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A complete expansion of three-body final-state kinematics yields self-consistent O(v²) corrections that suppress J/ψ + open-charm by ~15% and enhance the non-charm channel by ~12%.

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2026-07-31 15:01 UTC pith:PR5HSHVT

load-bearing objection Solid technical fix to three-body O(v²) expansions—scheme independence and fragmentation matching are real—but they drop a same-order phase-space Jacobian without a number, and the headline percentages are internally inconsistent. the 3 major comments →

arxiv 2607.24432 v1 pith:PR5HSHVT submitted 2026-07-27 hep-ph

Revisiting relativistic corrections to inclusive J/psi production at B factories: Complete expansion for three-body quarkonium production

Sheng-Juan Jiang , Sai Cui , Guang-Zhi Xu , Kui-Yong Liu This is my paper
classification hep-ph
keywords NRQCDJ/ψ productionrelativistic correctionsB factoriesthree-body phase spacefragmentation functionscolor-singletO(v²) expansion
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.

Inclusive J/ψ production at B-factory energies has long shown tension between NRQCD theory and Belle measurements. Earlier relativistic corrections for the three-body quark processes omitted the O(v²) dependence of the final-state energy and momentum variables, so the answers depended on how phase space was integrated. This paper expands every final-state kinematic parameter, obtaining cross sections that are independent of the integration variables and that match fragmentation-function results in both magnitude and momentum-line-shape at high energy. The new O(v²) pieces cut the J/ψ + X_c¯c rate by roughly 14.55% and raise the J/ψ + X_non-c¯c rate by about 12%. After all published QCD corrections, color-octet terms, two-photon channels and feed-down are added, the discrepancy with experiment remains, pointing to the need for complete O(α_s, v²) or higher-order prompt calculations.

Core claim

For three-body quark-level subprocesses that produce J/ψ, expanding every final-state kinematic variable—not only the relative momentum q and the quark energy E_q—produces O(v²) relativistic corrections that are theoretically self-consistent and independent of the choice of phase-space integration variables. Numerically these corrections suppress the J/ψ + X_c¯c cross section by ~14.55% and enhance the J/ψ + X_non-c¯c cross section by ~12%, while both the high-energy cross-section ratios and the shapes of the momentum distributions agree with fragmentation-function calculations.

What carries the argument

The first-order expansion coefficients z_i^(1) of the dimensionless final-state energy variables, together with the new term they generate in the squared amplitude; that term restores integration-scheme independence and high-energy consistency with fragmentation.

Load-bearing premise

Outgoing particle angles are treated as independent of the relative quark momentum squared, three-momenta are assumed to scale uniformly, and the O(v²) piece of the three-body phase-space Jacobian is dropped as negligible.

What would settle it

A fixed-order calculation that keeps the phase-space Jacobian and does not force angular independence of q², or a complete O(α_s v²) evaluation of both channels whose result either closes or deepens the remaining gap with Belle’s measured cross sections.

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

If this is right

  • Earlier O(v²) numbers that kept only q and E_q derivatives must be replaced by the full-expansion values (~–14.55% and ~+12%).
  • High-energy O(v²) fragmentation results for S-wave quarkonium are now corroborated by fixed-order three-body calculations.
  • Residual theory–experiment tension shows that incomplete O(α_s, v²) prompt sums are still insufficient.
  • Color-octet matrix elements fitted at hadron colliders remain inconsistent with B-factory data once the new corrections are included.
  • Analogous full kinematic expansions are required for other three-body quarkonium processes such as inclusive h_c production.

Where Pith is reading between the lines

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

  • The same z_i expansion should become standard for any three-body NRQCD process at moderate energies where 1/s suppression is not tiny.
  • If future O(α_s v²) results still undershoot the double-charm channel, production mechanisms beyond ordinary color-singlet plus color-octet may be required.
  • The sign change of the O(v²) correction across the J/ψ momentum spectrum is a differential signature that improved Belle-II spectra could test directly.
  • Scheme independence under the full expansion can serve as a practical diagnostic for incomplete relativistic calculations in other exclusive and inclusive quarkonium channels.

