Bubble nucleation and growth at a baryon-number-producing electroweak phase transition

We discuss the nucleation and growth of bubbles of the broken-symmetry phase of matter at the electroweak phase transition. We show that the bubble walls propagate with a mildly relativistic speed. The Lorentz $\gamma$ factor depends on the ratio of the Higgs to gauge-boson mass—for parameters allowing for baryogenesis at the transition (i.e., a fairly light Higgs mass) we find $\gamma v\sim 1$. We show that the bubble wall is mainly slowed by interactions with low-momentum gauge-boson pairs, and compute the damping rate due to these interactions. The width of the bubble wall is significantly larger than the typical wave-length of particles which are reflected from it, which allows us to use a WKB approximation for the particle scattering. The width is also larger than the mean free path for these particles, which means that the gauge boson fluid remains close to local thermal equilibrium. This situation results in mildly relativistic motion of the bubble wall. As a result, the baryon-number excess produced on the bubble wall is not much diluted by subsequent diffusion. We compute the effective equation of motion for the Higgs field, and the approximate shape of the moving bubble wall.