Study on the possible molecular states composed of $$\Lambda _c{\bar{D}}^*$$ Λ c D ¯ ∗ , $$\Sigma _c{\bar{D}}^*$$ Σ c D ¯ ∗ , $$\Xi _c{\bar{D}}^*$$ Ξ c D ¯ ∗ and $$\Xi _c'{\bar{D}}^*$$ Ξ c ′ D ¯ ∗ in the Bethe–Salpeter frame based on the pentaquark states $$P_c(4440)$$ P c ( 4440 ) , $$P_c(4457)$$ P c ( 4457 ) and $$P_{cs}(4459)$$ P cs ( 4459 )

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