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Six-dimensional light-front Wigner distributions of the proton

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Pith's one-line read The paper claims to compute, for the first time, all 16 leading-twist six-dimensional light-front Wigner distributions of the proton in a spectator-diquark model, covering every quark–proton polarization combination.

desk verdict A sound spectator-model calculation whose 'all 16 Wigner distributions' claim is not supported by its own equations, which define only 14; the missing two are the symmetric-traceless transverse-transverse functions. read the letter →

arxiv 2505.06615 v1 pith:4NYRDWEW submitted 2025-05-10 hep-ph

Yirui Yang , Tianbo Liu , Bo-Qiang Ma This is my paper · ORCID
classification hep-ph
keywords light-frontWignerdistributionsprotonstructurespectator-diquarkmodelGTMDsquarkorbitalangularmomentumboost-invariantlongitudinalcoordinatehelicitytensorcharge
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to show that the proton's internal quark structure can be described in full six-dimensional phase space: three position coordinates (one longitudinal, two transverse) plus three momentum coordinates, including how quark and proton spins correlate. It does this for all 16 independent leading-twist Wigner distributions, from unpolarized to longitudinal and transverse polarizations. If correct, this gives a complete tomographic picture of quarks inside the proton that reduces to known functions like PDFs, TMDs, GPDs, and GTMDs in the right limits and carries spin-orbit information. The payoff would be new observables for future scattering experiments.

What carries the argument

The central object is the six-dimensional light-front Wigner distribution, defined by a quark bilocal operator with a gauge link, averaged between proton states carrying longitudinal and transverse momentum transfers; a Fourier transform over the skewness $\xi$ and transverse transfer $\Delta_\perp$ converts those transfers into the boost-invariant longitudinal coordinate $\tilde z = b^- P^+$ and the transverse impact parameter $b_\perp$. The calculation expresses these distributions as overlaps of light-front wave functions for an active quark and a spectator diquark, with Melosh-Wigner rotation matrices supplying the relativistic spin structure and a BHL Gaussian wavefunction supplying the momentum dependence.

What would settle it

Compute the quark GTMDs of the proton from lattice QCD at nonzero skewness, Fourier-transform them to the same six-dimensional Wigner variables, and compare all 16 distributions (or their $\tilde z$- and $b_\perp$-moments) with these predictions; a mismatch in the sign, location, or relative strength of the central peaks and spin-orbit dipole lobes would show the model's proton structure is wrong.

Watch

Extended reading notes

Core claim

Within the light-front quark spectator-diquark model, the paper computes all 16 twist-two six-dimensional Wigner distributions of the proton as functions of the boost-invariant longitudinal coordinate $\tilde z$, momentum fraction $x$, transverse position $b_\perp$, and transverse momentum $k_\perp$. The calculation is a wave-function overlap built from the BHL Gaussian light-front wave function with Melosh-Wigner rotation matrices, using scalar and axial-vector diquark spectators to separate flavors. The paper shows that these distributions satisfy normalization identities yielding quark number, helicity $\Delta q$, and tensor charge $\Delta_T q$ moments, and that integrating over $\tilde z$ returns the known five-dimensional Wigner distributions at $\xi=0$. It further claims that the $\tilde z$-dependence reveals longitudinal localization, dipole patterns tied to spin-orbit coupling, and a quasi-probability character with non-positive-definite values.

Load-bearing premise

The load-bearing premise is that a proton can be faithfully described as one active quark plus a spectator diquark (a bound two-quark cluster) with the chosen Gaussian wavefunction, effective masses, and mixing angle giving the true shape of the proton's quark distributions.

