Unsubtracted Dispersion Relations and the Renormalization of the Weak Axial-Vector Coupling Constants

Assuming that the equal-time commutation rules for the vector and axial-vector-current octets proposed by Gell-Mann are valid and that the divergence of the $\Delta S=0\mathrm{}$, $\Delta I=1\mathrm{}$ axial current is a strongly convergent operator obeying unsubtracted dispersion relations and dominated by low-frequency contributions, we derive a sum rule for the renormalization of the neutron axial $\beta$-decay constant $G_A$, by the strong interactions. The result agrees with that previously obtained from the assumption that the axial-current divergence is proportional to the pion field. The results are generalized to the strangeness-changing leptonic decays in the context of Cabibbo theory and generalized Goldberger-Treiman relations, and are used to compute the $\frac{d}{f}$ ratio for the weak baryon axial-current coupling and an independent value of $G_A$.