Electroweak Bubble Nucleation, Nonperturbatively

TL;DR

This work presents a nonperturbative lattice framework to compute bubble nucleation rates during radiatively induced first-order electroweak phase transitions, including the dynamical prefactor and avoiding saddle-point approximations. It combines dimensional reduction to a 3D SU(2) Higgs theory for thermodynamics with Langevin dynamics for real-time evolution, employing multicanonical Monte Carlo to access exponentially rare critical-bubble configurations. Applied to the minimal Standard Model with an unphysical Higgs mass yielding a reasonably strong transition, the method reveals that conventional two-loop perturbation theory overestimates supercooling (by about a factor of 2) despite reasonably predicting the broken-phase Higgs value; including Higgs wave-function corrections improves the accuracy to roughly 25%. The study validates a nonperturbative benchmark for bubble nucleation and suggests straightforward extensions to MSSM/NMSSM scenarios, with significant implications for electroweak baryogenesis predictions.

Abstract

We present a lattice method to compute bubble nucleation rates at radiatively induced first order phase transitions, in high temperature, weakly coupled field theories, nonperturbatively. A generalization of Langer's approach, it makes no recourse to saddle point expansions and includes completely the dynamical prefactor. We test the technique by applying it to the electroweak phase transition in the minimal standard model, at an unphysically small Higgs mass which gives a reasonably strong phase transition (lambda/g^2 =0.036, which corresponds to m(Higgs)/m(W) = 0.54 at tree level but does not correspond to a positive physical Higgs mass when radiative effects of the top quark are included), and compare the results to older perturbative and other estimates. While two loop perturbation theory slightly under-estimates the strength of the transition measured by the latent heat, it over-estimates the amount of supercooling by a factor of 2.

Figures (16)

• Figure 1: Cartoon of how the constrained free energy = $-\log$ (probability of ${\phi^2_{\rm av}}$) varies with ${\phi^2_{\rm av}}$ at the equilibrium temperature in a large volume. The vertical axis gives minus the log of the fraction of states in the canonical ensemble with the given value of ${\phi^2_{\rm av}}$. The dotted line gives the free energy of a spatially homogeneous configuration with that value of ${\phi^2_{\rm av}}$; the truly most probable configurations at intermediate values are mixed phase configurations.
• Figure 2: Same as Fig. \ref{['maxwell']}, but in a finite volume where the free energy of the interface between phases is not considered negligible. The free energy of a mixed phase state is higher than either pure phase because of the surface tension of the phase boundary. The figure also illustrates the physical appearances of the states which dominate the ensemble at intermediate values of ${\phi^2_{\rm av}}$.
• Figure 3: Cartoon showing how free energy in a finite box changes when we lower the temperature. For small temperature changes, both minima survive, but one (A) is no longer globally stable. The least likely configuration on the way to the stable minimum is (C) the critical bubble: its unlikelihood restrains the rate at which configurations near (A) go to the true minimum.
• Figure 4: Area, and hence free energy, as a function of volume fraction, in the thin wall approximation. The solid line is the minimum over interface geometries; the large volume free energy curve would follow the solid line. Dotted lines are the metastable extensions of the sphere geometry (sloping) or the planar boundary geometry (flat), while the dashed lines show metastable extensions of the cylindrical geometry.
• Figure 5: Cartoon of how a 3+1 dimensional spacetime, $\Re^3 \times S^1$, drawn here as 2+1 dimensional, can look effectively 3 dimensional for long distances.