Three-loop corrections to the Fermi decay constant in the $\overline{\rm{MS}}$ scheme

TL;DR

This paper calculates the leading three-loop corrections to the Fermi decay constant within the Standard Model using the er scheme, significantly reducing theoretical uncertainties and refining the Higgs VEV estimate.

Contribution

It provides the first exact three-loop calculation of the Fermi constant in the er scheme, improving precision and reducing scale dependence compared to previous two-loop results.

Findings

Three-loop corrections decrease the Higgs VEV estimate by about 3.5 MeV.

Renormalization scale dependence is reduced to less than d7 10^{-6}.

Theoretical uncertainty in the VEV is now under 1 MeV.

Abstract

I present the leading 3-loop contributions to the Fermi decay constant in the Standard Model, using the tadpole-free pure $\overline{\rm{MS}}$ scheme. The calculation is exact in the limit in which the QCD coupling, the top-quark Yukawa coupling, and the square root of the Higgs self-coupling are all treated as large compared to the electroweak gauge couplings. The effect of the 3-loop contribution is to decrease the estimate of the Higgs VEV, for fixed on-shell inputs, by about 3.5 MeV. The renormalization scale dependence of the computed Fermi constant is greatly reduced compared to the previously known complete 2-loop order result, and as a fraction is now less than $\pm 2 \times 10^{-6}$ for renormalization scales between 90 and 250 GeV. This corresponds to a theoretical uncertainty in the VEV that is well under 1 MeV, and much smaller than the current parametric uncertainties coming mostly from the top-quark mass. I also comment on the impact on the precise prediction of the $W$-boson mass.