Implications of conformal invariance in momentum space

TL;DR

This work develops a momentum-space framework for conformal field theory correlators, introducing a minimal tensor-decomposition that reduces any 3-point function to a small set of scalar form factors and solving the resulting conformal Ward identities via triple-$K$ integrals. It provides a systematic recipe to derive the full correlators (including transverse and trace Ward identities) from a conserved, traceless, transverse-traceless sector, with explicit results for scalar, TJJ, and TT O cases and detailed examples in $d=3$. The methodology unifies the scalar Appell/$F_4$ structure with the tensor decomposition, yielding unique solutions up to a handful of primary constants that are fixed by 2-point normalisations, and clarifies regularisation and anomaly issues in even dimensions. The paper further extends the formalism to helicity-based representations and discusses prospects for higher-point functions, offering a comprehensive toolkit for momentum-space CFT analyses and holographic cosmology applications.

Abstract

We present a comprehensive analysis of the implications of conformal invariance for 3-point functions of the stress-energy tensor, conserved currents and scalar operators in general dimension and in momentum space. Our starting point is a novel and very effective decomposition of tensor correlators which reduces their computation to that of a number of scalar form factors. For example, the most general 3-point function of a conserved and traceless stress-energy tensor is determined by only five form factors. Dilatations and special conformal Ward identities then impose additional conditions on these form factors. The special conformal Ward identities become a set of first and second order differential equations, whose general solution is given in terms of integrals involving a product of three Bessel functions (`triple-K integrals'). All in all, the correlators are completely determined up to a number of constants, in agreement with well-known position space results. We develop systematic methods for explicitly evaluating the triple-K integrals. In odd dimensions they are given in terms of elementary functions while in even dimensions the results involve dilogarithms. In some cases, the triple-K integrals diverge and subtractions are necessary and we show how such subtractions are related to conformal anomalies. This paper contains two parts that can be read independently of each other. In the first part, we explain the method that leads to the solution for the correlators in terms of triple-K integrals and how to evaluate these integrals, while the second part contains a self-contained presentation of all results. Readers interested only in results may directly consult the second part of the paper.