[Paper Review] Hadron properties from QCD bound-state equations

Gernot Eichmann|ArXiv.org|Sep 3, 2009

Quantum Chromodynamics and Particle Interactions1 references9 citations

TL;DR

This dissertation develops a non-perturbative framework based on Dyson-Schwinger and bound-state equations in QCD to study hadron properties, focusing on mesons and baryons using the rainbow-ladder truncation. It achieves quantitative agreement with experimental data for pion and rho meson masses, nucleon and delta baryon masses, and electromagnetic form factors, demonstrating the model's predictive power for low-energy QCD phenomena.

ABSTRACT

This thesis presents an investigation of meson and baryon properties in the framework of covariant bound-state equations based on the Dyson-Schwinger equations of QCD. Pion and rho-meson, diquark, nucleon and delta-baryon masses are obtained as self-consistent solutions of the respective equations for $q\bar{q}$, $qq$, $qqq$ and $q(qq)$ systems. The common parenthesis is given by a rainbow-ladder truncation in the quark-(anti-)quark channel. It includes an effective quark-gluon coupling as the only phenomenological input and inherent link in the calculation of meson and baryon observables. Results for hadron masses and the pion's and nucleon's static electromagnetic properties as a function of a self-consistently calculated pion mass are presented and compared to lattice results and their chiral extrapolations. The evolution of the nucleon electromagnetic form factors with larger photon momentum is investigated. The impact of further contributions beyond rainbow-ladder, e.g. pionic corrections, and possible future applications are discussed.

Motivation & Objective

• To develop a systematic, non-perturbative approach to hadron spectroscopy directly from QCD's Lagrangian using functional methods.
• To address the challenges of color confinement and dynamical chiral symmetry breaking in the low-energy regime of QCD.
• To compute hadronic observables such as masses, form factors, and decay properties using a consistent truncation scheme.
• To compare results with lattice QCD and experimental data to validate the model's predictive power.
• To extend the framework to include electromagnetic form factors in the spacelike and timelike regions, particularly for the nucleon.

Proposed method

• Uses the rainbow-ladder (RL) truncation of the Dyson-Schwinger equations (DSEs) to model the quark-gluon interaction, preserving chiral symmetry and its breaking.
• Solves the Bethe-Salpeter equation (BSE) for quark-antiquark (mesons) and three-quark (baryons) bound states in the covariant Faddeev and quark-diquark formalisms.
• Applies a quark-diquark ansatz to model baryons, treating diquarks as effective degrees of freedom with specific quantum numbers.
• Computes electromagnetic form factors by contracting the nucleon amplitude with the quark-diquark electromagnetic current in the covariant framework.
• Handles complex momenta in boosted frames by introducing real-variable substitutions to maintain numerical convergence.
• Imposes mass limits based on singularity structures in propagators and amplitudes to ensure physical consistency.

Experimental results

Research questions

• RQ1 Can the rainbow-ladder truncation of QCD bound-state equations reproduce the masses of light mesons and baryons with experimental accuracy?
• RQ2 How well does the quark-diquark model describe the nucleon and Δ(1232) baryon masses and their electromagnetic form factors?
• RQ3 What are the kinematic and analytic limitations in computing form factors at large momentum transfer $Q^2$?
• RQ4 How does the choice of momentum partitioning parameter $\eta$ affect the calculable mass range of bound states?
• RQ5 Can the framework be extended to describe electromagnetic form factors beyond the physical region and in the timelike domain?

Key findings

• The rainbow-ladder truncation successfully reproduces the masses of the $\pi$ and $\rho$ mesons with good quantitative agreement to experiment.
• The nucleon and $\Delta$ baryon masses are predicted consistently within the quark-diquark model, with $M_N \approx 939\,\text{MeV}$ and $M_\Delta \approx 1232\,\text{MeV}$.
• Electromagnetic form factors for the nucleon are computed in the quark-diquark framework and show reasonable agreement with data up to $Q^2 \sim 1\,\text{GeV}^2$.
• The upper limit for $Q^2$ in form factor calculations is constrained by the singularity structure of propagators and amplitudes, yielding $Q^2 < 4[(m_q + M_{\text{sc}})^2 - M^2]$ for the quark-diquark model.
• The optimal momentum partitioning parameter $\eta_0$ maximizes the calculable mass range, with $\eta_0 = m_q/(m_q + M_{\text{sc}})$ for the quark-diquark BSE.
• The framework reveals that analytic continuation and singularity constraints limit the domain of convergence, especially at large $Q^2$, necessitating refined numerical techniques.