Double Distributions and Evolution Equations

Applications of perturbative QCD to deeply virtual Compton scattering and hard exclusive meson electroproduction processes require a generalization of usual parton distributions for the case when long-distance information is accumulated in nonforward matrix elements 〈p | O(0, z) | p〉 of quark and gluon light-cone operators. In our previous papers we used two types of nonperturbative functions parametrizing such matrix elements: double distributions F (x, y; t) and nonforward distribution functions Fζ(X; t). Here we discuss in more detail the double distributions (DD’s) and evolution equations which they satisfy. We propose simple models for F (x, y; t = 0) DD’s with correct spectral and symmetry properties which also satisfy the reduction relations connecting them to the usual parton densities f(x). In this way, we obtain self-consistent models for the ζ-dependence of nonforward distributions. We show that, for small ζ, one can easily obtain nonforward distributions (in the X > ζ region) from the parton densities: Fζ(X; t = 0) ≈ f(X−ζ/2).