On Higher-Order Contributions to the Trilinear Higgs Self-Coupling in the SM and Beyond

We present a precision computation of the trilinear Higgs boson self-coupling \(\lambda_{hhh}\) in the Standard Model (SM) and the Type-II Two-Higgs-Doublet-Model (\text{THDM}) in the alignment limit, both with vanishing external momenta for the vertex, employing a fully on-shell renormalization scheme. Our analysis includes all one-loop corrections and, for the first time to our knowledge, the dominant two-loop top- and bottom-Yukawa contributions of \(\mathcal{O}(N_c^2)\). Analytic results consisting of the leading contributions for a heavy top quark are also derived.We evaluate the results in both the \(\alpha\)- and \(G_F\)-parametrizations. In the SM, \(\lambda_{hhh}\) is reduced from its tree-level value of 187.3\,GeV (\(\alpha\)) and 190.7\,GeV (\(G_F\)) to 175.479\,GeV and 175.715\,GeV, respectively, when including the full one-loop and the two-loop top- and bottom-Yukawa corrections of \(\mathcal{O}(N_c^2)\).In the \text{THDM}, the phenomenologically viable parameter space is investigated. In the alignment limit, heavy scalar masses in the 600--1000\,GeV range can enhance \(\lambda_{hhh}\) substantially, with values up to \(\sim 980\)\,GeV being consistent with current constraints.We further analyze the sensitivity of \(\lambda_{hhh}\) to beyond-the-Standard-Model (BSM) effects by parametrizing deviations with respect to the one-loop contributions in the Standard Model. The current experimental bounds \(\kappa_\lambda \in [-1,\,7]\) allow sizeable modifications to the trilinear Higgs coupling, which is investigated with respect to the new two-loop contributions in the Standard Model.