Eikonalization of Conformal Blocks

TL;DR

This paper introduces the concept of eikonalization in conformal field theories, showing how large spin Fock space states contribute to OPEs and can be summed into an exponential form, aiding the analysis of large spin behavior.

Contribution

It establishes a systematic framework for summing Fock space exchanges in CFTs, connecting them to classical fields and simplifying large spin OPE computations.

Findings

Sum over Fock space states forms an exponential of a single operator exchange.

Derived the leading logarithmic dependence of large spin OPE coefficients.

Computed OPE coefficients for triple-trace operators explicitly.

Abstract

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.