  Motivated by recent attempts to reconcile hints of direct dark matter
detection by the CoGeNT and DAMA experiments, we construct simple particle
physics models that can accommodate the constraints. We point out challenges
for building reasonable models and identify the most promising scenarios for
getting isospin violation and inelasticity, as indicated by some
phenomenological studies. If inelastic scattering is demanded, we need two new
light gauge bosons, one of which kinetically mixes with the standard model
hypercharge and has mass < 2 GeV, and another which couples to baryon number
and has mass 6.8 +/- 0.2 GeV. Their interference gives the desired amount of
isospin violation. The dark matter is nearly Dirac, but with small Majorana
masses induced by spontaneous symmetry breaking, so that the gauge boson
couplings become exactly off-diagonal in the mass basis, and the small mass
splitting needed for inelasticity is simultaneously produced. If only elastic
scattering is demanded, then an alternative model, with interference between
the kinetically mixed gauge boson and a hidden sector scalar Higgs, is adequate
to give the required isospin violation. In both cases, the light kinetically
mixed gauge boson is in the range of interest for currently running fixed
target experiments.
