nCTEQ Analyses: Nuclear PDF Determinations

• nCTEQ analyses are global determinations of nuclear parton distribution functions that directly parameterize bound-proton PDFs with explicit A-dependence.
• They progressively integrate diverse data sets—from charged‑lepton DIS and Drell–Yan to RHIC and LHC measurements—to constrain gluon, valence, and strange sectors.
• The method allows independent u₍ᵥ₎ and d₍ᵥ₎ nuclear corrections and extends to high‑x, low‑Q² regions, reducing uncertainties through a rigorous Hessian error analysis.

nCTEQ analyses are a sequence of global determinations of nuclear parton distribution functions (nPDFs) developed within the CTEQ framework. Their defining choice is to parameterize the PDFs of a proton bound in a nucleus, $f_i^{p/A}(x,Q)$, introduce explicit $A$-dependence in the fit parameters, obtain bound-neutron PDFs from isospin symmetry, and construct the full nuclear PDFs by proton–neutron averaging. Across the program, the fitted data evolved from charged-lepton DIS and Drell–Yan to RHIC pion production, LHC $W/Z$, single-inclusive hadrons, heavy-quark and quarkonium production, neutrino DIS and charm dimuons, and high-$x$ Jefferson Lab structure-function measurements; the corresponding goals evolved from establishing a CTEQ-style nuclear fit with Hessian uncertainties to extending flavor separation and kinematic reach, especially for the gluon and strange sectors [(Kovarik et al., 2013); (Kovarik et al., 2015); (Klasen et al., 2022); (Risse et al., 2023)].

1. Origins and fit philosophy

The early nCTEQ program was framed as a DGLAP-based global nPDF analysis that differed from EPS-style fits by parameterizing the initial bound-proton PDFs directly rather than nuclear modification factors, by using a CTEQ-like free-proton baseline, and by adopting a general-mass variable-flavor-number scheme for heavy quarks. In the preliminary CTEQ-style nuclear fit that became the methodological basis of nCTEQ, the input distributions at $Q_0=1.3$ GeV were written in a flexible CTEQ-inspired form with $A$-dependent coefficients, and the full nuclear PDFs were assembled from bound-proton and bound-neutron PDFs through isospin symmetry [(Eskola, 2012); (Kovarik et al., 2013)].

A standard nCTEQ parameterization used later in the program is

$$x f_i^{p/A}(x,Q_0)=c_0\,x^{c_1}(1-x)^{c_2}e^{c_3x}(1+e^{c_4}x)^{c_5},$$

with nuclear dependence introduced through

$$c_k \to c_{k,0} + c_{k,1}\left(1-A^{-c_{k,2}}\right).$$

The physically relevant nuclear PDFs are then constructed as

$$f_i^{(A,Z)}(x,Q)=\frac{Z}{A}f_i^{p/A}(x,Q)+\frac{A-Z}{A}f_i^{n/A}(x,Q).$$

This formalism encodes nuclear effects directly in the bound-proton PDFs rather than as an external correction applied after a proton fit (Kovarik et al., 2015).

The first preliminary fit using charged-lepton DIS and Drell–Yan data employed the standard CTEQ cuts $Q>2~\text{GeV}$ and $A$0, used 17 free parameters, and achieved $A$1 for 708 data points. Its methodological novelty was a full Hessian error analysis inside the CTEQ fitting framework, establishing the basis for later official nCTEQ error sets (Kovarik et al., 2013).

2. nCTEQ15 as the baseline release

nCTEQ15 was the first official nCTEQ release with uncertainty PDFs. It extended the earlier nuclear fits by combining DIS, Drell–Yan, and RHIC inclusive pion production in a global analysis and by providing Hessian error sets rather than only central curves. The data entering the release comprised 616 charged-lepton DIS points, 92 Drell–Yan points, and 32 RHIC pion-production points. The fit used 16 free PDF parameters—7 for the gluon, 4 for $A$2-valence, 3 for $A$3-valence, and 2 for $A$4—with two additional normalization parameters when pion data were included (Kovarik et al., 2015, Kusina, 2016).

