Universal anomalous dimensions at large spin and large twist

TL;DR

This work establishes a universal form for the anomalous dimensions of double-trace operators at large spin and large twist across dimensions $d\ge 3$, derived via analytic conformal bootstrap. By constructing an approximate closed-form for conformal blocks in arbitrary $d$ and performing a careful large-$\ell$ expansion, the authors show that the leading $n$-dependent piece of the anomalous dimension scales as $\gamma_n \propto n^d$, with the coefficient expressed solely through the stress-tensor central charge $c_T$. They confirm this universal CFT result through two independent holographic computations, mapping $G_N$ to $c_T$ and reproducing the same leading behavior in the eikonal limit, and via an AdS-Schwarzschild gravity setup that remains robust against $\alpha'$ corrections. The findings reinforce a deep link between boundary unitarity/causality and bulk gravitational dynamics and outline avenues for extending the results to general twists and finite-$n$ corrections, as well as cross-checks with numerical bootstrap methods.

Abstract

In this paper we consider anomalous dimensions of double trace operators at large spin ($\ell$) and large twist ($τ$) in CFTs in arbitrary dimensions ($d\geq 3$). Using analytic conformal bootstrap methods, we show that the anomalous dimensions are universal in the limit $\ell\gg τ\gg 1$. 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.

Figures (3)

• Figure 1: $\log (-\gamma_nP_m/\Delta_\phi^2)$ vs. $\log n$ plot showing the dependence of $\gamma_n$ on $n$ for different values of $\Delta_\phi$ in different dimensions. For $n\gg1$, the coincidence of the graphs of different $\Delta_\phi$ for each $d$ indicates the universal formula given in \ref{['gl']}.
• Figure 2: The blue lines are $\log (\ell_{orb}^{d/2 -1}\delta E^d_{orb}/\mu)$ vs $\log n$ for large values of $\ell_{orb}$ and the red lines are the log-log plot for $rhs$ of \ref{['poly']}, in odd $d$. The matching of both lines at large $n$ is the numerical proof of \ref{['poly']} in odd $d$.
• Figure 3: $\gamma_{n=1}(\Delta_{\phi}^2/P_m)$ vs $\Delta_{\phi}$ plot in $d=4$ (red curve) and $d=6$ (blue curve), showing that the anomalous dimension may take positive values if unitarity is violated. Unitarity demands $\Delta_{\phi}\ge (d-2)/2$.