Color Confinement and Random Matrices -- A random walk down group manifold toward Casimir scaling --

TL;DR

This paper explains the origin of linear confinement and Casimir scaling in confining gauge theories using Haar random Polyakov lines, linking random walk on group manifolds to confinement phenomena and lattice results.

Contribution

It demonstrates that Haar randomness of Polyakov lines leads to confinement and Casimir scaling, providing a microscopic explanation consistent with lattice simulations.

Findings

Haar randomness of Polyakov lines explains linear confinement.

Casimir scaling holds approximately at intermediate distances.

String breaking and loss of Casimir scaling occur at long distances.

Abstract

We explain the microscopic origin of linear confinement potential with the Casimir scaling in generic confining gauge theories. In the low-temperature regime of confining gauge theories such as QCD, Polyakov lines are slowly varying Haar random modulo exponentially small corrections with respect to the inverse temperature, as shown by one of the authors (M.~H.) and Watanabe. With exact Haar randomness, computation of the two-point correlator of Polyakov loops reduces to the problem of random walk on group manifold. Linear confinement potential with approximate Casimir scaling except at short distances follows naturally from slowly varying Haar randomness. With exponentially small corrections to Haar randomness, string breaking and loss of Casimir scaling at long distance follow. Hence we obtain the Casimir scaling which is only approximate and holds only at intermediate distance, which is precisely needed to explain the results of lattice simulations. For $(1+1)$-dimensional theories, there is a simplification that admits the Casimir scaling at short distances as well.