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arxiv: 2604.14049 · v1 · submitted 2026-04-15 · ✦ hep-ph

The origin of Bjorken-x dependence in DIS: a case for a z-dependent weight functional in the CGC

Benjamin Guiot This is my paper

Pith reviewed 2026-05-10 13:06 UTC · model grok-4.3

classification ✦ hep-ph
keywords deep inelastic scatteringcolor glass condensateBjorken-xsmall-x evolutionweight functionalk_t-factorizationdipole amplitude
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The pith

The standard CGC framework for DIS yields cross sections independent of Bjorken-x at all orders, requiring a z-dependent weight functional to restore physical energy dependence.

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

The paper identifies an ambiguity in the eikonal CGC description of DIS where Bjorken-x enters only as a rapidity cutoff, making the entire cross section independent of x_b after all orders. To resolve this, the authors introduce a weight functional that depends explicitly on the light-cone momentum fraction z, with limits tied to x_b. This change ensures that the non-perturbative gluon structure depends on the probe's energy, which is physically expected. Consequently, changes in x_b for the cross section are not driven only by small-x evolution equations. The modified approach describes existing DIS data as well as the standard one and maintains consistency with k_t-factorization.

Core claim

In the standard CGC framework for DIS, the Bjorken-x dependence enters solely through the rapidity cutoff in the weight functional, resulting in an all-order cross section that is independent of x_b. This is addressed by making the weight functional depend explicitly on z, with integration limits determined by x_b. This modification is consistent with the physical expectation that the observed non-perturbative structure depends on the probe energy. Analysis of DIS fits shows that this leads to the conclusion that x_b variation is not solely driven by small-x evolution, while preserving compatibility with k_t-factorization.

What carries the argument

The z-dependent weight functional in the CGC, which incorporates explicit dependence on the light-cone momentum fraction z and sets integration limits based on x_b to introduce energy dependence beyond the cutoff.

If this is right

  • The x_b variation of the DIS cross section receives contributions from the explicit z-dependence in the weight functional in addition to small-x evolution.
  • Existing DIS data can be described equally well using the modified weight functional.
  • The modified CGC formulation remains consistent with k_t-factorization, resolving an incompatibility in the standard version.

Where Pith is reading between the lines

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

  • This modification might require reevaluating how saturation scales are determined in other high-energy processes.
  • It could lead to different predictions for the energy dependence of cross sections in heavy-ion collisions.
  • Testing this would involve checking if the z-dependence affects the dipole amplitude in ways observable at future electron-ion colliders.

Load-bearing premise

The non-perturbative structure in DIS must depend explicitly on the probe energy, rendering an all-order x_b-independent cross section physically unacceptable.

What would settle it

A precise measurement of the DIS cross section's dependence on Bjorken-x at fixed virtuality Q squared, compared against predictions from the unmodified CGC which should show no additional x dependence beyond evolution, would distinguish if the explicit z-dependence is necessary.

Figures

Figures reproduced from arXiv: 2604.14049 by Benjamin Guiot.

Figure 1
Figure 1. Figure 1: Reduced cross section obtained with Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Reduced cross section obtained with Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Experimental reduced cross section [27] compared to the CGC dipole model and our model for Q2 =2.7, 4.5, 6.5, 10, and 15 GeV2 (from bottom to top). that we reached roughly the same precision. Since we have the same number of parameters, but a Λ-independent dipole amplitude, we have demonstrated that the small-x evolution is not a necessary condition to the description of these data. However, the difference… view at source ↗
read the original abstract

We discuss what is, at best, an ambiguity, and possibly an inconsistency of the eikonal Color Glass Condensate (CGC) description of Deep Inelastic Scattering (DIS). In this framework, the Bjorken-$x$ dependence enters the cross section solely through the rapidity cutoff $\Lambda=x_b$, leading to an all-order cross section independent of $x_b$. To address this issue, we explore a natural modification in which the weight functional depends explicitly on the light-cone momentum fraction $z$, with integration limits determined by $x_b$. This modification is consistent with the physical expectation that the observed non-perturbative structure depends on the probe energy. Our analysis implies that the $x_b$ variation of the cross section is not solely driven by small-$x$ evolution equations. We support this conclusion through an analysis of existing DIS fits and by demonstrating that a similarly good description of the data can be obtained within the modified framework. Finally, we show that the modified formulation is compatible with $k_t$-factorization, unlike the standard one.

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 manuscript claims that the eikonal CGC framework for DIS has an ambiguity because x_b enters only through the rapidity cutoff, leading to an all-order x-independent cross section. It proposes a z-dependent weight functional to address the physical expectation that non-perturbative structure depends on probe energy. Support is provided through analysis of existing DIS fits and a demonstration of similarly good data description, with additional compatibility to k_t-factorization.

