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arxiv: 2605.08940 · v1 · submitted 2026-05-09 · ✦ hep-ph

Recognition: 1 theorem link

· Lean Theorem

Direct determination of the structure functions F_L, F_S and G from F₂ and dF₂/dQ² to O(α_s²)

G.R. Boroun, Loyal Durand, Phuoc Ha

Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:07 UTC · model grok-4.3

classification ✦ hep-ph
keywords deep inelastic scatteringstructure functionssmall-x physicsperturbative QCDgluon distributionF2 structure functionlongitudinal structure function
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The pith

Expressions for the longitudinal, singlet and gluon structure functions can be obtained directly from measured F2 and its derivative with respect to Q squared at small x to order alpha_s squared.

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

The paper shows how to derive formulas for the longitudinal structure function FL, the singlet structure function FS, and the gluon structure function G straight from the measured F2 and dF2 over d ln of Q squared. These formulas hold in the small-x region of deep inelastic scattering after subtracting small non-singlet contributions that can be handled with known quark distributions. The derivation is carried through consistently at order alpha_s squared, fixing and extending earlier mixed-order results. A reader would care because these functions carry direct information on the gluon density and quark singlet inside the proton, quantities that control many high-energy predictions.

Core claim

By using the perturbative relations among the structure functions at next-to-next-to-leading order, it is possible to solve for FL, FS and G in terms of F2 and dF2/d ln(Q²), with the only additional input being the small non-singlet pieces that are subtracted using existing quark distributions. The resulting expressions are valid in the very small-x regime and are presented uniformly to O(alpha_s²).

What carries the argument

The perturbative QCD relations among structure functions at O(alpha_s²), obtained from coefficient functions and DGLAP splitting functions, that allow direct algebraic solution for FL, FS and G once F2 and its logarithmic Q² derivative are known.

If this is right

  • The gluon structure function G can be extracted from existing F2 data without assuming a parametric form for the gluon density.
  • The expressions provide a consistent next-to-next-to-leading-order framework rather than the mixed-order treatment used previously.
  • Non-singlet contributions are isolated and handled separately, leaving the small-x singlet and gluon pieces determined by F2 and its derivative alone.
  • The method supplies a practical route to obtain FL directly from data, which is otherwise harder to measure.

Where Pith is reading between the lines

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

  • The approach could be applied to HERA or future Electron-Ion Collider data sets to produce rapid estimates of small-x gluon distributions for use in other calculations.
  • If the formulas hold, they offer an independent cross-check on global parton-distribution fits that rely on the same data but different fitting assumptions.
  • Higher-order extensions or small-x resummation could be incorporated by repeating the same inversion procedure with the corresponding higher-order coefficient functions.

Load-bearing premise

Non-singlet corrections remain small at very small x and can be subtracted reliably using existing quark distributions.

What would settle it

Compare the values of FL, FS and G obtained from the formulas against independent extractions or global fits at the same small x and moderate Q²; a systematic discrepancy larger than the estimated non-singlet size would falsify the direct-determination claim.

Figures

Figures reproduced from arXiv: 2605.08940 by G.R. Boroun, Loyal Durand, Phuoc Ha.

Figure 1
Figure 1. Figure 1: FIG. 1. The longitudinal structure function [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
read the original abstract

We extend the results of Lappi {\em et al.}, Eur.~Phys.~J.~C {\bf 84}, 84 (2024), to show that it is possible to obtain expressions for the longitudinal, singlet and gluon structure functions $F_L$, $F_S$ and $G$ in deep inelastic scattering directly in terms of the measured functions $F_2$ and $dF_2/\ln(Q^2)$ {\em modulo} non-singlet corrections expected to be small at very small $x$. The latter can be treated at low $x$ using existing quark distributions. Our results are presented consistently to $O(\alpha_s^2)$, correcting and extending the mixed-order results of Lappi {\em et al.}.

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 manuscript extends the results of Lappi et al. to O(α_s²) by deriving explicit expressions for the longitudinal structure function F_L, the singlet quark structure function F_S, and the gluon structure function G in deep inelastic scattering. These are obtained directly as linear combinations of the measured F_2 and dF_2/d ln Q² after subtracting non-singlet quark contributions, which are argued to be small at very small x and treatable via existing quark distributions. The derivation inverts the O(α_s²) coefficient functions and DGLAP splitting functions while maintaining perturbative consistency.

