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arxiv: 2605.25900 · v2 · pith:KSMLD2M6new · submitted 2026-05-25 · ✦ hep-ph · hep-ex· hep-lat· nucl-ex

Electromagnetic form factors of the nucleon from the instanton vacuum

Hui-Jae Lee , Yongwoo Choi , Hyun-Chul Kim This is my paper

Pith reviewed 2026-06-29 21:43 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-latnucl-ex
keywords electromagnetic form factorsnucleoninstanton vacuumchiral effective theoryproton charge radiusdynamical quark massSachs form factorsform factor ratios
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The pith

An effective theory derived from the QCD instanton vacuum reproduces the proton's electromagnetic form factors, including a charge radius of 0.841 fm that matches the muonic-hydrogen measurement.

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

The paper shows that the electromagnetic form factors of the nucleon can be calculated in a parameter-free way within an effective chiral theory coming from the instanton vacuum of QCD. The dynamical quark mass that depends on momentum naturally cuts off divergences in the quark loops. With all instanton parameters fixed once and for all by the saddle-point condition, the computed proton charge radius and the ratio of electric to magnetic form factors agree closely with experiment, unlike results from the chiral quark-soliton model.

Core claim

Within the effective chiral theory derived from the QCD instanton vacuum with finite current quark mass, the Sachs electric and magnetic form factors of the proton and neutron are computed using the momentum-dependent dynamical quark mass as a regulator. The resulting proton charge radius is 0.841 fm, in agreement with muonic hydrogen data, and the Q² dependence of μ_p G_E^p / G_M^p is well reproduced, providing a consistent description of nucleon electromagnetic structure without adjustable parameters.

What carries the argument

The momentum-dependent dynamical quark mass generated by the instanton medium, which regulates quark-loop divergences and encodes the effects of the QCD vacuum.

If this is right

  • The model yields a proton charge radius of 0.841 fm matching muonic-hydrogen measurements.
  • The Q² dependence of the proton form-factor ratio μ_p G_E^p/G_M^p is reproduced accurately.
  • Neutron Sachs form factors, charge and magnetization radii, and magnetic moments are obtained consistently.
  • The approach requires no additional regularization or free parameters beyond those fixed by the vacuum saddle-point equation.

Where Pith is reading between the lines

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

  • The success with fixed parameters suggests the instanton vacuum captures essential nonperturbative QCD effects that could be tested in lattice calculations of nucleon form factors.
  • This framework may be extended to compute other nucleon observables such as axial-vector form factors or transition form factors to resonances.
  • Discrepancies at higher Q² could indicate the momentum scale where instanton effects give way to perturbative gluon exchanges.

Load-bearing premise

The instanton parameters determined from the saddle-point equation in the vacuum remain unchanged and sufficient when applied to nucleon form factors at nonzero momentum transfer.

What would settle it

An experimental value for the proton charge radius that differs significantly from 0.841 fm, or data showing the proton form-factor ratio deviating from the predicted Q² dependence at accessible momentum transfers.

Figures

Figures reproduced from arXiv: 2605.25900 by Hui-Jae Lee, Hyun-Chul Kim, Yongwoo Choi.

Figure 1
Figure 1. Figure 1: FIG. 1. Isoscalar and isovector electric form factors of the nucleon. The solid curve represents the result for the isoscalar [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Electric form factor of the proton. The solid curve depicts the present numerical result for the proton electric form [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Electric form factor of the neutron. The solid curve depicts the present numerical result for the neutron electric form [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Isoscalar and isovector magnetic form factors of the nucleon. The solid curve represents the result for the isoscalar [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Magnetic form factor of the proton normalized by its magnetic moment. The solid curve depicts the present numerical [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Magnetic form factor of the neutron normalized by its magnetic moment. The solid curve depicts the present numerical [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Ratio of the electric and magnetic form factors [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Ratio of the electric and magnetic form factors [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

