Time-Dependent Precision Measurement of $B_s^0\to \phi\mu^+\mu^-$ Decay at FCC-$ee$

We investigate the feasibility of the time-dependent $C\!P$ violation measurement of the rare Flavor-Changing Neutral Current decays, $B_s^0 \to \phi(\to K^+K^-)\mu^+\mu^-$ at the FCC-$ee$. In the Standard Model (SM), $C\!P$ violation is absent at the leading order and only enters at loop-order. However, such rare processes are sensitive to interactions beyond the SM referred to as New Physics (NP). Such a NP effect could introduce non-negligible $C\!P$ violation. The presence of $C\!P$ violation would reveal the complex structure of NP. This makes time-dependent $C\!P$ violation in $B_s^0\to\phi\mu^+\mu^-$ decays an attractive laboratory to probe NP. Future $Z$-factories offer an ideal setting for measuring this decay due to the large statistics, clean environment, particle identification, and excellent vertexing capabilities, which are needed for establishing the precise location, and therefore lifetime of the decaying $B$ meson. Studies were conducted using the IDEA detector concept, PYTHIA Monte Carlo, and DELPHES detector simulation and reconstruction. Our results indicate that achieving a relative precision of $<\mathcal{O}(1\%)$ on the branching ratio and a precision of $\mathcal{O}(10^{-2})$ on the time-integrated $C\!P$ asymmetry is feasible. We seek to determine the sensitiveity of an FCC-$ee$ analysis to $D_f, S_f$ and $C_f$, three inter-dependent NP observables that define the time-dependent $C\!P$ violation. In the untagged time-dependent measurement, the $D_{f}$ precision of $\mathcal{O}(10^{-1})$ can be reached. The precision on $C_f$ and $S_f$ are found to reach the $\mathcal{O}(10^{-2})$ level if the $B_s^0$ oscillation is resolved using a time-dependent measurement of the decay. These precision measurements are interpreted using Weak Effective Theory, providing a comprehensive understanding of the $C\!P$  properties of the potential NP in these rare decays.