Anomalous dimensions of spinning operators from conformal symmetry

TL;DR

This work uses conformal symmetry alone to determine the leading anomalous dimensions of a large class of spinning operators at Wilson–Fisher–type fixed points, avoiding any Lagrangian input. Central to the method is conformal multiplet recombination, which yields two independent matching conditions from four- and five-point functions that fix the spectrum as the theory deforms from the free field limit. The authors obtain explicit analytic expressions for the first-order and, in many cases, second-order anomalous dimensions for operators $\mathcal{O}_{p,\ell}$ of arbitrary spin, including the weakly broken higher-spin currents, and show consistency with known results at $d=4-\epsilon$, $d=3-\epsilon$, and multicritical dimensions $d_m$. The results extend the bootstrap program beyond scalar operators, provide new data for spinning families, and suggest avenues to connect with crossing symmetry and Mellin-space approaches.

Abstract

We compute, to the first non-trivial order in the $ε$-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 $ε\to 0$ 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.