Higher-order isospin-symmetry-breaking corrections to nuclear matrix elements of Fermiβdecays

Within the nuclear shell model, we derive the exact expression for the isospin-symmetry breaking correction to the nuclear matrix element of Fermi $\beta$ decays. Based on a perturbation expansion in small quantities, such as the deviation of the overlap integral between proton and neutron radial wave functions from unity and of the transition density from its isospin-symmetry value, we demonstrate that ${\delta }_C$ can be obtained as a sum of six terms. These terms comprise two leading order (LO) terms, two next-to-leading order (NLO) terms, one next-to-next-to-leading order (NNLO) term, and one next-to-next-to-next-to-leading order (NNNLO) term. While the first two terms have been considered in a series of shell-model calculations [J. C. Hardy and I. S. Towner, Phys. Rev. C 102, 045501 (2020), and references therein], the remaining four terms have been neglected. A numerical calculation has been carried out for 24 superallowed $0^{+}\to 0^{+}$ transitions (18 isotriplets and six isoquintets) and three non-$0^{+}\to 0^{+}$ transitions, across the $p$ to $pf$ shells. For most $0^{+}\to 0^{+}$ transitions, the higher-order contribution is of the order ${10}^{-3}\%$ or smaller, well below the typical theoretical errors quantified within the shell model with Woods-Saxon radial wave functions given in the reference cited above. However, for specific cases such as ${}^{70}\mathrm{Br}$ and ${}^{74}\mathrm{Rb}$, where weakly bound effect dominates, it increases considerably, becoming comparable to or even exceeding the errors in the isospin mixing component of the LO terms. In the cases of ${}^{20}\mathrm{Mg}$ and ${}^{48}\mathrm{Fe}$, as well as in non-$0^{+}\to 0^{+}$ transitions, the higher-order contribution becomes more substantial. Notably, it reaches as large as $-4.460\%$ in ${}^{31}\mathrm{Cl}$ and $-2.027\%$ in ${}^{32}\mathrm{Cl}$, due to the concurrent effect of the weakly bound and strong isospin mixing in their daughter nuclei. In contrast, for ${}^{26}\mathrm{P}$, the NLO terms, despite their substantial magnitude, effectively cancel each other out due to their opposite signs.