Baryonic Bound States in the Non-Local NJL Model

Arpan Chatterjee; Stefan Groote

arxiv: 2605.00084 · v1 · submitted 2026-04-30 · ✦ hep-ph · nucl-th

Baryonic Bound States in the Non-Local NJL Model

Arpan Chatterjee , Stefan Groote This is my paper

Pith reviewed 2026-05-09 20:30 UTC · model grok-4.3

classification ✦ hep-ph nucl-th

keywords baryon bound statesnon-local NJL modelFaddeev approachquark-diquarkBethe-Salpeter equationform factorsnonperturbative QCDintegral equations

0 comments

The pith

The non-local NJL model reduces the three-quark baryon problem to a solvable quark-diquark eigenvalue equation.

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

This paper reviews how baryons as three-quark bound states are handled covariantly in the non-local Nambu-Jona-Lasinio model. The relativistic Faddeev method simplifies the three-body problem to an effective quark-diquark system that includes both scalar and axial-vector channels. The resulting Bethe-Salpeter equation is recast as an eigenvalue problem whose solutions deliver the baryon mass and form factors through numerical integration. A reader would care because the method supplies concrete numbers for hadron properties in the intermediate-energy region where perturbative QCD does not apply.

Core claim

The non-local NJL framework, motivated by QCD-based nonlocal interactions and Dyson-Schwinger considerations, provides a compact description in which baryon masses and form factors are extracted from the numerical solution of coupled integral equations. This is achieved by reducing the three-body quark problem via the relativistic Faddeev approach to an effective quark-diquark bound-state problem, with the quark-diquark Bethe-Salpeter equation written as an eigenvalue problem for the baryon mass.

What carries the argument

The quark-diquark Bethe-Salpeter equation derived from the Faddeev reduction in the non-local NJL model, which functions as an eigenvalue problem to determine the baryon mass.

If this is right

Baryon masses are obtained directly as eigenvalues of the coupled integral equations.
Electromagnetic form factors follow from the same bound-state wave functions.
Both scalar and axial-vector diquark channels enter the description on equal footing.
The approach applies to the intermediate-energy regime where nonperturbative effects dominate.

Where Pith is reading between the lines

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

The same reduction could be used to compute masses and form factors of other three-quark states such as hyperons or excited baryons.
Direct comparison of the predicted form factors with lattice QCD data would test the specific non-local interaction chosen.
The method offers a route to connect effective quark models with full QCD descriptions of nuclear matter.

Load-bearing premise

The non-local NJL model captures the dominant nonperturbative correlations for baryon bound states.

What would settle it

A numerical solution for the nucleon mass or its charge radius that deviates from the measured value by more than the uncertainty allowed by parameter choice would falsify the framework's ability to describe baryons.

read the original abstract

Baryons, as three-quark bound states, require a covariant treatment in the intermediate-energy regime where perturbative QCD is no longer applicable and where nonperturbative correlations dominate. This article reformulates the content of the CERN Baltic Conference 2025 presentation on baryonic bound states in the non-local Nambu--Jona-Lasinio (NJL) model. We review how the relativistic Faddeev approach reduces the three-body quark problem to an effective quark--diquark bound-state problem, describe the scalar and axial-vector diquark channels, and show how the resulting quark--diquark Bethe--Salpeter equation can be written as an eigenvalue problem for the baryon mass. The non-local NJL framework, motivated by QCD-based nonlocal interactions and Dyson--Schwinger considerations, provides a compact description in which baryon masses and form factors are extracted from the numerical solution of coupled integral equations.

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.

