Unveiling Skewness Dependence of Quark Wigner Distributions

Sujit Jana; Vikash Kumar Ojha

REVIEW 4 major objections 5 minor 61 references

Unveiling Skewness Dependence of Quark Wigner Distributions

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read In the light-front dressed quark model, all leading-twist quark Wigner distributions become skewness-dependent: circular forward-limit patterns give way to dipole and quadrupole lobes, asymmetries, localization, and sign-changing regions…

desk verdict A useful numerical catalog of skewness-dependent quark Wigner distributions in the dressed quark model, but the novelty is overstated and the transform normalization needs a check before the quantitative claims are trusted. read the letter →

arxiv 2507.18168 v1 pith:T2TI4QBV submitted 2025-07-24 hep-ph

Sujit Jana , Vikash Kumar Ojha This is my paper · ORCID

classification hep-ph PACS 12.38.-t13.60.Hb14.20.Dh

keywords Wignerdistributionsskewnessgeneralizedtransversemomentumdependentlight-frontdressedquarkmodeldipoleandquadrupolepatternsspin-orbitcorrelationsorbitalangularhadrontomography

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 asks what happens to the quantum phase-space picture of a quark when the initial and final hadron states carry different longitudinal momentum, measured by the skewness $\xi$. Working in the analytically solvable light-front dressed quark model, the authors compute all nine leading-twist quark Wigner distributions for unpolarized, longitudinally polarized, and transversely polarized quark and target configurations, at $\xi = 0$, $0.25$, and $0.5$. They find that nonzero $\xi$ systematically breaks the circular symmetry of the forward limit, producing dipole and quadrupole patterns, momentum-space shifts, negative density regions, and increasing localization. They interpret these features as spin-orbit correlations and quantum interference between light-front wave function components that differ in orbital angular momentum. If the claim holds, the model gives concrete off-forward predictions for phase-space structure that connect generalized parton distributions and transverse-momentum-dependent distributions.

What carries the argument

The central object is the quark Wigner distribution, a quantum phase-space quasi-probability distribution over the quark's transverse position $\mathbf{b}_\perp$ and transverse momentum $\mathbf{k}_\perp$, defined as a Fourier transform of the fully unintegrated quark-quark correlator with respect to the rescaled transverse momentum transfer $\mathbf{D}_\perp = \boldsymbol{\Delta}_\perp/(1-\xi^2)$, including a $(1-\xi^2)^{-3/2}$ prefactor. The input is the two-particle light-front wave function of a quark dressed by a gluon, which enters the analytic expressions for all leading-twist GTMDs $F_{1i}, G_{1i}, H_{1j}$. The paper inserts those GTMDs into the correlator decompositions of Eqs. (13)-(15), combines them according to the polarization definitions of Eqs. (12), (19), and (23), and numerically Fourier-transforms the results to obtain the nine Wigner distributions, tracking how the phases and $(1-\xi^2)$ factors alter them as $\xi$ varies.

What would settle it

Recompute the nine Wigner distributions with $\boldsymbol{\Delta}_\perp$ itself (no $(1-\xi^2)$ rescaling in Eq. (9)) as the Fourier variable. If the dipole and quadrupole structures and the localization trend disappear or change sign as $\xi$ grows, the paper's skewness effects are artifacts of the rescaling. A second check: integrate $\rho_{UU}$ over $\mathbf{b}_\perp$ and $\mathbf{k}_\perp$ and verify the result is independent of $\xi$; a $\xi$-dependent normalization would show the distribution is not behaving as a phase-space density.

Watch

Extended reading notes

Core claim

The central discovery is that turning on $\xi$ alters every quark Wigner distribution in the dressed quark model, not just the size of the phase-space support. The unpolarized $\rho_{UU}$, approximately circular at $\xi=0$, distorts and shifts as $\xi$ grows; $\rho_{UL}$ and $\rho^j_{UT}$ show persistent dipole structures in both impact-parameter and momentum space; $\rho_{LL}$ starts as a quadrupole and develops a dipole-like momentum asymmetry; and $\rho^i_{TU}$ and $\rho^i_{TL}$ acquire multipole interference patterns with sign-changing regions. These patterns are shown in transverse impact-parameter space, transverse momentum space, and mixed $(b_x,k_y)$ space. The paper attributes the multipoles to spin-orbit correlations and to interference between light-front wave function components whose orbital angular momentum differs by one or two units.

