Constraints on the trilinear and quartic Higgs couplings from triple Higgs production at the LHC and beyond

Experimental information on the trilinear Higgs boson self-coupling κ${}_3$ and the quartic self-coupling κ${}_4$ will be crucial for gaining insight into the shape of the Higgs potential and the nature of the electroweak phase transition. While Higgs pair production processes provide access to κ${}_3$, triple Higgs production processes, despite their small cross sections, will provide valuable complementary information on κ${}_3$ and first experimental constraints on κ${}_4$. We investigate triple Higgs boson production at the HL-LHC, employing efficient Graph Neural Network methodologies to maximise the statistical yield. We show that it will be possible to establish bounds on the variation of both couplings from the HL-LHC analyses that significantly go beyond the constraints from perturbative unitarity. We also discuss the prospects for the analysis of triple Higgs production at future high-energy lepton colliders operating at the TeV scale.