nCTEQ26 nuclear parton distribution functions

A. Kusina; C. L\'eger; F.I. Olness; I. Schienbein; J.G. Morf\'in; J. Wissmann; J.Y. Yu; K. Greve; K. Kova\v{r}\'ik; M. Klasen

REVIEW

nCTEQ26 nuclear parton distribution functions

nCTEQ Collaboration, K. Greve, T. Ježo, A. Kusina, J. Wissmann, M. Klasen, K. Kovařík, C. Léger

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J.G. Morfín, F.I. Olness, P. Risse, I. Schienbein, J.Y. Yu

High Energy Physics - Phenomenology · Submitted 9 Oct 2026 · 298 references

Pith's verdict: nCTEQ26 is a solid new benchmark for nuclear PDFs, but the data-driven heavy-quark treatment carries a load-bearing assumption at low x that deserves scrutiny.

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arXiv ID2610.13633 (version 1)

Pith IDOIUQJZYJ, Pith's permanent record of this paper and its review

CitationA. Kusina, C. L\'eger, F.I. Olness, I. Schienbein, J.G. Morf\'in, J. Wissmann, J.Y. Yu, K. Greve, K. Kova\v{r}\'ik, M. Klasen, nCTEQ Collaboration, P. Risse, T. Je\v{z}o. "nCTEQ26 nuclear parton distribution functions." arXiv:2610.13633 (2026). https://pith.science/paper/OIUQJZYJ

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This is the new nCTEQ26 global nPDF fit, and it delivers what it promises: the largest dataset to date (3598 points), a more flexible parametrization (quadratic in log A, CJ15 baseline), and a careful Hessian analysis with Wilks-based tolerance. The result supersedes nCTEQ15 and its follow-ups and will become the standard for LHC, EIC, and DUNE phenomenology. The fit quality is documented in remarkable detail—per-dataset χ², SE distributions, normalization penalties, two-dimensional scans. The authors are transparent about tensions, notably the LHCb Run-II forward D0 data (χ²/Ndata = 2.5) and the CHORUS normalization penalty.

The soft spot is the Crystal-Ball parametrization for heavy-quark(onium) production. The five CB parameters are fitted to pp data and transferred unchanged to pPb, assuming leading-twist collinear factorization with no final-state effects (energy loss, saturation). This assumption is most fragile at low x (forward rapidity), precisely where the fit struggles most. The poor χ² for LHCb forward D0 and the similar tension in the alternative perturbative OHF fit suggest this is not just a technical detail but a load-bearing condition for the low-x gluon. The authors acknowledge saturation as a possible cause, but the CB framework cannot distinguish it from genuine nPDF effects. The circularity is mild—the CB parameters are fitted to pp, not pPb—but the assumption that the same shape holds for nuclear ratios is untested.

That said, the paper is careful. They provide an alternative fit (nCTEQ26OHF) using perturbative QCD for open heavy flavor, which gives consistent gluon PDFs, softening the concern. The methodology is rigorous, the documentation is exhaustive, and the limitations are discussed honestly. This is a genuine advance.

I would accept a peer-review invitation. This paper deserves a serious referee—it is important, methodically executed, and its one significant assumption is flagged for future work. I will cite it and likely bring it to our next reading group.

Referee report

Major objections (3)

