Inclusive productions of $J/\psi+\eta_c$ in $e^{+}e^{-}$ annihilation at Belle

Shu-Jian Qi; Ying-Zhao Jiang; Zhan Sun

REVIEW

Inclusive productions of $J/\psi+\eta_c$ in $e^{+}e^{-}$ annihilation at Belle

Zhan Sun, Shu-Jian Qi, Ying-Zhao Jiang

High Energy Physics - Phenomenology · Submitted 9 Sep 2026

Pith's verdict: Solid NLO NRQCD calculation for an inclusive double-charmonium process, but the central claim about energy dependence as a sensitive test of NRQCD is overstated at Belle energies.

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arXiv ID2609.09590 (version 2)

Pith ID7SSE7CKG, Pith's permanent record of this paper and its review

CitationShu-Jian Qi, Ying-Zhao Jiang, Zhan Sun. "Inclusive productions of $J/\psi+\eta_c$ in $e^{+}e^{-}$ annihilation at Belle." arXiv:2609.09590 (2026). https://pith.science/paper/7SSE7CKG

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Claim paper

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Editor's note

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The main thing to know: this paper gives the first complete NLO NRQCD prediction for inclusive J/ψ+η_c production in e+e− annihilation, including all relevant color-octet channels and QED interference. That is a genuine technical step forward. The calculation uses two independent frameworks as a cross-check, the scale uncertainties are shown, and the discussion of experimental prospects (based on Belle’s reconstruction strategy) is concrete. The authors deserve credit for filling a clear gap.

The soft spot is the headline claim. The abstract states that the energy dependence provides a “sensitive test of NRQCD,” but the stress-test concern lands: the ∼s⁻³ vs ∼s⁻⁴ scaling only separates cleanly above 15 GeV. At the B‑factory energies where data exist (10.52–10.87 GeV), the NRQCD/CS ratio is 1.0–1.3 and LDME uncertainties overlap with the CS prediction—especially for LDME set 1, where the ratio is essentially unity. The paper never quantifies whether a 20–30% enhancement at Υ(4S) is statistically distinguishable given the foreseen luminosity and LDME errors. Without that, the claim of a “sensitive test” is stronger than the evidence warrants. The energy extrapolation to 20 GeV is interesting but irrelevant to Belle data.

The LDME dependence is another caveat: the conclusions flip depending on which fit you use. That is not a fatal flaw—it is honest—but it means the paper tests specific LDME sets rather than NRQCD in a parameter-free way. The authors could have been more explicit about this.

Who benefits: quarkonium theorists and experimentalists planning double-charmonium analyses at Belle II. The paper provides a needed NLO baseline. It deserves peer review—the calculation is technically solid and the topic matters—but the conclusions about energy dependence need to be dialed back. A revision that adds a statistical significance estimate for the Υ(4S) point (or simply states the limitation clearly) would make it publishable.

Referee report

Major objections (4)

[Abstract; §3, Table 2 and Fig. 8] The claim that the energy dependence provides a 'sensitive test of NRQCD' is stronger than the presented evidence at B-factory energies. At the three energies with existing or planned data (10.52, 10.58, 10.87 GeV), the NRQCD-to-CS ratio is 1.0–1.3 and the LDME error bands, especially for sets 2–4, overlap the CS prediction; for set 1 the ratio is essentially unity. The approximately s^-3 versus s^-4 separation becomes unambiguous only above √s ≈ 15 GeV (Fig. 8), which is beyond Belle. The paper should either quantify the statistical distinguishability at the foreseen 50 ab^-1 luminosity or soften the claim about discriminating power at B-factory energies.
[§2.3, Eq. (2.2) and surrounding text] The selection of CO+CO channels is only partially systematic: the channels [1S0[8]]+[1P1[8]] and [3PJ[8]]+[1P1[8]] are neglected on v^8-counting grounds, but the retained channels include several at the same nominal order in v (two CO LDMEs) whose SDCs are shown to be large, some even negative. Because no numerical estimate for the neglected channels is given, the statement that they 'can be safely neglected' needs either an explicit order-of-magnitude estimate from representative diagrams or a direct justification that the v^8 suppression beats the enhanced SDCs shown in Fig. 6.
[§2.2, Eqs. (2.8)–(2.11)] The P-wave channels are regularized by the soft/hard cutoff δs, but no check of δs-independence is reported. Since the finite SDCs should be independent of this artificial cutoff, the paper should include at least one stability statement, showing that the numerical results for the subtracted SDCs (and hence the cross sections) do not depend on the chosen δs.
[§3, Table 1 and Eqs. (3.1)] For set 1, the quoted input ⟨OJ/ψ(3S1[8])⟩ = (0.312 ± 0.93)×10^-2 GeV^3 has an uncertainty much larger than the central value, as written implying a negative lower bound. The authors should clarify whether these numbers are central values with asymmetric errors or some other convention, and describe how the errors in Table 2 and Fig. 8 were propagated (linear error propagation versus scanning with positivity constraints).

