Zero mode effect in the $1^{-+}$ four quark states

We calculate the masses of the $1^{-+}$ four quark states which decay dominantly into $\rho\pi$ and $\eta\pi$ respectively by QCD sum rules approach. We include the zero mode contribution and find it plays an important role in the sum rules. We predict that the masses of the states $\eta\pi$ and $\rho\pi$ both are 1.4-1.5 GeV. This is close to the experimental candidates $\pi_1(1370)$ and $\pi_1(1440)$.