Introduction
============
In traditional quark models, meson consists of a quark and an antiquark, and baryon is made up of three quarks. Since the first exotic resonance $X(3872)$ was announced by the Bell Collaboration in 2003 [@Choi:2003], this situation has been changed. Actually, multiquark states were proposed from the beginning of quark model [@Dyson; @Jaffe1; @Jaffe2; @Jaffe3]. In the past years, the charmonium-like and bottomonium-like states, the so-called $XYZ$ states have been observed by experiments which provided us with good opportunities to study exotic states and help us understand the strong interactions. Especially for those charged states, they must be multiquark states consisted of two heavy quarks and two light quarks, If they do exist. In comparison with these systems, the tetraquark states composed of four heavy quarks, $QQ\bar{Q}\bar{Q}$, $Q=b$ quark or $c$ quark states are much simpler because long-distance effects from light quarks are expected not to be appreciable and the short-distance effects play an important role now.
In this paper we will focus on the heaviest tetraquark system involve four bottom flavored quarks, $bb\bar{b}\bar{b}$ state. Although it is still missing in experiment, recent studies indicate that this QCD bound state may be observable as a resonance at the LHC in the mass range $\sim$ $18-19$ GeV in the four lepton final state [@CMS2016]. In fact, a hint has been reported [@CMS2018]. By solving the nonrelativistic Schrödinger equation, the mass of $bb\bar{b}\bar{b}$ state is under the threshold of decay into a vector bottomonia pair [@prd86034004]. In Ref [@yangbai2016], Y. Bai *et al.* gave the mass of $bb\bar{b}\bar{b}$ state, which is around 100 MeV below twice the $\eta_{b}$ mass using a diffusion Monte Carlo method. E. Eichten gave the point that such a heavy state with a large branching fraction into $\Upsilon\Upsilon^*$ is likely discoverable at the LHC since CMS has given the observation of $\Upsilon$ pair production [@CMS2016; @Eichten2016]. M. N. Anwar *et al.* advocated the existence of $bb\bar{b}\bar{b}$ state with the predicted mass $18.72\pm0.02$ GeV [@Anwar2018]. Z. G. Wang calculated the mass of $bb\bar{b}\bar{b}$ state, $M(0^{++})=18.84\pm0.09$ GeV and $M(2^{++})=18.85\pm0.09$ GeV with the moment QCD sum rules [@ZGwang2017]. The study by W. Chen *et al.* showed that the $bb\bar{b}\bar{b}$ states lie below the threshold of two $\eta_b$ and they are probably stable and very narrow [@Weichen2018].
But there also some work argues to the contrary. Hughes *et al.* found no evidence of a QCD bound tetraquark below the lowest noninteracting thresholds using the first-principles lattice nonrelativistic QCD methodology [@Hughes2018]. J. Wu *et al.* estimated the masses of $bb\bar{b}\bar{b}$ states and they are above the lowest meson-meson threshold in the framework of the color-magnetic interaction [@prd97094015]. J. -M. Richard *et al.* also gave the results that the $bb\bar{b}\bar{b}$ states are found to be unbound in the constituent quark model [@prd95054019]. Ref. [@prd95034011] also obtained the negative results about these states. Recent work in Ref. [@epjc782018] show that $bb\bar{b}\bar{b}$ should be hardly visible at LHCb, given the current sensitivity. Nevertheless, it should become observable with higher statistics.
In this work, we try to study the ground states of the beauty-full system with the quantum numbers $IJ^P=00^+,01^+,02^+$ in a nonrelativistic chiral quark model with the help of Gaussian expansion method (GEM) [@Hiyama:2003cu]. The pure meson-meson and pure diquark-antidiquark structure, and the coupling of two structures are considered respectively, along with all possible color configurations. For the interaction between the heavy quarks, the short-range gluon exchange force is a dominant source. Besides, we discuss the possible employment of the effective meson exchange between heavy quarks in the context of chiral quark model and try to look for attractive mechanism in the system.
The paper is organized as follows. After the introduction, we will simply introduce the chiral quark model and how to construct the wave functions of four-quark states. In Sec. \[Numerical Results\], our numerical results and discussion are presented. In Sec. \[epilogue\], a brief summary is given.
Chiral quark model and wave functions of four-quark states {#GEM and chiral quark model}
==========================================================
The chiral quark model
----------------------
The chiral quark model has been successful both in describing the hadron spectra and hadron-hadron interactions. The details of the model can be found in Ref. [@094016chen; @Vijande:2005]. For $b\bar{b}b\bar{b}$ full-heavy system, the Hamiltonian of the chiral quark model consists of three parts: quark rest mass, kinetic energy, and potential energy: $$\begin{aligned}
H & = \sum_{i=1}^4 m_i +\frac{p_{12}^2}{2\mu_{12}}+\frac{p_{34}^2}{2\mu_{34}}
+\frac{p_{1234}^2}{2\mu_{1234}} \quad \nonumber \\
& + \sum_{i<j=1}^4 \left( V_{ij}^{C}+V_{ij}^{G}\right).\end{aligned}$$ The potential energy consists of pieces describing quark confinement (C); one-gluon-exchange (G). The forms of potentials are shown below (only central parts are presented) [@094016chen]:
$$\begin{aligned}
V_{ij}^{C}&= ( -a_c r_{ij}^2-\Delta ) \boldsymbol{\lambda}_i^c
\cdot \boldsymbol{\lambda}_j^c , \\
%
V_{ij}^{G}&= \frac{\alpha_s}{4} \boldsymbol{\lambda}_i^c \cdot \boldsymbol{\lambda}_{j}^c
\left[\frac{1}{r_{ij}}-\frac{2\pi}{3m_im_j}\boldsymbol{\sigma}_i\cdot
\boldsymbol{\sigma}_j
\delta(\boldsymbol{r}_{ij})\right], \\
%
\delta{(\boldsymbol{r}_{ij})} & =
\frac{e^{-r_{ij}/r_0(\mu_{ij})}}{4\pi r_{ij}r_0^2(\mu_{ij})}.
%
%V_{ij}^{\eta_b}& = \frac{g_{ch}^2}{4\pi}\frac{m_{\eta_b}^2}{12m_im_j}
% \frac{\Lambda_{\eta_b}^2}{\Lambda_{\eta_b}^2-m_{\eta_b}^2}m_{\eta_b} \nonumber \\
% & \left[ Y(m_\chi r_{ij})-
%\frac{\Lambda_{\chi}^3}{m_{\chi}^3}Y(\Lambda_{\chi} r_{ij})
%\right] \lambda_i^8 \lambda_j^8
%\boldsymbol{\sigma}_i \cdot\boldsymbol{\sigma}_j,\\
%%
%V_{ij}^{\sigma}&= -\frac{g_{ch}^2}{4\pi}
%\frac{\Lambda_{\sigma}^2}{\Lambda_{\sigma}^2-m_{\sigma}^2}m_\sigma \nonumber \\
%%
%& \quad \times \left[
% Y(m_\sigma r_{ij})-\frac{\Lambda_{\sigma}}{m_\sigma}Y(\Lambda_{\sigma} r_{ij})\right] ,\end{aligned}$$
$m$ is the constituent masse of quark/antiquark, and $\mu_{ij}$ is the reduced masse of two interacting quarks and $$\mu_{1234}=\frac{(m_1+m_2)(m_3+m_4)}{m_1+m_2+m_3+m_4};$$ $\mathbf{p}_{ij}=(\mathbf{p}_i-\mathbf{p}_j)/2$, $\mathbf{p}_{1234}= (\mathbf{p}_{12}-\mathbf{p}_{34})/2$; $r_0(\mu_{ij}) =s_0/\mu_{ij}$; $\boldsymbol{\sigma}$ are the $SU(2)$ Pauli matrices; $\boldsymbol{\lambda}$, $\boldsymbol{\lambda}^c$ are $SU(3)$ flavor, color Gell-Mann matrices, respectively; and $\alpha_s$ is an effective scale-dependent running coupling [@Vijande:2005], $$\alpha_s(\mu_{ij})=\frac{\alpha_0}{\ln\left[(\mu_{ij}^2+\mu_0^2)/\Lambda_0^2\right]}.$$ All the parameters are determined by fitting the meson spectrum, from light to heavy; and the resulting values are listed in Table \[modelparameters\].
-------------- ---------------------------- -------
Quark masses $m_u=m_d$ 313
(MeV) $m_s$ 536
$m_c$ 1728
$m_b$ 5112
Confinement $a_c$ (MeV fm$^{-2}$) 101
$\Delta$ (MeV) -78.3
OGE $\alpha_0$ 3.67
$\Lambda_0({\rm fm}^{-1})$ 0.033
$\mu_0$(MeV) 36.98
$s_0$(MeV) 28.17
-------------- ---------------------------- -------
: \[modelparameters\] Model parameters, determined by fitting the meson spectrum from light to heavy.
The wave functions of four-quark states
---------------------------------------
The wave functions of four-quark states for the two structures, diquark-antidiquark and meson-meson, can be constructed in two steps. For each degree of freedom, first we construct the wave functions for two-body clusters, then coupling the wave functions of two clusters to the wave functions of four-quark states.
\(1) Diqaurk-antidiquark structure.
For spin part, the wave functions for two-body clusters are, $$\begin{aligned}
&\chi_{11}=\alpha\alpha,~~
\chi_{10}=\frac{1}{\sqrt{2}}(\alpha\beta+\beta\alpha),~~
\chi_{1-1}=\beta\beta,\nonumber \\
&\chi_{00}=\frac{1}{\sqrt{2}}(\alpha\beta-\beta\alpha),\end{aligned}$$ then the wave functions for four-quark states are obtained,
\[spinwavefunctions\] $$\begin{aligned}
\chi_{0}^{\sigma
1}&=\chi_{00}\chi_{00},\\
\chi_{0}^{\sigma
2}&=\sqrt{\frac{1}{3}}(\chi_{11}\chi_{1-1}-\chi_{10}\chi_{10}+\chi_{1-1}\chi_{11}),\\
\chi_{1}^{\sigma
3}&=\chi_{00}\chi_{11},\\
\chi_{1}^{\sigma
4}&=\chi_{11}\chi_{00},\\
\chi_{1}^{\sigma
5}&=\frac{1}{\sqrt{2}}(\chi_{11}\chi_{10}-\chi_{10}\chi_{11}),\\
\chi_{2}^{\sigma 6}&=\chi_{11}\chi_{11},\end{aligned}$$
where the subscript of $\chi$ represents the total spin of four-quark states, it takes the values $S=0, 1, 2$, and only one component is shown for a given total spin $S$.
For flavor part, the wave function for $bb\bar{b}\bar{b}$ state is very simple $$\chi_{d0}^{f} =bb\bar{b}\bar{b},$$ the subscript of $\chi$ represents the isospin of $bb\bar{b}\bar{b}$, and it takes $I=0$.
For color part, the wave functions of four-quark states must be color singlet $[222]$ and it is obtained as below,
$$\begin{aligned}
\chi^{c}_{d1} & =
\frac{\sqrt{3}}{6}(rg\bar{r}\bar{g}-rg\bar{g}\bar{r}+gr\bar{g}\bar{r}-gr\bar{r}\bar{g} \nonumber \\
&~~~+rb\bar{r}\bar{b}-rb\bar{b}\bar{r}+br\bar{b}\bar{r}-br\bar{r}\bar{b} \nonumber \\
&~~~+gb\bar{g}\bar{b}-gb\bar{b}\bar{g}+bg\bar{b}\bar{g}-bg\bar{g}\bar{b}). \\
\chi^{c}_{d2}&=\frac{\sqrt{6}}{12}(2rr\bar{r}\bar{r}+2gg\bar{g}\bar{g}+2bb\bar{b}\bar{b}
+rg\bar{r}\bar{g}+rg\bar{g}\bar{r} \nonumber \\
&~~~+gr\bar{g}\bar{r}+gr\bar{r}\bar{g}+rb\bar{r}\bar{b}+rb\bar{b}\bar{r}+br\bar{b}\bar{r} \nonumber \\
&~~~+br\bar{r}\bar{b}+gb\bar{g}\bar{b}+gb\bar{b}\bar{g}+bg\bar{b}\bar{g}+bg\bar{g}\bar{b}).\end{aligned}$$
Where, $\chi_{d1}^{c}$ and $\chi_{d2}^{c}$ represents the color antitriplet-triplet ($\bar{3}\times3$) and sextet-antisextet ($6\times\bar{6}$) coupling, respectively. The detailed coupling process for the color wave functions can refer to our previous work [@054022chen].
\(2) Meson-meson structure.
For spin part, the wave functions are the same as those of the diquark-antidiquark structure, Eq. (\[spinwavefunctions\]).
The flavor wave function of $bb\bar{b}\bar{b}$ state takes as follows, $$\chi_{m0}^{f} =\bar{b}b\bar{b}b,$$ the subscript of $\chi_{m0}$ represents the isospin of four-quark states, $I=0$.
For color part, the wave functions of four-quark states in the meson-meson structure are,
$$\begin{aligned}
\chi_{m1}^{c}&=\frac{1}{3}(\bar{r}r+\bar{g}g+\bar{b}b)(\bar{r}r+\bar{g}g+\bar{b}b),\\
\chi_{m2}^{c}&=\frac{\sqrt{2}}{12}(3\bar{b}r\bar{r}b+3\bar{g}r\bar{r}g+3\bar{b}g\bar{g}b+3\bar{g}b\bar{b}g+3\bar{r}g\bar{g}r \nonumber \\
&~~~+3\bar{r}b\bar{b}r+2\bar{r}r\bar{r}r+2\bar{g}g\bar{g}g+2\bar{b}b\bar{b}b-\bar{r}r\bar{g}g \nonumber\\
&~~~-\bar{g}g\bar{r}r-\bar{b}b\bar{g}g-\bar{b}b\bar{r}r-\bar{g}g\bar{b}b-\bar{r}r\bar{b}b).\end{aligned}$$
Where, $\chi_{m1}^{c}$ and $\chi_{m2}^{c}$ represents the color singlet-singlet($1\times1$) and color octet-octet($8\times8$) coupling, respectively. The details refer to our previous work [@054022chen].
As for the orbital wave functions, they can be constructed by coupling the orbital wave function for each relative motion of the system, $$\begin{aligned}
\label{spatialwavefunctions}
\Psi_{L}^{M_{L}}=\left[[\Psi_{l_1}({\bf r}_{12})\Psi_{l_2}({\bf
r}_{34})]_{l_{12}}\Psi_{L_r}({\bf r}_{1234}) \right]_{L}^{M_{L}},\end{aligned}$$ where $l_1$ and $l_2$ is the angular momentum of two clusters, respectively. $\Psi_{L_r}(\mathbf{r}_{1234})$ is the wave function of the relative motion between two sub-clusters with orbital angular momentum $L_r$. $L$ is the total orbital angular momentum of four-quark states. Here for the low-lying $bb\bar{b}\bar{b}$ state, all angular momentum ($l_1, l_2, L_r, L$) are taken as zero. The Jacobi coordinates are defined as, $$\begin{aligned}
\label{jacobi}
{\bf r}_{12}&={\bf r}_1-{\bf r}_2, \nonumber \\
{\bf r}_{34}&={\bf r}_3-{\bf r}_4, \nonumber\\
{\bf r}_{1234}&=\frac{m_1{\bf r}_1+m_2{\bf
r}_2}{m_1+m_2}-\frac{m_3{\bf r}_3+m_4{\bf r}_4}{m_3+m_4}.\end{aligned}$$ For diquark-antidiquark structure, the quarks are numbered as $1, 2$, and the antiquarks are numbered as $3, 4$; for meson-meson structure, the quark and antiquark in one cluster are marked as $1, 2$, the other quark and antiquark are marked as $3, 4$. In the two structure coupling calculation, the indices of quarks, antiquarks in diquark-antidiquark structure will be changed to be consistent with the numbering scheme in meson-meson structure. In GEM, the spatial wave function is expanded by Gaussians [@Hiyama:2003cu]:
\[radialpart\] $$\begin{aligned}
\Psi_{l}^{m}(\mathbf{r}) & = \sum_{n=1}^{n_{\rm max}} c_{n}\psi^G_{nlm}(\mathbf{r}),\\
%
\psi^G_{nlm}(\mathbf{r}) & = N_{nl}r^{l}
e^{-\nu_{n}r^2}Y_{lm}(\hat{\mathbf{r}}),\end{aligned}$$
where $N_{nl}$ are normalization constants, $$\begin{aligned}
N_{nl}=\left[\frac{2^{l+2}(2\nu_{n})^{l+\frac{3}{2}}}{\sqrt{\pi}(2l+1)}
\right]^\frac{1}{2}.\end{aligned}$$ $c_n$ are the variational parameters, which are determined dynamically. The Gaussian size parameters are chosen according to the following geometric progression $$\label{gaussiansize}
\nu_{n}=\frac{1}{r^2_n}, \quad r_n=r_1a^{n-1}, \quad
a=\left(\frac{r_{n_{\rm max}}}{r_1}\right)^{\frac{1}{n_{\rm
max}-1}}.$$ This procedure enables optimization of the ranges using just a small number of Gaussians. Finally, the complete channel wave function for the four-quark system for diquark-antidiquark structure is written as $$\begin{aligned}
\label{diquarkpsi}
&\Psi_{IJ,i,j}^{M_IM_J}={\cal A}_1[\Psi_{L}\chi_S^{\sigma i}]_{J}^{M_J}\chi_{d0}^{f}\chi^{c}_{dj},\nonumber \\
&(i=1\sim6; j=1,2; S=0,1,2),\end{aligned}$$ where ${\cal A}_1$ is the antisymmetrization operator, $${\cal A}_1=\frac{1}{2}(1-P_{12}-P_{34}+P_{12}P_{34}).$$
For meson-meson structure, the complete wave function is written as $$\begin{aligned}
&\Psi_{IJ,i,j}^{M_IM_J}= {\cal A}_2[\Psi_{L}\chi_S^{\sigma
i}]_{J}^{M_J}\chi_{m0}^{f}\chi^{c}_{mj},\nonumber \label{mesonpsi}\\
&(i=1\sim6; j=1,2; S=0,1,2),\end{aligned}$$ where ${\cal A}_2$ is the antisymmetrization operator, $${\cal A}_2=\frac{1}{2}(1-P_{13}-P_{24}+P_{13}P_{24}).$$
Lastly, the eigenenergies of the four-quark system are obtained by solving a Schrödinger equation: $$H \, \Psi^{\,M_IM_J}_{IJ}=E^{IJ} \Psi^{\,M_IM_J}_{IJ},$$ where $\Psi^{\,M_IM_J}_{IJ}$ is the wave function of the four-quark states, which is the linear combinations of the above channel wave functions, Eq. (\[diquarkpsi\]) in the diquark-antidiquark structure or Eq. (\[mesonpsi\]) in the meson-meson structure, respectively.
Numerical Results and discussions {#Numerical Results}
=================================
In the framework of the chiral quark model, we calculated the masses of the four-quark state $bb\bar{b}\bar{b}$ with quantum numbers $IJ^P=00^+,01^+,02^+$. The meson-meson and diquark-antidiqaurk structure, and the coupling of these two structures are first considered. For each structure, all possible color configurations and their coupling are taken into account. i.e., for meson-meson structure ($\bar{b}b\bar{b}b$), two color configurations, color singlet-singlet ($1\times1$), octet-octet ($8\times8$) and the mixture of them are studied. For diquark-antidiquark structure ($bb\bar{b}\bar{b}$), also two color configurations, antitriplet-triplet ($\bar{3}\times3$), sextet-antisextet ($6\times\bar{6}$) and their coupling are considered.
![\[veff\] The lowest eigeneneries of $00^+$, $01^+$ and $02^+$ state as a function of the distance between the diquark and antidiquark in adiabatic approximation.](veff.eps)
No Goldstone Boson Exchanges
----------------------------
For beauty-full system, generally the Goldstone boson exchanges between $b$ quarks are not introduced because of the large mass of $b$-quark. The eigenvalues of $bb\bar{b}\bar{b}$ four-quark state in meson-meson and diquark-antidiqaurk structures are demonstrated in Table \[nomesonexchanges1\] and Table \[nomesonexchanges2\], respectively. In Table \[nomesonexchanges1\], the column with head “channel” represents the index of the antisymmetrized wave functions of $bb\bar{b}\bar{b}$ state. $E_0$ is the eigenenergy of each channel. $E_{cc1}$ gives the eigenenergy with the channel coupling of the two color configurations ($1\times1$ and $8\times8$), and from the table, we can see that the effect of the hidden color channel is too tiny to be visible. Coupling all different spin-color configurations, we get the eigenvalue ($E_{cc2}$) for each set of quantum numbers ($00^+$,$01^+$,$02^+$). The results indicate that the couplings effect are also very small. All the eigenvalues are higher than the theoretical thresholds. With increasing range, all the eigenvalues are approaching to the theoretical thresholds. So we found no bound states for $bb\bar{b}\bar{b}$ state in meson-meson structure.
For diquark-antidiquark structure, in Table \[nomesonexchanges2\], we can see that the couplings of the two color configurations ($\bar{3}\times3$ and $6\times\bar{6}$) are rather strong. But the eigenvalue of each state is still higher than the corresponding theoretical threshold. Because the colorful clusters cannot fall apart, there may be a resonance even with the higher eigenenergy. To check this possibility, we preform an adiabatic calculations for the $00^+$, $01^+$ and $02^+$ states. In this case, the number of the Gaussians used for the relative motion between the diquark and antidiquark subclusters is limited to 1. The lowest adiabatic eigenenergies of these states with different separation between two subclusters are shown in Fig \[veff\]. It reveals that the energies are increasing when the separation increases. And at the separation about $0.3$ fm, there comes the minimum energies for these states which manifests that the subclusters are not willing to be too close or falling apart. And in our calculations, in pure diquark-antidiquark structure the $bb\bar{b}\bar{b}$ tetraquark state may be a resonance state with the lowest mass 19177.5 MeV which is a little larger than the bound state values 18827 MeV [@prd97094015] and $18826\pm25$ MeV [@prd95034011].
When considering the coupling of two structures, the results are shown in Table \[nomesonexchanges-mix\]. $E_1$ represents the low-lying eigenvalues. For each state, we found that the low-lying energy tends to be the same with those in the pure meson-meson structure, which indicates that there is still no bound state after considering the mixtures of two structures. In order to looking for the possible resonances, we calculated the distance between $b$ and $\bar{b}$ quark, denotes as $R_{b\bar{b}}$ in Table \[nomesonexchanges-mix\], as well as the distance between $b$ and $b$ quark, denotes as $R_{bb}$ for each eigenstate. $E_{2}$ represents the eigenenergy of the first possible resonance state. From the table, we can find that the lowest mass of the possible resonance state is reduced to 18872.8 MeV, which is close to the previous results [@prd97094015; @prd95034011]. Compared with 19177.5 MeV in pure diquark-antidiquark structure, we can see that the coupling of the two structures plays an important role. The percentage of each structure is not meaningful because there is a large overlap between two structures [@Ji]. For $01^+$ and $02^+$ state, we cannot find the relative stable resonance states around 19 GeV.
$IJ^P$ channel $E_0$ $E_{cc1}$ $E_{cc2}$ $E_{th1}$ $E_{th2}$
----------- ------------------------------------------------- ------------- -------------- -------------- -------------- -----------
$00^+$ $\chi^{\sigma 1}_{0}\chi^{f}_{m0}\chi^{c}_{m1}$ 18669.6 18669.6 18669.6 18669.3 18798.0
$\chi^{\sigma 1}_{0}\chi^{f}_{m0}\chi^{c}_{m2}$ 19205.4
$\chi^{\sigma 2}_{0}\chi^{f}_{m0}\chi^{c}_{m1}$ 18928.3 18928.3 18927.8 18920.6
$\chi^{\sigma 2}_{0}\chi^{f}_{m0}\chi^{c}_{m2}$ 19194.9
$01^+$ $\chi^{\sigma 3}_{1}\chi^{f}_{m0}\chi^{c}_{m1}$ 18798.9 18798.9 18798.9 18798.6 18859.3
$\chi^{\sigma 3}_{1}\chi^{f}_{m0}\chi^{c}_{m2}$ 19179.2
$\chi^{\sigma 4}_{1}\chi^{f}_{m0}\chi^{c}_{m1}$ 18798.9 18798.9 18798.6 18859.3
$\chi^{\sigma 4}_{1}\chi^{f}_{m0}\chi^{c}_{m2}$ 19179.2
$02^+$ $\chi^{\sigma 6}_{2}\chi^{f}_{m0}\chi^{c}_{m1}$ 18928.3 18928.3 18928.3 18927.8 18920.6
$\chi^{\sigma 6}_{2}\chi^{f}_{m0}\chi^{c}_{m2}$ 19195.0
: \[nomesonexchanges-mix\] The eigenenergies of $bb\bar{b}\bar{b}$ state after considering the coupling of two structures with no Goldstone boson exchanges.
$IJ^P$ channel $E_0$ $E_{cc1}$ $E_{th1}$
----------- ------------------------------------------------- ------------- -------------- --------------
$00^+$ $\chi^{\sigma 1}_{0}\chi^{f}_{d0}\chi^{c}_{d2}$ 19191.1 19177.5 18669.3
$\chi^{\sigma 2}_{0}\chi^{f}_{d0}\chi^{c}_{d1}$ 19221.0
$01^+$ $\chi^{\sigma 5}_{1}\chi^{f}_{d0}\chi^{c}_{d1}$ 19226.8 19226.8 18927.8
$02^+$ $\chi^{\sigma 6}_{2}\chi^{f}_{d0}\chi^{c}_{d1}$ 19237.6 19237.6 18927.8
: \[nomesonexchanges-mix\] The eigenenergies of $bb\bar{b}\bar{b}$ state after considering the coupling of two structures with no Goldstone boson exchanges.
$IJ^P$ $E_1$(MeV) $E_2$(MeV) $R_{b\bar{b}}$(fm) $R_{bb}$(fm)
------------- ----------------- ----------------- ------------------------- --------------
$00^+$ 18669.6 18872.8 0.58 0.85
$01^+$ 18798.9 ... ... ...
$02^+$ 18928.3 ... ... ...
: \[nomesonexchanges-mix\] The eigenenergies of $bb\bar{b}\bar{b}$ state after considering the coupling of two structures with no Goldstone boson exchanges.
$IJ^P$ channel $E_{cc1}$ $E_{th1}$ $E_{th2}$
-------- -------------------------------------------------------------------------------------------------- ----------- ----------- -----------
$00^+$ $\chi^{\sigma 1}_{0}\chi^{f}_{m0}\chi^{c}_{m1}$, $\chi^{\sigma 1}_{0}\chi^{f}_{m0}\chi^{c}_{m2}$ 18704.3 18704.0 18798.0
$\chi^{\sigma 2}_{0}\chi^{f}_{m0}\chi^{c}_{m1}$, $\chi^{\sigma 2}_{0}\chi^{f}_{m0}\chi^{c}_{m2}$ 18924.1 18923.8 18920.6
$01^+$ $\chi^{\sigma 3}_{1}\chi^{f}_{m0}\chi^{c}_{m1}$, $\chi^{\sigma 3}_{1}\chi^{f}_{m0}\chi^{c}_{m2}$ 18814.2 18813.9 18859.3
$\chi^{\sigma 4}_{1}\chi^{f}_{m0}\chi^{c}_{m1}$, $\chi^{\sigma 4}_{1}\chi^{f}_{m0}\chi^{c}_{m2}$ 18814.2 18813.9 18859.3
$02^+$ $\chi^{\sigma 6}_{2}\chi^{f}_{m0}\chi^{c}_{m1}$, $\chi^{\sigma 6}_{2}\chi^{f}_{m0}\chi^{c}_{m2}$ 18924.1 18923.8 18920.6
: \[etab1\] The eigenenergies of $bb\bar{b}\bar{b}$ state for meson-meson structure only with heavy meson $\eta_{b}$ exchange (unit: MeV). $E_{th1}$ and $E_{th2}$ represents the theoretical and experimental threshold of each channel, respectively.
$IJ^P$ channel $E_{cc1}$ $E_{th1}$ $E_{th2}$
-------- -------------------------------------------------------------------------------------------------- ----------- ----------- -----------
$00^+$ $\chi^{\sigma 1}_{0}\chi^{f}_{d0}\chi^{c}_{d2}$, $\chi^{\sigma 2}_{0}\chi^{f}_{d0}\chi^{c}_{d1}$ 19177.9 18704.0 18798.0
$01^+$ $\chi^{\sigma 5}_{1}\chi^{f}_{d0}\chi^{c}_{d1}$ 19226.4 18923.8 18920.6
$02^+$ $\chi^{\sigma 6}_{2}\chi^{f}_{d0}\chi^{c}_{d1}$ 19235.5 18923.8 18920.6
: \[etab2\] The eigenenergies of $bb\bar{b}\bar{b}$ state for diquark-antidiquark structure only with heavy meson $\eta_{b}$ exchange (unit: MeV). $E_{th1}$ and $E_{th2}$ represents the theoretical and experimental threshold of each channel, respectively.
Inclusion of $\eta_b$ Exchange
------------------------------
If the bound $bb\bar{b}\bar{b}$ state is found by experiments, quark model has to be expanded to account for the challenge. On hadron level, to study mutliquark systems, the heavy meson exchanges are invoked. In studying hidden-charm pentaquark states $N^{*}$ and $\Lambda^{*}$, the SU(4) flavor symmetry was employed [@Wu]. The exchange of heavy vector mesons was also used in exploring the charmonium-like hadrons [@Aceti; @He]. Here we extend the chiral quark model by including the heavy meson exchange between $b(\bar{b})$ quarks, to check whether the heavy meson exchange can provide attractive mechanism. First, the $\eta_b$ exchange is introduced, $$\begin{aligned}
V_{ij}^{\eta_b} & = & \frac{g_{ch}^2}{4\pi}\frac{m_{\eta_b}^2}{12m_im_j}
\frac{\Lambda_{\eta_b}^2}{\Lambda_{\eta_b}^2-m_{\eta_b}^2}m_{\eta_b} \nonumber \\
& & \left[ Y(m_\chi r_{ij})-\frac{\Lambda_{\chi}^3}{m_{\chi}^3}Y(\Lambda_{\chi} r_{ij})
\right] \lambda_i^{15} \lambda_j^{15} \boldsymbol{\sigma}_i \cdot\boldsymbol{\sigma}_j,\end{aligned}$$ where $\lambda_j^{15}=diag(1,1,1,1,-4)/\sqrt{10}$ and $Y(x)=e^{-x}/x$. The results with the inclusion of the $\eta_b$ exchange are shown in Tables \[etab1\] and \[etab2\]. As aforementioned, the coupling effects of the different spin-color configurations are very small, the inclusion of $\eta_b$ exchange does not change the statement. So in Tables \[etab1\] and \[etab2\], we just give the eigenvalues $E_{cc1}$ for conciseness. In meson-meson structure (Table \[etab1\]), we can see that the effect of $\eta_b$ exchange is too small due to its large mass and the eigenvalue for each state is still higher than and approaching to the corresponding theoretical threshold. No bound states are found.
For diquark-antidiquark structure (Table \[etab2\]), the eigenvalues for these states are similar with those without $\eta_b$ exchange, a resonance may be exist, rather than a meson-meson molecular state.
