Structure of theη→3πAmplitude

Difficulties with the usual phenomenological theory of the $C$-conserving part of $\eta \to 3\pi$, with $\left|\Delta I\right|=1\mathrm{}$ and dominance by the nearly constant symmetric amplitude, are summarized. It is discussed why the difficulties are indicative of the presence of a significant asymmetric part in the amplitude. An attempt is made to find the possible reductions in the theoretical branching ratio $R=\frac{\Gamma (\eta \to 3{\pi }^0)}{\Gamma (\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0)}$ as a result of including some large asymmetric parts. We found that a branching ratio of the order $R\approx 1.1$ is the lower limit of what can be reasonably achieved in these theories. If the experimental value should turn out to be $R\simeq 0.5$, it will be extremely difficult to reconcile with the theory even when large energy variations in the decay amplitude are allowed for; and it is nearly certain that a part with $\left|\Delta I\right|>\mathrm{}2\mathrm{}$ is present in $\eta \to 3\pi$ in that case.