Pion-induced production of the $Z_c(3900)$ off a nuclear target

We investigate the possibility to study the charmoniumlike state $Z_c(3900)$ through the pion-induced production off a nuclear target. By using a high-energy pion beam, the $Z_c(3900)$ can be produced off a proton or nucleus though the Primakoff effect. The production amplitude is calculated in an effective Lagrangian approach combined with the vector dominance model. The total cross sections of the $p(\pi^-, Z^-_c(3900))$ and $p(\pi^-, Z^-_c(3900)\to J/\psi\pi^-)$ reactions are calculated, and their order of magnitude is about 0.1 and 0.01nb, respectively, with an assumption of branch ratio 10\% for the $Z_c(3900)$ decay in $J/\psi\pi$ channel. If the proton target is replaced by a nuclear target, the production of the $Z_c(3900)$ enhances obviously. The predicted total cross sections for the $A(\pi^-, Z^-_c(3900))$ and $A(\pi^-, Z^-_c(3900)\to J/\psi\pi^-)$ reactions with $A=^{12}$C or $^{208}$Pb are on the order of magnitude of 100 and 10 nb, respectively, which is about one thousand times larger than the cross sections off a proton target. Based on these the results, we suggest the experimental study of the $Z_c(3900)$ by using high-energy pion beams with a nuclear target at facilities such as COMPASS and J-PARC.