Static Coulomb Corrections to the Point-Nucleus Fermi Function in Nuclear Beta Decay

Abstract

In this paper we use a simple model to include the effects of nuclear charge change in $\beta$ decay. We develop the point-nucleus Fermi function for the daughter nucleus in perturbation theory to $Z^2{\alpha }^2$ and corrections to it arising from the change in nuclear charge. The model considers only Coulomb interactions between the static parent-daughter nucleus with the leptons and allows for pair creation. This model has been used by Chern et al. and by Halpern and Chern to obtain analytical model dependences of the finite nuclear size, screening, and static $\beta$-vertex correction to first order in $\alpha$. The divergences in the Fermi function are shown to have the same source as the divergences occuring in this model due to the change in nuclear charge, so that both types of divergence must be treated in the same manner, including the use of the same cutoff. We find that the $Z{\alpha }^2$ contribution to the transition probability per unit time calculated from this model is comparable to the screening correction, and thus must be included if one takes the screening into account. An uncertainty has arisen in the literature concerning the incorporation of the first-order radiative correction into the Fermi function. We find from this model that the approximate factorization previously used in the literature to order $\alpha$ is also valid to order ${\alpha }^2$ for nuclei with $Z$ less than that of ${\mathrm{Al}}^{26}$. The validity of this factorization for higher $Z$ is still unresolved.