[Paper Review] Baryon resonances in large N_c QCD

Nicolas Matagne|ArXiv.org|Jan 9, 2007

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper investigates baryon resonances in large-Nc QCD using the 1/Nc expansion and SU(6) symmetry, deriving mass relations and matrix elements for excited baryons. It identifies the [70, 1−] multiplet as a key state, computes spin-orbit matrix elements, and confirms consistency with known results from the literature, providing a systematic framework for understanding baryon spectroscopy in the large-Nc limit.

ABSTRACT

This thesis deals with the study of baryon spectra in the context of the $1/N_c$ expansion. The standard tool to study baryon properties is the constituent quark model. The results are naturally model dependent. The $1/N_c$ expansion generates a new perturbative approach to QCD, convenient for low momentum transfer. It provides a new theoretical method that is quantitative, systematic and predictive. In the first part of the thesis, the $1/N_c$ expansion is introduced as well as the baryon structure at large $N_c$. A summary of important results for ground-state baryons is provided. The second part of the thesis is devoted to excited baryon states. The symmetric orbital states are treated by analogy to the ground state. For mixed symmetric states, two approaches are presented. The traditional one starts from the decoupling of the wave function into an excited quark and a symmetric core. To make the problem tractable the wave function is treated approximately, justified by a Hartree scheme. This approach is applied to the study of the $[{\bf 70},\ell^+] (\ell=0,2)$ multiplets (nonstrange and strange cases) and of the $[{\bf 56},4^+]$ multiplet. An important physical result is the dependence of the spin dependent terms of the mass operator on the excitation energy. Recently we suggested a new approach based on a rigorous group theoretical treatment of the matrix elements of SU(4). No decoupling and no approximations are necessary. When applied to the $[{\bf 70},1^-]$ nonstrange multiplet, it is found that the leading corrections to the mass operator are of order $1/N_c$ instead of $N_c^0$, as predicted by the decoupling procedure.

Motivation & Objective

• To develop a systematic framework for understanding baryon resonances in the large-Nc limit of QCD.
• To extend the 1/Nc expansion to excited baryons, particularly focusing on the [70, 1−] and [56, 4+] multiplets.
• To compute matrix elements of spin-orbit and flavor-singlet operators for baryon states with non-trivial spin and flavor structure.
• To derive mass relations and test their consistency with experimental data and known results from the literature.
• To clarify the role of wave function symmetry and SU(4) generators in determining baryon mass splittings and mixing patterns.

Proposed method

• Uses the 1/Nc expansion to classify baryon states via irreducible representations of SU(6) and SU(3), focusing on symmetric and mixed-symmetric wave functions.
• Applies operator expansion techniques to decompose the baryon mass operator into spin, orbital, and flavor components, using SU(4) generators as tensor operators.
• Computes matrix elements of SU(4) generators and spin-orbit operators using explicit wave functions for [70, 1−] and [56, 4+] multiplets.
• Derives isoscalar factors and Clebsch-Gordan coefficients for SU(3) and SU(4) using group-theoretical techniques and tables of coefficients.
• Performs explicit calculations of matrix elements for Nc=3, comparing results with known values from the literature (e.g., Ref. [9]).
• Uses Hartree approximation to construct baryon wave functions and analyze decoupling patterns in the large-Nc limit.

Experimental results

Research questions

• RQ1 How do baryon masses scale with Nc in the large-Nc limit, and what are the leading-order contributions?
• RQ2 What is the role of spin-orbit and spin-flavor operators in splitting the [70, 1−] baryon multiplet?
• RQ3 How do matrix elements of SU(4) generators and flavor-singlet operators behave in the large-Nc limit?
• RQ4 What is the consistency of the derived mass relations with experimental data and previous theoretical results?
• RQ5 How do wave function symmetries (symmetric vs. mixed-symmetric) affect the structure and mass spectrum of excited baryons?

Key findings

• The matrix element ⟨N′₁/₂|ℓs|N′₁/₂⟩ for the [70, 1−] state with Nc=3 is −5/6, matching the value from Ref. [9].
• The matrix element ⟨N′₁/₂|ℓSc|N′₁/₂⟩ for the same state is −5/3, consistent with the literature.
• For the Δ(1232) state, the matrix element ⟨Δ₁/₂|ℓs|Δ₁/₂⟩ is found to be 1/3, and ⟨Δ₁/₂|ℓSc|Δ₁/₂⟩ is −4/3.
• All matrix elements of T₈ and t₈ operators are of order Nc, due to the scaling of isoscalar factors and coefficients with Nc.
• The [56, 4+] multiplet wave functions are constructed explicitly using spin and flavor states, with χλ and χρ states defined for S=1/2.
• The [70, 1−] wave function is expressed as a superposition of ψρ, ψλ with spin states χ₃/₂, and the matrix elements are computed using explicit group-theoretical decomposition.