  QCD sum-rule mass predictions for tetraquark states provide insights on the
interpretations and internal structure of experimentally-observed exotic
mesons. However, the overwhelming majority of tetraquark QCD sum-rule analyses
have been performed at leading order (LO), which raises questions about the
underlying theoretical uncertainties from higher-loop corrections. The impact
of next-to-leading order (NLO) perturbative effects are systematically examined
in scalar ($J^{PC}=0^{++}$) isoscalar light-quark tetraquark systems where
comprehensive LO sum-rule analyses have been performed and NLO perturbative
corrections to the correlators have previously been calculated. Using the
scalar-isoscalar state as a detailed case study to illustrate the differences
between LO and NLO analyses, it is shown that NLO effects in individual Laplace
sum-rules are numerically significant and have an important role in improving
the reliability of the sum-rule analyses by widening the Borel window. However,
ratios of sum-rules are found to be less sensitive to NLO effects with the
additional advantage of cancelling the anomalous dimension that emerges from
the NLO corrections. NLO mass predictions based on these sum-rule ratios are
thus remarkably robust despite the slow perturbative convergence of the
underlying correlator. The mass predictions $0.52\,\text{GeV}< m_{\sigma}<
0.77\,\text{GeV}$ for the lightest scalar-isoscalar $\sigma$ state are in good
agreement with the four-quark interpretation of the $f_0(500)$, and the
relative coupling strengths of the $f_{0}(980)$ and $f_0(500)$ to the
tetraquark current agree with the pattern found in chiral Lagrangian analyses.
Effects of the $\sigma$ resonance width are studied for different models,
including resonance shapes inspired by chiral Lagrangians.
