AdS/CFT 4-point functions: How to succeed at z-integrals without really trying
Eric D'Hoker, Daniel Z. Freedman, Leonardo Rastelli
TL;DR
This work introduces a streamlined method to evaluate the z-integral in AdS exchange diagrams by recasting it as a simple differential equation for a scale-invariant function after conformal inversion. The approach reproduces known results for scalar, vector, and graviton exchanges and yields new results for massive vector exchange, with a striking finding that, for AdS5×S5, many exchanges reduce to finite sums of scalar quartic graphs due to terminating series. The authors also extend the framework to higher-point functions, showing that the corresponding z-integrals admit polynomial solutions under the AdS5×S5 symmetry constraints. Overall, the method significantly simplifies AdS/CFT amplitude calculations and provides structural insight into when amplitudes condense to finite sums.
Abstract
A new method is discussed which vastly simplifies one of the two integrals over AdS(d+1) required to compute exchange graphs for 4-point functions of scalars in the AdS/CFT correspondence. The explicit form of the bulk-to-bulk propagator is not required. Previous results for scalar, gauge boson and graviton exchange are reproduced, and new results are given for massive vectors. It is found that precisely for the cases that occur in the AdS(5) X S(5) compactification of Type IIB supergravity, the exchange diagrams reduce to a finite sum of graphs with quartic scalar vertices. The analogous integrals in n-point scalar diagrams for n>4 are also evaluated.