Fate of the Tetraquark Candidate $Z_c(3900)$ from Lattice QCD

The possible exotic meson $Z_c(3900)$, found in $e^{+}{e}^{-}$ reactions, is studied by the method of coupled-channel scattering in lattice QCD. The interactions among $\pi J/\psi$, $\rho {\eta }_c$, and $\overline{D}{D}^{*}$ channels are derived from ($2+1$)-flavor QCD simulations at $m_{\pi }=410–700\mathrm{MeV}$. The interactions are dominated by the off-diagonal $\pi J/\psi \text{−}\overline{D}{D}^{*}$ and $\rho {\eta }_c\text{−}\overline{D}{D}^{*}$ couplings, which indicates that the $Z_c(3900)$ is not a usual resonance but a threshold cusp. Semiphenomenological analyses with the coupled-channel interaction are also presented to confirm this conclusion.

Figure 1

A possible decay scheme of the $Y(4260)$ through $Zc(3900)$, together with the relevant two-meson thresholds of the $Z_c(3900)$ decay at $m_{\pi }\simeq 140$ (Expt.), 410 (case I), 570 (case II), and 700 (case III) MeV. The arrows represent the observed decay modes in the experiments [2, 3, 4].

Figure 2

The $s$-wave potentials for the (a) $\overline{D}{D}^{*}\text{−}\overline{D}{D}^{*}$, (b) $\rho {\eta }_c\text{−}\overline{D}{D}^{*}$, (c) $\rho {\eta }_c\text{−}\rho {\eta }_c$, (d) $\pi J/\psi \text{−}\overline{D}{D}^{*}$, (e) $\pi J/\psi \text{−}\rho {\eta }_c$, and (f) $\pi J/\psi \text{−}\pi J/\psi$ channels. The coupled-channel potentials are obtained at time slice $t=13$ for case I (red circles), case II (blue squares), and case III (black triangles).

Figure 3

(a) The two-body invariant mass spectra in the $\pi J/\psi$ (red circles, scaled by 5), $\rho {\eta }_c$ (green triangles), and $\overline{D}{D}^{*}$ (blue squares) channels. The two-body $\pi J/\psi$ spectrum without the off-diagonal component of $V^{\alpha \beta }$ is also shown by $\mathrm{Im}{f}_0^{\pi J/\psi ,\pi J/\psi }$ (black crosses, scaled by 25). (b) The pole of the $S$ matrix on the [$bbb$] sheet in the notation of Ref. [33] for $\pi J/\psi$, $\rho {\eta }_c$, and $\overline{D}{D}^{*}$ channels ($z=m_1^{\alpha }+m_2^{\alpha }+p_{\alpha }^2/2{\mu }^{\alpha }$). Both figures correspond to the case I in Table 1. In Fig. 3, the inner error is statistical, while the outer one is statistical and systematic combined in quadrature.

Figure 4

The invariant mass spectra of (a) $Y(4260)\to \pi \pi J/\psi$ and (b) $Y(4260)\to \pi \overline{D}{D}^{*}$ below the ${\overline{D}}^{*}{D}^{*}$ threshold calculated with $V^{\alpha \beta }$ for case I in Table 1. The shaded areas show the statistical errors. The vertical arrows show the predicted peak positions from the calculations. The blue dashed lines show the invariant mass spectra of the $Y(4260)$ decay without the off-diagonal components of $V^{\alpha \beta }$. The experimental data are taken from Ref. [2].