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The impact of the TMD shape function on matching the transverse momentum spectrum in $J/\psi$ production at the EIC

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Including the process-dependent Bep term in the J/ψ TMD shape function preserves positivity and boundedness of matched predictions at the EIC.

desk verdict Plausible but provisional: the z≈1 approximation is weakest exactly where the Bep term matters, so the headline claim needs a subleading-correction study or a provisional label. read the letter →

arxiv 2504.19617 v2 pith:6RYABHBH submitted 2025-04-28 hep-ph

Luca Maxia , Daniël Boer , Jelle Bor This is my paper · ORCID
classification hep-ph
keywords TMDfactorizationJ/psiproductionsemi-inclusivedeepinelasticscatteringshapefunctionSudakovresummationinverse-errorweightinggluondistributionsElectron-IonCollider
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

The paper asks whether the transverse-momentum-dependent (TMD) shape function that describes J/ψ hadronization needs a process-dependent term in its perturbative Sudakov coefficient. Using the inverse-error weighting method to join the TMD description at low transverse momentum qT with the collinear description at high qT, the authors produce matched predictions for the J/ψ cross section and its cos(2φψ) asymmetry in electron–proton collisions at the future Electron-Ion Collider. They find that when the color-octet term Bψ is included alone (BCO = 1, Bep = 0), the matched cross section becomes negative in the region ΛQCD ≪ qT ≲ μ/2 and the asymmetry band can exceed unity. Adding the process-dependent term Bep from Ref. [1] removes both violations. If right, this means future EIC data can distinguish between competing definitions of the TMD shape function and thereby test TMD factorization in quarkonium production, which matters for extracting gluon TMD distributions.

What carries the argument

The load-bearing object is the TMD shape function Δ(z, kT²), which incorporates TMD effects in the quarkonium formation mechanism; its perturbative matching to NRQCD long-distance matrix elements feeds the Sudakov exponent through the coefficient BCO = Bψ + Bep in the W-term of the differential cross section. The matching of the low-qT TMD expression to the high-qT fixed-order result is performed with the inverse-error weighting (InEW) method [36], which averages the two descriptions with weights built from their estimated errors; this is what allows the cross section and asymmetry to be evaluated over the full qT range. For the cos(2φψ) asymmetry, the numerator involves the linearly polarized gluon TMD distribution $h_1^{{⊥g}}$, whose perturbative expansion is taken from Ref. [20]. The choice of BCO changes the shape and height of the matched curves and is what drives the positivity and boundedness violations that the paper reports.

What would settle it

Measure the J/ψ qT spectrum and cos(2φψ) asymmetry at the EIC at Q ≈ 14–25 GeV and compare with the two prescriptions: if the data follow the BCO = 1 (Bep = 0) band with a positive cross section and |⟨cos(2φψ)⟩| ≤ 1 everywhere, the claim that Bep is required is refuted. A second decisive test is to compute the order-qT²/Mψ² corrections to the z-integrated cross section and show whether they alter the sign of the difference between the two prescriptions in the region ΛQCD ≪ qT ≲ μ/2.

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Extended reading notes

Core claim

The central claim is that the physical constraints of positivity of the matched cross section and |⟨cos(2φψ)⟩| ≤ 1 act as a diagnostic for the correct perturbative tail of the TMD shape function in J/ψ production. In the TMD region the color-octet contribution to the Sudakov exponent is BCO = Bψ + Bep, where Bψ = -CA/2 is the universal term and Bep = (CA/2) log((Mψ²+Q²)/Mψ²) is the process-dependent term derived in Ref. [1]. Evaluating the InEW-matched cross section at √s = 140 GeV with Q = 14 and 25 GeV, the authors find that the choice BCO = 1 (i.e. Bep = 0, as in Ref. [29]) produces negative cross sections for ΛQCD ≪ qT ≲ μ/2 and matched asymmetry bands that overshoot unity, whereas the choice with Bep ≠ 0 satisfies both constraints. The paper therefore concludes that the numerical results support the analytic derivation of Bep, and that the presence or absence of this term is testable by EIC data.

