Prediction of a missing higher charmonium around 4.26 GeV in $J/\psi$ family

Inspired by the similarity of mass gaps of $J/\psi$ and $\Upsilon$ families, the prediction of missing higher charmonium with mass $4263$ MeV and very narrow width is made. In addition, the properties of two charmonium-like states, $X(3940)$ and $X(4160)$, and charmonium $\psi(4415)$ are discussed. Here, $X(3940)$ as $\eta_c(3S)$ is established while the explanation of $X(4160)$ to be $\eta_c(4S)$ is fully excluded and $\eta_c(4S)$ is typically a very narrow state. These predictions can be accessible at BESIII, Belle and BelleII in near future.