Prospects for $B_c^+$ and $B^+\rightarrow \tau ^+ \nu _\tau $ at FCC-ee

The prospects are presented for precise measurements of the branching ratios of the purely leptonic $B^+_c → τ^+ν_τ$ and $B^+ → τ^+ν_τ$ decays at the Future Circular Collider (FCC). This work is focused on the hadronic $τ^+ → π^+π^+π^− \bar{ν}_τ$ decay in both $B^+_c → τ^+ν_τ$ and $B^+ →τ^+ν_τ$ processes. Events are selected with two Boosted Decision Tree algorithms to optimise the separation between the two signal processes as well as the generic hadronic $Z$ decay backgrounds. The range of the expected precision for both signals are evaluated in different scenarios of non-ideal background modelling. This paper demonstrates, for the first time, that the $B^+ → τ^+ν_τ$ decay can be well separated from both $B^+_c → τ^+ν_τ$ and generic $Z → b \bar{b}$ processes in the FCC-ee collision environment and proposes the corresponding branching ratio measurement as a novel way to determine the CKM matrix element $|V_{ub}|$. The theoretical impacts of both $B^+ → τ^+ν_τ$ and $B^+_c → τ^+ν_τ$ measurements on New Physics cases are discussed for interpretations in the generic Two-Higgs-doublet model and leptoquark models.