Do We Understand the Sphaleron Rate?

TL;DR

The paper argues that Bödeker's effective theory correctly captures the infrared dynamics governing the sphaleron rate $\Gamma$ in the hot phase and provides a purely analytic, physically intuitive model to estimate $\Gamma$. It derives a crude but quantitatively reasonable expression for $\Gamma$ by linking magnetic screening and diffusion of the Chern-Simons number, and finds good agreement with lattice results, while making testable predictions for the $N_c$-dependence in SU($N_c$) gauge theory. The work connects the diffusion of topological charge to infrared gauge-field dynamics, highlighting a scale $l_{\rm mag}$ that governs the diffusion process and offering a physical picture of baryon-number violation in the thermal environment. Overall, it presents a parametric understanding and an approximate numeric match to simulations, with suggestions for further cross-checks at larger $N_c$.

Abstract

I begin by answering a different question, ``Do we know the sphaleron rate?'' and conclude that we do. Then I discuss a crude but purely analytic picture which provides an estimate of the sphaleron rate within the context of Bodeker's effective theory. The estimate, which comes surprisingly close to the numerically determined sphaleron rate, gives a physical picture of baryon number violation in the hot phase, and provides a conjecture of the Nc dependence of the sphaleron rate in SU(Nc) gauge theory.

Figures (3)

• Figure 1: $\Gamma$ for pure lattice classical Yang-Mills theory against lattice spacing. It does not show rapid convergence to a nonzero limit at zero spacing. Data from fine_latt.
• Figure 2: Sphaleron rate in Bödeker's effective theory, two lattice implementations of HTL effective theory particlesWfields, and pure lattice theory interpreted as HTL effective theory (see Arnold_latt).
• Figure 3: Volume dependence of $\Gamma$ (pure lattice theory, $g^2 aT=1/2$) illustrates that $\Gamma$ turns on for a box about $2l_{\rm mag}$ across.