Radiative corrections in theSU(2)L×U(1)theory: A simple renormalization framework

Abstract

A simple renormalization framework to carry out practical calculations in the $\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ 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: $\ensuremath{\mu}$ decay, $\ensuremath{\beta}$ decay, $\ensuremath{\nu}$-induced reactions, and other leptonic and semileptonic processes. The role played by ${cos\ensuremath{\theta}}_{W}$ is discussed in some detail. Using the results and methods of the current-algebra formulation of radiative corrections, an effective Lagrangian is derived for $\ensuremath{\mu}$ 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 ${G}_{\ensuremath{\mu}}$. The corrections to the relation ${m}_{W}=\frac{{(\frac{\ensuremath{\pi}\ensuremath{\alpha}}{\sqrt{2}{G}_{\ensuremath{\mu}}})}^{\frac{1}{2}}}{{sin\ensuremath{\theta}}_{W}}$ 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.