We propose a novel dispersive treatment of the so-called inner radiative correction to the neutron and nuclear $\beta$ decay. We show that it requires knowledge of the parity-violating structure function $F_3^{(0)}$ that arises from the interference of the axial vector charged current and the isoscalar part of the electromagnetic current. By isospin symmetry, we relate this structure function to the charged current inelastic scattering of neutrinos and antineutrinos. Applying this new data-driven analysis we obtain a new, more precise evaluation for the universal radiative correction ${\mathrm{\Delta }}_R^{V,\mathrm{new}}=0.02467(22)$ that supersedes the previous estimate by Marciano and Sirlin, ${\mathrm{\Delta }}_R^V=0.02361(38)$. The substantial shift in the central value of ${\mathrm{\Delta }}_R^V$ reflects in a respective shift of $V_{ud}$ and a considerable tension in the unitarity constraint on the first row of the Cabibbo-Kobayashi-Maskawa matrix which is used as one of the most stringent constraints on new physics contributions in the charged current sector. We also point out that dispersion relations offer a unifying tool for treating hadronic and nuclear corrections within the same framework. We explore the potential of the dispersion relations for addressing the nuclear structure corrections absorbed in the $\mathcal{F}t$ values, a crucial ingredient alongside ${\mathrm{\Delta }}_R^V$ in extracting $V_{ud}$ from superallowed nuclear decays. In particular, we estimate the quenching of the free neutron Born contribution in the nuclear environment, corresponding to a quasielastic single-nucleon knockout, and find a significantly stronger quenching effect as compared to currently used estimates based on the quenching of spin operators in nuclear transitions. This observation suggests that the currently used theoretical uncertainties of $\mathcal{F}t$ values might be underestimated and require a renewed scrutiny, while emphasizing the importance of new, more precise measurements of the free neutron decay where nuclear corrections are absent.

• Received 6 March 2019

DOI:https://doi.org/10.1103/PhysRevD.100.013001