The Operator Product Expansion of N=4 SYM and the 4-point Functions of Supergravity

TL;DR

This work develops a detailed Operator Product Expansion interpretation for the AdS/CFT-based 4-point functions of protected operators in ${ m N}=4$ SYM. It demonstrates that power-singular terms in bulk AdS exchanges precisely reproduce the conformal partial waves of the exchanged primary and descendants, while leading logarithms arise from $O(1/N^2)$ mixing of double-trace operators, signaling anomalous dimensions at strong coupling. A key result is the prediction, rooted in ${ m U}(1)_Y$ symmetry, that the anomalous dimensions of double-trace composites such as $[{ m O}_{ m O}_{ m C}]$ and $[{ m O}_{ m O}_{ m f} - { m O}_{ m C}{ m O}_{ m C}]$ are $-16/N^2$ in the large $N$, large $\\lambda$ limit, with a protected, non-mixing state $[{ m O}_{ m B}{ m O}_{ m B}]$ ensuring a consistent algebra. The results provide a quantitative bridge between 5d supergravity amplitudes and 4d CFT data, argue for decoupling of string states at strong coupling, and offer a framework for extending OPE analyses to other AdS/CFT dual pairs. Overall, the paper strengthens the interpretation of AdS computations as a convergent double-OPE in the protected sector and highlights the role of double-trace operators in shaping the strong-coupling dynamics of ${ m N}=4$ SYM.

Abstract

We give a detailed Operator Product Expansion interpretation of the results for conformal 4-point functions computed from supergravity through the AdS/CFT duality. We show that for an arbitrary scalar exchange in AdS(d+1) all the power-singular terms in the direct channel limit (and only these terms) exactly match the corresponding contributions to the OPE of the operator dual to the exchanged bulk field and of its conformal descendents. The leading logarithmic singularities in the 4-point functions of protected N=4 super-Yang Mills operators (computed from IIB supergravity on AdS(5) X S(5) are interpreted as O(1/N^2) renormalization effects of the double-trace products appearing in the OPE. Applied to the 4-point functions of the operators Ophi ~ tr F^2 + ... and Oc ~ tr FF~ + ..., this analysis leads to the prediction that the double-trace composites [Ophi Oc] and [Ophi Ophi - Oc Oc] have anomalous dimension -16/N^2 in the large N, large g_{YM}^2 N limit. We describe a geometric picture of the OPE in the dual gravitational theory, for both the power-singular terms and the leading logarithms. We comment on several possible extensions of our results.

Figures (6)

• Figure 1: Scalar exchange in the 't--channel'.
• Figure 2: Upper--half plane representation of an exchange integral in AdS space. In the limit $x_1 \to x_2$ the $z$--integral is supported in a small ball (in coordinate units) close to the boundary point $x_1 \approx x_2$.
• Figure 3: Quartic graph.
• Figure 4: A quartic graph in the upper--half plane representation of AdS space. Each annulus is $e$ times bigger in diameter than the one nested inside it. The $z$--integration receives an equal contribution from each annular region.
• Figure 5: Born diagram for the 2--point function $\langle \hat{{\cal O}_\phi} \hat{{\cal O}_\phi} \rangle$ in the double--line representation.