Constraining flavour patterns of scalar leptoquarks in the effective field theory

We investigate the viability of extending the Standard Model with $S_1$ and $S_3$ scalar leptoquarks when the flavour structure is parametrized in terms of Froggatt-Nielsen charges. In contrast to a similar analysis with a vector leptoquark, we find essentially two solutions for the charges that fit the experimental constraints, which are dominated by the current tensions in $B$ decays. These two scenarios differ in their estimate of the anomalous magnetic moment of the muon, $(g-2)$, but they both predict sizeable contributions to $\tau\to\mu\gamma$, $\bar B_s\to\tau^\pm\mu^\mp$ and $B^+\to K^+\tau^+\mu^-$ decays, whose branching ratios are close to the current experimental limits.