Measuring $A_{\text{FB}}^b$ and $R_b$ with exclusive $b$-hadron decays at FCC-ee

This paper presents a novel tagging technique to measure the beauty-quark partial decay-width ratio $R_b$ and its forward-backward asymmetry $A_{\text{FB}}^b$ at FCC-ee in the presence of $\mathcal{O}(10^{12})$ $Z$-boson decays. The method relies on the exclusive reconstruction of a selected list of $b$-hadron decay modes in $Z\to b\bar{b}$ events at the $Z$ pole which unambiguously identifies the flavour and charge of the hemisphere. This approach effectively eliminates the contamination; from light-quark physics events and reduces leading systematic uncertainties arising from background contamination, tagging-efficiency correlations, and gluon-radiation corrections by exploiting the geometric and kinematic properties of beauty hadrons. This results in a total relative uncertainty of the order of $0.01\,\%$ for both observables. Furthermore, this precision enables to reach a competitive precision on the weak mixing angle $\sin^2(\theta_{\text{W}})$ compared to the muon forward-backward asymmetry at the level of $0.002\,\%$.