Bubble wall velocities in the Standard Model and beyond

TL;DR

This work addresses the problem of determining bubble wall velocity $v_w$ and thickness $L_w$ during a cosmological first-order electroweak phase transition across broad SM extensions. The authors model friction from top quarks and $W$-bosons using a Boltzmann-fluid framework, linearize deviations from equilibrium, and employ the three-dimensional tunneling action $S_3$ to relate the pressure difference and frictional terms, reducing the problem to two dimensionless parameters: the transition strength $oldsymbol{\xi= ilde{}/T}$ and $oldsymbol{ / ilde{}^4}$ (via $ $ and $ ilde{}$). They produce contour predictions for $v_w$ and $L_w$ showing universality for mild transitions, but reveal significant deviations and a genuine runaway boundary in strong transitions where the fluid approximation breaks down; multi-scalar extensions further challenge simple mappings and generally yield thicker walls and higher velocities. The results offer a practical, model-dependent framework to assess electroweak phase-transition dynamics and have implications for baryogenesis and gravitational-wave signals in a wide class of theories.

Abstract

We present results for the bubble wall velocity and bubble wall thickness during a cosmological first-order phase transition in a condensed form. Our results are for minimal extensions of the Standard Model but in principle are applicable to a much broader class of settings. Our first assumption about the model is that only the electroweak Higgs is obtaining a vacuum expectation value during the phase transition. The second is that most of the friction is produced by electroweak gauge bosons and top quarks. Under these assumptions the bubble wall velocity and thickness can be deduced as a function of two equilibrium properties of the plasma: the strength of the phase transition and the pressure difference along the bubble wall.

Figures (4)

• Figure 1: The plot shows the function $X$ defined in (\ref{['eq:tunneling']}). The limiting cases are the models with a $\phi^3$ and $\phi^6$ terms in the free energy that should cover most of realistic potentials.
• Figure 2: The constraints (\ref{['eq:WVbound']}) and (\ref{['eq:fluidConstraint']}) as a function of the phase transition strength $\xi_n$ and the normalised vacuum energy $\Delta V/\phi_n^4$, for the (top) cubic toy-model and (bottom) dimension six extension.
• Figure 3: The plot shows the wall velocity as a function of the phase transition strength $\xi_n$ and the normalised vacuum energy $\Delta V/\phi_n^4$, for the (top) cubic toy-model and (bottom) dimension six extension. The stars denote the Standard Model with a very light Higgs boson with masses $m_H \in \{50,30,20\}$ GeV. The results in the shaded regions are unreliable as seen in Fig. \ref{['fig:constraints']}.
• Figure 4: The same as Fig. \ref{['fig:moneyplots']} for the wall thickness.