Negative anomalous dimensions in N=4 SYM

TL;DR

This paper investigates one-loop anomalous dimensions of operators in N=4 SYM at finite N_c, revealing negative anomalous dimensions, operator mixing, and a connection to symmetric polynomials, with implications for spectral degeneracies.

Contribution

It provides a detailed analysis of finite N_c corrections to operator spectra, including the lifting of degeneracies and the emergence of negative anomalous dimensions, and establishes a link to symmetric polynomials.

Findings

Large N_c zero modes acquire non-positive anomalous dimensions at order 1/N_c^2

Operators of length 2N_c have increasingly negative dimensions as N_c grows

A correspondence between zero modes and symmetric polynomials of Mandelstam variables

Abstract

We elucidate aspects of the one-loop anomalous dimension of $so(6)$-singlet multi-trace operators in $\mathcal{N}=4\ SU(N_c)$ SYM at finite $N_c$. First, we study how $1/N_c$ corrections lift the large $N_c$ degeneracy of the spectrum, which we call the operator submixing problem. We observe that all large $N_c$ zero modes acquire non-positive anomalous dimension starting at order $1/N_c^2$, and they mix only among the operators with the same number of traces at leading order. Second, we study the lowest one-loop dimension of operators of length equal to $2N_c$. The dimension of such operators becomes more negative as $N_c$ increases, which will eventually diverge in a double scaling limit. Third, we examine the structure of level-crossing at finite $N_c$ in view of unitarity. Finally we find out a correspondence between the large $N_c$ zero modes and completely symmetric polynomials of Mandelstam variables.