Gap equations, particle mixing and the U(1) problem in QCD

The gap equation for the chiral-limiting quark mass is shown to be the basis of the Goldberger-Treiman relation at the quark level, again a gap equation, for both the massless it and massive U(1) ηo states. The strength β of the isoscalar quark annihilation graphs is also demonstrated to be the origin of isoscalar qq particle mixing. Combining these three results, the perturbative second order U(1) anomaly contribution βp (2)/m2 ηo then generates a third U(1) gap equation. The latter resolves the U(1) problem in QCD because it explains why the η′ mass is significantly larger than the π,κ,η octet masses.