Classical Sphaleron Rate on Fine Lattices

TL;DR

This study measures the sphaleron rate in hot classical Yang-Mills theory on very fine lattices, revealing its dependence on lattice spacing and supporting the Arnold-Son-Yaffe scaling relation, with implications for continuum limits.

Contribution

The paper provides the first detailed lattice study of the sphaleron rate at extremely fine lattice spacings, demonstrating nontrivial scaling and challenging the existence of a nonzero continuum limit.

Findings

Sphaleron rate scales linearly with lattice spacing on fine lattices.

Topological susceptibility decreases with lattice spacing, consistent with linear dependence.

Results disfavor a nonzero sphaleron rate in the continuum limit.

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.