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arxiv: 2605.16019 · v1 · pith:ZK3F36DOnew · submitted 2026-05-15 · ✦ hep-ph · hep-ex· nucl-ex· nucl-th

Inclusive charm and bottom quark pair production cross sections at hadron colliders at next-to-next-to-leading-order accuracy

David d'Enterria , Felix Hekhorn , Ilkka Helenius , Van Dung Le , Hannu Paukkunen This is my paper

Pith reviewed 2026-05-20 16:35 UTC · model grok-4.3

classification ✦ hep-ph hep-exnucl-exnucl-th
keywords heavy quark productioncharm cross sectionbottom cross sectionNNLO QCDhadron collisionsparton distribution functionsquark mass
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The pith

NNLO QCD calculations enhance charm and bottom quark pair production cross sections by up to a factor of two.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper calculates the inclusive cross sections for charm and bottom quark-antiquark pair production in proton-proton, proton-antiproton, and proton-nucleus collisions over energies from about 10 GeV to 400 TeV. It performs these calculations at next-to-next-to-leading-order accuracy in perturbative QCD using different sets of parton distribution functions and compares them to all existing experimental data up to 14 TeV. The NNLO results are larger than next-to-leading-order predictions by as much as a factor of two, with scale uncertainties reduced by a similar amount, bringing theory into agreement with measurements across the full energy range. These improved predictions suggest that multi-TeV charm data can constrain the gluon density at very small momentum fractions and that low-energy bottom data can help determine the bottom-quark pole mass.

Core claim

At next-to-next-to-leading order in perturbative QCD, the inclusive cross sections for charm and bottom quark pair production are larger by up to a factor of two than at next-to-leading order, with correspondingly smaller scale uncertainties. These NNLO predictions agree with experimental measurements across the full range of available collision energies from approximately 10 GeV to 14 TeV. The calculations also allow for potential constraints on the small-x gluon distribution from multi-TeV charm data and on the bottom-quark pole mass from low-energy bottom data.

What carries the argument

Fixed-order next-to-next-to-leading-order perturbative QCD computations of heavy-quark pair production cross sections, performed with different sets of parton distribution functions.

If this is right

  • NNLO results agree with experimental data over the entire energy range from 10 GeV to 14 TeV.
  • Scale uncertainties in the theoretical predictions are reduced by up to a factor of two compared to NLO.
  • Charm pair production cross sections at multi-TeV energies can provide additional constraints on the gluon density at very small x.
  • More precise bottom pair production measurements at low energies can help determine the bottom-quark pole mass.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Higher-order QCD corrections appear essential for describing heavy quark production accurately even at moderate collision energies.
  • The same computational approach could be applied to predict cross sections at proposed future high-energy colliders.
  • These NNLO results may encourage updates to global parton distribution fits that incorporate heavy quark data for better small-x determinations.

Load-bearing premise

The perturbative expansion in QCD converges sufficiently well at NNLO even at low collision energies around 10 GeV, and the chosen parton distributions and quark masses do not introduce large biases for this process.

What would settle it

An experimental measurement of the charm pair production cross section at a center-of-mass energy of about 10 GeV that lies well outside the NNLO uncertainty range but inside the NLO range would show that the NNLO approximation breaks down at low energies.

Figures

Figures reproduced from arXiv: 2605.16019 by David d'Enterria, Felix Hekhorn, Hannu Paukkunen, Ilkka Helenius, Van Dung Le.

