Quark Wigner distribution in frame-independent 3-dimensional space

Sujit Janaa; Vikash Kumar Ojha

REVIEW 4 major objections 5 minor 34 references

Quark Wigner distribution in frame-independent 3-dimensional space

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A quark's 3D position map shows atomic-orbital-like nodes.

desk verdict The 3D Wigner maps are a nice new visualization, but the atomic-orbital claim rests on an unexplained Fourier measure factor and an untested momentum cutoff, so it's not yet trustworthy. read the letter →

arxiv 2506.00510 v1 pith:VTQVHUED submitted 2025-05-31 hep-ph

Sujit Janaa , Vikash Kumar Ojha This is my paper

classification hep-ph

keywords WignerdistributiondressedquarkmodelGTMDslight-frontcoordinatesboost-invariantimpactparameterskewnessspatialquantizationphase-spacedensity

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

The paper tries to establish that a quark in a dressed quark model has a structured spatial probability distribution when viewed in a frame-independent three-dimensional position space built from the transverse impact parameter $\mathbf{b}_\perp$ and the boost-invariant longitudinal impact parameter $\sigma$. The computed Wigner distributions peak at the center, fall off in both directions, and contain symmetrically placed regions where the quark probability density is exactly zero, a pattern the authors compare to atomic orbitals. The result matters because it suggests that a quark's phase-space image inside a target is not a featureless smear but carries a spatial quantization structure, and it offers a concrete model-based target for future hadron-structure calculations. The qualitative shape is stable across polarization choices, although polarization sharpens the peaks and troughs.

What carries the argument

The central object is the three-dimensional Wigner distribution built from two Fourier transforms: one taking the skewness $\xi$ to the boost-invariant longitudinal impact parameter $\sigma=\tfrac{1}{2}b^-P^+$, and one taking the transverse momentum transfer $\Delta_\perp/(1-\xi^2)$ to the transverse impact parameter $\mathbf{b}_\perp$. The integrand is the quark\textendash quark correlator, expressed through generalized transverse momentum\textendash dependent parton distributions (GTMDs) $F_{1,i}$, $G_{1,i}$, $H_{1,j}$ at nonzero skewness; in the dressed quark model these GTMDs descend from the two-particle light-front wavefunction of the target state. The named objects are GTMDs, the generalization of parton distributions that keep both transverse momentum and transverse position information. This double-Fourier machinery converts the model's momentum-space wavefunctions into a spatial map that can be read as a probability density for finding the quark at a given $\sigma$ and $\mathbf{b}_\perp$.

What would settle it

Recompute the Wigner distributions with the $k_\perp$ integral extended well beyond 0.5 GeV, or with a smooth cutoff, while scanning the quark mass and coupling; if the zero-density rings move, disappear, or lose their symmetry, the claimed spatial quantization is a numerical artifact rather than a feature of the dressed quark model.

Watch

Extended reading notes

Core claim

Within the dressed quark model, the authors report that the quark Wigner distributions $\rho_{XY}(x,\sigma,\mathbf{b}_\perp)$ in boost-invariant three-dimensional space, after integrating over the quark transverse momentum $k_\perp$, are concentrated near $\mathbf{b}_\perp=0$ and $\sigma=0$, symmetric under reflection in both coordinates, and interspersed with regions of zero density that are symmetrically distributed around the origin. They interpret these nodal regions as resembling atomic orbitals, where occupation probability is concentrated in lobes separated by nodes, and take the pattern as evidence of spatial quantization around the target center. Their plots are made at fixed $x=0.3$ with the $k_\perp$ integral truncated to the interval $(0, 0.5)$ GeV, and the same qualitative features appear for unpolarized, longitudinally polarized, and transversely polarized targets and struck quarks.

Load-bearing premise

The pattern of zero-density regions rests on a truncated transverse momentum integral ($k_\perp$ from 0 to 0.5 GeV) together with unreported quark mass and coupling parameters; if the truncation or parameter choice distorts the integrand, the orbital-like nodes could be artifacts rather than quark structure.

Editorial extensions

If this is right

If the reported pattern is real, the quark at fixed $x=0.3$ is most likely found at the center of the target, with equal probability on either side in both the longitudinal and transverse directions.
The symmetric zero-density regions imply that the spatial density has genuine nodes, not merely a smooth exponential falloff, which is a sharp qualitative signature of the dressed quark model.
Because the overall shape barely changes when the target or quark polarization is changed, the peak-and-node structure is a property of the unpolarized Wigner distribution, not an accident of one spin configuration.
The same construction can be applied to other targets, such as hadrons and mesons, and to gluons, using the corresponding light-front wavefunctions, as the authors propose at the end of the paper.

