I am taking an introductory particle physics course, and from having studied about the breit wigner cross section and propagator terms:
where $E$ is the energy in the centre of mass frame, $E_0$ is the rest energy of the propagator particle, and $\Gamma$ is its decay width. I was under the impression that divergences in transition rates are necessarily supressed because $\Gamma$ is always non-zero. However I have now come across a decay $\pi^0 \rightarrow \gamma \gamma$ where the intermediate particle is an up quark. From wat I am aware, these have a vanishing decay rate. How then does the rate for this process not diverge?
I am vaguiely reminded about the need for the introduction of the Z bososn and electroweak unification because otherwise there would be divergences from massless propagators. The propagator here isn't massless, but if the energy is tuned right then the propagator term could diverge?