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arxiv: 2606.23594 · v1 · pith:Y3PVSEIOnew · submitted 2026-06-22 · ✦ hep-ph

Next-to-next-to-leading-order QCD corrections to {}³S₁⁽⁸⁾ gluon fragmentation function for quarkonium

Feng Feng , Yu Jia , Wen-Long Sang This is my paper

Pith reviewed 2026-06-26 08:05 UTC · model grok-4.3

classification ✦ hep-ph
keywords NNLO QCD correctionsgluon fragmentation functionquarkoniumNRQCDcolor-octetJ/psi productionpolarization
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The pith

The first NNLO QCD calculation finds that corrections to the ³S₁⁽⁸⁾ gluon fragmentation function for quarkonium are positive and substantial over most of the momentum fraction range.

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

This paper performs the first calculation of next-to-next-to-leading-order QCD corrections to the gluon fragmentation function into the color-octet ³S₁ state of a quarkonium. The work is carried out in the nonrelativistic QCD factorization approach, keeping only the leading term in the velocity expansion. A reader would care because the result affects how we predict the rates and polarizations of particles like the J/ψ at large transverse momenta in proton collisions. The corrections turn out to be positive and sizable in most of the range of the momentum fraction z.

Core claim

We present the first computation of the next-to-next-to-leading-order (NNLO) QCD corrections to the ³S₁⁽⁸⁾ gluon fragmentation function for quarkonium within the nonrelativistic QCD (NRQCD) factorization framework, accurate to the lowest order in the velocity expansion. The calculation is performed with high numerical precision and encompasses both polarized and unpolarized cases. We find that the NNLO corrections are positive and substantial across most of the z region. Furthermore, the logarithmic singularities near the endpoint z→1 are fully reconstructed, providing essential inputs for future threshold resummation beyond leading-logarithmic accuracy.

What carries the argument

The ³S₁⁽⁸⁾ gluon fragmentation function in the NRQCD factorization framework at NNLO in the strong coupling constant.

Load-bearing premise

NRQCD factorization continues to hold for the gluon fragmentation function when going to NNLO in the coupling while staying at leading order in velocity.

What would settle it

A high-precision measurement of the J/ψ cross section or polarization at transverse momenta above several hundred GeV that deviates markedly from predictions incorporating this NNLO fragmentation function would indicate the calculation does not capture the physics.

Figures

Figures reproduced from arXiv: 2606.23594 by Feng Feng, Wen-Long Sang, Yu Jia.

Figure 1
Figure 1. Figure 1: Representative Feynman diagrams for g → cc¯( 3S (8) 1 ) gluon fragmentation function up to NNLO in αs. The thin double line represents an eikonal line. with β0 = 11 6 Nc − 1 3 nL (where Nc = 3 is the number of colors and nL = 3 is the number of light quarks). Note that d (1) L (z, µ) is independent of µ. Physically, the NNLO SDCs are expected to take the form: d (2) λ (z, µ) = C λ 2 (z) ln2 µ 2 m2 + C λ 1 … view at source ↗
Figure 2
Figure 2. Figure 2: SDCs dλ(z, µ) for λ = T (left), L (middle) and unpolarized (right) at NLO and NNLO. The NLO and NNLO predictions are defined as d NLO λ (z, µ) ≡ α 2 s (µ) d (1) λ (z, µ) and d NNLO λ (z, µ) ≡ α 2 s (µ) d (1) λ (z, µ) + α 3 s (µ) d (2) λ (z, µ), respectively. The shaded bands represent scale uncertainties obtained by varying µ ∈ [2m, 4m]. The strong coupling αs is evaluated at three-loop accuracy using RunD… view at source ↗
Figure 3
Figure 3. Figure 3: Ratios ξλ(z, µ) ≡ dλ(z,µ) dT (z,µ)+dL(z,µ) for λ = T (left) and L (right) at NLO and NNLO. The definitions of the NLO and NNLO predictions follow those in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

We present the first computation of the next-to-next-to-leading-order (NNLO) QCD corrections to the ${}^3S_1^{(8)}$ gluon fragmentation function for quarkonium within the nonrelativistic QCD (NRQCD) factorization framework, accurate to the lowest order in the velocity expansion. The calculation is performed with high numerical precision and encompasses both polarized and unpolarized cases. We find that the NNLO corrections are positive and substantial across most of the $z$ region. Furthermore, the logarithmic singularities near the endpoint $z\to 1$ are fully reconstructed, providing essential inputs for future threshold resummation beyond leading-logarithmic accuracy. Combined with threshold-resummed formulas in the large-$z$ region, our results yield phenomenologically viable inputs for the $^3S_1^{(8)}$ gluon fragmentation function. This enables a more reliable description of large-$p_T$ $J/\psi$ ($\psi'$) and $\chi_{cJ}$ production and polarization at hadron colliders, representing a crucial step toward a definitive test of the color-octet mechanism.

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

0 major / 2 minor

Summary. The paper presents the first computation of the next-to-next-to-leading-order (NNLO) QCD corrections to the ^3S_1^(8) gluon fragmentation function for quarkonium within the NRQCD factorization framework at leading order in the velocity expansion. The calculation covers both polarized and unpolarized cases with high numerical precision, finds positive and substantial corrections across most of the z region, fully reconstructs the logarithmic singularities near the endpoint z→1, and combines the results with threshold resummation to yield phenomenological inputs for high-p_T J/ψ, ψ', and χ_cJ production and polarization.

Significance. If the central results hold, this constitutes a significant advance by supplying the first NNLO short-distance coefficients for the ^3S_1^(8) gluon fragmentation function. The explicit reconstruction of endpoint logarithms supplies essential inputs for threshold resummation beyond leading-log accuracy, and the high numerical precision is a clear strength that supports more reliable collider phenomenology for testing the color-octet mechanism.

minor comments (2)
  1. [Abstract] Abstract and §1: the claim of providing 'phenomenologically viable inputs' would benefit from a short quantitative comparison to NLO results to illustrate the size of the NNLO shift.
  2. [§2] Notation for the polarized versus unpolarized fragmentation functions should be introduced with an explicit definition in §2 before the NNLO calculation begins.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the accurate summary of its contributions, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper reports an explicit perturbative computation of NNLO short-distance coefficients for the gluon fragmentation function in the NRQCD framework at leading order in velocity. The derivation proceeds via standard Feynman-diagram evaluation, ultraviolet and infrared renormalization, and numerical integration; none of these steps reduce by the paper's own equations to a fitted parameter, a self-citation chain, or a renamed input. The NRQCD factorization premise is the conventional external assumption used for such calculations and is not internally derived or self-referential within the work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; specific free parameters, renormalization schemes, and numerical cutoffs are not visible. The central claim rests on the standard NRQCD factorization assumption and the truncation to leading velocity order.

axioms (2)
  • domain assumption NRQCD factorization applies to the gluon fragmentation function at NNLO in alpha_s
    Invoked as the framework for the entire calculation in the abstract.
  • domain assumption Lowest order in the velocity expansion is sufficient for the fragmentation function
    Explicitly stated as the accuracy level of the computation.

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