$H \rightarrow \tau^+ \tau^- \gamma$ as a Probe of the $\tau$ Magnetic Dipole Moment

Low energy observables involving the Standard Model fermions which are chirality-violating, such as anomalous electromagnetic moments, necessarily involve an insertion of the Higgs in order to maintain $SU(2) \times U(1)$ gauge invariance. As the result, the properties of the Higgs boson measured at the LHC impact our understanding of the associated low-energy quantities. We illustrate this feature with a discussion of the electromagnetic moments of the $\tau$-lepton, as probed by the rare decay $H \rightarrow \tau^+ \tau^- \gamma$. We assess the feasibility of measuring this decay at the LHC, and show that the current bounds from lower energy measurements imply that $13~\rm{TeV}$ running is very likely to improve our understanding of new physics contributing to the anomalous magnetic moment of the tau.