Molecular states from $\Sigma^{(*)}_c\bar{D}^{(*)}-\Lambda_c\bar{D}^{(*)}$ interaction

In this work, we systemically investigate the molecular states from the $\Sigma^{(*)}_c\bar{D}^{(*)}-\Lambda_c\bar{D}^{(*)}$ interaction with the help of the Lagrangians with heavy quark and chiral symmetries in a quasipotential Bethe-Salpeter equation (qBSE) approach. The molecular states are produced from isodoublet (I=1/2) $\Sigma_c\bar{D}$ interaction with spin parity $J^P=1/2^-$ and $\Sigma_c\bar{D}^*$ interaction with $1/2^-$ and $3/2^-$. Their masses and widths are consistent with the $P_c(4312)$, $P_c(4440)$ and $P_c(4457)$ observed at LHCb. The states, $\Sigma_c^*\bar{D}^*(1/2^-)$, $\Sigma_c^*\bar{D}^*(3/2^-)$ and $\Sigma^*_c\bar{D}(3/2^-)$, are also produced with the same parameters. The isodoublet $\Sigma_c^*\bar{D}^*$ interaction with $5/2^-$, as well as the isoquartet (I=3/2) $\Sigma_c\bar{D}^*$ interactions with $1/2^-$ and $3/2^-$, $\Sigma_c^*\bar{D}^*$ interaction with $3/2^-$ and $5/2^-$, are also attractive while very large cutoff is required to produce a molecular state. We also investigate the origin of the widths of these molecular states in the same qBSE frame. The $\Lambda\bar{D}^*$ channel is dominant in the decays of the states, $\Sigma_c\bar{D}^*(1/2^-)$, $\Sigma_c\bar{D}^*(3/2^-)$, $\Sigma_c^*\bar{D}(3/2^-)$, and $\Sigma_c\bar{D}(1/2^-)$. The $\Sigma^*_c\bar{D}^*(1/2^-)$ state has large coupling to $\Sigma_c\bar{D}$ channel while the $\Sigma_c\bar{D}^*$, $\Sigma^*_c\bar{D}$ and $\Lambda_c\bar{D}^*$ channels provide similar contributions to the width of the $\Sigma^*_c\bar{D}^*(3/2^-)$ state. These results will be helpful to understand the current LHCb experimental results, and the three predicted states and the decay pattern of these hidden-charmed molecular pentaquarks can be checked in future experiments.