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A new open-source code shows that higher-order perturbative effects on proton PDFs are essentially independent of the fitting methodology.

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2026-08-03 03:40 UTC pith:2PVO2BEV

load-bearing objection A genuinely useful public tool for matched-input PDF comparisons, with a solid main result and one overreach: the paper's attribution of published aN3LO gluon differences to settings rather than methodology goes beyond what a single-baseline design can show. the 3 major comments →

arxiv 2602.07118 v2 pith:2PVO2BEV submitted 2026-02-06 hep-ph hep-ex

Assessing the Impact of Fitting Methodology at aN³LO with FPPDF: an Open Source Tool for Extracting Parton Distribution Functions in the Hessian Approach

J. M. Cruz-Martinez , T. Giani , L. A. Harland-Lang This is my paper
classification hep-ph hep-ex
keywords parton distribution functionsHessian approachneural networksN3LOmissing higher order uncertaintiesPDF fittingopen sourceluminosities
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper introduces FPPDF, an open-source code that performs global fits of parton distribution functions (PDFs) using a fixed polynomial parameterisation with Hessian error propagation, while taking data and theory inputs from the same libraries used by neural-network fits. The authors use it to compare PDFs extracted at next-to-next-to-leading order (NNLO) and approximate next-to-next-to-next-to-leading order (aN3LO), each with and without missing-higher-order uncertainty (MHOU) estimates, across the two fitting methodologies. They find that the relative impact of moving to the higher perturbative order and of including MHOU is largely insensitive to whether the PDF is parameterised by a fixed polynomial or a neural network. They conclude that differences seen between the previously published aN3LO gluon PDFs are driven by theory and data settings, not by the choice of fitting methodology. If right, this means the aN3LO shifts are genuine perturbative effects, and it gives the community a benchmark platform for methodology-controlled PDF studies.

Core claim

The central claim is that the effect of including approximate N3LO corrections and missing-higher-order uncertainties on PDFs and on benchmark cross sections is essentially the same whether the PDFs are extracted with a fixed-polynomial Hessian method or with a neural-network Monte Carlo method, provided the data and theory inputs are identical. Fitting both ways to the same dataset and theory settings, the authors observe the same trends in parton luminosities and in the gluon and singlet PDFs when moving from NNLO to aN3LO: gluon-gluon luminosity is suppressed at small and intermediate invariant masses and enhanced at large masses, and the quark channels shift similarly. The previously obs

What carries the argument

The engine of the paper is FPPDF, a public code that combines a Chebyshev-polynomial parameterisation of PDFs (with an input scale, sum rules imposed analytically, and 52 or 61 free parameters) with Hessian error propagation, while reusing the publicly available data grids, evolution, and theory calculations used by the neural-network fitting code. This makes the only difference between the two compared fits the parameterisation and error-propagation methodology. The paper also uses a dynamic tolerance criterion (roughly Δχ² = T² with T≈3) to define Hessian uncertainties, which produces generally larger PDF errors than the Monte Carlo replica approach.

Load-bearing premise

The comparison assumes that the shared NNPDF-style theory settings—fitted charm at the default input scale, the 7-point MHOU covariance matrix, and the aN3LO theory grids—are equally suitable for both fitting methods, so that any remaining differences are purely due to parameterisation and error propagation.

What would settle it

A repeat of the NNLO-vs-aN3LO comparison in which the two fitting methods disagreed on the sign or size of the gluon luminosity shift at some mX, or in which changing from fitted charm to perturbative charm reversed the agreement, would undermine the claim of methodology insensitivity.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the aN3LO shifts are methodology-independent, then the suppression of the gluon-gluon luminosity at small/intermediate mX and enhancement at large mX is a genuine perturbative feature, not an artefact of a particular fitter.
  • The agreement implies that the previously published aN3LO gluon differences can be traced to theory/data settings; adopting common settings should bring independent sets into better agreement.
  • The open-source nature of FPPDF allows any group to run Hessian polynomial fits and neural-network fits on identical inputs, making methodology comparisons reproducible and extensible.
  • Benchmark cross sections (Higgs, W, Z) show the same relative NNLO-to-aN3LO trends for both methodologies, so predictions like the moderate reduction in Higgs production from aN3LO PDFs are robust to fitting methodology.
  • The code provides a standardised environment to study the role of tolerance and other error-propagation choices in PDF fits, decoupled from data and theory differences.

