  In this paper, we introduce a novel program of fixed-target searches for
thermal-origin Dark Matter (DM), which couples inelastically to the Standard
Model. Since the DM only interacts by transitioning to a heavier state,
freeze-out proceeds via coannihilation and the unstable heavier state is
depleted at later times. For sufficiently large mass splittings, direct
detection is kinematically forbidden and indirect detection is impossible, so
this scenario can only be tested with accelerators. Here we propose new
searches at proton and electron beam fixed-target experiments to probe sub-GeV
coannihilation, exploiting the distinctive signals of up- and down-scattering
as well as decay of the excited state inside the detector volume. We focus on a
representative model in which DM is a pseudo-Dirac fermion coupled to a hidden
gauge field (dark photon), which kinetically mixes with the visible photon. We
define theoretical targets in this framework and determine the existing bounds
by reanalyzing results from previous experiments. We find that LSND, E137, and
BaBar data already place strong constraints on the parameter space consistent
with a thermal freeze-out origin, and that future searches at Belle II and
MiniBooNE, as well as recently-proposed fixed-target experiments such as LDMX
and BDX, can cover nearly all remaining gaps. We also briefly comment on the
discovery potential for proposed beam dump and neutrino experiments which
operate at much higher beam energies.