Invoking of an effective $\sigma$ exchange
------------------------------------------
The chiral partner, scalar $\sigma$ meson provide a universal attraction in the $u,d$ systems. Extended to $u,d,s$ three-flavor world, the scalar nonet states are used. For the heavy quark systems, it is a possible way to introduce the scalar meson exchange to increase the attraction between quarks. Clearly to find $n^2$ scalar mesons, which associated with SU(n) flavor symmetry, is impractical. Instead often an effective $\sigma$ meson exchange is used [@sigma1; @sigma2]. Now we calculated the eigenenergies of the $bb\bar{b}\bar{b}$ state by considering an effective $\sigma$ meson exchange with the mass of $\sigma$ taking a series of values, 1.0 GeV, 1.5 GeV, 2.5 GeV, 3.5 GeV. The results for meson-meson and diquark-antidiquark structures are demonstrated in Tables \[sigma1\] and \[sigma2\], respectively. In the tables, the $E_B$ gives the binding energy of the state. For meson-meson structure, from Table \[sigma1\], the eigenenergies of the states are all lower than the corresponding thresholds. The binding energies are getting smaller with the increasing of the $\sigma$ mass. For diquark-antidiquark structure, the results are shown in table \[sigma2\], where “..." means that the eigenvalue is higher than the corresponding threshold, which indicates there’s no bound state. From the table, we can find that for $00^+$ state, no matter what the mass of $\sigma$ is, no bound states are found. For $01^+$ and $02^+$ state, only when the mass of $\sigma$ takes the small values, 1.0 GeV, 1.5 GeV, we can find bound states.
From the above results, we can see that extra attractive potential must be introduced if bound $bb\bar{b}\bar{b}$ states exist. Introducing an effective $\sigma$ meson exchange is a common and economic way to increase the attraction between clusters. However, too few scalar are found experimentally for the heavy flavor systems, and the use of effective $\sigma$ meson exchange is too artificial.
[ccccccccccc]{} $IJ^P$ &channel &$E_{cc1}$ &$E_{th_1}$ & $E_B$ \
& &\
$00^+$ &$\chi^{\sigma 1}_{0}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 1}_{0}\chi^{f}_{m0}\chi^{c}_{m2}$ &18107.5 &18337.8 &$-230.3$\
&$\chi^{\sigma 2}_{0}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 2}_{0}\chi^{f}_{m0}\chi^{c}_{m2}$ &18381.8 &18650.4 &$-268.6$\
$01^+$ &$\chi^{\sigma 3}_{1}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 3}_{1}\chi^{f}_{m0}\chi^{c}_{m2}$ &18288.2 &18494.1 &$-205.9$\
&$\chi^{\sigma 4}_{1}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 4}_{1}\chi^{f}_{m0}\chi^{c}_{m2}$ &18288.2 &18494.1 &$-205.9$\
$02^+$ &$\chi^{\sigma 6}_{2}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 6}_{2}\chi^{f}_{m0}\chi^{c}_{m2}$ &18425.4 &18650.4 &$-225.0$\
& &\
$00^+$ &$\chi^{\sigma 1}_{0}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 1}_{0}\chi^{f}_{m0}\chi^{c}_{m2}$ &18136.9 &18321.0 &$-184.1$\
&$\chi^{\sigma 2}_{0}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 2}_{0}\chi^{f}_{m0}\chi^{c}_{m2}$ &18429.6 &18654.0 &$-224.4$\
$01^+$ &$\chi^{\sigma 3}_{1}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 3}_{1}\chi^{f}_{m0}\chi^{c}_{m2}$ &18330.9 &18487.5 &$-156.6$\
&$\chi^{\sigma 4}_{1}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 4}_{1}\chi^{f}_{m0}\chi^{c}_{m2}$ &18330.9 &18487.5 &$-156.6$\
$02^+$ &$\chi^{\sigma 6}_{2}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 6}_{2}\chi^{f}_{m0}\chi^{c}_{m2}$ &18475.2 &18654.0 &$-178.8$\
& &\
$00^+$ &$\chi^{\sigma 1}_{0}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 1}_{0}\chi^{f}_{m0}\chi^{c}_{m2}$ &18255.1 &18335.2 &$-80.1$\
&$\chi^{\sigma 2}_{0}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 2}_{0}\chi^{f}_{m0}\chi^{c}_{m2}$ &18571.7 &18692.8 &$-121.1$\
$01^+$ &$\chi^{\sigma 3}_{1}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 3}_{1}\chi^{f}_{m0}\chi^{c}_{m2}$ &18461.1 &18514.0 &$-52.9$\
&$\chi^{\sigma 4}_{1}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 4}_{1}\chi^{f}_{m0}\chi^{c}_{m2}$ &18461.1 &18514.0 &$-52.9$\
$02^+$ &$\chi^{\sigma 6}_{2}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 6}_{2}\chi^{f}_{m0}\chi^{c}_{m2}$ &18619.1 &18692.8 &$-73.7$\
& &\
$00^+$ &$\chi^{\sigma 1}_{0}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 1}_{0}\chi^{f}_{m0}\chi^{c}_{m2}$ &18358.5 &18371.8 &$-13.3$\
&$\chi^{\sigma 2}_{0}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 2}_{0}\chi^{f}_{m0}\chi^{c}_{m2}$ &18688.8 &18736.0 &$-47.2$\
$01^+$ &$\chi^{\sigma 3}_{1}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 3}_{1}\chi^{f}_{m0}\chi^{c}_{m2}$ &18553.3 &18553.9 &$-0.6$\
&$\chi^{\sigma 4}_{1}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 4}_{1}\chi^{f}_{m0}\chi^{c}_{m2}$ &18553.3 &18553.9 &$-0.6$\
$02^+$ &$\chi^{\sigma 6}_{2}\chi^{f}_{m0}\chi^{c}_{m1}$,$\chi^{\sigma 6}_{2}\chi^{f}_{m0}\chi^{c}_{m2}$ &18728.4 &18736.0 &$-7.6$\
$IJ^P$ channel $E_{cc1}$ $E_{th_1}$ $E_B$
------------- ------------------------------------------------------------------------------------------------- --------------- ---------------- ---------- -- -- -- -- -- --
$00^+$ $\chi^{\sigma 1}_{0}\chi^{f}_{d0}\chi^{c}_{d2}$,$\chi^{\sigma 2}_{0}\chi^{f}_{d0}\chi^{c}_{d1}$ 18457.6 18337.8 ...
$01^+$ $\chi^{\sigma 5}_{1}\chi^{f}_{d0}\chi^{c}_{d1}$ 18526.6 18650.4 $-123.8$
$02^+$ $\chi^{\sigma 6}_{2}\chi^{f}_{d0}\chi^{c}_{d1}$ 18551.2 18650.4 $-99.2$
$00^+$ $\chi^{\sigma 1}_{0}\chi^{f}_{d0}\chi^{c}_{d2}$,$\chi^{\sigma 2}_{0}\chi^{f}_{d0}\chi^{c}_{d1}$ 18488.3 18321.0 ...
$01^+$ $\chi^{\sigma 5}_{1}\chi^{f}_{d0}\chi^{c}_{d1}$ 18566.3 18654.0 $-87.7$
$02^+$ $\chi^{\sigma 6}_{2}\chi^{f}_{d0}\chi^{c}_{d1}$ 18595.3 18654.0 $-58.7$
$00^+$ $\chi^{\sigma 1}_{0}\chi^{f}_{d0}\chi^{c}_{d2}$,$\chi^{\sigma 2}_{0}\chi^{f}_{d0}\chi^{c}_{d1}$ 18626.9 18335.2 ...
$01^+$ $\chi^{\sigma 5}_{1}\chi^{f}_{d0}\chi^{c}_{d1}$ 18712.5 18692.8 ...
$02^+$ $\chi^{\sigma 6}_{2}\chi^{f}_{d0}\chi^{c}_{d1}$ 18745.3 18692.8 ...
$00^+$ $\chi^{\sigma 1}_{0}\chi^{f}_{d0}\chi^{c}_{d2}$,$\chi^{\sigma 2}_{0}\chi^{f}_{d0}\chi^{c}_{d1}$ 18767.6 18371.8 ...
$01^+$ $\chi^{\sigma 5}_{1}\chi^{f}_{d0}\chi^{c}_{d1}$ 18849.4 18736.0 ...
$02^+$ $\chi^{\sigma 6}_{2}\chi^{f}_{d0}\chi^{c}_{d1}$ 18880.7 18736.0 ...
: \[sigma2\] The eigenenergies of $bb\bar{b}\bar{b}$ state for diquark-antidiquark structure only with an effective $\sigma$ exchange (unit: MeV).
Summary {#epilogue}
=======
In the chiral quark model, we calculated the eigenenergies of the low-lying $bb\bar{b}\bar{b}$ states with quantum numbers $IJ^P=00^+, 01^+, 02^+$ using the Gaussian expansion method. Two structures: meson-meson and diquark-antidiquark, and the coupling of them are investigated. For the beauty-full system, the interaction from the exchange of Goldstone bosons is absent generally, and we found that the energies of $bb\bar{b}\bar{b}$ states with both structures are all higher than the corresponding thresholds, leaving no space for a bound state in this situation. Gluon exchange, as a short range force, it helps to form compact tetraquarks rather than meson-meson molecules if bound four-quark states do exit. In our calculations, $bb\bar{b}\bar{b}$ tetraquark state may be a resonance state with the lowest mass $18872.8$ MeV when considering the coupling of the meson-meson and diquark-antidiquark structures. It is expected that the exotic tetraquark states composed of four heavy quarks may be observed at LHC.
As a test, the heavy meson $\eta_b$ exchange between the $b$ quarks is introduced, its effect is too small to change the situation. However, the employment of an effective scalar $\sigma$ will produce bound states for $bb\bar{b}\bar{b}$ system, because of the universal attractive property of the $\sigma$ meson exchange. The binding energy is smaller and smaller with the increasing the mass of $\sigma$. Hopefully, our study will be helpful to searching for the exotic tetraquark states composed of four heavy quarks.
This work is supported partly by the National Science Foundation of China under Contract No. 11847145.
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|
---
abstract: |
We have determined the relation between the in-medium $\omega $ meson mass and quark condensate in the framework of a Nambu Jona-Lasinio model constrained by some recent experimental data on the meson properties in nuclei. In addition to the usual four-quark interactions, we have included eight-quark terms in the Lagrangian. The parameters of this model have been determined using the meson properties in the vacuum as well as in the medium. More particularly, we have constrained both the in-medium pion decay constant to the value measured in experiments on deeply bound pionic atoms and the in-medium $\omega $ meson mass to the experimental value obtained either by the TAPS collaboration or by the E325 experiment at KEK. Our results are compared to several scaling laws and in particular to that of Brown and Rho.\
address: |
Université Bordeaux 1 ; CNRS/IN2P3 ;\
Centre d’Etudes Nucléaires de Bordeaux-Gradignan, UMR 5797\
Chemin du Solarium, BP120, 33175 Gradignan, France
author:
- 'R. Huguet, J.C. Caillon and J. Labarsouque'
title: 'In-medium omega meson mass and quark condensate in a Nambu Jona-Lasinio model constrained by recent experimental data'
---
Introduction
============
These last years, much attention has been focussed on the modification of hadron properties in nuclear environment and more particularly in the sector of the light vector mesons. The hope is that this modification could shed some light on prominent features of QCD at low energy. In particular, the knowledge of the dependence of the in-medium vector meson mass on the quark condensate is essential to a better understanding of the role played by the chiral structure of the QCD vacuum.
Experimentally, an indirect indication of the modification of hadron properties in the medium has been provided by the dilepton production measurements in relativistic heavy-ion collisions, like for example, experiments from CERES[@cer] and HELIOS[@hel] collaborations. However, the interpretation in terms of a reduction of the $\rho $ mass is still controversial. Recently, new experiments using proton-induced nuclear reactions[@nar], or $\gamma -A$ reactions[@tap] have provided a more clear experimental signature of the in-medium modifications of the $\omega $ mesons. In particular, the modification in nuclei of the $\omega $ meson has been investigated in photoproduction experiments by the TAPS collaboration[@tap] and its mass was found to be $m_{\omega }^{*}=722_{-4}^{+4}$ (stat)$_{-5}^{+35}$(syst) MeV at 0.6 times the saturation density of nuclear matter. The same order of magnitude, a 9% decrease of the in-medium $\omega $ mass at saturation, has been observed by Naruki et al.[@nar] in 12 GeV proton-nucleus reactions (E325/KEK).
On the other hand, experimental indications of the in-medium modification of the quark condensate, $\left\langle \overline{q}q\right\rangle $, can be obtained, for example, in experiments on deeply bound pionic atoms. Indeed, by deducing the isovector $\pi N$ interaction parameter in the pion-nucleus potential from the binding energy and width of deeply bound 1s states of $\pi ^{-}$ in heavy nuclei, the in-medium pion decay constant, $f_{\pi }^{*}$, can be extracted[@gei; @suz]. The quark condensate is then connected to $%
f_{\pi }^{*}$ through the Gell-Mann-Oaks-$%
%TCIMACRO{\func{Re}}
%BeginExpansion
\mathop{\rm Re}%
%EndExpansion
$nner relation. The observed enhancement of the isovector $\pi N$ interaction parameter over the free $\pi N$ value indicates a reduction of the pion decay constant in the medium which was found[@suz] to be $f_{\pi }^{*2}/f_{\pi }^{2}=0.64$ at saturation density of nuclear matter.
>From a theoretical point of view, starting from the assumption of Harada and Yamawaki[@h.y] on the ”vector manifestation” of chiral symmetry in which a hidden local symmetry theory is matched to QCD, Brown and Rho proposed[@br.] that, up to the saturation density, the vector meson mass in medium, $m_{V}^{*}$, scales according to the approximative relation : $%
m_{V}^{*}/m_{V}\sim \left[ \left\langle \overline{q}q\right\rangle
/\left\langle \overline{q}q\right\rangle _{0}\right] ^{1/2}$ (where $%
\left\langle \overline{q}q\right\rangle $ and $\left\langle \overline{q}%
q\right\rangle _{0}$ are respectively the in-medium and vacuum quark condensates). In quite different frameworks, like, for example, in finite density QCD sum rule calculations[@hat; @asa; @koi; @jin; @kli] or in the Nambu Jona-Lasinio model (NJL)[@b.m], the relation between the in-medium vector meson mass and quark condensate is not so clear and thus more complicated to handle.
The recent experimental data, like those previously mentioned, should provide stringent tests for the models and for the relation between the in-medium $\omega $ meson mass and quark condensate. An indication on the consequences of these new constraints could be obtained by enforcing them in quark models incorporating the most prominent features of QCD. In this context, the NJL model[@njl] appears as a good candidate since it allows a dynamical description of both the breaking of chiral symmetry and of the modification of the in-medium $\omega $ meson mass.
In this work, we have determined the dependence of the in-medium $\omega $ meson mass on the quark condensate in a NJL model constrained by in-medium meson properties in accordance with recent experimental data. In addition to the usual four-quark interactions, we have included eight-quark terms in the NJL Lagrangian. The parameters of this model have been determined using the meson properties in the vacuum as well as in the medium through the pion decay constant and $%
\omega $ meson mass. More particularly, the in-medium pion decay constant has been constrained by the value obtained in an experiment on deeply bound pionic atoms[@suz] and we have fixed the in-medium $\omega $ meson mass to the experimental values determined by the TAPS collaboration[@tap] or by the E325 experiment at KEK[@nar]. These in-medium changes of meson properties arise from dynamical chiral symmetry restoration at the quark mean-field-RPA level as well as from more complicated quark-gluon excitations usually parametrized in terms of many-body hadronic interactions. Considering the importance of the role played by the chiral structure of the QCD vacuum, concerning the $\omega $ meson mass and the pion decay constant, we made the assumption that, to leading order in nuclear density, the main contribution comes from dynamical chiral symmetry restoration at the quark mean-field-RPA level. Our results will be compared to several scaling laws and in particular to that of Brown and Rho.
Formalism
=========
We consider the following chirally invariant two-flavor NJL Lagrangian[@njl] up to eight-quark interaction terms :
$$\begin{aligned}
{\cal \ L} &=&\overline{q}\left[ i\gamma _{\mu }\partial ^{\mu
}-m_{0}\right] q+g_{1}\left[ (\overline{q}q)^{2}+(\overline{q}i\gamma _{5}%
{\bf \tau }q)^{2}\right] -g_{2}(\overline{q}\gamma _{\mu }q)^{2} \label{njl}
\\
&&+g_{3}\left[ (\overline{q}q)^{2}+(\overline{q}i\gamma _{5}{\bf \tau }%
q)^{2}\right] (\overline{q}\gamma _{\mu }q)^{2}+g_{4}\left[ (\overline{q}%
q)^{2}+(\overline{q}i\gamma _{5}{\bf \tau }q)^{2}\right] ^{2}, \nonumber\end{aligned}$$
where $q$ denotes the quark field with two flavor ($N_{f}=2$) and three color ($N_{c}=3$) degrees of freedom and $m_{0}$ is the diagonal matrix of the current quark masses (here in the isospin symmetric case). The second and third terms of Eq.\[njl\] represent local four-quark interactions while those proportional to $g_{3}$ and $g_{4}$ are eight-quark interactions. Let us recall that, in two-flavor models, the t’Hooft six fermion interaction term can be rewrited in terms of the four-quark interactions considered here[@kle]. We have not considered the term $(%
\overline{q}\gamma _{\mu }q)^{4}$ since, as in the nucleonic NJL model[@mis], it leads to a violation of the causality at high density.
The Dirac equation for a quark in mean-field approximation is given by :
$$\left[ i\gamma _{\mu }\partial ^{\mu }-m_{0}-2g_{2}\gamma _{0}\left\langle
\overline{q}\gamma _{0}q\right\rangle +2g_{1}\left\langle \overline{q}%
q\right\rangle +2g_{3}\left\langle \overline{q}q\right\rangle \left\langle
\overline{q}\gamma _{0}q\right\rangle ^{2}+4g_{4}\left\langle \overline{q}%
q\right\rangle ^{3}\right] q=0, \label{dir}$$
which defines a dynamical constituent-quark mass :
$$m=m_{0}-2g_{1}\left( 1+\frac{g_{3}N_{f}^{2}N_{c}^{2}\rho _{B}^{2}}{4g_{1}}+%
\frac{2g_{4}}{g_{1}}\left\langle \overline{q}q\right\rangle ^{2}\right)
\left\langle \overline{q}q\right\rangle , \label{gap}$$
generated by a strong scalar interaction of the quark with the Dirac vacuum. In the gap equation (Eq.\[gap\]), the quark condensate $%
\left\langle \overline{q}q\right\rangle $ can be written as :
$$\left\langle \overline{q}q\right\rangle =-i\int \frac{d^{4}k}{\left( 2\pi
\right) ^{4}}{\tt Tr}S(k), \label{qqbs}$$
where here Tr denotes traces over color, flavor and spin. In Eq.\[qqbs\], $%
S(k)$ represents the in-medium quark propagator defined as :
$$S(k)=\frac{1}{\gamma _{\mu }k^{\mu }-m+i\varepsilon }+i\pi \frac{\gamma
_{\mu }k^{\mu }+m}{E_{k}}\delta \left( k_{0}-E_{k}\right) \theta \left(
k_{F}-\left| {\bf k}\right| \right) , \label{pro}$$
where $E_{k}=\sqrt{{\bf k}^{2}+m^{2}}$ and $k_{F}$ is the quark Fermi momentum. The baryonic density is related to the total quark density by $\rho _{B}=\frac{1}{3}\rho _{q}$. The quark condensate is divergent due to the loop integrals and requires an appropriate regularization procedure. As many authors[@b.m; @bub], we introduce a three-momentum cutoff $\Lambda
$ which has the least impact on medium parts of the regularized integrals, in particular at zero temperature[@bub]. Thus, after the regularization procedure, the quark condensate is given at each density by :
$$\left\langle \overline{q}q\right\rangle =-\frac{N_{f}N_{c}}{\pi ^{2}}%
\int_{k_{F}}^{\Lambda }\frac{mk^{2}dk}{E_{k}}. \label{qqb}$$
As usual, the $\omega$ meson mass is obtained by solving the Bethe-Salpeter equation in the quark-antiquark channel. First, we define the quark-antiquark polarization operator in the $\omega $ channel by :
$$\begin{aligned}
\Pi _{\omega }^{\mu \nu }(q^{2}) &=&-i\int \frac{d^{4}p}{\left( 2\pi \right)
^{4}}{\tt Tr}\left[ i\gamma ^{\mu }iS(p+q/2)i\gamma ^{\nu }iS(p-q/2)\right]
\label{pola} \\
&=&\left( -g^{\mu \nu }+\frac{q^{\mu }q^{\upsilon }}{q^{2}}\right) \Pi
_{\omega }(q^{2}), \nonumber\end{aligned}$$
where $S(k)$ is given by Eq.\[pro\]. Using the same regularization procedure as for the quark condensate, we obtain : $$\Pi _{\omega }(q^{2})=\frac{N_{f}N_{c}}{12\pi ^{2}}q^{2}%
\int_{4(p_{F}^{2}+m^{2})}^{4(\Lambda ^{2}+m^{2})}\frac{\sqrt{1-4m^{2}/\kappa
^{2}}}{q^{2}-\kappa ^{2}}\left( 1+\frac{2m^{2}}{\kappa ^{2}}\right) d\kappa
^{2}. \label{polw}$$
The in-medium $\omega $ meson mass, $m_{\omega }^{*}$, is then determined by the pole structure of the $T$-matrix, i.e. by the condition :
$$1-2\left( g_{2}-g_{3}\left\langle \overline{q}q\right\rangle ^{2}\right) \Pi
_{\omega }(q^{2}=m_{\omega }^{*2})=0. \label{mw}$$
We also need the pion mass and decay constant to adjust the model parameters. In the pseudo-scalar channel, the polarization reads : $$\Pi _{\pi }(q^{2})=\frac{\left\langle \overline{q}q\right\rangle }{m}%
+N_{c}N_{f}q^{2}I(q^{2}), \label{ppi}$$ with
$$I(q^{2})=\frac{1}{8\pi ^{2}}\int_{4(\Lambda ^{2}+m^{2})}^{4(p_{F}^{2}+m^{2})}%
\frac{1}{q^{2}-\kappa ^{2}}\sqrt{1-\frac{4m^{2}}{\kappa ^{2}}}d\kappa ^{2}.
\label{iq2}$$
The in-medium $\pi $ meson mass, $m_{\pi }^{*}$, and decay constant, $f_{\pi
}^{*}$, are then given respectively by :
$$1-2\left( g_{1}+\frac{g_{3}N_{f}^{2}N_{c}^{2}\rho _{B}^{2}}{4}%
+2g_{4}\left\langle \overline{q}q\right\rangle ^{2}\right) \Pi _{\pi
}(q^{2}=m_{\pi }^{*2})=0, \label{mpi}$$
$$f_{\pi }^{*}=N_{c}N_{f}g_{\pi qq}^{*}mI(q^{2}=m_{\pi }^{*2}), \label{fpi}$$
where the in-medium pion-quark-quark coupling constant, $g_{\pi qq}^{*}$, is obtained by :
$$g_{\pi qq}^{*2}=\left[ \frac{d\Pi _{\pi }(q^{2})}{dq^{2}}\right]
_{q^{2}=m_{\pi }^{*2}}^{-1}. \label{res}$$
Results
=======
We have six free parameters : the cutoff $\Lambda $, the bare quark mass $%
m_{0}$, and the coupling constants $g_{1}$, $g_{2}$, $g_{3}$ and $g_{4}$. As usual, we have used for the fitting procedure the three following constraints : the pion mass $m_{\pi }=135$ MeV, the pion decay constant $%
f_{\pi }=92.4$ MeV and the $\omega $ meson mass $m_{\omega }=782$ MeV in vacuum. Since additional constraints like, for example, the value of the quark condensate in vacuum, do not allow to determine the cutoff $\Lambda $ unambiguously, as often done[@bub], we have considered several values for $\Lambda $ or equivalently several values of the in-vacuum constituent quark mass, $m_{vac}$, values ranging between $400$ MeV and $500$ MeV. Such a rather large mass prevents the $\omega $ meson to be unstable against decay into a quark-antiquark pair since $m_{\omega }^{*}$ is always lower than $2m$ for every density. Note that smaller constituent quark masses can be obtained in NJL models which take the confinement into account by including Polyakov loops[@rtw] or using a confining interaction[@chs] in the Lagrangian. As already mentioned, in addition to these in-vacuum constraints, we have also chosen to take into account recent experimental results which provide constraints in the medium. In particular, we have fixed $f_{\pi }^{*2}(\rho _{B}=\rho _{0})/f_{\pi }^{2}=0.64$ (where $%
\rho _{0}$ is the saturation density of nuclear matter) in accordance with what is obtained in experiments on deeply bound pionic atoms[@suz]. Moreover, the in-medium $\omega $ meson mass has been constrained to reproduce the experimental central value obtained either by the TAPS collaboration[@tap], $%
m_{\omega }^{*}(\rho _{B}=0.6\rho _{0})=722$ MeV, or by the E325 experiment at KEK[@nar], $m_{\omega }^{*}(\rho _{B}=\rho _{0})=711$ MeV. Thus, two families of parametrization sets denoted respectively by TAPS and KEK will be considered. Note that once $m_{\pi }$, $f_{\pi }$, $m_{\omega }$, $f_{\pi }^{*}$, $m_{\omega
}^{*}$ and $m_{vac}$ are fixed, all the parameters are determined unambiguously.
The results are shown on Fig.1 where we have plotted $m_{\omega
}^{*}/m_{\omega }$ as a function of $\left\langle \overline{q}q\right\rangle
/\left\langle \overline{q}q\right\rangle _{0}$ for the two parametrization sets TAPS and KEK. Note that $\left\langle \overline{q%
}q\right\rangle /\left\langle \overline{q}q\right\rangle _{0}=0.8$ corresponds to a baryonic density close to the saturation one. The shaded areas correspond to values of the $\omega $ meson mass for a constituent quark mass ranging from 400 MeV to 500 MeV. As we can see, these areas are rather narrow and the results are thus only weakly dependent on the value of $m_{vac}$ used.
In order to determine an approximate form for the relation between the vector meson mass and the quark condensate, we have considered scaling laws of the general form : $$\frac{m_{\omega }^{*}}{m_{\omega }}=\left[ \left\langle \overline{q}%
q\right\rangle /\left\langle \overline{q}q\right\rangle _{0}\right] ^{\alpha
}. \label{sca}$$ Any value of $\alpha $ can be considered but we have chosen to show here the results for $\alpha =1/2$ which corresponds to the Brown and Rho scaling and for neighboring values: $\alpha =1/3$ and $\alpha =1$.
The full lines on Fig.1 represent the scaling laws given by Eq.\[sca\] for $\alpha =1/3$, $1/2$ and $1$. A rather good agreement with the case $\alpha =1/2$ corresponding to the Brown and Rho scaling law is obtained for the TAPS parametrization set while the KEK result clearly favours $\alpha =1/3$. Assuming Eq.\[sca\] for the scaling law, this result can be understood since to leading order $\left( \left\langle \overline{q}q\right\rangle
/\left\langle \overline{q}q\right\rangle _{0}\right) ^{\alpha }\simeq
1-\alpha \left( 1-\left\langle \overline{q}q\right\rangle /\left\langle
\overline{q}q\right\rangle _{0}\right) $ and $\left\langle \overline{q}%
q\right\rangle /\left\langle \overline{q}q\right\rangle _{0}\simeq 1-\beta
\rho /\rho _{0}$ with $\beta =\sigma _{N}\rho _{0}/f_{\pi }^{2}m_{\pi }^{2}$ where $\sigma _{N}$ is the $\pi N$ sigma term. The TAPS or KEK results can then be used to determine the product $\alpha \beta $ obtained by eliminating the quark condensate in $m_{\omega }^{*}/m_{\omega }$. Using the value $\sigma _{N}\simeq 35$ MeV obtained in the NJL model for $m\simeq 450$ MeV, the TAPS and KEK results provide respectively $\alpha$ close to $1/2$ and $1/3$ in agreement with our full calculation.
On the other hand, the experimental values of the in-medium $\omega $ meson mass are not determined unambiguously. In particular, the TAPS collaboration obtained a rather large uncertainty including statistical and systematical errors. By taking into account such an uncertainty in our calculation, we have found that the scaling law (Eq.\[sca\]) with $\alpha =1$ is clearly ruled out but the case $\alpha =1/3$ is not totally excluded. However, note that the central value of the TAPS result clearly favours $\alpha =1/2$.
Let us now determine how the quark condensate and the $\omega $ meson mass depend on the baryonic density for central values of the constituent quark mass, i.e. for $m_{vac}=450$ MeV, respectively for the TAPS and KEK parametrizations. The parameters of the NJL model are then $\Lambda =575$ MeV, $m_{0}=5.6$ MeV, $g_{1}\Lambda ^{2}=2.53$, $g_{2}\Lambda ^{2}=5.20$, $g_{3}\Lambda
^{8}=62.5$ and $g_{4}\Lambda ^{8}=2.27$ for the TAPS parametrization and $\Lambda =575$ MeV, $m_{0}=5.6$ MeV, $g_{1}\Lambda
^{2}=2.58$, $g_{2}\Lambda ^{2}=4.12$, $g_{3}\Lambda ^{8}=12.4$ and $%
g_{4}\Lambda ^{8}=1.22$ for the KEK one. Let us mention that, whatever the parametrization considered, the model provides a quark condensate in the vacuum $\left\langle \overline{u}u\right\rangle ^{1/3}=-240$ MeV in good agreement with the lattice calculations: $\left\langle \overline{u}%
u\right\rangle ^{1/3}=-(231\pm 4\pm 8\pm 6)$ MeV [@lat] and a critical density close to four times the saturation one. On Fig.2, we have plotted $%
m_{\omega }^{*}/m_{\omega }$ (solid curves) for the two parametrizations TAPS and KEK as well as $\left[ \left\langle \overline{q}q\right\rangle
/\left\langle \overline{q}q\right\rangle _{0}\right] ^{\alpha }$ for $\alpha
=1/2$ (dashed curve) and $\alpha =1/3$ (dot-dashed curve), as a function of the dimensionless baryonic density $\rho _{B}/\rho _{0}$. As we can see, the density dependences of the in-medium $\omega $ meson mass obtained with the TAPS and KEK parametrizations lead to a drop close to 10% at saturation density (9% for KEK and 12% for TAPS). As already discussed, to a good level of approximation, the in-medium $\omega $ meson mass determined using the KEK result varies as a function of the baryonic density like the third root of the quark condensate. On the other hand, using the TAPS result, the density dependence of the in-medium $\omega $ meson mass is not very much different from the square root of the quark condensate reflecting the fact that the results follow approximately the Brown and Rho scaling law up to the saturation density.
Conclusion
==========
We have determined the in-medium $\omega $ meson mass and quark condensate in a NJL model with eight quark interaction terms. The parameters of this model have been determined using the meson properties in the vacuum but also in the medium through the value of the pion decay constant obtained in experiments on deeply bound pionic atoms as well as the $\omega $ meson mass measured either by the TAPS collaboration or by the E325/KEK experiment. When the in-medium $\omega $ meson mass is constrained to the experimental data obtained by the TAPS collaboration, the Brown and Rho scaling law is approximately recovered. On the other hand, when the KEK result is used, the in-medium $\omega $ meson mass varies rather like the third root of the quark condensate. However, in both cases, this corresponds to a drop of the $\omega $ meson mass at saturation density close to 10%, a result which is lower than those found in QCD sum rule calculations where a decrease close to 15-25% is generally obtained[@hat; @jin; @kli].
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|
---
abstract: 'Convolutional neural networks have become state-of-the-art in a wide range of image recognition tasks. The interpretation of their predictions, however, is an active area of research. Whereas various interpretation methods have been suggested for image classification, the interpretation of image segmentation still remains largely unexplored. To that end, we propose , a gradient-based method for interpreting semantic segmentation. Our method is an extension of the widely-used Grad-CAM method, applied locally to produce heatmaps showing the relevance of individual pixels for semantic segmentation.'
author:
- |
Kira Vinogradova, Alexandr Dibrov, Gene Myers\
Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany\
Center for Systems Biology Dresden, Germany\
`vinograd@mpi-cbg.de`
title: |
Towards Interpretable Semantic Segmentation via\
Gradient-weighted Class Activation Mapping
---
Introduction {#sec:intro}
============
Approaches based on deep learning, and convolutional neural networks (CNNs) in particular, have recently substantially improved the performance for various image understanding tasks, such as image classification, object detection, and image segmentation. However, our understanding of *why* and *how* CNNs achieve state-of-the-art results is rather immature.
