Gauge field marginal of an Abelian Higgs model

TL;DR

This paper analyzes the gauge field marginal in an Abelian Higgs model on a 2D lattice, providing a loop expansion and demonstrating ultraviolet stability of moments in a fixed gauge.

Contribution

It introduces a generalized loop expansion for the gauge field marginal and proves ultraviolet stability, extending previous results to non-Abelian structures and arbitrary graphs.

Findings

Loop expansion of the Radon--Nikodym derivative for gauge field marginals.

Ultraviolet stability of moments of H{"o}lder--Besov norms.

Quantitative diamagnetic inequality derived from the loop expansion.

Abstract

We study the gauge field marginal of an Abelian Higgs model with Villain action defined on a 2D lattice in finite volume. Our first main result, which holds for gauge theories on arbitrary finite graphs and does not assume that the structure group is Abelian, is a loop expansion of the Radon--Nikodym derivative of the law of the gauge field marginal with respect to that of the pure gauge theory. This expansion is similar to the one of Seiler but holds in greater generality and uses a different graph theoretic approach. Furthermore, we show ultraviolet stability for the gauge field marginal of the model in a fixed gauge. More specifically, we show that moments of the H{\"o}lder--Besov-type norms introduced in arXiv:1808.09196 are bounded uniformly in the lattice spacing. This latter result relies on a quantitative diamagnetic inequality that in turn follows from the loop expansion and elementary properties of Gaussian random variables.