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arxiv: 2606.31708 · v1 · pith:TYMESCEEnew · submitted 2026-06-30 · ✦ hep-ph

Energy-Flow Moments for Elliptic Gluon Wigner Tomography

Lei Wang This is my paper

Pith reviewed 2026-07-01 04:41 UTC · model grok-4.3

classification ✦ hep-ph
keywords elliptic gluon Wigner distributionsmall-x dijet factorizationenergy-flow momentsDIS dijet productioncos2φ momentSudakov factorHankel transformsWigner tomography
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The pith

The normalized cos2φ moment of an azimuthal energy-flow observable in DIS dijet production projects the elliptic small-x gluon Wigner distribution after kinematic subtraction.

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

This paper proposes an azimuthal energy-flow moment in deep-inelastic-scattering dijet production to access the elliptic gluon Wigner distribution at small x. The construction uses final-state energy weighting that remains infrared and collinear safe, unlike exclusive diffractive dijets whose recoil is sensitive to soft radiation. Within leading-power small-x dijet factorization the normalized cos2φ component becomes a linear projection of the elliptic Wigner harmonic after a calculable kinematic subtraction. In recoil space the Sudakov factor evolves the isotropic and elliptic channels through a fixed J0/J2 Hankel pair without leakage between them. The observable therefore turns elliptic Wigner tomography into a moment-level energy-flow measurement whose statistical reach follows from angular-moment counting.

Core claim

Within leading-power small-x dijet factorization the normalized cos2φ moment of the proposed azimuthal energy-flow observable is a linear projection of the elliptic Wigner harmonic after calculable kinematic subtraction. The final-state energy weighting is infrared and collinear safe. In conjugate recoil space a rotationally scalar Sudakov factor evolves the isotropic and elliptic channels through a fixed J0/J2 Hankel pair without W0 to W1 leakage. A proof-of-principle calculation gives a per-mille Sudakov-level moment in an unoptimized conservative window, while auxiliary perturbative-window scans reach several per mille.

What carries the argument

Azimuthal energy-flow moment in DIS dijet production whose normalized cos2φ component projects the elliptic Wigner harmonic after subtraction and evolves via a J0/J2 Hankel pair in recoil space.

If this is right

  • The energy weighting guarantees infrared and collinear safety of the observable.
  • Sudakov evolution separates isotropic and elliptic channels through a fixed Hankel pair without mixing.
  • Statistical reach of the measurement can be assessed by simple angular-moment counting.
  • Proof-of-principle yields per-mille level moments in conservative kinematic windows.

Where Pith is reading between the lines

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

  • The same moment construction could be tested in other small-x processes if the leading-power factorization assumption continues to hold.
  • Experimental extraction at electron-ion colliders would rely on the infrared safety to reduce systematic uncertainties from soft radiation.
  • The absence of channel leakage suggests the method cleanly isolates elliptic effects from isotropic ones across a range of recoil scales.

Load-bearing premise

The proposed energy-flow moment must lie inside leading-power small-x dijet factorization so that its normalized cos2φ component directly projects the elliptic Wigner harmonic after subtraction.

What would settle it

A calculation or measurement in which the normalized cos2φ moment deviates from the predicted linear projection of the elliptic Wigner harmonic after the kinematic subtraction, or in which the Sudakov evolution exhibits W0 to W1 leakage.

Figures

Figures reproduced from arXiv: 2606.31708 by Lei Wang.

Figure 1
Figure 1. Figure 1: shows the normalized AEEC distribution at Y = 5. Before soft resummation, the energy-weighted elliptic component produces a percent-level cos 2ϕ pat￾tern. The Sudakov factor suppresses the modulation to C2 = 5.9 × 10−4 in this conservative full-window exam￾ple, but the extrema remain locked to the same harmonic. This size is smaller than the local elliptic ratio W1/W0 because the measured moment also conta… view at source ↗
Figure 2
Figure 2. Figure 2: compares C2(Y ) for representative evolution kernels while keeping the same amplitude projection, en￾ergy weight, and fiducial cuts. In the running-coupling baseline, the moment crosses zero earlier and becomes negative at larger rapidity. Collinear improvements de￾lay this transition and reduce the late-rapidity response. We treat this as a model diagnostic rather than a claimed discriminator: the sign ca… view at source ↗
read the original abstract

The elliptic small-$x$ gluon Wigner distribution correlates transverse momentum with impact parameter, but it is usually accessed through exclusive diffractive dijets whose recoil is sensitive to soft radiation. We propose instead an azimuthal energy-flow moment in DIS dijet production. Within leading-power small-$x$ dijet factorization, its normalized $\cos2\phi$ moment is a linear projection of the elliptic Wigner harmonic after a calculable kinematic subtraction, while the final-state energy weighting is infrared and collinear safe. In conjugate recoil space a rotationally scalar Sudakov factor evolves the isotropic and elliptic channels through a fixed $J_0/J_2$ Hankel pair without $W_0\to W_1$ leakage. A proof-of-principle calculation gives a per-mille Sudakov-level moment in an unoptimized conservative window, while auxiliary perturbative-window scans reach several per mille. The observable therefore formulates elliptic Wigner tomography as a moment-level energy-flow measurement whose statistical reach can be assessed by simple angular-moment counting.

