Dolma
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allenai's Dolma dataset as native Hugging Face datasets
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\section{Introduction}
Our main result is the following.
\begin{theorem}\label{thm:general_three_col}
For any planar convex body $C$ there is a positive integer $m=m(C)$ such that any finite point set $P$ in the plane can be three-colored in a way that there is no translate of $C$ containing at least $m$ points of $P$, all of the same color.
\end{theorem}
This result closes a long line of research about coloring points with respect to planar range spaces that consist of translates of a fixed set, a problem that was proposed by Pach over forty years ago \cite{Pach80}.
In general, a pair $(P, \Sc)$, where $P$ is a set of points in the plane and $\Sc$ is a family of subsets of the plane, called the \emph{range space}, defines a \emph{primal} hypergraph $\Hc(P,\Sc)$ whose vertex set is $P$, and for each $S\in\Sc$ we add the edge $S\cap P$ to the hypergraph.
Given any hypergraph $\Hc$, a planar realization of $\Hc$ is defined as a pair $(P, \Sc)$ for which $\Hc(P,\Sc)$ is isomorphic to $\Hc$.
If $\Hc$ can be realized with some pair $(P, \Sc)$ where $\Sc$ is from some family $\Fc$, then we say that $\Hc$ is realizable with $\Fc$.
The dual of the hypergraph $\Hc(P,\Sc)$, where the elements of the range space $\Sc$ are the vertices and the points $P$ define the edges, is known as the \emph{dual} hypergraph and is denoted by $\Hc(\Sc,P)$.
If $\Hc=\Hc(\Sc,P)$ where $\Sc$ is from some family $\Fc$, then we say that $\Hc$ has a dual realization with $\Fc$.
Pach observed \cite{Pach80,surveycd} that if $\Fc$ is the family of translates of some set, then $\Hc$ has a dual realization with $\Fc$ if and only if $\Hc$ has a (primal) realization with $\Fc$.
Pach proposed to study the chromatic number of hypergraphs realizable with different geometric families $\Fc$.
It is important to distinguish between two types of hypergraph colorings that we will use, the \emph{proper} coloring and the \emph{polychromatic} coloring.
\begin{definition}
A hypergraph is \emph{properly $k$-colorable} if its vertices can be colored with $k$ colors such that each edge contains points from at least two color classes. Such a coloring is called a \emph{proper $k$-coloring}.
If a hypergraph has a proper $k$-coloring but not a proper $(k-1)$-coloring, then it is called \emph{$k$-chromatic}.
A hypergraph is \emph{polychromatic $k$-colorable} if its vertices can be colored with $k$ colors such that each edge contains points from each color class. Such a coloring is called a \emph{polychromatic $k$-coloring}.
\end{definition}
Note that for a polychromatic $k$-coloring to exist, it is necessary that each edge of the underlying hypergraph has at least $k$ vertices.
More generally, we say that a hypergraph is \emph{$m$-heavy} if each of its edges has at least $m$ vertices.
The main question that Pach raised can be rephrased as follows.
\begin{question}
For which planar families $\Fc$ is there an $m_k=m(\Fc,k)$ such that any $m_k$-heavy hypergraph realizable with $\Fc$ has a proper/polychromatic $k$-coloring?
\end{question}
Initially, this question has been mainly studied for polychromatic $k$-colorings (known in case of a dual range space as \emph{cover-decomposition} problem), and it was shown that such an $m_k$ exists if $\Fc$ is the family of translates of some convex polygon \cite{Pach86,TT07,PT10}, or the family of all halfplanes \cite{wcf2,MR2844088}, or the homothetic\footnote{A \emph{homothetic copy}, or \emph{homothet}, is a scaled and translated (but non-rotated) copy of a set. We always require the scaling factor to be positive. Note that this is sometimes called a positive homothet.} copies of a triangle \cite{octants} or of a square \cite{homotsquare}, while it was also shown that not even $m_2$ exists if $\Fc$ is the family of translates of some appropriate concave polygon \cite{MR2364757,MR2679054} or any body\footnote{By \emph{body}, we always mean a compact subset of the plane with a non-empty interior, though our results (and most of the results mentioned) also hold for sets that are unbounded, or that contain an arbitrary part of their boundary, and are thus neither open, nor closed. This is because a realization of a hypergraph can be perturbed slightly to move the points off from the boundaries of the sets realizing the respective edges of the hypergraph.} with a smooth boundary \cite{unsplittable}.
It was also shown that there is no $m_k$ for proper $k$-colorings if $\Fc$ is the family of all lines \cite{MR2364757} or all axis-parallel rectangles \cite{Chen}; for these families, the same holds in case of dual realizations \cite{MR2364757,PT08}.
For homothets of convex polygons other than triangles, it is known that there is no $m_2$ for dual realizations \cite{kovacs}, unlike for primal realizations.
Higher dimensional variants \cite{octants,CKMU13} and improved bounds for $m_k$ have been also studied \cite{Alou,MR2812512,MR3151767,MR3216669,MR3126347,CKMPUV20}.
For other results, see also the decade old survey \cite{surveycd}, or the up-to-date website \url{https://coge.elte.hu/cogezoo.html}.
If $\Fc$ is the translates or homothets of some planar convex body, it is an easy consequence of the properties of generalized Delaunay-triangulations and the
Four Color Theorem that any hypergraph realizable with $\Fc$ is proper 4-colorable if every edge
contains at least two vertices.
We have recently shown that this cannot be improved for homothets.
\begin{theorem}[Dam\'asdi, Pálvölgyi \cite{fourchromatic}]
Let $C$ be any convex body in the plane that has two parallel supporting lines such that $C$ is strictly convex in some neighborhood of the two points of tangencies. For any positive integer $m$, there exists a 4-chromatic $m$-uniform hypergraph that is realizable with homothets of $C$.
\end{theorem}
For translates, we recall the following result.
\begin{theorem}[Pach, Pálvölgyi \cite{unsplittable}]\label{thm:unsplittable}
Let $C$ be any convex body in the plane that has two parallel supporting lines such that $C$ is strictly convex in some neighborhood of the two points of tangencies.\footnote{This condition can be relaxed to require only one smooth neighborhood on the boundary. Since this is not the main topic of our paper, we just give a sketch of the construction in Appendix \ref{sec:halfdisk}.} For any positive integer $m$, there exists a 3-chromatic $m$-uniform hypergraph that is realizable with translates of $C$.
\end{theorem}
This left only the following question open: Is there for any planar convex body $C$ a positive integer $m$ such that that no 4-chromatic $m$-uniform hypergraph is realizable with translates of $C$?
Our Theorem \ref{thm:general_three_col} answers this question affirmatively for all $C$ by showing that all realizable $m$-heavy hypergraphs are three-colorable for some $m$.
This has been hitherto known to hold only when $C$ is a polygon (in which case 2 colors suffice \cite{PT10}, and 3 colors are known to be enough even for homothets \cite{3propercol}) and pseudodisk families that intersect in a common point \cite{MR4012917} (which generalizes the case when $C$ is unbounded, in which case 2 colors suffice \cite{unsplittable}).
Note that the extended abstract of our first proof attempt appeared recently in the proceedings of EuroComb 2021 \cite{threechromaticdisk}.
That proof, however, only worked when $C$ was a disk, and while the generalization to other convex bodies with a smooth boundary seemed feasible, we saw no way to extend it to arbitrary convex bodies.
The proof of Theorem \ref{thm:general_three_col} relies on a surprising connection to two other famous results, the solution of the two dimensional case of the Illumination conjecture \cite{MR76368}, and a recent solution of the Erdős-Sands-Sauer-Woodrow conjecture by Bousquet, Lochet and Thomassé~\cite{esswproof}.
In fact, we need a generalization of the latter result, which we prove with the addition of one more trick to their method; this can be of independent interest.\\
The rest of the paper is organized as follows.\\
In Section \ref{sec:tools} we present the three main ingredients of our proof:
\begin{itemize}
\item the Union Lemma (Section \ref{sec:unionlemma}),
\item the Erdős-Sands-Sauer-Woodrow conjecture (Section \ref{sec:essw})---the proof of our generalization of the Bousquet-Lochet-Thomassé theorem can be found in Appendix \ref{app:essw},
\item the Illumination conjecture (Section \ref{sec:illum}), which is a theorem of Levi in the plane.
\end{itemize}
In Section \ref{sec:proof} we give the detailed proof of Theorem \ref{thm:general_three_col}.\\
In Section \ref{sec:overview} we give a general overview of the steps of the algorithm requiring computation to show that we can find a three-coloring in randomized polynomial time.\\
Finally, in Section \ref{sec:open}, we pose some problems left open.
\section{Tools}\label{sec:tools}
\subsection{Union Lemma}\label{sec:unionlemma}
Polychromatic colorability is a much stronger property than proper colorability. Any polychromatic $k$-colorable hypergraph is proper $2$-colorable. We generalize this trivial observation to the following statement about unions of polychromatic $k$-colorable hypergraphs.
\begin{lemma}[Union Lemma]\label{lem:combine} Let $\Hc_1=(V,E_1),\dots, \Hc_{k-1}=(V,E_{k-1})$ be hypergraphs on a common vertex set $V$. If $\Hc_1,\dots, \Hc_{k-1}$ are polychromatic $k$-colorable, then the hypergraph $\bigcup\limits_{i=1}^{k-1} \Hc_i=(V,\bigcup\limits_{i=1}^{k-1} E_i)$ is proper $k$-colorable.
\end{lemma}
\begin{proof}
Let $c_i:V\rightarrow \{1,\ldots,k\}$ be a polychromatic $k$-coloring of $\Hc_i$.
Choose $c(v)\in \{1,\ldots,k\}$ such that it differs from each $c_i(v)$.
We claim that $c$ is a proper $k$-coloring of $\bigcup\limits_{i=1}^{k-1} \Hc_i$.
To prove this, it is enough to show that for every edge $H\in\Hc_i$ and for every color $j\in\{1,\ldots,k-1\}$, there is a $v\in H$ such that $c(v)\ne j$.
We can pick $v\in H$ for which $c_i(v)=j$.
This finishes the proof.
\end{proof}
Lemma \ref{lem:combine} is sharp in the sense that for every $k$ there are $k-1$ hypergraphs such that each is polychromatic $k$-colorable but their union is not properly $(k-1)$-colorable.\\
We will apply the Union Lemma combined with the theorem below.
A \emph{pseudoline arrangement} is a collection of simple curves, each of which splits $\mathbb R^2$ into two unbounded parts, such that any two curves intersect at most once.
A \emph{pseudohalfplane} is the region on one side of a pseudoline in such an arrangement.
For hypergraphs realizible by pseudohalfplanes the following was proved, generalizing a result of Smorodinsky and Yuditsky \cite{MR2844088} about halfplanes.
\begin{theorem}[Keszegh-P\'alv\"olgyi \cite{abafree}]\label{thm:pseudohalfplane}
Any $(2k-1)$-heavy hypergraph realizable by pseudohalfplanes is polychromatic $k$-colorable, i.e., given a finite set of points and a pseudohalfplane arrangement in the plane, the points can be $k$-colored such that every pseudohalfplane that contains at least $2k-1$ points contains all $k$ colors.
\end{theorem}
Combining Theorem \ref{thm:pseudohalfplane} with Lemma \ref{lem:combine} for $k=3$, we obtain the following.
\begin{corollary}\label{cor:pseudohalfplane}
Any $5$-heavy hypergraph realizable by two pseudohalfplane families is proper $3$-colorable, i.e., given a finite set of points and two different pseudohalfplane arrangements in the plane, the points can be $3$-colored such that every pseudohalfplane that contains at least $5$ points contains two differently colored points.
\end{corollary}
\subsection{Erdős-Sands-Sauer-Woodrow conjecture}\label{sec:essw}
Given a quasi order\footnote{A quasi order $\prec$ is a reflexive and transitive relation, but it is not required to be antisymmetric, so $p\prec q\prec p$ is allowed, unlike for partial orders.} $\prec$ on a set $V$, we interpret it as a digraph $D=(V,A)$, where the vertex set is $V$ and a pair $(x,y)$ defines an arc in $A$ if $x \prec y$.
The \emph{closed in-neighborhood} of a vertex $x\in V$ is $N^-(x)=\{x\}\cup \{y|(y,x)\in A \}$. Similarly the \emph{closed out-neighborhood} of a vertex $x$ is $N^+(x)=\{x\}\cup \{y|(x,y)\in A \}$. We extend this to subsets $S\subset V$ as $N^-(S) = \bigcup\limits_{ x\in S } N^-(x)$ and $N^+(S) = \bigcup\limits_{ x\in S } N^+(x)$.
A set of vertices $S$ such that $N^+(S) = V$ is said to be \emph{dominating}.
For $A,B\subset V$ we will also say that \emph{$A$ dominates $B$ } if $B\subset N^+(A)$.
A \emph{complete multidigraph} is a digraph where parallel edges are allowed and in which there is at least one arc between each pair of distinct vertices. Let
$D$ be a complete multidigraph whose arcs are the disjoint union of $k$ quasi orders $\prec_1, \dots , \prec_k$ (parallel arcs are allowed). Define $N^-_i(x)$ (resp.\ $N^+_i(x)$) as the closed in-neighborhood (resp.\ out-neighborhood) of the digraph induced by $\prec_i$.
Proving a conjecture of Erdős, and of Sands, Sauer and Woodrow \cite{sandssauer}, Bousquet, Lochet and Thomassé recently showed the following.
\begin{theorem}[Bousquet, Lochet, Thomassé~\cite{esswproof}]\label{thm:multi_essw_old}
For every $k$, there exists an integer $f(k)$ such that if $D$ is a complete multidigraph whose arcs are the union of $k$ quasi orders, then $D$ has a dominating set of size at most $f(k)$.
\end{theorem}
We show the following generalization of Theorem \ref{thm:multi_essw_old}.
\begin{theorem}\label{thm:multi_essw_new}
For every pair of positive integers $k$ and $l$, there exist an integer $f(k,l)$ such that if $D=(V,A)$ is a complete multidigraph whose arcs are the union of $k$ quasi orders $\prec_1,\dots, \prec_k$, then $V$ contains a family of pairwise disjoint subsets $S_{i}^j$ for $i\in [k]$, $j\in [l]$ with the following properties:
\begin{itemize}
\item $|\bigcup\limits_{i,j}S_{i}^j|\le f(k,l)$
\item For each vertex $v\in V\setminus \bigcup\limits_{i,j}S_{i}^j$ there is an $i\in [k]$ such that for each $j\in [l]$ there is an edge of $\prec_i$ from a vertex of $S_{i}^j$ to $v$.
\end{itemize}
\end{theorem}
Note that disjointness is the real difficulty here, without it the theorem would trivially hold from repeated applications of Theorem \ref{thm:multi_essw_old}.
We saw no way to derive Theorem \ref{thm:multi_essw_new} from Theorem \ref{thm:multi_essw_old}, but with an extra modification the proof goes through.
The full proof of Theorem \ref{thm:multi_essw_new} can be found in Appendix \ref{app:essw}.
\subsection{Hadwiger's Illumination conjecture and pseudolines}\label{sec:illum}
Hadwiger's Illumination conjecture has a number of equivalent formulations and names.\footnote{These include names such as Levi–Hadwiger Conjecture, Gohberg–Markus Covering Conjecture, Hadwiger Covering Conjecture, Boltyanski–Hadwiger Illumination Conjecture.} For a recent survey, see \cite{MR3816868}. We will use the following version of the conjecture.
Let $\mathbb{S}^{d-1}$ denote the unit sphere in $\mathbb R^d$.
For a convex body $C$, let $\partial C$ denote the boundary of $C$ and let $int(C)$ denote its interior.
A direction $u\in \mathbb{S}^{d-1}$ \emph{illuminates} $b\in \partial C$ if $\{b+\lambda u:\lambda>0 \}\cap int (C)\ne \emptyset$.
\begin{conjecture}
The boundary of any convex body in $\mathbb{R}^d$ can be illuminated by $2^d$ or fewer directions. Furthermore, the $2^d$ lights are necessary if and only if the body is a parallelepiped.
\end{conjecture}
The conjecture is open in general. The $d=2$ case was settled by Levi \cite{MR76368} in 1955. For $d=3$ the best result is due to Papadoperakis \cite{MR1689273}, who showed that 16 lights are enough.
In the following part we make an interesting connection between the Illumination conjecture and pseudolines. Roughly speaking, we show that the Illumination conjecture implies that for any convex body in the plane the boundary can be broken into three parts such that the translates of each part behave similarly to pseudolines, i.e., we get three pseudoline arrangements from the translates of the three parts.
To put this into precise terms, we need some technical definitions and statements.
Fix a body $C$ and an injective parametrization of $\partial C$, $\gamma:[0,1]\rightarrow \partial C$, that follows $\partial C$ counterclockwise.
For each point $p$ of $\partial C$ there is a set of possible tangents touching at $p$. Let $g(p)\subset \mathbb{S}^1$ denote the Gauss image of $p$, i.e., $g(p)$ is the set of unit outernormals of the tangent lines touching at $p$. Note that $g(p)$ is an arc of $\mathbb{S}^1$ and $g(p)$ is a proper subset of $\mathbb{S}^1$.
Let $g_+:\partial C\rightarrow\mathbb{S}^1$ be the function that assigns to $p$ the counterclockwise last element of $g(p)$. (See Figure \ref{fig:gauss_tan} left.) Similarly let $g_-$ be the function that assigns to $p$ the clockwise last element of $g(p)$. Thus, $g(p)$ is the arc from $g_-(p)$ to $g_+(p)$. Let $|g(p)|$ denote the length of $g(p)$.
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(5.824158146704215,1.6939909822621093) rectangle (10.8,4.4022061705050906);
\draw [shift={(7.,2.)},line width=1.0] plot[domain=0.5235987755982986:1.5707963267948966,variable=\t]({1.*2.*cos(\t r)+0.*2.*sin(\t r)},{0.*2.*cos(\t r)+1.*2.*sin(\t r)});
\draw [shift={(7.,4.)},line width=1.0] plot[domain=4.71238898038469:5.759586531581288,variable=\t]({1.*2.*cos(\t r)+0.*2.*sin(\t r)},{0.*2.*cos(\t r)+1.*2.*sin(\t r)});
\draw [shift={(7.,3.)},line width=1.0] plot[domain=1.5707963267948966:4.71238898038469,variable=\t]({1.*1.*cos(\t r)+0.*1.*sin(\t r)},{0.*1.*cos(\t r)+1.*1.*sin(\t r)});
\draw [line width=1.0,domain=5.824158146704215:10.274171485061972] plot(\x,{(--18.124355652982153-1.7320508075688785*\x)/1.});
\draw [line width=1.0,domain=5.824158146704215:10.274171485061972] plot(\x,{(-12.12435565298215--1.7320508075688785*\x)/1.});
\draw [->,line width=1.pt] (8.732050807568879,3.) -- (9.832456454322593,3.6353194963710407);
\draw [->,line width=1.pt] (8.732050807568879,3.) -- (9.844045808452828,2.3579893869021333);
\draw (8.15,3.25) node[anchor=north west] {$p$};
\draw (9.7,2.95) node[anchor=north west] {$g_-(p)$};
\draw (9.7,3.8) node[anchor=north west] {$g_+(p)$};
\end{tikzpicture}
~~~~~~~~
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.6cm,y=0.6cm]
\clip(1.010405779360095,0.7779725023481115) rectangle (9.945938145792228,4.969084292744436);
\draw (4.75,3.7) node[anchor=north west] {$p$};
\draw (4.9,1.4) node[anchor=north west] {$q$};
\draw [line width=1.0,dash pattern=on 3pt off 3pt] (5.52700500582115,3.969615338937309)-- (5.701093102682993,1.3323544637799518);
\draw [line width=1.0,dash pattern=on 3pt off 3pt] (4.691398477047223,3.8957218914504184)-- (4.86141088560568,1.3202036253261604);
\draw [line width=1.0] (5.516440449569205,1.8865088084802843)-- (4.995922845299596,1.0326578708773406);
\draw [line width=1.0] (8.392512263686044,1.2568699665550929)-- (8.913029867955656,2.1107209041580446);
\draw [line width=1.0] (3.0820370497082816,4.613210333135324)-- (4.995922845299596,4.032657870877341);
\draw [line width=1.0] (8.39251226368604,4.256869966555094)-- (6.478626468094716,4.837422428813083);
\draw [shift={(3.495922845299593,2.532657870877341)},line width=1.0] plot[domain=-0.30951591373703113:0.7853981633974475,variable=\t]({1.*2.1213203435596446*cos(\t r)+0.*2.1213203435596446*sin(\t r)},{0.*2.1213203435596446*cos(\t r)+1.*2.1213203435596446*sin(\t r)});
\draw [shift={(3.495922845299593,2.532657870877341)},line width=1.0] plot[domain=1.7671635199760698:3.9269908169872414,variable=\t]({1.*2.121320343559645*cos(\t r)+0.*2.121320343559645*sin(\t r)},{0.*2.121320343559645*cos(\t r)+1.*2.121320343559645*sin(\t r)});
\draw [shift={(6.892512263686043,2.756869966555093)},line width=1.0] plot[domain=-0.3095159137370276:0.7853981633974495,variable=\t]({1.*2.121320343559643*cos(\t r)+0.*2.121320343559643*sin(\t r)},{0.*2.121320343559643*cos(\t r)+1.*2.121320343559643*sin(\t r)});
\draw [shift={(6.892512263686043,2.756869966555093)},line width=1.0] plot[domain=1.7671635199760762:3.9269908169872414,variable=\t]({1.*2.1213203435596544*cos(\t r)+0.*2.1213203435596544*sin(\t r)},{0.*2.1213203435596544*cos(\t r)+1.*2.1213203435596544*sin(\t r)});
\draw (8.676673546001679,4.6) node[anchor=north west] {$J_2$};
\draw (0.8,4.6) node[anchor=north west] {$J_1$};
\end{tikzpicture}
\caption{Extremal tangents at a boundary point (on the left) and parallel tangents on two intersecting translates (on the right).}
\label{fig:gauss_tan}
\end{figure}
\begin{obs}\label{obs:continuity}
$g_+\circ \gamma$ is continuous from the right and $g_-\circ \gamma$ is continuous from the left.
\end{obs}
For $t_1<t_2$ let $\gamma_{[t_1,t_2]}$ denote the restriction of $\gamma$ to the interval $[t_1,t_2]$. For $t_1>t_2$ let $\gamma_{[t_1,t_2]}$ denote the concatenation of $\gamma_{[t_1,1]}$ and $\gamma_{[0,t_2]}$.
When it leads to no confusion, we identify $\gamma_{[t_1,t_2]}$ with its image, which is a connected part of the boundary $\partial C$.
