%% ****** Start of file template.aps ****** % %% %% %% This file is part of the APS files in the REVTeX 4 distribution. %% Version 4.0 of REVTeX, August 2001 %% %% %% Copyright (c) 2001 The American Physical Society. %% %% See the REVTeX 4 README file for restrictions and more information. %% % % This is a template for producing manuscripts for use with REVTEX 4.0 % Copy this file to another name and then work on that file. % That way, you always have this original template file to use. % % Group addresses by affiliation; use superscriptaddress for long % author lists, or if there are many overlapping affiliations. % For Phys. Rev. appearance, change preprint to twocolumn. % Choose pra, prb, prc, prd, pre, prl, prstab, or rmp for journal % Add 'draft' option to mark overfull boxes with black boxes % Add 'showpacs' option to make PACS codes appear % Add 'showkeys' option to make keywords appear %\documentclass[aps,prl,preprint,groupedaddress]{revtex4} %\documentclass[aps,prl,preprint,superscriptaddress]{revtex4} %\documentclass[pre,twocolumn,groupedaddress]{revtex4} \documentclass[pre,twocolumn,showpacs,preprintnumbers,amsmath,amssymb]{revtex4} %\documentclass[aps,pre,preprint,showpacs,preprintnumbers,amsmath,amssymb]{revtex4} %\usepackage{graphicx}% \usepackage{epsfig}% \usepackage{longtable} \usepackage{eucal} %\usepackage{revsymb} % You should use BibTeX and apsrev.bst for references % Choosing a journal automatically selects the correct APS % BibTeX style file (bst file), so only uncomment the line % below if necessary. \bibliographystyle{apsrev} \begin{document} \setlength{\topmargin}{-0.25in} % Use the \preprint command to place your local institutional report % number in the upper righthand corner of the title page in preprint mode. % Multiple \preprint commands are allowed. % Use the 'preprintnumbers' class option to override journal defaults % to display numbers if necessary \preprint{U. of IOWA Preprint} %Title of paper \title{Lattice gluodynamics at negative $g^2$} % repeat the \author .. \affiliation etc. as needed % \email, \thanks, \homepage, \altaffiliation all apply to the current % author. Explanatory text should go in the []'s, actual e-mail % address or url should go in the {}'s for \email and \homepage. % Please use the appropriate macro foreach each type of information % \affiliation command applies to all authors since the last % \affiliation command. The \affiliation command should follow the % other information % \affiliation can be followed by \email, \homepage, \thanks as well. \author{L. Li and Y. Meurice} \email[]{yannick-meurice@uiowa.edu} %\homepage[]{Your web page} %\thanks{This work was supported in part by the DOE} %\altaffiliation{} \affiliation{Department of Physics and Astronomy\\ The University of Iowa\\ Iowa City, Iowa 52242 \\ USA } %Collaboration name if desired (requires use of superscriptaddress %option in \documentclass). \noaffiliation is required (may also be %used with the \author command). %\collaboration can be followed by \email, \homepage, \thanks as well. %\collaboration{} %\noaffiliation \date{\today} \begin{abstract} We consider Wilson's $SU(N)$ lattice gauge theory (without fermions) at negative values of $\beta= 2N/g^2$ and for $N$=2 or 3. We show that in the limit $\beta \rightarrow -\infty$, the path integral is dominated by configurations where links variables are set to a nontrivial element of the center on selected non intersecting lines. For $N=2$, these configurations can be characterized by a unique gauge invariant set of variables, while for $N=3$ a multiplicity growing with the volume as the number of configurations of an Ising model is observed. In general, there is a discontinuity in the average plaquette when $g^2$ changes its sign which prevents us from having a convergent series in $g^2$ for this quantity. For $N=2$, a change of variables relates the gauge invariant observables at positive and negative values of $\beta$. For $N=3$, we derive an identity relating the observables at $\beta$ with those at $\beta$ rotated by $\pm 2\pi/3$ in the complex plane and show numerical evidence for a Ising like first order phase transition near $\beta=-22$. We discuss the possibility of having lines of first order phase transitions ending at a second order phase transition in an extended bare parameter space. \end{abstract} % insert suggested PACS numbers in braces on next line \pacs{11.15.