Study of the feasibility of the observation of $B^0 \to K^*(892) \tau^+ \tau^-$ at FCC-$ee$ and related vertex detector performance requirements

The aim of the work described in this analysis note is to study the capabilities of experiments at the Future Circular Collider FCC-$ee$ to unravel the quark transitions $b\to s\tau^+\tau^-$, unobserved to date. The decay $B^0_d\to K^* (892) \tau^+\tau^-$ considering the three-prongs transition $\tau\to\pi\pi\pi\nu_{\tau}$ is studied with a method to reconstruct explicitly the kinematics of the two undetected neutrinos. Detector requirements to observe this decay are evaluated; in particular several vertexing resolution performances are emulated and tested against this physics case.  An actual detector design (IDEA) has been as well tested with FCC-$ee$ simulations to fix a reference of the state-of-the-art performance. Analyses of simulated signal events together with simulated dominant backgrounds has been done in order to draw the precision of the $B^0_d\to K^*\tau^+\tau^-$ branching fraction measurement as function of the vertexing resolution performances. The vertexing performance to observed this specific decay at FCC-$ee$ fis provided.

This note serves as input to the "Mid-term review report, detector requirements". It is yet a very preliminary version and will be developed further. The results are final though.