The paper by Anastasiou and Melnikov focuses on computing the total cross-section for direct Higgs boson production in hadron collisions at next-to-next-to-leading order (NNLO) in perturbative Quantum Chromodynamics (QCD). They introduce a new technique based on the Cutkosky rules, integration by parts, and the differential equation method to evaluate inclusive phase-space integrals algorithmically. The authors discuss the numerical impact of the $\mathcal{O}(\alpha_s^2)$ QCD corrections on the Higgs boson production cross-section at the Large Hadron Collider (LHC) and the Tevatron. They present the full analytic result for the NNLO corrections and show that the corrections significantly reduce the scale dependence of the cross-section, improving its theoretical accuracy. The paper also includes a detailed derivation of the partonic cross-sections up to NNLO and their numerical evaluation using parton distribution functions. The results demonstrate the convergence properties of the perturbative series and highlight the importance of including NNLO corrections for more precise predictions.The paper by Anastasiou and Melnikov focuses on computing the total cross-section for direct Higgs boson production in hadron collisions at next-to-next-to-leading order (NNLO) in perturbative Quantum Chromodynamics (QCD). They introduce a new technique based on the Cutkosky rules, integration by parts, and the differential equation method to evaluate inclusive phase-space integrals algorithmically. The authors discuss the numerical impact of the $\mathcal{O}(\alpha_s^2)$ QCD corrections on the Higgs boson production cross-section at the Large Hadron Collider (LHC) and the Tevatron. They present the full analytic result for the NNLO corrections and show that the corrections significantly reduce the scale dependence of the cross-section, improving its theoretical accuracy. The paper also includes a detailed derivation of the partonic cross-sections up to NNLO and their numerical evaluation using parton distribution functions. The results demonstrate the convergence properties of the perturbative series and highlight the importance of including NNLO corrections for more precise predictions.