Improved calculation of the isospin-symmetry-breaking corrections to superallowed Fermi β decay

Abstract

We report new shell-model calculations of the isospin-symmetry-breaking correction ${\ensuremath{\delta}}_{C}$ to superallowed ${0}^{+}\ensuremath{\rightarrow}{0}^{+}$ nuclear \ensuremath{\beta} decay. The most important improvement is the inclusion of core orbitals, which are demonstrated to have a significant impact on the mismatch in the radial wave functions of the parent and daughter states. We determine which core orbitals are important to include from an examination of measured spectroscopic factors in single-nucleon pickup reactions. In addition, where new sets of effective interactions have become available since our last calculation, we now include them; this leads to small changes in ${\ensuremath{\delta}}_{\mathrm{NS}}$ as well. We also examine the new radiative-correction calculation by Marciano and Sirlin and, by a simple reorganization, show that it is possible to preserve the conventional separation into a nucleus-independent ``inner'' radiative term, ${\ensuremath{\Delta}}_{R}^{V}$, and a nucleus-dependent ``outer'' term, ${\ensuremath{\delta}}_{R}^{'}$. We tabulate the new values for ${\ensuremath{\delta}}_{C},{\ensuremath{\delta}}_{\mathrm{NS}}$, and ${\ensuremath{\delta}}_{R}^{'}$ for 20 superallowed transitions, including the 13 currently well-studied cases. With these new correction terms, the corrected $\mathcal{F}t$ values for the 13 cases are statistically consistent with one another, and the anomalousness of the $^{46}\mathrm{V}$ result disappears. These new calculations lead to a lower average $\overline{\mathcal{F}t}$ value and a higher value for ${V}_{\mathrm{ud}}$. The sum of squares of the top-row elements of the Cabibbo-Kobayashi-Maskawa matrix now agrees exactly with unitarity.