Longitudinal subtleties in diffusive Langevin equations for non-Abelian plasmas

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.