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REVIEW 3 major objections 5 minor 80 references

Tree-level NLO corrections to inclusive $\psi'$ production in High Energy Factorization

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Adding tree-level NLO corrections to high-energy factorization lets one global fit describe all unpolarized LHC psi(2S) data, including the previously problematic low-transverse-momentum and forward-rapidity regions.

desk verdict A credible phenomenological extension of the kT-factorized NLO† scheme to psi(2S), with a real low-pT improvement that is partly hostage to an unvalidated double-counting cut. read the letter →

arxiv 2509.09416 v1 pith:UMSYS6BK submitted 2025-09-11 hep-ph

A.A. Prokhorov , S.P. Baranov , A.V. Lipatov , M.A. Malyshev , X. Chen This is my paper
classification hep-ph
keywords psi(2S)productionhighenergyfactorizationkT-factorizationNRQCDTMDgluondensitiesLHCcoloroctetLDMENLOcorrections
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the long-standing failure of kT-factorization to describe psi(2S) production at low transverse momentum is resolved by merging leading-order 2-to-1 amplitudes with tree-level next-to-leading-order 2-to-2 amplitudes using a strengthened double-counting cut. The authors extend an earlier matching scheme to forward rapidities and low pT, then fit three color-octet long-distance matrix elements to all unpolarized LHC data at 5.02, 7, 8, and 13 TeV. With the A0 TMD gluon density the full kinematic range is described without any pT cutoff, while a competing gluon density fails below pT of 6 GeV. This is presented as the first simultaneous global description of psi(2S) production in the full accessible LHC range within high-energy factorization, and as validation of the improved merging scheme for future use.

What carries the argument

The central object is the double-counting-exclusion (DCE) cut, a phase-space condition that separates LO 2-to-1 gluon fusion from NLO 2-to-2 amplitudes. The strengthened form, |p_gT| > max(|k1T|, |k2T| + |k^init_2T|), accounts for the primordial transverse momentum of the lower-x gluon and prevents emissions already contained in the CCFM evolution from being counted again as hard NLO radiation. This cut carries the argument because the claimed improvement at low pT and forward rapidity depends entirely on correctly identifying which gluon emissions are genuinely new.

What would settle it

Take the global fit and replace the strengthened DCE cut by the older condition |p_gT| > max(|k1T|, |k2T|) while keeping all other settings. If the chi2/n.d.f. for the LHCb 7 and 13 TeV and ALICE forward datasets stays near the improved level, the improvement attributed to the NLO merging is an artifact of the cut. A complementary check: sample explicit CCFM emission histories from the A0 and JH'2013 set 2 TMD densities and test whether the inequality |p_gT^emission| < |k2T| + |k^init_2T| is ever violated.

Watch

Extended reading notes

Core claim

The paper's central claim is that tree-level NLO corrections, merged with LO contributions through a modified double-counting-exclusion (DCE) condition, bring the kT-factorization plus NRQCD framework into agreement with all unpolarized LHC psi(2S) measurements. The modification strengthens the cut to |p_gT| > max(|k1T|, |k2T| + |k^init_2T|), where k^init_2T is the primordial transverse momentum of the lower-x gluon read from the TMD density at the starting scale. With A0 gluons the global fit gives chi2/n.d.f. = 2.15 with no pT restriction; with JH'2013 set 2 gluons the same fit gives 7.29 and only becomes acceptable when pT > 6 GeV is imposed. The authors conclude that including the NLO te

Load-bearing premise

The load-bearing premise is that the primordial transverse momentum of the softer gluon, taken from the TMD density at the starting scale, correctly bounds every unresolved initial-state emission, so that the strengthened cut |p_gT| > max(|k1T|, |k2T| + |k^init_2T|) removes exactly the double-counted events and nothing else.

