Third order corrections to the semi-leptonic \boldmath{$b\to c$} and the muon decays

Abstract

We compute corrections of order ${\alpha }_s^3$ to the decay $b\to c\ell \bar{\nu }$ taking into account massive charm quarks. In the on-shell scheme large three-loop corrections are found. However, in the kinetic scheme the three-loop corrections are below 1\% and thus perturbation theory is well under control. We furthermore provide results for the order ${\alpha }_s^3$ corrections to $b\to u\ell \bar{\nu }$ and the third-order QED corrections to the muon decay which will be important input for reducing the uncertainty of the Fermi coupling constant $G_F$.

7 pages, v2: matches published version