Sphaleron rate in the symmetric electroweak phase

TL;DR

This work extends Bödeker’s infrared effective theory for hot Yang–Mills dynamics to include a light Higgs, enabling a nonperturbative study of the sphaleron rate in the symmetric phase near the electroweak phase transition. By formulating an infrared theory with a nonlocal Higgs-induced effective action $H_{\rm eff}$ and implementing a practical Langevin-based lattice scheme (with an $\eta \to \infty$ limit for Higgs evolution), the authors quantify how Higgs thermodynamics suppress the baryon-number-violating rate. They find the symmetric-phase rate is reduced by about 20% relative to pure Yang–Mills for a representative parameter set, and that in analytic crossover regimes the rate turns on smoothly around the $\phi^2$ susceptibility peak. These results imply that Yang–Mills-based estimates remain a reasonable baseline (within ~20%) for baryogenesis calculations in the Standard Model and its MSSM-like extensions, and they provide a framework for exploring Higgs effects in more viable models.

Abstract

Recently Bodeker has presented an effective infrared theory for the dynamics of Yang-Mills theory, suitable for studying the rate of baryon number violation in the early universe. We extend his theory to include Higgs fields, and study how much the Higgs affects the baryon number violation rate in the symmetric phase, at the phase coexistence temperature of a first order electroweak phase transition. The rate is about 20% smaller than in pure Yang-Mills theory. We also analyze the sphaleron rate in the analytic crossover regime. Our treatment relies on the ergodicity conjecture for 3-D scalar $φ^4$ theory.

Figures (5)

• Figure 1: gauge field self-energy insertions generated by the Higgs fields.
• Figure 2: Log of probability distribution for $\phi^2 \equiv 2 \Phi^\dagger \Phi$ ($\overline{\rm MS}$ renormalization point $\overline{\mu}=g^2T$) for cubic boxes of size $24^3$ (left) and $32^3$ (right), at lattice spacing $4/9g^2T$ ($\beta=9$). Inverting the $y$ axis roughly gives the free energy dependence on $\phi^2$. The left peak is the symmetric phase and the right peak is the broken phase; the dip between them is a free energy barrier separating the phases, which grows as the volume is increased (it is not free energy per volume). The asymmetric shape is typical for this order parameter.
• Figure 3: Left: probability distribution on supercooling, $m_H^2 = m_{\rm eq}^2 - .0093 g^4T^2$. Right: Langevin time history of $N_{\rm CS}$ starting in the symmetric phase, under such supercooling. After a tunneling event to the broken phase, $N_{\rm CS}$ stopped diffusing.
• Figure 4: Left: $\phi^2$ susceptibility; Right: $\phi^2$ susceptibility (peak with small errors) and sphaleron rate (larger errors, square plotting symbols) scaled to their maxima, when there is a smooth crossover. $m_H^2(T)$ plotted is the 3D theory value, $\overline{\rm MS}$ renormalized with $\overline{\mu}=g^2T$.
• Figure 5: Diagrams considered in Appendix \ref{['App_HTL']}. Solid lines are scalars, wiggly lines are gauge bosons, double lines are fermions. Diagrams (a) and (b) are a self-energy and a vertex correction considered in the text. Diagrams (c) and (d) are generic diagrams with external gauge and scalar lines and either a boson (alternating Higgs and gauge, diagram (c)) or a fermion (diagram (d)) in the loop. As discussed, (b), and all diagrams of type (c) and (d), do not give rise to hard thermal loops.