In the $V-A$ theory of weak interactions with coupling constant $G$, whether or not it is mediated by an intermediate vector meson, it is shown to order $\alpha G$ that the contribution of the vector hadron current to the Fermi part of the amplitude for $\beta$ decay involving hadrons in the same isomultiplet is independent of the details of the strong interactions. This contribution to the radiative corrections to $\beta$ decay has been re-evaluated, with results that agree in form with those obtained previously with the use of zero-order perturbation theory in the strong interactions. The contribution of the axial-vector current is evaluated approximately for various models of the strong interactions. These results are compared with the predictions of universality, and generally there is good agreement with the Cabibbo angle determined from $K_{e3\mathrm{}}$ decay. The factors which influence the cutoff dependence of the theory are discussed, and among other things, it is pointed out that with the neglect of certain "small" quantities, the ratio of the rates for $\mu$ decay and $\beta$ decay is cutoff-independent in the theory with an intermediate vector meson. The effect of Schwinger terms is studied, as well as the relationship between the cancellation occurring in the vector contribution to $\beta$ decay and the cancellation in a Yang-Mills theory which occurs as a consequence of the Ward identity.

• Received 21 August 1967

DOI:https://doi.org/10.1103/PhysRev.167.1461