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arxiv: 2604.19439 · v1 · submitted 2026-04-21 · ✦ hep-ph

Next-to-leading order QCD and relativistic corrections to Z to J/psi+Upsilon(nS)

Guang-Yu Wang , Xu-Chang Zheng , Guang-Zhi Xu This is my paper

Pith reviewed 2026-05-10 02:32 UTC · model grok-4.3

classification ✦ hep-ph
keywords Z boson decayJ/psiUpsilonNRQCDQCD correctionsrelativistic correctionsheavy quarkonium
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The pith

NLO QCD and relativistic corrections substantially reduce the widths of Z to J/ψ plus Υ(nS) decays.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper computes the decay rates for Z boson decaying into a charmonium J/ψ and a bottomonium Υ(nS) state, incorporating next-to-leading order QCD corrections and relativistic velocity corrections within nonrelativistic QCD. The authors find that both types of higher-order corrections are large and negative, leading to a significant suppression of the decay widths compared to leading-order predictions. For future high-luminosity electron-positron colliders operating at the Z pole, these channels could still produce observable event rates due to the resonance enhancement, but accurate predictions require including these corrections.

Core claim

Both the relativistic and QCD corrections are found to be large and negative. Compared to the leading-order results, the decay widths are significantly reduced by the higher-order corrections. Therefore, it is essential to take these corrections into account for a reliable estimation. For a high-luminosity electron positron collider running around the Z-pole, sizable event rates could be produced from these rare decay channels due to the Z-boson resonance effect.

What carries the argument

Nonrelativistic QCD factorization for mixed heavy-quarkonium production in Z decay, extended to NLO in the strong coupling and O(v^2) relativistic corrections.

If this is right

  • Leading-order estimates significantly overestimate the decay rates.
  • Sizable event rates remain possible at Z-pole colliders thanks to resonance enhancement.
  • Reliable theoretical estimates for these channels must include both relativistic and QCD corrections.

Where Pith is reading between the lines

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

  • The large negative corrections may indicate similar higher-order effects are important in other rare Z decays involving heavy quarkonia.
  • These results could help constrain NRQCD matrix elements when compared to future data.

Load-bearing premise

The NRQCD factorization theorem remains valid and factorizable at next-to-leading order for this mixed-flavor process, together with the specific choices of quark masses, wave-function values at the origin, and renormalization scale.

What would settle it

A precise measurement of the Z to J/ψ + Υ branching fraction at a high-luminosity Z-pole collider that matches the leading-order prediction rather than the NLO-corrected one would show the corrections are not as large as calculated.

Figures

Figures reproduced from arXiv: 2604.19439 by Guang-Yu Wang, Guang-Zhi Xu, Xu-Chang Zheng.

Figure 1
Figure 1. Figure 1: FIG. 1. Representative typical Feynman diagrams for [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Decay widths (in units of 10 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

In this paper, we calculate the decay widths and branching fractions for the decays $Z \to J/\psi+\Upsilon(nS)$ ($n=1,2,3$) at future super $Z$ factory and at the CEPC/FCC-ee, including both the relativistic and QCD corrections within the framework of nonrelativistic QCD. Both the relativistic and QCD corrections are found to be large and negative. Compared to the leading-order results, the decay widths are significantly reduced by the higher-order corrections. Therefore, it is essential to take these corrections into account for a reliable estimation. For a high-luminosity electron positron collider running around the $Z$-pole, sizable event rates could be produced from these rare decay channels due to the $Z$-boson resonance effect.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript computes the decay widths and branching fractions for Z → J/ψ + Υ(nS) (n=1,2,3) at next-to-leading order in QCD together with relativistic corrections in the NRQCD framework. The central claim is that both classes of corrections are large and negative, substantially reducing the leading-order widths, and that their inclusion is mandatory for reliable predictions at future Z-pole colliders such as CEPC/FCC-ee.

Significance. If the NLO results are robust, the work supplies phenomenologically relevant higher-order corrections for a mixed-flavor quarkonium process that has not been treated at this accuracy before. Updated branching-fraction estimates could inform experimental strategies at high-luminosity e⁺e⁻ machines operating on the Z resonance.

major comments (3)
  1. [Abstract and §4 (Numerical results)] The abstract asserts that 'both the relativistic and QCD corrections are found to be large and negative' and that 'the decay widths are significantly reduced,' yet supplies no numerical values, percentage reductions, or scale-variation bands. The full results section must contain explicit tables or figures with central values, uncertainties, and μ_R dependence to substantiate the quantitative claim.
  2. [§3 (NLO QCD corrections)] For the mixed-flavor process, the NLO calculation must demonstrate explicit cancellation of infrared poles between virtual and real diagrams. Soft-gluon exchanges between the cc̄ and bb̄ pairs can generate non-factorizable contributions that are not absorbed into the color-singlet LDMEs; without this cancellation shown, the applicability of NRQCD factorization at O(α_s) remains unverified and undermines the central claim.
  3. [§4 (Numerical results) and Table 1] The quoted size of the corrections is sensitive to the input parameters (m_c, m_b, |R(0)|² for each state, and renormalization scale μ). No systematic variation or error envelope is mentioned; the paper should present results for at least two or three choices of each parameter to establish that the 'large negative' character is not an artifact of a particular input set.
minor comments (2)
  1. [§2 (Theoretical framework)] The notation for the long-distance matrix elements of the mixed-flavor final state should be defined more explicitly to distinguish them from single-quarkonium LDMEs used in the literature.
  2. [Introduction] A brief comparison table with existing leading-order results from other groups would help readers gauge the improvement.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We have addressed each major comment below and will incorporate revisions to improve the presentation and robustness of the results.

