$\phi (2170)$ decaying to $\phi \pi \pi$ and $\phi K\bar{K}$

Within the framework of dispersion theory, we study the the processes $e^{+}{e}^{-}\to \phi (2170)\to \phi \pi \pi (K\bar{K})$. The strong pion-pion final-state interactions, especially the $K\bar{K}$ coupled channel in the $S$-wave, are taken into account in a model-independent way using the Omnès function solution. Through fitting the experimental data of the $\pi \pi$ and $\phi \pi$ invariant mass distributions of the $e^{+}{e}^{-}\to \phi (2170)\to \phi {\pi }^{+}{\pi }^{-}$ process, the low-energy constants in the chiral Lagrangian are determined. The theoretical prediction for the cross sections ratio $\sigma (e^{+}{e}^{-}\to \phi (2170)\to \phi K^{+}{K}^{-})\mathrm{/}\sigma (e^{+}{e}^{-}\to \phi (2170)\to \phi {\pi }^{+}{\pi }^{-})$ is given, which could be useful for selecting the physical solution when the fit to the $e^{+}{e}^{-}\to \phi K^{+}{K}^{-}$ cross section distribution is available in the future. Our results suggest that above the kinematical threshold of $\phi K\bar{K}$, the mechanism $e^{+}{e}^{-}\to \phi K^{+}{K}^{-}$ with the kaons rescattering to a pion pair plays an important role in the $e^{+}{e}^{-}\to \phi {\pi }^{+}{\pi }^{-}$ transition.