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arxiv: 2408.03600 · v2 · submitted 2024-08-07 · ✦ hep-ph

T-odd Wigner Distributions in boost-invariant longitudinal position space and Spin-momentum correlation in proton

Tanmay Maji This is my paper

Pith reviewed 2026-05-23 22:23 UTC · model grok-4.3

classification ✦ hep-ph
keywords T-odd GTMDsWigner distributionsboost-invariant position spaceSivers functionBoer-Mulders functionproton spin-momentum correlationDGLAP regionmomentum transfer dependence
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The pith

T-odd GTMDs transformed to boost-invariant σ-space display oscillatory patterns sensitive to total momentum transfer squared -t, with interference from transverse momenta.

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

The paper examines the skewness sensitivity of T-odd leading twist GTMDs in the DGLAP region when momentum transfers occur in both longitudinal and transverse directions. Transforming these distributions to the boost-invariant longitudinal position variable σ, the Fourier conjugate of skewness, produces oscillatory patterns that depend on the square of the total momentum transfer -t. These patterns resemble diffraction scattering of waves. An extra modulation of the diffraction arises from interference between the transverse momentum transfer and the transverse momentum of quarks. The work additionally maps the correlations between proton spin and constituent transverse momenta using the Sivers and Boer-Mulders Wigner distributions expressed in the same σ-space.

Core claim

In boost-invariant longitudinal position space the T-odd Wigner distributions exhibit oscillatory patterns whose period and amplitude depend on the square of the total momentum transfer -t, with an additional modulation caused by interference between the transverse momentum transfer bfd and the quark transverse momentum bfp; the same σ-space framework also displays the spin-momentum correlations encoded in the Sivers and Boer-Mulders functions.

What carries the argument

The Fourier transform of T-odd GTMDs to the boost-invariant longitudinal position variable σ (conjugate to skewness ξ), performed in the DGLAP region with both longitudinal and transverse momentum transfers present.

If this is right

  • The oscillatory σ-space patterns provide a direct probe of the momentum-transfer dependence of T-odd GTMDs.
  • Interference between bfd and bfp introduces a measurable correction to the diffraction-like pattern.
  • Sivers and Boer-Mulders Wigner distributions in σ-space encode explicit spin-transverse-momentum correlations inside the proton.
  • The optical analogy suggests that position-space distributions can be interpreted as wave diffraction in the proton.

Where Pith is reading between the lines

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

  • The same σ-space method could be applied to other twist-two GTMDs to extract longitudinal position information.
  • Future collider data sensitive to skewness and transverse momentum transfer could test the predicted interference term.
  • If the diffraction analogy holds, it may link to wave-like features in other QCD observables such as diffraction in exclusive processes.

Load-bearing premise

The chosen parametrization of the T-odd GTMDs remains valid throughout the DGLAP region once both longitudinal and transverse momentum transfers are included.

What would settle it

A measurement of the σ-space distributions that shows no oscillatory dependence on -t or no additional modulation from the interference between bfd and bfp would falsify the reported patterns.

Figures

Figures reproduced from arXiv: 2408.03600 by Tanmay Maji.

Figure 1
Figure 1. Figure 1: FIG. 1: SIDIS process at (a) tree level and (b)including final-state interaction (FSI) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Model results of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Model results of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Spin density for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Sivers Wigner distribution in boost invariant longitudinal space at [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Boer-Mulders Wigner distribution in boost invariant longitudinal space at [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The flavor dependent Sivers and Boer-Mulders Wigner Distribution in transverse momentum plane (upper row) with [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

The boost invariant longitudinal position space variable $\sigma$, which is the Fourier conjugate to skewness $\xi$, unravels the longitudinal impact parameter information in a proton. Here, we investigate the skewness sensitivity of T-odd leading twist GTMDs within the Dokshitzer Gribov Lipatov Altarelli Parisi (DGLAP) region, considering a momentum transfer to longitudinal as well as transverse direction. The $\sigma$-space distributions of the T-odd sector show oscillatory patterns that are sensitive to the square of the total momentum transfer $-t$, which is analogous to the diffraction scattering of waves in Optics. An additional effect on the diffraction pattern is reported, caused by interference between transverse momentum transfer $\bfd$ to the transverse momentum $\bfp$ of quarks. We also present the correlation of proton spin to the transverse momentum of constituents through Sivers and Boer-Mulders Wigner Distributions in boost invariant longitudinal position space.

