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arxiv: 2605.28747 · v2 · pith:42UNHE3Lnew · submitted 2026-05-27 · ✦ hep-ph

Maximum phase-space density of linearly polarized gluon TMDs in the saturation region

Lei Wang This is my paper

Pith reviewed 2026-06-29 11:23 UTC · model grok-4.3

classification ✦ hep-ph
keywords linearly polarized gluon TMDsaturation regionphase-space densitydipole distributionWeizsacker-Williams distributionMueller occupancy argumentsmall-x physicsCollins-Soper evolution
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The pith

The linearly polarized gluon TMD reaches twice the unpolarized maximum phase-space density in the dipole saturation region.

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

The paper calculates the Sudakov-limited maximum phase-space density for the linearly polarized gluon TMD coefficient h1 perpendicular g inside saturated matter at small x. It combines Mueller's occupancy limit with the Weizsacker-Williams and dipole gluon distributions of Metz and Zhou. For the dipole case the resulting maximum equals twice the unpolarized gluon density and scales as two times alpha_s to the minus three-halves in the same normalization. The Weizsacker-Williams case lacks the needed logarithmic enhancement, so its maximum sits at the saturation boundary instead. A numerical Collins-Soper evolution check shows the J2 Bessel weighting lowers the observable peak to roughly 6.6 to 7.1 at typical EIC scales.

Core claim

Using Mueller's occupancy argument together with the small-x Weizsacker-Williams and dipole gluon distributions, the maximum phase-space density n_h,DP^max equals 2 n_g^max and is approximately 2 alpha_s^{-3/2} for the dipole distribution in the same phase-space normalization. This dipole result is a process-dependent TMD proxy, not a literal gluon number density. For the WW distribution the deep-saturation tensor coefficient lacks the logarithmic enhancement needed for the Mueller saddle, so the maximum is pushed toward the saturation boundary.

What carries the argument

Mueller's occupancy argument applied to the linearly polarized gluon TMD coefficient h1^⊥g together with the small-x WW and dipole distributions of Metz and Zhou.

If this is right

  • The dipole TMD proxy saturates at twice the unpolarized gluon maximum density.
  • The WW distribution maximum is shifted to the saturation boundary because it lacks the required logarithmic enhancement.
  • Numerical Collins-Soper evolution with the J2 Bessel weight reduces the resolved peak height to c_h^num approximately 6.6 to 7.1 at representative EIC scales.
  • The dipole result remains a process-dependent proxy rather than a literal number density.

Where Pith is reading between the lines

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

  • Future collider data could distinguish whether different TMD observables saturate at different maximum densities.
  • The factor-of-two enhancement may appear in other tensor TMD coefficients at small x if the same occupancy logic applies.
  • Extending the argument beyond the dipole and WW cases could map out how polarization affects saturation limits across processes.

Load-bearing premise

Mueller's occupancy argument applies directly to the linearly polarized coefficient h1^⊥g when combined with the small-x WW and dipole gluon distributions.

What would settle it

A direct calculation or EIC measurement showing that the phase-space density of h1^⊥g does not reach twice the unpolarized gluon value inside the saturation region would falsify the central scaling result.

Figures

Figures reproduced from arXiv: 2605.28747 by Lei Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Limiting behavior of WW and dipole linear polar [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Direct polarized phase-space density obtained by re [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Effective Bessel-weight coefficient extracted from the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Direct numerical Collins-Soper evolution of the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Sensitivity of the fitted Bessel-weight coefficient to the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Coupling dependence of the maximum phase-space [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Dipole-sensitive benchmark for the EIC [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: shows the dependence on xg. As xg decreases, the saturation scale increases and the available polarized occupancy is enhanced over most of the range shown. The increase is not a simple power law, because a fixed value of k⊥ probes a changing ratio k⊥/Qs(xg). The com￾parison between ch = 1, ch = 3, and the numerical value ch = 6.7 shows that the xg dependence is qualitatively stable, whereas the normalizati… view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Two-dimensional EIC-oriented benchmark scan of [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Robustness scan for the [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
read the original abstract

We calculate the Sudakov-limited maximum phase-space density associated with the linearly polarized gluon TMD coefficient $h_1^{\perp g}$ in the saturation region. Using Mueller's occupancy argument together with the small-$x$ Weizs\"acker-Williams (WW) and dipole gluon distributions of Metz and Zhou, we find $n_{h,{\rm DP}}^{\rm max}=2n_g^{\rm max}\sim2\alpha_s^{-3/2}$ for the dipole distribution in the same phase-space normalization. This dipole result is a process-dependent TMD proxy, not a literal gluon number density. For the WW distribution, the deep-saturation tensor coefficient lacks the logarithmic enhancement needed for the Mueller saddle, so the maximum is pushed toward the saturation boundary. We also perform a numerical Collins-Soper evolution study and find that the $J_2$ Bessel weight in the tensor TMD definition reduces the resolved peak, giving $c_h^{\rm num}\simeq6.6$--$7.1$ for representative EIC scales.

