Prospect for measurement of the CP-violating phase $\phi_s$ in the $B_{s}\rightarrow J/\psi \phi$ channel at a future Z factory

The CP-violating phase $\phi_s$, the \Bs decay width ($\Gamma_s$), and the \Bs decay width difference ($\Delta\Gamma_s$) are sensitive probes to new physics and can constrain the heavy quark expansion theory. The potential for the measurement at future $Z$ factories is studied in this manuscript. It is found that operating at Tera-$Z$ mode, the expected precision can reach: $\sigma(\phi_s) = 4.6~\mathrm{mrad}$, $\sigma(\Delta\Gamma_s) = 2.4~\mathrm{ns^{-1}}$ and $\sigma(\Gamma_s) = 0.72~\mathrm{ns^{-1}}$. The precision of $\phi_s$ is 40\% larger than the expected precision with the LHCb experiment at HL-LHC. If operating at 10-Tera-$Z$ mode, the precision of $\phi_s$ can be measured at 45\% of the precision obtained from the LHCb experiment at HL-LHC. However, the measurement of $\Gamma_s$ and $\Delta\Gamma_s$ cannot benefit from the excellent time resolution and tagging power of the future $Z$-factories. Only operating at 10-Tera-$Z$ mode can the $\Gamma_s$ and $\Delta\Gamma_s$ reach an 18\% larger precision than the precision expected to be obtained from LHCb at HL-LHC. The control of penguin contamination at the future Z-factories is also discussed.