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Parton distributions confront LHC Run II data: a quantitative appraisal

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that, once PDF, strong-coupling and missing-higher-order uncertainties are all included, the major PDF sets describe LHC Run II and HERA data equally well and none generalises better to unseen data.

desk verdict A comprehensive, transparent PDF benchmark whose main claim is plausible but rests on a regularisation choice that needs a robustness test. read the letter →

arxiv 2501.10359 v2 pith:DJOCY72Z submitted 2025-01-17 hep-ph hep-ex

Amedeo Chiefa , Mark N. Costantini , Juan Cruz-Martinez , Emanuele R. Nocera , Tanjona R. Rabemananjara , Juan Rojo , Tanishq Sharma , Roy Stegeman
show 1 more author
Maria Ubiali
This is my paper · ORCID
classification hep-phhep-ex
keywords partondistributionfunctionsPDFbenchmarkingNNLOQCDpredictionsLHCRunIIdataDrell-Yanproductiontopquarkpairjetcrosssectionstheoreticaluncertainties
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether the leading parton distribution function (PDF) sets can be told apart by how well they predict high-precision measurements that were, with one exception, not included in their fits. It compares NNLO QCD predictions for LHC Run II Drell-Yan, top-quark pair, single-inclusive jet and di-jet cross sections, and HERA jet cross sections, using ABMP16, CT18 (and variants), MSHT20, NNPDF3.1, NNPDF4.0, PDF4LHC15 and PDF4LHC21. The central finding is that CT18, MSHT20, NNPDF4.0 and PDF4LHC21 describe all the datasets comparably well once PDF, strong-coupling and missing-higher-order uncertainties are all included in the comparison. The paper concludes that no major PDF set generalises better to unseen data. This matters because PDF choice is often the dominant uncertainty in LHC determinations of Standard Model parameters such as the W mass, the strong coupling and the weak mixing angle.

What carries the argument

The engine of the comparison is the reduced $\chi^2$ figure of merit of Eq. (3.1), evaluated with a total covariance matrix that adds four independent sources of uncertainty: experimental, missing higher orders (estimated by 7-point renormalisation/factorisation scale variations), PDF (from Hessian eigenvectors or Monte Carlo replicas), and $\alpha_s(m_Z)=0.118\pm0.001$. Predictions are computed at NNLO QCD accuracy and stored as PineAPPL interpolation grids, which lets every PDF set be evaluated at no extra cost. Because several experimental covariance matrices are ill-conditioned, the paper regularises them by clipping singular values below a threshold $Z=4$, following Ref. [109]. The $\Delta\chi^2$ and $\Delta n_\sigma$ estimators then convert raw $\chi^2$ differences into units of the expected statistical fluctuation of the $\chi^2$ distribution.

What would settle it

Recompute the $\chi^2_{\rm exp+th}$ spread for the CMS W rapidity and ATLAS/CMS single-inclusive jet datasets using the experiments' own alternative correlation or decorrelation models instead of the universal $Z=4$ clipping. If any PDF set moves by more than about one standard deviation of the $\chi^2$ distribution relative to the others, the claim that all PDF sets generalise equally well to those data would fail.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the apparent differences in predictive power between modern PDF sets largely disappear when all sources of theoretical uncertainty are treated on the same footing. Computed with the $\chi^2$ per data point and a covariance matrix that adds experimental, missing-higher-order, PDF and $\alpha_s$ uncertainties, the CT18, MSHT20, NNPDF4.0 and PDF4LHC21 sets give statistically equivalent descriptions of every dataset examined. The one exception is the ATLAS 8 TeV inclusive Z rapidity distribution, which NNPDF4.0 describes poorly even though an earlier version of the same measurement was used in its fit; the paper traces this to a tension between that dataset and other Drell-Yan data, and shows that including it in a fit with missing-higher-order uncertainties restores an acceptable description. From this the paper concludes that all major PDF sets have similar predictive power and that the spread between sets should not be read as evidence that one set is systematically better.

Load-bearing premise

The load-bearing assumption is that cutting the smallest singular values of the experimental covariance matrices (the same threshold $Z=4$ for all datasets) does not change which PDF set looks best, even though for the CMS W and LHC jet data this procedure shifts the $\chi^2$ by several standard deviations, and the paper asserts this rather than demonstrating it.

