Reconciling hadronic and partonic analyticity in $b\to s\ell\ell$ transitions

Rare $B$-meson decays mediated by $b\to s\ell\ell$ transitions constitute sensitive probes of physics beyond the Standard Model, and have triggered considerable interest due to hints for deviations from the Standard-Model prediction. To establish a discrepancy beyond a reasonable doubt, control over the nonlocal matrix elements involving charm loops is essential, which, for large spacelike virtualities, can be constrained by an operator product expansion with coefficients known at two-loop order. We observe that the analytic structure of this partonic calculation, whose understanding is important to put forward rigorous parameterizations, follows from simple triangle topologies and demonstrate explicitly how dispersion relations are fulfilled even in the case of anomalous thresholds. Crucially, these anomalous contributions match onto the ones expected when considering hadronic degrees of freedom, proving that the partonic calculation does not miss anomalous effects and justifying its use in regions of parameter space in which a perturbative description applies.