Treating $Z_c(3900)$ and $Z(4430)$ as the ground state and first radially excited tetraquarks

Exploration of the resonances $Z_c(3900)$ and $Z(4430)$ are performed by assuming that they are the ground state and first radial excitation of the same tetraquark with $J^P=1^{+}$. The mass and current coupling of the $Z_c(3900)$ and $Z(4430)$ states are calculated using the QCD two-point sum rule method by taking into account vacuum condensates up to eight dimensions. We investigate the vertices $Z_c{M}_h{M}_l$ and $Z{M}_h{M}_l$, with $M_h$ and $M_l$ being the heavy and light mesons and evaluate the strong couplings $g_{Z_c{M}_h{M}_l}$ and $g_{Z{M}_h{M}_l}$ using the QCD sum rule on the light cone. The extracted couplings allow us to find the partial width of the decays $Z_c(3900)\to J/\psi \pi$, ${\psi }^{\prime }\pi$, ${\eta }_c\rho$ and $Z(4430)\to {\psi }^{\prime }\pi$, $J/\psi \pi$, ${\eta }_c^{\prime }\rho$, ${\eta }_c\rho$, which may help in comprehensive investigation of these resonances. We compare the width of the decays of $Z_c(3900)$ and $Z(4430)$ resonances with available experimental data as well as existing theoretical predictions.

Figure 1

The mass of the state $Z(4430)$ as a function of the Borel parameter $M^2$ at fixed $s_0$ (left panel) and as a function of the continuum threshold $s_0$ at fixed $M^2$ (right panel).

Figure 2

The dependence of the current coupling $f_Z$ of the $Z(4430)$ resonance on the Borel parameter at chosen values of $s_0$ (left panel) and on $s_0$ at fixed $M^2$ (right panel).

Figure 3

The coupling $g_{Z{\psi }^{\prime }\pi }$ as a function of the Borel parameter $M^2$ at fixed $s_0$ (left panel) and as a function of the continuum threshold $s_0$ at fixed $M^2$ (right panel).