Toward ab-initio nuclear theory calculations of ${\delta }_C$

We propose a theory framework to study the isospin-symmetry breaking correction ${\delta }_{\text{C}}$ in superallowed nuclear $\beta$ decays, crucial for the precise determination of $|V_{ud}|$. Based on a general assumptions of the isovector dominance in isospin-symmetry breaking interactions, we construct a set of functions $F_{T_z}$ which involve nuclear matrix elements of isovector monopole operators and 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 ${\delta }_{\text{C}}$. We outline a strategy to perform ab initio calculations of $F_{T_z}$ based on the Lanczos algorithm, and discuss its similarity with other nuclear-structure-dependent inputs in nuclear $\beta$ decays.