Level crossing along sphaleron barriers

In the electroweak sector of the standard model topologically inequivalent vacua are separated by finite energy barriers, whose height is given by the sphaleron. For large values of the Higgs boson mass there exist several sphaleron solutions and the barriers are no longer symmetric. We construct paths of classical configurations from one vacuum to a neighboring one and solve the fermion equations in the background field configurations along such paths, choosing the fermions of a doublet degenerate in mass. As in the case of light Higgs boson masses we observe the level crossing phenomenon also for large Hibbs boson masses