pith. sign in

arxiv: 2606.22231 · v1 · pith:KYXQHIRGnew · submitted 2026-06-20 · ✦ hep-ph

Rare Exclusive Decays of the Z-boson into S-wave Quarkonia within the Bethe-Salpeter Formalism

Asif Ali , Yi-Jie Li , Guang-Zhi Xu , Kui-Yong Liu This is my paper

Pith reviewed 2026-06-26 11:33 UTC · model grok-4.3

classification ✦ hep-ph
keywords Z boson decaysquarkoniaBethe-Salpeter formalismcharmoniumbottomoniumNRQCDradiative decaysdouble production
0
0 comments X

The pith

Z-boson decays to charmonium pairs and radiative charmonium yield larger branching fractions in the Bethe-Salpeter formalism than in NRQCD, while bottomonium fractions are smaller.

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

The paper computes rare Z-boson decays into pairs of S-wave quarkonia (Z to VV and Z to VP) and into a single S-wave quarkonium plus a photon using the Bethe-Salpeter formalism for the bound states. Leading-order QCD and QED amplitudes are included, along with a new mixed bottomonium-charmonium channel via virtual photon. Results exceed NRQCD predictions for all charmonium modes but fall below them for bottomonium modes. The authors conclude that charmonium must be treated as a relativistic system while bottomonium behaves as a non-relativistic one. The heavy-quark limit is used throughout to reduce the algebra.

Core claim

Within the Bethe-Salpeter formalism and the heavy-quark limit, the calculated branching fractions for double-charmonium and radiative-charmonium Z decays exceed the corresponding NRQCD values, whereas the bottomonium branching fractions are smaller than NRQCD results. This pattern demonstrates that charmonium is a relativistic particle while bottomonium is non-relativistic.

What carries the argument

Bethe-Salpeter wave functions for S-wave quarkonia in the heavy-quark limit, which supply the relativistic bound-state amplitudes for both QCD and QED transition diagrams.

If this is right

  • The mixed QED channel Z to J/psi plus Upsilon(1S) and its charge-conjugate processes receive explicit leading-order predictions.
  • Radiative modes Z to eta_c gamma, Z to J/psi gamma, Z to eta_b gamma and Z to Upsilon gamma obtain new numerical estimates that differ from NRQCD.
  • Charmonium production rates require relativistic corrections beyond the NRQCD approximation.
  • Bottomonium rates remain consistent with a non-relativistic treatment.

Where Pith is reading between the lines

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

  • If the pattern holds, Bethe-Salpeter calculations would be preferred for any process involving the lighter charmonium system.
  • Precision measurements at a future Z factory could directly discriminate between the two formalisms for these rare decays.
  • The same Bethe-Salpeter amplitudes could be reused for related rare decays such as Z to quarkonium plus light meson.

Load-bearing premise

The heavy quark limit is adopted to simplify calculations.

What would settle it

A measured branching ratio for Z to J/psi J/psi or Z to J/psi gamma that matches the NRQCD number within experimental precision while lying outside the Bethe-Salpeter band would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.22231 by Asif Ali, Guang-Zhi Xu, Kui-Yong Liu, Yi-Jie Li.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: The radial wave functions of the ground states ( [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Lowest-order QCD Feynman diagrams for the Z [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Lowest-order QED Feynman diagrams for the Z [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Lowest-order Feynman diagrams for Z → X(QQ¯)γ process. Here ψ(q⊥) is the 3D-BS wavefunction of vector or pseudoscalar quarkonium, and k and ǫγ denote momentum and polarization vector of radiative photon. After substituting the BS wavefunction of the corresponding quarkonium in Eq. 28 and taking the Dirac trace, we obtained their amplitudes: MZ→P γ = − i 2 3 √ 3eeQggv cos θw(M2 Z − M2 p ) Z d 3~q (2π) 3 φp(… view at source ↗
read the original abstract

