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Recent high-precision $e^+e^-\to c\bar{c}$ data from the BESIII and Belle arehighly useful to understand the vector charmonium pole structure and puzzlinglineshapes due to the exotic hadron candidates $Y$. We thus conduct a globalcoupled-channel analysis of most of the available data (9 two-body, 8three-body, and 1 four-body final states) in $\sqrt{s}=3.75-4.7$ GeV. Not onlycross sections but also invariant mass distributions of subsystems are fitted.Our model includes dozens of (quasi) two-body states that nonperturbativelycouple with each other through bare charmonium excitations andparticle-exchange mechanisms required by the three-body unitarity. Theamplitudes obtained from the fits are analytically continued to vectorcharmonium and $Z_c$ poles. We do not find a $\psi(4160)$ pole that has beenconsidered well-established. Instead, we find two poles of $\sim 4230$ MeV;$\psi(4230)$ with $\Gamma= 36$ MeV and a broader one with $\Gamma= 114$ MeV.Two $Z_c$ poles are found as virtual states $\sim 40$ MeV below the$D^*\bar{D}^{(*)}$ thresholds, being consistent with lattice QCD results. Thiswork presents the first global analysis to determine the vector charmonium and$Z_c$ poles, thereby paving the way to extracting detailed properties of theprominent exotic hadron candidates from data.

Global coupled-channel analysis of $e^+e^-\to c\bar{c}$ processes in $\sqrt{s}=3.75-4.7$ GeV