We investigate the electromagnetic form factors of the nucleon within an effective chiral theory derived from the QCD instanton vacuum, taking into account the finite current quark mass. The momentum-dependent dynamical quark mass, generated by the instanton-antiinstanton medium, naturally plays the role of a regulator, so that no additional regularization is required to tame the divergences arising from quark loops. The instanton parameters, the average instanton size $\bar{\rho}=0.35$ fm and the average interdistance $\bar{R}=0.86$ fm, together with the dynamical quark mass at zero virtuality $M_0=385$ MeV, are all fixed by the saddle-point equation beyond the chiral limit, leaving no adjustable free parameter in the present calculation. We compute the Sachs electric and magnetic form factors of the proton and neutron, the nucleon charge and magnetization radii, the magnetic moments, and the ratios $\mu_{p,n} G_E^{p,n}(Q^2)/G_M^{p,n}(Q^2)$. The present results are compared with the experimental data, the chiral quark-soliton model ($\chi$QSM), and the Kelly parametrization. The proton charge radius, $\sqrt{\langle r^2 \rangle_\mathrm{ch}^p}=0.841$ fm, is in remarkable agreement with the recent muonic-hydrogen value, and the $Q^2$ dependence of the proton form-factor ratio $\mu_p G_E^p/G_M^p$ is reproduced very well, in clear contrast to the $\chi$QSM. The overall agreement with the experimental data confirms that the effective chiral theory derived from the QCD instanton vacuum provides a consistent and predictive framework for describing the electromagnetic structure of the nucleon.

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

1 major / 2 minor

Summary. The manuscript computes nucleon electromagnetic Sachs form factors in an effective chiral theory from the QCD instanton vacuum with finite current quark mass. The momentum-dependent dynamical mass M(k) regulates quark loops without additional cutoffs. Parameters ar ρ=0.35 fm, ar R=0.86 fm and M_0=385 MeV are stated to be fixed solely by the saddle-point equation beyond the chiral limit, yielding parameter-free predictions for G_E, G_M, charge/magnetization radii, magnetic moments and the ratios μ_{p,n} G_E^{p,n}/G_M^{p,n}. Results are compared to data, the χQSM and the Kelly parametrization, with highlighted agreement for √⟨r²⟩_ch^p=0.841 fm and the Q² dependence of μ_p G_E^p/G_M^p.

Significance. If the saddle-point fixing is shown to be unique and free of auxiliary inputs, the work supplies a genuinely predictive, regulator-free framework for nucleon structure directly tied to the instanton vacuum. The reported numerical agreement on the proton charge radius with the muonic-hydrogen value and the improved description of the proton form-factor ratio relative to χQSM would constitute a non-trivial success for the approach.

major comments (1)
  1. [Abstract] Abstract (and the section presenting the saddle-point solution): the central claim that ar ρ, ar R and M_0 are determined solely by the saddle-point equation with finite current quark mass, leaving no adjustable parameters for the subsequent quark-loop integrals that yield the Sachs form factors, is load-bearing. The manuscript must demonstrate explicitly that the finite-mass extension of the saddle-point equation is solved without reference to other observables (e.g., f_π or ⟨qq⟩) or model-specific choices that are then reused in the form-factor computation; otherwise the parameter-free status and the predictive character of the radius and ratio results remain unverified.
minor comments (2)
  1. [Method] The text should supply the explicit functional form or numerical procedure used to obtain M(k) from the saddle-point solution so that the regulator role can be reproduced independently.
  2. [Results] Table or figure captions listing the computed radii and moments should include the corresponding experimental uncertainties for direct comparison.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the importance of verifying the parameter-free character of the calculation. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the section presenting the saddle-point solution): the central claim that ar ρ, ar R and M_0 are determined solely by the saddle-point equation with finite current quark mass, leaving no adjustable parameters for the subsequent quark-loop integrals that yield the Sachs form factors, is load-bearing. The manuscript must demonstrate explicitly that the finite-mass extension of the saddle-point equation is solved without reference to other observables (e.g., f_π or ⟨qq⟩) or model-specific choices that are then reused in the form-factor computation; otherwise the parameter-free status and the predictive character of the radius and ratio results remain unverified.

    Authors: We agree that an explicit demonstration is required to substantiate the claim. The saddle-point equation is extended to finite current quark mass and solved numerically in the instanton vacuum framework using only the current quark mass as additional input; the resulting values ar ρ = 0.35 fm, ar R = 0.86 fm and M_0 = 385 MeV are obtained directly from this solution. These parameters enter the quark-loop integrals for the Sachs form factors without further adjustment, with the momentum dependence of M(k) supplying the regularization. No reference is made to f_π, the quark condensate or other observables during the solution of the saddle-point equation. To make this independence fully transparent, we will revise the relevant section (and, if needed, add a short appendix) to display the explicit numerical procedure and confirm that no auxiliary inputs or model choices are reused in the form-factor computation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; parameters fixed by saddle-point equation prior to form-factor computation