For such a $J=\gamma_{[t_1,t_2]}$, let $g(J)=\bigcup\limits_{p\in J}g(p)$. Clearly, $g(J)$ is an arc of $\mathbb{S}^1$ from $g_-(t_1)$ to $g_+(t_2)$; let $|g(J)|$ denote the length of this arc.
\begin{lemma}
Let $C$ be a convex body and assume that $J$ is a connected part of $\partial C$ such that $|g(J)|<\pi$. Then there are no two translates of $J$ that intersect in more than one point.
\end{lemma}
\begin{proof}
Suppose $J$ has two translates $J_1$ and $J_2$ such that they intersect in two points, $p$ and $q$. Now both $J_1$ and $J_2$ has a tangent that is parallel to the segment $pq$. (See Figure \ref{fig:gauss_tan} right.) This shows that $J$ has two different tangents parallel to $pq$ and therefore $g(J)\ge \pi$.
\end{proof}
\begin{lemma}\label{lemma:our_illumination}
For a convex body $C$, which is not a parallelogram, and an injective parametrization $\gamma$ of $\partial C$, we can pick $0\le t_1<t_2<t_3\le 1$ such that $|g(\gamma_{[t_1,t_2]})|,|g(\gamma_{[t_2,t_3]})|$ and $|g(\gamma_{[t_3,t_1]})|$ are each strictly smaller than $\pi$.
\end{lemma}
\begin{proof}
We use the 2-dimensional case of the Illumination conjecture (proved by Levi \cite{MR76368}). If $C$ is not a parallelogram, we can pick three directions, $u_1,u_2$ and $u_3$, that illuminate $C$. Pick $t_1$ such that $\gamma(t_1)$ is illuminated by both $u_1$ and $u_2$. To see why this is possible, suppose that the parts illuminated by $u_1$ and $u_2$ are disjoint. Each light illuminates a continuous open ended part of the boundary. So in this case there are two disjoint parts of the boundary that are not illuminated. If $u_3$ illuminates both, then it illuminates everything that is illuminated by $u_1$ or everything that is illuminated by $u_2$. But two lights are never enough for illumination, a contradiction.
Using the same argument, pick $t_2$ and $t_3$ such that $\gamma(t_2)$ is illuminated by both $u_2$ and $u_3$ and $\gamma(t_3)$ is illuminated by both $u_3$ and $u_1$.
Note that $u_1$ illuminates exactly those points for which $g_+(p)<u_1+\pi/2$ and $g_-(p)>u_1-\pi/2$. Therefore, $|g(\gamma_{[t_1,t_3]})|<u_1+\pi/2-(u_1-\pi/2)=\pi$. Similarly $|g(\gamma_{[t_1,t_2]})|<\pi$ and $|g(\gamma_{[t_2,t_3]})|<\pi$.
\end{proof}
Observation \ref{obs:continuity} and Lemma \ref{lemma:our_illumination} immediately imply the following statement.
\begin{lemma}\label{lemma:our_illumination_epsilon}
For a convex body $C$, which is not a parallelogram, and an injective parametrization $\gamma$ of $\partial C$, we can pick $0\le t_1<t_2<t_3\le 1$ and $\varepsilon>0$ such that $|g(\gamma_{[t_1-\varepsilon,t_2+\varepsilon]})|$, $|g(\gamma_{[t_2-\varepsilon,t_3+\varepsilon]})|$ and $|g(\gamma_{[t_3-\varepsilon,t_1+\varepsilon]})|$ are each strictly smaller than $\pi$.
\end{lemma}
\section{Proof of Theorem \ref{thm:general_three_col}}\label{sec:proof}
\subsection{Quasi orderings on planar point sets}
Cones provide a natural way to define quasi orderings on point sets (see \cite{TT07} for an example where this idea was used). A \emph{cone} is a closed region in the plane that is bounded by two rays that emanate from the origin. For a cone $K$ let $-K$ denote the cone that is the reflection of $K$ across the origin and let $p+K$ denote the translate of $K$ by the vector $p$.
\begin{obs}\label{obs:cones}
For any $p,q\in \mathbb{R}^2$ and cone $K$, the following are equivalent (see Figure \ref{fig:basic_cones}):
\begin{itemize}
\item $p\in q+K$
\item $q \in p+(-K)$
\item $p+K\subset q+K$
\end{itemize}
\end{obs}
\begin{figure}[!ht]
\centering
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\begin{scriptsize}
\draw [fill=black] (21.914471376040254,4.093030278329455) circle (2.5pt);
\draw [fill=black] (18.,2.) circle (2.5pt);
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\draw [fill=black] (8.339322178085318,5.977538969999748) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}}\caption{Basic properties of cones.}
\label{fig:basic_cones}
\end{figure}
For a cone $K$ let $\prec_K$ denote the quasi ordering on the points of the plane where a point $p$ is bigger than a point $q$ if and only if $p+K$ contains $q$, i.e., when interpreted as a digraph, $qp$ is an edge of $\prec_K$.
By Observation \ref{obs:cones}, this ordering is indeed transitive.
\begin{figure}[!ht]
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\label{fig:ordering}
\end{figure}
Suppose the cones $K_1, K_2, K_3$ correspond to the three corners of a triangle, in other words the cones $K_1,-K_3,K_2,-K_1,K_3,-K_2$ partition the plane around the origin in this order. Then we will say that $K_1, K_2, K_3$ is a \emph{set of tri-partition} cones. In this case the intersection of any translates of $K_1, K_2, K_3$ forms a (sometimes degenerate) triangle.
\begin{obs}
Let $K_1,K_2,K_3$ be a set of tri-partition cones and let $P$ be a planar point set. Then any two distinct points of $P$ are comparable in either $\prec_{K_1}$, $\prec_{K_2}$ or $\prec_{K_3}$. (See Figure \ref{fig:ordering}.)
\end{obs}
In other words, when interpreted as digraphs, the union of $\prec_{K_1}$, $\prec_{K_2}$ and $\prec_{K_3}$ forms a complete multidigraph on $P$. As a warm up for the proof of Theorem \ref{thm:general_three_col}, we show the following theorem.
\begin{theorem}\label{thm:three_cones}
There exists a positive integer $m$ such that for any point set $P$, and any set of tri-partition cones $K_1,K_2,K_3$, we can three-color $P$ such that no translate of $K_1$, $K_2$ or $K_3$ that contains at least $m$ points of $P$ is monochromatic.
\end{theorem}
\begin{proof}
We set $m$ to be $f(3,2)+13$ according to Theorem \ref{thm:multi_essw_new}.
Consider the three quasi orders $\prec_{K_1}$, $\prec_{K_2}$ or $\prec_{K_3}$. Their union gives a complete multidigraph on $P$, hence we can apply Theorem \ref{thm:multi_essw_new} with $k=3$ and $l=2$, resulting in subsets $S_i^j$ for $i\in[3],j\in [2]$. Let $S=\bigcup\limits_{i\in [3],j\in[2]}S_i^j$. For each point $p\in P\setminus S$ there is an $i$ such that $\prec_{K_i}$ has an edge from a vertex of $S_{i,1}$ and $S_{i,2}$ to $p$. Let $P_1,P_2,P_3$ be the partition of $P\setminus S$ according to this $i$ value.
We start by coloring the points of $S$. Color the points of $S_{1,1}\cup S_{2,1} \cup S_{3,1}$ with the first color and color the points of $S_{1,2}\cup S_{2,2}\cup S_{3,2}$ with the second color.
Any translate of $K_1$, $K_2$ or $K_3$ that contains $f(3,2)+13$ points of $P$, must contain $5$ points from either $P_1,P_2$ or $P_3$ by the pigeonhole principle. (Note that the cone might contain all points of $S$.) Therefore, it is enough to show that for each $i\in [3]$ the points of $P_i$ can be three-colored such that no translate of $K_1$, $K_2$, or $K_3$ that contains at least $5$ points of $P_i$ is monochromatic.
Consider $P_1$; the proof is the same for $P_2$ and $P_3$. Take a translate of $K_1$ and suppose that it contains a point $p$ of $P_1$. By Theorem \ref{thm:multi_essw_new}, there is and edge of $\prec_{K_1}$ from a vertex of $S_{1,1}$ to $p$ and another edge from a vertex of $S_{1,2}$ to $p$. Thus any such translate contains a point from $S_{1,1}$ and another point from $S_{1,2}$, and hence it cannot be monochromatic.
Therefore, we only have to consider the translates of $K_2$ and $K_3$. Two translates of a cone intersect at most once on their boundary. Hence, the translates of $K_2$ form a pseudohalfplane arrangement, and so do the translates of $K_3$. Therefore, by Corollary \ref{cor:pseudohalfplane}, there is a proper three-coloring for the translates of $K_2$ and $K_3$ together.
\end{proof}
\begin{remark}
From Theorem \ref{thm:three_cones}, it follows using standard methods (see Section \ref{sec:proofend}) that Theorem \ref{thm:general_three_col} holds for triangles.
This was of course known before, even for two-colorings of homothetic copies of triangles.
Our proof cannot be modified for homothets, but a two-coloring would follow if instead of Corollary \ref{cor:pseudohalfplane} we applied a more careful analysis for the two cones.
\end{remark}
\subsection{Proof of Theorem \ref{thm:general_three_col}}\label{sec:proofend}
If $C$ is a parallelogram, then our proof method fails.
Luckily, translates of parallelograms (and other symmetric polygons) were the first for which it was shown that even two colors are enough \cite{Pach86}; in fact, by now we know that two colors are enough even for homothets of parallelograms \cite{homotsquare}.
So from now on we assume that $C$ is not a parallelogram.
The proof of Theorem \ref{thm:general_three_col} relies on the same ideas as we used for Theorem \ref{thm:three_cones}. We partition $P$ into several parts, and for each part $P_i$, we divide the translates of $C$ into three families such that two of the families each form a pseudohalfplane arrangement over $P_i$, while the third family will only contain translates that are automatically non-monochromatic. Then Corollary \ref{cor:pseudohalfplane} would provide us a proper three-coloring. As in the proof of Theorem \ref{thm:three_cones}, this is not done directly. First, we divide the plane using a grid, and then in each small square we will use Theorem \ref{thm:multi_essw_new} to discard some of the translates of $C$ at the cost of a bounded number of points.\\
The first step of the proof is a classic divide and conquer idea \cite{Pach86}. We chose a constant $r=r(C)$ depending only on $C$ and divide the plane into a grid of squares of side length $r$. Since each translate of $C$ intersects some bounded number of squares, by the pidgeonhole principle we can find for any positive integer $m$ another integer $m'$ such that the following holds: each translate $\hat C$
of $C$ that contains at least $m'$ points intersects a square $Q$ such that $\hat C\cap Q$ contains at least $m$ points.
For example, choosing $m'=m(diam(C)/r+2)^2$ is sufficient, where $diam(C)$ denotes the diameter of $C$.
Therefore, it is enough to show the following localized version of Theorem \ref{thm:general_three_col}, since applying it separately for the points in each square of the grid provides a proper three-coloring of the whole point set.
\begin{theorem}\label{thm:local_three_col}
There is a positive integer $m$ such that for any convex body $C$ there is a positive real $r$ such that any finite point set $P$ in the plane that lies in a square of side length $r$ can be three-colored in a way that there is no translate of $C$ containing at least $m$ points of $P$, all of the same color.
\end{theorem}
We will show that $m$ can be chosen to be $f(3,2)+13$ according to Theorem \ref{thm:multi_essw_new}, independently of $C$.
\begin{proof}
We pick $r$ the following way. First we fix an injective parametrization $\gamma$ of $\partial C$ and then fix $t_1,t_2,t_3$ and $\varepsilon$ according to Lemma \ref{lemma:our_illumination_epsilon}. Let $\ell_1,\ell_2,\ell_3$ be the tangents of $C$ touching at $\gamma(t_1),\gamma(t_2)$ and $\gamma(t_3)$. Let $K_{1,2}$, $K_{2,3}$, $K_{3,1}$ be the set of tri-partition cones bordered by $\ell_1,\ell_2,\ell_3$, such that $K_{i,i+1}$ is bordered by $\ell_i$ on its counterclockwise side, and by $\ell_{i+1}$ on its clockwise side (see Figure \ref{fig:cone_in_C} left, and note that we always treat $3+1$ as 1 in the subscript).
For a translate $\hat{C}$ of $C$ we will denote by $\hat{\gamma}$ the translated parametrization of $\partial \hat{C}$, i.e., $\hat{\gamma}(t)=\gamma(t)+v$ if $\hat{C}$ was translated by vector $v$. Our aim is to choose $r$ small enough to satisfy the following two properties for each $i\in [3]$.
\begin{enumerate}[label=(\Alph*)]
\item Let $\hat C$ be a translate of $C$, and $Q$ be a square of side length $r$ such that $\partial \hat C\cap Q\subset \hat{\gamma}_{[t_i+\varepsilon/2,t_{i+1}-\varepsilon/2]}$ (see Figure \ref{fig:cone_in_C} right). Then for any translate $K$ of $K_{i,i+1}$ whose apex is in $Q\cap \hat C$, we have $K\cap Q\subset \hat C$. (I.e., $r$ is small with respect to $C$.)
\item Let $\hat C$ be a translate of $C$, and $Q$ be a square of side length $r$ such that $\hat{\gamma}_{[t_i-\varepsilon/2,t_{i+1}+\varepsilon/2]}$ intersects $Q$. Then $\partial \hat C\cap Q\subset \hat{\gamma}_{[t_i-\varepsilon,t_{i+1}+\varepsilon]}$. (I.e., $r$ is small compared to $\varepsilon$.)
\end{enumerate}
\begin{figure}[!ht]
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\caption{Selecting the cones (on the left) and Property (A) (on the right).}
\label{fig:cone_in_C}
\end{figure}
We show that an $r$ satisfying properties (A) and (B) can be found for $i=1$. The argument is the same for $i=2$ and $i=3$, and we can take the smallest among the three resulting $r$-s.
First, consider property (A). Since the sides of $K$ are parallel to $\ell_1$ and $\ell_2$, the portion of $K$ that lies ``above'' the segment $\overline{\hat{\gamma}(t_1)\hat{\gamma}(t_2)}$ is in $\hat{C}$. Hence, if we choose $r$ small enough so that $Q$ cannot intersect $\overline{\hat{\gamma}(t_1)\hat{\gamma}(t_2)}$, then property (A) is satisfied. For example, choosing $r$ to be smaller than $\frac{1}{\sqrt{2}}$ times the distance of the segments $\overline{\hat{\gamma}(t_1)\hat{\gamma}(t_2)}$ and $\overline{\hat{\gamma}(t_1+\varepsilon/2)\hat{\gamma}(t_2-\varepsilon/2)}$ works.
Using that $\gamma$ is a continuous function on a compact set, we can pick $r$ such that property (B) is satisfied.
Therefore, there is an $r$ satisfying properties (A) and (B).
\bigskip
The next step is a subdivision of the point set $P$ using Theorem \ref{thm:multi_essw_new}, like we did in the proof of Theorem \ref{thm:three_cones}.
The beginning of our argument is exactly the same.
Apply Theorem \ref{thm:multi_essw_new} for the graph given by the union of $\prec_{K_{1,2}}$, $\prec_{K_{2,3}}$ and $\prec_{K_{3,1}}$. By Observation \ref{obs:cones}, this is indeed a complete multidigraph on $P$.
We apply Theorem \ref{thm:multi_essw_new} with $k=3$ and $l=2$, resulting in subsets $S_i^j$ for $i\in[3],j\in [2]$. Let $S=\bigcup\limits_{i\in [3],j\in[2]}S_i^j$. For each point $p\in P\setminus S$ there is an $i$ such that $\prec_{K_{i,i+1}}$ has an edge from a vertex of $S_{i,1}$ and $S_{i,2}$ to $p$. Let $P_1,P_2,P_3$ be the partition of $P\setminus S$ according to this $i$ value.
We start by coloring the points of $S$. Color the points of $S_{1,1}\cup S_{2,1} \cup S_{3,1}$ with the first color and color the points of $S_{1,2}\cup S_{2,2}\cup S_{3,2}$ with the second color.
Note that $m$ is at least $f(3,2)+13$. Any translate of $C$ that contains $f(3,2)+13$ points of $P$ must contain $5$ points from either $P_1,P_2$ or $P_3$. (Note that the cone might contain all points of $S$). Therefore it is enough to show that for each $i\in [3]$ the points of $P_i$ can be colored with three color such that no translate of $C$ that contains at least $5$ points of $P_i$ is monochromatic.\\
Consider $P_1$, the proof is the same for $P_2$ and $P_3$. We divide the translates of $C$ that intersect $Q$ into four groups. Let $\mathcal{C}_0$ denote the translates where $\hat{C}\cap Q=Q$. Let $\mathcal{C}_1$ denote the translates for which $\partial \hat{C}\cap Q\subset \hat{\gamma}_{[t_1+\varepsilon/2,t_{2}-\varepsilon/2]}$. Let $\mathcal{C}_2$ denote the translates for which $\partial \hat{C}\cap Q\cap \hat{\gamma}_{[t_2-\varepsilon/2,t_{3}]}\ne \emptyset$. Let $\mathcal{C}_3$ denote the remaining translates for which $\partial \hat{C}\cap Q\cap \hat{\gamma}_{[t_3,t_{1}+\varepsilon/2]}\ne \emptyset$.
We do not need to worry about the translates in $\mathcal{C}_0$, as $Q$ itself will not be monochromatic.
Take a translate $\hat C$ from $\mathcal{C}_1$ and suppose that it contains a point $p\in P_1$. By Theorem \ref{thm:multi_essw_new}, there is an edge of $\prec_{K_{1,2}}$ from a vertex of $S_{1,1}$ to $p$ and another edge from a vertex of $S_{1,2}$ to $p$. I.e., the cone $p+K_{1,2}$ contains a point from $S_{1,1}$ and another point from $S_{1,2}$, and hence it is not monochromatic. From property (A) we know that every point in $(p+K_{1,2})\cap P$ is also in $\hat C$. Therefore, $\hat C$ is not monochromatic.
Now consider the translates in $\mathcal{C}_2$. From property (B) we know that for these translates we have $\partial \hat C\cap Q\subset \hat{\gamma}_{[t_2-\varepsilon,t_3+\varepsilon]}$. By the definition of $t_1,t_2$ and $t_3$, we know that this implies that any two translates from $\mathcal{C}_2$ intersect at most once on their boundary within $Q$, i.e., they behave as pseudohalfplanes. To turn the translates in $\mathcal{C}_2$ into a pseudohalfplane arrangement as defined earlier, we can do as follows. For a translate $\hat{C}$, replace it with the convex set whose boundary is $\hat{\gamma}_{[t_2-\varepsilon,t_3+\varepsilon]}$ extended from its endpoints with two rays orthogonal to the segment $\overline{\hat{\gamma}(t_2-\varepsilon)\hat{\gamma}(t_3+\varepsilon)}$. This new family provides the same intersection pattern in $Q$ and forms a pseudohalfplane arrangement. We can do the same with the translates in $\mathcal{C}_3$. Therefore, by Corollary \ref{cor:pseudohalfplane} there is a proper three-coloring for the translates in $\mathcal{C}_2\cup \mathcal{C}_3$.
\end{proof}
\section{Overview of the computational complexity of the algorithm}\label{sec:overview}
In this section we show that given a point set $P$ and a convex set $C$, we can determine some $m=m(C)$ and calculate a three-coloring of $P$ efficiently if $C$ is given in a natural way, for example, if $C$ is a disk.
Our algorithm is randomized and its running time is a polynomial of the number of points, $n=|P|$.
\begin{itemize}
\item First, we need to fix three points on the boundary, $\tau_1,\tau_2,\tau_3\subset \partial C$ such that Lemma \ref{lemma:our_illumination_epsilon} is satisfied with
$\tau_i=\gamma(t_i)$
for some $t_i$ and $\varepsilon>0$ for each $i$.
Note that we do not need to fix a complete parametrization $\gamma$ of $\partial C$ or $\varepsilon>0$; instead, it is enough to choose some points $\tau_i^{\scalebox{0.6}{$--$}}$ and $\tau_i^{\scalebox{0.6}{$++$}}$ that satisfy the conclusion of Lemma \ref{lemma:our_illumination_epsilon} if we assume $\tau_i^{\scalebox{0.6}{$--$}}=\gamma(t_i-\varepsilon)$ and $\tau_i^{\scalebox{0.6}{$++$}}=\gamma(t_i+\varepsilon)$ for each $i$.
If $C$ has a smooth boundary, like a disk, we can pick $\tau_1,\tau_2,\tau_3$ to be the touching points of an equilateral triangle with $C$ inscribed in it.
If the boundary of $C$ contains vertex-type sharp turns, the complexity of finding these turns depends on how $C$ is given, but for any reasonable input method, this should be straight-forward.
After that, one can follow closely the steps of the proof of the Illumination conjecture in the plane to get an algorithm, but apparently, this has not yet been studied in detail.
\item To pick $r$, the side length of the squares of the grid, we can fix some arbitrary points $\tau_i^{\scalebox{0.6}{$-$}}$ between $\tau_i^{\scalebox{0.6}{$--$}}$ and $\tau_i$, and points $\tau_i^{\scalebox{0.6}{$+$}}$ between $\tau_i$ and $\tau_i^{\scalebox{0.6}{$++$}}$, to play the roles of $\gamma(t_i-\varepsilon/2)$ and $\gamma(t_i+\varepsilon/2)$, respectively, for each $i$.
It is sufficient to pick $r$ so that $r\sqrt{2}$, the diameter of the square of side length $r$, is less than
\begin{itemize}
\item the distance of $\tau_i^{\scalebox{0.6}{$+$}}$ and $\tau_{i+1}^{\scalebox{0.6}{$-$}}$ from the segment $\overline{\tau_i\tau_{i+1}}$,
\item the distance of $\tau_i^{\scalebox{0.6}{$-$}}$ from $\tau_i^{\scalebox{0.6}{$--$}}$, and
\item the distance of $\tau_i^{\scalebox{0.6}{$+$}}$ from $\tau_i^{\scalebox{0.6}{$++$}}$,
\end{itemize}
for each $i$, to guarantee that properties (A) and (B) are satisfied.
\item Set $m=f(3,2)+13$, which is an absolute constant given by Theorem \ref{thm:multi_essw_new}.
We need to construct the complete multidigraph given by the tri-partition cones determined by $\tau_1,\tau_2,\tau_3$, which needs a comparison for each pair of points.
To obtain the subsets $S_i^j\subset P$ for $i\in[3],j\in [2]$, where $P$ is the set of points that are contained in a square of side length $r$, we randomly sample the required number of points from each of the constantly many $T_{j_1,\dots, j_i}$ according to the probability distributions $w_{j_1,\dots, j_i}$ given by Lemma \ref{lemma:prob_dist}.
