Two-body strong decays of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>2</mml:mn><mml:mi>P</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>3</mml:mn><mml:mi>P</mml:mi></mml:math> charmonium states
Zhi-Hui Wang, Guo‐Li Wang
Abstract
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 ${\ensuremath{\chi}}_{c0}(2P)$, the total strong decay width is smaller than the experimental data. But the strong decay width depends on the parameter $\ensuremath{\gamma}$ 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 ${\ensuremath{\chi}}_{c0}(2P)$. The $X(4160)$ is a good candidate for the ${\ensuremath{\chi}}_{c0}(3P)$, not only the strong decay width of the ${\ensuremath{\chi}}_{c0}(3P)$ is same as the experimental data, but the relative ratios $\frac{\mathrm{\ensuremath{\Gamma}}({\ensuremath{\chi}}_{c0}(3P)\ensuremath{\rightarrow}D\overline{D})}{\mathrm{\ensuremath{\Gamma}}({\ensuremath{\chi}}_{c0}(3P)\ensuremath{\rightarrow}{D}^{*}{\overline{D}}^{*})}\ensuremath{\approx}0.0019<0.09$, and $\frac{\mathrm{\ensuremath{\Gamma}}({\ensuremath{\chi}}_{c0}(3P)\ensuremath{\rightarrow}D{\overline{D}}^{*})}{\mathrm{\ensuremath{\Gamma}}({\ensuremath{\chi}}_{c0}(3P)\ensuremath{\rightarrow}{D}^{*}{\overline{D}}^{*})}=0<0.22$ are consistent with the experimental results of the $X(4160)$. Taking the $X(4274)$ as the ${\ensuremath{\chi}}_{c1}(3P)$, the strong decay width is consistent with the experimental data, so the $X(4274)$ is a good candidate for the ${\ensuremath{\chi}}_{c1}(3P)$. Assigning the $X(4350)$ as the ${\ensuremath{\chi}}_{c2}(3P)$, the corresponding strong decay width is slightly larger than the experimental data. To identify if the $X(4350)$ is ${\ensuremath{\chi}}_{c2}(3P)$, 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.