We derive the corrections induced by the $W$-boson propagator to the differential rates of the leptonic decay of a polarized muon and $\tau$ lepton. Results are presented both for decays inclusive of inner bremsstrahlung as well as for radiative ones, when a photon emitted in the decay process is measured. The numerical effect of these corrections is discussed. The definition of the Fermi constant $G_F$ is briefly reviewed.

• Received 23 October 2013

DOI:https://doi.org/10.1103/PhysRevD.88.093011