Load-bearing premise

The entire comparison relies on the cross section being dominated by z close to 1 over the whole qT range, so the TMD shape function can be evaluated at z = 1 and corrections of order qT²/Mψ² can be dropped; if those corrections matter in the intermediate-qT region, the relative behavior of the BCO = 1 and Bep ≠ 0 predictions could change.

Editorial extensions

If this is right

  • At Q ≈ 14–25 GeV the matched cross section differs substantially between the Bep = 0 and Bep ≠ 0 prescriptions, so high-Q EIC data can discriminate between them even with current LDME uncertainties.
  • The cos(2φψ) asymmetry develops a node at intermediate qT for a negative sign of the linearly polarized gluon TMD, while no node appears for a positive sign; measuring the node would settle the sign.
  • The BCO = 1 (Bep = 0) prediction fails basic physical constraints in the intermediate-qT region, so fits to future data should not use that prescription for the TMD shape function's perturbative tail.
  • If Bep is confirmed, TMD factorization for J/ψ production must include process-dependent soft-gluon effects, extending the usual universal-TMD picture.
  • The matching method itself, with its error-based weights, provides a practical route to predict full-qT spectra for quarkonia without relying on the full z-dependence of the TMD shape function.

Reading between the lines

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

  • If the z ≈ 1 dominance assumption fails at intermediate qT, the size of Bep's effect could be different, but the qualitative mechanism — a process-dependent single logarithm in the Sudakov factor — would likely survive in a full z-dependent treatment.
  • The same positivity-and-boundedness diagnostic could be applied to other quarkonium states such as Υ or ψ′ in SIDIS and photoproduction, where the analogue of Bep depends on the mass ratio and could be predicted before data arrive.
  • The double-peak structure seen in the matched cross section at high Q is a matching artifact candidate; it may serve as a sensitive probe of the TMD-to-collinear transition and could be used to tune the InEW weights once data exist.
  • The paper's method suggests that checking whether the TMD W-term turns negative in the intermediate-qT region is a cheap, model-independent test that any future TMD-ShF parametrization should pass.
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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

2 major / 5 minor

Summary. This paper studies J/ψ production in semi-inclusive deep inelastic scattering at the EIC, matching TMD and collinear factorizations with the inverse-error weighting (InEW) method. The authors compute the qT-dependent cross section and the cos(2φ) asymmetry, comparing three choices of the color-octet coefficient in the TMD shape function: no TMD-ShF (BCO=0), the universal term only (BCO=1, Bep=0), and the process-dependent term of Ref. [1] (Bep≠0). They find that for high Q the BCO=1 case leads to a negative TMD cross section and to an asymmetry band overshooting unity, while the Bep≠0 case does not, and interpret this as numerical support for the process-dependent term. They also propose the presence or absence of a node in the cos(2φ) asymmetry as a way to determine the sign of the linearly polarized gluon TMD.

Significance. If the central comparison is robust, the paper would provide one of the first numerical indications for a process-dependent soft factor in quarkonium TMD factorization, and the node prediction offers a falsifiable experimental test for the sign of h⊥g1. The analysis is systematic: it uses two LDME sets, several Q values, and explicit positivity and bound checks, and it displays the unphysical outputs rather than hiding them. However, the central claim is conditional: the discriminating region in qT is exactly where the z≈1 approximation is expected to be poor, and the matching error was modified in a way inspired by the term under test. The significance therefore rests on a point that the authors themselves acknowledge needs further study.

