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We have studied the masses for fully open-flavor tetraquark states$bc\bar{q}\bar{s}$ and $sc\bar{q}\bar{b}$ with quantum numbers$J^{P}=0^{+},1^{+}$. We systematically construct all diquark-antiquarkinterpolating currents and calculate the two-point correlation functions andspectral densities in the framework of QCD sum rule method. Our calculationsshow that the masses are about $7.1-7.2$ GeV for the $bc\bar{q}\bar{s}$tetraquark states and $7.0-7.1$ GeV for the $sc\bar{q}\bar{b}$ tetraquarks. Themasses of $bc\bar{q}\bar{s}$ tetraquarks are below the thresholds of$\bar{B}_{s}D$ and $\bar{B}_{s}^{*}D$ final states for the scalar andaxial-vector channels respectively. The $sc\bar{q}\bar{b}$ tetraquark stateswith $J^{P}=1^{+}$ lie below the $B_{c}^{+}K^{*}$ and $B_{s}^{*}D$ thresholds.Such low masses for these possible tetraquark states indicate that they canonly decay via weak interaction and thus are very narrow and stable.

Fully open-flavor tetraquark states $bc\bar{q}\bar{s}$ and $sc\bar{q}\bar{b}$ with $J^{P}=0^{+},1^{+}$