Momentum space conformal three-point functions of conserved currents and a general spinning operator

TL;DR

This work advances the momentum-space conformal bootstrap by constructing three-point functions involving a general spinning operator and conserved currents (spin-1 and spin-2). The authors solve the conformal Ward-Takahashi identities by projecting onto helicity structures and expressing the results in terms of triple-$K$ integrals, with a differential operator linking integrals of different indices. For cases with no or one conserved current, they obtain closed forms without the operator, and they demonstrate a consistent, parameter-reduced structure for correlators with two currents. The framework lays groundwork for higher-point functions and cosmological applications in momentum space.

Abstract

We construct conformal three-point functions in momentum space with a general tensor and conserved currents of spin $1$ and $2$. While conformal correlators in momentum space have been studied especially in the connection with cosmology, correlators involving a tensor of general spin and scaling dimension have not been studied very much yet. Such a direction is unavoidable when we go beyond three-point functions because general tensors always appear as an intermediate state. In this paper, as a first step, we solve the Ward-Takahashi identities for correlators of a general tensor and conserved currents. In particular we provide their expression in terms of the so-called triple-$K$ integrals and a differential operator which relates triple-$K$ integrals with different indices. For several correlators, closed forms without the differential operator are also found.