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

Real-time anomalous fermion number violation is investigated for massless chiral fermions in spherically0e 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 $\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}$ 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) Lüscher-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.