Radiative Corrections to the Goldberger-Treiman Relation: High-Frequency Contributions

The second-order electromagnetic corrections to the Goldberger-Treiman relation are discussed. It is shown that when this relation is expressed in terms of physical coupling constants and masses, renormalized by electromagnetism, the residual electromagnetic corrections are finite. The derivation assumes, among other things, the existence of operator-product expansions at short distances and Wilson's enumeration of the fields of low dimensionality. Furthermore, the hadronic matrix elements in the corrections of order $\alpha$ are treated in the PCAC (partial conservation of axial-vector current) approximation. Some observations are made concerning the applicability of PCAC to the study of radiative corrections. A new physical decay constant, $f_{\pi }^{\prime }$, renormalized by electromagnetism, is introduced and determined from experiment. Some salient features of the theory of $\beta$ decay to first order in $\alpha$ are reviewed, with particular emphasis on the role played by the renormalized constants.