Langevin dynamics of lattice Yang-Mills-Higgs and applications

TL;DR

The paper analyzes Langevin dynamics on lattice Yang-Mills-Higgs models with both bounded and unbounded Higgs fields, proving exponential ergodicity, Poincaré and log-Sobolev inequalities, mass gap, and large-N factorization under small-coupling regimes. It introduces a disintegration strategy to handle unbounded Higgs fields by conditioning on the Yang–Mills field Q, establishing functional inequalities for μ_Q and its marginal ν, then transferring them to the full infinite-volume measure μ. The results extend prior pure YM dynamical analyses to YM-Higgs systems, providing uniform-in-N mass-gap and large-N factorization results for compact Higgs targets, and offering gauge-fixing improvements to strengthen mass-gap conditions. These findings contribute rigorous dynamical insights into ergodicity, spatial mixing, and continuum-limit implications for lattice YM-Higgs theories, with potential relevance to gauge theories and Higgs mechanisms in higher dimensions.

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 $β$ and a Higgs parameter $κ$. 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 $(β,κ)$ are small in the first case or $(β,κ/m)$ 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.

Figures (3)

• Figure 1: The edge variables $Q$, the vertex variables $\Phi$, the plaquette variable $Q_p=\prod_{i=1}^4Q_{e_i}$, and the discrete covariant derivative $Q_e \Phi_y - \Phi_x$ for $e=(x,y)$.
• Figure 2: Illustration of conditioning in Step 1.
• Figure 3: Illustration of $\Lambda_{i,k}$. $F_{i,k}$ depends on $\Phi$ outside $\Lambda_{i,k}$, and $f_{x_{i+1}}$ depends on $\Phi$ near $x_{i+1}$, which leads to a decay $e^{-c_Nk}$ in the correlation analysis

Theorems & Definitions (73)

• Theorem 1.1: Well-posedness and invariant measure
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Mass gap
• Remark 1.5
• Lemma 2.1
• proof
• Remark 2.2
• Remark 2.3
• Lemma 3.1