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A scaling limit of $\mathrm{SU}(2)$ lattice Yang-Mills-Higgs theory

Sourav Chatterjee

TL;DR

This work constructs rigorous scaling limits for lattice Yang--Mills--Higgs theories in dimensions $d\ge 2$, proving that under a precise scaling of the gauge coupling $g$ and Higgs length $\alpha$, with $\alpha g$ tied to the lattice spacing $\varepsilon$, the gauge-fixed lattice fields converge to Euclidean Proca fields. In the Abelian $\mathrm{U}(1)$ case, the stereographic projections of link variables yield a single massive Gaussian field in the continuum limit with mass determined by $c^2$, while in the non-Abelian $\mathrm{SU}(2)$ case the limit is a triple of independent massive Gaussian fields, evidencing mass generation via the Higgs mechanism at the scaling limit. These are the first rigorous constructions of Gaussian scaling limits for non-Abelian lattice YM theories in $d>2$, and they provide a concrete step toward the elusive non-Gaussian continuum Yang--Mills scaling limit, leaving open the full non-Gaussian regime and extensions to other gauge groups and potentials. The results highlight the central role of the product $\alpha g$ in controlling the continuum limit and offer a quantitative framework for mass generation arising from the Higgs mechanism in lattice gauge theories. Further work is needed to understand slower decay regimes of $g$, general Higgs potentials, and the existence of non-Gaussian scaling limits.

Abstract

The construction of non-Abelian Euclidean Yang-Mills theories in dimension four, as scaling limits of lattice Yang-Mills theories or otherwise, is a central open question of mathematical physics. This paper takes the following small step towards this goal. In any dimension $d\ge 2$, we construct a scaling limit of $\mathrm{SU}(2)$ lattice Yang-Mills theory coupled to a Higgs field (under the degenerate potential) transforming in the fundamental representation of $\mathrm{SU}(2)$. After unitary gauge fixing and taking the lattice spacing $\varepsilon\to 0$, and simultaneously taking the gauge coupling constant $g\to 0$ and the Higgs length $α\to \infty$ in such a manner that $αg$ is always equal to $c\varepsilon$ for some fixed $c$ and $g= O(\varepsilon^{50d})$, a stereographic projection of the gauge field is shown to converge to a scale-invariant massive Gaussian field. This gives the first construction of a scaling limit of a non-Abelian lattice Yang-Mills theory in a dimension higher than two, as well as the first rigorous proof of mass generation by the Higgs mechanism in such a theory. Analogous results are proved for $\mathrm{U}(1)$ theory as well. The question of constructing a non-Gaussian scaling limit remains open.

A scaling limit of $\mathrm{SU}(2)$ lattice Yang-Mills-Higgs theory

TL;DR

This work constructs rigorous scaling limits for lattice Yang--Mills--Higgs theories in dimensions , proving that under a precise scaling of the gauge coupling and Higgs length , with tied to the lattice spacing , the gauge-fixed lattice fields converge to Euclidean Proca fields. In the Abelian case, the stereographic projections of link variables yield a single massive Gaussian field in the continuum limit with mass determined by , while in the non-Abelian case the limit is a triple of independent massive Gaussian fields, evidencing mass generation via the Higgs mechanism at the scaling limit. These are the first rigorous constructions of Gaussian scaling limits for non-Abelian lattice YM theories in , and they provide a concrete step toward the elusive non-Gaussian continuum Yang--Mills scaling limit, leaving open the full non-Gaussian regime and extensions to other gauge groups and potentials. The results highlight the central role of the product in controlling the continuum limit and offer a quantitative framework for mass generation arising from the Higgs mechanism in lattice gauge theories. Further work is needed to understand slower decay regimes of , general Higgs potentials, and the existence of non-Gaussian scaling limits.

Abstract

The construction of non-Abelian Euclidean Yang-Mills theories in dimension four, as scaling limits of lattice Yang-Mills theories or otherwise, is a central open question of mathematical physics. This paper takes the following small step towards this goal. In any dimension , we construct a scaling limit of lattice Yang-Mills theory coupled to a Higgs field (under the degenerate potential) transforming in the fundamental representation of . After unitary gauge fixing and taking the lattice spacing , and simultaneously taking the gauge coupling constant and the Higgs length in such a manner that is always equal to for some fixed and , a stereographic projection of the gauge field is shown to converge to a scale-invariant massive Gaussian field. This gives the first construction of a scaling limit of a non-Abelian lattice Yang-Mills theory in a dimension higher than two, as well as the first rigorous proof of mass generation by the Higgs mechanism in such a theory. Analogous results are proved for theory as well. The question of constructing a non-Gaussian scaling limit remains open.
Paper Structure (32 sections, 31 theorems, 214 equations, 3 figures)

This paper contains 32 sections, 31 theorems, 214 equations, 3 figures.

Table of Contents

  1. Lattice Yang--Mills theory with a Higgs field
  2. Definition
  3. Gauge transforms and gauge fixing
  4. Example: $\mathrm{U}(1)$ lattice Yang--Mills theory with a Higgs field
  5. Example: $\mathrm{SU}(2)$ lattice Yang--Mills theory with a Higgs field
  6. Random distributional $1$-forms
  7. Definition
  8. The Euclidean Proca field
  9. Scaling and translation of the Euclidean Proca field
  10. The mass of the Euclidean Proca field
  11. Main results
  12. A scaling limit of $\mathrm{U}(1)$ Yang--Mills--Higgs theory
  13. A scaling limit of $\mathrm{SU}(2)$ Yang--Mills--Higgs theory
  14. Comparison with prior work
  15. Sketch of the proof
  16. ...and 17 more sections

Key Result

Lemma 2.1

The operator $K_\lambda$ is well-defined on $\mathcal{S}(\mathbb{R}^d)$. Moreover, it is a bijection from $\mathcal{S}(\mathbb{R}^d)$ onto itself, with $K_\lambda^{-1} = -\Delta +\lambda I$, where $\Delta$ is the Laplacian operator and $I$ is the identity operator. Lastly, $K_\lambda$ commutes with

Figures (3)

  • Figure 1: A plaquette bounded by four directed edges joined end-to-end.
  • Figure 2: Stereographic projection $\sigma_1:S^1 \to \mathbb{R}$.
  • Figure 3: A plaquette bounded by four positively oriented edges.

Theorems & Definitions (56)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • proof
  • ...and 46 more