major comments (2)
  1. [Sec. III, paragraph after Eq. (32); footnote 7; Sec. V] The z≈1 approximation is used to evaluate the TMD-ShF and Sudakov coefficients at z=1 for all qT, with corrections of order qT^2/Mψ^2 neglected. The unphysical features used to discriminate BCO=1 from Bep≠0 — negative TMD cross section and asymmetry overshoot — occur for ΛQCD ≪ qT ≲ μ/2 (Sec. III and Fig. 5), where for Q=14 and 25 GeV the ratio qT^2/Mψ^2 ranges from roughly 5 to 16, so the neglected corrections are not small. The Bep term enters as a Q-dependent coefficient in the Sudakov exponent; if z<1 contributions carry different relative weight for the two BCO choices, the comparison is biased. The authors acknowledge in footnote 7 that a definitive exclusion of BCO=1 requires a study of subleading z≈1 terms, but the abstract and Sec. V state the conclusion without this qualification. The manuscript should either quantify the uncertainty from the z<1 contributions (for example, by using a model for the full z-dependence of the TMD-ShF or by limiting the claim to qT ≪ Mψ) or soften the central claim accordingly.
  2. [Sec. II, Eq. (30)] The modification of the InEW error term ΔFO is introduced as 'inspired by the divergent behavior found in Ref. [1]' — that is, by the very Bep term whose inclusion the paper aims to support. Since the InEW weights in Eq. (29) are constructed from these errors evaluated with the BCO-dependent cross sections, the numerical support for Bep is in part a self-consistency check. The comparison with BCO=0 and BCO=1 provides some independent grounding, but the robustness of the conclusion should be tested with an alternative matching prescription (e.g., the conventional W+Y formalism) or by a sensitivity scan over the functional form of ΔFO, including the original squared-log version of Ref. [36].
minor comments (5)
  1. [Fig. 4 caption] In the caption of Fig. 4, the sentence 'Figs. 3a and 3b: Q = 3 GeV...' should refer to panels (a) and (b) of Fig. 4, not Fig. 3.
  2. [Eq. (15)] In Eq. (15), the expansion of Cnn′ is written with C(k) aa′(z) in the second line; the subscript should be nn′.
  3. [Fig. 6 caption] In the caption of Fig. 6, 'SV12 is used for the lower ones' should read 'SV13'.
  4. [Sec. V, last paragraph] The last paragraph of Sec. V states that the presence (absence) of a node is related to the positive (negative) sign of h⊥g1; this is the reverse of what is shown in Figs. 6 and 7 and stated in Sec. IV.
  5. [Sec. III] The sentence in Sec. III about negative cross sections 'also observed for other sets, e.g., those reported in [61]' is vague; please specify which sets or remove the clause.

Circularity Check

1 steps flagged · score 4.0 of 10

The numerical 'support' for Bep is partly self-consistency: the InEW matching error was modified with a hard-scale logarithm taken from the same authors' earlier result that the paper claims to confirm.

  1. ansatz smuggled in via citation [Sec. II, Eq. (30) and the paragraph following it (matching weights)]
    "However, in contrast to Ref. [36] and inspired by the divergent behavior found in Ref. [1], we modified this term by including a logarithm of hard scales, which becomes relevant for high-Q values, and removed the square from the last term (log sqrt(Mψ^2+Q^2)/Mψ + log sqrt(μ^2+qT^2)/qT)."

    The InEW weights ω1 and ω2 are constructed from ΔFO, and ΔFO was modified by adding the hard-scale logarithm log(sqrt(Mψ^2+Q^2)/Mψ). Up to the color factor, this is exactly the process-dependent Bep term of Ref. [1], Bep = (CA/2) log((Mψ^2+Q^2)/Mψ^2), the same self-cited result that the paper later claims its physical constraints support. The matched curves used to discriminate BCO=1 from Bep≠0 therefore contain a matching bias toward Bep-like behavior, so the conclusion that physical constraints require Bep is in part a self-consistency check of an ansatz imported from the authors' own earlier paper rather than an independent test.

full rationale

The rest of the calculation is self-contained and not circular: no parameter is fitted to the target observables; Bep is an analytic input from Ref. [1]; and the BCO=0 and BCO=1 alternatives provide independent comparisons. The TMD-level negativity for BCO=1 arises in the TMD expression itself, before matching, and is independent of the matching-error modification, so the central claim has genuine non-circular content. The score of 4 reflects the one load-bearing contamination: the InEW error was deliberately modified 'inspired by' Ref. [1], biasing the matched comparison toward the Bep case. The paper's own footnote 7 concedes that a definitive exclusion of BCO=1 requires a careful study of the subleading z≈1 terms; that is a robustness/correctness limitation, not an additional circularity, but it underscores that the 'physical constraints select Bep' conclusion is conditional.

