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Based on the updated data of charmoniumlike state $Y(4220)$ reported in thehidden-charm %and open-charm channels of the $e^+e^-$ annihilation, we proposea $4S$-$3D$ mixing scheme to categorize $Y(4220)$ into the $J/\psi$ family. Wefind that the present experimental data can support this charmonium assignmentto $Y(4220)$. Thus, $Y(4220)$ plays a role of a scaling point in constructinghigher charmonia above 4 GeV. To further test this scenario, we provide moreabundant information on the decay properties of $Y(4220)$, and predict itscharmonium partner $\psi(4380)$, whose evidence is found by analyzing the$e^+e^-\to \psi(3686)\pi^+\pi^-$ data from BESIII. If $Y(4220)$ is indeed acharmonium, we must face how to settle the established charmonium $\psi(4415)$in the $J/\psi$ family. In this work, we may introduce a $5S$-$4D$ mixingscheme, and obtain the information of the resonance parameters and partialopen-charm decay widths of $\psi(4415)$, which do not contradict the presentexperimental data. Additionally, we predict a charmonium partner $\psi(4500)$of $\psi(4415)$, which can be accessible at future experiments, especially,BESIII and BelleII. The studies presented in this work provide new insights toestablish the higher charmonium spectrum.

Constructing $J/\psi$ family with updated data of charmoniumlike $Y$ states