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

3 major / 7 minor

Summary. The manuscript revisits O(v²) relativistic corrections to inclusive J/ψ production in e⁺e⁻ annihilation at B-factory energies, focusing on the three-body quark-level subprocesses e⁺e⁻ → J/ψ + c c̄ and e⁺e⁻ → J/ψ + gg. The central technical claim is that previous calculations were incomplete because they expanded the amplitude only in the relative momentum q and E_q while holding final-state kinematic parameters fixed; the authors derive closed-form expansion coefficients for the dimensionless energy fractions z_i (Eqs. 12–17) and add the corresponding ∂M/∂z_i terms to the amplitude expansion (Eqs. 19–21). They demonstrate numerically (Table I footnote) that with these terms the total O(v²) correction is invariant under the choice of phase-space integrand, whereas the q, E_q-only result is not. They find the full O(v²) correction suppresses σ(J/ψ + X_cc̄) by ~14.55% and enhances σ(J/ψ + X_non-cc̄) by ~12%, show high-energy agreement with fragmentation-function calculations (Fig. 1), and conclude that theory–experiment tension persists after combining O(α_s), color-octet, two-photon, and feed-down contributions.

Significance. If the result holds, it resolves a genuine and long-standing technical defect in the literature: prior O(v²) calculations for three-body quarkonium production gave scheme-dependent answers, and this paper identifies the missing z_i-derivative terms as the cure. The strengths are concrete: closed-form kinematic identities valid to arbitrary order in q² (Eqs. 16–17); a direct numerical demonstration of integrand independence, with the partial corrections (−0.692 vs +1.389 fb; −25.942 vs −28.023 fb) canceling to a scheme-independent total (Table I footnote); a falsifiable cross-check against fragmentation functions in both normalization and line shape (Figs. 1–2); and channel-resolved quantitative predictions relevant to the Belle tension. The downward revision of the X_cc̄ relativistic correction from prior values and the reduction of the X_non-cc̄ enhancement from 20–30% to ~12% would materially change the phenomenological ledger for prompt J/ψ production.