One avenue to remedy this is to visually indicate which regions of an input image are (especially) important for the decision made by a CNN. These so-called *heatmaps* can thus be useful to understand a CNN, for example to check that it does not focus on idiosyncratic details of the training images that will not generalize to unseen images.
Gradient-based heatmap methods have generally been popular in the context of image classification. A simple approach are *saliency maps* [@simonyan2013deep], which are obtained via the derivative of the logit $y^c$ (the score of class $c$ before the softmax) with respect to all pixels of the input image. Hence, they highlight pixels whose change would affect the score of class $c$ the most. A more recent and widely-used method by @selvaraju2017grad [-@selvaraju2017grad] is *gradient-weighted class activation mapping* (Grad-CAM). It first uses the aggregated gradients of logit $y^c$ with respect to chosen feature layers to determine their general relevance for the decision of the network. Based on this relevance, a heatmap is obtained as a weighted average of the activations of the respective feature layers (feature maps). Grad-CAM can be seen as a generalization of CAM [@zhou2016learning] , which could only produce class activation mappings for CNNs with a special architecture.
![ for a single pixel (white dot) and class *Flat*. The heatmap is obtained with respect to a convolutional layer at the bottleneck (i.e. end of contracting path) of a U-Net [@ronneberger2015u]. []{data-label="fig:example_cityscape"}](./Fig1_pixel.png){width="1\linewidth"}
Methods that provide visual explanations for the decisions of neural networks have predominantly focused on the task of image classification. In this work, we go beyond that and are interested in explaining the decisions of CNNs for semantic image segmentation. To that end, we propose , an extension of Grad-CAM for semantic segmentation, which can produce heatmaps that explain the relevance for the decision of individual pixels or regions in the input image. We demonstrate that our approach produces reasonable visual explanations for the commonly-used Cityscapes datasets [@Cordts2016Cityscapes].
Concurrent to our work, @hoyer2019grid have independently proposed a method for the visual explanation of semantic segmentation CNNs [@hoyer2019grid]. They assume co-occurences of some classes are important for their segmentation. However, their approach is not based on Grad-CAM, but on perturbation analysis, and is rather different from ours since it focuses on identification of contextual biases. To the best of our knowledge, we present the first approach to produce visual explanations of CNNs for semantic segmentation, specifically by extending Grad-CAM.
Method {#sec:method}
======
As mentioned above, our approach is based on Grad-CAM [@selvaraju2017grad], which we first briefly explain. Let $\{A^k\}_{k=1}^K$ be selected feature maps of interest ($K$ kernels of the last convolutional layer of a classification network), and $y^c$ the logit for a chosen class $c$. Grad-CAM averages the gradients of $y^c$ with respect to all $N$ pixels (indexed by $u,v$) of each feature map $A^k$ to produce a weight $\alpha_c^k$ to denote its importance. The heatmap $$\label{eq:grad-cam}
L^c = \mathrm{ReLU}\biggl(\sum_k \alpha_c^k A^k \biggr)
\ \ \text{with}\ \
\alpha_c^k = \frac{1}{N} \sum_{u,v} \frac{\partial y^c}{\partial A_{uv}^k}$$ is then generated by using these weights to sum the feature maps; finally, $\mathrm{ReLU}$ is applied pixel-wise to clip negative values at zero, to only highlight areas that positively contribute to the decision for class $c$.
Whereas a classification network predicts a single class distribution per input image $x$, a CNN for semantic segmentation typically produces logits $y_{ij}^c$ for every pixel $x_{ij}$ and class $c$.
Hence, we propose by replacing $y^c$ by $\sum_{(i,j) \in \mathcal{M}} y_{ij}^c$ in \[eq:grad-cam\], where $\mathcal{M}$ is a set of pixel indices of interest in the output mask. This allows to adapt Grad-CAM to a semantic segmentation network in a flexible way, since $\mathcal{M}$ can denote just a single pixel, or pixels of an object instance, or simply all pixels of the image. Furthermore, we explore using feature maps from intermediate convolutional layers, not only the last one as used in @selvaraju2017grad [-@selvaraju2017grad].
Experiments {#sec:experiments}
===========
We demonstrate our approach by training a U-Net [@ronneberger2015u] for semantic segmentation of the popular *Cityscapes* dataset [@Cordts2016Cityscapes]. We generally find that the convolutional layers of the U-Net bottleneck (end of the encoder before upsampling) are more informative than the layers close to the end of the U-Net decoder, which would be more similar to those inspected by @selvaraju2017grad [-@selvaraju2017grad]. As a sanity check, we do observe (not shown) that heatmaps produced from the initial convolutional layers exhibit edge-like structures, which does agree with common knowledge that early convolutional layers pick up on low-level image features. Feature maps located between the bottleneck and last layer successively give rise to heatmaps that look more and more similar to the logits of the selected class and the output segmentation mask.
\[fig:example\_cityscape\] shows a heatmap produced by for a bottleneck layer of the U-Net when $\mathcal{M}$ denotes a single pixel. The visually highlighted region seems plausible, mostly indicating similar pixels of the selected class. Note that the heatmap shows the weighted sum of feature maps activated for the whole image (cf. Eq. \[eq:grad-cam\]), and can thus go beyond the receptive field of the CNN for the selected pixel, whose relevance is only for determining the weights $\alpha_c^k$. Furthermore, \[fig:example\_cityscape2\] shows a heatmap for class *Sky* when $\mathcal{M}$ indicates all pixels of the image; it most strongly highlights pixels of a tree (class *Nature*), which may be highly informative to predict *Sky* pixels.
Discussion and Future Work {#sec:discussion}
==========================
Our initial results seem promising, and we would like to systematically investigate the generated heatmaps of our method in the future. Concretely, we want to compare and reason about different intermediate feature maps that can be chosen for visualization. Furthermore, it might be helpful to truncate the extent of the heatmap only to regions that are directly relevant for the prediction at pixels contained in $\mathcal{M}$. For a fixed class $c$, it would also be interesting to compare the weights $\{\alpha_c^k\}_{k=1}^K$ as obtained at different locations. Finally, we aim to explore other interpretation approaches [@montavon2018methods] and plan to demonstrate the merits of our method quantitatively, based on a suitable synthetic dataset.
![ for all pixels and class *Sky*. The heatmap is obtained with respect to a convolutional layer at the bottleneck (i.e. end of contracting path) of a U-Net [@ronneberger2015u]. []{data-label="fig:example_cityscape2"}](./Fig2_channel.png){width="1\linewidth"}
|
---
author:
- |
S. M. Sergeev\
Branch Institute for Nuclear Physics, Protvino 142284, Russia.\
E-mail: sergeev\_ms@mx.ihep.su
date: 'October, 1998'
title: QUANTUM 2+1 EVOLUTION MODEL
---
addtoreset[equation]{}[section]{}
-20pt -20pt 43
u v c ¶ Ł
[*PACS:*]{} 05.50; 02.10; 02.20.\
[*Mathematics Subject Classifications (1991):*]{} 47A60, 47A67, 22D25.\
[*Keywords:*]{} Discrete space – time evolution models; $2+1$ integrability; Tetrahedron equation
Introduction
============
3D integrable models
--------------------
In 3D integrable models the Tetrahedron equation (TE) takes place of the Yang – Baxter equation (YBE) in 2D. Having got a solution of TE, one may hope to construct a 3D integrable model. In the case of finite number of states one may construct usual layer – to – layer transfer matrices $T$, so that TE provides the commutability of them [@zam-solution; @bs-dsimpl; @baxter-pf]. Such finite states models are interpreted usually as statistical mechanics models. Really only one such model still exists, the Zamolodchikov – Bazhanov – Baxter model [@zam-solution; @baxter-pf; @bb-first; @mss-vertex]. The uniqueness does not mean that 3D world has no interest.
When 3D $\R$-matrices have infinitely many states, which is more usual in 3D, very natural is to investigate a kind of transfer matrices that has no hidden space. We denote such transfer matrices as $\U$ [*versus*]{} the notation for usual transfer matrix $T$. Matrices $\U$ commute with the set of $T$-s, but have no degrees of freedom when the set of $T$ is fixed. Thus $\U$ resemble a hamiltonian. Conventionally models with infinitely many states are regarded as field theory ones.
The structure of $\U$ is clarified in Fig. \[fig-ut\] for 2D case. Here $p$ and $q$ stand for the spectral parameters of the vertices of $T$, $p/q$ consequently is the argument of the vertices of $U$, and $\sigma_j$ and $\sigma_j'$ are the indices taking values in a finite set. This $2D$ picture we give just an example for the sake of clearness. The $1+1$ evolution models, connected with classical or quantum bilinear Hirota or Hirota-Miwa equitations on the lattice, are always formulated in terms of $U$-type evolution operators, see for example [@fv-hirota; @fv-quantization] and references therein.
In 3D, $\U$-matrix appears as an element of a cubic lattice included between two nearest inclined planes. We do not draw the graphical representation of 3D $\U$ here, we will consider sections of the cubic lattice by two, in- and out-, inclined planes mentioned. A two dimensional lattice appearing in such sections is called the kagome lattice and we will consider it in details below. The first who considered $\U$ – matrices in $3D$, constructed with a help of finite – state $R$ – matrix, and constructed some eigenvectors for it, was I. Korepanov [@korepanov-diss; @korepanov-u; @korepanov-BA].
3D integrability: usual approach
--------------------------------
So, the origin of 3D integrability is a solution of TE. Those who dealt with it know that it is practically impossible to find it directly. For example, even to prove TE analytically for an ansatz given and tested numerically is bloody complicated [@baxter-proof; @kms-stsq]. This means, we guess, there must be an alternative way of a 3D Boltsmann weights’ derivation.
Primitive way is to find a solution of TE is to consider the intertwining relation for a triple sets of 2D $L$-matrices, $$\ds
\sum_{j_1,j_2,j_3}\;\;
R_{i_1,i_2,i_3}^{j_1,j_2,j_3}\;
\bigl( L_{j_1}^{k_1}\bigr)_{1,2}\;
\bigl( L_{j_2}^{k_2}\bigr)_{1,3}\;
\bigl( L_{j_3}^{k_3}\bigr)_{2,3}\;=\;
\sum_{j_1,j_2,j_3}\;\;
\bigl( L_{i_3}^{j_3}\bigr)_{2,3}\;
\bigl( L_{i_2}^{j_2}\bigr)_{1,3}\;
\bigl( L_{i_1}^{j_1}\bigr)_{1,2}\;
R_{j_1,j_2,j_3}^{k_1,k_2,k_3}\;,$$ where the structure of $L_{1,2}L_{1,3}L_{2,3}$ [*versus*]{} $L_{1,2}L_{1,3}L_{2,3}$ is the Yang – Baxter structure, and the extra indices correspond to the possibility to consider coefficients $R_{i_1,i_2,i_3}^{k_1,k_2,k_3}$ as 3D $\R$ – matrix. TE appears as the admissibility condition for $$\ds
L_{1,2}\;L_{1,3}\;L_{2,3}\;L_{1,4}\;L_{2,4}\;L_{3,4}
\;\mapsto\;
L_{3,4}\;L_{2,4}\;L_{1,4}\;L_{2,3}\;L_{1,3}\;L_{1,2}\;.$$ Another scenario is the Zamolodchikov tetrahedral algebra $$\ds
\Psi_{1,2}^{k_1}\;\Psi_{1,3}^{k_2}\;\Psi_{2,3}^{k_3}\;=\;
\sum_{j_1,j_2,j_3}\;\;
\Psi_{2,3}^{j_3}\;\Psi_{1,3}^{\j_3}\;\Psi_{1,2}^{\j_1}\;
R_{j_1,j_2,j_3}^{k_1,k_2,k_3}\;.$$ In the compact form, introducing the formal basis for the indices of $R$, $e(i,j)\;\equiv\;|i><j|$, $$\ds R_{a,b,c}\;=\;\sum_{j,k}\;\;
R_{j_1,j_2,j_3}^{k_1,k_2,k_3}\;\; e_a(j_1,k_1)\; e_b(j_2,k_2)\;
e_c(j_3,k_3)\;.$$ TE looks like $$\label{te}
\ds
\R_{1,2,3}\*\R_{1,4,5}\*\R_{2,4,6}\*\R_{3,5,6}\;=\;
\R_{3,5,6}\*\R_{2,4,6}\*\R_{1,4,5}\*\R_{1,2,3}\;,$$ where the alphabetical indices, labelling the numbers of the spaces, are conventionally changed to the numerical indices, such change we will make frequently.
(450,300) (0,0)
(450,300) (0,100)[(1,0)[350]{}]{} (50,50)(50,0)[6]{}[(0,1)[100]{}]{} (50,150)(100,0)[3]{}[(1,2)[50]{}]{} (100,150)(100,0)[3]{}[(-1,2)[50]{}]{} (60,110)[$p$]{}(110,110)[$q$]{} (160,110)[$p$]{}(210,110)[$q$]{} (260,110)[$p$]{}(310,110)[$q$]{} (45,30)[$\sigma_1$]{}(95,260)[$\sigma_1'$]{} (95,30)[$\sigma_2$]{}(145,260)[$\sigma_2'$]{} (145,30)[$\sigma_3$]{}(195,260)[$\sigma_3'$]{} (195,30)[$\sigma_4$]{}(245,260)[$\sigma_4'$]{} (375,200)[$\ds \rightarrow \U({p\over q})$]{} (375,100)[$\ds \rightarrow T(p,q)$]{}
Most amusing thing is that all these really give a 3D $R$-matrix: Korepanov’s $R$-matrix. [@korepanov-diss; @korepanov]. Korepanov’s $R$-matrix as well as Hietarinta’s one are some special cases of more general, complete $R$-matrix derived by Sergeev, Mangazeev and Stroganov [@mss-vertex], and complete $R$-matrix is equivalent to Zamolodchikov – Bazhanov – Baxter’s weights in the thermodynamic limit.
3D integrability: functional approach
-------------------------------------
A way to get something else in 3D is to refuse the finite number of states in the previous approach. Namely, 3D models appear in the local Yang – Baxter equation (LYBE) approach. LYBE (i.e. a Yang – Baxter equation with different “spectral” parameters in the left and right hand sides) can be adapted to a discrete space – time evolution of the triangulated two dimensional oriented surface as a kind of zero curvature condition [@maillet-nijhoff-talk; @maillet-nijhoff; @maillet].
In few words, if a matrix $L_{i,j}(\x)$, acting as usual in a tensor product of two finite dimensional spaces labelled by numbers $i$ and $j$, with some fixed functional structure and depending on a set of parameters $\x$, obeys the equation $$\label{lybe}
\ds
L_{1,2}(\x_a)\; L_{1,3}(\x_b)\; L_{2,3}(\x_c)\;\;=\;\;
L_{2,3}(\x_c')\;L_{1,3}(\x_b')\;L_{1,2}(\x_a')\;,$$ called the local Yang–Baxter equation, so that parameters $\x_a$, $\x_b$ and $\x_c$ are independent and $\x_a'$, $\x_b'$ and $\x_c'$ can be restored from (\[lybe\]) without any ambiguity, $$\label{fun-map}
\ds
\x_a'\;=\;f_a(\x_a,\x_b,\x_c)\;,\;\;\;
\x_b'\;=\;f_b(\x_a,\x_b,\x_c)\;,\;\;\;
\x_c'\;=\;f_c(\x_a,\x_b,\x_c)\;,$$ then the functional map $\R$ is introduced: $$\ds
\R_{a,b,c}^{}\*\varphi(\x_a,\x_b,\x_c)\*\R_{a,b,c}^{-1}\;=\;
\varphi(\x_a',\x_b',\x_c')\;\;\;\;\forall\;\;\;\varphi(...)\;.$$ Due to the difference of the “spectral” parameters in the left and right hand sides of LYBE, any shift of a line of a two dimensional lattice, constructed with a help of $L_{i,j}(\x_{i,j})$, changes the set of parameters $\x_{i,j}$. Partially, if any shift of the lines can be decomposed into primitive shifts like (\[lybe\]) in different ways, then corresponding different products of $\R$-s coincide. The basic example of this is the functional Tetrahedron equation.
Suppose we move all the lines of the lattice in some regular way, conserving a structure of the lattice. Then the change of parameters $\x_{i,j}$ can be considered as an one-step evolution of the dynamical variables $\x_{i,j}$ governed by an appropriately defined evolution operator $\U\;=\;\prod_{\mbox{\scriptsize
triangles}}\;\R$. This evolution is integrable due to uniqueness of LYBE and (\[fun-map\]). The partition function for the lattice becomes the natural integral of motion. In terms of the transfer matrices, the partition function is the $T$-type transfer matrix. Being functional, these $\R$-operators correspond to something infinitely dimensional. Contrary to the previous finite dimensional $R$-matrices, there are known a lot of such $\R$-operators. The reader can find an interesting set of such simplest functional $\R$-s in [@oneparam].
A quantization of known functional $\R$-s is still open problem. Simplest functional $\R$-s are to be regarded as some functional limits of multivariable $\R$-s with a symplectic structure conserving. The problem is to rise known $\R$-s to the complete phase space case, this is done just for a couple of $\R$-s.
3D integrability: general concept of evolution
----------------------------------------------
Here we discuss, what else can be invented to get a 3D integrability.
The main observation is that the relations like tetrahedral Zamolodchikov algebra and LYBE have usual graphical interpretation as the equality of the objects assigned to two similar graphs. These graphs are the triangles, and we will denote them briefly as $\lgr$ and $\rgr$. Left hand side type graph $\lgr$ corresponds to a product like $L_{1,2}\;L_{1,3}\;L_{2,3}$, and right hand side type graph $\rgr$ corresponds to $L_{2,3}\;L_{1,3}\;L_{1,2}$. Algebraic objects are assigned to the elements of these graphs. In the case of LYBE these algebraical objects are matrix $L$ with the indices assigned to the edges (in the form of subscript, for $L_{1,2}$ $1$ and $2$ stand for the edges), and parameter $\x$ assigned to the vertex. From 3D point of view $\x$-s are the dynamical variables, whereas $L$ and its indices are auxiliary objects. The equality of l.h.s. of LYBE and r.h.s. of LYBE gives the notion of [**the algebraic equivalence**]{} of $\lgr$ and $\rgr$. Note, this form of the algebraic equivalence is [**not obligatory**]{} !
[**3D integrability we can get from any other decent definition of an algebraic equivalence.**]{}
In this paper we consider a system, associated with a set of equivalent planar graphs. We propose another notion of an algebraic equivalence of equivalent graphs.
We will deal with all elements of the cw-complex, so we start from recalling the relationship between the elements of a planar graph and repeating some definitions.
Consider a graph $G_n$ formed by $n$ straight intersecting lines. The elements of its cw-complex are the vertices, the edges and the sites. $G_n$ consists on $\ds N_V\;=\;{n(n-1)\over 2}$ vertices, $\ds N_S\;=\;{(n-1)(n-2)\over 2}$ closed inner sites and $\ds
N_S^*\;=\;2 n $ outer open sites, $\ds N_E\;=\;n(n-2)$ closed inner edges and $\ds N_E^*\;=\; 2 n$ outer edges. If two graphs $G_n$ and $G_n'$ have the same outer structure, i.e. $G_n'$ can be obtained from $G_n$ by appropriate shift of the lines, then call $G_n'$ and $G_n$ equivalent.
Suppose we assign to the elements of a graph some elementary algebraic (maybe, the term “arithmetical” is more exact) objects. These objects are divided into two classes: dynamical variables and auxiliary objects (see the interpretation of $L_{1,2}(\x)$ above in this subsection). Dynamical variables are parameters of graph $G_n$, and auxiliary objects give some two dimensional rules of a game (like the summation over all intermediate indices in the product of $L$-s). Dynamical variables [*plus*]{} a rule of game give an algebraic object corresponding to the whole graph (like the partition function for $L$-s). This algebraic (arithmetical) object for whole $G_n$ we call the observable object. Denote it $O(G_n)$. It depends on the set of the dynamical variables.
Consider now two equivalent graphs, $G_n$ and $G_n'$. The algebraic problem of the equivalence arises, $$\label{oo}
\ds O(G_n)\;=\;O(G_n')\;.$$ If, according to the two dimensional rules of the game, we can get (\[oo\]) choosing the dynamical variables for $G_n'$ appropriately for the variables of $G_n$ given, then the algebraic equivalence makes a sense. If, moreover, parameters of $G_n'$ can be restored from the algebraic equivalence condition (\[oo\]) without any ambiguity, then this equivalence is decent and the integrability is undoubted.
The algebraic equivalence usually called zero curvature, and LYBE as the zero curvature condition as well as functional evolution models was considered in [@maillet-nijhoff-talk; @maillet-nijhoff; @maillet]. Another formulation of the algebraic equivalence, different to the LYBE approach, is Korepanov’s matrix model (see [@korepanov-diss; @kks-fte] and references therein). The formulation of the matrix model differs from the usual assigning the vertex – type Boltsmann weights to the vertices of a lattice, but functional evolution models probably are the same.
[**We chose another way.**]{}
The method we use was formulated originally in [@electric], the classical (i.e. functional) evolution model was described in [@3d-symp], and the quasiclassical case was investigated in [@ks-boson]. This paper contains the overview of the method, and the description of the quantum evolution model. The main new result is the generating function for integrals of motion for this evolution.
Auxiliary Linear Problem
========================
In this section we give some rules allowing one to assign an algebraic system to a graph. The elements to which we assign something are vertices and sites. First, we give most general rules, which do not give an algebraic equivalence of equivalent graphs in general, due to a sort of a “gauge ambiguity”. As a special case we find the rules which contain not a gauge ambiguity, and so a notion of an algebraic equivalence will be introduced. Then we describe the map of the dynamical variables given by the equivalence of 2-simplices, and discuss other similar approaches giving this map.
General approach
----------------
Choose as a game the following rules:
- Assign to each oriented vertex $V$ an auxiliary “internal current” $\phi$. Suppose this current produces four “site currents” flowing from the vertex into four adjacent faces, and proportional to the internal current with some coefficients $\a,\b,\c,\d$, called the dynamical variables, as it is shown in Fig. \[fig-abcd-currents\]. All this variables, $\phi$ and $\a,...,\d$ for different vertices are independent for a while.
- Define the complete site current as an algebraic sum of the contributions of vertices surrounding this site.
- For any closed site of a lattice let its complete current is zero. Such zero relations we regard as the linear equations for the internal currents.
- For any graph $G_n$ the site currents assigned to outer (open) sites we call the “observable currents”. In part, two equivalent graphs $G_n$ and $G_n'$ must have the same observable currents – this is the algebraic meaning of the equivalence.
(450,200) (125,0)
(200,200) ( 0 , 0 )[( 1,1)[200]{}]{} ( 200 , 0 )[(-1,1)[200]{}]{} ( 100 , 100 ) ( 85 , 160 )[$\b\*\phi$]{} ( 85 , 30 )[$\c\*\phi$]{} ( 25 , 95 )[$\a\*\phi$]{} ( 145 , 95 )[$\d\*\phi$]{}
Clarify these rules on the example of equivalence of $G_3$. As it was mentioned, this is usual Yang – Baxter equivalence graphically, $\lgr=\rgr$, shown in Fig. \[fig-YBE\]. Assign to the vertices $W_j$ of the left hand side graph $\lgr$ the currents $\phi_j$ and the dynamical variables $\a_j,\b_j,\c_j,\d_j$, and to the vertices $W_j'$ of the right hand side graph $\rgr$ – the currents $\phi_j'$ and the dynamical variables $\a_j',\b_j',\c_j',\d_j'$. Six currents of outer sites denote as $\phi_b,...,\phi_g$, and two zero valued currents of closed sites – as $\phi_h$ and $\phi_a$ as it is shown in Fig. \[fig-YBE\]. Then, using the rules described above, we obtain the following system of eight linear (with respect to the currents) relations:
(450,200) (00,00)
(200,200) ( 10 , 70 )[(1,0)[180]{}]{} ( 40 , 10 )[(1,2)[90]{}]{} ( 160 , 10 )[(-1,2)[90]{}]{} ( 70 , 70 )(50,80)[$W_3$]{} ( 130 , 70 )(135,80)[$W_2$]{} ( 100 , 130 )(110,125)[$W_1$]{} ( 95 , 90 )[$\phi_h$]{} ( 95 , 190 )[$\phi_e$]{} ( 95 , 20 )[$\phi_b$]{} ( 35 , 120 )[$\phi_c$]{} ( 155 , 120 )[$\phi_d$]{} ( 0 , 20 )[$\phi_g$]{} ( 185 , 20 )[$\phi_f$]{}
(250,00)
(200,200) ( 10 , 130 )[(1,0)[180]{}]{} ( 70 , 10 )[(1,2)[90]{}]{} ( 130 , 10 )[(-1,2)[90]{}]{} ( 70 , 130 )(35,140)[$W'_2$]{} ( 130 , 130 )(145,140)[$W'_3$]{} ( 100 , 70 )(110,65)[$W'_1$]{} ( 95 , 105 )[$\phi_a$]{} ( 95 , 185 )[$\phi_e$]{} ( 95 , 20 )[$\phi_b$]{} ( 0 , 175 )[$\phi_c$]{} ( 185 , 175 )[$\phi_d$]{} ( 30 , 60 )[$\phi_g$]{} ( 160 , 60 )[$\phi_f$]{}
(210,90)
(30,50) (10,10)[$\sim $]{}
$$\label{phi-h}
\ds \phi_h\;\equiv\;\c_1^{}\*\phi_1^{}\;+\;\a_2^{}\*\phi_2^{}\;+\;
\b_3^{}\*\phi_3^{}\;=\;0\;,$$
$$\label{phi-bcd}
\ds\left\{\begin{array}{ccccc}
\ds\phi_b & \equiv &\ds\c_1'\*\phi_1' & = &
\ds\c_2^{}\*\phi_2\;+\;\d_3^{}\*\phi_3^{}\;,\\
&&\\
\ds\phi_c & \equiv &\ds\a_2'\*\phi_2' & = &
\ds\a_1^{}\*\phi_1\;+\;\a_3^{}\*\phi_3^{}\;,\\
&&\\
\ds\phi_d & \equiv &\ds\b_3'\*\phi_3' & = &
\ds\d_1^{}\*\phi_1\;+\;\b_2^{}\*\phi_2^{}\;,
\end{array}\right.$$
$$\label{phi-efg}
\ds\left\{\begin{array}{ccccc}
\ds\phi_e & \equiv &
\ds\b_2'\*\phi_2'\;+\;\a_3'\*\phi_3' & = & \b_1^{}\*\phi_1^{}\;,\\
&&\\
\ds\phi_f & \equiv &
\ds\d_1'\*\phi_1'\;+\;\d_3'\*\phi_3' & = & \d_2^{}\*\phi_2^{}\;,\\
&&\\
\ds\phi_g & \equiv &
\ds\a_1'\*\phi_1'\;+\;\c_2'\*\phi_2' & = & \c_3^{}\*\phi_3^{}\;,
\end{array}\right.$$
$$\label{phi-a}
\ds \phi_a \;\equiv\;\b_1'\*\phi_1'\;+\;\d_2'\*\phi_2'\;+\;
\c_3'\*\phi_3'\;=\;0\;.$$
Given are the currents and the dynamical variables for the left hand side graph. Due to $\phi_h=0$, eq. (\[phi-h\]), only two currents are independent, let them be $\phi_1$ and $\phi_3$. All the variables for the right hand side graph we try to restore via the linear system. First, use $\phi_b$, $\phi_c$ and $\phi_d$ (\[phi-bcd\]) to express all $\phi_j'$. Substitute $\phi_j'$ into relations for $\phi_e$, $\phi_f$ and $\phi_g$ (\[phi-efg\]), then it will appear three homogeneous linear relations for two arbitrary $\phi_1$ and $\phi_3$, so six coefficients of $\phi_1$ and $\phi_3$ must vanish. Solving this six equations with respect to the primed variables, we obtain $$\ds\left.
\begin{array}{cc}
\ds\b_2'\;\a_2^{\prime-1}\;=\;
\Lambda_1^{-1}\*\b_3^{}\;\a_3^{-1}\;,&
\ds\a_3'\;\b_3^{\prime-1}\;=\;
\Lambda_1^{-1}\*\a_2^{}\;\b_2^{-1}\;,\\
&\\
\ds\d_1'\;\c_1^{\prime-1}\;=\;
\Lambda_2^{-1}\*\b_3^{}\;\d_3^{-1}\;,&
\ds\d_3'\;\b_3^{\prime-1}\;=\;
\Lambda_2^{-1}\*\c_1^{}\;\d_1^{-1}\;,\\
&\\
\ds\a_1'\;\c_1^{\prime-1}\;=\;
\Lambda_3^{-1}\*\a_2^{}\;\c_2^{-1}\;,&
\ds\c_2'\;\a_2^{\prime-1}\;=\;
\Lambda_3^{-1}\*\c_1^{}\;\a_1^{-1}\;,
\end{array}\right.
\label{raz}$$ where three polynomials arisen: $$\label{lambda-general}
\ds\left.\begin{array}{ccc}
\ds \Lambda_1^{} & = &
\ds\b_3^{}\;\a_3^{-1}\;\a_1^{}\;\b_1^{-1}\;-\;
\c_1^{}\;\b_1^{-1}\;+\;\a_2^{}\;\b_2^{-1}\;
\d_1^{}\;\b_1^{-1}\;,\\
&&\\
\ds \Lambda_2^{} & = &
\ds\b_3^{}\;\d_3^{-1}\;\c_2^{}\;\d_2^{-1}\;-\;
\a_2^{}\;\d_2^{-1}\;+\;\c_1^{}\;
\d_1^{-1}\;\b_2^{}\;\d_2^{-1}\;,\\
&&\\
\ds \Lambda_3^{} & = &
\ds\a_2^{}\;\c_2^{-1}\;\d_3^{}\;\c_3^{-1}\;-\;
\b_3^{}\;\c_3^{-1}\;+\;\c_1^{}\;
\a_1^{-1}\;\a_3^{}\;\c_3^{-1}\;.
\end{array}\right.$$ Substituting $\phi_j'$ into $\phi_a=0$ (\[phi-a\]), we obtain the homogeneous linear equation for $\phi_1$, $\phi_3$ again, and the coefficients of them vanish if $$\label{dva}
\ds\left.\begin{array}{ccc}
\ds\b_1'\;\c_1^{\prime-1} & = & \ds\Lambda_a\;\Lambda_1\;\bigl(
\c_2^{}\;\b_2^{-1}\;\d_1^{}\;\b_1^{-1}\;+\;
\d_3^{}\;\a_3^{-1}\;\a_1^{}\;\b_1^{-1}\bigr)^{-1}\;,\\
&&\\
\ds\d_2'\;\a_2^{\prime-1} & = & \ds\Lambda_a\;\Lambda_2\;\bigl(
\a_1^{}\;\d_1^{-1}\;\b_2^{}\;\d_2^{-1}\;+\;
\a_3^{}\;\d_3^{-1}\;\c_2^{}\;\d_2^{-1}\bigr)^{-1}\;,\\
&&\\
\ds\c_3'\;\b_3^{\prime-1} & = & \ds\Lambda_a\;\Lambda_3\;\bigl(
\d_1^{}\;\a_1^{-1}\;\a_3^{}\;\c_3^{-1}\;+\;
\b_2^{}\;\c_2^{-1}\;\d_3^{}\;\c_3^{-1}\bigr)^{-1}\;,
\end{array}\right.$$ where $\Lambda_a$ is arbitrary. The origin of $\Lambda_a$ technically is $\phi_a\;=\;\Lambda_a\*\phi_h$.
