Probing $\tau$ lepton dipole moments at future Lepton Colliders

The electric and magnetic dipole moments of the electron and of the muon provide stringent tests of the Standard Model and sensitive probes of new physics. By contrast, the corresponding dipole moments of the $\tau$ lepton remain weakly constrained. This study explores the potential of future lepton colliders, focusing on the $e^+e^-$ Future Circular Collider and a multi-TeV muon collider, to probe $\tau$ dipole moments. We consider multiple channels, including $\ell^+\ell^- \to \tau^+\tau^-$ ($\ell=e,\mu$), associated Higgs production $\mu^+\mu^- \to \tau^+\tau^- H$, radiative Higgs decays $H \to \tau^+\tau^-\gamma$, and vector-boson scattering $\ell^+\ell^- \to \ell^+\ell^-\tau^+\tau^-$ and $\mu^+\mu^- \to \bar\nu\nu\tau^+\tau^-$. Our results show that these facilities are highly complementary and can extend existing bounds by several orders of magnitude.