Eur. Phys. J. C (2023) 83: 1074
https://doi.org/10.1140/epjc/s10052-023-12254-8

Regular Article - Theoretical Physics

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

¹ School of Science, Tianjin University, 300072, Tianjin, China
² School of Physics, Nankai University, 300071, Tianjin, China

^a khw020056@tju.edu.cn

Received: 13 October 2023
Accepted: 10 November 2023
Published online: 24 November 2023

Abstract

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