μ → e γ μ → e γ 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 μ → e γ 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/M² 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.