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Double heavy quarkonia production with color-octet channels at Z factory and at the CEPC/FCC-ee

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper shows that color-octet channels can dominate exclusive double heavy quarkonium production at Z-factory energies, making future electron-positron colliders direct probes of the color-octet mechanism.

desk verdict A competent NRQCD calculation of color-octet double quarkonium production at the Z pole, but the central CO-dominance claim lacks a robustness test against LDME uncertainty and the event numbers are reported inconsistently. read the letter →

arxiv 2501.15575 v1 pith:KJC65MS2 submitted 2025-01-26 hep-ph

Xiao-Peng Wang , Guang-Zhi Xu , Kui-Yong Liu This is my paper · ORCID
classification hep-ph
keywords doubleheavyquarkoniumcolor-octetmechanismNRQCDZfactoryCEPCFCC-eerelativisticcorrectionsgluonfragmentation
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 color-octet mechanism of nonrelativistic QCD (NRQCD) shows up in exclusive double heavy quarkonium production at future Z factories. It computes cross sections for double charmonium and double bottomonium through $\gamma^*/Z^*$ exchange, adding color-octet channels to the usual color-singlet ones. The central finding is that the octet channels are substantial or dominant for many channels at Z-pole energies, with gluon fragmentation into intermediate $^3S_1^{[8]}$ states the largest contributor. If this holds, Z-factory measurements would directly test the color-octet mechanism and constrain the corresponding long-distance matrix elements, especially $\langle\mathcal{O}(^3S_1^{[8]})\rangle$. The paper also evaluates relativistic corrections and gives event-rate estimates for CEPC and FCC-ee, concluding that several channels are experimentally accessible.

What carries the argument

The machinery is NRQCD factorization: each cross section is written as a sum over $Q\bar Q$ intermediate states of short-distance coefficients times long-distance matrix elements (LDMEs), the non-perturbative quantities that describe hadronization of the pair into a quarkonium state. Color-octet channels are included at tree level for both QCD and electroweak diagrams, and the key dynamical feature is gluon fragmentation into $^3S_1^{[8]}$ pairs. Relativistic corrections are implemented by expanding amplitudes to $O(v^2)$, with $\langle v^2\rangle$ fixed through the Gremm-Kapustin relation.

What would settle it

Measure $e^+e^- \to J/\psi+\eta_c$ at $\sqrt{s}=m_Z$ with 16 ab$^{-1}$ (CEPC) or 150 ab$^{-1}$ (FCC-ee). The color-singlet-only prediction is about 10 or 93 events after $v^2$ and $\alpha_s$ corrections, while the full color-singlet plus octet prediction is 22 or 206 events. A measured rate consistent with the lower number would rule out the adopted octet matrix elements; a rate near the higher number would support them.

Watch

Extended reading notes

Core claim

The paper's claim is that, at $\sqrt{s}\simeq m_Z$, the exclusive reactions $e^+e^- \to H_1+H_2$ for double charmonium and double bottomonium receive color-octet contributions that are comparable to or larger than the color-singlet ones for most final states; for $\eta_c+\eta_c$ and $\eta_b+\eta_b$ the color-singlet contribution vanishes in the $\gamma^*/Z^*$-propagated channels, making those processes pure octet probes. Gluon fragmentation into $^3S_1^{[8]}$ intermediate states is the dominant octet mechanism. Relativistic corrections to order $v^2$ suppress the charmonium cross sections by roughly a factor of 0.5 and bottomonium by 0.7-0.8. With NLO QCD $K$ factors and the planned luminosities, the paper predicts, for example, 22 and 206 events for $J/\psi+\eta_c$ at CEPC (16 ab$^{-1}$) and FCC-ee (150 ab$^{-1}$), respectively, with larger numbers for $J/\psi+J/\psi$.

Load-bearing premise

The claimed octet dominance rests on the adopted color-octet long-distance matrix elements, especially $\langle\mathcal{O}^{J/\psi}(^3S_1^{[8]})\rangle = (0.0013 \pm 0.0013)$ GeV$^3$ and the corresponding bottomonium values from a photoproduction fit; if those matrix elements are smaller or negative, the octet contributions shrink and the dominance could disappear.

