Longitudinal subtleties in diffusive Langevin equations for non-Abelian plasmas

TL;DR

The paper critically analyzes Bödeker’s diffusive effective theory for soft non-Abelian gauge fields in hot plasmas, focusing on how longitudinal electric fields behave and whether a transverse-only description can suffice for the transverse dynamics. It shows that, within leading-log accuracy, Bödeker’s theory correctly captures both longitudinal and transverse fluctuations, but a careful treatment of the zero mode of the collision operator and projection issues is required. The authors demonstrate that a truly transverse theory can be equivalent to the full theory only after accounting for noise discretization effects, which introduce a centrifugal drift that is absorbed by an effective potential term linked to the gauge orbit volume, V_eff = V − (1/2)T Tr ln(−D^2). This leads to the correct equilibrium distribution and clarifies the stochastic-field-theory structure of non-Abelian plasmas, with implications for simulations of topological transitions and baryon-number violation.

Abstract

Boedeker has recently argued that non-perturbative processes in very high temperature non-Abelian plasmas (such as electroweak baryon number violation in the very hot early Universe) can be quantitatively described, to leading logarithmic accuracy, by a simple diffusive effective theory. Boedeker's effective theory is intended to describe the long-distance transverse electric and magnetic fields which are responsible for non-perturbative dynamics. His effective theory, however, also contains long-wavelength longitudinal electric fields. We discuss several subtleties in the treatment of longitudinal dynamics which were not closely examined in Boedeker's original treatment. Somewhat to our surprise, we find that within its domain of validity Boedeker's effective theory does correctly describe both longitudinal and transverse fluctuations. We also show that, as far as the transverse dynamics of interest is concerned, Boedeker's effective theory could be replaced by a transverse-only theory that removes the longitudinal dynamics altogether. In the process, we discuss several interesting aspects of stochastic field theories.

Figures (1)

• Figure 1: Radial motion generated by (\ref{['eq:discreteth']}) depending on whether $\hat{\theta}$ is defined by the (a) beginning, (b) end, or (c) middle of a discrete time jump $t \to t+\epsilon$.