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 F_ζ(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: F_ζ(X;t=0)≈f(X-ζ/2).