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REVIEW 3 major objections 7 minor 52 references

Vacuum-polarization flavor mixing shifts charged pion and kaon valence PDFs in proportion to quark mass differences and improves the distributions at both 4 and 27 GeV².

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-30 20:43 UTC pith:GRQDOOEZ

load-bearing objection Real but tiny mixing residuals on pion/kaon PDFs inside NJL; useful extension, undercut by no refit (mπ→104 MeV) and a flagged normalization convention. the 3 major comments →

arxiv 2607.23497 v1 pith:GRQDOOEZ submitted 2026-07-26 hep-ph hep-exhep-latnucl-th

Effects of flavor-mixings on charged kaon and pion parton distribution functions

Fabio L. Braghin , Parada T. P. Hutauruk This is my paper
classification hep-ph hep-exhep-latnucl-th
keywords parton distribution functionspionkaonNambu–Jona-Lasinio modelflavor mixingvacuum polarizationDGLAP evolutioncharge symmetry
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper asks how a flavor-mixing interaction generated by vacuum polarization, rather than by instantons, changes the valence-quark parton distributions of the charged pion and kaon. Working inside the U(3) Nambu–Jona-Lasinio model with proper-time regularization, the authors recompute dynamical quark masses, meson masses and meson–quark couplings once the mixing is turned on, then evolve the resulting valence PDFs to the experimental scales 4 and 27 GeV². They find that the size of the mixing correction tracks the effective quark masses and their differences, producing the largest residuals near x ≈ 0.3 and 0.8 for the pion and near x ≈ 0.2 and 0.7 for the kaon. The same pattern appears at both scales, and the mixed distributions sit closer to existing data and global analyses than the unmixed baseline. A sympathetic reader cares because forthcoming electron–ion collider measurements will finally have the precision to test whether such vacuum-induced mixing is visible in light-meson structure.

Core claim

The strength of vacuum-polarization flavor-mixing effects on charged-pion and charged-kaon valence PDFs is proportional to the quark effective masses and their differences; the dominant residuals lie near x ≃ 0.3 and 0.8 (pion) and x ≃ 0.2 and 0.7 (kaon) at both μ² = 4 and 27 GeV², and the mixing improves the PDFs relative to the unmixed NJL baseline.

What carries the argument

Implicit flavor mixing: diagonal couplings G_ff acquire dependence on the constituent masses of the other flavors through vacuum-polarization loops, normalized so that the charged-pion coupling remains the reference value G_11 = G_0.

Load-bearing premise

The authors keep the original model parameters fixed and simply insert the new flavor-dependent couplings, even though this shifts the pion mass well below its physical value.

What would settle it

A high-precision measurement of the charged-pion or kaon valence PDF near x ≈ 0.3 and x ≈ 0.8 (or 0.2 and 0.7 for the kaon) at a few-GeV scale that shows no residual of the size and shape predicted by the mixed versus unmixed NJL curves would falsify the claimed improvement.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Vacuum-polarization mixing, not only the ’t Hooft determinant, must be included when extracting light-meson PDFs from effective models.
  • The same mass-difference scaling implies larger mixing residuals for heavier flavor combinations once data become available.
  • Lattice calculations of pion and kaon PDFs can isolate the quark-mass-difference piece of flavor mixing by comparing isospin-symmetric and broken ensembles.
  • Upcoming EIC, EicC, J-PARC and AMBER Sullivan/Drell–Yan data will be able to test the predicted residual shapes directly.

Where Pith is reading between the lines

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

  • Because the residual strength is essentially scale-independent between 4 and 27 GeV², the mixing signature is already fixed at the model scale and is only mildly diluted by DGLAP evolution.
  • Refitting the ultraviolet cutoff and bare couplings after mixing is turned on would likely restore the physical pion mass while preserving the shape of the PDF residuals, offering a clean next calculation.
  • The same implicit-mixing mechanism should generate analogous shifts in the pion and kaon electromagnetic form factors and generalized parton distributions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. The authors compute charged pion and kaon valence PDFs in the U(3) NJL model with proper-time regularization, adding flavor-dependent four-fermion couplings generated by vacuum polarization (Eqs. 4–15) rather than the 't Hooft determinant. The modified couplings enter the gap equations and Bethe–Salpeter equations, producing shifted constituent masses, meson masses, and meson–quark couplings (Tables I–II), which feed the standard NJL PDF calculation of Ref. [3] followed by NLO-DGLAP evolution to μ² = 4 and 27 GeV². The paper reports that mixing-induced residuals in the valence distributions are O(10⁻³), peaking near x ≃ 0.3 and 0.8 for the pion and x ≃ 0.2–0.3 and 0.7 for the kaon, and asserts that the mixing "certainly improves" the PDFs relative to the unmixed baseline when compared to E615 data and the JAM analysis.

