Constraints on the quartic Higgs self-coupling from double-Higgs production at future hadron colliders

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 $gg \to hh$ amplitudes that involve a modified $h^4$ vertex. Based on our novel results, we estimate the reach of a $pp$ collider operating at $27 \, {\rm TeV}$ and $100 \, {\rm TeV}$ 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.