Minor objections (5)

[§3, Table 2] The cross sections at 10.52 GeV are very close to those at Υ(4S), so the abstract and the summary should make explicit that the quoted '20%–30% enhancement at Υ(4S)' refers to the 10.58 GeV resonance point and that the enhancement at 10.52 GeV is comparable but not identical.
[§3, Fig. 8 and Eq. (2.6)] The notation for the CS-only curves is inconsistent: Fig. 8 uses σα2 and σα2+α3, while Figs. 5–7 and Table 2 simply say 'CS'. These definitions should be harmonized so the reader can tell exactly which contributions are included in each curve.
[§2.1, Eq. (2.7)] The renormalization constants are written in a conventional form, but the MS-scheme constants omit the log(4π)−γE terms in some entries; the authors should state the exact convention (for example, whether the MS or MS-bar scheme is being used after subtracting 1/ϵUV poles) so that the calculation is fully reproducible.
[§3, Eq. (3.2)] The event-yield estimates quote 'a few' events at 10.52 GeV and Υ(5S) and 'about a dozen' at Υ(4S) without explicitly listing the efficiency ϵ, luminosity L, and branching-fraction product from ref. [91]. These inputs should be stated so the estimates can be checked.
[Throughout] There are multiple typos and LaTeX artifacts in the text, including 'productions ofJ/ψ+η c', 'thee +e−', and inconsistent symbol formatting for μΛ. A careful proofreading pass is needed before submission.

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The paper

The paper claims

This paper predicts a 20-30% enhancement in inclusive J/ψ+η_c production at B-factory energies due to color-octet contributions, with a distinctive energy dependence that can test NRQCD.

The reading

The paper calculates inclusive e+e- → J/ψ+η_c+X at next-to-leading order in α_s within the NRQCD framework, including all relevant color-octet channels. At the Υ(4S) energy, the color-octet terms increase the cross section by 20-30% over the color-singlet-only prediction, and this enhancement grows rapidly with collision energy. The predicted cross section scales as approximately $s^{-3}$, clearly different from the $s^{-4}$ scaling of the color-singlet result. This difference provides a sensitive experimental test of the NRQCD formalism at low energies. The calculation also accounts for QED interference and ψ(2S) feed-down, which further boosts the predicted signal by about 40%.

What carries the argument

The central machinery is the short-distance coefficient (SDC) for color-octet channels, especially the $c\bar{c}[^1S_0^{[8]}]+c\bar{c}[^3S_1^{[8]}]$ and $c\bar{c}[^3P_J^{[8]}]+c\bar{c}[^3S_1^{[8]}]$ processes. These SDCs are found to be over an order of magnitude larger than the color-singlet SDC, compensating for the smaller color-octet long-distance matrix elements. The NLO QCD corrections further enhance these channels by about a factor of three. Renormalization of the LDMEs absorbs infrared divergences from P-wave states, yielding finite predictions.

Extended reading notes

Core claim

This work demonstrates that at center-of-mass energies around the Υ(4S) resonance, the inclusive production of J/ψ+η_c receives a substantial 20-30% enhancement from color-octet channels when computed at NLO in α_s within NRQCD factorization. The enhancement is driven by large short-distance coefficients for processes such as c̄c[1S0[8]]+c̄c[3S1[8]], which were previously omitted. The resulting cross section scales as $s^{-3}$, in contrast to the $s^{-4}$ behavior of the color-singlet prediction. This difference in energy dependence offers a clear experimental signature. The calculation also includes non-negligible QED interference contributions to the color-singlet channel and estimates a ~40% increase from ψ(2S) feed-down, making the process more accessible to observation by Belle and Belle II.

Load-bearing premise

The load-bearing premise is that the long-distance matrix elements (LDMEs) extracted from hadroproduction data are universal and apply at B-factory energies; if non-universal effects or non-factorizable contributions alter these LDMEs at low energies, the predicted NRQCD enhancement may not materialize.

