\sin{\omega} \left( \kappa \sin^2{\omega} \tan^2{\omega} 
  + 4 \lambda \cos^2{\omega} \right) \nonumber \\
  &=& \frac{1}{2v \sin{\omega}} \left[(M^2_H - 2 M^2_{\Pi_t})\cos{2\omega} 
    + M^2_H \right].\end{aligned}$$ These couplings are important for top-Higgs decays when $M_H > 2
M_{\Pi_t}$.

Having worked out the physical top-pion combination, all that remains is to express the interactions in the Lagrangian in terms of mass eigenstates. The top-pion combination is given above, while the gauge boson and fermion mass eigenstates are given in Ref. [@Chivukula:2009ck] and are summarized in Appendix A. This conversion is straight-forward, but tedious, so we just summarize the results for the three-point couplings in Tables \[tab:pipiV\]–\[tab:H-couplings\]. We write the couplings in terms of $x$, $\sin\theta$, and $g_{0}$, with the latter two defined as in Eq. . The results in this section are given as an expansion in powers of $x$ and include terms up to order $x^2$.

  ------------------------------------------------------------------------------------------------
                 Vertex                                         Strength
  ------------------------------------ -----------------------------------------------------------
        $A_{\mu}\Pi_t^+\Pi_t^-$                     $e\,(p_{\Pi^+}-p_{\Pi^-})_{\mu}$

        $Z_{\mu}\Pi_t^+\Pi_t^-$                         $\frac{g_0}{\cos\theta} 
                                               \left[\left(\frac{1}{2} - \sin^2\theta \right)
                                                       + \frac{x^2}{16} \sec^2\theta
                                            \left(2 + \cos2\theta \,\sec^2\omega \right) \right]
                                                        (p_{\Pi^-}-p_{\Pi^+})_{\mu}$

    $Z^{\prime}_{\mu}\Pi_t^+\Pi_t^-$               $\frac{g_0}{2x} \left[\sin^2\omega
                                           - \frac{x^2}{16}(7+\cos 2\omega) \sec^2\theta \right] 
                                                       (p_{\Pi^-}-p_{\Pi^+})_{\mu}$

      $W_{\mu}^{+}\Pi_t^0\Pi_t^-$                       $-\frac{g_{0}}{2} \left[1
                                           + \frac{x^{2}}{8}(2+\cos 2\omega) \sec^2\omega \right]
                                                        (p_{\Pi^0}-p_{\Pi^-})_{\mu}$

   $W_{\mu}^{\prime +}\Pi_t^0\Pi_t^-$                       $-\frac{g_0}{2x}
                                                             \left[\sin^2\omega
                                            - \frac{x^2}{16}\left(7+\cos 2\omega \right) \right]
                                                        (p_{\Pi^0}-p_{\Pi^-})_{\mu}$
  ------------------------------------------------------------------------------------------------

  : \[tab:pipiV\] Couplings of two top-pions 