Group Theory Analysis of $P_c$ Resonances in Molecular Picture

Hidden-charm pentaquarks are analysed in Group Theory in the baryon-meson molecule picture within the framework of the constituent quark model. The investigation of strong decays of pentaquark resonances reveals that pentaquark states of all quark configurations can decay through the $N J/\psi$ channel while only five pentaquark states may decay in open-charm modes. The partial decay widths in the $p\eta_c$ channel are in the same order as in the $p J/\psi$ channel for $J^P=1/2^-$ baryon-meson pentaquark states, but the $p\eta_c$ channel is forbidden for $J^P=3/2^-$ states. The $p \eta_c$, $\Sigma_c\bar{D}$ and $\Lambda_c^+\bar{D}$ channels open for $J^P=1/2^-$ $P_c(4440)$ and $P_c(4457)$ while the $\Sigma_c^*\bar{D}$ channel opens for $J^P=3/2^-$ $P_c(4440)$ and $P_c(4457)$. For $P_c(4312)$, due to the mass threshold of $\Sigma_c\bar{D}$ and $\Sigma_c^*\bar{D}$ higher than 4312 MeV, the $p \eta_c$ and $\Lambda_c^+\bar{D}$ channels are open for $J^P=1/2^-$ but no open-charm channel open for $J^P=3/2^-$. We strongly suggest that the spin of the $P_c$ resonances may be determined in experiments by investigating the $p \eta_c$ and open-charm channels.