Loop inequalities and confinement

TL;DR

This paper explores correlation inequalities derived from loop equations in lattice gauge theory and spin systems, providing a simple method to analyze long-range correlations and phase transitions, with applications to confinement in 4D gauge theories.

Contribution

It introduces a technique based on loop inequalities that simplifies the analysis of correlation decay and confinement, applicable to both spin models and lattice gauge theories.

Findings

Demonstrates exponential decay of correlations in 2D O(N) models for all β

Provides a framework to analyze area law and confinement in 4D lattice gauge theories

Reproduces known results with less effort using correlation inequalities

Abstract

We consider correlation inequalities that follow from the well-known loop equations of LGT, and their analogues in spin systems. They provide a way of bounding long range by short or intermediate range correlations. In several cases the method easily reproduces results that otherwise require considerable effort to obtain. In particular, in the case of the 2-dimensional O(N) spin model, where large N analytical results are available, the absence of a phase transition and the exponential decay of correlations for all $\beta$ is easily demonstrated. We report on the possible application of this technique to the analogous 4-dimensional problem of area law for the Wilson loop in LGT at large $\beta$.