I will present a framework in the language of effective field theory (EFT) to describe DM-SM interactions and combine the experimental limits from nuclear energies at direct detection experiments to the TeV-scale probed at the LHC.
To improve the high energy-validity of conventional DM EFT a dynamical (pseudo-) scalar is added serving as mediator to the dark sector, where richer new-physics sectors can be consistently included via higher-dimensional operators. The model is formulated in a gauge-invariant way and allows to confront classical DM observables with Higgs sector measurements. Interestingly the leading effects originate at dimension-five, allowing to capture them with a rather small set of parameters. Constraints on these arising from collider mono-X searches, the relic abundance, indirect and direct detection experiments are presented. The "model-independent" approach allows to apply the results to different UV-complete models such as 2HDM+a, extended fermion sectors and the NMSSM. With a small set of eDMEFT operators, motivated by a flavor symmetry, we explore the reach of the HL-LHC and CLIC for a new final state with DM plus correlated di fermions.