Reading between the lines

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

The zero-density rings are demonstrated only under a transverse momentum cutoff at 0.5 GeV with no sensitivity study; a natural next step is to check whether the nodal pattern survives as the cutoff is raised or replaced by a smooth regulator.
If the nodes are physical, they should also appear as zeros in the DVCS amplitude in the same $(\sigma,\mathbf{b}_\perp)$ conjugate space, giving an indirect experimental avenue to test the model.
The analogy to atomic orbitals suggests that each node may trace back to a zero of the two-particle light-front wavefunction; locating those zeros explicitly would turn the resemblance into a quantitative prediction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a competent but incremental follow-up to the authors' own GTMD calculations. The new piece is the joint (σ, b⊥) Wigner maps in a boost-invariant 3D space, which I haven't seen elsewhere. The qualitative features – concentration near the center and inversion symmetry – are solid and likely robust. The paper's headline claim, however, is the appearance of zero-density regions that resemble atomic orbitals. That claim rests on two wobbly supports: an unexplained factor in the Fourier measure and an arbitrary transverse-momentum cutoff. Neither is adequately supported, so the orbital analogy is currently unverified.

What the paper does well: it systematically covers all polarization configurations for both target and struck quark, and it clearly explains the kinematic setup. The plots are readable and the central-peak behavior is consistent with physics expectations. The approach is a natural extension of [31–33], and the authors are honest that the GTMD inputs come from their earlier work.

The soft spots are real. First, the measure. Eq. (10) defines the Fourier transform over D⊥ = Δ⊥/(1−ξ²). Substituting D⊥ in terms of Δ⊥ should give a Jacobian (1−ξ²)^{-2}. Eq. (12) writes the Δ⊥ integral with a single (1−ξ²)^{-1} and no (2π)^{-2}, and Eqs. (17)–(24) use (1−ξ²)^{-3/2}. No derivation is given for these powers. The difference matters because the zeros of the oscillatory integrals are controlled by the phase weight; a wrong power can shift or wipe out the zeros. This is the load-bearing issue for the orbital claim.

Second, the numerics. The quark transverse momentum is integrated over (0, 0.5) GeV with no physical justification and no sensitivity test. The GTMD denominators from [32] don't obviously make the k⊥ integral converge fast, so the hard cutoff can impose artificial oscillations. The quark mass and coupling are not stated, so the plots are not reproducible without guessing. The orbital analogy is also purely qualitative – there's no comparison to actual orbital nodal patterns or any quantitative metric.

That said, the paper isn't a waste of time. The central-peak and symmetry observations would survive even if the measure factor is corrected. But the atomic-orbital conclusion, the part that makes the paper interesting, depends on exactly those unverified ingredients.

Who's this for? People working on light-front models and Wigner distributions of hadrons. It won't change experimental or phenomenological practice. Still, it's a reasonable model study that could be worth publishing after revision. I'd send it to a referee, not desk-reject. The referee should ask for a clean derivation of the measure factors, a cutoff sensitivity study, and explicit parameter values.

Referee Report

4 major / 5 minor

Summary. The manuscript computes quark Wigner distributions in a boost-invariant three-dimensional position space (transverse impact parameter b_perp and longitudinal coordinate sigma) within the dressed quark model. The distributions are obtained by Fourier transforming GTMDs with respect to the skewness xi and the transverse momentum transfer Delta_perp, using GTMD expressions from the authors' earlier paper [32]. Numerical plots are presented for various target and quark polarizations, showing central concentration, reflection symmetry, and zero-density regions that the authors compare to atomic orbitals. The claimed novelty is the extension of Wigner distribution studies to a frame-independent 3D coordinate space.

Significance. If the derivation and numerics were sound, the paper would provide a useful model calculation of quark phase-space structure in a boost-invariant 3D setting, extending prior 2D impact-parameter studies. The apparent 'atomic orbital' zero-density pattern, if genuine, would be an interesting prediction of spatial quantization. However, the central claim is not currently established because of an inconsistent Fourier measure and an untested transverse-momentum cutoff. The paper also benefits from a clear connection to the existing GTMD literature, and the qualitative symmetry and central-peak features are robust expectations for such model distributions.

major comments (4)

