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arxiv: 2509.02115 · v3 · submitted 2025-09-02 · ✦ hep-ph · nucl-th

Probing nuclear structure with the Balitsky-Kovchegov equation in full impact-parameter dependence

J. Cepila , M. Matas , M. Vaculciak This is my paper

Pith reviewed 2026-05-18 19:48 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords Balitsky-Kovchegov equationgluon saturationnuclear targetsdeep-inelastic scatteringvector meson productionimpact-parameter dependenceEIC
0
0 comments X

The pith

The full impact-parameter Balitsky-Kovchegov equation applied to nuclei generates predictions for deep-inelastic scattering and diffractive vector-meson production that can reveal gluon saturation.

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

This paper takes the newly solved Balitsky-Kovchegov equation that tracks parton evolution with complete dependence on impact parameter and applies it to nuclear targets instead of protons. It computes results for deep-inelastic scattering and for diffractive production of vector mesons on several nuclei. The authors also run the linearised version of the same equation to isolate a channel where saturation effects should stand out most clearly. They further replace the usual round Woods-Saxon nuclear shape with a tetrahedral model for oxygen to look for geometry-driven differences. The calculations are intended to support planning for the Electron-Ion Collider and to interpret existing nuclear data from the LHC.

Core claim

Using the full impact-parameter solution of the Balitsky-Kovchegov equation, predictions are obtained for deep-inelastic scattering and diffractive vector-meson production on a range of nuclear targets. Comparison of the nonlinear evolution against its linearised form identifies a promising observable for gluon saturation. A tetrahedral model for the oxygen nucleus is introduced to test deviations from the standard isotropic Woods-Saxon nuclear density.

What carries the argument

The full impact-parameter dependence solution of the Balitsky-Kovchegov equation, which evolves the dipole scattering amplitude while retaining both transverse momentum and spatial position information.

If this is right

  • Predictions become available for deep-inelastic scattering on multiple nuclear targets.
  • Diffractive vector-meson production cross sections are calculated for the same nuclei.
  • The linearised Balitsky-Kovchegov equation isolates an observable sensitive to the onset of gluon saturation.
  • The tetrahedral oxygen model produces measurable deviations from the standard Woods-Saxon density profile in nuclear collisions.

Where Pith is reading between the lines

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

  • The nuclear predictions can help prioritise observables for saturation searches once the Electron-Ion Collider begins operation.
  • The same framework may be used to reinterpret existing LHC measurements of vector-meson production in ultraperipheral nuclear collisions.
  • The method opens a route to test more realistic nuclear geometries by replacing simple density profiles with microscopic nuclear models.

Load-bearing premise

The Balitsky-Kovchegov equation that was derived for protons can be used for nuclei without any extra nuclear corrections even when full impact-parameter dependence is kept.

What would settle it

A measurement of the ratio of nuclear to proton vector-meson production cross sections at small x that lies significantly outside the band spanned by the full nonlinear BK predictions and the linearised BK predictions would test the central claim.

Figures

Figures reproduced from arXiv: 2509.02115 by J. Cepila, M. Matas, M. Vaculciak.

Figure 1
Figure 1. Figure 1: An example density profile obtained from a single random orientation of the oxygen [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Gaussian profile of the proton (shown in black), and partially integrated [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the dipole amplitudes at the initial condition (full) and evolved to [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The F2(x, Q2 ) structure functions of proton (left panel) and nuclear (right panel) targets from the full-impact-parameter dependent BK equation. The available data from HERA [4] are shown for the proton target. Results for the tetrahedral oxygen are not shown in these plots as they are indistinguishable from the Woods-Saxon profile. The full curves correspond to the standard BK equation (κ = 1), the dashe… view at source ↗
Figure 5
Figure 5. Figure 5: The nuclear modification factor for virtualities [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Differential cross-sections of diffractive coherent [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Differential cross-sections of the diffractive [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Total (left) and differential (right) cross sections for [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
read the original abstract

Building on the newly available solution of the Balitsky-Kovchegov (BK) equation with the full impact-parameter dependence, we extend the study of parton evolution from proton to nuclear targets. Since a key part of the scientific programme for future experimental facilities such as the EIC is to study gluon dynamics and shed new light on the phenomenon of parton saturation, we present predictions for key processes, such as deep-inelastic scattering or the diffractive production of vector mesons, on a variety of nuclear targets. Besides the standard BK equation, we employ its linearised version to identify a promising channel to search for gluon saturation in the nuclear collisions. Furthermore, we implement a tetrahedral model of oxygen to search for deviations from the standard, isotropic, Woods-Saxon approach. In addition to the future colliders, the presented results are also of interest for the current studies of nuclear vector meson production at the LHC.

