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

LHC searches with $\tau$ leptons in the final state are always inclusive in missing-energy sources. A signal in the flavor-violating Higgs decay search, $h\to \tau \mu$, 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 $h\to \tau \mu \phi$, where $\phi$ 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, $(g-2)\mu$, 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 $h\to \tau \mu$ searches.