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arxiv: 2604.15122 · v1 · submitted 2026-04-16 · ✦ hep-ph · hep-th

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Symmetry Preserving Contact Interaction Approaches: An Overview of Meson and Diquark Form Factors

L.X. Guti\'errez-Guerrero , Roger Jos\'e Hern\'andez-Pinto
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Pith reviewed 2026-05-10 10:35 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords contact interactionmeson form factorsdiquark form factorsnonperturbative QCDhadron structuresymmetry preservinglattice QCDquark models
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The pith

The contact interaction framework provides a consistent and simple description of meson and diquark form factors that aligns with lattice QCD.

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

This paper reviews the symmetry-preserving Contact Interaction model for hadrons composed of light and heavy quarks. It demonstrates that the model computes mass spectra and elastic form factors for forty mesons with various spins and parities along with their diquark partners, including both ground and excited states. Updated comparisons with recent lattice QCD results and other approaches refine earlier conclusions and support the model as a practical tool for nonperturbative studies. The review identifies potential extensions to baryons and multiquark states while noting connections to upcoming experiments. A reader should care because the approach supplies testable predictions for hadron structure where full dynamical calculations remain intensive.

Core claim

The authors present an updated overview showing that the Contact Interaction has evolved to treat ground and excited states, yielding a simple yet consistent approach to nonperturbative QCD. They examine the mass spectrum and elastic form factors of forty mesons and their diquark partners, update comparisons with recent literature including lattice results, and conclude that the framework remains a practical tool for hadron structure studies with applications to baryons and multiquark states.

What carries the argument

The symmetry-preserving Contact Interaction approximation, which replaces the full quark-quark interaction with a momentum-independent form while preserving key QCD symmetries to enable bound-state calculations.

If this is right

  • The model supplies predictions that can guide studies of baryons and multiquark states.
  • Updated comparisons clarify the model's performance, strengths, and limitations across quark sectors.
  • Forthcoming data from FAIR will supply indirect constraints through spectroscopy and in-medium properties.
  • High-precision form-factor measurements at Jefferson Lab and the Electron Ion Collider will directly test the predictions.

Where Pith is reading between the lines

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

  • If the framework continues to match data, it could serve as a computationally light baseline for exploring more complex systems before committing to full lattice simulations.
  • Systematic discrepancies in particular channels might indicate where momentum-dependent gluon effects must be restored.
  • The same machinery could be adapted to compute transition form factors between ground and excited states for direct comparison with planned experiments.
  • Consistent success across forty states suggests the approach may generalize to in-medium modifications relevant for heavy-ion environments.

Load-bearing premise

The contact interaction approximation remains sufficiently accurate for the internal structure and form factors of mesons and diquarks across light and heavy sectors when benchmarked against full QCD or lattice results.

What would settle it

High-precision lattice QCD or experimental measurements of electromagnetic form factors for excited heavy mesons that deviate markedly from Contact Interaction predictions while agreeing among themselves would show the approximation has broken down.

Figures

Figures reproduced from arXiv: 2604.15122 by L.X. Guti\'errez-Guerrero, Roger Jos\'e Hern\'andez-Pinto.

