The parity-odd structure function of nucleon from the Compton amplitude

The dominant contribution to the theoretical uncertainty in the extracted weak parameters of the Standard Model comes from the hadronic uncertainties in the electroweak boxes, i.e. $\gamma-W^\pm/Z$ exchange diagrams. A dispersive analysis relates the box diagrams to the parity-odd structure function, $F_3$, for which the experimental data either do not exist or belong to a separate isospin channel. Therefore a first-principles calculation of $F_3$ is highly desirable. In this contribution, we report on the QCDSF/UKQCD Collaboration's progress in calculating the moments of the $F_3^{\gamma Z}$ structure function from the forward Compton amplitude at the SU(3) symmetric point.