Abstract
In this paper we consider anomalous dimensions of double trace operators at large spin ($\ell$) and large twist ($\tau$) in CFTs in arbitrary dimensions ($d\geq 3$). Using analytic conformal bootstrap methods, we show that the anomalous dimensions are universal in the limit $\ell\gg \tau\gg 1$. In the course of the derivation, we extract an approximate closed form expression for the conformal blocks arising in the four point function of identical scalars in any dimension. We compare our results with two different calculations in holography and find perfect agreement.
Highlights
Using analytic conformal bootstrap methods, we show that the anomalous dimensions are universal in the limit τ 1
In the course of the derivation, we extract an approximate closed form expression for the conformal blocks arising in the four point function of identical scalars in any dimension
One of the key results which will enable us to perform this calculation is the derivation of a closed form expression for the conformal blocks in arbitrary dimensions in a certain approximation
Summary
For general d dimensional CFT, the minimal twist τm = d − 2. In the limit of the large spin, gτ(d, )(v, u) undergo significant simplification as given in appendix A of [19]. In the limit of large and fixed τ and for u → 0, following appendix A of [19], gτ(d, )(v, u) = k2 (1 − u)vτ/2F (d)(τ, v) + O. To derive a form of the function F (d)(τ, v), we start by writing down the recursion relation relating the conformal block in d dimension to the ones in d−2 dimensions (see [19, 25]), z − z (1−z)(1−z) g∆(d,) (v, u) =. Using properties of gamma functions the above series simplifies to the one given in (2.11)..
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