Editorial extensions

If this is right

  • The complete set of 16 distributions provides a unified phase-space picture that reduces, by integration, to the unpolarized TMD $f_1$, helicity TMD $g_{1L}$, worm-gear functions, and the GPDs $H$ and $\tilde H$ in their respective limits.
  • The longitudinal coordinate $\tilde z$ becomes an accessible dimension in proton imaging: its Fourier link to skewness means $\tilde z$-dependent distributions could be constrained by diffractive patterns in deeply virtual Compton scattering.
  • The spin-orbit dipole structures in $\rho_{LL}$ and $\rho_{LT}$ give a concrete route to quark orbital angular momentum and its correlation with quark spin, quantified by moments such as $\ell_q^z$ and $C_q^z$.
  • The normalization checks tie the six-dimensional distributions to quark number, helicity, and tensor charge, providing sum rules that any future model or extraction must respect.
  • The non-positive-definite nature of these distributions signals genuine quantum interference in the proton's phase space, distinguishing them from classical probability densities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the model's quark-diquark picture is close to the proton, the same 16-fold classification and normalization identities should survive in lattice QCD or other first-principles approaches; a lattice computation of the underlying GTMDs at nonzero skewness would be the natural quantitative check.
  • The paper's qualitative claims about $\tilde z$ as a thickness coordinate could be tested by comparing its predicted diffraction-like longitudinal pattern with DVCS data or with a model-independent extraction of $\xi$-dependent GPDs.
  • Because the model omits T-odd contributions, the dipole and spin-orbit structures here are T-even predictions; adding gauge-link or final-state-interaction effects is a direct extension that could change the sign or shape of those lobes.
  • The parameter set (quark mass 0.33 GeV, spectator masses 0.6 and 0.8 GeV, mixing angle $\pi/4$) is the main knob: a systematic scan over spectator masses and wavefunction forms would show which features are robust proton phenomenology and which are model artifacts.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper presents a model calculation of six-dimensional light-front quark Wigner distributions of the proton in the light-front quark spectator-diquark model, with scalar and axial-vector spectators, a Brodsky-Huang-Lepage Gaussian wavefunction, and Melosh-Wigner rotations. It defines a set of polarization-projected Wigner distributions (Eqs. 20-29), gives an overlap representation (Eqs. 40-42), checks normalization identities (Eqs. 30-39), plots u- and d-quark distributions in several ~ z–b⊥ and ~ z–k⊥ planes for selected x values, and shows that integrating over ~ z reduces the six-dimensional functions to the known five-dimensional ones at ξ = 0 (Eqs. 61-64). The paper claims to provide the first complete calculation of all 16 leading-twist six-dimensional Wigner distributions of the proton.

Significance. If completed as claimed, this would be a useful phenomenological benchmark: the overlap representation is standard, the polarization sums in Eqs. (30)-(39) correctly reproduce quark number, helicity, and tensor charges, and the reduction to the five-dimensional Wigner distribution is a genuine consistency check. The model parameters are taken from the literature rather than fitted to the Wigner results, so the computation is not circular. These are real strengths. However, the manuscript as written does not deliver the advertised complete set of 16 distributions, so its central novelty claim is not yet supported.

major comments (2)
  1. [Sec. III.A, Eqs. (20)-(29)] The manuscript's central claim, stated in the abstract, Sec. I, and Sec. V, is the calculation of all 16 leading-twist six-dimensional Wigner distributions. The definitions in Eqs. (20)-(29), however, define only 14 independent scalar functions. The transverse-transverse block is projected only onto the trace (δ_{ij}) and antisymmetric (ε_{ij}) combinations, leaving the two symmetric traceless combinations (e.g., ρ_TT^{11}-ρ_TT^{22} and ρ_TT^{12}+ρ_TT^{21}) undefined. For a spin-1/2 quark in a spin-1/2 hadron the TT block is a 2×2 matrix with four independent components, so a complete 16-function enumeration requires these two additional distributions. This is an internal inconsistency between the stated result and the equations presented, and the sum rules (30)-(39) and the numerical figures likewise cover only the 14 defined structures.
  2. [Sec. III.A, Eq. (42)] The overlap representation for the transverse quark projector does not specify how the j=2 components are obtained. The text states that ↑ and ↓ denote transverse polarization along êe x and -êe x, and the right-hand side of Eq. (42) contains no dependence on the index j. As written, this defines only the j=1 projection; the j=2 projection, which enters ρ_UT^2, ρ_LT^2, and both ρ_TT and ρ_TT^⊥ through the sums in Eqs. (28)-(29), requires a different combination of the x-basis overlaps. The authors should provide the explicit j-dependence or state clearly that the transverse quantization axis is aligned with j for each component.
minor comments (3)
  1. [Sec. IV.C and Figs. 9-12] The text in Sec. IV.C says the three columns correspond to x = 0.25, 0.50, and 0.75, while all four figure captions (Figs. 9-12) say x = 0.10, 0.25, and 0.40; these must be reconciled.
  2. [Sec. IV.A, Eq. (39)] Equation (39) writes the argument of ρ_TT^⊥ as (~ z, b⊥, k⊥, x), omitting x from the integration on the left and listing x after b⊥ and k⊥; the argument order should be made uniform with the other distributions.
  3. [Sec. IV, opening paragraph] The text says the remaining polarization configurations are collected in “Appendix V,” but Section V is the Summary; either the label is a typo or the appendix is missing from the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Wigner-distribution calculation follows from an externally parameterized spectator-diquark model, and no prediction reduces to a fitted input or to a self-citation chain.

