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We compute the dominant two-loop corrections to the Higgs trilinear coupling$\lambda_{hhh}$ and to the Higgs quartic coupling $\lambda_{hhhh}$ in modelswith extended Higgs sectors, using the effective-potential approximation. Weprovide in this paper all necessary details about our calculations, and presentgeneral $\overline{\text{MS}}$ expressions for derivatives of the integralsappearing in the effective potential at two loops. We also consider threeparticular Beyond-the-Standard-Model (BSM) scenarios -- namely a typicalscenario of an Inert Doublet Model (IDM), and scenarios of a Two-Higgs-DoubletModel (2HDM) and of a Higgs Singlet Model (HSM) without scalar mixing -- and weinclude all the necessary finite counterterms to obtain (in addition to$\overline{\text{MS}}$ results) on-shell scheme expressions for the correctionsto the Higgs self-couplings. With these analytic results, we investigate thepossible magnitude of two-loop BSM contributions to the Higgs self-couplingsand the fate of the non-decoupling effects that are known to appear at oneloop. We find that, at least as long as pertubative unitarity conditions arefulfilled, the size of two-loop corrections remains well below that of one-loopcorrections. Typically, two-loop contributions to $\lambda_{hhh}$ amount toapproximately 20% of those at one loop, implying that the non-decouplingeffects observed at one loop are not significantly modified, but also meaningthat higher-order corrections need to be taken into account for the futureperspective of precise measurements of the Higgs trilinear coupling.

Leading two-loop corrections to the Higgs boson self-couplings in models with extended scalar sectors