High-gradient operators in the N-vector model

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 + ǫ 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 + ǫ expansion can lead to erroneous interpretations for high-gradient operators. While this makes us cautious against over-interpreting such expansions (either 2 + ǫ or 1/N ), the stability problem in the N -vector model persists also in first order in 1/N below three dimensions.

Integrability of Three-Particle Evolution Equations in QCD

“…The advantage of the ' ψ-representation' is that in this basis the latter condition is simply shift-invariance [13]. Therefore, eigenfunctions of two-particle Hamiltonians are given by simple powers ψ l = (z i − z k ) l instead of Jacobi polynomials in standard variables [14].…”

Stability Problem in theO(N)Nonlinear Sigma Model

“…Moreover, the pure combinatorical analysis shows that the k-th order term in the series (14) behaves as s k+1 at large s. Thus to obtain the answer for x s which were sensible for large sǫ/N one need sum up all order corrections (see for further discussion Refs. [17,18]). Even taking into account the recent progress in the higher order calculations [12] the feasibility of this program causes the great doubts.…”

High-gradient operators in thepsl(2|2)Gross–Neveu model

“…Atypical irreducibles of Lie superalgebras can form indecomposables. If one is not interested in the precise form in which such indecomposables are built from their constituents, all tensor products of finite-dimensional g representations may be determined by restricting the factors to the even subalgebra, tensoring the associated g (0) representations and combining the resulting products back into representations of g. The first and last step require no more than our decomposition formulas (4) and (5). The tensor products of irreducibles, including the indecomposable structures, have been worked out in [11].…”

“…From these formulas one can determine the branching functions into representations of the superalgebra g with the help of Eqs. (4) and (5). For the first few levels, the resulting decomposition of the vacuum character χ 0 reads…”

“…In the existing literature on the subject, the predominant attitude is to consider the RG-relevance of high-gradient operators as a bug or an artifact, see e.g. [4,5]. Polyakov, on the other hand, has argued [6] that it could be a desirable feature in order to establish a general pattern of dualities between NLSMs and WZNW models.…”