Deconfining Phase Transition as a Matrix Model of Renormalized Polyakov Loops

TL;DR

This paper develops a method to extract and analyze renormalized Polyakov loops in SU(N) gauge theories at finite temperature, demonstrating their relation to matrix models and large N behavior.

Contribution

It introduces a multiplicative renormalization procedure for Polyakov loops in various representations and connects the results to an SU(3) matrix model and large N expansion insights.

Findings

Renormalized loops in lowest representations measured numerically.

Condensates for higher representations relate to fundamental loops.

Deconfining transition resembles large N matrix model behavior.

Abstract

We discuss how to extract renormalized from bare Polyakov loops in SU(N) lattice gauge theories at nonzero temperature in four spacetime dimensions. Single loops in an irreducible representation are multiplicatively renormalized without mixing, through a renormalization constant which depends upon both representation and temperature. The values of renormalized loops in the four lowest representations of SU(3) were measured numerically on small, coarse lattices. We find that in magnitude, condensates for the sextet and octet loops are approximately the square of the triplet loop. This agrees with a large $N$ expansion, where factorization implies that the expectation values of loops in adjoint and higher representations are just powers of fundamental and anti-fundamental loops. For three colors, numerically the corrections to the large $N$ relations are greatest for the sextet loop, $\leq 25%$; these represent corrections of $\sim 1/N$ for N=3. The values of the renormalized triplet loop can be described by an SU(3) matrix model, with an effective action dominated by the triplet loop. In several ways, the deconfining phase transition for N=3 appears to be like that in the $N=\infty$ matrix model of Gross and Witten.