Eikonalization of conformal blocks

Classical field configurations such as the Coulomb potential and Schwarzschild solution are built from the t-channel exchange of many light degrees of freedom. We study the CFT analog of this phenomenon, which we term the 'eikonalization' of conformal blocks. We show that when an operator T appears in the OPE O (x) O (0), then the large spin Fock space states [TT ... T](l) also appear in this OPE with a computable coefficient. The sum over the exchange of these Fock space states in an < OOOO > correlator build the classical 'T field' in the dual AdS description. In some limits the sum of all Fock space exchanges can be represented as the exponential of a single T exchange in the 4-pt correlator of O. Our results should be useful for systematizing 1/l perturbation theory in general CFTs and simplifying the computation of large spin OPE coefficients. As examples we obtain the leading log 'dependence of Fock space conformal block coefficients, and we directly compute the OPE coefficients of the simplest 'triple-trace' operators.