Possible $C$-Noninvariant Effects in the $3\pi$ Decay Modes of ${\eta }^0$ and ${\omega }^0$

In this paper, the observed $\mathrm{CP}$ noninvariance in $K_2^{}{}_{}{}^0\to {\pi }^{+}+{\pi }^{-}$ is assumed to be due to the existence of a new strangeness-conserving but $C$-noninvariant and $T$-noninvariant interaction called $H_F$, whose coupling constant $F$ is $\sim {10}^{3}G$ where $G$ is the Fermi coupling constant in the usual $\mathrm{CP}$-invariant weak interaction. A phenomenological analysis of the possible forms of the energy asymmetry between ${\pi }^{+}$ and ${\pi }^{-}$ in the $3\pi$ decay modes of ${\eta }^0$ and ${\omega }^0$ is made. It is pointed out that a study of such an asymmetry can be used to test the possible existence of $H_F$ as well as its isotopic-spin selection rules.