This $\Lambda_a$ is a sort of a gauge. The origin of it is that due to $\phi_a\;\equiv\;0$ we may change it $\phi_a\mapsto\lambda_a\phi_a$, this gives $\Lambda_a\mapsto\lambda_a\Lambda_a$, or equivalent $$\label{lambda-a}
\ds
\b_1'\mapsto\lambda_a\,\b_1'\;,\;\;\;
\d_2'\mapsto\lambda_a\,\d_2'\;,\;\;\;
\c_3'\mapsto\lambda_a\,\c_3'\;.$$ Analogous degree of freedom is lost in the map $W_1,W_2,W_3$ $\mapsto$ $W_1',W_2',W_3'$: the system of the observables is not changed when $\phi_h\mapsto\lambda_h\phi_h$, i.e. when $$\label{lambda-h}
\ds
\c_1\mapsto\lambda_h\,\c_1\;,\;\;\;
\a_2\mapsto\lambda_h\,\a_2\;,\;\;\;
\b_3\mapsto\lambda_h\,\b_3\;,$$ and the formulae for $W_j'$ do not change with (\[lambda-h\]). Call such type invariance of the system of the observables [**the site projective invariance**]{} (correspondingly, the site ambiguity of the dynamical variables).
The other obvious invariance (ambiguity) is [**the vertex projective**]{} one. As the consequence of simple re-scaling of the currents almost nothing changes if $$\label{projective}
\ds
\a\mapsto\a\,\lambda\;,\;\;\;
\b\mapsto\b\,\lambda\;,\;\;\;
\c\mapsto\c\,\lambda\;,\;\;\;
\d\mapsto\d\,\lambda\;$$ partially in all vertices $W_j$ and $W_j'$ with six different $\lambda_j$ and $\lambda_j'$.
Thus in the most general interpretation: the map $W_1,W_2,W_3$ $\mapsto$ $W_1',W_2',W_3'$ is defined up to projective ambiguity $\lambda_1,\lambda_2,\lambda_3,\lambda_h$ $\mapsto$ $\lambda_1',\lambda_2',\lambda_3',\lambda_a$.
Very important feature of all these calculations is that
Return to a general case of graph $G_n$. $4\,N_V\;=\;2\,n\,(n-1)$ free invertible variables $\a_V,\b_V,\c_V,\d_V$, assigned to the vertices $V$ of $G_n$, we regard as the generators of a body ${\cal B}(G_n)$. Let ${\cal B}_P(G_n)$ be the set of functions on ${\cal B}(G_n)$ invariant with respect to the vertex ambiguity (\[projective\]). Note in general, for an open graph $G_n$ one may consider ${\cal B}'_P(G_n)$ – set of functions invariant with respect to both vertex and closed site ambiguities. But such general considerations of ${\cal B}'_P$ for the closed graphs, i.e. the graphs defined on the torus, needs a notion of a trace (or of a characteristic polynomials), or equivalent, of an algebra. The algebra will be introduced in the subsequent section.
Consider a little change of $G_n$, so that only one $\lgr$ in $G_n$ transforms into $\rgr$. Call the resulting graph $G_n'$. Let the vertices involved into this change are marked as $W_1,W_2,W_3$ for $\lgr$ and $W_1',W_2',W_3'$ for $\rgr$ arranged as in Fig. \[fig-YBE\]. Introduce a functional operator $\R\;=\;\R_{1,2,3}$ making the corresponding map on ${\cal B}_P$: $$\label{themap}
\ds
\R_{1,2,3}^{}\cdot\varphi(W_1,W_2,W_3,...)\cdot
\R_{1,2,3}^{-1}\;=\;
\varphi(W_1',W_2',W_3',...)\;,\;\;\;\;
\varphi\in{\cal B}_P\;,$$ where $W_j$ stands for $\{\a_j,\b_j,\c_j,\d_j\}$ forever, and all other vertices except $W_1,W_2,W_3$ and their variables remain untouched. This $\R$ we call the [**fundamental map**]{}.
Let now $G_n'$ be an arbitrary graph equivalent to $G_n$. $G_n'$ can be obtained from $G_n$ by different sequences of elementary $\lgr\mapsto\rgr$ in general. Thus the corresponding different sequences of $\R$-s must coincide, this is natural admissibility (or associability) condition for $G_n\mapsto G_n'$.
Note that in terms of functional operators the sequence of naïve geometrical transformations is antihomomorphic to the sequence of corresponding functional maps.
The simplest case is the equivalence of two quadrilaterals, $G_4$, and the admissibility condition is nothing but the Tetrahedron equation (\[te\]). And due to the ambiguity of $\R$, (\[lambda-a\],\[lambda-h\]), any admissibility condition is still equation for $\Lambda_a$-th involved. Note, ${\cal
B}_P'(G_n)$ introduced previously, is gauge invariant subspace of ${\cal B}_P$. $\R$ acts on ${\cal B}_P'$ uniquely. Unfortunately the basis of ${\cal B}_P'$ is not local, and it is simpler to introduce an algebra constraints removing the projective ambiguities then to consider ${\cal B}'_P$ formally.
[**A way to remove $\Lambda_a$ – ambiguity from the definition of $\R$, (\[raz\],\[dva\]), is to impose some additional conditions for the elements of $W$, $\a,\b,\c,\d$, such that (\[dva\]) would become a consequence of (\[raz\]) and the additional conditions.**]{}
Complete classification of these additional conditions is still the open problem, and this is the main mathematical problem of this approach.
Local case: the Weyl algebra
----------------------------
Here we consider a special [*local*]{} case: suppose first that the elements of two different $W_i$ and $W_j$ for the given $G_n$ commute. Destroy also the vertex projective invariance choosing $\a_j\equiv 1$ for any $j$ forever. Then (\[raz\]) give the expressions for $\b_2',\b_3'$, $\d_1'\c_1^{\prime-1},\d_3'\b_3^{\prime-1}$, $\c_1',\c_2'$. Suppose also any pair of the variables from $W$ are linearly independent, then
- the commutability of the elements for different $W_j'$ from $\rgr$ gives (after some calculations) $\b\c=q\c\b$ with the same ${\cal C}$-number $q$ for any vertex,
- these relations conserve by the map $\R$, i.e. $\b'\,\c'\;=\;q\;\c'\,\b'$.
- Also $\b^{-1}\c^{-1}\d$ appear to be centres, depending on the vertex.
The gauge ambiguity becomes the ambiguity for these centres. We are going to get a sort of quantum theory, $\b$ and $\c$ are already quantized, so we have to keep all centres to be invariant, $\b_j^{-1}\c_j^{-1}\d_j^{}\;=\;\b_j^{\prime-1}
\c_j^{\prime-1}\d_j^\prime$. This is possible, and further we will threat these centres as a kind of spectral parameters.
Change now notations for the dynamical variables to more conventional, and write down the resulting expressions for the map $\R$. New notations for the site currents are shown in Fig.\[fig-weyl-vertex\].
(450,200) (125,0)
(200,200) ( 0 , 0 )[( 1,1)[200]{}]{} ( 200 , 0 )[(-1,1)[200]{}]{} ( 100 , 100 ) ( 70 , 160 )[$q^{1/2}\;\;\u\*\phi$]{} ( 90 , 30 )[$\w\*\phi$]{} ( 25 , 95 )[$\phi$]{} ( 135 , 95 )[$\kappa\;\;\u\*\w\*\phi$]{}
[**Proposition.**]{} $\bullet$ Let the vertex dynamical variables are given by $$\ds\a\;=\;1\;,\;\;\;\;
\b\;=\;q^{1/2}\;\u\;,\;\;\;
\c\;=\;\w\;,\;\;\;\;
\d\;=\;\kappa\;\u\;\w\;,$$ here $\u,\w$ obey the local Weyl algebra relation, $$\ds \u\*\w\;\;=\;\;q\;\;\w\*\u\;,$$ and $\u$ and $\w$ for different vertices commute, and number $\kappa$ is the invariant of the vertex, i.e. $\kappa_{i,j}$, assigned to the intersection of lines $i$ and $j$, is the same for all equivalent graphs.
Then the problem of the algebraic equivalence (i.e. equality of the outer currents) of two graphs: $G$ with the data $\phi,\u,\w$, and $G'$ with the data $\phi',\u',\w'$, can be solved [**without any ambiguity**]{} with respect to all $\phi',\u',\w'$, and the local Weyl algebra structure for the set of $\u',\w'$ is the consequence of the local Weyl algebra relations for the set of $\u,\w$. $\bullet$
Write the fundamental simplex map for $\lgr=\rgr$ explicitly. The map $\R=\R_{1,2,3}$ : $W_1,W_2,W_3$ $\mapsto$ $W_1',W_2',W_3'$, $$\label{R-action}
\R\*\u_j\;=\;\u_j'\*\R\;,\;\;\;
\R\*\w_j\;=\;\w_j'\*\R\;,\;\;\;j=1,2,3\;,$$ is given by $$\fbox{$\;\;\;
\ds\left.\begin{array}{ll}
&\\
\ds\w_1'\;=\;\ds\w_2^{}\* \Lambda_3^{}\;, &
\ds\u_1'\;=\;\ds\Lambda_2^{-1}\* \w_3^{-1}\;,\\
&\\
\ds\w_2'\;=\;\ds\Lambda_3^{-1}\* \w_1^{}\;, &
\ds\u_2'\;=\;\ds\Lambda_1^{-1}\* \u_3^{}\;,\\
&\\
\ds\w_3'\;=\;\ds\Lambda_2^{-1}\* \u_1^{-1}\;, &
\ds\u_3'\;=\;\ds\u_2^{}\* \Lambda_1^{}\;,\\
&
\end{array}\right.
\;\;\;$}
\label{ev}$$ where $$\label{lambda}
\fbox{$\;\;\;
\ds\left.\begin{array}{ccl}
&&\\
\ds\Lambda_1 & = & \ds
\u_1^{-1}\*\u_3^{}\;-\;
q^{1/2}\;\u_1^{-1}\*\w_1^{}\;+\;
\kappa_1\;\;\w_1^{}\*\u_2^{-1}\;,\\
&&\\
\ds\Lambda_2 & = & \ds
{\kappa_1\over\kappa_2}\;\;\u_2^{-1}\*\w_3^{-1}\;+\;
{\kappa_3\over\kappa_2}\;\;\u_1^{-1}\*\w_2^{-1}\;-\;
q^{-1/2}\;\;{\kappa_1\;\kappa_3\over\kappa_2}\;\;
\u_2^{-1}\*\w_2^{-1}\;,\\
&&\\
\ds\Lambda_3 & = & \ds
\w_1^{}\*\w_3^{-1}\;-\;
q^{1/2}\;\;\u_3^{}\*\w_3^{-1}\;+\;
\kappa_3\;\;\w_2^{-1}\*\u_3\;.\\
&&
\end{array}\right.
\;\;\;$}$$ Reverse formulae, giving $\R^{-1}$, look similar: $$\ds
\left.\begin{array}{ccl}
\ds \Lambda_1^{-1} & = & \ds
{\kappa_1\over\kappa_2}\;\;\u_1^{\prime}\*\u_3^{\prime-1}\;-\;
q^{1/2}\;\;{\kappa_3\over\kappa_2}\;\;\u_1^{\prime}\*\w_1^{\prime-1}
\;+\;\kappa_3\;\;\w_1^{\prime-1}\*\u_2^{\prime}\;,\\
&&\\
\ds \Lambda_2^{-1} & = & \ds
\u_2^\prime\*\w_3^{\prime}\;+\;\u_1^\prime\*\w_2^\prime
\;-\;q^{-1/2}\;\;\kappa_2\;\;\u_2^\prime\*\w_2^\prime\;,\\
&&\\
\ds \Lambda_3^{-1} & = & \ds
{\kappa_3\over\kappa_2}\;\;\w_1^{\prime-1}\*\w_3^{\prime}
\;-\;q^{1/2}\;\;
{\kappa_1\over\kappa_2}\;\;\u_3^{\prime-1}\*\w_3^{\prime}
\;+\;\kappa_1\;\;\w_2^{\prime}\*\u_3^{\prime-1}\;.
\end{array}\right.$$
The conservation of the Weyl algebra structure $$\ds
\u_j\*\w_j\;=\;q\;\;\w_j\*\u_j\;\;\;\mapsto\;\;\;
\u_j'\*\w_j'\;=\;q\;\;\w_j'\*\u_j'$$ means that $\R$ is the canonical map, hence $\R_{1,2,3}$ can be regarded as an usual operator depending on $\u_1,\w_1,\u_2,\w_2,\u_3,\w_3$. The structure of $\R$ will be described in the next subsection.
Now the projective ambiguity is removed, and the current system game gives the unique correspondence between the elements of equivalent graphs. This is well defined meaning of the algebraic equivalence. Hence all the admissibility conditions (and surely the Tetrahedron relation) become trivial consequences of this umambiguity, and we get them gratis !
Mention now a couple of useful limits of our fundamental map $\R_{1,2,3}$. The first one is the limit when $\kappa_1=\kappa_2=\kappa_3=\kappa$, and then $\kappa\mapsto 0$. Denote such limiting procedure via $$\label{planar_conditions}
\ds
\kappa_1\;=\;\kappa_2\;=\;\kappa_3\;\;<<\;\;1\;.$$ Corresponding map we denote $\R^{pl}_{1,2,3}$. The conditions for $\kappa$-s are uniform for whole Tetrahedron relation, $$\ds
\kappa_1\;=\;\kappa_2\;=\;\kappa_3\;=\;\kappa_4\;=\;\kappa_5\;=\;
\kappa_6\;\;<<\;\;1\;,$$ so $\R^{pl}$ obeys TE. The other case is the limit of $\R_{1,2,3}$ when $$\label{one_conditions}
\ds
\kappa_1\;\;<<\;\;\kappa_2\;=\;\kappa_3\;\;<<\;\;1\;.$$ These conditions are uniform for TE again, $$\ds
\kappa_1\;\;<<\;\;\kappa_2\;=\;\kappa_3\;\;<<\;\;
\kappa_4\;=\;\kappa_5\;=\;\kappa_6\;\;<<\;\;1\;.$$ Corresponding map we call $\mbox{\sf r}_{1,2,3}$, and due to the uniformness it also obeys TE. Recall, all these maps, $\R$ with $\kappa_1\;=\;\kappa_2\;=\;\kappa_3\;=\;1$, $\R^{pl}$ and $\mbox{\sf r}$ were derived previously as a hierarchy of $\R$ – operators solving TE, see [@sbm-qd; @s-qd; @ks-fun; @ms-modified].
Structure of $\R$
-----------------
Remarkable feature of $\R$ is its spatial invariance. Change a little the operators on which $\R$ depends: $$\label{gammas}
\ds
\Gamma_1^{}\;=\;
\kappa_1^{-1}\;\;\u_2^{}\*\u_3^{-1}\*\Lambda_1^{}\;,\;\;\;\;
\Gamma_2^{}\;=\;
\kappa_2^{}\;\;\u_1^{}\*\w_3^{}\*\Lambda_2\;,\;\;\;\;
\Gamma_3^{}\;=\;
\kappa_3^{-1}\;\;\w_1^{-1}\*\w_2^{}\*\Lambda_3^{}\;.$$ Then for $\ds\alpha,\beta,\gamma$ being the cyclic permutations of $1,2,3$, $$\ds\begin{array}{crc}
&\ds(\Gamma_\beta\cdot\Gamma_\alpha\;-\;q\;
\Gamma_\alpha\cdot\Gamma_\beta)\cdot
\Gamma_\gamma\;-\;\Gamma_\gamma\cdot
(\Gamma_\beta\cdot\Gamma_\alpha\;-\;q\;
\Gamma_\alpha\cdot\Gamma_\beta)& \\
&&\\
&\ds -\; q^{-1}\;(1-q)\;(1-q^2)\;
(\Gamma_\alpha\;-\;\Gamma_\beta) & =\;0\;,
\end{array}$$ and $$\ds\begin{array}{crc}
& \ds
q\;\Gamma_\alpha\cdot\Gamma_\beta\;-\;q^{-1}\;
\Gamma_\beta\cdot\Gamma_\alpha\;-\;
\Gamma_\gamma\;(q^{1/2}\;
\Gamma_\alpha\cdot\Gamma_\beta\;-\;q^{-1/2}\;
\Gamma_\beta\cdot\Gamma_\alpha) &\\
&&\\
& \ds +\;q^{-1/2}\;(1-q)\;(q^{-1}\;\Gamma_\alpha\;+\;q\;
\Gamma_\beta\;-\;\Gamma_\gamma) & =\;0\;.
\end{array}$$ It resembles $SO(3)$ invariance.
Give now a realisation of $\R$ in terms of more simple functions. First, recall the definition and properties of the quantum dilogarithm. Let conventionally $$\ds (\x;q)_n\;=\;(1-\x)\;(1-q\x)\;(1-q^2\x)\;...\;(1-q^{n-1}\x)\;.$$ Then the quantum dilogarithm (by definition) [@fk-qd; @br-qd] $$\ds
\psi(\x)\;\stackrel{def}{=}\; (q^{1/2}\x;q)_\infty\;=\;
\sum_{n=0}^\infty\;\; {(-1)^n\,q^{n^2/2}\over (q;q)_n}\;\x^n\;,$$ and $$\ds\psi(\x)^{-1}\;=\;\sum_{n=0}^\infty\;\;{q^{n/2}\over
(q;q)_n}\;x^n\;.$$ This function is useful for the rational transformations of the Weyl algebra: $$\ds
\psi(q\x)\;=\;(1-q^{1/2}\x)^{-1}\;\psi(\x)\;,\;\;\;\;
\psi(q^{-1}\x)\;=\;(1-q^{-1/2}\x)\;\psi(\x)\;,$$ hence $$\ds
\psi(\u)\*\w\;=\;\w\*(1-q^{1/2}\u)^{-1}\*\psi(\u)\;,\;\;\;\;
\psi(\w)\*\u\;=\;\u\*(1-q^{-1/2}\w)\*\psi(\w)\;.$$ $\psi$ is called the quantum dilogarithm due to the pentagon identity [@fk-qd] $$\ds
\psi(\w)\*\psi(\u)\;=\;\psi(\u)\*\psi(-q^{-1/2}\;\u\;\w)\*\psi(\w)\;,$$ this corresponds to Roger’s five term relation for the usual dilogarithm. From the other side $\psi$ is the quantum exponent due to $$\ds \psi(\u)\*\psi(\w)\;=\;\psi(\u+\w)\;.$$ Recall, everywhere the Weyl algebra relation $\u\,\w\;=\;q\,\w\,\u$ is implied.
Introduce now a generalised permutation function. Let $\P(\x,\y)$, $\x\,\cdot\,\y\;=\;q^2\;\y\,\cdot\,\x$, is defined by the following relations: $$\ds\left\{\begin{array}{ccccc}
\ds\P(q\;\x,\y) & = & \ds\y^{-1}\;\P(\x,\y) & = & \ds\P(\x,\y)\;\y\;,\\
&&&&\\
\ds\P(\x,q\;\y) & = & \ds\P(\x,\y)\;\x^{-1} & = & \ds\x\;\P(\x,\y)\;,
\end{array}\right.$$ and $$\ds\P(\x,\y)^2\;=\;1\;.$$ For $\z$ obeying $$\ds
\x\*\z\;=\;q^{f_x}\;\z\*\x\;,\;\;\;\;
\y\*\z\;=\;q^{f_y}\;\z\*\y$$ it follows $$\ds
\P(\x,\y)\*\z\;=\;q^{f_x\,f_y}\;\;\z\*\x^{f_y}\*\y^{-f_x}\*\P(\x,\y)\;.$$ This function we call the generalised permutation because of usual permutation operator of the tensor product is $$\ds
\P\;\equiv\;\P(\u\otimes\u^{-1},\w\otimes\w^{-1})\;\;.$$
Considering three independent $\Gamma_\alpha$ (\[gammas\]), $\alpha=1,2,3$, one may see that all them depends on three operators $\U$, $\W$ and $\s$: $$\ds
\U\;=\;\w_{2}^{-1}\*\w_{3}^{}\;,\;\;\;\;
\W\;=\;\w_{1}^{}\*\u_{3}^{-1}\;,\;\;\;\;
-\;q^{1/2}\;\;\s\*\U\*\W^{-1}\;=\;\u_{1}^{}\*\u_{2}^{-1}\;.$$ $\U\;\W\;=\;q\;\;\W\;\U$ and $\s$ is the center. One can directly verify that $$\label{R-op}
\ds
\fbox{$\ds
\R\;=\;\psi(\kappa_3\;\U)\*\psi(\W^{-1})\*
\P\bigl(\sqrt{\kappa_3\over\kappa_2}\;\;\U\;,\;\s^{-1}\*\W^{\,2}\bigr)\*
\psi({\kappa_1\over\kappa_3}\;\W)^{-1}\*
\psi(\kappa_2\;\U^{-1})^{-1}\;,
$}$$ being substituted into (\[R-action\]), gives (\[ev\],\[lambda\]). On $\U$ and $\W$ , $\R$ acts as follows. $$\ds\left\{\begin{array}{ccl}
\ds\R\*\U\*\R^{-1} & = & \ds
{\kappa_2\over\kappa_3}\;\;\U^{-1}\*
\bigl(\;\W\;-\;q^{-1/2}\;+\;\kappa_3\;\;\U\;\bigr)\*\\
&&\\
&&\ds\*
\bigl(\;\W\;-\;q^{-1/2}\;\;{\kappa_1\over\kappa_3}\;\;\s\;+\;
\kappa_1\;\;\;\s\*\U\;\bigr)^{-1}\;,\\
&&\\
\ds\R\*\W\*\R^{-1} & = & \ds
\s\*\W^{-1}\*
\bigl(\;\W\;-\;q^{1/2}\;+\;\kappa_3\;\;\U\;\bigr)\*
\bigl(\;\W\;-\;q^{1/2}\;+\;\kappa_1\;\;\s\*\U\;\bigr)^{-1}\;.
\end{array}\right.$$
When $\kappa_1=\kappa_2=\kappa_3=1$, expression (\[R-op\]) for $\R$ coincides with the operator solution of the Tetrahedron equation from [@s-qd; @ms-modified]. This is the generalisation of the finite dimensional 3D $R$-matrix from $q^N=1$ to general $q$, and the finite dimensional $R$-matrix corresponds to the Zamolodchikov–Bazhanov–Baxter model.
We don’t discuss this correspondence here, the reader may find the details concerning Zamolodchikov – Bazhanov – Baxter model in [@zam-solution; @baxter-pf; @bb-first; @kms-stsq], the details concerning the finite $R$-matrix – in [@mss-vertex], the details concerning the quantum dilogarithm – in original papers [@fk-qd; @br-qd], and operator valued $\R$ as the generalisation of finite $R$ – in [@sbm-qd; @s-qd; @ms-modified; @double].
Few words concerning the meaning of (\[R-op\]). All $\psi$-s can be decomposed into the seria with respect to their arguments. Substitute these $\R$-s into the Tetrahedron relation (\[te\]) and move all the generalized permutations $\P$ out. $\P$-s theirself obey the Tetrahedron equation and so can be cancelled from TE for $\R$-s. Then twelve $\psi$-s rest in the left hand side of TE, and twelve $\psi$-s rest in the right hand side. The Tetrahedron equation in this case becomes a relation resembling the braid group relation in $2D$. This twenty-four terms relation can be proved [**directly**]{} via the seria decomposition of all $24$ quantum dilogarithms. The proof is based on several [**finite**]{} $q$-re-summations (like $q$-binomial theorems). This is the first value of the formula (\[R-op\]). The second one is that relation (\[R-op\]) gives a nice way to derive the finite dimensional complete $R$-matrix (just replacing $\psi$-s and $\P$ by their finite dimensional counterparts, [@br-qd; @s-qd; @ms-modified]).
Generalised permutation $\P(\x,\y)$ has no good series realization. Note, if we abolish condition $\P^2=1$ for a moment, then formally $$\ds
\P(\x,\y)\;\sim\;\sum_{\alpha,\beta\in Z}\;\;
q^{-\alpha\,\beta}\;\x^\alpha\*\y^\beta\;.$$ This $\P$ obeys $\P^2=1$ if one takes the Euler definition $\ds\sum_{n\in Z}\;\;q^{n\;m}\;=\;\delta_{m,0}$ [@euler; @struik]. Note that in the manipulations with $q$-seria the Euler principle “A sum of any infinite series is the value of an expression, which expansion gives this series” was never failed. Actually $\P(\x,\y)$ is to be defined specially for every realisation of the Weyl algebra. As an example mention Kashaev and Faddeev’s non invariant realisation of the Weyl generators as shifts on the space of appropriately defined functions $\varphi([t])$: $$\ds
\w_j\*\varphi([t])\;=\;[t]_j\;\phi([t])\;,\;\;\;\;
\u_j\*\phi([t])\;=\;-\;q^{-1/2}\;[t]_j\;
\varphi([t]:[t]_j\,\mapsto\,q^{-1}\,[t]_j)\;.$$ Where $[t]$ is a list of the arguments of $\varphi$ and $[t]_j$ is its $j$-th component. Thus $\u_j$ and $\w_j$ refer to the $j$-th “pointer” of the list of arguments and hence are not functional operators in usual sense. Actually the action of $\u_j$, $\w_j$ on $\varphi([t])$ would be given symbolically by the following correspondence: $$\ds
\varphi([t])\;\;\leftrightarrow\;\;
|t_1>\otimes |t_2>\otimes |t_3>\otimes\;...\;,$$ if the eigenvectors $|t_j>$ of the operators $\w_j$ might be defined.
Generalised permutation introduced $$\ds
\P_{1,2,3}\;=\;
P(\sqrt{\kappa_3\over\kappa_2}\;\U\;,\;\s^{-1}\,\W^2)\;=\;
\P(\sqrt{\kappa_3\over\kappa_2}\;\w_2^{-1}\,\w_3^{}\;,\;
-\,q^{-1/2}\,\u_1^{-1}\w_1^{}\,\w_2^{-1}\,\u_2^{}\,
\w_3^{}\u_3^{-1})$$ act of $\u_j$, $\w_j$, $j=1,2,3$, as follows: $$\ds\left\{
\begin{array}{ccl}
\ds\P_{1,2,3}\*\w_1 & = & \ds\sqrt{\kappa_2\over\kappa_3}\;
\w_1^{}\,\w_2^{}\,\w_3^{-1}\*\P_{1,2,3}\;,\\
&&\\
\ds\P_{1,2,3}\*\w_2 & = & \ds\sqrt{\kappa_3\over\kappa_2}\;
\w_3^{}\*\P_{1,2,3}\;,\\
&&\\
\ds\P_{1,2,3}\*\w_3 & = & \ds\sqrt{\kappa_2\over\kappa_3}\;
\w_2^{}\*\P_{1,2,3}\;,
\end{array}\right.$$ and $$\ds\left\{
\begin{array}{ccl}
\ds\P_{1,2,3}\*\u_1 & = & \ds\sqrt{\kappa_2\over\kappa_3}\;
\u_1^{}\,\w_2^{}\,\w_3^{-1}\*\P_{1,2,3}\;,\\
&&\\
\ds\P_{1,2,3}\*\u_2 & = &
\ds -\;q^{1/2}\;\sqrt{\kappa_3\over\kappa_2}\;
\u_1^{}\,\w_1^{-1}\,\u_3^{}\*\P_{1,2,3}\;,\\
&&\\
\ds\P_{1,2,3}\*\u_3 & = &
\ds-\;q^{1/2}\;\sqrt{\kappa_2\over\kappa_3}\;
\u_1^{-1}\,\w_1^{}\,\u_2^{}\*\P_{1,2,3}\;.
\end{array}\right.$$ This gives the following action of $\P$ on $\varphi(t_1,t_2,t_3)$: $$\ds
\P_{1,2,3}\*\varphi(t_1,t_2,t_3)\;=\;
\varphi(\sqrt{\kappa_2\over\kappa_3}\;{t_1\,t_2\over t_3}\;,\;
\sqrt{\kappa_3\over\kappa_2}\; t_3\;,\;
\sqrt{\kappa_2\over\kappa_3}\; t_2)\;,$$ where $t_1,t_2,t_3$ stand on the positions corresponding $1,2,3$ of $\R_{1,2,3}$.
Another thing to be mentioned is the case of $|q|=1$. In this case the quantum dilogarithmic functions should be replaced by Faddeev’s integral [@f-modular]. In few words, it appears when one considers the Jacoby imaginary transformation of an argument of $\psi$ and $q$: $$\ds
\u\;=\;\mbox{\large e}^{i\,z}\;,\;\;\;\;
-\;q^{1/2}\;=\;\mbox{\large e}^{i\,\pi\,\theta}\;\;\;\mapsto\;\;\;
\widetilde{\u}\;=\;\mbox{\large e}^{i\,z/\theta}\;,\;\;\;\;
-\widetilde{q}^{1/2}\;=\;\mbox{\large e}^{-i\,\pi/\theta}\;.$$ Then $$\ds
\psi_{F}(\u)\;=\;{\ds(\;q^{1/2}\;\u\;;\;q\;)_\infty\over\ds
(\;\widetilde{q}^{1/2}\;\widetilde{\u}\;;
\;\widetilde{q}\;)_\infty}\;,$$ and the following expression for $\psi_F(\u)$ is valid in the limit of real $\theta$ [@f-modular]: $$\ds
\psi_F(\u)\;(=\;s(z))\;=\;
\exp\;{1\over 4}\;\int_{\infty}^{\infty}
{\ds \mbox{\large e}^{z\,\xi}\over\ds\sinh \pi\xi\;
\sinh \pi\theta\xi}\;{\ds d\;\xi\over\ds \xi}\;,$$ where the singularity at $\xi=0$ is circled from above.
Return now to map (\[R-op\]). The map $\R$ conserves four independent operators: $$\ds
\w_1\cdot\w_2\;,\;\;\;\;\;\;
\u_2\cdot\u_3\;,\;\;\;\;\;\;\s$$ and $$\label{H-op}
\ds\begin{array}{ccl}
\H & = &
\w_1^{}\*\u_3^{-1}
\;\;-\;\;
q^{1/2}\;\;\u_1^{}\*\u_2^{-1}\*\w_2^{}\*\w_3^{-1}\\
&&\\
& - &
\kappa_1\;\;q^{-1/2}\;\;\u_1^{}\*\w_1^{}\*\u_2^{-1}\*\u_3^{-1}
\;\;+\;\;
\kappa_3\;\;\u_1^{}\*\u_2^{-1}\\
&&\\
& - &
\kappa_2\;\;q^{-1/2}\;\;\w_1^{}\*\w_2^{}\*\u_3^{-1}\*\w_3^{-1}
\;\;+\;\;
\kappa_2\;\;\w_2^{}\*\w_3^{-1}\\
&&\\
& = & \ds
\biggl(\W^{-1}\;+\;\kappa_1\;\U\;-\;
q^{1/2}\;\kappa_3\;\U\;\W^{-1}\biggr)
\;+\;\s^{-1}\;
\biggl(\W\;+\;\kappa_2\;\U^{-1}\;-\;
q^{1/2}\;\kappa_2\;\U^{-1}\;\W\biggr)\;.
\end{array}$$ Actually $\R$ depends only on two of them, $\s$ and $\H$.