full rationale

The paper determines the instanton parameters (ar ρ, ar R, M_0) via the saddle-point equation of the instanton vacuum model beyond the chiral limit. These fixed values then enter the quark-loop integrals for the Sachs form factors, with the momentum-dependent M(k) acting as regulator. The resulting radii, moments, and Q² ratios are computed and compared to external data (muonic-hydrogen radius, Kelly parametrization, experiment). No parameter is adjusted to the form-factor observables themselves, no self-citation chain carries the central claim, and no ansatz or uniqueness theorem is imported from prior author work to force the outcome. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the instanton vacuum generating a dynamical quark mass that regulates loops and on the saddle-point equation determining all parameters; no free parameters are introduced for the form-factor stage.

axioms (1)
  • domain assumption The QCD instanton vacuum provides an effective chiral theory whose parameters are fixed by the saddle-point equation beyond the chiral limit.
    Invoked in the abstract to set ρ-bar, R-bar, and M_0 without further adjustment.

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discussion (0)

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Reference graph

Works this paper leans on

124 extracted references · 82 canonical work pages · 64 internal anchors

  1. [1]

    C. F. Perdrisat, V. Punjabi, and M. Vanderhaeghen, Prog. Part. Nucl. Phys.59, 694 (2007), arXiv:hep-ph/0612014

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  2. [2]

    Nucleon Form Factors - A Jefferson Lab Perspective

    J. Arrington, K. de Jager, and C. F. Perdrisat, J. Phys. Conf. Ser.299, 012002 (2011), arXiv:1102.2463 [nucl-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  3. [3]

    The Structure of the Nucleon: Elastic Electromagnetic Form Factors

    V. Punjabi, C. F. Perdrisat, M. K. Jones, E. J. Brash, and C. E. Carlson, Eur. Phys. J. A51, 79 (2015), arXiv:1503.01452 [nucl-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  4. [4]

    Hofstadter, Rev

    R. Hofstadter, Rev. Mod. Phys.28, 214 (1956)

    1956

  5. [5]

    D. R. Yennie, M. M. L´ evy, and D. G. Ravenhall, Rev. Mod. Phys.29, 144 (1957). 16

    1957

  6. [6]

    Bartel, F

    W. Bartel, F. W. Busser, W. r. Dix, R. Felst, D. Harms, H. Krehbiel, P. E. Kuhlmann, J. McElroy, J. Meyer, and G. Weber, Nucl. Phys. B58, 429 (1973)

    1973

  7. [7]

    M. E. Christyet al.(E94110), Phys. Rev. C70, 015206 (2004), arXiv:nucl-ex/0401030

    work page arXiv 2004

  8. [8]

    J. C. Bernaueret al.(A1), Phys. Rev. C90, 015206 (2014), arXiv:1307.6227 [nucl-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  9. [9]

    Meyerhoffet al., Phys

    M. Meyerhoffet al., Phys. Lett. B327, 201 (1994)

    1994

  10. [10]

    Edenet al., Phys

    T. Edenet al., Phys. Rev. C50, R1749 (1994)

    1994

  11. [11]

    Beckeret al., Eur

    J. Beckeret al., Eur. Phys. J. A6, 329 (1999)

    1999

  12. [12]

    Herberget al., Eur

    C. Herberget al., Eur. Phys. J. A5, 131 (1999)

    1999

  13. [13]

    Ostricket al., Phys

    M. Ostricket al., Phys. Rev. Lett.83, 276 (1999)

    1999

  14. [14]

    The Charge Form Factor of the Neutron from the Reaction \pol{2H}(\pol{e},e'n)p

    I. Passchieret al., Phys. Rev. Lett.82, 4988 (1999), arXiv:nucl-ex/9907012

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  15. [15]

    Roheet al., Phys

    D. Roheet al., Phys. Rev. Lett.83, 4257 (1999)

    1999

  16. [16]

    A Measurement of the Electric Form Factor of the Neutron through $\vec{d}(\vec{e},e'n)p$ at $Q^2 = 0.5$ (GeV/c)$^2$

    H. Zhuet al.(E93026), Phys. Rev. Lett.87, 081801 (2001), arXiv:nucl-ex/0105001

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  17. [17]

    The neutron charge form factor and target analyzing powers from pol.3He(pol e,e'n)-scattering

    J. Bermuthet al., Phys. Lett. B564, 199 (2003), arXiv:nucl-ex/0303015

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  18. [18]