The uncertainty analysis was based on a Hessian expansion of the global $A$5,

$A$6

followed by diagonalization into eigenvector directions. For an observable $A$7, the standard symmetric estimate was

$A$8

In the full nCTEQ15 treatment, the tolerance was $A$9, and the release introduced an eigenvector rescaling procedure so that the Hessian approximation better matched the true $W/Z$0 scans at the chosen tolerance (Kovarik et al., 2015).

The original nCTEQ15 fit reported $W/Z$1 for 740 points, corresponding to $W/Z$2. A comparison fit without pion data gave $W/Z$3. The added RHIC pion data materially altered the gluon: including them decreases the lead gluon PDF at larger $W/Z$4, roughly $W/Z$5, increases it at smaller $W/Z$6, and reduces the gluon uncertainty band in the intermediate-to-large-$W/Z$7 region. This was the first major nCTEQ demonstration that collider hadron production could supply a direct constraint on the nuclear gluon, which DIS and DY alone cannot provide (Kusina, 2016).

A second defining feature of nCTEQ15 was the decision to allow independent nuclear corrections for $W/Z$8 and $W/Z$9. Earlier discussions in the nCTEQ program emphasized that there is no strong physical reason to force universal valence corrections, even though the available data have limited sensitivity to separate them cleanly. In the nCTEQ15 phenomenology, that extra freedom reduced $x$0 by about $x$1, but the resulting differences were most visible at the bound-proton level rather than in full nuclear observables (Olness, 2015).

3. Collider extensions: $x$2, single-inclusive hadrons, and heavy flavor

After nCTEQ15, the program expanded through a sequence of fits that added LHC proton–nucleus data. The main branch was nCTEQ15WZ, which incorporated $x$3 boson production; nCTEQ15WZSIH, which added single-inclusive hadron production; and nCTEQ15HQ, which further included heavy-quark and quarkonium data. In the stepwise summary of the program, the data volume increased from 740 points in nCTEQ15 to 860 in nCTEQ15WZ, 948 in nCTEQ15WZSIH, and 1484 in nCTEQ15HQ, while the later fits used 19 free parameters rather than 16 (Klasen et al., 2022).

Fit	New ingredients	Data points
nCTEQ15	DIS, DY, RHIC $x$4	740
nCTEQ15WZ	adds LHC $x$5 in $x$6	860
nCTEQ15WZSIH	adds single-inclusive hadron production in $x$7	948
nCTEQ15HQ	adds heavy-quark and quarkonium production in $x$8	1484

The $x$9 measurements in nCTEQ15WZ significantly reduced the uncertainty of the gluon PDF, most notably in the region $Q_0=1.3$0 to $Q_0=1.3$1 at $Q_0=1.3$2 GeV, where the gluon uncertainty was reduced by about a factor of two. They also pulled the strange-quark density upward, although the strange uncertainty remained large. nCTEQ15WZ was enabled by nCTEQ++, a C++ implementation of the fitting code that interfaced to HOPPET for DGLAP evolution, APPLgrid for fast interpolation, and MCFM for NLO hard-process calculations; normalization freedom for the $Q_0=1.3$3 datasets was treated with a D’Agostini-type $Q_0=1.3$4 prescription (Kusina et al., 2020).

Single-inclusive hadron data were then incorporated systematically in nCTEQ15WZSIH using RHIC and ALICE nuclear modification ratios together with a modified and optimized version of INCNLO. The calculations were restricted to $Q_0=1.3$5, used precomputed grids, and propagated fragmentation-function uncertainties from DSS14 and DSS17 by adding them to the experimental systematic uncertainties. Relative to the DIS/DY-only baseline, the SIH data raise the gluon PDF at low $Q_0=1.3$6, suppress it in the medium-$Q_0=1.3$7 region, and flatten the gluon modification around $Q_0=1.3$8; compared with nCTEQ15WZ, they reduced the gluon uncertainty again by roughly a factor of two and yielded a best overall combined fit $Q_0=1.3$9 for nCTEQ15WZ+SIH (Duwentäster et al., 2021).

