How quark confinement solves theη→3πproblem

Gauge theories of quarks which have chiral SU(3)×SU(3) symmetries necessarily also have a ninth axial-vector symmetry. Empirically this symmetry is not manifest in the particle spectrum in either the conventional or Goldstone form. We argue on the basis of an exactly solvable model that the apparent lack of the ninth axial-vector symmetry is due to the same long-range forces which confine quarks in infrared-unstable Yang-Mills theories. A related problem is the violation of Sutherland's low-energy theorem for the decay $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$. We show how the same phenomenon naturally ruins the validity of Sutherland's conclusion for soft neutral pions. In the course of this work we discuss the Schwinger model with massive fermions. On the basis of mass perturbation theory and simple energy considerations, it appears that this is a well-behaved theory of confinement.