cLFV leptophilic $Z^\prime$ as a dark matter portal: prospects for colliders

Extensions of the Standard Model featuring light vector bosons have been explored with the goal of resolving certain tensions between theory and experiment, among them the discrepancy in the anomalous magnetic moment of the muon, $\Delta a_{\mu}$. In particular, this is the case of a minimal construction including a leptophilic, strictly flavour violating, vector boson $Z^\prime$. These new vector bosons are also well-motivated dark matter portals, with non-trivial couplings to stable, weakly interacting states which can account for the correct dark matter density. Here we study the prospects of a Standard Model extension (via a vector boson and a fermionic dark matter candidate) concerning signatures at the LHC, and at future lepton and hadron colliders.We discuss the cross-sections of several processes leading to same- and opposite-sign muon-tau lepton pairs in the final state, as well as final states with missing energy (in the form of neutrinos and/or dark matter). Our findings suggest that a future muon collider offers the best prospects to probe this model (together with searches for dilepton pairs and missing energy signatures at the FCC-ee running at the $Z$-pole); moreover, the complementarity of the different future high-energy colliders is also paramount to probing distinct $Z^\prime$ mass regimes.