$φ(2170)$ decaying to $φππ$ and $φK\bar{K}$

Abstract: 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 Omn`es function solution. Through fitting the experimental data of the $\pi\pi$ and $\phi\pi$ invariant mass distributions of the $$e^{+e^-} \to \phi(2170) \to \phi \pi<sup>+\pi<sup>-$$ process, the low-energy constants in the chiral Lagrangian are determined. The theoretical prediction for the cross sections ratio $${\sigma(e^{+e^-} \to \phi(2170)\to \phi K⁺ K^{-)}/{\sigma(e^{+e^-}} \to \phi(2170)\to \phi\pi<sup>+\pi<sup>-)}$$ is given, which could be useful for selecting the physical solution when the fit to the $$e^{+e^-} \to \phi K⁺ K<sup>-$$ cross section distribution is available in the future. Our results suggest that above the kinematical threshold of $\phi K\bar{K}$, the mechanism $$e^{+e^-} \to \phi K⁺ K<sup>-$$ with the kaons rescattering to a pion pair plays an important role in the $$e^{+e^-} \to \phi\pi<sup>+\pi<sup>-$$ transition.