Editorial extensions

If this is right

  • If the claim holds, precision Standard Model measurements at the LHC do not need to identify a single best PDF set; any of the major sets yields equally reliable predictions once its uncertainties are propagated.
  • A PDF set with small uncertainties, such as NNPDF4.0, does not thereby predict unseen data better; precision and predictive power are separate properties.
  • The ATLAS 8 TeV Z rapidity measurement, which underlies a precise $\alpha_s(m_Z)$ extraction, is in tension with other Drell-Yan data in NNPDF4.0, so PDF uncertainties quoted from a single baseline set may understate the spread.
  • LHC jet and top-quark pair measurements, not HERA jets or Drell-Yan rapidity shapes, are the datasets most able to discriminate PDF sets and to constrain the large-$x$ gluon.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the same methodology could be applied to the newer aN3LO and MHOU-equipped PDF sets to test whether comparable predictive power survives another order in perturbation theory.
  • Beyond the paper: the finding that PDF set choice rarely moves $\chi^2_{\rm exp+th}$ by more than one standard deviation suggests that future PDF benchmarking should report agreement including theory covariance matrices as standard, rather than comparing experimental $\chi^2$ only.
  • Beyond the paper: a direct test of the weakest assumption would be to recompute the CMS W and jet conclusions using experiment-specific decorrelation models; if the regularisation changes relative PDF rankings in any of those datasets, the comparable-predictive-power claim would need to be restricted to the remaining datasets.
  • Beyond the paper: if the conclusion holds, the spread between PDF sets quoted in $\alpha_s(m_Z)$, $m_W$, and $\sin^2\theta_{\rm eff}$ analyses should be interpreted as methodological spread rather than as evidence that one set is wrong.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper benchmarks nine modern PDF sets (ABMP16, CT18/CT18A/CT18Z, MSHT20, NNPDF3.1, NNPDF4.0, PDF4LHC15, PDF4LHC21) against high-precision LHC Run II and HERA datasets for Drell-Yan, top-quark pair, inclusive jet, dijet, and HERA jet production. Theoretical predictions are computed at NNLO QCD using PineAPPL interpolation grids, and the data-theory agreement is quantified through reduced chi-square statistics that include experimental, PDF, alpha_s, and missing-higher-order uncertainties. The central claim is that, once all theory uncertainties are properly accounted for, the CT18, MSHT20, NNPDF4.0, and PDF4LHC21 sets provide a comparable description of the data and generalise similarly well to unseen measurements.

Significance. If the conclusions hold, this is a useful and timely quantitative comparison. The paper provides a transparent methodology, makes the NNLO interpolation grids publicly available, and explicitly documents the effect of covariance-matrix regularisation in Appendix A. The most nontrivial finding, that NNPDF4.0 with its markedly smaller PDF uncertainties does not describe the data worse on average than less precise sets, is an important input for the PDF-user community. The study of the ATLAS 8 TeV Z rapidity measurement, including dedicated refits and comparisons with earlier versions of the data, is careful and informative. The main caveat is that the central conclusion of 'comparable predictive power' rests on the regularisation of ill-conditioned experimental covariance matrices, whose threshold is not varied or validated.

major comments (3)
  1. [Sect. 3.2, Appendix A] The conclusion that no PDF set is significantly better on the CMS W and LHC jet datasets is not robustly established because it depends on the singular-value clipping threshold Z=4. Table A.1 shows that regularisation changes chi2_exp+th for CMS W+ from an unregularised spread of 10.2-14.6 (about 13 sigma) to a regularised spread of 0.85-1.56 (about 2 sigma), with analogous reductions for the jet datasets. The paper asserts in Appendix A that the regularisation does not alter the relative pattern of chi2 values across PDF sets, but it does not demonstrate that the statistical significance of the PDF-to-PDF differences is stable to the choice of Z or to alternative physically motivated decorrelation models (e.g., the model of Ref. [111] for jets). Please provide a scan over the clipping threshold (for instance Z=3, 4, 6, 8) or an alternative decorrelation prescription, and show how the Delta-n_sigma estimators and the final conclusions change.
  2. [Eq. (3.7)] The 7-point MHO prescription is mis-specified: the definition of Delta0- is repeated with the same arguments (1,1/2), so the independent variation (mu_R=1, mu_F=2) is missing. Since the MHO covariance matrix Eq. (3.5) is a key ingredient of the theory covariance matrix that drives the central conclusion, please correct this definition. If the numerical results were obtained with the correct set of scale variations, state so explicitly; if they were obtained using Eq. (3.7) as written, the affected chi2 values must be recomputed and the conclusions checked.
  3. [Eq. (3.18), Sect. 4.6] The estimator Delta-n_sigma uses sqrt(2/ndat) as the standard deviation of the spread of chi2 values across PDF sets. This is the expected fluctuation of a single chi2 statistic under the null hypothesis, but the chi2 values for different PDF sets are not independent: they are computed from the same data and from theory predictions that are correlated through the underlying PDFs and the same theoretical framework. The statement that PDF-to-PDF differences are 'almost always within Delta-n_sigma = 1' is therefore not a statistically rigorous significance statement. Please replace this heuristic with a more appropriate test, for example by constructing the distribution of chi2 differences under a bootstrap or by using the covariance of theory predictions across PDF sets.
minor comments (5)
  1. [Table 3.1] The number of data points for LHCb 13 TeV Z is listed as 17 in Table 3.1 but as 18 in Tables 2.1 and 4.1; similarly, the H1 low-Q2 single-inclusive jet and dijet datasets are listed with 48 points in Table 3.1 but with 37 points in Tables 2.1 and 4.6. Please correct these inconsistencies.
  2. [Fig. 4.1 caption] The caption states sqrt(2/ndat)=0.23 for the CMS W datasets with ndat=18; the correct value is sqrt(2/18)=0.33, as stated in Table 4.1.
  3. [Sect. 3.2] The sentence in Sect. 3.2 that the regularisation 'does not alter our judgement' on the relative ability of PDF sets is an assertion rather than a demonstrated result; after the requested stability test is added, this statement should be either substantiated or qualified.
  4. [Sect. 4.4, Sect. 5] The phrase 'similar predictive power' may overstate the case because including PDF uncertainties in the covariance matrix makes agreement easier for sets with large uncertainties; the paper does show chi2_exp separately, but the summary would benefit from explicitly noting that the comparable description holds only after the PDF uncertainties of each set are folded into the figure of merit.
  5. [Sect. 2.3-2.4] For LHC and HERA jet production, the computations are at NNLO in the leading-color approximation and do not include NLO electroweak corrections; the possible impact of these missing terms on the chi2 values, especially at high pT, should be commented on explicitly, even if the effect is expected to be PDF-independent.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: external benchmark data are compared against published PDF sets, and the conclusion is conditional on explicitly included theory uncertainties.

