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Lepton flavour violating Higgs and tau to mu gamma

We update phenomenological constraints on a Two Higgs Doublet Model with lepton flavour non-conserving Yukawa couplings. We review that $\tan β$ is ambiguous in such "Type III" models, and define it from the $τ$ Yukawa coupling. The neutral scalars $ϕ$ could be searched for at hadron colliders in $ ϕ\to τ\barμ$, and are constrained by the rare decay $τ\to μγ$. The Feynman diagrams for the collider process, with Higgs production via gluon fusion, are similar to the two-loop "Barr-Zee" diagrams which contribute to $τ\to μγ$. Some "tuning" is required to obtain a collider cross-section of order the Standard Model expectation for $σ(gg \to h_{SM} \to τ^+ τ^-)$, while agreeing with the current bound from $τ\to μγ$.