[Paper Review] The general QCD parametrization and the 1/N(c) expansion: A comparison

G. Dillon, G. Morpurgo|ArXiv.org|Nov 16, 2000

Particle physics theoretical and experimental studies3 references3 citations

TL;DR

This paper compares the general QCD parametrization (GP) and the 1/Nc expansion for describing hadron properties, showing that GP—derived directly from QCD with three colors—provides an exact spin-flavor parametrization, while the 1/Nc method, though widely used, lacks a rigorous derivation from real QCD. The key result is that higher-order terms in the GP hierarchy (e.g., β and γ parameters) are essential for accurate predictions, and their omission in the 1/Nc approach leads to incorrect results, especially in magnetic moments and electromagnetic transitions.

ABSTRACT

A comparison is presented of the two methods mentioned in the title for treating hadron properties in QCD.While the general parametrization is derived exactly from real QCD,the equivalence of the large N(c) description to real QCD with 3 colors,is questionable.The reason why in some cases the large N(c) method approximately works (while in others does not) is clarified.

Motivation & Objective

• To clarify why the 1/Nc expansion, despite its popularity, often fails to accurately describe hadron properties in real QCD with Nc=3.
• To demonstrate that the general QCD parametrization (GP) is an exact consequence of QCD's structure, based on spin-flavor factorization and general Lagrangian properties.
• To show that terms of second order in the GP hierarchy—previously neglected in 1/Nc approaches—are essential for fitting experimental data on magnetic moments and electromagnetic transitions.
• To expose the fundamental flaw in assuming that Nc=∞ QCD approximates Nc=3 QCD, especially when higher-order corrections are non-negligible.

Proposed method

• The GP method uses exact unitary transformations to map quark wave functions into the full QCD eigenstates, preserving quantum numbers and including configuration mixing via V|ϕB⟩.
• It derives a parametrized mass operator for baryons as a spin-flavor expansion: M_B = ⟨WB|parametrized mass|WB⟩, with terms ordered by decreasing coefficients.
• The method relies on the factorization of baryon wave functions into spatial (L=0) and spin-flavor parts, enabling exact parametrization without invoking SU(6) symmetry.
• It identifies a hierarchy of parameters (M0, B, C, D, E, a, b, c) in the mass and magnetic moment operators, with lower-order terms dominating.
• The 1/Nc method is contrasted by analyzing its omission of second-order terms (β, γ), which are shown to be crucial for fitting data.
• Explicit calculations of magnetic moments and Δ→pγ transitions are performed using both GP and 1/Nc frameworks to compare predictions.

Experimental results

Research questions

• RQ1 Why does the 1/Nc expansion sometimes fail to reproduce experimental hadron properties despite its widespread use?
• RQ2 What is the role of higher-order terms in the general QCD parametrization hierarchy, and why are they essential for accurate predictions?
• RQ3 How does the general QCD parametrization differ fundamentally from the 1/Nc approach in its derivation and physical basis?
• RQ4 To what extent is the assumption that Nc=3 QCD resembles Nc=∞ QCD justified, especially when higher-order corrections are significant?
• RQ5 Why does the observed p/n magnetic moment ratio deviate only 3% from -3/2, and what does this imply about the underlying quark dynamics?

Key findings

• The general QCD parametrization is an exact consequence of QCD's structure, derived from the Lagrangian and factorization of baryon wave functions, without assuming SU(6) symmetry.
• The 1/Nc method lacks a rigorous derivation from real QCD and assumes that Nc=3 QCD is similar to the Nc=∞ limit, a questionable assumption.
• Second-order terms in the GP hierarchy—specifically β and γ parameters—are essential for fitting the magnetic moments of p, n, and Δ, and the Δ→pγ transition matrix element.
• The experimental p/n ratio deviates only 3% from -3/2 not due to symmetry, but due to a near-cancellation between δ and (β+4γ), a fine-tuned effect requiring four independent parameters.
• The 1/Nc approach, which omits β and γ terms, fails to reproduce the data, as shown by the discrepancy in (Δ→pγ)₀ predictions when higher-order terms are neglected.
• The parametrized magnetic moment expression μ(B) = ⟨WB|∑perm[αQ₁+δ(Q₂+Q₃)]σ₁z + [βQ₁+γ(Q₂+Q₃)]σ₁z(σ₂·σ₃)|WB⟩ requires four independent parameters, contradicting the 1/Nc method’s use of only two or three.