Hard exclusive electroproduction of two pions and their resonances

We study the hard exclusive production of two pions in the virtual photon fragmentation region with various invariant masses including the resonance region. The amplitude is expressed in terms of two-pion light cone distribution amplitudes ($2\pi$DA's). We derive dispersion relations for these amplitudes, which enables us to fix them completely in terms of $\pi\pi$ scattering phases and a few low-energy subtraction constants determind by the effective chiral lagrangian. Quantitative estimates of the resonance as well $\pi\pi$ background DA's at low normalization point are made. We also prove certain new soft pion theorem relating two-pion DA's to the one-pion DA. Crossing relations between $2\pi$DA's and parton distributions in a pion are discussed. We demontrate that by studying the shape of the $\pi\pi$ mass spectra (not absolute cross section!) in a diffractive electroproduction one can extract the deviation of the meson ($\pi,\rho,$ etc.) wave functions from their asymptotic form $6 z(1-z)$ and hence to get important information about the structure of mesons. Also we discuss how the moments of quark distributions in a pion can be measured in the hard diffraction. We suggest (alternative to S\"oding's) parametrization of $\pi\pi$ spectra which is suitable for large photon virtuality.