My first question on this site concerned experimental evidence about the ratio of $U\bar{U}$ to $D\bar{D}$ content of the neutral pion, or in other words, its isospin purity. But what does theory say? I have a vision of a mass matrix with unequal $Q\bar{Q}$ masses (squared) on the diagonal and roughly equal mixing elements everywhere. Its eigenvalues would represent observed meson masses (squared).
‘t Hooft tried to explain the mechanism that gives the $\eta '$ mass non-perturbatively, via instantons and the triangle anomaly, but the experimental value of the theta parameter is consistent with zero. Is his explanation still viable?
Isn’t there a significant perturbative contribution from an intermediate state of two (or more) gluons? Imagine two triangle diagrams joined back-to-back by gluon propagators. My intuition says the mixing element should scale like ${{p}^{2}}{{g}^{4}}(p)\log (?)$, where $p$ is the momentum that flows through, but the algebra says there’s a quadratic divergence. Huh? How should one deal with the naughty gluon loop: $\left\langle 0 \right|{{g}^{4}}\int{{{d}^{4}}x}\ \exp (ipx)\ {{\varepsilon }^{....}}{{F}_{..}}{{F}_{..}}(x)\ {{\varepsilon }^{....}}{{F}_{..}}{{F}_{..}}(0)\left| 0 \right\rangle $?
The coupling of mesons to the triangle diagram raises a further question. The accepted calculation of the neutral pion’s decay rate uses ${{f}_{\pi }}^{-1}$ and gets the right answer, but what about heavier $Q\bar{Q}$ states?
If the lightest neutral meson were “overweight” in the lightest quarks, the proportion would affect the neutral pion’s decay rate. A hypothetical $U\bar{U}$ meson should decay to two photons 3.55 times faster than an ideal $(U\bar{U}-D\bar{D})/\sqrt{2}$ meson of similar mass.