Consider the following product $$\label{sigma-psi}
\ds\sigma\;=\;
\psi(a\;\w^{-1})\*\psi(b\;\u)\*
\psi(-\;q^{-1/2}\;c\;\u\;\w)\*
\psi(a'\;\w)\*\psi(b'\;\u^{-1})\;.$$ Let $$\ds
\chi\;=\;a\w^{-1}+a'\w+b\u+b'\u^{-1}-q^{-1/2}c\;\u\w
-q^{-1/2}ab'\;\u^{-1}\w^{-1}\;.$$ It is easy to check $\sigma\*\chi\;=\;\chi\*\sigma$. Hence $\sigma$ as an operator is a function on $\chi$: $$\label{sigma-chi}
\ds
\sigma\;=\;\sigma(\;aa'\;,\;bb'\;,\;{c\over a'b}\;\;|\;\chi\;)\;,$$ I did not find explicit form of function $\sigma$, only a special case of $\sigma$ when $c=b'=0$, then $$\ds
\psi(a\,\w^{-1})\;\psi(b\,\u)\;\psi(a'\,\w)\;=\;
\psi(a\,\theta^{-1})\;\psi(a'\,\theta)$$ where $$\ds a\,\theta^{-1}\;+\;a'\,\theta\;=\;a\,\w^{-1}\;+\;b\,\u\;+\;a'\,\w\;.$$ Nevertheless direct calculations give $\R^2$ in terms of $\sigma$ introduced. First, it is convenient to rewrite $\R$: $$\ds
\R\;=\;\psi(\W^{-1})\,
\psi(-q^{1/2}\kappa_3\U\W^{-1})\,
\P(\sqrt{\kappa_3\over\kappa_2}\,\U,\s^{-1}\W^2)\,
\psi(-q^{1/2}{\kappa_1\kappa_2\over\kappa_3}\,\U^{-1}\W)^{-1}
\,\psi({\kappa_1\over\kappa_3}\,\W)^{-1}\;.$$ Then $$\ds
\R^2\;=\;\mbox{\cal N}\*\mbox{\cal D}^{-1}\;,$$ where $$\ds
\mbox{\cal N}\;=\;
\psi(\W^{-1})\,
\psi(-q^{1/2}\kappa_3\U\W^{-1})\,
\psi(\kappa_1\U)\,
\psi(\s^{-1}\W)\,
\psi(-q^{1/2}\kappa_2\s^{-1}\U^{-1}\W)\;,$$ and $$\ds
\mbox{\cal D}\;=\;
\psi({\kappa_1\over\kappa_3}\W)\,
\psi(-q^{1/2}{\kappa_1\kappa_2\over\kappa_3}\U^{-1}\W)\,
\psi({\kappa_1\kappa_2\over\kappa_3}\U^{-1})\,
\psi({\kappa_1\over\kappa_3}\s\W^{-1})\,
\psi(-q^{1/2}\kappa_1\s\U\W^{-1})\;.$$ Comparing these with the definition of $\sigma$, we obtain $$\ds
\mbox{\cal N}\;=\;
\sigma(\s^{-1},\kappa_2\kappa_3\s^{-1},
{\kappa_1\over\kappa_3}\s|\,\H\,)\;,\;\;\;\;
\mbox{\cal D}\;=\;
\sigma({\kappa_1^2\over\kappa_3^2}\s,
{\kappa_1^2\kappa_2\over\kappa_3}\s,
{\kappa_3\over\kappa_1}\s^{-1}|\,
{\kappa_1\over\kappa_3}\s\H\,)\;,$$ where $\H$ is given by (\[H-op\]).
Fusion
------
One more remarkable feature of the current model is a sort of a fusion. As an example consider a planar graph formed by two pairs of the parallel lines. Four vertices arise as the intersection points of these two pairs of the lines. This is shown in Fig. \[fig-fusion\].
The vertices are labelled by the pairs of the indices, $W_{1,1}$, $W_{1,2}$, $W_{2,1}$ and $W_{2,2}$. Single closed site means that there are three independent currents. Let them be the internal current $\phi_{1,1}$, assigned to north-west corner, and two currents $x$ and $y$ assigned to southern and western semi-strips, $x$ and $y$ are observable currents for this cross considered as an alone graph.
Applying the linear system rules, we obtain step by step $$\ds\begin{array}{ccl}
\ds\phi_{1,1} & = & \ds\phi\;,\\
&&\\
\ds\phi_{1,2} & = & \ds y \;-\; \w_{1,1}^{}\*\phi\;,\\
&&\\
\ds\phi_{2,2} & = & \ds \w_{2,2}^{-1}\*x\;-\;
\kappa_{1,2}^{}\;\u_{1,2}^{}\;\w_{1,2}^{}\;\w_{2,2}^{-1}\*y\;+\;
\kappa_{1,2}^{}\;\w_{1,1}^{}\;\u_{1,2}^{}\;\w_{1,2}^{}\;\w_{2,2}^{-1}
\*\phi\;,\\
\end{array}$$ and from zero value of the closed site current $$\ds\begin{array}{ccl}
\ds\phi_{2,1} & = & \ds
-\;\w_{2,1}^{-1}\;\w_{2,2}^{-1}\*x\;+\;
(\kappa_{1,2}^{}\;\u_{1,2}^{}\;\w_{1,2}^{}\;\w_{2,1}^{-1}\;\w_{2,2}^{-1}-
q^{1/2}\;\u_{1,2}^{}\;\w_{2,1}^{-1})\*y\\
&&\\
\ds & + & \ds
(q^{1/2}\;\w_{1,1}^{}\;\u_{1,2}^{}\;\w_{2,1}^{-1}-\kappa_{1,1}^{}\;
\u_{1,1}^{}\;\w_{1,1}^{}\;\w_{2,1}^{-1}-
\kappa_{1,2}^{}\;\w_{1,1}^{}\;\u_{1,2}^{}\;\w_{1,2}^{}\;
\w_{2,1}^{-1}\;\w_{2,2}^{-1})\*\phi\;.
\end{array}$$
(450,200) (110,0)
(200,200) (0,50)[(1,0)[200]{}]{}(0,150)[(1,0)[200]{}]{} (50,0)[(0,1)[200]{}]{}(150,0)[(0,1)[200]{}]{} (50,50)(50,150) (150,50)(150,150) (55,55)[$1,2$]{}(55,155)[$1,1$]{} (155,55)[$2,2$]{}(155,155)[$2,1$]{} (0,190)[$\phi$]{} (0,95)[$y$]{} (95,0)[$x$]{} (187,95)[$-y'$]{} (92,190)[$-x'$]{}
Let further $-x'$ and $-y'$ are the edge variables assigned to the northern and eastern semistips. In general they are $$\label{matrixmap}
\ds \left\{\begin{array}{ccc}
\ds x' & = & \ds \alpha\* x\;+\;\beta\* y\;+\;\mbox{\sf f}_x\*\phi\;,\\
&&\\
\ds y' & = & \ds \gamma\* x\;+\;\delta\* y\;+\;\mbox{\sf f}_y\*\phi\;,
\end{array}\right.$$ where $$\ds\left\{\begin{array}{ccl}
\ds\alpha & = & \ds \w_{2,1}^{-1}\;\w_{2,2}^{-1}\;,\\
&&\\
\ds\beta & = & \ds q^{1/2}\;\u_{1,2}^{}\;\w_{2,1}^{-1}\;-\;
\kappa_{1,2}^{}\;\u_{1,2}^{}\w_{1,2}^{}\;\w_{2,1}^{-1}\;\w_{2,2}^{-1}\;,\\
&&\\
\ds\gamma & = & \ds -\;q^{1/2}\;\u_{2,2}^{}\;\w_{2,2}^{-1}\;+\;
\kappa_{2,1}^{}\;\u_{2,1}^{}\;\w_{2,2}^{-1}\;,\\
&&\\
\ds\delta & = & \ds q^{1/2}\;\kappa_{2,1}^{}\;\u_{1,2}^{}\;\u_{2,1}^{}
\;+\;q^{1/2}\;\kappa_{1,2}^{}\;
\u_{1,2}^{}\;\w_{1,2}^{}\;\u_{2,2}^{}\;\w_{2,2}^{-1}\\
&&\\
&&\ds -\;
\kappa_{1,2}^{}\;\kappa_{2,1}^{}\;
\u_{1,2}^{}\;\w_{1,2}^{}\;\u_{2,1}^{}\w_{2,2}^{-1}\;,
\end{array}\right.$$ and $$\label{feshki}
\ds\left\{\begin{array}{ccl}
\ds\mbox{\sf f}_x & = & \ds
-\;q^{1/2}\;\u_{1,1}^{}\;-\;
q^{1/2}\;\w_{1,1}^{}\;\u_{1,2}^{}\;\w_{2,1}^{-1}\\
&&\\
&& +\;\kappa_{1,1}^{}\;\u_{1,1}^{}\;\w_{1,1}^{}\;\w_{2,1}^{-1}
\;+\;\kappa_{1,2}^{}\;\w_{1,1}^{}\;\u_{1,2}^{}\;
\w_{1,2}^{}\;\w_{2,1}^{-1}\;\w_{2,2}^{-1}\;,\\
&&\\
\ds\mbox{\sf f}_y & = & \ds
-\;q^{1/2}\;\kappa_{2,1}^{}\;\w_{1,1}^{}\;\u_{1,2}^{}\;\u_{2,1}^{}\;-\;
q^{1/2}\;\kappa_{1,2}\;\w_{1,1}^{}\;\u_{1,2}^{}\;\w_{1,2}^{}\;
\u_{2,2}^{}\;\w_{2,2}^{-1}\\
&&\\
&&\ds+\;\kappa_{1,1}^{}\;\kappa_{2,1}^{}\;
\u_{1,1}^{}\;\w_{1,1}^{}\;\u_{2,1}^{}\;+\;
\kappa_{1,2}^{}\;\kappa_{2,1}^{}\;\w_{1,1}^{}\;\u_{1,2}^{}\;\w_{1,2}^{}\;
\u_{2,1}^{}\;\w_{2,2}^{-1}\;.
\end{array}\right.$$
Curents $x,y,x',y'$ become the edge currents when we rewrite the cross in Fig. \[fig-fusion\] as a single vertex with modified (thick) lines; denote it as $\widetilde{W}\sim\{W_{1,1},W_{1,2},W_{2,1},W_{2,2}\}$. Suppose we combine such crosses $\widetilde{W}$ (vertices with thick lines) in any way, then zero value conditions for the restricted strips (closed thick edges) look very simply: due to the signs $(-)$ in the definition of outgoing $x'$ and $y'$ these zero value conditions becomes “outgoing edge current of one thick vertex $=$ incoming edge current of another thick vertex”. Thus the strip variables just transfer from one combined (thick) vertex to another, and therefore they look like [**edge variables**]{} of the thick vertices.
In the case when for any thick vertex the map $x,y$ $\mapsto$ $x',y'$ (\[matrixmap\]) does not contain extra $\phi$, i.e. $\f_x=\f_y=0$ in any sense, then the part of the linear system corresponding to the edge variables factorises from the whole current system. [**If such factorisation exists for a graph $G$ then it exists for any equivalent graph $G'$, so sub-manifold of ${\cal B}_P$ given by $\f_x=\f_y=0$ is invariant of a map $G\mapsto G'$.**]{}
On this sub-manifold we can delete all the edge currents $x=y=...=0$. In this case all corner currents of cross $\widetilde{W}$ are proportional to $\phi=\phi_{1,1}$, and the structure of the “thick” vertex becomes the structure of usual vertex. Thus one may define “thick” analogies of $\u,\w$ and $\kappa$. This phenomenon resembles the usual two-dimensional fusion.
Write now explicit formulae. Introduce $$\ds\begin{array}{ccl}
\ds\mbox{\sf K}^{-1} & = & \ds
{1\over\kappa_{1,2}^{}\kappa_{2,1}^{}\kappa_{2,2}^{}}\;\;\bigl(
\u_{1,1}^{}\;\u_{1,2}^{-1}\;+\;\w_{1,1}^{}\;\w_{2,1}^{-1}\;-\;
q^{-1/2}\;\;\kappa_{1,1}^{}\;
\u_{1,1}^{}\;\u_{1,2}^{-1}\;\w_{1,1}^{}\;\w_{2,1}^{-1}\bigr)\;,\\
&&\\
\ds\mbox{\sf k} & = & \ds q^{1/2}\;\;
\kappa_{2,1}^{}\;\kappa_{2,2}^{}\;
\w_{1,1}^{-1}\;\w_{1,2}^{-1}\;\w_{2,1}^{}\;\w_{2,2}^{}\;,\\
&&\\
\ds\widetilde{\mbox{\sf k}} & = & \ds q^{-1/2}\;\;
\kappa_{1,2}^{}\;\kappa_{2,2}^{}\;
\u_{1,1}^{-1}\;\u_{1,2}^{}\;\u_{2,1}^{-1}\;\u_{2,2}^{}\;.
\end{array}$$ Without mentioning of a representation of the Weyl algebra, its right module etc., suppose $\phi$ in (\[matrixmap\]) obeys $\mbox{\sf f}_x\*\phi\;=\;\mbox{\sf f}_y\*\phi\;=\;0$. From this, it follows that $$\ds\mbox{\sf K}^{-1}\*\phi\;=\;\mbox{\sf k}^{-1}\*\phi\;=\;
\widetilde{\mbox{\sf k}}^{\;-1}\*\phi\;=\;K^{-1}\;\;\phi,$$ where $K$ is introduced as an “eigenvalue”. On this “subspace” the fusion is defined as $$\ds
\Delta(\w)\;=\;-\;\w_{1,1}^{}\;\w_{1,2}^{}\;,\;\;\;\;
\Delta(\u)\;=\;-\;q^{1/2}\;\u_{1,1}^{}\;\u_{2,1}^{}\;,\;\;\;\;
\Delta(\kappa)\;=\; K\;.$$
The meaning of all these is the following. Consider three “thick” crosses $\widetilde{W}_j\mapsto\widetilde{W}_j'$, $j=1,2,3$, arranged into “thick” Yang – Baxter – type graphs, $\lgr$ and $\rgr$. Solving the complete problem of the equivalence (with twelve vertices in each hand side) one obtains the set of relations like $$\Delta(\w_j')\*\phi_j'\;-\;\Delta(\w_j)'\*\phi_j'\;=\;
\sum_{k=1}^3\;\;X_k\*\f_{x,k}\;\phi_k+Y_k\*\f_{y,k}\;\phi_k\;,$$ etc., with some $X_k$ and $Y_k$, $\f_{x,k}$ and $\f_{y,k}$ given by (\[feshki\]). $\Delta(\w_j')$ we obtain from $\Delta(\w_j)$ applying all eight $\R$-s repeatedly, and $\Delta(\w_j)'$ is the result of the application of single $\R$ in terms of $\Delta(\u_j)$, $\Delta(\w_j)$ and $\Delta(K)$.
Matrix part
-----------
Few words concerning the matrix variables $\alpha,\beta,\gamma,\delta$ in (\[matrixmap\]). This remark is not important for our current approach, but the structure of matrix variables is very interesting. First, the map of edge auxiliary variables $$\label{korepanovmap}
\ds \left\{\begin{array}{ccc}
\ds x' & = & \ds \alpha\* x\;+\;\beta\* y\;,\\
&&\\
\ds y' & = & \ds \gamma\* x\;+\;\delta\* y\;,
\end{array}\right.$$ appeared in Korepanov’s matrix models [@korepanov-diss; @kks-fte]. The Yang – Baxter equivalence in Korepanov’s interpretation is the Korepanov equation: admissibility of the map of three edge variables $(x,y,z)$ assigned to three lines of the Yang – Baxter graph. Let $$\ds
X_1^{} \;=\;\left(\begin{array}{ccc}
\alpha_1 & \beta_1 & 0 \\ \gamma_1 & \delta_1 & 0 \\ 0 & 0 & 1
\end{array}\right)\;,\;\;\;\;
X_2^{} \;=\;\left(\begin{array}{ccc}
\alpha_2 & 0 & \beta_2 \\ 0 & 1 & 0 \\ \gamma_2 & 0 & \delta_2
\end{array}\right)\;,\;\;\;\;
X_3^{} \;=\;\left(\begin{array}{ccc}
1 & 0 & 0 \\ 0 & \alpha_3 & \beta_3 \\ 0 & \gamma_3 & \delta_3
\end{array}\right)\;,$$ Then the admissibility is $$\label{KE}
\ds X_1^{}\* X_2^{}\* X_3^{}\;=\; X_3'\* X_2'\* X_1'\;,$$ where the primed $X$-s consist on primed $\alpha,\beta,\gamma,\delta$. Korepanov’s equation is equivalent to the usual local Yang – Baxter equation for the so-called ferroelectric weights: $$\ds X\;=\;\left(\begin{array}{cc} \alpha & \beta \\
\gamma & \delta\end{array}\right)
\;\;\;\;\mapsto\;\;\;\;
L\;=\;\left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \alpha & \beta & 0 \\
0 & \gamma & \delta & 0\\
0 & 0 & 0 & \zeta
\end{array}\right)\;,$$ where in the numeric case $\zeta\;=\;\alpha\;\delta \;-\;\beta\;\gamma$, and the conventional $2^2\times 2^2$ $=$ $4\times 4$ matrix form for Yang – Baxter matrix $L$ is used (for the equivalence see [@oneparam] for example). In our case the elements of different $X$-s commute, and the elements of one $X$ obey the algebra $$\label{algebra}
\ds\left.\begin{array}{clc}
&\ds \alpha\*\beta\;=\;\beta\*\alpha\;,\;\;\;\;
\ds\gamma\*\delta\;=\;\delta\*\gamma\;, &\\
&&\\
&\ds\alpha\*\gamma\;=\;q\;\;\gamma\*\alpha\;,\;\;\;\;
\ds\alpha\*\delta\;=\;q\;\;\delta\*\alpha\;,
\;\;\;\;
\ds\beta\*\delta\;=\;q\;\;\delta\*\beta\;,
\end{array}\right.$$ and $$\label{z-eq}
\ds
\zeta\;\stackrel{def}{=}\;
\alpha\*\delta\;-\;\beta\*\gamma\;=\;
\delta\*\alpha\;-\;\gamma\*\beta\;.$$ From (\[algebra\]) and (\[z-eq\]) it follows that $\zeta\;\beta=\beta\;\zeta$, $\zeta\;\gamma=\gamma\;\zeta$, and consequently $$\ds\left.\begin{array}{clc}
&\ds\beta^2\*\gamma\;+\;q\;\;\gamma\*\beta^2\;-\;
(1+q)\;\;\beta\*\gamma\*\beta\;=\;0\;,&\\
&&\\
&\ds\beta\*\gamma^2\;+\;q\;\;\gamma^2\*\beta\;-\;
(1+q)\;\;\gamma\*\beta\*\gamma\;=\;0\;.
\end{array}\right.$$ Hence $$\label{ad-bc}
\ds\alpha\*\delta\;=\;-\;{q\over 1-q}\;\;
(\beta\*\gamma\;-\;\gamma\*\beta)\;,$$ and $$\label{z-bc}
\ds\zeta\;=\;-\;{1\over 1-q}\;\;
(\beta\*\gamma\;-\;q\;\;\gamma\*\beta)\;.$$ Call the algebra of $\alpha,\beta,\gamma,\delta$, given by (\[algebra\]) and (\[z-eq\]), as ${\cal X}$. Interesting is the following
[**Proposition.**]{} $\bullet$ Korepanov’s equations are nine equation for twelve variables, so $X_j'$ are defined ambiguously: in general one can’t fix one element from $\alpha_1^\prime,\alpha_2^\prime$, one from $\delta_2^\prime,\delta_3^\prime$, and one from $\alpha_3^\prime,\delta_2^\prime$. Impose on these three arbitrary elements the simple part of ${\cal X}$, (\[algebra\]). Then all other relations of ${\cal X}$, namely relations (\[algebra\]) for the other primed elements and all three relations (\[z-eq\]) (or, equivalent, (\[ad-bc\])) for the elements of $\{X_1',X_2',X_3'\}$, hold automatically as the consequence of Korepanov’s equation. $\bullet$
This observation, we guess, is a way of a quantization of Korepanov’s matrix model. Remarkably is that ${\cal X}$ is the nontrivial algebra.
As an example consider the case when $\delta\;=\;0$. Corresponding algebra, ${\cal X}_{\delta=0}$, contains only one nontrivial relation, $\alpha\*\gamma\;=\;q\;\;\gamma\*\alpha$, and $\beta$ is a center. Being a ${\cal C}$ – number, $\beta_j$ are to be conserved by the map $X_j^{}\mapsto X_j^\prime$. Equations (\[KE\]) contain $\beta_2\;=\;\beta_1\;\beta_3$. Hence $\beta$ is the pure gauge and one may put $\beta\equiv 1$. The solution of (\[KE\]) is: $$\ds\left.\begin{array}{clc}
\ds & \left\{\begin{array}{ccl}
\ds\alpha_1^\prime & = & \ds
(\alpha_3^{}\;+\;\alpha_1^{}\*\gamma_3^{})^{-1}
\*\alpha_1^{}\*\alpha_2^{}\\
&&\\
\ds\gamma_1^\prime & = & \ds
f\*\gamma_1^{}\*\alpha_3^{}\*
(\alpha_3^{}\;+\;\alpha_1^{}\*\gamma_3^{})^{-1}
\end{array}\right. & \\
&\\
\ds & \left\{\begin{array}{ccl}
\ds\alpha_2^\prime & = & \ds
\alpha_3^{}\;+\;\alpha_1^{}\*\gamma_3^{}\\
&&\\
\ds\gamma_2^\prime & = & \ds\gamma_1^{}\*\gamma_3^{}
\end{array}\right. & \\
&\\
\ds & \left\{\begin{array}{ccl}
\ds\alpha_3^\prime & = & \alpha_2^{}\* f^{-1}\\
&&\\
\ds\gamma_3^\prime & = &
(\alpha_3^{}\;+\;\alpha_1^{}\*\gamma_3^{})
\*\gamma_1^{-1}\*\gamma_2^{}\*\alpha_3^{-1}\* f^{-1}
\end{array}\right. &
\end{array}\right.$$ where $f$ is not fixed by (\[KE\]), this is the ambiguity mentioned. Permutation relations for the combinations of the primed elements, which do not contain $f$, namely for $\alpha_1^\prime$, $\alpha_2^\prime$, $\gamma_2^\prime$, $\alpha_3^\prime\;\gamma_1^\prime$ and $\gamma_3^\prime\;\gamma_1^\prime$, do not contradict the set of the local Weyl algebrae $\alpha_j^\prime\*\gamma_j^\prime\;=\;q\;\;
\gamma_j^\prime\*\alpha_j^\prime$. This corresponds to the statement of the proposition above. Consider now the Weyl algebrae for all primed elements. From this, it follows immediately $$f\*\alpha_j\;=\;\alpha_j\* f\;,\;\;\;\;
f\*\gamma_j\;=\;\gamma_j\* f\;.$$ Hence $f$ is a ${\cal C}$ – number, and therefore we may put $f\;=\;1$. Thus the conservation of ${\cal X}_{\delta=0}$ fixes the ambiguity.
The map $\alpha_j,\gamma_j \mapsto \alpha_j^\prime,\gamma_j^\prime$ we’ve obtained is nothing but $\mbox{\sf r}_{1,2,3}$, given by the limiting procedure (\[one\_conditions\]). The identification is $\alpha\;=\;\w^{-1}$ and $\gamma\;=\;-\;q^{1/2}\;\u^{}\cdot\w^{-1}$. This case, $$\ds X\;=\;\left(\begin{array}{rcr}
\w^{-1} &,& 1 \\ -\;q^{1/2}\;\u^{}\cdot\w^{-1} &,& 0
\end{array}\right)$$ is the quantization of the case $(\eta)$ from the list of simple functional maps in [@oneparam].
The case of general ${\cal X}$ is rather complicated technically, it is a subject of a separate investigation.
Co-current system and $\L$-operator
-----------------------------------
In this subsection we give another form of the current approach.
Consider the whole linear system for a graph $G$ defined on a torus (boundary conditions assumed). This system is the set of zero equations $$\label{site-eq}
\ds
\phi_{\mbox{site}}\;\;\stackrel{def}{=}\;\;
\sum_{\mbox{vertices}}\;\;
W_{\mbox{vertex}}\*\phi_{\mbox{vertex}}\;\;=\;\;0\;,$$ where such equation we write for each site of $G$, the sum is taken over all vertices surrounded this site, and contribution from vertex $V$, denoted as $W_V\cdot\phi_V$, is one of $\phi_V$, $q^{1/2}\u_V\cdot\phi_V$, $\w_V\cdot\phi_V$ or $\kappa_V\u_V\w_V\cdot\phi_V$ according to Fig. \[fig-weyl-vertex\] and the orientation of $V$. The toroidal structure means that all the sites are closed and the number of the sites equals to the number of the vertices. Gathering all zero equations (\[site-eq\]) together, we obtain the matrix form of them, $$\label{LP}
\ds{\bf L}\*\Phi\;=\;0\;,$$ where we combine the internal currents $\phi_V$ into the column $\Phi$ and the matrix of the coefficients $\bf L$ consists of $$\ds
1\;,\;\;\; q^{1/2}\;\u_V\;,\;\;\; \w_V\;\;\;\mbox{and}\;\;\;
\kappa_V\;\u_V\;\w_V$$ for all vertices $V$ of the lattice. ${\bf L}$ is the square matrix, and explicit form of it depends on the geometry $G$.
(\[LP\]) can be interpreted as an equation of motion for the action ${\cal A}\;=\;\Phi^*\*{\bf L}\*\Phi$, where the row co-current vector $\Phi^*$ does not depend on $\Phi$ and its components $\phi^*_S$ are assigned to the sites $S$ of $G$. The equation of motion for $\Phi^*$ is $\Phi^*\*{\bf L}\;=\;0$. Corresponding zero equations now are assigned to the vertices, and each such equation connects the site co-currents from the sites surrounding this vertex. The problem of the equivalence of $\lgr$ and $\rgr$ in terms of co-currents can be formulated as follows: the co-current system for the left hand side graph $\lgr$ of Fig. \[fig-YBE\] is $$\label{co-lhs}
\ds\left\{\begin{array}{ccccc}
\ds\phi^*_1 & \equiv & \ds\phi^*_c\;+
\;\phi^*_e\*q^{1/2}\;\u_1^{}\;+\;
\phi^*_h\*\w_1^{}\;+
\;\phi^*_d\*\kappa_1^{}\;\u_1^{}\;\w_1^{} & = & 0\;,\\
&&&&\\
\ds\phi^*_2 & \equiv & \ds\phi^*_h\;+
\;\phi^*_d\*q^{1/2}\;\u_2^{}\;+\;
\phi^*_b\*\w_2^{}\;+
\;\phi^*_f\*\kappa_2^{}\;\u_2^{}\;\w_2^{} & = & 0\;,\\
&&&&\\
\ds\phi^*_3 & \equiv & \ds\phi^*_c\;+
\;\phi^*_h\*q^{1/2}\;\u_3^{}\;+\;
\phi^*_g\*\w_3^{}\;+
\;\phi^*_b\*\kappa_3^{}\;\u_3^{}\;\w_3^{} & = & 0\;,
\end{array}\right.$$ and co-current system for the right hand side graph $\rgr$ is $$\label{co-rhs}
\ds\left\{\begin{array}{ccccc}
\ds\phi^{*\prime}_1 & \equiv & \ds
\phi^*_g\;+\;\phi^*_a\*q^{1/2}\;\u_1'\;+\;
\phi^*_b\*\w_1'\;+\;\phi^*_f\*\kappa_1^{}\;\u_1'\;\w_1'
& = & 0\;,\\ &&&&\\
\ds\phi^{*\prime}_2 & \equiv & \ds
\phi^*_c\;+\;\phi^*_e\*q^{1/2}\;\u_2'\;+\;
\phi^*_g\*\w_2'\;+\;\phi^*_a\*\kappa_2^{}\;\u_2'\;\w_2'
& = & 0\;,\\ &&&&\\
\ds\phi^{*\prime}_3 & \equiv & \ds
\phi^*_e\;+\;\phi^*_d\*q^{1/2}\;\u_3'\;+\;
\phi^*_a\*\w_3'\;+\;\phi^*_f\*\kappa_3^{}\;\u_3'\;\w_3' & = & 0\;.
\end{array}\right.$$ The equivalence means that when we remove $\phi^*_h$ from (\[co-lhs\]) and $\phi^*_a$ from (\[co-rhs\]), then the resulting systems as the systems for $\phi^*_b,...,\phi^*_f$ are equivalent.
Consider now co-current equation for single vertex, as in Figs. \[fig-weyl-vertex\] or \[fig-abcd-currents\]. Let the co-currents be $\phi^*_a$, $\phi^*_b$, $\phi^*_c$ and $\phi^*_d$, where the indices $a,b,c,d$ are arranged as in Fig. \[fig-abcd-currents\]. The co-current equation for this vertex is $$\label{co-eq}
\ds\phi^*\;\equiv\;\phi^*_a\;+\;\phi^*_b\* q^{1/2}\;\u\;+\;
\phi^*_c\*\w\;+\;\phi^*_d\*\kappa\;\u\;\w\;\;=\;\;0\;.$$ Suppose we have solved a part of such equations for whole graph $G$, and obtain $\phi^*_a$, $\phi^*_c$, $\phi^*_d$ in the form usual for homogeneous linear equations: $$\ds
\phi^*_a\;=\;-\;\phi^*_c\* q^{1/2}\;\y\;,\;\;\;\;
\phi^*_c\;=\;-\;\phi^*_d\* q^{1/2}\;\x$$ with some multipliers $\x$ and $\y$. Then from (\[co-eq\]) we get $$\ds\phi^*_b\;=\;-\;\phi^*_d\* q^{1/2}\;\y'\;,\;\;\;
\mbox{or}\;\;\;\phi^*_a\;=\;-\;\phi^*_b\* q^{1/2}\;\x'\;,$$ where $$\ds\x'\;=\;\omega^{-1}\*\y\;,\;\;\mbox{and}\;\;\y'\;=\;\x\*\omega$$ with $$\label{omega}
\ds\omega\;=\;\omega(\,\x\,,\,\y\,|\,\u\,,\,\w\,)\;=\;
\y\*\u^{-1}\;-\;q^{1/2}\;\;\u^{-1}\*\w\;+\;\kappa\;\;\x^{-1}\*\w
\;.$$ Now we may change the interpretation completely. Assign $\x$, $\y$, $\x'$, $\y'$ to the edges which separates corresponding sites. These edge variables are shown in Fig. \[fig-edge\].
(200,200) ( 50 , 50 )[(1,1)[100]{}]{} ( 150 , 50 )[(-1,1)[100]{}]{} ( 100 , 100 ) ( 30 , 20 )[$\y$]{} ( 30 , 160 )[$\x'$]{} ( 160 , 20 )[$\x$]{} ( 160 , 160 )[$\y'$]{} ( 110 , 95 )[$\kappa,\u,\w$]{}
Now we can introduce the auxiliary functional operator $\L$, giving the map $\x,\y$ $\mapsto$ $\x',\y'$, as we used to be: $$\label{L-op}
\ds\fbox{$\ds\;\;\;
\left.\begin{array}{ccc}
&&\\
\ds \L_{\x,\y}(\kappa,\u,\w)\*\x & = &
\omega(\,\x\,,\,\y\,|\,\u\,,\,\w\,)^{-1}\*
\y\*\L_{\x,\y}(\kappa,\u,\w)\;,\\
&&\\
\ds \L_{\x,\y}(\kappa,\u,\w)\*\y & = &
\;\;\x\*
\omega(\,\x\,,\,\y\,|\,\u\,,\,\w\,)\*
\;\L_{\x,\y}(\kappa,\u,\w)\;.\\
&&
\end{array}\right.
\;\;\;$}$$ With the definition (\[R-action\]), $\L$ operators obey $$\label{RLLL}
\ds\begin{array}{ccc}
& \ds
\L_{\y,\z}(\kappa_3,\u_3,\w_3)\*\L_{\x,\z}(\kappa_2,\u_2,\w_2)\*
\L_{\x,\y}(\kappa_1,\u_1,\w_1)\*\R_{1,2,3}\;= &\\
&&\\
&\ds =\;
\R_{1,2,3}\*\L_{\x,\y}(\kappa_1,\u_1,\w_1)\*
\L_{\x,\z}(\kappa_2\u_2,\w_2)\*
\L_{\y,\z}(\kappa_3,\u_3,\w_3) & \;.
\end{array}$$ Moreover, Local Yang-Baxter relation $$\label{LLL}
\ds\begin{array}{ccc}
&\ds
\L_{\y,\z}(\kappa_3,\u_3,\w_3)\*\L_{\x,\z}(\kappa_2,\u_2,\w_2)\*
\L_{\x,\y}(\kappa_1,\u_1,\w_1)\;=&\\
&&\\
&\ds =\;
\L_{\x,\y}(\kappa_1,\u_1',\w_1')\*
\L_{\x,\z}(\kappa_2,\u_2',\w_2')\*
\L_{\y,\z}(\kappa_3,\u_3',\w_3')
\end{array}$$ as a set of relations for $\u_k',\w_k'$, with $\u_k,\w_k$ given and with $\x,\y,\z$ arbitrary, gives again the map (\[ev\]) uniquely ! Thus the kind of the local Yang – Baxter relation appears and for our current approach.
