Wilson line in finite-temperature gauge theories

Abstract

The phases of $\mathrm{SU}(N)$ gauge theories at finite temperature can be analyzed in terms of the Wilson line $L(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})=\mathrm{Tr}P\mathrm{exp}[ig \ensuremath{\int}{0}^{\ensuremath{\beta}}{A}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},\ensuremath{\tau})d\ensuremath{\tau}]$. The question of confinement is related to the spontaneous breakdown of a $Z(N)$ symmetry: $〈L〉\ensuremath{\ne}0$ corresponds to symmetry breaking and deconfinement, $〈L〉=0$ implies symmetry restoration and confinement. A previous calculation of the one-loop effective potential for $L$ is discussed in $\mathrm{SU}(N)$ theories, both with and without fermions. A possible mechanism for the confining-deconfining transition in terms of the formation of "$Z(N)$ bubbles" is discussed. For SU(2) theories, the $Z(2)$ bubbles are simply finite-temperature instantons. However, $Z(N)$ bubbles are not finite-temperature instantons for $\mathrm{SU}(N)$. It is shown that a bag picture of hadronic structure similar to that of Callan, Dashen, and Gross may emerge from this picture.