Editorial extensions

If this is right

  • At the Z pole, several channels such as $J/\psi+\chi_{c1}$ and $\eta_c+\eta_c$ become color-octet dominated, so measuring them provides a direct test of the color-octet mechanism.
  • Comparing predicted and measured rates, especially for $J/\psi+\eta_c$, would give a direct constraint on $\langle\mathcal{O}(^3S_1^{[8]})\rangle$ for charmonium and on the corresponding bottomonium matrix element.
  • Relativistic corrections must be included in such comparisons: for charmonium they reduce leading-order rates by about half, so omitting them would bias extracted matrix elements.
  • The estimated event counts, such as 22 and 206 $J/\psi+\eta_c$ events at CEPC and FCC-ee, indicate these measurements are feasible with the planned integrated luminosities.
  • Double-$J/\psi$ production is almost purely color-singlet, so it can serve as a normalization channel when extracting octet matrix elements from other final states.

Reading between the lines

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

  • Because the adopted octet LDMEs carry large uncertainties and come from a photoproduction fit, the absolute event counts are more fragile than the qualitative conclusion; ratios such as $\sigma(J/\psi+\eta_c)/\sigma(J/\psi+J/\psi)$ would cancel most of the parameter sensitivity.
  • The paper's differential cross sections show different angular shapes for CS and CO channels, so a $\cos\theta$ or $p_t$ cut could isolate the octet component experimentally.
  • The same gluon-fragmentation enhancement should appear in inclusive $Z\to$ quarkonium plus light-hadron decays, allowing a cross-check of whether the extracted $\langle\mathcal{O}(^3S_1^{[8]})\rangle$ is universal.
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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 / 4 minor

Summary. This paper studies exclusive production of double heavy quarkonia (double charmonia and double bottomonia) in e+e- annihilation at the Z pole and at future CEPC/FCC-ee, working in the NRQCD factorization framework. The authors include color-singlet and color-octet channels at tree level, add relativistic O(v^2) corrections, and adopt NLO alpha_s K factors for the CS channels from the literature. Their central claim is that color-octet contributions, especially those mediated by gluon fragmentation into the ^3S1[8] state, are significant or dominant for many double quarkonium channels at Z-factory energies, and they provide event-rate estimates for CEPC and FCC-ee. The paper also presents cross sections as functions of center-of-mass energy, CO/CS ratios, differential distributions, and several uncertainty studies.

Significance. If the central claim is robust, the paper identifies new channels in which future Z-factory measurements could discriminate between color-singlet and color-octet mechanisms and could constrain the ^3S1[8] LDMEs. The work is useful in scope: it covers a broad set of double charmonium and bottomonium final states, includes relativistic corrections, and compares the CS part of the calculation with existing results in Refs. [12, 36, 40, 43, 74]. The main weakness is that the headline conclusion of CO dominance depends linearly on CO LDMEs whose uncertainties are not propagated; Section IV.C explicitly declines to discuss LDME uncertainties. The reader's stress-test concern about this point therefore lands. The paper is not circular in the fitting sense, since no observable is fitted and the LDMEs are taken from prior extractions, but the central quantitative claim is parameter-sensitive and lacks a demonstrated robustness interval.

