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Relativistic corrections to gluon fragmentation explain the high-pT psi(2S) surplus without color-octet states.

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2026-08-04 11:09 UTC pith:ATZASENV

load-bearing objection First combined study of O(v^2) relativistic corrections to prompt psi(2S) fragmentation at high pT, with honest uncertainty treatment; but the 'no CO needed' conclusion outruns an uncontrolled v^2 expansion. the 4 major comments →

arxiv 2510.06456 v2 pith:ATZASENV submitted 2025-10-07 hep-ph hep-exhep-thnucl-th

Impact of relativistic corrections to high-pT prompt-psi(2S) production at hadron colliders

Valerio Bertone , Jean-Philippe Lansberg , Kate Lynch This is my paper
classification hep-ph hep-exhep-thnucl-th
keywords quarkonium productionNRQCDfragmentation functionsrelativistic correctionspsi(2S)color singlethadron colliderspolarization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to show that the long-standing psi(2S) surplus at high transverse momentum can be accounted for by including O(v^2) relativistic corrections to the fragmentation functions in NRQCD leading-power factorization. The O(v^2) correction to gluon fragmentation into psi(2S) grows like log^2(1-z) near z~1, so at the large momentum fractions that dominate high-pT production its Mellin moment is about 15. This boosts the gluon channel by factors of 3.3 or 6.6 depending on the value of , bringing the computed cross section into agreement with the latest ATLAS and CMS measurements for pT above roughly 60 GeV, and bringing the polarization anisotropy closer to CMS data. If correct, this removes the need for fitted color-octet long-distance matrix elements, making inclusive quarkonium production predictive from color-singlet physics alone.

Core claim

On the paper's own terms: a leading-power fragmentation calculation of prompt psi(2S) hadroproduction, with fragmentation functions evolved to next-to-leading logarithmic accuracy and short-distance coefficients at next-to-leading order, now includes relativistic O(v^2) corrections. The v^2 correction to the gluon->psi(2S)gg fragmentation function has a strong threshold enhancement; its N~6 Mellin moment at mu0=2m_c is of order 15<v^2>, and after evolution to mu=100 GeV it remains large. As a result the gluon contribution to dsigma/dp_T rises by a factor 3.3 for <v^2>=0.25 and 6.6 for <v^2>=0.5. Using two scenarios for the wave function and velocity parameter, the calculation agrees with ATL

What carries the argument

The load-bearing object is the O(v^2) coefficient of the gluon fragmentation function for g->psi(2S)gg, denoted d(3,2)_g. It is a parametrized function containing terms with log^2(1-z), which grow as the momentum fraction z approaches 1. Since the high-pT hadronic cross section is sensitive to the N~6 Mellin moment of the fragmentation function, this threshold growth multiplies into a large ~15<v^2> correction at the initial scale and survives NLL evolution; that is what carries the argument. The machinery also includes coupled DGLAP evolution of charm and gluon fragmentation functions at NLL, NLO short-distance coefficients, and the Gremm-Kapustin relation to set <v^2> from the meson and qu

Load-bearing premise

The entire surplus resolution rests on the assumed accuracy of the O(v^2) gluon fragmentation function coefficient near z=1; if that parametrization is wrong, or if O(v^4) corrections (which mix with color-octet states) are significant at <v^2>=0.5, the agreement with LHC data and the 'no color-octet needed' conclusion collapse.

What would settle it

A direct next-to-leading-order (O(alpha_s^4)) computation of the gluon->psi(2S) fragmentation function, or a measurement of the z-distribution of psi(2S) inside jets, would test the large-z threshold growth. If the NLO result does not reproduce the ~15<v^2> Mellin moment, or the jet z-spectrum lacks the predicted enhancement, the central claim is falsified.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If confirmed, the color-singlet leading-power picture suffices for high-pT psi(2S); color-octet matrix elements would no longer be needed there.
  • Predictions at FCC-hh energies (100 TeV, up to about 1 TeV) become available, where NLL rather than LL evolution shifts the rate by about 20%.
  • The known depolarization from missing NLO corrections to the gluon channel is expected to improve the already approximate agreement with CMS lambda_theta.
  • The same formalism applies to quarkonia inside jets, so jet-based measurements should reflect the same large O(v^2) threshold enhancement.
  • For J/psi, the same effect should be present, but the much larger role of chi_c feeddown must be controlled first.

