A PNJL Model for Adjoint Fermions with Periodic Boundary Conditions

TL;DR

This paper develops a PNJL model to study how adjoint fermions with periodic boundary conditions influence confinement and chiral symmetry breaking in gauge theories on small compactified spaces, aligning with recent lattice results.

Contribution

It introduces a combined perturbative and chiral symmetry breaking PNJL model for adjoint QCD on $R^{3} imes S^{1}$, revealing a connected phase structure at small and large $L$.

Findings

The model reproduces lattice results showing no direct link between small and large $L$ confined phases.

Chiral symmetry breaking effects are crucial for the phase structure at small $L$.

Small-$L$ and large-$L$ confined phases are connected through an extended field theory with four-fermion interactions.

Abstract

Recent work on QCD-like theories has shown that the addition of adjoint fermions obeying periodic boundary conditions to gauge theories on $R^{3}\times S^{1}$ can lead to a restoration of center symmetry and confinement for sufficiently small circumference $L$ of $S^{1}$. At small $L$, perturbation theory may be used reliably to compute the effective potential for the Polyakov loop $P$ in the compact direction. Periodic adjoint fermions act in opposition to the gauge fields, which by themselves would lead to a deconfined phase at small $L$. In order for the fermionic effects to dominate gauge field effects in the effective potential, the fermion mass must be sufficiently small. This indicates that chiral symmetry breaking effects are potentially important. We develop a Polyakov-Nambu-Jona Lasinio (PNJL) model which combines the known perturbative behavior of adjoint QCD models at small $L$ with chiral symmetry breaking effects to produce an effective potential for the Polyakov loop $P$ and the chiral order parameter $\bar{\psi}\psi$. A rich phase structure emerges from the effective potential. Our results are consistent with the recent lattice simulations of Cossu and D'Elia, which found no evidence for a direct connection between the small-$L$ and large-$L$ confining regions. Nevertheless, the two confined regions are connected indirectly if an extended field theory model with an irrelevant four-fermion interaction is considered. Thus the small-$L$ and large-$L$ regions are part of a single confined phase.