Bubble Friction in Symmetry-Restoring Transitions

TL;DR

The paper extends the Bödeker–Moore framework to symmetry-restoring first-order phase transitions, showing that the leading 1→1 thermal pressure is antifriction and that the 1→2 transition-radiation pressure can be negative at small-to-intermediate wall speeds. At large wall Lorentz factors the 1→2 pressure becomes positive and scales linearly with γ, aligning with symmetry-breaking results, but the negative regime persists over a substantial range of γ, yielding a larger terminal velocity γ_t. The authors introduce γ_* and γ_90 to characterize the transition from antifriction to friction and from intermediate to standard high-γ behavior, and provide an empirical fit for P_{1→2} as a function of particle masses. These findings imply stronger bubble-wall dynamics, potentially enhancing gravitational-wave signals and the production of heavy relic states in the early universe, with implications for baryogenesis and dark matter scenarios.

Abstract

In standard (symmetry-breaking) first-order phase transitions, the frictional pressure on expanding bubble walls can be dominated by transition radiation -- the emission of a gauge boson with phase-dependent masses as particles present in the thermal plasma pass through bubble walls. This process is enhanced in the soft limit, and is known to produce a significant frictional effect that is proportional to the Lorentz factor $γ$ of the bubble wall, thereby prohibiting runaway behavior. We calculate the analogous pressure for phase transitions with symmetry restoration. In such transitions, we show that the pressure due to this process can be $\textit{negative}$, producing the opposite effect. However, when the Lorentz factor of the wall gets very large, the result approaches the same scaling as the standard scenarios. Therefore, phase transitions with symmetry restoration can feature an intermediate negative friction regime even in the presence of significant interactions with the plasma, and the bubble wall terminal Lorentz factor can be significantly larger (by more than an order of magnitude) than in the corresponding symmetry-breaking scenarios. This can carry important implications for various phenomenological applications, from gravitational waves to physics beyond-the-Standard-Model.

Figures (5)

• Figure 1: Illustration of the 1-to-2 transition, the main process studied in this paper. Left: For the symmetry-breaking scenario, massless particles ($m_{a,\mathrm{s}} = 0$) are incident on the bubble wall from the symmetric phase, and they transition into pairs of massive particles ($m_{b,\mathrm{h}}, m_{c,\mathrm{h}} > 0$) in the broken phase. Right: For the symmetry-restoring scenario, the phases are switched, so that incident particles are massive and emergent particles are massless. In the rest frame of the plasma, the wall moves left with Lorentz factor $\gamma$; in the rest frame of the wall, the particles have a thermal distribution of momenta boosted to the right by Lorentz factor $\gamma$.
• Figure 2: Dependence of the thermal pressure on the speed of the bubble wall. We plot the 1-to-2 thermal pressure $\mathcal{P}_{1 \to 2}$ as a function of the bubble wall's Lorentz factor $\gamma$. Orange curves labeled $\mathrm{s} \to \mathrm{h}$ indicate the symmetry-breaking scenario, and blue curves labeled $\mathrm{h} \to \mathrm{s}$ indicate the symmetry-restoring scenario. In the first panel we fix $m_{a,\mathrm{h}} = m_{c,\mathrm{h}} = T$ and we vary $m_{b,\mathrm{h}}$, and in the second panel we fix $m_{b,\mathrm{h}} = 0.1 T$ and we vary $m_{a,\mathrm{h}} = m_{c,\mathrm{h}}$. We take the gauge factors $g^2 C_2[R] = 1$, and more generally the pressure scales linearly with this factor.
• Figure 3: The ratio of the pressure in the symmetry-restoring scenario to that in the symmetry-breaking scenario as a function of the wall's Lorentz factor for several choices of masses.
• Figure 4: Special Lorentz factors across the parameter space of particle masses. In both panels we show the parameter space in which the broken-phase masses ($m_{a,\mathrm{h}}=m_{c,\mathrm{h}}$ and $m_{b,\mathrm{h}}$) are varied. The first panel shows contours of $\gamma_\ast$ (defined in eq. (\ref{['eq:gamma0']})), and the second panel shows $\gamma_{90}$ (defined in eq. (\ref{['eq:gamma90']})).
• Figure 5: Terminal Lorentz factor $\gamma_t$ for various choices of particle masses for symmetry-restoring scenarios ($\mathrm{h}\to \mathrm{s}$, blue curves) and symmetry-breaking scenarios ($\mathrm{s}\to \mathrm{h}$, orange curves) as a function of the difference in vacuum energy $\Delta V / T^4$. We take $g^2 C_2[R] = 1$.