⚛️ nuclear theory

Impact of Open Heavy-Flavour Production on nCTEQ Nuclear PDFs

The paper introduces nCTEQ26OHF, a global analysis of nuclear parton distribution functions that replaces the Crystal-Ball function with an SACOT GM-VFNS calculation for open heavy-flavor production, demonstrating that the resulting PDFs remain consistent with the standard nCTEQ26 release and confirming that open heavy-flavor data alone provides sufficient constraints.

Original authors: Jan Wissmann, Tomáš Ježo, Michael Klasen, Ingo Schienbein, Hubert Spiesberger

Published 2026-09-30

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CC BY 4.0

Original authors: Jan Wissmann, Tomáš Ježo, Michael Klasen, Ingo Schienbein, Hubert Spiesberger

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

Inside the heart of an atom, protons and neutrons are not solid, unchanging spheres. Instead, they are bustling clouds of smaller particles called quarks and gluons, constantly interacting and exchanging energy. When these protons and neutrons bind together to form a nucleus, their internal structures do not simply sit side by side; they influence one another, shifting the way their constituent particles are distributed. This phenomenon, known as the EMC effect, means that the rules governing a particle inside a heavy nucleus are slightly different from those governing a free-floating particle. To understand high-energy collisions, such as those occurring at the Large Hadron Collider where scientists smash protons into heavy lead nuclei, researchers need a precise map of these internal structures. These maps are called nuclear parton distribution functions, and they describe how likely it is to find a specific particle carrying a certain amount of momentum within a nucleus. Without these maps, physicists cannot accurately interpret the debris from these cosmic-scale collisions or test the fundamental laws of nature.

A team of researchers has recently refined these maps, focusing on a specific type of data that had been handled with a simplified approach in previous studies. In their latest work, they replaced a mathematical shortcut with a more rigorous calculation to see if the results would change significantly. The shortcut they used previously was a flexible formula designed to fit existing data from proton collisions, which was then applied to predict what happens in lead collisions. While effective, this method relied on assumptions about how heavy particles are created. To test the strength of these assumptions, the team performed a new global analysis, creating a fresh set of nuclear maps where they swapped the shortcut for a detailed, first-principles calculation of how heavy particles, specifically those containing charm or bottom quarks, are produced. They also removed a specific category of data involving bound states of heavy quarks to isolate the effect of the production mechanism itself.

The researchers found that the new, more complex calculation produced nuclear maps that were virtually identical to the old ones, within the margins of error. This was a surprising and reassuring result. It suggested that the heavy-particle production data, even when analyzed with a more demanding theoretical framework, still provides a powerful and independent constraint on the distribution of gluons, the particles that carry the strong force holding the nucleus together. In the previous analysis, the team had included data on quarkonium, which are particles made of a heavy quark and its antiparticle bound together, to help pin down the behavior of gluons at very low momentum. By removing this specific data and relying solely on the open heavy-flavor data, the team expected the uncertainty in their gluon map to grow larger. Instead, the data proved robust enough to keep the uncertainty bands tight, demonstrating that the open heavy-flavor measurements alone are sufficient to guide the shape of the nuclear structure.

The study also revealed that the simplified formula used in earlier work was broadly compatible with the more rigorous calculation. While the new analysis resulted in a slightly higher statistical mismatch with some of the experimental data points, this was largely because the previous method had included extra uncertainties from the fitting process that artificially lowered the mismatch score. When the researchers accounted for this, the two approaches were found to be in good agreement. The team analyzed nearly 3,600 data points from various experiments, a significant increase from the 740 points used in their previous major release, allowing them to probe the nucleus with much greater precision. Their work confirms that the current understanding of how heavy particles are created in nuclear collisions is solid, and that the nuclear maps derived from them are reliable. This gives physicists confidence that they are looking at the true structure of the atomic nucleus, rather than an artifact of the mathematical tools used to describe it.