This paper investigates the rare decays of the Z boson into S-wave quarkonia within the Bethe-Salpeter formalism. Both the production of double S-wave quarkonia and radiative decays into single S-wave quarkonium are analyzed. For double quarkonia production, i.e., $Z\to VV$ and $Z\to VP$ (where $V$ and $P$ represent vector and pseudoscalar quarkonia, respectively), we consider the leading order contribution from both QCD as well as electromagnetic transition via virtual photon (QED) amplitudes. The heavy quark limit is adopted to simplify calculations. Additionally, we have introduced another possible leading order channel for Z-boson decays to double S-wave quarkonia, where the Z boson decays into bottomonium plus charmonium via QED amplitude. Such processes may include $Z\to J/\psi+\Upsilon(1S)$, $Z\to\eta_{b}+J/\psi$ and $Z\to\eta_{c}+\Upsilon(1S)$. Moreover, we have also studied radiative Z boson decays to S-wave quarkonium, namely $Z\to X(Q\bar{Q})\gamma$, where $X(Q\bar{Q})=J/\psi$, $\Upsilon$, $\eta_{c}$, and $\eta_{b}$. Interestingly, for double charmonium and radiative charmonium production, our results are larger than the NRQCD finding, while for the bottomonium case, our findings are comparatively smaller. This shows that charmonium is a relativistic particle, while bottomonium is a non-relativistic particle.

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

1 major / 2 minor

Summary. The manuscript investigates rare exclusive decays of the Z-boson into S-wave quarkonia using the Bethe-Salpeter formalism. It analyzes double S-wave quarkonia production (Z→VV, Z→VP) including QCD and QED contributions, introduces a new QED channel for mixed bottomonium-charmonium states, and studies radiative decays Z→X(QQ̄)γ. The heavy quark limit is used to simplify calculations. Results for charmonium processes are larger than NRQCD, while for bottomonium they are smaller, interpreted as indicating that charmonium is relativistic and bottomonium is non-relativistic.

Significance. If the results hold, the work offers a relativistic treatment of these rare decays via the Bethe-Salpeter approach, providing a comparison to NRQCD that could highlight the importance of relativistic effects in lighter quarkonia. The introduction of the mixed flavor production channel is a novel contribution. Credit is given for considering both QCD and QED amplitudes and for exploring the implications for quarkonium dynamics.

major comments (1)
  1. [Abstract and §2] Abstract and §2: The heavy quark limit is adopted to simplify the Bethe-Salpeter amplitudes for both charmonium and bottomonium. Yet the central interpretive claim uses larger BS results (vs NRQCD) for charmonium processes to conclude that charmonium is relativistic while bottomonium is non-relativistic. This is load-bearing for the conclusion but internally inconsistent: if the limit suffices to trust the numerics for both systems, the deviation for charm should be small; the observed deviation instead questions the limit's applicability to charm without an independent check (e.g., extracted v² or explicit 1/m correction size) referenced in the text.
minor comments (2)
  1. The abstract would benefit from quoting at least one or two explicit branching-ratio values (or ratios to NRQCD) to make the size of the reported deviations concrete.
  2. Ensure all NRQCD comparison references are explicitly cited with paper numbers in the text and reference list.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment. We address the major point below.

read point-by-point responses

  1. Referee: [Abstract and §2] Abstract and §2: The heavy quark limit is adopted to simplify the Bethe-Salpeter amplitudes for both charmonium and bottomonium. Yet the central interpretive claim uses larger BS results (vs NRQCD) for charmonium processes to conclude that charmonium is relativistic while bottomonium is non-relativistic. This is load-bearing for the conclusion but internally inconsistent: if the limit suffices to trust the numerics for both systems, the deviation for charm should be small; the observed deviation instead questions the limit's applicability to charm without an independent check (e.g., extracted v² or explicit 1/m correction size) referenced in the text.

    Authors: The heavy quark limit is applied uniformly to simplify the propagators and reduce the BS amplitudes to their leading covariant structures for both systems. Within this framework the BS equation remains relativistic, so the resulting amplitudes differ from the NRQCD expansion even after the limit is taken. The larger rates obtained for charmonium therefore indicate that relativistic kinematics and binding effects retained by the BS approach are numerically important for charm, while the smaller rates for bottomonium show closer agreement with the non-relativistic expectation. The direct numerical comparison to NRQCD thus functions as the consistency check on the applicability of the limit. We will add a clarifying paragraph in §2 and a corresponding sentence in the abstract stating that the observed deviations themselves test the validity of the heavy-quark approximation for each system. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation remains independent of its inputs

full rationale

The paper applies the Bethe-Salpeter formalism to Z-boson decay amplitudes, explicitly adopting the heavy-quark limit as an approximation to simplify the wave functions and vertices for both charmonium and bottomonium channels. Numerical results are then compared to existing NRQCD literature values, with the difference interpreted as evidence of relativistic effects in charm systems. No equation in the provided text reduces a computed observable to a parameter fitted from the same observable; the heavy-quark limit is stated as an external modeling choice rather than derived from the target decay rates; and no self-citation chain is invoked to justify uniqueness or to rename a fitted quantity as a prediction. The central interpretive claim therefore rests on an external benchmark (NRQCD) rather than on any internal redefinition or tautological substitution, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities listed. Relies on standard Bethe-Salpeter wave functions and the heavy quark limit stated as an approximation.