Load-bearing premise

The result stands on the convention, inherited from earlier off-forward studies, that the Fourier conjugate to the transverse impact parameter is $\mathbf{D}_\perp = \boldsymbol{\Delta}_\perp/(1-\xi^2)$ with a $(1-\xi^2)^{-3/2}$ prefactor; if that is not the correct $\xi$-generalization of the Wigner phase-space definition, every plotted pattern and localization trend changes.

Editorial extensions

If this is right

At nonzero $\xi$, all nine leading-twist quark Wigner distributions lose the rotational symmetry of the forward limit, so any observable sensitive to off-forward phase space should see dipole or quadrupole lobes and negative regions.
The transverse localization that grows with $\xi$ means large-skewness kinematics sharply reduce the overlap between initial and final state wave functions, redistributing the quark density into a narrower phase-space region.
The dipole in $\rho_{UL}$ and the quadrupole in $\rho_{LL}$ provide model-level signatures of spin-orbit correlation and quark orbital angular momentum that can be compared with other hadron models.
The momentum-space asymmetries, analogous to T-odd transverse-momentum-dependent effects, are enhanced by skewness, so off-forward kinematics amplify spin-momentum correlations.
Because the GTMDs carry explicit $\xi$ factors such as $(1-\xi^2)$ and $(x^2-\xi^2)$, the model makes quantitative predictions for how each multipole moment of the Wigner distribution scales with skewness, testable in lattice QCD or in other light-front models.

Reading between the lines

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

The paper does not compare its rescaled Fourier convention against the alternative of Fourier transforming directly in $\boldsymbol{\Delta}_\perp$; that comparison would show whether the reported multipoles survive under a different off-forward phase-space definition.
A natural next step is to apply the same skewness-dependent machinery to gluon Wigner distributions in the same dressed quark model; if the gluon patterns mimic the quark ones, the multipole mechanism is generic rather than quark-specific.
One could also test the paper's localization claim by computing the variance $\langle \mathbf{b}_\perp^2\rangle$ as a function of $\xi$ from the plotted distributions; a monotonic decrease would confirm the squeezing, while a plateau would call the interpretation into question.

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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

Short version: the paper takes the known analytic GTMDs at nonzero skewness and computes all nine leading-twist quark Wigner distributions in b_perp, k_perp, and mixed space. The figures are clean and the organization is good. If you work with this model, this is a handy reference.

What is actually new: the systematic numerical survey of xi dependence across polarization channels, and the observation that dipoles, quadrupoles, and localization grow with xi. That is a legitimate extension of the earlier analytic work in refs [39,40], and the qualitative message—off-forward phase space is richer than the forward limit—is plausible.

The soft spots are in the framing and the normalization. The abstract says previous studies were largely forward-limit, but refs [39,40], including one by the same group, already derived the non-zero-xi GTMDs and the Wigner master formulas. The new content here is re-expression and visualization of those results. Calling it 'one of the most comprehensive analyses' is a stretch.

More seriously, the transform in Eqs. (9)–(16) uses a (1-xi^2)^(-3/2) prefactor. Under the substitution D_perp = Delta_perp/(1-xi^2), the phase-space Jacobian alone gives (1-xi^2)^(-2); getting to (-3/2) requires a compensating (1-xi^2)^(1/2) from the spinor factors in Eq. (13), which is never shown. The prefactor depends on xi but not on Delta_perp, so the shape in any single panel is likely unaffected, but the relative normalization across xi—and any quantitative localization claim—depends on this convention. It should be checked against the original derivation in refs [39,40].