Where Pith is reading between the lines

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

  • The size of the boost hinges on the d(3,2)_g parametrization near z->1. If a future direct NLO calculation of the gluon FF softens that threshold behavior, the claimed surplus resolution could weaken or shift to other channels.
  • With <v^2>=0.5, O(v^4) corrections are about 0.25 and mix with color-octet transitions; the paper does not quantify this mixing, so the effective absence of CO contributions is only argued indirectly.
  • A natural test is psi(2S) production inside jets: the large-z enhancement implies an unusual z-distribution in the jet fragmenting into the psi(2S), which jet substructure data could confirm or rule out.
  • The same v^2 resummation logic should apply to Upsilon states, where <v^2> is smaller, predicting a milder but still measurable boost.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This Letter studies prompt ψ(2S) production at high pT using leading-power single-parton fragmentation with NRQCD-based initial-scale FFs. The authors include the O(v^2) relativistic corrections to the g→ψ(2S)gg fragmentation function, the charm-quark fragmentation function up to NLO in αs and O(v^2), NLL DGLAP evolution in the VFNS, and NLO hard-scattering coefficients. They compare their predictions with ATLAS and CMS cross-section data at 13 TeV and with CMS polarization data. The central claim is that the O(v^2) gluon correction, which boosts the g→ψ(2S)gg contribution by a factor 3.3 or 6.6 for the adopted ⟨v²⟩ values, brings the color-singlet fragmentation prediction into agreement with the high-pT data, so that higher-order color-octet contributions are not needed. They also present an uncertainty breakdown and FCC-hh predictions.

Significance. If the central claim holds, the paper offers a significant step toward resolving the long-standing ψ(2S) surplus puzzle: it suggests that at LHC energies the high-pT prompt yield and polarization may be understood within color-singlet LP fragmentation once O(v²) relativistic corrections are included, without fitting color-octet LDMEs. The paper is transparent about its inputs: no parameters are fitted to the ψ(2S) cross-section data; ⟨v²⟩ and |R(0)|² are taken from potential models and the Gremm-Kapustin relation, and the uncertainty treatment is unusually detailed, with supplementary tables separating all scale and PDF uncertainties. The confirmation of Bodwin, Kim and Lee's large-N Mellin-moment enhancement at N≈6.2 is also explicitly reported. However, the significance is currently limited by two issues: the O(v²) correction is numerically dominant rather than a small perturbation, and the total theory uncertainty (about a factor of five) makes the data comparison a weak discriminator.