[Section 5, Eqs. (10)-(24)] The Fourier measure is inconsistent between Eq. (10), Eq. (12), and Eqs. (17)-(24). Since D_perp = Delta_perp/(1-xi^2), the measure in Eq. (10) is d^2 Delta_perp / [(2 pi)^2 (1-xi^2)^2], but Eq. (12) writes d^2 Delta_perp / (1-xi^2), and Eqs. (17)-(24) contain a factor (1-xi^2)^(-3/2) without derivation. This factor multiplies the phase of the oscillatory integrals and therefore controls the positions of the reported zero-density regions, so the central claim is not supported by a self-consistent derivation; please derive the Jacobian and any spinor-normalization factors explicitly.
[Section 6] The transverse momentum integral is truncated to |k_perp| in (0, 0.5) GeV with no sensitivity check. The GTMD denominators from Ref. [32] are not manifestly convergent in k_perp, so this hard cutoff can produce artificial oscillations and zero-density nodes. To establish that the orbital-like pattern is a property of the model rather than of the cutoff, show convergence as the cutoff increases and provide a physical justification for the chosen scale.
[Section 6] The numerical results are not reproducible because the quark mass m and coupling g are never stated. Since the nodal structure of the Wigner maps may depend on these parameters, provide the values used in Figs. 1-3 and study the parameter dependence of the claimed zero-density regions.
[Sections 4-6] The GTMD inputs are taken from Ref. [32] without being reproduced or summarized in this manuscript. Because the Wigner distributions are obtained as Fourier transforms of those GTMDs, the paper is not self-contained and the reader cannot verify the transformation without consulting the prior work; please include the relevant GTMD formulas (or an appendix) and specify the numerical integration grid and domains.

minor comments (5)

[Section 6] The paper refers to 'quark probability density' and 'zero-density regions,' but the Wigner distribution is a quasi-probability that can be negative; the atomic orbital analogy should be qualified accordingly.
[Equations (17)-(24)] Several formulas have ambiguous typesetting of fractions, e.g., '-i/m2(1-xi^2)^(3/2)'; please ensure all expressions are typeset unambiguously.
[Section 6] The integration range is stated as '(0 -> 0.5) GeV'; please specify that this is |k_perp| and give the full integration domains for xi and Delta_perp.
[Section 4] The symbol rho is used for both the Wigner distribution and the correlator; please use distinct notation or define the conventions clearly.
[Introduction] The definition of sigma = (1/2) b^- P^+ appears without derivation; it would help to explain why this variable is boost-invariant.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Wigner maps are Fourier transforms of independently derived GTMD inputs, with no fitted parameter renamed as a prediction.

full rationale

The derivation chain in this paper is: dressed-quark light-front wavefunctions (Eq. 5) lead to GTMDs, whose analytical expressions for non-zero skewness are taken from the authors' prior work [32]; the paper then constructs Wigner distributions by explicit Fourier transforms over the skewness and transverse momentum transfer (Eqs. 17-24) and plots the resulting functions. No parameter appearing in the plotted Wigner distributions is fitted to those distributions or to the claimed zero-density orbital-like pattern; the GTMDs are analytic inputs with stated model assumptions, and the Fourier transform is a fixed mathematical operation that is not adjusted to reproduce the output. The self-citations ([11], [32], [33]) supply the GTMD input, but they do not themselves assert the zero-density result, so the central claim does not reduce to a self-citation chain. Concerns raised in the manuscript review about the unexplained (1-xi^2)^{-3/2} prefactor, the k_perp integration cutoff, and unreported quark mass and coupling values are correctness, reproducibility, or numerical-stability issues, not circularity: altering these would change the numerical output but would not make the output an input of the calculation. The central claim is therefore a model-based prediction obtained by a well-defined transform of independent inputs, with no definitional equivalence or fitted-input-renamed-as-prediction step. Accordingly, no circular step is found and the score is 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

No new entities are postulated. The free parameters are model inputs and numerical choices; the axioms are standard light-front model assumptions plus reliance on the same group's earlier GTMD results.

free parameters (4)

quark mass m
Mass parameter of the dressed quark model; the numerical value used in the plots is not stated in the paper, though the GTMD expressions and all plotted distributions depend on it.
quark-gluon coupling g
Coupling in the one-loop dressed quark Fock state; the value used for the figures is not reported.
transverse momentum integration cutoff = 0.5 GeV
The paper integrates k_perp over (0, 0.5) GeV (Section 6) without stating a physical justification or testing the sensitivity of the plotted features to the upper limit.
longitudinal momentum fraction x = 0.3
All plots fix x = 0.3; no scan over x is presented, so the robustness of the orbital-like pattern in x is untested.

assumptions (4)

domain assumption One-loop Fock expansion of the dressed quark state is sufficient.
The target state in Eq. (3) is truncated at the two-particle sector; higher Fock states are neglected without error estimate.
standard math GTMD parametrization of the quark-quark correlator in Eqs. (7)-(9) is complete.
This parametrization follows [30] and is standard for spin-1/2 targets.
domain assumption Analytic GTMD expressions at nonzero skewness from [32] are correct.
The central numerical results are Fourier transforms of these external expressions; they are not reproduced in this paper and are authored by the same group (Ojha, Jana, Maji).
domain assumption The boost-invariant longitudinal coordinate sigma and transverse impact parameter b_perp give a frame-independent 3D position space.
This is the hadron-optics formalism from [25, 26]; the paper adopts it without critical discussion.