Significance. If the central expressions hold with controlled errors, the work would enable more direct extraction of F_L, F_S, and G from data at small x without requiring complete global PDF fits, which is potentially useful for testing small-x QCD dynamics and for future collider analyses. The consistent O(α_s²) treatment corrects the mixed-order limitation of the prior work and provides a falsifiable framework once explicit formulas and numerical validations are supplied.

major comments (2)
  1. [non-singlet subtraction procedure (following the system inversion)] The non-singlet subtraction step (invoked throughout the derivation to isolate the singlet and gluon pieces) relies on pre-existing quark distributions. The manuscript does not demonstrate that these distributions are evaluated at precisely the same O(α_s²) perturbative order and with matching small-x accuracy as the coefficient and splitting functions used for the inversion; any mismatch would propagate directly into the extracted F_S and G and undermine the 'modulo small corrections' claim.
  2. [main results section] No explicit analytic expressions for the O(α_s²) coefficients multiplying F_2 and dF_2/d ln Q² are provided in the text, nor are numerical error estimates or comparisons against known small-x limits or existing PDF sets shown. This absence prevents direct verification that the claimed parameter-free character survives after the non-singlet subtraction.
minor comments (2)
  1. [abstract and results] The abstract states the results are 'presented consistently to O(α_s²)', but the manuscript should include a brief table or appendix listing the exact coefficient functions and splitting-function combinations retained at this order for transparency.
  2. [discussion of small-x applicability] A short discussion of the expected numerical size of the non-singlet remainder at the smallest x values accessible at HERA or the EIC would strengthen the practical utility claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which help clarify important aspects of the non-singlet treatment and the presentation of results. We address each major comment below and indicate the revisions planned for the next version of the manuscript.

read point-by-point responses

  1. Referee: [non-singlet subtraction procedure (following the system inversion)] The non-singlet subtraction step (invoked throughout the derivation to isolate the singlet and gluon pieces) relies on pre-existing quark distributions. The manuscript does not demonstrate that these distributions are evaluated at precisely the same O(α_s²) perturbative order and with matching small-x accuracy as the coefficient and splitting functions used for the inversion; any mismatch would propagate directly into the extracted F_S and G and undermine the 'modulo small corrections' claim.

    Authors: We agree that explicit consistency between the perturbative order of the non-singlet subtraction and the O(α_s²) inversion is necessary to maintain the claimed accuracy. In the revised manuscript we will add a dedicated paragraph specifying that the pre-existing quark distributions are to be taken from NNLO (O(α_s²)) global fits. We will further note that, because the non-singlet pieces are power-suppressed at small x, any residual mismatch in small-x resummation enters only at higher order and does not affect the leading small-x behavior of the extracted F_S and G at the working accuracy. revision: yes

  2. Referee: [main results section] No explicit analytic expressions for the O(α_s²) coefficients multiplying F_2 and dF_2/d ln Q² are provided in the text, nor are numerical error estimates or comparisons against known small-x limits or existing PDF sets shown. This absence prevents direct verification that the claimed parameter-free character survives after the non-singlet subtraction.

    Authors: The inversion procedure that yields the coefficients is described in the text, but we accept that the final explicit O(α_s²) expressions are not written out in closed form, which impedes immediate verification. In the revision we will insert the complete analytic expressions for the coefficients multiplying F_2 and dF_2/d ln Q². We will also add a short numerical section that compares the resulting F_L, F_S and G against known small-x limits and against extractions from existing NNLO PDF sets, thereby demonstrating that the parameter-free character is preserved after the non-singlet subtraction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation inverts standard perturbative relations

full rationale

The paper solves the coupled O(α_s²) DGLAP equations and coefficient functions (C_{2,S}, C_{L,S}, C_{2,g}, C_{L,g} and P_{ij}) to algebraically express F_L, F_S and G as linear combinations of the measured F_2 and dF_2/dlnQ² after subtracting a non-singlet quark term. This is a direct inversion of the standard perturbative QCD framework rather than a self-referential definition or fit. The cited Lappi et al. result is from independent authors and serves only as the base case being extended to consistent order; no self-citation chain or uniqueness theorem is invoked to force the result. The non-singlet subtraction uses external PDFs as an approximation justified by small-x suppression and is not part of the core derivation loop. No step reduces by construction to the target quantities themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on perturbative QCD evolution at small x and the assumption that non-singlet terms remain negligible.

axioms (2)
  • domain assumption DGLAP evolution equations relate the Q² derivative of F2 to other structure functions at small x
    Invoked to obtain direct expressions for FL, FS, and G from F2 and dF2/dln(Q²)
  • domain assumption Non-singlet quark contributions are small at very small x
    Stated explicitly in the abstract as the basis for neglecting them in the direct determination

pith-pipeline@v0.9.0 · 5456 in / 1392 out tokens · 57500 ms · 2026-05-12T02:07:07.253410+00:00 · methodology

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Lean theorems connected to this paper

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Reference graph

Works this paper leans on

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