Scattering in anti–de Sitter space and operator product expansion

Abstract

We develop a formalism to evaluate generic scalar exchange diagrams in anti--de Sitter $(\mathrm{AdS}{)}_{d+1}$ relevant for the calculation of four-point functions in AdS conformal field theory (CFT) correspondence. The result may be written as an infinite power series of functions of cross ratios. Logarithmic singularities appear in all orders whenever the dimensions of involved operators satisfy certain relations. We show that the ${\mathrm{AdS}}_{d+1}$ amplitude can be written in a form recognizable as the conformal partial wave expansion of a four-point function in ${\mathrm{CFT}}_{d}$ and identify the spectrum of intermediate operators. We find that, in addition to the contribution of the scalar operator associated with the exchanged field in the ${\mathrm{AdS}}_{d+1}$ diagram, there are also contributions of some other operators which may possibly be identified with two-particle bound states in ${\mathrm{AdS}}_{d+1}.$ The ${\mathrm{CFT}}_{d}$ interpretation also provides a useful way to ``regularize'' the logarithms appearing in ${\mathrm{AdS}}_{d+1}$ amplitude.