Electroweak axial structure functions and improved extraction of the $V_{ud}$ CKM matrix element

We present a comprehensive analysis of the $gamma W$ interference radiative correction to the neutron $beta$-decay matrix element. Within a dispersion relations approach, we compute 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. 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 $Box^{gamma W}_A = 3.90(9) times 10^{-3}$. This improved correction gives a theoretical estimate of the CKM matrix element $|V_{ud}|^2=0.94805(26)$, which represents a 4$sigma$ violation of unitarity.