Langevin dynamics of lattice Yang-Mills-Higgs and applications

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.

💡 Research Summary

This paper develops a comprehensive stochastic quantization framework for several lattice formulations of the Yang–Mills–Higgs (YMH) model and establishes a suite of rigorous results concerning its long‑time behavior, functional inequalities, and spatial correlation decay. The authors consider a finite hypercubic lattice Λ⊂ℤᵈ and a gauge group G=SO(N) (N>2). The lattice gauge field Q assigns a group element to each oriented edge, while the Higgs field Φ lives on vertices and takes values in a target manifold M chosen from three prototypical families: the Euclidean space ℝⁿ (unbounded), the unit sphere Sⁿ⁻¹, or the compact Lie group G itself. The discrete action (1.6) contains three coupling parameters: the inverse Yang–Mills coupling β, the Higgs–gauge interaction strength κ, and, when M=ℝⁿ, a Higgs mass term m>0. The associated Gibbs measure μ_Λ (1.5) is defined via the Haar measure on G and a G‑invariant reference measure on M.

The central dynamical object is the Langevin stochastic differential equation (SDE) d(Q,Φ)=∇S dt+√2 dB (1.11), where ∇ denotes the gradient with respect to the product Riemannian structure on the configuration space and B is a cylindrical Brownian motion. The authors first prove global well‑posedness of this SDE for all three choices of M (Theorem 1.1). Moreover, μ_Λ is shown to be invariant, and the family {μ_Λ}_Λ is tight as Λ exhausts ℤᵈ, yielding invariant measures for the infinite‑volume dynamics.

A major technical challenge arises when M=ℝⁿ because the Higgs field is unbounded, precluding a direct Bakry–Émery curvature lower bound that would otherwise give log‑Sobolev (LSI) and Poincaré inequalities. To overcome this, the authors disintegrate the Gibbs measure as μ_Λ(dQ,dΦ)=ν(dQ) μ_Q(dΦ) (1.12). For each fixed gauge configuration Q, the conditional Higgs measure μ_Q is shown to satisfy LSI and a Poincaré inequality with constants depending only on the mass m (Lemma 4.3). This is achieved by establishing uniform moment bounds for Φ (Lemma 4.2) via the dynamics, which control the growth of the potential and allow the use of a modified Bakry–Émery argument.

The next step is to lift these conditional inequalities to the full joint measure. The authors prove that the marginal law ν on the gauge field also satisfies a Poincaré inequality, but this does not follow from curvature arguments because ν involves averaging over the Higgs field. Instead, they exploit the moment bounds and the exponential ergodicity of the dynamics to control correlations between distant regions of the lattice (see diagram (1.14)). Combining the conditional Poincaré inequality with a delicate analysis of correlation functions yields a uniform (lattice‑size independent) Poincaré constant for μ_Λ (Theorem 4.7, Corollary 4.8). Consequently, the infinite‑volume dynamics is exponentially ergodic in L²(μ).

For compact target spaces M=Sⁿ⁻¹ or G, the underlying Riemannian manifolds have positive Ricci curvature, and the Bakry–Émery condition can be applied directly. The authors verify that the curvature term dominates the Hessian of the action in the strong‑coupling regime (small |β|, κ), leading to LSI and Poincaré inequalities for the full measure without the need for disintegration.

Having established functional inequalities, the paper turns to spatial mixing. Using the standard relationship between temporal mixing (exponential ergodicity) and spatial correlation decay, the authors prove a mass gap: for any pair of local observables f and g supported at distance |x−y|, the covariance under μ satisfies |Cov_μ(f,g)| ≤ C e^{-c|x−y|}. The proof follows the strategy of writing Cov(f,g)=Cov(P_t f,P_t g)+Comm, choosing t≈|x−y|, and showing that both terms decay exponentially. The commutator term requires a careful decomposition of the lattice into boxes and the use of the conditional mass gap for μ_Q (Lemma 4.4), which holds even without any restriction on β or κ when m>0.

Finally, the authors investigate the large‑N limit for compact target spaces. By rescaling observables appropriately, they demonstrate factorization of correlations as N→∞, extending earlier results for pure Yang–Mills (SZZ22) to the coupled YMH setting. This indicates that the presence of the Higgs field does not destroy the emergent independence of distant Wilson loops in the large‑N regime.

In summary, the paper makes three principal contributions: (i) a novel method to obtain functional inequalities for lattice gauge theories with unbounded Higgs fields via conditional measures and moment bounds; (ii) a rigorous bridge from temporal ergodicity of the Langevin dynamics to a spatial mass gap for the full YMH model; and (iii) an extension of large‑N factorization results to models with coupled Higgs sectors. These results deepen the mathematical understanding of non‑abelian gauge theories with scalar matter and provide powerful tools for future work on lattice field theory, stochastic quantization, and rigorous quantum field theory.