There is also an internal inconsistency: Sec. IV A describes rho_UU at xi=0 as having 'approximate rotational symmetry and a quadruple-like structure,' while the conclusion calls it 'circular symmetry.' Those are not the same; the baseline story should be fixed.

Minor points: no derivation of the GTMDs (appendix just lists them), no code or data, and the numerical choices (m=0.0033 GeV, Delta_max=1 GeV, fixed slices) are stated but not convergence-checked. These are minor for a model study.

Bottom line: the paper is a useful resource and the central qualitative claim holds up as a model-level prediction. The Jacobian convention and the xi=0 description need a referee's attention, but they are fixable. I would send it to review rather than desk-reject; with the normalization sorted out, it can be a citeable reference. If you are in the subfield, you might want it on your reading list, but it is not a major advance.

Referee Report

4 major / 5 minor

Summary. This manuscript studies the skewness (ξ) dependence of the nine leading-twist quark Wigner distributions in the light-front dressed quark model. Starting from published analytic GTMDs with nonzero ξ (Appendix A, taken from Ref. [39]) and from Wigner master formulas quoted from Refs. [39,40], it evaluates ρ for unpolarized, longitudinally, and transversely polarized quarks in targets of each polarization, plotting the distributions in transverse impact-parameter space, transverse momentum space, and a mixed (bx,ky) space for ξ=0, 0.25, 0.5, and for ρ_UU as a function of ξ from 0 to 0.5. The central claim is that increasing ξ distorts the forward-limit distributions, produces dipole and quadrupole patterns, breaks rotational/time-reversal symmetry, and localizes the phase-space support through reduced initial-final overlap and interference between light-front wave function components of different orbital angular momentum. The paper contains no data fitting; all results are numerical evaluations of closed-form model expressions.

Significance. The paper's qualitative claims—ξ-dependent localization, symmetry breaking, and multipole formation in every leading-twist channel—are, if correct, a useful model-level benchmark for off-forward phase-space tomography and could guide lattice QCD and exclusive-process phenomenology. The manuscript deserves credit for covering all nine polarization channels systematically, for working with analytic input, and for not introducing any fit parameters. In its present form, however, the contribution is essentially a numerical elaboration of self-cited analytic results [39,40], and the two load-bearing conventions it imports (the D⊥-space Wigner definition and the form of the GTMDs) are not derived or checked here. The significance is therefore conditional on resolving the technical issues below; the qualitative pattern claims are falsifiable and would be interesting if the definitions are confirmed.

major comments (4)

