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The rare $b\to s\nu\bar{\nu}$ decays are sensitive to contributions of newphysics (NP) and helpful to resolve the puzzle of multiple $B$ flavoranomalies. In this work, we propose to study the $b\to s\nu\bar{\nu}$transition at a future lepton collider operating at the $Z$ pole through the$B_s \to \phi\nu\bar{\nu}$ decay. Using the $B_s\to\phi$ decay form factorsfrom lattice simulations, we first update the SM prediction of BR($B_s \to\phi\nu\bar{\nu})_{\mathrm{SM}}=(9.93\pm 0.72)\times 10^{-6}$ and thecorresponding $\phi$ longitudinal polarization fraction$F_{L,{\mathrm{SM}}}=0.53\pm 0.04$. Our analysis uses the full CEPC simulationsamples with a net statistic of $\mathcal{O}(10^9)$ $Z$ decays. Precise $\phi$and $B_s$ reconstructions are used to suppress backgrounds. The results showthat BR($B_s \to \phi\nu\bar{\nu})$ can be measured with a statisticaluncertainty of $\mathcal{O}(\%)$ and an $S/B$ ratio of $\mathcal{O}(1)$ at theCEPC. The quality measures for the event reconstruction are also derived. Bycombining the measurement of BR($B_s \to \phi\nu\bar{\nu})$ and $F_L$, theconstraints on the effective theory couplings at low energy are given.

Analysis of $B_s\to\phi\nu\bar{\nu}$ at CEPC