The classical sphaleron transition rate exists and is equal to $1.1(α_w T)^4$

TL;DR

The paper shows that at high temperatures the diffusion of Chern–Simons number in the SU(2) electroweak sector can be computed from classical lattice gauge theory, yielding a real-time rate Γ = κ (α_w T)^4 with κ = 1.09 ± 0.04. By sampling a constrained thermal ensemble and tracking ⟨B^2(t)⟩_T, the authors extract Γ from the linear growth regime, and demonstrate that Γ is UV-insensitive and governed by long-wavelength magnetic physics. The results support using classical simulations to approximate the high-temperature quantum regime, with dimensional-reduction reasoning suggesting α_w can be effectively replaced by α_w(T). This provides a nonperturbative handle on baryon-number violating processes in the early universe and connects lattice results to the magnetic-scale dynamics of the hot standard model.

Abstract

Results of a large scale numerical simulation show that the high temperature Chern-Simons number diffusion rate in the electroweak theory has a classical limit $Γ= κ(α_w T)^4$, where $κ= 1.09\pm 0.04$ and $α_w$ is the weak fine structure constant.

Figures (4)

• Figure 1: Diffusion of $N_{\rm cs}$: mean square of $B(t)$ as a function of the lag $t$ for $\beta=12$, $L_L=32$.
• Figure 2: An enhanced region of $0\leq t\leq 25$ of Figure 1. The perturbative estimate of $\left\langle B^2(t)\right\rangle_T$ is shown by the solid curve.
• Figure 3: The rate prefactor $\kappa$ dependence on the dimensionless parameter $L_L/\beta$. The data points are for $\beta=10$ (diamonds), 12 (squares), and 14 (triangles).
• Figure 4: Diffusion constant per unit volume $\Gamma_\eta$ of $\eta(t)$ plotted against the dimensionless parameter $L_L/\beta$. The data points are for $\beta=10$ (diamonds), 12 (squares), and 14 (triangles).