Electroweak phase transition with composite Higgs models: calculability, gravitational waves and collider searches

We study the strong first order electroweak phase transition (SFOEWPT) with the $SO(6)/SO(5)$ composite Higgs model, whose scalar sector contains one Higgs doublet and one real singlet. Six benchmark models are built with fermion embeddings in $\textbf{1}$, $\textbf{6}$, and $\textbf{15}$ of $SO(6)$. We show that SFOEWPT cannot be triggered under the minimal Higgs potential hypothesis, which assumes the scalar potential is dominated by the form factors from the lightest composite resonances. To get a SFOEWPT, the contributions from local operators induced by physics above the cutoff scale are needed. We take the $\textbf{6}+\textbf{6}$ model as an example to investigate the gravitational waves prediction and the related collider phenomenology.