{m_q+m_{\bar{q}} } } \vec{q}\cdot \vec{r}} X^i_{q(\bar{q})} | M^*(s) \rangle\end{aligned}$$

where $\vec{\sigma}_{q(\bar{q})}/2$ and $\vec{r}_{q(\bar{q})}$ are the spin and position of the quark (antiquark), $\overleftarrow{p}'= -i \overleftarrow{\nabla}$ is the momentum operator acting on the final-state wave function and the upper (lower) sign refers to the $q (\bar{q})$ case. The $X^i_{q(\bar{q})}$ are flavour operators. The calculations are most readily performed by taking $\vec{q}=q\hat{z}$ thereby calculating helicity amplitudes $H_m$ where $m=s'=s$ and then tranforming to the usual partial-wave basis. The details of this approach are given in Ref. [@godfrey85xj].

In our calculations we use harmonic oscillator wave functions with the oscillator parameter obtained by fitting the rms radius of the harmonic oscillator wavefunction to the rms radius of the full wavefunction. Because the harmonic oscillator parameter, $\beta$, enters the decay expressions in combinations, we averaged the initial and final state $\beta$’s to obtain the effective $\beta_{c\bar{s}}$ used in our numerical results. This allows us to calculate the amplitudes analytically, revealing relations between them that lets us classify the resulting amplitudes into two classes: “structure independent” amplitudes which have only momentum dependence dictated by angular momentum considerations along with the elastic form factor $F(q^2)$ defined below, and “structure dependent” amplitudes which have additional polynomial momentum 