What is the scale of new physics behind the B-flavour anomalies?

Motivated by the recent hints of lepton flavour non-universality in B-meson semi-leptonic decays, we study the constraints of perturbative unitarity on the new physics interpretation of the anomalies in b → cl anti ν and b → sl anti l transitions. Within an effective field theory approach we find that 2 → 2 fermion scattering amplitudes saturate the unitarity bound below 9 and 80 TeV, respectively for b → cl anti ν and b → sl anti l transitions. Stronger bounds, up to few TeV, are obtained when the leading effective operators are oriented in the direction of the third generation, as suggested by flavour models. We finally address unitarity constraints on simplified models explaining the anomalies and show that the new physics interpretation is ruled out in a class of perturbative realizations. (orig.)