Significance. If the results hold, the paper provides a controlled estimate of a genuinely parameter-conservative mixing mechanism: the vacuum-polarization-induced couplings ΔGij introduce no new energy scales or free parameters beyond those of the standard NJL model, and the work correctly preserves the valence and momentum sum rules (Eqs. 33–34) through the full pipeline. The predicted residual sizes and x-locations are, in principle, falsifiable with forthcoming EIC/EicC/J-PARC/AMBER data, and the comparison to both experimental data and the JAM global analysis is the right external benchmark. The gap/BSE + proper-time + NLO-DGLAP machinery is standard and internally consistent. The significance is tempered, however, by the fact that the advertised effects are ~10⁻³ in xq(x) — far below current experimental and global-fit uncertainties — so the practical impact rests entirely on the robustness of the residual calculation, which is where the manuscript is currently weakest.

major comments (3)
  1. [§II.C and Table I (G19-Set4, G10-Set4 rows)] The parameters (G0, ΛUV, ΛIR, current masses) are fitted to physical mπ and mK in the unmixed model and then held fixed when the mixing is switched on. As a result the pion mass collapses from 140 to 104 MeV (G19-Set4) and 121 MeV (G10-Set4), and mK shifts by ~15 MeV. The PDFs in Eq. (29) depend explicitly on m²ps and on the constituent masses that are themselves shifted; a comparison to data and JAM performed with a 104 MeV pion cannot support the claim (abstract and §IV) that mixing 'certainly improves' the PDFs. The improvement claim is load-bearing for the paper's conclusion. The authors should either (a) refit at least one parameter set with mixing active so that mπ and mK remain physical, and show whether the residuals and the improvement survive, or (b) remove the improvement claim and present the residuals strictly as a sensitivity estimate at fixed parameters.
  2. [§II, Eq. (16)] All quantitative results (residual magnitudes, signs, and peak locations) are generated after imposing the normalization G11 = G0, which the text itself describes as one choice among several ('Other normalization prescriptions can be explored in future studies'). Because the normalized couplings are ratios (G0+ΔGii)/(G0+ΔG11), choosing a different reference channel (e.g., G44, or a flavor-symmetric average) redistributes the shifts among Mu, Ms, mπ, mK, and the meson–quark couplings, and hence among the PDF residuals, which are differences of O(10⁻³) between two full calculations. Given that the residuals are tiny compared to both the PDF values and the data uncertainties, even a modest prescription-induced reshuffling could move or partially cancel the advertised peaks at x ≃ 0.3/0.8 (pion) and 0.2/0.7 (kaon). A robustness check — recomputing Tables I–II and one representative residual
  3. [§IV (discussion of Figs. 1–2) and abstract] The statement that mixing 'certainly improves' the pion and kaon PDFs is not supported by any quantitative criterion in the manuscript. No χ², no comparison against the JAM uncertainty band, and no residual-with-respect-to-data is shown; the figures plot only the difference between the two model variants. With residuals of O(10⁻³) and data/JAM uncertainties orders of magnitude larger, an improvement at this level is indistinguishable from noise by any standard metric. The authors should either provide a quantitative goodness-of-fit comparison demonstrating the improvement or soften the claim to a statement about the size and location of the mixing effect.
minor comments (7)
  1. [Abstract vs. §IV] The abstract states the dominant kaon mixing effects appear at x ≃ 0.2 and x ≃ 0.7, but the body text (discussion of Figs. 4–6) repeatedly identifies the dominant kaon residual at x ≃ 0.3 and describes the x ≃ 0.8 region as 'significantly suppressed'. The abstract and body should be reconciled.
  2. [§IV, paragraph on Fig. 6(b)] The text reads 'we show the difference of the kaon up valence quark distribution at scale μ² = 27 GeV² for fixed G0 = 10 GeV⁻²' while referring to Fig. 6(b), which is labeled μ² = 4 GeV². Please correct the scale reference.
  3. [§IV, kaon vs. pion residuals] The statement that the kaon PDF difference is 'one order of magnitude larger' than the pion's is not borne out by the figure axes: pion residuals peak at ~±0.0015–0.002 (Figs. 1, 3) and kaon residuals at ~±0.006 (Fig. 6), i.e., a factor of roughly 3–4. Please correct.
  4. [§IV, multiple occurrences] The coupling is written as 'G0 = 10 GeV²' in several places; the correct unit is GeV⁻² (as used elsewhere). Please make the units uniform.
  5. [§III, Eq. (39)] The NLO running-coupling expression is written in a nonstandard compact notation (β3, ln ln(QΛ)/ln(QΛ)); please check for typos and consider giving the standard two-loop form with β1/(4πβ0) coefficient for clarity.
  6. [Figs. 1–9] The residual panels would be more informative with the JAM uncertainty band (or E615 error bars) overlaid, so the reader can directly see that the mixing residuals are below current sensitivity; this would also discipline the 'improvement' language.
  7. [§II, Eqs. (18)–(20)] In the isospin-symmetric limit used here, Eqs. (18) and (19) are identical; it would help the reader to state explicitly that Guu = Gdd by construction and that all u/d asymmetry in the PDFs comes from the meson–quark vertex and mass terms, not from the couplings.

