Hot Gauge Theories and $Z_{N}$ Phases

TL;DR

This paper examines the behavior of $Z_{N}$ symmetry in high-temperature gauge theories, highlighting issues with metastable bubbles and demonstrating, via a toy model, that $Z_{N}$ symmetry may remain unbroken at high temperatures.

Contribution

It provides a detailed analysis of $Z_{N}$ symmetry in gauge theories and shows that certain metastable bubbles are unphysical, supported by a toy model study.

Findings

Metastable $Z_{N}$ bubbles have negative thermodynamic properties.

Such bubbles are unphysical in Minkowski space.

In the toy model, $Z_{N}$ symmetry remains unbroken at high temperature.

Abstract

In this paper the several aspects of the $Z_{N}$ symmetry in gauge theories at high temperatures are discussed. The metastable $Z_{N}$ bubbles in the $SU(N)$ gauge theories with fermions may have, generically, unacceptable thermodynamic behavior. Their free energy $F \propto T^4$ with a positive proportionality constant. This leads not only to negative pressure but also to negative specific heat and, more seriously, to negative entropy. We argue that although such domains are important in the Euclidean theory, they cannot be interpreted as physical domains in Minkowski space. The related problem is connected with the analysis of the high-temperature limit of the confining phase. Using the two-dimensional QCD with adjoint fermions as a toy model we shall demonstrate that in the light fermion limit in this theory there is no breaking of the $Z_{N}$ symmetry in the high-temperature limit and thus there are no $Z_{N}$ bubbles.