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 $ψ$ states listed in the PDG, and also several states near open-charm thresholds. The compositeness of the near-threshold poles suggests that $ψ(4040)$ could mainly consist of a $D^*\bar{D}^*$-molecule component, rather than a conventionally accepted quark-model $ψ(3S)$ state. Also, $ψ(4230)$ and $ψ(4360)$ might be substantial mixtures of $D_1(2420)\bar{D}$, $D_1(2420)\bar{D}^*$, $D_s^*\bar{D}_s^*$, and $c\bar{c}$ components.