axioms (1)
  • domain assumption Heavy quark limit simplifies calculations
    Explicitly adopted in the abstract to simplify the Bethe-Salpeter treatment of quarkonia.

pith-pipeline@v0.9.1-grok · 5827 in / 1251 out tokens · 21069 ms · 2026-06-26T11:33:46.728640+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

39 extracted references · 5 linked inside Pith

  1. [1]

    Search for Hi ggs and Z boson decays to J/ψ or Υ pairs in the four-muon final state in proton-proton collisio ns at √ s = 13 TeV

    A.M. Sirunyan et al. (CMS Collaboration), “Search for Hi ggs and Z boson decays to J/ψ or Υ pairs in the four-muon final state in proton-proton collisio ns at √ s = 13 TeV”, Phys. Lett. B 797, 134811 (2019). 17

    2019

  2. [2]

    Search fo r Higgs boson decays into Z and J/ψ and for Higgs and Z boson decays into J/ψ or Υ pairs in p-p collisions at √ s = 13 TeV

    Tumasyan, Armen, et al. (CMS Collaboration), “Search fo r Higgs boson decays into Z and J/ψ and for Higgs and Z boson decays into J/ψ or Υ pairs in p-p collisions at √ s = 13 TeV”, Physics Letters B 842, 137534 (2023)

    2023

  3. [3]

    Aad et al

    G. Aad et al. [ATLAS], Searches for exclusive Higgs and Z b oson decays into a vector quarkonium state and a photon using 139 fb − 1 of ATLAS √ s = 13 TeV proton–proton collision data, Eur. Phys. J. C 83, 781 (2023)

    2023

  4. [4]

    Aad et al

    G. Aad et al. [ATLAS], Search for Higgs and Z Boson Decays t o J/ψγ and Υ(nS)γ with the ATLAS Detector, Phys. Rev. Lett. 114, 121801 (2015)

    2015

  5. [5]

    Baer et al

    H. Baer et al. [ILC], The International Linear Collider T echnical Design Report- Volume 2: Physics, arXiv:1306.6352 [hep-ph]

    Pith/arXiv arXiv

  6. [6]

    J. P. Ma and Z. X. Zhang (The super Z-factory group), Prefa ce, Sci. China: Phys., Mech. Astron. 53, 1947(2010)

    1947

  7. [7]

    J. B. Guimarães da Costa et al. [CEPC Study Group], CEPC Co nceptual Design Report: Volume 2- Physics & Detector, arXiv:1811.10545 [hep-ex]

    Pith/arXiv arXiv

  8. [8]

    Abada et al

    A. Abada et al. [FCC], FCC-ee: The Lepton Collider: Futur e Circular Collider Conceptual Design Report Volume 2, Eur. Phys. J. ST 228, 261 (2019)

    2019

  9. [9]

    On the rare decays Z → VV and Z → VP

    L. Bergström and R.W. Robinett, “On the rare decays Z → VV and Z → VP ”, Phys. Rev. D 41, 3513 (1990)

    1990

  10. [10]

    Note on rare Z-boson decays to dou ble heavy quarkonia

    D.-N. Gao and X. Gong, “Note on rare Z-boson decays to dou ble heavy quarkonia”, Chin. Phys. C 47, 043106 (2023)

    2023

  11. [11]

    Analysis of double- J/ψ production in Z decay at next-to- leading-order QCD accuracy

    C. Li, Z. Sun, and G.-Y. Zhang, “Analysis of double- J/ψ production in Z decay at next-to- leading-order QCD accuracy”, J. High Energ. Phys. 10, 001 (2 023)

  12. [12]

    Double charmonia pr oduction in exclusive Z-boson decays

    A.K. Likhoded and A.V. Luchinsky, “Double charmonia pr oduction in exclusive Z-boson decays”, Mod. Phys. Lett. A 33, 1850078 (2018)

    2018

  13. [13]

    Rare D ecays of the Z 0

    B. Guberina, J.H. Kühn, R.D. Peccei and R. Röckl, “Rare D ecays of the Z 0”, Nucl. Phys. B 174, 317 (1980)

    1980

  14. [14]