These probability distributions can be computed by LP.
With high probability, all the $S_i^j$-s will be disjoint---otherwise, we can resample until we obtain disjoint sets.
\item To find the three-coloring for the two pseudohalfplane arrangements, given by Corollary \ref{cor:pseudohalfplane}, it is enough to determine the two-coloring given by Theorem \ref{thm:pseudohalfplane} for one pseudohalfplane arrangement.
While not mentioned explicitly in \cite{abafree}, the polychromatic $k$-coloring can be found in polynomial time if we know the hypergraph determined by the range space, as this hypergraph can only have a polynomial number of edges, and the coloring algorithm only needs to check some simple relations among a constant number of vertices and edges.
\item Finally, to compute a suitable $m'$ for Theorem \ref{thm:general_three_col} from the $m$ of Theorem \ref{thm:local_three_col}, it is enough to know any upper bound $B$ for the diameter of $C$, and let $m'=m(B/r+2)^2$.
\end{itemize}
\section{Open questions}\label{sec:open}
It is a natural question whether there is a universal $m$ that works for all convex bodies in Theorem \ref{thm:general_three_col}, like in Theorem \ref{thm:local_three_col}.
This would follow if we could choose $r$ to be a universal constant.
While the $r$ given by our algorithm can depend on $C$, we can apply an appropriate affine transformation to $C$ before choosing $r$; this does not change the hypergraphs that can be realized with the range space determined by the translates of $C$.
To ensure that properties (A) and (B) are satisfied would require further study of the Illumination conjecture.
Our bound for $m$ is quite large, even for the unit disk, both in Theorems \ref{thm:general_three_col} and \ref{thm:local_three_col}, which is mainly due to the fact that $f(3,2)$ given by Theorem \ref{thm:multi_essw_new} is huge.
It has been conjectured that in Theorem \ref{thm:multi_essw_old} the optimal value is $f(3)=3$, and a similarly small number seems realistic for $f(3,2)$ as well.
While Theorem \ref{thm:general_three_col} closed the last question left open for primal hypergraphs realizable by translates of planar bodies, the respective problem is still open in higher dimensions.
While it is not hard to show that some hypergraphs with high chromatic number often used in constructions can be easily realized by unit balls in $\mathbb{R}^5$, we do not know whether the chromatic number is bounded or not in $\mathbb{R}^3$.
From our Union Lemma (Lemma \ref{lem:combine}) it follows that to establish boundedness, it would be enough to find a polychromatic $k$-coloring for pseudohalfspaces, whatever this word means.
| proofpile-arXiv_065-54 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
One of the most significant incentives for recent research on movement assessment, is the availability of affordable 3D skeleton recognition devices, such as Kinect, which redefine the target audience of applications that are based on user pose and movement. Addressing this problem is considered a hard task, as it requires paying attention to timings, performances and low-level details.
In the recent decade, different solutions have been proposed for dealing with automatic assessment of movements, based on machine-learning algorithms. In this work, we review these solutions and compare them.
We divide the assessment problem into two typical problems of detecting abnormalities in repetitive movements and predicting scores in structured movements. We then list the existing works and their features and the existing public datasets. We elaborate on the secondary problems that take part in the algorithmic flow of typical movement assessment solutions and list the methods and algorithms used by the different works. Finally, we discuss the findings in a high level.
The outline of this review is as follows. In the next chapter, we first present the main types of movement assessment problems, list the features of existing works and list the used public datasets. In addition, we elaborate on the secondary problems and list the methods that were implemented to solve them. The last two chapters include a discussion and conclusions, respectively.
\section{Movement Assessment}
\label{lbl:movementAssessment}
There are generally two main types of movement assessment solutions. The first type focuses on detecting abnormalities in relatively long, repetitive movements~\cite{paiement2014online,jun2020feature,chaaraoui2015abnormal,nguyen2016skeleton,devanne2016learning,baptista2018deformation,nguyen2018estimating,nguyen2018skeleton,khokhlova2018kinematic}, such as gait, as visualized in Figure~\ref{fig:gait}. The second type of movements, on the other hand, focuses on assessing structured movements~\cite{parisi2016human,su2013personal,eichler20183d,eichler2018non,hakim2019mal,hakim2019malf,masalha2020predicting,dressler2019data,dressler2020towards,hu2014real,capecci2016physical,capecci2018hidden,osgouei2018objective,al2019quantifying,cao2019novel,williams2019assessment,yu2019dynamic,lei2020learning}, such as movements from the Fugl-Meyer Assessment (FMA)~\cite{fugl1975post} or Berg Balance Scale (BBS)~\cite{bbs} medical assessments, as visualized in Figure~\ref{fig:fma_and_bbs}, which usually have clear definitions of starting positions, ending positions, objectives and constraints.
\begin{figure}[]
\centering
\includegraphics[width=0.75\linewidth,keepaspectratio]{images/gait.png}
\caption[]{A walking-up-stairs movement with 3D skeleton joints detected from a Kinect RGB-D video~\cite{paiement2014online}.}
\label{fig:gait}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=0.75\linewidth,keepaspectratio]{images/fma_and_bbs.png}
\caption[]{An FMA assessment~\cite{eichler2018non} and a BBS assessment~\cite{masalha2020predicting}.}
\label{fig:fma_and_bbs}
\end{figure}
While most of the works deal with assessing known-in-advance, limited range of movement types, only a few works try to provide general solutions, which aim to be adaptive to new types of movements. Such works, which were evaluated on multiple types of movements~\cite{parisi2016human,su2013personal,eichler20183d,eichler2018non,hakim2019mal,hakim2019malf,masalha2020predicting,hu2014real,capecci2016physical,capecci2018hidden,al2019quantifying,cao2019novel,williams2019assessment,lei2020learning}, may therefore assume no prior knowledge on a learned movement type, such that they may need to automatically extract its most important properties from the training set, or use learning algorithms that are adaptive in their nature.
A typical movement assessment algorithm will need to address the following fundamental problems: capturing or detecting human skeleton joint positions, geometric normalization, temporal alignment, feature extraction, score prediction and feedback generation. In this chapter, we review the solutions existing works implemented for each of these problems.
\subsection{Movement Domains and Solution Features}
Most of the works that deal with structured movements, mainly deal with predicting the quality of a performance and sometimes producing feedback. On the contrary, most of the works that deal with repetitive movements, such as gait, give more focus to detecting abnormalities and computing scores that are based on similarity to normal movements.
Table~\ref{tbl:features} summarizes the features of each of the works that deal with structured movements. When a solution produces a quality score on a continuous scale, then we consider the numerical score feature as existing. When a solution classifies performances into a discrete scale of qualities, then we consider the quality classification feature as existing. When a solution produces unbound textual feedback or presents describable features that can be directly translated into textual feedback, then we consider the unbound feedback feature as existing. When a training set that only consists of proper performances is sufficient for a solution to work, then we consider the trains-on-proper-movements feature as existing.
\begin{table}
\centering
\resizebox{0.98\linewidth}{!}{%
\begin{tabular}{ |c|c|c|c|c|c|c| }
\hline
\textbf{} & \textbf{Movement} & \textbf{No. Movement} & \textbf{Numerical} & \textbf{Quality} & \textbf{Unbound} & \textbf{Trains on} \\
\textbf{Work} & \textbf{Domain} & \textbf{Types Evaluated} & \textbf{Score} & \textbf{Classification} & \textbf{Feedback} & \textbf{Proper Movements} \\
\hline
\cite{parisi2016human} & Powerlifting & 3 & \checkmark & & \checkmark & \checkmark \\
\hline
\cite{su2013personal} & Rehabilitation & - & \checkmark & \checkmark & & \checkmark \\
\hline
\cite{eichler20183d,eichler2018non} & FMA & 2 & & \checkmark & & \\
\hline
\cite{hakim2019mal,hakim2019malf} & FMA & 3 & \checkmark & \checkmark & \checkmark & \checkmark \\
\hline
\cite{masalha2020predicting} & BBS & 14 & & \checkmark & & \\
\hline
\cite{dressler2019data,dressler2020towards} & Deep Squats & 1 & \checkmark & & - & \checkmark\\
\hline
\cite{hu2014real} & Qigong+others & 4+6 & \checkmark & & & \checkmark\\
\hline
\cite{capecci2016physical,capecci2018hidden} & Physiotherapy & 5 & \checkmark & & & \checkmark\\
\hline
\cite{osgouei2018objective} & Shoulder Abduction & 1 & \checkmark & & & \\
\hline
\cite{al2019quantifying} & General & 3 & & \checkmark & & \checkmark \\
\hline
\cite{cao2019novel} & Brunnstrom Scale & 9 & & \checkmark & & \\
\hline
\cite{williams2019assessment} & Rehabilitation & 2 & \checkmark & & & \\
\hline
\cite{yu2019dynamic} & Tai Chi & 1 & \checkmark & & & \checkmark \\
\hline
\cite{lei2020learning} & Olympic Sports & 9 & \checkmark & & & \\
\hline
\end{tabular}}
\\
\caption[]{Features of works that deal with assessing structured movements. The minus sign represents missing information.}
\label{tbl:features}
\end{table}
\subsection{Public Datasets}
Many of the works used existing public datasets for evaluating their solutions, while others created their own datasets, for different assessment tasks. The used datasets have either been kept private or made public~\cite{paiement2014online,nguyen2018estimating,nguyen2018skeleton,chaaraoui2015abnormal}. Some of the works used both public and private datasets. Table~\ref{tbl:datasets} lists the public datasets used by existing works.
\begin{table}
\centering
\resizebox{0.9\linewidth}{!}{%
\begin{tabular}{ |c|c|c| }
\hline
\textbf{Dataset} & \textbf{Movement Types} & \textbf{Used by} \\
\hline
SPHERE-staircase 2014,2015~\cite{paiement2014online} & Gait on stairs & \cite{paiement2014online,chaaraoui2015abnormal,devanne2016learning,baptista2018deformation,khokhlova2018kinematic} \\
\hline
DGD: DAI gait dataset~\cite{chaaraoui2015abnormal} & Gait & \cite{chaaraoui2015abnormal,devanne2016learning,khokhlova2018kinematic} \\
\hline
Walking gait dataset~\cite{nguyen2018walking} & Gait, under 9 different conditions & \cite{jun2020feature,nguyen2018estimating,nguyen2018skeleton,khokhlova2018kinematic} \\
\hline
UPCV Gait K2~\cite{kastaniotis2016pose} & Gait - normal walking & \cite{khokhlova2018kinematic} \\
\hline
Eyes. Mocap data~\cite{eyesmocapdata} & Gait captured by a Mocap system & \cite{nguyen2016skeleton} \\
\hline
HMRA~\cite{hmra} & Qigong and others & \cite{hu2014real} \\
\hline
UI-PRMD~\cite{vakanski2018data} & Physical therapy & \cite{williams2019assessment} \\
\hline
MIT Olympic Scoring Dataset~\cite{mitolympic} & Olympic scoring on RGB videos & \cite{lei2020learning} \\
\hline
UNLV Olympic Scoring Dataset~\cite{unlvoplymic} & Olympic scoring on RGB videos & \cite{lei2020learning} \\
\hline
\end{tabular}}
\\
\caption[]{Public movement assessment datasets.}
\label{tbl:datasets}
\end{table}
\subsection{Methods and Algorithms}
\subsubsection{Skeleton Detection.}
The majority of the works use 3D cameras, such as Kinect1 or Kinect2, with the Microsoft Kinect SDK~\cite{shotton2011real} or OpenNI for detection of 3D skeletons. Sometimes, marker-based motion-capture (Mocap) systems are used~\cite{nguyen2016skeleton,al2019quantifying,williams2019assessment}. Lei \textit{et al.}~\cite{lei2020learning} used 2D skeletons that were extracted from RGB videos, using OpenPose~\cite{cao2017realtime}, as visualized in Figure~\ref{fig:openpose}.
\begin{figure}[]
\centering
\includegraphics[width=0.2\linewidth,keepaspectratio]{images/openpose.png}
\caption[]{An OpenPose 2D skeleton~\cite{lei2020learning}.}
\label{fig:openpose}
\end{figure}
\subsubsection{Geometric Normalization.}
People perform movements in different distances and angles in respect to the 3D camera that captures their motion. Additionally, different people have different body dimensions, which have to be addressed by either pre-normalizing the skeleton dimensions and coordinates, as demonstrated in Figure~\ref{fig:geometric}, or extracting features that are inherently invariant to the camera location and body-dimensions, such as joint angles. This step therefore, may be considered an either independent or integral part of the feature-extraction process. Table~\ref{tbl:geometric} summarizes the geometric normalization methods used by existing works.
\begin{figure}[]
\centering
\includegraphics[width=0.35\linewidth,keepaspectratio]{images/geometric.png}
\caption[]{A geometric normalization step~\cite{hakim2019mal,hakim2019malf}.}
\label{fig:geometric}
\end{figure}
\begin{table}
\centering
\resizebox{0.98\linewidth}{!}{%
\begin{tabular}{ |c|l| }
\hline
\textbf{Work} & \textbf{Implementation} \\
\hline
\cite{paiement2014online} & Translation, rotation and scaling due to varying heights of the subjects. \\
\hline
\cite{jun2020feature} & Implementing the method from~\cite{paiement2014online}. \\
\hline
\cite{chaaraoui2015abnormal} & Translation, rotation by shoulder and hip joints, scaling.\\
\hline
\cite{nguyen2016skeleton} & Using features that are invariant to camera location and angle. \\
\hline
\cite{devanne2016learning} & - \\
\hline
\cite{baptista2018deformation} & Projection on the main direction of the motion variation. \\
\hline
\cite{nguyen2018estimating,nguyen2018skeleton} & Scaling the coordinates to the range between 0 and 1. \\
\hline
\cite{khokhlova2018kinematic} & Using features that are invariant to camera location and angle. \\
\hline
\cite{parisi2016human} & Translation. \\
\hline
\cite{su2013personal} & Geometric calibration as a system initialization step, before capturing skeleton videos. \\
\hline
\cite{eichler20183d,eichler2018non} & Using features that are invariant to camera location and angle. \\
\hline
\cite{hakim2019mal,hakim2019malf} & Projection on spine-shoulders plane, translation and equalizing skeleton edge lengths. \\
\hline
\cite{masalha2020predicting} & Using features that are invariant to camera location and angle. \\
\hline
\cite{dressler2019data,dressler2020towards} & Using features that are invariant to camera location and angle. \\
\hline
\cite{hu2014real} & Using features that are invariant to camera location and angle. \\
\hline
\cite{capecci2016physical,capecci2018hidden} & Using features that are invariant to camera location and angle. \\
\hline
\cite{osgouei2018objective} & Using features that are invariant to camera location and angle. \\
\hline
\cite{al2019quantifying} & Using features that are invariant to camera location and angle. \\
\hline
\cite{cao2019novel} & - \\
\hline
\cite{williams2019assessment} & Using features that are invariant to camera location and angle. \\
\hline
\cite{yu2019dynamic} & Projection on arm-leg-based coordinate system. \\
\hline
\cite{lei2020learning} & Scaling of the 2D human body. \\
\hline
\end{tabular}}
\caption[]{Geometric normalization methods.}
\label{tbl:geometric}
\end{table}
\subsubsection{Temporal Alignment.}
In order to produce reliable assessment outputs, a tested movement, which is a temporal sequence of data, usually has to be well-aligned in time with movements it will be compared to. For that purpose, most works either use models that inherently deal with sequences, such as HMMs and RNNs, as illustrated in Figure~\ref{fig:hmm}, or use temporal alignment algorithms, such as the DTW algorithm or its variants, as illustrated in Figure~\ref{fig:dtw}.
\begin{figure}[]
\centering
\includegraphics[width=0.45\linewidth,keepaspectratio]{images/hmm.png}
\caption[]{A Hidden Markov Model (HMM), which defines states, observations and probabilities of state transitions and observations~\cite{nguyen2016skeleton}.}
\label{fig:hmm}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=0.65\linewidth,keepaspectratio]{images/dtw.png}
\caption[]{Dynamic Time Warping (DTW) for alignment of two series of scalars, by matching pairs of indices~\cite{simpledtw}.}
\label{fig:dtw}
\end{figure}
Hakim and Shimshoni~\cite{hakim2019mal,hakim2019malf} introduced a novel warping algorithm, which was based on the detection of temporal points-of-interest (PoIs) and on linearly warping the sequences between them, as illustrated in Figure~\ref{fig:warp}. Dressler \textit{et al.}~\cite{dressler2019data,dressler2020towards} introduced a novel DTW variation with skips, similarly to Hu \textit{et al.}~\cite{hu2014real}. Other novel approaches were introduced by Devanne \textit{et al.}~\cite{devanne2016learning}, by Baptista \textit{et al.}~\cite{baptista2018deformation} and by Yu and Xiong~\cite{yu2019dynamic}.
Another less mentioned algorithm is the Correlation Optimized Warping
(COW) algorithm~\cite{tomasi2004correlation}.
Palma \textit{et al.}~\cite{palma2016hmm} and Hagelb\"{a}ck \textit{et al.}~\cite{hagelback2019variants} elaborated more on the topic of temporal alignment algorithms in the context of movement assessment. Table~\ref{tbl:temporal} summarizes the alignment methods used by existing works.
\begin{figure}[]
\centering
\includegraphics[width=0.8\linewidth,keepaspectratio]{images/warp.png}
\caption[]{A continuous warping by scaling between detected pairs of temporal points-of-interest~\cite{hakim2019mal,hakim2019malf}.}.
\label{fig:warp}
\end{figure}
\begin{table}
\centering
\resizebox{0.8\linewidth}{!}{%
\begin{tabular}{ |c|l| }
\hline
\textbf{Work} & \textbf{Implementation} \\
\hline
\cite{paiement2014online} & Inherently solved by the choice to use an HMM statistical model. \\
\hline
\cite{jun2020feature} & Inherently solved by the choice to use an RNN Autoencoder. \\
\hline
\cite{chaaraoui2015abnormal} & Discrete warping using the Dynamic Time Warping (DTW) algorithm. \\
\hline
\cite{nguyen2016skeleton} & Inherently solved by the choice to use an HMM statistical model. \\
\hline
\cite{devanne2016learning} & Riemannian shape analysis of legs shape evolution within a sliding window. \\
\hline
\cite{baptista2018deformation} & Key-point detection with deformation-based curve alignment~\cite{demisse2017deformation}. \\
\hline
\cite{nguyen2018estimating,nguyen2018skeleton} & Inherently solved by the choice to use a recurrent neural network.\\
\hline
\cite{khokhlova2018kinematic} & - \\
\hline
\cite{parisi2016human} & Inherently solved by the choice to use a recurrent neural network. \\
\hline
\cite{su2013personal} & Discrete warping using the Dynamic Time Warping (DTW) algorithm. \\
\hline
\cite{eichler20183d,eichler2018non} & - \\
\hline
\cite{hakim2019mal,hakim2019malf} & Detecting mutual temporal PoIs and continuously warping between them. \\
\hline
\cite{masalha2020predicting} & - \\
\hline
\cite{dressler2019data,dressler2020towards} & A novel DTW variant, with skips. \\
\hline
\cite{hu2014real} & A novel DTW variant with tolerance to editing. \\
\hline
\cite{capecci2016physical,capecci2018hidden} & DTW and Hidden Semi-Markov Models (HSMM). \\
\hline
\cite{osgouei2018objective} & DTW and HMM. \\
\hline
\cite{al2019quantifying} & - \\
\hline
\cite{cao2019novel} & Inherently solved by the choice to use a recurrent neural network. \\
\hline
\cite{williams2019assessment} & - \\
\hline
\cite{yu2019dynamic} & A novel DTW variant that minimizes angles between pairs of vectors. \\
\hline
\cite{lei2020learning} & - \\
\hline
\end{tabular}}
\caption[]{Temporal alignment methods.}
\label{tbl:temporal}
\end{table}
\subsubsection{Feature Extraction.}
The assessment of different types of movements requires paying attention to different details, which may include joint angles, pairwise joint distances, joint positions, joint velocities and event timings. Many of the feature extraction methods are invariant to the subject's skeleton scale and to the camera location and angle, as illustrated in Figure~\ref{fig:feature}, while others are usually preceded by a geometric normalization step. In the recent years, some works used deep features, which were automatically learned and were obscure, rather than using explainable handcrafted features.
It is notable that while some works were designated for specific domains of movements and exploited their prior knowledge to choose their features, other works were designated to be more versatile and adaptive to many movement domains, and therefore used general features. Table~\ref{tbl:feature} summarizes the feature extraction methods used by existing works.
\begin{figure}[]
\centering
\includegraphics[width=0.6\linewidth,keepaspectratio]{images/features.png}
\caption[]{Angles as extracted skeleton features, which are invariant to the camera location and to body dimension differences~\cite{nguyen2016skeleton}.}.
\label{fig:feature}
\end{figure}
\begin{table}
\centering
\resizebox{0.98\linewidth}{!}{%
\begin{tabular}{ |c|l| }
\hline
\textbf{Work} & \textbf{Implementation} \\
\hline
\cite{paiement2014online} & Applying Diffusion Maps~\cite{coifman2006diffusion} on the normalized 3D skeleton joint positions. \\
\hline
\cite{jun2020feature} & Deep features learned by training RNN Autoencoders. \\
\hline
\cite{chaaraoui2015abnormal} & Joint Motion History (JMH): spatio-temporal joint 3D positions. \\
\hline
\cite{nguyen2016skeleton} & Lower-body joint angles and the angle between the two feet. \\
\hline
\cite{devanne2016learning} & Square-root-velocity function (SRVF)~\cite{joshi2007novel} on temporal sequences of joint positions. \\
\hline
\cite{baptista2018deformation} & Distances between the projections of the two knees on the movement direction. \\
\hline
\cite{nguyen2018estimating,nguyen2018skeleton} & Deep features learned by Autoencoders. \\
\hline
\cite{khokhlova2018kinematic} & Covariance matrices of hip and knee flexion angles. \\
\hline
\cite{parisi2016human} & 13 joint 3D positions and velocities. \\
\hline
\cite{su2013personal} & Joint 3D positions and velocities. \\
\hline
\cite{eichler20183d,eichler2018non} & Joint angles, distances and heights from the ground. \\
\hline
\cite{hakim2019mal,hakim2019malf} & Joint 3D positions and velocities, distances and edge angles. Sequence timings. \\
\hline
\cite{masalha2020predicting} & Relative joint positions, joint distances, angles and height of joints from the ground. \\
\hline
\cite{dressler2019data,dressler2020towards} & Joint positions and NASM features (a list of selected skeleton angles). \\
\hline
\cite{hu2014real} & Torso direction and joint relative position represented by elevation and azimuth. \\
\hline
\cite{capecci2016physical,capecci2018hidden} & Selected features varying between movement types. \\
\hline
\cite{osgouei2018objective} & Shoulder and arm angles. \\
\hline
\cite{al2019quantifying} & Autoencoder embeddings of manually-extracted attributes. \\
\hline
\cite{cao2019novel} & Raw skeleton 3D data. \\
\hline
\cite{williams2019assessment} & GMM encoding of Autoencoder dimensionality-reduced joint angle data. \\
\hline
\cite{yu2019dynamic} & Angles of selected joints. \\
\hline
\cite{lei2020learning} & Self-similarity descriptors of joint trajectories and a joint displacement sequence.\\
\hline
\end{tabular}}
\caption[]{Feature extraction methods.}
\label{tbl:feature}
\end{table}
\subsubsection{Score Prediction.}
The prediction of an assessment score refers to one or more of the following cases:
\begin{enumerate}
\item Classifying a performance into a class from a predefined set of discrete quality classes.
