Radial overlap correction to superallowed $0^+ \to 0^+$ nuclear $\beta$ decays using the shell model with Hartree-Fock radial wave functions

The radial overlap correction, $\delta_{RO}$ is re-examined for 30 superallowed $\beta$ decays using the shell model with Hartree-Fock (HF) radial wave functions. Our calculation is based on the Skyrme interaction including Coulomb, Charge-Symmetry-Breaking (CSB) and Charge-Independence-Breaking (CIB) terms. The corrections due to the gradient of charge density, vacuum polarization, Coulomb spin-orbit and finite size of nucleons are also considered. To avoid the spurious isospin mixing, the local equivalent potential is constructed from the solution of a HF calculation for the $Z=N$ nucleus without charge-dependent forces. The obtained mean field is solved non-iteratively for the parent and daughter nuclei. It turns out that the CIB term has no significant impact on $\delta_{RO}$. On the other hand, the CSB term makes $\delta_{RO}$ increasing between 10 and 30%. The gradient density leads to a further increase of $\delta_{RO}$ between 2 and 14%, while other electromagnetic corrections are negligible. The effect of the suppression of isospin spuriosity is somewhat complicated. In general, it produces a larger $\delta_{RO}$ value, however there are few cases for which $\delta_{RO}$ is mostly unaffected or even reduced, especially the light even-even emitters. All these improvements partly explain the long-standing discrepancy between the correction values obtained with Woods-Saxon (WS) and HF radial wave functions. Nevertheless, the remaining discrepancy is still significant, except for the emitters with $A\le 26$. In addition, the staggering presented on the results obtained with the WS potential are not well reproduced by our calculations. This subsequently leads to a large difference in the predicted mirror $ft$ ratios. These problems might relate to the errors of the calculated radii as well as correlation in the data used for constraining the asymptotic radial wave functions.