    A. V. Luchinsky, Leading order NRQCD and Light-Cone Ana lysis of Exclusive Charmonia Pro- duction in Radiative Z-boson Decays, arXiv:1706.04091

    Pith/arXiv arXiv

  15. [15]

    T. C. Huang and F. Petriello, Rare exclusive decays of th e Z-boson revisited, Phys. Rev. D 92, 014007 (2015). 18

    2015

  16. [16]

    Z-boso n decays into S-wave quarkonium plus a photon up to O(αsv2) corrections

    G.-Y. Wang, X.-C. Zheng, X.-G. Wu, and G.-Z. Xu, “Z-boso n decays into S-wave quarkonium plus a photon up to O(αsv2) corrections”, Phys. Rev. D 109, 074004 (2024)

    2024

  17. [17]

    A dynamical principle f or 3D–4D interlinkage in Salpeter-like equations

    A.N. Mitra and B.M. Sodermark, “A dynamical principle f or 3D–4D interlinkage in Salpeter-like equations”, Nucl. Phys. A 695, 328 (2001)

    2001

  18. [18]

    Mass Corrections to the Fine Structure of Hydrogen-like Atoms

    E.E. Salpeter, “Mass Corrections to the Fine Structure of Hydrogen-like Atoms”, Phys. Rev. 87, 328 (1952)

    1952

  19. [19]

    Decays of the meson BC to a P-wave charmonium state χc or hc

    C. H. Chang, Y. Q. Chen, G. L. Wang, and H. S. Zong, “Decays of the meson BC to a P-wave charmonium state χc or hc”, Phys. Rev. D 65(1), 014017 (2001)

    2001

  20. [20]

    Decay constants of heavy vector mesons in r elativistic Bethe-Salpeter method

    G.-L. Wang, “Decay constants of heavy vector mesons in r elativistic Bethe-Salpeter method”, Phys. Lett. B 633, 492 (2006)

    2006

  21. [21]

    Some of semileptonic and nonleptonic decays of Bc meson in a Bethe-Salpeter relativistic quark model

    C. Chang, H.-F. Fu, G.-L. Wang, and J.-M. Zhang, “Some of semileptonic and nonleptonic decays of Bc meson in a Bethe-Salpeter relativistic quark model”, Sci. C hina Phys. Mech. Astron. 58, 011001 (2015)

    2015

  22. [22]

    (γ,χ cJ(J = 0, 1)) and (γ,η c) production in electron-positron annihilation at √ s = 10.6 GeV and 4.6 GeV in the framework of Bethe-Salpeter equa tion

    S. Bhatnagar and V. Guleria, “ (γ,χ cJ(J = 0, 1)) and (γ,η c) production in electron-positron annihilation at √ s = 10.6 GeV and 4.6 GeV in the framework of Bethe-Salpeter equa tion”, Nucl. Phys. A 1041, 122783 (2024)

    2024

  23. [23]

    Double heav y-quarkonium production from electron-positron annihilation in the Bethe-Salpeter for malism

    X.-H. Guo, H.-W. Ke, X.-Q. Li, and X.-H. Wu, “Double heav y-quarkonium production from electron-positron annihilation in the Bethe-Salpeter for malism”, arXiv:0804.0949 [hep-ph] (2008)

    Pith/arXiv arXiv 2008

  24. [24]

    Double Charmonium Product ions in Electron-Positron Annihila- tion Using Bethe-Salpeter Approach

    H. Negash and S. Bhatnagar, “Double Charmonium Product ions in Electron-Positron Annihila- tion Using Bethe-Salpeter Approach”, Adv. High Energy Phys . 2019, 4029356 (2019)

    2019

  25. [25]

    Spectrum for heavy quarkonia a nd mixture of the relevant wave functions within the framework of Bethe-Salpeter equation

    C. Chang and G.-L. Wang, “Spectrum for heavy quarkonia a nd mixture of the relevant wave functions within the framework of Bethe-Salpeter equation ”, Sci. China Phys. Mech. Astron. 53, 2005 (2010)

    2005

  26. [26]

    Next-to-leading order QCD and relativistic corrections to Z → J/ψ + Υ(nS)

    G.-Y. Wang, X.-C. Zheng and G.-Z. Xu, “Next-to-leading order QCD and relativistic corrections to Z → J/ψ + Υ(nS)”, arXiv:2604.19439 [hep-ph] (2026)

    Pith/arXiv arXiv 2026

  27. [27]

