On the Fermion Level Crossing in the Electroweak Instanton Background

Abstract

We investigate fermion level crossing in the electroweak instanton background, taking into account the Euclidean-time dependence of the fermion energy throughout our analysis, from the field equations to the spectral flow of fermion energy levels. Modifying the standard fermion ansatzes, we show that, irrespective of the model parameters, the duration over which the fermion energy spectrum flows from one continuum to another corresponds to exactly one unit change in the Chern--Simons number of the instanton. We further demonstrate that incorporating this time dependence is essential for establishing a one-to-one correspondence between the number of fermion zero modes and the instanton winding number, providing a numerical confirmation of the index theorem in the context of instanton backgrounds.