major comments (3)
  1. [§II, Eq. (18) vs. Eqs. (16), (20)–(21)] Eq. 18 and the paragraph following it: the O(q²) phase-space Jacobian term (∂z₃⁽¹⁾/∂z₃⁽⁰⁾ + ∂z₄⁽¹⁾/∂z₄⁽⁰⁾)𝒒² is dropped as 'suppressed by the s term in the denominators of z_i⁽¹⁾, and ... negligible.' This justification is internally inconsistent: by Eq. (16), z_i⁽¹⁾ = 8ba_i/s, so the z_i-derivative amplitude terms retained in Eqs. (20)–(21) carry exactly the same 1/s suppression. The paper's own Table I footnote shows those equally-suppressed terms contribute −25.9 to −28.0 fb — the dominant piece of the total correction — which refutes '1/s suppressed ⇒ negligible' as a power-counting argument for this class of terms. A consistent O(v²) expansion of ∫dΦ₃(q²)|M(q²)|² contains both the measure term and the amplitude terms at the same order. No estimate, bound, or integrated value of the Jacobian contribution is given; at B-factory energies r = 4m_c²/s ≈ 0.08 this is not parametrically ti
  2. [Footnote 3 / Ref. [49]; §III] Footnote 3 cites Ref. [49] (Li, Liu, Huang, Sang), described as giving semi-analytical results for J/ψ + X_non-cc̄ through O(α_s v²) with the 'conventional expansion approach' that 'yield negligible contributions to the O(v²) and O(α_s v²) corrections.' If accurate, an independent calculation finding a negligible LO O(v²) correction for the non-cc̄ channel directly contradicts the ~12% enhancement reported here — one of the paper's two headline numbers. Burying this in a footnote is not adequate. The manuscript needs a dedicated comparison: do the authors reproduce/disagree with Ref. [49]'s LO v² SDC, and if they disagree, where (e.g., the z_i-derivative terms, the treatment of the phase-space measure)? This is a correctness-risk issue for the central quantitative claim, not a consensus issue.
  3. [Abstract; §III first paragraph; §IV; Table I] The two headline percentages are stated inconsistently. The abstract gives −14.55% (X_cc̄) and +12.18% (X_non-cc̄). The first paragraph of §III states the reverse assignment ('increases the LO cross section by 14.6%' for X_non-cc̄ and 'suppresses the LO yield by 12.2%' for X_cc̄). §IV gives 14.55% and 12.54%. Meanwhile Table I implies 0.027/0.183 ≈ 14.8% and 0.018/0.121 ≈ 14.9% for X_cc̄, and 0.038/0.329 ≈ 11.6% and 0.025/0.218 ≈ 11.5% for X_non-cc̄ — none of which equals 14.55%, 12.18%, or 12.54% exactly. For a paper whose deliverable is precision percentages, the channel assignment in §III must be corrected and every quoted number reconciled with Table I (specifying μ_r and rounding).
minor comments (7)
  1. [§II, paragraph preceding Eq. (8)] The statement that the final-state scattering angles 'are Lorentz scalars, which ensures manifest covariance order-by-order' is misleading: CM-frame angles are frame-defined quantities, not Lorentz invariants. The expansion procedure (fixed angles, uniformly scaled three-momenta) is standard and defensible on its own; please reword rather than claim manifest covariance.
  2. [§III, Eq. (23)] The value ⟨v²⟩_J/ψ = 0.23 is introduced without a citation or provenance (potential-model estimate? which one?). Please cite the source, and likewise state the source for ⟨v²⟩_χcJ = 0.23 used for Eq. (25).
  3. [Table I footnote; Fig. 1 caption] Table I footnote: 'amount to −0.692 fb for the J/ψ + X_cc̄ channels, respectively' — 'respectively' with a single channel; also state which μ_r the fb-level values correspond to. In the caption of Fig. 1, 'NLOa' is defined via a superscript footnote on the figure, which is easy to miss; define it in the caption text.
  4. [§III, Table I footnote] The scheme-independence claim rests on a numerical cancellation between two specific integrands (Eq. 8 vs. Refs. [9,13]). An analytic argument (e.g., that the z_i-derivative terms are precisely the chain-rule completion making the expanded integral a total derivative under reparametrization) would substantially strengthen §III and would also clarify whether a retained Jacobian term preserves the invariance.
  5. [Eqs. (20)–(21); various] Notation: m_Q appears in Eqs. (20)–(21) while m_c is used everywhere else; Π^{αβ} = −g^{αβ} + p^αp^β/p² uses p where P (the meson momentum) is presumably meant. 'form the derivative terms' → 'from the derivative terms' (Table I footnote); 'both at both LO and NLO' (§III); 'One may tune the values of m_c and the renormalization scale can bring...' (§III) needs grammatical repair.
  6. [Eq. (25); §IV] The χ_cJ cross sections in Eq. (25) appear with minimal setup (inputs only 'same as Ref. [10]'); please state the P-wave LDMEs used and whether the suppression factors 0.661/0.542/0.697 carry any scale dependence. Also, Ref. [48] is cited as 'in preparation' while being invoked in §IV as supporting evidence for the importance of complete expansions in P-wave production — soften this claim or wait for a citable version.
  7. [Fig. 2; §III] Fig. 2 would benefit from stating the μ_r choice and from indicating the fragmentation-calculation curve for the non-cc̄ channel if available, parallel to what is shown for the cc̄ channel; currently the 'twofold validation' statement in §III references fragmentation only for the cc̄ line shape.

