Next-to-Leading Order QCD Correction to $\bm{e^+ e^- \to J/\psi + \eta_c}$ at $\sqrt {s}=10.6$GeV}

K.T.Chao; Y.J.Gao; Y.J.Zhang

REVIEW

Next-to-Leading Order QCD Correction to $\bm{e^+ e^- \to J/\psi + \eta_c}$ at $\sqrt {s}=10.6$GeV}

Y.J.Zhang, Y.J.Gao, K.T.Chao

High Energy Physics - Phenomenology · Submitted 7 Jun 2005 · 21 references

Pith's verdict: First NLO correction to e+e- -> J/psi + eta_c, a real calculation with a robust K factor but an absolute cross section that is quoted at one scale and should carry the caveat.

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arXiv IDhep-ph/0506076 (version 3)

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CitationK.T.Chao, Y.J.Gao, Y.J.Zhang. "Next-to-Leading Order QCD Correction to $\bm{e^+ e^- \to J/\psi + \eta_c}$ at $\sqrt {s}=10.6$GeV}." arXiv:hep-ph/0506076 (2005). https://pith.science/paper/WC3K552Y

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Editor's note

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Worth your time if you follow the double-charmonium discrepancy: this is the first NLO QCD correction to e+e- -> J/psi + eta_c, and the correction is large — K of about 1.8-2.1, roughly doubling the LO rate and moving theory toward the Belle/BaBar numbers. That is the paper's real contribution and it holds up as a ratio, which is what most people will cite it for. What is actually new is the full one-loop set — self-energy, triangle, box, pentagon — with the Coulomb singularity factored into |R_S(0)|^2 in the standard NRQCD way. The singularity bookkeeping is spelled out rather than hand-waved: renormalization constants in Eq. (7) with OS for the quark mass and field and MS-bar for the gluon, pentagon N10 reduced explicitly in Eqs. (9)-(11), and the IR pieces shown to collapse onto C0[p1c, -p1cbar, 0, m, m]. That is a real calculation, not an estimate. Where it is soft. First, the letter defers the full derivation to a forthcoming paper, so the multi-loop reduction cannot be checked from what is here. For a letter that is normal, but the referee is being asked to trust the algebra. Second, and this is the one I would press on: the abstract's '18.9 fb reaches the lower bound of experiment' is a single point at mu = 2m_c. Fig. 3 shows the NLO curve running from about 27 fb down to about 11 fb over mu = 1.5-5.5 GeV, and the text itself quotes 11.2 fb at mu = sqrt(s)/2 — below both experimental lower bounds. The K factor survives this because it is a ratio; the absolute number does not. The abstract should carry the scale caveat. Third, |R_S(0)|^2 = 0.978 GeV^3 is 1.21x the potential-model value used in Refs. [7,8], and since the cross section scales as |R|^4 that alone inflates the absolute rate by about 1.46. The choice is defensible — extracting it from the NLO leptonic width is the consistent thing to do — but it belongs in the abstract as a stated choice, not only in the body. On the reader's worry about the v -> 0 limit: the authors flag it themselves, and it cuts the other way, since relativistic corrections would push theory up toward data, not down. It is not a threat to the headline claim. Bottom line: solid, honest, technically credible, and the number people will cite. Send it out.

Referee report

Major objections (3)

[Abstract (s5) and Conclusion] The abstract and conclusion state that with m_c = 1.4 GeV and μ = 2m_c the NLO cross section is 18.9 fb, 'which reaches to the lower bound of experiment.' This is a single point at one scale. Fig. 3 and the text show that the NLO cross section spans roughly 11-27 fb over μ = 1.5-5.5 GeV, and at μ = √s/2 the text quotes 11.2 fb, which is below both the Belle (25.6 fb) and BaBar (17.6 fb) lower bounds. The abstract and conclusion should either quote the scale band or explicitly qualify the 'reaches the lower bound' statement, since it is load-bearing for the claim that the NLO result 'greatly reduces' the discrepancy. The authors also acknowledge that relativistic corrections may further enhance the cross section, which is a further caveat on the absolute number.
[Text after Eq. (12), 'Details of the calculation can be found in a forthcoming paper'] The central technical claim is that the NLO correction is UV-, IR-, and Coulomb-finite. The cancellation pattern is asserted rather than demonstrated: the text states that Box N3, N6, N7 and Pentagon N12 are IR-finite, that Box N1+N2+N4+N9 is IR-finite, that the IR divergence of Pentagon N11 is canceled by vertex diagrams, and that the IR divergences of Box N5+N8+Pentagon N10 are canceled by counterterms. A compact table listing the individual diagram divergences (UV, IR, Coulomb) would make the finiteness verifiable without requiring the reader to consult the forthcoming paper.
[Numerical results, extraction of |R_S(0)|^2] The wave function squared is extracted as |R_S(0)|^2 = 0.978 GeV^3, which is 1.21 times the potential-model value 0.810 GeV^3 used in Refs. [7,8]. Since the cross section scales as |R_S(0)|^4, this choice alone enhances the absolute rate by about 1.46 relative to those works. The paper is transparent about this, but the abstract's 'reaches the lower bound' depends on it, and the comparison with Refs. [7,8] ('our LO result is also consistent with Ref. [7] if we take their smaller value') should be made explicit. In addition, using the same |R_S(0)| for J/ψ and ηc ignores the O(v^2) difference between the two wave functions.