Circularity Check

1 steps flagged

Mild author self-citation supplies the flavor-mixing interaction; PDF residuals and external data comparisons are not forced by construction.

specific steps
  1. self citation load bearing [Sec. I (Introduction) and opening of Sec. II]
    "More recently, it has been shown that vacuum polarization or gluon exchanges can also induce flavor mixing due to flavor symmetry breaking (FSB), which has been identified by means of flavor-dependent effective quark interactions [34, 35]. ... In this section, we describe the U(3) NJL model with the flavor mixing interactions that have been extensively discussed in Refs. [34–37]."

    The central premise that a vacuum-polarization (non-’t Hooft) flavor-mixing interaction of the stated form exists and should be inserted into the NJL couplings is justified only by citations to the present authors’ prior work. That premise selects the entire calculation (ΔGij kernels, implicit mixing in Guu/Gss/G44, and the subsequent PDF residuals). It is not an external theorem or independent measurement; without those self-citations the “novel” interaction is undefined. The PDF-vs-data comparison itself remains non-circular.

full rationale

The derivation is a standard NJL pipeline: bare parameters (G0, cutoffs, current masses) are fixed in the unmixed model to physical mπ and mK; vacuum-polarization corrections ΔGij are then computed from the quark loops, normalized by the explicit convention G11=G0 (Eq. 16), gap and BSE equations are re-solved without refitting, and valence PDFs are obtained from the resulting masses and meson–quark couplings, evolved with NLO DGLAP, and compared to external Conway data and JAM fits. That external comparison is independent of the fit targets, so the PDF shapes and the claim of improvement are not predictions that reduce to the inputs by definition. The only circularity-adjacent element is that the functional form and motivation of the “novel” vacuum-polarization flavor mixing are taken from the authors’ own prior papers rather than from an external derivation; this is load-bearing for the setup but not for the numerical PDF residuals themselves. The acknowledged normalization choice (G11 fixed to G0) and the decision not to refit when mπ drops to 104 MeV are model-prescription and robustness issues, not self-definitional or fitted-input-as-prediction circularity under the stated criteria. Score 2 reflects one non-decisive self-citation chain; no step makes the residual peaks or the data comparison tautological.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 1 invented entities

The calculation rests on the standard NJL effective Lagrangian plus one previously introduced vacuum-polarization correction to the couplings, regulated by proper-time cut-offs that simultaneously serve as a confinement proxy. All dimensionful inputs (G0, ΛUV, ΛIR, current masses, initial evolution scale) are fixed to meson masses/decay constants in the unmixed theory and then held fixed when mixing is switched on. No new dynamical entity is postulated in this paper; the mixing interaction is imported from the authors’ earlier work.

