Conformal Regge theory

Abstract

We generalize Regge theory to correlation functions in conformal field theories. This is done by exploring the analogy between Mellin amplitudes in AdS/CFT and S-matrix elements. In the process, we develop the conformal partial wave expansion in Mellin space, elucidating the analytic structure of the partial amplitudes. We apply the new formalism to the case of four point correlation functions between protected scalar operators in N=4 Super Yang Mills, in cases where the Regge limit is controlled by the leading twist operators associated to the pomeron-graviton Regge trajectory. At weak coupling, we are able to predict to arbitrary high order in the 't Hooft coupling the behaviour near J=1 of the OPE coefficients C_{OOJ} between the external scalars and the spin J leading twist operators. At strong coupling, we use recent results for the anomalous dimension of the leading twist operators to improve current knowledge of the AdS graviton Regge trajectory - in particular, determining the next and next to next leading order corrections to the intercept. Finally, by taking the flat space limit and considering the Virasoro-Shapiro S-matrix element, we compute the strong coupling limit of the OPE coefficient C_{LLJ} between two Lagrangians and the leading twist operators of spin J.

Figures (7)

• Figure 1: Chew-Frautschi plot of the spectrum of exchanged particles in the Virasoro-Shapiro amplitude.
• Figure 2: Integration contours in the $J$-plane involved in the Sommerfeld-Watson transform. Initially, the sum is written as an integral over the contour $C$ encircling all positive integers. Then, the contour is deformed to the contour $C'$ plus the contribution of the Regge poles.
• Figure 3: Leading Regge trajectory in the dimension--spin plane for various values of the 't Hooft coupling. The physical operators have even spin $J$ and positive dimension, and are represented by blue dots along the curves. The horizontal dashed line $J=2$ corresponds to the strong coupling limit $g\to \infty$. The weak coupling limit $g \to 0$, is described by the other dashed line, with 3 branches: $J=1$, $\Delta-2=J$ and $2-\Delta=J$. The intercept $j(0)$ moves from $1$ to $2$, as the coupling $g$ goes from 0 to $\infty$.
• Figure 4: Weak (in blue) and strong (in red) coupling expansions of the BFKL intercept $j(0)$. The plot suggests a smooth interpolation, like the black dashed curve, from 1 at $g=0$ to 2 at $g=\infty$.
• Figure 5: Witten diagrams of (a) exchange of a dimension $\Delta$ and spin $J$ field in AdS and (b) contact interaction.