$b\to s\tau^+\tau^-$ Physics at Future $Z$ Factories

$b\to s\tau^+\tau^-$ measurements are highly motivated for addressing lepton-flavor-universality (LFU)-violating puzzles such as $R_{K^{(\ast)}}$ anomalies. The anomalies of $R_{D^{(*)}}$ and $R_{J/\psi}$ further strengthen their necessity and importance, given that the LFU-violating hints from both involve the third-generation leptons directly. $Z$ factories at the future $e^-e^+$ colliders stand at a great position to conduct such measurements because of their relatively high production rates and reconstruction efficiencies for $B$ mesons at the $Z$ pole. To fully explore this potential, we pursue a dedicated sensitivity study in four $b\to s\tau^+\tau^-$ benchmark channels, namely $B^0\to K^{\ast 0} \tau^+ \tau^-$, $B_s\to\phi \tau^+ \tau^-$, $B^+ \to K^+ \tau^+ \tau^- $ and $B_s \to \tau^+ \tau^-$, at the future $Z$ factories. We develop a fully tracker-based scheme for reconstructing the signal $B$ mesons and introduce a semi-quantitative method for estimating their major backgrounds. The simulations indicate that branching ratios of the first three channels can be measured with a precision $\sim \mathcal O(10^{-7} - 10^{-6})$ and that of $B_s \to \tau^+ \tau^-$ with a precision $\sim \mathcal O(10^{-5})$ at Tera-$Z$. The impacts of luminosity and tracker resolution on the expected sensitivities are explored. The interpretations of these results in effective field theory are also presented.