Anomalous dimensions of spinning operators from conformal symmetry

TL;DR

This paper calculates anomalous dimensions of spinning operators in scalar field theories using conformal symmetry constraints, without relying on a Lagrangian, and provides both known and new results across various dimensions.

Contribution

It introduces a bootstrap-based method to compute anomalous dimensions of spinning operators in arbitrary dimensions without reference to a specific Lagrangian.

Findings

Reproduces known anomalous dimensions for higher-spin currents.

Provides new anomalous dimensions for other spinning operators.

Demonstrates the method's independence from Lagrangian or equations of motion.

Abstract

We compute, to the first non-trivial order in the $\epsilon$-expansion of a perturbed scalar field theory, the anomalous dimensions of an infinite class of primary operators with arbitrary spin $\ell=0,1,..$, including as a particular case the weakly broken higher-spin currents, using only constraints from conformal symmetry. Following the bootstrap philosophy, no reference is made to any Lagrangian, equations of motion or coupling constants. Even the space dimensions d are left free. The interaction is implicitly turned on through the local operators by letting them acquire anomalous dimensions. When matching certain four-point and five-point functions with the corresponding quantities of the free field theory in the $\epsilon\to 0$ limit, no free parameter remains. It turns out that only the expected discrete d values are permitted and the ensuing anomalous dimensions reproduce known results for the weakly broken higher-spin currents and provide new results for the other spinning operators.