Momentum space spinning correlators and higher spin equations in three dimensions

TL;DR

The paper computes explicit momentum-space correlators for scalars and spinning operators in three-dimensional free bosonic and free fermionic CFTs, including parity-even and parity-odd structures. It develops a parity-odd projector basis and employs momentum-inversion and Schouten identities to efficiently evaluate loop integrals, enabling closed-form expressions for several three- and four-point functions, as well as a five-point scalar function. It then explores higher-spin Ward identities in momentum space, showing that some three-point spinning correlators can be determined without conformal invariance, while four-point HS constraints require conformal input for full resolution; the results are checked against direct computations. The work highlights the utility of HS equations in momentum space as algebraic constraints and points to future directions, including interacting CS-matter theories and potential double-copy structures in CFT correlators.

Abstract

In this article, we explicitly compute in momentum space the three and four-point correlation functions involving scalar and spinning operators in the free bosonic and the free fermionic theory in three dimensions. We also evaluate the five-point function of the scalar operator in the free bosonic theory. We discuss techniques which are more efficient than the usual PV reduction to evaluate one loop integrals. Our techniques can be easily generalised to momentum space correlators of complicated spinning operators and to higher point functions. The three dimensional fermionic theory has the interesting feature that the scalar operator $\barψψ$ is odd under parity. To account for this, we develop a parity odd basis which is useful to write correlation functions involving spinning operators and an odd number of $\barψψ$ operators. We further study higher spin (HS) equations in momentum space which are algebraic in nature and hence simpler than their position space counterparts. We use them to solve for three-point functions involving spinning operators without invoking conformal invariance. However, at the level of four-point functions, solving the HS equation requires additional constraints that come from conformal invariance and we could only verify that our explicit results solve the HS equation.