I am uncomfy with the calculation of the neutral pion’s decay rate via the triangle anomaly diagram, which gets touted as evidence of three colors. The calculation invokes PCAC in the guise of the occult Goldberger-Treiman relation, which says that ${{f}_{\pi }}$ (the charged pion’s proportionality to the divergence of the axial current) is inversely related to ${{g}_{\pi NN}}$ (the pion’s effective coupling to a nucleon line). The charged pion’s decay amplitude is directly proportional to ${{f}_{\pi }}$by definition, but the neutral pion’s is found to be inversely proportional to it.

Since I think of pions as $Q\bar{Q}$ bound states that could in principle be described by a wave function, I would intuitively have expected both decay amplitudes to be directly proportional to aspects of the elusive wave function. The axial current operator is strictly local, so the charged pion’s decay amplitude should depend on the value of the wave function at zero separation. But what about the triangle diagram? One would have to project the wave function onto something, but what? The quark propagator between the two electromagnetic vertices suggests something extended, on the order of the Compton wavelength of a quark, whatever that might mean for confined quarks. (I’ve never seen a colored quark, and I never hope to see one, but I can tell you anyhow, I’d rather see than free one.) What is the numerical success of the G.-T. relation telling us about the pion’s (and maybe the nucleon’s) wave function?