A Three-Coupled-Channel Analysis of $Z_c(3900)$ Involving $D\bar{D}^*$, $πJ/ψ$, and $ρη_c $

In this work, we conduct a three-coupled-channel analysis of the $Z_c(3900)$ structure, focusing on the $D\bar{D}^*$, $J/\psi \pi$, and $\rho \eta_c$ channels, based on the one-boson exchange model. Drawing from previous study on the exotic state $T_{cc}$, we only utilize one more parameter to construct the interactions between the channels. Our model successfully reproduces the experimental line shapes of the invariant mass distribution at $\sqrt{s} = 4.23$ and $4.26$ GeV for the three channels. Additionally, the finite-volume energy levels in our model show agreement with current LQCD conclusion. Detailed analysis suggests that the $Z_c(3900)$ peaks in the $\pi J/\psi$ and $\rho \eta_c$ distributions primarily arise from the triangle loop involving the $D_1 \bar{D} D^*$ intermediate system. In the $D\bar{D}^*$ distribution, the threshold peak is generated by the cascade decay mechanism enhanced by a triangle diagram. Moreover, we find a virtual pole located far from the threshold, indicating that $Z_c(3900)$ peaks are not associated with physical pole. We conclude that the $Z_c(3900)$ peaks are predominantly caused by the threshold cusp.