    Measurement of the Electric Form Factor of the Neutron at Q^2=0.5 and 1.0 (GeV/c)^2

    G. Warrenet al.(Jefferson Lab E93-026), Phys. Rev. Lett.92, 042301 (2004), arXiv:nucl-ex/0308021

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  19. [19]

    D. I. Glazieret al., Eur. Phys. J. A24, 101 (2005), arXiv:nucl-ex/0410026

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  20. [20]

    The Charge Form Factor of the Neutron at Low Momentum Transfer from the $^{2}\vec{\rm H}(\vec{\rm e},{\rm e}'{\rm n}){\rm p}$ Reaction

    E. Geiset al.(BLAST), Phys. Rev. Lett.101, 042501 (2008), arXiv:0803.3827 [nucl-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  21. [21]

    Measurements of GEn/GMn from the ^2H(vec{e},e'vec{n})^1H Reaction to Q^2=1.45 (GeV/c)^2

    R. Madeyet al.(E93-038), Phys. Rev. Lett.91, 122002 (2003), arXiv:nucl-ex/0308007

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  22. [22]

    Measurements of the neutron electric to magnetic form factor ratio GEn/GMn via the ^2H(\vec{e},e'\vec{n})^1H reaction to Q^2 = 1.45 (GeV/c)^2

    B. Plasteret al.(Jefferson Laboratory E93-038), Phys. Rev. C73, 025205 (2006), arXiv:nucl-ex/0511025

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  23. [23]

    C. B. Crawfordet al., Phys. Rev. Lett.98, 052301 (2007), arXiv:nucl-ex/0609007

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  24. [24]

    The Proton Elastic Form Factor Ratio at Low Momentum Transfer

    G. Ronet al., Phys. Rev. Lett.99, 202002 (2007), arXiv:0706.0128 [nucl-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  25. [25]

    Low Q^2 measurements of the proton form factor ratio $mu_p G_E / G_M$

    G. Ronet al.(Jefferson Lab Hall A), Phys. Rev. C84, 055204 (2011), arXiv:1103.5784 [nucl-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  26. [26]

    High Precision Measurement of the Proton Elastic Form Factor Ratio $\mu_pG_E/G_M$ at low $Q^2$

    X. Zhanet al., Phys. Lett. B705, 59 (2011), arXiv:1102.0318 [nucl-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  27. [27]

    J. J. Kelly, Phys. Rev. C70, 068202 (2004)

    2004

  28. [28]

    W. M. Alberico, S. M. Bilenky, C. Giunti, and K. M. Graczyk, Phys. Rev. C79, 065204 (2009), arXiv:0812.3539 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  29. [29]

    I. A. Qattan and J. Arrington, Phys. Rev. C86, 065210 (2012), arXiv:1209.0683 [nucl-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  30. [30]

    Nucleon electromagnetic form factors in two-flavour QCD

    S. Capitani, M. Della Morte, D. Djukanovic, G. von Hippel, J. Hua, B. J¨ ager, B. Knippschild, H. B. Meyer, T. D. Rae, and H. Wittig, Phys. Rev. D92, 054511 (2015), arXiv:1504.04628 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  31. [31]

    Y.-C. Jang, R. Gupta, H.-W. Lin, B. Yoon, and T. Bhattacharya, Phys. Rev. D101, 014507 (2020), arXiv:1906.07217 [hep-lat]

    work page arXiv 2020

  32. [32]

    Alexandrou, S

    C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, K. Hadjiyiannakou, K. Jansen, G. Koutsou, and A. Vaquero Aviles-Casco, Phys. Rev. D100, 014509 (2019), arXiv:1812.10311 [hep-lat]

    work page arXiv 2019

  33. [33]

    Alexandrou, S

    C. Alexandrou, S. Bacchio, G. Koutsou, B. Prasad, and G. Spanoudes, (2025), arXiv:2507.20910 [hep-lat]

    work page arXiv 2025

  34. [34]

    Pohlet al., Nature466, 213 (2010)

    R. Pohlet al., Nature466, 213 (2010)

    2010

  35. [35]

    Antogniniet al., Science339, 417 (2013)

    A. Antogniniet al., Science339, 417 (2013)

    2013

  36. [36]

    Evaluation of the strength of electron-proton scattering data for determining the proton charge radius

    M. Horbatsch and E. A. Hessels, Phys. Rev. C93, 015204 (2016), arXiv:1509.05644 [nucl-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  37. [37]

