The Fermion Determinant in External Scalar Fields: Exact Results and Large Mass Approximation

We present results for the fermion determinant in an external Yukawa-coupled scalar field, a quantity relevant to instanton and sphaleron transition rates as well as bubble nucleation occuring in the electroweak phase transition when the scalar field is identified with the Higgs field. We calculate the determinant exactly and find an approximation (essentialy of the gradient expansion type) that reproduces the exact results very well if the fermion mass $m_F$ becomes larger than the inverse of the size $R$ of the background field configuration. The results are given for 2, 3 and 4 Euclidean dimensions.