We study the indirect constraints on the quartic Higgs self-coupling that arise from double-Higgs production at future hadron colliders. To this purpose, we calculate the two-loop contributions to the gg → hh amplitudes that involve a modified h4 vertex. Based on our results, we estimate the reach of a pp collider operating at 27 TeV and 100 TeV centre-of-mass energy in constraining the cubic and quartic Higgs self-couplings by measurements of double-Higgs and triple-Higgs production in gluon-fusion
