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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&#39;| 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 F_ζ(X;t). Here we discuss in more detail the double distributions (DD&#39;s) and evolution equations which they satisfy. We propose simple models for F(x,y;t=0) DD&#39;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 > ζregion) from the parton densities: F_ζ(X;t=0) \approx f(X-ζ/2).