full rationale

The paper is an external benchmark, not a derivation. It takes published PDF sets from independent groups and compares them against LHC and HERA datasets, mostly not included in those PDF determinations. The chi2 metric of Eq. (3.13) includes, by explicit construction, the PDF, alpha_s, and MHO uncertainties of the same set being tested; this is a stated condition of the comparison ('once all sources of theoretical uncertainty are taken into account'), not a concealed fit or a renamed prediction. The closest element to a self-citation concern is the use of the covariance-matrix regularisation of Ref. [109], whose author list overlaps with this paper. However, the paper reports the unregularised chi2 values in Table A.1 and argues that the relative pattern across PDF sets is preserved; whether that argument is fully convincing is a robustness question, not a circular reduction. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no central claim is justified solely by a self-citation. The conclusion that no PDF set is systematically better on these data is thus an empirical finding of the benchmark, contingent on the stated methodology, rather than an artifact of the construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new free parameters fitted to data, but relies on several hand-chosen prescriptions: the MHO covariance normalization, the Z=4 regularisation threshold, and kinematic cuts. The axioms are standard QCD domain assumptions plus one ad hoc threshold adopted from prior literature. No invented physical entities appear.

free parameters (3)
  • MHO covariance normalization factor = 1/3 (prescription of Refs. [105,106])
    Chosen by hand in Eq. (3.5) to estimate missing higher order uncertainties from 7-point scale variations; scales the MHO covariance matrix.
  • Covariance regularisation clipping constant Z = Z = 4
    Threshold for clipping singular values of experimental correlation matrices, taken from Ref. [109]; determines which datasets are regularised and how strongly.
  • Low-pT cut for ATLAS Z pT spectrum = 30 GeV
    Kinematic cut applied in Sect. 2.1 to remove the resummation region; hand-chosen.
assumptions (6)
  • domain assumption Missing higher order uncertainties are estimated from 7-point renormalisation and factorisation scale variations with a 1/3 normalization.
    Invoked in Eq. (3.5) to construct the MHO covariance matrix; a standard but non-unique prescription.
  • domain assumption Experimental correlated uncertainties are 100% correlated when reconstructing the covariance matrix.
    Assumed in Eq. (3.4); the paper notes decorrelation is only applied via the regularisation procedure.
  • domain assumption PDF and alpha_s uncertainties are Gaussian and mutually independent, so they add in quadrature.
    Used in Eqs. (3.8)-(3.12) to build the total theory covariance; Gaussianity is an approximation for PDF sets with non-Gaussian tails.
  • domain assumption Data points with scale mu <= 10 GeV are excluded for HERA jets to stay above the b-quark mass in the massless-jet scheme.
    Stated in Sect. 2.4; a kinematic cut required by the zero-mass variable-flavour-number scheme.
  • ad hoc to paper The condition number threshold Z=4 defines when an experimental covariance matrix is ill-conditioned.
    Adopted from Ref. [109] in Sect. 3.2; determines which datasets are regularised and affects the reported chi2 values.
  • standard math The chi2 per point is approximately normal with standard deviation sqrt(2/ndat).
    Used in Eqs. (3.16)-(3.18) to gauge the statistical significance of PDF-set differences.

