Bounds for OPE coefficients on the Regge trajectory

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 $j(\nu)$, where $\nu$ 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 $C_{{\cal J}{\cal J}j(\nu)}$ and $C_{TTj(\nu)}$ 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 $\nu=0$, 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 $\Delta_g$ as $\Delta_g^{-2}$ or $\Delta_g^{-4}$. 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 $j(\nu)$ is convex, extending this property to the continuation to complex spin.