Editorial extensions

If this is right

  • Inclusion of tree-level NLO corrections changes the shape of the pT spectrum, reducing its slope and improving agreement with data across the full pT range compared with LO alone.
  • The strengthened DCE cut removes the low-pT forward overestimate and smoothly converges to the previous cut at pT above about 6 GeV.
  • The extracted color-octet matrix elements show a strong negative correlation between the 1S0[8] and 3PJ[8] contributions, so only the linear combination is well determined; the 3S1[8] matrix element is extracted with good accuracy.
  • A0 and JH'2013 set 2 TMD gluon densities give similar results at moderate and high pT, but the low-pT region clearly favors A0, providing a discriminating observable for TMD densities.
  • The merging scheme is validated for use in future calculations and is to be implemented in the authors' Monte-Carlo event generator.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural testable extension is to apply the same LO+NLO merging with the strengthened DCE cut to J/psi and Upsilon production; if the low-pT fit quality transfers, the scheme is a general feature of kT-factorization rather than a psi(2S)-specific tuning.
  • The strong correlation between 1S0[8] and 3PJ[8] matrix elements suggests that current unpolarized data cannot separately pin down these parameters; future polarization or associated-production measurements may break the degeneracy.
  • The low-pT forward sensitivity to the TMD choice turns these measurements into a clean discriminator of initial-state evolution models, so adding more differential observables such as psi(2S)+jet production could strengthen the conclusion.
  • Because the strengthened cut depends on the value of k^init_2T taken from the TMD density, predictions inherit an extra sensitivity to the starting-scale treatment; this could be tested by comparing fits using different prescriptions for the primordial gluon momentum.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper presents a kT-factorization/NRQCD calculation of inclusive ψ(2S) production at LHC energies, merging leading-order 2→1 gluon-fusion amplitudes with tree-level 2→2 NLO amplitudes. To avoid double counting, a previous DCE cut |p_gT| > max(|k1T|, |k2T|) is strengthened to |p_gT| > max(|k1T|, |k2T| + |k^init_2T|) for forward/low-pT kinematics. The authors fit two color-octet LDME parameters (a linear combination ⟨O_l.c.⟩ and ⟨O[3S1[8]]⟩) to ψ' data from ATLAS, CMS, LHCb and ALICE at 5.02–13 TeV, using A0 and JH'2013 set 2 TMD gluon densities, and report that NLO corrections improve the χ2 and that A0 gluons describe the full pT range better than JH'2013.

Significance. The paper is a serious phenomenological step: it extends an existing LO+NLO matching scheme to the low-pT forward region and provides the first simultaneous kT-factorization fit to all available unpolarized ψ' LHC data. If the scheme is reliable, it demonstrates that tree-level NLO corrections can cure the longstanding low-pT overestimate of the LO approach and offers a way to discriminate between TMD gluon sets. The study is transparent about its free parameters and uses public TMD tools. However, the novelty reduces essentially to one modified cut whose derivation is not established, and the global fit quality is moderate (χ2/n.d.f. ≈ 2.15 for A0), so the strength of the conclusions exceeds what the analysis demonstrates.

major comments (3)
  1. [Sec. 2.2] The strengthened DCE cut |p_gT| > max(|k1T|, |k2T| + |k^init_2T|) is the only new ingredient, but its justification is incomplete. The inequality |p^emission_gT| < |k2T| + |k^init_2T| does not follow from k^init_2T = Σ p^emission_gTi + k2T: with several emissions, the hardest emission can exceed this bound if the others cancel. No CCFM-based derivation is given, and |k^init_2T| is only a starting-scale average (1.12 GeV A0, 1.77 GeV JH'2013). Since this cut is the only change from Ref. [41], the NLO improvement is entangled with it. Provide a test of the inequality and a sensitivity scan over |k^init_2T|.
  2. [Sec. 3, Table 3] The claims of 'good/reasonable simultaneous description' are overstated. The A0 fit yields χ2/n.d.f. = 2.15 overall, with 5.54 (ATLAS 13 TeV) and 3.27/3.44 (LHCb 7/13 TeV). The LO vs LO+NLO comparison does show improvement (2.15 vs 2.54), so 'better' is supported, but 'simultaneously describe' is not. Since the LDMEs are fitted to the same data, absolute normalization is not predictive. Please report shape-only χ2 and temper the conclusion.
  3. [Sec. 2.4, Eqs. (8)-(9), Table 1] The linear-combination reduction uses r defined as a pT-dependent cross-section ratio and averaged over rapidity. Table 1 gives r = 0.51 vs 0.64 (A0) and 0.56 vs 0.65 (JH'2013) for central vs forward. A single combination cannot represent two independent LDMEs unless the 1S0[8] and 3PJ[8] shapes are proportional, which is not shown. This could bias the fitted LDMEs and the pT shapes. Please test the dependence of the fit on r.
minor comments (5)
  1. [Throughout] Typos: 'strenghtened' (Sec. 2.2 and Fig. 1), 'trasformation' (Sec. 2.1), 'Bycling' (Ref. [68]), 'Szcurek' (Ref. [67]).
  2. [Fig. 7 caption] The caption says 'CMS [56], [57], [58]' but the CMS 5.02 TeV data are Ref. [52]; [56–58] are ALICE. Please correct the citation.
  3. [Fig. 1] The axis label 'c(cT)p2' appears garbled; it should likely be p_T(c cbar) in GeV. The meaning of the '100×' panels is not explained in the caption.
  4. [Sec. 2.2] The notation 'NLO*' and 'NLO†' is used without a crisp definition of how NLO† differs from the usual NLO*. Please state explicitly that NLO† denotes the tree-level matching scheme with the DCE cut.
  5. [Sec. 2.3] The sentence 'we shift the scale as μ_F → μ_F = m_T^ψ' is ambiguous; clarify that μ_F is set to m_T^ψ for 2→2 CO subprocesses.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: LDMEs are explicitly fitted parameters, and the independent content is in the shapes and relative comparisons; the DCE cut is a heuristic modeling assumption, not a fitted input.

