Sum Rules in Nuclear Beta Decay

Abstract

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