From Boltzmann equations to steady wall velocities

TL;DR

The paper develops a relativistic, gradient-expanded Boltzmann framework based on the Schwinger–Keldysh/Kadanoff–Baym formalism to compute electroweak bubble-wall velocities and thicknesses during first-order phase transitions. It identifies two contributions to the Higgs equation of motion: the out-of-equilibrium friction from species driven by the wall and a background friction from latent-heat redistribution, and derives coupled transport equations in the wall frame for SM-like plasmas. The results show that for thick walls ("L T ≫ 20"), a phenomenological friction model reproduces the Boltzmann results, while for thinner walls or near the speed of sound, full Boltzmann treatment yields notable differences and can even disallow static wall solutions in some regimes. Applications to the SM with a light Higgs, the SM with a low cutoff, and a singlet-extended SM illustrate both agreements and model-dependent deviations, informing predictions for gravitational wave signals and baryogenesis. This framework thus provides a principled, first-principles method to predict bubble dynamics across a range of beyond-Standard-Model scenarios.

Abstract

By means of a relativistic microscopic approach we calculate the expansion velocity of bubbles generated during a first-order electroweak phase transition. In particular, we use the gradient expansion of the Kadanoff-Baym equations to set up the fluid system. This turns out to be equivalent to the one found in the semi-classical approach in the non-relativistic limit. Finally, by including hydrodynamic deflagration effects and solving the Higgs equations of motion in the fluid, we determine velocity and thickness of the bubble walls. Our findings are compared with phenomenological models of wall velocities. As illustrative examples, we apply these results to three theories providing first-order phase transitions with a particle content in the thermal plasma that resembles the Standard Model.

Figures (8)

• Figure 1: The friction components as functions of the wall thickness $L \, T$ for $\phi_0/T= 1$ and two different wall velocities $v_w= 0.1$ and $v_w=0.8$. The friction components of the background for subsonic and supersonic wall velocities are shown in separate plots since they differ greatly.
• Figure 2: Dependence of the friction components on the strength of the phase transition $\phi_0 / T$. The lines for the different velocities are normalized to unity at $\phi_0/T = 1$. The wall thickness is $L \, T = 30$.
• Figure 3: The velocity dependence of the friction components in comparison to the fit (\ref{['fit1']}) - (\ref{['fit4']}) (solid lines). Different colors represent different strengths of phase transition for the fluid parts ($\phi_0/T= \{ 1, 2 \}$), while the background components scale as $(\phi_0 / T)^4$. The wall thickness in all plots is $L \, T=30$.
• Figure 4: Example for the fluctuations in the fluid and background fields; $v_w= 0.5,LT= 30, \phi_0/T=1$.
• Figure 5: Example for the fluctuations in the fluid and background fields; $v_w= 0.7,LT= 30, \phi_0/T=1$.