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 / 0 minor

Summary. The paper investigates the skewness sensitivity of T-odd leading-twist GTMDs in the DGLAP region (|x| > |ξ|) for a proton, incorporating momentum transfers in both longitudinal (via skewness ξ) and transverse directions. Using the boost-invariant longitudinal position-space variable σ (Fourier conjugate to ξ), it reports that the resulting σ-space distributions of the T-odd sector exhibit oscillatory patterns whose amplitude and form depend on the squared total momentum transfer −t, with an analogy drawn to optical diffraction. An additional modulation of these patterns is attributed to interference between the transverse momentum transfer Δ⊥ and the quark transverse momentum p⊥. The work also computes the Sivers and Boer-Mulders Wigner distributions in σ-space to illustrate spin-momentum correlations.

Significance. If the reported oscillatory behavior and interference survive changes in parametrization, the results would supply a concrete illustration of how longitudinal position-space information encoded in GTMDs can produce diffraction-like signatures, potentially guiding future phenomenological studies of T-odd distributions at facilities sensitive to skewness and transverse momentum. The explicit construction of σ-space Wigner distributions for Sivers and Boer-Mulders functions adds a useful visualization tool, though its impact remains tied to the underlying model assumptions.

major comments (1)
  1. [Abstract] Abstract and main text: The oscillatory patterns in σ-space and the bfd–bfp interference are obtained after Fourier transformation of a chosen parametrization of the T-odd GTMDs. The manuscript assumes this parametrization remains valid across the DGLAP region when both ξ and t are simultaneously non-zero, yet supplies no model-independent argument or general property of GTMDs demonstrating that the diffraction-like oscillations and interference term persist under alternative functional forms for the transverse-momentum dependence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the major comment below.

read point-by-point responses

  1. Referee: The oscillatory patterns in σ-space and the bfd–bfp interference are obtained after Fourier transformation of a chosen parametrization of the T-odd GTMDs. The manuscript assumes this parametrization remains valid across the DGLAP region when both ξ and t are simultaneously non-zero, yet supplies no model-independent argument or general property of GTMDs demonstrating that the diffraction-like oscillations and interference term persist under alternative functional forms for the transverse-momentum dependence.

    Authors: We agree that the reported oscillatory patterns and interference arise from the Fourier transform of our chosen parametrization of the T-odd GTMDs (a standard dipole-like form commonly employed for such phenomenological studies). No model-independent proof is provided because the work is explicitly phenomenological. The oscillations are a direct consequence of the Fourier conjugate relation between ξ and σ combined with the support properties in the DGLAP region; the bfd–bfp interference follows from the bilinear structure in transverse momenta. To address the point we have added explicit statements in the abstract, Section 3, and conclusions acknowledging the model dependence and noting that while amplitudes may change with alternative forms, the qualitative diffraction-like features are tied to the Fourier transform itself. We have also referenced related literature using different parametrizations that exhibit similar behavior. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses explicit model inputs without self-referential reduction

full rationale

The provided abstract and skeptic summary describe Fourier transformation of a chosen parametrization of T-odd GTMDs to obtain σ-space distributions and interference effects. No quoted equations or sections exhibit self-definition (e.g., a quantity defined in terms of its own output), fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claim to prior author work by construction. The DGLAP-region assumption and parametrization validity are stated as model inputs rather than derived results. The oscillatory patterns and bfd–bfp interference are direct computational outputs from those inputs. This is the common case of a self-contained model calculation; no load-bearing step collapses to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; ledger left empty pending full text.

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Lean theorems connected to this paper

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  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
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    Relation between the paper passage and the cited Recognition theorem.

    The σ-space distributions of the T-odd sector show oscillatory patterns that are sensitive to the square of the total momentum transfer −t, analogous to the diffraction scattering of waves in Optics. An additional effect on the diffraction pattern is reported, caused by interference between transverse momentum transfer Δ⊥ to the transverse momentum p⊥ of quarks.

  • IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The Fourier transform of GTMDs with respect to the skewness variable ξ yields the Wigner distributions in the longitudinal position space σ.

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Reference graph

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