Constraining conformal field theory with higher spin symmetry in four dimensions

TL;DR

This work analyzes how conformal invariance plus higher spin symmetry constrain four-dimensional CFT correlation functions of the stress-energy tensor. Using a biharmonic scalar bi-field $V_2$ within a Global Conformal Invariance (GCI) framework, it shows all $n$-point functions have at most double pole singularities and that the 4-, 5-, and 6-point functions reduce to linear combinations of free-field structures for scalar, fermion, and Maxwell fields. The findings provide strong evidence that HS-symmetric 4D CFTs are effectively free, with a universal set of coefficients across $n$-point functions, and they outline open questions about the generality of rationality, extension to all $n$, and odd-spin sectors. Overall, the paper advances a compelling argument that higher-spin constraints enforce a free-theory-like behavior in 4D CFTs under the GCI paradigm.

Abstract

We analyze the constraints on the general form and the singularity structure of the correlation functions of the symmetric, traceless and conserved stress-energy tensor implied by conformal invariance and higher spin symmetry in four dimensions. In particular, we show that all these correlation functions will have at most double pole singularities. We then compute the 4-, 5- and 6-point functions of the stress-energy tensor and find that they are linear combinations of the three free field expressions (scalar, fermion and Maxwell field). This is a strong indication that all such theories are essentially free.