_{\omega}$ and $\phi _{\omega}$ define the orientation of the $\omega$ decay plane in the $\omega$ rest frame. Note that the $A$ decay plane has an azimuthal angle $\phi _A$ both in the $x_Ay_Az_A$ and in the $x_{\omega}y_{\omega}z_{\omega}$ references. As the angle $\phi _A$ is arbitrary, the only angle that has a physical meaning is $\chi = 
\phi _A - \phi _{\omega}$, the opening angle between the $A$ decay plane and the $\omega$ decay plane.

Both the $B$ meson and the $D$ meson are pseudoscalar, therefore their helicity is 0. Thus $A$ will be longitudinally polarized independently of its spin. In order to calculate the decay amplitude for this process, we need to sum over the $\omega$ helicity states:

$${\cal A} = \Sigma _{\lambda _{\omega}} D^{\star J_A}_{0\lambda
_{\omega}}(\phi _A,\theta _A, -\phi _A)
 D^{\star 1}_{\lambda_{\omega}0}(\phi _{\omega},\theta_{\omega}, -
\phi _
{\omega}) B_{\lambda_{\omega}0},$$

here $D^{\star 1}_{\lambda_{\omega}}(\phi _A,\theta _A,-\phi _A)$ is the rotation matrix that relates the $x_Ay_Az_A$ and the $x_{\omega}y_{\omega}z_{\omega}$ frames and $ D^{\star 1}_{\lambda_{\omega}}(\phi 
_{\omega},\theta_{\omega}, -\phi
_{\omega})$ is the rotation matrix relating the $x_{\omega}y_{\omega}z_{\omega}$ and the direction of the normal to the $\omega$ decay plane $\hat{n}(\theta _{\omega},\phi_{\omega})$.

In general, there are three helicity amplitudes that contribute to this decay: $B_{10}$ and $B_{-10}$, corresponding 