Where Pith is reading between the lines

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

  • If the insensitivity holds more generally, then PDF uncertainties quoted by a single methodology may be more trustworthy for capturing perturbative shifts, since the shift itself is method-independent; the main methodological differences are in the size of uncertainties, not in the direction of the effect.
  • One testable extension: apply the same comparison at full N3LO when complete splitting functions and hard cross sections become available; if the two methods still agree, the conclusion would extend beyond the 'approximate' corrections.
  • The code could be used to isolate the role of the tolerance parameter by repeating the Hessian fits with fixed T² = 1, which prior work suggests brings the Hessian and Monte Carlo uncertainty estimates into closer agreement; that would sharpen which uncertainty differences are methodological.
  • The valence quark dip seen in the fixed-polynomial fits suggests that the data allow genuine shape freedom in the valence distribution; this could be probed with future measurements targeting medium-x valence quarks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents FPPDF, a new open-source code for PDF global fits that uses the MSHT-like fixed-polynomial parameterisation and Hessian error propagation, but takes data and theory inputs (including theory covariance matrices) from the public NNPDF libraries. As a first application, the authors produce fits at NNLO, NNLO+MHOU and aN3LO+MHOU with both the FPPDF (fixed parameterisation) and NNPDF (neural network) methodologies under identical data/theory settings, and compare central values, uncertainties, parton luminosities, PDFs and benchmark cross sections. Their central claim is that the relative impact of going to higher perturbative order and/or including MHOU is rather insensitive to the PDF parameterisation methodology. They further argue that differences between the published MSHT and NNPDF aN3LO gluon and luminosity results are driven by data/theory settings rather than by methodological choices.

Significance. If the central claim holds, the paper provides a valuable controlled comparison of two major PDF fitting methodologies and strengthens the interpretation of aN3LO PDF shifts as genuine perturbative effects. The release of FPPDF as an open-source tool, with shared runcards and NNPDF-compatible inputs, is a concrete and useful contribution to the community: it lowers the barrier for methodology-controlled PDF studies and enables closure tests and extensions to new datasets. The paper is careful to use identical data, theory grids, heavy-quark treatment and MHOU covariance matrices for both methodologies, which is a clear strength. The main weakness is that the insensitivity claim is assessed visually rather than with a quantitative metric, and the stronger causal attribution about published MSHT/NNPDF differences rests on a single baseline without probing method-settings interactions.

major comments (3)
  1. [Sec. 3.2.3, Fig. 3] The conclusion that the published MSHT/NNPDF aN3LO gluon and luminosity differences are 'driven by the specific theory and data settings ... and not by the specific methodological choices' is underdetermined by the presented evidence. The design varies the methodology at one fixed set of NNPDF-like settings (fitted charm at Q0=1.65 GeV, 7-point MHOU covariance, NNPDF theory grids). This cannot exclude a method x settings interaction: the polynomial basis may respond differently than the neural network to, e.g., fitted charm or the MHOU covariance, so an additive decomposition of methodology vs settings is not justified. To support the attribution, the authors would need to run both methods under at least one alternative settings point (e.g., perturbative charm, Q0=1 GeV, MSHT-style MHOU) or provide closure tests validating the fixed-polynomial extraction under the chosen baseline. Short
  2. [Abstract; Secs. 3.2.1-3.2.3; Figs. 1-2, 7] The central claim that the impact of aN3LO and MHOU corrections is 'rather insensitive' to the fitting methodology is supported only by visual inspection of ratios. Since 'insensitive' is a quantitative statement, please provide a quantitative measure, e.g., the difference between the two methods' NNLO -> aN3LO shifts in luminosities and cross sections, with uncertainties, or a correlation/chi2-type agreement metric. Without this, the claim is not falsifiable and the reader cannot judge the size of the residual methodology dependence relative to the effects being studied.
  3. [Secs. 3.2.2-3.2.3; Figs. 1-7] The uncertainty comparisons mix different prescriptions: the fixed-parameterisation sets use the dynamic tolerance (T~3) while the NNPDF sets use the MC replica 68% confidence level. The authors acknowledge this, but it affects any quantitative reading of 'impact' in terms of uncertainties, e.g., the larger fixed-parameterisation bands in Figs. 1-7. Please separate statements about central-value shifts from statements about uncertainty size, and/or show results for the fixed-parameterisation fits with T^2=1 in an appendix or supplementary figure, so that the methodology comparison is not conflated with the tolerance choice.
minor comments (5)
  1. [Fig. 3 caption] The caption says 'for the (left) fixed parameterisation and (right) NNPDF methodologies', but all panels appear to show both methodologies overlaid with ratio to aN3LO MHOU nnpdf. Please correct the caption to match the actual layout.
  2. [Fig. 2] The middle-row y-axis label appears to be missing the symbol for the singlet distribution; it is displayed as ' at 100 GeV'.
  3. [Fig. 4 right panel and Sec. 3.2.3] The caption refers to 'MSHT and NNPDF methodologies' for the charm comparison, but the fits are produced with FPPDF (fixed parameterisation) and NNPDF. Please use consistent terminology to avoid implying the results come from the published MSHT code.
  4. [Sec. 4 vs Sec. 3.2.3] The opening of Sec. 4 states that the authors 'have not aimed to explain the differences observed between PDFs produced by different methodologies', which seems inconsistent with the causal attribution made in Sec. 3.2.3 (that differences are driven by data/theory settings and not by methodology). Please clarify the intended scope.
  5. [Sec. 3.2.1] The bullet 'the fit quality in the fixed parameterisation case is somewhat lower in the no MHOU fits' could be misread as worse fit quality; since lower chi2 is better, consider rephrasing to 'the chi2 is lower (better)' for clarity.

