Study on the possible molecular states composed of $\Lambda_c\bar D^*$, $\Sigma_c\bar D^*$, $\Xi_c\bar D^*$ and $\Xi_c'\bar D^*$ in the Bethe-Salpeter frame based on the pentaquark states $P_c(4440)$, $P_c(4457)$ and $P_{cs}(4459)$

TL;DR

This study uses the Bethe-Salpeter framework to explore possible molecular states of certain charmed baryon-anticharmed meson systems, providing insights into the structure of observed pentaquark states and predicting new bound states.

Contribution

It systematically investigates molecular states of $ ext{Lambda}_car D^*$, $ ext{Sigma}_car D^*$, $ ext{Xi}_car D^*$, and $ ext{Xi}_c'ar D^*$, offering explanations for observed pentaquarks and predicting new bound states.

Findings

$ ext{Sigma}_car D^*$ can form a bound state corresponding to $P_c(4440)$.

Weak attraction prevents $ ext{Sigma}_car D^*$ from forming $P_c(4457)$ as a bound state.

Existence of two $ ext{Xi}_car D^*$ bound states supports the hypothesis of two states $P_{cs}(4455)$ and $P_{cs}(4468)$.

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 $\Sigma_c\bar D^*$, in the earlier works, they were regarded as molecular states of $\Sigma_c\bar D^*$ 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 $\Xi_c\bar D^*$ bound state with $I=0$. Within the Bethe-Salpeter (B-S) framework we systematically study the possible bound states of $\Lambda_c\bar D^*$, $\Sigma_c\bar D^*$, $\Xi_c\bar D^*$ and $\Xi_c'\bar D^*$. Our results indicate that $\Sigma_c\bar D^*$ 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 $\bar D^*$ is too weak to constitute a molecule, so $P_{c}(4457)$ may not be a bound state of $\Sigma_c\bar D^*$ with $I(J^P)=\frac{1}{2}(\frac{3}{2}^-)$. As $\Xi_c\bar D^*$ and $\Xi_c'\bar D^*$ systems we take into account of the mixing between $\Xi_c$ and $\Xi'_c$ and the eigenstets should include two normal bound states $\Xi_c\bar D^*$ and $\Xi_c'\bar D^*$ with $I(J^P)=\frac{1}{2}(\frac{1}{2}^-)$ and a loosely bound state $\Xi_c\bar D^*$ with $I(J^P)=\frac{1}{2}(\frac{3}{2}^-)$. The conclusion that two $\Xi_c\bar D^*$ 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 $\Xi_c'\bar D^*$ 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.