major comments (4)
  1. [Main text, "Phenomenological parameters..." and Fig. 1(c)] The load-bearing observation is that the O(v²) correction to the gluon fragmentation moment is O(15⟨v²⟩) at N≈6.2. For the two adopted ⟨v²⟩ values this is 3.75 and 7.5 times the leading g→ψ(2S)gg moment at μ0=2m_c. The O(v²) term is therefore not a small relativistic correction but the dominant contribution, and truncating at O(v²) is not obviously valid. The Introduction itself notes that at O(v⁴) the CS v⁴ corrections mix with the leading 3S1[8] CO contribution, and that CO contributions are "in essence higher-order relativistic corrections." No numerical estimate of the O(v⁴) Mellin moment or of the leading CO contribution at N≈6.2 is provided, even though the known O(v⁴) expressions in Ref. [37] are cited. Without such an estimate, the apparent agreement with ATLAS/CMS data could be an artifact of an accidentally large O(v²) coefficient compensating an O(v⁴)/CO term of similar magnit
  2. [Main text, polarization discussion and Fig. 1(b)] The paper states that the total theoretical uncertainty is at least a factor of five, dominated by the μ_R0^g scale variation of the gluon FF, which alone is close to a factor of four (α_s(m_c)/α_s(4m_c)≈1.6, cubed). Figure 1(c) shows the full theory band spanning roughly 0.25–2.25 times the ATLAS data. With such a wide band, the observed "agreement" with the cross-section data, especially for pT≳60 GeV, is a consistency check rather than a discriminating test of the color-singlet vs. color-octet mechanism. The abstract and conclusions describe the agreement as evidence against the need for color-octet contributions, but the scale uncertainty alone allows the leading-order, v²=0 prediction to be compatible with the data at the upper end of its band. The authors should either present a more quantitative measure of the discrimination (e.g., show the v²=0 curve within the same uncertainty b
  3. [Supplementary Eq. (7) and Table I] The abstract claims the polar anisotropy is "close to CMS data," but the comparison is made with polarized FFs D_{T,L}^c and D_{T,L}^g known only at LO in αs and LO in v², while the unpolarized cross section uses NLO SDCs and NLL evolution. The authors themselves note that further conclusions would require NLO corrections to the polarized FFs. Given that the polarization prediction is thus on a much less complete footing than the cross-section prediction, the abstract's wording overstates the strength of the polarization agreement. This is a minor-to-moderate issue, but because the abstract advertises it as a result, the paper should either add a caveat in the abstract or downplay the polarization claim to "consistent with CMS data within the large theoretical uncertainties."
  4. [Main text, "Phenomenological parameters..."] The gluon O(v²) parametrization d(3,2)_g in Supplementary Eq. (7), taken from Bodwin et al., contains ln²(1−z) terms that grow at large N. The N≈6.2 Mellin moment is the key quantity driving the paper's central mechanism, yet the paper relies on the external parametrization without an independent check beyond confirming the reported O(15⟨v²⟩) value. Since the high-z behavior of this parametrization is the origin of the large moment, the authors should provide, or at least comment on, a direct numerical verification of the Mellin transform of Eq. (7) against the original analytic expressions, and discuss the sensitivity of the N=6.2 moment to the assumed large-z form. This is not a demand for new physics, but a robustness check that would materially strengthen the claim.
minor comments (5)
  1. [Main text, page 3] The caption of Fig. 1(c) reads "Same as (b) but for the cross section ratio to ATLAS data," which is confusing: (b) shows polarization and (c) shows the cross-section ratio. Probably "Same as (a)" or "Same line/color scheme as (b), but for the cross-section ratio" is intended.
  2. [Main text, p. 3] The sentence "We note that the unknown NLO corrections to the gluon channel are likely to produce depolarization, which would further improve the agreement" is speculative; it is fine as a heuristic but should be labeled as such.
  3. [References] In the supplementary material, Eq. (3) writes d(2,0)_c→ψ(z/y, μ0) inside the integral; the notation is slightly inconsistent with the definition of d(2,0)_c/Q̄Q and may confuse a reader. A minor clarifying subscript or comment would help.
  4. [Main text, p. 2] The code INCNLO1.4 is referenced only by a website (Ref. [53]) without version or DOI; given that the results depend on it, a fuller citation would be useful. Also, the benchmark against FMNLO (Ref. [54]) is mentioned but no numerical comparison is shown.
  5. [Supplementary material] The table separates contributions (A) diagonal and (B) off-diagonal, but the column headers are easy to misread; explicitly labeling whether each column is multiplied by 10^4 in the header, or using exponent notation, would improve clarity.

Circularity Check

0 steps flagged

No significant circularity: the O(v^2) enhancement is an external NRQCD input, not fitted to the psi(2S) data, and the paper's comparisons are benchmarked against independent experimental results.

full rationale

The paper's central input, the O(v^2) gluon fragmentation function d(3,2)_g, is not derived or fitted here; it is imported from Bodwin, Kim and Lee. The supplementary material states: 'Equation 7 gives the alpha_s^3 v^2 corrections to the g->psi(2S)gg contribution and we make use of the parametric form presented in the Appendix of [6].' The reported boost factors of 3.3 and 6.6 are Mellin moments of that external parametrization evolved and convolved with PDFs and SDCs; no parameter is adjusted to the ATLAS/CMS psi(2S) cross-section points. The choices <v^2>=0.25 and 0.5 and the anti-correlated pairing with |R(0)|^2 are motivated by the Gremm-Kapustin relation and potential-model values, and are presented as bracketing scenarios rather than fits. The only self-citations in the paper (e.g., a review and a paper on leptonic-width instabilities) are contextual or used for an external physics argument, and the derivation does not reduce to them. The paper itself flags the possible O(v^4)/color-octet mixing and the need for NLO gluon FFs; these are convergence and accuracy concerns, not circularity of the derivation chain.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The paper introduces no new degrees of freedom; its freedom lies in the choices of |R(0)|^2, <v^2>, and m_c. These are not fitted to the target psi(2S) data but are selected from external potential models and prior NRQCD estimates. The two scenarios are anti-correlated to keep the gluon product roughly constant, which is a reasonable bracketing strategy rather than a fit to the LHC points.

