Bounds for OPE coefficients on the Regge trajectory

TL;DR

This work develops a conformal Regge theory framework for four-point functions, connecting Regge-limit data to AdS bulk phase shifts. By proving AdS unitarity from CFT, the authors derive positivity constraints on analytic continuations of OPE coefficients along the leading Regge trajectory, yielding bounds on couplings to the current and stress-tensor operators. In large-gap theories, non-minimal bulk couplings are shown to be suppressed by powers of the gap, and the approach provides a CFT derivation of conformal collider bounds. The results hold for both scalar and spinning external operators, with a detailed treatment of intercept and convexity properties, and they illuminate finite-$N$ caveats and potential tests in weakly coupled regimes.

Abstract

We consider the Regge limit of the CFT correlation functions $\langle {\cal J} {\cal J} {\cal O}{\cal O}\rangle$ and $\langle TT {\cal O}{\cal O}\rangle$, where ${\cal J}$ is a vector current, $T$ is the stress tensor and ${\cal O}$ is some scalar operator. These correlation functions are related by a type of Fourier transform to the AdS phase shift of the dual 2-to-2 scattering process. AdS unitarity was conjectured some time ago to be positivity of the imaginary part of this bulk phase shift. This condition was recently proved using purely CFT arguments. For large $N$ CFTs we further expand on these ideas, by considering the phase shift in the Regge limit, which is dominated by the leading Regge pole with spin $j(ν)$, where $ν$ is a spectral parameter. We compute the phase shift as a function of the bulk impact parameter, and then use AdS unitarity to impose bounds on the analytically continued OPE coefficients $C_{{\cal J}{\cal J}j(ν)}$ and $C_{TTj(ν)}$ that describe the coupling to the leading Regge trajectory of the current ${\cal J}$ and stress tensor $T$. AdS unitarity implies that the OPE coefficients associated to non-minimal couplings of the bulk theory vanish at the intercept value $ν=0$, for any CFT. Focusing on the case of large gap theories, this result can be used to show that the physical OPE coefficients $C_{{\cal J}{\cal J}T}$ and $C_{TTT}$, associated to non-minimal bulk couplings, scale with the gap $Δ_g$ as $Δ_g^{-2}$ or $Δ_g^{-4}$. Also, looking directly at the unitarity condition imposed at the OPE coefficients $C_{{\cal J}{\cal J}T}$ and $C_{TTT}$ results precisely in the known conformal collider bounds, giving a new CFT derivation of these bounds. We finish with remarks on finite $N$ theories and show directly in the CFT that the spin function $j(ν)$ is convex, extending this property to the continuation to complex spin.

Figures (3)

• Figure 1: Shape of the leading Regge trajectory $J=j(\nu)$ with vacuum quantum numbers in a CFT. The dimension of operators $\Delta$ is related to the spectral parameter $\nu$ by $\Delta=h+i\nu$ where $h=d/2$. The function $j(\nu)$ is even and convex. The minimum (for imaginary $\nu$) is the intercept $j(0)\equiv j_0$.
• Figure 2: Regge kinematics requires $y_{13}^2,y_{24}^2>0$ and $y_{14}^2,y_{23}^2<0$. The left panel shows the Regge kinematics for $y_{12}^2,y_{34}^2>0$, but that is not necessary since we may allow $y_2$ to cross the light cone of $y_1$ and $y_4$ the light cone of $y_3$. The right panel shows the path of the cross ratios $z,\bar{z}$ as we analytically continue from the Euclidean region to the Lorentzian one.
• Figure 3: The CFT can be defined on the Lorentzian cylinder (left figure). By a conformal transformation one can move to a Poincaré patch, defining the theory on Minkowski space, that covers only a portion of the cylinder. The central Poincaré patch, where operator insertions are close to null infinity, is shown in blue. One may instead consider Poincaré patches whose origins, shown as white dots, are at null infinity of the central Poincaré patch. The operator ${\cal O}_i$ is then inserted very close to the origin of the Poincaré patch ${\cal P}_i$, where we use coordinates $x_i$. To visualize the different Poincaré patches it is convenient to open the cylinder (right figure). The red lines are identified in this picture.