Minor objections (8)

[Numerical results, K factor] The K factor at μ = m is 27.5/16.2 ≈ 1.7, slightly outside the quoted 1.8-2.1 range; the range should be given as 1.7-2.1 or the μ = m point should be noted explicitly.
[Fig. 3 caption] It is not clear which of the four curves are NLO and which are LO, and which mass each curve corresponds to; labeling the curves directly in the figure would help.
[Numerical results, meson masses] The paper sets m_J/ψ = m_ηc = 2m; the physical masses differ by about 110 MeV, and the ηc and J/ψ wave functions differ at O(v^2). A sentence estimating the size of this effect would be useful.
[Abstract and comparison with data] The comparison with experiment uses σ × B(ηc → ≥ 2 charged tracks), which is a lower bound on σ rather than σ itself. The text acknowledges this, but the abstract should be explicit that the comparison is to a lower bound.
[Renormalization scheme, nf = 4] The use of nf = 4 at μ ≈ 3 GeV, below the charm threshold, is a scheme choice; its numerical effect on α_s and on the cross section should be estimated or justified.
[Color-octet contribution] The statement that 'the color-octet c cbar contribution is of higher order in v and therefore negligible' could be quantified, for example by stating the power of v by which it is suppressed.
[Abstract, wording] The sentence 'which reaches to the lower bound of experiment' should read 'which reaches the lower bound of experiment'.
[Ref. [21]] Ref. [21], cited for the possible enhancement from relativistic corrections, is listed as 'to be submitted'; since this is invoked as a caveat on the NLO result, a published or more accessible reference would be preferable.

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The paper

The paper claims

A one-loop QCD correction roughly doubles the predicted rate for $e^+e^-$ producing a $J/ψ$ plus an $η_c$, bringing theory down to the edge of the measured window.

The reading

Belle and BaBar measured the process $e^+e^-$ → $J/ψ + η_c$ at 10.6 GeV to have a cross section of order 18–26 fb, while leading-order theory gave only about 4–5 fb. This paper asks whether the gap is an artifact of stopping at leading order in the strong coupling. The authors compute the full next-to-leading order QCD correction — every one-loop self-energy, triangle, box and pentagon diagram — and find it multiplies the leading-order answer by roughly 1.8 to 2.1. At $m_c = 1.4$ GeV and $μ = 2m_c$ the corrected cross section comes out at 18.9 fb, which reaches the lower edge of the experimental range. If the calculation holds, the long-standing factor-of-five discrepancy in double charmonium production is largely a higher-order effect rather than a failure of the underlying framework.

What carries the argument

The load-bearing device is the nonrelativistic QCD factorization formula, which separates the short-distance creation of a $c̄c$ pair from its long-distance hadronization into $J/ψ$ or $η_c$ and lets the whole one-loop amplitude be computed in the nonrelativistic limit. The Coulomb singularity — arising from longitudinal gluon exchange between the quark and antiquark inside a meson, and showing up as a $2πα_sC_F/v$ term in the cross section — is regularized by a small relative velocity $v$ and then factored into the wave function, with the two meson operators together supplying exactly the factor of 2 needed at order $α_s$. Ultraviolet divergences are removed by on-shell renormalization of the quark field and mass and modified-minimal-subtraction renormalization of the gluon field and coupling, with the two-loop running of $α_s$ used for consistency. The five-point integral of the pentagon diagram is reduced to box-type integrals plus a finite remainder computed in four dimensions.

