to a vector boson. These have been calculated to $\mathcal{O}(x^{2})$. Here $p_{\Pi}$ is the outgoing momentum of particle $\Pi$.

  ----------------------------------------------------------------------------------------------------------------------
                     Vertex                                                     Strength
  --------------------------------------------- ------------------------------------------------------------------------
            $\Pi_t^-A_{\mu}W_{\nu}^+$                                             $0$

       $\Pi_t^-A_{\mu}W^{\prime +}_{\nu}$                                         $0$

            $\Pi_t^-Z_{\mu}W_{\nu}^+$                                    $-\frac{i e^2 x^2}{16}
                                                                      v\,\sec^3\theta\,\tan\,\omega$

       $\Pi_t^-Z_{\mu}W_{\nu}^{\prime +}$                                $-\frac{i\,g_0^2}{4x} 
                                                                        v\,\sec\theta\,\sin2\omega 
                                                          \left[1 + \frac{x^2}{16}\left(3+5\,\cos2\theta \right)
                                                                            \sec^2\theta \right]$

       $\Pi_t^-Z^{\prime}_{\mu}W^+_{\nu}$                          $\frac{i\,g_0^2}{4x} v \sin2\omega
                                                                                  \left[1
                                                    + \frac{x^2}{16}\left(5+3\,\cos2\theta \right)\sec^2\theta \right]$

   $\Pi_t^-Z^{\prime}_{\mu}W^{\prime +}_{\nu}$                         $-\frac{i\,g_0^2}{8} v \,
                                                                      \tan^2\theta \left[\sin 2\omega
                                                                 + \frac{x^2}{32}\left(10-2\,\cos 2\theta
                                                              + 3\,\cos [2(\theta-\omega)] + 2\,\cos 2\omega
                                                           + 3\,\cos [2(\theta+\omega)] \right) \sec^2 \theta\,
                                                                            \tan\omega \right]$

      $\Pi_t^0W^+_{\mu}W^{\prime -}_{\nu}$                              $\frac{ig_0^2}{4x} v \,
                                                               \sin 2\omega \left(1 + \frac{x^2}{2}\right)$
  ----------------------------------------------------------------------------------------------------------------------

  : \[tab:piVV\] Couplings of a top-pion to a pair of gauge bosons. The corresponding Feynman rule is obtained by inserting a $g_{\mu\nu}$ in each coupling. These have been calculated to $\mathcal{O}(x^{2})$.

  ----------------------------------------------------------------------------------------------------------------------
                    Vertex                                                     Strength
  ------------------------------------------- --------------------------------------------------------------------------
            $H_tW^+_{\mu}W^-_{\nu}$                               $\frac{g_{0}^{2}}{2} v \sin\omega 
                                                                    \left(1+\frac{3\,x^2}{4} \right)$

       $H_tW^{\prime +}_{\mu}W^-_{\nu}$                         $-\frac{g_{0}^{2} x}{4} v \sin\omega$

   $H_tW^{\prime +}_{\mu}W^{\prime -}_{\nu}$                         $\frac{g_{0}^{2} x^{2}}{8} v
                                                                               \sin\omega$

              $H_tZ_{\mu}Z_{\nu}$                                  $\frac{g_0^2}{4 \cos^2\theta} v
                                                    \sin\omega \left[1 + \frac{x^2}{4}\left(1 + 2\cos 2\theta \right)
                                                                         \sec^2 \theta \right]$

         $H_tZ^{\prime}_{\mu}Z_{\nu}$                          $-\frac{g_{0}^{2} x}{4 \cos^2\theta} v 
                                                                   \sec\theta \cos 2\theta \sin\omega$

     $H_tZ^{\prime}_{\mu}Z^{\prime}_{\nu}$                     $\frac{g_{0}^{2} x^2}{16 \cos^2\theta} 
                                                                v \sec^2\theta \cos^2 2\theta \sin\omega$

            $H_t \Pi_t^- W^+_{\mu}$                                   $\frac{g_0}{2} \cos\omega 
                                                         \left(1+\frac{3 x^2}{8} \right)(p_H - p_{\Pi^-})_{\mu}$

       $H_t \Pi_t^- W^{\prime +}_{\mu}$                             $-\frac{g_{0} x}{4} \cos\omega
                                                                        (p_H - p_{\Pi^-})_{\mu}$

             $H_t \Pi_t^0 Z_{\mu}$                             