Two-body open charm strong decays of the $2P$ and $3P$ charmonium states are studied by the Bethe-Salpeter method combined with the ${}^3{P}_0$ model. The wave functions and mass spectra of the $2P$ and $3P$ charmonium states are obtained by solving the Bethe-Salpeter equation with the relativistic correction. The strong decay widths and relative ratios of the $2P$ and $3P$ charmonium states are calculated. Comparing our results with the experimental data, we obtain some interesting results. Considering the $X^{*}(3860)$ as the ${\chi }_{c0}(2P)$, the total strong decay width is smaller than the experimental data. But the strong decay width depends on the parameter $\gamma$ in the ${}^3{P}_0$ model, and the mass and width of the $X^{*}(3860)$ have large errors, we cannot rule out the possibility that the $X^{*}(3860)$ is the ${\chi }_{c0}(2P)$. The $X(4160)$ is a good candidate for the ${\chi }_{c0}(3P)$, not only the strong decay width of the ${\chi }_{c0}(3P)$ is same as the experimental data, but the relative ratios $\frac{\mathrm{\Gamma }({\chi }_{c0}(3P)\to D\overline{D})}{\mathrm{\Gamma }({\chi }_{c0}(3P)\to D^{*}{\overline{D}}^{*})}\approx 0.0019<0.09$, and $\frac{\mathrm{\Gamma }({\chi }_{c0}(3P)\to D{\overline{D}}^{*})}{\mathrm{\Gamma }({\chi }_{c0}(3P)\to D^{*}{\overline{D}}^{*})}=0<0.22$ are consistent with the experimental results of the $X(4160)$. Taking the $X(4274)$ as the ${\chi }_{c1}(3P)$, the strong decay width is consistent with the experimental data, so the $X(4274)$ is a good candidate for the ${\chi }_{c1}(3P)$. Assigning the $X(4350)$ as the ${\chi }_{c2}(3P)$, the corresponding strong decay width is slightly larger than the experimental data. To identify if the $X(4350)$ is ${\chi }_{c2}(3P)$, many more investigations are needed. All of the strong decay widths and relative ratios of the $2P$ and $3P$ charmonium states can provide the useful information to discover and confirm these particles in the future.

• Received 20 April 2022
• Accepted 13 September 2022

DOI:https://doi.org/10.1103/PhysRevD.106.054037

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Published by the American Physical Society