# On two-loop corrections to the Higgs trilinear coupling in models with extended scalar sectors

## TL;DR

This paper calculates two-loop radiative corrections to the Higgs trilinear coupling in extended scalar sector models, finding these corrections are smaller than one-loop effects and do not significantly change the predicted deviations from the Standard Model.

## Contribution

It provides the first detailed two-loop analysis of the Higgs trilinear coupling in 2HDM and IDM models, including all necessary renormalization scheme conversions.

## Key findings

- Two-loop corrections are 10-20% of one-loop corrections.
- Large deviations from the Standard Model prediction are not significantly affected.
- The study introduces a consistent on-shell prescription for the 2HDM soft-breaking mass.

## Abstract

We investigate the possible size of two-loop radiative corrections to the Higgs trilinear coupling $\lambda_{hhh}$ in two types of models with extended Higgs sectors, namely in a Two-Higgs-Doublet Model (2HDM) and in the Inert Doublet Model (IDM). We calculate the leading contributions at two loops arising from the additional (heavy) scalars and the top quark of these theories in the effective-potential approximation. We include all necessary conversion shifts in order to obtain expressions both in the $\overline{\text{MS}}$ and on-shell renormalisation schemes, and in particular, we devise a consistent "on-shell" prescription for the soft-breaking mass of the 2HDM at the two-loop level. We illustrate our analytical results with numerical studies of simple aligned scenarios and show that the two-loop corrections to $\lambda_{hhh}$ remain smaller than their one-loop counterparts, with a typical size being 10-20% of the one-loop corrections, at least while perturbative unitarity conditions are fulfilled. As a consequence, the existence of a large deviation of the Higgs trilinear coupling from the prediction in the Standard Model, which has been discussed in the literature at one loop, is not altered significantly.