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 $\mathcal{O}(x) \mathcal{O}(0)$, then the large spin $\ell$ Fock space states $[TT \cdots T]_{\ell}$ also appear in this OPE with a computable coefficient. The sum over the exchange of these Fock space states in an $\langle \mathcal{O} \mathcal{O} \mathcal{O} \mathcal{O} \rangle$ 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 $\mathcal{O}$. Our results should be useful for systematizing $1/\ell$ perturbation theory in general CFTs and simplifying the computation of large spin OPE coefficients. As examples we obtain the leading $\log \ell$ dependence of Fock space conformal block coefficients, and we directly compute the OPE coefficients of the simplest `triple-trace' operators.