# Uniqueness of Gauge Covariant Renormalisation of Stochastic 3D Yang–Mills–Higgs

## TL;DR

This paper proves that there is only one way to adjust the mass of a Yang–Mills field to ensure solutions remain gauge covariant in a 3D stochastic model.

## Contribution

The novelty is proving uniqueness of the mass renormalisation leading to gauge covariant solutions in 3D Yang–Mills–Higgs stochastic quantisation.

## Key findings

- Uniqueness of mass renormalisation ensuring gauge covariance in 3D Yang–Mills–Higgs solutions is established.
- Short-time expansions of singular stochastic PDEs and regularised Wilson loops are used in the proof.
- State spaces introduced by Cao are strengthened for better control of line integrals in Wilson loop expansions.

## Abstract

Local solutions to the 3D stochastic quantisation equations of Yang–Mills–Higgs were constructed in Chandra (Invent Math 237:541–696, 2024), 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 the uniqueness of the mass renormalisation that leads to gauge covariant solutions. This strengthens the main result of Chandra (Invent Math 237:541–696, 2024), 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 of Cao (Comm Part Diff Equ 48:209–251, 2023); Cao (Comm Math Phys 405:3, 2024); Chandra (Invent Math 237:541–696, 2024) to allow for finer control on line integrals appearing in expansions of Wilson loops.