Conclude this section by few remarks concerning the functional maps. All the maps introduced are connected to several graphical manipulations. Usually we combine such manipulations ($\lgr$ $\mapsto$ $\rgr$ of $\x,\y$ $\mapsto$ $\x',\y'$ etc.), and write the sequence of the dynamical variables’ sets obtained $\Sigma$ $\mapsto$ $\Sigma'$, in the direct form $$\ds
\Sigma\;=\;\Sigma_0\;
\stackrel{\ds A_1}{\mapsto}\;\Sigma_1\;
\stackrel{\ds A_2}{\mapsto}\;\Sigma_2\;\dots\;\Sigma_{n-1}\;
\stackrel{\ds A_n}{\mapsto}\;\Sigma_n\;,$$ where $A_j$ stands for $j$-th manipulation, which allows us to calculate $\Sigma_j$ in terms of previous variables $\Sigma_{j-1}$. The same result, $\Sigma_0\mapsto\Sigma_n$, can be obtained as $$\ds
\A_1\;\A_2\;...\;\A_n\*\Sigma_0\;=
\;\Sigma_n\*\A_1\;\A_2\;...\;\A_n\;,$$ where $\A_j$ is a functional operator corresponding the manipulation $A_j$. Remarkable is the reverse order of the operators with respect to the naïve manipulations. Note that the direct order we obtain considering the “pointer” action of the operators, as it was mentioned in the previous subsection, but the “pointer” action is not suitable for the quantization.
Evolution system
================
In this section we apply operator $\R$ defined in the previous section to construct an evolution model explicitly. Due to the current system’s background we formulate this model in terms of the regular lattice defined on the torus, its motion, its current system and so on.
The main result of our paper is the generating function for the integrals of motion for the evolution. The derivation of the integrals is based on the auxiliary linear problem.
Kagome lattice on the torus
---------------------------
An example of a regular lattice which contains both $\lgr$ and $\rgr$ – type triangles is so-called kagome lattice. As it was mentioned in the introduction, the kagome lattices appear in the sections of the regular 3D cubic lattices by inclined planes. Thus the kagome lattice and its evolution corresponds actually to the rectangular 3D lattice and thus is quite natural. The kagome lattice consists on three sets of parallel lines, usual situation shown in Fig. \[fig-kagome\]. The sites of the lattice are both $\lgr$ and $\rgr$ triangles, and hexagons.
For given lattice introduce the labelling for the vertices. Mark the $\lgr$ triangles by the point notation $P$, and let $a$ and $b$ are the multiplicative shifts in the northern and eastern directions, so that the elementary shift in the south-east direction is $c=a^{-1}b$. Nearest to triangle $P$ are triangles $aP$, $bP$, $cP$, $a^{-1}P$, $b^{-1}P$ and $c^{-1}P$. Some of them are shown in Fig \[fig-kagome\].
For three vertices surrounding the $\lgr$-type triangle $P$ introduce the notations $(1,P)$, $(2,P)$ and $(3,P)$. These notations we will use as the subscripts for everything assigned to the vertices.
(450,280) (85,0)
(280,280) (20,0)(120,0)[3]{}[(0,1)[280]{}]{} (0,20)(0,120)[3]{}[(1,0)[280]{}]{} (0,100)[(1,-1)[100]{}]{} (0,220)[(1,-1)[220]{}]{} (60,280)[(1,-1)[220]{}]{} (180,280)[(1,-1)[100]{}]{} (147,147)[$1$]{} (82,147)[$3$]{} (147,82)[$2$]{} (117,117)[$P$]{} (112,237)[$aP$]{} (232,117)[$bP$]{} (228,237)[$abP$]{}
This kagome lattice we define on the torus of size $M$, formally this means the following equivalence: $$\ds a^M\;P\;\sim\;b^M\;P\sim\;c^M\;P\;\sim\;P\;.$$
Since the notion of the equivalence, we may consider the shifts af all inclined lines through the rectangular vertices into north-eastern direction as it is shown in Fig. \[fig-U\]. It is easy to see that Fig. \[fig-U\] is equivalent to Fig. \[fig-YBE\]. The structure of the kagome lattice conserves by such shifts being made simultaneously for all $\lgr$-s, but the marking of the vertices changes a little. This is visible in Fig. \[fig-U\].
(450,200) (00,0)
(200,200) (0,150)[(1,0)[200]{}]{}(50,150) (0,200)[(1,-1)[200]{}]{}(150,50) (150,0)[(0,1)[200]{}]{}(150,150) (160,160)[$1,P$]{} (55,160)[$3,P$]{} (160,55)[$2,P$]{} (105,110)[$\u_j,\w_j$]{}
(220,80)[$\ds\stackrel{\U}{\mapsto}$]{} (250,0)
(200,200) (50,0)[(0,1)[200]{}]{}(50,50) (0,50)[(1,0)[200]{}]{}(50,150) (0,200)[(1,-1)[200]{}]{}(150,50) (20,30)[$1,P$]{} (60,155)[$2,a\,P$]{} (155,60)[$3,b\,P$]{} (60,75)[$\u_j',\w_j'$]{}
Give now pure algebraic definition of the evolution. The phase space of the system is the set of $3\;M^2$ Weyl pairs $\u_{j,P}$ and $\w_{j,P}$, $j=1,2,3$, $P = a^\alpha\,b^\beta\,P_0$, where $P_0$ is some frame of the reference’s distinguished point, and the toroidal boundary conditions mean $$\ds\left.\begin{array}{ccccc}
\ds\u_{j,a^M\,P} & = & \ds\u_{j,b^M\,P} & = & \ds\u_{j,P}\;,\\
&&&&\\
\ds\w_{j,a^M\,P} & = & \ds\w_{j,b^M\,P} & = & \ds\w_{j,P}\;.
\end{array}\right.$$ The phase space is quantized by the definition. Let $\u_{j,P}',\w_{j,P}'$ for $P$ fixed are given by (\[ev\]), so that the map $\{\u_{j,P},\w_{j,P}\}$ $\mapsto$ $\{\u_{j,P}',\w_{j,P}'\}$ is given by the operator $$\ds
\ds{\cal R}\;=\;\prod_{P}\;\;\R_P\;,$$ where $\R_{P'}$ acts trivially on the variables of any triangle $P\neq P'$. Note, we suppose $\kappa_{j,P}$ do not depend on $P$, $$\ds\kappa_{j,P}\;=\;\kappa_{j}\;,$$ so that with respect to $\kappa$-s the translation invariance of the lattice is assumed. Define the superscript ‘$\smb$’ as follows: $$\label{smb}
\ds \left\{\begin{array}{ll}
\ds\u_{1,P}^\smb\;=\;\u_{1,P}'\;,&\ds\w_{1,P}^\smb
\;=\;\u_{1,P}'\;,\\ &\\
\ds\u_{2,aP}^\smb\;=\;\u_{2,P}'\;,&\ds\w_{2,aP}^\smb
\;=\;\w_{2,P}'\;,\\ &\\
\ds\u_{3,bP}^\smb\;=\;\u_{3,P}'\;,&\ds\w_{3,bP}^\smb
\;=\;\w_{3,P}'\;.
\end{array}\right.$$ This identification means following: $\u_{j,P}^\smb,\w_{j,P}^\smb$ are the variables which appear on the places of previous $\u_{j,P},\w_{j,P}$ according to Fig. \[fig-U\]. The evolution operator $\U$ : $\{\u_{j,P},\w_{j,P}\}$ $\mapsto$ $\{\u_{j,P}^\smb,\w_{j,P}^\smb\}$ we define as usual: $$\label{u-star}
\ds
\U\*\u_{j,P}^{}\*\U^{-1}\;=\;\u_{j,P}^\smb\;,\;\;\;\;
\U\*\w_{j,P}^{}\*\U^{-1}\;=\;\w_{j,P}^\smb\;.$$ Regard the primary variables $\{\u_{j,P},\w_{j,P}\}$ of the given lattice as the initial data for the discrete time evolution, $$\ds\u_{j,P}\;=\;\u_{j,P}(0)\;,\;\;\;\;
\w_{j,P}\;=\;\w_{j,P}(0)\;.$$ The evolution from $t=n$ to $t=n+1$ is just $$\label{u-nn}
\ds\u_{j,P}(n+1)\;=\;\U\*\u_{j,P}(n)\*\U^{-1}\;,\;\;\;\;
\w_{j,P}(n+1)\;=\;\U\*\w_{j,P}(n)\*\U^{-1}\;.$$ Surely, the map $\U$ is the canonical map for the Weyl algebrae, so that $\U$ is the quantum evolution operator. Further we’ll consider mainly the situation for $t=0$ and the map from $t=0$ to $t=1$. We will omit the time variable and write $f$ instead of $f(0)$ and $f^\smb=\U\* f\* U^{-1}$ instead of $f(1)$ for any object $f$. Due to the homogeneity of evolution (\[u-nn\],\[u-star\],\[smb\]) our considerations appear to be valid for a situation with $t=n$ and the map from $t=n$ to $t=n+1$.
Linear system
-------------
Investigate now the linear system for the quantum system obtained.
Assign to the vertex $(j,P)$ of the primary ($t=0$) kagome lattice the internal current $\phi_{j,P}$. The linear system is the set of $3\,M^2$ linear homogeneous equation for $3\,M^2$ internal currents $$\label{f-values}
\ds\left\{\begin{array}{ccccc}
\ds f_{1,P} & \equiv & \ds
\w_{1,P}\*\phi_{1,P}+\phi_{2,P}+q^{1/2}\u_{3,P}\*\phi_{3,P}
& = & 0\;,\\
&&&&\\
\ds f_{2,P} & \equiv & \ds
q^{1/2}\u_{1,P}\*\phi_{1,P}+
\kappa_2\u_{2,aP}\w_{2,aP}\*\phi_{2,aP}
+\w_{3,bP}\*\phi_{3,bP} & = & 0\;,\\
&&&&\\
\ds f_{3,P} & \equiv & \ds
\phi_{1,a^{-1}P}+
\kappa_1\u_{1,b^{-1}P}\w_{1,b^{-1},P}\*\phi_{1,b^{-1}P}
+\w_{2,P}\*\phi_{2,P}&&\\
&&&&\\
&&\ds +q^{1/2}\u_{b,b^{-1}P}\*\phi_{2,b^{-1}P}
+\phi_{3,a^{-1}P}+\kappa_3\u_{3,P}\w_{3,P}\*\phi_{3,P} & = & 0\;.
\end{array}\right.$$ Here we have introduced absolutely unessential notations $f_{j,P}$ just in order to distinguish these equations. $f_{j,P}$ are assigned to the sites. Due to the homogeneity we may impose [**the quasiperiodical boundary conditions**]{} for $\phi_{j,P}$: $$\label{quasiperiod}
\ds
\phi_{j,a^M\,P}\;=\;A\;\phi_{j,P}\;,\;\;\;\;
\phi_{j,b^M\,P}\;=\;B\;\phi_{j,P}\;.$$ It is useful to rewrite this system in the matrix form as (\[LP\]), $F\;\equiv\;{\bf L}\*\Phi\;=\;0$. First combine $\phi_{j,P}$ with the same $j$ into the column vector $\Phi_j$ with $M^2$ components, so as $(\Phi_j)_P\;=\;\phi_{j,P}$. Introduce matrices $T_a$ and $T_b$ as $$\ds
(T_a\*\Phi_j)_P\;=\;\phi_{j,a\,P}\;,\;\;\;
(T_b\*\Phi_j)_P\;=\;\phi_{j,b\,P}\;.$$ Due to (\[quasiperiod\]) $$\ds T_a^M\;=\;A\;,\;\;\; T_b^M\;=\;B\;.$$ Combine further $\u_{j,P}$ and $\w_{j,P}$ with the same $j$ into diagonal matrices $\u_j$ and $\w_j$ with the same ordering of $P$ as in the definition of $\Phi_j$, $$\ds
\u_{j}\;=\;\mbox{diag}_P\;\; u_{j,P}\;\;,\;\;\;\;
\w_{j}\;=\;\mbox{diag}_P\;\; w_{j,P}\;\;.$$ Obviously, $$\ds
(T_a^{}\*\u_j\* T_a^{-1})_P\;=\;\u_{j,a\,P}\;,\;\;\;\;
(T_b^{}\*\u_j\* T_b^{-1})_P\;=\;\u_{j,b\,P}\;,$$ and the same for $\w_j$.
Combine further $\Phi_1$, $\Phi_1$, $\Phi_3$ into $3\,M^2$ column $\Phi$. Then from (\[f-values\]) the matrix ${\bf L}$ can be extracted in the $3\times 3$ $M^2\times M^2$ block form: $$\label{L-matrix}
\ds\fbox{
$\ds
{\bf L}\;=\;
\left(\begin{array}{rcrcr}
\w_1 &,& 1 &,& q^{1/2}\;\u_3\\
&&&&\\
q^{1/2}\;\u_1 &,& T_a\;\kappa_2\;\u_2\;\w_2 &,& T_b\;\w_3 \\
&&&&\\
T_a^{-1}\+ T_b^{-1}\;\kappa_1\;\u_1\;\w_1 &,&
\w_2\+T_b^{-1}\;q^{1/2}\;\u_2 &,&
T_a^{-1}\+\kappa_3\;\u_3\;\w_3
\end{array}\right)
$
}$$ Recall, system ${\bf L}\*\Phi\;=\;0$ is $3\,M^2$ equations for $3\,M^2$ components of $\Phi$.
Introduce now co-currents. As it was mentioned, ${\bf L}\*\Phi\;=\;0$ we regard as the equations of motion for 2D system with the action $$\ds {\cal A}\;\equiv\;\Phi^*\*{\bf L}\*\Phi\;.$$ The block form of the co-currents $\Phi^*$ is thus fixed from the form of ${\bf L}$, or from (\[f-values\]). Equations of motion for $\Phi^*$ are $F^*\;\equiv\;\Phi^*\*{\bf L}\;=\;0$, and in the component form $$\label{co-f-prime-values}
\ds\left\{\begin{array}{ccc}
\ds f^{*}_{1,P} & \equiv & \ds
\phi^{*}_{1,P}\*q^{1/2}\;\u_{1,P}^{}+
\phi^*_{2,b^{-1}P}+
\phi^*_{2,a^{-1}P}\*\kappa_1\;\u_{1,P}^{}\;\w_{1,P}^{}+
\phi^*_{3,P}\*\w_{1,P}^{}\;,\\
&&\\
\ds f^{*}_{2,P} & \equiv & \ds
\phi^{*}_{1,P}\*\kappa_2\;\u_{2,P}^{}\;\w_{2,P}^{}+
\phi^*_{2,P}\*q^{1/2}\;\u_{2,P}^{}+
\phi^*_{2,b^{-1}P}\*\w_{2,P}^{}+
\phi^*_{3,aP}\;,\\
&&\\
\ds f^{*}_{3,P} & \equiv & \ds
\phi^{*}_{1,P}\*\w_{3,P}^{}+
\phi^*_{2,P}+
\phi^*_{2,a^{-1}P}\*\kappa_3\;\u_{3,P}^{}\;\w_{3,P}^{}+
\phi^*_{3,bP}\*q^{1/2}\;u_{3,P}^{}\;.
\end{array}\right.$$ Here $f_{j,P}^*$ corresponds to $(j,P)$-th vertex. The assignment of the co-currents is shown in Fig. \[fig-cocurrents\].
(450,200) (00,0)
(200,200) (0,150)[(1,0)[200]{}]{}(50,150) (0,200)[(1,-1)[200]{}]{}(150,50) (150,0)[(0,1)[200]{}]{}(150,150) (160,160)[$1,P$]{} (55,160)[$3,P$]{} (160,55)[$2,P$]{} (105,120)[$\phi^*_{1,P}$]{} (180,180)[$\phi^*_{2,P}$]{} (50,50)[$\phi^*_{3,P}$]{} (10,200)[$\z_P$]{} (0,140)[$\y_P$]{} (130,0)[$\x_P$]{} (190,20)[$\z_{a^{-1}bP}$]{} (190,135)[$\y_{bP}$]{} (120,190)[$\x_{aP}$]{}
(220,80)[$\ds\stackrel{\U}{\mapsto}$]{} (250,0)
(200,200) (50,0)[(0,1)[200]{}]{}(50,50) (0,50)[(1,0)[200]{}]{}(50,150) (0,200)[(1,-1)[200]{}]{}(150,50) (20,30)[$1,P$]{} (60,155)[$2,a\,P$]{} (155,60)[$3,b\,P$]{} (70,75)[$\phi^{*\smb}_{2,P}$]{} (10,10)[$\phi^{*\smb}_{1,P}$]{} (140,140)[$\phi^{*\smb}_{3,abP}$]{} (0,60)[$\y_P$]{} (0,175)[$\z_P$]{} (60,0)[$\x_P$]{} (190,60)[$\y_{bP}$]{} (150,0)[$\z_{a^{-1}bP}$]{} (60,190)[$\x_{aP}$]{}
Elements of $F^*\;=\;\Phi^*\*{\bf L}$ have the following remarkable feature: coefficients in $f^*_{j,P}$ belong to the algebra of $\u_{j,P}$, $\w_{j,P}$ only. We will use this in the next subsection.
Properties of ${\bf L}$ and the quantum determinant
---------------------------------------------------
Consider first the general properties of equation $\Phi^*\*{\bf V}\;=\;0$ for a matrix ${\bf V}$ similar to ${\bf L}$ introduced: $$\ds {\bf V}\;=\;||\v_{j,k}||\;\;,$$ with the commutative columns, $$\label{commutativity}
\ds \forall j,j'\;:\;\;\;\;
\v_{j,k}\* \v_{j',k'}\- \v_{j',k'}\* \v_{j,k}\;=\;0\;\;\;
\mbox{if}\;\;\; k'\neq k\;.$$ Such matrices have the following properties.
[**Property**]{} 1: Consider a system $$\ds \sum_{j}\;z_j\*\v_{j,k}\;=\;\alpha_k$$ with $\alpha_k$ being $C$ – numbers, as the system for $z_j$. Then for $k\neq k'$ $$\ds
\alpha_k\;\alpha_{k'}\-\alpha_{k'}\;\alpha_k\;=\;
\sum_{j'}\; z_{j'}\;\alpha_k\;\v_{j',k'}\-
\sum_j\; z_j\;\alpha_{k'}\;\v_{j,k}\;=\;
\sum_{j,j'}\; (z_{j'}z_{j}-z_{j}z_{j'})\;\v_{j,k}\;\v_{j',k'}
\;=\;0\;.$$ Matrix $||\v_{j,k}\*\v_{j',k'}||$ is non-degenerative in general, so the last equality gives immediately $$\label{z-com}\ds
z_{j}\* z_{j'}\;=\; z_{j'}\* z_{j}\;.$$
[**Consequence**]{}: Let $||\widetilde{\v}_{i,j}||$ is the inverse to $||\v_{j,k}||$ matrix: $$\ds \sum_{j}\;\widetilde{\v}_{i,j}\* \v_{j,k}\;=\;
\sum_{j}\;\v_{i,j}\*\widetilde{\v}_{j,k}\;=\;\delta_{i,k}\;,$$ then $$\ds \forall i\;\;\;\;
\widetilde{\v}_{i,j}\*\widetilde{\v}_{i,j'}\;-\;
\widetilde{\v}_{i,j'}\*\widetilde{\v}_{i,j}\;=\;0\;.$$
[**Property**]{} 2: Because of in $||\v_{j,k}||$ non-commutative elements belong to the same column, the algebraic supplements $V_{k,l}$ as well as the quantum determinant $\mbox{\sf det}\;( \v)$ are well defined. Here we’ve used the notation “$\mbox{\sf det}$” as the formal operator-valued determinant $$\ds \mbox{\sf det}\;||\v_{i,j}||\;=\;
\sum_{\sigma}\;(-1)^\sigma\;\prod_{j}\;\v_{j,\sigma(j)}\;.$$ $V_{i,j}$ and $\mbox{\sf det}\;(\v)$ are polynomials of $\v_{j,k}$ such that in each summand all multipliers belong to different columns and thus commute. Moreover, if in $||\v||$ two rows coincide, then $\mbox{\sf det}\;(\v)\;\equiv\;0$. Hence $$\ds
\sum_{k}\;\; \v_{j,k}\* V_{k,l}\;=\;\delta_{j,l}\;\;
\mbox{\sf det}\;(\v)\;.$$ Note, $\ds \v_{j,k}\* V_{k,l}\;=\;V_{k,l}\*\v_{j,k}$.
As it was mentioned previously, sometimes it is useful to introduce formally a module for the body of $||\v_{j,k}||$. Here we do this, introducing $\phi^*_j$ and $\phi^*_0$ which belong to such formal module. This allows us to formulate the following
[**Consequence**]{}: Consider now the system of co-vector equations $$\label{veceq}
\ds
(\Phi^*\*{\bf V})_k\;=\;\sum_{j}\;\;\phi^*_j\*\v_{j,k}\;=\;0\;.$$ Due to property 2 all $\phi^*_j$ belong to the null space of $\mbox{\sf det}\;(\v)$: $$\ds\phi^*_j\*\mbox{\sf det}\;(\v)\;=\;0\;.$$ From the other hand side, $\phi^*_{j}$-s are connected by $z_{j,j'}$ – some rational functions of $\v_{j,k}$: $$\ds
\phi^*_{j'}\;=\;\phi^*_{j}\* z_{j,j'}\;.$$ Property 1 provides the commutability of $z_{j,j'}$, hence a solution of (\[veceq\]) can be written as $$\label{properties}
\ds
\phi^*_j\;=\;\phi^*_0\*z_j\;,\;\;\;
z_j\*z_{j'}\;=\;z_{j'}\*z_j\;,\;\;\;
\phi^*_0\*\mbox{\sf det}\;(\v)\;=\;0\;,\;\;\;
z_j\*\mbox{\sf det}\;(\v)\;=\;\mbox{\sf det}\;
(\v)\*\widetilde z_j\;,$$ where in general $z_j\;\neq\;\widetilde z_j$.
Apply now both properties and their consequences to ${\bf L}$ given by (\[L-matrix\]). First, for any representation of the Weyl algebrae the null subspace $\phi^*_0$ of whole Gilbert space is defined, $$\ds \phi^*_0\*\mbox{\sf det}\;({\bf L})\;=\;0\;.$$ The existence of $\phi^*_0$ means the solvability of $\Phi^*\*{\bf L}\;=\;0$. Corresponding $z_j$ have the lattice structure, $z_{j,P}$. These commutative elements are assigned to the sites of the kagome lattice, and observable are $z_{j,P}^{}\;z_{j'.P'}^{-1}$. These operators connect the co-currents in different sites, and thus $z_{j,P}$ actually give the realisation of the path group on the kagome lattice.
Another important thing is that due to $T_a^M\;=\;A$ and $T_b^M\;=\;B$, $\mbox{\sf det}({\bf L})$ is a Laurent polynomial with respect to the quasimomenta $A$ and $B$.
Evolution of the co-currents and integrals of motion
----------------------------------------------------
Consider now the shift of the inclined lines giving the evolution. The internal currents as well as the co-currents change, and we can trace these changes.
Introduce two extra matrices, ${\bf K}$ and ${\bf M}$: $$\label{K-matrix}
\ds
{\bf K}\;=\;
\left(\begin{array}{rcrcr}
0 &,& \Lambda_0 &,& 0\\
&&&&\\
0 &,& 0 &,& T_a\;T_b \\
&&&&\\
1 &,& K_{3,2} &,& 0
\end{array}\right)\;,$$ where $$\ds
\Lambda_0\;=\;{\kappa_1\over\kappa_2}
\;q^{-1/2}\;\w_1^{}\,\u_2^{-1}\,\w_3^{-1}\;+\;
{\kappa_3\over\kappa_2}\;\u_1^{-1}\,\w_2^{-1}\,\u_3^{}\;,$$ $$\ds
K_{3,2}\;=\;
T_a^{-1}\;\;q^{-1/2}\;\;\Lambda_2\+{\kappa_3\over\kappa_2}
\;\;\Lambda_1\+
T_b^{-1}\;\;{\kappa_1\over\kappa_2}\;\;\Lambda_3\;.$$ with $\Lambda_j$ standing for the diagonal matrices with the entries given by(\[ev\]) correspondingly, and $$\label{M-matrix}
\ds
{\bf M}\;=\;
\left(\begin{array}{rcrcr}
0 &,& \u_1^{-1}\;\u_2'\;T_a &,& q^{-1/2}\;\u_1^{-1}\;T_b\\
&&&&\\
\ds{\kappa_1\over\kappa_2}\;\w_2^{-1}\;
\u_2^{-1}\;\u_1'\;\w_1' &,& 0 &,&
\ds{\kappa_3\over\kappa_2}\;\w_2^{-1}\;
\u_2^{-1}\;\u_3'\;\w_3'\;T_b\\
&&&&\\
\w_3^{-1} &,& \w_3^{-1}\;\w_2'\;T_a &,& 0
\end{array}\right)\;.$$ Apply the evolution operator $\U$ to $\bf L$: $\ds {\bf L}^\smb\;\equiv\;\U\*{\bf L}\*\U^{-1}$, $$\ds
{\bf L}^\smb\;=\;
\left(\begin{array}{rcrcr}
\w_1' &,& 1 &,& q^{1/2}T_b^{-1}\u_3'T_b^{}\\
&&&&\\
q^{1/2}\u_1' &,& \kappa_2\u_2'\w_2'T_a^{} &,& \w_3'T_b^{} \\
&&&&\\
T_a^{-1}+ T_b^{-1}\kappa_1\u_1'\w_1' &,&
T_a^{-1}(\w_2'+T_b^{-1}q^{1/2}\u_2')T_a^{} &,&
T_a^{-1}+T_b^{-1}\kappa_3\u_3'\w_3'T_b^{}
\end{array}\right).$$ The following relation can be verified directly: $$\label{KLM}
\ds
{\bf K}\*{\bf L}^\smb\;=\;
{\bf L}\* {\bf M}\;.$$ $\bf M$ in general is the matrix making $\phi_{k,P}^\smb\mapsto\phi_{k,P}$, and $\bf K$ makes $\phi^*_{k,P}\mapsto
\phi^{*\smb}_{k,P}$. Also ${\bf K}$ and ${\bf M}$ admit $$\ds
{\bf K}\;\mapsto\;{\bf K}\+ {\bf L}\* {\bf N}\;,\;\;\;\;
{\bf M}\;\mapsto\;{\bf M}\+ {\bf N}\* {\bf L}^\smb$$ with arbitrary ${\bf N}$. One can prove the following
[**Proposition**]{}: $\bullet$ ${\bf K}\*\mbox{\sf det}({\bf L})\;=\;
\mbox{\sf det}({\bf L})\*\widetilde{\bf K}$ $\bullet$
One can understand this in other terms. Since $\Phi^*\*{\bf L}\;=\;0$ can be solved for $t=0$, then for $t=1$ equation $\Phi^{*\smb}\*{\bf L}^\smb\;=\;0$ must also be solved because they are bounded by simple linear relations. Hence subspace $\phi^*_0$ must coincide with $(\phi^*_0)^\smb$, i.e. $$\label{dets}
\ds
\mbox{\sf det}({\bf L}^\smb)\;=\;\mbox{\sf det}({\bf L})\*D\;,$$ with some operator $D$. One may hope, $D$ is not too complicated, and (\[dets\]) is not trivial.
Careful analysis of ${\bf K}$ and ${\bf M}$ shows that this $D$ does not depend on the quasimomenta $A$ and $B$. In the functional limit $q^{1/2}\mapsto\pm 1$ one may easily calculate the determinants of ${\bf K}$ and ${\bf M}$, both them are proportional to $A^M\;B^M$, and this term cancels from the determinants of the left and right hand sides of (\[KLM\]). This is so and in the quantum case.
Hence $D$ in (\[dets\]) is a ratio of any $A,B$ – monomials from $\mbox{\sf det}({\bf L})$ and $\mbox{\sf det}({\bf L}^\smb)$. Element $D$ can be extracted, say, from $A^M\;B^{-M}$ component of $\mbox{det} ({\bf L})$: $$\ds
D\;=\;\prod_{P}\;\;\u_{1,P}^{-1}\*
\prod_{P}\;\;\u_{1,P}^\smb\;.$$ This means that we can introduce a simple operator $d$: $$\ds D\;=\;d\*d^{\smb -1}\;.$$ Thus $$\ds\J\;=\;\mbox{\sf det} \;(\;{\bf L}\;)\* d$$ is the invariant of the evolution, $\J^\smb\;=\;\J$, i.e. $$\ds\U\*\J\;=\;\J\*\U\;.$$ Decompose $\J$ as a series of $A$ and $B$, $$\ds\J\;=\;\sum_{\alpha,\beta\in\Pi}
\;\;A^\alpha\;B^\beta\;\J_{\alpha,\beta}\;,$$ where $\alpha$ and $\beta$ are integers and their domain (Newton’s polygon) $\Pi$ is defined by $|\alpha|\leq M$, $|\beta|\leq M$ and $|\alpha+\beta|\leq M$. Quasimomenta $A$ and $B$ are arbitrary ${\cal C}$ – numbers, and the invariance of $\J$ means the invariance of each $\J_{\alpha,\beta}$. From the other side, $\J$ is a functional of the dynamical variables of the lattice, i.e. $$\ds
\J_{\alpha,\beta}\;=\;\J_{\alpha,\beta}(
\{\u_{j,P}\,,\,\w_{j,P}\})\;.$$ Surely, due to the homogeneity of the lattice these functionals does not depend on time layer, and hence the conservation of $\J$, $\J^\smb\;=\;\J$, means $$\ds
\J_{\alpha,\beta}(\{\u_{j,P},\w_{j,P}\})\;=\;
\J_{\alpha,\beta}(\{\u_{j,P}^\smb,\w_{j,P}^\smb\})\;,$$ i.e. functionals $\J_{\alpha,\beta}$ give the integrals of motion in usual sense.
Note further, $$\label{U-on-z}
\ds\Phi^*\*{\bf K}\;\sim\;\Phi^*\*\U^{-1}\;,$$ where it is supposed $\phi^*_0\*\U^{-1}\;\sim\;\phi^*_0$, and (\[U-on-z\]) gives the linear action of $\U$ on $z_{j,P}$. To get the equality from (\[U-on-z\]), one has to normalize only one component of $\Phi^*$.
Some elements of $\mbox{\sf det}({\bf L})$, corresponding to the border of the Newton polygon $\Pi$ of $\J(A,B)$, can be easily calculated. Operator $d$ introduced is defined up to any integral of motion. The simplest integrals are $$\ds
j_1\;=\;\prod_P\;\u_{2,P}\;\u_{3,P}\;,\;\;\;\;
j_2\;=\;\prod_P\;\u_{1,P}\;\w_{3,P}^{-1}\;,\;\;\;
j_3\;=\;\prod_P\;\w_{1,P}\;\w_{2,P}\;,$$ and the convenient choice of $d$ is $$\label{mu}
\ds d\;=\;
\prod_{P}\; \biggl( q^{1/2}\;
\u_{2,P}\*\u_{3,P}\*\w_{3,P}\biggr)^{-1}\;.$$ $d$ can be absorbed into det, $$\ds\J\;\;=\;\;\mbox{\sf det}\;({\bf L}^{(0)})\;,$$ where $$\label{L0-matrix}
\ds
{\bf L}^{(0)}\;\;=\;\;
\left(\begin{array}{rcrcr}
\w_1^{} &,& q^{-1/2}\;\u_2^{-1} &,& q^{-1/2}\;\w_3^{-1}\\
&&&&\\
q^{1/2}\;\u_1^{} &,& T_a\;q^{1/2}\;\kappa_2\;\w_2^{} &,&
T_b\;\u_3^{-1} \\
&&&&\\
T_a^{-1}\+ T_b^{-1}\;\kappa_1\;\u_1^{}\;\w_1^{} &,&
q^{1/2}\;\u_2^{-1}\;\w_2^{}\+T_b^{-1} &,&
T_a^{-1}\;\w_3^{-1}\;\u_3^{-1}\+\kappa_3
\end{array}\right)\;,$$ Whole number of $\J_{\alpha,\beta}$ is $3M^2+3M+1$, and there are $3M^2+1$ independent between them, and between these one can choose only $3M^2$ commutative, so $\J$ gives the complete set of integrals. (As to whole number of summands in $\J$, e. g. for $M=2$ it is $1536\;=\;2^9\; 3$.) The existence of $3\,M^2$ abelian integrals is the hypothesis tested for small $M$-s.
