Chern-Simons diffusion rate near the electroweak phase transition for mH ≈ mW

Abstract

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

Lattice Chern-Simons number without ultraviolet problems

“…We have used our technique to study the diffusion constant for N CS in the broken electroweak phase and find that the rate is very substantially suppressed with respect to the symmetric phase rate; for the Higgs self-coupling we used, the rate at the equilibrium temperature is consistent with zero and is less than κ = 0.005, in agreement with the expectations of the "Sphaleron approximation" estimate of the rate but in contradiction to previous results, which were contaminated by ultraviolet diffusion due to the non-topological definition of N CS used there [10]. We have also measured the diffusion constant in Yang-Mills theory at a range of lattice spacings, using an O(a) improved match between the lattice spacing and the physical length scale, and we find a nonzero dependence on lattice spacing, which implies that the infrared dynamics relevant to N CS diffusion depends in an important way on the physics of hard thermal loops.…”

“…There has already been substantial work on finding κ numerically [7,13,9,14,15,10,11]. All this work has been based on a lattice implementation of N CS as an integral over a local operator, usually…”

Chern-Simons number diffusion with hard thermal loops

“…4 The first four conditions for a valid update rule have specified the update uniquely except for the freedom to perform rotations of the form shown in (22). It would seem natural to define the magnetic field B a i at a point in terms of the 3 nearest plaquettes 5 and to rotate momenta according toṗ…”

Classical field dynamics of the electroweak phase transition

“…This is directly relevant to the study of baryon number violation because the axial anomaly relates the motion of baryon number to the motion of N CS . In fact the classical technique was originally proposed to investigate this question [17], and since then it has been used to investigate N CS violation in Yang-Mills theory [20,21,22], in the symmetric phase of Yang-Mills Higgs theory [18,19], and in the broken phase of Yang-Mills Higgs theory [23,46]. It has yet to be applied to perhaps the most relevant problem, which is the rate of N CS motion in the out of equilibrium environment of a moving bubble wall during the phase transition.…”