Toward a First-Principles Calculation of Electroweak Box Diagrams

We derive a Feynman-Hellmann theorem relating the second-order nucleon energy shift resulting from the introduction of periodic source terms of electromagnetic and isovector axial currents to the parity-odd nucleon structure function $F_3^N$. It is a crucial ingredient in the theoretical study of the $\gamma W$ and $\gamma Z$ box diagrams that are known to suffer from large hadronic uncertainties. We demonstrate that for a given $Q^2$, one only needs to compute a small number of energy shifts in order to obtain the required inputs for the box diagrams. Future lattice calculations based on this approach may shed new light on various topics in precision physics including the refined determination of the Cabibbo-Kobayashi-Maskawa matrix elements and the weak mixing angle.