  We analyze the universal radiative correction $\Delta_R^V$ 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
$\Delta_R^V = 0.02467(22)$, wherein the hadronic uncertainty is reduced.
Assuming other Standard Model theoretical calculations and experimental
measurements remain unchanged, we obtain an updated value of $|V_{ud}| =
0.97366(15)$, raising tension with the first row CKM unitarity constraint. We
comment on ways current and future experiments can provide input to our
dispersive analysis.
