From hard thermal loops to Langevin dynamics

TL;DR

This paper derives an effective Langevin equation for soft gauge field modes in hot non-Abelian gauge theories, enabling calculation of baryon number violation rates in the early universe.

Contribution

It extends the hard thermal loop effective theory by integrating out the scale gT, resulting in a Langevin equation that describes soft modes including noise and collision effects.

Findings

Derived the Langevin equation for soft gauge fields.

Calculated the noise and collision term in leading logarithmic approximation.

Provided a framework to compute baryon violation rates on the lattice.

Abstract

In hot non-Abelian gauge theories, processes characterized by the momentum scale $g^2 T$ (such as electroweak baryon number violation in the very early universe) are non-perturbative. An effective theory for the soft ($|\vec{p}|\sim g^2 T$) field modes is obtained by integrating out momenta larger than $g^2 T$. Starting from the hard thermal loop effective theory, which is the result of integrating out the scale $T$, it is shown how to integrate out the scale $gT$ in an expansion in the gauge coupling $g$. At leading order in $g$, one obtains Vlasov-Boltzmann equations for the soft field modes, which contain a Gaussian noise and a collision term. The 2-point function of the noise and the collision term are explicitly calculated in a leading logarithmic approximation. In this approximation the Boltzmann equation is solved. The resulting effective theory for the soft field modes is described by a Langevin equation. It determines the parametric form of the hot baryon number violation rate as $\Gamma = \kappa g^{10} \log(1/g) T^4$, and it allows for a calculation of $\kappa$ on the lattice.