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Large N QCD

Aneesh V. Manohar

TL;DR

The work surveys the large-$N$ expansion as a unifying framework for nonperturbative QCD, showing how planar diagrams dominate and how a semiclassical effective theory of color-singlet mesons and baryons emerges. It connects the four-dimensional QCD dynamics to tractable toy models (Gross-Neveu, ’t Hooft model), develops diagrammatic $N$-counting and master-field concepts, and applies these ideas to meson and baryon phenomenology. Key results include Zweig’s rule and eta' mass behavior, factorization in non-leptonic decays, and a contracted SU(2N_f) spin-flavor symmetry that yields predictive mass relations and coupling patterns with controlled $1/N$ corrections. The framework links QCD to chiral perturbation theory and Skyrme-type pictures, providing a coherent, testable set of predictions that align with much of the observed hadron spectrum and interactions while highlighting areas needing further refinement. Overall, the large-$N$ program offers a powerful, systematically improvable lens for interpreting low-energy QCD.

Abstract

1. Introduction 2. The Gross-Neveu Model 3. QCD 3.1 N-Counting Rules for Diagrams 3.1.1 U(1) Ghosts 3.2 The 't Hooft Model 3.3 $N$-Counting Rules for Correlation Functions 3.4 The Master Field 4. Meson Phenomenology 4.1 Zweig's Rule 4.2 Exotics 4.3 Chiral Perturbation Theory 4.4 Non-leptonic K Decay 4.5 $K-\bar K$ mixing 4.6 Axial U(1) and the eta' Mass 4.7 Resonances and 1/N 5 Baryons 5.1 N-Counting Rules for Baryons 5.2 The Non-Relativistic Quark Model 6 Spin-Flavor Symmetry for Baryons 6.1 Consistency Conditions 6.2 1/N Corrections 6.3 Solution of Consistency Conditions 7 Masses with SU(3) Breaking 8 Other Results for Baryons 9 Large N and Chiral Perturbation Theory

Large N QCD

TL;DR

The work surveys the large- expansion as a unifying framework for nonperturbative QCD, showing how planar diagrams dominate and how a semiclassical effective theory of color-singlet mesons and baryons emerges. It connects the four-dimensional QCD dynamics to tractable toy models (Gross-Neveu, ’t Hooft model), develops diagrammatic -counting and master-field concepts, and applies these ideas to meson and baryon phenomenology. Key results include Zweig’s rule and eta' mass behavior, factorization in non-leptonic decays, and a contracted SU(2N_f) spin-flavor symmetry that yields predictive mass relations and coupling patterns with controlled corrections. The framework links QCD to chiral perturbation theory and Skyrme-type pictures, providing a coherent, testable set of predictions that align with much of the observed hadron spectrum and interactions while highlighting areas needing further refinement. Overall, the large- program offers a powerful, systematically improvable lens for interpreting low-energy QCD.

Abstract

1. Introduction 2. The Gross-Neveu Model 3. QCD 3.1 N-Counting Rules for Diagrams 3.1.1 U(1) Ghosts 3.2 The 't Hooft Model 3.3 -Counting Rules for Correlation Functions 3.4 The Master Field 4. Meson Phenomenology 4.1 Zweig's Rule 4.2 Exotics 4.3 Chiral Perturbation Theory 4.4 Non-leptonic K Decay 4.5 mixing 4.6 Axial U(1) and the eta' Mass 4.7 Resonances and 1/N 5 Baryons 5.1 N-Counting Rules for Baryons 5.2 The Non-Relativistic Quark Model 6 Spin-Flavor Symmetry for Baryons 6.1 Consistency Conditions 6.2 1/N Corrections 6.3 Solution of Consistency Conditions 7 Masses with SU(3) Breaking 8 Other Results for Baryons 9 Large N and Chiral Perturbation Theory

Paper Structure

This paper contains 29 sections, 168 equations, 47 figures, 8 tables.

Table of Contents

  1. Large N QCD
  2. Introduction
  3. The Gross-Neveu Model
  4. QCD
  5. $N$-Counting Rules for Diagrams
  6. $U(1)$ Ghosts
  7. The 't Hooft Model
  8. $N$-Counting Rules for Correlation Functions
  9. The Master Field
  10. Meson Phenomenology
  11. Zweig's Rule
  12. Exotics
  13. Chiral Perturbation Theory
  14. Non-leptonic $K$ Decay
  15. $K-\overline K$ mixing
  16. ...and 14 more sections

Figures (47)

  • Figure 1: The four-Fermi vertex of the Gross-Neveu model. $a,b$ are flavor labels.
  • Figure 2:
  • Figure 3: Feynman rules for the Gross-Neveu Lagrangian eq. (\ref{['2.4']}). The $\psi$ field is represented by a solid line, and the auxiliary field $\sigma$ by a dashed line.
  • Figure 4: The graphs of fig. \ref{['fig:2']}(a,c) using the auxiliary field form of the Lagrangian.
  • Figure 5: Diagrammatic expansion for $L_{\rm eff}\left( \sigma \right)$.
  • ...and 42 more figures