Quantum Fluctuations around the Electroweak Sphaleron

TL;DR

This work delivers a self-contained computation of the one-loop quantum corrections to the electroweak sphaleron rate by evaluating the fluctuation determinant around the sphaleron using a Euclidean Green-function approach with a partial-wave decomposition in the background gauge. The authors carefully treat zero and unstable modes and perform renormalization within a three-dimensional high-temperature effective theory, ultimately showing that quantum fluctuations enhance the sphaleron transition rate relative to the classical saddle point estimate and that their results differ dramatically from prior analyses. Through detailed numerical analysis across the Higgs-to-W mass ratio $\xi$, the study clarifies the reliability of the DPY approximation and emphasizes gauge-invariant aspects of the sphaleron rate, with implications for baryon-number violation in the early universe. The methodology and findings provide a robust benchmark for finite-temperature electroweak theory and its role in early-universe cosmology.

Abstract

We present an analysis of the quantum fluctuations around the electroweak sphaleron and calculate the associated determinant which gives the 1--loop correction to the sphaleron transition rate. The calculation differs in various technical aspects from a previous analysis by Carson et al. so that it can be considered as independent. The numerical results differ also -- by several orders of magnitude -- from those of this previous analysis; we find that the sphaleron transition rate is much less suppressed than found previously.