free parameters (4)
  • G0 (two values: 19.04 and 10 GeV^{-2}) = 19.04 GeV^{-2} and 10 GeV^{-2}
    Four-fermion coupling of the baseline NJL model; fixed by fitting unmixed pion/kaon masses and then reused without refit when flavor mixing is turned on.
  • ΛUV / ΛIR = 645 MeV / 240 MeV
    Ultraviolet and infrared proper-time cut-offs; IR cutoff is interpreted as a confinement scale. Taken from prior NJL PDF papers and held fixed.
  • current quark masses m0u=m0d, m0s = (16.4, 356) MeV and (7.5, 260) MeV
    Bare masses adjusted (together with G0 and cut-offs) so that the unmixed model reproduces mπ and mK; two different sets appear in Table I.
  • initial PDF scale μ0² = 0.18 GeV²
    Hadronic scale at which the pure-valence NJL distributions are defined before NLO-DGLAP evolution; chosen by hand to match the earlier Hutauruk et al. setup.
axioms (5)
  • domain assumption The U(3) NJL contact interaction with proper-time regularization adequately captures the non-perturbative valence structure of light pseudoscalar mesons.
    Stated throughout Secs. I–II; standard in the NJL-PDF literature the paper builds on.
  • ad hoc to paper Vacuum-polarization corrections generate flavor-dependent diagonal couplings ΔGij (Eqs. 4–15) that may be normalized to G11=G0 (Eq. 16) without introducing new free parameters.
    Normalization choice is made explicitly so that charged-pion mass shifts arise only from mass changes in the gap equation; other normalizations are deferred to future work.
  • domain assumption Explicit off-diagonal flavor mixings G_i≠j and the ’t Hooft determinant can be set to zero for charged-meson PDFs.
    Sec. II A–B: “such mixing effects are neglected, corresponding to the approximation G_f≠g=0.”
  • domain assumption NLO DGLAP evolution from a pure-valence initial condition at μ0²=0.18 GeV² correctly generates the sea and glue that are compared with data at 4 and 27 GeV².
    Sec. III; standard but non-trivial because the initial scale lies near the edge of the perturbative regime.
  • domain assumption Charge symmetry (mu=md) may be kept while still generating useful strange–light mixing through Ms≠Mu.
    Stated in the Introduction; CSB is deferred to earlier work.
invented entities (1)
  • Vacuum-polarization-induced implicit flavor-mixing couplings (ΔGij / Gff) no independent evidence
    purpose: Provide a parameter-free, non-instanton source of flavor mixing that feeds into gap equations, meson masses, couplings, and ultimately PDFs.
    Not invented in this paper; introduced in Braghin, Phys. Rev. D 103 (2021) 094028 and follow-ups. Here they are merely applied to PDFs. independent_evidence remains limited to meson-spectrum phenomenology in those earlier works.

pith-pipeline@v1.2.0-grok45-kimik3 · 23513 in / 4104 out tokens · 74223 ms · 2026-07-30T20:43:39.943217+00:00 · methodology

0 comments
read the original abstract

We investigate the charged kaon and pion parton distribution functions (PDFs) in the U(3) Nambu--Jona-Lasinio (NJL) model, considering a novel flavor-mixing interaction arising from vacuum polarization that differs from the instanton-induced 't~Hooft interaction. In this work, the proper-time regularization scheme is employed, and, effectively, it might take quark confinement into account. The gap equations, the meson masses, and the meson--quark coupling constants are calculated in the presence of flavor mixing interactions. The valence-quark distributions of the charged kaon and pion are calculated and compared with existing experimental data and JAM analysis results at scales $\mu^{2} =$ 4 and 27 GeV$^{2}$. The strength of mixing effects on the PDFs is found to be proportional to the quark effective masses and their differences. We further find that the dominant flavor mixing effects in the pion valence up-quark distribution at scales $\mu^{2} =$ 27 and 4 GeV$^2$ are around $x \simeq 0.3$ and $x \simeq 0.8$, while for the kaon, the effects are shown at around $x \simeq 0.2$ and $x \simeq 0.7$. We also find similar strength of mixing effects on the PDFs at both scales $\mu^{2} =4$ and 27 GeV$^{2}$; however, it certainly improves the pion and kaon PDFs.

Figures

Figures reproduced from arXiv: 2607.23497 by Fabio L. Braghin, Parada T. P. Hutauruk.

