Decoding Dark Matter at future $e^+ e^-$ colliders

We explore the potential of the $e^+ e^-$ colliders to discover dark matter and determine its properties such as mass and the spin. For this purpose we study spin zero and spin one-half cases of dark matter, $D$ which belongs to $SU(2)$ weak doublet and therefore has the charged doublet partner, $D^+$. For the case of scalar dark matter we chose Inert Doublet Model, while for the case of fermion dark matter we suggest the new minimal fermion dark matter model with only three parameters. We choose two benchmarks for the models under study which provide the correct amount of observed DM relic density and consistent with the current DM searches. We focus on the particular process $e^+ e^- \to D^+ D^- \to D D W^+ W^- \to DD(q \bar{q})(\mu^\pm\nu)$ at 500 GeV ILC collider which gives rise to the "di-jet +$\mu$ + missing $E_T$" signature and study it at the level of fast detector simulation, taking into account Bremsstrahlung and ISR effects. We have found that two kinematical observables -- the energy of the muon, $E_\mu$, and the angular distribution of $W$-boson, reconstructed from di-jet, $\cos\theta_{jj}$ are very powerful in determination of DM mass and spin, respectively. In particular we have demonstrated that in case of fermion DM, the masses can be measured with a few percent accuracy already at 500 fb$^{-1}$ integrated luminosity. At the same time, the scalar DM model which has about an order of magnitude lower signal, requires about factor of 40 higher luminosity to reach the same accuracy in the mass measurement. We have found that one can distinguish fermion and scalar DM scenarios with about 2 ab$^{-1}$ total integrated luminosity or less without using the information on the cross sections for benchmarks under study.