The rate of B-violation in the standard model at finite temperature is closely related to the diffusion rate Γ of Chern-Simons number. We compute this rate for m H ≈ m W in the classical approximation in an effective SU(2)-Higgs model, usingKrasnitz's algorithm. The parameters in the effective hamiltonian are determined by comparison with dimensional reduction. In the high temperature phase we find Γ/V (α W T ) 4 ≈ 1, neglecting a finite renormalization. In the low temperature phase near the transition we find the rate to be much larger than might be expected from previous analytic calculations based on the sphaleron. 1

New results for the rate are presented using the canonical ensemble in the classical approximation on a spatial lattice. We nd that the rate at high temperatures is proportional to T 2 , and strongly dependent on the lattice spacing a. We conclude that a better e ective action is needed for the classical approximation.

We compute the sphaleron transition rate in the 1+1 dimensional abelian Higgs model at nite temperature, by real time simulation using the classical canonical ensemble.

We give arguments that in the 1+1 dimensional abelian Higgs model the classical approximation can be good for the leading high temperature behavior of real time processes. The Chern-Simons diffusion rate ('sphaleron rate') is studied numerically in this approximation. New results at high temperature show a T 2/3 behavior of the rate at sufficiently small lattice spacing.

Using the (effective) classical approximation, we compute numerically time-dependent correlation functions in the SU(2)-Higgs model around the electroweak phase transition, for $m_H \approx m_W$. The parameters of the classical model have been determined previously by the dimensional reduction relations for time-independent correlators. The $H$ and $W$ correlation functions correspond to gauge invariant fields. They show damped oscillatory behavior from which we extract frequencies $\om$ and damping rates $\gm$. In the Higgs phase the damping rates have roughly the values obtained in analytic calculations in the quantum theory. In the plasma phase (where analytic estimates for gauge invariant fields are not available), the damping rate associated with $H$ is an order of magnitude larger than in the Higgs phase, while the $W$ correlator appears to be overdamped, with a small rate. The frequency $\om_H$ shows a clear dip at the transition. The results are approximately independent of the lattice spacing, but this appears to be compatible with the lattice spacing dependence expected from perturbation theory.Comment: 27 pages, latex, 13 figures. Improved presentation, data unchanged. New is the conclusion that the apparent lattice spacing independence of the `plasmon' frequency is compatible with the divergencies expected from perturbation theory. (Version resubmitted to NPB on 17 July 1997.