Lattice Chern-Simons Number Without Ultraviolet Problems
Guy D. Moore, Neil G. Turok
TL;DR
The authors present a topological lattice method to measure Chern-Simons number diffusion in real-time evolution that avoids ultraviolet contamination by using a slave field $S(x)\in SU(2)$ whose minimal-energy winding encodes the topology via $-N_S$. This approach yields a UV-insensitive estimate of the diffusion rate, showing strong suppression of $N_{CS}$ changes in the broken electroweak phase and revealing a nontrivial lattice-spacing dependence in pure YM theory. Comparisons with a cooled-field definition confirm infrared agreement and expose UV artifacts in traditional definitions, while results support Arnold–Yaffe-type scaling with finite $(l_d/l_n)^2$ corrections. Limitations include volume-dependent breakdowns and Gribov ambiguities, but the framework offers a robust route toward relating lattice diffusion rates to the physical, quantum theory, potentially by including hard thermal loop physics via additional degrees of freedom. The work advances topological diagnostics for nonperturbative gauge dynamics and baryon-number violation in the early universe.
Abstract
We develop a topological method of measuring Chern-Simons number change in the real time evolution of classical lattice SU(2) and SU(2) Higgs theory. We find that the Chern-Simons number diffusion rate per physical 4-volume is very heavily suppressed in the broken phase, and that it decreases with lattice spacing in pure Yang-Mills theory, although not as quickly as predicted by Arnold, Son, and Yaffe.