Lepton polarization asymmetries in rare semi-tauonic $ b \rightarrow s $ exclusive decays at FCC-$ee$

International audience; We consider measurements of exclusive rare semi-tauonic b-hadron decays, mediated by the $b \rightarrow s \tau ^+ \tau ^-$ transition, at a future high-energy circular electron–positron collider (FCC-ee). We argue that the high boosts of b-hadrons originating from on-shell Z boson decays allow for a full reconstruction of the decay kinematics in hadronic $\tau $ decay modes (up to discrete ambiguities). This, together with the potentially large statistics of $Z\rightarrow b\bar{b}$ , opens the door for the experimental determination of $\tau $ polarizations in these rare b-hadron decays. In the light of the current experimental situation on lepton flavor universality in rare semileptonic B decays, we discuss the complementary short-distance physics information carried by the $\tau $ polarizations and suggest suitable theoretically clean observables in the form of single- and double- $\tau $ polarization asymmetries.