Graviton exchange and complete 4-point functions in the AdS/CFT correspondence

TL;DR

<3-5 sentence high-level summary>We compute the graviton-exchange contribution to 4-point functions in the AdS/CFT framework, focusing on dilaton and axion operators in IIB supergravity on AdS5×S5. By reducing the graviton exchange to a finite sum of scalar quartic graphs via integrals over bulk points, we obtain explicit expressions for the 4-point amplitudes in terms of $D_{ ext{Δ1Δ3Δ2Δ4}}$ functions and their conformal variables. The short-distance (direct-channel) behavior reveals the expected leading contribution from the stress-energy tensor in a double OPE, while the complete correlators exhibit infinite logarithmic singularities at subleading orders, signaling renormalization effects from composite operators. These results provide a detailed, nonperturbative check of the AdS/CFT correspondence at large $N$ and large $rac{}{ olinebreak} olinebreak ext{λ}$ and sharpen the understanding of the OPE structure in the dual ${ m N}=4$ SYM theory.

Abstract

The graviton exchange diagram for the correlation function of arbitrary scalar operators is evaluated in anti-de Sitter space, AdS(d+1). This enables us to complete the computation of the 4-point amplitudes of dilaton and axion fields in IIB supergravity on AdS5 X S5. By the AdS/CFT correspondence, we obtain the 4-point functions of the marginal operators Tr (F^2+...) and Tr(F F~ +...) in N=4, d=4 SU(N) SYM at large N, large g^2 N. The short distance asymptotics of the amplitudes are studied. We find that in the direct channel the leading power singularity agrees with the expected contribution of the stress-energy tensor in a double OPE expansion. Logarithmic singularities occur in the complete 4-point functions at subleading orders.

Figures (6)

• Figure 1: Disconnected contribution to $\langle{\cal O}_\phi {\cal O}_C {\cal O}_\phi {\cal O}_C \rangle$. $a$: $O(N^4)$; $b$: $O(N^2)$.
• Figure 2: Connected $O(N^2)$ contributions to $\langle{\cal O}_\phi {\cal O}_C {\cal O}_\phi {\cal O}_C \rangle$.
• Figure 3: Disconnected $O(N^4)$ and $O(N^2)$ contributions to $\langle{\cal O}_C {\cal O}_C {\cal O}_C {\cal O}_C \rangle$.
• Figure 4: Connected $O(N^2)$ contributions to $\langle{\cal O}_C {\cal O}_C {\cal O}_C {\cal O}_C \rangle$.
• Figure 5: Definition of $D_{\Delta_1 \Delta_3 \Delta_2 \Delta_4}$.