The critical O(N) CFT: Methods and conformal data

TL;DR

This work surveys the critical ${O(N)}$ vector model across $2<d<4$, compiling a comprehensive dataset of conformal data from perturbative expansions ($4-\varepsilon$, large $N$, and $2+\tilde{\varepsilon}$ NLσM) and exact $d=2$ results, then organizing them into an extensible Mathematica package ONdata.m. It details the conformal data framework—operator spectra, representations, OPE coefficients, and central charges—along with systematic methods (diagrammatic RG, dilatation operators, and analytic bootstrap) used to compute and cross-check dimensions and OPE data. The report emphasizes spectrum continuity, operator mixing, and multiplet recombination, linking perturbative results to nonperturbative bootstrap and large-$N$ insights, and it provides extensive numerical data for the Ising case ($N=1$) and general $N$. The ancillary data file enables programmatic access to hundreds of operator dimensions, OPE coefficients, and central charges, facilitating precise crosschecks and further developments in conformal bootstrap and RG analyses. Overall, the paper centralizes the O(N) CFT as a versatile testing ground for conformal methods, delivering a dense, cross-validated map of conformal data across dimensions and $N$.

Abstract

The critical $O(N)$ CFT in spacetime dimensions $2 < d < 4$ is one of the most important examples of a conformal field theory, with the Ising CFT at $N=1$, $2 \leq d < 4$, as a notable special case. Apart from numerous physical applications, it serves frequently as a concrete testing ground for new approaches and techniques based on conformal symmetry. In the perturbative limits - the $4-\varepsilon$ expansion, the large $N$ expansion and the $2+\tildeε$ expansion - a lot of conformal data have been computed over the years. In this report, we give an overview of the critical $O(N)$ CFT, including some methods to study it, and present a large collection of conformal data. The data, extracted from the literature and supplemented by many additional computations of order $\varepsilon$ anomalous dimensions, are made available through an ancillary data file.

Figures (2)

• Figure 1: Qualitative plot of the critical ${\mathrm O(N)}$ CFT as a continuous family of interacting conformal field theories parametrised by $d$ and $N$. Here we use $\sqrt{1-\frac{C_T}{C_{T,\mathrm{free}}}}$ for the central charge $C_T$ (see \ref{['eq:CTNscalars']}--\ref{['eq:OPEwithTandJ']}) as a proxy for the "coupling strength" and indicate the different expansion regimes. The non-linear sigma model in $2+{\tilde{\epsilon}}$ is a weakly coupled description of the theory using $N-1$ fields, hence in the limit ${\tilde{\epsilon}}\to0$, $\frac{C_T}{C_{T,\mathrm{free}}}$ approaches $\frac{N-1}{N}$ and not $1$ (c.f. \ref{['eq:CTnlsm']}). For $N>2$ one must have $d>2$ strictly. The extension down to $N=-2$ is not displayed.
• Figure 2: Anomalous dimensions of operators Op[S,4,$i$], $i=2,3,4,5$ as functions of $N$.