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Gaussian limits of lattice Higgs models with complete symmetry breaking

Frederick Rajasekaran, Oren Yakir, Yanxin Zhou

Abstract

Given any compact connected matrix Lie group $G$ and any lattice dimension $d\ge 2$, we construct a massive Gaussian scaling limit for the $G$-valued lattice Yang-Mills-Higgs theory in the "complete breakdown of symmetry" regime. This limit arises as the lattice spacing tends to zero and the (inverse) gauge coupling constant tends to infinity sufficiently fast, causing the theory to "abelianize" and yield a Gaussian limit. This complements a recent work by Chatterjee (arXiv:2401.10507), which obtained a similar scaling limit in the special case $G= SU(2)$.

Gaussian limits of lattice Higgs models with complete symmetry breaking

Abstract

Given any compact connected matrix Lie group and any lattice dimension , we construct a massive Gaussian scaling limit for the -valued lattice Yang-Mills-Higgs theory in the "complete breakdown of symmetry" regime. This limit arises as the lattice spacing tends to zero and the (inverse) gauge coupling constant tends to infinity sufficiently fast, causing the theory to "abelianize" and yield a Gaussian limit. This complements a recent work by Chatterjee (arXiv:2401.10507), which obtained a similar scaling limit in the special case .

Paper Structure

This paper contains 21 sections, 11 theorems, 154 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The lattice Yang-Mills-Higgs model
  3. Logarithmic coordinates
  4. The Proca field
  5. Main result
  6. Complete breakdown of symmetry
  7. Related works
  8. Breakdown of the proof
  9. Gibbs measures on the lattice
  10. Discrete to continuum
  11. Proca approximation on the lattice
  12. Some preparations
  13. An application of the Baker-Campbell-Hausdorff formula
  14. Density comparison on $\mathfrak{g}$
  15. Concluding the proof of Proposition \ref{['prop:YMH_is_close_to_Proca']}
  16. ...and 6 more sections

Key Result

Theorem 1

For $m,\beta,{\varepsilon}>0$ let $\nu_{\beta,{\varepsilon} m}$ be an infinite volume limit of eq:def_of_YMH_measure_on_torus with inverse gauge coupling $\beta>0$ and mass ${\varepsilon} m$. Assume that $\beta\to \infty$ and ${\varepsilon} \to 0$ simultaneously such that for some $C_{d,n}>0$. Then the random generalized $\mathfrak{g}$-valued 1-form $Z^{{\varepsilon}}$ converges in distribution t

Figures (1)

  • Figure 1: A plaquette bounded by the positively oriented edges $e_1,e_2,e_3,e_4$.

Theorems & Definitions (40)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Remark 3
  • Proposition 4
  • Proposition 5
  • Lemma 6
  • ...and 30 more