Extended reading notes

Core claim

The paper reports the first complete next-to-leading order QCD correction to $e^+e^- → J/ψ + η_c$ at $√s = 10.6$ GeV in the nonrelativistic QCD factorization formalism, treating the charmonia as color-singlet $c̄c$ pairs. All one-loop self-energy, triangle, box and pentagon diagrams are included, and the result is ultraviolet, infrared and Coulomb finite after renormalization and after the Coulomb-singular term is absorbed into the $c̄c$ wave function at the origin. The correction is large and positive: the ratio of next-to-leading to leading order is about 1.8–2.1. With $|R_S(0)|^2 = 0.978$ GeV$^3$ extracted from the $J/ψ$ leptonic width at next-to-leading order, $m = 1.5$ GeV gives 15.7 fb against a leading-order 8.0 fb, and $m_c = 1.4$ GeV with $μ = 2m_c$ gives 18.9 fb against 9.2 fb. The authors conclude that this substantially reduces the discrepancy with the Belle and BaBar measurements, reaching the lower bound of the experimental range.

Load-bearing premise

The whole comparison rests on the nonrelativistic limit being good enough: the calculation sets the relative velocity of the charm quarks to zero and folds the Coulomb-singular piece into the wave function, while the authors themselves note that relativistic corrections may further significantly enhance the cross section.

Editorial extensions

If this is right

The factor-of-five gap between leading-order theory and the Belle/BaBar measurements of $e^+e^- → J/ψ + η_c$ is mostly a higher-order QCD effect, not evidence against the nonrelativistic QCD description of double charmonium production.
The predicted cross section varies with the renormalization scale from roughly 11 fb at $μ = √s/2$ to about 27 fb at $μ = m$, so scale choice alone spans a large part of the experimental window.
Using the wave function extracted from the $J/ψ$ leptonic width at next-to-leading order, rather than potential-model values, raises $|R_S(0)|^2$ by a factor of about 1.21 and pushes the prediction upward further.
The same class of one-loop corrections may be sizable in related double-charmonium channels, so leading-order comparisons there should be treated with similar caution.

Reading between the lines

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

Because the correction is large and positive at one loop, the natural next question is whether the two-loop term is similarly large; the scale band from 11 to 27 fb is wide enough that a well-behaved perturbative series is not yet demonstrated.
The authors note that relativistic corrections may further enhance the cross section by lowering the gluon virtuality from its nonrelativistic maximum of $s/4$; a full $O(v^2)$ treatment would test whether the agreement with data is genuine or accidental.
If the correction is confirmed, the same machinery should be applied to the inclusive $e^+e^- → J/ψ + c̄c$ channel, where the discrepancy was even larger, to see whether the same higher-order enhancement appears.
A cleaner experimental separation of the $J/ψ + η_c$ final state from other double-charmonium and two-photon backgrounds would sharpen what the theory number must actually match.

Original abstract

One of the most challenging open problems in heavy quarkonium physics is the double charm production in $e^+e^-$ annihilation at B factories. The measured cross section of $e^+ e^- \to J/\psi + \eta_c$ is much larger than leading order (LO) theoretical predictions. With the nonrelativistic QCD factorization formalism, we calculate the next-to-leading order (NLO) QCD correction to this process. Taking all one-loop self-energy, triangle, box, and pentagon diagrams into account, and factoring the Coulomb-singular term into the $c\bar c$ bound state wave function, we get an ultraviolet and infrared finite correction to the cross section of $e^+e^-\to J/\psi + \eta_c$ at $\sqrt{s} =10.6$ GeV. We find that the NLO QCD correction can substantially enhance the cross section with a K factor (the ratio of NLO to LO) of about 1.8-2.1; hence it greatly reduces the large discrepancy between theory and experiment. With $m_c=1.4{\rm GeV}$ and $\mu=2m_c$, the NLO cross section is estimated to be 18.9 fb, which reaches to the lower bound of experiment.

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.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.29 · human participants 0.01 · controlled comparison 0.19 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.95 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

One of the most challenging open problems in heavy quarkonium physics is the double charm production in $e^+e^-$ annihilation at B factories. The measured cross section of $e^+ e^- \to J/\psi + \eta_c$ is much larger than leading order (LO) theoretical predictions. With the nonrelativistic QCD factorization formalism, we calculate the next-to-leading order (NLO) QCD correction to this process. Taking all one-loop self-energy, triangle, box, and pentagon diagrams into account, and factoring the Coulomb-singular term into the $c\bar c$ bound state wave function, we get an ultraviolet and infrared finite correction to the cross section of $e^+e^-\to J/\psi + \eta_c$ at $\sqrt{s} =10.6$ GeV. We find that the NLO QCD correction can substantially enhance the cross section with a K factor (the ratio of NLO to LO) of about 1.8-2.1; hence it greatly reduces the large discrepancy between theory and experiment. With $m_c=1.4{\rm GeV}$ and $\mu=2m_c$, the NLO cross section is estimated to be 18.9 fb, which reaches to the lower bound of experiment.

