Data taking strategy for $\psi (3770)$ and $\mathrm{{\rm Y}}(4S)$ branching fraction measurements at $e^{+}{e}^{-}$ colliders

The $\psi (3770)$ and $\mathrm{{\rm Y}}(4S)$ states predominantly decay into open-flavor meson pairs, whereas the decays of $\psi (3770)\to \text{non-}D\bar{D}$ and $\mathrm{{\rm Y}}(4S)\to \text{non-}B\bar{B}$ are rare but crucial for elucidating the inner structure and decay dynamics of heavy quarkonium states. To achieve precise branching fraction measurements for $\psi (3770)\to \text{non-}D\bar{D}$ and $\mathrm{{\rm Y}}(4S)\to \text{non-}B\bar{B}$ decays at the high luminosity $e^{+}{e}^{-}$ annihilation experiments, we employed Monte Carlo simulations and Fisher information to evaluate various data taking scenarios, ultimately determining the optimal scheme. The consistent results of both methodologies indicate that the optimal energy points for studies of $\psi (3770)\to \text{non-}D\bar{D}$ decays are $3.769\text{gev}$ and $3.781\text{gev}$, whereas those for $\mathrm{{\rm Y}}(4S)\to \text{non-}B\bar{B}$ decays are $10.574\text{gev}$ and $10.585\text{gev}$. In addition, we studied the dependence of the precision in branching fraction measurements on the integrated luminosity, with the branching fractions spanning several orders of magnitude.