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We propose a new theory framework to study the isospin-symmetry breakingcorrection $\delta_\text{C}$ in superallowed nuclear beta decays, crucial forthe precise determination of $|V_{ud}|$. Based on a general assumptions of theisovector dominance in ISB interactions, we construct a set of functions$F_{T_z}$ which involve nuclear matrix elements of isovector monopole operatorsand the nuclear Green's function. Via the functions $F_{T_z}$, a connection of$\delta_\text{C}$ to measurable electroweak nuclear radii is established,providing an experimental gauge of the theory accuracy of $\delta_\text{C}$. Weoutline a strategy to perform ab-initio calculations of $F_{T_z}$ based on theLanczos algorithm, and discuss its similarity with othernuclear-structure-dependent inputs in nuclear beta decays.

Towards $\it{ab}$-$\it{initio}$ nuclear theory calculations of $\delta_\mathrm{C}$