[Paper Review] Contemporary applications of Dyson-Schwinger equations

M. B. Hecht, Craig D. Roberts+1 more|ArXiv.org|Oct 6, 2000

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper demonstrates the application of Dyson-Schwinger equations (DSEs) in quantum chromodynamics (QCD) to calculate pseudoscalar meson masses and nucleon properties using a renormalisation-group-improved rainbow-ladder truncation. It shows that the model-independent interplay of the gap equation, Bethe-Salpeter equation, and axial-vector Ward-Takahashi identity explains dynamical chiral symmetry breaking, while pion-loop corrections contribute only a 1% increase to the nucleon mass, indicating their minor role in mass generation.

ABSTRACT

We illustrate the contemporary application of Dyson-Schwinger equations using two examples: the calculation of pseudoscalar meson masses, an associated model-independent mass formula and the approach to the heavy-quark limit; and the study of nucleon observables, including a calculation of its mass, $M$, via a covariant Fadde'ev equation and an estimate of pion-loop contributions to $M$.

Motivation & Objective

• To apply Dyson-Schwinger equations (DSEs) to nonperturbative QCD phenomena, particularly meson and nucleon properties.
• To investigate the role of dynamical chiral symmetry breaking in pseudoscalar mesons using the interplay of the gap equation, Bethe-Salpeter equation, and axial-vector Ward-Takahashi identity.
• To assess the contribution of pion loops to nucleon mass via a self-energy model and determine their quantitative impact.
• To explore the consistency of nucleon and Δ resonance masses using scalar and pseudovector diquark correlations in a Fadde’ev equation framework.
• To establish a phenomenologically reliable, model-independent framework for understanding hadron structure within the DSE approach.

Proposed method

• Solving the renormalised homogeneous Bethe-Salpeter equation (BSE) for pseudoscalar mesons using a rainbow-ladder truncated quark-antiquark kernel.
• Employing the axial-vector Ward-Takahashi identity to ensure model-independent results in the context of dynamical chiral symmetry breaking.
• Using a covariant Fadde’ev equation to describe the nucleon as a bound state of quarks and diquark correlations, with scalar and pseudovector diquark components.
• Estimating pion-loop contributions to the nucleon self-energy using a momentum-dependent πNN coupling derived from on-shell data and an off-shell suppression Ansatz.
• Applying a translationally invariant regularisation with a cutoff scale Λ, and taking the limit Λ→∞ to ensure cutoff independence.
• Using a product Ansatz for the off-shell πNN coupling to approximate the angular average in the loop integral, ensuring consistency with the nucleon’s off-shell nature.

Experimental results

Research questions

• RQ1 How do Dyson-Schwinger equations reproduce pseudoscalar meson masses and their evolution with current-quark mass?
• RQ2 What is the origin of the kernel enhancement in the QCD gap equation that supports dynamical chiral symmetry breaking?
• RQ3 To what extent do pion loops contribute to the nucleon’s mass, and how does this compare to diquark correlations?
• RQ4 Can a consistent description of the nucleon and Δ resonance be achieved using only scalar and pseudovector diquark correlations?
• RQ5 How does the nucleon mass depend on diquark parameters such as mass and coupling strength?

Key findings

• The kernel in the QCD gap equation must exhibit significant enhancement in the domain ΛQCD² ≤ k² ≤ 2 GeV² to reproduce observed meson spectra.
• Pseudoscalar meson masses are calculated reliably using the rainbow-ladder truncation, with results independent of the momentum partitioning parameter ηP.
• Pion-loop corrections to the nucleon self-energy contribute only +10 MeV to the nucleon mass, amounting to a 1% increase, indicating a minor role in mass generation.
• The nucleon mass decreases by 19% when diquark masses are reduced by 21%, demonstrating strong sensitivity to diquark correlations.
• Increasing the diquark mass parameter ω1+ can reduce the nucleon mass by enhancing the support of the pseudovector binding contribution, even though it reduces the coupling g1+.
• A well-constrained scalar diquark model requires the parameter r to be in the range ∼0.5–0.7 for consistency with nucleon and Δ mass data.