full rationale

The paper's central numerical results are self-contained evaluations of a model, not predictions forced by construction. The model inputs, m = 0.33 GeV, beta_D = 0.33 GeV, m_S = 0.6 GeV, m_V = 0.8 GeV, and theta = pi/4, are stated as adopted from the literature (Sec. IV), not fitted to the six-dimensional Wigner distributions. The distributions themselves are defined through overlap expressions in Eqs. (40)-(42) using the light-front wave functions of Sec. II, with the kinematics of Eqs. (43)-(46); the figures are numerical integrations of these definitions. The sum rules in Eqs. (30)-(39) are derived integrals of the defined distributions: for example, Eq. (30) equals the quark number N_q by normalization, and Eqs. (34) and (38) give model values of helicity and tensor charge after integration, not inputs imposed beforehand. The five-dimensional reduction in Eqs. (61)-(64) is a mathematical identity following from the Fourier transform of the z-tilde dependence, which the paper uses as a consistency check against earlier five-dimensional results; this is not circular because the six-dimensional objects are independently defined first. The formalism of the six-dimensional operator and the boost-invariant ztilde coordinate is credited to refs. [49]-[53]; even if those references include the current authors, they provide a parameter-free definition, and the present contribution is a model calculation within that framework rather than a result whose validity is made to rest on an unverified self-citation. No uniqueness theorem is invoked to forbid alternative choices, and no fitted parameter is renamed as a prediction. A separate issue, noted for completeness rather than as circularity, is that the abstract and Sec. V claim 16 independent Wigner distributions while Eqs. (20)-(29) explicitly define only 14 scalar functions, omitting the two symmetric traceless transverse-transverse components. That mismatch is a completeness or counting problem in the presentation, not a circular reduction of the derived results to their inputs.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The computation uses established machinery: the spectator-diquark decomposition, the BHL wavefunction, Melosh-Wigner rotations, and the six-dimensional Wigner operator from ref. [49]. No new particles, forces, or conserved quantities are introduced; the spectator diquarks are borrowed quasi-particles, and z-tilde is explicitly attributed to Brodsky and Miller [50-52]. The free parameters are all effective inputs adopted from prior literature rather than fitted to the Wigner distributions themselves. The dominant unstated assumptions are physical modeling choices, not mathematical identities.

free parameters (5)
  • quark mass m = 0.33 GeV
    Effective constituent quark mass used in the invariant mass (Eq. 9) and the Melosh rotation (Eq. 5); adopted from refs. [87, 94, 95]. It controls the width and location of the distributions.
  • harmonic oscillator scale beta_D = 0.33 GeV
    Width of the BHL wavefunction (Eq. 8); adopted from refs. [87, 94, 95]. It sets the transverse spatial and momentum scales of all plotted results.
  • scalar diquark mass m_S = 0.6 GeV
    Effective spectator mass for the scalar-diquark component (Sec. IV); the paper explicitly notes spectator masses are treated as effective parameters.
  • axial-vector diquark mass m_V = 0.8 GeV
    Effective spectator mass for the vector-diquark component (Sec. IV), which is needed for flavor separation.
  • mixing angle theta = pi/4
    Spin-flavor SU(6) breaking parameter in Eq. (2); chosen by hand, with precedent in refs. [63-66, 74-76].
assumptions (5)
  • domain assumption Proton state = scalar plus axial-vector diquark Fock decomposition (Eq. 2), with no explicit gluon degrees of freedom.
    This is the basis of the spectator model [40, 73]. The paper acknowledges gluon polarization at small x is omitted (Sec. IV.C) and that T-odd final-state interactions are not included (Secs. III.B, IV.D).
  • domain assumption The BHL Gaussian form phi_{qD} = A_{qD} exp(-M^2/8 beta_D^2) (Eq. 8) is the momentum-space wavefunction.
    Phenomenological choice from refs. [79-81, 86, 87]; other prescriptions (TK, CCP, VSGL) are listed but not used, and the paper admits different wavefunctions would alter the features (Sec. IV.B).
  • domain assumption The six-dimensional light-front Wigner operator (Eq. 18) with z-tilde = b^- P^+ (Eq. 49) is a valid phase-space quasi-distribution carrying longitudinal spatial information.
    Definition taken from ref. [49]; the paper argues z-tilde offers information beyond being the Fourier conjugate of xi (Sec. III.A) and concedes the functions are not positive definite (Sec. III.B).
  • standard math The Melosh-Wigner rotation matrices (Eqs. 5, 7) correctly map instant-form spin states to light-front helicities.
    Established results [60-62, 78]; used to build the Fock amplitudes in Eqs. (12)-(14), with the paper noting agreement with light-front field theory [77].
  • domain assumption The leading-twist overlap representation (Eqs. 40-42) with xi-integration over [-x, x] captures the full twist-two content.
    Follows the GTMD/Wigner literature [30-33, 49]; the xi-integration range is stated in Sec. III.A as the DGLAP region, and the model omits gauge-link and T-odd effects.

