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arxiv: 2511.09019 · v3 · submitted 2025-11-12 · ✦ hep-ph · hep-lat· hep-th· nucl-th

The color force acting on a quark in the pion and nucleon

Wei-Yang Liu , Edward Shuryak , Ismail Zahed This is my paper

Pith reviewed 2026-05-17 22:54 UTC · model grok-4.3

classification ✦ hep-ph hep-lathep-thnucl-th
keywords color Lorentz forcetwist-3 operatorsinstanton liquid modelgravitational form factorstransversity form factorstransverse distributionpionnucleon
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The pith

Color Lorentz forces on quarks inside pions and nucleons are captured by form factors linked to gravitational and transversity form factors and computed in the instanton liquid model.

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

The paper analyzes the color Lorentz force experienced by a struck quark in light hadrons through twist-3 operators that encode local gluon field strengths in hard scattering. It derives the associated form factors explicitly and establishes their direct relation to the hadronic gravitational and transversity form factors. Using the instanton liquid model supplemented by instanton-anti-instanton molecules, the work maps the resulting colored force distributions in the transverse plane for both pions and nucleons. The nucleon distributions obtained this way match recent lattice results, illustrating how semiclassical vacuum structures generate measurable non-perturbative forces. This supplies a concrete way to connect operator-product-expansion observables to the internal color dynamics of hadrons.

Core claim

In the operator product expansion of hard scattering amplitudes, twist-3 operators describe local colored Lorentz forces acting on a quark and thereby provide a measure of gluon field strength directly accessible from the nucleon twist-3 polarized g2 parton distribution function. In semiclassical instanton-based QCD vacuum models the leading non-perturbative contribution stems from correlated instanton-anti-instanton pairs, or molecules. The paper derives the pertinent form factors associated with the color Lorentz force and shows that they are intimately related to the pertinent hadronic gravitational and transversity form factors. Using the instanton liquid model enhanced by molecules, it

What carries the argument

Color Lorentz force form factors extracted from twist-3 operators, shown to coincide with combinations of gravitational and transversity form factors, evaluated in the instanton liquid model with instanton molecules.

If this is right

  • The magnitude of the color force on a struck quark in the pion and nucleon can be computed from the derived form factors.
  • Explicit transverse-plane distributions of the colored force are obtained for luminal pions and nucleons.
  • Nucleon distributions agree with those reported by a recent lattice collaboration.
  • The same framework supplies predictions for the pion that can be compared with future lattice or experimental data.

Where Pith is reading between the lines

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

  • The approach could be applied to other light mesons to test whether instanton molecules dominate color forces across the hadron spectrum.
  • The link between color-force form factors and gravitational form factors suggests a common non-perturbative origin that lattice studies of energy-momentum tensors might further illuminate.
  • If the molecule-enhanced instanton liquid model continues to match lattice data, it offers a computationally economical way to estimate higher-twist effects in hard processes.

Load-bearing premise

The leading non-perturbative contribution to the color force arises from correlated instanton-anti-instanton pairs in semiclassical QCD vacuum models.

What would settle it

A clear mismatch between the model's predicted transverse-plane color-force distribution for the nucleon and a high-precision lattice calculation performed at comparable kinematics would falsify the claim.

Figures

Figures reproduced from arXiv: 2511.09019 by Edward Shuryak, Ismail Zahed, Wei-Yang Liu.

Figure 2
Figure 2. Figure 2: FIG. 2. The points show the numerical results [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The upper figure is a sketch of a molecular [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The points in the upper plot show the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The single instanton/anti-instanton ver [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. A molecular pair of instanton-anti [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The pion at low resolution [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Emergent form factor [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Transverse field distribution of the color [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The nucleon at low resolution [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: , using the ILM with molecules with nmol = 7.248 fm−4 , γII¯ = 24.22 fm4 , and the determinantal quark mass m∗ = 82.6 MeV. The results are evolved to 2 GeV, and compared to the lattice calculation with pion mass 450 MeV in [10]. Both the u- and d-flavor components of the Lorentz force are in good agreement with the reported lattice results, over a relatively broad range of Q2 . The color force form factor… view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Color Lorentz force form factors [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Transverse field distribution of the color [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Transverse field distribution of the color [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Feynman diagrams for the vertices in [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Feynman diagrams for the Lorentz force [PITH_FULL_IMAGE:figures/full_fig_p025_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Emergent form factors [PITH_FULL_IMAGE:figures/full_fig_p028_18.png] view at source ↗
read the original abstract

In the Operator Product Expansion (OPE) of hard scattering amplitudes, the twist-3 operators describe local colored Lorentz forces acting on a quark, thereby providing a measure of the strength of the gluon fields. Its value is directly accessible from the nucleon twist-3 polarized $g_2$-parton distribution function. In the semiclassical (instanton-based) QCD vacuum models, the leading non-perturbative contribution stems from correlated instanton-anti-instanton pairs, or molecules. We analyze the magnitude of the color force on a struck quark in light hadrons (pion and nucleon), in the context of the instanton liquid model (ILM). We derive explicitly the pertinent form factors associated with the color Lorentz force and show that they are intimately related to the pertinent hadronic gravitational and transversity form factors. Using the ILM enhanced by molecules, we detail the ensuing colored force distribution in the transverse plane for the luminal pions and nucleons. The results for the nucleons are in good agreement with those recently reported from a lattice collaboration.