[Sec. III, Eqs. (9), (10), (16)-(26)] The off-forward Wigner transform is not derived. With D⊥ = Δ⊥/(1−ξ²), the Jacobian is d²D⊥ = (1−ξ²)^{-2} d²Δ⊥, so Eq. (10) combined with the change of variables gives a prefactor (1−ξ²)^{-2}, not (1−ξ²)^{-3/2}, unless the spinor contraction in Eq. (13) supplies an additional (1−ξ²)^{1/2}. The paper simply cites [39,40] for Eqs. (16)–(26), but those equations are the paper's own central definitions and the extra factor is never exhibited. This matters because the overall (1−ξ²)-dependent prefactor rescales every ξ column in Figs. 1–7, so the claimed 'more pronounced' multipole patterns and localization with increasing ξ may be partly an artifact of the normalization convention. The authors should derive one representative Wigner formula from Eqs. (11)–(15), or state explicitly the convention used in [39,40] for the spinor factor.
[Appendix A, Eqs. (A2), (A3), (A9)–(A16), and (A19)] The auxiliary function β in Eq. (A19) is α/[(1−x)(k2Δ1−k1Δ2)], and several GTMDs contain terms without a cross-product factor in the numerator. For example, F1,2 and the second square bracket in F1,3 are proportional to β times quantities such as 4m²ξ(1+x)k⊥·Δ⊥ or 8m²(1−x)²[2(1+x)ξk⊥−(1−x)xΔ⊥]·k⊥, which do not vanish when k∥Δ. As written, these GTMDs therefore have singularities along Δ⊥∥k⊥, and the subsequent 2D Fourier integrals over Δ⊥ would be at least logarithmically divergent unless regularized. This is a load-bearing issue for the numerical analysis because the Wigner plots of ρ_TU, ρ_TL, ρ_TT, and ρ_UL inherit such terms. Please verify these expressions against the original derivation, correct any transcription errors, and state how the numerical integration avoids the singular locus.
[Sec. IV.A and Sec. V (Conclusion)] There is a direct contradiction about the ξ=0 baseline. Section IV A states that at ξ=0 in impact-parameter space ρ_UU has 'approximate rotational symmetry and a quadruple-like structure,' while the concluding paragraph says it 'shows circular symmetry at ξ = 0.' A quadrupole pattern is not circular, and with k⊥ fixed to 0.4 y-hat one would not expect exact circular symmetry at ξ=0. The authors should specify the precise zero-skewness baseline and then formulate the 'emergence' of multipoles with ξ relative to that baseline; as written, the central qualitative claim is internally inconsistent.
[Sec. IV, Numerical Analysis] The numerical setup is under-specified in ways that affect reproducibility and interpretation. The text says 'x was integrated over the DGLAP region (ξ < x < 1),' but ρ in Eqs. (16)–(26) is a function of x and no x-integrated quantity is defined; if x was integrated, the plots should be labeled or the equations changed. In addition, the values of m=0.0033 GeV and Δ_max=1 GeV, the fixed transverse coordinates k⊥=0.4 y-hat and b⊥=0.4 y-hat GeV^-1, and the mixed-space integration ranges [0,0.4] are stated without justification, and no convergence check with respect to Δ_max or the grid is reported. Since the conclusions are purely qualitative statements about the shapes of the plots, a sensitivity check is needed to establish that the observed lobes and localizations are not numerical artifacts.

minor comments (5)

[Sec. III, Eq. (10)] Eq. (10) writes W[Γ](x,ξ,Δ⊥,k⊥;S) but the integration variable is D⊥; after the substitution the argument should be written consistently as a function of D⊥ (or the relation should be stated).
[Appendix A, Eq. (A18)] The quantities q⊥, y, q′⊥, and x′ entering D(q⊥,y) and D∗(q′⊥,x′) are never defined in the manuscript, so the appendix is not self-contained; define them in terms of x, ξ, k⊥, and Δ⊥ or state the mapping to Ref. [39] explicitly.
[Appendix A, p. 24] The phrase 'The and auxiliary functions α, and D are defined as:' contains a typo and should read 'The auxiliary functions α and D are defined as:'.
[Abstract and Sec. IV] In the abstract, 'using the analytical expression all leading-twist GTMDs' is missing 'of'; and in Sec. IV, 'The parton longitudinal momentum fraction x was integrated over' conflicts with the x-dependent definitions in Eqs. (16)–(26).
[Fig. 1] Figure 1's caption uses 'left, middle, and right panels' but does not label the rows; given that the text refers to top/middle/bottom rows, adding row labels would aid readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the skewness-dependent Wigner distributions are Fourier transforms of previously published GTMDs, not fitted parameters or definitional identities.

full rationale

The paper's derivation chain is straightforward: it takes the analytic skewness-dependent GTMDs for the light-front dressed quark model (Appendix A, with the computation attributed to ref. [39]), inserts them into the defining Fourier transform of the off-forward Wigner distribution (Eqs. (9)-(10), with master formulas cited from refs. [39,40]), and numerically evaluates the resulting integrals. No parameter is fitted to a subset of data and then renamed as a prediction, and no uniqueness theorem is invoked to force the chosen Wigner convention. The observed dipole/quadrupole/localization features are literal Fourier transforms of the input GTMDs; they are consequences of the stated model input, not identities that presuppose the conclusions. The use of refs. [39,40] is indeed a self-citation chain, since the GTMD input and the Wigner master formulas are not rederived in this paper, but the cited GTMD computation is parameter-free within the stated dressed-quark assumptions and those assumptions do not include the Wigner-distribution phenomenology being reported. The cited result is therefore real prior evidence rather than a circular justification. Concerns about the (1-xi^2)^(-3/2) normalization factor in Eqs. (16)-(26) and about the wording 'quadruple-like' versus 'circular symmetry' at xi=0 are correctness or consistency issues, not instances of a derivation reducing to its own inputs.

