Effective potential for the order parameter of gauge theories at finite temperature

Abstract

$\mathrm{SU}(n)$ gauge theories at finite temperature $T={\ensuremath{\beta}}^{\ensuremath{-}1}$ are analyzed in terms of the spontaneous breakdown of a $Z(n)$ symmetry corresponding to the order parameter $L(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})=(\frac{1}{n})\mathrm{Tr}P\mathrm{exp}[ig\ensuremath{\int}{0}^{\ensuremath{\beta}}{A}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},t)dt]$. An "effective potential" for $L$ is evaluated in the one-loop approximation for both the continuum and the lattice gauge theories. It is shown that the $Z(n)$ symmetry is broken, so that the continuum theory does not confine for high temperatures. Similarly the lattice theory does not confine for sufficiently weak coupling if the number of time sites ${N}_{t}$ is finite. It is argued that as ${N}_{t}\ensuremath{\rightarrow}\ensuremath{\infty}$ the $Z(n)$ symmetry is restored and the theory will confine for all values of the coupling.