Investigation of the ratio $\frac{\sigma_{r}}{F_{2}}(Q^2/s,Q^2)$ in the momentum-space approach

G.R.Boroun

arxiv: 2511.04245 · v2 · submitted 2025-11-06 · ✦ hep-ph

Investigation of the ratio frac{σ_(r)}{F₂}(Q²/s,Q²) in the momentum-space approach

G.R.Boroun This is my paper

Pith reviewed 2026-05-18 01:15 UTC · model grok-4.3

classification ✦ hep-ph

keywords reduced cross sectionproton structure functionmomentum spaceHERA datahigher twistcolor dipole modelLHCFCC

0 comments

The pith

The ratio of reduced cross section to proton structure function is computed in momentum space using the Block-Durand-Ha parameterization and matches HERA data.

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

The paper calculates the ratio of the reduced cross section σ_r to the structure function F2 both at general x and at the specific point x = Q²/s using a momentum-space approach and the Block-Durand-Ha parameterization of F2. These calculations are compared directly to H1 and HERA measurements across a range of Q² values, including extensions to high inelasticity and low x. The results at fixed √s are also checked against color dipole model bounds. The work shows that the method remains consistent with existing data when a simple higher-twist correction is added at low Q². This consistency supports using the same approach for analyses at the LHC and Future Circular Collider.

Core claim

Using the Block-Durand-Ha parameterization of F2(x, Q²) in momentum space yields values for the ratio σ_r/F2(x, Q²) and for the ratio σ_r/F2(Q²/s, Q²) that agree with HERA data over the measured range; adding the higher-twist term F2 * H2/Q² improves agreement at low x and low Q², confirming that the momentum-space method can be applied to future collider projects.

What carries the argument

The Block-Durand-Ha parameterization of F2(x, Q²) evaluated in momentum space at the kinematic point x = Q²/s to obtain the ratio σ_r/F2.

If this is right

The momentum-space ratio at fixed Q²/s can be used directly in analyses of LHC data at high inelasticity.
The same parameterization supports predictions for the ratio at Future Circular Collider energies.
Adding a higher-twist term of the form F2 * H2/Q² improves the description of the ratio at low Q² and low x.
The calculated ratio remains bounded by color dipole model expectations across the examined kinematic range.

Where Pith is reading between the lines

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

The momentum-space method may reduce the need for explicit integration over parton distributions when extracting the ratio from future collider measurements.
Agreement at the kinematic boundary x = Q²/s suggests the ratio could serve as a simple observable for testing saturation models at smaller x.
Extending the higher-twist addition to even lower Q² could provide a testable correction for very forward physics at the LHC.

Load-bearing premise

The Block-Durand-Ha parameterization of F2 remains accurate when extrapolated to high-inelasticity and low-x, low-Q² regions.

What would settle it

A measurement of σ_r/F2 at a new collider energy or at x values below current HERA reach that lies outside the calculated band including the higher-twist term would show the extrapolation fails.

Figures

Figures reproduced from arXiv: 2511.04245 by G.R.Boroun.

**Figure 2.** Figure 2: FIG. 2: We plot the ratio [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

read the original abstract

We present a calculation of the ratio $\frac{\sigma_{r}}{F_{2}}(x, Q^2)$ in momentum-space approach using the Block-Durand-Ha (BDH) parameterization of the proton structure function $F_{2}(x,Q^2)$. The results are compared with H1 data and extended to high inelasticity. We also examine the ratio $\frac{\sigma_{r}}{F_{2}}(\frac{Q^2}{s}, Q^2)$ obtained at a fixed $\sqrt{s}$ and $Q^2$ to the minimum value of $x$ given by $Q^2/s$, comparing them with both the HERA data and the color dipole model bounds. These results and comparisons with HERA data demonstrate that the suggested method for the ratio $\frac{\sigma_{r}}{F_{2}}$ can be applied in analyses of the Large Hadron Collider and Future Circular Collider projects. The effect of adding a simple higher twist term of the form $F_{2}{\ast}H_{2}/Q^2$ to the description of the ratio $\frac{\sigma_{r}}{F_{2}}(\frac{Q^2}{s}, Q^2)$ at low-$x$ and low-$Q^2$ values for comparison with the color dipole bounds and the HERA data is investigated.

