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

Abstract

Within the framework of dispersion theory, we study the the processes $e^{+}{e}^{-}\to \varphi (2170)\to \varphi \pi \pi (K\bar{K})$. 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 $\varphi \pi$ invariant mass distributions of the $e^{+}{e}^{-}\to \varphi (2170)\to \varphi {\pi }^{+}{\pi }^{-}$ process, the low-energy constants in the chiral Lagrangian are determined. The theoretical prediction for the cross sections ratio $\sigma (e^{+}{e}^{-}\to \varphi (2170)\to \varphi K^{+}{K}^{-})/\sigma (e^{+}{e}^{-}\to \varphi (2170)\to \varphi {\pi }^{+}{\pi }^{-})$ is given, which could be useful for selecting the physical solution when the fit to the $e^{+}{e}^{-}\to \varphi K^{+}{K}^{-}$ cross section distribution is available in the future. Our results suggest that above the kinematical threshold of $\varphi K\bar{K}$, the mechanism $e^{+}{e}^{-}\to \varphi K^{+}{K}^{-}$ with the kaons rescattering to a pion pair plays an important role in the $e^{+}{e}^{-}\to \varphi {\pi }^{+}{\pi }^{-}$ transition.

14 pages, 3 figures. arXiv admin note: text overlap with arXiv:1902.10957, arXiv:2104.01752

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