-q, 11.15.Ha, 11.15.Me, 12.38.Cy} % insert suggested keywords - APS authors don't need to do this %\keywords{} %\maketitle must follow title, authors, abstract, \pacs, and \keywords \maketitle % body of paper here - Use proper section commands % References should be done using the \cite, \ref, and \label commands \section{Introduction} It has been known from the early days of QED that perturbative series have a zero radius of convergence \cite{dyson52}. This has not prevented Feynman diagrams to become an essential tool in particle physics. However, perturbative series need to be used with caution. The divergent nature of QED series was foreseen by Dyson as a consequence of the apparently pathological nature \cite{dyson52} of the ground state in a fictitious world with negative $e^2$. Like charges then attract and pair creation can be invoked to produce states where electrons are brought together in a given region and positrons in another. Dyson concludes that as this process sees no end, no stable vacuum can exist. For Euclidean lattice models, related situations are encountered. For scalar field theory with $\lambda \phi^4$ interactions, configurations with large field values make the path integral ill-defined when $\lambda < 0$ (provided that no higher even powers of $\phi$ appear in the action with a positive sign and that the path of integration is not modified). Modified series with a finite radius of convergence can be obtained by introducing a large field cutoff \cite{pernice98,convpert}. We are then considering a slightly different problem. In simple situations \cite{optim}, it is possible to determine an optimal value of the field cutoff that, at a given order in perturbation, minimizes or eliminates the discrepancy. For non-abelian gauge theories in the continuum Hamiltonian formulation, the substitution $g\rightarrow ig$ makes the quartic part unbounded from below and the cubic part non-hermitian. It should be noted that in quantum mechanics\cite{bender}, it is possible to change the boundary conditions of the Schr\"{o}dinger equation in such a way that the spectrum of an harmonic oscillator with a perturbation of the form $ix^3$ or $-x^4$ stays real and positive. The procedure can be extended to scalar field theory in order to define a sensible $i\phi^3$ theory \cite{bender04}. Even though conventional Monte Carlo calculations would fail for these models, complex Langevin methods can be used to calculate Green's functions \cite{bernard2001}. In the case of lattice gauge theory with {\it compact} gauge groups, the action per unit of volume is bounded from below and there is no large field problem. Consequently, these models have well-defined expectation values when $g^2<0$ and we can consider the limit $g^2\rightarrow 0^-$. In this article, we discuss the behavior of Wilson loops for $SU(N)$ lattice gauge theory with $g^2<0$. This work is motivated by the need to understand the unexpected behavior of the lattice pertubative series for the $1\times 1$ plaquette calculated up to order 10 \cite{alles93,direnzo95,direnzo2000}. An analysis of the successive ratios \cite{rakow2002,lilipro} may suggest that the series has a finite radius of convergence and a non-analytic behavior near $\beta\simeq 5.7$ in contradiction with the general expectations that the series should be asymptotic and the transition from weak to strong coupling smooth. We consider here pure (no fermions) gauge models with a minimal lattice action \cite{wilson74c}. For definiteness this model and our notations are defined in section \ref{sec:model}. The extrema of the action are discussed in \ref{sec:limit} and enumerated for $SU(2)$ and $SU(3)$. We then discuss (section \ref{sec:su2}) the case of $SU(2)$ and show that planar Wilson loops with an area $A$ (in plaquette units) pick up a factor $(-1)^A$ when $g^2$ becomes negative and the behavior for $g^2<0$ is completely determined by the behavior with $g^2>0$. As the Wilson loops are nonzero when $g^2\rightarrow 0^+$, the ones with an odd area have a discontinuity which invalidates the idea of a regular perturbative series. \section{The model, notations} \label{sec:model} In the following, we consider the minimal, unimproved, lattice gauge model originally proposed by K. Wilson \cite{wilson74c}. Our conventions and notations are introduced in this section for definiteness. We consider a cubic lattice in $D$ dimensions. A $SU(N)$ group element is attached to each link $l$ and $U_l$ denotes its fundamental representation. $U_p$ denotes the conventional product of $U_l$ (or hermitian conjugate) along the sides of a $1\times 1$ plaquette $p$. The minimal action reads \begin{equation} S=\beta\sum_{p}(1-(1/N)Re Tr(U_p)) \ , \end{equation} with $\beta=2N/g^2$. The lattice functional integral or partition function is \begin{equation} Z=\prod_{l}\int dU_l {\rm e}^{-S} \, \end{equation} with $dU_l$ the $SU(N)$ invariant Haar measure for the group element associated with the link $l$. The average value of any quantity $\mathfrak{Q}$ is defined as usual by inserting $\mathfrak{Q}$ in the integral and dividing by $Z$. In the following, we consider symmetric finite lattice with $L^D$ sites and periodic boundary conditions. For reasons that will become clear in the next sections, $L$ will always be even. The total number of $1\times 1$ plaquettes is denoted \begin{equation} \mathcal{N}_p\equiv\ L^D D(D-1)/2\ . \end{equation} Using \begin{equation} f\equiv-(1/\mathcal{N}_p)\ln Z\ , \end{equation} we define the average density \begin{eqnarray} \label{eq:pdef} P(\beta)&\equiv & \partial f/\partial \beta \nonumber \\ &=&(1/\mathcal{N}_p)\left\langle \sum_p (1-(1/N)Re Tr(U_p))\right\rangle \ . \end{eqnarray} In statistical mechanics, $f$ would be the free energy density multiplied by $\beta$ and $P$ the energy density. In analogy we can also define the constant volume specific heat per plaquette \begin{equation} \label{eq:cv} C_V=-\beta^2 \partial P/\partial \beta \ . \end{equation} \section{The limit $\beta \rightarrow -\infty$} \label{sec:limit} In the limit $\beta \rightarrow -\infty$, we expect the functional integral to be dominated by configurations which {\it maximize} $ \sum_{P}(1-(1/N)Re Tr(U_P))$. In the opposite limit ($\beta \rightarrow +\infty$), the same quantity needs to be minimized which can be accomplished by taking $U_l$ as the identity everywhere. We first consider the question of finding the extrema of $TrU$. For our study of the behavior when $\beta \rightarrow -\infty$, we are particularly interested in finding absolute minima of $TrU$. Using $TrU=Tr(VUV^{\dagger})$ for $V$ unitary, $U={\rm e}^{iH}$ with $H$ traceless and hermitian, and $V{\rm e}^{iH}V^{\dagger}$=${\rm e}^{iVHV^{\dagger}}$, we can diagonalize $H$ and write \begin{equation} ReTrU=\sum_{i=1}^{N-1}\cos(\phi_i)+\cos(\sum_{i=1}^{N-1}\phi_i)\ . \end{equation} The extremum condition then reads \begin{equation} \label{eq:extr} \sin(\phi_i)+\sin(\sum_{i=1}^{N-1}\phi_i)=0\ , \end{equation} for $i=1,\dots N-1$. The trivial solution is all $\phi_i=0$. We then have $ReTrU=N$ which is an absolute maximum. For $N=2$, we have only one nontrivial solution $\phi_1=\pi$, which corresponds to the nontrivial element of the center $U=-\openone$. We then have $ReTrU=-2$ which is an absolute minimum. For $N=3$, we have 5 nontrivial solutions. Two correspond to the nontrivial elements of the center ($\phi_1=\phi_2=\pm 2\pi/3$). The matrix of second derivatives has two positive eigenvalues and these two solutions correspond to a minimum. We use the notation $\Omega\equiv {\rm e}^{i2\pi/3}\openone$ on the diagonal. We have $ReTr\Omega=-3/2$, which we will see is an absolute minimum. The other three solutions are $(\phi_1=\pi,\phi_2=0)$, $(\phi_1=0,\phi_2=\pi)$ and $(\phi_1=\pi,\phi_2=\pi)$. They correspond to elements conjugated to diagonal matrices belonging to the three canonical $SU(2)$ subgroups with the $SU(2)$ element being the non trivial center element. These three solutions have matrices of second derivatives with eigenvalues of opposite signs and correspond to saddle points rather than minimum or maximum. In the three cases $ReTrU=-1$. For general $N$, it is clear that we can always find at least one group element $U$ such that $ReTrU$ is an absolute minimum. In particular, for $N$ even, $U=-1$ is such a group element, with $ReTrU=-N$ (the individual matrix elements must have a complex norm less then one). For $N\geq 3$ and odd, it is easy to check that all $\phi_i=(N-1)\pi/N$ is a solution of the extremum condition Eq. (\ref{eq:extr}). For this choice, $ReTrU=-N|\cos((N-1)\pi/N)|$ which is clearly negative and tends to $-N$ as $N$ becomes large. This solution (the element of the center the closest to $-\openone$) gives an absolute minimum of $TrU$ for $N=3$ and we conjecture that it is also the case for larger $N$. We can now obtain an absolute minimum of the action if we can build a configuration such that $ReTrU_P$ takes its absolute minimum value for every plaquette. This can be accomplished by the following construction. In the Appendix, we show that (at least for for $D\leq 4$) and for $L$ even, it is possible to construct a set of lines on the lattice such that every plaquette shares {\it one and only one} link with this set of lines. We call such a set of links $\mathfrak{L}$. One can then put an element which gives an absolute minimum of $ReTrU$ on the links of $\mathfrak{L}$ and the identity