Analysis of the $P_c(4312)$, $P_c(4440)$, $P_c(4457)$ and related hidden-charm pentaquark states with QCD sum rules

In this article, we restudy the ground state mass spectrum of the diquark-diquark-antiquark type $uudc\bar{c}$ pentaquark states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of $13$ in a consistent way. The predicted masses support assigning the $P_c(4312)$ to be the hidden-charm pentaquark state with $J^P={\frac{1}{2}}^{-}$, assigning the $P_c(4440)$ to be the hidden-charm pentaquark state with $J^P={\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$ or ${\frac{5}{2}}^{-}$, assigning the $P_c(4457)$ to be the hidden-charm pentaquark state with $J^P={\frac{1}{2}}^{-}$ or ${\frac{3}{2}}^{-}$.