% Ramon G. Rodriguez 8/16/76
% DPF Proceeding
% preprint UMN-TH-1507/96
% preprint NUC-MINN-96/14-T
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\begin{document}

\rightline{\small UMN-TH-1507/96}
\rightline{\small NUC-MINN-96/14-T}
\vglue .5cm

\title{Aspects of Confinement and Chiral Dynamics in 2-d QED at Finite 
Temperature\footnote{~To appear in the Proceedings of {\it DPF96} 
University of Minnesota, August 10-15}}
\author{R. RODRIGUEZ and Y. HOSOTANI}

\address{School of Physics and Astronomy, University of Minnesota\\
Minneapolis, MN 55455, USA}

\author{J.E. HETRICK}

\address{Department of Physics, University of Arizona,  Tucson,
AZ 85721, USA}

\author{S. ISO}

\address{Theory Division, KEK, Tsukuba,
Ibaraki 305, Japan}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
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\maketitle\abstracts{We evaluate the Polyakov loop and string tension at 
zero and finite temperature in $QED_2$. Using bozonization the problem 
is reduced to solving the Schr\"odinger equation  with a particular potential 
determined by the ground state . In the presence of two sources of 
opposite charges the vacuum angle parameter $\theta $ changes by 
$2\pi (q/e)$, independent of the number of flavors. This, in turn, 
alters the chiral condensate. Particularly, in the one 
flavor case through a simple computer algorithm, 
we explore the chiral dynamics of a heavy fermion.}


The Schwinger model, quantum electrodynamics in $1+1$ dimensions, has 
expediently been used as a metaphor of color screening 
in gauge theories.\cite{Casher}
In its massless form it is reduced to a free field theory and, consequently, 
solved exactly.\cite{Sch}
With massive fermions the theory is no longer solvable.\cite{CJS} 
Here, fractionally charged test particles are confined quite analogously 
to quark confinement in QCD.\cite{HNZ,Gross,Grignani} 
%The Lagrangian is 
%\beeq
%{\cal L} = - \hbox{$1\over 4$} \, F_{\mu\nu} F^{\mu\nu} + 
%\sum_{a=1}^N \psibar_a \Big\{ \gamma^\mu (i \dd_\mu - e A_\mu) -
%  m_a  \Big\} \psi_a ~~.
%\eneq

In the Matsubara formalism the model at finite temperature in 
equilibrium is equivalent to an Euclidean field theory of compact 
imaginary time $\tau .$ The strategy is to solve the model on a circle 
$S^1$ where we use the powerfull machinery of bosonization and Wick 
rotate by the replacement $L \rightarrow \beta , it \rightarrow x ,
x \rightarrow \tau .$ The Wilson line on the circle corresponds to 
the Polyakov loop in Euclidean space of compact $\tau .$

The bosonization is carried out in the interaction picture defined by free 
massless fermions: $i\gamma \partial \psi =0.$ 
%The Hamiltonian becomes
%\begin{figure}
\begin{figure}[t,b]
\hskip 1.8cm
\epsfxsize= 8.cm    % changed from 10 cm  to 8.5 cm.  9cm is too big.
\epsffile[137 539 399 730]{procfig1.ps}
%\epsffile[100 240 450 500]{P-F.ps}
%\epsffile[162 539 399 723]{dpff1.ps}
\caption{$T$-dependence of the Polyakov loop and free energy in
the $N=3$ model with $m/\mu=.01$.  The free energy stays finite at all $T$.
\label{fig:Polyakov}}
\end{figure}
%\beqn
%&&H_\tot = H_0 + H_\phi + H_\mass\cr 
%\noalign{\kern 8pt}
%&&H_0 = {\pi \mu^2 L\over 2N} P_W^2 
%+ {1\over 2\pi L} \sum_{a=1}^N (\Theta_W+2\pi p_a)^2 \cr
%\noalign{\kern 5pt}
%&&H_\phi = \int_0^L dx \, {1\over 2} 
%\bigg\{  \sum_{a=1}^N \Big( {\Pi_a}^2 +\phi_a'^2 \Big)  
%+ \mu^2 \,  \Big( {1\over \sqrt{N}}\sum_a \phi_a \Big)^2
% \bigg\}
%\label{Hamiltonian1}
%\eeqn
%Expression (\ref{Hamiltonian1}) posseses a residual gauge symmetry. 
The Hamiltonian posseses a residual gauge symmetry. 
In view of this degeneracy the 
ground state of the theory is the theta vacumm, quite analogously
to QCD. 
%It is more convenient to use a coherent state basis with respect
%to the eigenvalues of $p_a.$ 
When the fermion masses are small 
we express the vacumm wave function in terms of 
$f(\vphi_1, \cdots, \vphi_{N-1}; \theta)$.\cite{RH} 
Then, the problem is reduced to 
the solution of a Schr\"odinger equation of $N-1$ degrees of freedom. 
\beqn
&&\Big\{ -\LapN + \potN \Big\} ~ f(\vphi_1, \cdots, \vphi_{N-1})
= \ep ~ f(\vphi_1, \cdots, \vphi_{N-1}) \cr
\noalign{\kern 6pt}
&&\LapN = 
\sum_{a=1}^{N-1} {\dd^2\over \dd\vphi_a^2} 
-{2\over N-1} \sum_{a<b}^{N-1} {\dd^2\over \dd\vphi_a\dd\vphi_b} \cr
&&\potN = -
~  \sum_{a=1}^N   m_a A_a  \cos \vphi_a    \hskip .5cm   
\Big( \sum_{a=1}^N \vphi_a =\theta \Big) 
\label{Sch1}
\eeqn
where $A_a$ is determined by the boson masses $\mu_\alpha$. 
The boson masses themselves are to be determined self-consistently 
by $f(\vphi_1, \cdots, \vphi_{N-1}; \theta).$ Schematically the following
process proceeds until convergence: 
$A_a \go f(\vphi) \go \mu_\alpha \go A_a ~.$ 
%\begin{figure}
\begin{figure}[t,b]
\hskip 1.2cm
\epsfxsize= 8.cm    % changed from 10 cm  to 8.5 cm.  9cm is too big.
\epsffile[137 539 399 730]{procfig2.ps}
%\epsffile[162 539 399 723]{dpf2v2.ps}
\caption{$T$-dependence of the chiral condensates 
with fixed $m/\mu $ and $\theta $
\label{fig:cc}}
\end{figure}
We use the vacumm wave function to compute the Wilson line, from 
which we obtain the Polyakov loop.\cite{RH} Results are displayed in fig. 1.
The Polyakov loop vanishes for fractional charge due to gauge invariance.

