Analysis of ${\eta }^0$, $X^0\to {\pi }^{+}{\pi }^{-}\gamma$ with a Possible $C$ Violation

The asymmetry of the ${\pi }^{+}$, ${\pi }^{-}$ energy spectra in ${\eta }^0$, $X^0\to {\pi }^{+}{\pi }^{-}\gamma$ decays is analyzed under the assumption of a possible $C$ violation in electromagnetic interactions. It is assumed that the $P$- and $D$-wave pion-pion phase shifts satisfy an effective-range expansion whose parameters are determined by the position and width of the ${\eta }^0$ and $f^0$ resonances. Within the validity of this assumption the upper limit for the asymmetry parameter in ${\eta }^0$ decay is found to be 1.1%, corresponding to a ratio $R_{\eta }=\frac{\Gamma ({\eta }_0\to {\pi }^0{\pi }^0\gamma )}{\Gamma ({\eta }^0\to {\pi }^{+}{\pi }^{-}\gamma )}=0.25\mathrm{}$; the upper limit for the asymmetry parameter in $X^0$ decay is 18%, corresponding to a ratio $\Gamma (X^0\to {\pi }^{+}{\pi }^{-}\gamma )=0.25\mathrm{}$. For any given values of the "decay radius" and the ratio of the $C$-violating to $C$-conserving coupling constants, the ${\pi }^{+}{\pi }^{-}$ asymmetry is much larger in $X^0$ decay than in ${\eta }^0$ decay; hence the notion of maximal $C$ violation in electromagnetic processes can be suitably tested in this decay. This is, moreover, a more sensitive test than a search for the neutral mode $X^0\to {\pi }^0{\pi }^0\gamma$, since the ${\pi }^{+}{\pi }^{-}$ asymmetry can be as large as 10% for low values of $R_{\mathrm{X}}=0.04\mathrm{}$.