All integrals corresponding to the boundary of domain $\Pi$, $|\alpha|=M$, $|\beta|=M$, $|\alpha+\beta|=M$, are equivalent to the following $3M$ elements: $$\label{overline}
\ds\left\{\begin{array}{ccl}
\ds \overline{\u}_j & = & \ds
\prod_{\sigma}\;\;
\w^{-1}_{1,a^\sigma b^j P_0}\;
\w^{-1}_{2,a^\sigma b^j P_0}\;,\\
&&\\
\ds \overline{\v}_j & = & \ds
\prod_{\sigma}\;\;
\u^{}_{2,a^{j+\sigma} b^{-\sigma} P_0}\;
\u^{}_{3,a^{j+\sigma} b^{-\sigma} P_0}\;,\\
&&\\
\ds \overline{\w}_j & = & \ds
\prod_{\sigma}\;\;
\u^{}_{1,a^j b^{\sigma} P_0}\;
\w^{-1}_{3,a^j b^{\sigma} P_0}\;,
\end{array}\right.$$ where $P_0$ is some frame of reference’s point as previously. Note, $\overline{\v}_j$ are not $T_a,T_b$ – invariant, but restoring this invariance in any way, one obtains the invariants of $\U$. Between $\overline{\w}_j,\overline{\u}_j,\overline{\v}_j$ one may choose $3M-1$ commutative elements. Inner part of $\Pi$ gives $3M^2-3M+1$ highly complicated independent integrals, which gives $g\;=\;3M^2-3M+1$ commutative (up to (\[overline\])) independent elements. Note, $g$ is the formal genus of the curve $\J(A,B)=const$.
Walks on the lattice and the integrals of motion
------------------------------------------------
Give now a geometrical interpretation of the integrals of motion. This interpretation follows directly from the analysis of the determinant. Every integral of motion is a sum of monomials associated with walks on the lattice such that all the walks have the same homotopy class with respect to the torus on which the kagome lattice is defined.
It is useful to formulate the walks in terms of general variables $\a$, $\b$, $\c$ and $\d$ as in Fig. \[fig-abcd-currents\]. Recall the shorter notation $W\;=\;\{\a,\b,\c,\d\}$ for the dynamical variables’ set. Consider the matrix ${\bf L}$ in this general case. Each row in ${\bf L}$ corresponds to a vertex of the lattice, and each column of ${\bf L}$ corresponds to a polygon (i.e. to a site) of the lattice. Thus $\mbox{\sf det} ({\bf L})$ consists on the monomials, each of them corresponds (up to a sign) to a product of different $W_{j,P}$ such that:
- for any vertex $(j,P)$ of $\a_{j,P}$, $\b_{j,P}$, $\c_{j,P}$, $\d_{j,P}$ is taken in this monomial, and
- for any site of surrounding $\a$, $...$, $\d$ is taken in this monomial.
Take the lattice and mark the places of the vertex variables $\a$, $...$, $\d$, corresponding to the monomial, by the arrows, ingoing to the corresponding vertices. Thus for any site and for any vertex we have painted only one arrow.
In order to get a purely invariant functional, we have to multiply $\mbox{\sf det} ({\bf L})$ by an integrating monomial, in general case this monomial is, for example, $\ds\prod_{P}\;\a_{1,P}^{-1}\;\d_{2,P}^{-1}\;\b_{3,P}^{-1}$. This choice of the integrating multiplier corresponds to element $d$ given by (\[mu\]). It is easy to see that this monomial has the same structure as described above. But due to the power $-1$ we may interpret geometrically this monomial as the set of outgoing arrows.
The system of the outgoing arrows is thus fixed and shown in Fig. \[fig-outlet\] for each $\lgr$ – type triangle of the lattice. For the system of the outgoing arrows and any system of ingoing arrows the following is valid:
- for any site there exist exactly one outgoing arrow and exactly one ingoing arrow, and they may touch the same vertex, and
- for any vertex there exist exactly one outgoing arrow and exactly one ingoing arrow, and they may belong to the same site.
Hence there is the unique way to connect all the arrows inside each site so that a walk appears.
(450,200) (125,0)
(200,200) ( 10 , 70 )[(1,0)[180]{}]{} ( 40 , 10 )[(1,2)[90]{}]{} ( 160 , 10 )[(-1,2)[90]{}]{} ( 70 , 70 )(70,55)[$3,P$]{} ( 130 , 70 )(136,73)[$2,P$]{} ( 100 , 130 )(106,128)[$1,P$]{} (100,130)[(-1,0)[25]{}]{} (70,70)[(2,1)[20]{}]{} (130,70)[(2,-1)[20]{}]{}
So, the walks we consider, obey the following demands:
- the system of outlets of the walk is fixed and given by Fig. \[fig-outlet\],
- the walk visits any site only once,
- the walk must visit all the sites and
- the walk must visit all the vertices.
For any walk ${\cal W}$ let $\sigma({\cal W})$ be the number of the components of the connectedness (i.e. the number simply connected subwalks).
Let now walk ${\cal W}$ belongs to a given homotopy class $\alpha\;{\cal A}\;+\;\beta\;{\cal B}$ of the torus, where cycle ${\cal A}$ corresponds to $T_a^M$ and cycle ${\cal B}$ corresponds to $T_b^M$, and denote such walk as ${\cal W}_{\alpha,\beta}$.
To a walk given assign a monomial according to the following rules: let the walk pass through vertex $(j,P)$ so that the walk ingoes the vertex form the side $\x$ $\in$ $W_{j,P}$, and outgoes the vertex from the side $\y$ $\in$ $W_{j,P}$. Then the multiplier corresponding to $(j,P)$ is $\ds\x\*\y^{-1}$. The monomial $\J_{\cal W}$ is the product of such multipliers corresponding to all the vertices. Thus the reader may see that each monomial we construct gains the structure of an element of ${\cal B}^\prime_P$, described in section “Auxiliary linear problem”, subsection “General approach”: monomial $\J_{\cal W}$, $$\ds
\J_{\cal W}\;=\;
\cdots\;\;\x\*\y^{-1}\*\x^\prime\*\y^{\prime-1}\;\;\cdots\;,$$ $\x$ and $\y$ are assigned to a same vertex, so $\x\*\y^{-1}$ does not contain the vertex projective ambiguity, and $\y$ and $\x^\prime$ belong to a same site, so $\y^{-1}\*\x^\prime$ does not contain the site ambiguity. Finally we have to provide the projective invariance of $\J_{\cal W}$ with respect to the start and end points of each simply connected subwalk. In our case of the local Weyl algebrae this invariance is obvious, because of elements $\x\*\y{-1}$ for different vertices commute.
With the structure of the walks introduced, the simple analysis of the determinant gives immediately $$\ds
\J_{\alpha,\beta}\;\;=\;\;
\sum_{\ds\mbox{all}\;\;{\cal W}_{\alpha,\beta}}
\;\;\; (-)^{\ds\sigma({\cal W}_{\alpha,\beta})}
\;\*\; \J_{\ds{\cal W}_{\alpha,\beta}}\;,$$ where the sum is taken over all the walks of the homotopy class $\alpha\;{\cal A}\;+\;\beta\;{\cal B}$ given and the system of the outlets of the walks fixed.
Monodromy operator
------------------
Consider now another interpretation of the two dimensional kagome lattice.
Let now to each vertex of the lattice the local $\L$-operator is assigned. Instead of $\omega$ (\[omega\]) in the definition of $\L$, use $$\ds \x'\;=\;\C^{-1}\*\x\;,\;\;\;\;\;
\y'\;=\;\C^{}\*\y\;,$$ where $$\ds\C\;=\;\C(\x^{-1}\u , \y^{-1}\w)\;$$ and $$\ds
\C(\u,\w)\;=\;\u^{-1}\;-\;q^{1/2}\;\u^{-1}\w\;+\;
\kappa\;\w\;,$$
For the $\lgr$-type triangle $P$ let the in - edge variables be $\x_P$, $\y_P$, $\z_P$ so that out edge variables are $\x_{aP}$, $\y_{bP}$ and $\z_{cP}$, $c\;=\;a^{-1}\, b$. These notations are shown in Fig. \[fig-edge-triangle\]. Surely LYBE for $\L$-operators means that for $\x_P$, $\y_P$, $\z_P$ given, $\x_{aP}$, $\y_{bP}$ and $\z_{cP}$ are the same for the right hand side YBE graph $\rgr$.
(450,200) (125,0)
(200,200) ( 10 , 70 )[(1,0)[180]{}]{} ( 40 , 10 )[(1,2)[90]{}]{} ( 160 , 10 )[(-1,2)[90]{}]{} ( 70 , 70 )(76,73)[$3,P$]{} ( 130 , 70 )(136,73)[$2,P$]{} ( 100 , 130 )(106,128)[$1,P$]{} (0,60)[$\z_P$]{} (80,50)[$\C_{3,P}\z_P$]{} (195,60)[$\z_{cP}=\C_{2,P}\C_{3,P}\z_P$]{} (30,0)[$\y_P$]{} (30,100)[$\C_{3,P}^{-1}\y_P$]{} (135,190)[$\y_{bP}=\C_{1,P}\C_{3,P}^{-1}\y_P$]{} (165,0)[$\x_P$]{} (120,100)[$\C_{2,P}^{-1}\x_P$]{} (-50,190)[$\x_{aP}=\C_{1,P}^{-1}\C_{2,P}^{-1}\x_P$]{}
Consider now the whole toroidal kagome lattice. We are going to assign $\x$, $\y$ and $\z$ to some minimal set of the edges so that the variables of all other edges can be restored via functions $\C_{j,P}$.
To do this, cut the torus along some line, shown as the dashed line in Fig. \[fig-monodromy\]. Call this line ‘the string’. The edge variables along the string we’ll denote as $\x_j$, $\y_j$ and $\z_j$. It is useful to draw the string so that it intersects all $\x$ and $\z$ lines once, and $\y$-lines twice (i.e. the homotopy class of the string is $\pm(2\;{\cal A}-{\cal B})$, an orientation of the string and so a sign are unessential) Note, $\x_j$, $\y_j$ and $\z_j$ introduced we assign to the edges which are right-touched to the string.
(400,400) (0,0)
(400,400) (100,400)(0,-20)[10]{}[(0,-1)[15]{}]{} (100,200)(20,-20)[10]{}[(1,-1)[15]{}]{} (300,400)(0,-20)[10]{}[(0,-1)[15]{}]{} (300,200)(20,-20)[5]{}[(1,-1)[15]{}]{} (0,100)(20,-20)[5]{}[(1,-1)[15]{}]{} (100,220)[(1,0)[200]{}]{} (100,200)[(1,0)[200]{}]{} (120,180)[(1,0)[200]{}]{} (85,215)[$\y_2$]{}(85,195)[$\y_1$]{}(105,175)[$\y_0$]{} (180,120)[(0,1)[280]{}]{}(180,0)[(0,1)[100]{}]{} (200,100)[(0,1)[300]{}]{}(200,0)[(0,1)[80]{}]{} (220,80)[(0,1)[320]{}]{}(220,0)[(0,1)[60]{}]{} (175,105)[$\x_0$]{}(195,85)[$\x_1$]{}(215,65)[$\x_2$]{} (100,320)[(1,-1)[300]{}]{}(20,400)[(1,-1)[60]{}]{} (100,300)[(1,-1)[300]{}]{}(0,400)[(1,-1)[80]{}]{} (100,280)[(1,-1)[280]{}]{}(0,380)[(1,-1)[80]{}]{} (0,20)[(1,-1)[20]{}]{}(380,400)[(1,-1)[20]{}]{} (85,329)[$\z_2$]{}(85,309)[$\z_1$]{}(85,289)[$\z_0$]{} (0,0)[(1,0)[400]{}]{}(0,0)[(0,1)[400]{}]{} (0,400)[(1,0)[400]{}]{}(400,0)[(0,1)[400]{}]{}
Now switch on the $\L$-operator game with the edge variables. We interpret it as the shift of the string. We enumerate the lines so that the triangle $P\;=\;a^{j}\;b^{i}\;P_0$ is surrounded by the lines $\x_{i}$, $\y_{j}$ and $\z_{i+j}$, so as the $\L$ – operators are $$\label{L-on-lattice}
\ds
\L_{\x_i,\y_j}(\{\u,\w\}_{1,a^{j}\,b^{i}\,P_0})\;,\;\;\;\;
\L_{\x_i,\z_k}(\{\u,\w\}_{2,a^{k-i}\,b^{i}\,P_0})\;,\;\;\;\;
\L_{\y_j,\z_k}(\{\u,\w\}_{3,a^{j}\,b^{k-i}\,P_0})\;.$$ The $\L$-operator game allows us to restore all the edge variables for the lattice, including the left-touched to the original string variables $\x_j'$, $\y_j'$, $\z_j'$.
Thus for given variables from the right side of the string we obtain the analogous values from the left side of the string as functionals of the given variables. Thus the map corresponding to the kagome lattice and the string chosen appears: $$\ds T\;(\L)\;:\;\{\x_i,\y_j,\z_k\}\;\;\mapsto\;\;
\{\x'_i,\y'_j,\z'_k\}\;.$$
As the operator, $T(\L)$ is ordered product of all $\L$ (\[L-on-lattice\]). Define $A<B$ if the ordered product of $A$ and $B$ is $A\* B$. Then in $T(\L)$ $$\begin{aligned}
&\ds\L_{\y_j,\z_{i+j}}\;<
\;\L_{\x_i,\z_{i+j}}\;<
\;\L_{\x_i,\y_j}\;,&\\
&\ds\L_{\x_i,\y_j}\;<
\;\L_{\x_i,\z_{i+j+1}}\;<
\;\L_{\y_j,\z_{i+j+1}}\;.&\end{aligned}$$ These relations are enough to restore $T(\L)$. Operator $T(\L)$ resembles the monodromy matrix in $2D$. The difference is that instead of the distinguished point in $2D$ monodromy matrix (i.e. the point where the transfer matrix is torn), in $3D$ we have the distinguished string.
Now, what should stand for a “trace” of $T(\L)$. Consider the system $$\label{trace}
\x_j'\;=\;\x_j\;,\;\;\;\;\y_j'\;=\;\y_j\;,\;\;\;\;\z_j'\;=\;\z_j$$ on some left module element $\phi^*_0$. Here are $3\,M$ equations, $3\,M-1$ from them are independent due to $$\ds\prod_j\;\;\x_j\;\y_j\;\z_j\;=
\;\prod_j\;\;\x_j'\;\y_j'\;\z_j'\;,$$ where it is implied that all $\x_{P},\y_{P},\z_{P}$ $\forall P$ are commutative, this is the consequence of the commutability of primary $\x_j,\y_j,\z_j$. Then solve $3\,M-2$ equations of (\[trace\]) leaving two variables, up to unessential signs and powers of $q$ : $$\ds
A\;=\;\prod_{j}\;\;q\;\y_j^{}\;\z_j^{}\;,\;\;\;\;
B\;=\;\prod_{j}\;\;\x_j^{-1}\;\z_j^{}\;.$$ A single equation rests for $A$ and $B$, and amusingly this equation coincides with the quantum determinant relation $\phi_0^*\*\J(A,B)=0$. So in this sense $\J(A,B)\;=\;0$ is the trace of the monodromy operator.
Note however, $\J(A,B)$ is [**the invariant curve**]{}, this was established in the previous section, so it is not necessary to consider $\phi^*_0\*\J(A,B)\;=\;0$. The actual problem for the further investigations is to diagonalize $\J(A,B)$ for $A$ and $B$ arbitrary.
Discussion
==========
Conclude this paper by an overview of the problems to be solved and the aims to be reached. The approach proposed gives a way for their solution.
First, mention the problems of the classification of the map $$\ds
\R\;\;\;:\;\;\;\{\a_j,\b_j,\c_j,\d_j\}\;\;\mapsto\;\;
\{\a_j',\b_j',\c_j',\d_j'\}\;,
\;\;\;\;j=1,2,3,$$ in general. The aim is to classify all conserving symplectic structures of the body ${\cal B}$. We have discussed only the local case, when the variables, assigned to different vertices commute, and the scalars (spectral parameters) are conserved. We suppose, such case is not unique, and there are another ways to remove the projective ambiguity. The simplest case to be investigated is to consider all the variables $\a,\b,\c,\d$ for each vertex as matrices with, for example, non-commutative entries, but with this entries commutative for any two vertices. The matrix structure may be common for all vertices, and thus we would have no commutation between different vertices in general. Another simple possibility is another kind of locality, the case when the dynamical variables commute while do not belong to a same site.
Note, once our locality is imposed, the Weyl structure appears immediately. Thus the Weyl algebra is the consequence of the locality technically, but a principal origin of the Weyl algebra is mysterious.
Next fundamental problem is the quantization of Korepanov’s matrix model mentioned above. The conservation of complete algebra ${\cal X}$, (\[algebra\],\[z-eq\]), means that we can not use (\[ad-bc\]) to fix all the ambiguity of Korepanov’s equation. Analysis of (\[KE\]) [*plus*]{} some extra (but natural) symmetry conditions allows to fix the functional map $r$ : $X_j^{}\mapsto X_j'$ up to one unknown function of three variables. The problem of the Tetrahedron equation for these $r$ is open. All these are a subject of a separate paper.
Pure technical problem to be mentioned is the investigation of $q$ - hypergeometrical function $\sigma$, eqs. (\[sigma-psi\],\[sigma-chi\]).
Another interesting thing is functional $\L$ – operators and LYBE related to them. The map given by $\L$, eqs. (\[omega\],\[L-op\]), is a bi-rational one. Note, the case of linear $\L$ coincides with Korepanov’s $X$. Thus the rational case of it as well as the general case of the bi-rational transformation have good perspectives for the investigation.
The main set of problems for further investigations is connected with the integrals of $\U$. $\J(A,B)$ seems to be not constructive. The aim is to calculate the spectrum of it, and to calculate $\U$ as a function of its integrals. Possible approach is functional equations for the integrals of motion, that should follow from the determinant or topological representation of $\J$.
Another possibility is a way resembling the Bethe ansatz in 2D should exist in 3D, i.e. a way of a triangulation of $\U$ with a help of some artificial operators. If such way exists, it must based on the linear problem derived.
[**Acknowledgements**]{} I would like to thank sincerely Rinat Kashaev, Igor Korepanov and Alexey Isaev for their interest to this work and many fruitful discussions. Many thanks also for Yu. Stroganov, G. Pronko, V. Mangazeev and H. Boos.
The work was partially almost supported by the RFBR grant No. 98-01-00070.
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---
abstract: 'We present a concrete picture of spoof surface plasmons (SSPs) combined with cavity resonance to clarify the basic mechanism underlying extraordinary light transmission through metal films with subwavelength slits or holes. This picture may indicate a general mechanism of metallic nanostructure optics: When light is incident on a non-planar conducting surface, the free electrons cannot move homogeneously in response to the incident electric field, i.e., their movement can be impeded at the rough parts, forming inhomogeneous charge distributions. The oscillating charges/dipoles then emit photons (similar to Thomson scattering of x rays by oscillating electrons), and the interference between the photons may give rise to anomalous transmission, reflection or scattering.'
author:
- 'X. R. Huang'
- 'R. W. Peng'
- 'Z. Wang'
- 'F. Gao'
- 'S. S. Jiang'
title: 'Charge oscillation-induced light transmission through subwavelength slits and holes'
---
Since the discovery of extraordinary light transmission through metal films perforated by subwavelength hole arrays [@r1], tremendous theoretical and experimental work has been carried out to understand the underlying physics. Several mechanisms, particularly surface plasmons (SPs), have been proposed as the possible origins [@r2; @r3; @r4; @r5; @r6; @r7; @r8]. However, no universal understanding has been reached to date. Here based on first-principle calculations, we present a charge oscillation-induced light emission mechanism, which gives the origin of enhanced transmission in subwavelength systems and may also shed light on the fundamental of interactions between light and metallic nanostructures.
In non-magnetic media, the electric and magnetic fields are coupled by Maxwell’s equations $\nabla \times {\mathbf E} = -iK{\mathbf H}$ and $\nabla \times {\mathbf H} = iK\varepsilon{\mathbf E}$ ($K = 2\pi/\lambda$, $\lambda$ the wavelength in vacuum, and $\varepsilon$ the effective permittivity). For varying $\varepsilon({\mathbf r}$) (i.e., $\nabla\varepsilon\neq 0$), the divergence of the second equation generally gives $\nabla \cdot {\mathbf E} = -[(\nabla \varepsilon)
\cdot {\mathbf E}]/\varepsilon = 4\pi \rho \neq 0$, where $\rho$ is the charge density. Consider in Fig. 1 the free-standing one-dimensional (1D) gold grating illuminated by a plane wave, where Maxwell’s equations can be solved by the rigorous coupled-wave analysis (RCWA), a first-principle method [@r9]. For simplicity, we only discuss normal incidence in this Letter. Figure 1 shows the zero-order transmittance ($T_0$) spectra of both TM (${\mathbf H} \, \| \, \hat{\mathbf y}$) [@r4; @r5] and TE (${\mathbf E} \, \| \, \hat{\mathbf y}$) waves [@r10]. Here RCWA correctly reveals the cutoff wavelength $\lambda_c \simeq 2W$, above which transmission of TE waves is forbidden. The reason is that TE waves in the slit approximately take the waveguide modes $E_y \propto \sin(m\pi x/W )\exp[\pm \, i\pi z (4 /\lambda^2 - m^2/W^2)^{1/2}]$ ($m \neq 0$ being integers) [@r11], where for $\lambda > 2W$, all the modes are evanescent. The drastic difference between the TE and TM spectra stems from the fact that TE waves satisfy $\nabla \cdot {\mathbf E} \equiv 0$ while TM waves may induce electric charge oscillation $\rho({\mathbf r})e^{i\omega t}$ ($\omega$ the frequency), where $\rho({\mathbf r}) = \nabla \cdot {\mathbf E}({\mathbf r})/4\pi$.
![(color online) Transmission spectra of a gold grating with period $d = 3.5$, slit width $W = 0.5$, and thickness $h = 4$ $\mu$m. $\hat{\mathbf x}$, $\hat{\mathbf y}$, $\hat{\mathbf z}$ are unit vectors. The TM$'$ curve was calculated with Re\[$\varepsilon_c(\omega)$\] ($< 0$) for gold replaced by $-$Re\[$\varepsilon_c(\omega)$\].](Figure1.eps)
For TM polarization, the vertical component $E_z(x,z)$ of the electric field (invariant with $y$) has abrupt discontinuity across the surfaces at $z_s=0$ and $h$, from which one obtains the *surface charge density* $\tilde{\rho}_s(x, z_s) = \delta E_z(x, z_s)/4\pi$. In Fig. 2(a), the calculated $\tilde{\rho}_s(x, 0)$ profile correctly shows that charges only exist on the metal surface \[$\tilde{\rho}_s(x,0)=0$ for $0\!<\!x\!<\!W$\], and the charges tend to accumulate at the metal corners. At resonant wavelengths, $\tilde{\rho}_s(x,0)\simeq (-1)^N \tilde{\rho}_s(x,h)$, i.e., the charge patterns on the two surfaces are nearly the same, but they have opposite signs for odd resonant numbers $N$ (defined below). The *bulk charge density* is given by $\rho_v = \nabla \cdot {\mathbf E}/4\pi$. In Fig. 2(b), the $\rho_v (x, z = \mbox{const})$ curve calculated from the internal ${\mathbf E}$ eigenmodes reveals that there is a strong peak exactly centered at each slit wall, $x_w = 0$ or $W$ (plus any multiple of $d$). When sufficient diffraction orders (1601 orders in Fig. 2) are retained in RCWA, $\rho_v (x, z = \mbox{const})$ approaches a delta function across $x_w$ \[see Inset of Fig. 2(b)\], i.e. charges inside the grating also appear as surface charges on the walls.
The profile $\rho_v(0, z)$ \[$\equiv -\rho_v(W,z)$\] plotted in Fig. 2(c) shows that in the central range the charge density on the wall is nearly a standing wave (with wavevector $k_z \simeq 2\pi /\lambda$). This indicates that the electric field in the slit is also a standing wave [@r12; @r13] consisting of a forward wave ${\mathbf E}_a \simeq E_a \exp(-ik_z z) \hat{\mathbf x}$ and a backward wave ${\mathbf E}_b \simeq E_b \exp(ik_z z) \hat{\mathbf x}$ in Fig. 3. Note that $|\rho_v(x_w,z)|$ increases sharply when $z\rightarrow z_s$ in Figs. 2(c). Mathematically, the charge density at each corner consists of both $\tilde{\rho}_s(x_w,z_s)$ and $\rho_v(x_w, z\! \rightarrow \! z_s)$. Since they are always in phase, these two contributions are superimposed constructively. This makes the total charge densities at the corners much higher than in other regions, which is a typical edge effect. Consequently, there exist two large oscillating dipoles at the two ends of the slit.
Now we may obtain a clear picture about the light scattering process. As shown in Fig. 3, the incident electric field ${\mathbf E}_{in}$ drives (mainly) free electrons on the metal surface to move, but the movement is blocked at one corner of the metal, resulting in accumulation of electrons there. Meanwhile, extra positive charges appear at the other corner since some of the electrons have moved away. Thus, two dipoles ${\mathbf P}_a$ and ${\mathbf P}_r$ are formed. Since they oscillate with the incident wave, these dipoles act as *light sources* emitting new wavelets (photons), which form scattered waves (Thomson scattering). If we consider each period of the array as an overall scattering unit, the wavelets emitted from two adjacent units have a path difference $d\sin\theta$ along an arbitrary direction $\theta$. Note that for a *subwavelength* slit array with $d < \lambda$, we have $d\sin\theta \leq d <\lambda$, so the *phase difference* can never reach $2\pi$. This means that the oblique wavelets always tend to cancel each other out in the far fields (destructive interference, similar to the absence of x-ray diffraction at non-Bragg angles). Thus, they form evanescent waves near the surface. Mathematically, these evanescent waves can be expressed as ${\mathbf E}_m \exp[-(G_m^2-K^2)^{1/2}|z|-iG_m x]$, where $G_m = 2m\pi /d$ ($m \neq 0$ being integers) and $K = 2\pi /\lambda$ (note that $|G_m|>K$ for $d<\lambda$) [@r9].
One may prove that for any wavelength $\lambda>d$, the charge patterns on the upper surface are *always* the same as that in Fig. 2(a) except that the peak heights vary with $\lambda$. Therefore, the period of the charge density wave always equals the lattice constant $d$ and generally does not satisfy the dispersion relation $k_{sp} = K[\varepsilon_c/(1+\varepsilon_c)]^{1/2}$ of a classical SP (CSP) [@r14] at all ($\varepsilon_c$ the conductor’s permittivity and $k_{sp}$ the wavevector of the CSP). However, the evanescent waves are indeed surface-bound modes with their strengths decaying exponentially along $-z$. So they are *spoof* SPs (SSPs) [@r15], but their formation is the result of *charge oscillation-induced light emission and destructive interference*. The SSP model has been conceptually proposed by Pendry *et al*. [@r3] (also see [@r16]). Here we provide a concrete picture illustrating its origin.
The wavelets propagating along the backward direction ($\theta=0$) are always *in phase* (the path difference is zero). Therefore, the back reflected wave is not evanescent but a propagating mode. However, since ${\mathbf E}_R$ and ${\mathbf E}_r$ always have opposite directions, they tend to offset each other, thus reducing the overall back reflection.
In Fig. 3, dipole ${\mathbf P}_a$ also emits a wavelet ${\mathbf E}_a$ inside the slit. Similarly, *electrons on the slit walls also move in response to the electric field in the slit* (represented by the current ${\mathbf J}$). Then the charge movement is disrupted again at the exit edges, giving rise to another large dipole ${\mathbf P}_b$. Through this tunneling process, the charge patterns on the upper surface are duplicated on the lower surface. Oblique wavelets emitted from these duplicated light sources then form SSPs again below the grating. The sub-wavelets ${\mathbf E}_T$ and ${\mathbf E}_t$ along the forward direction form the zero-order transmitted wave. Overall, the grating only emits two propagating modes, the reflected and transmitted waves, that share the incident energy, while all the other modes are SSPs.
The oscillating dipole ${\mathbf P}_b$ can give a strong feedback to the upper surface by emitting a wavelet ${\mathbf E}_b$ propagating upward. If ${\mathbf E}_b$ is *not* in phase with ${\mathbf E}_a$ (and ${\mathbf E}_{in}$) at $z = 0$, it suppresses the strengths of ${\mathbf P}_a$ and ${\mathbf E}_R$. Then ${\mathbf E}_r$, *which includes specular reflection from the metal surface*, becomes dominant, leading to strong backward reflection. The weakened ${\mathbf E}_a$ meanwhile reduces ${\mathbf P}_b$. However, if ${\mathbf E}_b$ is in phase with ${\mathbf E}_a$ at $z = 0$, it enhances ${\mathbf P}_a$. The enhanced ${\mathbf P}_a$ subsequently strengthens ${\mathbf E}_a$, ${\mathbf P}_b$, ${\mathbf E}_b$, and so on. Then a Fabry-Perot-like resonant state is formed with the strengths of all the dipoles and wavelets maximized. Under this condition, ${\mathbf E}_r$ is largely offset by ${\mathbf E}_R$ in the far-field regions, leading to minimized backward reflection. Below the grating, wavelet ${\mathbf E}_t$ also partly offsets ${\mathbf E}_T$, but $|{\mathbf E}_T|$ can be much larger than $|{\mathbf E}_t|$ (unlike ${\mathbf E}_r$, ${\mathbf E}_t$ does not include any specular reflection). Therefore, when ${\mathbf E}_T$ is maximized, it maximizes $T_0$. The resonant wavelength is always slightly longer than $2h /N$, the ideal Fabry-Perot wavelength, where $N$ is the resonant number (number of the standing wave nodes). Although the Fabry-Perot-like resonance has been recognized before [@r4; @r5; @r12; @r13; @r17; @r18], here we explictily illustrated its origin. Particularly, Figure 3 shows that it is dipoles ${\mathbf P}_a$ and ${\mathbf P}_b$ that act as the two “reflecting planes” required to form a vertical resonator [@r13].
Note that when $\lambda \leq d$, some of the oblique waves satisfying $|G_m|\leq K$ become propagating diffracted waves (i.e., they are no longer SSP modes). The sharing of the incident energy by these diffracted waves significantly reduces $T_0$ (and backward reflection). This explains the much lower transmittance in the $\lambda \leq d$ range in Fig. 1.
Light scattering from a 2D hole array has a similar picture. The incident electric field drives electrons on the upper surface to oscillate. The charge movement is blocked at the hole edges, giving rise to oscillating dipoles at the entrance openings. Thus, the holes act as a 2D array of light sources. For a subwavelength lattice with $\max(d, d_2)< \lambda$ (Fig. 4), the phase difference between wavelets emitting from adjacent holes again is less than $2\pi$ along any oblique directions. So these wavelets form SSPs above the film except that the wavelets along $-z$ constitute a non-evanescent reflected wave (which offsets specular backward reflection). Meanwhile, the charge patterns are tunneled onto the lower surface due to the charge movement on the hole walls. Consequently, a similar set of SSPs and a forward transmitted wave are formed below the film.
