Analysis of the strong decays of the $P_c(4312)$ as a pentaquark molecular state with QCD sum rules

In this article, we tentatively assign the $P_c(4312)$ to be the $\bar{D}\Sigma_c$ pentaquark molecular state with the spin-parity $J^P={\frac{1}{2}}^-$, and discuss the factorizable and non-factorizable contributions in the two-point QCD sum rules for the $\bar{D}\Sigma_c$ molecular state in details to prove the reliability of the single pole approximation in the hadronic spectral density. We study its two-body strong decays with the QCD sum rules, special attentions are paid to match the hadron side with the QCD side of the correlation functions to obtain solid duality. We obtain the partial decay widths $\Gamma\left(P_c(4312)\to \eta_c p\right)=0.255\,{\rm{MeV}}$ and $\Gamma\left(P_c(4312)\to J/\psi p\right)=9.296^{+19.542}_{-9.296}\,\,{\rm{MeV}}$, which are compatible with the experimental value of the total width, and support assigning the $P_c(4312)$ to be the $\bar{D}\Sigma_c$ pentaquark molecular state.