Di-jet/$e^+e^-$+ MET to Probe $Z_2-$Odd Mediators to the Dark Sector

We explore a scenario where Dark Matter (DM) couples to the Standard Model mainly via a scalar mediator ${\cal S}$ that is odd under a $Z_2$ symmetry, leading to interesting collider signatures. In fact, if linear interactions with the mediator are absent the most important DM production mechanisms at colliders could lead to final states with missing transverse energy (MET) in association with at least two fermions, such as di-jet or di-electron signatures. The framework we consider is model-independent, in a sense that it is only based on symmetry and formulated in the (extended) DM Effective Field Theory (eDMEFT) approach. Moreover, it allows to address the smallness of first-generation fermion masses via suppressed $Z_2$ breaking effects. From a di-jet +MET analysis at the LHC, we find rather loose bounds on the effective ${\cal S}$-${\cal S}$-DM-DM interactions, unless the mediator couples very strongly to SM fermions, while a future $e^+ e^-$ collider, such as CLIC, could deliver tighter constraints on the corresponding model parameters, given the mediator is leptophilic. We finally highlight the parameter space that allows to produce the observed DM density, including constraints from direct-detection experiments.