Circularity Check

0 steps flagged

No significant circularity: O(v²) kinematic expansion is derived from Lorentz constraints and NRQCD matching, with external LDMEs and independent fragmentation cross-checks.

full rationale

The paper’s central result—the complete O(v²) expansion of three-body final-state kinematics z_i (Eqs. 12–21) and the resulting scheme-independent cross-section corrections—is obtained from on-shell and three-momentum conservation identities plus the standard NRQCD amplitude expansion; nothing in that chain is defined in terms of the output percentages or the Belle data. LDMEs (|R_S(0)|², ⟨v²⟩, color-octet values) are taken from external literature and used as fixed inputs. High-energy agreement with fragmentation-function calculations is an a-posteriori consistency check against an independent two-body method, not a fit or a definitional identity. Self-citations (e.g., prior relativistic-correction factors for color-octet channels) supply conventional numerical inputs and are not load-bearing uniqueness theorems for the new z_i terms. The residual theory–experiment tension is reported rather than absorbed by retuning. No step reduces a claimed prediction to its own inputs by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The calculation rests on standard NRQCD factorization, color-singlet projectors, and externally fixed LDMEs and quark-mass parameters. No new particles or forces are introduced. The only paper-specific modeling choices are the uniform three-momentum scaling ansatz, the neglect of the O(q²) phase-space Jacobian, and the numerical values assigned to m_c, ⟨v²⟩ and the color-octet matrix elements taken from global fits.

free parameters (5)
  • m_c = 1.5 GeV (default)
    Charm-quark pole mass fixed by hand to 1.5 GeV (and varied to 1.4 GeV in figures); directly controls phase space and short-distance coefficients.
  • ⟨v²⟩_J/ψ = 0.23
    Ratio of O(v²) to LO LDMEs; multiplies every relativistic correction. Taken as 0.23 from prior phenomenology, not derived here.
  • |R_S(0)|² = 1.01 GeV³
    Radial wave function at the origin that sets the LO color-singlet LDME; taken from earlier B-factory analyses.
  • color-octet LDMEs = 3.04e-2, 1.68e-3, -9.08e-3 GeV^(3/5); 0.215e-2 GeV³
    ⟨O(¹S₀⁸)⟩, ⟨O(³S₁⁸)⟩, ⟨O(³P₀⁸)⟩ from Butenschoen–Kniehl global NLO fit and χ_c0 matrix element from Ma–Wang–Chao; used to build the prompt totals that are compared with Belle.
  • renormalization scale μ_r = 2m_c or √s/2
    Chosen as 2m_c or √s/2; α_s and the size of O(α_s) pieces depend on it, affecting the final theory–data comparison.
axioms (5)
  • domain assumption NRQCD factorization of the inclusive J/ψ cross section into SDCs × LDMEs through O(v²) (Eq. 3).
    Standard Bodwin–Braaten–Lepage framework assumed valid at B-factory energies; not re-derived.
  • ad hoc to paper Final-state angular directions are independent of q²; three-momenta scale uniformly (Eq. 12).
    Key modeling step that allows analytic z_i^(1) coefficients; not forced by Lorentz invariance alone once three-body phase space is considered.
  • ad hoc to paper O(q²) contribution to the three-body phase-space measure is negligible and may be dropped (after Eq. 18).
    Justified only by 1/s suppression; never quantified numerically at √s = 10.58 GeV.
  • domain assumption Color-singlet Lorentz-covariant spin/color projectors with standard Dirac normalization (Eq. 6).
    Textbook NRQCD projection technique used throughout the amplitude expansion.
  • domain assumption Fragmentation-function relativistic corrections for ³S₁ states equal −(11/6)⟨v²⟩, etc., taken from high-energy literature.
    Used to build the prompt NLO(α_s,v²) totals for color-octet and feed-down pieces without recomputing them in the three-body formalism.

