Conserved Currents, Consistency Relations and Operator Product Expansions in the Conformally Invariant O(N) Vector Model

TL;DR

This paper analyzes conserved currents and operator product expansions in the conformally invariant O(N) vector model, providing explicit expressions, a graphical evaluation method, and insights into the model's duality and critical parameters.

Contribution

It introduces an alternative graphical approach to four-point functions and determines key parameters of the O(N) model up to next-to-leading order in 1/N.

Findings

Explicit OPE expressions for four-point functions.

A graphical expansion method for evaluating correlators.

Next-to-leading order correction for the energy-momentum tensor normalization.

Abstract

We discuss conserved currents and operator product expansions (OPE's) in the context of a $O(N)$ invariant conformal field theory. Using OPE's we find explicit expressions for the first few terms in suitable short-distance limits for various four-point functions involving the fundamental $N$-component scalar field $\phi^{\alpha}(x)$, $\alpha=1,2,..,N$. We propose an alternative evaluation of these four-point functions based on graphical expansions. Requiring consistency of the algebraic and graphical treatments of the four-point functions we obtain the values of the dynamical parameters in either a free theory of $N$ massless fields or a non-trivial conformally invariant $O(N)$ vector model in $2<d<4$, up to next-to-leading order in a $1/N$ expansion. Our approach suggests an interesting duality property of the critical $O(N)$ invariant theory. Also, solving our consistency relations we obtain the next-to-leading order in $1/N$ correction for $C_{T}$ which corresponds to the normalisation of the energy momentum tensor two-point function.