From hard thermal loops to Langevin dynamics

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.

42 pages, 2 figures, uses elsart.sty; explanatory paragraph added to the introduction, 4 references added, a few quotations added; to appear in Nucl. Phys. B

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