General Scalar Exchange in AdS_d+1

TL;DR

This paper evaluates scalar exchange diagrams in AdS space for four scalar operators with arbitrary dimensions, deriving simplified expressions under certain conditions and analyzing short-distance behavior and singularities.

Contribution

It provides a general method for calculating scalar exchange amplitudes in AdS with arbitrary operator dimensions, extending previous gauge boson exchange techniques.

Findings

Amplitude simplifies for integer dimensions and specific inequalities.

Short-distance singularities match the double operator product expansion.

Logarithmic and squared logarithmic singularities are identified in different channels.

Abstract

The scalar field exchange diagram for the correlation function of four scalar operators is evaluated in anti-de Sitter space, $AdS_{d+1}$. The conformal dimensions $\Delta_i$, $i=1,...,4$ of the scalar operators and the dimension $\Delta$ of the exchanged field are arbitrary, constrained only to obey the unitarity bound. Techniques similar to those developed earlier for gauge boson exchange are used, but results are generally more complicated. However, for integer $\Delta_i, \Delta$, the amplitude can be presented as a multiple derivative of a simple universal function. Results simplify if further conditions hold, such as the inequalities, $\Delta< \Delta_1+\Delta_3$ or $\Delta<\Delta_2+\Delta_4$. These conditions are satisfied, with $<$ replaced by $\le$, in Type IIB supergravity on $AdS_5\times S_5$ because of selection rules from SO(6) symmetry. A new form of interaction is suggested for the marginal case of the inequalities. The short distance asymptotics of the amplitudes are studied. In the direct channel the leading singular term agrees with the double operator product expansion. Logarithmic singularities occur at sub-leading order in the direct channel but at leading order in the crossed channel. When the inequalities above are violated, there are also $(\log)^2$ singularities in the direct channel.