Circularity Check

0 steps flagged

Empirical methodology comparison with no circular reduction; self-citations are contextual, not load-bearing.

full rationale

The paper's central claim is a comparative statement about fits produced by two independent parameterisation methodologies (fixed Chebyshev polynomial vs neural network) using identical data and theory inputs. The NNLO-to-aN3LO shifts in PDFs and luminosities are read off from the fits; no equation defines the claimed result in terms of a fitted value, and no fitted parameter is relabeled as a prediction. FPPDF's adoption of the MSHT polynomial parameterisation and dynamic tolerance is an explicit import of the methodology being tested, not a hidden derivation of the conclusion. Self-citations are present (e.g., Refs. [4,8], which involve one of the current authors), but they are used as background, motivation, and methodological definitions; the new fits performed in this paper are the actual evidence for the central claim. The Sec. 3.2.3 attribution that published MSHT/NNPDF aN3LO differences are driven by theory/data settings rather than methodology is an interpretation of the controlled single-baseline comparison and may be broader than the design strictly proves, but underdetermination of that attribution is a strength-of-evidence limitation, not circularity. The Sec. 4 statement that the authors have not aimed to explain all methodology differences reinforces that the paper frames itself as a tool and comparison study. No step in the derivation chain reduces to its own inputs by construction, so no significant circularity is identified.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

Most of the physics content is inherited from the NNPDF libraries and the MSHT parameterisation; the paper contributes the code and the comparison. The central claim rests on several domain assumptions about the theory and data, and on specific methodological choices (polynomial form, dynamic tolerance) that are not independently validated here.

free parameters (5)
  • Fixed-polynomial PDF parameters (A, eta, delta, a_i) in Eq. (1) = 52 (perturbative charm) or 61 (fitted charm) free parameters; values not reported
    Fitted to the global dataset; they determine the Hessian central values and eigenvectors.
  • Dynamic tolerance T^2 (Delta chi^2 = T^2) = T ~ 3
    Enlarges the Hessian uncertainties in the fixed-parameterisation fits; set by a dynamic procedure based on fit quality (Sec. 2).
  • Chebyshev polynomial truncation and k = 0.5 = Not specified; 'exact same polynomial parameterisation used in the MSHT20 fit'
    The functional basis (Eq. 1) is an ad hoc modelling choice inherited from MSHT; it limits the space of allowed PDF shapes for the Hessian method.
  • Input scale Q0 for fitted charm = 1.65 GeV
    Chosen to match NNPDF settings; the comparison assumes this is a good common baseline for both methods.
  • NNPDF neural-network weights and stopping criterion = Not specified in this paper
    These are the NNPDF method's internal fitted parameters and hyperparameters; the paper treats them as fixed inputs from the public NNPDF framework.
axioms (5)
  • domain assumption QCD factorisation and the universality of PDFs
    The whole fitting framework assumes PDFs exist and can be extracted from a global dataset (Introduction, Sec. 1).
  • domain assumption Correctness of the NNPDF theory libraries (DGLAP evolution, hard matrix elements, heavy-quark treatment, approximate N3LO splitting functions)
    FPPDF takes all theory grids from the public NNPDF code; the approximations at aN3LO are described in Sec. 3.1.
  • domain assumption Validity of the NNPDF4.0 dataset and experimental covariance matrices
    Both fits use the same public datasets; any experimental or statistical error enters the comparison symmetrically.
  • ad hoc to paper The dynamic tolerance procedure is a valid uncertainty prescription
    Used only in the Hessian fits to inflate uncertainties to T~3; it is not used in the NNPDF method, so uncertainty comparisons depend on this choice.
  • domain assumption Sufficient flexibility of the fixed polynomial basis (Eq. 1)
    If the Chebyshev polynomial basis is too rigid, the Hessian fit could be biased; the paper assumes 52/61 parameters are enough to describe the data.