\item Performing a regression that will map a performance into a number on a predefined continuous scale.
\item Producing scores that reflect the similarity between given model and performance, unbound to ground-truth or predefined scales.
\end{enumerate}
\noindent The two first types of scoring capabilities are mainly essential for formal assessments, such as medical assessments or Olympic performance judgements. The third type of scoring is mainly useful for comparing subject performances, which can be either a certain subject whose progress is monitored over time, or different subjects who compete.
Table~\ref{tbl:score} lists the algorithms used to produce quality scores. It is notable that score prediction was not implemented in many works, as it was irrelevant for them, since they only addressed normal/abnormal binary classifications.
\begin{table}
\centering
\resizebox{0.98\linewidth}{!}{%
\begin{tabular}{ |c|l| }
\hline
\textbf{Work} & \textbf{Implementation} \\
\hline
\cite{paiement2014online} & Pose and dynamics log likelihoods. \\
\hline
\cite{jun2020feature} & - \\
\hline
\cite{chaaraoui2015abnormal} & - \\
\hline
\cite{nguyen2016skeleton} & - \\
\hline
\cite{devanne2016learning} & Mean log-probability over the segments of the test sequence. \\
\hline
\cite{baptista2018deformation} & Distance between time-aligned feature sequences with reflection of time-variations. \\
\hline
\cite{nguyen2018estimating,nguyen2018skeleton} & - \\
\hline
\cite{khokhlova2018kinematic} & - \\
\hline
\cite{parisi2016human} & Difference between actual and RNN-predicted next frames. \\
\hline
\cite{su2013personal} & Handcrafted classification using Fuzzy Logic~\cite{zadeh1965fuzzy}. \\
\hline
\cite{eichler20183d,eichler2018non} & SVM, Decision Tree and Random Forest quality classification using handcrafted features. \\
\hline
\cite{hakim2019mal,hakim2019malf} & Thresholded weighted sum of normalized, time-filtered active/inactive joint and timing scores. \\
\hline
\cite{masalha2020predicting} & SVM and Random Forest quality classification using handcrafted features. \\
\hline
\cite{dressler2019data,dressler2020towards} & Weighted sum of selected feature differences. \\
\hline
\cite{hu2014real} & Average of frame cross-correlations. \\
\hline
\cite{capecci2016physical,capecci2018hidden} & Normalized log-likelihoods or DTW distances. \\
\hline
\cite{osgouei2018objective} & Difference from proper performance divided by difference between worst and proper performances. \\
\hline
\cite{al2019quantifying} & Classification using One-Class SVM. \\
\hline
\cite{cao2019novel} & Classification using a hybrid LSTM-CNN model. \\
\hline
\cite{williams2019assessment} & Normalized log-likelihoods. \\
\hline
\cite{yu2019dynamic} & DTW similarity. \\
\hline
\cite{lei2020learning} & Regression based on high-level features combined with joint trajectories and displacements. \\
\hline
\end{tabular}}
\caption[]{Score prediction methods.}
\label{tbl:score}
\end{table}
\subsubsection{Feedback Generation.}
There are two main types of feedback that can be generated: bound feedback and unbound feedback. Feedback is bound when it can only consist of predefined mistakes or abnormalities that can be detected. Feedback is unbound when it is a generated natural language text that can describe any type of mistake, deviation or abnormality. The generation of unbound feedback usually requires the usage of describable low-level features, so that when a performance is not proper, it will be possible to indicate the most significant features that reduced the score and translate them to natural language, such that the user can use the feedback to learn how to improve their next performance. Such feedback may include temporal sequences that deviate similarly, as visualized in Figure~\ref{fig:ParameterTimeSegmentation}.
Table~\ref{tbl:feedback} summarizes the types of feedback and generation methods used by the works. It is notable that: 1) Most of the works do not generate feedback. 2) There are no works that produce feedback while not predicting quality scores, for the same reason of only focusing on binary detection of abnormalities.
\begin{figure}[]
\centering
\includegraphics[width=0.75\linewidth,keepaspectratio]{images/segmentation.png}
\caption[]{Temporal segmentation of parameter deviations for feedback generation~\cite{hakim2019mal,hakim2019malf}.}
\label{fig:ParameterTimeSegmentation}
\end{figure}
\begin{table}
\centering
\resizebox{0.9\linewidth}{!}{%
\begin{tabular}{ |c|l| }
\hline
\textbf{Work} & \textbf{Implementation} \\
\hline
\cite{paiement2014online} & - \\
\hline
\cite{jun2020feature} & - \\
\hline
\cite{chaaraoui2015abnormal} & - \\
\hline
\cite{nguyen2016skeleton} & - \\
\hline
\cite{devanne2016learning} & Visualizing the deviations of the body parts. \\
\hline
\cite{baptista2018deformation} & - \\
\hline
\cite{nguyen2018estimating,nguyen2018skeleton} & - \\
\hline
\cite{khokhlova2018kinematic} & - \\
\hline
\cite{parisi2016human} & Sequences of parameter deviations and detection of predefined typical mistakes. \\
\hline
\cite{su2013personal} & Three quality classifications indicating trajectory similarity and right speed. \\
\hline
\cite{eichler20183d,eichler2018non} & - \\
\hline
\cite{hakim2019mal,hakim2019malf} & Translation of worst class-segmented parameter temporal deviations into text. \\
\hline
\cite{masalha2020predicting} & - \\
\hline
\cite{dressler2019data,dressler2020towards} & Indication of weak links according to angle differences. \\
\hline
\cite{hu2014real} & - \\
\hline
\cite{capecci2016physical,capecci2018hidden} & - \\
\hline
\cite{osgouei2018objective} & - \\
\hline
\cite{al2019quantifying} & - \\
\hline
\cite{cao2019novel} & - \\
\hline
\cite{williams2019assessment} & - \\
\hline
\cite{yu2019dynamic} & - \\
\hline
\cite{lei2020learning} & - \\
\hline
\end{tabular}}
\caption[]{Feedback generation methods.}
\label{tbl:feedback}
\end{table}
\section{Discussion}
From the reviewed works, we can learn that a typical movement assessment solution may deal with detection of abnormal events or predicting quality scores, using classification, regression or computation of a normalized similarity measure. The task of detecting abnormal events is usually associated with repetitive movements, while the task of predicting scores is usually associated with structured movements.
We can learn that while public skeleton datasets exist and are used by some of the works, most of the works use private datasets that were acquired for the sake of a specific work. It is notable that while many novelties are proposed in the different works, the absence of common datasets and evaluation metrics leads to different works dealing with different problems, evaluating themselves on different datasets of different movement domains, using different metrics.
It is notable that temporal alignment is a key-problem in movement assessment. From the reviewed works, we can learn that around a half of the works base their temporal alignment solutions on models that are designated for sequential inputs, such as Hidden Markov Models and recurrent neural networks, while others use either the Dynamic Time Warping algorithm, sometimes with novel improvements, or other novel warping and alignment approaches.
We can learn that while a few works use features that were automatically learned by neural networks, most of the works make use of handcrafted skeleton features. In many of those works, the use features are invariant to the camera location and angle and to the body-dimensions of the performing subjects. Other works that make use of handcrafted features usually have to apply a geometric normalization step before continuing to the next steps. It is worth to mention that while some of the works were designed to deal with a specific type of movement, other works were designed to be adaptive and deal with many types of movements, a choice that is usually clearly reflected in the feature extraction step.
We can learn that a quality score is produced by most of the works. While works that deal with medical assessments mainly focus on classification into predefined discrete scoring scales, other works predict scores on continuous scales. Such scores are rarely learned as a regression problem and are usually based on a normalized similarity measure. Finally, we can learn that only a few works deal with producing feedback, which can be bound or unbound.
In the future, the formation of a large, general public dataset and a common evaluation metric may help define the state-of-the-art and boost the research on the topic of movement assessment. In addition, the improvement of mobile-device cameras, as well as computer vision algorithms that detect skeletons in RGB-D or even RGB videos, may raise the interest in researching this topic.
\section{Conclusions}
We have provided a review of the existing works in the domain of movement assessment from skeletal data, which gives a high level picture of the problems addressed and approaches implemented by existing solutions.
We divided the types of assessment tasks into two main categories, which were detection of abnormalities in long, repetitive movements and scoring structured movements, sometimes while generating textual feedback. The objectives and challenges of the assessment task were discussed and the ways they were addressed by each of the works were listed, including skeleton joint detection, geometric normalization, temporal alignment, feature extraction, score prediction and feedback generation. The existing public datasets and evaluated movement domains were listed. Finally, a high level discussion was provided. We hope that this review will provide a good starting point for new researchers.
\bibliographystyle{splncs}
| proofpile-arXiv_065-150 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
\label{sec: intro}
Gauge models with anomaly are interesting from different points of view.
First, there is a problem of consistent quantization for these models.
Due to anomaly some constraints change their nature after quantization:
instead of being first-class constraints, they turn into second-class
ones. A consistent canonical quantization scheme clearly should take
into account such a change \cite{jack85}-\cite{sarad91}.
Next is a problem of the relativistic invariance. It is known that in the
physical sector where the local gauge invariance holds the relativistic
invariance is broken for some anomalous models, namely the chiral
Schwinger model (CSM) and chiral $QCD_2$ \cite{niemi86}-\cite{sarad96}.
For both models the Poincare
algebra commutation relations breaking term can be constructed
explicitly \cite{sarad96}.
In the present paper we address ourselves to another aspect of anomalous
models: the Berry phase and its connection to anomaly. A common topological
nature of the Berry phase, or more generally quantum holonomy, and gauge
anomalies was noted in \cite{alva85},\cite{niemi85}. The former was shown
to be crucial in the hamiltonian interpretation of anomalies.
We consider a general version of the CSM with a ${\rm U}(1)$ gauge field
coupled with different charges to both chiral components of a fermionic
field. The non-anomalous Schwinger model (SM) where these charges are
equal is a special case of the generalized CSM. This will allow us
to see any distinction between the models with and without anomaly.
We suppose that space is a circle of length ${\rm L}$,
$-\frac{\rm L}{2} \leq x < \frac{\rm L}{2}$, so space-time manifold
is a cylinder ${\rm S}^1 \otimes {\rm R}^1$. We work in the temporal
gauge $A_0=0$ and use the system of units where $c=1$. Only matter
fields are quantized, while $A_1$ is handled as a classical
background field. Our aim is to calculate the Berry phase and the
corresponding ${\rm U}(1)$ connection and curvature for the fermionic
Fock vacuum as well as for many particle states constructed over the
vacuum and to show explicitly a connection between the nonvanishing
vacuum Berry phase and anomaly.
Our paper is organized as follows. In Sect.~\ref{sec: quant}, we
apply first and second quantization to the matter fields and obtain
the second quantized fermionic Hamiltonian. We define the Fock
vacuum and construct many particle Fock states over the vacuum. We
use a particle-hole interpretation for these states.
In Sect.~\ref{sec: berry} , we first derive a general formula for
the Berry phase and then calculate it for the vacuum and many
particle states. We show that for all Fock states the Berry phase
vanishes in the case of models without anomaly. We discuss a connection
between the nonvanishing vacuum Berry phase, anomaly and effective
action of the model.
Our conclusions are in Sect.~\ref{sec: con}.
\newpage
\section{Quantization of matter fields}
\label{sec: quant}
The Lagrangian density of the generalized CSM is
\begin{equation}
{\cal L} = - {\frac{1}{4}} {\rm F}_{\mu \nu} {\rm F}^{\mu \nu} +
\bar{\psi} i {\hbar} {\gamma}^{\mu} {\partial}_{\mu} \psi +
e_{+} {\hbar} \bar{\psi}_{+} {\gamma}^{\mu} {\psi_{+}} A_{\mu} +
e_{-} {\hbar} \bar{\psi}_{-} {\gamma}^{\mu} {\psi_{-}} A_{\mu} ,
\label{eq: odin}
\end{equation}
where ${\rm F}_{\mu \nu}= \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$ ,
$(\mu, \nu) = \overline{0,1}$ , $\gamma^{0}={\sigma}_1$,
${\gamma}^{1}=-i{\sigma}_2$, ${\gamma}^0 {\gamma}^1={\gamma}^5=
{\sigma}_3$, ${\sigma}_i (i=\overline{1,3})$ are Pauli matrices.
The field $\psi$ is $2$--component Dirac spinor, $\bar{\psi} =
\psi^{\dagger} \gamma^0$ and $\psi_{\pm}=\frac{1}{2} (1 \pm \gamma^5)
\psi$.
In the temporal gauge $A_0=0$, the Hamiltonian density is
\begin{equation}
{\cal H} = \frac{1}{2}{\rm E}^2 + {\cal H}_{+} + {\cal H}_{-},
\label{eq: dva}
\end{equation}
with ${\rm E}$ momentum canonically conjugate to $A_1$, and
\[
{\cal H}_{\pm} \equiv \hbar \psi_{\pm}^{\dagger} d_{\pm} \psi_{\pm} =
\mp \hbar \psi_{\pm}^{\dagger}(i{\partial}_{1}+e_{\pm}A_1)\psi_{\pm}.
\]
On the circle boundary conditions for the fields must be specified.
We impose the periodic ones
\begin{eqnarray}
{A_1} (- \frac{\rm L}{2}) & = & {A_1} (\frac{\rm L}{2}) \nonumber \\
{\psi_{\pm}} (- \frac{\rm L}{2}) & = & {\psi_{\pm}} (\frac{\rm L}{2}).
\label{eq: tri}
\end{eqnarray}
The Lagrangian and Hamiltonian densities
are invariant under local time-independent gauge
transformations
\begin{eqnarray*}
A_1 & \rightarrow & A_1 + {\partial}_{1} \lambda,\\
\psi_{\pm} & \rightarrow & \exp\{ie_{\pm} \lambda\} \psi_{\pm},
\end{eqnarray*}
$\lambda$ being a gauge function.
For arbitrary $e_{+},e_{-}$, the gauge transformations do not respect
the boundary conditions ~\ref{eq: tri}.
The gauge transformations compatible with the boundary conditions
must be either of the form
\[
\lambda (\frac{\rm L}{2})=\lambda (- \frac{\rm L}{2}) +
{\frac{2\pi}{e_{+}}}n,
\hspace{5 mm}
{\rm n} \in \cal Z.
\]
with $e_{+} \neq 0$ and
\begin{equation}
\frac{e_{-}}{e_{+}} = {\rm N},
\hspace{5 mm}
{\rm N} \in \cal Z,
\label{eq: cet}
\end{equation}
or of the form
\[
\lambda(\frac{\rm L}{2}) = \lambda(-\frac{\rm L}{2}) +
\frac{2\pi}{e_{-}} n ,
\hspace{5 mm}
{\rm n} \in \cal Z,
\]
with $e_{-} \neq 0$ and
\begin{equation}
\frac{e_{+}}{e_{-}} = \bar{\rm N},
\hspace{5 mm}
\bar{\rm N} \in \cal Z.
\label{eq: pet}
\end{equation}
Eqs. ~\ref{eq: cet} and ~\ref{eq: pet} imply a quantization condition
for the charges. Without loss of generality, we choose ~\ref{eq: cet}.
For ${\rm N}=1$, $e_{-}=e_{+}$ and we have the standard Schwinger model.
For ${\rm N}=0$, we get the model in which only the positive chirality
component of the Dirac field is coupled to the gauge field.
We see that the gauge transformations under consideration are divided
into topological classes characterized by the integer $n$. If
$\lambda(\frac{\rm L}{2}) = \lambda(-\frac{\rm L}{2})$, then the
gauge transformation is topologically trivial and belongs to the
$n=0$ class. If $n \neq 0$ it is nontrivial and has winding number $n$.
The eigenfunctions and the eigenvalues of the first quantized
fermionic Hamiltonians are
\[
d_{\pm} \langle x|n;{\pm} \rangle = \pm \varepsilon_{n,{\pm }}
\langle x|n;{\pm } \rangle ,
\]
where
\[
\langle x|n;{\pm } \rangle = \frac{1}{\sqrt {\rm L}}
\exp\{ie_{\pm} \int_{-{\rm L}/2}^{x} dz{A_1}(z) +
i\varepsilon_{n,{\pm}} \cdot x\},
\]
\[
\varepsilon_{n,{\pm }} = \frac{2\pi}{\rm L}
(n - \frac{e_{\pm}b{\rm L}}{2\pi}).
\]
We see that the spectrum of the eigenvalues depends on the zero
mode of the gauge field:
\[
b \equiv \frac{1}{\rm L} \int_{-{\rm L}/2}^{{\rm L}/2} dx
A_1(x,t).
\]
For $\frac{e_{+}b{\rm L}}{2\pi}={\rm integer}$, the spectrum contains
the zero energy level. As $b$ increases from $0$ to
$\frac{2\pi}{e_{+}{\rm L}}$, the energies of
$\varepsilon_{n,+}$ decrease by $\frac{2\pi}{\rm L}$, while the energies
of $(-\varepsilon_{n,-})$ increase by $\frac{2\pi}{\rm L} {\rm N}$.
Some of energy levels change sign. However, the spectrum at the
configurations $b=0$ and $b=\frac{2\pi}{e_{+}{\rm L}}$
is the same, namely, the integers, as it must be since these gauge-field
configurations are gauge-equivalent. In what follows, we
will use separately the integer and fractional parts of
$\frac{e_{\pm}b{\rm L}}{2\pi}$, denoting them as
$[\frac{e_{\pm}b{\rm L}}{2\pi}]$ as $\{\frac{e_{\pm}b{\rm L}}{2\pi}\}$
correspondingly.
Now we introduce the second quantized right-handed and
left-handed Dirac fields. For the moment, we will assume that $d_{\pm}$
do not have zero eigenvalues. At time $t=0$, in terms of the
eigenfunctions of the first quantized fermionic Hamiltonians the second
quantized ($\zeta$--function regulated) fields have the expansion
\cite{niese86} :
\[
\psi_{+}^s (x) = \sum_{n \in \cal Z} a_n \langle x|n;{+} \rangle
|\lambda \varepsilon_{n,+}|^{-s/2},
\]
\begin{equation}
\psi_{-}^s (x) = \sum_{n \in \cal Z} b_n \langle x|n;{-} \rangle
|\lambda \varepsilon_{n,-}|^{-s/2}.
\label{eq: vosem}
\end{equation}
Here $\lambda$ is an arbitrary constant with dimension of length
which is necessary to make $\lambda \varepsilon_{n,\pm}$ dimensionless,
while $a_n, a_n^{\dagger}$ and $b_n, b_n^{\dagger}$ are correspondingly
right-handed and left-handed fermionic creation and annihilation
operators which fulfil the commutation relations
\[
[a_n , a_m^{\dagger}]_{+} = [b_n , b_n^{\dagger}]_{+} =\delta_{m,n} .
\]
For $\psi_{\pm }^{s} (x)$, the equal time anticommutators are
\begin{equation}
[\psi_{\pm}^{s}(x) , \psi_{\pm}^{\dagger s}(y)]_{+}=\zeta_{\pm} (s,x,y),
\label{eq: devet}
\end{equation}
with all other anticommutators vanishing, where
\[
\zeta_{\pm} (s,x,y) \equiv \sum_{n \in \cal Z} \langle x|n;{\pm} \rangle
\langle n;{\pm}|y \rangle |\lambda \varepsilon_{n,\pm}|^{-s},
\]
$s$ being large and positive. In the limit, when the regulator
is removed, i.e. $s=0$, $\zeta_{\pm}(s=0,x,y) = \delta(x-y)$ and
Eq.~\ref{eq: devet} takes the standard form.
The vacuum state of the second quantized fermionic Hamiltonian
\[
|{\rm vac};A \rangle = |{\rm vac};A;+ \rangle \otimes
|{\rm vac};A;- \rangle
\]
is defined such that all negative energy
levels are filled and the others are empty:
\begin{eqnarray}
a_n|{\rm vac};A;+\rangle =0 & {\rm for} & n>[\frac{e_{+}b{\rm L}}{2\pi}],
\nonumber \\
a_n^{\dagger} |{\rm vac};A;+ \rangle =0 & {\rm for} & n \leq
[\frac{e_{+}b{\rm L}}{2\pi}],
\label{eq: deset}
\end{eqnarray}
and
\begin{eqnarray}
b_n|{\rm vac};A;-\rangle =0 & {\rm for} & n \leq
[\frac{e_{-}b{\rm L}}{2\pi}], \nonumber \\
b_n^{\dagger} |{\rm vac};A;- \rangle =0 & {\rm for} & n >
[\frac{e_{-}b{\rm L}}{2\pi}].
\label{eq: odinodin}
\end{eqnarray}
In other words, in the positive chirality vacuum all the levels
with energy lower than ${\varepsilon}_{[\frac{e_{+}b{\rm L}}
{2\pi}]+1,+}$ and in the negative chirality one all the levels
with energy lower than $(-{\varepsilon}_{[\frac{e_{-}b{\rm L}}
{2\pi}],-})$ are filled:
\begin{eqnarray*}
|{\rm vac}; A;+ \rangle & = & \prod_{n=\infty}^{[\frac{e_{+}b
{\rm L}}{2\pi}]} a_m^{\dagger} |0;+ \rangle, \\
|{\rm vac}; A;- \rangle & = & \prod_{n=[\frac{e_{-}b{\rm L}}
{2\pi}]+1}^{+\infty} b_n^{\dagger} |0;- \rangle, \\
\end{eqnarray*}
where $|0 \rangle = |0,+ \rangle \otimes |0,- \rangle$ is the state
of "nothing" with all the energy levels empty.
The Fermi surfaces which are defined to lie halfway between the highest
filled and lowest empty levels are
\[
{\varepsilon}_{\pm}^{\rm F} = \pm \frac{2\pi}{\rm L}
(\frac{1}{2} - \{\frac{e_{\pm}b{\rm L}}{2\pi}\}).
