Deconfining phase transition as a matrix model of renormalized Polyakov loops

Abstract

We discuss how to extract renormalized loops from bare Polyakov loops in $\mathrm{SU}(N)$ lattice gauge theories at nonzero temperature. Single loops in an irreducible representation are multiplicatively renormalized, without mixing, through mass renormalization. The values of renormalized loops in the four lowest representations of $\mathrm{SU}(3)$ 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 powers of fundamental and antifundamental loops. The corrections to the large N relations at three colors are greatest for the sextet loop, $\ensuremath{\sim}1/N,$ and are found to be $<~25%.$ The values of the renormalized triplet loop can be described by a matrix model, with an effective action dominated by the triplet loop: the deconfining phase transition for $N=3$ is close to the Gross-Witten point at $N=\ensuremath{\infty}.$