Polyakov Loops, Z(N) Symmetry, and Sine-Law Scaling

TL;DR

This paper develops an effective action for Polyakov loops based on eigenvalues, demonstrating how sine-law scaling naturally arises from nearest-neighbor interactions and analyzing the implications for confinement and string tensions.

Contribution

It introduces a novel effective action framework using Polyakov loop eigenvalues, connecting Z(N) symmetry, sine-law scaling, and confinement properties.

Findings

Sine-law scaling emerges from nearest-neighbor eigenvalue interactions.

The effective action predicts non-zero string tensions for all non-trivial N-ality representations.

The model captures the transition from confined to deconfined phases with a smooth connection to perturbation theory.

Abstract

We construct an effective action for Polyakov loops using the eigenvalues of the Polyakov loops as the fundamental variables. We assume Z(N) symmetry in the confined phase, a finite difference in energy densities between the confined and deconfined phases as $T\to 0$, and a smooth connection to perturbation theory for large $T$. The low-temperature phase consists of $N-1$ independent fields fluctuating around an explicitly Z(N) symmetric background. In the low-temperature phase, the effective action yields non-zero string tensions for all representations with non-trivial $N$-ality. Mixing occurs naturally between representations of the same $N$-ality. Sine-law scaling emerges as a special case, associated with nearest-neighbor interactions between Polyakov loop eigenvalues.