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A detailed analysis to find a missing $\psi(4S)$ is made by utilizing therecent precise mesurements of the cross section for the process $e^+e^-\to\psi(2S)\pi^+\pi^-$ by Belle. Assuming three resonances $Y(4360)$, $Y(4660)$,and $\psi(4S)$ to fit the data, we obtain the resonance parameters for$\psi(4S)$ as $m=4243$ MeV and $\Gamma=16 \pm 31$ MeV showing a narrow state aspredicted before. A combined fit to the data $e^+ e^- \to \psi(2S) \pi^+ \pi^-,\ h_c \pi^+ \pi^-$, and $\chi_{c0} \omega$ is also performed to obtain thesimilar resonance parameters of $\psi(4S)$. The upper limit of the branchingratio is fitted to be $\mathcal{B}(\psi(4S)\to\psi(2S)\pi^+\pi^-) < 3\times10^{-3}$, which can be understood by hadronic loop contributions in reasonableparameters range. In addition, the ratios of the branching ratios of the$\psi(4S)$ dipion transition to the one of $\psi(4S)\to\chi_{c0}\omega$ arefitted, which can be further measured by BESIII and forthcoming BelleII toconfirm the existence of $\psi(4S)$.

Search for missing $\psi(4S)$ in the $e^+e^-\to \pi^+\pi^-\psi(2S)$ process