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 $〈p^{\prime }|\mathcal{O}\mathrm{}(0,z)\mathrm{}|p〉$ of quark and gluon light-cone operators. In our previous papers we used two types of nonperturbative functions parametrizing such matrix elements: double distributions $F(x,y;t)\mathrm{}$ and nonforward distribution functions ${\mathcal{F}}_{\zeta }\mathrm{}(X;t).$ Here we discuss in more detail the double distributions (DD’s) and evolution equations which they satisfy. We propose simple models for $F(x,y;t=0\mathrm{})\mathrm{}$ DD’s with correct spectral and symmetry properties which also satisfy the reduction relations connecting them to the usual parton densities $f\left(x\right).$ In this way, we obtain self-consistent models for the $\zeta$ dependence of nonforward distributions. We show that, for small $\zeta ,$ one can easily obtain nonforward distributions (in the $X>\mathrm{}\zeta$ region) from the parton densities: ${\mathcal{F}}_{\zeta }\mathrm{}(X;t=0\mathrm{})\mathrm{}\approx f(X-\zeta /2).\mathrm{}$