    Evaluation of the proton charge radius from e-p scattering

    J. Arrington and I. Sick, J. Phys. Chem. Ref. Data44, 031204 (2015), arXiv:1505.02680 [nucl-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  38. [38]

    D. W. Higinbotham, A. A. Kabir, V. Lin, D. Meekins, B. Norum, and B. Sawatzky, Phys. Rev. C93, 055207 (2016), arXiv:1510.01293 [nucl-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  39. [39]

    New measurement of the $1S-3S$ transition frequency of hydrogen: contribution to the proton charge radius puzzle

    H. Fleurbaey, S. Galtier, S. Thomas, M. Bonnaud, L. Julien, F. Biraben, F. Nez, M. Abgrall, and J. Gu´ ena, Phys. Rev. Lett.120, 183001 (2018), arXiv:1801.08816 [physics.atom-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  40. [40]

    Bezginov, T

    N. Bezginov, T. Valdez, M. Horbatsch, A. Marsman, A. C. Vutha, and E. A. Hessels, Science365, 1007 (2019)

    2019

  41. [41]

    Xionget al., Nature575, 147 (2019)

    W. Xionget al., Nature575, 147 (2019)

    2019

  42. [42]

    Grinin, A

    A. Grinin, A. Matveev, D. C. Yost, L. Maisenbacher, V. Wirthl, R. Pohl, T. W. H¨ ansch, and T. Udem, Science370, abc7776 (2020)

    2020

  43. [43]

    H. Atac, M. Constantinou, Z. E. Meziani, M. Paolone, and N. Sparveris, Nature Commun.12, 1759 (2021), arXiv:2103.10840 [nucl-ex]

    work page arXiv 2021

  44. [44]

    A. D. Brandt, S. F. Cooper, C. Rasor, Z. Burkley, D. C. Yost, and A. Matveev, Phys. Rev. Lett.128, 023001 (2022), arXiv:2111.08554 [physics.atom-ph]

    work page arXiv 2022

  45. [45]

    Scheidegger and F

    S. Scheidegger and F. Merkt, Phys. Rev. Lett.132, 113001 (2024)

    2024

  46. [46]

    P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, Rev. Mod. Phys.97, 025002 (2025), arXiv:2409.03787 [hep-ph]

    work page arXiv 2025

  47. [47]

    Maisenbacher, V

    L. Maisenbacher, V. Wirthl, A. Matveev, A. Grinin, R. Pohl, T. W. H¨ ansch, and T. Udem, Nature650, 845 (2026), arXiv:2602.14980 [physics.atom-ph]

    work page arXiv 2026

  48. [48]

    R. Pohl, R. Gilman, G. A. Miller, and K. Pachucki, Ann. Rev. Nucl. Part. Sci.63, 175 (2013), arXiv:1301.0905 [physics.atom-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  49. [49]

    Gao and M

    H. Gao and M. Vanderhaeghen, Rev. Mod. Phys.94, 015002 (2022), arXiv:2105.00571 [hep-ph]

    work page arXiv 2022

  50. [50]

    Antognini, F

    A. Antognini, F. Hagelstein, and V. Pascalutsa, Ann. Rev. Nucl. Part. Sci.72, 389 (2022), arXiv:2205.10076 [nucl-th]

    work page arXiv 2022

  51. [51]

    Nakamuraet al.(Particle Data Group), J

    K. Nakamuraet al.(Particle Data Group), J. Phys. G37, 075021 (2010)

    2010

  52. [52]

    Navaset al.(Particle Data Group), Phys

    S. Navaset al.(Particle Data Group), Phys. Rev. D110, 030001 (2024)

    2024

  53. [53]

    Wave Functions, Evolution Equations and Evolution Kernels from Light-Ray Operators of QCD

    D. M¨ uller, D. Robaschik, B. Geyer, F. M. Dittes, and J. Hoˇ rejˇ si, Fortsch. Phys.42, 101 (1994), arXiv:hep-ph/9812448

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  54. [54]

    Deeply Virtual Compton Scattering

    X.-D. Ji, Phys. Rev. D55, 7114 (1997), arXiv:hep-ph/9609381

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  55. [55]

    A. V. Radyushkin, Phys. Lett. B385, 333 (1996), arXiv:hep-ph/9605431

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  56. [56]

    Hard Exclusive Reactions and the Structure of Hadrons

    K. Goeke, M. V. Polyakov, and M. Vanderhaeghen, Prog. Part. Nucl. Phys.47, 401 (2001), arXiv:hep-ph/0106012

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  57. [57]

