Sensitivity to triple Higgs couplings via di-Higgs production in the RxSM at the (HL-)LHC and future e$^+$e$^−$ colliders

The real Higgs singlet extension of the Standard Model (SM) without Z$_2$ symmetry, the RxSM, is the simplest extension of the SM that features a First Order Electroweak Phase Transition (FOEWPT) in the early universe. The FOEWPT is one of the requirements needed for electroweak baryogenesis to explain the baryon asymmetry of the universe (BAU). Thus, the RxSM is a perfect example to study features related to the FOEWPT at current and future collider experiments. The RxSM has two CP-even Higgs bosons, h and H, with masses m$_h$ < m$_H$, where we assume that h corresponds to the Higgs boson discovered at the LHC. Our analysis is based on a benchmark plane that ensures the occurence of a strong FOEWPT, where m$_H$ > 2m$_h$ is found. In a first step we analyze the di-Higgs production at the (HL-)LHC, gg → hh, with a focus on the impact of the trilinear Higgs couplings (THCs), λ$_{hhh}$ and λ$_{hhH}$. The interferences of the resonant H-exchange diagram involving λ$_{hhH}$ and the non-resonant diagrams result in a characteristic peak-dip (or dip-peak) structure in the mhh distribution. We analyze how λ$_{hhH}$ can be accessed, taking into account the experimental smearing and binning. We also demonstrate that the approximation used by ATLAS and CMS for the resonant di-Higgs searches may fail to capture the relevant effects and lead to erroneous results. In a second step we analyze the benchmark plane at a future high-energy e$^+$e$^−$ collider with $\sqrt{s}$= 1000 GeV (ILC1000). We analyze the potential sensitivity to λ$_{hhH}$ via an experimental determination at the ILC1000 and demonstrate that it is nearly independent of the heavy Higgs-boson mass, as long as it is kinematically accessible.