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

The prospects are presented for precise measurements of the branching ratios of the purely leptonic $B_c^{+}\to {\tau }^{+}{\nu }_{\tau }$ and $B^{+}\to {\tau }^{+}{\nu }_{\tau }$ decays at the Future Circular Collider (FCC). This work is focused on the hadronic ${\tau }^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}{\overline{\nu }}_{\tau }$ decay in both $B_c^{+}\to {\tau }^{+}{\nu }_{\tau }$ and $B^{+}\to {\tau }^{+}{\nu }_{\tau }$ 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 $B^{+}\to {\tau }^{+}{\nu }_{\tau }$ decay can be well separated from both $B_c^{+}\to {\tau }^{+}{\nu }_{\tau }$ and generic $Z\to b\overline{b}$ 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 $B^{+}\to {\tau }^{+}{\nu }_{\tau }$ and $B_c^{+}\to {\tau }^{+}{\nu }_{\tau }$ measurements on New Physics cases are discussed for interpretations in the generic Two-Higgs-doublet model and leptoquark models.