Approximate computation of the sphaleron prefactor. Application to the two-dimensional Abelian Higgs model

The determinant $\kappa$ of thermal fluctuations around the sphaleron configuration of the (1+1)-dimensional Abelian Higgs model is computed in the limit of large temperatures using approximations based on Schwinger's proper-time representation. When the gauge-boson mass $M_W$ exceeds the scalar-boson mass $M_H$ these approximations give results which agree well with the known closed-form expressions for the determinant. For $M_W\le M_H$, however, the approximations fail to give accurate estimates. Implications for estimating the sphaleron prefactor of electroweak theory are briefly discussed.