Higgs in the composite sector). The elementary sector has a \[SU(3)$_c\times$SU(2)$_L\times$U(1)$_Y$\]$^{el}$ gauge symmetry and the composite sector has a \[SU(3)$_c\times$SU(2)$_L\times$SU(2)$_R\times$U(1)$_X$\]$^{cp}$ global symmetry, with the Higgs transforming as a bidoublet of \[SU(2)$_L\times$SU(2)$_R$\]$^{cp}$. The elementary fermions have the same quantum numbers as the SM fermions, whereas the composite fermions fulfill complete representations of the composite symmetry. From now on we closely follow the general description of Ref.  [@Contino:2006nn], see also Refs. [@Contino:2008hi; @Pomarol:2008bh]. Considering the terms up to dimension four, the Lagrangian can be written as: $${\cal L}={\cal L}_{el}+{\cal L}_{comp}+{\cal L}_{mix}$$ with $$\begin{aligned}
{\cal L}_{el}&=&-\frac{1}{4}F^{el\ 2}_{\mu\nu}+\bar\psi^{el}_L i {{\not \!\!D}}\psi^{el}_L+\bar{\tilde\psi}^{el}_R i {{\not \!\!D}}\tilde\psi^{el}_R \ , \\
{\cal L}_{cp}&=&-\frac{1}{4}F^{cp\ 2}_{\mu\nu}+\frac{m_A^2}{2}A^{cp\ 2}_\mu+\bar\psi^{cp}(i{{\not \!\!D}}^{cp}-m_\psi)\psi^{cp}+\bar{\tilde\psi}^{cp}(i{{\not \!\!D}}^{cp}-\tilde m_{\tilde\psi})\tilde\psi^{cp} \nonumber\\
& & +|D^{cp}_\mu \Sigma|^2-V(\Sigma)-y_{cp}\bar\psi^{cp}\Sigma\tilde\psi^{cp}+{\rm h.c.} \ , \\
{\cal L}_{mix}&=&\frac{m_A^2}{2}\left(-2\frac{g_{el}}{g_{cp}}A^{el}_\mu {\cal P}_AA^{cp}_\mu+\frac{g_{el}^2}{g_{cp}^2}A^{el\ 2}_\mu\right)+\bar\psi^{el}_L \Delta_\psi {\cal P}_\psi \psi^{cp}_R+\bar{\tilde\psi}^{el}_R \tilde\Delta_{\tilde\psi} {\cal P}_{\tilde\psi} \tilde\psi^{cp}_L +{\rm h.c.} \ \label{Lmix},\end{aligned}$$ where the indices $el$ and $cp$ denote elementary and composite fields and couplings, $D$ ($D^{cp}$) is the covariant derivative with respect to $A^{el}$ ($A^{cp}$) and the composite Higgs is $\Sigma=(\tilde H,H)$. A sum over $\psi$ and $\tilde\psi$ is understood, this sum is over the SM Left- and Right-handed fermions, respectively. The elementary fermions are coupled linearly with the composite fermions, and the operators ${\cal 