Shedding light on the nature of $\phi(2170)$ state in the $e^+e^- \to \phi \pi^+\pi^-$ reaction

We investigate the production of $\phi(2170)$ state in the $e^+e^- \to \phi \pi^+\pi^-$ reaction with the effective Lagrangian approach. In addition to the tree level contributions from the $\phi(1680)$ meson and a possible X(1750) state, we consider also the $K_1$-$K$-$\bar{K}$ intermediate state process from the perspective of triangular singularity. Based on the one-photon exchange approximation, a pair of $K_1 \bar{K}$ mesons was firstly produced, and then the $K_1$ meson subsequently decays into $\phi$ and $K$, and the $K\bar{K}$ pair produce the $\pi^+ \pi^-$ through the final state interactions, in which the scalar meson $f_0(980)$ is dynamically generated. We show that the inclusion of the triangle loop diagrams leads to a good description of the new BESIII measurements, especially for the structure of $\phi(2170)$. This provides a novel interpretation of the $\phi(2170)$ state, offering new insights into its fundamental nature which is still unclear. Furthermore, it is found that these measurements on the $e^+e^- \to \phi \pi^+\pi^-$ reaction can be used to determine some of the properties of two $K_1$ mesons with masses around 1610 MeV and 1895 MeV, which are crucial to reproduce the experimental data.