major comments (3)
  1. [Sec. III, Eq. (26), and Sec. IV.C] The central claim that CO contributions are 'significant or dominant' (abstract and Table I) is not robust because the adopted CO LDMEs are varied at all. In particular, Eq. (26) gives <O^{J/psi}(3S1[8])> = 0.0013 +/- 0.0013 GeV^3, whose 1-sigma lower endpoint is zero, and the bottomonium value <O^{Upsilon}(3S1[8])> = 0.0477 +/- 0.0334 GeV^3 has a similarly large fractional error. Since the CO cross sections are linear in these LDMEs, the CO fractions quoted in Table I (for example 66.1% for J/psi+eta_c and 92.4% for Upsilon+chi_b1) would be drastically reduced or vanish at the lower endpoints. Section IV.C states 'we won't discuss the LDMEs uncertainty,' and the only robustness check varies the CS potential-model set, not the CO LDMEs. Please propagate the LDME uncertainties or, at minimum, show the cross sections obtained with an alternative hadroproduction-fit LDME set; without this the CO-dominance conclusion is not established.
  2. [Sec. IV.B, Table III, and abstract] The event numbers are internally inconsistent. The text in Section IV.B says that the total cross sections at O(v0) are (32.7, 2738.9, 73.1, 53.8) x 10^-4 fb and that 'the final events would be (52, 4382, 117, 86) and (491, 41083, 1096, 806)' for CEPC and FCC-ee, respectively. Table III, by contrast, reports NLO(v2) events (22, 570, 71, 61) for CEPC and (206, 5343, 665, 576) for FCC-ee for the same four channels, and the abstract quotes the latter numbers. The reader cannot tell which set is the actual prediction. Please reconcile the text with Table III and the abstract and state explicitly which perturbative order is used for the final event counts.
  3. [Sec. IV.B] The NLO alpha_s K factors from Ref. [40] are applied to the CS cross sections, whereas the CO cross sections are kept at tree level in alpha_s. Since the adopted CS K factors are large (for example 3.75 and 3.9 for J/psi+eta_c and J/psi+J/psi), the CO channels could receive comparably large NLO QCD corrections, and the hierarchy between CO and CS may change. The paper does state that the CO channels are treated at tree level, but given that the CO-dominance conclusion is the paper's main message, please add a discussion of the expected size of NLO corrections to the CO channels, or at least an explicit caveat that the CO predictions are leading-order in alpha_s.
minor comments (4)
  1. [Sec. IV.A and Fig. 11 caption] The text refers to 'Appendix VI' and 'Appendix VII,' but the appendices are labeled 'APPENDIX. A' and 'APPENDIX. B'; please align the cross-references.
  2. [Sec. IV.B] The K factors quoted in Section IV.B (3.75, 3.9, 2.55, 2.5 and 1.08, 1.01, 0.775, 0.908) are presented without the corresponding m_c or m_b values; please state the quark masses used when applying the Ref. [40] results so the reader can reproduce the numbers.
  3. [References] References [32] and [34] are the same paper (Erler et al., 'Physics impact of GigaZ') and should be merged or renumbered.
  4. [Sec. IV.C and Tables IV/V] The R+ and R- ratios in Tables IV and V are computed at E_cm = 97% and 103% of m_Z, but the text says '15% to 20% of its peak values,' which is only true for some channels; please state that the reduction is channel-dependent.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: cross sections are NRQCD SDCs multiplied by externally fitted CO and CS LDMEs; no input observable is redefined as a prediction.

full rationale

Walking the derivation from Eq. (1) through the projection-operator amplitudes to Tables I-III, every input (masses, couplings, v^2 values, LDMEs) is specified in Sec. III from external sources, chiefly Ref. [52] for CO LDMEs and potential-model values for CS wave functions. The computed cross sections are thus genuine predictions conditional on those inputs, not quantities that have been fitted to themselves. The CO-dominance claim is parameter-sensitive: because Eq. (1) is linear in the LDMEs, a different CO LDME extraction could reduce the CO fraction, and Sec. IV.C explicitly declines to discuss LDME uncertainty. That is a robustness limitation, not circularity. Self-citations to Refs. [49,50,86,87] are used to verify the relativistic-correction formalism and its high-energy ratios (Table VIII), but the central CO-significance claim is made already at LO O(v^0) in Table I and does not rest on those citations. The internal inconsistency between event numbers in Sec. IV.B and Table III/abstract is a correctness concern, not a circular derivation. No equation is defined in terms of the quantity it is used to predict.

Assumptions & free parameters 10 free parameters · 6 assumptions · 0 invented entities

The calculation rests on standard NRQCD factorization and velocity power counting, plus a set of non-perturbative LDMEs taken from prior extractions. The paper's new claim (CO dominance) is not a first-principles result; it is a prediction conditional on those LDMEs. Relativistic corrections use the conventional <v^2> expansion with values from the Gremm-Kapustin relation. No new entities are introduced.

