Recently observed $P_c$ as molecular states and possible mixture of $P_c(4457)$

Recently observed spectrum of $P_c$ states exhibits a strong link to $\Sigma_c \bar{D}^{(*)}$ thresholds. In spite of successful molecular interpretations, we still push forward to wonder whether there exist finer structures. Utilizing the effecitve lagrangians respecting heavy quark symmetry and chiral symmetry, as well as instantaneous Bethe-Salpeter equations, we investigate the $\Sigma_c \bar{D}^{(*)}$ interactions and three $P_c$ states. We confirm that $P_c(4312)$ and $P_c(4440)$ are good candidates of $\Sigma_c \bar{D}$ and $\Sigma_c \bar{D}^{*}$ molecules with spin-$\frac12$, respectively. Unlike other molecular calculations, our results indicate $P_c(4457)$ signal might be a mixture of spin-$\frac32$ and spin-$\frac12$ $\Sigma_c \bar{D}^{*}$ molecules, where the latter one appears to be an excitation of $P_c(4440)$. Therefore we conclude that, confronting three LHCb $P_c$ signals, there may exist not three, but four molecular states.