One pion exchange and the quantum numbers of the Pc(4440) and Pc(4457) pentaquarks

The LHCb collaboration has recently discovered three pentaquark-like states --- the $P_c(4312)$, $P_c(4440)$ and $P_c(4457)$ --- close to the $\bar{D} \Sigma_c$ and the $\bar{D}^* \Sigma_c$ meson-baryon thresholds. The standard interpretation is that they are heavy antimeson-baryon molecules. Their quantum numbers have not been determined yet, which implies two possibilities for the $P_c(4440)$ and $P_c(4457)$: $J^P = \tfrac{1}{2}^{-}$ and $J^P = \tfrac{3}{2}^{-}$. The preferred interpretation within a contact-range effective field theory is that the $P_c(4440)$ is the $J^P = \tfrac{1}{2}^-$ molecule, while the $P_c(4457)$ is the $J^P = \tfrac{3}{2}^-$ one. Here we show that when the one pion exchange potential between the heavy-antimeson and heavy-baryon is taken into account, this conclusion changes, with the contrary identification being as likely as the original one. The identification is however cutoff dependent, which suggests that improvements of the present description (e.g. the inclusion of subleading order corrections, like two-pion exchanges) are necessary in order to disambiguate the spectroscopy of the molecular pentaquarks.