The decay widths of the $Z_{cs}(3985/4000)$ based on rigorous quark-hadron duality

In this work, we explore the hadronic coupling constants $G_{ZJ/\psi K}$, $G_{Z\eta_c K^*}$, $G_{ZD^* \bar{D}_s}$ of the exotic states $Z_{cs}(3985/4000)$ both in the pictures of the tetraquark states and molecular states with the tentative assignments $J^{PC}=1^{+-}$ based on the rigorous quark-hadron duality. Then we obtain the total widths $\Gamma_{Z_{cs}}^T =15.31\pm 2.06\,\rm{MeV}$ and $\Gamma_{Z_{cs}}^M=83.51\pm21.09\,\rm{MeV}$, which are consistent with the experimental data $13.8^{+8.1}_{-5.2}\pm4.9\,\rm{MeV}$ from the BESIII collaboration and $131 \pm 15 \pm 26\,\rm{MeV}$ from the LHCb collaboration, respectively, and support assigning the $Z_{cs}(3985)$ and $Z_{cs}(4000)$ to be the hidden-charm tetraquark state and molecular state with the $J^{PC}=1^{+-}$, respectively.