Dark sectors and enhanced $h\to \tau \mu$ transitions

LHC searches with τ leptons in the final state are always inclusive in missing-energy sources. A signal in the flavor-violating Higgs decay search, h → τ μ, could therefore equally well be due to a flavor conserving decay, but with an extended decay topology with additional invisible particles. We demonstrate this with the three-body decay h → τ μφ, where φ is a flavorful mediator decaying to a dark-sector. This scenario can give thermal relic dark matter that carries lepton flavor charges, a realistic structure of the charged lepton masses, and explain the anomalous magnetic moment of the muon, (g − 2)μ, while simultaneously obey all indirect constraints from flavor-changing neutral currents. Another potentially observable consequence is the broadening of the collinear mass distributions in the h → τ μ searches.