The $\tau$ Magnetic Dipole Moment at Future Lepton Colliders

The magnetic moment of the $\tau$ lepton is an interesting quantity that is potentially sensitive to physics beyond the Standard Model. Electroweak gauge invariance implies that a heavy new physics contribution to it takes the form of an operator which involves the Higgs boson, implying that rare Higgs decays are able to probe the same physics as $a_\tau$. We examine the prospects for rare Higgs decays at future high energy lepton (electron or muon) colliders, and find that such a project collecting a few ab$^{-1}$ would be able to advance our understanding of this physics by roughly a factor of 10 compared to the expected reach of the high luminosity LHC.