Expanded regimes of area law for lattice Yang-Mills theories

TL;DR

This paper broadens the parameter regimes where the area law is proven for pure U(N) lattice Yang-Mills theories, especially at large N, by developing a novel approximation method for the master loop equation.

Contribution

It introduces a truncated model approach to handle the merger term in the master loop equation, extending the proven regimes of the area law for lattice Yang-Mills theories.

Findings

Extended the proven parameter regimes for the area law.

Developed a truncated model that approximates the original model.

Improved upon classical results from 1978.

Abstract

We extend the parameter regimes for which area law is proven for pure $\mathrm{U}(N)$ lattice Yang-Mills theories, in particular when $N$ is large. This improves on a classical result of Osterwalder-Seiler from 1978. To do so, we view the master loop equation as a linear inhomogeneous equation for Wilson string expectations, and then prove an a priori bound for solutions to the equation. The main novelty is in how we deal with the merger term in the master loop equation. This is done by introducing a truncated model for which the merger term is unproblematic, and then showing that the truncated model well approximates the original model.