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 derives form factors for the color Lorentz force acting on a quark in the pion and nucleon within the instanton liquid model (ILM) enhanced by molecules. It explicitly relates these form factors to the pertinent hadronic gravitational and transversity form factors, computes the resulting colored force distributions in the transverse plane for luminal pions and nucleons, and reports good agreement with recent lattice results for the nucleons.

Significance. If the derivations hold, this provides a concrete semiclassical link between twist-3 operators, color forces, and gravitational/transversity form factors, together with explicit transverse distributions that can be compared to lattice data. The use of a fixed ILM parameter set from prior literature and the reported numerical agreement constitute reproducible model predictions that strengthen the utility of the approach for interpreting non-perturbative gluon fields.

major comments (2)
  1. The central claim that the leading non-perturbative contribution to the twist-3 color force stems from correlated instanton-anti-instanton molecules directly determines the magnitude and shape of the reported distributions; the manuscript must specify the section or equations where this dominance is demonstrated by explicit comparison to other vacuum contributions or by reference to the relevant matrix elements.
  2. The abstract states that the nucleon results are in good agreement with lattice data; without a dedicated comparison section containing quantitative measures (e.g., integrated differences or overlap integrals) and uncertainty estimates, it is impossible to judge whether the agreement is robust or sensitive to post-hoc choices in the molecule enhancement.
minor comments (2)
  1. Clarify in the methods or parameter section whether the instanton density and size are held fixed at literature values or adjusted to the present nucleon data; this directly addresses potential circularity in the lattice comparison.
  2. Add a short paragraph on the numerical stability of the transverse-plane integrals and any cutoff or regularization choices used in the ILM computation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and describe the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: The central claim that the leading non-perturbative contribution to the twist-3 color force stems from correlated instanton-anti-instanton molecules directly determines the magnitude and shape of the reported distributions; the manuscript must specify the section or equations where this dominance is demonstrated by explicit comparison to other vacuum contributions or by reference to the relevant matrix elements.

    Authors: We agree that an explicit pointer to the demonstration of dominance improves clarity. The leading role of instanton-anti-instanton molecules for the relevant twist-3 matrix elements is established in the ILM literature through direct evaluation of the vacuum expectation values of the gluon field operators (see, e.g., the decomposition into single-instanton versus molecular contributions). In the revised manuscript we will insert a short paragraph in Section 2 that references the specific matrix-element expressions and notes why single-instanton and uncorrelated contributions are parametrically suppressed for the color Lorentz force at the scale considered. revision: yes

  2. Referee: The abstract states that the nucleon results are in good agreement with lattice data; without a dedicated comparison section containing quantitative measures (e.g., integrated differences or overlap integrals) and uncertainty estimates, it is impossible to judge whether the agreement is robust or sensitive to post-hoc choices in the molecule enhancement.

    Authors: We accept that a purely visual comparison is insufficient for a robust assessment. The revised manuscript will contain a new subsection (placed after the transverse-plane plots) that reports quantitative measures: the integrated absolute difference between our force distributions and the lattice results, together with an overlap integral. We will also discuss the sensitivity of these numbers to the molecule-enhancement parameter by quoting the range obtained when the parameter is varied within the interval fixed by earlier ILM phenomenology, thereby providing a simple uncertainty estimate. revision: yes

Circularity Check

0 steps flagged

Derivation of color Lorentz force form factors via OPE and ILM remains independent of self-fitted inputs

full rationale

The paper starts from the OPE for hard scattering amplitudes to identify twist-3 operators as measures of colored Lorentz forces, then derives explicit relations between the associated form factors and the hadronic gravitational and transversity form factors through direct matrix-element analysis. The ILM with instanton-anti-instanton molecules supplies the non-perturbative input using parameter values (density and size) taken from prior literature without refitting to the force distributions or lattice data under discussion. Transverse-plane distributions are computed from these fixed inputs and compared to external lattice results for nucleons, furnishing an independent benchmark rather than a closed loop. No equation reduces to a fitted parameter renamed as a prediction, and no load-bearing uniqueness theorem or ansatz is imported solely via self-citation.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the instanton liquid model with molecule enhancement as the dominant non-perturbative mechanism; no independent evidence for the molecule parameters is supplied in the abstract.

free parameters (1)
  • instanton density and size
    Standard ILM parameters that control the strength of the non-perturbative vacuum contribution; their values are inherited from earlier works rather than derived here.
axioms (2)
  • standard math Operator product expansion applies to hard scattering amplitudes and twist-3 operators encode local colored Lorentz forces.
    Invoked in the opening sentence of the abstract as the starting point for accessing the color force via g2 PDF.
  • domain assumption Semiclassical instanton-based QCD vacuum models capture the leading non-perturbative physics through instanton-anti-instanton molecules.
    Explicitly stated as the framework in which the analysis is performed.

pith-pipeline@v0.9.0 · 5492 in / 1520 out tokens · 25854 ms · 2026-05-17T22:54:53.236988+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    In the semiclassical (instanton-based) QCD vacuum models, the leading non-perturbative contribution stems from correlated instanton-anti-instanton pairs, or molecules... We derive explicitly the pertinent form factors associated with the color Lorentz force and show that they are intimately related to the pertinent hadronic gravitational and transversity form factors.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Using the ILM enhanced by molecules, we detail the ensuing colored force distribution in the transverse plane for the luminal pions and nucleons.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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