100 TeV proton collider within the Future-Circular-Collider (FCC) project  [@Abada:2019lih; @Benedikt:2018csr]. Special emphasis will be given to the study of the theoretical uncertainties affecting the results and in particular the scale and scheme uncertainty related to the top-quark mass. We will also study the variation of the trilinear Higgs coupling and show that the NLO mass effects shift the minimum of the total cross section as a function of $\lambda_{H^3}$. They vary substantially over the range of $\lambda_{H^3}$ values.

The paper is organized as follows. We present the notation of our calculation in Section \[sc:lo\] and discuss the results at LO. In Section \[sc:nlo\] we move to the NLO QCD corrections. We discuss the details of the calculation of the virtual corrections in Section \[sc:virtuals\]. We describe the derivation of the real corrections in Section \[sc:reals\]. Our numerical analysis is performed in Section \[sc:results\]. Finally, the conclusions are given in Section \[sc:conclusions\].

Leading-order cross section \[sc:lo\]
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At LO, Higgs-boson pair production via gluon fusion is mediated by the generic diagrams of Fig. \[fg:hhdia\], including all permutations of the external lines. There are triangle and box diagrams with the former involving the trilinear Higgs coupling through an $s$-channel Higgs exchange. 