Precision Tests in $b\to s\ell^+\ell^-$ ($\ell=e,\mu$) at FCC-ee

The rare semi-leptonic decays $B\to K^\ast \ell^+\ell^-$, with $\ell=e, \mu$, are highly sensitive to new physics (NP) due to their suppression in the Standard Model (SM). Current LHCb measurements in the muon channel exhibit a significant tension with state-of-the-art SM theory predictions. The proposed tera-$Z$ run at FCC-ee provides a unique opportunity to untangle the origin of this tension by producing a very large sample of $B$-mesons in a clean $e^+e^-$ environment. We explore the expected precision of $B\to K^\ast \ell^+\ell^-$ ($\ell=e, \mu$) measurements at FCC-ee, complementing existing studies with $\tau^+\tau^-$ in the final state, and compare with HL-LHC projections. For the case of muons in the final state, we show that HL-LHC and FCC-ee are expected to deliver a similar number of events, while the latter performs much better in the case of final state electrons. Regardless of the lepton flavour, we expect the FCC-ee environment to be much cleaner than at HL-LHC, with subleading systematics. We also find that a significant reduction in theory uncertainties on the SM predictions is required to capitalize on the advantage going from HL-LHC to FCC-ee. We demonstrate the power of such measurements at FCC-ee to extract information on the long-distance contribution to these decays, and to reveal evidence for new physics even if no deviations are seen in electroweak precision tests.