Study on the resonant parameters of $Y(4220)$ and $Y(4390)$

Many vector charmonium-like states have been reported recently in the cross sections of $e^+e^- \rightarrow \omega\chi_{c0}$, $\pi^{+}\pi^{-}h_c$, $\pi^{+}\pi^{-}J/\psi$, $\pi^{+}\pi^{-}\psi(3686)$ and $\pi^{+}D^{0}D^{*-}+c.c.$ To better understand the nature of these states, a combined fit is performed to these cross sections by using three resonances $Y(4220)$, $Y(4390)$ and $Y(4660)$. The resonant parameters for the three resonances are obtained. We emphasize that two resonances $Y(4220)$ and $Y(4390)$ are sufficient to explain these cross sections below 4.6 GeV. The lower limits of $Y(4220)$ and $Y(4390)$'s leptonic decay widths are also determined to be $(36.4\pm2.0\pm4.2)$ and $(123.8\pm6.5\pm9.0)$ eV.