Neutral Pion Decay

Question

While studying C-symmetry, a question about neutral pion decay came up. The most probable channels in which neutral pion $\pi^0$ decays are:

• $\pi^0\longrightarrow\gamma+\gamma$ (98%)
• $\pi^0\longrightarrow\gamma+e^{+}+e^{-}$ (1.3%)

that are permitted as C-symmetry is conserved ($C_{\pi^0}=+1$). The process $$\pi^0\longrightarrow\gamma+\gamma+\gamma$$ is forbidden, as $C_\gamma\gamma\gamma=-1$. Can anyone explain me how does the hypothetic forbidden process arise from theory (i.e. from Feynman Diagrams)? The book says that if I study the rate $$R={{\Gamma(\pi^0\rightarrow3\gamma)}\over{\Gamma(\pi^0\rightarrow2\gamma)}}$$ I expect that it is of the order $\alpha \approx {1\over137}$, while it is actually smaller (being a C-symmetry violating process). Why should I expect the rate to be $\alpha$?

Answer 1

Roughly: in a Feynman diagram each vertex of a gamma comes with an alpha factor. If three gammas were allowed from CP( which they are not) there would be 3 alpha factors in the nominator, and two in the denominator, so the ratio would be alpha.