The structure of the spectrum of anomalous dimensions in the N-vector model in 4-epsilon dimensions

TL;DR

This work analyzes the one-loop spectrum of anomalous dimensions for composite operators in the $N$-vector model in $d=4-\epsilon$ dimensions, framing the problem around the spectrum-generating operator $V_N$ that acts on the conformal-invariant operator space. It develops an explicit conformal representation on the operator algebra, derives exact results for $n\le 3$ fields and $l\le 5$ gradients, and explores the large-$N$ limit where the spectrum simplifies to products of derivative polynomials acting on $\phi^2$ with a spherical-model-like scaling. The spectrum is found to be highly structured for simple sectors, with rational ground-state eigenvalues and a large degenerate zero-eigenvalue subspace that grows with spin, while in more complex sectors the spectrum is typically irrational and dominated by operator mixing, unlike the algebraic structure seen in two dimensions. A local interaction interpretation and a Laughlin-like construction for zero modes illuminate deeper connections to quantum Hall-like physics on a curved configuration space, though many questions remain about higher-loop lifting and the full operator-content complexity.

Abstract

In a recent publication we have investigated the spectrum of anomalous dimensions for arbitrary composite operators in the critical N-vector model in 4-epsilon dimensions. We could establish properties like upper and lower bounds for the anomalous dimensions in one-loop order. In this paper we extend these results and explicitly derive parts of the one-loop spectrum of anomalous dimensions. This analysis becomes possible by an explicit representation of the conformal symmetry group on the operator algebra. Still the structure of the spectrum of anomalous dimensions is quite complicated and does generally not resemble the algebraic structures familiar from two dimensional conformal field theories.