Hot B violation, color conductivity, and log(1/alpha) effects

TL;DR

This paper provides a transparent, physically intuitive derivation of Bödeker's logarithmically enhanced rate for non-perturbative processes in hot non-Abelian plasmas by linking the dynamics to color conductivity. Using a Boltzmann-Waldmann-Snider framework, it identifies the adjoint color channel and shows that the leading-log behavior originates from semi-hard momentum transfers and collisional damping, yielding σ ∼ m_pl^2/γ_g with γ_g ∼ α C_A T ln(1/g). Consequently, the slow time scale is t ∼ 1/(g^4 T ln(1/g)) and the non-perturbative rate Γ ∼ α^5 T^4 ln(1/α). The work also clarifies the ultraviolet insensitivity of Bödeker's effective theory, supporting its use in lattice simulations and resolving historical ambiguities about color conductivity.

Abstract

Bodeker 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) are logarithmically enhanced over previous estimates and take place at a rate per unit volume of order $α^5 T^4 \ln(1/α)$ for small coupling. We give a simple physical interpretation of Bodeker's qualitative and quantitative results in terms of Lenz's Law -- the fact that conducting media resist changes in the magnetic field -- and earlier authors' calculations of the color conductivity of such plasmas. In the process, we resolve some confusions in the literature about the value of the color conductivity and present an independent calculation. We also discuss the issue of whether the classical effective theory proposed by Bodeker has a good continuum limit.

Figures (4)

• Figure 1: The dominant scattering process: $t$-channel gauge boson exchange. The solid lines represent any sort of hard particles, including gauge bosons themselves. The labels $a,b,c,d$ show our convention for naming color indices of the various lines.
• Figure 2: The Feynman diagrams (assuming finite-temperature Feynman rules) that produce the conductivity due to hard excitations at leading-log order. Specifically, the ladder diagrams are for the self-energy of the soft fields, whose imaginary part is proportional to the conductivity at low frequency (it's $\omega\sigma$ in $A_0 = 0$ gauge). The external lines have soft momentum ($g^2 T$) and softer frequencies, the solid lines correspond to any type of colored particle with hard momentum ($T$), and the rungs have semi-hard momentum ($g^2 T\ln g^{-1} \ll q \ll g T$). The double lines indicate that the dominant one-loop contributions to the self-energies have been included in the propagators. The analog of the two-loop chain diagram of $\phi^3$ theory JeonJeonYaffe is not included because we only integrate out hard and semi-hard, but not soft, fields to obtain the effective theory of interest. Other diagrams relevant to $\phi^3$ theory ( e.g., non-pinching boxes and chain diagrams) have been dropped because they do not correspond to $t$-channel scattering and so should be sub-leading in the gauge-theory case.
• Figure 3: The Schwinger-Keldysh closed-time-path contour.
• Figure 4: The leading contribution to $\Sigma_{12}$.