  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.
