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

TL;DR

This paper uses dispersion theory to analyze the decay processes of the $ ext{phi}(2170)$ resonance into $ ext{phi} ext{pi} ext{pi}$ and $ ext{phi} ext{K}ar{ ext{K}}$, fitting experimental data to determine low-energy constants and predicting cross section ratios.

Contribution

It introduces a model-independent dispersive approach to study $ ext{phi}(2170)$ decays, incorporating coupled-channel effects and providing predictions for future experimental tests.

Findings

The low-energy constants in the chiral Lagrangian are determined from data.

The ratio of cross sections for $ ext{e}^+ ext{e}^- o ext{phi}(2170) o ext{phi} ext{K}^+ ext{K}^-$ and $ ext{phi} ext{pi}^+ ext{pi}^-$ is predicted.

Kaon rescattering significantly influences the $ ext{phi} ext{pi} ext{pi}$ transition above the $ ext{phi} ext{K}ar{ ext{K}}$ threshold.

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 $e^+e^- \to \phi(2170) \to \phi \pi^+\pi^-$ process, the low-energy constants in the chiral Lagrangian are determined. The theoretical prediction for the cross sections ratio ${\sigma(e^+e^- \to \phi(2170)\to \phi K^+ K^-)}/{\sigma(e^+e^- \to \phi(2170)\to \phi\pi^+\pi^-)}$ 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.