  Lepton Flavor Violating (LFV) observables such as $\mu\to e\gamma$, $\mu\to
3e$ and $\mu N \to eN$ are among the best probes for new physics at the TeV
scale. In the near future the bounds on these observables will improve by many
orders of magnitude. In this work we use the SM EFT to understand the impact of
these measurements. The precision reach is such that the interpretation of the
bounds requires an analysis of the dimension-six operator mixing up to the
two-loop level. Using on-shell amplitude techniques, which make transparent
many selection rules, we classify and calculate the different operator mixing
chains. At the leading order, on-shell techniques allow to calculate anomalous
dimensions of SM EFT operators from the product of tree-level amplitudes, even
for two-loop renormalization group mixings. We illustrate the importance of our
EFT approach in models with extra vector-like fermions.
