Level Crossing for Hot Sphalerons

TL;DR

This paper investigates how high-temperature sphaleron fluctuations affect the Dirac spectrum in the early universe, providing evidence that lattice sphaleron configurations correspond to continuum solutions.

Contribution

It demonstrates the connection between sphaleron-induced eigenvalue crossings and changes in Chern-Simons number using lattice simulations.

Findings

Eigenvalues cross zero during Chern-Simons number change

Sphaleron fluctuations cause sign changes in generalized chirality

Supports continuum interpretation of lattice sphaleron configurations

Abstract

We study the spectrum of the Dirac Hamiltonian in the presence of high temperature sphaleron-like fluctuations of the electroweak gauge and Higgs fields, relevant for the conditions prevailing in the early universe. The fluctuations are created by numerical lattice simulations. It is shown that a change in Chern-Simons number by one unit is accompanied by eigenvalues crossing zero and a change of sign of the generalized chirality $\tGf= (-1)^{2T+1} \gf$ which labels these modes. This provides further evidence that the sphaleron-like configurations observed in lattice simulations may be viewed as representing continuum configurations.