$P^\Lambda_{\psi s}(4459)$ and $P^\Lambda_{\psi s}(4338)$ as molecular states in $ J/\psi \Lambda$ invariant mass spectra

Recently, the LHCb Collaboration has reported two strange hidden-charm pentaquark states named $P^\Lambda_{\psi s}(4459)$ and $P^\Lambda_{\psi s}(4338)$ in the $ J/\psi \Lambda$ invariant mass spectra of decays $\Xi_b^-$$\to$$ J/\psi \Lambda K^-$ and $B^-$$\to$$J/\psi \Lambda \bar{p}$,respectively. In this work, we perform a coupled-channel study of the interactions $\Xi_c^*\bar{D}^*$, $\Xi'_c\bar{D}^*$, $\Xi^*_c\bar{D}$, $\Xi_c\bar{D}^*$, $\Xi'_c\bar{D}$, $\Lambda_c\bar{D}_s^{*}$, $\Xi_c\bar{D}$, $\Lambda_c\bar{D}_s$, and $\Lambda J/\psi$ in the quasipotential Bethe-Salpeter equation approach to estimate the $ J/\psi \Lambda$ invariant mass spectra. With the help of effective Lagrangians, the potential kernel can be constructed by meson exchanges to obtain the scattering amplitudes, from which the poles of the bound states and the invariant mass spectra can be reached. The coupled-channel calculation results in that the width of state $\Xi_c\bar{D}^{*}(1/2^-)$ is about $18$~MeV and that of state $\Xi_c\bar{D}^{*}(3/2^-)$ is only about $1.6$~MeV. By comparison with experimental data, it indicates that the structure $P^{\Lambda}_{\psi s}(4459)$ is mainly from the contribution from the $\Xi_c \bar{D}^{*}(1/2^-)$ state while the role of state $\Xi_c \bar{D}^{*}(3/2^-)$ cannot be excluded. The line shape of the structure $P^{\Lambda}_{\psi s}(4338)$ can be reproduced roughly by a narrow molecular state from the $\Xi_c \bar{D}$ interaction with $J^P$=$1/2^-$, which is extremely close to the threshold, with a large interference effect. Besides, an additional state $\Xi'_c \bar{D}(1/2^-)$ is suggested to be observed as a dip structure in the $J/\psi \Lambda$ invariant mass spectrum.