Gauge Theories Based on Quarks and theη→3πProblem

If chiral $\mathrm{U}{\mathrm{}\left(3\right)\mathrm{}}_L\times \mathrm{U}{\mathrm{}\left(3\right)\mathrm{}}_R$ 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 $U_3$ term to the $\eta \to 3\pi$ 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 $\sqrt{2}{\epsilon }_0+{\epsilon }_8=0\mathrm{}$ or $\sqrt{2}{\epsilon }_0+{\epsilon }_8=\pm \sqrt{3}{\epsilon }_3$ were satisfied exactly, which, however, does not seem to be permitted by nature. Thus $\eta \to 3\pi$ decay remains a major problem for the class of models mentioned above, even in the presence of the $U_3$ term.