Three-loop corrections to the Fermi decay constant in the MS¯ scheme

I present the leading 3-loop contributions to the Fermi decay constant in the Standard Model, using the tadpole-free pure <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mover accent="true"><a:mi>MS</a:mi><a:mo stretchy="true">¯</a:mo></a:mover></a:math> scheme. The calculation is exact in the limit in which the quantum chromodynamics 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 vacuum expectation value (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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mo>±</e:mo><e:mn>2</e:mn><e:mo>×</e:mo><e:msup><e:mn>10</e:mn><e:mrow><e:mo>−</e:mo><e:mn>6</e:mn></e:mrow></e:msup></e:math> 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>W</g:mi></g:math>-boson mass.