Time-Dependent Precision Measurement of $B_s^0\to \ddot{I}•\hat{I}{¼}^{+}\hat{I}{¼}^{-}$ Decay at FCC-$ee$

Abstract

We study the feasibility of measuring time-dependent $CP$ violation in the rare flavor-changing neutral current (FCNC) decay $B_s^0\to \ddot{I}•(\to K^{+}{K}^{-})\hat{I}{¼}^{+}\hat{I}{¼}^{-}$ at the FCC-$ee$. In the Standard Model (SM), $CP$ violation in this mode arises only at higher orders and is highly suppressed. Extensions of the SM, collectively referred to as New Physics (NP), can introduce additional $CP$-violating phases that enhance such effects. The decay $B_s^0\to \ddot{I}•\hat{I}{¼}^{+}\hat{I}{¼}^{-}$, mediated by the $b\to s{\ell }^{+}{\ell }^{-}$ transition, is therefore a promising probe of NP. The FCC-$ee$, operating as a high-luminosity $Z$-factory, offers an optimal environment for this measurement due to its large event yield, clean conditions, efficient particle identification, and excellent vertex resolution. We perform a Monte Carlo study using Pythia and Delphes with the IDEA detector concept. A relative precision better than $\mathcal{O}(1\%)$ on the branching ratio and $\mathcal{O}(10^{-2})$ on the time-integrated $CP$ asymmetry is found to be achievable. We determine the projected sensitivities to the observables $D_f$, $C_f$, and $S_f$, which parameterize time-dependent $CP$ violation. In the untagged analysis, a precision of $\mathcal{O}(10^{-1})$ on $D_f$ can be reached. With flavor tagging, sensitivities to $C_f$ and $S_f$ improve to $\mathcal{O}(10^{-2})$. These measurements remain inaccessible to current flavor experiments. Interpreting the results within the Weak Effective Theory provides model-independent constraints on $CP$-violating NP. This study demonstrates that FCC-$ee$ enables first-time access to $CP$-sensitive observables previously beyond experimental reach.

38 pages, 20 figures