Bodeker's Effective Theory: From Langevin Dynamics to Dyson-Schwinger Equations

TL;DR

This paper develops an analytic framework using Dyson-Schwinger equations and stochastic quantisation to study Bodeker's effective theory for soft gauge field dynamics at high temperature, complementing previous lattice approaches.

Contribution

It introduces a field theoretic path integral formulation of Bodeker's Langevin equation, incorporating gauge ghosts and deriving Dyson-Schwinger equations for the theory.

Findings

Formulation of Bodeker's theory as a BRST symmetric path integral.

Derivation of Dyson-Schwinger equations for soft gauge fields.

Establishment of Ward identities related to gauge invariance and stochastic origin.

Abstract

The dynamics of weakly coupled, non-abelian gauge fields at high temperature is non-perturbative if the characteristic momentum scale is of order |k|~ g^2 T. Such a situation is typical for the processes of electroweak baryon number violation in the early Universe. Bodeker has derived an effective theory that describes the dynamics of the soft field modes by means of a Langevin equation. This effective theory has been used for lattice calculations so far. In this work we provide a complementary, more analytic approach based on Dyson-Schwinger equations. Using methods known from stochastic quantisation, we recast Bodeker's Langevin equation in the form of a field theoretic path integral. We introduce gauge ghosts in order to help control possible gauge artefacts that might appear after truncation, and which leads to a BRST symmetric formulation and to corresponding Ward identities. A second set of Ward identities, reflecting the origin of the theory in a stochastic differential equation, is also obtained. Finally Dyson-Schwinger equations are derived.