Hidden-charm pentaquark states $qqqc\bar{c}$ $(q = u,d)$ in the chiral SU(3) quark model

In this work, we systematically calculate the spectrum of hidden-charm pentaquark states $qqqc\bar{c}$ $(q = u,d)$ in the chiral SU(3) quark model, which has been quite successful in reproducing consistently the energies of octet and decuplet baryon ground states, the binding energy of deuteron, and the nucleon-nucleon ($NN$) scattering phase shifts and mixing parameters for partial waves with total angular momentum up to $J=6$. The Hamiltonian contains the one-gluon-exchange (OGE) potential, the Goldstone-boson-exchange (GBE) potential, the confinement potential, and the kinetic energy of the system. We solve the Schr\"odinger equation by use of the variational method. It is found that the masses of all the experimentally observed $P_c(4312)$, $P_c(4380)$, $P_c(4440)$, and $P_c(4457)$ states are much overestimated, indicating that these states are not compact pentaquark states in the chiral SU(3) quark model. All other $qqqc\bar{c}$ $(q = u,d)$ states are found to lie much above the corresponding baryon-meson thresholds, and thus are not suggested as stable pentaquark states due to their fall-apart decays. A detailed comparison of the results with those obtained in the OGE model and the chromomagnetic interaction (CMI) model is further given.