  We introduce a Minimal Simplified Dark Matter (MSDM) framework to
quantitatively characterise dark matter (DM) searches at the LHC. We study two
MSDM models where the DM is a Dirac fermion which interacts with a vector and
axial-vector mediator. The models are characterised by four parameters: the DM
and mediator masses, and the mediator couplings to DM and quarks. The MSDM
models accurately capture the full event kinematics, and the dependence on all
masses and couplings can be systematically studied. The interpretation of
mono-jet searches in this framework can be used to establish an equal-footing
comparison with direct detection experiments. For theories with a vector
mediator, LHC mono-jet searches possess better sensitivity than direct
detection searches for light DM masses (less than 5 GeV). For axial-vector
mediators, LHC and direct detection searches generally probe orthogonal
directions in the parameter space. We explore the projected limits of these
searches from the ultimate reach of the LHC and multi-ton xenon direct
detection experiments, and find that the complementarity of the searches
remains. Finally, we provide a comparison of limits in the MSDM and effective
field theory (EFT) frameworks to highlight the deficiencies of the EFT
framework, particularly when exploring the complementarity of mono-jet and
direct detection searches.
