Model Dependence of Corrections to the Fermi Function and ft Values of Some 0+ → 0+ Super-Allowed β Decays, and the Value of the Vector Coupling Constant

Abstract

This paper presents a consistent field-theoretic perturbative calculation of the Fermi function and the analytical model dependences of corrections to it. The screening, finite nuclear size, and radiative corrections are considered in a unified treatment. We are able to separate out the point-nucleus Fermi function $F_0$ from the corrections, and thus only treat the corrections in a power-series expansion. The proper way of incorporating the radiative correction into $F_0$ is discussed. Detailed analytical results for the model dependence of the finite-nuclear-size and radiative corrections are presented. The most realistic model chosen employs the nuclear charge distributions as given by Hofstadter and the single-particle harmonic-oscillator model for the nuclear matrix elements. The model-dependent results are used to calculate the $\mathrm{ft}$ values of the super-allowed decays of ${\mathrm{O}}^{14}$, ${\mathrm{Al}}^{26}$, ${\mathrm{Cl}}^{34}$, ${\mathrm{Sc}}^{42}$, ${\mathrm{V}}^{46}$, ${\mathrm{Mn}}^{50}$, and ${\mathrm{Co}}^{54}$. The values of the vector coupling constant $G_V$ are calculated for the above nuclei using the most realistic model, and the variation of the $G_V$ is discussed in the light of the predictions of the conserved-vector-current theory of weak interactions.