General scalar exchange in AdS(d+1)

The scalar field exchange diagram for the correlation function of four scal ar operators is evaluated in anti-de Sitter space, AdS(d+1) The conformal d imensions Delta(i), i = 1,...,4 of the scalar operators and the dimension D elta of the exchanged field are arbitrary, constrained only to obey the uni tarity bound. Techniques similar to those developed earlier for gauge boson exchange are used, but results are generally more complicated. However, fo r integer Delta i, Delta, the amplitude can be presented as a multiple deri vative of a simple universal function. Results simplify if further conditio ns hold, such as the inequalities, Delta < Delta(1) + Delta(3) or Delta < D elta(2) + Delta(4). These conditions are satisfied, with < replaced by less than or equal to in Type IIB supergravity on AdS(5) x S-5 because of selec tion rules from SO(6) symmetry. A new form of interaction is suggested for the marginal case of the inequalities. The short distance asymptotics of th e amplitudes are studied. In the direct channel the leading singular term a grees with the double operator product expansion. Logarithmic singularities occur at sub-leading order in the direct channel but at leading order in t he crossed channel. When the inequalities above are violated, there are als o (log)(2) singularities in the direct channel. (C) 1999 Published by Elsev ier Science B.V. All rights reserved.