A novel way to probe distribution amplitudes of neutral mesons in e^+e^- annihilation

We derive the amplitude for the process $e^+e^-\to \pi^0\pi^0$ at large invariant energy. The process goes through the two-photon exchange and its amplitude is expressed in terms of the convolution integral which depends on the shape of the pion distribution amplitude (DA) and the centre of mass scattering angle. Remarkable feature of the integral is that it is very sensitive to the end-point behaviour of the pion DA -- it starts to diverge if pion DA nullifies at the end-point as $\sqrt x$ or slower. That makes the $e^+e^-\to \pi^0\pi^0$ process unique probe of the shape of the meson DAs. The estimated cross section is rather small, for $\sqrt s = 3$ GeV it ranges from a fraction of femtobarn (for the asymptotic DA) to couple of femtobarn (for the Chernyak-Zhitnitsky DA). The observation of the process $e^+e^-\to\pi^0\pi^0$ with the cross section higher as estimated here would imply very unusual form of the pion DA, e.g. the flat one. The derived amplitude can be easily generalized to other processes like $e^+e^-\to \sigma\sigma, K_SK_S, \eta\eta, \eta^\prime\eta, \pi^0 f_2$, etc.