$Z_c(3900)$ as a resonance from the $D\bar{D}^*$ interaction

In this paper it is proposed that the charged charmonium-like state $Z_c(3900)$ is a resonance above the threshold from the $D\bar{D}^*$ interaction. The $D\bar{D}^*$ interaction is described by the one-boson exchange model with light meson exchanges plus a short-range $J/\psi$ exchange. The scattering amplitude is calculated within a Bethe-Salpeter equation approach and the poles near the $D\bar{D}^*$ threshold are searched. In the isoscalar sector, two poles found under the $D\bar{D}^*$ threshold, i.e., bound states, have the quantum numbers $I^G(J^{PC})=0^-(1^{+-})$ and $0^+(1^{++})$. The latter can be related to the $X(3872)$. In the isovector sector, a bound state with $I^G(J^P)=1^+(1^+)$ is found with a large cutoff at about 3 GeV. If a cutoff at about 2 GeV is adopted with which a pole carrying the quantum number of the $X(3872)$ is produced at an energy of about 3871 MeV, the pole for the bound state with $1^+(1^+)$ runs across the threshold to a second Rienman sheet and becomes a resonance above the $D\bar{D}^*$ threshold, which can be identified with the $Z_c(3900)$. With such a cutoff, the $D\bar{D}^{*}$ invariant mass spectrum is also investigated and the experimental results found by BESIII can be reproduced.