The Higgs boson trilinear and quartic self-couplings are directly related to the shape of the Higgs potential; measuring them with precision is extremely important, as they provide invaluable information on the electroweak symmetry breaking and the electroweak phase transition.
In this paper, we perform a detailed analysis of double Higgs boson production, through the gluon-gluon fusion process, in the most promising decay channels $b\bar{b} \gamma\gamma$, $b\bar{b} \tau\tau$, and $b\bar{b}b\bar{b}$ for several future colliders: the HL-LHC at 14 TeV and the FCC-hh at 100 TeV, assuming respectively 3 $ab^{-1}$ and 30 $ab^{-1}$ of integrated luminosity.
In the HL-LHC scenario, we expect an upper limit on the di-Higgs cross-section production of 0.76 at 95\% confidence level, corresponding to a significance of 2.8 $\sigma$.
In the FCC-hh scenario, depending on the assumed detector performance and systematic uncertainties, we expect that the Higgs self-coupling will be measured with a precision in the range 4.8-8.5\% at 95\% confidence level.