    Generalized Parton Distributions

    M. Diehl, Phys. Rept.388, 41 (2003), arXiv:hep-ph/0307382

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  58. [58]

    G. A. Miller, Phys. Rev. Lett.99, 112001 (2007), arXiv:0705.2409 [nucl-th]. 17

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  59. [59]

    Lorc´ e, Charge Distributions of Moving Nucleons, Phys

    C. Lorc´ e, Phys. Rev. Lett.125, 232002 (2020), arXiv:2007.05318 [hep-ph]

    work page arXiv 2020

  60. [60]

    Kim and H.-C

    J.-Y. Kim and H.-C. Kim, Phys. Rev. D104, 074003 (2021), arXiv:2106.10986 [hep-ph]

    work page arXiv 2021

  61. [61]

    The angular momentum controversy: What's it all about and does it matter?

    E. Leader and C. Lorc´ e, Phys. Rept.541, 163 (2014), arXiv:1309.4235 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  62. [62]

    Spin-orbit correlations in the nucleon

    C. Lorc´ e, Phys. Lett. B735, 344 (2014), arXiv:1401.7784 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  63. [63]

    Kim, H.-Y

    J.-Y. Kim, H.-Y. Won, H.-C. Kim, and C. Weiss, Phys. Rev. D110, 054026 (2024), arXiv:2403.07186 [hep-ph]

    work page arXiv 2024

  64. [64]

    E. V. Shuryak, Nucl. Phys. B203, 93 (1982)

    1982

  65. [65]

    Diakonov and V

    D. Diakonov and V. Y. Petrov, Nucl. Phys. B245, 259 (1984)

    1984

  66. [66]

    Diakonov and V

    D. Diakonov and V. Y. Petrov, Nucl. Phys. B272, 457 (1986)

    1986

  67. [67]

    Instantons in QCD

    T. Sch¨ afer and E. V. Shuryak, Rev. Mod. Phys.70, 323 (1998), arXiv:hep-ph/9610451

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  68. [68]

    Instantons at work

    D. Diakonov, Prog. Part. Nucl. Phys.51, 173 (2003), arXiv:hep-ph/0212026

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  69. [69]

    Liu, (2025), arXiv:2501.07776 [hep-ph]

    W.-Y. Liu, (2025), arXiv:2501.07776 [hep-ph]

    work page arXiv 2025

  70. [70]

    M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B147, 385 (1979)

    1979

  71. [71]

    Hadronic matrix elements of gluon operators in the instanton vacuum

    D. Diakonov, M. V. Polyakov, and C. Weiss, Nucl. Phys. B461, 539 (1996), arXiv:hep-ph/9510232

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  72. [72]

    H.-C. Kim, M. M. Musakhanov, and M. Siddikov, Phys. Lett. B633, 701 (2006), arXiv:hep-ph/0508211

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  73. [73]

    1/N_c corrections to the magnetic susceptibility of the QCD vacuum

    K. Goeke, H.-C. Kim, M. M. Musakhanov, and M. Siddikov, Phys. Rev. D76, 116007 (2007), arXiv:0708.3526 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  74. [74]

    H.-C. Kim, M. V. Polyakov, and K. Goeke, Phys. Rev. D53, 4715 (1996), arXiv:hep-ph/9509283

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  75. [75]

    M. V. Polyakov and H.-D. Son, Phys. Rev. D102, 114005 (2020), arXiv:2008.06270 [hep-ph]

    work page arXiv 2020

  76. [76]

    Improved Effective Action for Light Quarks Beyond Chiral Limit

    M. Musakhanov, Eur. Phys. J. C9, 235 (1999), arXiv:hep-ph/9810295

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  77. [77]

    Musakhanov, Nucl

    M. Musakhanov, Nucl. Phys. A699, 340 (2002)

    2002

  78. [78]

    Choi and H.-C

    Y. Choi and H.-C. Kim, Phys. Rev. D111, 074023 (2025), arXiv:2501.12114 [hep-ph]

    work page arXiv 2025

  79. [79]

    Diakonov, V

    D. Diakonov, V. Y. Petrov, and P. V. Pobylitsa, Nucl. Phys. B306, 809 (1988)

    1988

  80. [80]

    Wakamatsu and H

    M. Wakamatsu and H. Yoshiki, Nucl. Phys. A524, 561 (1991)

    1991

Showing first 80 references.