Mellin space bootstrap for global symmetry

TL;DR

This work develops and applies a Mellin-space analytic bootstrap for conformal field theories with $O(N)$ symmetry and cubic anisotropy, deriving crossing- and OPE-consistency constraints and solving them to obtain anomalous dimensions and OPE coefficients for operators quadratic in the fields, up to $O(\epsilon^3)$. It also analyzes the large-$N$ limit and the cubic-anisotropy case, delivering new OPE data and confirming known results where applicable. The approach yields explicit results for $\Delta_\phi$, scalar and higher-spin data in the $\epsilon$-expansion and provides leading $1/N$ corrections consistent with the literature, while highlighting simplifications that enable analytic progress. The results enhance analytic control over critical $O(N)$ models and their symmetry-breaking variants, with potential cross-checks against numerics and connections to AdS/CFT.

Abstract

We apply analytic conformal bootstrap ideas in Mellin space to conformal field theories with $O(N)$ symmetry and cubic anisotropy. We write down the conditions arising from the consistency between the operator product expansion and crossing symmetry in Mellin space. We solve the constraint equations to compute the anomalous dimension and the OPE coefficients of all operators quadratic in the fields in the epsilon expansion. We reproduce known results and derive new results up to $O(ε^3)$. For the $O(N)$ case, we also study the large $N$ limit in general dimensions and reproduce known results at the leading order in $1/N$.

Figures (1)

• Figure 1: Feynman diagrams with $\lambda_{ijkl}$ parameter. The first diagram on left is a two loop diagram. The second and third on the right are three loop diagrams. The colour indices are indicated at each vertex. The blobs denote a composite operator ($T$ or $J$) insertion.