[Paper Review] On the structure of the pion: A QCD--inspired point of view

Hans–Christian Pauli|ArXiv.org|Nov 5, 2001

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper proposes a QCD-inspired light-cone Hamiltonian model using the up-down (↑↓) framework to describe the pion as a highly relativistic bound state of quarks and antiquarks. By applying the method of iterated resolvents to derive an effective interaction, the authors compute the pion's light-cone wave function analytically and derive its mean-square radius as 0.33 fm—half the empirical value—revealing that pion constituents move at relativistic speeds due to strong binding, challenging non-relativistic potential models.

ABSTRACT

The effective interaction between a quark and an anti-quark as obtained previously with by the method of iterated resolvents is replaced by the so called up-down-model and applied to flavor off-diagonal mesons including the pion. The only free parameters are those of canonical quantum chromo-dynamics (QCD), particularly the coupling constant and the masses of the quarks. The so obtained light-cone wave function can be used to calculate the pion's form factor, particularly its mean-square radius can be computed analytically. The results allow for the exciting conclusion that the pion is built by highly relativistic constituents, in strong contrast to composite systems like atoms or nuclei with non-relativistic constituents.

Motivation & Objective

• To develop a covariant, QCD-inspired model for the pion's internal structure using a simplified but symmetry-preserving approach.
• To address the failure of non-relativistic potential models in describing the pion by incorporating relativistic dynamics from the start.
• To compute the pion’s light-cone wave function and form factor analytically using only canonical QCD parameters: quark masses and the strong coupling constant.
• To demonstrate that the pion is an ultra-strongly bound system where binding energy accounts for over 80% of the constituent quark mass.
• To provide a unified framework capable of describing all mesons—from pions to bottomonia—within a single relativistic, covariant formalism.

Proposed method

• The study employs the light-cone Hamiltonian derived from the QCD Lagrangian in the light-cone gauge, excluding zero modes.
• It uses the method of iterated resolvents to systematically reduce the full many-body Fock space problem to an effective one-body problem in the q̄q sector.
• The effective interaction is derived by expressing higher Fock-state amplitudes (e.g., gg, q̄qg) as functionals of the q̄q wave function, preserving all Lagrangian symmetries.
• The wave function is transformed from light-cone variables (x, k⊥) to momentum space (p) via a change of variables, enabling numerical solution of the integral equation.
• The resulting wave function φ(p) is fitted to a closed-form analytical expression Φa(p) = N(1 + p²/pa²)⁻² with pa = 1.338m.
• The pion’s form factor and mean-square radius are computed via overlap integrals (Eq. 1), leading to an analytical expression for ⟨r²⟩ (Eq. 23).

Experimental results

Research questions

• RQ1 Can a simplified QCD-inspired model reproduce the pion’s mass and structure using only canonical QCD parameters?
• RQ2 Why do non-relativistic potential models fail to describe the pion, and what relativistic features are essential?
• RQ3 What is the intrinsic momentum distribution of quarks inside the pion, and how relativistic are they?
• RQ4 How does the binding energy in the pion compare to the constituent quark mass, and what does this imply for its structure?
• RQ5 Can a single, unified framework describe both light and heavy mesons starting from QCD principles?

Key findings

• The pion’s mean-square radius is computed as ⟨r²⟩¹ᐟ² = 0.33 fm, which is half the empirical value of 0.67 fm, indicating a need for further corrections or higher Fock states.
• The effective Bohr momentum of the quark-antiquark constituents is pa = 1.338m, implying their average momentum exceeds their mass by 40%, confirming highly relativistic motion.
• Over 80% of the constituent quark mass is attributed to binding energy, a feature unique to the pion among known hadrons.
• The light-cone wave function ψ(x, k⊥) is derived in closed form (Eq. 22), enabling analytical computation of the form factor and structure functions.
• The model predicts that the pion is an ultra-strongly bound system due to a large coupling constant and strong hyperfine interaction, distinguishing it from atomic or nuclear systems.
• The analytical wave function Φa(p) = N(1 + p²/pa²)⁻² provides a viable baseline for computing higher Fock-state amplitudes and testing consistency with experiments like Ashery’s.