Figure 1
Figure 1. Figure 1: Diagrammatic representation of the CI, employing the simplified interaction of Eq. (1). • The quark propagator for a quark of flavor f is S −1 f (p) = iγ · p + mf + 16π 3 αˆ IR Z d 4 q (2π) 4 γµ Sf (q) γµ , (2) where mf is the current-quark mass. The integral possesses quadratic and logarithmic divergences and we regularize them in a Poincaré covariant manner. The solution of this equation is : S −1 f (p) … view at source ↗
Figure 2
Figure 2. Figure 2: S mesons (such as the σ) can be interpreted as parity partners of PS mesons (e.g., the π), while V mesons (e.g., the ρ) form chiral partner pairs with AV mesons (e.g., the a1 ). In this work, S states are treated exclusively as quark–antiquark systems. Consequently, S masses obtained below the 1–1.5 GeV region correspond to hypothetical qq¯ bound states. The physical nature of these low-lying S states rema… view at source ↗
Figure 3
Figure 3. Figure 3: Diagrammatic representation of the BSE. Blue (solid) circles represent dressed quark propagators S, red (solid) circle is the meson BSA Γ while the blue (solid) rectangle is the dressed￾quark-antiquark scattering kernel K. The equation takes the form [69] ΓH(k; P)tu = Z q χ(q; P)rs K rs tu(q, k; P), χ(q; P) = S(q+) ΓH(q; P) S(q−), (10) q+ = q + ηPP, q− = q − (1 − ηP)P, (11) where ΓH(k; P) represents the Be… view at source ↗
Figure 4
Figure 4. Figure 4: The multiplet for the PS mesons made of u, d, s and c quarks. The lightest states lie in the central orange region, while the outer vertices correspond to heavy–light mesons. The ηc appears inside the same region but belongs to the heavy sector. hadronic systems examined to date and provides a useful reference point for the results discussed in the sections that follow [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 5
Figure 5. Figure 5: Structure of the V-meson multiplet composed of u, d, s, and c quarks. The diagram illustrates the quark–antiquark content of each state and highlights the organization of light, strange, and charm V mesons within the SU(4) flavor-symmetry framework. that emerge within the CI framework, including the behavior of chiral partners and the resulting pattern of mass splittings. In parallel with CI treatment of t… view at source ↗
Figure 6
Figure 6. Figure 6: Triangle diagram representing the impulse approximation to the MγM vertex.The red circles denote the Bethe–Salpeter amplitudes associated with the incoming and outgoing mesons, which encode the internal quark–antiquark correlations. The blue circle corresponds to the dressed quark–photon vertex, describing the interaction between the photon and the quark. The arrows indicate the momentum flow through the d… view at source ↗
Figure 7
Figure 7. Figure 7: Dressing function associated with the transverse component of the quark–photon vertex in Eq. (45), evaluated using the parameter set defined in CI. The pole associated with the ground-state vector meson is clearly evident. 4.1. Form Factors of Pseudoscalar Mesons The EFF of the pion, the most extensively studied meson in this channel, was first investigated within CI in Ref. [32], with further developments… view at source ↗
Figure 8
Figure 8. Figure 8: EFFs of PS mesons computed within the CI framework. Left panel: charged mesons formed by quarks of different flavors. Center panel: quarkonia states, including a hypothetical ground-state strangeonium (ss¯). Right panel: neutral mesons composed of quarks of various flavors. The results are taken from Ref. [35]. expression: F PS(Q 2 ) = eM + aPS Q2 + bPS Q4 1 + cPS Q2 + dPS Q4 , (52) where eM ≡ F PS(Q2 = 0)… view at source ↗
Figure 9
Figure 9. Figure 9 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Within constituent-quark-based approaches, S mesons can be viewed as excited states of the corresponding PS mesons. The figure is adapted from Ref. [108] [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: EFFs for S mesons in the CI model. Left panel: electrically charges mesons composed of quarks of different flavors. Central panel: quarkonia including a hypothetical ground state strangeonium (ss¯). Right panel: electrically neutral mesons composed of quarks of different flavors. EFFs of electrically neutral but flavored mesons have been normalized to F S (0) = 0. numbers, we can immediately infer the lar… view at source ↗
Figure 13
Figure 13. Figure 13: V mesons electric, magnetic, and quadrupole FFs are displayed in the top, middle, and bottom rows, respectively. FFs are depicted on the left, center, and right columns for the charged V mesons, neutral V mesons consisting of identical flavored quarks and the neutral V mesons composed of different flavored quarks, respectively. We can now readily compute the charge, magnetic and quadrupole radii, using r … view at source ↗
Figure 14
Figure 14. Figure 14: Electromagnetic (blue), Magnetic (green), and quadrupole (red) FFs for ρ-meson. The central curve in each case is obtained using the τUV value from Tab. 11. The width of the band represents a 5% variation in the charge radius. The meson life is so short that it is challenging to carry out experimental measurements of its EFFs. We compare CI results with those obtained from lattice QCD [113,114] and the SD… view at source ↗
Figure 15
Figure 15. Figure 15: As we anticipated and expected, the CI results are harder due to its point-like interaction. This behaviour is typical of the CI model and it is irrespective of the nature of the mesons under study. EFFs for S diquarks in the CI model. Left panel: diquarks composed of different flavored quarks with total charge of the system of 1/3. Central panel: diquark system composed of the same flavored quarks. Right… view at source ↗
Figure 16
Figure 16. Figure 16: EFFs for PS diquarks in the CI. Left panel: diquarks composed of different flavored quarks with total charge of the system of 1/3. Central panel: diquark system composed of the same flavored quarks. Right panel: diquarks composed of different flavored quarks with total charge of the system of −2/3 and 4/3. distribution inside the diquark. In [PITH_FULL_IMAGE:figures/full_fig_p032_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: AV diquarks electric, magnetic, and quadrupole FFs are displayed in the top, middle, and bottom rows, respectively. Left panel: diquarks composed of different flavored quarks with total charge of the system of 1/3. Central panel: diquark system composed of the same flavored quarks. Right panel: diquarks composed of different flavored quarks with total charge of the system of −2/3 and 4/3. magnetic and qua… view at source ↗
Figure 18
Figure 18. Figure 18: Charge radii of PS (left), S (center) and V (right) ground state mesons in the CI framework. It is clear from this plot that the radii tend to decrease as the constituent quarks’ masses increase. • In contrast to mesons, diquarks exhibit a more extended spatial structure. Our analysis shows that S and PS diquarks display larger charge radii than their meson partners in most flavor configurations, with dif… view at source ↗
read the original abstract

