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The 2 Higgs Doublet Model (2HDM) of type III has renormalisable LeptonFlavour-Violating couplings, and its one and two-loop ("Barr-Zee")contributions to $\mu \to e \gamma$ are known. In the decoupling limit, wherethe mass scale M of the second doublet is much greater than the electroweakscale, the model can be parametrised with an Effective Field Theory(EFT)containing dimension six operators. The $1/M^2$ terms of the exact calculationare reproduced in the EFT, provided that the four-fermion operator basis belowthe weak scale is enlarged with respect to the SU(2)-invariant Buchmuller-Wylerlist. The two-loop "Barr-Zee" contributions are located in the EFT, showingthat two-loop matching and running would be required to obtain the mostimportant contributions, and that dimension eight operators can be numericallyrelevant.

Mu to e gamma in the 2 Higgs Doublet Model: an exercise in EFT