Charmonium pair production in ultraperipheral collision

Bingwei Long; Hao Yang; Zi-Qiang Chen

REVIEW 3 major objections 5 minor 37 references

Charmonium pair production in ultraperipheral collision

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper predicts that exclusive J/ψ-pair and ηc-pair production in ultraperipheral Pb-Pb collisions has measurable rates at NLO in NRQCD, with opposite-sign NLO corrections for the two channels.

desk verdict A legitimate but modest extension of an existing NLO NRQCD calculation to UPC; the central cross sections are not robust because the NLO expansion itself signals a breakdown at the chosen scale. read the letter →

arxiv 2504.14850 v1 pith:O7REIUZM submitted 2025-04-21 hep-ph

Hao Yang , Zi-Qiang Chen , Bingwei Long This is my paper · ORCID

classification hep-ph

keywords charmoniumpairproductionultraperipheralcollisionNRQCDNLOQCDcorrectionsphoton-photonfusionJ/ψηcX(6900)

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

This paper aims to establish that exclusive double charmonium production via photon-photon fusion in ultraperipheral heavy-ion collisions at the HL-LHC and FCC can serve as a clean test of NRQCD factorization. Using next-to-leading-order QCD corrections in the color-singlet sector, it predicts total cross sections of 28.0 nb for J/ψ pairs and 65.1 nb for ηc pairs in Pb-Pb collisions at 5.52 TeV, with large negative corrections for J/ψ and positive for ηc. The resulting event rates are small but topologically clean because UPC events lack pileup and have large rapidity gaps, so backgrounds from QCD interactions are suppressed. The paper also estimates X(6900) production through two-photon fusion using the two-photon-width formula for resonance production.

What carries the argument

The central machinery is NRQCD (non-relativistic QCD) factorization at next-to-leading order in QCD, applied to the color-singlet γγ→H+H amplitudes using covariant spin and color projectors, combined with the equivalent-photon approximation for ultraperipheral collisions. The NLO amplitudes are regularized dimensionally, with on-shell renormalization for the heavy quark field and mass and MS renormalization for the strong coupling. The observable UPC cross section is obtained by convolving the NLO parton-level cross section with the ion photon spectral function for heavy ions, and this factorization is what carries the argument from a QCD calculation to measurable rates.

What would settle it

Measure the rapidity-difference distribution of exclusive J/ψ-pair production in Pb-Pb UPC at the HL-LHC; if the bins with |Δy|>2 show positive rates near the leading-order size, or if the total rate differs from 28 nb by more than the quoted uncertainties, the NLO NRQCD prediction is falsified. The ηc-pair rate of 65.1 nb offers an independent check.

Watch

Extended reading notes

Core claim

The paper's central claim is that at next-to-leading order in QCD, the photon-photon fusion cross sections for exclusive J/ψ-J/ψ and ηc-ηc production in ultraperipheral Pb-Pb collisions are 28.0 nb and 65.1 nb respectively, and that the NLO corrections have opposite signs: large negative for J/ψ pairs and positive for ηc pairs. These predictions follow from NRQCD factorization with color-singlet matrix elements, and the paper argues that UPC event topologies suppress QCD backgrounds enough that the rates are measurable at the HL-LHC and FCC. The J/ψ-pair K-factor is about 0.25 and the ηc-pair about 2.35, and differential distributions in pT, mγγ, and |Δy| are provided. At the default scale, the J/ψ-pair differential cross section becomes negative for |Δy|>2, which the authors attribute to large negative loop corrections and suggest may be cured by scale resetting or resummation.

Load-bearing premise

The central numbers depend on the assumption that the next-to-leading-order QCD correction is a small, trustworthy correction; the paper's own J/ψ-pair distribution turning negative at large rapidity differences is a warning that this may not hold.