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	One of the most challenging open problems in heavy quarkonium physics is the double charm production in $e^+e^-$ annihilation at B factories.	background 0.89	0.04	0.02	asserted 1.00
s1	The measured cross section of $e^+ e^- \to J/\psi + \eta_c$ is much larger than leading order (LO) theoretical predictions.	background 0.79	0.07	0.04	asserted 1.00
s2	With the nonrelativistic QCD factorization formalism, we calculate the next-to-leading order (NLO) QCD correction to this process.	contribution 0.60	0.96	0.05	asserted 0.99
s3	Taking all one-loop self-energy, triangle, box, and pentagon diagrams into account, and factoring the Coulomb-singular term into the $c\bar c$ bound state wave function, we get an ultraviolet and infrared finite correction to the cross section of $e^+e^-\to J/\psi + \eta_c$ at $\sqrt{s} =10.6$ GeV.	method 0.93	0.95	0.09	asserted 1.00
s4	We find that the NLO QCD correction can substantially enhance the cross section with a K factor (the ratio of NLO to LO) of about 1.8-2.1; hence it greatly reduces the large discrepancy between theory and experiment.	result 0.68	0.96	0.85	asserted 1.00
s5	With $m_c=1.4{\rm GeV}$ and $\mu=2m_c$, the NLO cross section is estimated to be 18.9 fb, which reaches to the lower bound of experiment.	result 0.99	0.93	0.21	asserted 0.90

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.58.

What was examined, and what was not

sources read

arXiv listing abstract v3

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

Figures

**Figure 2.** FIG. 2: Twelve of the twenty-four box and pentagon diagrams fo

**Figure 3.** FIG. 3: Cross sections as functions of the renormalization

Figures reproduced from arXiv:hep-ph/0506076 by the authors.

References

Works cited (21)

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Forward citations

Cited by 13 Pith papers

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

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The authors present the first analytical two-loop NRQCD amplitude for e+e- to J/psi+eta_c as an expansion in m_c^2/s, with cross-section predictions consistent (within uncertainties) with B-factory data.
Inclusive productions of $J/\psi+\eta_c$ in $e^{+}e^{-}$ annihilation at Belle
hep-ph 2026-09 accept novelty 6.0 of 10

The inclusive cross section for e+e- -> J/psi + eta_c + X is computed at NLO in NRQCD, finding that color-octet contributions increase the rate by 20-30% at the Upsilon(4S) and alter the energy dependence.
Revisiting relativistic corrections to inclusive $J/\psi$ production at B factories: Complete expansion for three-body quarkonium production
hep-ph 2026-07 conditional novelty 6.0 of 10

Full final-state kinematic expansion yields scheme-independent O(v²) corrections of about −14.6% (cc̄) and +12% (non-cc̄) for inclusive J/ψ at B factories, yet tension with Belle data remains.
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hep-ph 2026-06 unverdicted novelty 6.0 of 10

Semi-analytical NNLO QCD corrections to e+e- -> J/psi + chi_cJ are computed via asymptotic expansions, yielding large corrections that reduce scale dependence for some channels and a discrepancy with Belle data on the...
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NLO QCD corrections enhance radiative decay widths of fully heavy tetraquarks T4Q → VQ/ηQ + γ by factors of 2–3 and reduce scale dependence.

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The paper reports a leading-order NRQCD estimate of roughly 3 fb for exclusive Xi_cc + anti-Xi_cc production at B-factory energies and concludes the channel is unobservable at Belle II and BaBar.
Reanalysis of exclusive charmonium production at the $B$ factories
hep-ph 2026-09 accept novelty 5.0 of 10

Amplitude-level truncation of NNLO QCD corrections yields better perturbative convergence and reduced scale dependence for eight exclusive charmonium production processes at B factories.
Statistical analysis of the azimuthal asymmetry in the $J/\psi$ leptoproduction in unpolarized $ep$ collisions
hep-ph 2019-08 conditional novelty 5.0 of 10

In a selected kinematic region, the measured cos(psi) and cos(2psi) azimuthal asymmetries of J/psi leptoproduction can separate the color-singlet model, the 1S0[8]-dominant NRQCD picture, and the equal-color-octet NRQ...
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QCD radiative correction to color-octet $J/\psi$ inclusive production at B Factories
hep-ph 2009-11 conditional novelty 5.0 of 10

