Dispersive formalism for the nuclear structure correction ${\delta }_{\mathrm{N}\mathrm{S}}$ to the $\beta$ decay rate

We analyze the axial $\gamma W$-box diagram for $I(J^P)=1(0^{+})$ nuclei and provide a dispersion representation of the nuclear-structure correction ${\delta }_{\text{NS}}$ including its energy-dependent part. We also summarize useful isospin rotation formula and representations in nuclear theory that could facilitate the calculation of the parity-odd nuclear structure function $F_3(\nu ,Q^2)$. They provide a rigorous theory framework for the future, high-precision calculation of the nuclear structure correction ${\delta }_{\text{NS}}$ necessary for the extraction of the Cabibbo-Kobayashi-Maskawa matrix element $\mathrm{\mid }{V}_{ud}\mathrm{\mid }$ from superallowed nuclear $\beta$ decays.