The Epsilon-Expansion from Conformal Field Theory

TL;DR

This work demonstrates that analytic results of the $\epsilon$-expansion at the Wilson-Fisher fixed point can be derived from conformal field theory principles, specifically via multiplet recombination between $\varphi$ and $\varphi^3$ and a minimal axiom set. By analyzing three-point functions and conformal OPEs under a consistent operator mapping to the free theory, the authors reproduce the leading anomalous dimensions $y_{n,1}=\frac{1}{6}n(n-1)$ and $y_{1,2}=\frac{1}{108}$ for the scalar sector, and extend the framework to the $O(N)$ model with closed-form results. The approach shows the analytic bootstrap can capture WF data without perturbative Lagrangian techniques and generalizes to broader operator families and symmetries. These results lay groundwork for higher-order $\epsilon$ terms and potentially for fermionic theories via similar recombination constraints.

Abstract

Conformal multiplets of $φ$ and $φ^3$ recombine at the Wilson-Fisher fixed point, as a consequence of the equations of motion. Using this fact and other constraints from conformal symmetry, we reproduce the lowest nontrivial order results for the anomalous dimensions of operators, without any input from perturbation theory.