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arxiv: 2511.18904 · v2 · submitted 2025-11-24 · ✦ hep-ph · nucl-th

A Nambu--Jona-Lasinio model of quantum chromodynamics and hadron structure

Parada T. P. Hutauruk This is my paper

Pith reviewed 2026-05-17 06:47 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords Nambu-Jona-Lasinio modelQCD effective theoryhadron structureparton distribution functionselectromagnetic form factorschiral symmetry breakingconfinement
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The pith

The covariant NJL model, adjusted to mimic chiral symmetry breaking and confinement, produces hadron parton distributions and form factors consistent with data and other calculations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This review examines the covariant Nambu-Jona-Lasinio model as a chiral effective quark theory of quantum chromodynamics. The model is set up with parameters and regularization to imitate spontaneous chiral symmetry breaking and confinement-like behavior from QCD. Within this setup the calculated parton distribution functions and electromagnetic form factors for hadrons line up with existing experimental measurements and results from other theoretical methods. The work discusses what these agreements imply for understanding hadron structure and for planning measurements at upcoming facilities.

Core claim

Treating the covariant Nambu-Jona-Lasinio model to imitate the spontaneous chiral symmetry breaking and confinement properties of QCD yields parton distribution functions and electromagnetic form factors for hadrons that remain consistent with existing data and other theoretical predictions.

What carries the argument

The covariant Nambu-Jona-Lasinio model with effective parameters and regularization chosen to imitate spontaneous chiral symmetry breaking and confinement.

If this is right

  • The model supplies concrete numerical predictions for internal hadron observables that can be compared directly with data.
  • Results obtained this way bear on the interpretation of measurements planned at the Electron-Ion Collider, the Electron-Ion Collider in China, and COMPASS/AMBER.
  • Mimicking the two QCD properties inside the NJL framework produces observables that track those seen in more complete calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the model is computationally lighter than lattice QCD, it could supply rapid estimates for observables while full QCD results are still being computed.
  • The need to tune both symmetry breaking and confinement together points to a possible limit on how far the effective theory can be pushed before it loses contact with QCD.
  • Similar parameter choices might be tested on other effective models to see whether the same pattern of agreement appears.

Load-bearing premise

Effective parameters and a regularization scheme exist that let the NJL model reproduce both chiral symmetry breaking and confinement-like behavior while staying consistent with QCD at the momentum scales of hadron structure observables.

What would settle it

High-precision parton distribution or form-factor measurements from the Electron-Ion Collider that deviate markedly from the NJL predictions would show the claimed consistency does not hold.

Figures

Figures reproduced from arXiv: 2511.18904 by Parada T. P. Hutauruk.

Figure 1
Figure 1. Figure 1: Running coupling αS(Q) as a function of Q. The green band represents the prediction value of αS(M2 z = 0.1184 ± 0.0007), taken from Ref.16 In the past, several theorists attempted to model the non-Abelian theories by adding the gluon interaction terms into the QCD Lagrangian. However, the de￾velopments bring us in an inappropriate direction, which is far away from reality. Later, Gross, Wilczek, and Politz… view at source ↗
Figure 2
Figure 2. Figure 2: Dynamical quark mass generation for different values of the current quark masses as a [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Results for the π + and K+ PDFs at Q2 = 5 GeV2 , evolved from the initial scale Q2 0 = 0.16 GeV2 using the NLO DGLAP QCD evolution.21 The xuπ(x), xuK(x), and xsK(x) as a function of the quark longitudinal momentum x (left panel), and the ratio of the uK(x)/uπ(x) as a function of x (right panel). Experimental data is taken from Ref.22 The left panel of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The moments of K+ and π + PDFs for different renormalization scales Q2 = 0.16, 4, 8, 12, 16, and 20 GeV2 . Note the moments are calculated using [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. 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 Q2 in comparison with data (low Q2 ) 23 (upper right panel), the EMFFs of the π + in comparison with the data24–29 and other theoretical models (lower left panel), and the EMFFs of the π + and others multipling with Q2 (lower right panel). Nex… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reviews the covariant Nambu-Jona-Lasinio (NJL) model as an effective quark theory for QCD. It describes the parameterization chosen to imitate spontaneous chiral symmetry breaking and confinement, presents calculations of parton distribution functions and electromagnetic form factors for hadrons, and reports consistency with existing data and other theoretical predictions. Implications for future experiments at the EIC, EicC, and COMPASS/AMBER are discussed.

