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Motivated by the recent hints of lepton flavour non-universality in $B$-mesonsemi-leptonic decays, we study the constraints of perturbative unitarity on thenew physics interpretation of the anomalies in $b \to c \ell \bar \nu$ and $b\to s \ell \bar \ell$ transitions. Within an effective field theory approach wefind that $2 \to 2$ fermion scattering amplitudes saturate the unitarity boundbelow 9 TeV and 80 TeV, respectively for $b \to c \ell \bar \nu$ and $b \to s\ell \bar \ell$ transitions. Stronger bounds, up to few TeV, are obtained whenthe leading effective operators are oriented in the direction of the thirdgeneration, as suggested by flavour models. We finally address unitarityconstraints on simplified models explaining the anomalies and show that the newphysics interpretation is ruled out in a class of perturbative realizations.

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