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Cite this review

Pith. "Pith review of Parton distributions confront LHC Run II data: a quantitative appraisal." pith.science (2026). https://pith.science/paper/DJOCY72Z

@misc{pith2026250110359,
  author       = {Pith},
  title        = {Pith review of: Parton distributions confront LHC Run II data: a quantitative appraisal},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DJOCY72Z}},
  note         = {Machine review of arXiv:2501.10359}
}
read the original abstract

We present a systematic comparison of theoretical predictions and various high-precision experimental measurements, specifically of differential cross sections performed by the LHC run II for Drell-Yan gauge boson, top-quark pair, single-inclusive jet and di-jet production, and by HERA for single-inclusive jet and di-jet production. Theoretical predictions are computed at next-to-next-to-leading order (NNLO) accuracy in perturbative Quantum Chromodynamics. The most widely employed sets of Parton Distribution Functions (PDFs) are used, and PDF, strong coupling, and missing higher order uncertainties are taken into account. We quantitatively assess the predictive power of each PDF set and the contribution of the different sources of experimental and theoretical uncertainty to the agreement between data and predictions. We show that control over all of these aspects is crucial to precision physics studies, such as the determination of Standard Model parameters at the LHC.

Figures

Figures reproduced from arXiv: 2501.10359 by the authors.

Figure 3.1
Figure 3.1. The quark-quark (top left), quark-antiquark (top right), gluon-quark (bottom left), and gluon-gluon (bottom right) partonic luminosities, Eqs. (1–4) of [126], as a function of the invariant mass of the final state mX at a centre-of-mass energy √ s = 13 TeV obtained with the ABMP16, CT18, MSHT20, NNPDF4.0, and PDF4LHC21 PDF sets. Results are normalised to PDF4LHC21. collated in a single set. The number of replicas is… view at source ↗
Figure 4.1
Figure 4.1. The breakdown of χ 2 exp+th, Eq. (3.13), into χ 2 exp+mho, Eq. (3.15), and χ 2 exp, Eq. (3.14), for the ATLAS (ndat = 38, p 2/ndat = 0.23), CMS (each set made of ndat = 18, p 2/ndat = 0.23), and LHCb (ndat = 18, p 2/ndat = 0.33) Drell-Yan gauge boson production measurements at the LHC 13 TeV. Note that the inclusion of MHOUs has a negligible impact on the χ 2 values of the CMS 13 TeV W distributions when theoretical… view at source ↗
Figure 4.2
Figure 4.2. Data-theory comparison for the ATLAS, CMS, and LHCb Drell-Yan gauge boson production measure￾ments at the LHC 13 TeV of [PITH_FULL_IMAGE:figures/full_fig_p017_4_2.png] view at source ↗
Figures from the paper (11 more)
Figure 4.3
Figure 4.3. Figure 4.3: Same as Figs. 4.1 (left) and 4.2 (right) for the ATLAS Drell-Yan gauge boson production measurements at the LHC 8 TeV [42] (ndat = 7, p 2/ndat = 0.53). of the data description of the CMS measurement would have been rather worse, at face value, had the regularisation …
Figure 4.4
Figure 4.4. Figure 4.4: Same as [PITH_FULL_IMAGE:figures/full_fig_p022_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: Same as [PITH_FULL_IMAGE:figures/full_fig_p023_4_5.png]
Figure 4.6
Figure 4.6. Figure 4.6: Same as [PITH_FULL_IMAGE:figures/full_fig_p025_4_6.png]
Figure 4.7
Figure 4.7. Figure 4.7: Same as [PITH_FULL_IMAGE:figures/full_fig_p025_4_7.png]
Figure 4.8
Figure 4.8. Figure 4.8: Same as [PITH_FULL_IMAGE:figures/full_fig_p026_4_8.png]
Figure 4.9
Figure 4.9. Figure 4.9: Same as [PITH_FULL_IMAGE:figures/full_fig_p026_4_9.png]
Figure 4.10
Figure 4.10. Figure 4.10: Same as [PITH_FULL_IMAGE:figures/full_fig_p028_4_10.png]
Figure 4.11
Figure 4.11. Figure 4.11: Same as [PITH_FULL_IMAGE:figures/full_fig_p029_4_11.png]
Figure 4.12
Figure 4.12. Figure 4.12: The relative change in the total χ 2 exp+th due to the change of i-th input PDF set, ∆χ 2(i) , with respect to the average χ 2 exp+th over PDF sets, see Eq. (3.16). The PDF sets considered here are ABMP16, CT18, MSHT20, NNPDF4.0, and PDF4LHC21. The datasets are grou…
Figure 4.13
Figure 4.13. Figure 4.13: Same as [PITH_FULL_IMAGE:figures/full_fig_p031_4_13.png]

Discussion (0). Continue with ORCID to comment.

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Reviewed August 10, 2026 · model on record in the stance chip above.