Analytic bootstrap at large spin

TL;DR

This paper develops analytic bootstrap technology to compute anomalous dimensions $\gamma(n,\ell)$ and corrected OPE coefficients for large-spin towers in 4d CFTs with a scalar of dimension $\Delta_{\phi}$. Under a twist gap, it shows $\tau=2\Delta_{\phi}+2n$ for the towers and derives the leading $\ell^{-2}$ suppression of $\gamma(n,\ell)$, obtaining explicit $n$-dependent forms for two minimal-twist exchanges: a twist-2 scalar and a twist-2 graviton, with the latter yielding a universal $\gamma_n \sim -4 n^4$ at large $n$ (independent of $\Delta_{\phi}$) when properly normalized. Subleading $1/\ell$ corrections are computed for stress-tensor exchange, yielding $\gamma(n,\ell)=\gamma^0_n(1-2n/\ell)$ and associated corrections to OPE data, and the results are shown to be consistent with AdS/CFT eikonal predictions in the gravity-dominated regime. In the large-twist, large-spin limit the results reproduce AdS/CFT expectations for both $\ell\gg n\gg1$ and $n\gg\ell\gg1$, including the graviton-dominated cases, providing a robust analytic bridge between bootstrap and holography and highlighting the role of unitarity and bulk causality in shaping the sign and scaling of $\gamma(n,\ell)$. These findings offer a universal, gravity-consistent picture of large-spin dynamics in 4d CFTs and pave the way for further analytic explorations at subleading orders and in broader settings.

Abstract

We use analytic conformal bootstrap methods to determine the anomalous dimensions and OPE coefficients for large spin operators in general conformal field theories in four dimensions containing a scalar operator of conformal dimension $Δ_φ$. It is known that such theories will contain an infinite sequence of large spin operators with twists approaching $2Δ_φ+2n$ for each integer $n$. By considering the case where such operators are separated by a twist gap from other operators at large spin, we analytically determine the $n$, $Δ_φ$ dependence of the anomalous dimensions. We find that for all $n$, the anomalous dimensions are negative for $Δ_φ$ satisfying the unitarity bound. We further compute the first subleading correction at large spin and show that it becomes universal for large twist. In the limit when $n$ is large, we find exact agreement with the AdS/CFT prediction corresponding to the Eikonal limit of a 2-2 scattering with dominant graviton exchange.

Figures (4)

• Figure 1: The variation of the anomalous dimensions $\gamma_n$ with $\Delta_\phi$ showing that some of the anomalous dimensions become positive when $\Delta_\phi<(d-2)/2$.
• Figure 2: $\log |\gamma_n|$ vs. $\log n$ plot showing the dependence of $\gamma_n$ on $n$ for $n\gg1$. $\gamma_T$ is the anomalous dimension for the spin-$2$ operator exchange and $\gamma_S$ for the scalar operator exchange. The slope of the blue straight line for spin-$2$ exchange is $3.998$ while the red line denotes the scalar exchange for which $\gamma_n$ is constant for all $n$. We have used $\Delta_\phi=2$ in the above plots.
• Figure 3: Plot for ${\mathcal{C}}_n$ for three cases. The blue curve is for ${\mathcal{N}}=4$, the red curve for the twist-$2$, spin-$2$ operator exchange and the yellow for the twist-$2$ scalar. We have scaled down the OPE coefficients by a factor $10^8$ in this figure.
• Figure 4: Plot for the numerical estimate of the exponent of $n$ for $\tau_m=4$ for a range of $\Delta_\phi$.