The strongest small-$A$0 gluon constraint entered with nCTEQ15HQ. Heavy-quark and quarkonium measurements from ALICE, ATLAS, CMS, and LHCb probe a broad range in $A$1 and $A$2, with a kinematic estimate

$A$3

reaching $A$4. In the programmatic summary, these data reduce the gluon uncertainty almost by an order of magnitude relative to nCTEQ15WZSIH and extend the reliable gluon constraint down to $A$5. The fit quality for nCTEQ15HQ was reported as $A$6, and the fast fitting strategy relied on a data-driven ansatz for the partonic amplitude validated against NLO QCD and NRQCD calculations (Klasen et al., 2022).

4. Neutrino DIS, dimuon production, and the strange sector

Neutrino DIS and charm dimuon production were always central to the broader nCTEQ motivation because they are highly sensitive to the strange-quark density. The later dedicated compatibility studies revisited the long-standing question of whether charged-current neutrino data can be described consistently together with the charged-lepton DIS, Drell–Yan, and collider data used in the reference nPDF fits. The baseline for these studies was nCTEQ15WZSIHdeut, which incorporated nuclear effects in the deuteron structure function using CJ15 deuteron corrections (Muzakka et al., 2022).

The neutrino analysis found that tensions persist even after improved deuteron treatment, larger Hessian tolerance, checks of proton-baseline dependence, and explicit treatment of correlations and normalization uncertainties. In the hypothesis-testing language of the study, the baseline $A$7 increased from 735 to 866 when the full neutrino sample was added, far exceeding the effective tolerance $A$8. The most severe discrepancies came from low $A$9, especially for the iron-target datasets, while CHORUS on lead behaved more like the charged-lepton expectation (Muzakka et al., 2022).

A kinematic cut $$x f_i^{p/A}(x,Q_0)=c_0\,x^{c_1}(1-x)^{c_2}e^{c_3x}(1+e^{c_4}x)^{c_5},$$0 can reduce the tension, but only at the cost of worsening the global description; the later summary quantified this as an increase of $$x f_i^{p/A}(x,Q_0)=c_0\,x^{c_1}(1-x)^{c_2}e^{c_3x}(1+e^{c_4}x)^{c_5},$$1 by 46 for 4644 neutrino data points. A more successful strategy was selective inclusion: the fit called BaseDimuChorus, built from CHORUS neutrino DIS plus charm dimuon data, gave a much better fit with no additional kinematic cut and only $$x f_i^{p/A}(x,Q_0)=c_0\,x^{c_1}(1-x)^{c_2}e^{c_3x}(1+e^{c_4}x)^{c_5},$$2 for 974 neutrino data points. In the resulting flavor decomposition, valence and antiquark parameters remained mainly constrained by neutral-current DIS, gluon parameters by LHC $$x f_i^{p/A}(x,Q_0)=c_0\,x^{c_1}(1-x)^{c_2}e^{c_3x}(1+e^{c_4}x)^{c_5},$$3 and SIH data, and strange-quark parameters mainly by CHORUS and dimuon data (Klasen et al., 2022).

These studies established a characteristic nCTEQ position on neutrino data. Neutrino DIS is indispensable for strange-quark separation, but not all historical neutrino datasets are equally compatible with the broader nuclear dataset. A plausible implication is that the strange sector in unified global nPDF fits depends not only on statistical power but also on careful dataset selection and on the treatment of correlated systematics, normalization, and the proton baseline.

5. Extension into the high-$$x f_i^{p/A}(x,Q_0)=c_0\,x^{c_1}(1-x)^{c_2}e^{c_3x}(1+e^{c_4}x)^{c_5},$$4, low-$$x f_i^{p/A}(x,Q_0)=c_0\,x^{c_1}(1-x)^{c_2}e^{c_3x}(1+e^{c_4}x)^{c_5},$$5 regime