Desk Editor's Note private letter to a colleague

This paper is mostly a recap of a conference talk on the non-local NJL model for baryons, without new calculations or tests.

read the letter

The main thing to know is that this is a reformulation of a 2025 conference presentation rather than a new piece of research. It describes the non-local NJL model for baryons but stops short of any actual computations. The paper does a decent job explaining the standard reduction from the three-quark Faddeev equation to an effective quark-diquark Bethe-Salpeter problem. It covers the scalar and axial-vector diquark channels and sets up the eigenvalue problem for the baryon mass. The motivation from nonlocal QCD interactions and Dyson-Schwinger equations is clear and follows common practice in the literature. If someone needs a compact overview of how to extract masses and form factors from those integral equations, this lays it out plainly. Where it falls short is the lack of any numerical work. There are no solved masses, no form factor plots, no fits to data, and no error estimates. The abstract claims the framework provides a compact description, but without results it's impossible to see if that holds or how it compares to other models like the standard NJL or lattice calculations. Parameter fitting could introduce circularity, though the stress test didn't flag any internal problems. This kind of paper is mainly for people already deep in hadron effective models who might use it as a quick reference for the formalism. A general reader or someone outside the subfield won't get much from it. It doesn't seem to warrant a full peer review in its current form because the core contribution is expository. I wouldn't send it to referees yet. The authors need to include concrete calculations and comparisons to make it a real contribution.

Referee Report

0 major / 1 minor

Summary. The paper reformulates a conference presentation on baryonic bound states in the non-local Nambu--Jona-Lasinio (NJL) model. It reviews the relativistic Faddeev reduction of the three-quark problem to an effective quark--diquark Bethe--Salpeter eigenvalue problem, incorporating scalar and axial-vector diquark channels, and explains how baryon masses and form factors are obtained from the numerical solution of the resulting coupled integral equations within a framework motivated by QCD-inspired nonlocal interactions and Dyson--Schwinger considerations.

Significance. The non-local NJL framework, as outlined, supplies a compact covariant description for extracting baryon properties in the nonperturbative regime. The manuscript aligns with standard practice in Dyson--Schwinger studies of baryons and provides a clear methodological overview without introducing internal inconsistencies or unsubstantiated claims. While the reformulation of an existing presentation adds limited novelty, the explicit motivation from QCD-based interactions and the reduction to an eigenvalue problem constitute a useful reference for the field.

minor comments (1)

The abstract states that baryon masses and form factors are extracted from numerical solutions but provides no sample results, parameter values, or comparisons; if the manuscript is intended as a self-contained journal article rather than a proceedings summary, adding a brief illustrative result or reference to prior numerical work would improve completeness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the accurate summary of its content, and the recommendation to accept. The report correctly identifies the reformulation of the conference presentation and the methodological focus on the quark-diquark reduction within the non-local NJL framework.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's core procedure reduces the three-quark Faddeev equation to a quark-diquark Bethe-Salpeter eigenvalue problem whose solution yields baryon masses and form factors as direct numerical outputs. This is a standard, non-circular computational step: the model Lagrangian and interaction kernel are fixed inputs motivated by QCD, the integral equations are solved for the bound-state amplitudes, and the resulting masses are predictions rather than redefinitions of the inputs. No self-definitional relations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the described chain. The framework remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no specific free parameters, axioms, or invented entities are identifiable. The non-local NJL model and Faddeev reduction likely rely on parameters and assumptions from prior literature, but details are absent.

pith-pipeline@v0.9.0 · 5454 in / 1169 out tokens · 34109 ms · 2026-05-09T20:30:46.174507+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · 1 internal anchor

[1]

L. D. Faddeev and S. P. Merkuriev,Quantum Scattering Theory for Several Particle Systems, Springer, Dordrecht, 1993

work page 1993
[2]

Production processes as a tool to study parameterizations of quark confinement,

S. Ahlig, R. Alkofer, C. S. Fischer, M. Oettel, H. Reinhardt and H. Weigel, “Production processes as a tool to study parameterizations of quark confinement,” Phys. Rev. D64, 014004 (2001)

work page 2001
[3]

Relativistic three quark bound states in separable two quark approximation,

M. Oettel, L. von Smekal and R. Alkofer, “Relativistic three quark bound states in separable two quark approximation,” Comput. Phys. Commun.144, 63 (2002)

work page 2002
[4]

