  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 the
related various configuration states in the framework of the QCD sum rules.
Particularly, to ensure the quality of the analysis, condensates up to
dimension $12$ are included to inspect the convergence of operator product
expansion (OPE) and improve the final results of the studied states. We note
that some condensate contributions could play an important role on the OPE
side. By releasing the rigid OPE convergence criterion, we arrive at the
numerical value $5.57^{+0.35}_{-0.23}~\mbox{GeV}$ for the scalar-scalar
diquark-antidiquark $0^{+}$ state, which agrees with the experimental data for
the $X(5568)$ and could support its interpretation in terms of a $0^{+}$
tetraquark state with the scalar-scalar configuration. The corresponding result
for 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 a
tetraquark state with the axial-axial configuration therefore cannot be
definitely excluded. For the pseudoscalar-pseudoscalar and the vector-vector
cases, their unsatisfactory OPE convergence make it difficult to find
reasonable work windows to extract the hadronic information.
