# Chiral Symmetry Restoration and $Z_N$ Symmetry

## TL;DR

The paper shows that chiral symmetry restoration in quenched finite temperature QCD depends on the $Z_3$ phase of the Polyakov loop, linking deconfinement and chiral symmetry in a model incorporating the Polyakov loop effects.

## Contribution

It introduces a model coupling chiral symmetry breaking to the Polyakov loop, explaining the dependence of chiral restoration on $Z_3$ phases and matching recent simulation results.

## Key findings

- Chiral symmetry restoration depends on the $Z_3$ phase of the Polyakov loop.
- In certain $Z_3$ phases, chiral symmetry is not restored at deconfinement.
- The mechanism likely applies to full QCD, affecting metastable phase lifetimes.

## Abstract

We demonstrate that chiral symmetry restoration in quenched finite temperature QCD depends crucially on the $Z_3$ phase of the Polyakov loop ${\cal P}$. This dependence is a general consequence of the coupling of the chiral order parameter to the Polyakov loop. We construct a model for chiral symmetry breaking and restoration which includes the effect of a nontrivial Polyakov loop by calculating the effective potential for the chiral condensate of a Nambu-Jona-Lasinio model in a uniform temperature dependent $A_0$ gauge field background. Above the deconfinement temperature there are three possible phases corresponding to the $Z_3$ symmetric phases of the Polyakov loop in the pure gauge theory. In the phase in which ${\rm tr_c}({\cal P})$ is real and positive the first order deconfining transition induces chiral symmetry restoration in agreement with simulation results. In the two phases where $Re[{\rm tr_c}({\cal P})] < 0$ the sign of the leading finite temperature correction to the effective potential is reversed from the normal phase, and chiral symmetry is not restored at the deconfinement transition; this agrees with the recent simulation studies of Chandrasekharan and Christ. In the case of $SU(N)$ a rich set of possibilites emerges. The generality of the mechanism makes it likely to occur in full QCD as well; this will increase the lifetimes of metastable $Z_3$ phases.