We have considered the problem of the asymmetry observed in the decay $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$. Bernstein, Feinberg, and Lee suggested that an asymmetry in the average energies of the charged pions in this decay would be proof of $C$ violation in the electromagnetic interaction among hadrons. Recall, however, that the $\eta$ is identified as a peak in the invariant-mass distribution of three pions from a reaction like ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}{\pi }^{0}n$. Yuta and Okubo proposed that the observed effect is due to interference between the $\eta$ channel for this reaction and direct $3\pi$ production background processes. We have calculated the background size and its possible contributions to the asymmetry using a simple model which takes into account only tree diagrams (no closed loops) with constant vertices and ignores the effects of spin. We can account for 40% of the background observed in the experiment of Gormley et al. but find that the largest asymmetry which can be produced through interference is ${10}^{-5\mathrm{}}$, as compared to the experimentally observed 1.5×${10}^{-2\mathrm{}}$. We conclude that if the observed asymmetry persists, then charge-conjugation invariance is violated by the decay. These results agree with the analysis of Stein of the experimental background. He concluded that the background could account for an asymmetry of at most 2.7×${10}^{-3\mathrm{}}$.

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