A QFT scalar toy model analogous to positronium and pion decays

In the framework of a scalar QFT, we evaluate the decay of an initial massive state into two massless particles through a triangle-shaped diagram in which virtual fields propagate. Under certain conditions, the decaying state can be seen as a bound state, thus it is analogous to the neutral pion (quark-antiquark pair) and to the positronium (electron-positron pair), which decay into two photons. While the pion is a relativistic composite object, the positronium is a non-relativistic compound close to threshold. We exam similarities and differences between these two types of bound states.
Comment: 6 pages, 4 figures, 2 tables, Prepared for the proceedings of the 18th Symposium on new trends in Nuclear and Medical Physics, Cracow, Poland, 18-20 October 2023