Exploring the molecular scenario of $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$

In the present work, we assign the newly observed $P_c(4312)$ as a $I(J^P)= \frac{1}{2} (\frac{1}{2})^-$ molecular state composed of $\Sigma_c \bar{D}$, while $P_c(4440)$ and $P_c(4457)$ as $\Sigma_c \bar{D}^\ast$ molecular states with $I(J^P)= \frac{1}{2} (\frac{1}{2})^-$ and $\frac{1}{2} (\frac{3}{2})^-$, respectively. In this molecular scenario, we investigate the $P_c\to J/\psi p$ process of these three states and further predict the ratios of the $\mathcal B(P_c\to J/\psi p)$ and those of $\mathcal B(\Lambda_b\to P_c K)$ between these three states, which could serve as a crucial test of the present molecular scenario.