Electroweak axial structure functions and CKM unitarity

The $\gamma W$ box radiative correction is the largest source of hadronic uncertainty in the determination of the CKM matrix element $V_{ud}$ from super-allowed nuclear $\beta$-decay. The recent development of computational methods using dispersion relations allows for a systematic improvement to the calculation of the axial-vector part of the $\gamma W$ box amplitude $\Box^{\gamma W}_{A}$ in terms of the isoscalar part of the $F_3^{\gamma W}$ interference structure function, with improved and quantifiable estimates of the hadronic uncertainties. Using the latest available phenomenology for $F_3^{\gamma W}$ from the nucleon elastic, resonance, deep-inelastic, and Regge regions, we find the real part of the box correction to be \mbox{$\Box^{\gamma W}_A = 3.90(9) \times 10^{-3}$}. This gives a theoretical estimate of the CKM matrix element $|V_{ud}|^2=0.94805(26)$, which represents a 4$\sigma$ violation of unitarity. Implications for the $\gamma Z$ interference radiative corrections applicable to parity-violating electron-proton scattering are also discussed.