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arxiv: 2604.22332 · v1 · submitted 2026-04-24 · ✦ hep-ph · nucl-th

Recognition: unknown

Confronting Color Glass Condensate at next-to-leading order with HERA data

Carlisle Casuga , Heikki M\"antysaari
Authors on Pith no claims yet

Pith reviewed 2026-05-08 11:13 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords Color Glass CondensateBalitsky-Kovchegov equationHERA datadipole amplitudesmall-x physicsBayesian analysisNLO calculations
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The pith

Global fit to HERA data determines the non-perturbative initial condition for the next-to-leading order Balitsky-Kovchegov equation.

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

The paper performs a global analysis of HERA inclusive cross section and charm production data to fix the starting values for the dipole amplitude that enters the next-to-leading-order Balitsky-Kovchegov evolution equation. It combines NLO deep-inelastic scattering impact factors with the NLO BK equation that includes resummation of large double and single logarithms, then uses a Bayesian framework to extract the posterior distribution over the initial-condition parameters. This posterior supplies a quantitative estimate of the theoretical uncertainty attached to the dipole amplitude. Readers care because the procedure supplies a practical route for carrying those uncertainties forward into any NLO calculation performed inside the color glass condensate framework.

Core claim

We perform a global analysis of HERA total inclusive cross section and charm quark production data to extract the non-perturbative initial condition for the next-to-leading order Balitsky-Kovchegov (BK) equation. We extend our previous analyses to full next-to-leading order plus next-to-leading logarithm accuracy by combining the NLO DIS impact factors with the NLO BK equation that includes the resummation of large double and single logarithms. The developed Bayesian setup extracts the posterior that describes the distribution of the parameters that best fit the data.

What carries the argument

The Bayesian posterior distribution over the parameters of the non-perturbative initial condition for the NLO Balitsky-Kovchegov equation, which directly supplies the dipole amplitude and its uncertainty.

If this is right

  • NLO color-glass-condensate calculations of scattering processes can now carry a documented uncertainty band coming from the initial condition.
  • The same initial-condition posterior can be reused for predictions at the future Electron-Ion Collider without refitting.
  • Consistency checks between the fitted dipole amplitude and independent small-x observables become possible at the same perturbative order.
  • The method supplies a template for propagating initial-condition uncertainties into nuclear or heavy-ion calculations performed at NLO.

Where Pith is reading between the lines

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

  • The posterior can be sampled to generate an ensemble of dipole amplitudes for use in Monte-Carlo event generators that incorporate NLO CGC dynamics.
  • If the fit quality remains high when additional HERA or LHC data are included, it would indicate that the resummed NLO framework already captures the dominant small-x dynamics in the measured region.
  • The extracted initial condition provides a concrete benchmark against which future analytic or lattice determinations of the same non-perturbative input can be compared.

Load-bearing premise

The NLO Balitsky-Kovchegov equation with resummation of large logarithms, when used together with NLO deep-inelastic scattering impact factors, describes the HERA data in the measured kinematic range without large missing higher-order or non-CGC contributions.

What would settle it

A clear mismatch between the model's predictions for a new observable, such as forward hadron or dijet production at the LHC, and measured cross sections that lies outside the uncertainty band derived from the posterior would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.22332 by Carlisle Casuga, Heikki M\"antysaari.

Figure 1
Figure 1. Figure 1: FIG. 1: Comparison between emulator prediction and view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Z-score distribution, vertical lines correspond to the median and one standard deviation of the distribution view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: One-dimensional projections of posterior distributions from closure tests for the Bayesian setup, constraining view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: 1- (diagonal) and 2- (off-diagonal) projections of the 4-dimensional posterior distribution obtained for the view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Reduced total cross section and charm production as a function of view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: 1- (diagonal) and 2- (off-diagonal) dimensional projections of the 5-dimensional posterior distribution found view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: 1- (diagonal) and 2- (off-diagonal) dimensional projections of the 5-dimensional posterior distribution found view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Reduced total cross section and charm production as a function of view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Reduced total cross section and charm production as a function of view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Initial and evolved dipole amplitudes as a function of dipole size, view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Longitudinal structure function calculated us view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 view at source ↗
read the original abstract

We perform a global analysis of HERA total inclusive cross section and charm quark production data to extract the non-perturbative initial condition for the next-to-leading order Balitsky-Kovchegov (BK) equation. We extend our previous analyses to full next-to-leading order + next-to-leading logarithm (NLO+NLL) accuracy by combining the NLO DIS impact factors with the NLO BK equation that includes the resummation of large double and single logarithms. The developed Bayesian setup extracts the posterior that describes the distribution of the parameters that best fit the data. The posterior distribution allows for estimates of the theoretical uncertainty of the dipole amplitude and, hence, offers a streamlined method for propagating uncertainties in the BK initial condition in NLO CGC calculations.

