Exact computation of the small-fluctuation determinant around a sphaleron

We compute exactly, in the high-temperature limit, the determinant $\kappa$ 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 $\frac{\lambda }{g^2}$ of scalar four-point coupling $\lambda$ to gauge coupling $g^2$ near unity, we find that $\kappa$ is 0.03. For $\frac{\lambda }{g^2}$ large corresponding to a strongly coupled Higgs phase, or for $\frac{\lambda }{g^2}$ very small tending to the Coleman-Weinberg limit, we find that the determinant strongly suppresses the rate of baryon-number-changing processes.