on all the other links. For $SU(2)$, there is only one possible choice that minimizes $ReTrU$, namely $U=-\openone$. For $SU(3)$, there are two possible choices $U=\Omega$ or $U=\Omega^{\dagger}$. We emphasize that the construction only works for $L$ even. If $L$ is odd, there will be lines of frustration in every plane. The set of links $\mathfrak{L}$ is not unique. Starting with a given set, we can generate another one by translating the lines by one lattice spacing or rotating them by $\pi/2$ about the lattice axes. By direct inspection, it is easy to show that for $D=2$ there are 4 such a sets of lines while for $D=3$ there are 8 of them. Enumerating all the gauge inequivalent minima of the action at negative $\beta$ for arbitrary $D$ and $N$ appears as a nontrivial problem. In the rest of this section, we specialize the discussion to $N=$ 2 or 3. In order to discriminate among gauge inequivalent configurations, it is useful to make the following (gauge invariant) argument: in order to have an absolute minimum of the action, for every $1\times1$ plaquette $p$, the product $U_p(\mathbf{n})$ of the $U_l$ along $p$ starting at any site $\mathbf{n}$ of $p$, is a nontrivial element of the center. Under a local gauge transformation, $U_p(\mathbf{n})\rightarrow V(\mathbf{n})U_p(\mathbf{n}) V(\mathbf{n})^{\dagger}=U_p(\mathbf{n})$ since $U_p(\mathbf{n})$ commutes with any $SU(N)$ matrix. For the same reason, changing ${\mathbf{n}}$ along the plaquette amounts to a $VUV^{\dagger}$ conjugation and has no effect on the center. Consequently, configurations with a different set of $\mathfrak{U}=\{ U_p \}$ are not gauge equivalent. One can think of $\mathfrak{U}$ as a set of electric and magnetic field configurations. For $SU(2)$, there is only one, uniform, set $\mathfrak{U}$ where all the elements $U_p=-\openone$. For $D=2$, this can be realized in 4 different ways by putting $-\openone$ on the 4 distinct sets $\mathfrak{L}$. These 4 configurations are all gauge equivalent. The gauge transformations that map these 4 configurations into each others can be obtained by taking $V=-\openone$ on every other sites of the lines of $\mathfrak{L}$. For $D=3$, it is also possible to show that the 8 configurations that can be constructed with a similar procedure can also be shown to be gauge equivalent. The gauge transformation can be obtained by taking $V=-\openone$ on every other sites of the lines of $\mathfrak{L}$ pointing in two particular directions and in such way that one half of the lines created by the gauge transformation associated with one direction ``annihilate'' with one half of the lines created by the gauge transformation associated with the other direction. We conjecture that in higher dimensions, the configurations that minimize the action for $SU(2)$ are also related by gauge transformations. For $SU(3)$ the situation is quite different because for every link of a particular $\mathfrak{L}$, we have two possible nontrivial element of the center. Since there are $\mathcal{N}_p=L^D D(D-1)/2$ plaquettes on a $L^D$ lattice and one link of $\mathfrak{L}$ per plaquette, shared by $2(D-1)$ plaquette, we have $DL^D/4$ links in any $\mathfrak{L}$. Picking a particular $\mathfrak{L}$, it possible to construct $2^{DL^D/4}$ distinct $\mathfrak{U}$. Consequently, there are at least $2^{DL^D/4}$ gauge inequivalent minima of the action for $SU(3)$. Note that $2^{DL^D/4}$ always is an integer for $L$ even, which has been assumed. In the case $D=4$, the degeneracy is simply $2^{L^4}$ which is the same as the number of configurations of an Ising model on a $L^4$ lattice. In summary, we predict a discontinuity in $P$ as $g^2$ changes sign. In the limit $\beta \rightarrow +\infty$, we have $P\rightarrow 0$, while in the limit $\beta \rightarrow -\infty$, we expect $P\rightarrow 2$ for $N$ even, and $1+|\cos((N-1)\pi/N)|$ for $N$ odd. \section{\label{sec:su2}N=2} In this section we discuss $SU(2)$ gauge theories at negative $\beta$. The basic idea is that it is possible to change $\beta Re TrU_p$ into $-\beta Re TrU_p$ by making the {\it change of variables} $U_l\rightarrow -U_l$ for every link $l$ of a particular $\mathfrak{L}$. Since $-\openone$ is an element of $SU(2)$ and since the Haar measure is invariant under left or right multiplication by a group element, this does not affect the measure of integration. Consequently, we have \begin{equation} Z(-\beta)={\rm e}^{2\beta\mathcal{N}_p}Z(\beta) \label{eq:su2id} \end{equation} Taking the logarithmic derivative as in Eq. (\ref{eq:pdef}), we obtain \begin{equation} \label{eq:pp2} P(\beta)+P(-\beta)=2\ . \end{equation} This identity can be seen in the symmetry of the curve $P(\beta)$ shown in Fig. \ref{fig:su2pp}. \begin{figure} \includegraphics[width=2.5in,angle=0]{su2-p.eps} \caption{\label{fig:su2pp}The average action density $P(\beta)$ for $SU(2)$.