Quantum chromodynamics and hadron structure

TL;DR

This work tackles hadron structure in nonperturbative QCD by employing the covariant Nambu–Jona-Lasinio (NJL) model with Schwinger proper time regularization to mimic confinement and dynamical chiral symmetry breaking. It computes $π^+$ and $K^+$ PDFs and EMFFs, including their quark-sector decompositions, and demonstrates consistency with existing data and other theoretical approaches. The results, including the ratio $u_K(x)/u_π(x)$ and the $Q^2$ evolution of moments, show the model’s ability to describe hadron structure at moderate $Q^2$ and provide predictions relevant for future facilities such as the EIC, EicC, and COMPASS/AMBER, while complementing lattice QCD and global analyses for constraining gluon distributions. Overall, the work highlights the value of QCD-inspired effective theories for connecting microscopic quark dynamics to observable hadron structure.

Abstract

In this review paper, I present a study of the structure of the hadrons computed in the covariant Nambu-Jona-Lasinio model as the chiral effective quark theory of QCD. I describe how the NJL model is treated to imitate the spontaneous chiral symmetry breaking and confinement QCD properties. The consistency for the parton distribution functions and electromagnetic form factors, as internal structure observables, in comparison with existing data and other theoretical predictions, is also shown. The implications of mimicking the QCD properties in the NJL model for hadron structure observables, as well as the relevance of the results to EIC, EicC, and COMPASS/AMBER future experiments, are discussed.

Figures (5)

• Figure 1: Running coupling $\alpha_S (Q)$ as a function of $Q$. The green band represents the prediction value of $\alpha_S (M_z^2 = 0.1184 \pm 0.0007)$, taken from Ref. ParticleDataGroup:2012pjm.
• Figure 2: Dynamical quark mass generation for different values of the current quark masses as a function of $G_\pi /G_{\mathrm{critical}}$.
• Figure 3: Results for the $\pi^+$ and $K^+$ PDFs at $Q^2 =$ 5 GeV$^2$, evolved from the initial scale $Q_0^2 =$ 0.16 GeV$^2$ using the NLO DGLAP QCD evolution Miyama:1995bd. The $xu_\pi (x)$, $xu_K (x)$, and $xs_K (x)$ as a function of the quark longitudinal momentum $x$ (left panel), and the ratio of the $u_K (x)/u_\pi (x)$ as a function of $x$ (right panel). Experimental data is taken from Ref. E615:1989bda.
• Figure 4: The moments of $K^+$ and $\pi^+$ PDFs for different renormalization scales $Q^2 =$ 0.16, 4, 8, 12, 16, and 20 GeV$^2$. Note the moments are calculated using $< x^{n-1} > = \int_0^1 dx x^{n-1} q_v (x)$, where $n=1, 2, \cdot \cdot \cdot$ is integer, and for $n=1$ (first moment) is the PDFs normalization, $\int_0^1 dx q_v (x) = 1$.
• Figure 5: Results for the dressed EMFFs of the $K^+$ and its quark sector form factors (upper left panel), the EMFFs of the $K^+$ and their quark sector form factor multiplying with $Q^2$ in comparison with data (low $Q^2)$Amendolia:1986ui (upper right panel), the EMFFs of the $\pi^+$ in comparison with the data Amendolia:1984nzNA7:1986vavJeffersonLabFpi-2:2006yshJeffersonLabFpi:2007virJeffersonLab:2008jveJeffersonLab:2008gyl and other theoretical models (lower left panel), and the EMFFs of the $\pi^+$ and others multipling with $Q^2$ (lower right panel).