Perturbative Unitarity Bounds in Composite 2-Higgs Doublet Models

We study bounds from perturbative unitarity in a Composite 2-Higgs Doublet Model (C2HDM) based on the spontaneous breakdown of a global symmetry $SO(6)\to SO(4)\times SO(2)$ at the compositeness scale $f$. The eight pseudo Nambu-Goldstone Bosons (pNGBs) emerging from such a dynamics are identified as two isospin doublet Higgs fields. We calculate the $S$-wave amplitude for all possible 2-to-2-body elastic (pseudo)scalar boson scatterings at energy scales $\sqrt{s}$ reachable at the Large Hadron Collider (LHC) and beyond it, including the longitudinal components of weak gauge boson states as the corresponding pNGB states. In our calculation, the Higgs potential is assumed to have the same form as that in the Elementary 2-Higgs Doublet Model (E2HDM) with a discrete $Z_2$ symmetry, which is expected to be generated at the one-loop level via the Coleman-Weinberg (CW) mechanism. We find that the $S$-wave amplitude matrix can be block-diagonalized with maximally $2\times 2$ submatrices in a way similar to the E2HDM case as long as we only keep the contributions from ${\cal O }(\xi s)$ and ${\cal O }(\xi^0 s^0)$ in the amplitudes, where $\xi=v_{\text{SM}}^2/f^2$ and $v_{\text{SM}}^{}\simeq 246$ GeV, which is an appropriate approximation for our analysis. By requiring the C2HDM to satisfy perturbative unitarity at energies reachable by the LHC, we derive bounds on its parameters such as $\xi$ and the masses of extra Higgs bosons present in the scenario alongside the Standard Model (SM)-like Higgs state discovered in 2012.