A comprehensive theory framework for perturbative calculations of $δ_C$ in superallowed beta decays

Abstract: The present high precision determination of $|V_{ud}|$ from superallowed beta decays of $J<sup>P=0<sup>+$, $T=1$ nuclei is largely built on top of the standard'' calculation of the isospin breaking correction $δ_C$ to the Fermi matrix element, which assumes the splitting $δ_C=δ_{C1}+δ_{C2}$, where $δ_{C1}$ and $δ_{C2}$ represent theisospin mixing'' correction and the radial mismatch'' correction, respectively. In this paper I show that this formalism violates the rule of nucleon basis independence, similar to the gauge invariance requirement in quantum field theory, therefore its outcome is inevitably subject to uncontrolled systematic errors. I further argue that the perturbative formalism is the only approach that allows the computation of $δ_C$ at the precision level required for the test of the Cabibbo unitarity. Advancing from existing literature, I develop the fullgenerating function approach'' to compute $δ_C$ perturbatively, with connections to nuclear mass splittings and the isospin breaking in nuclear charge radii as theory benchmarks.