High-gradient operators in the N-vector model

TL;DR

This paper studies high-gradient operators in the N-vector model using 1/N-expansion, revealing potential errors in 2+eps expansions and confirming the stability problem persists below three dimensions.

Contribution

It establishes an asymptotic naive addition law for anomalous dimensions, highlighting limitations of 2+eps and 1/N expansions in analyzing high-gradient operators.

Findings

2+eps expansions may misinterpret high-gradient operators

Stability issues persist in the N-vector model below three dimensions

First-order 1/N-expansion confirms ongoing stability problems

Abstract

It has been shown by several authors that a certain class of composite operators with many fields and gradients endangers the stability of nontrivial fixed points in 2+eps expansions for various models. This problem is so far unresolved. We investigate it in the N-vector model in an 1/N-expansion. By establishing an asymptotic naive addition law for anomalous dimensions we demonstrate that the first orders in the 2+eps expansion can lead to erroneous interpretations for high--gradient operators. While this makes us cautious against over--interpreting such expansions (either 2+eps or 1/N), the stability problem in the N-vector model persists also in first order in 1/N below three dimensions.