Generic two-loop results for trilinear and quartic scalar self-interactions

Reconstructing the shape of the Higgs potential realised in Nature is a central part of the physics programme at the LHC and future colliders. In this context, accurate theoretical predictions for trilinear and quartic Higgs couplings are becoming increasingly important. In this paper, we present results that enable significant progress in the automation of these calculations at the two-loop level in a wide range of models. Specifically, we calculate the generic two-loop corrections for scalar n-point functions with n<=4 assuming that all external scalars are identical. Working in the zero-momentum approximation, we express the results in terms of generic couplings and masses. Additionally, by exploiting permutation invariances, we reduce the number of Feynman diagrams appearing to a substantially smaller set of basis diagrams. To ease the application of our setup, we also provide routines that allow to map our generic results to scalar two-loop amplitudes generated with the package FeynArts. We perform a series of calculations to cross-check our results with existing results in the literature. Moreover, we present new two-loop results for the trilinear Higgs coupling in the general singlet extension of the Standard Model. We also present the public Python package Tintegrals, which allows for fast and stable evaluations of all relevant two-loop integrals with vanishing external momenta.