Revisiting the dilatation operator of the Wilson-Fisher fixed point

TL;DR

This work demonstrates that the order-$\varepsilon$ dilatation operator at the Wilson-Fisher fixed point can be fixed by symmetry alone. Starting from $D(\varepsilon)=D_0+\varepsilon D_1$ with $[J_0,D_1]=0$, the authors show that $H_{ij}=\sum_{\ell}a(\ell)P_{\ell}$ and, leveraging recent CFT results that higher-spin currents have vanishing anomalous dimensions at order $\varepsilon$, all $a(\ell>0)=0$, leaving $H_{12}=\tfrac{1}{2}P_0$. The harmonic-action formalism and an $SU(1,1)$ subsector lifting confirm this structure and reproduce known anomalous dimensions, while discussions of multiplet recombination ensure consistency with conformal representation theory. The approach extends naturally to the $O(N)$ model and suggests pathways to higher-order corrections, evanescent-sector issues, and applications to other fixed points. Overall, the paper provides a symmetry-centric derivation of the WF spectrum at order $\varepsilon$ that matches perturbative results and aligns with bootstrap insights.

Abstract

We revisit the order $\varepsilon$ dilatation operator of the Wilson-Fisher fixed point obtained by Kehrein, Pismak, and Wegner in light of recent results in conformal field theory. Our approach is algebraic and based only on symmetry principles. The starting point of our analysis is that the first correction to the dilatation operator is a conformal invariant, which implies that its form is fixed up to an infinite set of coefficients associated with the scaling dimensions of higher-spin currents. These coefficients can be fixed using well-known perturbative results, however, they were recently re-obtained using CFT arguments without relying on perturbation theory. Our analysis then implies that all order-$\varepsilon$ scaling dimensions of the Wilson-Fisher fixed point can be fixed by symmetry.