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read the original abstract

We revisit the calculations of relativistic corrections to inclusive $J/\psi$ production at B factories. For quark-level subprocesses with three-body final states, we carry out a full expansion of the final-state kinematic parameters. The resulting cross sections are theoretically self-consistent and independent of the choice of integration variables. In the high-energy limit, both cross-section magnitudes and the line shapes of correction curves for energy and momentum distributions agree remarkably well with fragmentation-function calculations. We find that $\mathcal{O}(v^2)$ corrections suppress the cross section by roughly $14.55\%$ in the $J/\psi + X_{c\bar{c}}$ channel and enhance it by approximately $12.18\%$ for the $J/\psi + X_{\text{non-}c\bar{c}}$ channel. After incorporating published $\mathcal{O}(\alpha_s)$ corrections, color-octet contributions, two-photon production channels, and feed-down effects, tension persists between theoretical predictions and experimental measurements. This indicates that complete prompt-quarkonium calculations through $\mathcal{O}(\alpha_s, v^2)$, or evaluations incorporating higher-order corrections, are required to resolve this tension.

Figures

Figures reproduced from arXiv: 2607.24432 by Guang-Zhi Xu, Kui-Yong Liu, Sai Cui, Sheng-Juan Jiang.

Figure 1
Figure 1. Figure 1: Ratios of the cross section 𝜎(𝑒 + 𝑒 − → 𝐽/𝜓 + 𝑋𝑐𝑐¯) to 𝜎(𝑒 + 𝑒 − → 𝑐𝑐¯) as a function of the CM energy 𝐸𝑐𝑚. ”Frag LO” and ”Frag NLO” denote the leading-order and next-to-leading-order results in the fragmentation approximation, while ”Full LO” and ”Full NLO” denote the corresponding results obtained from the full fixed-order calculation. a The results retaining only the contributions form the derivative te… view at source ↗
Figure 2
Figure 2. Figure 2: the differential cross sections at LO and NLO(𝑣 2 ) as functions of the magnitude of the 𝐽/𝜓 three-momentum 𝑃𝜓. The red short-dashed line labeled RC stands for the result of the O (𝑣 2 ) relativistic corrections. The left and right panels correspond to the processes 𝑒 + 𝑒 − → 𝐽/𝜓 + 𝑋𝑐𝑐¯ and 𝑒 + 𝑒 − → 𝐽/𝜓 + 𝑋non-𝑐𝑐¯ , respectively. 𝐽/𝜓 + 𝑋non-𝑐𝑐¯ channels once O (𝑣 2 ) corrections are included. When contras… view at source ↗
Figure 3
Figure 3. Figure 3: Prompt cross sections of the processes 𝐽/𝜓 + 𝑋𝑐𝑐¯ + 𝑋 and 𝐽/𝜓 + 𝑋non-𝑐𝑐¯ at NLO(𝛼𝑠) and NLO(𝛼𝑠, 𝑣2 ) compared with the Belle measurements. The left and right panels correspond to 𝑚𝑐 = 1.4 GeV and 𝑚𝑐 = 1.5 GeV, respectively. the total NLO(𝛼𝑠) prompt cross sections are expressed as 𝜎 NLO(𝛼𝑠 ) prompt (𝐽/𝜓 + 𝑋𝑐𝑐¯ + 𝑋) = 1.355 𝜎 NLO(𝛼𝑠 ) direct (𝐽/𝜓 + 𝑋𝑐𝑐¯) + 49 fb, 𝜎 NLO(𝛼𝑠 ) prompt (𝐽/𝜓 + 𝑋non−𝑐𝑐¯) = 1.355𝜎 N… view at source ↗
Figure 4
Figure 4. Figure 4: Prompt cross sections of the processes 𝐽/𝜓 + 𝑋𝑐𝑐¯ + 𝑋 and 𝐽/𝜓 + 𝑋non-𝑐𝑐¯ at NLO(𝛼𝑠) and NLO(𝛼𝑠, 𝑣2 ) compared with the Belle measurements. Compared with [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

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Forward citations

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Reference graph

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