a small further modification of the cross section [@Banerjee:2018lfq; @Chen:2019lzz; @Chen:2019fhs]. This calculation uses the N$^3$LO corrections to the effective Higgs and Higgs-pair couplings to gluons in the heavy-top limit (HTL) [@Spira:2016zna]. The higher-order QCD corrections increase the total LO cross section by about a factor of two. Recently, the full NLO results have been matched to parton showers [@Heinrich:2017kxx; @Jones:2017giv] and the full NNLO results in the limit of heavy top quarks have been merged with the NLO mass effects and supplemented by the additional top-mass effects in the double-real corrections [@Grazzini:2018bsd].

![\[fg:hhdia\] *Generic diagrams contributing to Higgs-boson pair production via gluon fusion. The contribution of the trilinear Higgs coupling is marked in red.*](./plots/dia_gg2hh.pdf){width="110.00000%"}

The goal of this paper is to present in detail the calculation of Ref. [@Baglio:2018lrj] of the full NLO corrections to Higgs pair production in gluon fusion. We rely on a direct numerical integration of the Feynman diagrams, without any tensor reduction. We extend the results presented in Ref. [@Baglio:2018lrj] and study not only the LHC at center-of-mass energies of 13 and 14 TeV, but also present numbers for a potential high-energy upgrade of the LHC (HE-LHC) at 27 TeV [@Abada:2019ono] and for a provisional 