Spin-Flavor Structure of Large N Baryons

TL;DR

This work develops a comprehensive operator-based $1/N_c$ expansion for large-$N_c$ baryons, deriving all two-body quark operator identities and a universal operator reduction rule that collapses the expansion to a finite, predictive set of operators. It applies the framework to axial currents, baryon masses, magnetic moments, and hyperon non-leptonic decays, for both two and three light flavors, including perturbative and completely broken SU(3) symmetry cases. The authors explicitly connect the quark-model and Skyrme-model realizations, establishing equivalences at leading order and detailing subleading differences, with explicit mappings between $G^{ia}$ and Skyrme operators $X^{ia}$. The results explain the observed spin-flavor patterns in baryons, yield controlled $1/N_c$ corrections, and provide a unified view of baryon structure across symmetry limits and breaking patterns.

Abstract

The spin-flavor structure of large N baryons is described in the 1/N expansion of QCD using quark operators. The complete set of quark operator identities is obtained, and used to derive an operator reduction rule which simplifies the 1/N expansion. The operator reduction rule is applied to the axial currents, masses, magnetic moments and hyperon non-leptonic decay amplitudes in the $SU(3)$ limit, to first order in $SU(3)$ breaking, and without assuming $SU(3)$ symmetry. The connection between the Skyrme and quark representations is discussed. An explicit formula is given for the quark model operators in terms of the Skyrme model operators to all orders in $1/\N$ for the two flavor case.

Figures (6)

• Figure 1: $SU(2F)$ representation for ground-state baryons. The Young tableau has $N_c$ boxes.
• Figure 2: Weight diagram for the $SU(3)$ flavor representation of the spin-${1 \over 2}$ baryons. The top of the weight diagram has baryons with zero strange quarks. The long side of the weight diagram contains ${1 \over 2}\left( N_c + 1 \right)$ weights. The numbers denote the multiplicity of the weights.
• Figure 3: Weight diagram for the $SU(3)$ flavor representation of the spin-${3 \over 2}$ baryons. The top of the weight diagram has baryons with zero strange quarks. The long side of the weight diagram contains ${1 \over 2}\left( N_c - 1 \right)$ weights. The numbers denote the multiplicity of the weights.
• Figure 4: Feynman diagrams depicting the insertion of a one-quark QCD operator on the $N_c$ quark lines of the baryon. Graphs (b) contain additional planar gluons, and are of the same order as (a).
• Figure 5: The hyperfine mass splittings within a baryon tower. Splittings at the bottom of the tower are of order $1/N_c$, whereas splittings at the top of the tower are of order $1$. There are $(N_c+1)/2$ energy levels in the flavor symmetry limit.