Higher spin 3-point functions in 3d CFT using spinor-helicity variables
Sachin Jain, Renjan Rajan John, Abhishek Mehta, Amin A. Nizami, Adithya Suresh
TL;DR
The paper develops a spinor-helicity formalism for three-dimensional CFTs to compute three-point functions involving scalar operators and conserved spin-$s$ currents, revealing a close link between parity-even and parity-odd structures in this language. By solving conformal Ward identities in spinor-helicity variables and then translating to momentum space, the authors express all relevant correlators in terms of a compact set of conformally invariant structures (including triple-$K$ integrals) and systematically separate homogeneous and non-homogeneous pieces, with contact terms identified for parity-odd sectors. Renormalization is carefully treated for divergent cases (e.g., $\langle JJO_{\Delta}\rangle$, $\langle TTO_{\Delta}\rangle$), and weight-shifting operators provide cross-checks and alternative derivations of many results. A set of momentum-space invariants is introduced to streamline the representation of higher-spin correlators, and several key observations—such as the parity-even/odd equivalence of homogeneous parts and the explicit structure of contact terms—emerge from the spinor-helicity framework. The work advances momentum-space CFT techniques, with implications for cosmology and holography, and opens avenues for extending to higher-point functions and non-conserved operators.
Abstract
In this paper we use the spinor-helicity formalism to calculate 3-point functions involving scalar operators and spin-$s$ conserved currents in general 3d CFTs. In spinor-helicity variables we notice that the parity-even and the parity-odd parts of a correlator are related. Upon converting spinor-helicity answers to momentum space, we show that correlators involving spin-$s$ currents can be expressed in terms of some simple conformally invariant conserved structures. This in particular allows us to understand and separate out contact terms systematically, especially for the parity-odd case. We also reproduce some of the correlators using weight-shifting operators.