Technical Summary: Impact of Open Heavy-Flavour Production on nCTEQ Nuclear PDFs

Problem Statement
Nuclear parton distribution functions (nPDFs) are essential for describing high-energy proton-nucleus collisions, as nuclear structure deviates from a simple sum of free nucleons (the EMC effect). While global QCD analyses like nCTEQ15HQ and the recent nCTEQ26 utilize Open Heavy-Flavor (OHF) and quarkonium production data to constrain the gluon PDF at low momentum fraction $x$ , the theoretical treatment of OHF in nCTEQ26 relies on the Crystal-Ball (CB) function. This function is an effective matrix element parametrized to fit proton-proton ($pp$) data and subsequently applied to proton-lead ( $p$ Pb) predictions. The validity of this effective approach versus a rigorous perturbative QCD (pQCD) calculation within the global fit framework requires scrutiny.

Methodology
The authors present nCTEQ26OHF, a global analysis of nPDFs designed to test the compatibility of the CB function approach with a full pQCD calculation.

Baseline Framework: The analysis builds upon the nCTEQ26 setup, which significantly expands the dataset from 740 points (nCTEQ15) to 3,598 points. This includes JLab neutral-current DIS, $W/Z$ and heavy-quark production, charged-current DIS, and LHC OHF and quarkonium data. The baseline proton PDF uses the CJ15 parametrization with an $A$ -dependent modification to parameters, employing a flexible functional form at the initial scale $Q_0 = 1.3$ GeV.
The nCTEQ26OHF Variation: In this specific fit, the authors retain the nCTEQ26 methodology and dataset selection for all processes except quarkonium production (which is excluded) and OHF predictions.
- Replacement of Theory: The CB function is replaced by a General-Mass Variable Flavor Number Scheme (GM-VFNS) calculation using the SACOT (Simplified Axiomatic Collinear-Operator Technique) scheme.
- Implementation: The SACOT predictions utilize KKKS08 fragmentation functions and are implemented via PineAPPL grids.
- Scale Choices: The renormalization, initial-state factorization, and final-state factorization scales are tuned to the same $pp$ baseline data used for the CB parameters in the standard nCTEQ26 fit. The scale coefficients are set to $(\xi_r, \xi_f, \xi'_f) = (1, 0.6, 2.5)$ .
Fitting Procedure: The analysis is performed at NLO QCD. The error analysis utilizes the Hessian method with a tolerance $T=50$ and dynamic rescaling. Of the 68 available $A$ -dependent parameters, 35 are opened for fitting, a significant increase from nCTEQ15, allowing for a more detailed description of nuclear modifications across 19 nuclei.

Key Results

PDF Compatibility: The resulting nCTEQ26OHF PDFs are compatible with the standard nCTEQ26 PDFs within uncertainties for all observables (valence quarks, sea quarks, and gluons).
Gluon Constraints: While the gluon uncertainty bands in nCTEQ26OHF are slightly larger than in nCTEQ26 (due to the exclusion of quarkonium data), they remain well-constrained. The authors attribute the tightness of the gluon constraint in nCTEQ26OHF to the strong constraining power of the OHF data alone, particularly at low $x$ .
$\chi^2$ Analysis:
- The $\chi^2$ values for the OHF datasets are generally higher in the nCTEQ26OHF fit compared to nCTEQ26.
- The authors note that the lower $\chi^2$ in nCTEQ26 is partly an artifact of including the uncertainties of the fitted CB parameters, which effectively lowers the $\chi^2$ even if the data description is not objectively superior.
- The $\chi^2$ scan along the gluon parameter $a_1$ (dominant at low $x$ ) reveals a "flanking" behavior: OHF data constrains the parameter in the negative direction, while $W/Z$ production data constrains it in the positive direction, narrowing the overall uncertainty.

Significance and Claims
The paper claims that the OHF data alone provides ample constraint on the nuclear gluon PDF, even without the inclusion of quarkonium data or the use of the effective CB function. The primary significance of the work is demonstrating that the CB function, used in previous nCTEQ releases, is broadly compatible with the more rigorous SACOT pQCD calculation within the context of global nPDF fits. Consequently, the nCTEQ26OHF results validate the robustness of the nCTEQ26 gluon extraction, confirming that the effective parametrization does not introduce significant bias compared to the pQCD approach, despite the latter's higher $\chi^2$ values in the absence of CB parameter uncertainties.

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