free parameters (3)
  • |R(0)|^2 (radial wavefunction at origin) = 0.53 GeV^3 (default), 0.93 GeV^3 (alternate)
    Input from potential models (Buchmüller-Tye and Cornell respectively); not fitted here, but the normalization of the cross section is directly proportional to it, and the two values are used as bracketing scenarios.
  • <v^2> (average squared quark velocity) = 0.50 (default), 0.25 (alternate)
    Chosen from Gremm-Kapustin estimate (~0.5 for m_c=1.5 GeV) and a smaller value; the v^2 correction scales linearly with this, so the central result's magnitude is controlled by it.
  • m_c (charm quark pole mass) = 1.5 GeV
    Fixed value; the paper states mass variations are accounted for via LDME and <v^2> variations, but m_c itself enters the FFs and the Gremm-Kapustin estimate.
axioms (5)
  • domain assumption NRQCD factorization conjecture: production expands in alpha_s and v, with color-singlet 3S1[1] dominating at leading v power
    Used throughout the computation of color-singlet FFs; the central prediction relies on the hierarchy of NRQCD velocity scaling.
  • domain assumption Leading-power fragmentation factorization (Eq. 1) is valid up to M^2/pT^2 corrections; gluon and charm FFs dominate at high pT
    The whole framework neglects power corrections in M^2/pT^2, which are expected to be small only at high pT; the paper explicitly restricts to this regime.
  • domain assumption Gremm-Kapustin relation: <O_2(3S1[1])> = <O_0(3S1[1])> and <v^2> = (M^2-4m_c^2)/(4m_c^2)
    Used to convert the O(v^2) LDME into <v^2> times the LO LDME; standard in NRQCD but holds only up to O(v^2) corrections.
  • domain assumption Vacuum saturation approximation: <O_0(3S1[1])> = 2N_c(2J+1)|R(0)|^2/(4 pi)
    Relates the color-singlet LDME to the potential-model wave function at the origin; an approximation that ignores higher Fock states.
  • standard math CT18NLO PDFs, APFEL++ evolution, and INCNLO1.4 SDCs are reliable at NLO/NLL
    External computational inputs; their uncertainties are propagated via PDF error sets and scale variations, but their correctness is not questioned.

pith-pipeline@v1.3.0-alltime-deepseek · 25143 in / 14044 out tokens · 106063 ms · 2026-08-04T11:09:15.671539+00:00 · methodology

0 comments
read the original abstract

We study relativistic corrections to prompt psi(2S) production at high pT in hadron colliders. Our calculation employs leading-power factorization with Fragmentation Functions (FFs) computed in nonrelativistic QCD and evolved to next-to-leading-logarithmic accuracy. Relativistic corrections increase the cross section significantly for the gluon channel, but moderately for charm. We perform a full analysis of uncertainties. We observe a good agreement with both ATLAS and CMS cross sections without the need of higher-order color-octet contributions. The polar anisotropy is found to be close to CMS data, partly due to charm fragmentation.

Figures

Figures reproduced from arXiv: 2510.06456 by Jean-Philippe Lansberg, Kate Lynch, Valerio Bertone.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Prompt ATLAS and CMS [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. FCC-hh predictions for prompt [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Coupled charm (green) and gluon (gray-blue) fragmentation functions at (a,c,e) [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: compares prompt ATLAS ψ(2S) PT -differential cross section data [7] in three bins of rapidity at √ s = 13 TeV to our results for coupled c & g channels using |R(0)ψ(2S ) | 2 = 0.53 GeV3 , ⟨v 2 ψ(2S ) ⟩ = 0.50 (a) and |R(0)ψ(2S ) | 2 = 0.93 GeV3 , ⟨v 2 ψ(2S ) ⟩ = 0.25 (b). 1 1 1 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: compares prompt CDF ψ(2S) PT -differential cross section at √ s = 1.96 TeV to our results for coupled c & g (hatched red), c (solid green), and g (solid light blue) channels using |R(0)ψ(2S ) | 2 = 0.53 GeV3 , ⟨v 2 ψ(2S ) ⟩ = 0.5. 5 5 5 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Three-gluon decays of radially excited quarkonia $\psi(2S)$ and $\Upsilon(2S)$ with both relativistic and QCD radiative corrections

    hep-ph 2026-03 conditional novelty 5.0

    Phenomenologically improved Bethe-Salpeter calculations of ψ(2S)/Υ(2S) → ggg, including relativistic and QCD corrections, match experiment and extract lower β_V values reflecting nodal sensitivity.

Reference graph

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