Higgs trilinear coupling in the standard model effective field theory at the high luminosity LHC and the FCC-ee

Motivated by the updated HL-LHC projections for Higgs pair production from ATLAS and CMS and by the release of the FCC-ee Feasibility Study, we critically revisit the sensitivity of the global SMEFT analysis to deformations of the Higgs self-coupling modifier $\kappa_3$. To this end, we quantify the impact of SMEFT operators modifying double Higgs production at the LHC and single Higgs production, including loop corrections, at the FCC-ee, and include Renormalisation Group Evolution throughout. We demonstrate that significantly improving on the legacy HL-LHC constraints on $\kappa_3$ at the FCC-ee is not possible without the $\sqrt{s}=365$ GeV run; that individual and marginalised determinations are similar at the HL-LHC while differing by up to a factor 3 at the FCC-ee; and that quadratic EFT corrections cannot be neglected. Overall, the combination of HL-LHC and FCC-ee data offers unique potential to pin down the Higgs self-coupling with $\sim$$15\%$ precision.