Editorial extensions

If this is right

At √s=10.58 GeV, the NRQCD inclusive cross section is 15–16 fb, about 20–30% higher than the color-singlet value of ~12 fb.
The cross section scales as s^-3 under NRQCD, versus s^-4 for the color-singlet model; this difference grows with energy, reaching a factor of 2–4 at 15–20 GeV.
Including ψ(2S) feed-down boosts the NRQCD yield by an additional 40%, improving the observability at Belle and Belle II.
The NLO QED interference term in the color-singlet channel contributes 10–20% and must be included in comparisons with experimental data.

Reading between the lines

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

A precise measurement of the inclusive J/ψ+η_c cross section at Υ(4S) could discriminate among the four LDME sets: Set 1 gives results close to the color-singlet prediction, while Sets 2–4 predict larger enhancements.
The same NLO calculation can be extended to other inclusive double-charmonium pairs (e.g., J/ψ+χ_c or η_c+η_c) to provide additional tests of NRQCD.
The predicted s^-3 scaling could be tested at SuperKEKB by running at different energies, or at future colliders like the FCC-ee.

Original abstract

We study $e^{+}e^{-} \to J/\psi+\eta_c+X$ at next-to-leading order (NLO) in $\alpha_s$ within the nonrelativistic QCD (NRQCD) framework to assess the color-octet (CO) mechanism at low energies. QCD corrections significantly enhance both color-singlet (CS) and CO leading-order results, with non-negligible QED-diagram contributions to the CS channel. At $\Upsilon(4S)$, CO terms increase the inclusive cross section by $20\%$--$30\%$, and the enhancement grows rapidly with $\sqrt{s}$, yielding an energy dependence distinct from CS predictions and providing a sensitive test of NRQCD. For the central LDME choice, including $\psi(2S)$ feed-down further amplifies the NRQCD prediction by about $40\%$ at $\Upsilon(4S)$. With Belle's reconstruction of $J/\psi \to \mu^+\mu^-$ and six $\eta_c$ decay channels, the process shows promising potential for observation.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,495 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.15 · human participants 0.02 · controlled comparison 0.16 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.78 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study $e^{+}e^{-} \to J/\psi+\eta_c+X$ at next-to-leading order (NLO) in $\alpha_s$ within the nonrelativistic QCD (NRQCD) framework to assess the color-octet (CO) mechanism at low energies. QCD corrections significantly enhance both color-singlet (CS) and CO leading-order results, with non-negligible QED-diagram contributions to the CS channel. At $\Upsilon(4S)$, CO terms increase the inclusive cross section by $20\%$--$30\%$, and the enhancement grows rapidly with $\sqrt{s}$, yielding an energy dependence distinct from CS predictions and providing a sensitive test of NRQCD. Including $\psi(2S)$ feed-down further amplifies the NRQCD prediction by about $40\%$ at $\Upsilon(4S)$. With Belle's reconstruction of $J/\psi \to \mu^+\mu^-$ and six $\eta_c$ decay channels, the process shows promising potential for observation.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study $e^{+}e^{-} \to J/\psi+\eta_c+X$ at next-to-leading order (NLO) in $\alpha_s$ within the nonrelativistic QCD (NRQCD) framework to assess the color-octet (CO) mechanism at low energies.	contribution 0.88	0.89	0.07	asserted 0.93
s1	QCD corrections significantly enhance both color-singlet (CS) and CO leading-order results, with non-negligible QED-diagram contributions to the CS channel.	result 1.00	0.80	0.12	asserted 1.00
s2	At $\Upsilon(4S)$, CO terms increase the inclusive cross section by $20\%$--$30\%$, and the enhancement grows rapidly with $\sqrt{s}$, yielding an energy dependence distinct from CS predictions and providing a sensitive test of NRQCD.	result 0.87	0.83	0.62	asserted 1.00
s3	Including $\psi(2S)$ feed-down further amplifies the NRQCD prediction by about $40\%$ at $\Upsilon(4S)$.	result 0.99	0.79	0.10	asserted 0.99
s4	With Belle's reconstruction of $J/\psi \to \mu^+\mu^-$ and six $\eta_c$ decay channels, the process shows promising potential for observation.	interpretation 0.64	0.17	0.10	hedged possibility 0.75

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from simulation or computation 0.76.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Discussion

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