[Sec. III F, Eq. (41)] The Crystal-Ball (CB) parametrization for heavy-quark(onium) production assumes that the five parameters (λ, κ, ⟨pT⟩, n, a) fitted to pp data (Table X) can be transferred unchanged to pPb collisions, so that only the nPDFs generate the nuclear dependence. This is a load-bearing assumption because the heavy-quark data provide the only direct constraints on the gluon at x < 10⁻⁴ (Fig. 1). The paper itself reports χ²/Ndata = 2.5 for the LHCb Run-II forward D0 dataset (ID 3133, 66 points), the largest outlier, and notes (Sec. IV B) that this tension peaks at forward rapidity (low x) and speculates about saturation. The authors do not quantify how the low-x gluon PDF changes if these forward data are excluded, if a more stringent kinematic cut is applied, or if an A-dependent term is allowed in the CB parametrization. Since the alternative OHF fit (nCTEQ26OHF) shows similar tension in the same region (Fig. 12c discussion), the concern that beyond-leading-twist effects are absorbed into the nPDFs is not addressed. I request a test of the robustness of the gluon PDF at x < 10⁻⁴ under at least one of these variations, with the resulting shift in the gluon and its uncertainty documented.
[Sec. III F] The heavy-quark data are fitted with a data-driven Crystal-Ball parametrization that assumes 2→2 gg dominance and a specific functional form. The paper states that the approach 'assumes that the gluon-gluon channel contributes the dominant fraction of the total cross section' and that it is 'sufficient to consider subprocesses with 2→2 kinematics.' This approximation is not validated against existing NLO+PS calculations for these processes. While the χ²/Ndof for the pp baseline is good (Table X), the assumption could introduce a systematic bias that is not captured by the Hessian uncertainty. I ask the authors to provide a quantitative estimate of the theoretical uncertainty from this approximation, e.g., by comparing nCTEQ26 predictions to a fixed-order NLO calculation for open heavy flavor in a limited kinematic range, or by quoting the shift in the gluon PDF when the CB fit is omitted altogether (the alternative OHF fit provides part of this, but it excludes quarkonium entirely). Without such an estimate, it is unclear whether the strikingly small gluon uncertainties at low x are realistic.
[Sec. VI B] The comparison between the main nCTEQ26 fit and the alternative nCTEQ26OHF fit (which uses perturbative GM-VFNS for open heavy flavor) is presented only briefly (Fig. 12c discussion and Sec. VI B). The paper states that nCTEQ26OHF produces a 'less satisfactory description' of the OHF data, especially in the forward D0 region. Given that the main fit relies on the CB parametrization for quarkonia, while the alternative excludes them, a direct comparison of the resulting gluon PDFs at low x between the two fits is essential. I request a plot comparing the gluon PDFs (ratios to central or absolute) at Q = 2 GeV and Q = 10 GeV for lead from nCTEQ26 and nCTEQ26OHF, with uncertainty bands, to allow the reader to assess the impact of the CB assumption on the gluon.

Minor objections (7)

[Sec. II A, Eq. (6)] The notation for the A-dependence parameters uses a_j and b_j, but in Table XI the columns are labelled 'a' and 'b'. For clarity, please label the columns consistently with the text, e.g., 'a_j' and 'b_j'.
[Sec. III F, Table VIII] The dataset IDs 3127 (ψ(2S) 5.02 TeV ALICE) report χ²/Ndata = 3.23 with only 2 data points. This is a statistical fluctuation due to low statistics, but the large χ²/Ndata might be misinterpreted. Consider adding a note that datasets with few points (Ndata < 5) can have inflated χ²/Ndata values.
[Sec. IV C, Fig. 14] The eigenvalue comparison between double-precision and 256-bit precision shows agreement to better than 10⁻⁷ (relative). Please specify whether the relative precision applies to all eigenvalues or only the largest, and if the smallest eigenvalues (some negative) also satisfy this criterion.
[Sec. IV B, Fig. 12] The y-axis ranges for the D0 data in panels (a), (c), (d) differ by large factors. The vertical offsets (×100, ×101, etc.) are helpful, but the actual values are not printed on the y-axis. Adding a few numerical labels (e.g., 10^2, 10^4) would improve readability.
[Sec. V A] The comparison with EPPS21 and nNNPDF3.0 uses the nuclear modification ratio R^A_i. Since this ratio depends on the proton baseline used, the paper's note that denominators come from the corresponding analyses is appreciated. However, the discussion would be enhanced by showing also ratios where all groups use the same proton baseline (e.g., CJ15 or CT18) to separate proton-PDF effects from nuclear effects.
[Sec. VI A, Fig. 32] The asymmetric uncertainty band (Eq. 35) shows a much tighter lower bound at low x for the strange quark. The paper states that this is not data-driven and recommends symmetric uncertainties. Please add a sentence clarifying that the symmetric prescription is the authors' recommended choice for the nCTEQ26 release.
[Sec. VII] Predictions for future experiments (NOMAD dimuons, CMS dijets, etc.) are an important validation. For the dijet and top-quark pair predictions, provide a numerical estimate of the expected uncertainty reduction compared to previous nPDF sets (e.g., nCTEQ15) to demonstrate the improved constraining power.

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The paper

The paper claims

nCTEQ26 provides the most precise nuclear PDFs using 3598 data points.

The reading

The nCTEQ Collaboration presents nCTEQ26, a new set of nuclear parton distribution functions (nPDFs) that aims to be the most accurate and comprehensive description of the internal partonic structure of nuclei. The work incorporates the largest collection of experimental data ever assembled in a global nPDF analysis, including neutral- and charged-current deep-inelastic scattering, fixed-target Drell-Yan, W/Z boson production, single-inclusive hadron production, and heavy-quark(onium) production from RHIC and LHC Runs I and II. The analysis uses a more flexible parametrization in both x and nuclear mass number A, target-mass and deuteron corrections, and a data-driven approach for heavy-quark(onium) data. If correct, nCTEQ26 will serve as a critical baseline for identifying non-perturbative nuclear effects in high-energy collider experiments and for future facilities such as the HL-LHC, EIC, and DUNE.

