Langevin dynamics of lattice Yang-Mills-Higgs and applications

TL;DR

This paper studies the Langevin dynamics of lattice Yang-Mills-Higgs models, proving ergodicity, decay of correlations, and mass gap properties, extending previous pure Yang-Mills results to include Higgs fields with both bounded and unbounded target spaces.

Contribution

It establishes exponential ergodicity and correlation decay for lattice Yang-Mills-Higgs models, including cases with unbounded Higgs fields, using new analytical techniques.

Findings

Proved exponential ergodicity of the dynamics.

Established decay of correlations and mass gap.

Extended results to models with unbounded Higgs fields.

Abstract

In this paper, we investigate the Langevin dynamics of various lattice formulations of the Yang--Mills--Higgs model, with an inverse Yang--Mills coupling $\beta$ and a Higgs parameter $\kappa$. The Higgs component is either a bounded field taking values in a compact target space, or an unbounded field taking values in a vector space in which case the model also has a Higgs mass parameter $m$. We study the regime where $(\beta,\kappa)$ are small in the first case or $(\beta,\kappa/m)$ are small in the second case. We prove the exponential ergodicity of the dynamics on the whole lattice via functional inequalities. We establish exponential decay of correlations for a broad class of observables, namely, the infinite volume measure exhibits a strictly positive mass gap. Moreover, when the target space of the Higgs field is compact, appropriately rescaled observables exhibit factorized correlations in the large $N$ limit. These extend the earlier results \cite{SZZ22} on pure lattice Yang--Mills to the case with a coupled Higgs field. Unlike pure lattice Yang--Mills where the field is always bounded, in the case where the coupled Higgs component is unbounded, the control of its behavior is much harder and requires new techniques. Our approach involves a disintegration argument and a delicate analysis of correlations to effectively control the unbounded Higgs component.