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.
Highlights
Introduction and summaryThe correlation functions of local operators in Conformal Field Theories (CFTs) must satisfy fundamental consistency conditions encoding conformal symmetry and quantum mechanical unitarity
We show that when an operator T appears in the operator product expansion (OPE) O(x)O(0), the large spin Fock space states [T T · · · T ] appear in this OPE with a computable coefficient
The non-universal behavior of the [T T ]n, OPE coefficients dictates the existence of other operators in the O1(x)O2(0) OPE
Summary
The correlation functions of local operators in Conformal Field Theories (CFTs) must satisfy fundamental consistency conditions encoding conformal symmetry and quantum mechanical unitarity. These results hold in the limit → ∞ with the external dimensions ∆i fixed, which has the AdS interpretation of the exchange of well-separated light mediators between two objects with fixed energy. The universal contributions to exchanged conformal blocks that we derive here arise instead from the approximate Fock space structure of CFT operators at large spin We refer to this behavior as the ‘eikonalization’ of conformal blocks because of its exponentiated structure at large ∆i, suggesting a similar interpretation in terms of classical background fields in AdS.
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