The minimal conformal O(N) vector sigma model at d=3

TL;DR

This work analyzes the minimal conformal $O(N)$ vector sigma model in $d=3$, focusing on the auxiliary field $\alpha$ and its $n$-point functions at $O(1/N)$. Employing a conformal skeleton expansion, a Legendre transform from the free $O(N)$ current $J$ to $\alpha$, and conformal bootstrap, the authors determine the fixed-point coupling $z$ and anomalous dimensions, obtaining explicit leading results and showing the special simplifications at $d=3$. The $\alpha$ four-point function at the interacting fixed point is computed and shown to be logarithm-free, decomposing into exchange amplitudes of currents $(2\phi)_{l,t}$ and composites $(2\alpha)_{l,t}$, with composite dimensions $\delta((n\alpha)_{l,t})=n(2+\eta(\alpha))+l+2t+O(1/N^2)$. The analysis reveals that all $n$-point functions up to $O(1/N)$ are free of logarithms and that certain operator dimensions simplify (e.g., the scalar family $ (n\alpha)_{0,0}$ becomes linear in $n$ at $d=3$), providing a clear, log-free structure supportive of AdS$_4$/CFT$_3$ and higher-spin dual descriptions.

Abstract

For the minimal O(N) sigma model, which is defined to be generated by the O(N) scalar auxiliary field alone, all n-point functions, till order 1/N included, can be expressed by elementary functions without logarithms. Consequently, the conformal composite fields of m auxiliary fields possess at the same order such dimensions, which are m times the dimension of the auxiliary field plus the order of differentiation.