On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space

TL;DR

<3-5 sentence high-level summary> The paper analyzes scalar 4-point functions in momentum space under conformal Ward identities in general $d$, revealing that dual conformal symmetry (DC) together with conformal invariance pins the solution to a unique hypergeometric form. It derives exact DCC solutions using dual-conformal ans"atze, showing these can be expressed as Appell $F_4$ functions or equivalently as 3K integrals, and demonstrates a deep link between 3- and 4-point correlators (as in ladder diagrams) that collapses to the box diagram and melons in $d=4$. In a complementary analysis, it studies high-energy fixed-angle limits, obtaining factorized solutions described by Lauricella $F_C$ functions and generalized 4K integrals, capturing the asymptotic behavior with fixed $-t/s$. Overall, the work establishes a unified hypergeometric framework for CWI solutions in momentum space and highlights their connections to perturbative diagrams and higher-point generalizations relevant to conformal bootstrap and holographic contexts.

Abstract

We discuss specific hypergeometric solutions of the conformal Ward identities (CWI's) of scalar 4-point functions of primary fields in momentum space, in $d$ spacetime dimensions. We determine such solutions using various dual conformal ansätze (DCA's). We start from a generic dual conformal correlator, and require it to be conformally covariant in coordinate space. The two requirements constrain such solutions to take a unique hypergeometric form. They describe correlators which are at the same time conformal and dual conformal in any dimension. These specific ansätze also show the existence of a link between 3- and 4-point functions of a CFT for such class of exact solutions, similarly to what found for planar ladder diagrams. We show that in $d=4$ only the box diagram and its melonic variants, in free field theory, satisfies such conditions, the remaining solutions being nonperturbative. We then turn to the analysis of some approximate high energy fixed angle solutions of the CWI's which also in this case take the form of generalized hypergeometric functions. We show that they describe the behaviour of the 4-point functions at large energy and momentum transfers, with a fixed $-t/s$. The equations, in this case, are solved by linear combinations of Lauricella functions of 3 variables and can be rewritten as generalized 4K integrals. In both cases the CWI's alone are sufficient to identify such solutions and their special connection with generalized hypergeometric systems of equations.

Figures (2)

• Figure 1: The box with its dual (left) and its higher scaling version (right). While the first is conformal for $d\ne2$ in ordinary conformal coordinates and for $d=4$ in dual coordinates, the right one is not conformal in coordinate and dual coordinate space at the same time.
• Figure 2: The two loop box digram with its dual. This diagram is not conformal in coordinate and dual coordinate space at the same time.