Wilson line in finite-temperature gauge theories

Abstract

The phases of $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ gauge theories at finite temperature can be analyzed in terms of the Wilson line $L\left(\overrightarrow{\mathrm{x}}\right)=\mathrm{Tr}P\exp \left[ig \int 0^{\beta }{A}_0(\overrightarrow{\mathrm{x}},\tau )d\tau \right]$. The question of confinement is related to the spontaneous breakdown of a $Z\left(N\right)\mathrm{}$ symmetry: $〈L〉\ne 0$ corresponds to symmetry breaking and deconfinement, $〈L〉=0\mathrm{}$ implies symmetry restoration and confinement. A previous calculation of the one-loop effective potential for $L$ is discussed in $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ theories, both with and without fermions. A possible mechanism for the confining-deconfining transition in terms of the formation of "$Z\left(N\right)\mathrm{}$ bubbles" is discussed. For SU(2) theories, the $Z\left(2\right)\mathrm{}$ bubbles are simply finite-temperature instantons. However, $Z\left(N\right)\mathrm{}$ bubbles are not finite-temperature instantons for $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$. It is shown that a bag picture of hadronic structure similar to that of Callan, Dashen, and Gross may emerge from this picture.