One-Loop QCD Spin Chain and its Spectrum

TL;DR

This work analyzes the one-loop renormalization of gauge-invariant operators in large-$N_c$ QCD by mapping the planar dilatation operator to a spin-chain Hamiltonian. Using conformal symmetry and a complete operator basis, the authors derive the full one-loop anomalous-dimension matrix and show it contains a large integrable sector governed by a chiral $SO(4,2)$ spin chain; the ground state corresponds to an anti-ferromagnetic XXX$_1$ chain and gapless excitations split into chiral (relativistic spinons) and anti-chiral (non-relativistic) types. Derivative operators extend the integrable sector to an $SO(4,2)$ chain with three Bethe-root types, yielding BMN-like states and a rich spectrum including open chains with integrable boundaries. The results connect to prior light-cone and $ rm{N}=4$ SYM analyses, providing a non-supersymmetric benchmark for the QCD string dynamics and highlighting spin-separation phenomena in the anti-ferromagnetic background. The findings offer a structured framework to study the QCD operator spectrum and its potential string-theoretic dual through integrable techniques.

Abstract

We study the renormalization of gauge invariant operators in large Nc QCD. We compute the complete matrix of anomalous dimensions to leading order in the 't Hooft coupling and study its eigenvalues. Thinking of the mixing matrix as the Hamiltonian of a generalized spin chain we find a large integrable sector consisting of purely gluonic operators constructed with self-dual field strengths and an arbitrary number of derivatives. This sector contains the true ground state of the spin chain and all the gapless excitations above it. The ground state is essentially the anti-ferromagnetic ground state of a XXX1 spin chain and the excitations carry either a chiral spin quantum number with relativistic dispersion relation or an anti-chiral one with non-relativistic dispersion relation.

Figures (5)

• Figure 1: Closed and open "spin chains" corresponding to purely gluonic operators or mesonic operators respectively.
• Figure 2: Renormalization of a four gluon composite operator (denoted by a cross) in double line notation. To the left is depicted a diagram connecting two adjacent gluon legs containing a closed color loop. To the right is shown a diagram connecting two non-adjacent gluon legs, not containing any color loop and thus subleading in $1/N_{\mathrm{c}}$.
• Figure 3: Typical 't Hooft diagrams contributing to the one-loop scaling dimension of a four gluon operator. There are also "degenerate" diagrams where one of the vertices is connected directly to the gluon field appearing in a covariant derivative or a commutator inside the operator.
• Figure 4: Reduction of the distinguished Dynkin diagram of $SU(4|2,2)$ to $SU(2,2)$. Above the diagrams we have indicated the Dynkin labels of the spin representation and below the excitation numbers of each simple root.
• Figure 5: Spectrum of anomalous dimensions constructed over the chiral anti-ferromagnetic ground state. An identical tower of excitations can be constructed over the anti-chiral anti-ferromagnetic ground state.