We analyze the universal radiative correction ${\mathrm{\Delta }}_R^V$ to neutron and superallowed nuclear $\beta$ decay by expressing the hadronic $\gamma W$-box contribution in terms of a dispersion relation, which we identify as an integral over the first Nachtmann moment of the $\gamma W$ interference structure function $F_3^{(0)}$. By connecting the needed input to existing data on neutrino and antineutrino scattering, we obtain an updated value of ${\mathrm{\Delta }}_R^V=0.02467(22)$, wherein the hadronic uncertainty is reduced. Assuming other standard model theoretical calculations and experimental measurements remain unchanged, we obtain an updated value of $|V_{ud}|=0.97370(14)$, raising tension with the first row Cabibbo-Kobayashi-Maskawa unitarity constraint. We comment on ways current and future experiments can provide input to our dispersive analysis.

• Received 16 August 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.241804

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Published by the American Physical Society