C_{T} and C_{J} up to next-to-leading order in 1/N in the Conformally Invariant O(N) Vector Model for 2<d<4

TL;DR

The paper computes the next-to-leading order corrections in $1/N$ for the conformal central charges $C_{T}$ and $C_{J}$ at the IR fixed point of the conformally invariant $O(N)$ vector model in the range $2<d<4$. It employs an operator-product-expansion framework for the four-point function of $oldsymbol{\phi^{oldsymbol{eta}}}$, augmented by a conformally invariant skeleton-graph expansion with a scalar exchange, and uses the DEPP integral to extract the $1/N$ corrections to operator dimensions and three-point couplings. The authors obtain explicit expressions for the anomalous dimensions $oldsymbol{ ilde{eta}}$ and $oldsymbol{ ilde{eta}_o}$, the vertex coupling $g_{oldsymbol{eta}oldsymbol{eta}O}$, and the central charges $C_{T}$ and $C_{J}$ up to $oldsymbol{O(1/N)}$, finding that UV values exceed IR values, in line with generalized $C$- and $k$-theorems. The results agree with known $ ext{O}(N)$ results in various limits, are consistent with $oldsymbol{4-oldsymbol{ ilde{eta}}}$-expansion, and indicate that $C_{T}$ in $d=3$ is not a simple rational number at finite $oldsymbol{N}$, highlighting nontrivial finite-size and dimensional effects. These findings provide a nonperturbative probe of conformal data and offer benchmarks for higher-dimensional CFT generalizations of the $C$- and $k$-theorems.

Abstract

Using Operator Product Expansions and a graphical ansatz for the four-point function of the fundamental field φ^α(x) in the conformally invariant O(N) vector model, we calculate the next-to-leading order in 1/N values of the quantities C_{T} and C_{J}. We check the results against what is expected from possible generalisations of the C- and k-theorems in higher dimensions and also against known three-loop calculations in a O(N) invariant φ^{4} theory for d=4-ε.

Figures (3)

• Figure 1: The Graphical Expansion for $F_{f}(u,v)$
• Figure 2: The Skeleton Graph Expansion for $H(x,\eta)\,F^{\tilde{\eta}_{o}}(u,v)$
• Figure 3: $C_{T,1}$ and $C_{J,1}$ for $2<d<4$.