Gauge theories based on quarks and the eta → 3π problem

If chiral $U{\left(3\right)}_L\times U{\left(3\right)}_R$ symmetry is broken only in the (3,3*) + (3*,3) manner (ignoring weak and electromagnetic interactions), as is the case for the class of renormalizable gauge theories based on quarks and the simple quark-gluon model, then it is shown that the contribution of the U₃ term to the η→3π decay treated to all orders in perturbation theory leads to a grossly incorrect slope and perhaps a suppressed magnitude contrary to usual expectations. Exceptions to the above result might have arisen if either of the interesting relations $\sqrt{2}{ϵ}_0+{ϵ}_8=0$ or $\sqrt{2}{ϵ}_0+{ϵ}_8=\pm \sqrt{3}{ϵ}_3$ were satisfied exactly, which, however, does not seem to be permitted by nature. Thus η→3π decay remains a major problem for the class of models mentioned above, even in the presence of the U₃ term.