Figure 1
Figure 1. Figure 1: FIG. 1. Valence quark distributions of the pion and their PDF differences at scales (a) [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Same as in Fig [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Differences of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

52 extracted references · 4 canonical work pages

  1. [1]

    arXiv:2507.12664

    Parton Distribution Functions and their Generalizations. arXiv:2507.12664

    Pith/arXiv arXiv

  2. [2]

    Gross, et al., 50 Years of Quantum Chromodynamics, Eur

    F. Gross, et al., 50 Years of Quantum Chromodynamics, Eur. Phys. J. C 83 (2023) 1125.arXiv:2212.11107,doi: 10.1140/epjc/s10052-023-11949-2

    Pith/arXiv arXiv 2023

  3. [3]

    P. T. P. Hutauruk, I. C. Cloet, A. W. Thomas, Flavor dependence of the pion and kaon form factors and par- ton distribution functions, Phys. Rev. C 94 (3) (2016) 035201.arXiv:1604.02853,doi:10.1103/PhysRevC.94. 035201

    Pith/arXiv arXiv 2016

  4. [4]

    Watanabe, T

    A. Watanabe, T. Sawada, C. W. Kao, Kaon quark distri- bution functions in the chiral constituent quark model, Phys. Rev. D 97 (7) (2018) 074015.arXiv:1710.09529, doi:10.1103/PhysRevD.97.074015

    Pith/arXiv arXiv 2018

  5. [5]

    Frederico, G

    T. Frederico, G. A. Miller, Deep inelastic structure func- tion of the pion in the null plane phenomenology, Phys. Rev. D 50 (1994) 210–216.doi:10.1103/PhysRevD.50. 210

    doi:10.1103/physrevd.50 1994

  6. [6]

    Bentz, T

    W. Bentz, T. Hama, T. Matsuki, K. Yazaki, NJL model on the light cone and pion structure function, Nucl. Phys. A 651 (1999) 143–173.arXiv:hep-ph/9901377,doi:10. 1016/S0375-9474(99)00130-X

    Pith/arXiv arXiv 1999

  7. [7]

    Avila, J

    C. Avila, J. C. Sanabria, J. Magnin, Pion and kaon par- ton distribution functions in a meson cloud model, Phys. Rev. D 67 (2003) 034022, [Erratum: Phys.Rev.D 68, 079902 (2003)].arXiv:hep-ph/0210421,doi:10.1103/ PhysRevD.68.079902

    Pith/arXiv arXiv 2003

  8. [8]

    Nguyen, A

    T. Nguyen, A. Bashir, C. D. Roberts, P. C. Tandy, Pion and kaon valence-quark parton distribution func- tions, Phys. Rev. C 83 (2011) 062201.arXiv:1102.2448, doi:10.1103/PhysRevC.83.062201

    Pith/arXiv arXiv 2011

  9. [9]

    P. T. P. Hutauruk, Pseudoscalar Meson Parton Distri- butions Within Gauge-Invariant Nonlocal Chiral Quark Model, Symmetry 17 (6) (2025) 971.arXiv:2505.06726, doi:10.3390/sym17060971

    Pith/arXiv arXiv 2025

  10. [10]

    P. T. P. Hutauruk, S.-i. Nam, Parton Distribution Func- tions for Pseudoscalar Mesons in the COnfining Effective Chiral Quark Theory, Makara J. Sci. 26 (4) (2022) 296– 303.arXiv:2303.01652,doi:10.7454/mss.v26i4.1402

    Pith/arXiv arXiv 2022

  11. [11]

    J. Chen, X. Wang, Y. Cai, X. Chen, Q. Wang, Valence quark distributions of pions: insights from Tsallis en- tropy*, Chin. Phys. C 50 (1) (2026) 013103.arXiv: 2408.03068,doi:10.1088/1674-1137/ae0998

    arXiv 2026

  12. [12]

    Aicher, A

    M. Aicher, A. Schafer, W. Vogelsang, Soft-gluon resum- mation and the valence parton distribution function of the pion, Phys. Rev. Lett. 105 (2010) 252003.arXiv: 1009.2481,doi:10.1103/PhysRevLett.105.252003

    Pith/arXiv arXiv 2010

  13. [13]

    G. A. Miller, A. K. Opper, E. J. Stephenson, Charge symmetry breaking and QCD, Ann. Rev. Nucl. Part. Sci. 56 (2006) 253–292.arXiv:nucl-ex/0602021,doi:10. 1146/annurev.nucl.56.080805.140446

    Pith/arXiv arXiv 2006

  14. [14]

    J. T. Londergan, J. C. Peng, A. W. Thomas, Charge Symmetry at the Partonic Level, Rev. Mod. Phys. 82 (2010) 2009–2052.arXiv:0907.2352,doi:10.1103/ RevModPhys.82.2009