\]
For $e_{+}=e_{-}$, ${\varepsilon}_{+}^{\rm F}=-{\varepsilon}_{-}^{\rm F}$.
Next we define the fermionic parts of the second-quantized Hamiltonian as
\[
\hat{\rm H}_{\pm}^s = \int_{-{\rm L}/2}^{{\rm L}/2} dx
\hat{\cal H}_{\pm}^s(x)= \frac{1}{2} \hbar \int_{-{\rm L}/2}^{{\rm L}/2}dx
(\psi_{\pm}^{\dagger s} d_{\pm} \psi_{\pm}^s
- \psi_{\pm}^s d_{\pm}^{\star} \psi_{\pm}^{\dagger s}).
\]
Substituting ~\ref{eq: vosem} into this expression, we get
\begin{equation}
\hat{\rm H}_{\pm} = \hat{\rm H}_{0,\pm} \mp
e_{\pm} b \hbar :\rho_{\pm}(0): + {\hbar} \frac{\rm L}{4\pi}
({\varepsilon}_{\pm}^{\rm F})^2,
\label{eq: hamil}
\end{equation}
where double dots indicate normal ordering with respect to
$|{\rm vac},A \rangle$ ,
\begin{eqnarray*}
\hat{\rm H}_{0,+} & = & \hbar \frac{2 \pi}{\rm L} \lim_{s \to 0}
\{ \sum_{k >[\frac{e_{+}b{\rm L}}{2 \pi}]} k a_k^{\dagger} a_k
|\lambda \varepsilon_{k,+}|^{-s} - \sum_{k \leq [\frac{e_{+}b{\rm L}}
{2 \pi}]} k a_k a_k^{\dagger} |\lambda \varepsilon_{k,+}|^{-s} \},\\
\hat{\rm H}_{0,-} & = & \hbar \frac{2 \pi}{\rm L} \lim_{s \to 0}
\{ \sum_{k>[\frac{e_{-}b{\rm L}}{2 \pi}]} k b_{k} b_{k}^{\dagger}
|\lambda \varepsilon_{k,-}|^{-s} - \sum_{k \leq [\frac{e_{-}b{\rm L}}
{2 \pi}]} k b_{k}^{\dagger} b_{k} |\lambda \varepsilon_{k,-}|^{-s} \}
\end{eqnarray*}
are free fermionic Hamiltonians, and
\begin{eqnarray*}
:\rho_{+} (0): & = & \lim_{s \to 0} \{ \sum_{k >[\frac{e_{+}b{\rm L}}
{2 \pi}]} a_k^{\dagger} a_k |\lambda \varepsilon_{k,+}|^{-s} -
\sum_{k \leq [\frac{e_{+}b{\rm L}}{2 \pi}]} a_k a_k^{\dagger}
|\lambda \varepsilon_{k,+}|^{-s} \}, \\
:\rho_{-} (0): & = & \lim_{s \to 0} \{ \sum_{k \leq [\frac{e_{-}b{\rm L}}
{2 \pi}]} b_{k}^{\dagger} b_{k} |\lambda \varepsilon_{k,-}|^{-s} -
\sum_{k>[\frac{e_{-}b{\rm L}}{2 \pi}]} b_{k} b_{k}^{\dagger}
|\lambda \varepsilon_{k,-}|^{-s} \}
\end{eqnarray*}
are charge operators for the positive and negative chirality fermion
fields respectively. The fermion momentum operators constructed
analogously are
\[
\hat{\rm P}_{\pm} = \hat{\rm H}_{0,\pm}.
\]
The operators $:\hat{\rm H}_{\pm}:$, $:\rho_{\pm}(0):$
and $\hat{\rm P}_{\pm}$ are
well defined when acting on finitely excited states which have only a
finite number of excitations relative to the Fock vacuum.
For the vacuum state,
\[
:\hat{\rm H}_{\pm}:|{\rm vac}; A;\pm \rangle =
:{\rho}_{\pm}(0):|{\rm vac}; A;\pm \rangle =0.
\]
Due to the normal ordering, the energy of the vacuum which is at the
same time the ground state of the fermionic Hamiltonians turns out
to be equal to zero ( we neglect an infinite energy of the filled
levels below the Fermi surfaces ${\varepsilon}_{\pm}^{\rm F}$).
The vacuum state can be considered also as a state of the zero charge.
Any other state of the same charge will have some of the levels above
${\varepsilon}_{+}^{\rm F}$ (${\varepsilon}_{-}^{\rm F}$) occupied
and some levels below ${\varepsilon}_{+}^{\rm F}$ (${\varepsilon}_{-}
^{\rm F}$) unoccupied. It is convenient to use the vacuum state
$|{\rm vac}; A \rangle$ as a reference, describing the removal
of a particle of positive (negative) chirality from one of the levels
below ${\varepsilon}_{+}^{\rm F}$ (${\varepsilon}_{-}^{\rm F}$) as
the creation of a "hole" \cite{dirac64},\cite{feyn72}.
Particles in the levels above
${\varepsilon}_{+}^{\rm F}$ (${\varepsilon}_{-}^{\rm F}$) are still
called particles. If a particle of positive (negative) chirality
is excited from the level $m$ below the Fermi surface to the level
$n$ above the Fermi surface, then we say that a hole of positive
chirality with energy $(-{\hbar}{\varepsilon}_{m,+})$ and
momentum $(-{\hbar}\frac{2\pi}{\rm L} m)$ ( or of negative chirality
with energy ${\hbar}{\varepsilon}_{m,-}$ and momentum
${\hbar}\frac{2\pi}{\rm L} m$) has been created as well as the
positive chirality particle with energy ${\hbar}{\varepsilon}_{n,+}$
and momentum ${\hbar}\frac{2\pi}{\rm L}n$ ( or the negative chirality
one with energy $(-{\hbar}{\varepsilon}_{n,-})$ and momentum
$(-{\hbar}\frac{2\pi}{\rm L}n)$ ). The operators $a_k (k \leq
[\frac{e_{+}b{\rm L}}{2\pi}])$ and $b_k (k>[\frac{e_{-}b{\rm L}}{2\pi}])$
behave like creation operators for the positive and negative chirality
holes correspondingly.
In the charge operator a hole counts as $-1$, so that, for example,
any state with one particle and one hole as well as the vacuum state
has vanishing charge.
The number of particles and holes of positive and negative chirality
outside the vacuum state is given by the operators
\begin{eqnarray*}
{\rm N}_{+} & = & \lim_{s \to 0} \{ \sum_{k>[\frac{e_{+}b{\rm L}}
{2\pi}]} a_k^{\dagger} a_k + \sum_{k \leq [\frac{e_{+}b{\rm L}}
{2\pi}]} a_k a_k^{\dagger} \} |{\lambda}{\varepsilon}_{k,+}|^{-s}, \\
{\rm N}_{-} & = & \lim_{s \to 0} \{ \sum_{k \leq [\frac{e_{-}b{\rm L}}
{2\pi}]} b_k^{\dagger} b_k + \sum_{k>[\frac{e_{-}b{\rm L}}{2\pi}]}
b_k b_k^{\dagger} \} |{\lambda}{\varepsilon}_{k,-}|^{-s},\\
\end{eqnarray*}
which count both particle and hole as $+1$.
Excited states are constructed by operating creation operators
on the vacuum. We start with $1$-particle states. Let us define the
states $|m; A;\pm \rangle$ as follows
$$
|m; A;+ \rangle \equiv \left\{
\begin{array}{cc}
a_m^{\dagger}|{\rm vac}; A;+
\rangle & {\rm for} \hspace{5 mm} m>[\frac{e_{+}b{\rm L}}{2\pi}], \\
a_m |{\rm vac}; A;+
\rangle & {\rm for} \hspace{5 mm} m \leq [\frac{e_{+}b{\rm L}}{2\pi}]
\end{array}
\right.
$$
and
$$
|m; A;- \rangle \equiv
\left\{ \begin{array}{cc}
b_m^{\dagger} |{\rm vac}; A;-
\rangle & {\rm for} \hspace{5 mm} m \leq [\frac{e_{-}b{\rm L}}{2\pi}],\\
b_m |{\rm vac}; A;- \rangle & {\rm for} \hspace{5 mm} m>[\frac{e_{-}b{\rm
L}}{2\pi}]. \end{array}
\right .
$$
The states $|m; A;\pm \rangle$ are orthonormalized,
\[
\langle m; A;\pm |n, A; \pm \rangle = \delta_{mn},
\]
and fulfil the completeness relation
\[
\sum_{m \in \cal Z} |m; A;\pm \rangle \cdot
\langle m; A;\pm| =1.
\]
It is easily checked that
\begin{eqnarray*}
:\hat{\rm H}_{\pm}: |m; A;\pm \rangle & = & {\hbar}{\varepsilon}
_{m,\pm} |m; A;\pm \rangle, \\
\hat{\rm P}_{\pm} |m; A;\pm \rangle & = & {\hbar}\frac{2\pi}{\rm L}
m |m; A;\pm \rangle, \\
:{\rho}_{\pm}(0): |m; A;\pm \rangle & = & \pm |m; A;\pm \rangle
\hspace{5 mm}
{\rm for}
\hspace{5 mm}
m > [\frac{e_{\pm}b{\rm L}}{2\pi}]
\end{eqnarray*}
and
\begin{eqnarray*}
:\hat{\rm H}_{\pm}: |m; A;\pm \rangle & = & - {\hbar}{\varepsilon}
_{m,\pm} |m; A;\pm \rangle, \\
\hat{\rm P}_{\pm} |m; A;\pm \rangle & = & -{\hbar} \frac{2\pi}{\rm L}
m |m; A;\pm \rangle, \\
:{\rho}_{\pm}(0): |m; A;\pm \rangle & = & \mp |m; A;\pm \rangle
\hspace{5 mm}
{\rm for}
\hspace{5 mm}
m \leq [\frac{e_{\pm}b{\rm L}}{2\pi}].
\end{eqnarray*}
We see that $|m; A;+ \rangle$ is a state with one particle of
positive chirality with energy ${\hbar}{\varepsilon}_{m,+}$ and
momentum ${\hbar}\frac{2\pi}{\rm L} m$ for $m>[\frac{e_{+}b{\rm L}}
{2\pi}]$ or a state with one hole of the same chirality with energy
$(-{\hbar}{\varepsilon}_{m,+})$ and momentum $(-\hbar \frac{2\pi}{\rm L}
m)$ for $m \leq [\frac{e_{+}b{\rm L}}{2\pi}]$. The negative chirality
state $|m; A;- \rangle$ is a state with one particle with energy
$(-\hbar {\varepsilon}_{m,-})$ and momentum $(-\hbar \frac{2\pi}{\rm L}
m)$ for $m \leq [\frac{e_{-}b{\rm L}}{2\pi}]$ or a state with one hole
with energy $\hbar {\varepsilon}_{m,-}$ and momentum $\hbar
\frac{2\pi}{\rm L} m$ for $m >[\frac{e_{-}b{\rm L}}{2\pi}]$. In any
case,
\[
{\rm N}_{\pm}|m; A;\pm \rangle = |m; A;\pm \rangle,
\]
that is why $|m; A;\pm \rangle$ are called $1$-particle states.
By applying $n$ creation operators to the vacuum states $|{\rm vac};
A;\pm \rangle$ we can get also $n$-particle states
\[
|m_1;m_2;...;m_n; A;\pm \rangle
\hspace{5 mm}
(m_1 < m_2 < ... <m_n),
\]
which are orthonormalized:
\[
\langle m_1;m_2;...;m_n; A;\pm |\overline{m}_1;\overline{m}_2;
...;\overline{m}_n; A;\pm \rangle =
{\delta}_{m_1 \overline{m}_1} {\delta}_{m_2 \overline{m}_2} ...
{\delta}_{m_n \overline{m}_n}.
\]
The completeness relation is written in the following form
\begin{equation}
\frac{1}{n!} \sum_{m_1 \in \cal Z} ... \sum_{m_n \in \cal Z}
|m_1;m_2;...;m_n; A;\pm \rangle \cdot
\langle m_1;m_2;...;m_n; A;\pm| =1.
\label{eq: polnota}
\end{equation}
Here the range of $m_i$ ($i=\overline{1,n}$) is not restricted by
the condition $(m_1<m_2<...<m_n)$, duplication of states being taken care
of by the $1/n!$ and the normalization. The $1$ on the right-hand side
of Eq.~\ref{eq: polnota} means the unit operator on the space of
$n$-particle states.
The case $n=0$ corresponds to the zero-particle states. They form a
one-dimensional space, all of whose elements are proportional to
the vacuum state.
The multiparticle Hilbert space is a direct sum of an infinite
sequence of the $n$-particle Hilbert spaces. The states of different
numbers of particles are defined to be orthogonal to each other.
The completeness relation in the multiparticle Hilbert space has the
form
\begin{equation}
\sum_{n=0}^{\infty} \frac{1}{n!} \sum_{m_1,m_2,...m_n \in \cal Z}
|m_1;m_2;...;m_n; A;\pm \rangle \cdot
\langle m_1;m_2;...;m_n; A;\pm| = 1,
\label{eq: plete}
\end{equation}
where "1" on the right-hand side means the unit operator on the
whole multiparticle space.
For $n$-particle states,
\[
:\hat{\rm H}_{\pm}: |m_1;m_2;...;m_n; A;\pm \rangle =
\hbar \sum_{k=1}^{n} {\varepsilon}_{m_k,\pm} \cdot {\rm sign}
({\varepsilon}_{m_k,\pm}) |m_1;m_2;...;m_n; A;\pm \rangle
\]
and
\[
:{\rho}_{\pm}(0): |m_1;m_2;...;m_n; A;\pm \rangle =
\pm \sum_{k=1}^{n} {\rm sign}({\varepsilon}_{m_k,\pm})
|m_1;m_2;...;m_n; A;\pm \rangle.
\]
\newpage
\section{Calculation of Berry phases}
\label{sec: berry}
In the adiabatic approach \cite{schiff68}-\cite{zwanz}, the
dynamical
variables are divided into two sets, one which we call fast variables
and the other which we call slow variables. In our case, we treat the
fermions as fast variables and the gauge fields as slow variables.
Let ${\cal A}^1$ be a manifold of all static gauge field
configurations ${A_1}(x)$. On ${\cal A}^1$ a time-dependent
gauge field ${A_1}(x,t)$ corresponds to a path and a periodic gauge
field to a closed loop.
We consider the fermionic part of the second-quantized Hamiltonian
$:\hat{\rm H}_{\rm F}:=:\hat{\rm H}_{+}: + :\hat{\rm H}_{-}:$
which depends on $t$ through the background
gauge field $A_1$ and so changes very slowly with time. We consider
next the periodic gauge field ${A_1}(x,t) (0 \leq t <T)$ . After a
time $T$ the periodic field ${A_1}(x,t)$ returns to its original
value: ${A_1}(x,0) = {A_1}(x,T)$, so that $:\hat{\rm H}_{\pm}:(0)=
:\hat{\rm H}_{\pm}:(T)$ .
At each instant $t$ we define eigenstates for $:\hat{\rm H}_{\pm}:
(t)$ by
\[
:\hat{\rm H}_{\pm}:(t) |{\rm F}, A(t);\pm \rangle =
{\varepsilon}_{{\rm F},\pm}(t) |{\rm F}, A(t);\pm \rangle.
\]
The state $|{\rm F}=0, A(t);\pm \rangle \equiv |{\rm vac}; A(t);\pm \rangle$
is a ground state of $:\hat{\rm H}_{\pm}:(t)$ ,
\[
:\hat{\rm H}_{\pm}:(t) |{\rm vac}; A(t);\pm \rangle =0.
\]
The Fock states $|{\rm F}, A(t) \rangle = |{\rm F},A(t);+ \rangle
\otimes |{\rm F},A(t);- \rangle $
depend on $t$ only through
their implicit dependence on $A_1$. They are assumed to be
orthonormalized,
\[
\langle {\rm F^{\prime}}, A(t)|{\rm F}, A(t) \rangle =
\delta_{{\rm F},{\rm F^{\prime}}},
\]
and nondegenerate.
The time evolution of the wave function of our system (fermions
in a background gauge field) is clearly governed by the Schrodinger
equation:
\[
i \hbar \frac{\partial \psi(t)}{\partial t} =
:\hat{\rm H}_{\rm F}:(t) \psi(t) .
\]
For each $t$, this wave function can be expanded in terms of the
"instantaneous" eigenstates $|{\rm F}, A(t) \rangle$ .
Let us choose ${\psi}_{\rm F}(0)=|{\rm F}, A(0) \rangle$, i.e.
the system is initially described by the eigenstate
$|{\rm F},A(0) \rangle$ . According to the adiabatic approximation,
if at $t=0$ our system starts in an stationary state $|{\rm F},A(0)
\rangle $ of $:\hat{\rm H}_{\rm F}:(0)$, then it will remain,
at any other instant of time $t$, in the corresponding eigenstate
$|{\rm F}, A(t) \rangle$ of the instantaneous Hamiltonian
$:\hat{\rm H}_{\rm F}:(t)$. In other words, in the adiabatic
approximation transitions to other eigenstates are neglected.
Thus, at some time $t$ later our system will be described up to
a phase by the same Fock state $|{\rm F}, A(t) \rangle $:
\[
\psi_{\rm F}(t) = {\rm C}_{\rm F}(t) \cdot |{\rm F},A(t) \rangle,
\]
where ${\rm C}_{\rm F}(t)$ is yet undetermined phase.
To find the phase, we insert $\psi_{\rm F}(t)$ into the
Schrodinger equation :
\[
\hbar \dot{\rm C}_{\rm F}(t) = -i {\rm C}_{\rm F}(t)
(\varepsilon_{{\rm F},+}(t) + \varepsilon_{{\rm F},-}(t))
- \hbar {\rm C}_{\rm F}(t)
\langle {\rm F},A(t)|\frac{\partial}{\partial t}|{\rm F},A(t) \rangle.
\]
Solving this equation, we get
\[
{\rm C}_{\rm F}(t) = \exp\{- \frac{i}{\hbar} \int_{0}^{t} d{t^{\prime}}
({\varepsilon}_{{\rm F},+}({t^{\prime}}) +
{\varepsilon}_{{\rm F},-}({t^{\prime}}) ) - \int_{0}^{t} d{t^{\prime}}
\langle {\rm F},A({t^{\prime}})|\frac{\partial}{\partial{t^{\prime}}}|
{\rm F},A({t^{\prime}}) \rangle \}.
\]
For $t=T$, $|{\rm F},A(T) \rangle =|{\rm F},A(0) \rangle$ ( the
instantaneous eigenfunctions are chosen to be periodic in time)
and
\[
{\psi}_{\rm F}(T) = \exp\{i {\gamma}_{\rm F}^{\rm dyn} +
i {\gamma}_{\rm F}^{\rm Berry} \}\cdot {\psi}_{\rm F}(0),
\]
where
\[ {\gamma}_{\rm F}^{\rm dyn} \equiv - \frac{1}{\hbar}
\int_{0}^{T} dt \cdot ({\varepsilon}_{{\rm F},+}(t)
+ {\varepsilon}_{{\rm F},-}(t)),
\]
while
\begin{equation}
{\gamma}_{\rm F}^{\rm Berry} = {\gamma}_{\rm F,+}^{\rm Berry} +
{\gamma}_{\rm F,-}^{\rm Berry},
\label{eq: summa}
\end{equation}
\[
{\gamma}_{{\rm F},\pm}^{\rm Berry} \equiv \int_{0}^{T} dt \int_{-{\rm L}/2}^
{{\rm L}/2} dx \dot{A_1}(x,t) \langle {\rm F},A(t);\pm|i \frac{\delta}
{\delta A_1(x,t)}|{\rm F},A(t);\pm \rangle
\]
is Berry's phase \cite{berry84}.
If we define the $U(1)$ connections
\begin{equation}
{\cal A}_{{\rm F},\pm}(x,t) \equiv \langle {\rm F},A(t);\pm|i \frac{\delta}
{\delta A_1(x,t)}|{\rm F},A(t);\pm \rangle,
\label{eq: dvatri}
\end{equation}
then
\[
{\gamma}_{{\rm F},\pm}^{\rm Berry} = \int_{0}^{T} dt \int_{-{\rm L}/2}^
{{\rm L}/2} dx \dot{A}_1(x,t) {\cal A}_{{\rm F},\pm}(x,t).
\]
We see that upon parallel transport around a closed loop on
${\cal A}^1$ the Fock states $|{\rm F},A(t);\pm \rangle$ acquire an
additional phase which is integrated exponential of ${\cal A}_{{\rm F},\pm}
(x,t)$. Whereas the dynamical phase ${\gamma}_{\rm F}^{\rm dyn}$
provides information about the duration of the evolution, the
Berry's phase reflects the nontrivial holonomy of the Fock states
on ${\cal A}^1$.
However, a direct computation of the diagonal matrix elements of
$\frac{\delta}{\delta A_1(x,t)}$ in ~\ref{eq: summa} requires a
globally single-valued basis for the eigenstates $|{\rm F},A(t);\pm \rangle$
which is not available. The connections ~\ref{eq: dvatri} can be
defined only locally on ${\cal A}^1$, in regions where
$[\frac{e_{+}b{\rm L}}{2 \pi}]$ is fixed. The values of $A_1$ in regions
of different $[\frac{e_{+}b{\rm L}}{2 \pi}]$ are connected by
topologically nontrivial gauge transformations.
If $[\frac{e_{+}b{\rm L}}{2 \pi}]$ changes, then
there is a nontrivial spectral flow , i.e. some of energy levels
of the first quantized fermionic Hamiltonians cross zero and change
sign. This means that the definition of the Fock vacuum of the second
quantized fermionic Hamiltonian changes (see Eq.~\ref{eq: deset}
and ~\ref{eq: odinodin}). Since the creation and annihilation operators
$a^{\dagger}, a$ (and $b^{\dagger}, b$ ) are
continuous functionals of $A_1(x)$, the definition of all excited
Fock states $|{\rm F},A(t) \rangle$ is also discontinuous. The
connections ${\cal A}_{{\rm F},\pm}$ are not therefore well-defined
globally.
Their global characterization necessiates the usual introduction of
transition functions.
Furthermore, ${\cal A}_{{\rm F},\pm}$ are not invariant under
$A$--dependent
redefinitions of the phases of the Fock states: $|{\rm F},A(t);\pm \rangle
\rightarrow \exp\{-i {\chi}_{\pm}[A]\} |{\rm F},A(t);\pm \rangle$, and
transform like a $U(1)$ vector potential
\[
{\cal A}_{{\rm F},\pm} \rightarrow {\cal A}_{{\rm F},\pm} +
\frac{\delta {\chi}_{\pm}[A]}{\delta A_1}.
