Application of the Goldberger-Treiman Relation to the Beta Decay of Complex Nuclei

Abstract

The theory of the beta decay of complex nuclei, ${N}_{i}\ensuremath{\rightarrow}{N}_{f}+{e}^{\ensuremath{-}}+{\overline{\ensuremath{\nu}}}_{e}$, is developed on the basis of a treatment which considers the nuclei involved (${N}_{i}$) as "elementary" particles and applies the hypotheses of the conserved polar-vector hadron weak current (CVC) and the partially conserved axial-vector hadron weak current (PCAC) to determine the effective polar-vector and axial-vector weak coupling constants ${G}_{V}({N}_{i}\ensuremath{\rightarrow}{N}_{f})$ and ${G}_{A}({N}_{i}\ensuremath{\rightarrow}{N}_{f})$; the numerical values of ${G}_{V}({N}_{i}\ensuremath{\rightarrow}{N}_{f})$ and ${G}_{A}({N}_{i}\ensuremath{\rightarrow}{N}_{f})$ reflect in this treatment the complexity of internal nuclear structure. Using CVC, and supposing that $|{N}_{i}〉$ and $|{N}_{f}〉$ are sufficiently pure isospin eigenstates, we can immediately calculate ${G}_{V}({N}_{i}\ensuremath{\rightarrow}{N}_{f})$, while PCAC, together with a suitable pion-pole-dominance assumption, implies the Goldberger-Treiman (G-T) relation which expresses ${G}_{A}({N}_{i}\ensuremath{\rightarrow}{N}_{f})$ in terms of the pion-initial-nucleus-final-nucleus coupling constant ${f}_{\ensuremath{\pi}{N}_{i}{N}_{f}}$; this coupling constant can be found from a pological analysis of $n+{N}_{f}\ensuremath{\rightarrow}p+{N}_{i}$ nucleon charge-exchange scattering experiments. Since such experiments are not as yet available, we calculate the values of the ${f}_{\ensuremath{\pi}{N}_{i}{N}_{f}}$ in terms of the known magnetic moments of ${N}_{i}$ and ${N}_{f}$ by means of a very crude theory, and compare these values with the values of the ${f}_{\ensuremath{\pi}{N}_{i}{N}_{f}}$ calculated by means of the G-T relation from the ${G}_{A}({N}_{i}\ensuremath{\rightarrow}{N}_{f})$ deduced from observed beta-decay rates. The agreement is, in general, somewhat better than that found between calculated and observed rates in the customary impulse-approximation theory of beta decay.