full rationale

The paper's derivation chain is not circular. The CO LDMEs are explicitly treated as free parameters fitted to the data (Sec. 2.4), and the absolute normalization is never claimed as a prediction; the independent content lies in the pT/rapidity shapes, the LO-versus-NLO† comparison, and the A0-versus-JH'2013 comparison, none of which reduces by construction to the fitted constants. The TMD gluon densities used are fitted to HERA F2 data, which is external to the psi' data analyzed here. The matching/DCE scheme is inherited from the authors' previous work [41], but that work was validated against chi_c data, which is external to the present psi' fit, so the citation is independent support rather than a self-referential load. The strengthened DCE cut in Sec. 2.2 is a heuristic modeling assumption: its inequality |p_emission_gT| < |k2T| + |k^init_2T| is asserted rather than rigorously derived, and no sensitivity scan is shown. That is a robustness/correctness concern, not circularity, because the cut is not fitted to the psi' data and the NLO amplitudes themselves are calculated from independent off-shell matrix elements. No step exhibits Eq. X = Eq. Y by construction or a fitted parameter renamed as a prediction.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The calculation rests on standard NRQCD and kT-factorization hypotheses, on CCFM-based TMD gluon densities, on the HQSS relations and on the multipole-radiation hadronization model from earlier work. The genuinely new element is the strengthened DCE cut, an ad hoc kinematic assumption that is not rigorously derived. No new particles or forces are invented, but the soft-gluon energy and the primordial gluon momentum are model inputs that carry uncertainty.

free parameters (4)
  • ⟨O_l.c.⟩ = ⟨O[3P0[8]]⟩ + r⟨O[1S0[8]]⟩ (linear combination of color-octet LDMEs) = A0: (6.06±0.07)×10^-3 GeV^5; JH'2013 set 2: (6.17±0.06)×10^-3 GeV^5; JH'2013 (pT>6): (9.95±0.09)×10^-3 GeV^5
    Fitted to all unpolarized LHC psi(2S) data; carries most of the normalization of the color-octet contribution.
  • ⟨O[3S1[8]]⟩ = A0: (10.2±0.2)×10^-4 GeV^3; JH'2013 set 2: (10.7±0.1)×10^-4 GeV^3; JH'2013 (pT>6): (7.46±0.13)×10^-4 GeV^3
    Second free LDME fitted in the global chi2 minimization.
  • Soft-gluon energy E_g in the CO to psi(2S) transition = Lambda_QCD ± 50 MeV (varied for uncertainty bands)
    Model parameter from the multipole-radiation scenario [40], not determined by theory; contributes to the normalization uncertainty.
  • Average primordial gluon transverse momentum |k^init_T| = 1.12 GeV (A0), 1.77 GeV (JH'2013 set 2)
    Extracted from the TMD gluon distribution at the starting scale mu0; sets the strengthened DCE cut.
assumptions (7)
  • domain assumption NRQCD factorization: the psi(2S) cross section is a linear sum of LDMEs times partonic cross sections (Eq. 7)
    Standard NRQCD working hypothesis; without it the LDME extraction is meaningless.
  • domain assumption kT-factorization prescription (Eqs. 4-5): cross sections are convolutions of off-shell amplitudes with TMD gluon densities
    Foundation of the calculation, cited to [14,15].
  • domain assumption CCFM evolution at LLA for the A0 and JH'2013 set 2 TMD gluon densities
    The TMD densities are solutions of CCFM at leading-logarithmic accuracy; NLL corrections are not known [71].
  • ad hoc to paper DCE cut |p_gT| > max(|k1T|, |k2T|+|k^init_2T|) and the estimate |p^emission_gT| < |k2T|+|k^init_2T|
    New merging assumption introduced in Section 2.2; no rigorous proof, only a kinematic estimate.
  • domain assumption Off-shell amplitudes are gauge invariant through effective vertices ([65,66]); amplitudes are taken from [20,41]
    Relies on earlier results; gauge invariance was tested in those papers.
  • domain assumption HQSS relations ⟨O[3PJ[8]]⟩ = (2J+1)⟨O[3P0[8]]⟩ (Eq. 6)
    Reduces the 3PJ parameters to one; standard in NRQCD analyses.
  • domain assumption Multipole E1 transitions with finite-energy soft gluons for CO to psi(2S) evolution
    Non-perturbative hadronization model taken from [40].

