Universality of the Weak Vector Coupling Constant

A previous calculation of the radiative corrections to the rate for pure Fermi $\beta$ decay in the $V-A$ theory, computed to first order in $\alpha$ and to zero order in the lepton momenta, is extended here to include, in addition, the corrections linear in the product of a lepton momentum and a nuclear charge radius. These additional contributions are shown to modify the Coulomb correction to the positron wave function to account for the finite distribution of electric and isotopic charge within the nucleus. They are evaluated and taken together with the other electromagnetic corrections, the origin of which are reviewed in the text, to reassess the question of the universality of the weak vector coupling constant. The conclusions obtained, although basically in agreement with the results of others, differ in detail from previous surveys of universality, generally in a direction to strengthen the concept of universality in the sense of Cabibbo.