Topics in quantum field theory: Renormalization groups in Hamiltonian framework and baryon structure in a non-local QCD model,

A. H. Rezaeian, “Topics in quantum field theory: Renormalization groups in Hamiltonian framework and baryon structure in a non-local QCD model,” arXiv:hep-ph/0507304

work page internal anchor Pith review arXiv
[5]

Baryon structure in a quark-confining non-local NJL model,

A. H. Rezaeian, N. R. Walet and M. C. Birse, “Baryon structure in a quark-confining non-local NJL model,” Phys. Rev. C70, 065203 (2004)

work page 2004
[6]

A nonlocal, covariant generalization of the NJL model,

R. D. Bowler and M. C. Birse, “A nonlocal, covariant generalization of the NJL model,” Nucl. Phys. A582, 655 (1995)

work page 1995
[7]

Meson properties in an extended nonlocal NJL model,

R. S. Plant and M. C. Birse, “Meson properties in an extended nonlocal NJL model,” Nucl. Phys. A628, 607 (1998)

work page 1998
[8]

A chiral theory of nucleons,

D. Diakonov, V. Y. Petrov and P. V. Pobylitsa, “A chiral theory of nucleons,” Nucl. Phys. B306, 809 (1988)

work page 1988
[9]

Nambu–Jona-Lasinio model correlation functions from QCD,

M. Frasca, A. Ghoshal and S. Groote, “Nambu–Jona-Lasinio model correlation functions from QCD,” Nucl. Part. Phys. Proc.318–323, 138 (2022). 6

work page 2022

[1] [1]

L. D. Faddeev and S. P. Merkuriev,Quantum Scattering Theory for Several Particle Systems, Springer, Dordrecht, 1993

work page 1993

[2] [2]

Production processes as a tool to study parameterizations of quark confinement,

S. Ahlig, R. Alkofer, C. S. Fischer, M. Oettel, H. Reinhardt and H. Weigel, “Production processes as a tool to study parameterizations of quark confinement,” Phys. Rev. D64, 014004 (2001)

work page 2001

[3] [3]

Relativistic three quark bound states in separable two quark approximation,

M. Oettel, L. von Smekal and R. Alkofer, “Relativistic three quark bound states in separable two quark approximation,” Comput. Phys. Commun.144, 63 (2002)

work page 2002

[4] [4]

Topics in quantum field theory: Renormalization groups in Hamiltonian framework and baryon structure in a non-local QCD model,

A. H. Rezaeian, “Topics in quantum field theory: Renormalization groups in Hamiltonian framework and baryon structure in a non-local QCD model,” arXiv:hep-ph/0507304

work page internal anchor Pith review arXiv

[5] [5]

Baryon structure in a quark-confining non-local NJL model,

A. H. Rezaeian, N. R. Walet and M. C. Birse, “Baryon structure in a quark-confining non-local NJL model,” Phys. Rev. C70, 065203 (2004)

work page 2004

[6] [6]

A nonlocal, covariant generalization of the NJL model,

R. D. Bowler and M. C. Birse, “A nonlocal, covariant generalization of the NJL model,” Nucl. Phys. A582, 655 (1995)

work page 1995

[7] [7]

Meson properties in an extended nonlocal NJL model,

R. S. Plant and M. C. Birse, “Meson properties in an extended nonlocal NJL model,” Nucl. Phys. A628, 607 (1998)

work page 1998

[8] [8]

A chiral theory of nucleons,

D. Diakonov, V. Y. Petrov and P. V. Pobylitsa, “A chiral theory of nucleons,” Nucl. Phys. B306, 809 (1988)

work page 1988

[9] [9]

Nambu–Jona-Lasinio model correlation functions from QCD,

M. Frasca, A. Ghoshal and S. Groote, “Nambu–Jona-Lasinio model correlation functions from QCD,” Nucl. Part. Phys. Proc.318–323, 138 (2022). 6

work page 2022