Higher spin 3-point functions in 3d CFT using spinor-helicity variables

Amin A. Nizami,Sachin Jain,Renjan Rajan John,Abhishek Mehta,Adithya Suresh

doi:10.1007/jhep09(2021)041

Amin A. Nizami, Sachin Jain + Show 3 more

Open Access

https://doi.org/10.1007/jhep09(2021)041

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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.

Highlights

The study of CFT correlation functions in momentum space was initiated systematically in [1, 2]
In appendix B we describe in detail our terminology of homogeneous and non-homogeneous contributions to a correlator and discuss how they differ from the usual splitting of a correlation function into transverse and longitudinal pieces
We will convert these expressions into the momentum space and see that the non-homogeneous contribution to the parity-odd correlator becomes a contact term

Summary

Introduction

The study of CFT correlation functions in momentum space was initiated systematically in [1, 2]. Our goal in this paper is to study 3-point correlators of scalar operators and conserved spin-s currents in d = 3 CFTs in spinor-helicity variables as well as in momentum space. In [2], parity-even 3-point momentum space correlators involving scalar operator and conserved currents up to spin-2 (stress tensor) were computed by solving the conformal Ward identities (CWI). These correlators were later obtained using weight-shifting operators in [11, 12]. Upon converting spinor-helicity answers to momentum space, we see that the results for correlators involving spin-s currents can be expressed in terms of some simple conformally invariant conserved structures.

Conformal correlators in spinor-helicity variables

Conformal generators

Two-point functions

Three-point functions: general discussion

Homogeneous and non-homogeneous solutions

Degeneracy structure

Three-point functions: explicit solutions in spinor-helicity variables

JsJsJs

JsJsJs for general spin: the homogeneous part

Js1 JsJs

Conformal correlators in momentum space

Two point function