\begin{document}$ Qq\bar{Q}\bar{q} $\end{document} (\begin{document}$ Q=c,b $\end{document}, and \begin{document}$ q=u,d,s $\end{document}) with different quantum numbers: \begin{document}$ J^{PC}=0^{+(+)} $\end{document}, \begin{document}$ 1^{+(\pm)} $\end{document}, and \begin{document}$ 2^{+(+)} $\end{document}. The mass spectra of tetraquark states are predicted, and the possible decay channels are analyzed by considering both the angular momentum and \begin{document}$ \mathcal{C} $\end{document}-parity conservation. The recently observed hidden-charm tetraquark states with strangeness, such as \begin{document}$ Z_{cs}(3985)^- $\end{document}, \begin{document}$ X(3960) $\end{document}, and \begin{document}$ Z_{cs}(4220)^+ $\end{document}, can be well explained in our model. Additionally, according to the wave function of each tetraquark state, we find that the low-lying states of each \begin{document}$ Qq\bar{Q}\bar{q} $\end{document} configuration have a large overlap to the \begin{document}$ Q\bar Q $\end{document} and \begin{document}$ q\bar q $\end{document} meson basis, instead of the \begin{document}$ Q\bar q $\end{document} and \begin{document}$ q\bar Q $\end{document} meson basis. This indicates that one can search these tetraquark states in future experiments via the channel of \begin{document}$ Q\bar Q $\end{document} and \begin{document}$ q\bar q $\end{document} mesons."> Investigation of the tetraquark states <inline-formula><tex-math id="M1">\begin{document}${\boldsymbol Qq\bar{\boldsymbol Q} \bar{\boldsymbol q}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/0f99660e-ff67-44da-8530-eb192ab4b1d4/CPC-2023-0051_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/0f99660e-ff67-44da-8530-eb192ab4b1d4/CPC-2023-0051_M1.png"/></alternatives></inline-formula> in the improved chromomagnetic interaction model -
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