To study confinement we pose the question: 
What is the energy of a pair of external sources of charge $q$ and
$-q$ ? This in fact, is related to the Polyakov loop correlator and
can be computed in mass perturbation theory. 
Perturbation theory cannot be employed when 
$N\ge 2$ as physical quantities are not analytic in $m$ at $T=0.$
\cite{Coleman}
Nevertheless, the change in the energy due to the introduction of external 
sources, when $m_a=m\ll \mu $ to $O(d/L)$ ($d$ is the separation), can 
be found.\cite{RH} One finds that the string tension 
$\sigma = Nm \Big\{ \la \psibar\psi\ra_{\theta_\eff}
 - \la\psibar\psi\ra_\theta \Big\} ~~,~~
\theta_\eff = \theta-2\pi q/e ~.$ 
%\beeq
%\sigma = Nm \Big\{ \la \psibar\psi\ra_{\theta_\eff}
% - \la\psibar\psi\ra_\theta \Big\} ~~~,~~~
%\theta_\eff = \theta-\myfrac{2\pi q}{e} ~, 
%\label{string1}
%\eneq
%essentially, a generalization of 
%the one flavor case.

Removing the restriction of small masses in the one flavor case, we have 
computed the chiral condensate (fig.2). Polyakov loops have also been
computed.\cite{HR}

%\end{document}

\section*{Acknowledgments}
This work was supported in part by the U.S.\ Department of Energy
under contracts  DE-AC02-83ER-40105
(Y.H.),   DE-FG02-87ER-40328 (R.R.), and  DE-FG03-95ER-40906 (J.H.)



\section*{References}
\begin{thebibliography}{99}

\bibitem{Casher}A. Casher, J. Kogut and L. Susskind,
\Journal{\PRL}{31}{792}{1973};
\Journal{\PRD}{11}{3594}{1975}.

\bibitem{Sch}J. Schwinger,
\Journal{\PR}{128}{2425}{1962}.

\bibitem{CJS}
S. Coleman, R. Jackiw, and L. Susskind,  \Journal{\AP}{93}{267}{1975}.

\bibitem{Coleman} S. Coleman, \Journal{\AP}{101}{239}{1976}.

\bibitem{HNZ}  T.H. Hansson, H.B. Nielsen and I. Zahed,
        \Journal{\NPB}{451}{162}{1995}.

\bibitem{Gross} D.J.\ Gross, I.R.\ Klebanov, A.V.\ Matytsin, A.V.\ Smilga, 
       \Journal{\NPB}{461}{109}{1996}.

\bibitem{Grignani} G.\ Grignani, G.\ Semenoff, P.\ Sodano, O.\ Tirkkonen, 
      {\tt hep-th/9511110}.

\bibitem{RH}R. Rodriguez and Y. Hosotani, \Journal{\PLB}{375}{273}{1996}.

\bibitem{HR}Y. Hosotani and R. Rodriguez, {UMN-TH-1504/96}. 

\end{thebibliography}

\end{document}


