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In order to investigate the possibility of the recently observed $X(5568)$being a $0^{+}$ tetraquark state, we make an improvement to the study of therelated various configuration states in the framework of the QCD sum rules.Particularly, to ensure the quality of the analysis, condensates up todimension $12$ are included to inspect the convergence of operator productexpansion (OPE) and improve the final results of the studied states. We notethat some condensate contributions could play an important role on the OPEside. By releasing the rigid OPE convergence criterion, we arrive at thenumerical value $5.57^{+0.35}_{-0.23}~\mbox{GeV}$ for the scalar-scalardiquark-antidiquark $0^{+}$ state, which agrees with the experimental data forthe $X(5568)$ and could support its interpretation in terms of a $0^{+}$tetraquark state with the scalar-scalar configuration. The corresponding resultfor the axial-axial current is calculated to be$5.77^{+0.44}_{-0.33}~\mbox{GeV}$, which is still consistent with the mass of$X(5568)$ in view of the uncertainty. The feasibility of $X(5568)$ being atetraquark state with the axial-axial configuration therefore cannot bedefinitely excluded. For the pseudoscalar-pseudoscalar and the vector-vectorcases, their unsatisfactory OPE convergence make it difficult to findreasonable work windows to extract the hadronic information.

$0^{+}$ tetraquark states from improved QCD sum rules: delving into $X(5568)$