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

The central claim rests on five hand-chosen numerical parameters and several imported modeling assumptions. No new particles, forces, or conserved quantities are introduced. The largest burden is the Fourier-transform prescription and the borrowed GTMD expressions, which are not independently verified in this manuscript.

free parameters (5)

quark mass m = 0.0033 GeV
Chosen model input in Sec. IV; controls phase-space scales and denominators of the light-front wave functions.
transverse momentum transfer cutoff Delta_max = 1 GeV
Used as the upper limit for numerical Fourier integrals in Sec. IV; no convergence study is provided.
fixed transverse momentum k_perp = 0.4 y-hat GeV = 0.4 y-hat GeV
Slice choice for impact-parameter-plane plots; hand-picked and not varied.
fixed impact parameter b_perp = 0.4 y-hat GeV^-1 = 0.4 y-hat GeV^-1
Slice choice for transverse-momentum-plane plots; hand-picked and not varied.
mixed-space integration ranges kx, by in [0, 0.4] = kx and by in [0, 0.4]
Ad hoc integration window used in mixed-space panels; can alter the apparent ridge and lobe structures.

assumptions (5)

domain assumption Dressed quark Fock-state truncation to the quark plus one gluon sector
Invoked in Eq. (3); all results depend on this truncation and no higher Fock states are included.
domain assumption Leading-twist GTMD parametrization of the quark-quark correlator in Eqs. (13)-(15)
Adopted from refs [25,39,40]; the paper does not derive or re-justify this decomposition at nonzero xi.
domain assumption Wigner distributions obtained by Fourier transform over D_perp = Delta_perp/(1-xi^2) with prefactor (1-xi^2)^(-3/2)
Stated in Eqs. (9)-(10) and (16); this off-forward generalization of the standard xi=0 definition is not justified in the text.
domain assumption Quark longitudinal fraction x integrated only over the DGLAP region (xi < x < 1)
Sec. IV; the ERBL region is excluded without presenting an alternative result.
standard math Light-front gauge A+ = 0 makes the Wilson line unity
Used after Eq. (11); a standard simplification in perturbative light-front models.

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Cite this review

Pith. "Pith review of Unveiling Skewness Dependence of Quark Wigner Distributions." pith.science (2026). https://pith.science/paper/T2TI4QBV

@misc{pith2026250718168,
  author       = {Pith},
  title        = {Pith review of: Unveiling Skewness Dependence of Quark Wigner Distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T2TI4QBV}},
  note         = {Machine review of arXiv:2507.18168}
}

abstract

We present a detailed investigation of the skewness dependence of quark Wigner distributions within the light-front dressed quark model. While previous studies have largely focused on the forward limit, we explore the impact of nonzero longitudinal momentum transfer ($\xi \neq 0$) on the full set of leading-twist quark Wigner distributions across various polarization configurations. We observe characteristic distortions in the spatial and momentum correlations with increasing skewness, including the emergence of dipole and quadrupole patterns, asymmetries, and localization effects. These features reflect spin-orbit correlations and quantum interference between light-front wave function components with differing orbital angular momentum.

Figures

Figures reproduced from arXiv: 2507.18168 by the authors.

**Figure 2.** FIG. 2: The evolution of the quark Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** FIG. 3: The quark Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: displays the Wigner distribution ρ x UT , which corresponds to a transversely polarized quark (polarized along the ˆx direction) in an unpolarized target. In the impact parameter space (a) (b) (c) (d) (e) (f) (g) (h) (i) FIG. 4: The quark Wigner distribution ρ x UT in …

**Figure 5.** Figure 5: FIG. 5: The quark Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: FIG. 6: The quark Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: FIG. 7: The quark Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

Discussion (0). Continue with ORCID to comment.

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