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Uniqueness of gauge covariant renormalisation of stochastic 3D Yang-Mills-Higgs

Ilya Chevyrev, Hao Shen

TL;DR

The paper establishes the intrinsic uniqueness of the gauge-covariant renormalisation for the 3D stochastic Yang-Mills-Higgs Langevin dynamics by showing that any two renormalisations yielding gauge-covariant limits must coincide as the mollification parameter ε tends to 0. The authors develop a rigorous short-time expansion framework for singular SPDEs, introduce a refined YM heat-flow state space to control leading singularities, and leverage regularised Wilson loops to detect potential non-covariance. A finite-term expansion with carefully chosen initial data and YM-heat regularisation scale is used to produce lower-bound distinctions that would contradict gauge covariance if the renormalisation differed, thereby proving intrinsic uniqueness. This work strengthens the connection between continuum gauge-covariant dynamics and potential lattice universality and provides tools to identify the continuum limit for related approximations.

Abstract

Local solutions to the 3D stochastic quantisation equations of Yang-Mills-Higgs were constructed in (arXiv:2201.03487), and it was shown that, in the limit of smooth mollifications, there exists a mass renormalisation of the Yang-Mills field such that the solution is gauge covariant. In this paper we prove uniqueness of the mass renormalisation that leads to gauge covariant solutions. This strengthens the main result of (arXiv:2201.03487), and is potentially important for the identification of the limit of other approximations, such as lattice dynamics. Our proof relies on systematic short-time expansions of singular stochastic PDEs and of regularised Wilson loops. We also strengthen the recently introduced state spaces to allow finer control on line integrals appearing in expansions of Wilson loops.

Uniqueness of gauge covariant renormalisation of stochastic 3D Yang-Mills-Higgs

TL;DR

The paper establishes the intrinsic uniqueness of the gauge-covariant renormalisation for the 3D stochastic Yang-Mills-Higgs Langevin dynamics by showing that any two renormalisations yielding gauge-covariant limits must coincide as the mollification parameter ε tends to 0. The authors develop a rigorous short-time expansion framework for singular SPDEs, introduce a refined YM heat-flow state space to control leading singularities, and leverage regularised Wilson loops to detect potential non-covariance. A finite-term expansion with carefully chosen initial data and YM-heat regularisation scale is used to produce lower-bound distinctions that would contradict gauge covariance if the renormalisation differed, thereby proving intrinsic uniqueness. This work strengthens the connection between continuum gauge-covariant dynamics and potential lattice universality and provides tools to identify the continuum limit for related approximations.

Abstract

Local solutions to the 3D stochastic quantisation equations of Yang-Mills-Higgs were constructed in (arXiv:2201.03487), and it was shown that, in the limit of smooth mollifications, there exists a mass renormalisation of the Yang-Mills field such that the solution is gauge covariant. In this paper we prove uniqueness of the mass renormalisation that leads to gauge covariant solutions. This strengthens the main result of (arXiv:2201.03487), and is potentially important for the identification of the limit of other approximations, such as lattice dynamics. Our proof relies on systematic short-time expansions of singular stochastic PDEs and of regularised Wilson loops. We also strengthen the recently introduced state spaces to allow finer control on line integrals appearing in expansions of Wilson loops.

Paper Structure

This paper contains 18 sections, 22 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Background and setup
  3. Main result
  4. Idea of proof
  5. Notation
  6. Short-time estimates for SPDEs
  7. Lifting to the space of modelled distributions
  8. Short time expansion
  9. Short-time estimates for the YM heat flow
  10. State space
  11. Perturbation of the YM heat flow
  12. Regularised Wilson loops
  13. Estimates on quadratic terms
  14. Deterministic estimates
  15. Stochastic estimates
  16. ...and 3 more sections

Key Result

Theorem 1.3

(CCHS_3D) Let $\chi$ be a non-anticipative mollifier. There exists a unique $\mathring{C}_{\mathsf{A}} \in L_{G}(\mathfrak{g})$, independent of mollifier $\chi$, with the following property. Let $C = \{C^\varepsilon\}_{\varepsilon\in(0,1)}$ be as in eq:def-C-family and eq:C_eps. Then for all $g(0) \ where $g$ is the solution to with initial condition $g(0)$, where $A$ is the $\mathfrak{g}^3$-comp

Theorems & Definitions (44)

  • Remark 1.1
  • Definition 1.2: Path space and solution
  • Theorem 1.3
  • Definition 1.4: YM heat flow
  • Definition 1.5: Wilson loop
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Proposition 1.9
  • Remark 1.10
  • ...and 34 more