The hot baryon violation rate is $O(\alpha_W^5 T^4)$

The rate per unit volume for anomalous electroweak baryon number violation at high temperatures, in the symmetric phase, has been estimated in the literature to be $O(\alpha_W^4 T^4)$ based on simple scaling arguments. We argue that damping effects in the plasma suppress the rate by an extra power of $\alpha_W$ to give $O(\alpha_W^5 T^4)$. We show how to understand this effect in a variety of ways ranging from an effective description of the long-distance modes responsible for baryon number violation, to a microscopic picture of the short-distance modes responsible for damping. In particular, we resolve an old controversy as to whether damping effects are relevant. Finally, we argue that similar damping effects should occur in numerical simulations of the rate in classical thermal field theory on a spatial lattice, and we point out a potential problem with simulations in the literature that have not found such an effect.