We present an updated overview of the symmetry preserving Contact Interaction model in hadronic physics, developed a little over a decade ago to describe the mass spectrum and internal structure of mesons and diquarks composed of light and heavy quarks. Over the years, the Contact Interaction has evolved into a framework capable of treating both ground and excited states, providing a simple yet consistent approach to nonperturbative QCD. In this review, we examine the mass spectrum and elastic form factors of forty mesons with different spins and parities, together with their corresponding diquark partners. Importantly, we update the comparison of Contact Interaction predictions using recent results from the literature, offering a fresh perspective on the model's performance, strengths, and limitations. The analysis presented here refines previous conclusions and supports the Contact Interaction as a practical tool for hadron structure studies, with potential applications to baryons and multiquark states. We also present comparisons with other theoretical models and approaches, including lattice quantum chromodynamics, and comment on future prospects in view of ongoing and planned hadron structure experimental programs. In particular, forthcoming measurements at FAIR, together with future studies at Jefferson Lab and the Electron Ion Collider, are expected to provide key insights into hadron structure, with FAIR offering indirect constraints via hadron spectroscopy, hadronic interactions, and in-medium properties, while high-precision data on meson structure and form factors from Jefferson Lab and the Electron Ion Collider will provide valuable benchmarks to confront Contact Interaction based predictions.

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 is an updated overview of the symmetry-preserving contact interaction (CI) model in hadronic physics. It examines the mass spectra and elastic form factors of forty mesons (ground and excited states with various spins and parities, light and heavy quarks) and their diquark partners. The review updates comparisons to lattice QCD and other models, refines prior conclusions on the framework's consistency, discusses strengths and limitations, and outlines potential extensions to baryons/multiquark states along with prospects for experiments at FAIR, Jefferson Lab, and the EIC.

Significance. If the updated comparisons demonstrate acceptable accuracy, the review would establish the CI model as a computationally simple yet symmetry-preserving tool for nonperturbative QCD studies of hadron structure. It could usefully complement lattice QCD for excited states and heavy sectors, serving as a practical benchmark for interpreting data from upcoming hadron physics programs.

