The $Z_c$ structures in a coupled-channels model

Abstract: The $Z_{c}(3900)^\pm/Z_c(3885)^\pm$ and $Z_{c}(4020)^\pm$ are two charmonium-like structures discovered in the $\pi J/\psi$ and $D^\ast\bar D^{(\ast)}+h.c.$ invariant mass spectra. Their nature is puzzling due to their charge, which forces its minimal quark content to be $c\bar c u\bar d$ ($c\bar c d\bar u$). Thus, it is necessary to explore four-quark systems in order to understand their inner structure. Additionally, their strong coupling to channels such as $\pi J/\psi$ and the closeness of their mass to $D^\ast\bar D^{(\ast)}$-thresholds stimulates both a molecular interpretation or a coupled-channels threshold effect. In this work we perform a coupled-channels calculation of the $I^G(J^{PC})=1^+(1^{+-})$ sector including $D^{(\ast)}\bar D^{\ast}+h.c.$, $\pi J/\psi$ and $\rho\eta_c$ channels in the framework of a constituent quark model which satisfactorily describes a wide range of properties of (non-)conventional hadrons containing heavy quarks. The meson-meson interactions are dominated by the non-diagonal $\pi J/\psi-D^\ast\bar D^{(\ast)}$ and $\rho\eta_c-D^\ast\bar D^{(\ast)}$ couplings which indicates that the $Z_{c}(3900)^\pm/Z_c(3885)^\pm$ and $Z_{c}(4020)^\pm$ are unusual structures. The study of the analytic structure of the $S$-matrix allows us to conclude that the point-wise behavior of the line shapes in the $\pi J/\psi$ and $D\bar D^*$ invariant mass distributions is due to the presence of two virtual states that produce the $Z_c$ peaks.