free parameters (10)
  • <O^{J/psi}(3S1[8])> = 0.0013 +/- 0.0013 GeV^3
    CO LDME from Ref [52]; is the gluon-fragmentation input that drives the claimed CO dominance.
  • <O^{J/psi}(1S0[8])> = 0.0180 +/- 0.0087 GeV^3
    CO LDME from Ref [52]; contributes to several channels.
  • <O^{J/psi}(3P0[8])> = (0.0180 +/- 0.0087) m_c^2 GeV^3
    CO LDME from Ref [52]; enters P-wave associated channels.
  • <O^{eta_c}(3S1[8])> = 0.0180 +/- 0.0087 GeV^3
    CO LDME related to J/psi 1S0[8] by HQSS; used for eta_c final states.
  • <O^{Upsilon}(3S1[8])> = 0.0477 +/- 0.0334 GeV^3
    Bottomonium CO LDME from Refs [55,56,62-64]; selected in Section III.
  • v^2_{c cbar} = 0.23
    From Gremm-Kapustin relation (Eq. 25); controls the size of the relativistic suppression.
  • v^2_{b bbar} = 0.1
    Same relation for bottomonia.
  • NLO alpha_s K factors from Ref [40] = K_QCD=3.75,3.9,2.55,2.5; K_EW=1.08,1.01,0.775,0.908
    Used only in Section IV.B to produce the alternative event numbers that conflict with Table III.
  • <O^{J/psi}(3S1[1])> = 1.2 GeV^3
    CS LDME from potential model; affects the size of the color-singlet background and the CO/CS ratio.
  • <O^{Upsilon}(3S1[1])> = 10.9 GeV^3
    CS LDME for bottomonia; from Ref [55,56].
assumptions (6)
  • domain assumption The cross section factorizes into short-distance coefficients and universal long-distance matrix elements (Eq. 1).
    Standard NRQCD framework; not proven for these exclusive processes.
  • domain assumption The Fock-state expansion and velocity power counting (Eq. 2) justify keeping only the CS and CO channels up to O(v^2).
    Truncation of the NRQCD velocity expansion; higher-order terms are neglected.
  • domain assumption The adopted CO LDMEs from Ref [52] (charmonium) and Refs [55,56,62-64] (bottomonium) are universal.
    CO LDME universality is contested in the literature; the paper's central CO-dominance result is directly proportional to these inputs.
  • domain assumption The Gremm-Kapustin relation (Eq. 25) gives the values v^2_c = 0.23 and v^2_b = 0.1 for the relativistic corrections.
    Used for both CS and CO channels; different estimates would alter the K factors.
  • domain assumption The conventional relativistic correction expansion (Section II.B, Eqs. (19)-(20)) is convergent at O(v^2).
    No resummation is attempted; for double charmonium this expansion is known to give large corrections and may not converge rapidly.
  • domain assumption The t-channel EW contributions are negligible except for J/psi pair and Upsilon pair production.
    Modeling choice stated in Section IV.A; if wrong, some cross sections would change.

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Cite this review

Pith. "Pith review of Double heavy quarkonia production with color-octet channels at Z factory and at the CEPC/FCC-ee." pith.science (2026). https://pith.science/paper/KJC65MS2

@misc{pith2026250115575,
  author       = {Pith},
  title        = {Pith review of: Double heavy quarkonia production with color-octet channels at Z factory and at the CEPC/FCC-ee},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KJC65MS2}},
  note         = {Machine review of arXiv:2501.15575}
}
abstract

Within the NRQCD framework, we calculate the exclusive production of double heavy quarkonium(double charmonium and double bottomonium) at future super $Z$ factory and at the CEPC/FCC-ee. The color-octet(CO) channels in the $\gamma^*/Z^*$-propagated process are considered along with the color-singlet(CS) channels. We found that the contributions of CO states to the total cross section are significant or dominant for many processes within energy region at $Z$ factory and at the CEPC/FCC-ee. The experimental measurements will help us to verify the CO mechanism. Among these CO channels, the gluon fragmentation into $^3S_1^{8}$ states is most important. Thus, the comparison between the theoretical results and future data will give a strong constraint to the matrix elements $\langle\mathcal{O}\left(^3S_1^{[8]}\right)\rangle$. Additionally, we consider the relativistic corrections to both the CS and CO channels which decrease the cross sections significantly. Specially, the $K$ factors are about $0.5$ for most charmonium channels. We get estimates of the events for double heavy quarkonium production. The final events of $J/\psi+\eta_c$, $J/\psi+J/\psi$, $\Upsilon+\eta_b$, $\Upsilon+\Upsilon$ production would be (22, 570, 71, 61) and (206, 5343, 665, 576) at the CEPC (2-year) and at the FCC-ee (4-year) for the $Z$ factory mode, respectively.

Figures

Figures reproduced from arXiv: 2501.15575 by the authors.

Figure 1
Figure 1. FIG. 1: CS/CO Feynman diagrams for [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (Color online) Cross sections ( [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (Color online) Cross sections ( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: FIG. 4: (Color online) The ratios ( [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: (Color online) The ratios ( [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: (Color online) The cross sections with uncertainties of [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: (Color online) The differential cross sections [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: (Color online) The differential cross sections [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: (Color online) The differential cross sections ( [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: (Color online) The differential cross section ( [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: (Color online) Cross sections ( [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: (Color online) Cross section ( [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: (Color online) The [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: (Color online) The [PITH_FULL_IMAGE:figures/full_fig_p026_14.png]

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