Eikonal approximation in AdS/CFT:: Conformal partial waves and finite N four-point functions

We introduce the impact parameter representation for conformal field theory correlators of the form A similar to (O1O2O1O2). This representation is appropriate in the eikonal kinematical regime, and approximates the conformal partial wave decomposition in the limit of large spin and dimension of the exchanged primary. Using recent results on the two-point function (O1O1)shock in the presence of a shock wave in anti-de Sitter, and its relation to the discontinuity of the four-point amplitude A across a kinematical branch cut, we find the high spin and dimension conformal partial wave decomposition of all tree-level anti-de Sitter Witten diagrams. We show that, as in flat space, the eikonal kinematical regime is dominated by the T-channel exchange of the massless particle with highest spin (graviton dominance). We also compute the anomalous dimensions of the high spin O1O2 composites. Finally, we conjecture a formula re-summing crossed-ladder Witten diagrams to all orders in the gravitational coupling. (C) 2007 Elsevier B.V. All rights reserved.