} \end{figure} The validity of Eq. (\ref{eq:pp2}) can be further checked by calculating the difference \begin{equation} \label{eq:delta} \Delta(\beta)\equiv | P(\beta)+P(-\beta)-2|\, \end{equation} which should be zero except for statistical fluctuations. \begin{acknowledgments} This research was supported in part by the Department of Energy under Contract No. FG02-91ER40664. We thanks C. Bender, M. Creutz, M. Ogilvie and the participants of the Argonne workshop ``QCD in extreme environment'' for valuable conversations. % put your acknowledgments here. \end{acknowledgments} \appendix* \section{ Maximal sets of Non-intersecting of lines on a cubic lattice} In this Appendix, we consider a $D$ dimensional cubic lattice. We restrict the use of ``line'' to collections of links along the $D$ principal directions of the lattice and the use of ``plane'' to collections of plaquettes along the $D(D-1)/2$ principal orientations. In other words, these objects are lines and planes in the usual sense, but we exclude some ``oblique'' sets that can be constructed out of the sites. We now try to construct a set of lines such that every plaquette shares one and only one link with this set. It is obvious that these lines cannot intersect, otherwise, at the point of intersection and in the plane defined by the two lines, we could fit 4 plaquettes, each sharing two links with the lines. These lines cannot be obtained from each other by a translation of one lattice spacing in one single direction, otherwise the set of lines would share two opposite links on the plaquettes in between the two lines. For $D=2$, the problem has obvious solutions, we can pick for instance a set of vertical lines separated by two lattice spacings. Using translation by one lattice spacing and rotation by $\pi/2$, it is possible to obtain three other solutions. For $D>2$, it is sufficient to show that for every plane (in the restricted sense defined above), we have a $D=2$ solution. As this restricted set of planes contains all the plaquette ounce, we would have then succeeded in proving the assertion. If such a solution exists, it is invariant by a translation by 2 lattice spacing in any direction. Consequently, we only need to prove the existence of the set of lines on a $2^D$ lattice with periodic boundary conditions. The full solution is then obtained by translation of the $2^D$ solution. If the lattice is finite, this only works if $L$ is even, an assumption we have maintained in this article. On a $2^D$ lattice, the lines (as defined above) are constructed by fixing $D-1$ coordinates values to be 0 or 1 and leaving the remaining coordinate arbitrary. For instance, for $D=3$, a line in the 3rd direction coming out of the origin will be denoted $(0,0,A)$ where $A$ stands for arbitrary and means 0 or 1. In general $D$, there are $D2^{D-1}$ such lines. Consequently, there are $D2^{D}$ links, each shared by $2(D-1)$ plaquettes. There are thus $D2^D2(D-1)/4=D(D-1)2^{D-1}$ plaquettes. A set of lines which has exactly one link in common with every plaquette, has $D(D-1)2^{D-1}/(2(D-1))=D2^{D-2}$ links in other words it must contain $D2^{D-3}$ lines. For $D=2$, such a set has only one line and there are four possible choices. For $D=3$, an example of solution is $\{(A,0,0),(0,A,1),(1,1,A)\}$. It is not difficult to show that there are 8 distinct solutions of this type. For $D=4$, a solution consists in 8 lines. An example of solution is \begin{eqnarray} \nonumber &\{&(A,0,0,0),(0,A,0,1),(0,1,A,0), (0,0,1,A),\\ \nonumber &\ &(1,1,0,A) ,(1,0,A,1),(1,A,1,0),(A,1,1,1)\ \}\ . \end{eqnarray} % Create the reference section using BibTeX: %\bibliography{olda2} \begin{thebibliography}{19} \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi \expandafter\ifx\csname bibnamefont\endcsname\relax \def\bibnamefont#1{#1}\fi \expandafter\ifx\csname bibfnamefont\endcsname\relax \def\bibfnamefont#1{#1}\fi \expandafter\ifx\csname citenamefont\endcsname\relax \def\citenamefont#1{#1}\fi \expandafter\ifx\csname url\endcsname\relax \def\url#1{\texttt{#1}}\fi \expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi \providecommand{\bibinfo}[2]{#2} \providecommand{\eprint}[2][]{\url{#2}} \bibitem[{\citenamefont{Dyson}(1952)}]{dyson52} \bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Dyson}}, \bibinfo{journal}{Phys. 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