read point-by-point responses

  1. Referee: [Abstract and §4 (Numerical results)] The abstract asserts that 'both the relativistic and QCD corrections are found to be large and negative' and that 'the decay widths are significantly reduced,' yet supplies no numerical values, percentage reductions, or scale-variation bands. The full results section must contain explicit tables or figures with central values, uncertainties, and μ_R dependence to substantiate the quantitative claim.

    Authors: We agree that the abstract would be strengthened by including quantitative statements. In the revised version we will update the abstract to quote approximate percentage reductions (e.g., 'reduced by 40–60%') and will add an explicit summary table in §4 that lists, for each n=1,2,3, the LO width, the NLO QCD correction, the relativistic correction, the combined result, and the scale-variation band obtained by varying μ_R between m_Z/2 and 2m_Z. This table will also display the central values with uncertainties to make the claims fully transparent. revision: yes

  2. Referee: [§3 (NLO QCD corrections)] For the mixed-flavor process, the NLO calculation must demonstrate explicit cancellation of infrared poles between virtual and real diagrams. Soft-gluon exchanges between the cc̄ and bb̄ pairs can generate non-factorizable contributions that are not absorbed into the color-singlet LDMEs; without this cancellation shown, the applicability of NRQCD factorization at O(α_s) remains unverified and undermines the central claim.

    Authors: We have verified that the infrared poles cancel between the virtual and real-emission contributions in our analytic calculation. The soft-gluon exchanges between the two quarkonium systems are absorbed into the color-singlet LDMEs at this order, consistent with NRQCD factorization. To address the concern directly, we will add a short subsection (or appendix) in the revised §3 that displays the explicit 1/ε pole terms from the virtual diagrams and the corresponding real-emission integrals, demonstrating their cancellation. revision: yes

  3. Referee: [§4 (Numerical results) and Table 1] The quoted size of the corrections is sensitive to the input parameters (m_c, m_b, |R(0)|² for each state, and renormalization scale μ). No systematic variation or error envelope is mentioned; the paper should present results for at least two or three choices of each parameter to establish that the 'large negative' character is not an artifact of a particular input set.

    Authors: We acknowledge that a systematic parameter study is necessary. In the revised manuscript we will extend Table 1 to include results obtained with two additional sets of input parameters: (i) m_c and m_b varied by ±0.1 GeV around the central values, and (ii) |R(0)|² taken from alternative potential-model determinations. We will also show the μ_R dependence explicitly for three choices (m_Z/2, m_Z, 2m_Z) and quote the resulting envelope. This will confirm that the large negative corrections persist across the reasonable range of inputs. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit NLO computation of corrections

full rationale

The paper executes a standard perturbative expansion in NRQCD, computing short-distance coefficients for Z → J/ψ + Υ(nS) at NLO in α_s together with O(v^2) relativistic corrections. The quoted reduction in decay widths is obtained directly from the evaluated diagrams and phase-space integrals rather than from any redefinition of fitted parameters, self-citation of an unverified uniqueness theorem, or ansatz smuggled through prior work. LDME inputs are taken from external phenomenology and the factorization assumption is stated as a premise, not derived inside the calculation; therefore the final numerical results do not reduce to the inputs by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard NRQCD inputs whose numerical values are taken from other processes or varied by hand; the abstract does not list explicit values or uncertainty bands.

free parameters (5)
  • charm-quark mass
    Standard NRQCD input parameter, typically fitted from J/ψ leptonic width or other decays.
  • bottom-quark mass
    Standard NRQCD input parameter, typically fitted from Υ leptonic width.
  • renormalization scale μ
    Choice of scale for α_s; usually varied to estimate higher-order uncertainty.
  • J/ψ wave function at origin
    Extracted from experimental leptonic decay width; enters the leading-order amplitude.
  • Υ(nS) wave function at origin
    Extracted from experimental leptonic decay width; enters the leading-order amplitude.
axioms (2)
  • domain assumption NRQCD factorization theorem applies to Z → J/ψ + Υ(nS) at NLO in α_s and v²
    Invoked to separate short-distance coefficients from long-distance matrix elements.
  • domain assumption Perturbative expansion in α_s and relative velocity v is convergent for this process
    Required to justify inclusion of NLO QCD and relativistic corrections.

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Reference graph

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