Computing the Strong Sphaleron Rate

TL;DR

This work investigates the strong sphaleron rate during the electroweak transition by computing the diffusion constant of Chern-Simons number in classical lattice SU(3) Yang-Mills theory. Extending methods developed for SU(2) to SU(3) and carefully matching lattice thermodynamics to the continuum, the study provides a quantitative estimate of the strong sphaleron diffusion constant $\Gamma_{ss}$ and its ratio to the weak sphaleron rate $\Gamma_{ws}$ via hard thermal loop matching. The key result is that $\Gamma_{ss}$ for SU(3) is substantially larger than SU(2), yielding a decay time for chiral quark number of about $\tau \sim 80/T$, though with systematic uncertainties of order a factor of two. These findings imply that strong sphalerons can efficiently damp chiral quark number during the electroweak phase transition, potentially affecting baryogenesis scenarios, while highlighting theoretical and lattice-to-quantum mapping uncertainties that warrant further study.

Abstract

We measure the diffusion constant for Chern-Simons number for classical, lattice SU(3) Yang-Mills theory, using a generalization of the topological definition of Chern-Simons number developed recently by Moore and Turok. The diffusion constant is much larger than that for SU(2), even before the ratio of coupling constants has been accounted for, which implies that chiral quark number is efficiently destroyed by strong processes during the electroweak phase transition. For the physical value of α_s we estimate the decay time for chiral quark number to be about 80/T, although various systematics make this number uncertain by about a factor of 2.

Figures (2)

• Figure 1: Examples of diagrams contributing to the renormalization of the gauge field. The full effect of a gauge line propagating between scalar lines must match between theories; besides self-energy corrections, there are also corrections at the vertex, and corrections because the vertex is renormalized at the same time the scalar propagator is.
• Figure 2: The value of the integral, Eq. (\ref{['NCSintegral']}), versus the actual winding number for gauge transformations in an SU(2) evolution at $\beta_L = 8$ on a $20^3$ lattice, at left, shows that the integral can be used to unambiguously reconstruct the original winding number. At right, Eq. (\ref{['NCSintegral']}) for gauge transformations in an SU(3) evolution on a $24^3$ grid at $\beta_L = 16$, plotted against the reconstructed winding number. The values of the integral are clustered with large breaks which makes the reconstruction unambiguous, even though we have no direct integer measure of the winding number.