Significance. Should the central claim hold after clarification, the work would highlight that x_b dependence in DIS is not exclusively from small-x evolution, potentially impacting how CGC is applied to high-energy processes. The compatibility with k_t-factorization is a positive aspect if demonstrated rigorously.

major comments (2)
  1. [Abstract] The key premise that the eikonal CGC yields an all-order x_b-independent cross section is not consistent with the standard CGC setup, in which the weight functional evolves with rapidity Y = ln(1/x_b) according to the JIMWLK or BK equations. This evolution makes the dipole scattering amplitude explicitly dependent on x_b, so the motivation for introducing a z-dependent weight functional requires re-examination or explicit restriction to the unevolved case.
  2. [Abstract] The statement that 'a similarly good description of the data can be obtained within the modified framework' is not supported by sufficient detail in the provided abstract; without quantitative metrics (e.g., fit quality, parameter counts, or specific functional form), it is unclear whether this constitutes a prediction or an adjustment that restores x-dependence by construction.
minor comments (1)
  1. The abstract could benefit from a brief mention of the specific form of the z-dependent weight functional to aid reader understanding.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major points below and have revised the manuscript to provide the requested clarifications and additional details.

read point-by-point responses

  1. Referee: [Abstract] The key premise that the eikonal CGC yields an all-order x_b-independent cross section is not consistent with the standard CGC setup, in which the weight functional evolves with rapidity Y = ln(1/x_b) according to the JIMWLK or BK equations. This evolution makes the dipole scattering amplitude explicitly dependent on x_b, so the motivation for introducing a z-dependent weight functional requires re-examination or explicit restriction to the unevolved case.

    Authors: We thank the referee for this observation. In the manuscript, 'all-order' refers exclusively to the resummation of the eikonal dipole scattering amplitude to all orders in the strong background field, while the weight functional itself is held fixed (i.e., unevolved with respect to the rapidity cutoff). The JIMWLK/BK evolution constitutes a separate resummation of perturbative corrections in α_s Y and is not part of the basic eikonal framework under discussion. We will revise the manuscript to explicitly restrict the analysis to the unevolved case, where x_b indeed enters the cross section only through the cutoff Λ = x_b, rendering the result independent of x_b. This clarification preserves the motivation for a z-dependent weight functional as a means to restore physically expected x_b dependence without relying on small-x evolution. revision: yes

  2. Referee: [Abstract] The statement that 'a similarly good description of the data can be obtained within the modified framework' is not supported by sufficient detail in the provided abstract; without quantitative metrics (e.g., fit quality, parameter counts, or specific functional form), it is unclear whether this constitutes a prediction or an adjustment that restores x-dependence by construction.

    Authors: We agree that the abstract lacks sufficient quantitative support for this claim. The full manuscript contains a detailed comparison, including the explicit functional form of the z-dependent weight, the number of parameters, and numerical fit results to DIS data. We will revise the abstract to incorporate key quantitative metrics (e.g., χ²/dof values and parameter counts) demonstrating that the modified framework achieves a description of the data comparable to existing fits. This will make clear that the z-dependence is introduced on physical grounds rather than as an ad hoc adjustment. revision: yes

Circularity Check

1 steps flagged

x_b dependence in modified CGC restored by fitting z-dependent weight functional

specific steps
  1. fitted input called prediction [Abstract]
    "We support this conclusion through an analysis of existing DIS fits and by demonstrating that a similarly good description of the data can be obtained within the modified framework."

    The modified framework adds explicit z-dependence to the weight functional (with integration limits set by x_b). Demonstrating a 'similarly good description' requires fitting this new functional to DIS data whose x_b variation is already present. The x_b dependence is therefore reintroduced by the fit itself rather than predicted from first principles or shown to be independent of evolution.

full rationale

The paper's core claim is that standard eikonal CGC yields an all-order x_b-independent DIS cross section (x_b only sets the rapidity cutoff), so x_b variation must come from elsewhere. They introduce a z-dependent weight functional and support the conclusion by showing 'a similarly good description of the data can be obtained within the modified framework' after analyzing existing DIS fits. This validation step reduces to fitting the new z-dependent parameters to data that already encodes x_b dependence, making the claimed non-evolution origin of x_b variation tautological rather than independently derived. This matches the fitted-input-called-prediction pattern. The central support for the implication therefore collapses to the fit by construction. No other load-bearing self-definitional or self-citation reductions were identifiable from the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The central claim rests on the domain assumption that non-perturbative structure must vary with probe energy and on the standard CGC eikonal approximation whose inconsistency is being highlighted.

axioms (2)
  • domain assumption Non-perturbative structure in the target must depend on the probe energy (i.e., an all-order x-independent cross section is unacceptable)
    Invoked in the abstract to motivate the modification
  • domain assumption The eikonal CGC framework is otherwise valid except for the identified x-dependence issue
    The paper works within the CGC but modifies only the weight functional
invented entities (1)
  • z-dependent weight functional no independent evidence
    purpose: To restore explicit Bjorken-x dependence in the DIS cross section while preserving other CGC features
    New functional form introduced to fix the identified inconsistency; no independent evidence provided in abstract

pith-pipeline@v0.9.0 · 5491 in / 1495 out tokens · 40867 ms · 2026-05-10T13:06:45.991656+00:00 · methodology

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Reference graph

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