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 performs a global Bayesian analysis of HERA total inclusive cross section and charm quark production data to extract the non-perturbative initial condition parameters for the next-to-leading order Balitsky-Kovchegov (BK) equation including resummation of large double and single logarithms, combined with NLO DIS impact factors. The resulting posterior distribution on these parameters is used to estimate theoretical uncertainties on the dipole amplitude, providing a method to propagate uncertainties in the BK initial condition for NLO CGC calculations.

Significance. If the NLO+NLL BK evolution accurately describes the HERA data, this work supplies a practical Bayesian framework for uncertainty quantification in small-x phenomenology, which is a clear strength for propagating non-perturbative input uncertainties in future CGC predictions.

major comments (1)
  1. [numerical results and Bayesian analysis] The central claim that the posterior furnishes reliable data-driven uncertainties on the dipole amplitude requires that the NLO BK equation plus NLO impact factors yields a statistically acceptable description of the HERA data. However, no chi-squared values, degrees of freedom, or goodness-of-fit metrics are reported for the Bayesian fit (see abstract and the description of the numerical analysis). Without these, it is impossible to rule out that the posterior width is dominated by model mismatch or the prior rather than data constraints, undermining the advertised uncertainty-propagation method.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of goodness-of-fit metrics in validating the Bayesian analysis. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [numerical results and Bayesian analysis] The central claim that the posterior furnishes reliable data-driven uncertainties on the dipole amplitude requires that the NLO BK equation plus NLO impact factors yields a statistically acceptable description of the HERA data. However, no chi-squared values, degrees of freedom, or goodness-of-fit metrics are reported for the Bayesian fit (see abstract and the description of the numerical analysis). Without these, it is impossible to rule out that the posterior width is dominated by model mismatch or the prior rather than data constraints, undermining the advertised uncertainty-propagation method.

    Authors: We agree that explicit goodness-of-fit metrics are necessary to substantiate the reliability of the extracted posterior and the associated uncertainty estimates. In the revised manuscript we will report the chi-squared value evaluated at the maximum a posteriori (MAP) point together with the number of degrees of freedom for both the inclusive and charm data sets. We will also quote the reduced chi-squared and briefly discuss its interpretation in the context of the Bayesian sampling. In addition, we will add a short paragraph describing posterior predictive checks performed on a subset of the HERA data points to demonstrate that the model predictions are statistically consistent with the measurements. These additions will directly address the concern that the posterior width might be dominated by prior or model mismatch rather than data constraints. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard data-driven extraction of non-perturbative input

full rationale

The paper performs an explicit global fit of the BK initial-condition parameters to external HERA inclusive and charm data using the NLO BK equation plus NLO impact factors. This is a phenomenological extraction whose output (posterior on the dipole amplitude) is defined by the fit to independent data rather than by any internal self-definition or renaming. No load-bearing step reduces by construction to the inputs: the evolution equation and impact factors are taken as given (with resummation), the data are external, and the Bayesian posterior propagates fit uncertainties in the standard way. Self-citations to prior analyses exist but are not load-bearing for the central claim, which adds NLO+NLL accuracy and charm data. The result is self-contained against the HERA benchmark and does not manufacture predictions from fitted quantities.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the NLO BK framework with chosen resummation and on the ability of a parametrized initial condition to capture all relevant non-perturbative physics when fitted to HERA data.

free parameters (1)
  • Parameters of the non-perturbative BK initial condition
    Multiple parameters (e.g., saturation scale, anomalous dimension) are fitted to HERA data to determine the starting dipole amplitude.
axioms (1)
  • domain assumption NLO BK equation with NLL resummation of large double and single logarithms accurately models small-x evolution for HERA kinematics
    Invoked when combining NLO DIS impact factors with the evolution equation to fit the data.

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discussion (0)

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