Pion Annihilation: Can Neutral Pions Really Convert to Photons?

[TriTertButoxy]

I'm running into a dilemma:

I've recently worked out the Feynman rules for Chiral perturbation theory for 2 flavors, and discovered that the term

$$\mathcal{L}=\frac{f_\pi^2}{4} \Tr[(D_\mu U)^\dag (D^\mu U)]$$​

seems to contain the term $~ e^2 A_\mu A^\mu \pi^0 \,\pi^0$ describing a direct interaction of two photons with two neutral pions. This is weird since the neutral pions don't carry electric charge to which photons can couple.

This term would allow the two neutral pions to annihilate into two photons. Can that happen, or did I make a mistake in computing the Feynman rules? Also, is there a reference containing all the Feynman rules for Chiral perturbation theory?

 

[Meir Achuz]

Pions have no charge but they do have a charge form factor due to their quark structure.
pi0 + pi0 --> 2 photons is conceivable, but extremely unlikely.
Of course, a single pi0 decays to two photons via its electromagnetic interaction.