Effect of $\epsilon (J^P=0^{+},I=0)$ Resonance on $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}\gamma$

The product of the coupling constants for $\epsilon \to \pi \pi$ and $\epsilon \to 2\gamma$ is estimated from a super-convergent sum rule obtained from the helicity-flip amplitude of pion-Compton scattering. It is then used to show that the contribution of $\epsilon$ (even $C$) to the process $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}\gamma$ (to order ${\alpha }^3$) is about two orders of magnitude larger than that when the pions are in odd-$C$ states.