Heavy pentaquark states $P_c(4380)$ and $P_c(4450)$ in the $J/\psi$ production induced by pion beams off the nucleon

In this study, we investigate the $J/\psi$ production induced by pion beams off the nucleon, particularly the heavy pentaquarks $P_c(4380)$ and $P_c(4450)$ in intermediate states, based on a hybridized Regge model. The process involving $\rho$ and $\pi$ meson exchange in the $t$ channel is considered as background, and the heavy pentaquark exchange is included in the $s$ channel. The coupling constants such as the $\rho NN$ and $\pi NN$ vertices are taken from the $NN$ potentials, whereas those for the $J/\psi\rho\pi$ and $J/\psi\pi\pi$ vertices are determined by using experimental data based on the branching ratios. In order to estimate the $P_c(4380)$ and $P_c(4450)$ coupling constants, we use the experimental upper limit on the total cross section as a guide for the $\pi N\to J/\psi N$ reaction. The background total cross section is the order of $10^{-4}-10^{-3}$ nb. In the vicinity of the heavy pentaquark masses, the total cross section reaches about $1$ nb.