\]
For these reasons, to calculate ${\gamma}_{\rm F}^{\rm Berry}$ it
is more convenient to compute first the $U(1)$ curvature tensors
\begin{equation}
{\cal F}_{\rm F}^{\pm}(x,y,t) \equiv \frac{\delta}{\delta A_1(x,t)}
{\cal A}_{{\rm F},\pm}(y,t) - \frac{\delta}{\delta A_1(y,t)}
{\cal A}_{{\rm F},\pm}(x,t)
\label{eq: dvacet}
\end{equation}
and then deduce ${\cal A}_{{\rm F},\pm}$.
i) $n$-particle states $(n \geq 3)$.
For $n$-particle states $|m_1;m_2;...;m_n; A;\pm \rangle$
$(m_1<m_2<...<m_n)$, the ${\rm U}(1)$ curvature tensors are
\[
{\cal F}_{m_1,m_2,...,m_n}^{\pm}(x,y,t)
= i \sum_{k=0}^{\infty}
\frac{1}{k!} \sum_{\overline{m}_1, \overline{m}_2, ...,
\overline{m}_k \in \cal Z} \{ \langle m_1;m_2:...;m_n; A;\pm|
\frac{\delta}{\delta {A}_1(x,t)}
\]
\[
|\overline{m}_1;\overline{m}_2;
...;\overline{m}_k; A;\pm \rangle
\cdot \langle \overline{m}_1; \overline{m}_2; ...; \overline{m}_k;
A;\pm| \frac{\delta}{\delta A_1(y,t)}|
m_1;m_2;...;m_n; A;\pm \rangle - (x \leftrightarrow y) \}
\]
where the completeness condition ~\ref{eq: plete} is inserted.
Since
\[
\langle m_1;m_2;...;m_n; A;\pm |\frac{\delta
:\hat{\rm H}_{\pm}:}{\delta A_1(x,t)}|
\overline{m}_1;\overline{m}_2;...;\overline{m}_k; A;\pm \rangle
= {\hbar} \{ \sum_{i=1}^k {\varepsilon}_{\overline{m}_i,\pm} \cdot
{\rm sign}({\varepsilon}_{\overline{m}_i,\pm})
\]
\[
-\sum_{i=1}^n {\varepsilon}_{m_i,\pm} \cdot
{\rm sign}({\varepsilon}_{m_i,\pm}) \} \cdot
\langle m_1;m_2;...;m_n;A;\pm|\frac{\delta}{\delta A_1(x,t)}|
\overline{m}_1; \overline{m}_2;...;\overline{m}_k; A;\pm \rangle
\]
and $:\hat{\rm H}_{\pm}:$ are quadratic in the positive and negative
chirality creation and annihilation operators, the matrix elements
$\langle m_1;m_2;...;m_n; A;\pm|\frac{\delta}{\delta A_1(x,t)}|
\overline{m}_1;\overline{m}_2;...;\overline{m}_k; A;\pm \rangle$
and so the corresponding curvature tensors
${\cal F}_{m_1,m_2,...,m_n}^{\pm}$ and Berry phases
${\gamma}_{m_1,m_2,...,m_n;\pm}^{\rm Berry}$ vanish for all values
of $m_i (i=\overline{1,n})$ for $n \geq 3$.
ii) $2$-particle states.
For $2$-particle states $|m_1;m_2; A;\pm \rangle$ $(m_1<m_2)$,
only the vacuum state survives in the completeness condition inserted
so that the curvature tensors ${\cal F}_{m_1m_2}^{\pm}$ take the form
\[
{\cal F}_{m_1m_2}^{\pm}(x,y,t) = \frac{i}{{\hbar}^2} \frac{1}
{({\varepsilon}_{m_1,\pm} \cdot {\rm sign}({\varepsilon}_{m_1,\pm}) +
{\varepsilon}_{m_2,\pm} \cdot {\rm sign}({\varepsilon}_{m_2,\pm}))^2}
\]
\[
\cdot \{ \langle m_1;m_2;A;\pm| \frac{\delta :\hat{\rm H}_{\pm}:}
{\delta A_1(y,t)}|{\rm vac}; A;\pm \rangle
\langle {\rm vac};A;\pm|\frac{\delta :\hat{\rm H}_{\pm}:}
{\delta A_1(x,t)}|m_1;m_2;A;\pm \rangle -
(x \leftrightarrow y) \}.
\]
With $:\hat{\rm H}_{\pm}:(t)$ given by ~\ref{eq: hamil},
${\cal F}_{m_1m_2}^{\pm}$ are evaluated as
$$
{\cal F}_{m_1m_2}^{\pm}= \left \{
\begin{array}{cc}
0 & \mbox{for $m_1,m_2 >[\frac{e_{\pm}b{\rm L}}
{2\pi}] \hspace{3 mm} {\rm and} \hspace{3 mm} m_1,m_2 \leq
[\frac{e_{\pm}b{\rm
L}}{2\pi}]$},\\ \mp \frac{e_{\pm}^2}{2{\pi}^2} \frac{1}{(m_2-m_1)^2}
\sin\{\frac{2\pi}{\rm L}(m_2-m_1)(x-y)\} & \mbox{for
$m_1 \leq [\frac{e_{\pm}b{\rm L}}{2\pi}], m_2>[\frac{e_{\pm}b{\rm L}}
{2\pi}]$},
\end{array}\right.
$$
i.e. the curvatures are nonvanishing only for states with one
particle and one hole.
The corresponding connections are easily deduced as
\[
{\cal A}_{m_1m_2}^{\pm}(x,t) =
-\frac{1}{2} \int_{-{\rm L}/2}^{{\rm L}/2} dy
{\cal F}_{m_1m_2}^{\pm}(x,y,t) A_1(y,t).
\]
The Berry phases become
\[
{\gamma}_{m_1m_2,\pm}^{\rm Berry} = - \frac{1}{2} \int_{0}^{\rm T} dt
\int_{-{\rm L}/2}^{{\rm L}/2} dx \int_{-{\rm L}/2}^{{\rm L}/2} dy
\dot{A}_1(x,t) {\cal F}_{m_1m_2}^{\pm}(x,y,t) A_1(y,t).
\]
If we introduce the Fourier expansion for the gauge field
\[
A_1(x,t) =b(t) + \sum_{\stackrel{p \in \cal Z}{p \neq 0}}
e^{i\frac{2\pi}{\rm L} px} {\alpha}_p(t),
\]
then in terms of the gauge field Fourier components the Berry phases
take the form
\[
{\gamma}_{m_1m_2,\pm}^{\rm Berry} =
\mp \frac{e_{\pm}^2{\rm L}^2}{8{\pi}^2} \frac{1}{(m_2-m_1)^2}
\int_{0}^{\rm T} dt i ({\alpha}_{m_2-m_1} \dot{\alpha}_{m_1-m_2}
- {\alpha}_{m_1-m_2} \dot{\alpha}_{m_2-m_1})
\]
for $m_1 \leq [\frac{e_{\pm}b{\rm L}}{2\pi}],
m_2>[\frac{e_{\pm}b{\rm L}}{2\pi}]$,
vanishing for $m_1,m_2 >[\frac{e_{\pm}b{\rm L}}{2\pi}]$ and
$m_1,m_2 \leq [\frac{e_{\pm}b{\rm L}}{2\pi}]$.
Therefore, a parallel transportation of the states $|m_1;m_2;A;\pm
\rangle$ with two particles or two holes around a closed loop in
$({\alpha}_p,{\alpha}_{-p})$-space $(p>0)$ yields back the same states,
while the states with one particle and one hole are multiplied by
the phases ${\gamma}_{m_1m_2,\pm}^{\rm Berry}$.
For the Schwinger model when ${\rm N}=1$ and $e_{+}=e_{-}$
as well as for axial electrodynamics when ${\rm N}=-1$ and
$e_{+}=-e_{-}$, the nonvanishing
Berry phases for the positive and negative chirality $2$-particle states
are opposite in sign,
\[
{\gamma}_{m_1m_2,+}^{\rm Berry} = - {\gamma}_{m_1m_2,-}^{\rm Berry},
\]
so that for the states $|m_1;m_2;A \rangle =
|m_1;m_2;A;+ \rangle \otimes |m_1;m_2;A;- \rangle$
the total Berry phase is zero.
iii) $1$-particle states.
For $1$-particle states $|m;A;\pm \rangle$, the ${\rm U}(1)$ curvature
tensors are
\[
{\cal F}_{m}^{\pm}(x,y,t) = i
\sum_{\stackrel{\overline{m} \in \cal Z}{\overline{m} \neq m}}
\frac{1}{{\hbar}^2}
\frac{1}{({\varepsilon}_{\overline{m},\pm} \cdot {\rm sign}
({\varepsilon}_{\overline{m},\pm}) - {\varepsilon}_{m,\pm} \cdot
{\rm sign}({\varepsilon}_{m,\pm}))^2}
\]
\[
\cdot \{ \langle m;A;\pm|
\frac{\delta : \hat{\rm H}_{\pm}:}{\delta A_1(y,t)}
|\overline{m};A;\pm \rangle
\langle \overline{m};A;\pm|
\frac{\delta :\hat{\rm H}_{\pm}:} {\delta A_1(x,t)}
|m;A;\pm \rangle - (x \longleftrightarrow y) \}. \\
\]
By a direct calculation we easily get
\begin{eqnarray*}
{\cal F}_{m>[\frac{e_{\pm}b{\rm L}}{2\pi}]}^{\pm} & = &
\sum_{\overline{m}=m-[\frac{e_{\pm}b{\rm L}}{2\pi}]}^{\infty}
{\cal F}_{0\overline{m}}^{\pm}, \\
{\cal F}_{m \leq [\frac{e_{\pm}b{\rm L}}{2\pi}]}^{\pm} & = &
\sum_{\overline{m}= [\frac{e_{\pm}b{\rm L}}{2\pi}] - m+1}^{\infty}
{\cal F}_{0\overline{m}}^{\pm},
\end{eqnarray*}
where ${\cal F}_{0\overline{m}}^{\pm}$ are curvature tensors for the
$2$-particle states $|0;\overline{m};A;\pm \rangle$ $(\overline{m}>0)$.
The Berry phases acquired by the states $|m;A;\pm \rangle$ by their
parallel transportation around a closed loop in $({\alpha}_p,
{\alpha}_{-p})$-space $(p>0)$ are
\begin{eqnarray*}
{\gamma}_{\pm}^{\rm Berry}(m>[\frac{e_{\pm}b{\rm L}}{2\pi}]) & = &
\sum_{\overline{m}=m - [\frac{e_{\pm}b{\rm L}}{2\pi}]}^{\infty}
{\gamma}_{0\overline{m};\pm}^{\rm Berry}, \\
{\gamma}_{\pm}^{\rm Berry}(m \leq [\frac{e_{\pm}b{\rm L}}{2\pi}]) & = &
\sum_{\overline{m}=[\frac{e_{\pm}b{\rm L}}{2\pi}] -m+1}^{\infty}
{\gamma}_{0\overline{m};\pm}^{\rm Berry},
\end{eqnarray*}
where ${\gamma}_{0\overline{m};\pm}^{\rm Berry}$ are phases
acquired by the states $|0;\overline{m};A;\pm \rangle$ by the same
transportation.
For the ${\rm N}=\pm 1$ models, the total $1$-particle curvature
tensor ${\cal F}_m ={\cal F}_m^{+} + {\cal F}_m^{-}$ and total Berry
phase ${\gamma}^{\rm Berry} ={\gamma}_{+}^{\rm Berry} +
{\gamma}_{-}^{\rm Berry}$ vanish.
iv) vacuum states.
For the vacuum case, only $2$-particle states contribute to the sum
of the completeness condition, so the vacuum curvature tensors are
\[
{\cal F}_{\rm vac}^{\pm}(x,y,t) = - \frac{1}{2}
\sum_{\overline{m}_1; \overline{m}_2 \in \cal Z}
{\cal F}_{\overline{m}_1 \overline{m}_2}(x,y,t).
\]
Taking the sums, we get
\begin{equation}
{\cal F}_{\rm vac}^{\pm} =
\pm \frac{e_{+}^2}{2{\pi}} \sum_{n>0}
( \frac{1}{2} \epsilon(x-y)
- \frac{1}{\rm L} (x-y) ).
\label{eq: dvasem}
\end{equation}
The total vacuum curvature tensor
\[
{\cal F}_{\rm vac} = {\cal F}_{\rm vac}^{+} + {\cal F}_{\rm vac}^{-}=
(1-{\rm N}^2) \frac{e_{+}^2}{2\pi} (\frac{1}{2} \epsilon(x-y) -
\frac{1}{\rm L} (x-y))
\]
vanishes for ${\rm N}=\pm 1$.
The corresponding ${\rm U}(1)$ connection is deduced as
\[
{\cal A}_{\rm vac}(x,t) = - \frac{1}{2} \int_{-{\rm L}/2}^{{\rm L}/2}
dy {\cal F}_{\rm vac}(x,y,t) A_1(y,t),
\]
so the total vacuum Berry phase is
\[
{\gamma}_{\rm vac}^{\rm Berry} = - \frac{1}{2} \int_{0}^{T} dt
\int_{-{\rm L}/2}^{{\rm L}/2} dx \int_{-{\rm L}/2}^{{\rm L}/2} dy
\dot{A_1}(x,t) {\cal F}_{\rm vac}(x,y,t) A_1(y,t),
\]
For ${\rm N}=0$ and in the limit ${\rm L} \to \infty$,
when the second term in ~\ref{eq: dvasem} may be neglected,
the $U(1)$ curvature tensor
coincides with that obtained in \cite{niemi86,semen87},
while the Berry phase becomes
\[
{\gamma}_{\rm vac}^{\rm Berry} = \int_{0}^{T} dt
\int_{- \infty}^{\infty} dx {\cal L}_{\rm nonlocal}(x,t),
\]
where
\[
{\cal L}_{\rm nonlocal}(x,t) \equiv - \frac{e_{+}^2}{8 {\pi}^2}
\int_{- \infty}^{\infty}
dy \dot{A_1}(x,t) \epsilon(x-y) A_1(y,t)
\]
is a non-local part of the effective Lagrange density of the CSM
\cite{sarad93}. The effective Lagrange density is a sum of the
ordinary Lagrange density of the CSM and the nonlocal part
${\cal L}_{\rm nonlocal}$. As shown in \cite{sarad93}, the effective
Lagrange density is equivalent to the ordinary one in the sense that
the corresponding preliminary Hamiltonians coincide on the constrained
submanifold ${\rm G} \approx 0$. This equivalence is valid at the
quantum level, too. If we start from the effective Lagrange density
and apply appropriately the Dirac quantization procedure, then we
come to a quantum theory which is exactly the quantum theory
obtained from the ordinary Lagrange density. We get therefore
that the Berry phase is an action and that the CSM can be defined
equivalently by both the effective action with the Berry phase
included and the ordinary one without the Berry phase.
In terms of the gauge field Fourier components, the connection
${\cal A}_{\rm vac}$ is rewritten as
\[
\langle {\rm vac};A(t)|\frac{d}{db(t)}|{\rm vac};A(t)\rangle =0,
\]
\[
\langle {\rm vac};A(t)|\frac{d}{d{\alpha}_{\pm p}(t)}|{\rm vac};A(t)\rangle
\equiv {\cal A}_{{\rm vac};\pm}(p,t)= \pm (1-{\rm N}^2)
\frac{e_{+}^2{\rm L}^2}{8{\pi}^2} \frac{1}{p} {\alpha}_{\mp p},
\]
so the nonvanishing vacuum curvature is
\[
{\cal F}_{\rm vac}(p) \equiv \frac{d}{d{\alpha}_{-p}}
{\cal A}_{{\rm vac};+} - \frac{d}{d{\alpha}_p}
{\cal A}_{{\rm vac};-} =
(1-{\rm N}^2) \frac{e_{+}^2{\rm L}^2}{4{\pi}^2} \frac{1}{p}.
\]
The total vacuum Berry phase becomes
\[
{\gamma}_{\rm vac}^{\rm Berry} = \int_{0}^{\rm T} dt
\sum_{p>0} {\cal F}_{\rm vac}(p) {\alpha}_p \dot{\alpha}_{-p}.
\]
For the ${\rm N} \neq \pm 1$ models where the local gauge symmetry
is known to be realized projectively \cite{sarad91},
the vacuum Berry phase is
non-zero. For ${\rm N}=\pm 1$ when the representation is unitary,
the curvature ${\cal F}_{\rm vac}(p)$ and the vacuum Berry phase
vanish.
The projective representation of the local gauge symmetry is
responsible for anomaly. In the full quantized theory of the
CSM when the gauge fields are also quantized the physical states
respond to gauge transformations from the zero topological class
with a phase \cite{sarad91}. This phase contributes to the
commutator of the Gauss law generators by a Schwinger term and
produces therefore an anomaly.
A connection of the nonvanishing vacuum Berry phase to the
projective representation can be shown in a more direct way.
Under the topologically trivial gauge transformations,
the gauge field Fourier components
${\alpha}_p, {\alpha}_{-p}$ transform as follows
\begin{eqnarray*}
{\alpha}_p & \stackrel{\tau}{\rightarrow} & {\alpha}_p - ip{\tau}_{-}(p),\\
{\alpha}_{-p} & \stackrel{\tau}{\rightarrow} & {\alpha}_{-p} -ip{\tau}_{+}(p),\\
\end{eqnarray*}
where ${\tau}_{\pm}(p)$ are smooth gauge parameters.
The nonlocal Lagrangian
\[
{\rm L}_{\rm nonlocal}(t) \equiv \int_{-{\rm L}/2}^{{\rm L}/2} dx
{\cal L}_{\rm nonlocal}(x,t) =
\sum_{p>0} {\cal F}_{\rm vac}(p)
i{\alpha}_{p} \dot{\alpha}_{-p}
\]
changes as
\[
{\rm L}_{\rm nonlocal}(t) \stackrel{\tau}{\rightarrow}
{\rm L}_{\rm nonlocal}(t) - 2{\pi} \frac{d}{dt} {\alpha}_1(A;{\tau}),
\]
where
\[
{\alpha}_1(A;{\tau}) \equiv - \frac{1}{4\pi}
\sum_{p>0} p{\cal F}_{\rm vac}(p) ({\alpha}_{-p} {\tau}_{-}
- {\alpha}_{p} {\tau}_{+})
\]
is just $1$--cocycle occuring in the projective
representation of the gauge group. This examplifies a connection
between the nonvanishing vacuum Berry phase and the fact that the local
gauge symmetry is realized projectively.
\newpage
\section{Conclusions}
\label{sec: con}
Let us summarize.
i) We have calculated explicitly the Berry phase and the corresponding
${\rm U}(1)$ connection and curvature for the fermionic vacuum and many
particle Fock states. For the ${\rm N} \neq \pm 1$ models, we get that
the Berry phase is non-zero for the vacuum, $1$-particle and $2$-particle
states with one particle and one hole. For all other many particle states
the Berry phase vanishes. This is caused by the form of the second
quantized fermionic Hamiltonian which is quadratic in the positive
and negative chirality creation and annihilation operators.
ii) For the ${\rm N}= \pm 1$ models without anomaly, i.e. for the SM and
axial electrodynamics, the Berry phases acquired by the negative and
positive chirality parts of the Fock states are opposite in sign
and cancel each other , so that
the total Berry phase for all Fock states is zero.
iii) A connection between the Berry phase and anomaly becomes more
explicit for the vacuum state. We have shown that for our model
the vacuum Berry phase contributes to the effective action, being
that additional part of the effective action which differs it from the
ordinary one. Under the topologically trivial gauge transformations
the corresponding addition in the effective Lagrangian changes by a
total
time derivative of the gauge group $1$-cocycle occuring in the projective
representation. This demonstrates an interrelation between the Berry
phase, anomaly and effective action.
\newpage
| proofpile-arXiv_065-366 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{ Introduction }
There has been a strong revival of interest, recently, in the physics
of magnetic vortices in type II and high-temperature superconductors
\cite{reviews}. Most research efforts have been devoted to phenomena
relating to the nature of the mixed phase of a superconductor in some
externally applied magnetic field and supercurrent. Issues connected
with the pinning of the flux lines by defects have been widly studied.
We \cite{ieju}, as well as Ao and Thouless \cite{aoth} and Stephen
\cite{stephen}, have addressed the problem of the quantum dynamics of
vortices in the absence of an external field but in the presence of
an externally driven supercurrent, quantum dissipation and pinning.
This leads to the decay of a supercurrent, or a residual zero-temperature
resistance in the superconductor. Whilst most of the dissipation seems
to be ascribed to vortices tunneling in the sample from the edge, an
interesting novel possibility also explored by us in depth is that of
a residual resistance arising from spontaneous vortex-antivortex pair
creation in the bulk of a thin film. This is the mesoscopic counterpart
of electron-positron pair production of two-dimensional (2D) quantum
electrodynamics (QED) in the presence of static e.m. fields, which in a
superconductor arise from the static and velocity-dependent components of
the Magnus force acting on the vortices. Exploiting this analogy with QED,
a powerful ``relativistic'' quantum field theory approach has been
set up to study vortex nucleation in the 2D geometry in the presence of
quantum dissipation and of pinning potentials. The central result is that
the nucleation rate $\Gamma$ has a strong exponential dependence on the
number current density $J$, given by
\begin{equation}
\Gamma{\propto}\eta^{1/2}\eta_{eff}J^{-1}
\exp\{-\eta_{eff}{\cal E}_{0R}^2/4\pi J^2\}
\label{rate0}
\end{equation}
\noindent
Here $\eta_{eff}$ is an effective viscosity coefficient as renormalised by
the magnetic-like part of the Magnus force, and ${\cal E}_{0R}$ is the rest-
or nucleation-energy of a single vortex as renormalized by screened
Coulomb interactions and (fake) Landau-level corrections. This
exponential dependence would make the vortex nucleation (folded, e.g.,
into the sample's resistance) observable in a rather narrow range of
$J$-values. Thus, normally the superconductor is essentially resistance-free.
However, the high values of $J$ that can be reached in the high-$T_c$
materials make the possibility of observing pair creation in static fields
within reach for thin films. One particular feature that would uniquely
relate the residual resistance to the phenomenon of spontaneous vortex-pair
creation is the presence of {\em oscillations} in the $J$-dependence of
$\Gamma(J)$ in case a {\em periodic} pinning potential is artificially
created in the film. These oscillations are in fact strictly connected to
the pinning-lattice spacing $d=2\pi/k$ of the periodic potential (we assume
a square lattice), e.g.
\begin{equation}
U({\bf q}(t))=U_0 \sum_{a=1}^2 \left [ 1 - \cos \left ( kq_a(t)
\right ) \right ]
\label{potent}
\end{equation}
\noindent
acting on the nucleating vortex-pairs described by a coordinate ${\bf q}$.
