μ → eγ in the 2HDM: an exercise in EFT

The 2 Higgs Doublet Model of type III has renormalisable Lepton Flavour-Violating couplings, and its one- and two-loop (Barr-Zee) contributions to mu -> 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. It is found that the dominant two-loop Barr-Zee contributions arise mostly in two-loop matching and running, and that dimension-eight operators might be numerically relevant.