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

Stephen P. Martin

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arXiv · 2507.15946

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

Stephen P. Martin

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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.

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Stephen P. Martin. 2025-07-21. Three-loop corrections to the Fermi decay constant in the $\overline{\rm{MS}}$ scheme. https://arxiv.org/abs/2507.15946

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