Study on the possible molecular states composed of $Λ_c\bar D^*$, $Σ_c\bar D^*$, $Ξ_c\bar D^*$ and $Ξ_c'\bar D^*$ in the Bethe-Salpeter frame based on the pentaquark states $P_c(4440)$, $P_c(4457)$ and $P_{cs}(4459)$
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 $Σ_c\bar D^*$, in the earlier works, they were regarded as molecular states of $Σ_c\bar D^*$ 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 $Ξ_c\bar D^*$ bound state with $I=0$. Within the Bethe-Salpeter (B-S) framework we systematically study the possible bound states of $Λ_c\bar D^*$, $Σ_c\bar D^*$, $Ξ_c\bar D^*$ and $Ξ_c'\bar D^*$. Our results indicate that $Σ_c\bar D^*$ 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 $Σ_c$ and $\bar D^*$ is too weak to constitute a molecule, so $P_{c}(4457)$ may not be a bound state of $Σ_c\bar D^*$ with $I(J^P)=\frac{1}{2}(\frac{3}{2}^-)$. As $Ξ_c\bar D^*$ and $Ξ_c'\bar D^*$ systems we take into account of the mixing between $Ξ_c$ and $Ξ'_c$ and the eigenstets should include two normal bound states $Ξ_c\bar D^*$ and $Ξ_c'\bar D^*$ with $I(J^P)=\frac{1}{2}(\frac{1}{2}^-)$ and a loosely bound state $Ξ_c\bar D^*$ with $I(J^P)=\frac{1}{2}(\frac{3}{2}^-)$. The conclusion that two $Ξ_c\bar D^*$ 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 $Ξ_c'\bar D^*$ 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.