Regge Theory and Multi–Reggeon Exchange

We analyze conformal field theory 4–point functions of the form A ∼ 〈O1 (x1)O2 (x2)O1 (x3)O2 (x4)〉, where the operators Oi are scalar primaries. We show that, in the Lorentzian regime, the limit x1 → x3 is dominated by the exchange of conformal partial waves of highest spin. When partial waves of arbitrary spin contribute to A, the behavior of the Lorentzian amplitude for x1 → x3 must be analyzed using complex–spin techniques, leading to a generalized Regge theory for CFT’s. Whenever the CFT is dual to a string theory, the string tree–level contribution Atree to the amplitude A presents a Regge pole corresponding the a gravi– reggeon exchange. In this case, we apply the impact parameter representation for CFT amplitudes, previously developed, to analyze multiple reggeon exchanges in the eikonal limit. As an example, we apply these general techniques to N = 4 super–Yang–Mills theory in d = 4 in the limit of large ’t Hooft coupling, including the leading string corrections to pure graviton exchange.