$P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ productions in $e^{+} e^{-}$ collisions

In the present work, we propose to investigate the productions of $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ in the $e^{+} e^{-} \rightarrow p \bar{p} J/\psi$ process. By using an effective Lagrangian approach, we estimate the cross section for the processes under consideration. For the $e^{+} e^{-} \rightarrow \bar{p} P_{c}(4312)$, $e^{+} e^{-} \rightarrow \bar{p} P_{c}(4440)$, and $e^{+} e^{-} \rightarrow \bar{p} P_{c}(4457)$ processes, the cross sections are evaluated to be ($46.6^{+88.4}_{-34.5}$) fb, ($57.4^{+109}_{-42.5}$) fb, and ($55.1^{+105}_{-40.8}$) fb at $\sqrt{s}=6$ GeV, respectively, where the central values are estimated with $\Lambda_{r}=3.0$ GeV, and the uncertainties are resulted from the variations of $\Lambda_{r}$ from 2.5 to 3.5 GeV. Considering that the $P_{c}$ states can decay into $J/\psi p$, we estimate the cross sections for $e^{+} e^{-} \rightarrow p \bar{p} J/\psi$ and the differential cross sections depending on the $J/\psi p$ invariant mass. At $\sqrt{s}=6$ GeV, the cross sections for $e^{+} e^{-} \rightarrow p \bar{p} J/\psi$ are estimated to be ($34.6^{+68.0}_{-26.0}$) fb. Moreover, our estimations indicate that the cross section resulted from the $P_{c}(4457)$ and $\bar{P}_{c}(4457)$ intermediate states is dominant. In the $J/\psi p$ invariant mass spectrum, a splitting signal can be observed around 4.45 GeV, with the two peaks corresponding to $P_{c}(4440)$ and $P_{c}(4457)$.