The Sphaleron Rate in the Minimal Standard Model

TL;DR

Large-scale lattice simulations are used to compute the rate of baryon number violating processes (the sphaleron rate), the Higgs field expectation value, and the critical temperature in the standard model across the electroweak phase transition temperature.

Abstract

We use large-scale lattice simulations to compute the rate of baryon number violating processes (the sphaleron rate), the Higgs field expectation value, and the critical temperature in the Standard Model across the electroweak phase transition temperature. While there is no true phase transition between the high-temperature symmetric phase and the low-temperature broken phase, the cross-over is sharply defined at $T_c = (159\pm 1)$\,GeV. The sphaleron rate in the symmetric phase ($T> T_c$) is $Γ/T^4 = (18\pm 3)α_W^5$, and in the broken phase in the physically interesting temperature range $130\mbox{\,GeV} < T < T_c$ it can be parametrized as $\log(Γ/T^4) = (0.83\pm 0.01)T/{\rm GeV} - (147.7\pm 1.9)$. The freeze-out temperature in the early Universe, where the Hubble rate wins over the baryon number violation rate, is $T_* = (131.7\pm 2.3)$\,GeV. These values, beyond being intrinsic properties of the Standard Model, are relevant for e.g. low-scale leptogenesis scenarios.

Figures (3)

• Figure 1: The parameters of the effective theory (\ref{['effective']}) as functions of the temperature.
• Figure 2: The Higgs expectation value as a function of temperature, compared with the perturbative result KLRS.
• Figure 3: The measured sphaleron rate and the fit to the broken phase rate, Eq. (\ref{['fit']}), shown with a shaded error band. The perturbative result is from Burnier et al. burnier with the non-perturbative correction used there removed; see main text. Pure gauge refers to the rate in hot SU(2) gauge theory puregauge. The freeze-out temperature $T_*$ is solved from the crossing of $\Gamma$ and the appropriately scaled Hubble rate, shown with the almost horizontal line.