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The 2 Higgs Doublet Model (2HDM) of type III has renormalisable Lepton Flavour-Violating couplings, and its one and two-loop ("Barr-Zee") contributions to $\mu \to e \gamma$ are known.
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R. Harnik, J. Kopp and J. Zupan, “Flavor Violating Higgs Decays,” JHEP 1303
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A. J. Buras, “Weak Hamiltonian, CP violation and rare decays,” hep-ph/9806471
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