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

RESUMO

The present high precision determination of $|V_{ud}|$ from superallowed beta decays of $J^P=0^+$, $T=1$ nuclei is largely built on top of the ``standard'' calculation of the isospin breaking correction $\delta_C$ to the Fermi matrix element, which assumes the splitting $\delta_C=\delta_{C1}+\delta_{C2}$, where $\delta_{C1}$ and $\delta_{C2}$ represent the ``isospin 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 $\delta_C$ at the precision level required for the test of the Cabibbo unitarity. Advancing from existing literature, I develop the full ``generating function approach'' to compute $\delta_C$ perturbatively, with connections to nuclear mass splittings and the isospin breaking in nuclear charge radii as theory benchmarks.