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.

Desk Editor's Note private letter to a colleague

Applies BDH parameterization to the sigma_r over F2 ratio in momentum space and extends to collider kinematics at x=Q2/s, but the extrapolation claim rests on unverified assumptions outside the fit range.

read the letter

The punchline here is that the paper applies the existing Block-Durand-Ha parameterization of the proton structure function to compute the ratio of the reduced cross section to F2 in a momentum-space framework, with extensions to high inelasticity and specific collider kinematics at x equal to Q squared over s. It shows reasonable agreement with H1 data in accessible regions and explores the impact of a simple higher-twist addition when comparing to color-dipole bounds. What the work does well is provide explicit numerical evaluations and direct comparisons to both experimental data from HERA and theoretical bounds from the color dipole model. The momentum-space approach seems to allow straightforward calculation of the ratio at fixed center-of-mass energy, which could be practical for estimating behaviors at future machines. Adding the higher-twist term F2 times H2 over Q squared and checking its effect at low x and low Q2 gives a sense of how corrections might influence the ratio in those regimes. The soft spots are more about the foundations than the execution. Since the BDH parameterization comes from a fit to data, the comparisons to HERA measurements carry some built-in dependence, even if the color-dipole bounds offer an independent check. The key claim that this method can be applied to analyses at the LHC and Future Circular Collider hinges on the reliability of the extrapolation when x is set to Q2/s and when inelasticity gets high. The abstract does not detail any additional validation, such as testing against alternative F2 fits or data from other experiments in the extrapolated domain. If the BDH form starts to deviate there, the ratio values and the collider applicability statement would need revisiting. This paper is really for phenomenologists and experimentalists who deal with deep-inelastic scattering at high energies and want a computational shortcut for this particular ratio. Someone planning analyses for next-generation colliders might pull the numbers or the method for quick estimates, but it is unlikely to change how people think about QCD dynamics or structure functions at a fundamental level. Overall, the calculations appear solid enough on their own terms to warrant a serious look from referees. I would recommend sending it out for peer review, with the expectation that the authors clarify the extrapolation assumptions and perhaps include more cross-checks to strengthen the collider part.

Referee Report

2 major / 3 minor

Summary. The manuscript calculates the ratio σ_r / F_2(x, Q²) in a momentum-space approach employing the Block-Durand-Ha (BDH) parameterization of the proton structure function F_2. Results are compared to H1 data, extended to high inelasticity y, and evaluated specifically at the minimum x = Q²/s for fixed √s, with comparisons to HERA data and color-dipole model bounds. The effect of an optional higher-twist term of the form F_2 * H_2 / Q² is examined at low x and low Q². The authors conclude that the method can be applied to analyses of the LHC and Future Circular Collider projects.

Significance. If the BDH parameterization remains reliable when extrapolated to the high-y, low-x, low-Q² domain accessed by collider kinematics, the work supplies a concrete momentum-space procedure for estimating the reduced-cross-section to structure-function ratio. This could be useful for planning measurements at the LHC and FCC. The comparisons with color-dipole bounds provide a modest independent cross-check, but the overall significance is limited by the absence of direct validation of the extrapolation.

major comments (2)

[Abstract and BDH parameterization discussion] Abstract and the paragraph on BDH use and higher-twist addition: the claim that the suggested method 'can be applied in analyses of the Large Hadron Collider and Future Circular Collider projects' rests on the BDH parameterization remaining accurate at x = Q²/s and y approaching 1. No cross-check against independent F_2 fits or direct data in the extrapolated domain is described, which is load-bearing for the central collider-applicability statement.
[Results for ratio at Q²/s] The section presenting the ratio at fixed √s: the comparison with HERA data and color-dipole bounds is shown, yet the manuscript provides no quantitative uncertainty band arising from the BDH extrapolation itself. This omission weakens the robustness of the ratio values used to support the LHC/FCC conclusion.

