Charge asymmetry in the process $e^+e^-\to \pi^+\pi^-\pi^0\gamma$

The charge asymmetry in the process $e^+e^-\to \pi^+\pi^-\pi^0\gamma$ is studied for photons with energies $\nu\ll\sqrt{s}$. The longitudinal polarization of electrons (positrons) is taken into account. The asymmetry arises due to interference of the amplitudes for production of pion states with opposite $C$ parities. The contribution of $a_1^0(1260)$ meson in the intermediate state to the charge asymmetry is discussed. This contribution is not the leading one, however it is not negligible. It is shown that the charge asymmetry can reach several tens of percent.