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

Abstract

The theory of the beta decay of complex nuclei, $N_i\to N_f+e^{-}+{\overline{\nu }}_e$, 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 $G_V(N_i\to N_f)$ and $G_A(N_i\to N_f)$; the numerical values of $G_V(N_i\to N_f)$ and $G_A(N_i\to N_f)$ 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 $G_V(N_i\to N_f)$, while PCAC, together with a suitable pion-pole-dominance assumption, implies the Goldberger-Treiman (G-T) relation which expresses $G_A(N_i\to N_f)$ in terms of the pion-initial-nucleus-final-nucleus coupling constant $f_{\pi N_i{N}_f}$; this coupling constant can be found from a pological analysis of $n+N_f\to p+N_i$ nucleon charge-exchange scattering experiments. Since such experiments are not as yet available, we calculate the values of the $f_{\pi N_i{N}_f}$ 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 $f_{\pi N_i{N}_f}$ calculated by means of the G-T relation from the $G_A(N_i\to N_f)$ 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.