Determination of the nature of Dark Matter (DM) is one of the most fundamental problems of particle physics and cosmology. If DM is light enough and interacts with Standard Model particles directly or via some mediators with a strength beyond the gravitational one, it can be directly produced at particle accelerators. The study of the complete set of dimension 5 and 6 effective DM operators which we present here demonstrates the LHC potential to distinguish operators with different spin of DM: they have a different energy dependence and respectively different distributions of the invariant mass of the DM pair which consequently leads to different missing transverse energy distributions. This study can be directly applied beyond the EFT paradigm, as we demonstrate for three cases -- Supersymmetry
(DM with spin 1/2), inert two Higgs doublet model (i2HDM)
(spin 0 of DM) and minimal isotriplet of vector dark matter model (MVDM).
We also stress an importance of complementarity of the collider searches, DM direct detection experiments as well as relic density and CMB data. For several appealing DM models as an example, we show that the interplay between high and low energy data has unique power for an identification of DM nature. We also highlight prospects of new signature from DM theories such as disappearing charge tracks which are characteristic for wide class of DM theories.
Finally, we advocate the importance of the framework which would combine the experience of HEP community and would allow to effectively identify the underlying theory of DM.