Improved determination of the classical sphaleron transition rate

TL;DR

This study investigates the classical sphaleron transition rate using real-time lattice simulations with an improved, cooling-based lattice topological charge definition to reduce discretization artifacts. By examining both Yang-Mills-Higgs theory (broken phase) and pure Yang-Mills theory across small and large volumes, and by analyzing gauge-field dynamics in the Coulomb gauge, it finds strong suppression of sphaleron transitions in the Higgs phase and a rate in large volumes that decreases with lattice spacing, leaving open multiple possible continuum behaviors. The work highlights the sensitivity of rate extraction to lattice discretization and finite-volume effects and provides a practical framework (cooling-based topological charge, Coulomb-gauge correlators) to assess continuum limits of topology-changing processes in thermal gauge theories. Overall, it clarifies where the classical approximation may or may not yield a finite continuum rate and informs the interpretation of electroweak baryon-number violation in high-temperature contexts.

Abstract

We determine the sphaleron transition rate using real time lattice simulations of the classical system. An improved definition of the lattice topological charge allows us to obtain a more reliable estimate of the transition rate. For an SU(2) Yang-Mills-Higgs system in the broken phase we find the transition rate to be strongly suppressed, and we have observed no sphaleron transitions in the range of coupling constants used. For a pure SU(2) Yang-Mills system in large volumes the rate behaves as $κ(α_w T)^4$, with $κ$ slightly decreasing as the lattice spacing is reduced. If the lattice size is reduced to about twice the magnetic screening length, the rate is suppressed by finite-size effects, and $κ$ is approximately proportional to the lattice spacing. Our rate measurements are supplemented by analysis of gauge field correlation functions in the Coulomb gauge.

Figures (7)

• Figure 1: A $\Pi$-shaped trajectory in the $t,\tau$ plain used for determination of topological charge with cooling. The endpoints of a $\tau=0$ real-time trajectory, shown by a dotted line, are denoted by diamonds.
• Figure 2: The "string bit" order parameter (diamonds) and the average energy per degree of freedom normalized by the energy of a free theory (pluses) for the Yang-Mills-Higgs theory with the tree-level $m_H/m_W=0.423$. Note the discontinuity of both quantities at $\beta=11.25$. The error bars are smaller than the plotting symbols. The dotted lines are to guide the eye.
• Figure 3: Time history of $N_{\rm CS}$ in the Yang-Mills-Higgs theory with (the upper curve) and without (the lower curve) the cooling improvement.
• Figure 4: Time history of $N_{\rm CS}$ in the Yang-Mills and without the cooling improvement at $\beta=L=12$. Both curves exhibit rapid fluctuations and a sphaleron transition at $t\approx 2000$, but, for the naive definition of $N_{\rm CS}$, there also is a slow diffusion upwards.
• Figure 5: Effective $\kappa$ as a function of $T\equiv 1/\beta$ in the Yang-Mills theory for $\beta=L$. The dotted line is a linear fit through the origin using all the data except the $\beta=20$ point.