  We investigate new-physics contributions to $b\to s \ell\ell$ transitions in
the context of an effective field theory extension of the Standard Model,
including operator mixing at one loop. We identify the few scenarios where a
single Wilson coefficient, $C/\Lambda^2 \sim 1/{\rm TeV}^2$, induces a
substantial shift in the lepton flavour universality ratios $R_K$ and $R_{K^*}$
at one loop, while evading $Z$-pole precision tests, collider bounds, and other
flavour constraints. Good fits to the present data are achieved by a
left-handed current operator with quark-flavour indices $(2,2)$ or $(3,3)$,
hitherto overlooked. Interestingly, the running of the Standard Model Yukawa
matrices gives the dominant effect for these scenarios. We match the favoured
effective-theory scenarios to minimal, single-mediator models, which are
subject to additional stringent constraints. Notably, we recognise three viable
instances of a leptoquark with one coupling to fermions only. If the anomalies
were confirmed, it appears that one-loop explanations have good prospects of
being directly tested at the LHC.