Editorial extensions

If this is right

At the HL-LHC, the predicted yields are 140-194 J/ψ-pair events and 325-456 ηc-pair events per run from heavy-ion UPCs, with 835 and 1980 from p-p UPCs, before decay branching ratios.
Using the branching fractions quoted in the paper, J/ψ→l+l− at 12% and ηc→K\bar Kπ at 7.3%, reconstructed candidates would be 2-12 per year for J/ψ pairs and 1-10 for ηc pairs at the HL-LHC; at the FCC the J/ψ-pair yield grows to 180-200.
The opposite signs of the NLO K-factors, about 0.25 for J/ψ pairs and 2.35 for ηc pairs, mean the two channels respond very differently to higher-order QCD, so a simultaneous measurement would be a sensitive NRQCD test.
The differential cross sections in pT, mγγ, and |Δy| are given without cuts, providing specific shapes that UPC experiments can compare directly with data.
The X(6900) production cross section via two-photon fusion is estimated at 6×10^3 to 2×10^5 nb depending on the two-photon width model, implying a potentially large di-charmonium signal if the state decays predominantly into charmonium pairs.

Reading between the lines

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

Not stated in the paper: if the negative NLO differential cross section at |Δy|>2 reflects a genuine breakdown of fixed-order perturbation theory rather than a scale artifact, the 28.0 nb total may be unreliable; a resummed or next-to-next-to-leading-order calculation would settle this.
A test that follows from the sign asymmetry but is not proposed by the authors: measuring the ratio σ(ηc-ηc)/σ(J/ψ-J/ψ) as a function of pT would isolate the spin dependence of the NLO corrections and could expose missing relativistic or color-octet effects.
The X(6900) estimates span two orders of magnitude across the three two-photon-width models; a UPC measurement of the di-J/ψ invariant mass near 6.9 GeV would effectively measure Γ(X→γγ) and discriminate among those models, going beyond the paper's tabulated estimates.
The same NLO machinery could be applied to other exclusive channels, such as J/ψ plus ψ(2S), to test whether the opposite-sign correction pattern is specific to the spin-triplet versus spin-singlet ground states.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does exactly what it says: it takes the NLO gamma-gamma -> charmonium-pair amplitudes from the authors' previous work, convolutes them with equivalent-photon fluxes for a range of ion species at the HL-LHC and FCC, and produces total and differential cross sections. That is a real extension and the numerical results are new. The treatment of the EPA is standard, the input parameters are documented, and the authors are transparent about the large scale dependence and even flag the negative differential bins. I appreciate that honesty.

The problem is not the calculation, which looks sound in structure; it is the interpretation of the central numbers. The J/psi-pair NLO total cross section changes from 11.0 nb to 83.8 nb when the scale goes from 2m_c to sqrt(s_γγ), and the chosen "central" scale sqrt(4m_c^2+p_T^2) gives a K-factor around 0.25 and negative dsigma/d|Δy| for |Δy| > 2. The paper acknowledges this and suggests scale resetting or resummation, but then still quotes 28.0 nb as the headline result. That is not a reliable absolute prediction. If the expansion breaks down at the central scale, the number should carry a strong caveat or a different scale should be used. The eta_c channel is less dramatic, but the same issue of a large positive NLO correction and scale sensitivity applies.

The X(6900) section is the weakest part. The estimated cross sections range from 6e3 to 2e5 nb, which the authors themselves call abnormal and attribute to overestimated branching fractions. That is a self-admitted unphysical result, and it does not strengthen the paper. I would remove it or reframe it as a rough upper-limit illustration with the caveat front and center.

The citation pattern is fine; the relevant NRQCD and UPC literature is covered. No code or data is shipped, so the reproducibility rests on the formulas in the paper, which are sufficiently detailed for a specialist to reproduce.

Bottom line: this is a useful phenomenological paper for people working on UPC and double-charmonium searches, but the headline cross sections should not be used as solid numbers until the scale issue is dealt with. The flaws are fixable with a careful revision, so I would send it to peer review rather than reject it outright. A serious referee should push the authors to either choose a scale where the NLO expansion is well-behaved or present the scale sensitivity as the main uncertainty, and to remove or heavily qualify the X(6900) estimates.

