Classical Sphaleron Rate on Fine Lattices

TL;DR

The paper investigates how the hot classical Yang-Mills sphaleron rate Γ scales with lattice spacing, testing the Arnold–Son–Yaffe (ASY) scaling against prior small-volume ambiguities. Using a topological N_CS definition with calibrated cooling on extremely fine lattices, it analyzes large and small volumes, showing that ASY scaling (Γ ∝ a with logarithmic corrections) only emerges at very small lattice spacings and that finite-a corrections are substantial. A careful fixed-volume reanalysis of small-volume data reconciles previous discrepancies with ASY, reinforcing the ASY picture in both large and small volumes. The results yield MSM-consistent estimates for Γ and align with HTL-augmented analyses, providing a robust lattice-based validation of the ASY scenario for baryon-number violating dynamics in the electroweak context.

Abstract

We measure the sphaleron rate for hot, classical Yang-Mills theory on the lattice, in order to study its dependence on lattice spacing. By using a topological definition of Chern-Simons number and going to extremely fine lattices (up to beta=32, or lattice spacing a = 1 / (8 g^2 T)) we demonstrate nontrivial scaling. The topological susceptibility, converted to physical units, falls with lattice spacing on fine lattices in a way which is consistent with linear dependence on $a$ (the Arnold-Son-Yaffe scaling relation) and strongly disfavors a nonzero continuum limit. We also explain some unusual behavior of the rate in small volumes, reported by Ambjorn and Krasnitz.

Figures (5)

• Figure 1: Right: $N_{\rm CS}$ measured with and without 2-fold blocking on a $32^3$ lattice at $\beta=12$ ($a=1/3 g^2T$). The smearing depth is the same, $7.5 a^2=0.83/g^4 T^2$ for the two curves, with blocking (upper curve) and without (lower curve, shifted down for clarity). Left: the difference between the curves, which is small and spectrally white; note the difference in scale between the plots.
• Figure 2: Volume dependence of the sphaleron rate in cubic volumes $L$ on a side with periodic boundary conditions and $a = 1/2g^2 T$ ($\beta=8$). Large volume behavior is obtained by $L=8/g^2T$. Around $L=5/g^2T$ the sphaleron rate falls off abruptly and it is virtually zero already at $L=3 / g^2T$.
• Figure 3: Chern-Simons number diffusion constant plotted against lattice spacing $a$, and a linear fit for the last 5 points. Around $a \sim 0.5/g^2T$ the dependence is somewhat weak, but at larger $a$ a rapid falloff is evident.
• Figure 4: $\Gamma/a$ against $a$. The straight line fit assumes the ASY scaling argument, the curved fit includes Bödeker's logarithmic correction to this scaling argument; each are based on the 5 highest points.
• Figure 5: Dependence of $\Gamma$ on $a$ in a small volume, $L \simeq 4 / g^2 T$. Left, the results Ambjørn and Krasnitz reported in AmbKras2. Right, results when more care is taken to keep the lattice volume fixed in physical units. The overly strong $a$ dependence in the Ambjørn-Krasnitz data is due to small changes in the physical volume, which $\Gamma$ depends on very strongly.