Deconfining Phase Transition as a Matrix Model of Renormalized Polyakov Loops

TL;DR

This work develops a nonperturbative framework for renormalizing Polyakov loops in SU(N) gauge theories at finite temperature, extracting representation- and temperature-dependent renormalization constants from lattice data. It finds that higher-representation condensates factorize as powers of the fundamental loop in the large-N limit, with finite-N corrections, and shows that renormalized SU(3) triplet loops can be described by a triplet-dominated matrix model whose mean-field behavior captures the deconfining transition, reminiscent of the Gross–Witten point. Lattice results for SU(3) reveal a clear hierarchy in the renormalized loops above Td and small, spike-like deviations in the difference loops near Td, supporting approximate large-N factorization and guiding the construction of effective theories. The paper connects renormalization, large-N factorization, and matrix-model approaches to provide a cohesive picture of deconfinement and offers a path to extend these ideas to more colors and dynamical quarks.

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.

Figures (12)

• Figure 1: A Polyakov loop with two cusps, at $\tau = 0$ and $\tau = 1/(2T)$. The dotted lines denote $\tau=0$ and $1/T$.
• Figure 2: The bare triplet Polyakov loop as a function of temperature.
• Figure 3: The bare sextet Polyakov loop as a function of temperature.
• Figure 4: The bare adjoint Polyakov loop as a function of temperature.
• Figure 5: The bare decuplet Polyakov loop as a function of temperature.