Onium-Onium scattering at fixed impact parameter: exact equivalence between the color dipole model and the BFKL Pomeron

We compute the onium-onium scattering amplitude at fixed impact parameter in the framework of the perturbative QCD dipole model. Relying on conformal properties of the dipole cascade and of the elementary dipole-dipole scattering amplitude, we obtain an exact result for this onium-onium scattering amplitude, which is proven to be identical to the BFKL result, and which exhibits the frame invariance of the calculation. The asymptotic expression for this amplitude and for the dipole distribution in an onium at fixed impact parameter agree with previous numerical simulations. We show how it is possible to describe onium-$e^{\pm}$ deep inelastic scattering in the dipole model, relying on k_T-factorization properties. The elementary scattering amplitudes involved in the various processes are computed using eikonal techniques.