The Sphaleron Rate: Bodeker's Leading Log

TL;DR

The paper determines the leading-log coefficient for the high-temperature sphaleron rate Γ in the SU(2) sector, showing Γ ≃ κ' [log(m_D / g^2 T) + O(1)] (g^2 T^2 / m_D^2) α_W^5 T^4 with κ' = 10.8 ± 0.7, via Bödeker's UV-safe Langevin effective theory tied to hard thermal loop dynamics. It presents two complementary derivations—conductivity-based and HTL/Wilson-line—demonstrating how infrared magnetic-field diffusion controls CS number diffusion at scale k ∼ g^2 T, and derives an exponential decay for Wilson lines with a characteristic length λ^-1 ∝ log(m_D / g^2 T). The numerics show strong lattice-volume and lattice-spacing stability, yielding a robust leading-log coefficient, while systematic, nonlog corrections and Higgs-sector effects remain sizable and require careful handling. These results provide a solid benchmark for IR dynamics in hot gauge theories and inform extrapolations of lattice results to physical parameters, with implications for electroweak baryogenesis and beyond.

Abstract

Bodeker has recently shown that the high temperature sphaleron rate, which measures baryon number violation in the hot standard model, receives logarithmic corrections to its leading parametric behavior; Gamma = kappa' [log(m_D / g^2 T) + O(1)] (g^2 T^2 / m_D^2) α_W^5 T^4. After discussing the physical origin of these corrections, I compute the leading log coefficient numerically; kappa' = 10.8 pm 0.7. The log is fairly small relative to the O(1) ``correction;'' so nonlogarithmic contributions dominate at realistic values of the coupling.

Figures (4)

• Figure 1: "It's quite simple, really $\ldots \;$." A scorecard of the scales involved in the problem and the approximations which are valid in each. None of these scales are distinct if we do not take $g \ll 1$.
• Figure 2: (a) Emission of a gluon below the light cone; (b) Self-energy diagram which, cut, describes the emission; (c) The cut must split an HTL insertion, so the physical process is scattering with a small exchange momentum.
• Figure 3: A simple 2-D example of how a Wilson line (diagonal) on a lattice (dashed lines) is replaced by the sequence of links which stay closest to it (solid, jagged line).
• Figure 4: Small lattice spacing extrapolation of data in pure classical lattice Yang-Mills theory, taken from slavepaper. The data have been corrected to absorb the leading log dependence on $m_D^2 \propto 1/a$ determined here. They show a substantial linear correction to the predicted $\Gamma \propto a$ behavior. This is evidence of nonnegligible corrections to the parametric $m_D \gg g^2 T$ limit.