Confinement, Screening and the Center on S^3 x S^1

TL;DR

This paper calculates the one-loop effective potential for the Polyakov loop on S^3 x S^1 in various gauge theories, revealing phase structures and symmetry behaviors consistent with lattice results and elucidating confinement and deconfinement phenomena.

Contribution

It provides a general computation of the Polyakov loop potential for arbitrary gauge groups and matter content on S^3 x S^1, and applies it to analyze phase transitions and symmetry breaking.

Findings

Polyakov loop is zero at zero temperature due to kinematics.

At high temperatures, the theory transitions to a deconfined phase with a non-zero Polyakov loop.

The phase structure aligns with lattice calculations and depends on center symmetry and matter content.

Abstract

We compute the one-loop effective potential for the Polyakov loop on $S^3 times S^1$ for an asymptotically free gauge theory of arbitrary group $G$ and a generic matter content. We apply this result to study the phase structures of $G_2$, SO(N) and $Sp(2N)$ gauge theories which turn out to be in qualitative agreement with the results of lattice calculations. On $S^3 \times S^1$, at zero temperature, the Polyakov loop is zero for kinematical reasons. For small but non-zero temperature, the Polyakov loop is still zero if the gauge theory has an unbroken center, while it acquires a small vacuum expectation value for gauge theories whose center is trivial or explicitly broken by the presence of dynamical matter fields. At high temperatures, the saddle point structure of the effective potential is different from the low temperature case suggesting that the theory is in the deconfined phase. At finite $N$, the Polyakov loop is non-zero in the high temperature phase only in theories with no unbroken center symmetry, consistent with screening of the external charge introduced by the Polyakov loop and Gauss law on a compact space.