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 / 2 minor

Summary. The manuscript extends the recently solved Balitsky-Kovchegov (BK) equation with full impact-parameter dependence from proton to nuclear targets. It presents predictions for deep-inelastic scattering and diffractive vector-meson production on several nuclei, contrasts the full non-linear BK evolution against its linearised version to isolate saturation signals, and introduces a tetrahedral model for oxygen to test deviations from the standard Woods-Saxon initial condition. The results are framed as relevant for gluon-saturation searches at the EIC and for ongoing LHC vector-meson studies.

Significance. Should the central assumptions prove robust, the work supplies concrete, impact-parameter-resolved predictions that could guide experimental searches for gluon saturation in nuclei. The use of the newly available full-IP BK solution, the explicit linear/non-linear comparison, and the alternative nuclear geometry constitute clear strengths that would be valuable for EIC planning if the nuclear applicability is established.

major comments (1)
  1. [Abstract and the description of the nuclear initial conditions] The central claim rests on solving the BK equation (originally derived for proton targets) for nuclei by changing only the initial condition (Woods-Saxon or tetrahedral oxygen) while leaving the evolution kernel unchanged. This assumption directly supports the DIS and vector-meson predictions and the identification of a saturation-search channel; a dedicated justification or sensitivity test for possible nuclear modifications to the kernel (e.g., collective effects or modified dipole-nucleus scattering in the dense environment) is required to substantiate the quantitative distinction between linear and non-linear regimes.
minor comments (2)
  1. [Numerical results section] Provide explicit error estimates or validation against known proton limits for the numerical BK solutions when extended to nuclei.
  2. [Nuclear geometry subsection] Clarify the precise definition and parameter values of the tetrahedral oxygen model versus the Woods-Saxon profile.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive major comment. We address the point directly below, providing a substantive response and indicating the planned revision.

read point-by-point responses

  1. Referee: [Abstract and the description of the nuclear initial conditions] The central claim rests on solving the BK equation (originally derived for proton targets) for nuclei by changing only the initial condition (Woods-Saxon or tetrahedral oxygen) while leaving the evolution kernel unchanged. This assumption directly supports the DIS and vector-meson predictions and the identification of a saturation-search channel; a dedicated justification or sensitivity test for possible nuclear modifications to the kernel (e.g., collective effects or modified dipole-nucleus scattering in the dense environment) is required to substantiate the quantitative distinction between linear and non-linear regimes.

    Authors: We agree that explicit justification is warranted. The BK evolution kernel is universal within the leading-logarithmic high-energy QCD framework and does not depend on the target; it describes the rapidity evolution of the dipole scattering amplitude for any color source. Nuclear modifications enter exclusively through the initial condition at the starting rapidity, where the Woods-Saxon distribution (or the tetrahedral geometry for oxygen) encodes the higher transverse gluon density and possible deviations from spherical symmetry. This is the standard procedure used throughout the saturation literature for nuclear targets. To address the referee's concern, we will add a dedicated paragraph in the introduction and in the section on nuclear initial conditions. The paragraph will (i) state the universality of the kernel, (ii) cite prior works that employ the same approach for nuclei, and (iii) note that, in the kinematic window relevant for EIC and LHC vector-meson studies, collective effects beyond the initial condition remain sub-leading. We will also include a brief sensitivity discussion by varying the nuclear radius parameter in the initial condition. A full re-derivation or numerical scan of a modified kernel lies outside the present scope but is not required to substantiate the linear versus non-linear comparison within the adopted framework. revision: partial

Circularity Check

0 steps flagged

No significant circularity in BK extension to nuclei

full rationale

The paper numerically solves the Balitsky-Kovchegov equation (and its linearised version) with full impact-parameter dependence, extending from proton to nuclear targets solely by substituting standard initial conditions (Woods-Saxon or tetrahedral oxygen). Predictions for DIS and diffractive vector-meson production are direct outputs of this evolution, not reductions of fitted parameters or self-referential definitions within the same work. No load-bearing self-citations, ansatz smuggling, or uniqueness theorems imported from the authors' prior results are required for the central claims; the derivation remains self-contained as an application of an established evolution equation to new targets and geometries.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claims rest on the validity of the BK equation for nuclei, the choice of initial conditions for the evolution, and the geometric model of the nucleus; these are standard domain assumptions rather than new axioms invented in the paper.

free parameters (1)
  • initial saturation scale for nuclei
    Typical free parameter in BK initial conditions that must be chosen or fitted before evolution begins.
axioms (1)
  • domain assumption Balitsky-Kovchegov equation remains applicable to nuclear targets without additional nuclear corrections
    Invoked when the proton-derived equation is directly extended to nuclei in the abstract.
invented entities (1)
  • tetrahedral model of oxygen no independent evidence
    purpose: To test deviations from isotropic Woods-Saxon nuclear density
    New geometric ansatz introduced to probe nuclear-structure sensitivity; no independent evidence supplied in the abstract.

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