Sum Rules in Nuclear Beta Decay

The previously developed "elementary-particle" treatment of nuclear beta decay, $N_i\to N_j+e^{\mp }+{\nu }_e$, is combined with the Adler-Weisberger procedure, to obtain sum rules relating axial-vector coupling constants for nuclear beta decays, $G_A(N_i\to N_f)$, to integrals over pion energy of pion-final-nucleus total cross sections, $\mathcal{g}({\pi }^{\pm },N_f)$. It is shown that the ratio $\left[\frac{\mathcal{g}({\pi }^{\pm },N_f)}{\mathcal{g}({\pi }^{\pm },p)}\right]=-\left[\frac{\mathcal{g}({\pi }^{\pm },N_f)}{\mathcal{g}({\pi }^{\pm },n)}\right]$ is just equal to the difference between the number of protons and the number of neutrons in $N_f$, [$Z-(A-Z)$], multiplied by a factor which becomes unity in the approximation of neglect of meson-exchange corrections to $G_A(N_i\to N_f)$.