The leading nuclear-structure electrostatic correction in arbitrary $\beta$ decays
We develop a systematic theoretical framework to improve theoretical predictions for nuclear $\beta$ decays of arbitrary angular momentum $J$, leading to a model-independent nuclear-structure electrostatic correction to the Coulomb interaction between the emitted lepton and the nuclear charge distribution, useful for ongoing and future precision searches for physics beyond the Standard Model. The formalism is based on nuclear matrix elements expanded in multipole operators, as commonly used in \emph{ab initio} calculations. First-order Coulomb corrections are derived from one-photon exchange preserving the full multipole and angular structure of the decay rate, and are subsequently expanded in the relevant small parameters of the nuclear problem, suppressing the leading nuclear structure correction to a few per-mills for medium mass nuclei with natural beta decay properties. We show that within this formalism, the leading Coulomb correction originates from three modifications of the original weak-only interaction: a modification of the nuclear charge form factor, which yields a correction similar to the known Fermi function, a shift of the momentum transfer within the lepton traces, and the same shift but inside the nuclear multipole operators. We additionally provide explicit results for allowed Gamow--Teller and unique first-forbidden transitions.