Towards an all-orders calculation of the electroweak bubble wall velocity

TL;DR

This work derives an all-orders leading-logarithmic calculation of the thermal pressure exerted by Standard Model particles on ultrarelativistic Higgs-phase bubble walls during a first-order electroweak phase transition. It combines analytic Sudakov-type resummation with a broken-phase parton shower to account for multiple soft gauge-boson emissions, showing that the pressure scales as $P \sim \gamma^2 T^4$ and that the terminal velocity is considerably slower than previous fixed-order estimates. A detailed momentum-transfer analysis yields a LL estimate for $\langle\Delta p_z/(\gamma T)\rangle$ and is corroborated by numerical shower simulations that agree within ~10-20%. The slower walls shift the dominant gravitational-wave sources from bubble collisions to fluid phenomena and have important implications for electroweak baryogenesis and primordial magnetic fields, with Extensions to beyond-Standard-Model plasmas a natural avenue for future work.

Abstract

We analyze Higgs condensate bubble expansion during a first-order electroweak phase transition in the early Universe. The interaction of particles with the bubble wall can be accompanied by the emission of multiple soft gauge bosons. When computed at fixed order in perturbation theory, this process exhibits large logarithmic enhancements which must be resummed to all orders when the wall velocity is large. We perform this resummation both analytically and numerically at leading logarithmic accuracy. The numerical simulation is achieved by means of a particle shower in the broken phase of the electroweak theory. The two approaches agree to the 10\% level. For fast-moving walls, we find the scaling of the thermal pressure exerted against the wall to be $P\sim γ^2T^4$, independent of the particle masses, implying a significantly slower terminal velocity than previously suggested.

Figures (5)

• Figure 1: Sketch of the real-emission kinematics. The bubble wall is shown in blue and moves in the $-z$-direction with speed $v$ in the rest frame of the plasma in front of the wall. The Higgs vacuum expectation value is denoted as $\langle \phi \rangle$. The incoming particle $a$ has a light-like four momentum $p_a$ in the wall's rest frame. The scattered particle $b$ and the soft emission, $c$, have four momenta $p_b$ and $p_c$.
• Figure 2: Comparison between fixed-order and resummed result for the relative momentum transfer distribution in the full Standard Model using the parton-shower approximation. The number of emissions is limited to one. Note that the fixed-order result is not normalized to the total rate, as the rate tends to infinity in this case.
• Figure 3: Cutoff-dependence of the relative momentum transfer in the parton-shower approximation.
• Figure 4: Boost factor dependence of the relative momentum transfer in the parton-shower approximation.
• Figure 5: Comparison of the terminal Lorentz factor from our analysis (black) compared to those inferred from a fixed order calculation (blue). We show predictions for different values of the order parameter $\langle\phi\rangle/T$, with $\langle\phi\rangle/T=1$, 3, 5 for the set of black curves from top to bottom, and similarly for the blue curves. For large $\alpha_{\theta}$ (strong transitions), our results predict significantly slower walls than implied by previous analyses, with the terminal velocity independent of the particle masses. For an electroweak-scale phase transition in a SM-like plasma, several of the approximations made break down for $\gamma_{\rm eq} \lesssim 10$, rendering the predictions in the shaded pink region unreliable. There, our result should be interpreted as an upper bound on the terminal Lorentz factor, indicated by the dashed black line and the arrows.