  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. In the decoupling limit, where
the mass scale M of the second doublet is much greater than the electroweak
scale, the model can be parametrised with an Effective Field Theory(EFT)
containing dimension six operators. The $1/M^2$ terms of the exact calculation
are reproduced in the EFT, provided that the four-fermion operator basis below
the weak scale is enlarged with respect to the SU(2)-invariant Buchmuller-Wyler
list. The two-loop "Barr-Zee" contributions are located in the EFT, showing
that two-loop matching and running would be required to obtain the most
important contributions, and that dimension eight operators can be numerically
relevant.