major comments (2)
  1. [mass spectrum and form factor results for the forty mesons] The sections presenting the mass spectrum and form factor results for the forty mesons (including excited states): the claim that the CI provides a 'consistent approach' across light and heavy sectors requires explicit quantification of agreement with lattice QCD (e.g., average relative deviations or tabulated differences for radial excitations), as the momentum-independent approximation may introduce larger systematic effects in these cases that are not fully addressed by the overview.
  2. [updated comparisons for heavy-quark mesons and diquarks] The updated comparisons for heavy-quark mesons and diquarks: without a dedicated analysis of how deviations from lattice results scale with quark mass or for excited states, it is difficult to confirm that the framework remains sufficiently accurate to support its use as a practical tool, given the known limitations of contact interactions in capturing momentum-dependent nonperturbative effects.
minor comments (2)
  1. The abstract states that forty mesons are examined but does not enumerate the specific states or quantum numbers covered; adding a summary table or explicit list would improve clarity for readers assessing the scope.
  2. Figure captions and tables comparing CI predictions to lattice data should explicitly reference the sources of the lattice results and note any post-hoc adjustments or error treatments used in the comparisons.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the positive overall assessment of our overview. We address each major point below and outline the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [mass spectrum and form factor results for the forty mesons] The sections presenting the mass spectrum and form factor results for the forty mesons (including excited states): the claim that the CI provides a 'consistent approach' across light and heavy sectors requires explicit quantification of agreement with lattice QCD (e.g., average relative deviations or tabulated differences for radial excitations), as the momentum-independent approximation may introduce larger systematic effects in these cases that are not fully addressed by the overview.

    Authors: We agree that explicit quantification will improve the clarity and rigor of our claims regarding consistency. In the revised manuscript we will add tables that report relative deviations (and, where appropriate, average deviations) between the contact-interaction results and available lattice-QCD data for both masses and form factors. These tables will be organized by light versus heavy sectors and by ground versus radially excited states. We will also expand the accompanying text to discuss the possible systematic impact of the momentum-independent approximation on excited-state observables. revision: yes

  2. Referee: [updated comparisons for heavy-quark mesons and diquarks] The updated comparisons for heavy-quark mesons and diquarks: without a dedicated analysis of how deviations from lattice results scale with quark mass or for excited states, it is difficult to confirm that the framework remains sufficiently accurate to support its use as a practical tool, given the known limitations of contact interactions in capturing momentum-dependent nonperturbative effects.

    Authors: We acknowledge that a quantitative scaling study would be desirable. Because the present work is an overview that compiles and updates existing literature results rather than performing new calculations, a full dedicated scaling analysis lies outside the scope of this review. Nevertheless, we will add a concise discussion subsection that extracts and comments on the observed trends in deviations as a function of quark mass and excitation level from the updated comparisons already presented. This addition will make the limitations of the contact-interaction framework more explicit while still supporting its utility as a practical, symmetry-preserving tool. revision: partial

Circularity Check

1 steps flagged

Minor self-citation to prior model development present but not load-bearing; external lattice QCD benchmarks provide independent validation

specific steps
  1. self citation load bearing [Abstract]
    "We present an updated overview of the symmetry preserving Contact Interaction model in hadronic physics, developed a little over a decade ago to describe the mass spectrum and internal structure of mesons and diquarks composed of light and heavy quarks. Over the years, the Contact Interaction has evolved into a framework capable of treating both ground and excited states, providing a simple yet consistent approach to nonperturbative QCD."

    The overview's premise of the model having 'evolved' into a consistent framework is justified by reference to its prior development, with overlapping author involvement implied by the review context; however, this is mitigated by the paper's explicit update of comparisons to independent external results like lattice QCD, preventing full load-bearing circularity.

full rationale

The paper is an overview that references the Contact Interaction model's development 'a little over a decade ago' and its evolution into a framework for ground and excited states. It updates comparisons of predictions against recent literature results, explicitly including lattice QCD. No derivation chain within the review reduces by construction to self-defined inputs, fitted parameters renamed as predictions, or unverified self-citations as the sole support. The central claim that the model is a practical tool rests on these external benchmarks rather than circular reduction to prior author works.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This is a review paper; the underlying contact interaction model rests on domain assumptions about symmetry preservation and a constant interaction approximating gluon exchange, with parameters typically fitted to meson spectra in prior literature.

axioms (1)
  • domain assumption The contact interaction preserves chiral symmetry and other QCD symmetries in the Bethe-Salpeter framework for bound states.
    Invoked throughout the model description as the basis for consistent treatment of mesons and diquarks.

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Forward citations

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Reference graph

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