The problem of quantum dissipation for a particle moving in a periodic
potential has some interesting features in its own right
\cite{schmid,ghm,fizw}. It is characterised by a localization phase
transition driven by dissipation; accordingly, two phases can occur
depending on whether the dissipation coefficient \cite{cale} $\eta$ is
greater (confined phase) or smaller (mobile phase) than a critical
value $\eta_c=k^2/2\pi=2\pi/d^2$. This localization transition is described
by a Kosterlitz-type renormalization group (RG) approach, yet with some
important differences that will be recalled below. We have implemented
the RG approach for the evaluation of the dependence of the spontaneous
nucleation rate of vortex-antivortex pairs on the external parameters for
our own quantum dynamical system. A remnant of the dissipation-driven
phase transition is observed and the pair production rate $\Gamma$ can
be derived in both phases by means of a frequency-space RG procedure
leading to observable current-oscillations if $\eta > \eta_c$.
\section{ RG approach to dissipative localization transition }
First, we briefly recall the RG description of the localization
transition driven by quantum dissipation \cite{fizw}. The effective
action for a particle diffusing in a periodic potential and subject to
quantum dissipation of the Caldeira-Leggett type \cite{cale} is, in
Fourier frequency space:
\begin{equation}
{\cal S}=\int_0^{\tau}{\cal L}({\bf q})=\tau
\sum_n \{ \frac{1}{2}m\omega_n^2+\frac{1}{2}\eta |\omega_n| \}
\bar{q}_a(\omega_n)\bar{q}_a(-\omega_n)+\int_0^{\tau} dt U({\bf q})
\label{action0}
\end{equation}
\noindent
where $m$ is the mass of the quantum particle and $\eta$ the
phenomenological friction coefficient. In the low-frequency limit the
effects of inertia can be neglected and the problem would acquire the same
phenomenology as for the sine-Gordon model (in (0+1)-dimensions),
except for the peculiar $\omega_n$-dependence of the propagator reflecting
the broken time-reversal symmetry of quantum dissipation. When the RG
procedure is applied to Eq. (\ref{action0}) a renormalization of the
potential amplitude $U_0$ occurs, but not of the friction coefficient
$\eta$ since only local operators in the time variable can be generated
within a RG transformation. In terms of the dimensionless parameters
($\Omega$ is a large frequency-cutoff) ${\cal U}=U_0/\Omega$ and
$\alpha=2\pi\eta/k^2$, the RG recursion relations read
\begin{equation}
\frac{d{\cal U}}{d\ell}=\left ( 1-\frac{1}{\alpha} \right ) {\cal U}
+ \cdots, \qquad
\frac{d\alpha}{d\ell}=0
\label{recrel}
\end{equation}
\noindent
with $e^{-\ell}$ the frequency-scale renormalization parameter. These have
the simple solution
\begin{equation}
{\cal U}(\ell)={\cal U}(0)e^{(1-\eta_c/\eta)\ell}, \qquad
\alpha(\ell)=\alpha(0)
\label{rgflow}
\end{equation}
\noindent
displaying the localization transition for $\eta=\eta_c=k^2/2\pi=2\pi/d^2$.
The potential's amplitude vanishes
under a RG change of time scale for $\eta < \eta_c$, but for
$\eta > \eta_c$ it tends to diverge and the RG procedure must be
interrupted. Unlike in the Kosterlitz RG scheme, this cannot be done
unequivocally in the present situation, for there is no true characteristic
correlation time or frequency owing to the fact that one never moves away
from the neighbourhood of the critical point $\eta_c$. An alternative
strategy for the confined phase is to resort to a variational treatment
\cite{fizw}, which dynamically generates a correlation time.
In this procedure the action of Eq. (\ref{action0}) is replaced by a
trial Gaussian form (neglecting inertia)
\begin{equation}
{\cal S}_{tr}=\frac{\eta}{4\pi} \int_0^{\tau} dt \int_{-\infty}^{+\infty} dt'
\left ( \frac{{\bf q}(t)-{\bf q}(t')}{t-t'} \right )^2 + \frac{1}{2} M^2
\int_0^{\tau} dt {\bf q}(t)^2
\label{actiontr}
\end{equation}
\noindent
where $M^2$ is determined by minimising self-consistently the free energy
$F_{tr}+\langle S-S_{tr} \rangle_{tr}$. This leads to the equation
\begin{equation}
M^2=U_0k^2 \exp \left \{ -\frac{k^2}{2\tau} \sum_n \frac{1}{\eta|\omega_n|
+M^2} \right \}
\end{equation}
\noindent
having a solution $M^2{\neq}0$ only in the confined phase ($\eta > \eta_c$),
since introducing the cutoff $\Omega$ in the (continuous) sum over frequency
modes $\omega_n=2{\pi}n/\tau$, the equation for $M^2$ leads to (for
$M^2\rightarrow 0$)
\begin{equation}
M^2=\eta\Omega \left ( \frac{2\pi U_0}{\Omega} \frac{\eta_c}{\eta}
\right )^{\eta/(\eta-\eta_c)}{\equiv}\eta\Omega\mu
\label{mass}
\end{equation}
\noindent
This spontaneously generated ``mass'' interrupts the divergent
renormalization of the periodic potential amplitude $U_0$, which in the
RG limit $\ell{\rightarrow}{\infty}$ tends to
\begin{equation}
U_0(\ell)=U_0 \left ( \frac{e^{-\ell}+\mu}{1+\mu} \right )^{\eta_c/\eta}
{\rightarrow}U_0 \left ( \frac{\mu + 1/n^{*}}
{\mu + 1} \right )^{\eta_c/\eta}
\end{equation}
\noindent
Here, we have put $\Omega=2\pi n^{*}/\tau=n^{*}\omega_1$ and
$\mu=M^2/\Omega\eta$.
\section{ RG treatment of vortex-antivortex pair-creation in the presence
of a periodic pinning potential }
We begin by recalling the need for a relativistic description of the
process. This leads \cite{ieju} to a Schwinger-type formula for the decay
of the ``vacuum'', represented by a thin superconducting film in which static
e.m.-like fields arise when a supercurrent is switched on. The quantum
fluctuations of these fields are vortex-antivortex pairs, nucleating at a
rate given by
\begin{equation}
\frac{\Gamma}{L^2}=\frac{2}{L^2T} Im \int_{\epsilon}^{\infty}
\frac{d\tau}{\tau} e^{-{\cal E}_0^2\tau} \int
{\cal D}q(t) \exp\{ -\int_0^{\tau} dt {\cal L}_E \}
\label{rate}
\end{equation}
\noindent
where $L^2T$ is the space-time volume of the sample and ${\cal E}_0$ the
vortex-nucleation energy (suitably renormalised by vortex-screening effects).
Also
\begin{eqnarray}
{\cal L}_E&=&\frac{1}{2}m_{\mu}\dot{q}_{\mu}\dot{q}_{\mu}-\frac{1}{2}i
\dot{q}_{\mu}F_{\mu\nu}q_{\nu} + V({\bf q}) \nonumber \\
&+&\sum_k \left \{ \frac{1}{2}m_k\dot{\bf x}_k^2
+\frac{1}{2}m_k\omega_k^2 \left( {\bf x}_k+\frac{c_k}{m_k\omega_k^2}{\bf q}
\right )^2 \right \}
\label{lagran}
\end{eqnarray}
\noindent
is the Euclidean single-particle relativistic Lagrangian, incorporating the
pinning potential $V({\bf q})=2{\cal E}_0U({\bf q})$ and the Caldeira-Leggett
mechanism \cite{cale}. In the absence of the pinning potential, the
relativistic action is quadratic and the path integral in Eq. (\ref{rate})
can be evaluated exactly. The leading term in the expression for $\Gamma$
follows from the lowest pole in the $\tau$-integral and this can be obtained
exactly in the (non-relativistic) limit in which
$m_1=m_2=\frac{\gamma}{2}{\rightarrow}0$, with
$\frac{1}{\gamma}={\cal E}_0/m{\rightarrow}{\infty}$ playing the role of the
square of the speed of light. The result \cite{ieju} is Eq. (\ref{rate0}).
We now come to the evaluation of $\Gamma$ in the presence of the periodic
potential, which calls for the RG approach of Section 2. Integrating out the
Euclidean ``time''-like component $q_3(t)$, we reach a formulation in which
the electric-like and the magnetic-like Magnus field components are
disentangled. In terms of Fourier components, dropping the magnetic-like part
and for $\gamma{\rightarrow}0$:
\begin{equation}
\int_0^{\tau} dt {\cal L}_E({\bf q})=\tau\sum_{n\neq 0} \{ \frac{1}{2}\eta
|\omega_n| - E^2{\delta}_{a1} \} \bar{q}_a(\omega_n) \bar{q}_a(-\omega_n)
+\int_0^{\tau} dt V({\bf q})
\label{lagranr}
\end{equation}
\noindent
with $E=2\pi J$ the electric-like field due to the supercurrent donsity $J$.
We have shown \cite{ieju} that the only role of the magnetic-like field is
to renormalize the nucleation energy and the friction coefficient, hence our
problem amounts to an effective one-dimensional system in the presence of
${\bf E}$ and dissipation. The evaluation of the Feynman Path Integral (FPI)
proceeds by means of integrating out the zero-mode, $\bar{q}_0$, as well as
the high-frequency modes $\bar{q}_n$ with $n>1$, since again the leading
term for $\Gamma$ in Eq. (\ref{rate}) comes from the divergence of the FPI
associated with the lowest mode coupling to ${\bf E}$. The effect of
$\bar{q}_n$ with $n > 1$ is taken into account through the frequency-shell
RG method of Section 2, leading to a renormalization of the amplitude
$V_0=2{\cal E}_0U_0$ of the (relativistic) pinning potential. The
renormalization has to be carried out from the outer shell of radius $\Omega$
to $\omega_1=2\pi/\tau$. In the mobile phase ($\eta < \eta_c$) this implies
$e^{\ell}=\Omega\tau/2\pi=n^{*}$ in Eq. (\ref{rgflow}), with (from the leading
pole of the FPI) $\tau=\pi\eta/E^2$ (${\rightarrow}\infty$ for relatively
weak currents). In the more interesting confined phase ($\eta > \eta_c$) we
must integrate out the massive $n > 1$ modes with a Lagrangian
${\cal L}(\bar{q}_n)=\tau \left ( \frac{1}{2}\eta |\omega_n|+\frac{1}{2}
M^2-E^2 \right ) \bar{q}_n\bar{q}_n^{*}$. This leads to an additional,
entropy-like renormalization of the activation energy ${\cal E}_{0R}$, beside
the renormalization of $V_0$. We are therefore left with the integration
over the modes $\bar{q}_0$ and $\bar{q}_1$, with a renormalised potential
\begin{eqnarray}
&&\int_0^{\tau} dt V_R(q_0,q_1(t)) = V_0\tau - V_{0R}\int_0^{\tau} dt
\cos ( k(q_0+q_1(t)) ) \nonumber \\
&&{\simeq} V_0\tau - V_{0R}\tau J_0(2k|\bar{q}_1|)\cos ( kq_0 )
\label{potentr}
\end{eqnarray}
\noindent
Here, $J_0$ is the Bessel function and the renormalised amplitude $V_{0R}$ is
\begin{eqnarray}
V_{0R}= \left \{ \begin{array}{ll}
V_0 \left ( \frac{\Omega\tau}{2\pi} \right )^{-\eta_c/\eta}
& \mbox{if $\eta < \eta_c$} \\
V_0 \left ( \frac{\mu +1/n^{*}}{ \mu +1} \right )^{\eta_c/\eta}
& \mbox{if $\eta > \eta_c$}
\end{array} \right.
\label{amplitr}
\end{eqnarray}
\noindent
In Eq. (\ref{potentr}) the phase of the $\bar{q}_1$ mode has been integrated
out, allowing us to integrate out the $\bar{q}_0$-mode exactly; this leads
to the expression
\begin{equation}
\frac{\Gamma}{2L^2}= Im \int_{\epsilon}^{\infty} d{\tau} {\cal N}(\tau)
e^{-({\cal E}_{0R}^2+V_0)\tau}
\int_0^{\infty} d|\bar{q}_1|^2 e^{-(\pi\eta-E^2\tau)|\bar{q}_1|^2}
I_0 \left (
V_{0R}\tau J_0(2k|\bar{q}_1|) \right )
\label{rate1}
\end{equation}
\noindent
where $I_0$ is the modified Bessel function. It is clear that the singularity
from the $\bar{q}_1$-integral occurs at $\tau=\pi\eta/E^2$; evaluating the
normalization factor ${\cal N}(\tau)$, we finally arrive at
\begin{eqnarray}
&&\Gamma=\Gamma_0K(J) \\
\label{final}
&&K(J)=e(1+\mu) \left ( 1+\frac{\mu\Omega\eta}{8\pi^2 J^2} \right )
I_0 \left ( \frac{V_{0R}\eta}{4\pi J^2} J_0(2k{\ell}_N) \right ) \nonumber
\end{eqnarray}
\noindent
where $\Gamma_0$ is given by Eq. (\ref{rate0}), there is a further
renormalization ${\cal E}_{0R}^2{\rightarrow}{\cal E}_{0R}^2+V_0$ and
we have set $E=2\pi J$. $\ell_N$ is a nucleation length, which is in first
approximation given by
\begin{equation}
{\ell}_N^2{\simeq}\frac{ {\cal E}_{0R}^2}{4\pi^2 J^2}
-\frac{V_{0R}}{4\pi^2 J^2} \left | J_0 \left ( k
\frac{ {\cal E}_{0R} } {\pi J} \right ) \right |
\label{nuclen}
\end{equation}
\noindent
and corresponds physically to the distance a vortex and antivortex
pair must travel to acquire the nucleation energy ${\cal E}_{0R}$.
The presence of the $J_0(2k{\ell}_N)$ argument in the correction factor
$K(J)$ due to the pinning lattice thus gives rise to oscillations in
$\Gamma (J)$ (hence in the sample's resistance) through the parameter
$2k{\ell}_N=4\pi{\ell}_N/d$. Vortex nucleation is therefore
sensitive to the corrugation of the pinning substrate. However, these
oscillations should be observable only in the confined phase, $\eta > \eta_c$,
where interrupted-renormalization prevents the prefactor in front of the
$J_0(x)$ oscillating function from becoming too small for relatively
small current densities.
\section*{References}
| proofpile-arXiv_065-369 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The precision data collected to date have confirmed the
Standard Model to be a good description of physics below
the electroweak scale \cite{Schaile}.
Despite of its great success, there are many reasons to believe
that some kind of new physics must exist. On the other hand, the
non-abelian structure of the gauge
boson self-couplings is still poorly tested and one of the most sensitive
probes for new physics is provided by the trilinear gauge boson couplings
(TGC) \cite{TGC}.
Many studies have been devoted to the $WW\gamma$ and $WWZ$ couplings.
At hadron colliders and $e^+e^-$ colliders, the present bounds
(Tevatron \cite{Errede}) and prospects (LHC, LEP2 and
NLC \cite{TGC,LEP2}) are mostly based on diboson production ($WW$,
$W\gamma$ and $WZ$).
In $ep$ collisions, HERA could provide
further information
analyzing single $W$ production ($ep\to eWX$ \cite{ABZ})
and radiative charged current scattering
($ep\to\nu\gamma X$ \cite{hubert}). There is also some
literature on $WW\gamma$ couplings in $W$-pair production at future
very high energy photon colliders (bremsstrahlung photons in peripheral
heavy ion collisions \cite{HIC} and Compton backscattered laser
beams \cite{gg}).
Only recently, attention has been paid to the $Z\gamma Z$, $Z\gamma\g$ and
$ZZZ$ couplings. There is a detailed analysis of $Z\gamma V$
couplings ($V=\gamma,Z$) for hadron colliders in \cite{BB}.
CDF \cite{CDF} and D\O\ \cite{D0} have obtained bounds on the
$Z\gamma Z$ and $Z\gamma\g$ anomalous couplings, while L3 has studied
only the first ones \cite{L3}. Studies on the sensitivities to
these vertices in future $e^+e^-$ colliders,
LEP2 \cite{LEP2} and NLC \cite{Boudjema}, have been performed during
the last years.
Some proposals have been made to probe these neutral boson gauge
couplings at future photon colliders in $e\gamma\to Ze$ \cite{eg}.
In this work we study the prospects for measuring the
TGC in the process $ep\to
e\gamma X$. In particular, we will concentrate on the $Z\gamma\g$ couplings,
which can be more stringently bounded than the $Z\gamma Z$ ones
for this process.
In Section 2, we present the TGC. The next section deals with the
different contributions to the process $ep\to e\gamma X$ and the cuts
and methods we have employed
in our analysis. Section 4 contains our results
for the Standard Model total cross section and distributions and
the estimates of the sensitivity of these quantities to the
presence of anomalous couplings. Finally, in the last section we
present our conclusions.
\section{Phenomenological parametrization of the neutral TGC}
A convenient way to study deviations from the standard model predictions
consists of considering the most general lagrangian compatible with
Lorentz invariance, the electromagnetic U(1) gauge symmetry, and
other possible gauge symmetries.
For the trilinear $Z\gamma V$ couplings ($V=\gamma,Z)$ the most general vertex
function invariant under Lorentz and electromagnetic gauge transformations
can be described in terms of four independent dimensionless form
factors \cite{hagiwara}, denoted by $h^V_i$, i=1,2,3,4:
\begin{eqnarray}
\Gamma^{\a\b\mu}_{Z\gamma V} (q_1,q_2,p)=\frac{f(V)}{M^2_Z}
\{
h^V_1 (q^\mu_2 g^{\a\b} - q^\a_2 g^{\mu\b})
+\frac{h^V_2}{M^2_Z} p^\a (p\cdot q_2g^{\mu\b}-q^\mu_2 p^\b)
\nonumber \\
+h^V_3 \varepsilon^{\mu\a\b\r}q_{2_\r}
+\frac{h^V_4}{M^2_Z}p^\a\varepsilon^{\mu\b\r\sigma}p_\r q_{2_\sigma}
\}. \hspace{3cm}
\label{vertex}
\end{eqnarray}
Terms proportional to $p^\mu$, $q^\a_1$ and $q^\b_2$ are omitted as long as
the scalar components of all three vector bosons can be neglected
(whenever they couple to almost massless fermions) or they are zero
(on-shell condition for $Z$ or U(1) gauge boson character of the photon).
The overall factor, $f(V)$, is $p^2-q^2_1$ for $Z\gamma Z$ or $p^2$ for $Z\gamma\g$
and is a result of Bose symmetry and electromagnetic gauge invariance.
These latter constraints reduce the familiar seven form factors
of the most general $WWV$ vertex to only these four for the
$Z\gamma V$ vertex. There still remains a global factor that can be fixed,
without loss of generality, to $g_{Z\gamma Z}=g_{Z\gamma\g}=e$. Combinations
of $h^V_3 (h^V_1)$ and $h^V_4 (h^V_2)$ correspond to electric
(magnetic) dipole and magnetic (electric) quadrupole transition
moments in the static limit.
All the terms are $C$-odd. The terms proportional to $h^V_1$ and $h^V_2$
are $CP$-odd while the other two are $CP$-even. All the form factors
are zero at tree level in the Standard Model. At the one-loop level,
only the $CP$-conserving $h^V_3$ and $h^V_4$ are nonzero \cite{barroso}
but too small (${\cal O}(\a/\pi$)) to lead to any observable
effect at any present or planned experiment. However, larger effects
might appear in theories or models beyond the Standard Model,
for instance when the gauge bosons are composite objects
\cite{composite}.
This is a purely phenomenological, model independent parametrization.
Tree-level unitarity restricts the $Z\gamma V$ to the Standard Model
values at asympotically high energies \cite{unitarity}. This
implies that the couplings $h^V_i$ have to be described by form factors
$h^V_i(q^2_1,q^2_2,p^2)$ which vanish when $q^2_1$, $q^2_2$ or $p^2$
become large. In hadron colliders, large values of $p^2=\hat{s}$
come into play and the energy dependence has to be taken into
account, including unknown dumping factors \cite{BB}.
A scale dependence appears as an additional parameter (the scale
of new physics, $\L$). Alternatively,
one could introduce a set of operators invariant under SU(2)$\times$U(1)
involving the gauge bosons and/or additional would-be-Goldstone bosons
and the physical Higgs. Depending on the new physics dynamics,
operators with dimension $d$ could be generated at the scale $\L$,
with a strength which is generally suppressed by factors like
$(M_W/\L)^{d-4}$ or $(\sqrt{s}/\L)^{d-4}$ \cite{NPscale}.
It can be shown that $h^V_1$ and $h^V_3$ receive contributions from
operators of dimension $\ge 6$ and $h^V_2$ and $h^V_4$ from
operators of dimension $\ge 8$.
Unlike hadron colliders, in $ep\to e\gamma X$ at HERA energies, we can ignore
the dependence of the form factors on the scale. On the other
hand, the anomalous couplings are tested in a different kinematical region,
which makes their study in this process complementary to the ones
performed at hadron and lepton colliders.
\section{The process $ep\to e\gamma X$}
The process under study is $ep\to e\gamma X$, which is described in the
parton model by the radiative neutral current electron-quark and
electron-antiquark scattering,
\begin{equation}
\label{process}
e^- \ \stackrel{(-)}{q} \to e^- \ \stackrel{(-)}{q} \ \gamma .
\end{equation}
There are eight Feynman diagrams contributing to this process in
the Standard Model and three additional ones if one includes anomalous vertices:
one extra diagram for the $Z\gamma Z$ vertex and two for the $Z\gamma\g$
vertex (Fig. \ref{feyndiag}).
