In this paper it is proposed that the charged charmonium-like state $Z_c(3900)$ is a resonance above the threshold from the $D{\overline{D}}^{*}$ interaction. The $D{\overline{D}}^{*}$ 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 $D{\overline{D}}^{*}$ threshold are searched. In the isoscalar sector, two poles found under the $D{\overline{D}}^{*}$ threshold—i.e., bound states—have the quantum numbers $I^G(J^{\mathrm{PC}})=0^{-}(1^{+-})$ 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 Riemann sheet and becomes a resonance above the $D{\overline{D}}^{*}$ threshold, which can be identified with the $Z_c(3900)$. With such a cutoff, the $D{\overline{D}}^{*}$ invariant mass spectrum is also investigated and the experimental results found by BESIII can be reproduced.

• Received 22 May 2015

DOI:https://doi.org/10.1103/PhysRevD.92.034004