What carries the argument

The central machinery is the nPDF parametrization Eq. (2)-(6) and (10): each parton flavor f_k at the initial scale Q0 = 1.3 GeV is parametrized as x $f^{{p/A}}$_k(x, Q0) = c_{k0} $x^{{c_{k1}}$} (1-x)^{c_{k2}} (1 + c_{k3} sqrt{x} + c_{k4} x) $e^{{c_{k5}}$ sqrt{x}}, with the coefficients c_j(A) = p_j + a_j ln A + b_j $ln^{2}$ A for A>1. This links all nuclei through a common analytic form, pooling constraining power across different nuclear targets while allowing non-monotonic A-dependence. Sum rules fix normalization parameters, and the gluon and strange normalizations are fitted. Hessian error propagation with a tolerance T=50 (based on Wilks theorem) provides uncertainties.

Extended reading notes

Core claim

The nCTEQ26 nPDF set significantly advances the state of the art by simultaneously fitting 3598 data points from 21 different nuclei with a total chi2/Ndof = 0.943. The new parametrization combines a flexible x-dependence (extended from the proton baseline CJ15) with a novel quadratic-logarithmic A-dependence: c_j(A) = p_j + a_j ln A + b_j $ln^{2}$ A. This reduces parameter correlations and allows the uncertainties to reflect data constraints more faithfully, avoiding artificially smooth A-dependence. Target-mass corrections and deuteron corrections are included, enabling the reliable inclusion of high-x JLab data. Heavy-quark(onium) production is treated via a data-driven Crystal-Ball parametrization fitted to pp data, and the consistency of this approach is verified by an alternative fit using perturbative predictions for open heavy-flavor alone. The resulting nPDFs show good agreement with other global sets (EPPS21, nNNPDF3.0) within uncertainties, with notable improvements in the gluon and strange distributions.

Load-bearing premise

Heavy-quark(onium) production in proton-lead collisions can be described by a Crystal-Ball parametrization with the same five parameters (lambda, kappa, <pT>, n, a) fitted to proton-proton data, implying that all nuclear effects beyond leading-twist collinear factorization (e.g., energy loss, saturation) are negligible in the fitted region, especially at low x and forward rapidity where the fit is weakest.

Editorial extensions

If this is right

nCTEQ26 will serve as the standard baseline for distinguishing cold nuclear matter effects from hot QCD matter (e.g., quark-gluon plasma) in pA and AA collisions at the LHC.
The improved constraints on the gluon at small x (down to about 10^-5) directly impact predictions for heavy-quark production, dijet, and direct-photon processes at current and future colliders.
The strange-quark distribution remains the most uncertain, but the inclusion of CC DIS and dimuon data provides tighter constraints around x~0.01-0.1 for iron and lead; the fitted strange-to-light-sea ratio varies with A and scale, which can be tested by future neutrino experiments (e.g., DUNE).
The A-dependence results (e.g., uncertainties that peak at medium A for some flavors) provide quantitative tests for models of nuclear structure, such as those based on short-range correlated pairs.

Reading between the lines

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

The poor fit (chi2/Ndata = 2.5) of the LHCb Run-II forward D0 data in pPb collisions at 8.16 TeV, despite good description of Run-I data, may hint at missing nuclear effects (e.g., saturation) at very low x, which could be clarified by future EIC measurements in the same kinematic region.
The Crystal-Ball parametrization assumption that the same 5 parameters fit to pp data transfer directly to pPb without any final-state nuclear modifications is most vulnerable at forward rapidities and low pT; a dedicated study of direct-photon pPb data in this region would provide an independent cross-check.
The observation that the symmetric uncertainty definition gives much larger lower bounds for the strange distribution at small x than the asymmetric definition suggests that the low-x strange PDF is genuinely unconstrained by current data, not just asymmetric. Future dimuon measurements with extended coverage could dramatically reduce this uncertainty.

