Minimal Vector-like leptonic Dark Matter and Signatures at the LHC

We propose a minimal vector-like leptonic dark matter (DM) with renormalisable interaction in a beyond the Standard Model (SM) scenario, where the SM is augmented with a vector-like doublet and a singlet leptons. The additional fermions are odd under a discrete $Z_2$ symmetry, while the rest of the SM particles are singlets, and thus providing stability to the DM. In this scenario, we show that, the DM emerges as an admixture of the neutral component of the vector-like doublet and the singlet leptons. The singlet-doublet mixing ($\sin \theta$) plays a crucial role in yielding the correct relic density as well as in obtaining null direct DM search results through an interplay of interactions via $Z$ and Higgs mediation. The mixing is also strongly constrained from the invisible Z and Higgs decay width. We found that the correct relic abundance of DM can be obtained in a large region of parameter space for DM-mass larger than $ M_Z/2$ and $\sin \theta \le 0.1$. The details of model phenomenology with collider signatures at the Large hadron Collider (LHC) are discussed. In particular, we show that for $\sin \theta \le 0.01$, the charged companion of the DM can give rise to an observable displaced vertex signature, marking a significant departure from other fermionic DM scenarios, while keeping the relic abundance intact.