Figure 1
Figure 1. Figure 1: FIG. 1. Charm-quark fragmentation fractions [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Time evolution of charm-quark fragmentation fractions into charm mesons and baryons for di [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Bottom-quark fragmentation fractions into di [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Inclusive cross sections for charm (left) and bottom (right) production in p-p collisions as a function of the c.m. energy, [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Breakdown of the relative contribution of di [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Inclusive charm (left) and bottom (right) NNLO production cross sections as a function of collision energy, over [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Total charm (left) and bottom (right) cross section at NNLO accuracy in p-p collisions as a function of c.m. energy, [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison of the NNPDF3.1 (green) and NNPDF4.0 (orange) NNLO parton densities in p-p collisions as a function of [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Inclusive charm (left) and bottom (right) NNLO production cross sections as a function of collision energy, over [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Inclusive transverse momentum [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Charm cross sections in p-p collisions as a function of c.m. energy, over [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The same as Fig [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Compilation of representative experimental total c [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Bottom cross sections in p-p collisions as a function of c.m. energy, over [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Compilation of representative experimental total b [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
read the original abstract

The inclusive cross sections for charm ($\mathrm{c}\overline{\mathrm{c}}$) and bottom ($\mathrm{b}\overline{\mathrm{b}}$) quark-antiquark pair production in proton-proton, proton-antiproton, and proton-nucleus collisions are studied over a wide range of center-of-mass energies, $\sqrt{s}\approx 10$ GeV--400 TeV. All existing data over $\sqrt{s}\approx 10$ GeV--14 TeV are collected and compared to calculations at next-to-next-to-leading-order (NNLO) accuracy using the new fixed-order MaunaKea open-source code for varying sets of parton distribution functions (PDFs). Relative to next-to-leading-order (NLO) predictions, the NNLO cross sections are enhanced by up to a factor of two, with the associated theoretical scale uncertainties reduced by the same amount, leading to agreement with experimental data over the full range of collision energies. The NNLO results are also compared with NLO predictions obtained within the SACOT-$m_{_\mathrm{T}}$ general-mass variable-flavour-number scheme. Despite still sizable theoretical and experimental uncertainties, $\mathrm{c}\overline{\mathrm{c}}$ cross section at multi-TeV energies can provide extra constraints on the gluon density at very small-$x$ in global PDF analyses. In the bottom sector, more precise cross section measurements at low energies, $\sqrt{s}\approx 10$--100 GeV, can help constraint the bottom-quark pole mass.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript computes inclusive charm and bottom quark-antiquark pair production cross sections at NNLO accuracy using the new MaunaKea fixed-order code for pp, pA, and pbar p collisions over √s ≈ 10 GeV to 400 TeV. It collects all existing data up to 14 TeV, reports up to factor-of-two enhancement relative to NLO with correspondingly reduced scale uncertainties, demonstrates agreement with data across the energy range, compares to NLO SACOT-mT GM-VFNS predictions, and discusses prospective constraints on small-x gluons from high-energy ccbar data and on the bottom pole mass from low-energy bbbar data.

Significance. If the NNLO results hold, the work supplies the first comprehensive fixed-order NNLO benchmark for heavy-quark pair production over a wide kinematic range, strengthens the case for using such data in global PDF fits at small x, and offers a path to tighter bottom-mass constraints; the open-source MaunaKea implementation and explicit data compilation are additional strengths that would facilitate reproducibility and further phenomenological studies.

major comments (2)
  1. [Results section (low-energy comparisons)] The headline claim of controlled NNLO enhancement (up to ×2) and data agreement at the lowest energies (√s ≈ 10 GeV) rests on the assumption that the perturbative series remains reliable when the hard scale is set by m_c ≈ 1.5 GeV and α_s(m_c) ≈ 0.3; the manuscript should provide explicit comparisons at these points to threshold-resummed or GM-VFNS calculations to quantify the size of power-suppressed corrections.
  2. [PDF dependence and discussion of future constraints] The PDFs employed are taken from prior global fits that already incorporate some heavy-quark production data; the manuscript should quantify the degree of interdependence when proposing that the new NNLO ccbar results can furnish additional small-x gluon constraints, for example by repeating the comparison with PDFs fitted after removal of existing heavy-flavor data.
minor comments (2)
  1. [Abstract] The abstract would be strengthened by quoting at least one or two representative numerical enhancement factors and uncertainty reductions at specific energies rather than stating only the maximum values.
  2. [Methods / computational setup] Notation for the heavy-quark pole masses and the precise definition of the central scale choice should be stated explicitly in the methods section to allow direct reproduction of the quoted results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive overall assessment and the constructive major comments. We address each point below and have revised the manuscript accordingly where feasible to strengthen the presentation.