nCTEQ15HIX extended the program into the large-$$x f_i^{p/A}(x,Q_0)=c_0\,x^{c_1}(1-x)^{c_2}e^{c_3x}(1+e^{c_4}x)^{c_5},$$6, low-to-intermediate-$$x f_i^{p/A}(x,Q_0)=c_0\,x^{c_1}(1-x)^{c_2}e^{c_3x}(1+e^{c_4}x)^{c_5},$$7 DIS region probed by Jefferson Lab. The standard nCTEQ15 cuts $$x f_i^{p/A}(x,Q_0)=c_0\,x^{c_1}(1-x)^{c_2}e^{c_3x}(1+e^{c_4}x)^{c_5},$$8 and $$x f_i^{p/A}(x,Q_0)=c_0\,x^{c_1}(1-x)^{c_2}e^{c_3x}(1+e^{c_4}x)^{c_5},$$9 were relaxed to $$c_k \to c_{k,0} + c_{k,1}\left(1-A^{-c_{k,2}}\right).$$0 and $$c_k \to c_{k,0} + c_{k,1}\left(1-A^{-c_{k,2}}\right).$$1, increasing the usable data from 708 points to 1564 points. The new kinematic coverage required additional phenomenology: target-mass corrections, higher-twist corrections, deuteron corrections, and a more flexible large-$$c_k \to c_{k,0} + c_{k,1}\left(1-A^{-c_{k,2}}\right).$$2 PDF parameterization (Segarra et al., 2020).

The target-mass treatment used the Nachtmann variable and a leading prefactor,

$$c_k \to c_{k,0} + c_{k,1}\left(1-A^{-c_{k,2}}\right).$$3

while the higher-twist correction followed the CJ15-inspired form

$$c_k \to c_{k,0} + c_{k,1}\left(1-A^{-c_{k,2}}\right).$$4

The deuteron correction was implemented through

$$c_k \to c_{k,0} + c_{k,1}\left(1-A^{-c_{k,2}}\right).$$5

Among the added ingredients, deuteron corrections had the largest numerical impact on fit quality (Segarra et al., 2020).

The fit-quality sequence illustrates the logic of the extension. Evaluating the old nCTEQ15 PDFs on the relaxed-cut kinematics gave $$c_k \to c_{k,0} + c_{k,1}\left(1-A^{-c_{k,2}}\right).$$6 for 1564 points, with $$c_k \to c_{k,0} + c_{k,1}\left(1-A^{-c_{k,2}}\right).$$7. A refit with relaxed cuts but no extra phenomenology reduced this to $$c_k \to c_{k,0} + c_{k,1}\left(1-A^{-c_{k,2}}\right).$$8, $$c_k \to c_{k,0} + c_{k,1}\left(1-A^{-c_{k,2}}\right).$$9. Adding higher twist gave $$f_i^{(A,Z)}(x,Q)=\frac{Z}{A}f_i^{p/A}(x,Q)+\frac{A-Z}{A}f_i^{n/A}(x,Q).$$0, $$f_i^{(A,Z)}(x,Q)=\frac{Z}{A}f_i^{p/A}(x,Q)+\frac{A-Z}{A}f_i^{n/A}(x,Q).$$1; adding deuteron corrections gave $$f_i^{(A,Z)}(x,Q)=\frac{Z}{A}f_i^{p/A}(x,Q)+\frac{A-Z}{A}f_i^{n/A}(x,Q).$$2, $$f_i^{(A,Z)}(x,Q)=\frac{Z}{A}f_i^{p/A}(x,Q)+\frac{A-Z}{A}f_i^{n/A}(x,Q).$$3; and combining both in nCTEQ15HIX yielded $$f_i^{(A,Z)}(x,Q)=\frac{Z}{A}f_i^{p/A}(x,Q)+\frac{A-Z}{A}f_i^{n/A}(x,Q).$$4, $$f_i^{(A,Z)}(x,Q)=\frac{Z}{A}f_i^{p/A}(x,Q)+\frac{A-Z}{A}f_i^{n/A}(x,Q).$$5. The paper summarized the improvements as roughly $$f_i^{(A,Z)}(x,Q)=\frac{Z}{A}f_i^{p/A}(x,Q)+\frac{A-Z}{A}f_i^{n/A}(x,Q).$$6 from higher twist, $$f_i^{(A,Z)}(x,Q)=\frac{Z}{A}f_i^{p/A}(x,Q)+\frac{A-Z}{A}f_i^{n/A}(x,Q).$$7 from deuteron corrections, and $$f_i^{(A,Z)}(x,Q)=\frac{Z}{A}f_i^{p/A}(x,Q)+\frac{A-Z}{A}f_i^{n/A}(x,Q).$$8 from both together (Segarra et al., 2020).