Referee Report

3 major / 5 minor

Summary. The paper computes next-to-leading-order (NLO) QCD corrections, in the NRQCD color-singlet framework, to exclusive γγ→J/ψJ/ψ and γγ→ηcηc production in ultraperipheral collisions, using the equivalent-photon approximation. It reports total cross sections of 28.0 nb for J/ψ-pair and 65.1 nb for ηc-pair production in Pb-Pb UPC at √sNN=5.52 TeV, along with differential distributions in transverse momentum, diphoton invariant mass, and rapidity difference, and estimates of event rates at the HL-LHC and FCC. The paper also estimates two-photon production of the fully charmed tetraquark X(6900) using the Low formula with several model assumptions for its diphoton width.

Significance. If the central NLO predictions were robust, this study would offer a clean test of NRQCD factorization in an exclusive two-photon process, with color-octet contributions strongly suppressed and UPC event topologies providing large rapidity gaps. The paper usefully extends the authors' earlier NLO calculation to heavy-ion UPCs, provides explicit flux-convolution formulas, and tabulates cross sections for many collision systems. The X(6900) discussion, though speculative, points to a possible two-photon production window for fully charmed tetraquarks. However, as detailed in the major comments, the scale dependence of the NLO series and the occurrence of negative differential cross sections at the chosen central scale mean that the headline numbers are not yet reliable as quantitative predictions.

major comments (3)

[§III, Table I and Table III] The central NLO predictions are not robust because the central scale choice is both ambiguous for integrated observables and strongly scale-dependent. For the J/ψ pair, the NLO total cross section varies from 11.0 nb at μ=2mc to 83.8 nb at μ=√ŝγγ (Table I), a factor of about 7.6, while the quoted charm-mass uncertainty on the central value is only +18.4/−11.1 nb. At the central scale μ=√(4mc^2+p_T^2), the NLO correction is approximately −75% of the LO result (28.0 nb versus 111 nb), and Table III shows that the NLO differential cross section dσ/d|Δy| becomes negative for |Δy|>2, e.g., −0.47 nb in the 2.5–3 bin. The text acknowledges this and defers to scale resetting or resummation, but no such improved calculation is provided. The paper also never defines which p_T value is used when this scale is applied to the total cross section, since p_T is integrated over. These issues directly affect the abstract's headline value of 28.0 nb, so they are load-bearing rather than cosmetic.
[§III, Table I and Abstract] The qualitative claim that NLO corrections are large and negative for J/ψ pairs and positive for ηc pairs is not scale-invariant. Table I shows that for J/ψ pairs the NLO correction is negative at μ=2mc (11.0 nb versus 120 nb LO) but positive at μ=√ŝγγ (83.8 nb versus 62.6 nb LO). The K-factors are approximately 0.25 for J/ψ and 2.35 for ηc, both far from unity, indicating that the fixed-order expansion is not a small perturbation. Therefore, the sign and magnitude of the NLO corrections, and hence the claimed qualitative NRQCD test, depend strongly on an arbitrary renormalization-scale choice. The abstract and conclusions should be revised to present these results as scale-sensitive fixed-order estimates rather than definitive predictions.
[§III, Table I and text after Table III] The quoted theoretical uncertainties are incomplete: Table I gives only the variation with mc at fixed renormalization scale, while the scale dependence itself is an order of magnitude larger than the mass uncertainty. For a phenomenological claim based on total cross sections, the scale variation should be included as an uncertainty band, and the negative bins in Table III should be addressed by a concrete prescription rather than a statement that the problem 'could be cured' by resetting the scale. Without such a treatment, the event-rate projections derived from the central values (e.g., 140–194 J/ψ-pair events in heavy-ion UPCs) inherit an uncontrolled systematic uncertainty.

minor comments (5)

