Superconformal operators in N=4 super-Yang-Mills theory

TL;DR

This work analyzes twist-two conformal operators in ${\cal N}=4$ SYM, showing that all two-particle quasipartonic operators with different $SU(4)$ quantum numbers reside in a single supermultiplet whose components share a universal anomalous dimension. By projecting onto the light-cone and employing the collinear $SL(2|4)$ symmetry, the authors construct the complete operator basis, organize it into a supermultiplet, and compute the one-loop anomalous-dimension matrix, demonstrating its eigenstates and confirming a universal γ(j). They also relate the one-loop dilatation operator to the Hamiltonian of an integrable $SL(2|4)$ Heisenberg spin chain, emphasizing the deep connection between renormalization, supersymmetry, and integrability in ${\cal N}=4$ SYM, with implications for AdS/CFT and high-spin dynamics. The results provide a framework for extending integrability insights from maximal-helicity subsectors to the full set of quasipartonic operators and highlight the role of light-cone superfields in encoding the spectrum of anomalous dimensions.

Abstract

We construct, in the framework of the N=4 SYM theory, a supermultiplet of twist-two conformal operators and study their renormalization properties. The components of the supermultiplet have the same anomalous dimension and enter as building blocks into multi-particle quasipartonic operators. The latter are determined by the condition that their twist equals the number of elementary constituent fields from which they are built. A unique feature of the N=4 SYM is that all quasipartonic operators with different SU(4) quantum numbers fall into a single supermultiplet. Among them there is a subsector of the operators of maximal helicity, which has been known to be integrable in the multi-color limit in QCD, independent of the presence of supersymmetry. In the N=4 SYM theory, this symmetry is extended to the whole supermultiplet of quasipartonic operators and the one-loop dilatation operator coincides with a Hamiltonian of integrable SL(2|4) Heisenberg spin chain.

Figures (5)

• Figure 1: A diagram representing the supermultiplet of twist-two conformal operators in ${\cal N} = 4$ super-Yang-Mills theory. The Lorentz, flavor and conformal-spin indices are omitted for brevity. For explicit transformation rules, see appendix \ref{['SUSYofComponents']}.
• Figure 2: One loop diagrams contributing to the anomalous dimension of $[ \bar{\cal T}^{sg}_{jl} ]^\mu_{AB}$ in the light-cone gauge. The self-energy diagrams [the last two graphs] are multiplied by the symmetry factor $1/2$. The dashed and wiggly lines represent the scalar and gluon fields, respectively.
• Figure 3: One-loop diagrams for diagonal ${\cal T}^{qq, \hbox{\boldmath$\bar{6}$}} \to {\cal T}^{qq, \hbox{\boldmath$\bar{6}$}}$ transitions. The self-energy diagrams are multiplied by the combinatoric factor $1/2$. The solid line represent the gaugino.
• Figure 4: One-loop transitions changing the particle content of operators and contributing to the off-diagonal elements of the mixing matrix.
• Figure 5: One-loop corrections to the scalar, gluon and gaugino propagators.