  We present a treatment of the high energy scattering of dark Dirac fermions
from nuclei, mediated by the exchange of a light vector boson. The dark
fermions are produced by proton-nucleus interactions in a fixed target and,
after traversing shielding that screens out strongly interacting products,
appear similarly to neutrino neutral current scattering in a detector. Using
the Fermilab experiment E613 as an example, we place limits on a secluded dark
matter scenario. Visible scattering in the detector includes both the familiar
regime of large momentum transfer to the nucleus ($Q^2$) described by deeply
inelastic scattering, as well as small $Q^2$ kinematics described by the
exchanged vector mediator fluctuating into a quark-antiquark pair whose
interaction with the nucleus is described by a saturation model. We find that
the improved description of the low $Q^2$ scattering leads to important
corrections, resulting in more robust constraints in a regime where a
description entirely in terms of deeply inelastic scattering cannot be trusted.
