  Many models of dark matter scattering with baryons may be treated either as a
simple contact interaction or as the exchange of a light mediator particle. We
study an alternative, in which a continuum of light mediator states may be
exchanged. This could arise, for instance, from coupling to a sector which is
approximately conformal at the relevant momentum transfer scale. In the
non-relativistic effective theory of dark matter-baryon scattering, which is
useful for parametrizing direct detection signals, the effect of such continuum
mediators is to multiply the amplitude by a function of the momentum transfer
q, which in the simplest case is just a power law. We develop the basic
framework and study two examples: the case where the mediator is a scalar
operator coupling to the Higgs portal (which turns out to be highly
constrained) and the case of an antisymmetric tensor operator ${\cal O}_{\mu
\nu}$ that mixes with the hypercharge field strength and couples to dark matter
tensor currents, which has an interesting viable parameter space. We describe
the effect of such mediators on the cross sections and recoil energy spectra
that could be observed in direct detection.
