I am taking an introductory course in particle physics and am quite confused about conservation of angular momentum at a vertex. This trouble arose specifically when considering the decay $\pi^0 \rightarrow \gamma \gamma$. I am asked to draw the Feynman diagram. I thought that it would most likely happen via a a) $\gamma$, b) $Z$ boson or c) gluon intermediate. I have now seen a diagram with a quark loop (I will upload an image when possible; for some reason all of my attempts are 'failing to upload' at the moment)
But I wanted to figure out why (if) my suggestions of an S diagram with a single propagator intermediate were wrong.
I discarded a) because I think it is not possible for a photon to decay into two photons. But I was also wondering if one can argue that angular momentum is not conserved at the first vertex. This would require that a photon cannot have angular momentum.
I am not at all sure about b) and c). I am not sure if it is permissible for Z bosons and gluons to couple to the photon. With the Z boson, on the one hand within the framework of electroweak unification, it is a linear combo of the same gauge bosons as the photon, so I would think it can interact. On the other hand, it is uncharged. The gluon is also uncharged, so maybe it cannot couple to the photon.
In any case, if it is impossible for a single paticle intermediate to have zero angular momentum, then the pseudoscalar mesons can never decay in this way!