Exact computation of the small-fluctuation determinant around a sphaleron

Abstract

We compute exactly, in the high-temperature limit, the determinant $\ensuremath{\kappa}$ of small fluctuations around the sphaleron configuration of electroweak theory, exploiting the symmetry of the sphaleron under spatial rotations combined with isospin and custodial SU(2) transformations. For the ratio $\frac{\ensuremath{\lambda}}{{g}^{2}}$ of scalar four-point coupling $\ensuremath{\lambda}$ to gauge coupling ${g}^{2}$ near unity, we find that $\ensuremath{\kappa}$ is 0.03. For $\frac{\ensuremath{\lambda}}{{g}^{2}}$ large corresponding to a strongly coupled Higgs phase, or for $\frac{\ensuremath{\lambda}}{{g}^{2}}$ very small tending to the Coleman-Weinberg limit, we find that the determinant strongly suppresses the rate of baryon-number-changing processes.