An efficient approach to electroweak bubble velocities

TL;DR

This work develops a fast, hydrodynamics-based framework to predict electroweak bubble wall velocities during a first-order phase transition in SM-like extensions. By modeling the Universe as a perfect relativistic fluid coupled to the Higgs via a friction term and calibrating this friction through a relaxation-time approach, the authors connect microphysical particle dynamics to macroscopic wall propagation. They apply the method to a dimension-6 extension of the SM, showing wall velocities typically subsonic for moderate phase-transition strength but capable of becoming supersonic or runaway for stronger transitions; friction calibrations from SM results and runaway criteria are shown to be consistent. The approach provides a practical tool for exploring baryogenesis viability and gravitational-wave signatures across a broad class of beyond-Standard-Model scenarios, with potential easy adaptation to other extensions.

Abstract

Extensions of the Standard Model are being considered as viable settings for a first-order electroweak phase transition which satisfy Sakharov's three conditions for the generation of the baryon asymmetry of the Universe. These extensions provide a sufficiently strong phase transition and remove the main obstacles which appear in the context of the Standard Model: A far-too-high lower bound on the Higgs mass, immediate wipeout of the newly-created baryon asymmetry, and insufficient CP violation. We describe the Universe hydrodynamically as a fluid coupled to the Higgs field via a phenomenological friction term, and study the time evolution of bubbles nucleated during the phase transition. We express the friction term in the hydrodynamic equations in terms of the particle content of the model, calibrate the friction on the basis of existing calculations for the Standard Model, and produce predictions for the velocity of the expanding bubble wall in the stationary regime. This way we develop a very efficient approach to compute bubble velocities. As an example, we apply our formalism to the first-order phase transition of a dimension-6 extension of the Standard Model which, within the present bounds on the Higgs mass, can reproduce the observed baryon asymmetry of the Universe. Depending on the strength of the phase transition, the wall velocity varies from about 0.3 to approaching the speed of light. Our method can easily be adapted to compute wall velocities in other interesting extensions of the Standard Model.

Figures (10)

• Figure 1: Finite-temperature effective potential for a first-order phase transition.
• Figure 2: Critical bubble solution for the $\phi^6$ model with $M=800$ GeV, $m_h=115$ GeV, $T=105.62$ GeV (for parameter definitions see section 5).
• Figure 3: Higgs VEV $\phi$, velocity and temperature profiles across the (deflagration) bubble wall for the $\phi^6$ model with $M=800$ GeV, $m_h=115$ GeV at the temperature of the universe (ahead of the shock front) $T_u=105.49$ GeV and a value of the friction coefficient $\eta = 0.398$. The broken symmetry phase is on the right and the bubble wall propagates from right to left. The profile has been obtained solving the system (\ref{['eqn:fluid1']})-(\ref{['eqn:fluid3']}) via a linearisation procedure.
• Figure 4: Velocity and temperature profiles across the intermediate region between the bubble wall and the shock front for the same deflagration bubble as in figure \ref{['fig:wall1']} with $M=800$ GeV, $m_h=115$ GeV at $T_u=105.49$ GeV. Here the position of the bubble wall is on the left end of the integration interval and that of the shock front on the right. If the sphericity of the bubble is neglected (and a less realistic planar approximation adopted instead), $v$ and $T$ do not vary across this region.
• Figure 5: Spatially varying part of the friction term according to the relaxation time approximation (slow wall limit), $\phi^2 \phi' \int\frac{d^3p}{(2\pi)^3 (2E)^2} \frac{e^{\beta \gamma (E-v_w p_z)}}{(e^{\beta \gamma (E-v_w p_z)}\pm1)^2}$ (solid line), and the same term with the momentum integral replaced by a fitted constant (dotted line), $C\;\phi^2 \phi'$, for fermions (left) and bosons (right) in an example case. We use the hyperbolic tangent Ansatz to approximate the bubble profile and assume $\phi_0 = 100$ GeV in the broken symmetry phase, $T = 100$ GeV, $L_w \cdot T = 15$, and a mass dependence $m \equiv \frac{1}{\sqrt{2}} \phi$.