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We study the shell-model formalism to include the isospin-symmetry-breakingcorrection ($\delta_{C}$) to nuclear matrix element of superallowed $0^+\to0^+$ Fermi $\beta$ decays of $T=1$ nuclei. Based on a perturbation expansion ina small quantity, such as the deviation of the overlap integral between protonand neutron radial wave functions from unity or of the transition density fromits isospin-symmetry value, we derive that $\delta_C$ can be obtained as a sumof six terms, including two leading order (LO) terms, two next-to-leading order(NLO) terms, one next-to-next-to-leading order (NNLO) term and onenext-to-next-to-next-to-leading order (NNNLO) term. The first two terms havebeen considered in a series of shell-model calculations of Towner and Hardy aswell as in the recent calculation of the present authors, while the remainingfour terms are usually neglected. A numerical calculation has been carried outfor 13 transitions in the $p$, $sd$ and the lower part of $pf$ shells. Ourresults indicate that the magnitude of the sum of all higher order terms is ofthe order of $10^{-3}$ %. This number is well below typical theoretical errorsquantified within the shell model with Woods-Saxon radial wave functions.

Higher-order isospin-symmetry breaking corrections to nuclear matrix elements of superallowed $0^+\to 0^+$ Fermi $\beta$ decay of $T=1$ nuclei