The $Z_c$ structures in a coupled-channels model

The $Z_c(3900{)}^{\pm }\mathrm{/}{Z}_c(3885{)}^{\pm }$ and $Z_c(4020{)}^{\pm }$ are two charmonium-like structures discovered in the $\pi J\mathrm{/}\psi$ 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 $\pi J\mathrm{/}\psi$ 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 $Z_c(3900{)}^{\pm }\mathrm{/}{Z}_c(3885{)}^{\pm }$ 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 $\pi J\mathrm{/}\psi$ and $D{\bar{D}}^{\ast }$ invariant mass distributions is due to the presence of two virtual states that produce the $Z_c$ peaks.