Structure of theη→3πAmplitude

Abstract

Difficulties with the usual phenomenological theory of the $C$-conserving part of $\ensuremath{\eta}\ensuremath{\rightarrow}3\ensuremath{\pi}$, with $|\ensuremath{\Delta}I|=1$ 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 $R=\frac{\ensuremath{\Gamma}(\ensuremath{\eta}\ensuremath{\rightarrow}3{\ensuremath{\pi}}^{0})}{\ensuremath{\Gamma}(\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0})}$ as a result of including some large asymmetric parts. We found that a branching ratio of the order $R\ensuremath{\approx}1.1$ is the lower limit of what can be reasonably achieved in these theories. If the experimental value should turn out to be $R\ensuremath{\simeq}0.5$, 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 $|\ensuremath{\Delta}I|>2$ is present in $\ensuremath{\eta}\ensuremath{\rightarrow}3\ensuremath{\pi}$ in that case.