Eikonalization of Conformal Blocks

TL;DR

The paper develops a CFT framework for the eikonalization of conformal blocks, showing that large-$\\ell$ multi-trace exchanges generated by a low-twist operator $T$ produce universal OPE data and can exponentiate to a simple AdS background in appropriate limits. By combining lightcone bootstrap, Mellin amplitudes, and Darboux-type arguments, it derives leading $\\log\\ell$ behavior for large-spin OPE coefficients, constructs explicit large-$\\ell$ multi-trace modes, and demonstrates consistency with an AdS field-theory interpretation. It also analyzes how additional operators in the ${\\cal O}_i(x) T(0)$ OPE modify the coefficients via cross-channel exchanges, and provides direct routes to extracting large-spin data via differential operators and Mellin bounds. The work offers a systematic approach to 1/\\ell perturbation theory in general CFTs and strengthens the bridge between CFT bootstrap and bulk AdS dynamics, with potential applications to sub-AdS locality and holographic universality.

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.

Figures (6)

• Figure 1: This figure indicates how one might sum over multiple virtual exchanges in order to construct an effective classical background. We would like to understand this process directly in the CFT, with minimal assumptions. When the first diagram determines the sum of the rest, we say that the conformal blocks 'eikonalize'.
• Figure 2: This figure indicates conformal partial waves that necessarily contribute to two different 4-pt CFT correlators, based on the assumed OPEs. We indicate the conformal block on the left as ${\cal O}_1 T \to {\cal O}_1 \to {\cal O}_1 T$.
• Figure 3: Large $\ell$ Fock space operators should have a universal behavior. This figure indicates how one might try to use known OPE coefficients to construct conformal blocks, and then take the OPE limit in a different channel to obtain new information about general Fock space states. By making a simple assumption about the CFT correlators in Mellin space, these OPE limits can to be shown to exist, and give a universal result for OPE coefficients with Fock space states.
• Figure 4: This figure illustrates terms that contribute to the lightcone OPE limit of the CFT bootstrap equation. The consequence of the first two terms on the left-hand side are reviewed in section \ref{['sec:Review']}, while the third term and its generalizations are discussed using this bootstrap equation in section \ref{['sec:Eikonalization']}.
• Figure 5: This figure suggests other contributions to the $\langle {\cal O}_1 {\cal O}_1 T T \rangle$ correlator from ${\cal O}'$ conformal blocks. These contributions affect the $\langle {\cal O}_1 {\cal O}_1 [TT]_{n, \ell} \rangle$ OPE coefficients, but they do not contaminate eikonalization unless $\tau_{{\cal O}'} < \tau_1$ and the two twists differ by an integer. Generically, they contribute to $[{\cal O}' {\cal O}']_{n, \ell}$ exchange at large $\ell$ in the cross-channel, as pictured, and also to $[{\cal O}_i {\cal O}']_{n,\ell}$.