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Cite this review

Pith. "Pith review of Tree-level NLO corrections to inclusive $\psi'$ production in High Energy Factorization." pith.science (2026). https://pith.science/paper/UMSYS6BK

@misc{pith2026250909416,
  author       = {Pith},
  title        = {Pith review of: Tree-level NLO corrections to inclusive $\psi'$ production in High Energy Factorization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMSYS6BK}},
  note         = {Machine review of arXiv:2509.09416}
}
abstract

We consider inclusive $\psi(2S)$ production in proton-proton collisions at collider energies in the framework of non-relativistic QCD and high-energy factorization beyond the low-order approximation. We utilise a matching scheme proposed earlier to merge the leading order and tree-level next-to-leading order production amplitudes and now extend it to low transverse momenta and/or forward rapidities. With the improved scheme, we examine the possibility to simultaneously describe all unpolarized LHC data in the full kinematic range accessible to experimental measurements and try different Transverse Momentum Dependent (TMD) gluon distributions in the proton. A global fit to the data is carried out to extract the color octet long-distance matrix elements for $\psi(2S)$ mesons. We show that taking the NLO corrections into account leads to better description of the data.

Figures

Figures reproduced from arXiv: 2509.09416 by the authors.

Figure 1
Figure 1. Comparison of dσ2→2/dpT contributions for color octer cc¯[ 1S [8] 0 ] (left panel) and cc¯[ 3P [8] 2 ] (right panel) with standard (dashed) and enhanced DCE cuts (solid) in the forward rapidity region 2 < y < 4.5 at √ s = 13 TeV. package tmdlib [77], which is a C++ library providing a framework and an interface to many different parametrizations. They are implemented also into the Monte-Carlo event generator pegasus… view at source ↗
Figure 2
Figure 2. Lowest values of χ 2/n.d.f. as function of minimal ψ ′ transverse momenta. LDME A0 JH’2013 set 2 JH’2013 set 2 (pT > 6) LO+NLO† ⟨Ol.c.⟩/GeV5 (6.06 ± 0.07) × 10−3 (6.17 ± 0.06) × 10−3 (9.95 ± 0.09) × 10−3 ⟨O[ 3S [8] 1 ]⟩/GeV3 (10.2 ± 0.2) × 10−4 (10.7 ± 0.1) × 10−4 (7.46 ± 0.13) × 10−4 LO ⟨Ol.c.⟩/GeV5 (10.4 ± 0.1) × 10−3 (8.61 ± 0.09) × 10−3 (18.9 ± 0.2) × 10−3 ⟨O[ 3S [8] 1 ]⟩/GeV3 (20.1 ± 0.3) × 10−4 (29.3 ± 0.2) × … view at source ↗
Figure 3
Figure 3. Ratios of differential cross sections for inclusive [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Ratios of differential cross sections for inclusive [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Ratios of differential cross sections for inclusive [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Ratios of differential cross sections for inclusive [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Ratios of differential cross sections for inclusive [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Ratios of differential cross sections for inclusive [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

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