pith-pipeline@v1.3.0-alltime-deepseek · 13899 in / 14984 out tokens · 142464 ms · 2026-08-03T03:40:13.244680+00:00 · methodology

0 comments
read the original abstract

We present a new public code, FPPDF, to perform global fits of parton distribution functions (PDFs). The fitting methodology follows that implemented by the MSHT collaboration, namely applying a fixed polynomial parameterisation of the PDFs and Hessian approach to error propagation, while for data and theory settings the libraries used by the NNPDF collaboration are taken. This therefore complements the already publicly available NNPDF fitting code to enable fits with both neural network and fixed polynomial PDF parameterisations to be performed by the community, with otherwise identical theoretical and experimental inputs. As a first application, we use the new code to compare the PDFs found from fits at both NNLO and aN$^3$LO perturbative orders, but applying these two fitting approaches. We assess the impact of the two different methodologies on the PDFs and their uncertainties, providing results that complement previous comparisons between published PDF sets at NNLO and aN$^3$LO. We in particular find that the relative impact of going to the higher perturbative order and/or including missing higher order uncertainties is rather insensitive to which of these PDF parameterisation methodologies are used.

Figures

Figures reproduced from arXiv: 2602.07118 by J. M. Cruz-Martinez, L. A. Harland-Lang, T. Giani.

Figure 1
Figure 1. Figure 1: Parton luminosities at NNLO, NNLO+MHOU and aN3 LO+MHOU for the (left) fixed parameterisation and (right) NNPDF methodologies. Fixed parameterisation uncertainties are calculated using the dynamic tolerance criterion. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Gluon and singlet distributions at NNLO, NNLO+MHOU and aN3 LO+MHOU for the (left) fixed parame￾terisation and (right) NNPDF methodologies. Fixed parameterisation uncertainties are calculated using the dynamic tolerance criterion. give somewhat different results, highlighting a tension around mX ∼ 102 GeV for the gluon￾gluon and gluon-quark channels [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between parton luminosities at aN3 LO+MHOU for the (left) fixed parameterisation and (right) NNPDF methodologies. Fixed parameterisation uncertainties are calculated using the dynamic tolerance criterion. 10 5 10 4 10 3 10 2 10 1 10 0 x 0.85 0.90 0.95 1.00 1.05 1.10 1.15 Ratio to aN3LO MHOU nnpdf g at 100 GeV aN3LO MHOU nnpdf (68% c.l.+1 ) aN3LO MHOU fixpar (68% c.l.) 0.2 0.4 0.6 0.8 x 0.000 0.0… view at source ↗
Figure 4
Figure 4. Figure 4: Left panel: comparison between the gluon PDF obtained using the fixed parameterisation and NNPDF methodologies. Right panel: comparison between the medium and large-x charm at 1.65 GeV obtained using the MSHT and NNPDF methodologies. Fixed parameterisation uncertainties are calculated using the dynamic tolerance criterion. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Valence (left) and down quark (right) PDFs found using the fixed parameterisation and NNPDF method￾ologies. Fixed parameterisation uncertainties are calculated using the dynamic tolerance criterion. 10 5 10 4 10 3 10 2 10 1 10 0 x 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 (R atio to a N 3 L O M H O U n n p df) g at 100 GeV aN3LO MHOU nnpdf aN3LO MHOU fixpar 0.2 0.4 0.6 0.8 x 0.00 0.05 0.10 0.15 0.20 … view at source ↗
Figure 6
Figure 6. Figure 6: Gluon PDF uncertainty found using the fixed parameterisation and NNPDF methodologies. Fixed parameterisation uncertainties are calculated using the dynamic tolerance criterion. Valence and down quark PDFs A non–negligible difference between the two sets of results presented here can be seen for the valence distribution, defined as V = P i (qi − q¯i) and plotted in the left panel of [PITH_FULL_IMAGE:figure… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison between selected cross section predictions at the √ s = 14 TeV LHC for the NNLO, NNLO+MHOU and aN3 LO+MHOU PDFs extracted using the fixed parameterisation and NNPDF methodologies. All cross sections are calculated as described in the text, and correspond to N3 LO matrix elements in all cases. PDF uncertainties alone are indicated. dominantly due to the presence of a tolerance in the fixed parame… view at source ↗

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Forward citations

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