The principal PDF effect was a hardening at large $$f_i^{(A,Z)}(x,Q)=\frac{Z}{A}f_i^{p/A}(x,Q)+\frac{A-Z}{A}f_i^{n/A}(x,Q).$$9: $Q>2~\text{GeV}$0 and $Q>2~\text{GeV}$1 increased at large $Q>2~\text{GeV}$2, the gluon increased somewhat, and the sea quarks were reduced. The large-$Q>2~\text{GeV}$3 uncertainties were reduced most strongly for up- and down-valence quarks. This extension therefore supplied a complementary constraint to the collider-driven low-$Q>2~\text{GeV}$4 program: JLab sharpened the high-$Q>2~\text{GeV}$5 valence sector, while LHC measurements sharpened the low-$Q>2~\text{GeV}$6 gluon and strange sectors.

6. Comparisons, methodological distinctions, and unified global releases

A recurrent point in nCTEQ comparisons with HKN07, EPS09, and DSSZ is that the largest visible differences often occur in the bound-proton valence PDFs, especially because nCTEQ allows independent $Q>2~\text{GeV}$7 and $Q>2~\text{GeV}$8 nuclear corrections while other groups often impose universal valence nuclear modifications. nCTEQ emphasized that there is no strong physical reason to force universality between $Q>2~\text{GeV}$9 and $A$00. At the same time, when the physically relevant full nuclear PDFs are formed as

$A$01

the bound-proton valence differences are diluted by proton–neutron averaging, and the full nuclear valence PDFs agree very well with other groups. For sea quarks and the gluon, the overall agreement is generally reasonable, with the gluon showing the largest spread because direct constraints remain limited (Kusina, 2016).

The uncertainty comparison follows the same pattern. nCTEQ15 error bands were reported to be comparable to EPS09 and generally larger than those from HKN07 and DSSZ, while both the original nCTEQ15 papers and later summaries cautioned that even these Hessian uncertainties are likely still underestimated because the fitted parameter space is restricted and several structural assumptions are unavoidable. This caution is consistent with the broader nCTEQ theme that data scarcity, heterogeneous nuclei, and process-dependent theory systematics make nuclear uncertainty estimation intrinsically harder than in free-proton PDF fits (Kovarik et al., 2015, Kusina, 2016).

The move toward a single unified release began with the 2023 programmatic proposal for a new global nCTEQ fit that would, for the first time, include neutrino DIS in a consistent manner together with JLab high-$A$02 DIS data and new LHC $A$03-Pb data. That proposal replaced the older CTEQ6-based proton reference with CJ, adopted a simpler proton baseline

$A$04

and changed the nuclear dependence to

$A$05

with the explicit aim of reducing parameter correlations and making the $A$06 landscape more transparent (Risse et al., 2023).

The 2025 review of the post-2020 nCTEQ sequence described the preliminary nCTEQ25 global analysis as the synthesis of the specialized branches: JLab neutral-current DIS from nCTEQ15HIX, neutrino DIS and dimuons, LHC electroweak bosons, single-inclusive hadrons, and heavy-quark production. In that status report, the new fit used CJ15 as proton baseline together with polynomial $A$07-dependence, logarithmic $A$08-dependence, a larger number of free parameters, and a larger tolerance. The reported qualitative outcome was a more precise gluon distribution, a more precise strange-quark distribution, and a single framework that combines the benefits of the earlier separate analyses (Klasen, 7 Oct 2025).

Taken together, nCTEQ analyses define a program rather than a single fit. The program’s characteristic elements are direct bound-proton parameterization, explicit $A$09-dependence, Hessian uncertainties, systematic exploration of data tensions, and progressive enlargement of the dataset to constrain different flavor sectors in different kinematic regions. The result is not a closed final answer on nuclear PDFs, but a structured sequence of global analyses that progressively sharpened the valence sector at high $A$10, the gluon at low $A$11, and the strange sector through a combination of electroweak, hadronic, and neutrino observables.