A Nonperturbative Measurement of the Broken Phase Sphaleron Rate

TL;DR

This study presents a nonperturbative, lattice-based approach to determine the broken-phase sphaleron rate in the electroweak theory by defining the Chern-Simons number $N_{CS}$ via a cooling path and measuring near-peak fluctuations with a multicanonical algorithm. It reports nonperturbatively computed diffusion rates $\Gamma_d$ for several Higgs self-couplings and finds them slower than perturbative estimates, enabling a quantitative baryon-number erasure bound. By integrating $\Gamma_d$ over cosmological evolution and accounting for the temperature dependence through $y = m_H^2/(g^4 T^2)$, the authors derive a bound of $\lambda/g^2 \lesssim 0.036$ (SM) on preserving baryon asymmetry, with a similar bound in the MSSM. The work highlights the importance of nonperturbative dynamics and potential corrections from dynamical prefactors and hard-thermal-loop effects, suggesting robustness of the bound against heavy degrees of freedom but sensitivity to light states or phase-transition reheating assumptions.

Abstract

We develop a method to compute the sphaleron rate in the electroweak broken phase nonperturbatively. The rate is somewhat slower than a perturbative estimate. In SU(2) X U(1) Higgs theory at the physical value of Theta_W, and assuming that the latent heat of the phase transition reheats the universe to the equilibrium temperature, baryon number erasure after the phase transition is prevented only when lambda/g^2 <= 0.036.

Figures (2)

• Figure 1: "Cartoon" of the free energy dependence on $N_{\rm CS}$.
• Figure 2: Free energy (left) and $( \phi^2_{\rm broken} - \phi^2_{\rm symm} ) / ( g^2 T^2 )$ as functions of $N_{\rm CS}$ at $x \equiv \lambda / g^2 = 0.039$, in a $(16/g^2 T)^3$ volume. The plot of $\phi^2$ is a check that the volume used was large enough to prevent the sphaleron from causing a transition to the symmetric phase.