Bounds for OPE coefficients on the Regge trajectory

Abstract

We consider the Regge limit of the CFT correlation functions J J OO and T T OO , where J is a vector current, T is the stress tensor and 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 J J j(ν) and C T T j(ν) that describe the coupling to the leading Regge trajectory of the current 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 J J T and C T T T , associated to non-minimal bulk couplings, scale with the gap ∆ g as ∆ −2 g or ∆ −4 g . Also, looking directly at the unitarity condition imposed at the OPE coefficients C J J T and C T T T 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.

Unitarity methods in AdS/CFT

“…It would also be interesting to understand in more detail how consistency conditions in AdS/CFT at higher loops, such as causality, unitarity, and crossing symmetry, constrain the space of theories. There has been great success in constraining graviton [78][79][80][81][82][83][84][85][86][87] and general higher derivative interactions [69,80,88,89] at leading order in 1/N , but the study of these consistency conditions at higher loops remains unexplored. Individual loop amplitudes are also sensitive to the existence of compact, extra dimensions [9,15].…”

Chaos in the butterfly cone

“…This is precisely the value where (4.19) gets saturated. It is reasonable to assume that if the stress tensor is the lightest operator on the leading Regge trajectory, there is no non-analicity in g(ν) closer than this to the 16 A very similar discussion can be found in [40], who uses the ability to shift the saddle to put bounds on analytically continued OPE data on the leading Regge trajectory from AdS unitarity. Here, we merely focus on the exponent and do not discuss its coefficient.…”

“…The precise form of it will be irrelevant for us, but it will be important that it contains an explicit factor of sin πj(ν) 2 −1 coming from the kinematical factors. Its explicit form is written down in [36,40].…”

“…Here, we merely focus on the exponent and do not discuss its coefficient. Another key difference is that the unitarity bounds considered in [40] require restricting to the real part of g(ν), from which the poles at even j coming from sin πj(ν) 2 −1 turn out to cancel. For us, the pole at j = 2 will be crucial for the bound (1.7) to be obeyed.…”

Scalar CFTs and their large N limits

“…At the free theory limit, the scaling dimension of the first O(N ) singlet operator is given by ∆[ i φ i φ i ] = 1, while at the critical O(N ) point, its dimension is given by D − 1 = 2 plus 1 N corrections. It is not yet clear what are the necessary and sufficient conditions for a CFT to have weakly coupled dual description in AdS [31,32,34,46,47,48]. As conjecture in [31], besides large N factorization, any CFT with Einstein like local bulk dual description must also have a large gap for all single trace operators with spin higher than 2.…”