Considerations Concerning the Radiative Corrections to Muon Decay in the Fermi and Standard Theories

The FAC, PMS, and BLM optimization methods are applied to the QED corrections to the muon lifetime in the Fermi V-A theory. The FAC and PMS scales are close to m_e, while the BLM scale nearly concides with the geometric average \sqrt{m_e m_\mu}. The optimized expressions are employed to estimate the third order coefficient in the \alpha(m_\mu) expansion and the theoretical error of the perturbative series. Using arguments based on effective field theory and a simple examination of Feynman diagrams, it is shown that, if contributions of O(\alpha m_\mu^2/M_W^2) are neglected, the corrections to muon decay in the SM factorize into the QED correction of the Fermi V-A theory and the electroweak amplitude g^2/(1-\Delta r), both of which are strictly scale-independent. We use the results to clarify how the QED corrections to muon decay and the Fermi constant G_F should be used in the SM, and what is the natural choice of scales if running couplings are employed.