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nCTEQ26 nuclear parton distribution functions

The nCTEQ collaboration presents nCTEQ26, a new set of nuclear parton distribution functions derived from a global analysis incorporating the largest selection of experimental data to date—including DIS, Drell-Yan, and collider measurements—alongside an advanced parametrization and data-driven approach to determine uncertainties and validate consistency.

Original authors: nCTEQ Collaboration, K. Greve, T. Ježo, A. Kusina, J. Wissmann, M. Klasen, K. Kovařík, C. Léger, J. G. Morfín, F. I. Olness, P. Risse, I. Schienbein, J. Y. Yu

Published 2026-10-12

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CC BY 4.0

Original authors: nCTEQ Collaboration, K. Greve, T. Ježo, A. Kusina, J. Wissmann, M. Klasen, K. Kovařík, C. Léger, J. G. Morfín, F. I. Olness, P. Risse, I. Schienbein, J. Y. Yu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep inside the heart of every atom lies a bustling, invisible world. Protons and neutrons, the building blocks of atomic nuclei, are not solid, indivisible spheres. Instead, they are dynamic clouds of even smaller particles called quarks and gluons, bound together by the strong force. When scientists smash these atoms together at nearly the speed of light, they do not just see the individual protons and neutrons; they see how these internal clouds behave when squeezed into the tight space of a heavy nucleus. This behavior is different from how a single, lonely proton acts in isolation. To make sense of the chaotic collisions happening in massive particle accelerators, physicists need a precise map of this internal landscape, known as a parton distribution function. This map tells them exactly how much momentum the quarks and gluons carry at different speeds and positions within the nucleus. Without this map, it is impossible to distinguish between the normal effects of nuclear matter and the exotic, high-energy phenomena scientists are hunting for, such as the creation of a super-hot, liquid-like state of matter known as the quark-gluon plasma.

A team of researchers has just released a significantly improved version of this map, called nCTEQ26. This new work represents a major leap forward in understanding the structure of atomic nuclei. For years, scientists have relied on older maps that were based on a relatively small collection of experimental data. The new analysis, however, digests a massive and diverse dataset, combining over 3,500 distinct measurements from experiments around the world. These measurements come from a wide variety of sources, including deep-inelastic scattering where electrons or neutrinos probe the nucleus, the production of heavy particles like the W and Z bosons, and the creation of heavy quarks and their bound states. By weaving together data from facilities like the Large Hadron Collider in Europe and the Relativistic Heavy Ion Collider in the United States, the team has constructed a picture that is far more detailed and covers a much wider range of conditions than ever before.

The researchers did not simply add more data; they also refined the tools used to interpret it. They introduced a new mathematical framework that allows the map to adapt more flexibly to the size of the nucleus. In the past, the models assumed that the changes in nuclear structure happened in a smooth, predictable way as the nucleus got heavier. The new approach recognizes that this relationship is more complex, allowing for sharper changes in lighter nuclei and more subtle shifts in heavier ones. They also incorporated corrections for the mass of the particles involved and the specific effects of the deuteron, a simple nucleus made of one proton and one neutron, ensuring that the baseline for their calculations is as clean and accurate as possible. This careful attention to detail prevents the team from double-counting information or missing subtle effects that could skew the results.

One of the most striking findings of this work is how the uncertainty of the map has changed. In previous versions, the errors on the map were surprisingly small, but this was partly because the models were too rigid to show the full range of possibilities. The new, more flexible model reveals that while the team is very confident about the behavior of the most common particles, there is still significant room for surprise in the distribution of strange quarks and the behavior of gluons at extremely low speeds. The data suggests that the strange quarks, which are heavier and less common than up and down quarks, are suppressed in heavy nuclei, but the exact amount of this suppression varies depending on the size of the nucleus and the energy of the collision. The team found that their new map is consistent with the vast majority of the experimental data, successfully describing the complex interactions observed in the lab.

