Discovering an invisible Z' at the muon collider

We show in this letter how a heavy $(\mathcal{O}(TeV))$ invisible $Z'$ gauge boson that will practically be out of reach of the Large Hadron Collider (LHC), can be discovered at the future muon collider. The new force carrier has a relatively stronger coupling with the beyond standard model (BSM) sector, while its interaction with the SM fields is much weaker. This weaker coupling is induced through mixing mechanisms, specifically via gauge kinetic mixing and the $Z-Z'$ mixing. We consider a scenario where the new gauge boson decays mostly to charge-neutral long-lived particles and/or dark matter (DM). We show how producing and detecting this heavier invisible $Z'$, that will be beyond the reach of even the very high luminosity LHC, becomes possible if it is produced in association with an energetic photon at the future muon collider. The on-shell production of the $Z'$ will lead to a peak in the photon energy distribution, following the so-called radiative return phenomena and can lead to the accurate determination of the $Z'$ mass and its interaction with SM particles.