Original abstract

We present the new release of nuclear parton distribution functions (nPDFs) from the nCTEQ collaboration: nCTEQ26. In this global analysis, we combine our recent, more selective studies of different experimental measurements and extend them to determine a modern set of nPDFs based on the largest selection of data considered to date. This selection includes neutral- and charged-current deep-inelastic scattering (DIS), fixed-target Drell-Yan, and a large variety of collider data from RHIC and LHC Runs I and II. On the theoretical side, we match this considerably extended kinematic range with a new nPDF parametrization that combines a flexible functional form of the x-dependence with a quadratic logarithmic nuclear A-dependence as well as target-mass and deuteron corrections for the reliable inclusion of high-x DIS data. Following our previous successful work, we partially employ a data-driven approach that allows us to not only include open heavy-quark data, but also quarkonium data. In addition, we establish the consistency of this approach by comparing it to an alternative fit based on perturbative predictions of open heavy-quark data alone. We determine the uncertainties of our nCTEQ26 nPDFs with a Hessian error analysis, test their reliability by comparing our nPDFs to other, similar recent analyses, and make predictions for current and future experimental measurements that have not been included in the analysis.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-12: 1 request, 5,831 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.15 · human participants 0.02 · controlled comparison 0.44 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.97 · mentions code or data availability 0.04.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present the new release of nuclear parton distribution functions (nPDFs) from the nCTEQ collaboration: nCTEQ26. In this global analysis, we combine our recent, more selective studies of different experimental measurements and extend them to determine a modern set of nPDFs based on the largest selection of data considered to date. This selection includes neutral- and charged-current deep-inelastic scattering (DIS), fixed-target Drell-Yan, and a large variety of collider data from RHIC and LHC Runs I and II. On the theoretical side, we match this considerably extended kinematic range with a new nPDF parametrization that combines a flexible functional form of the x-dependence with a quadratic logarithmic nuclear A-dependence as well as target-mass and deuteron corrections for the reliable inclusion of high-x DIS data. Following our previous successful work, we partially employ a data-driven approach that allows us to not only include open heavy-quark data, but also quarkonium data. In addition, we establish the consistency of this approach by comparing it to an alternative fit based on perturbative predictions of open heavy-quark data alone. We determine the uncertainties of our nCTEQ26 nPDFs with a Hessian error analysis, test their reliability by comparing our nPDFs to other, similar recent analyses, and make predictions for current and future experimental measurements that have not been included in the analysis.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present the new release of nuclear parton distribution functions (nPDFs) from the nCTEQ collaboration: nCTEQ26.	contribution 1.00	0.89	0.28	asserted 0.99
s1	In this global analysis, we combine our recent, more selective studies of different experimental measurements and extend them to determine a modern set of nPDFs based on the largest selection of data considered to date.	contribution 0.56	0.93	0.15	asserted 1.00
s2	This selection includes neutral- and charged-current deep-inelastic scattering (DIS), fixed-target Drell-Yan, and a large variety of collider data from RHIC and LHC Runs I and II.	method 0.93	0.53	0.05	asserted 1.00
s3	On the theoretical side, we match this considerably extended kinematic range with a new nPDF parametrization that combines a flexible functional form of the x-dependence with a quadratic logarithmic nuclear A-dependence as well as target-mass and deuteron corrections for the reliable inclusion of high-x DIS data.	method 0.96	0.93	0.08	asserted 1.00
s4	Following our previous successful work, we partially employ a data-driven approach that allows us to not only include open heavy-quark data, but also quarkonium data.	method 0.86	0.90	0.05	asserted 0.99
s5	In addition, we establish the consistency of this approach by comparing it to an alternative fit based on perturbative predictions of open heavy-quark data alone.	method 0.89	0.95	0.05	asserted 0.99
s6	We determine the uncertainties of our nCTEQ26 nPDFs with a Hessian error analysis, test their reliability by comparing our nPDFs to other, similar recent analyses, and make predictions for current and future experimental measurements that have not been included in the analysis.	method 0.60	0.97	0.19	asserted 1.00

What kind of paper the classifier reports this is: dataset or benchmark 0.81 · evidence from analysis of existing data 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 5.** In an ideal case, these values would follow again a normal distribution. The reason why we chose this metric instead of the normalization coefficient itself is that each coefficient follows its own normal distribution with the width corresponding to the normalization error. The first two moments are consistent with the Gaussian distribution, which is indicated by the close alignment of the fitted analytic curve (dashed line) and the ideal normal distribution (solid line). The central peak is more pronounced than expected, but the tails are symmetric. Most normalization adjustments…

**Figure 34.** and therefore suggests an important increase in flexibility of the parametrization through the inclusion of the c s+¯s 5 parameter, which then allows for a better exploration of the strange-quark uncertainty. Another noteworthy feature that we observed is a strong correlation between the a s+¯s 1 and b s+¯s 1 parameters leading to a χ 2 valley along the diagonal direction. Fits featuring different combinations of the values of these parameters along the diagonal do not differ appreciably in the data quality description as measured by χ 2 . However, they yield significantly

Showing 2 of the paper's 45 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.13633 by the authors.

References

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Discussion

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