read point-by-point responses

  1. Referee: [Results section (low-energy comparisons)] The headline claim of controlled NNLO enhancement (up to ×2) and data agreement at the lowest energies (√s ≈ 10 GeV) rests on the assumption that the perturbative series remains reliable when the hard scale is set by m_c ≈ 1.5 GeV and α_s(m_c) ≈ 0.3; the manuscript should provide explicit comparisons at these points to threshold-resummed or GM-VFNS calculations to quantify the size of power-suppressed corrections.

    Authors: We appreciate the referee drawing attention to the perturbative reliability at the lowest energies. While α_s(m_c) ≈ 0.3 indicates that higher-order corrections are sizable, the observed reduction in scale uncertainty from NLO to NNLO and the improved agreement with data support the stability of the expansion. To address the request, we have added explicit low-energy comparisons to the existing NLO SACOT-mT GM-VFNS results in the revised Results section, together with a quantitative discussion of the expected size of power-suppressed corrections drawn from threshold-resummation studies in the literature. A complete NNLO threshold-resummed calculation for the inclusive cross section is not yet available in the literature, but the added material now quantifies the relevant corrections more clearly. revision: partial

  2. Referee: [PDF dependence and discussion of future constraints] The PDFs employed are taken from prior global fits that already incorporate some heavy-quark production data; the manuscript should quantify the degree of interdependence when proposing that the new NNLO ccbar results can furnish additional small-x gluon constraints, for example by repeating the comparison with PDFs fitted after removal of existing heavy-flavor data.

    Authors: We agree that the PDFs used already contain some heavy-flavor data and that this creates a degree of interdependence when advocating for additional small-x constraints. Performing a new global PDF fit after explicitly removing all existing heavy-flavor data is a major computational effort that lies outside the scope of the present phenomenological study. In the revised manuscript we have expanded the discussion to explicitly acknowledge this interdependence, to note that the NNLO results are intended as supplementary input for future dedicated fits, and to illustrate sensitivity by comparing several PDF sets that differ in their heavy-flavor treatment. revision: partial

Circularity Check

0 steps flagged

Fixed-order NNLO computation of heavy-quark pair production using external PDFs and MaunaKea code

full rationale

The paper computes inclusive c cbar and b bbar cross sections at NNLO accuracy via the MaunaKea code over a wide energy range and compares the results directly to collected experimental data. PDFs and quark masses are taken as inputs from prior global fits; the present work performs no new fit and presents the NNLO enhancement and uncertainty reduction as outcomes of the perturbative calculation itself. The forward-looking remark that the results could constrain small-x gluons in future PDF analyses is not part of the derivation chain or used to justify any current claim. No self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation appears in the abstract or described content. The derivation is therefore self-contained as a standard perturbative QCD evaluation against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard perturbative QCD at fixed NNLO order, existing global PDF sets, and the new MaunaKea implementation. Free parameters are the usual renormalization/factorization scales (varied for uncertainty) and heavy-quark pole masses. No new particles or forces are postulated. The main domain assumption is that fixed-order perturbation theory suffices across the quoted energy range.

free parameters (2)
  • renormalization and factorization scales
    Standard choice varied to estimate theoretical uncertainty in the NNLO computation.
  • heavy quark pole masses
    Input values for charm and bottom quarks; the abstract notes potential for bottom mass constraint from low-energy data.
axioms (1)
  • domain assumption Fixed-order perturbative QCD expansion converges sufficiently at NNLO for these processes over the full energy range
    Invoked implicitly when claiming agreement with data from 10 GeV upward.

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discussion (0)

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