    Pith/arXiv arXiv 2010

  15. [15]

    J. T. Londergan, A. W. Thomas, The Validity of charge symmetry for parton distributions, Prog. Part. Nucl. Phys. 41 (1998) 49–124.arXiv:hep-ph/9806510,doi: 10.1016/S0146-6410(98)00055-6

    Pith/arXiv arXiv 1998

  16. [16]

    G. A. Miller, B. M. K. Nefkens, I. Slaus, Charge symme- try, quarks and mesons, Phys. Rept. 194 (1990) 1–116. doi:10.1016/0370-1573(90)90102-8

    work page doi:10.1016/0370-1573(90)90102-8 1990

  17. [17]

    Gasser, H

    J. Gasser, H. Leutwyler, Quark masses, Phys. Rept. 87 (1989) 77.doi:10.1016/0370-1573(82)90035-7

    work page doi:10.1016/0370-1573(82)90035-7 1989

  18. [18]

    Donoghue, Light quark masses and chiral symmetry, Annu

    J. Donoghue, Light quark masses and chiral symmetry, Annu. Rev. Nucl. Part. Sci. 39 (1989) 1.doi:10.1146/ annurev.ns.39.120189.000245

    arXiv 1989

  19. [19]

    X. G. Wang, A. W. Thomas, R. D. Young, Electromag- netic contribution to charge symmetry violation in parton distributions, Phys. Lett. B 753 (2016) 595–599.arXiv: 1512.04139,doi:10.1016/j.physletb.2015.12.062

    Pith/arXiv arXiv 2016

  20. [20]

    A. W. Thomas, X. G. Wang, R. D. Young, Electromag- netic Contribution to the Proton-Neutron Mass Splitting, Phys. Rev. C 91 (1) (2015) 015209.arXiv:1406.4579, doi:10.1103/PhysRevC.91.015209

    Pith/arXiv arXiv 2015

  21. [21]

    P. T. P. Hutauruk, W. Bentz, I. C. Clo¨ et, A. W. Thomas, Effects of charge symmetry breaking on form factors of the pion and kaon, JPS Conf. Proc. 26 (2019) 031005. arXiv:1901.06068,doi:10.7566/JPSCP.26.031005

    Pith/arXiv arXiv 2019

  22. [22]

    Sather, Isospin violating quark distributions in the nu- cleon, Phys

    E. Sather, Isospin violating quark distributions in the nu- cleon, Phys. Lett. B 274 (1992) 433–438.doi:10.1016/ 0370-2693(92)92011-5

    1992

  23. [23]

    Horsley, Y

    R. Horsley, Y. Nakamura, D. Pleiter, P. E. L. Rakow, G. Schierholz, H. Stuben, A. W. Thomas, F. Winter, R. D. Young, J. M. Zanotti, Charge Symmetry Break- ing in Parton Distribution Functions from Lattice QCD, Phys. Rev. D 83 (2011) 051501.arXiv:1012.0215,doi: 10.1103/PhysRevD.83.051501

    Pith/arXiv arXiv 2011

  24. [24]

    P. E. Shanahan, R. Horsley, Y. Nakamura, D. Pleiter, P. E. L. Rakow, G. Schierholz, H. St¨ uben, A. W. Thomas, 13 R. D. Young, J. M. Zanotti, Charge symmetry violation in the electromagnetic form factors of the nucleon, Phys. Rev. D 91 (11) (2015) 113006.arXiv:1503.01142,doi: 10.1103/PhysRevD.91.113006

    Pith/arXiv arXiv 2015

  25. [25]

    Arrington, et al., Revealing the structure of light pseu- doscalar mesons at the electron–ion collider, J

    J. Arrington, et al., Revealing the structure of light pseu- doscalar mesons at the electron–ion collider, J. Phys. G 48 (7) (2021) 075106.arXiv:2102.11788,doi:10.1088/ 1361-6471/abf5c3

    Pith/arXiv arXiv 2021

  26. [26]

    D. P. Anderle, et al., Electron-ion collider in China, Front. Phys. (Beijing) 16 (6) (2021) 64701.arXiv: 2102.09222,doi:10.1007/s11467-021-1062-0

    Pith/arXiv arXiv 2021

  27. [27]