[§III, Eq. (7) and Eq. (8)] Please clarify how the NLO-extracted radial wave function |R_NLO_J/ψ(0)|^2=0.907 GeV^3 is obtained from Eq. (8) with the two-loop running coupling, and state explicitly which value of μ0 is used in the numerical extraction beyond the statement μ0=2mc.
[§III, Fig. 2, Fig. 3, Fig. 4] In the submitted version many axis labels and panel annotations in Figures 2–4 are unreadable. Please ensure the final figures have clearly legible labels, legends, and units.
[§III, Table III] Table III is titled as the rapidity-difference distribution of the J/ψ pair, but the text and Figure 4 suggest the same distribution is shown for ηc pairs as well. Please clarify whether Table III applies to both channels or only to J/ψ pairs.
[§III, Table IV] The X(6900) cross-section estimates in Table IV are orders of magnitude larger than the direct double-charmonium yields, and the text itself states this may reflect overestimated diphoton and di-charmonium decay fractions. Please label these values as model-dependent upper-bound-like estimates or remove the quantitative comparison, since the current presentation invites an unjustified quantitative interpretation.
[§III, feasibility discussion] The feasibility conclusion is based on reconstructed event counts of 2–12 per year after branching ratios, but no background estimate or detection-efficiency assumption is given. A quantitative background assessment, even a rough one, is needed to support the claim that these channels are experimentally accessible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the NLO gamma-gamma to charmonium-pair cross sections come from a previous independent calculation and are convolved with external photon fluxes, while the wave-function inputs are taken from the measured leptonic width and spin symmetry, not from the target UPC cross sections.

full rationale

The derivation chain is self-contained and not circular. The total UPC cross section in Eq. (1) is a convolution of the equivalent-photon fluxes n_i(x) with the parton-level cross section sigma_hat(gamma gamma -> H H). The latter is taken from the authors' previous NLO calculation [11], which is a separate published computation of gamma gamma -> J/psi J/psi and does not use any UPC cross section or any of the 28.0 nb / 65.1 nb results as input. The wave-function input |R_{J/psi}(0)|^2 is extracted in Eqs. (7)-(8) from the measured leptonic width Gamma(J/psi -> e+e-) = 5.55 keV, an external experimental quantity, and R_eta_c(0) is set equal to R_{J/psi}(0) by the standard heavy-quark spin symmetry relation; neither is fitted to the target processes. The scale dependence shown in Table I and the negative NLO differential cross sections at large |Delta y| in Table III are genuine limitations of the fixed-order truncation, but the paper reports them explicitly and does not hide them behind a circular argument. The X(6900) estimates use the Low formula with external decay-width estimates from VMD, NRA, and chi_c0 approximations, again independent of the double-charmonium predictions. The only self-citation that is load-bearing, Ref. [11], is an independent published calculation with stated assumptions that do not include the present target observables, so it does not constitute circularity under the stated criteria.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central predictions depend on NRQCD factorization, color-singlet dominance, the equivalent photon approximation, and the choice of wave-function input. The least controlled input is the truncated NLO series itself, which the paper's own Table III shows going negative; the relativistic corrections mentioned in Refs. [22,23] are not included in the calculation.

free parameters (5)

charm quark mass m_c = 1.5 +/- 0.1 GeV
Set to roughly half the J/psi mass (Eq. 7); varied to estimate uncertainty and affects the cross sections strongly.
J/psi radial wave function at the origin |R_J/psi(0)|^2 = LO: 0.528 GeV^3; NLO: 0.907 GeV^3
Extracted from the measured leptonic width Gamma(J/psi to e+e-) = 5.55 keV via Eq. 8; the eta_c wave function is set equal to this by heavy quark spin symmetry (Eq. 7).
minimum impact parameter b_min = R_A from 7.1 fm (Pb) to 0.7 fm (proton)
Chosen as the nucleus radius in the equivalent photon spectrum Eq. 2; no uncertainty is assigned even though it directly affects the photon flux normalization.
renormalization scale mu = 2 m_c, sqrt(4 m_c^2 + p_T^2), or sqrt(s_hat)
The scale is varied, with central predictions in Table I using sqrt(4 m_c^2 + p_T^2). Results vary by up to a factor of about 8 for the J/psi pair, so this choice is numerically important.
X(6900) two-photon width Gamma(X(6900) to gamma gamma) = 67 keV (VMD), 10 keV (NRA), 2 keV (chi_c0 approximation)
Three model estimates from Refs. [33-36] enter the Low formula Eq. 10 and produce the Table IV cross sections; the authors call the resulting rates overestimated.

