[Paper Review] Semiperturbative construction for the quark-gluon vertex

Felipe J. Llanes–Estrada, Christian S. Fischer+1 more|ArXiv.org|Jul 29, 2004

Particle physics theoretical and experimental studies2 references4 citations

TL;DR

This paper proposes a semiperturbative model for the quark-gluon vertex in Landau gauge QCD by replacing tree-level propagators with dressed ones from Dyson-Schwinger equations and enhancing internal vertices with ghost dressing factors. The non-Abelian one-loop diagram dominates by a factor of $N_c$, and the resulting vertex shows good agreement with lattice data, particularly in reproducing infrared suppression and chiral symmetry breaking features.

ABSTRACT

We construct a model for the quark-gluon vertex of Landau gauge QCD. This is of twofold interest: on the one hand the quark-gluon interaction is at the heart of quark confinement, on the other hand it is a central element in hadron phenomenology based on QCD Greens functions. We employ the non-Abelian one-loop diagram in perturbation theory, which is of order Nc. As a novelty we replace the tree-level quark and gluon propagators in this diagram by their dressed counterparts solving the Dyson-Schwinger equations. The Nc-suppressed Abelian diagram is an order of magnitude smaller in various kinematics. We also study the effect of ghost dressing factors on the vertex obtaining a construction in good agreement with recent low-momentum lattice calculations.

Motivation & Objective

• To construct a quark-gluon vertex model that captures nonperturbative infrared dynamics essential for quark confinement and chiral symmetry breaking.
• To improve upon the rainbow truncation by incorporating one-loop corrections with dressed propagators and ghost dressing factors.
• To ensure consistency with the Slavnov-Taylor identity and reproduce lattice data for the vertex in various kinematic regimes.
• To investigate the mass dependence of the vertex and its role in heavy quark systems and chiral symmetry breaking.
• To enable a consistent construction of the quark scattering kernel within the Dyson-Schwinger equation framework.

Proposed method

• Construct the vertex using the non-Abelian one-loop diagram as the primary contribution, with the Abelian diagram treated as a subdominant correction.
• Replace bare quark and gluon propagators in the loop with their dressed counterparts obtained from solving the Dyson-Schwinger equations.
• Enhance internal $q\bar{q}g$ vertices with ghost dressing functions to reflect the Slavnov-Taylor identity and improve infrared behavior.
• Project the vertex into a tensor basis using the Dirac structure from the literature, enabling numerical evaluation in four dimensions.
• Perform numerical integration using Gauss-Legendre grids and analytic reduction to scalar integrals for precision.
• Fix the renormalization constant $Z_{1F}$ by requiring the $\gamma_\mu$ component to be unity at 2 GeV.

Experimental results

Research questions

• RQ1 How does the inclusion of dressed propagators and ghost dressing factors affect the infrared structure of the quark-gluon vertex?
• RQ2 To what extent does the non-Abelian one-loop diagram dominate over the Abelian one in different kinematic configurations?
• RQ3 Can the model reproduce lattice data for the quark-gluon vertex, especially in the low-momentum regime?
• RQ4 How does the vertex behave in the chiral limit and in the heavy quark mass regime?
• RQ5 What is the impact of the vertex model on the quark Dyson-Schwinger equation and chiral symmetry breaking?

Key findings

• The non-Abelian one-loop diagram dominates the vertex by a factor of $N_c^2$, remaining an order of magnitude larger than the Abelian diagram after loop integration across various kinematics.
• The Abelian diagram contributes less than 10% to the quark DSE kernel, justifying its neglect for precision studies below 10% error.
• The model successfully reproduces lattice data for the leading Dirac amplitude $\lambda_1$ and the scalar amplitude $\lambda_3$ at the asymmetric point ($p_g = 0$, $p_1 = p_2$).
• The amplitude $4p^2\lambda_2$ vanishes at low momenta in the model, while lattice data (with large errors) suggest a finite limit, indicating a potential divergence in $\lambda_2$.
• In the totally asymmetric kinematic region ($p_1 = p$, $p_2 = 2p$, $p_g = 3p$), the four leading Dirac amplitudes ($\lambda_1$, $\lambda_3$, $\lambda_4$, $\tau_1$) dominate, with the remaining eight amplitudes suppressed by up to two orders of magnitude.
• The $\lambda_3$ amplitude exhibits a maximum for intermediate quark masses, while $\lambda_1/\mathcal{A}$ deviates from unity in the chiral limit, signaling non-Abelian dynamics beyond the Ball-Chiu construction.