Helicity basis for three-dimensional conformal field theory

TL;DR

This work introduces a helicity-based basis for spinning three-point functions in 3d CFTs, leveraging the conformal-invariance of helicity to diagonalize OPE data and crossing. Using Euclidean and Lorentzian inversion formulas, it computes mean-field theory OPE data in the helicity basis and then applies the framework to AdS$_4$/CFT$_3$, deriving tree-level anomalous dimensions for conserved currents from bulk Yang–Mills interactions, including higher-derivative corrections. The large-twist/large-$n$ analysis reveals a precise match with flat-space gluon partial waves, strengthening the bulk-point/flat-space correspondence and yielding insights into helicity selection rules and potential simplifies for crossing kernels. The results provide analytic control over spinning correlators in 3d CFTs and illuminate how holographic and flat-space limits interrelate for higher-spin operators and gauge fields.

Abstract

Three-point correlators of spinning operators admit multiple tensor structures compatible with conformal symmetry. For conserved currents in three dimensions, we point out that helicity commutes with conformal transformations and we use this to construct three-point structures which diagonalize helicity. In this helicity basis, OPE data is found to be diagonal for mean-field correlators of conserved currents and stress tensor. Furthermore, we use Lorentzian inversion formula to obtain anomalous dimensions for conserved currents at bulk tree-level order in holographic theories, which we compare with corresponding flat-space gluon scattering amplitudes.

Figures (5)

• Figure 1: Conformal frame used for three-point functions: $\langle\mathcal{O}_1(0)\mathcal{O}_2(x)\mathcal{O}_3(\infty)\rangle$.
• Figure 2: Four-point function factorized into three-point functions.
• Figure 3: Spectrum of double-twist operators of the form $[JJ]_{n,J}$ and $[JT]_{n,J}$. Double circles indicate multiplicity: there is a single trajectory for $n=0$ and two for each $n\geq 1$.
• Figure 4: Witten diagram for $\langle VVVV\rangle$ with on-shell $t$-channel gluon exchange. Two even and one odd coupling can be used in each vertex; $u$-channel is similar with $1$ and $2$ swapped.
• Figure 5: Bulk-point kinematics in Lorentzian cylinder of AdS. $X_1$ and $X_2$ are at Lorentzian time $-\pi/2$, $X_3$ and $X_4$ are at Lorentzian time $\pi/2$, where particles are focused on the bulk-point $P$.