Spectral flow of chiral fermions in nondissipative Yang-Mills gauge field backgrounds

TL;DR

This paper studies the spectral flow of chiral fermions in spherically symmetric SU(2) Yang-Mills backgrounds, revealing how the flow relates to topological characteristics like the winding and a newly introduced twist factor, with implications for baryon number violation.

Contribution

It introduces a new twist factor related to the spectral flow of fermions and connects it with the winding factor, providing a comprehensive understanding of fermion number violation in gauge fields.

Findings

Spectral flow relates to the winding and twist factors of the gauge field.

Explicit calculations for Luscher-Schechter solutions show nontrivial spectral flow contributions.

Results have potential implications for electroweak baryon number violation.

Abstract

Real-time anomalous fermion number violation is investigated for massless chiral fermions in spherically symmetric SU(2) Yang-Mills gauge field backgrounds which can be weakly dissipative or even nondissipative. Restricting consideration to spherically symmetric fermion fields, the zero-eigenvalue equation of the time-dependent effective Dirac Hamiltonian is studied in detail. For generic spherically symmetric SU(2) gauge fields in Minkowski spacetime, a relation is presented between the spectral flow and two characteristics of the background gauge field. These characteristics are the well-known ``winding factor,'' which is defined to be the change of the Chern-Simons number of the associated vacuum sector of the background gauge field, and a new ``twist factor,'' which can be obtained from the zero-eigenvalue equation of the effective Dirac Hamiltonian but is entirely determined by the background gauge field. For a particular class of (weakly dissipative) Luscher-Schechter gauge field solutions, the level crossings are calculated directly and nontrivial contributions to the spectral flow from both the winding factor and the twist factor are observed. The general result for the spectral flow may be relevant to electroweak baryon number violation in the early universe.