Is $Y(4008)$ possibly a $1^{--}$ $ψ(3^3S_1)$ state?

The strong decays of the radially excited $ψ(3^3S_1)$ state are studied within the $^3P_0$ model. As a believed $ψ(3^3S_1)$, some strong decay widths and relevant ratios of $ψ(4040)$ are calculated in the model. The theoretical results are consistent with experiments. In a similar way, as a possible $ψ(3^3S_1)$, the same strong decay widths and relevant ratios of $Y(4008)$ are presented. Our study indicates that $Y(4008)$ is hard to be identified with a $ψ(3^3S_1)$ charmonium once it is confirmed under the $D^*\bar{D}^*$ threshold, but it is very possibly a $ψ(3^3S_1)$ charmonium once it is confirmed above the $D^*\bar{D}^*$ threshold by experiment.

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