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 $Γ= κg^{10} \log(1/g) T^4$, and it allows for a calculation of $κ$ on the lattice.

Figures (2)

• Figure 1: The interactions which are relevant to the leading order effective theory for the soft gauge fields. The hard modes dynamically screen the soft and semi-hard fields. The relevant hard--semi-hard interaction is small angle scattering which exchanges color charge between hard particles (see Sec. \ref{['sec.logarithmic']}). Soft--semi-hard interactions in the full quantum field theory can be neglected. Therefore, the relevant interactions within the hard thermal loop effective theory are due to hard thermal loop vertices.
• Figure 2: The soft gauge fields correspond to extended field configurations (shaded) with a typical size $R$ of order $(g^2 T)^{-1}$. Since the soft fields are changing in time, they are associated with a long wavelength electric field. The hard modes act like almost free particles moving on light-like trajectories (thick lines). They absorb energy from the long wavelength electric field which leads to the damping of the soft dynamics. The semi-hard field modes (thin lines) are responsible for color changing small angle scattering of the hard particles. In the leading log approximation, the typical distance between these scattering events is small compared with $R$, so that the damping becomes effectively local.