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 / 2 minor

Summary. The paper proposes an azimuthal energy-flow moment in DIS dijet production as a new observable for elliptic small-x gluon Wigner tomography. Within leading-power small-x dijet factorization, the normalized cos2φ moment is claimed to be a linear projection of the elliptic Wigner harmonic after a calculable kinematic subtraction. The final-state energy weighting is asserted to be infrared and collinear safe. In conjugate recoil space, a rotationally scalar Sudakov factor evolves the isotropic and elliptic channels through a fixed J0/J2 Hankel pair without W0→W1 leakage. A proof-of-principle calculation reports a per-mille Sudakov-level moment in a conservative window, with auxiliary scans reaching several per mille.

Significance. If the leading-power factorization holds for the weighted observable, the construction provides a statistically accessible moment-level probe of the elliptic Wigner distribution that avoids the soft-radiation sensitivity of exclusive diffractive dijets. Credit is due for the explicit IR/collinear safety argument, the rotationally scalar Sudakov with fixed Hankel pair, and the concrete per-mille numerical demonstration in the proof-of-principle. These elements make the proposal falsifiable via angular-moment counting once the factorization assumption is validated.

major comments (2)
  1. [Abstract, §2] Abstract and §2: The central claim that the normalized cos2φ moment equals a linear projection of the elliptic Wigner harmonic after kinematic subtraction rests on the assumption that the energy-flow weighted dijet cross section lies inside leading-power small-x dijet factorization. The manuscript invokes this without deriving the factorization theorem for the weighted observable or showing that the energy weighting introduces neither power-suppressed corrections nor mixing with other small-x distributions in the cos2φ channel. This is load-bearing for the projection relation.
  2. [§4] §4 (Sudakov evolution): The assertion of a rotationally scalar Sudakov that evolves via a fixed J0/J2 Hankel pair with no W0→W1 leakage must be shown to survive the specific energy weighting at the order needed for per-mille accuracy; the current argument appears to transplant the unweighted dijet Sudakov without an explicit check that the weighting preserves the rotational scalar property and channel decoupling.
minor comments (2)
  1. Notation for the elliptic Wigner harmonic (W2 or similar) should be defined explicitly on first use and kept consistent with standard small-x literature.
  2. The kinematic subtraction procedure in the projection relation would benefit from an explicit formula or algorithm in an appendix to allow independent verification of the per-mille result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and for highlighting the load-bearing assumptions in our proposal. We address each major comment below and will incorporate clarifications and explicit justifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract, §2] Abstract and §2: The central claim that the normalized cos2φ moment equals a linear projection of the elliptic Wigner harmonic after kinematic subtraction rests on the assumption that the energy-flow weighted dijet cross section lies inside leading-power small-x dijet factorization. The manuscript invokes this without deriving the factorization theorem for the weighted observable or showing that the energy weighting introduces neither power-suppressed corrections nor mixing with other small-x distributions in the cos2φ channel. This is load-bearing for the projection relation.

    Authors: We agree that the extension of leading-power small-x dijet factorization to the energy-weighted case is central and was invoked rather than re-derived. The manuscript already notes that the weighting is infrared and collinear safe, which protects against soft and collinear singularities, but a more explicit argument is needed to confirm absence of power corrections or channel mixing in the cos2φ projection. In the revision we will add a short subsection in §2 sketching why the final-state energy weighting (being a collinear-safe jet observable) preserves the leading-power structure of the unweighted dijet factorization without introducing new small-x distributions or power-suppressed terms at the order relevant for the normalized moment. revision: yes

  2. Referee: [§4] §4 (Sudakov evolution): The assertion of a rotationally scalar Sudakov that evolves via a fixed J0/J2 Hankel pair with no W0→W1 leakage must be shown to survive the specific energy weighting at the order needed for per-mille accuracy; the current argument appears to transplant the unweighted dijet Sudakov without an explicit check that the weighting preserves the rotational scalar property and channel decoupling.

    Authors: The Sudakov is formulated in conjugate recoil space after the hard scattering, where the energy weighting acts as a rotationally invariant multiplicative factor on the final-state jets. Because the weighting commutes with the azimuthal Fourier decomposition and does not introduce additional transverse-momentum dependence that breaks rotational invariance, the J0/J2 Hankel structure and absence of W0→W1 leakage are preserved. Nevertheless, an explicit verification at the per-mille level for the weighted case was not provided. In the revised §4 we will include a short perturbative check (or reference to an auxiliary calculation) confirming that the weighted Sudakov remains rotationally scalar with the same fixed Hankel pair, thereby supporting the per-mille accuracy claim. revision: yes

Circularity Check

0 steps flagged

New observable proposal grounded in standard small-x factorization; no circular reduction to inputs

full rationale

The paper proposes an azimuthal energy-flow moment in DIS dijet production. Its central relation—that the normalized cos2φ moment equals a linear projection of the elliptic Wigner harmonic after kinematic subtraction—is stated as holding within the existing leading-power small-x dijet factorization framework, which is treated as an external input rather than derived from the paper's own results. The IR/collinear safety and Sudakov evolution via J0/J2 Hankel pair are presented as consequences of that framework applied to the weighted observable. No equations reduce the claimed projection to a fit, self-definition, or self-citation chain; the factorization theorem is invoked as standard and independent. The derivation chain therefore remains self-contained against external benchmarks with no load-bearing step that collapses by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption of leading-power small-x dijet factorization; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Leading-power small-x dijet factorization holds for the energy-flow moment observable
    Invoked to establish that the normalized cos2φ moment is a linear projection of the elliptic Wigner harmonic.

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