minor comments (3)

[Introduction] The notation distinguishing the ratio evaluated at x = Q²/s from the general (x, Q²) case should be introduced more explicitly in the introduction to avoid reader confusion.
[Figures] Figure captions for the comparisons with H1 and HERA data could usefully state the precise kinematic cuts applied and whether any post-selection on y was performed.
[BDH parameterization section] A reference to the original Block-Durand-Ha paper should be added if not already present, together with a brief statement of the fit range in x and Q².

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 each major point below, providing clarifications on the scope of our momentum-space approach and proposing revisions where appropriate to strengthen the presentation.

read point-by-point responses

Referee: [Abstract and BDH parameterization discussion] the claim that the suggested method 'can be applied in analyses of the Large Hadron Collider and Future Circular Collider projects' rests on the BDH parameterization remaining accurate at x = Q²/s and y approaching 1. No cross-check against independent F_2 fits or direct data in the extrapolated domain is described.

Authors: We agree that the central claim of applicability to LHC and FCC analyses depends on the BDH parameterization's behavior under extrapolation to lower x and higher y. The manuscript demonstrates consistency with existing H1 data at high inelasticity and compares the ratio to color-dipole bounds, but does not include explicit comparisons to other independent F2 parameterizations in the extrapolated regime. The momentum-space method for computing the ratio is general and can be paired with any F2 description; BDH was chosen as a concrete, analytic example that reproduces HERA data well. We will revise the abstract and the relevant discussion paragraph to qualify the applicability statement with a brief note on the kinematic range of BDH validation from the literature and to emphasize that future data can be used to test the extrapolation. revision: partial
Referee: [Results for ratio at Q²/s] the manuscript provides no quantitative uncertainty band arising from the BDH extrapolation itself. This omission weakens the robustness of the ratio values used to support the LHC/FCC conclusion.

Authors: We concur that quantitative uncertainty estimates would improve the robustness of the presented ratio values. Because the original BDH fit does not supply a public covariance matrix, a full propagation of parameter uncertainties is not straightforward within the present scope. However, the manuscript already explores sensitivity through the optional higher-twist term. In the revised version we will add a short discussion of this sensitivity and include illustrative bands obtained by varying the higher-twist coefficient within the range that still describes the HERA data, thereby providing a quantitative indication of theoretical uncertainty on the ratio at fixed √s. revision: yes

Circularity Check

0 steps flagged

No significant circularity; external parameterization and data benchmarks

full rationale

The paper computes the ratio σ_r/F2 using the Block-Durand-Ha parameterization of F2(x,Q²) as an input model and compares the resulting values (including at x = Q²/s) directly to HERA/H1 data and color-dipole bounds. No derivation step reduces by construction to a fitted quantity renamed as a prediction, nor does any central claim rest on a self-citation chain or self-definitional loop. The applicability statement for LHC/FCC follows from these external comparisons rather than from re-deriving the input parameterization itself. The derivation is therefore self-contained against independent benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the accuracy of the pre-existing BDH parameterization across the extended kinematic range and on the validity of the momentum-space formulation for the ratio; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption BDH parameterization of F2(x,Q2) is sufficiently accurate for the ratio calculations at high inelasticity and low x
The entire computation is built on this parameterization as the input for F2.

pith-pipeline@v0.9.0 · 5781 in / 1194 out tokens · 43465 ms · 2026-05-18T01:15:48.192750+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We present a calculation of the ratio σ_r/F2(x,Q²) in momentum-space approach using the Block-Durand-Ha (BDH) parameterization of the proton structure function F2(x,Q²).
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The ratio σ_r/F2(Q²/s,Q²)|y=1 = 1 - FL/F2(Q²/s,Q²).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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G.R.Boroun and B.Rezaei, Phys.Rev.C 103, 065202 (2021)

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