Current algebra formulation of radiative corrections in gauge theories and the universality of the weak interactions

A current algebra formulation of the radiative corrections in gauge theories, with special applications to the analysis of the universality of the weak interactions, is developed in the framework of quantum chromodynamics. For definiteness, we work in the SU(2)×U(1) model with four quark flavors, but the methods are quite general and can be applied to other theories. The explicit cancellation of ultraviolet divergences for arbitrary semileptonic processes is achieved relying solely on the Ward identities and general considerations, both in the $W$ and Higgs sectors. The finite parts of order $G_F\alpha$ are then evaluated in the case of the superallowed Fermi transitions, including small effects proportional to $g_s^{-2\mathrm{}}\left({\kappa }^2\right)$, which are induced by the strong interactions in the asymptotic domain. We consider here both the simplest version of the Weinberg-Salam model in which the Higgs scalars transform as a single isospinor, as well as the case of general symmetry breaking. Except for the small effects proportional to $g_s^{-2\mathrm{}}\left({\kappa }^2\right)$, the results are identical to the answers previously found on the basis of heuristic arguments. The phenomenological verification of Cabibbo universality on the basis of these corrections and the superallowed Fermi transitions has been discussed before and found to be in very good agreement with present experimental evidence. The analogous calculation for the transition rate of pion $\beta$ decay is given. Theoretical alternatives to quantum chromodynamics as a framework for the evaluation of the radiative corrections are briefly discussed. The appendixes contain a generalization of an important result in the theory of radiative corrections due to L. S. Brown, G. Preparata, and W. I. Weisberger, an analysis of the hadronic contributions to the $W$ and $\phi$ propagators, mathematical methods for evaluating the $g_s^{-2\mathrm{}}\left({\kappa }^2\right)$ corrections, and discussions of quark mass renormalization and the absence of operator "seagulls" in the hadronic correlation functions. Some of the methods discussed in this paper can also be applied to the study of radiative corrections of order $G_F\alpha$ to other processes affected by the strong interactions.