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

2 major / 1 minor

Summary. The paper calculates the Sudakov-limited maximum phase-space density for the linearly polarized gluon TMD coefficient h_1^⊥g in the saturation region. Applying Mueller's occupancy argument to the small-x WW and dipole gluon distributions of Metz and Zhou, it reports n_{h,DP}^max = 2 n_g^max ∼ 2 α_s^{-3/2} for the dipole case (a process-dependent TMD proxy) while noting that the WW tensor coefficient lacks logarithmic enhancement and thus has its maximum pushed to the saturation boundary. A numerical Collins-Soper evolution study yields c_h^num ≃ 6.6–7.1 for representative EIC scales, with the J_2 Bessel weight reducing the resolved peak.

Significance. If the central extension of Mueller's argument holds, the work provides a concrete bound on the phase-space density of a polarized TMD in saturation, distinguishing WW and dipole cases and including a numerical evolution check. The explicit acknowledgment that the dipole result is a process-dependent proxy rather than a literal gluon density is a strength, as is the numerical Collins-Soper study that quantifies the effect of the J_2 weight.

major comments (2)
  1. [Abstract] Abstract (paragraph 2): the extension of Mueller's occupancy argument to the tensor coefficient h_1^⊥g is asserted without explicit demonstration that the saddle-point condition remains valid once the J_2-weighted definition and process-dependent normalization are inserted; the noted absence of log enhancement in the WW case already indicates that the saddle condition is distribution-dependent, so the factor-of-2 relation for the dipole case requires supporting algebra to establish it is not an artifact of the chosen proxy.
  2. [Abstract] Abstract (paragraph 2) and numerical study: the claim that the WW deep-saturation tensor coefficient lacks the logarithmic enhancement needed for the Mueller saddle is stated without visible derivation steps or error estimates against the paper's own equations; this directly affects whether the maximum is indeed pushed to the saturation boundary.
minor comments (1)
  1. The normalization convention for the phase-space density n_h should be stated explicitly when comparing the dipole and WW cases, as the process dependence is emphasized but the precise phase-space measure is not detailed in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each major comment below and will make revisions to improve the clarity of the derivations as suggested.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph 2): the extension of Mueller's occupancy argument to the tensor coefficient h_1^⊥g is asserted without explicit demonstration that the saddle-point condition remains valid once the J_2-weighted definition and process-dependent normalization are inserted; the noted absence of log enhancement in the WW case already indicates that the saddle condition is distribution-dependent, so the factor-of-2 relation for the dipole case requires supporting algebra to establish it is not an artifact of the chosen proxy.

    Authors: We agree that an explicit demonstration of the saddle-point condition for the J_2-weighted tensor coefficient is necessary to support the extension of Mueller's argument. In the revised manuscript, we will add the supporting algebra in the main text or an appendix, showing how the dipole distribution's form leads to the factor-of-2 relation while maintaining the saddle validity, and contrast it with the WW case to confirm it is not an artifact. revision: yes

  2. Referee: [Abstract] Abstract (paragraph 2) and numerical study: the claim that the WW deep-saturation tensor coefficient lacks the logarithmic enhancement needed for the Mueller saddle is stated without visible derivation steps or error estimates against the paper's own equations; this directly affects whether the maximum is indeed pushed to the saturation boundary.

    Authors: The claim follows directly from the functional forms of the WW and dipole distributions as defined in the paper. We will include explicit derivation steps comparing the tensor coefficients to the relevant equations, along with error estimates from the numerical Collins-Soper study, to substantiate that the maximum for WW is pushed to the saturation boundary. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives its central result n_{h,DP}^max=2n_g^max by applying Mueller's occupancy argument to the independent small-x WW and dipole distributions of Metz and Zhou, then supplements with a separate numerical Collins-Soper evolution study that produces c_h^num from evolution equations rather than any fit to the target maximum. No self-citations appear in the load-bearing steps, no parameter is fitted to a subset and renamed as a prediction, and no ansatz or uniqueness theorem is imported from the authors' own prior work. The derivation therefore remains self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on Mueller's occupancy argument applied to TMD coefficients and on the small-x distributions of Metz and Zhou; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Mueller's occupancy argument applies to the linearly polarized gluon TMD coefficient h1^⊥g in the saturation region
    Invoked in abstract to obtain the maximum density; no independent derivation supplied.
  • domain assumption The small-x Weizsacker-Williams and dipole gluon distributions of Metz and Zhou are valid inputs in the saturation regime
    Used directly to combine with the occupancy argument.

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discussion (0)

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