How fast can the wall move? A study of the electroweak phase transition dynamics

TL;DR

This work analyzes the dynamics of electroweak-bubble walls in the Minimal Standard Model, deriving a semi-classical equation of motion for the Higgs condensate and solving for wall velocity and thickness using a two-loop finite-temperature potential and a Boltzmann-fluid treatment of particle populations. It finds a robust subsonic wall with $v_w \approx 0.36$–$0.44$ and $L T \approx 23$–$29$ across $0<m_H<90$ GeV, with top quarks and thermal particles as the main sources of friction; infrared gauge bosons are estimated to enhance friction beyond the fluid approximation. The analysis further explores the impact of an infrared gauge condensate, which could push the wall toward supersonic speeds but not ultrarelativistic detonation, and discusses the limitations of the fluid approach, including potential underestimation of infrared boson friction and the regime where the wall shape deviates from the chosen Ansätze. Overall, the study provides a coherent, quantitative picture of wall dynamics and latent-heat effects during the electroweak phase transition, with implications for CP-violating transport and baryogenesis scenarios.

Abstract

We consider the dynamics of bubble growth in the Minimal Standard Model at the electroweak phase transition and determine the shape and the velocity of the phase boundary, or bubble wall. We show that in the semi-classical approximation the friction on the wall arises from the deviation of massive particle populations from thermal equilibrium. We treat these with Boltzmann equations in a fluid approximation. This approximation is reasonable for the top quarks and the light species while it underestimates the friction from the infrared $W$ bosons and Higgs particles. We use the two-loop finite temperature effective potential and find a subsonic bubble wall for the whole range of Higgs masses $0<m_H<90$GeV. The result is weakly dependent on $m_H$: the wall velocity $v_w$ falls in the range $0.36<v_w<0.44$, while the wall thickness is in the range $29> L T > 23 $. The wall is thicker than the phase equilibrium value because out of equilibrium particles exert more friction on the back than on the base of a moving wall. We also consider the effect of an infrared gauge condensate which may exist in the symmetric phase; modelling it simplemindedly, we find that the wall may become supersonic, but not ultrarelativistic.

Figures (3)

• Figure 1: All annihilation, scattering, and absorption re-emission diagrams considered in the text
• Figure 2: Plot of the solutions of Eq. (\ref{['constraints']}) for the case $\lambda_T = 0.03$, $A_F = 0$. The solid line is the velocity constraint and the dotted line is the thickness constraint.
• Figure 3: Wall shape; at left, for $\lambda_T = 0.04$, $A_F=0.1$, and at right, for $\lambda_T = 0.03$, $A_F = 0$. The solid line is the numerical value, the dotted line is the tanh Ansatz shape. The scale is the same for both; the full width of the pictures are $96/T$ and $64/T$.