However, the work also highlights where the current understanding is still incomplete. When the researchers tested their new map against certain datasets that were not included in the creation of the map, such as specific measurements of heavy quark production in the forward direction, they found some discrepancies. The data in these regions did not always match the theoretical predictions perfectly, suggesting that there might be additional physical effects at play that are not yet fully captured by the standard models. These mismatches are not failures of the work but rather signposts pointing toward the next frontier of discovery. They indicate that while the team has built a robust foundation, there are still subtle nuances in how nuclear matter behaves at the smallest scales that require further investigation.

The nCTEQ26 release serves as a critical benchmark for the future of high-energy physics. It provides the most precise and reliable reference available for interpreting current experiments and planning for the next generation of facilities, such as the High-Luminosity Large Hadron Collider and the proposed Electron-Ion Collider. By offering a clearer view of the nuclear landscape, this work allows scientists to better isolate the signals of new physics from the background noise of known nuclear effects. It confirms that the standard framework of particle physics holds up under intense scrutiny, even as it reveals the precise boundaries where our knowledge is still evolving. The result is a tool that is not just a static picture, but a dynamic, evolving resource that will guide physicists as they continue to probe the fundamental structure of matter.

Technical Summary of nCTEQ26: Nuclear Parton Distribution Functions

Problem and Motivation
The paper addresses the need for a precise, modern determination of nuclear parton distribution functions (nPDFs) to serve as a baseline for high-energy nuclear physics. While previous analyses, such as nCTEQ15, provided valuable pre-LHC references, they were limited by a smaller dataset (740 points), restricted kinematic coverage, and less flexible parametrizations. The authors argue that with the unprecedented wealth of precision data from the LHC (Runs I and II), JLab, and RHIC, a new global analysis is required to isolate "cold" nuclear effects from "hot" nuclear matter phenomena (like the quark-gluon plasma) and to provide robust Standard Model baselines for future experiments like the HL-LHC, EIC, and DUNE. A specific challenge highlighted is the tension between charged-lepton and neutrino deep-inelastic scattering (DIS) data, as well as the need to constrain the gluon distribution at very low $x$ ( $\sim 10^{-5}$ ) and the strange quark distribution, which remains poorly constrained.

Methodology
The nCTEQ26 analysis employs a global fit of experimental data using a Hessian error analysis. Key methodological components include:

Data Selection: The dataset is significantly expanded to 3,598 data points, incorporating neutral-current (NC) and charged-current (CC) DIS, fixed-target Drell-Yan, LHC W/Z boson production, single-inclusive hadron (SIH) production, and heavy-quark (open heavy flavor and quarkonium) production. Kinematic cuts were relaxed (e.g., $Q > 1.3$ GeV, $W > 1.7$ GeV) to include high- $x$ JLab data and low- $Q^2$ regions previously excluded.
Parametrization:
- Proton Baseline: The analysis adopts the CJ15 proton PDF set as a baseline. This choice is motivated by its careful treatment of high- $x$ regions, target-mass corrections (TMCs), and deuteron corrections, and crucially, its independence from nuclear data to avoid double-counting.
- Nuclear Dependence: A new parametrization for the nuclear mass number $A$ dependence is introduced. Instead of the previous exponential form, the coefficients $c_j(A)$ are modeled as quadratic functions of $\ln A$ : $c_j(A) = p_j + a_j \ln A + b_j \ln^2 A$ . This allows for more flexible evolution of uncertainties across different nuclei without imposing artificial smoothness.
- Flavor Structure: The $x$ -dependence follows the CJ15 functional form but includes additional flexibility, particularly for the strange quark (via an exponential factor) and the $\bar{d}/\bar{u}$ ratio.
Theoretical Modeling: Calculations are performed at Next-to-Leading Order (NLO) QCD.
- Heavy Quarks: For the main fit, a "data-driven" approach using a Crystal-Ball (CB) parametrization is employed for open heavy-flavor (OHF) and quarkonium production. This method fits the proton-proton baseline to extract CB parameters, which are then used to describe proton-lead data, effectively handling non-perturbative aspects of quarkonium production.
- Corrections: Target-mass corrections (TMCs) and deuteron corrections are included to reliably describe data in the shallow-inelastic scattering (SIS) and high- $x$ regions. Higher-twist (HT) corrections were investigated but found not to improve the description of the data significantly and were excluded from the final fit.
Uncertainty Determination: Uncertainties are determined using the Hessian method with a tolerance criterion $\Delta\chi^2 = T = 50$ , based on Wilks' theorem. Both symmetric and asymmetric uncertainty prescriptions are discussed, with the symmetric definition recommended for the strange quark to avoid underestimation in unconstrained regions.