Assumptions & free parameters 7 free parameters · 8 assumptions · 0 invented entities

The calculation rests on external fitted inputs (LDMEs, nonperturbative Sudakov parameters, PDFs) and on a set of modeling choices (z close to 1, gpsi=0, Bep form, same Sudakov for asymmetry numerator and denominator). No fundamentally new entity is introduced in this paper; Bep and the TMD-ShF are carried over from prior work. The counts below reflect the load-bearing external and ad hoc inputs.

free parameters (7)
  • Nonperturbative Sudakov coefficient ANP = central 0.414 GeV^2, range [0.05, 0.8] GeV^2
    Appears in Eq. (25) for SNP; varied to build error bands; originates from prior quark TMD fits, quoted from Refs. [52,53].
  • Aybat-Rogers parameters g1, g2, g3, xc = g1=0.201 GeV^2, g2=0.184 GeV^2, g3=-0.129 GeV^2, xc=0.009
    Used in BNP in Eq. (25) to fix the nonperturbative Sudakov for the proton TMD; values from Ref. [51].
  • gpsi (TMD-ShF nonperturbative parameter) = 0 (assumed)
    Unknown constant for the J/psi shape function; set to zero by assumption in Sec. II, affecting peak position and matching.
  • bmax = 1.5 GeV^-1
    Cutoff for bT in b* prescription, Eq. (22); chosen value affects the transition from perturbative to nonperturbative regime.
  • InEW matching parameters n and m = n=2, m=1 GeV
    Chosen in Sec. II to make TMD contribution negligible for qT >> mu; error weights and Delta_sigma depend on them.
  • Matching error threshold Delta_T = 15%
    Defines acceptable overlap region (Delta_sigma/sigma < 10.6%) in Fig. 2; used to present results.
  • Long-distance matrix elements (CM12 and SV13) = CM12 and SV13, values in Table I
    External extractions from Refs. [58] and [59]; their uncertainties drive the bands and the negative-cross-section issue.
assumptions (8)
  • domain assumption TMD factorization at low qT and collinear factorization at high qT for J/psi SIDIS, matched via W+Y/CSS or InEW
    Framework of the paper, Eq. (5) and Eqs. (26)-(28).
  • domain assumption NRQCD factorization with Fock state truncation at order v^4, retaining CS 3S1^(1) and CO 1S0^(8), 3PJ^(8)
    Sec. II, after Eq. (5); CO channels are essential to the Bep claim.
  • ad hoc to paper z close to 1 dominance for all qT, with TMD-ShF evaluated at z=1 and corrections O(qT^2/Mpsi^2) neglected
    Sec. III, after Eq. (32); acknowledged as an approximation that restricts the conclusion at intermediate qT (footnote 7).
  • domain assumption LDMEs evolve approximately diagonally, so <O[n']>(mu_b) is approximately <O[n']>(mu_psi)
    Comment 1 after Eq. (8); off-diagonal terms negligible for J/psi.
  • ad hoc to paper gpsi is constant and set to zero; no process-dependent nonperturbative Sudakov for the TMD-ShF
    Eq. (25) and footnote 3; explicitly flagged as an assumption that may be false.
  • ad hoc to paper The nonperturbative Sudakov for the numerator of the cos(2phi) asymmetry equals that of the denominator
    Sec. IV, paragraph on the asymmetry Sudakov; stated as an assumption to be improved.
  • ad hoc to paper Bep is given by Bep^(1)=C_A/2 log((Mpsi^2+Q^2)/Mpsi^2), obtained by setting the auxiliary scale mu_psi=Mpsi in the separation of logarithms
    Eq. (14) and Appendix A; the choice mu_psi=Mpsi removes scale dependence and defines Bep; central to the conclusion.
  • domain assumption Perturbative expansion of h_perp^g from Ref. [20] describes the linearly polarized gluon TMD
    Sec. IV, footnote 6; the sign is treated as an input.

