Vector Portals at Future Lepton Colliders

We assess the sensitivity of future lepton colliders to weakly coupled vector dark portals (aka ``$Z'$ bosons'') with masses ranging from tens of GeV to a few TeV. Our analysis focuses on dark photons and $L_{\mu} - L_{\tau}$ gauge bosons. We consider both visible and invisible decay channels. We demonstrate that both high energy $\mu$ colliders and future $e^+e^-$ colliders, using the FCC-ee $Z$-pole and $ZH$ operation modes as a benchmark, offer significant improvements in sensitivity. We find that both colliders can enhance the sensitivity to $L_{\mu} - L_{\tau}$ bosons (for both visible and invisible decays) and to invisibly decaying dark photons by 1--2 orders of magnitude across the relevant mass range. Furthermore, we study the impact of forward $\mu$ detectors at the $\mu$-collider on the sensitivity to both models.