In this article, we study the $\bar{ D}\Sigma_c$, $\bar{ D}\Sigma_c^*$, $\bar{ D}^{ *}\Sigma_c$ and $\bar{ D}^{ *}\Sigma_c^*$ pentaquark molecular states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension $13$ in a consistent way. The present calculations support assigning the $P_c(4312)$ to be the $\bar{ D}\Sigma_c$ pentaquark molecular state with $J^P={ \frac{ 1}{ 2}}^-$, assigning the $P_c(4380)$ to be the $\bar{ D}\Sigma_c^*$ pentaquark molecular state with $J^P={ \frac{ 3}{ 2}}^-$, assigning the $P_c(4440/4457)$ to be the $\bar{ D}^{ *}\Sigma_c$ pentaquark molecular state with $J^P={ \frac{ 3}{ 2}}^-$ or the $\bar{ D}^{ *}\Sigma_c^*$ pentaquark molecular state with $J^P={ \frac{ 5}{ 2}}^-$. Special attentions are payed to the operator product expansion.
Categories: hep-ph