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Pith. "Pith review of The impact of the TMD shape function on matching the transverse momentum spectrum in $J/\psi$ production at the EIC." pith.science (2026). https://pith.science/paper/6RYABHBH

@misc{pith2026250419617,
  author       = {Pith},
  title        = {Pith review of: The impact of the TMD shape function on matching the transverse momentum spectrum in $J/\psi$ production at the EIC},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6RYABHBH}},
  note         = {Machine review of arXiv:2504.19617}
}
abstract

The impact of the inclusion of TMD shape functions on the transverse momentum spectrum in $J/\psi$ production at the EIC is investigated by considering the matching of the TMD factorization description at low transverse momentum with the collinear factorization description at high transverse momentum by means of the inverse-error weighting method. Despite large uncertainties from scale variations and the $J/\psi$ long-distance matrix elements, predictions for the differential cross section and its $\cos(2\phi_\psi)$ modulation are obtained. We find that physical constraints are satisfied in case a process-dependent term is included for color octet production, but not in all cases when it is excluded. These numerical results support the analytic calculations in \cite{Boer:2023zit}. Future experimental data can thus test the validity of TMD factorization in $J/\psi$ production and explore the presence of nontrivial process-dependent effects in the soft-gluon resummation. This will be crucial in the extraction of gluon unpolarized and linearly polarized TMD distributions of the proton. In addition, we suggest how the sign of the latter can be determined by investigating the presence of a node in the $\cos(2\phi_\psi)$ modulation as a function of transverse momentum.

Figures

Figures reproduced from arXiv: 2504.19617 by the authors.

Figure 1
Figure 1. General representation of the SIDIS process. The gray circle denotes the hard process, the green the proton correlator, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Differential cross section with respect to qT for √ s = 140 GeV. Figs. 3a and 3b: Q = 3 GeV and xB = 10−3 for the CM12 and SV13 LDME sets, respectively; Figs. 3c and 3d: Q = 25 GeV and xB = 0.035 for the same sets. The hard scale is set as µ = q M2 ψ + Q2. The blue band corresponds to the FO result (valid at high qT ), the green bands to the TMD one (valid at low qT ), and the red bands to the matched one (valid for… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Differential cross section with respect to qT at the lowest nontrivial αs order for √ s = 45 GeV. Figs. 3a and 3b: Q = 3 GeV and xB = 10−2 for the CM12 and SV13 LDME sets, respectively; Figs. 3c and 3d: Q = 14 GeV and xB = 0.115 for the same sets. The hard scale is set…
Figure 5
Figure 5. Figure 5: Dependence of the matched differential cross section for different BCO = Bψ + Bep choices, with BCO = 1 from Ref. [29] and Bep ̸= 0 from Ref. [1]. Upper panels: √ s = 140 GeV for Q = 25 GeV and xB = 0.035 using CM12 (5a) and SV13 (5b). Lower panels: √ s = 45 GeV for Q …
Figure 6
Figure 6. Figure 6: Dependence of the matched ⟨cos(2ϕψ)⟩ asymmetry on qT , at √ s = 140 GeV and fixed kinematics: (a) and (d) Q = 3 GeV and xB = 10−3 ; (b) and (e) Q = 14 GeV and xB = 0.115 ; (c) and (f ) Q = 20 GeV and xB = 0.035. Upper panels are evaluated using CM12, whereas SV12 is us…
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Dependence of the ⟨cos(2ϕψ)⟩ asymmetry on the BCO choice, with BCO = 1 from Ref. [29] and Bep ̸= 0 from Ref. [1]. Asymmetries are shown with respect to qT at fixed kinematics: (a) and (c) Q = 14 GeV and xB = 0.115 ; (b) and (d) Q = 20 GeV and xB = 0.035. Upper panels a…

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

  1. The role of the soft scale for $J/\psi$ production in the transverse momentum dependent framework

    hep-ph 2025-09 conditional novelty 6.0 of 10

    The paper derives new TMD soft transition functions and shows they dominate J/psi production at small transverse momentum by a factor of 1/v over previously used shape functions.

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Reviewed August 16, 2026 · model on record in the stance chip above.