Motion of Chern-Simons Number at High Temperature Under a Chemical Potential

Abstract

I investigate the evolution of finite temperature, classical Yang-Mills field equations under the influence of a chemical potential for Chern Simons number $N_{CS}$. The rate of $N_{CS}$ diffusion, $\Gamma_d$, and the linear response of $N_{CS}$ to a chemical potential, $\Gamma_\mu$, are both computed; the relation $\Gamma_d = 2 \Gamma_\mu$ is satisfied numerically and the results agree with the recent measurement of $\Gamma_d$ by Ambjorn and Krasnitz. The response of $N_{CS}$ under chemical potential remains linear at least to $\mu = 6 T$, which is impossible if there is a free energy barrier to the motion of $N_{CS}$. The possibility that the result depends on lattice artefacts via hard thermal loops is investigated by changing the lattice action and by examining elongated rectangular lattices; provided that the lattice is fine enough, the result is weakly if at all dependent on the specifics of the cutoff. I also compare SU(2) with SU(3) and find $\Gamma_{\rm SU(3)} \sim 7 (\alpha_s/\alpha_w)^4 \Gamma_{\rm SU(2)}$.