However, 2D holes have a different tunneling mechanism. Consider the rectangular hole (a unit cell of an array) in Fig. 4. Except for the wave distortions near the ends, the electric fields in the hole roughly take the waveguide modes. For subwavelength holes, the basic 2D TE$_{1,0}$ modes dominate [@r3; @r11]. When $\lambda>2L$, the modes in the hole are evanescent and approximately take the forms ${\mathbf E}_{a} \propto \sin(\pi y /L) \exp(-\beta z)\hat{\mathbf x}$ and ${\mathbf E}_{b} \propto \sin(\pi y /L) \exp[-\beta (h-z)]\hat{\mathbf x}$, where $\beta = \pi(1/L^2-4/\lambda^2)^{1/2}$ [@r13]. ${\mathbf E}_a$ drives charges on the walls to move/oscillate, but the charge density decays with decaying ${\mathbf E}_a$ toward $+z$ due to the waveguide restriction. Nevertheless, the sharp discontinuity of the charge movements at the exit end can still cause significant accumulation of charges there, giving rise to a relatively large dipole ${\mathbf P}_b$, provided that $h$ is adequately small. Since ${\mathbf E}_a$ and ${\mathbf E}_b$ no longer have position-dependent phase factors, they are always in phase with each other and the incident wave, and they always resonate as the feedback ${\mathbf E}_b$ is always constructive. But the strengths of the dipoles and wavelets decrease with increasing $h$ due to the decaying effect. For fixed $h$ and in the absence of diffraction effects, the transmittance would increase monotonically with $\lambda$ decreasing toward $2L$ ($\beta$ decreasing). However, when $\lambda < \max(d, d_2)$, diffracted waves emerge, which reduces the zero-order transmittance in the short-wavelength range. Therefore, strongest transmission must occur for $\lambda > \max(d, d_2)$, the non-diffraction range, but meanwhile $\lambda$ should still be close to $\max(d, d_2)$ so that the damping coefficient $\beta$ is sufficiently small.
This picture agrees very well with the measured transmission spectra in the literature, and our finite-difference time-domain (FDTD) computations have unambiguously proved it. Figure 4(c) shows the calculated charge distribution $|\rho(x,y,0)|$ on a free-standing silver film with a hole array, where the charges again exist mainly on the two walls $x=x_w$ (perpendicular to the charge movement direction). The residual charges in other regions are surface charges and they disappear for $0<z<h$. (Here note that FDTD gives the overall charge densities.) The shape of the $\rho(0, y, z)$ \[$\equiv -\rho(W,y,z) \sim \sin(\pi y /L)$\] profile is independent of $z$, but the maximum density $\rho(0, L/2, z)$ changes with $z$ in inset II. Here our calculations indeed reveal the two large dipoles at the two openings of the hole (also see [@r19] for the single-hole case).
The above illustrations may indicate a general Thomson scattering mechanism of metallic nanostructures similar to x-ray scattering by oscillating electrons. That is, free electrons on a non-planar conducting surface cannot move homogeneously in response to the incident wave, thus forming inhomogeneous charge distributions. The oscillating charges then emit wavelets, and the interference between the wavelets may give rise to anomalous light scattering. The basic requirement here is free electrons, so this mechanism can explain scattering from various conducting nanostructures, including perfect conductors and conductors with Re$(\varepsilon_c)>0$ (see the TM$'$ curve in Fig. 1 and the experiments and simulations of tungsten hole arrays with positive Re$(\varepsilon_c)$ in [@r8; @r20]). It is also applicable to non-periodic structures. For example, Figure 3 indicates that an *isolated* slit (or hole) can still be a single light source emitting *divergent* light. If the slit is surrounded by grooves (in which the cavity resonance may be different though), the grooves provide additional light sources that suppress the oblique wavelets (forming SSPs). Meanwhile, wavelets emitted from the grooves along the backward direction offset specular reflection. Thus, transmission is enhanced. If grooves also exist on the exit surface, they again suppress oblique wavelets so that a collimated zero-order transmitted beam can be achieved [@r2]. As another example, it is known that when a nanowire is illuminated by a TM wave at one end, light can be “transferred” to the other end [@r21; @r22]. The common explanation is that light is transferred by CSPs on the wire, but the dispersion trend in [@r21] is quite different from that of CSPs. Based on our mechanism, the incident wave causes inhomogeneous charges at the illuminated end that tend to propagate away due to the charge-netural tendency. This is similar to normal electricitiy transmission over metallic wires. The charge movement is then discontinued at the other end, leading to charge accumulation and oscillation that emit new light. The electrons can be bounced back, resulting in a Fabry-Perot-like charge pattern (that obviously do not need to obey the CSP principle). Here high conductivity can enhance the transfer efficiency, which is opposite to the CSP prediction that (nearly) perfect conductors do not support CSPs. Overall, our mechanism reveals the fundamental of metallic nanostructure optics.
X.R.H is grateful to A. T. Macrander and S. K. Gray for invaluable suggestions and discussions. This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357, and by the National Nature Science Foundation of China (10625417, 50672035, 10021001) and the Ministry of Science and Technology of China (the State Key Program for Basic Research, 2004CB619005, 2006CB921804).
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---
abstract: 'We report NOrthern Extended Millimetre Array (NOEMA) observations of warm molecular gas traced by [CO($5-4$) ]{}in a $z \sim 3.2$ gas-rich main-sequence galaxy (MS), initially serendipitously detected in CO($3-2$) emission in ‘blind’ deep NOEMA observations. Our target shows a gas excitation consistent with that seen in $z \sim 1.5$ MS galaxies ($L''_{\rm CO( 5 - 4)}/L''_{\rm CO (3 - 2)} = 0.41 \pm 0.14$), albeit toward the low end, as well as a similar star formation efficiency based on the CO($3-2$) line luminosity and the $L_{\rm IR}$. However, it shows a high molecular gas fraction ($f_{\rm gas} = 0.9\pm 0.2$) as compared to $z\sim 1.5$ MS galaxies ($f_{\rm gas} \sim 0.42$), consistent with a cosmologically increasing gas fraction beyond $z\gtrsim3$ and our current understanding of scaling relations between $z$, $f_{\rm gas}$, the stellar mass $M_*$, and the specific star formation rate sSFR. Our results are consistent with recent findings by the COLDz and ASPECS molecular line scan surveys and suggest that deep searches for CO emission are a powerful means to identify gas-rich, star-forming galaxies at high redshift.'
author:
- Avani Gowardhan
- Dominik Riechers
- Riccardo Pavesi
- Emanuele Daddi
- Helmut Dannerbauer
- Roberto Neri
title: 'High gas fraction in a CO-selected main-sequence galaxy at $z > 3$'
---
Introduction
============
Observations of molecular gas - the fuel for star formation - in sizable galaxy samples at high-$z$ are essential to understanding the onset and evolution of the peak epoch of cosmic star formation and stellar mass assembly at $z \sim 1-3$ [see @carilli2013 for a review]. Star-forming galaxies (SFGs) at all cosmic epochs show a redshift-modulated correlation between the stellar mass and the star-formation rate (SFR) - the galaxy main-sequence - suggesting that the bulk of star formation takes place in quasi-steady state, with galaxies undergoing short-lived starburst activity lying significantly above the galaxy main-sequence at any redshift [e.g. @rodighiero2011; @speagle2014]. Observations of the molecular gas traced by CO as well as dust-based measurements of the total gas and dust mass suggest that the observed increase in star-formation rates in [high-$z$]{}$ $ SFGs is driven concurrently by increasing gas fraction ([$f_{\rm gas}$]{}) and star-formation efficiency (SFE) [e.g. @tacconi2013; @genzel2015; @scoville2016; @pavesi2018]. However, while there is a general agreement on the evolution of the molecular gas fraction and specific star formation rate (sSFR) up to $z \sim 2$, there is considerable debate about its evolution beyond that epoch. While some studies find a continuing increase in the molecular gas fraction at $z \gtrsim 3$ [@tan2013; @dz2015; @dz2017], as expected from theoretical models [e.g @obreschkow2009a; @lagos2011], other measurements indicate a plateauing or even a decline of the molecular gas fraction at the highest redshifts [e.g @saintonge2013; @troncoso2014; @bethermin2015; @dz2015; @schinnerer2016]. CO line stacking of 78 galaxies at a mean redshift of $z\sim 2.4$ also shows a lower molecular gas fraction than expected for massive main-sequence galaxies [@pavesi2018]. This disagreement can be attributed to the scarcity of molecular gas detections in MS galaxies at $z \gtrsim 3$. CO detections in SFGs at $z\gtrsim 3$ are currently largely restricted to highly lensed systems (magnified $30-60 \times$ @coppin2007 [@riechers2010a; @saintonge2013; @dz2017]). Searches in unlensed Lyman-Break galaxies (LBGs) at $z\sim 3$ have had limited success, with only two detections to date [@magdis2012; @tan2013; @magdis2017].
Observing both low-$J$ and high-$J$ CO lines in high-$z$ SFGs is important as they trace the cold and warm molecular gas phases respectively. While CO Spectral Line Energy Distributions (SLEDs) have been studied in FIR-bright submillimeter galaxies (SMGs) and quasars at [high-$z$]{}$ $ [e.g. @weiss2005; @riechers2006; @weiss2007; @riechers2011b; @riechers2011c; @danielson2011; @riechers2013; @bothwell2013; @yang2017; @strandet2017], these systems are undergoing intense star-formation, have small gas depletion timescales [@yang2017], and are unlikely to be representative of MS galaxies. The CO SLED has been only sparsely sampled for more ‘normal’ [high-$z$]{}$ $ star-forming galaxies, with observations limited to four BzK galaxies at $z \sim 1.5$ [@daddi2015] and one lensed source at $z\sim 3.6$ [@dz2017]. While low-$J$ ($J_{\rm upper} = 1,2,3$) CO line ratios in these systems resemble those of star-forming galaxies in the local universe, CO($5-4$) observations reveal the presence of an additional, warmer molecular gas component, demonstrating the necessity of sampling the CO SLED at multiple $J$s to accurately probe ISM properties [@daddi2015].
We here present observations of CO($5-4$) emission in EGSIRAC J141912.03+524924.0 (hereafter EGS141912), a gas-rich MS galaxy at $z \sim 3.2$, detected serendipitously in CO($3-2$) emission [@ag2017 hereafter ]. Our new observations confirm the target redshift and provide some of the first constraints on the molecular gas excitation and star formation efficiency in $z > 3$ MS galaxies.
The paper is organized as follows: we present the observations in and the spectral energy distribution (SED) fitting in . In and , we discuss our results and conclusions. We use a $\Lambda$CDM cosmology, with $H_{0} = 71$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\rm M} = 0.27$, and $\Omega_{\Lambda} = 0.73$ [@spergel2007].
Observations {#sec:obs}
============
CO observations
---------------
NOEMA observations of the CO($5-4$) line ($\nu_{\rm rest} = 576.26793$ GHz) in EGS141912 were conducted in April 2017 (Program ID W16DR), with 8 antennas in the compact D configuration, for a total on-source time of 9.2 hours split across two tracks. Weather conditions were good for both tracks, with a precipitable water vapor (pwv) of $2-15$ mm, with most of the observations taken in good weather. 3C273 was used as the absolute flux calibrator, and the source J1418+546 was used for phase and bandpass calibration. The WideX correlator (bandwidth $\sim 3.6$ GHz) was tuned to a frequency of $136.605$ GHz. Observations were carried out in a dual polarization mode, with a binned spectral resolution of $\sim 2.5$ MHz ($\sim 5.5$ km s$^{-1}$ at 136 GHz). All observations were calibrated using the IRAM PdBI data reduction pipeline in CLIC (Continuum and Line Interferometer Calibration), with subsequent additional flagging by hand. The reduced visibility data were imaged in the software , using the tasks and , using natural baseline weighting and the Hogbom cleaning algorithm. The final synthesized beam size is $3.''0 \times 2.''5$. The rms noise in the cube is $1.0$ mJy beam$^{-1}$ per $\sim 15.5$ [km s$^{-1}$]{}$ $ channel. Upon binning the line-free channels, we obtain an rms noise of $0.03$ mJy beam$^{-1}$ in the continuum map.
VLA observations
----------------
Radio continuum observations covering EGS141912 have been conducted using the NSF’s Karl G. Jansky Very Large Array (VLA) over 3 epochs in July-September 2013 (Program IDs 13B-289 and 13A-449). Observations were made in dual polarization using the X-band receivers in the C and CnB array configurations, with a 2 GHz bandwidth ($7.988 - 9.884$ GHz) sampled at a spectral resolution of 1 MHz. The total on-source time was 2.5 hours. 3C295 and J1419+5423 were used for absolute flux and phase calibration, respectively.
The VLA reduction pipeline in CASA$v 5.0.0$ was used to flag and calibrate the observations. The weights for the visibilities were calculated using [STATWT]{} for the reduced measurement sets from each observational epoch, and they were combined into a single measurement set using the task [CONCAT]{}. The final measurement set was imaged and cleaned using the CASA task [TCLEAN]{}, using natural weighting to maximize point source sensitivity, and a pixel size of $0.''5 \times 0.''5$. Primary beam correction was applied during the cleaning process. All channels were binned together during cleaning. The resulting cleaned image has an rms noise of 1.3 $\mu$Jy beam$^{-1}$ over the entire 2 GHz bandwidth, and a synthesized beam size of $3''.1 \times 2''.3$ (PA: $-76\degree$).
{width="49.00000%"} {width="48.00000%"}
Results {#sec:results}
=======
CO observations {#sec:co54_obs}
---------------
We detect CO($5-4$) emission from EGS141912 at $\sim 6 \sigma$ significance, where the moment-0 emission map () is created by binning the CO($5-4$) line over the same velocities as the CO($3-2$) emission in [^1].
Based on a 2-D Gaussian fitting to the moment-0 map, we find a velocity integrated line flux of $I_{\rm CO(5-4)} = 0.72 \pm 0.12 $ Jy km s$^{-1}$. This corresponds to a line luminosity of $L'_{\rm CO(5-4)} = (1.3 \pm 0.2) \times 10^{10}$ [K km s$^{-1}$ pc$^{-2}$]{}. Both CO($3-2$) and CO($5-4$) spectra are extracted from a circular aperture with radius $1.0''$ centred on the position in Table 2 in order to compare their line profiles, though we caution that this corresponds to a small fraction of the beam for the CO($3-2$) cube, given its $\sim4\times$ coarser spatial resolution (see ). We do not detect any continuum emission from EGS141912, giving a $3\sigma$ upper limit of $f_{\lambda} \leqslant 0.1$ mJy at $\lambda_{\rm obs} = 2.2$ mm.
We also create the rms-weighted average of the CO($3-2$) and CO($5-4$) spectra () and detect the combined emission at a $\sim 8\sigma$ significance, resulting in an improved $z_{\rm spec} = 3.2185 \pm 0.0002$. The total gas mass has been derived using the CO($3-2$) line (as in ) and the line luminosities are listed in Table 2.
Radio continuum observations
----------------------------
We do not detect 9 GHz radio continuum emission from EGS 141912 at the spatial position of the CO emission, and find a $3\sigma$ upper limit of $f_{\rm 9 GHz} \lesssim 3.9$ $\mu$Jy [^2]. We use this limit to constrain the 1.4 GHz luminosity ($L_{\rm 1.4 GHz}$) as follows
$$\label{eqn:eqn1}
L_{\rm 1.4 GHz} = \frac{4 \pi D_{L}^{2}}{(1+z)^{1+\alpha}} (\frac{1.4}{\nu_{\rm obs}})^{\alpha}S_{\nu_{\rm obs}}$$
where $D_{\rm L}$ is the luminosity distance in metres, $z$ is the source redshift, $\nu_{\rm obs}\sim 9$ GHz, and $\alpha$ is the radio spectral slope of $\alpha = -0.7$ (such that $S_{\nu} \propto \nu^{\alpha}$). This gives a $3\sigma$ upper limit on the 1.4 GHz luminosity of $L_{\rm 1.4 GHz} \lesssim 8.7 \times 10^{23}$ W Hz$^{-1}$.
SED fitting {#ssec:sedfitting}
-----------
To obtain the stellar mass, we adopt the spectral energy distribution (SED) fitting package Code for Investigating GALaxy Emission (CIGALE ; @burgarella2005 [@noll2009; @serra2011] as described in with minor changes (see for more details). We here only use those photometric data points where the emission is detected at SNR $\gtrsim 2$ as well as the upper limits on continuum emission based on our CO observations. The best-fit SED is shown in Figure 2, and the results of the SED fitting as well as all source properties are listed in Table 2. Based on the stellar mass based on the SED fit and gas mass based on the CO($3-2$) line strength, we find a gas mass fraction $f_{\rm gas} = M_{\rm gas}/(M_{\rm gas} + M_{*}) = 0.9 \pm 0.2$. The quoted uncertainty in the gas fraction does not include the systematic uncertainty associated with the stellar mass estimate due to assumptions about the star-formation history ($\sim 30\%$, see ), the uncertainties in the CO line luminosity ratio $L'_{\rm CO(3-2)}/L'_{\rm CO(1-0)}$, assumed to be $r_{\rm 31} = 0.42 \pm 0.07$ based on @daddi2015, or systematic uncertainties in the CO-$\rm H_{2}$ gas mass conversion factor $\alpha_{\rm CO}$ (see @bolatto2013 for a review).
There are large uncertainties associated with the $L_{\rm IR}$ for EGS141912. This is best demonstrated in Figure 2, where we compare the best-fit SED from CIGALE to high-$z$ SED templates, both for normal and starburst galaxies [@magdis2012][^3]. It is clear that in the absence of photometry sampling the peak of the IR emission, the shape of the SED - and therefore the integrated $L_{\rm IR}$ - is poorly constrained. Physically, this arises because a mixed dust/star system may look identical to a dimmer, dust-free system at optical/UV wavelengths, and the two can be distinguished only using far-IR photometry. This lack of far-IR coverage also results in relatively poorly constrained dust mass obtained through SED fitting, $M_{\rm dust} = (6.4 \pm 4.7) \times 10^{8} M_{\odot}$ (also see @berta2016). This corresponds to a gas-to-dust mass ratio of $\delta_{\rm GDR} = 400 \pm 300$, which is higher than but consistent with the expected $\delta_{\rm GDR} \sim 100$ for solar metallicities [@leroy2011] within the uncertainties.
Anchoring SED templates for $z\sim 3$ main-sequence galaxies to the 24[$\mu$m ]{}flux [@magdis2012], we infer an IR luminosity of $L_{\rm IR}^{\rm MS} = (2.1 \pm 0.3) \times 10^{12} L_{\odot}$. To get an upper limit on the $L_{\rm IR}$, we fit the upper limit on the NOEMA 2mm continuum flux with a Modified Blackbody function combined with a power-law mid-IR emission [see @pavesi2016 for details]. We here assume an uniform prior on the dust temperature of $T_{\rm dust} = 35 \pm 10$ K (as suitable for $z\sim 3$ galaxies, @magnelli2014) and a dust emissivity of $\beta = 1.7 \pm 0.2$ [@planck2014]. We find an upper limit of $L_{\rm IR} \lesssim 4.8 \times 10^{12} L_{\odot}$ with a 99.7% confidence limit. Overall, we treat the $L_{\rm IR}$ as lying between the $L_{\rm IR}^{\rm lower} = 2.1 \times 10^{12} L_{\odot}$ and $L_{\rm IR}^{\rm upper} = 4.8 \times 10^{12} L_{\odot}$. These limits on the $L_{\rm IR}$ are consistent with those derived using the upper limit on the 1.4 GHz luminosity $L_{\rm 1.4 GHz}$ when assuming a redshift-dependent radio-IR correlation [^4][@delhaize2017]. We find $q_{\rm IR} \sim 2.2$ for $z\sim 3.2$ (assuming $\alpha = -0.7$) as compared to $q_{\rm IR} \sim 2.6$ for a non-evolving radio-IR correlation [see Fig 3 @molnar2018]. These correspond to upper limits on the $L_{\rm IR} \lesssim 1.4 \times 10^{12} L_{\odot}$ and $L_{\rm IR} \lesssim 3.4 \times 10^{12} L_{\odot}$, respectively.
We use the limits on $L_{\rm IR}$ to get limits for the SFR$_{\rm IR} = 1.09 \times 10^{-10} L_{\rm IR}$ [@chabrier2003], finding SFR$_{\rm IR} = 230 - 520 M_{\odot}$ yr$^{-1}$. EGS141912 then has a specific star-formation rate of sSFR$ = 7.6 - 17.4 $ Gyr$^{-1}$ and gas depletion timescales of $\tau_{\rm dep} = 1.1 - 0.5 $ Gyr. The sSFR is thus $0.9 - 2.1 \times$ sSFR$_{\rm MS}$, where sSFR$_{\rm MS}$ is the sSFR expected from a galaxy lying on the MS at $z\sim 3.2$ [@speagle2014; @tacconi2018]. EGS141912 is therefore consistent with the MS at $z\sim 3.2$.
\[fig:plt22\] {width="48.00000%"}
Discussion {#sec:discussion}
==========
CO excitation at $z\sim 3$
--------------------------
In general, the CO excitation (measured by line luminosity ratio between high-$J$ and low-$J$ CO lines) is expected to increase at higher-$z$ due to the increased dust temperature [@magdis2012], and potentially due to higher dense gas fractions and star formation efficiencies [e.g. @daddi2010; @scoville2016]. Such a warm, highly excited molecular gas component is also expected based on simulations of gas excitation and feedback at higher redshifts [e.g. @narayanan2014; @bournaud2015]. We here quantify the CO excitation in EGS141912 using the CO($3-2$) and CO($5-4$) line detections. For EGS141912, we find a line luminosity ratio of $L'_{\rm CO(5-4)}/L'_{\rm CO(3-2)} = 1.3 \pm 0.2 / 3.0 \pm 0.5 = 0.41 \pm 0.10$. This is slightly lower than but consistent with the excitation observed for BzK galaxies ($L'_{\rm CO(5-4)}/L'_{\rm CO(3-2)} = 0.53 \pm 0.19$; @daddi2010 [@daddi2015]), and is lower than the observed excitation in submillimetre galaxies (SMGs ; $r_{53} = 0.61 \pm 0.20$; @bothwell2013).
The star-formation efficiency in EGS141912 ($L_{\rm IR}/M_{\rm gas} \sim (8.1 - 18.4) L_{\odot}/M_{\odot}$) is also consistent with those observed in BzK galaxies ($L_{\rm IR}/M_{\rm gas} \sim (13 \pm 3) L_{\odot}/M_{\odot}$ @daddi2015).
The CO-$L_{\rm IR}$ correlation
-------------------------------
{width="45.00000%"} {width="45.00000%"}
CO($5-4$) emission is a tracer of warm and dense molecular gas. $L'_{\rm CO(5-4)}$ has been observed to correlate linearly with star formation rates and with $L_{\rm IR}$ in galaxies ranging from local spirals and (U)LIRGs to high-$z$ star-forming galaxies, SMGs and QSOs [e.g. @liu2015; @daddi2015; @yang2017]. This correlation is somewhat indirectly driven, as the CO emission arises from warm molecular gas, potentially partially heated by mechanical feedback and winds from star-forming regions. A similar correlation also exists between the $L'_{\rm CO(3-2)}$ and the $L_{\rm IR}$ (see ). The observed $L'_{\rm CO(5-4)}$ and $L'_{\rm CO(3-2)}$ are consistent with these relations within the scatter.
Evolution of the cosmic gas fraction
------------------------------------
![The ratio of molecular gas mass to stellar mass (calculated using an [$\alpha_{\rm CO}$]{}$\sim 3.6$ [$M_{\odot} $(K km s$^{-1}$ pc$^{-2}$)$^{-1}$]{}$ $ for all sources) adapted from @carilli2013. Previous observations are from [@leroy2009; @riechers2010a; @daddi2010; @geach2011; @magnelli2012; @magdis2012; @tacconi2013; @dz2015; @decarli2016a; @dz2017; @dannerbauer2017; @ag2017; @pavesi2018]. The black line shows the scaling relation between $f_{\rm gas}$ and $z$, assuming a stellar mass of $M_{*} = 3 \times 10^{10} M_{\odot}$ and $\rm sSFR = 2 \rm sSFR_{\rm MS}$; the shaded regions show the 99.7% confidence regions. EGS141912 is consistent with an increasing gas fraction at $z \gtrsim 3$.[]{data-label="fig:plt4"}](fgas_z.png){width="49.00000%"}
The gas fraction $f_{\rm gas}$ in galaxies is a function of $M_{*}$, sSFR/sSFR$_{\rm MS}$, and $z$, with an increasing gas fraction at higher redshift, lower $M_*$, and in galaxies having lying above the MS [e.g. @bouche2010; @dave2011b; @saintonge2011a; @saintonge2012; @saintonge2017; @scoville2017; @tacconi2018]. We use the function for this evolution given by @scoville2017:
$$\begin{aligned}
f_{\rm gas} = (1.0 + (1.41 \pm 0.18) \times(1.0 + z)^{-1.84 \pm 0.14} \\
\times (\rm sSFR/sSFR_{\rm MS})^{-0.32 \pm 0.06} \\
\times (M_{*}/10^{10}M_{\odot})^{0.70 \pm 0.04})^{-1}. \\
\end{aligned}$$
We compare this against the gas fraction obtained for EGS141912 in . For a MS galaxy at $z\sim 3.2$ with a stellar mass of $M_* = 3 \times 10^{10} M_{\odot}$, the expected gas fraction is $f_{\rm gas} = 0.82$ for $\rm sSFR/sSFR_{\rm ms} = 1.0$, and $f_{\rm gas} = 0.85$ for $\rm sSFR/sSFR_{\rm ms} = 2.0$. EGS141912 shows a gas fraction of $f_{\rm gas} = M_{\rm gas}/(M_{*} + M_{\rm gas}) = 0.9 \pm 0.2$, which falls within a $99.7\%$ confidence interval of the above relation. Similarly high gas fractions have been found in two MS galaxies at $z\sim 2 - 2.5$ [@tacconi2013; @decarli2016a], with one showing comparable $M_*$ and $M_{\rm gas}$ to EGS141912, and the other having a significantly lower stellar mass ($M_{*} = 6 \times 10^{9} M_{\odot}$; @tacconi2013).
Conclusion {#sec:conclusions}
==========
We have presented molecular gas observations of EGS141912, one of the highest redshift unlensed MS galaxies detected in CO to date. Our observations of the CO($3-2$) and CO($5-4$) emission reveal that the gas excitation is consistent with that seen in $z\sim 1.5$ BzK galaxies, although toward the low end. EGS141912 also has a similar star formation efficiency as other high-$z$ MS galaxies between $z\sim 1.5-2.5$. We find EGS141912 to be gas-rich, with a gas fraction of $f_{\rm gas} \sim 0.9 \pm 0.2$, which is consistent with scaling relations for the gas fraction of MS galaxies derived using dust-based measurements of the total ISM mass [@scoville2017]. The uncertainties on the star formation efficiency and gas fraction for EGS141912 are driven by those on [$\alpha_{\rm CO}$]{}, $L_{\rm IR}$ and the unknown gas excitation, and we need both high spatial resolution observations of the CO($1-0$) emission as well as observations at the peak of the far-IR SED to improve our knowledge of the cold molecular gas, the molecular gas fraction and its star-formation efficiency. EGS141912 lies well within the attained CO sensitivities by blind surveys such as ASPECS-Pilot [@decarli2016a; @decarli2016b; @walter2016] and COLDz [@pavesi2018; @riechers2019].
While most gas-rich galaxies in the universe at $z > 2$ have optical/IR counterparts [@tacconi2013; @decarli2016a; @pavesi2018], our findings for EGS141912 show that some of the most gas-rich systems would not be *preferentially* selected for targeted CO follow-up studies at high redshift, either based on optical or far-IR selection criteria (e.g. PHIBBS, @tacconi2013). Molecular line scan surveys such as COLDz and ASPECS, which by design are ideal for picking up galaxies like EGS141912, thus provide a complementary probe of the distant universe, and thus, significantly contribute towards our understanding of the total cold gas content throughout cosmic history (e.g., @decarli2016a [@riechers2019]).
We thank the referee for excellent and helpful comments which have greatly improved the clarity of the work. A.G acknowledges support from the HST grant HST-GO-14938.003-A. D.R. and R.P acknowledge support from the National Science Foundation under grant number AST-1614213 to Cornell University. RP acknowledges support through the grant SOSPA3-008. This work is based on observations carried out under project number W16DR with the IRAM NOEMA Interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). This study makes use of data from AEGIS, a multiwavelength sky survey conducted with the Chandra, GALEX, Hubble, Keck, CFHT, MMT, Subaru, Palomar, Spitzer, VLA, and other telescopes and supported in part by the NSF, NASA, and the STFC. This work is based on observations taken by the 3D-HST Treasury Program (GO 12177 and 12328) with the NSAS/EST HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
Details of SED modelling {#appendix}
========================
We have used CIGALE to model the UV to IR SED of EGS141912. Although CIGALE can estimate a large number of galaxy physical properties (including dust attenuation, dust luminosity, $M_{*}$, SFR and $L_{\rm IR}$), given the lack of far-IR photometry for EGS141912, we do not consider the $L_{\rm IR}$ and SFR estimates to be highly reliable (see ).
The modelling and estimation of uncertainties performed by CIGALE have been discussed in greater detail in @noll2009 [@boq2018], but we briefly describe them as follows. CIGALE uses independent modules for modelling star-formation histories (SFHs), stellar emission from different population synthesis models [@bruzual2003; @maraston2005], dust attenuation [@calzetti2000], dust emission [e.g. @draine2007] and radio emission, which together create an integrated SED template. The code implicitly maintains energy balance between the UV attenuation and dust emission. CIGALE takes a range of parameters for each of these modules as input, and builds a model for each combination of parameters. After the grid of normalized models is computed. The models are scaled and compared against the provided photometry, CIGALE finds a likelihood for each of the models, defined as $e^{-\chi^{2}}$. These likelihoods are used to compute the likelihood-weighted mean of the physical parameters and their likelihood-weighted uncertainties, which are returned as the best-fit parameters.
We here focus on the uncertainties on the stellar mass $M_{*}$. For EGS141912, we find a stellar mass of $M_* = (3.0 \pm 0.1) \times 10^{10} M_{\odot}$, assuming a delayed exponential star-formation history, and the @bruzual2003 stellar population synthesis model. To test how robust $M_{*}$ is to our choice of SFH, we have explored the different possible SFHs allowed by CIGALE - a double exponential, a delayed star-formation, as well as a periodic bursts of star-formation. We find a $\sim 30\%$ variation in $M_{*}$ assuming different models, with $M_{*} = (3.0 \pm 0.1) \times 10^{10} M_{\odot}$ for a delayed exponential SFH, to $M_{*} = (3.9 \pm 0.5) \times 10^{10} M_{\odot}$ for periodic bursts of star formation. Assuming a delayed SFH results in the fit with the lowest reduced $\chi^{2} \sim 2.3$, as compared to $\chi^{2} \sim 2.7$ and $\chi^{2} \sim 3.0$ for double exponential SFH and a periodic SFH, respectively. We therefore assume a delayed SFH for the final best-fit SED.
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[^1]: We do not fit a 1-D Gaussian to the CO($5-4$) spectral line profile, as the line was observed close to the edge of the spectral band and we lack continuum coverage on one side of the band.
[^2]: We assume that the emission is not spatially resolved in the X-band observations, as it is not resolved in the CO($5-4$) emission, observed with a similar beam size.
[^3]: <http://georgiosmagdis.pbworks.com/w/page-revisions/59019974/SED%20Templates>
[^4]: The evolution of $q_{\rm IR}$ is an open question, with some studies finding a weak redshift evolution [@magnelli2015; @calistro2017; @delhaize2017], and with others finding differential evolution for for disc- vs spheroid- dominated galaxies [@molnar2018].
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