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 this work is to study the capabilities of the Futur Circular Collider (FCC) $ee$ phase to study the electroweak penguin transitions $b\to s\tau^+\tau^-$ unobserved to date. At meson scale the decay $B^0_d\to K^*\tau^+\tau^-$ with the transition $\tau\to\pi\pi\pi\nu_{\tau}$ is studied with a method to reconstruct explicitly the two undetected neutrinos. The detector requirements to study this decay are evaluated, in particular the vertexing resolution performance are emulated and compared to IDEA one simulated detector used for FCC$ee$ simulations. Analysis 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 and evaluate the feasibility of this measurement at FCC$ee$.