\phi=A,\psi,\tilde\psi\ ,$$ and have masses $$\label{custodianmass}
M_{\tilde{\cal P}\phi}=m_\phi=M_\phi \cos\theta_\phi,$$ where we have considered $m_\phi$ as a function of $M_\phi$. Hereafter we will consider $M_\phi$ as the fundamental parameter. The composite fermions $\tilde{\cal P}_\psi \psi^{cp}$ and $\tilde{\cal P}_{\tilde\psi} \tilde\psi^{cp}$ are usually called custodians.

If we fix the mass of the composite fields $\phi^*$, denoted as $M_\phi$ in Eq. (\[M\]), the mass of their SU(2)$_R$ partners $\tilde{\cal P}_\phi \phi^{cp}$ is controlled by the mixing angle $\theta_\phi$. We are interested in the limit $g\ll g_{cp}\ll 4\pi$, thus in general we will have $\sin\theta_A\ll1$, implying that the mass scale of all the composite vector bosons is of order $M_A$, no matter if they are of type ${\cal P}_A A^{cp}$ or $\tilde{\cal P}_A A^{cp}$. However, for the heavy SM fermions as the top, it will be necessary to consider the case $\sin\theta_\psi\sim1$. As previously noted in Ref. [@Pomarol:2008bh], in this limit, the custodians ${\cal P}_\psi \psi^{cp}$ become very light, and the corresponding massless state $\psi_L$ becomes mostly composite (a similar situation holds for $\psi_R$, exchanging $\psi\to\tilde\psi$).

The Yukawa couplings of the massless fermions $\psi_L$ and $\tilde\psi_R$ are given by: $$\label{yukawa}
y_\psi=y_{cp}\sin\theta_\psi\sin\theta_{\tilde\psi} \ .$$ From the top mass $m_t=y_t v/\sqrt{2}$ we obtain a lower bound on 