\bfi{htb}
\begin{center}
\bigphotons
\bpi{35000}{21000}
\put(4000,8000){(a)}
\put(200,17000){\vector(1,0){1300}}
\put(1500,17000){\vector(1,0){3900}}
\put(5400,17000){\line(1,0){2600}}
\drawline\photon[\S\REG](2800,17000)[5]
\put(200,\pbacky){\vector(1,0){1300}}
\put(1500,\pbacky){\vector(1,0){2600}}
\put(4100,\pbacky){\vector(1,0){2600}}
\put(6700,\pbacky){\line(1,0){1300}}
\put(0,13000){$q$}
\put(8200,13000){$q$}
\put(3300,\pmidy){$\gamma,Z$}
\drawline\photon[\SE\FLIPPED](4900,\pbacky)[4]
\put(0,18000){$e$}
\put(8200,18000){$e$}
\put(8200,\pbacky){$\gamma$}
\put(13000,8000){(b)}
\put(9500,17000){\vector(1,0){1300}}
\put(10800,17000){\vector(1,0){2600}}
\put(13400,17000){\vector(1,0){2600}}
\put(16000,17000){\line(1,0){1300}}
\drawline\photon[\S\REG](12100,17000)[5]
\put(9500,\pbacky){\vector(1,0){1300}}
\put(10800,\pbacky){\vector(1,0){3900}}
\put(14700,\pbacky){\line(1,0){2600}}
\drawline\photon[\NE\FLIPPED](14200,17000)[4]
\put(22000,8000){(c)}
\put(18500,17000){\vector(1,0){3250}}
\put(21750,17000){\vector(1,0){3250}}
\put(25000,17000){\line(1,0){1300}}
\drawline\photon[\S\REG](23700,17000)[5]
\put(18500,\pbacky){\vector(1,0){1300}}
\put(19800,\pbacky){\vector(1,0){2600}}
\put(22400,\pbacky){\vector(1,0){2600}}
\put(25000,\pbacky){\line(1,0){1300}}
\drawline\photon[\SE\FLIPPED](21100,\pbacky)[4]
\put(31000,8000){(d)}
\put(27500,17000){\vector(1,0){1300}}
\put(28800,17000){\vector(1,0){2600}}
\put(31400,17000){\vector(1,0){2600}}
\put(34000,17000){\line(1,0){1300}}
\drawline\photon[\S\REG](32700,17000)[5]
\put(27500,\pbacky){\vector(1,0){3250}}
\put(30750,\pbacky){\vector(1,0){3250}}
\put(33900,\pbacky){\line(1,0){1300}}
\drawline\photon[\NE\FLIPPED](30100,17000)[4]
\put(17800,0){(e)}
\put(17100,5500){$\gamma,Z$}
\put(17100,3000){$\gamma,Z$}
\put(14000,7000){\vector(1,0){1300}}
\put(15300,7000){\vector(1,0){3900}}
\put(19200,7000){\line(1,0){2600}}
\drawline\photon[\S\REG](16600,7000)[5]
\put(16750,\pmidy){\circle*{500}}
\put(14000,\pbacky){\vector(1,0){1300}}
\put(15300,\pbacky){\vector(1,0){3900}}
\put(19200,\pbacky){\line(1,0){2600}}
\drawline\photon[\E\REG](16750,\pmidy)[5]
\put(22300,\pbacky){$\gamma$}
\end{picture}
\end{center}
\caption{\it Feynman diagrams for the process $e^- q \to e^- q \gamma$.
\label{feyndiag}}
\end{figure}
Diagrams with $\gamma$ exchanged in the t-channel are dominant. Nevertheless,
we consider the whole set of diagrams in the calculation.
On the other side, u-channel fermion exchange poles appear, in the limit
of massless quarks and electrons (diagrams (c) and (d)).
Since the anomalous diagrams (e) do not present such infrared or
collinear singularities, it seems appropriate to avoid almost
on-shell photons exchanged and fermion poles by cutting the
transverse momenta of the final fermions (electron and jet) to
enhance the signal from anomalous vertices.
Due to the suppression factor coming from $Z$ propagator, the
anomalous diagrams are more sensitive to $Z\gamma\g$ than to $Z\gamma Z$ vertices.
In the following we will focus our attention on the former.
The basic variables of the parton level process are five. A
suitable choice is: $E_\gamma$ (energy of the final photon),
$\cos\th_\gamma$, $\cos\th_{q'}$ (cosines of the polar angles of the
photon and the scattered quark defined with respect to the proton direction),
$\phi$ (the angle between the transverse momenta of the photon and the
scattered quark in a plane perpendicular to the beam), and a
trivial azimuthal angle that is integrated out (unpolarized beams).
All the variables are referred to the laboratory frame. One needs
an extra variable, the Bjorken-x, to connect the partonic process
with the $ep$ process. The phase space integration over these six
variables is carried out by {\tt VEGAS} \cite{VEGAS} and has been
cross-checked with the {\tt RAMBO} subroutine \cite{RAMBO}.
We adopt two kinds of event cuts to constrain conveniently
the phase space:
\begin{itemize}
\item
{\em Acceptance and isolation} cuts. The former are to exclude
phase space regions
which are not accessible to the detector, because of angular or
efficiency limitations:\footnote{The threshold for the transverse
momentum of the scattered quark ensures that its kinematics can be
described in terms of a jet.}
\begin{eqnarray}
\label{cut1}
8^o < \theta_e,\ \theta_\gamma,\ \theta_{\rm jet} < 172^o; \nonumber\\
E_e, \ E_\gamma, \ p^{\rm q'}_{\rm T} > 10 \ {\rm GeV}.
\end{eqnarray}
The latter keep the final photon well separated
from both the final electron and the jet:
\begin{eqnarray}
\label{cut2}
\cos \langle \gamma,e \rangle < 0.9; \nonumber\\
R > 1.5,
\end{eqnarray}
where $R\equiv\sqrt{\Delta\eta^2+\phi^2}$ is the separation between
the photon and the jet in the rapidity-azimuthal plane, and $\langle \gamma,e \rangle$ is the angle between the photon and the scattered electron.
\item
Cuts for {\em intrinsic background suppression}. They consist of
strengthening some of the
previous cuts or adding new ones to enhance the signal of the anomalous
diagrams against the Standard Model background.
\end{itemize}
We have developed a Monte Carlo program for the simulation of the
process $ep\to e\gamma X$ where $X$ is the remnant of the proton plus one jet
formed by the scattered quark of the subprocess (\ref{process}). It
includes the Standard Model helicitity amplitudes computed using the {\tt HELAS} subroutines \cite{HELAS}. We added new code to account for the
anomalous diagrams. The squares of these anomalous amplitudes have been
cross-checked with their analytical expressions computed using {\tt FORM}
\cite{FORM}. For the parton distribution functions,
we employ both the set 1 of Duke-Owens' parametrizations \cite{DO}
and the modified MRS(A) parametrizations \cite{MRS}, with the scale chosen to
be the hadronic momentum transfer.
As inputs, we use the beam energies $E_e=30$ GeV and $E_p=820$ GeV,
the $Z$ mass $M_Z=91.187$ GeV, the weak angle $\sin^2_W=0.2315$
\cite{PDB} and the fine structure constant $\a=1/128$. A more correct choice
would be the running fine structure constant with $Q^2$ as the argument.
However, as we are interested in large $Q^2$ events, the value $\a(M^2_Z)$
is accurate enough for our purposes. We consider only
the first and second generations of quarks, assumed to be massless.
We start by applying the cuts (\ref{cut1}) and (\ref{cut2})
and examining the contribution to a set of observables of the
Standard Model and the anomalous diagrams, separately. Next, we
select one observable such that, when a cut on it is performed,
only Standard Model events are mostly eliminated. The procedure
is repeated with this new cut built in. After several runs, adding
new cuts, the ratio standard/anomalous cross sections is reduced
and hence the sensitivity to anomalous couplings is improved.
\section{Results}
\subsection{Observables}
The total cross section of $ep\to e\gamma X$ can be written as
\begin{equation}
\sigma=\sigma_{{\rm SM}} + \sum_{i} \t_i \cdot h^\gamma_i + \sum_{i}\sigma_i\cdot (h^\gamma_i)^2
+ \sigma_{12} \cdot h^\gamma_1 h^\gamma_2 + \sigma_{34} \cdot h^\gamma_3 h^\gamma_4.
\end{equation}
\bfi{htb}
\setlength{\unitlength}{1cm}
\bpi{8}{7}
\epsfxsize=11cm
\put(-1,-4){\epsfbox{eng_acciso.ps}}
\end{picture}
\bpi{8}{7}
\epsfxsize=11cm
\put(0.,-4){\epsfbox{ptg_acciso.ps}}
\end{picture}
\bpi{8}{6}
\epsfxsize=11cm
\put(-1,-5){\epsfbox{angge_acciso.ps}}
\end{picture}
\bpi{8}{6}
\epsfxsize=11cm
\put(0.,-5){\epsfbox{anggj_acciso.ps}}
\end{picture}
\bpi{8}{6}
\epsfxsize=11cm
\put(-1,-5){\epsfbox{angej_acciso.ps}}
\end{picture}
\bpi{8}{7}
\epsfxsize=11cm
\put(0.,-5){\epsfbox{q2e_acciso.ps}}
\end{picture}
\caption{\it Differential cross sections (pb) for the process $ep\to e\gamma X$ at
HERA, with only acceptance and isolation cuts.
The solid line is the Standard Model contribution and the dash (dot-dash) line
correspond to 10000 times the $\sigma_1$ ($\sigma_2$) anomalous contributions.\label{A}}
\end{figure}
The forthcoming results are obtained using the MRS'95
pa\-ra\-me\-tri\-za\-tion of the parton densities\footnote{The values
change $\sim 10$\% when using the (old) Duke-Owens' structure functions.}
\cite{MRS}.
The linear terms of the $P$-violating couplings $h^\gamma_3$
and $h^\gamma_4$ are negligible, as they mostly arise from the interference of
standard model diagrams with photon exchange ($P$-even) and anomalous
$P$-odd diagrams ($\t_3\simeq \t_4\simeq 0$). Moreover, anomalous diagrams with
different $P$ do not interfere either. On the other hand, the quadratic terms
proportional to $(h^\gamma_1)^2$ and $(h^\gamma_3)^2$ have identical expressions, and
the same for $h^\gamma_2$ and $h^\gamma_4$ ($\sigma_1=\sigma_3$, $\sigma_2=\sigma_4$). Only the
linear terms make their bounds different. The interference terms $\sigma_{12}$
and $\sigma_{34}$ are also identical.
\bfi{htb}
\setlength{\unitlength}{1cm}
\bpi{8}{7}
\epsfxsize=11cm
\put(-1,-4){\epsfbox{eng_bkgsup.ps}}
\end{picture}
\bpi{8}{7}
\epsfxsize=11cm
\put(0.,-4){\epsfbox{ptg_bkgsup.ps}}
\end{picture}
\bpi{8}{6}
\epsfxsize=11cm
\put(-1,-5){\epsfbox{angge_bkgsup.ps}}
\end{picture}
\bpi{8}{6}
\epsfxsize=11cm
\put(0.,-5){\epsfbox{anggj_bkgsup.ps}}
\end{picture}
\bpi{8}{6}
\epsfxsize=11cm
\put(-1,-5){\epsfbox{angej_bkgsup.ps}}
\end{picture}
\bpi{8}{7}
\epsfxsize=11cm
\put(0.,-5){\epsfbox{q2e_bkgsup.ps}}
\end{picture}
\caption{\it Differential cross sections (pb) for the process $ep\to e\gamma X$ at
HERA, after intrinsic background suppression.
The solid line is the Standard Model contribution and the dash (dot-dash) line correspond to 500 times the $\sigma_1$ ($\sigma_2$) anomalous contributions.\label{B}}
\end{figure}
We have analyzed the distributions of more than twenty
observables in the laboratory frame, including the energies, transverse
momenta and angular distributions of the jet, the photon and the final
electron, as well as their spatial, polar and azimuthal separations.
Also the bjorken-x, the leptonic and hadronic momenta transfer and other fractional energies are considered.
The process of intrinsic background suppression is illustrated
by comparing Figures \ref{A} and \ref{B}. For simplicity, only
the most interesting variables are shown: the energy $E(\gamma)$ and transverse
momentum $p_T(\gamma)$ of the photon; the angles between the photon and
the scattered electron $\langle \gamma,e \rangle$, the photon and the jet
$\langle \gamma,j \rangle$, and the scattered electron and the jet $\langle e,j
\rangle$; and the leptonic momentum transfer $Q^2(e)$.
In Fig.~\ref{A}, these variables
are plotted with only acceptance and isolation cuts
implemented.
All of them share the property of disposing of a range
where any anomalous effect is negligible, whereas the contribution
to the total SM cross section is large. The set of cuts
listed below were added to reach eventually the distributions of
Fig.~\ref{B}:
\begin{itemize}
\item
The main contribution to the Standard Model cross section comes from
soft photons with very low transverse momentum. The following cuts
suppress a 97$\%$ of these events, without hardly affecting the
anomalous diagrams which, conversely, enfavour high energy photons:
\begin{eqnarray}
E_\gamma > 30 \ {\rm GeV} \nonumber \\
p^\gamma_T > 20 \ {\rm GeV}
\label{cut3}
\end{eqnarray}
\item
Another remarkable feature of anomalous diagrams is the very different
typical momentum transfers. Let's concentrate on the leptonic momentum
transfer, $Q^2_e=-(p'_e-p_e)^2$. The phase space enhances high
$Q^2_e$, while the photon propagator of the Standard Model diagrams
prefer low values (above the threshold for electron detectability,
$Q^2_e>5.8$~GeV$^2$, with our required minimum energy and angle). On the
contrary, the anomalous diagrams have always a $Z$ propagator
which introduces a suppression factor of the order of $Q^2_e/M^2_Z$ and
makes irrelevant the $Q^2_e$ dependence, which is only determined by
the phase space. As a consequence, the following cut looks appropriate,
\begin{equation}
Q^2_e > 1000 \ {\rm GeV}^2
\label{cut4}
\end{equation}
\end{itemize}
It is important to notice at this point why usual form factors for the
anomalous couplings can be neglected at HERA. For our process, these
form factors should be proportional to $1/(1+Q^2/\L^2)^n$. With the scale of
new physics $\L=500$~GeV to 1~TeV, these factors can be taken to be one. This
is not the case in lepton or hadron high energy colliders where the diboson production in the s-channel needs dumping factors $1/(1+\hat{s}/\L^2)^n$.
The total cross section for the Standard Model with acceptance and isolation
cuts is $\sigma_{\rm SM}=21.38$~pb and is reduced to 0.37~pb when all the cuts are applied, while the quadratic contributions only change from
$\sigma_1=2\times10^{-3}$~pb, $\sigma_2=1.12\times10^{-3}$~pb to
$\sigma_1=1.58\times10^{-3}$~pb, $\sigma_2=1.05\times10^{-3}$~pb. The linear
terms are of importance and change from $\t_1=1.18\times10^{-2}$~pb, $\t_2=1.27\times10^{-3}$~pb to $\t_1=7.13\times10^{-3}$~pb, $\t_2=1.26\times10^{-3}$~pb. Finally, the interference term $\sigma_{12}=1.87\times10^{-3}$~pb changes to $\sigma_{12}=1.71\times10^{-3}$~pb.
The typical Standard Model events consist of soft and low-$p_T$ photons
mostly backwards, tending to go in the same direction of the scattered
electrons (part of them are emitted by the hadronic
current in the forward direction), close to the required angular separation ($\sim 30^o$). The low-$p_T$ jet goes opposite to both the photon and the scattered electron, also in the transverse plane.
On the contrary, the anomalous events have not so soft and high-$p_T$ photons,
concentrated in the forward region as it the case for the scattered electron
and the jet.
\subsection{Sensitivity to anomalous couplings}
In order to estimate the sensitivity to anomalous couplings, we
consider the $\chi^2$ function.
One can define the $\chi^2$, which is related to the likelihood
function ${\cal L}$, as
\begin{equation}
\label{chi2}
\chi^2\equiv-2\ln{\cal L}=
2 L \displaystyle\left(\sigma^{th}-\sigma^{o}+\sigma^{o}
\ln\displaystyle\frac{\sigma^{o}}{\sigma^{th}}\right)
\simeq L \displaystyle\dis\frac{(\sigma^{th}-\sigma^{o})^2}{\sigma^{o}},
\end{equation}
where $L=N^{th}/\sigma^{th}=N^o/\sigma^o$ is the integrated luminosity
and $N^{th}$ ($N^o$) is the number of theoretical (observed)
events. The last line of (\ref{chi2}) is a useful
and familiar approximation, only valid when $\mid \sigma^{th}-\sigma^o \mid/
\sigma^o \ll 1$.
This function is a measure of the probability that statistical
fluctuations can make undistinguisable the observed and the predicted
number of events, that is the Standard Model prediction. The well
known $\chi^2$-CL curve allows us to determine the corresponding
confidence level (CL).
We establish bounds on the anomalous couplings by fixing a
certain $\chi^2=\d^2$ and allowing for the $h^\gamma_i$
values to vary, $N^o=N^o(h^\gamma_i)$. The parameter $\d$ is often referred
as the number of
standard deviations or `sigmas'. A $95\%$ CL corresponds to almost
two sigmas ($\d=1.96$).
When $\sigma \simeq \sigma_{{\rm SM}} + (h^\gamma_i)^2 \sigma_i$ (case of the $CP$-odd
terms) and the anomalous contribution is small enough, the
upper limits present some useful, approximate scaling properties,
with the luminosity,
\begin{equation}
h^\gamma_i (L')\simeq\sqrt[4]{\frac{L}{L'}} \ h^\gamma_i (L).
\end{equation}
A brief comment on the interpretation of the results is in order.
As the cross section grows with $h^\gamma_i$, in the relevant range of
values, the $N^o$ upper limits can be regarded as the lowest number
of measured events that would discard the Standard Model, or the
largest values of $h^\gamma_i$ that could be bounded if no effect is
observed, with the given CL. This procedure approaches the
method of upper limits for Poisson processes when the
number of events is large ($\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 10$).
\bfi{htb}
\setlength{\unitlength}{1cm}
\bpi{8}{8}
\epsfxsize=12cm
\put(3.35,4.245){+}
\put(-2.5,-1.5){\epsfbox{conh1h2.nogrid.ps}}
\end{picture}
\bpi{8}{8}
\epsfxsize=12cm
\put(4.1,4.245){+}
\put(-1.75,-1.5){\epsfbox{conh3h4.nogrid.ps}}
\end{picture}
\caption{\it Limit contours for $Z\gamma\g$ couplings at HERA with an integrated luminosity of 10, 100, 250, 1000 pb$^{-1}$ and a 95\% CL.\label{contour}}
\end{figure}
In Fig. \ref{contour} the sensitivities for different luminosities are shown.
Unfortunately, HERA cannot compete with Tevatron, whose best
bounds, reported by the D\O\ collaboration \cite{D0}, are
\begin{eqnarray}
|h^\gamma_1|, \ |h^\gamma_3| &<& 1.9 \ (3.1), \nonumber
\\
|h^\gamma_2|, \ |h^\gamma_4| &<& 0.5 \ (0.8).
\end{eqnarray}
For the first value it was assumed that only one anomalous coupling contributes
(`axial limits') and for the second there are two couplings contributing (`correlated limits'). Our results are summarized in Table \ref{table}.
\begin{table}
\begin{center}
\begin{tabular}{|c|r|r|r|r|r|r|r|r|}
\hline
HERA & \multicolumn{2}{c|}{10 pb$^{-1}$} & \multicolumn{2}{c|}{100 pb$^{-1}$}
& \multicolumn{2}{c|}{250 pb$^{-1}$} & \multicolumn{2}{c|}{1 fb$^{-1}$} \\
\hline \hline
$h^\gamma_1$ & -19.0 & 14.5 & -11.5 & 7.0 & -9.5 & 5.5 & -8.0 & 3.5 \\
& -26.0 & 19.5 & -16.0 & 9.5 & -14.0 & 7.0 & -11.5 & 4.5 \\
\hline
$h^\gamma_2$ & -21.5 & 20.0 & -12.0 & 10.0 & - 9.5 & 8.0 & -7.0 & 6.0 \\
& -26.0 & 30.0 & -13.0 & 18.0 & -10.0 & 15.0 & - 7.5 & 12.0 \\
\hline
$h^\gamma_3$ & -17.0 & 17.0 & -9.0 & 9.0 & -7.5 & 7.5 & -5.5 & 5.5 \\
& -22.5 & 22.5 & -12.0 & 12.0 & -10.0 & 10.0 & -7.0 & 7.0 \\
\hline
$h^\gamma_4$ & -20.5 & 20.5 & -11.0 & 11.0 & -8.5 & 8.5 & -6.0 & 6.0 \\
& -27.5 & 27.5 & -14.5 & 14.5 & -12.0 & 12.0 & -8.5 & 8.5 \\
\hline
\end{tabular}
\end{center}
\caption{\it Axial and correlated limits for the $Z\gamma\g$ anomalous couplings
at HERA with different integrated luminosities and $95\%$ CL. \label{table}}
\end{table}
The origin of so poor results is the fact that, unlike diboson production
at hadron or $e^+e^-$ colliders, the anomalous diagrams of $ep\to e\gamma X$
have a $Z$ propagator decreasing their effect.
The process $ep\to eZX$ avoids this problem thanks to the absence
of these propagators: the Standard Model cross section is similar
to the anomalous one but, as a drawback, they are of the order of
femtobarns.
\section{Summary and conclusions}
The radiative neutral current process $ep\to e\gamma X$
at HERA has been studied. Realistic cuts have been applied in order to
observe a clean signal consisting of detectable and well separated
electron, photon and jet.
The possibility of testing the trilinear neutral gauge boson couplings
in this process has also been explored. The $Z\gamma Z$ couplings are
very suppressed by two $Z$ propagators. Only the $Z\gamma \gamma$ couplings
have been considered. A Monte Carlo program has been developed to
account for such anomalous vertex and further cuts have been implemented
to improve the sensitivity to this source of new physics.
Our estimates are based on total cross sections since the expected number
of events is so small that a distribution analysis is not possible.
The distributions just helped us to find the optimum cuts. Unfortunately,
competitive bounds on these anomalous couplings cannot be achieved at
HERA, even with the future luminosity upgrades.\footnote{We would like to
apologize for the optimistic but incorrect results that were presented
at the workshop due to a regrettable and unlucky mistake in our programs.}
As a counterpart, a different kinematical region is explored, in which
the form factors can be neglected.
\section*{Acknowledgements}
One of us (J.I.) would like to thank the Workshop organizers for financial
support and very especially the electroweak working group conveners
and the Group from Madrid at ZEUS for hospitality and useful conversations.
This work has been partially supported by the CICYT and the European Commission
under contract CHRX-CT-92-0004.
| proofpile-arXiv_065-424 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section*{References}
| proofpile-arXiv_065-562 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
"\\section{Introduction}\n\nSystems producing absorption in the spectra of distant quasars offer\nan(...TRUNCATED) | proofpile-arXiv_065-612 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
"\\section{Introduction}\n\nMany ``connection-dynamic'' theories of gravity with propagating torsion(...TRUNCATED) | proofpile-arXiv_065-650 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
"\\section{Introduction}\n\nBefore we can use our Galaxy as a tool for the interaction of cosmic ray(...TRUNCATED) | proofpile-arXiv_065-783 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
"\\section{Introduction}\nThe problem of transmission and storage of quantum states has received\na (...TRUNCATED) | proofpile-arXiv_065-865 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
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