    Sawada, W.-C

    T. Sawada, W.-C. Chang, S. Kumano, J.-C. Peng, S. Sawada, K. Tanaka, Accessing proton generalized par- ton distributions and pion distribution amplitudes with the exclusive pion-induced Drell-Yan process at J-PARC, Phys. Rev. D 93 (11) (2016) 114034.arXiv:1605.00364, doi:10.1103/PhysRevD.93.114034

    Pith/arXiv arXiv 2016

  28. [28]

    Adams, et al., Letter of Intent: A New QCD fa- cility at the M2 beam line of the CERN SPS (COM- PASS++/AMBER), – (8 2018).arXiv:1808.00848

    B. Adams, et al., Letter of Intent: A New QCD fa- cility at the M2 beam line of the CERN SPS (COM- PASS++/AMBER), – (8 2018).arXiv:1808.00848

    Pith/arXiv arXiv 2018

  29. [29]

    Z. Lu, Z. Yu, T. Lin, Y.-T. Liang, R. Wang, W. Chang, W. Xiong, Feasibility study of pion and kaon struc- ture via the Sullivan process at the EicC, Phys. Rev. D 113 (11) (2026) 114002.arXiv:2512.01720,doi: 10.1103/1s8f-9cny

    Pith/arXiv arXiv 2026

  30. [30]

    J. T. Londergan, G. T. Carvey, G. Q. Liu, E. N. Ro- dionov, A. W. Thomas, Drell-Yan processes as a probe of charge symmetry violation in the pion and the nu- cleon, Phys. Lett. B 340 (1994) 115–121.arXiv:hep-ph/ 9503221,doi:10.1016/0370-2693(94)91306-4

    work page doi:10.1016/0370-2693(94)91306-4 1994

  31. [31]

    H. T. Klest, Drell-Yan process at the Electron-Ion Col- lider, Phys. Rev. D 113 (9) (2026) 094029.arXiv: 2512.14690,doi:10.1103/cv19-g1f8

    arXiv 2026

  32. [32]

    M. Ding, K. Raya, D. Binosi, L. Chang, C. D. Roberts, S. M. Schmidt, Symmetry, symmetry breaking, and pion parton distributions, Phys. Rev. D 101 (5) (2020) 054014.arXiv:1905.05208,doi:10.1103/PhysRevD. 101.054014

    Pith/arXiv arXiv 2020

  33. [33]

    ’t Hooft, How Instantons Solve the U(1) Prob- lem, Phys

    G. ’t Hooft, How Instantons Solve the U(1) Prob- lem, Phys. Rept. 142 (1986) 357–387.doi:10.1016/ 0370-1573(86)90117-1

    1986

  34. [34]

    F. L. Braghin, Flavor-dependent U(3) Nambu–Jona- Lasinio coupling constant, Phys. Rev. D 103 (9) (2021) 094028.arXiv:2008.00346,doi:10.1103/PhysRevD. 103.094028

    Pith/arXiv arXiv 2021

  35. [35]

    F. L. Braghin, Mixing interactions and effects in the NJL- model, PoS CD2024 (2026) 075.arXiv:2501.07658,doi: 10.22323/1.479.0075

    Pith/arXiv arXiv 2026

  36. [36]

    F. L. Braghin, Strangeness content of the pion in the U(3) Nambu–Jona–Lasinio model, J. Phys. G 49 (5) (2022) 055101, [Erratum: J.Phys.G 50, 119501 (2023)].arXiv: 2108.02748,doi:10.1088/1361-6471/ac4d79

    Pith/arXiv arXiv 2022

  37. [37]

    F. L. Braghin, Quark-antiquark states of the lightest scalar mesons within the Nambu-Jona-Lasinio model with flavor-dependent coupling constants, J. Phys. G 50 (9) (2023) 095101.arXiv:2212.06616,doi:10.1088/ 1361-6471/acdaea

    Pith/arXiv arXiv 2023

  38. [38]

    F. L. Braghin, B. El-Bennich, F. E. Serna, Charge depen- dence of mesons with flavored contact-interaction cou- plings, Phys. Rev. D 113 (2026) 014010.doi:10.1103/ pmwg-8t5d

    2026

  39. [39]

    de Sousa, F

    W. de Sousa, F. L. Braghin, U(5) Nambu–Jona–Lasinio model with flavor dependent coupling constants: pseu- doscalar and scalar mesons masses, - The European Physical Journal A 59 (2026) 271.doi:10.1140/epja/ s10050-023-01190-6

    doi:10.1140/epja/ 2026

  40. [40]