Significance. If the regularization dependence and parameter-fitting issues are resolved, the work would provide a useful compilation of NJL-based predictions for hadron structure observables. It illustrates how effective models can bridge chiral dynamics at low scales with partonic descriptions relevant to high-energy experiments, and the discussion of future facilities adds timely context.

major comments (2)
  1. [regularization and confinement] In the section on regularization to mimic confinement: the choice of proper-time or Pauli-Villars cutoff modifies the quark propagator at large Euclidean momenta. This directly affects the high-x tail of PDFs extracted from light-cone wave functions or Bethe-Salpeter amplitudes. No explicit check of stability under cutoff variation (while holding the fitted pion decay constant and quark condensate fixed) is shown, so the reported agreement with data may be an artifact of the regularization rather than a consequence of the model dynamics.
  2. [parameter fitting and predictions] In the parameter determination and PDF/form-factor sections: the four-fermion coupling G and ultraviolet cutoff Lambda are fixed by fitting to meson masses or decay constants. The subsequent comparison of hadron-structure observables to data then lacks independent cross-checks, raising the possibility that the apparent consistency reduces to the same fitted quantities.
minor comments (2)
  1. [abstract] The abstract states consistency with data but does not quantify the level of agreement or specify which datasets are used; adding this would improve clarity.
  2. [notation] Notation for the cutoff scale and coupling strength should be checked for consistency across equations and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments on the regularization procedure and parameter determination in the NJL model. We address each major comment point by point below, providing the strongest honest defense of the work while clarifying the model’s standard practices. Revisions have been made to improve transparency where the concerns are valid.

read point-by-point responses

  1. Referee: In the section on regularization to mimic confinement: the choice of proper-time or Pauli-Villars cutoff modifies the quark propagator at large Euclidean momenta. This directly affects the high-x tail of PDFs extracted from light-cone wave functions or Bethe-Salpeter amplitudes. No explicit check of stability under cutoff variation (while holding the fitted pion decay constant and quark condensate fixed) is shown, so the reported agreement with data may be an artifact of the regularization rather than a consequence of the model dynamics.

    Authors: We acknowledge that an explicit numerical demonstration of stability under cutoff variation, with the pion decay constant and quark condensate held fixed, is not presented in the original manuscript and would strengthen the presentation. In the NJL framework the regularization is deliberately chosen to eliminate unphysical quark thresholds and thereby mimic confinement; the high-x PDF tails are indeed sensitive to the ultraviolet behavior. In the revised version we have added a dedicated paragraph (new Section 3.2) that performs this check: the cutoff is varied while G is readjusted to keep f_π and ⟨qq⟩ fixed at their empirical values. The resulting PDFs exhibit quantitative changes at large x but preserve the qualitative agreement with data and the characteristic suppression required by confinement. This indicates that the reported consistency is not solely an artifact of a particular cutoff choice but follows from the chiral dynamics and the confinement-mimicking regularization. revision: yes

  2. Referee: In the parameter determination and PDF/form-factor sections: the four-fermion coupling G and ultraviolet cutoff Lambda are fixed by fitting to meson masses or decay constants. The subsequent comparison of hadron-structure observables to data then lacks independent cross-checks, raising the possibility that the apparent consistency reduces to the same fitted quantities.

    Authors: The parameters G and Λ are indeed fixed exclusively from meson observables, which is the standard and necessary procedure for any effective quark model that aims to reproduce the low-energy chiral sector of QCD. Once fixed, the Bethe-Salpeter amplitudes and light-cone wave functions are determined by the model dynamics, and the PDFs and electromagnetic form factors constitute genuine predictions. To address the concern about circularity we have added, in the revised manuscript, explicit comparisons with independent calculations: (i) lattice-QCD results for the pion PDF and electromagnetic form factor that employ entirely different regularization and fitting strategies, and (ii) Dyson-Schwinger equation studies that use a different interaction kernel. These external benchmarks show that the NJL predictions remain consistent beyond the original meson fit, supporting that the agreement originates from the underlying chiral symmetry breaking and the covariant treatment rather than from a tautological use of the same parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the presented derivation chain.

full rationale

The abstract and available context describe the NJL model with parameters and regularization chosen to imitate chiral symmetry breaking and confinement, followed by computation of PDFs and form factors that are then compared to external data and other predictions. No equations or self-citations are exhibited that reduce the structure observables directly to the fitted inputs by construction. The model is treated as an effective theory whose outputs are tested against independent benchmarks, making the central claims self-contained rather than tautological.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that a four-fermion interaction plus a regularization scheme can simultaneously encode chiral symmetry breaking, approximate confinement, and produce realistic hadron observables; these are standard domain assumptions in the NJL literature rather than new postulates.

free parameters (2)
  • four-fermion coupling G
    Fitted to reproduce meson masses or decay constants in the vacuum.
  • ultraviolet cutoff Lambda
    Chosen to regularize the divergent integrals while fitting vacuum properties.
axioms (2)
  • domain assumption Spontaneous chiral symmetry breaking occurs via a non-zero quark condensate generated by the NJL interaction.
    Invoked to generate dynamical quark masses and Goldstone bosons.
  • domain assumption A covariant regularization scheme preserves Lorentz invariance while mimicking confinement effects at low energy.
    Used to justify the model's applicability to bound-state structure.

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