Study on the possible molecular states composed of ${\Lambda }_c{\bar{D}}^{\ast }$, ${\Sigma }_c{\bar{D}}^{\ast }$, ${\Xi }_c{\bar{D}}^{\ast }$ and ${\Xi }_c^{\prime }{\bar{D}}^{\ast }$ in the Bethe-Salpeter frame based on the pentaquark states $P_c(4440)$, $P_c(4457)$ and $P_{cs}(4459)$

Abstract

The measurements on a few pentaquarks states $P_c(4440)$, $P_c(4457)$ and $P_{cs}(4459)$ excite our new interests about their structures. Since the masses of $P_c(4440)$ and $P_c(4457)$ are close to the threshold of ${\Sigma }_c{\bar{D}}^{\ast }$, in the earlier works, they were regarded as molecular states of ${\Sigma }_c{\bar{D}}^{\ast }$ with quantum numbers $I(J^P)=\frac{1}{2}({\frac{1}{2}}^{-})$ and $\frac{1}{2}({\frac{3}{2}}^{-})$, respectively. In a similar way $P_{cs}(4459)$ is naturally considered as a ${\Xi }_c{\bar{D}}^{\ast }$ bound state with $I=0$. Within the Bethe-Salpeter (B-S) framework we systematically study the possible bound states of ${\Lambda }_c{\bar{D}}^{\ast }$, ${\Sigma }_c{\bar{D}}^{\ast }$, ${\Xi }_c{\bar{D}}^{\ast }$ and ${\Xi }_c^{\prime }{\bar{D}}^{\ast }$. Our results indicate that ${\Sigma }_c{\bar{D}}^{\ast }$ can form a bound state with $I(J^P)=\frac{1}{2}({\frac{1}{2}}^{-})$, which corresponds to $P_c(4440)$. However for the $I(J^P)=\frac{1}{2}({\frac{3}{2}}^{-})$ system the attraction between ${\Sigma }_c$ and ${\bar{D}}^{\ast }$ is too weak to constitute a molecule, so $P_c(4457)$ may not be a bound state of ${\Sigma }_c{\bar{D}}^{\ast }$ with $I(J^P)=\frac{1}{2}({\frac{3}{2}}^{-})$. As ${\Xi }_c{\bar{D}}^{\ast }$ and ${\Xi }_c^{\prime }{\bar{D}}^{\ast }$ systems we take into account of the mixing between ${\Xi }_c$ and ${\Xi }_c^{\prime }$ and the eigenstets should include two normal bound states ${\Xi }_c{\bar{D}}^{\ast }$ and ${\Xi }_c^{\prime }{\bar{D}}^{\ast }$ with $I(J^P)=\frac{1}{2}({\frac{1}{2}}^{-})$ and a loosely bound state ${\Xi }_c{\bar{D}}^{\ast }$ with $I(J^P)=\frac{1}{2}({\frac{3}{2}}^{-})$. The conclusion that two ${\Xi }_c{\bar{D}}^{\ast }$ bound states exist, supports the suggestion that the observed peak of $P_{cs}(4459)$ may hide two states $P_{cs}(4455)$ and $P_{cs}(4468)$. Based on the computations we predict a bound state ${\Xi }_c^{\prime }{\bar{D}}^{\ast }$ with $I(J^P)=\frac{1}{2}({\frac{1}{2}}^{-})$ but not that with $I(J^P)=\frac{1}{2}({\frac{3}{2}}^{-})$. Further more accurate experiments will test our approach and results.

24 pages, 2 figures and 12 tables. arXiv admin note: text overlap with arXiv:1909.12509

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