Reduced uncertainty of the axial $gamma Z$-box correction to the protons weak charge

We present the fully up-to-date calculation of the $gamma Z$-box correction which needs to be taken into account to determine the weak mixing angle at low energies from parity-violating electron proton scattering. We make use of neutrino and antineutrino inclusive scattering data to predict the parity-violating structure function $F_3^{gamma Z}$ by isospin symmetry. Our new analysis confirms previous results for the axial contribution to the $gamma Z$-box graph, and reduces the uncertainty by a factor of~2. In addition, we note that the presence of parity-violating photon-hadron interactions induces an additional contribution via $F_3^{gamma gamma}$. Using experimental and theoretical constraints on the nucleon anapole moment we are able to estimate the uncertainty associated with this contribution. We point out that future measurements are expected to significantly reduce this latter uncertainty.