Measuring the Broken Phase Sphaleron Rate Nonperturbatively

TL;DR

This work presents a fully nonperturbative framework to compute the broken-phase sphaleron rate using lattice methods that combine gradient-flow-inspired separatrices, a lattice definition of Chern-Simons-like observables, and multicanonical Monte Carlo sampling. By measuring the flux of configurations across a separatrix and a dynamical prefactor that accounts for prompt recrossings, the authors obtain the $N_{ m CS}$ diffusion constant and the rate $\Gamma_d$ (per unit volume) for hot baryon-number erasure, including hard thermal loop effects. The results indicate that the broken-phase erasure rate is slower than perturbative estimates by approximately $\exp(-3.6)$, and they provide a quantitative criterion $\lambda/g^2 < 0.037$ for preserving baryon number after a sufficiently reheated electroweak transition. The methodology relies on dimensional reduction to a 3D bosonic theory, a gradient-flow based separatrix, a robust lattice definition of $N_{ m CS}$ (with a pre-cooled variant to suppress UV noise), and multicanonical Monte Carlo to access rare topological fluctuations, together enabling a controlled nonperturbative assessment of electroweak baryon-number violation.

Abstract

We present details for a method to compute the broken phase sphaleron rate (rate of hot baryon number violation below the electroweak phase transition) nonperturbatively, using a combination of multicanonical and real time lattice techniques. The calculation includes the ``dynamical prefactor,'' which accounts for prompt recrossings of the sphaleron barrier. The prefactor depends on the hard thermal loops, getting smaller with increasing Debye mass; but for realistic Debye masses the effect is not large. The baryon number erasure rate in the broken phase is slower than a perturbative estimate by about exp(-3.6). Assuming the electroweak phase transition has enough latent heat to reheat the universe to the equilibrium temperature, baryon number is preserved after the phase transition if the ratio of (``dimensionally reduced'' thermal) scalar to gauge couplings (lambda / g^2) is less than .037.

Figures (11)

• Figure 1: Cartoon of periodic vacuum structure, separatrix, typical path which stays near a vacuum, and exceptional path which crosses the separatrix and leads to permanent $N_{\rm CS}$ change
• Figure 2: Cartoon of how a poor choice of separatrix can lead to overcounting the flux, and a small dynamical prefactor.
• Figure 3: Illustration of the pieces which make up a lattice gauge theory.
• Figure 4: Original lattice and the coarsened version, which is just the solid lines and filled vertices. The link matrix between two vertices of the coarsened lattice is the product of the matrices on the two original links which make it up.
• Figure 5: $(g^2/8\pi^2)\int E_i^a B_i^a d^3 x d \tau$ for a series of points on a Hamiltonian trajectory. It is clear where to adjust the integer part of Eq. (\ref{['Def_of_NCS']}) to keep $N_{\rm CS}$ (approximately) continuous.