Bound states and corresponding elementary fields

Question

Let's have some bound state, like positronium or meson. I need to
calculate an amplitude of a process which involves the bound state in an
in- or out-state. Is it necessary to introduce the corresponding
elementary field (for example, elementary "positronium" field) and its
interaction lagrangian for using perturbation theory? Or there are some
other methods for calculating amplitudes?

I.e., consider the matrix element $\langle\gamma\gamma|\hat S|Ps\rangle$
for process positronium decay, $Ps\to\gamma\gamma$.
Is it possible to use lagrangian $L_{int}=\bar\psi\gamma_\mu\psi A^\mu$
(i.e., positronium creation operator is simple $a_s^+(p)b_{s′}^+(k)$,
where $\hat a$, $\hat b$ correspond to electron and positron
destruction operators), or do I need to introduce a positronium field
$\phi_{Ps}$ and its interaction lagrangian to calculate such an amplitude?

[edited upon import]