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On Sphaleron Deformations Induced by Yukawa Interactions

Due to the presence of the chiral anomaly sphalerons, half a number of Chern-Simons (CS=1/2) are the only static configurations that allow a fermion level crossing in the two-dimensional Abelian-Higgs model with massless fermions, i.e. in the absence of Yukawa interactions. In the presence of fermion-Higgs interactions we demonstrate the existence of zero energy solutions to the one-dimensional Dirac equation at deformed sphalerons with CS ≠ 1/2. Induced level crossing due to Yukawa interactions illustrates a non-trivial generalization of the Atiyah-Patodi-Singer index theorem and of the equivalence between parity anomaly in odd and chiral anomaly in even dimensions. We discuss a subtle manifestation of this effect in the standard electroweak theory at finite temperatures.