Fluctuations Destroying Long-Range Order in SU(2) Yang-Mills Theory

TL;DR

This paper investigates how fluctuations in a heat bath affect the long-range order in SU(2) Yang-Mills theory, finding that fluctuations destroy order in the non-linear sigma model, unlike in U(1) gauge theory.

Contribution

It introduces a Langevin equation approach with Gaussian fluctuations to analyze the impact on long-range order in SU(2) Yang-Mills theory.

Findings

Fluctuations destroy long-range order in SU(2) Yang-Mills theory.

The phase transition persists in U(1) gauge theory despite fluctuations.

Abstract

We study lattice SU(2) Yang-Mills theory with dimension $d\ge 4$. The model can be expressed as a $(d-1)$-dimensional O(4) non-linear $\sigma$-model in a $d$-dimensional heat bath. As is well known, the non-linear $\sigma$-model alone shows a phase transition. If the quark confinement is a consequence of absence of a phase transition for the Yang-Mills theory, then the fluctuations of the heat bath must destroy the long-range order of the non-linear $\sigma$-model. In order to clarify whether this is true, we replace the fluctuations of the heat bath with Gaussian random variables, and obtain a Langevin equation which yields the effective action of the non-linear $\sigma$-model through analyzing the Fokker-Planck equation. It turns out that the fluctuations indeed destroy the long-range order of the non-linear $\sigma$-model within a mean field approximation estimating a critical point, whereas for the corresponding U(1) gauge theory, the phase transition to the massless phase remains against the fluctuations.