Single and double SM-like Higgs boson production at future electron-positron colliders in composite 2HDMs

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We investigate single- and double-$h$, the discovered Standard Model (SM)-like Higgs boson, production at future $e^+e^-$ colliders in Composite 2-Higgs Doublet Models (C2HDMs) and Elementary 2-Higgs Doublet Models (E2HDMs) with a softly-broken $Z_2$ symmetry. We first survey their parameter spaces allowed by theoretical bounds from perturbative unitarity and vacuum stability as well as by future data at the Large Hadron Collider (LHC) with an integrated luminosity up to 3000 fb$^{-1}$ under the assumption that no new Higgs boson is detected. We then discuss how different the cross sections can be between the two scenarios when $\kappa_V^{}$, the $hVV$ ($V=W^\pm,Z$) coupling normalised to the SM value, is taken to be the same value in the both scenario We find that if $\kappa_V^2$ is found to be, e.g., $0.98$, then the cross sections in C2HDMs with $f$ (the compositeness scale) in the TeV region can be maximally changed to be about $-15\%$, $-18\%$, $-50\%$ and $-35\%$ for the $e^+e^-\to t \bar t h$, $e^+e^-\to Zhh$, $e^+e^-\to e^+e^-hh$ and $e^+e^-\to t\bar{t}hh$ processes, respectively, with respect to those in E2HDMs. Thus, a future electron-positron collider has the potential to discriminate between E2HDMs and C2HDMs, even when only $h$ event rates are measured.