    Cross section for double charmonium production in electron- positron annihilation at energy √ s = 10. 6 GeV

    E. Mengesha and S. Bhatnagar, “Cross section for double charmonium production in electron- positron annihilation at energy √ s = 10. 6 GeV”, Int. J. Mod. Phys. E 20, 2521–2533 (2011)

    2011

  28. [28]

    Observation of do uble c¯c production in e+e− annihilation at √ s ≈ 10. 6 GeV

    K. Abe et al. (Belle Collaboration), “Observation of do uble c¯c production in e+e− annihilation at √ s ≈ 10. 6 GeV”, Phys. Rev. Lett. 89, 142001 (2002). 19

    2002

  29. [29]

    Study of Double Ch armonium Production in e+e− Annihi- lation at √ s ≈ 10. 6 GeV

    K. Abe et al. (Belle Collaboration), “Study of Double Ch armonium Production in e+e− Annihi- lation at √ s ≈ 10. 6 GeV”, Phys. Rev. D 70, 071102 (2004)

    2004

  30. [30]

    Measurement o f double charmonium production in e+e− annihilations at √ s = 10. 6 GeV

    B. Aubert et al. (BaBar Collaboration), “Measurement o f double charmonium production in e+e− annihilations at √ s = 10. 6 GeV”, Phys. Rev. D 72, 031101 (2005)

    2005

  31. [31]

    Exclusive Double Charmonium Pro duction from e+e− Annihilation into a Virtual Photon

    E. Braaten and J. Lee, “Exclusive Double Charmonium Pro duction from e+e− Annihilation into a Virtual Photon”, Phys. Rev. D 67, 054007 (2003) [Erratum: P hys. Rev. D 72, 099901 (2005)]

    2003

  32. [32]

    Problems of double c harm production in e+e− annihilation at √ s = 10. 6 GeV

    K.-Y. Liu, Z.-G. He and K.-T. Chao, “Problems of double c harm production in e+e− annihilation at √ s = 10. 6 GeV”, Phys. Lett. B 557, 45–54 (2003)

    2003

  33. [33]

    Next-to-leadin g order QCD correction to e+e− → J/ψ +ηc at √ s = 10. 6 GeV

    Y.-J. Zhang, Y.-J. Gao and K.-T. Chao, “Next-to-leadin g order QCD correction to e+e− → J/ψ +ηc at √ s = 10. 6 GeV”, Phys. Rev. Lett. 96, 092001 (2006)

    2006

  34. [34]

    QCD corrections to J/ψ plus ηc production in e+e− annihilation at√ s = 10. 6 GeV

    B. Gong and J.-X. Wang, “QCD corrections to J/ψ plus ηc production in e+e− annihilation at√ s = 10. 6 GeV”, Phys. Rev. D 77, 054028 (2008)

    2008

  35. [35]

    Relativistic correcti ons to J/ψ exclusive and inclusive double charm production at B factories

    Z.-G. He, Y. Fan and K.-T. Chao, “Relativistic correcti ons to J/ψ exclusive and inclusive double charm production at B factories”, Phys. Rev. D 75, 074011 (20 07)

  36. [36]

    Resummation of Relativis tic Corrections to e+e− → J/ψ +ηc

    G. T. Bodwin, J. Lee and C. Yu, “Resummation of Relativis tic Corrections to e+e− → J/ψ +ηc”, Phys. Rev. D 77, 094018 (2008)

    2008

  37. [37]

    O(αsv2) correction to J/ψ plusηc production in e+e− annihilation at√ s = 10. 6 GeV

    X.-H. Li and J.-X. Wang, “ O(αsv2) correction to J/ψ plusηc production in e+e− annihilation at√ s = 10. 6 GeV”, Chin. Phys. C 38, 043101 (2014)

    2014

  38. [38]

    Analytical two-loop ampli- tudes of e+e− → J/ψ +ηc at B factories

    X. Chen, X. Guan, C.-Q. He, Y.-Q. Ma, J. Wang and D.-J. Zha ng, “Analytical two-loop ampli- tudes of e+e− → J/ψ +ηc at B factories”, (2025)

    2025

  39. [39]

    Next-to-next-to- leading-order QCD corrections to J/ψ plus ηc production at the B factories

    X.-D. Huang, B. Gong and J.-X. Wang, “Next-to-next-to- leading-order QCD corrections to J/ψ plus ηc production at the B factories”, JHEP 02, 049 (2023). 20

    2023