Radiative corrections in the $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$ theory: A simple renormalization framework

A simple renormalization framework to carry out practical calculations in the $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$ theory is discussed. The basic counterterms associated with the mass matrix of the gauge bosons and their interactions with quarks and leptons are generated and determined in a straightforward manner. They can then be applied, in a systematic fashion, to study the radiative corrections to the various processes of interest: $\mu$ decay, $\beta$ decay, $\nu$-induced reactions, and other leptonic and semileptonic processes. The role played by ${cos\theta }_W$ is discussed in some detail. Using the results and methods of the current-algebra formulation of radiative corrections, an effective Lagrangian is derived for $\mu$ decay in which the contributions of the heavy particles and some of the photonic corrections are reduced to a renormalization factor of the zeroth-order amplitude. This analysis determines the connection between the renormalized constants of the present framework and $G_{\mu }$. The corrections to the relation $m_W=\frac{{\left(\frac{\pi \alpha }{\sqrt{2}{G}_{\mu }}\right)}^{\frac{1}{2}}}{{sin\theta }_W}$ are discussed. This leads to some relevant observations concerning certain model-dependent hadronic contributions to these corrections and their cancellation in the comparison between charged-current transitions and neutral-current processes occurring at large momentum transfers.