assumptions (6)

domain assumption NRQCD factorization: double charmonium production separates into a perturbative short-distance coefficient and long-distance matrix elements.
The whole calculation is built on this factorization, introduced in Section II and used to define the NLO cross section in Eqs. 3 and 4.
domain assumption Color-singlet contributions dominate; color-octet contributions are suppressed by v^8 and neglected.
Stated in the Introduction as the reason the J/psi pair process is simpler than inclusive J/psi production; if color-octet terms are not negligible, the rates change.
domain assumption Heavy quark spin symmetry gives R_eta_c(0) = R_J/psi(0) at leading order in the relative velocity expansion.
Invoked in Eq. 7 and the following sentence; spin-symmetry breaking is not quantified.
domain assumption The equivalent photon approximation with the photon flux of Eq. 2 and b_min = R_A describes UPC as purely electromagnetic photon-photon fusion.
This is the core model for converting the gamma-gamma cross section into ion-ion cross sections in Eq. 1 and neglects hadronic and nuclear-breakup effects.
ad hoc to paper The truncated NLO QCD series gives physical cross sections at the chosen scale, with negative bins treated as a scale artifact.
Table III shows negative J/psi pair cross sections for |Delta y| > 2 at mu = sqrt(4 m_c^2 + p_T^2); the paper asserts that resetting the scale or resummation may cure this, but the quoted NLO result is itself unphysical at that scale.
domain assumption The Low formula Eq. 10 relates the X(6900) photon-fusion cross section to its two-photon width and the effective photon luminosity.
Used for the X(6900) estimates in Table IV; relies on a narrow-resonance approximation that is questionable for a broad state like X(6900).

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Cite this review

Pith. "Pith review of Charmonium pair production in ultraperipheral collision." pith.science (2026). https://pith.science/paper/O7REIUZM

@misc{pith2026250414850,
  author       = {Pith},
  title        = {Pith review of: Charmonium pair production in ultraperipheral collision},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O7REIUZM}},
  note         = {Machine review of arXiv:2504.14850}
}

abstract

We study the exclusive double charmonium ($J/\psi \mbox{-} J/\psi$ and $\eta_c \mbox{-} \eta_c$) production through photon-photon fusion via ultraperipheral collision (UPC) at the HL-LHC and FCC with next-to-leading order (NLO) QCD predictions in the framework of non-relativistic QCD (NRQCD). Numerical results indicate that the NLO corrections for $J/\psi$ pair are large and negative, while positive for $\eta_c$ pair. The total cross section of $J/\psi \mbox{-} J/\psi$ ($\eta_c \mbox{-} \eta_c$) in Pb-Pb UPC is 28.0 (65.1) nb at nucleon-nucleon c.m. energy $\sqrt{s_{NN}} = 5.52$ TeV. Due to the backgrounds from various QCD interactions at UPC are highly suppressed and the event topologies for charmonium pair are easy to tag, the phenomenological studies at the LHC and FCC are feasible. The detailed transverse momentum $p_T$, diphoton invariant mass $m_{\gamma\gamma}$ and the rapidity difference $\Delta y$ distributions are given. The production for X(6900) is also discussed.

Figures

Figures reproduced from arXiv: 2504.14850 by the authors.

**Figure 2.** FIG. 2: The transverse momentum [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** FIG. 3: The invariant mass [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: The absolute value of rapidity difference of the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

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

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