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Cite this review

Pith. "Pith review of Quark Wigner distribution in frame-independent 3-dimensional space." pith.science (2026). https://pith.science/paper/VTQVHUED

@misc{pith2026250600510,
  author       = {Pith},
  title        = {Pith review of: Quark Wigner distribution in frame-independent 3-dimensional space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VTQVHUED}},
  note         = {Machine review of arXiv:2506.00510}
}

read the original abstract

We investigate the quark Wigner distribution in a frame-independent, three-dimensional position space within the framework of the dressed quark model. It is observed that the distributions are concentrated near the center of the target and gradually diminish as one moves away in both the longitudinal and transverse directions. The distribution exhibits symmetry along both axes, indicating an equal probability of locating the quark in either direction around the center. Interestingly, the spatial profile of the distribution resembles that of atomic orbitals, where the probability of finding an electron is highest in certain regions compared to others.

Figures

Figures reproduced from arXiv: 2506.00510 by the authors.

**Figure 2.** Quark Wigner distributions in the longitudinally polarized tar [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Quark Wigner distributions in the transversely polarized target [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [32]

V . K. Ojha, S. Jana, T. Maji, Quark generalized tmds at skewness and wigner distributions in boost invariant longitudinal space, Phys- ical Review D 107 (7) (2023) 074040

work page 2023

[2] [1]

Mirhosseini, O

M. Mirhosseini, O. S. Maga ˜na Loaiza, C. Chen, S. M. Hashemi Rafsanjani, R. W. Boyd, Wigner distribution of twisted photons, Phys. Rev. Lett. 116 (2016) 130402. doi:10.1103/PhysRevLett.116.130402. URL https://link.aps.org/doi/10.1103/PhysRevLett. 116.130402

work page doi:10.1103/physrevlett.116.130402 2016

[3] [2]

Simon, E

R. Simon, E. C. G. Sudarshan, N. Mukunda, Gaussian-wigner distributions in quantum mechanics and optics, Phys. Rev. A 36 (1987) 3868–3880. doi:10.1103/PhysRevA.36.3868. URL https://link.aps.org/doi/10.1103/PhysRevA.36. 3868

doi:10.1103/physreva.36.3868 1987

[4] [3]

Radhakrishnan, V

R. Radhakrishnan, V . K. Ojha, Wigner distribution of Sine-Gordon and Kink solitons, Mod. Phys. Lett. A 37 (37n38) (2022) 2250236. arXiv:2205.02531, doi:10.1142/S0217732322502364

work page arXiv 2022

[5] [4]

Van Herstraeten, N

Z. Van Herstraeten, N. J. Cerf, Quantum wigner entropy, Phys. Rev. A 104 (2021) 042211. doi:10.1103/PhysRevA.104.042211. URL https://link.aps.org/doi/10.1103/PhysRevA.104. 042211

work page doi:10.1103/physreva.104.042211 2021

[6] [5]

Raussendorf, D

R. Raussendorf, D. E. Browne, N. Delfosse, C. Okay, J. Bermejo- Vega, Contextuality and wigner-function negativity in qubit quantum computation, Phys. Rev. A 95 (2017) 052334. doi:10.1103/PhysRevA.95.052334. URL https://link.aps.org/doi/10.1103/PhysRevA.95. 052334

doi:10.1103/physreva.95.052334 2017

[7] [6]

A. Mari, J. Eisert, Positive wigner functions render classical simulation of quantum computation e fficient, Phys. Rev. Lett. 109 (2012) 230503. doi:10.1103/PhysRevLett.109.230503. URL https://link.aps.org/doi/10.1103/PhysRevLett. 109.230503

doi:10.1103/physrevlett.109.230503 2012

[8] [7]

Boashash, Note on the use of the wigner distribution for time-frequency signal analysis, IEEE Transactions on Acous- tics, Speech, and Signal Processing 36 (9) (1988) 1518–1521

B. Boashash, Note on the use of the wigner distribution for time-frequency signal analysis, IEEE Transactions on Acous- tics, Speech, and Signal Processing 36 (9) (1988) 1518–1521. doi:10.1109/29.90380

work page doi:10.1109/29.90380 1988