Divergent Part of the Second-Order Electromagnetic Corrections to the RatioGAGV

The divergent part of the second-order electromagnetic corrections to the axial-vector matrix elements is studied. The main tool in this work is a simple extension of the Bjorken limit to the case involving the timeordered product of three currents (or local operators.) Applying this generalization to theories in which all gradient terms are assumed to be $c$ numbers, one finds, in the over-all corrections to the axial-vector matrix elements, logarithmic divergences which are not present (to the same approximation in $\alpha$) in the Fermi amplitude. These divergences give a vanishing contribution if a relevant double commutator involves only a $\delta$ function and if the commutator [${\partial }_0{j}^{\mu }\left(\mathrm{x}\right), j_{\mu }\mathrm{}\left(0\right)\mathrm{}$] is a $c$ number, $j^{\mu }$ being the hadronic part of the electromagnetic current. If these two conditions hold, one can show, for example, that the second-order corrections to $\frac{G_A}{G_V}$ (in neutron $\beta$ decay) are finite in the simple intermediate-boson model previously discussed. In the usual case in which [${\partial }_0{j}^{\mu }\left(\mathrm{x}\right), j_{\mu }\mathrm{}\left(0\right)\mathrm{}$] is an operator, one finds, in general, uncompensated logarithmic divergences in the over-all corrections to $\frac{G_A}{G_V}$ with coefficients which appear to be highly model-dependent. The effect of operator gradient terms and the possible connection with the well-known divergences in the electromagnetic mass shifts is discussed. The case of the $S{U}_3\otimes S{U}_3$ algebra of fields is studied separately. Applying the generalization of the Bjorken limit, it is easy to identify in this theory the part of the (divergent) radiative corrections to $\beta$ decay which is associated with the mass shifts of the charged vector and axial-vector mesons. After this part is removed by a mass renormalization of such mesons, the remaining divergences maintain the chiral structure, are independent of momentum transfer, and amount to one-half the Bjorken correction factor. Thus, by applying the method of this paper, one readily recovers, in the case of the algebra of fields, Schwinger's interpretation of this factor, as well as his expressions for the divergent parts of the mass shifts of the charged vector and axial-vector mesons. It is pointed out that, in the broader context of current algebra, such identifications are model-dependent. This is illustrated by examining, on the basis of the approach of this work, the familiar example of muon decay.