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Pith. "Pith review of Six-dimensional light-front Wigner distributions of the proton." pith.science (2026). https://pith.science/paper/4NYRDWEW

@misc{pith2026250506615,
  author       = {Pith},
  title        = {Pith review of: Six-dimensional light-front Wigner distributions of the proton},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4NYRDWEW}},
  note         = {Machine review of arXiv:2505.06615}
}
abstract

We investigate six-dimensional quark Wigner distributions of the proton in a light-front quark spectator-diquark model. Benefiting from the light-front boost-invariant longitudinal variable $\tilde{z}$, these light-front Wigner distributions provide complete information of parton distribution in the phase space and the correlation with spins. At the leading twist, one can define 16 independent distribution functions in according to different combinations of quark and proton polarizations. Numerical results of all these Wigner distributions are presented, unraveling rich structures of the proton, which may potentially provide new observables to be explored at future experiments.

Figures

Figures reproduced from arXiv: 2505.06615 by the authors.

Figure 1
Figure 1. FIG. 1. Six-dimensional unpolarized light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Six-dimensional unpolarized light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Six-dimensional unpolarized light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (37 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Six-dimensional unpolarized light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Six-dimensional longitudinal light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Six-dimensional longitudinal light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Six-dimensional longitudinal light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Six-dimensional longitudinal light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Six-dimensional longitudinal-transverse light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Six-dimensional longitudinal-transverse light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Six-dimensional longitudinal-transverse light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Six-dimensional longitudinal-transverse light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Six-dimensional transverse-longitudinal light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Six-dimensional transverse-longitudinal light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Six-dimensional transverse-longitudinal light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Six-dimensional transverse-longitudinal light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Six-dimensional unpolarized-longitudinal light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Six-dimensional unpolarized-longitudinal light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. Six-dimensional unpolarized-longitudinal light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p024_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Six-dimensional unpolarized-longitudinal light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p024_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. Six-dimensional unpolarized-transverse light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p025_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. Six-dimensional unpolarized-transverse light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p026_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23. Six-dimensional unpolarized-transverse light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p026_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24. Six-dimensional unpolarized-transverse light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p027_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. Six-dimensional longitudinal-unpolarized light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p028_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26. Six-dimensional longitudinal-unpolarized light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p028_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27. Six-dimensional longitudinal-unpolarized light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p029_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28. Six-dimensional longitudinal-unpolarized light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p029_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29. Six-dimensional transverse-unpolarized light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p031_29.png]
Figure 30
Figure 30. Figure 30: FIG. 30. Six-dimensional transverse-unpolarized light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p031_30.png]
Figure 31
Figure 31. Figure 31: FIG. 31. Six-dimensional transverse-unpolarized light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p032_31.png]
Figure 32
Figure 32. Figure 32: FIG. 32. Six-dimensional transverse-unpolarized light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p032_32.png]
Figure 33
Figure 33. Figure 33: FIG. 33. Six-dimensional transverse light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p033_33.png]
Figure 34
Figure 34. Figure 34: FIG. 34. Six-dimensional transverse light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p034_34.png]
Figure 35
Figure 35. Figure 35: FIG. 35. Six-dimensional transverse light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p034_35.png]
Figure 36
Figure 36. Figure 36: FIG. 36. Six-dimensional transverse LF Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p035_36.png]
Figure 37
Figure 37. Figure 37: FIG. 37. Six-dimensional pretzelous light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p036_37.png]
Figure 38
Figure 38. Figure 38: FIG. 38. Six-dimensional pretzelous light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p036_38.png]
Figure 39
Figure 39. Figure 39: FIG. 39. Six-dimensional pretzelous light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p037_39.png]
Figure 40
Figure 40. Figure 40: FIG. 40. Six-dimensional pretzelous light-front Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p037_40.png]

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