Improved calculation of the isospin-symmetry-breaking corrections to superallowed Fermi $\beta$ decay
I. S. Towner and J. C. Hardy

We report new shell-model calculations of the isospin-symmetry-breaking correction, $\delta_C$, to superallowed $0^{+} \rightarrow 0^{+}$ nuclear $\beta$ 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 pick-up 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 $\delta_{NS}$ 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, $\DRV$, and a nucleus-dependent ``outer" term, $\delta_R^{\prime}$ We tabulate the new values for $\delta_C$, $\delta_{NS}$ and $\delta_R^{\prime}$ for twenty superallowed transitions, including the thirteen currently well-studied cases. With these new correction terms the corrected $\F t$ values for the thirteen cases are statistically consistent with one another and the anomalousness of the $^{46}$V result disappears. These new calculations lead to a lower average $\overline{\F t}$ value and a higher value for $V_{ud}$. The sum of squares of the top-row elements of the CKM matrix now agrees exactly with unitarity.

© 2008 The American Physical Society.