Critical O(N) model to order $ε^4$ from analytic bootstrap

TL;DR

This work determines the full CFT data for leading twist operators in the critical $\mathrm{O}(N)$ model to fourth order in the $\epsilon$-expansion using large spin perturbation theory within a bootstrap framework. It extends prior Wilson-Fisher results by delivering new OPE coefficients and $\epsilon^4$ corrections to the central charges $C_T$ and $C_J$, and provides Padé-based predictions for central charges in strongly coupled 3d regimes. The results pass consistency checks with known diagrammatic and large-$N$ expansions and improve 3d predictions relative to earlier truncations, illustrating the power of the Froissart–Gribov inversion approach for analytic CFT data. The methods offer a unified route to high-order CFT data from crossing symmetry, with potential applications to other symmetry groups and dimensions.

Abstract

We compute, using the method of large spin perturbation theory, the anomalous dimensions and OPE coefficients of all leading twist operators in the critical $ O(N) $ model, to fourth order in the $ ε$-expansion. This is done fully within a bootstrap framework, and generalizes a recent result for the CFT-data of the Wilson-Fisher model. The anomalous dimensions we obtain for the $ O(N) $ singlet operators agree with the literature values, obtained by diagrammatic techniques, while the anomalous dimensions for operators in other representations, as well as all OPE coefficients, are new. From the results for the OPE coefficients, we derive the $ ε^4 $ corrections to the central charges $ C_T $ and $ C_J $, which are found to be compatible with the known large $ N $ expansions. Predictions for the central charge in the strongly coupled 3d model, including the 3d Ising model, are made for various values of $ N $, which compare favourably with numerical results and previous predictions.

Figures (1)

• Figure 1: Padé approximants for $C_T/C_{T,\mathrm{free}}$ in three dimensions. For $N=1,2$, the crosses show the values from the corresponding Padé approximants of lower orders, as given in table \ref{['tab:padeCT']}.