Study of two--pion light--cone distribution amplitudes in the resonance region and at low energies

The dispersion relations which allow to express the two-pion light-cone distribution amplitudes (DA's) in a wide range of two-pion invariant masses (including the resonance region) in terms of $\pi\pi$ scattering phase shifts and of a few low-energy subtraction constants are derived. The corresponding subtraction constants can be determined in the low energy region--where the effective chiral theory is applicable. In this region we use an effective quark-pion chiral lagrangian derived from the instanton vacuum to make quantitative estimates of the subtraction constant and hence to fix completely two-pion DA's (both chirally even and odd) in a wide range of two-pion invariant masses including the resonance one. We show that the distribution amplitudes of the resonances ($\rho, f_2, \rho_3$, etc.) can be expressed in terms of the two-pion DA's at invariant mass of two pions close to the mass of resonance. The quantitative estimates of the resonance DA's (chirally even and odd) at low normalization point are made. Certain soft pion theorems relating the two-pion DA to the pion distribution amplitude are proven. Applications of $2\pi$DA's for a hard production of two pions in the reaction \gamma_L^* + N --> 2\pi+N with invariant energy of two pions below and in the resonance region are discussed. In particular, we argue that studying the shape of \pi\pi mass spectra (not the absolute cross section!) in diffractive pions production experiments one can extract the deviation of the meson (\pi and \rho) DA's from their asymptotic form 6z(1-z), and hence to obtain non-perturbative information about structure of mesons.