NLO QCD corrections roughly double the color-octet J/psi production rate at B factories, and Belle data then imply a color-octet matrix element below 2.0 x 10^-2 GeV^3.
Measurement of Double Charmonium Production in $e^+e^-$ Annihilations at $\sqrt{s}=10.6$ GeV
hep-ex 2005-06 conditional novelty 4.0 of 10

BABAR independently confirms Belle's observation of e+e- to J/psi plus eta_c(1S), chi_c0, and eta_c(2S), reporting cross sections of 17.6, 10.3, and 16.4 fb, roughly ten times above NRQCD predictions.
On the mechanisms of heavy-quarkonium hadroproduction
hep-ph 2008-11 conditional novelty 3.0 of 10

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QCD corrections to J/psi plus eta_c production in e+e- annihilation at sqrt{s}=10.6 GeV
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How the literature uses it

43 arXiv papers cite this in 70 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 48 sentences without a confident label

Uses its method or tool 2 papers

The method of resummation [ref] has been applied to resum a class of the relativistic corrections to the $e^+ e^- →J/ψ+ η_c$ [ref] in combination with the QCD NLO corrections in Ref. [ref] .

Recent Developments in Heavy-Quarkonium Phenomenology · Relativistic corrections to quarkonium processes · 0.88
The on-mass-shell (OS) scheme is chosen for $Z_2b$ and $Z_2c$ [ref] : [equation] where $μ$ is the renormalization scale, $γ_E$ is the Euler's constant.

$\Upsilon$ decay to two-charm quark jets as a Probe of the Color Octet Mechanism · Color-singlet piece $b b̅ (^3S_1,1) →c c̅$ · 0.93

Uses its data or values 3 papers

We compute the quantity $(σ^QCD_0, NLO- σ^QCD_0)/√(σ^QCD_0)$ by making use of results [ref] that have been provided by the authors of Ref. [ref] .

Resummation of Relativistic Corrections to e+ e- -> J/psi+eta_c · Results · 0.95
The $K$ factor from NLO corrections in $α_s$ is about 2, which depends on the charm-quark mass $m_c$ [ref] .

Higher-order corrections to exclusive production of charmonia at B factories · $e^+ e^- →J/ψ+ η_c$ · 0.80
The $K$ factor for the QCD NLO prediction to that of LO in both $α_s$ and $v$ was about a factor of 2 [ref] .

Recent Developments in Heavy-Quarkonium Phenomenology · Relativistic corrections to quarkonium processes · 0.89

Compares with it 3 papers

They confirmed the result given in [ref] , in addition, they also provided the fully analytical expression for $C^(1)$.

Bridging light-cone and NRQCD approaches: asymptotic behavior of $B_c$ electromagnetic form factor · Challenge of the refactorization program to $J/ψ-η_c$ EM form factor · 0.91
This $K$ factor is roughly consistent with what is obtained through actual higher-order NRQCD calculations [ref] .

Exclusive Double Charmonium Production from $\Upsilon$ Decay · Determination of $K$ from $B$ factories measurement · 0.84
We guess that the corrections may not be as large as those observed in the two-charmonium production from $e^+ e^-$ annihilation [ref] , where all such corrections are aligned to make a significant enhancement of the prediction for the cross section.

Exclusive heavy quarkonium + gamma production from e+ e- annihilation into a virtual photon · Uncertainties · 0.94

Cites it as support 1 paper

This is confirmed by the explicit one-loop calculations of Refs. [ref] , for the $J/ψ+ η_c$ and $J/ψ+ χ_c$ cases.

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Cites it as background 10 papers

An important recent theoretical development is the calculation of the corrections of next-to-leading order (NLO) in $α_s$ (Ref. [ref] ).

Resummation of Relativistic Corrections to e+ e- -> J/psi+eta_c · Introduction · 0.86
Over the past two decades, inclusive and exclusive $J/ψ$ production at $B$ factories with $√(s)≃10.58 GeV$ has provided a clean platform for probing both perturbative and nonperturbative mechanisms underlying quarkonium formation [ref] .

Revisiting relativistic corrections to inclusive $J/\psi$ production at B factories: Complete expansion for three-body quarkonium production · Introduction · 0.94
Over the past decades, the research on double charmonium at various high-energy colliders has been made considerable progress both experimentally and theoretically [ref] .

Next-to-leading-order QCD correction to the exclusive double charmonium production via $Z$ decays · Introduction · 0.99

Discussion

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