The Sphaleron Rate through the Electroweak Cross-over

TL;DR

This work nonperturbatively determines the sphaleron diffusion rate Γ_diff(T) through the electroweak cross-over for Higgs masses m_H = 115 and 160 GeV using a dimensionally reduced 3D SU(2)-Higgs lattice theory. It combines canonical Monte Carlo in the symmetric phase with multicanonical methods and real-time Langevin-like evolution (with calibrated cooling) to resolve both unsuppressed and exponentially suppressed regimes, extracting Γ_diff(T) and the Higgs expectation value v(T). The results show a rapid but gradual shutdown of sphaleron transitions across the cross-over, with similar slopes for both m_H values and a cross-over range tied to m_H, and demonstrate that accurately modeling the gradual suppression is important for Leptogenesis-based baryogenesis calculations. The findings provide essential input for baryogenesis studies and illustrate the importance of nonperturbative dynamics in early-Universe phenomenology, while validating the lattice methodology through overlap between canonical and multicanonical results.

Abstract

Using lattice simulations, we measure the sphaleron rate in the Standard Model as a function of temperature through the electroweak cross-over, for the Higgs masses m_H=115 and m_H=160 GeV. We pay special attention to the shutting off of the baryon rate as the temperature is lowered. This quantity enters computations of Baryogenesis via Leptogenesis, where non-zero lepton number is converted into non-zero baryon number by equilibrium sphaleron transitions. Combining existing numerical methods applicable in the symmetric and broken electroweak phases, we find the temperature dependence of the sphaleron rate at very high temperature, through the electroweak cross-over transition, and deep into the broken phase.

Figures (10)

• Figure 1: The values of $x$ (left) and $y$ (right) for $m_H=115$ and $160$ GeV in the temperature range of interest.
• Figure 2: Measurement of the Chern-Simons number evolution Moore:1998swa. The solid circles show the configurations generated by the real-time evolution using the Langevin/heat-bath method. At fixed intervals, the configurations are cooled by the same amount in order to construct a cooled trajectory, where the UV noise is almost completely eliminated, allowing to calculate $\delta N_{\rm CS}$ from \ref{['HTL6.3']}. The cooling from vacuum to vacuum works as a test for residual errors: $\delta N_{\rm CS}$ must then be close to an integer, the deviations from which are subtracted, thus avoiding the accumulation of errors.
• Figure 3: $N_{\rm CS}$ from a heat-bath trajectory (left), and the resulting probability distribution (right), folded into the interval $[0,1]$, at $m_H =115\,$GeV and $T =152$ (top), $145$ (middle) and $140$ GeV (bottom). At high temperature, in the symmetric phase, the sphaleron transitions are unsuppressed, whereas at low $T$ the transitions are so strongly suppressed that they do not happen in canonical simulations.
• Figure 4: A heat-bath trajectory for $N_{\rm CS}$, still for $m_H =$ 115 GeV and $T =$ 140 but now with multicanonical simulations (left), and the corresponding multicanonical probability distribution $P_{\textrm{muca}}$ (right).
• Figure 5: The physical distribution $P_{\textrm{can}}$ (left), after reweighting the result in figure \ref{['fig:ncs_broken_multi']} with the multicanonical weight function (right).