    U. Vogl, W. Weise, The Nambu and Jona Lasinio model: Its implications for hadrons and nuclei, Prog. Part. Nucl. Phys. 27 (1991) 195–272.doi:10.1016/0146-6410(91) 90005-9

    work page doi:10.1016/0146-6410(91 1991

  41. [41]

    S. P. Klevansky, The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys. 64 (1992) 649–708.doi:10.1103/RevModPhys.64.649

    doi:10.1103/revmodphys.64.649 1992

  42. [42]

    Hatsuda, T

    T. Hatsuda, T. Kunihiro, QCD phenomenology based on a chiral effective Lagrangian, Phys. Rept. 247 (1994) 221–367.arXiv:hep-ph/9401310,doi:10.1016/ 0370-1573(94)90022-1

    Pith/arXiv arXiv 1994

  43. [43]

    Bijnens, Chiral Lagrangians and Nambu-Jona-Lasinio - like models, Phys

    J. Bijnens, Chiral Lagrangians and Nambu-Jona-Lasinio - like models, Phys. Rept. 265 (1996) 369–446.arXiv: hep-ph/9502335,doi:10.1016/0370-1573(95)00051-8

    Pith/arXiv arXiv 1996

  44. [44]

    Miyama, S

    M. Miyama, S. Kumano, Numerical solution of Q**2 evo- lution equations in a brute force method, Comput. Phys. Commun. 94 (1996) 185–215.arXiv:hep-ph/9508246, doi:10.1016/0010-4655(96)00013-6

    Pith/arXiv arXiv 1996

  45. [45]

    J. S. Conway, et al., Experimental Study of Muon Pairs Produced by 252-GeV Pions on Tungsten, Phys. Rev. D 39 (1989) 92–122.doi:10.1103/PhysRevD.39.92

    doi:10.1103/physrevd.39.92 1989

  46. [46]

    P. C. Barry, C.-R. Ji, N. Sato, W. Melnitchouk, Global QCD Analysis of Pion Parton Distributions with Thresh- old Resummation, Phys. Rev. Lett. 127 (23) (2021) 232001.arXiv:2108.05822,doi:10.1103/PhysRevLett. 127.232001

    Pith/arXiv arXiv 2021

  47. [47]

    R. J. Crewther, Status of the U(1) Problem, Riv. Nuovo Cim. 2N8 (1979) 63–117

    1979

  48. [48]

    Fan, H.-W

    Z. Fan, H.-W. Lin, Gluon parton distribution of the pion from lattice QCD, Phys. Lett. B 823 (2021) 136778.arXiv:2104.06372,doi:10.1016/j.physletb. 2021.136778

    Pith/arXiv arXiv 2021

  49. [49]

    Salas-Chavira, Z

    A. Salas-Chavira, Z. Fan, H.-W. Lin, First glimpse into the kaon gluon parton distribution using lattice QCD, Phys. Rev. D 106 (9) (2022) 094510.arXiv:2112.03124, doi:10.1103/PhysRevD.106.094510

    Pith/arXiv arXiv 2022

  50. [50]

    Holligan, H.-W

    J. Holligan, H.-W. Lin, Pion valence quark distribution at physical pion mass of N f = 2 + 1 + 1 lattice QCD, J. Phys. G 51 (6) (2024) 065101.arXiv:2404.14525, doi:10.1088/1361-6471/ad3162

    Pith/arXiv arXiv 2024

  51. [51]

    Lin, J.-W

    H.-W. Lin, J.-W. Chen, Z. Fan, J.-H. Zhang, R. Zhang, Valence-Quark Distribution of the Kaon and Pion from Lattice QCD, Phys. Rev. D 103 (1) (2021) 014516.arXiv:2003.14128,doi:10.1103/PhysRevD. 103.014516

    Pith/arXiv arXiv 2021

  52. [52]

    P. T. P. Hutauruk, W. Bentz, I. C. Clo¨ et, A. W. Thomas, Charge Symmetry Breaking Effects in Pion and Kaon Structure, Phys. Rev. C 97 (5) (2018) 055210.arXiv: 1802.05511,doi:10.1103/PhysRevC.97.055210

    Pith/arXiv arXiv 2018