Probe of Electroweak Interference Effects in Non-Resonant Inelastic $\vec{e}p$ Scattering

The $Q_{weak}$ Experiment at Jefferson Lab made the first direct measurement of the proton's weak charge, $Q^p_W$, via a measurement of the parity-violating asymmetry in elastic $\vec{e}p$ scattering with low four-momentum transfer. Energy-dependent radiative corrections must be applied to the asymmetry. The most problematic of these is referred to as the $\gamma Z$ box, where a photon and a Z-boson are simultaneously exchanged. The asymmetry arising from the $\gamma Z$ box depends on the $\gamma Z$ interference structure functions, $F_{1,3}^{\gamma Z}$, for which there is almost no experimental world data. Using the $Q_{weak}$ apparatus, with modifications, a measurement of the parity-violating asymmetry of non-resonant inelastic $\vec{e}p$ scattering was made with 3.35 GeV incident electrons. Inelastically scattered electrons with $Q^2 \approx 0.09$ GeV$^2$ and $W \approx 2.23$ GeV were selected and focused on eight quartz Cerenkov detectors. This asymmetry measurement resides in a kinematic region that will provide additional validation of the theoretical models used to predict the $\gamma Z$ box and can provide insight on hadron structure. Analysis methods and preliminary results will be presented.