Key Contributions

Expanded Dataset: The inclusion of over 3,500 data points, nearly five times the size of nCTEQ15, significantly broadens the kinematic reach, particularly in low- $x$ (via heavy-quark data) and high- $x$ (via relaxed JLab cuts).
Theoretical Refinements: The implementation of a quadratic logarithmic $A$ -dependence and the adoption of the CJ15 baseline with consistent TMC and deuteron corrections represent a major step forward in modeling reliability.
Data-Driven Heavy Quark Treatment: The successful integration of quarkonium and OHF data via the Crystal-Ball parametrization allows for strong constraints on the nuclear gluon distribution at low $x$ without relying solely on perturbative QCD calculations that may struggle with quarkonium production mechanisms.
Consistency Check: An alternative fit, nCTEQ26OHF, was performed using only perturbative QCD (GM-VFNS) for open heavy flavor and excluding quarkonium. This confirmed the consistency of the main fit's central values and uncertainties, validating the data-driven approach.

Results

Fit Quality: The global fit yields a total $\chi^2/\text{dof} = 0.943$ , indicating a good description of the data. Most datasets are described well, though some tensions remain, particularly in the CHORUS neutrino CC DIS data ( $\chi^2/N \approx 1.3$ ) and specific LHCb Run-II D0 forward rapidity data.
Parton Distributions:
- Gluon: The nuclear gluon distribution is constrained down to $x \sim 10^{-5}$ by heavy-quark data. The uncertainty is smallest for heavy nuclei (Pb) and increases for lighter nuclei.
- Strange Quark: The strange distribution exhibits the largest uncertainties, particularly at low $x$ where data constraints are weak. The analysis highlights a strong correlation between fit parameters that creates flat $\chi^2$ valleys, leading to significant uncertainty in the strange PDF at low $x$ .
- Valence and Sea: Valence distributions are well-constrained in the mid-to-high $x$ region. The light sea quarks show standard shadowing and anti-shadowing patterns.
Nuclear Dependence: The new $A$ -dependence parametrization successfully captures the transition from light to heavy nuclei, showing that nuclear modifications grow with mass number $A$ , but with a non-monotonic evolution of uncertainties in regions lacking direct constraints.
Comparisons: Compared to nCTEQ15, the new uncertainties are generally larger (except for the gluon at low $x$ ), reflecting a more realistic assessment of errors by removing parametrization bias. Comparisons with EPPS21 and nNNPDF3.0 show general agreement in central values but highlight differences in uncertainty quantification, particularly for the strange quark.

Significance and Claims
The paper claims that nCTEQ26 represents a "new generation" of nuclear PDFs that establishes a critical benchmark for future theoretical and experimental studies. By integrating the full range of available experimental data and incorporating advanced theoretical treatments (TMCs, deuteron corrections, flexible parametrizations), the set provides a more reliable and precise foundation for QCD phenomenology. The authors emphasize that the improved constraints on the gluon and the realistic treatment of uncertainties are vital for interpreting data from the HL-LHC, EIC, and neutrino experiments like DUNE. The analysis also serves as a nontrivial test of QCD factorization in the nuclear environment, identifying regions where higher-twist effects or other nuclear phenomena might become relevant. The paper concludes that nCTEQ26 advances the field by enabling more stringent tests of nuclear structure and sharpening sensitivity to potential deviations from the Standard Model.

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