Coupled-channel analysis for vector charmonia and their nature

High-precision $e^{+}{e}^{-}\to c\bar{c}$ data (20 final states) from the BESIII and Belle in $\sqrt{s}=3.75-4.7$ GeV are analyzed with a semi three-body unitary coupled-channel model. Vector charmonium poles are extracted from the amplitudes obtained from the fit. We find well-known $\psi$ states listed in the PDG, and also several states near open-charm thresholds. The compositeness of the near-threshold poles suggests that $\psi (4040)$ could mainly consist of a $D^{\ast }{\bar{D}}^{\ast }$-molecule component, rather than a conventionally accepted quark-model $\psi (3S)$ state. Also, $\psi (4230)$ and $\psi (4360)$ might be substantial mixtures of $D_1(2420)\bar{D}$, $D_1(2420){\bar{D}}^{\ast }$, $D_s^{\ast }{\bar{D}}_s^{\ast }$, and $c\bar{c}$ components.