Prompt charmonia hadroproduction in High Energy Factorization with tree-level NLO corrections

A. A. Prokhorov, S. P. Baranov, A. V. Lipatov, M. A. Malyshev

Oct 21, 2026arXiv:2610.29297v1

hep-ph

Scorecard· 16/16

4.0/10 impact

Competent, comprehensive but incremental extension of the authors' own NLO† kT-factorization scheme to J/ψ and ηc, with niche impact and mediocre fit quality on several datasets.

Abstract

We consider prompt $J / ψ$ , $χ_{cJ}$ and $η_{c}$ production in high energy $p p$ collisions in the framework of non-relativistic QCD. We apply high-energy factorization with taking into account tree-level next-to-leading order corrections to the production amplitudes. Using several CCFM-evolved gluon densities in the proton and a previously proposed matching scheme to merge the leading and higher-order contributions, we extract the long-distance matrix elements for the entire charmonium family from a simultaneous fit to available LHC data. We provide predictions for the transverse momentum spectra and relative production rates $σ(ψ^\prime)/σ(J/ψ)$ and $σ(χ_{cJ})/σ(J/ψ)$ and find them to be in good agreement with the LHC data within the theoretical and experimental uncertainties across a wide kinematic region. We demonstrate that taking into account the tree-level NLO corrections improves the global description of the data.

AI Impact Assessments

(1 model)

Paper type: Phenomenological, empirical fit. It applies an existing theoretical framework (kT-factorization + NRQCD + the NLO† merging scheme) to a broad dataset.

Core Contribution

The paper extends the authors' earlier NLO† scheme, which merges tree-level 2→1 and 2→2 off-shell amplitudes with a double-counting-exclusion cut, from ψ′ and χcJ to prompt J/ψ and ηc. It fits color-octet LDMEs for the whole charmonium family to a large LHC dataset, about 983 J/ψ points and 59 χcJ points. The main technical devices are:

a two-step fit that avoids χ² bias from the dominant J/ψ data;

a linear combination of the 1S0[8] and 3P_J[8] LDMEs, which is needed because the two are strongly anticorrelated;

linear regression relations linking J/ψ direct and χcJ feed-down contributions;

HQSS relations used to predict ηc parameter-free;

an alternative scenario with independent |R′χc1|² and |R′χc2|².

The headline physics findings are:

Tree-level NLO† corrections improve the global description relative to LO.

Allowing HQSS violation in the χc1/χc2 wave functions describes σ(χc2)/σ(χc1) well. The required violation is about ±50%, well beyond expected O(v²) corrections, though NLO† reduces it by up to a factor of two relative to LO.

ηc production at LHCb is overestimated, which confirms the known ηc/HQSS puzzle. The paper also notes a discrepancy between the two LHCb 13 TeV analyses.

Methodological Rigor

Strengths:

Very broad dataset: 11 J/ψ datasets spanning 5–13 TeV and central and forward rapidity.

Two CCFM gluon densities (A0 and JH'2013 set 2).

Theory uncertainty bands that include the soft-gluon-energy and renormalization scale variations.

ψ′/J/ψ ratios used as an out-of-fit validation.

Weaknesses:

χ²/n.d.f. for J/ψ is poor. It is 3.8 for A0 and 3.3 for JH'2013 after a pT>6 GeV cut. Individual datasets reach 12–18 (LHCb 5 TeV and 13 TeV, ATLAS 13 TeV, CMS 5 TeV).

The claim of "good agreement" rests on theory uncertainty bands, and the fit χ² uses experimental errors only.

The JH'2013 gluon requires a pT>6 GeV cut to match A0, which shows gluon-density dependence at low kT.

The χcJ fit has few points (59, or 50 after cuts) and many free parameters in the neq scenario. The χ²/n.d.f. values well below 1 (e.g. 0.07) suggest overfitting or overestimated errors.

The ad hoc rule of dropping the spin-alignment uncertainty for χc1+χc2 sums is an assumption.

The ηc "prediction" is not truly parameter-free. The authors choose ⟨Oηc(3S1[8])⟩=0 and ⟨Oηc(1P1[8])⟩=3⟨l.c.⟩ to improve agreement, so the ηc result is a conditional range, not a clean test.

The linear-combination and regression tricks hide the separate CO contributions, so the LDMEs are not individually extractable and are hard to compare with collinear NLO NRQCD fits.

The claim that NLO† improves on LO is shown mainly in Fig. 1, as χ² versus pT cutoff. A systematic LO-versus-NLO† comparison is limited, and the scheme depends on a model-dependent soft-gluon emission picture.

The polarization puzzle is not addressed here, though the paper is explicit about deferring it.

Potential Impact

This is a specialized contribution within the small kT-factorization/CCFM quarkonium community, with some relevance to the NRQCD community more broadly. It provides a useful global LDME set within a framework that effectively resums high-energy logarithms and uses tree-level NLO amplitudes, which is an alternative to collinear NLO. The χc1/χc2 HQSS-violation finding and the ηc discrepancy feed ongoing debates (the heavy-quark spin symmetry puzzle). It is unlikely to change how the broader field approaches quarkonium production, since conclusions depend on the CCFM gluon choice and on the authors' soft-gluon model. Real-world application is minimal, beyond serving as input for Monte Carlo generators (CASCADE, PEGASUS) and for predictions at future colliders or in heavy-ion baselines.

Timeliness

Charmonium production remains an open issue. New LHC data (ATLAS 13 TeV 2024, LHCb ηc 2024) and the updated BESIII ηc→γγ width make a global reanalysis timely. The ηc data-tension observation is useful to experimentalists.

Strengths

The most comprehensive simultaneous fit of the whole charmonium family within kT-factorization.

A transparent discussion of limitations, including the ηc and HQSS tensions.

Consistent reuse of an established scheme.

Flags the inconsistency between the LHCb ηc measurements.

Limitations

Incremental over the authors' own prior papers (2024 and 2025). The new elements are the J/ψ and ηc application and the combined fit.

High χ² on key datasets, and a gluon-density-dependent low-pT cut.

Model-dependent soft-gluon treatment.

The "NLO" corrections are only tree-level real emission, with no loop corrections.

Inherently limited reach beyond a niche community.

No public code or data table is mentioned. The inputs are public, but the amplitude calculation is complex, so replication is difficult.

Overall: A solid but incremental phenomenology paper of modest, niche impact.

Rating:4/ 10

Significance 3.5Rigor 5Novelty 3.5Clarity 6

Generated Oct 22, 2026

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Scorecard· 16/16

4.0/10 impact

Competent, comprehensive but incremental extension of the authors' own NLO† kT-factorization scheme to J/ψ and ηc, with niche impact and mediocre fit quality on several datasets.