Radial Overlap Correction to Superallowed $0^+ \to 0^+$ $\beta $-decay revisited

TL;DR

This paper revisits the calculation of radial overlap corrections in superallowed $0^+ o 0^+$ beta decays using an optimized shell model approach, improving the accuracy of theoretical corrections for fundamental symmetry tests.

Contribution

It introduces a new method for calculating charge radii and refines the radial mismatch correction within the shell model, reducing dependence on parametrization choices.

Findings

New correction values generally agree with previous results.

Optimized potential parameters reduce model dependence.

Small differences observed in corrected $ar{ ext{Ft}}$ values.

Abstract

Within the nuclear shell model, we investigate the correction $\delta_{RO}$ to the Fermi matrix element due to a mismatch between proton and neutron single-particle radial wave functions. Eight superallowed $0^+ \to 0^+$ $\beta$ decays in the $sd$-shell, comprised of $^{22}$Mg, $^{26m}$Al, $^{26}$Si, $^{30}$S, $^{34}$Cl, $^{34}$Ar, $^{38m}$K and $^{38}$Ca are re-examined. The radial wave functions are obtained from a spherical Woods-Saxon potential whose parametrizations are optimized in a consistent adjustment of the depth and the length parameter to relevant experimental observables, such as nucleon separation energies and charge radii, respectively. The chosen fit strategy eliminates the strong dependence of the radial mismatch correction to a specific parametrization, except for calculations with an additional surface-peaked term. As an improvement, our model proposes a new way to calculate the charge radii, based on a parentage expansion which accounts for correlations beyond the extreme independent-particle model. Apart from the calculations with a surface-peak term and the cases where we used a different model space, the new sets of $\delta_{RO}$ are in general agreement with the earlier result of Towner and Hardy [1]. Small differences of the corrected average $\overline{\mathcal{F}t}$ value are observed.