The CP-violating asymmetry in ηtoπ^+ π^- e^+e^-

We study the CP-violating asymmetry {\cal A}_{\rm CP}, which arises, in \eta\to\pi^+\pi^- e^+e^-, from the angular correlation of the e^+ e^- and \pi^+\pi^- planes due to the interference between the magnetic and electric decay amplitudes. With the phenomenologically determined magnetic amplitude and branching ratio as input, the asymmetry, induced by the electric bremsstrahlung amplitude through the CP-violating decay \eta\to\pi^+\pi^-, and by an unconventional tensor type operator, has been estimated respectively. The upper bound of {\cal A}_{\rm CP} from the former is about 10^{-3}, and the asymmetry from the latter might be up to O(10^{-2}). One can therefore expect that this CP asymmetry would be an interesting CP-violating observable for the future precise measurements in the \eta factories.