Crossing and Radon Tomography for Generalized Parton Distributions

The crossing properties of the matrix elements of non-local operators, parameterized by Generalized Parton Distribution, are considered. They are especially simple in terms of the Double Distributions which are common for the various kinematical regions. As a result, Double Distributions may be in principle extracted from the combined data in these regions by making use of the inverse Radon transform, known as a standard method in tomography. The ambiguities analogous to the ones for the vetor potential in the two-dimensional magneto-statics are outlined. The possible generalizations are discussed. Introduction Deeply Virtual Compton Scattering (DVCS) [1, 2] is the cleanest hard process which is sensitive to the Generalized Parton Distributions (GPD), the most popular version of which is probably represented by the Skewed Parton Distributions (SPD), and has been the subject of extensive theoretical investigations for a few years. First experimental data became recently available (see e.g. [3, 4]) and much more data are expected from JLAB, DESY, and CERN in the near future. In the present note we shall limit ourself to the case of the pion target, although our consideration can be applicable to any target. The crossing version is then provided by the process γγ → ππ̄ with a highly virtual photon but small hadronic invariant mass W . It was recently investigated in the framework of QCD factorization[5, 6, 7]. It allows to study the pion pair produced in the isoscalar channel, where the huge ρ-meson peak is absent. This process is analogous to the single pion production, described by the pion transition form-factor, being the long time object of QCD studies [8, 9]. In particular, the generalized distribution amplitude (GDA), describing the non-perturbative stage of this process, is a natural counterpart of the pion light cone distribution amplitude. From the other side, it may be considered as a crossing counterpart of pion SPD [5] and related to it [10] by making use of the suitable polynomial basis. The alternative (and quite elegant) description of non-perturbative stage of DVCS and other hard exclusive processes is provided by Double Distributions (DD) [1, 11]. They