Pith. sign in

REVIEW 2 major objections 2 minor 37 references

Rare Radiative Decays of Z boson into S- and P-wave Charmonium within the Bethe-Salpeter Model

T0 review · 2 major / 2 minor · reviewed 2026-06-26 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Bethe-Salpeter calculations find a higher branching fraction for Z to J/ψ gamma than prior theory.

desk verdict BS model gives new branching fraction numbers for Z to charmonium gamma, but the claim that prior J/psi results were underestimated rests on untested approximations. read the letter →

arxiv 2606.24189 v1 pith:TM2ONPID submitted 2026-06-23 hep-ph

Asif Ali , Yi-Jie Li , Guang-Zhi Xu , Kui-Yong Liu This is my paper · ORCID
classification hep-ph
keywords Bethe-SalpeterequationZbosonradiativedecayscharmoniumbranchingfractionsJ/psiinstantaneousapproximationS-wavestatesP-wave
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

The paper applies the Bethe-Salpeter model in its instantaneous approximation to compute branching fractions for radiative decays of the Z boson into S-wave charmonium states psi(nS) and eta_c(nS) as well as P-wave states chi_c(mP) and h_c(mP). The authors evaluate all amplitudes from leading-order Feynman diagrams and obtain numerical results for n and m up to 3. Their key result is that the J/ψ channel branching fraction exceeds earlier theoretical estimates, which would make observation more feasible at future experiments. A reader would care because these rare processes test how bound-state dynamics in heavy quarks couple to electroweak currents.

What carries the argument

Instantaneous approximation to the Bethe-Salpeter equation, supplying the charmonium wave functions that enter the leading-order decay amplitudes.

What would settle it

A measured branching fraction for Z to J/ψ gamma that lies well below the Bethe-Salpeter prediction would show the calculation overestimates the rate.

Watch

Extended reading notes

Core claim

Within the instantaneous approximation of the Bethe-Salpeter equation the paper calculates the decay amplitudes for Z boson radiative transitions to charmonium using leading-order Feynman diagrams. For the J/ψ channel the resulting branching fraction is larger than previous theoretical results, indicating that those earlier estimates were likely too low.

Load-bearing premise

The instantaneous approximation of the Bethe-Salpeter equation together with leading-order Feynman diagrams suffice to compute the decay amplitudes for all listed channels.

Editorial extensions

If this is right

  • Branching fractions are now available for all listed S- and P-wave channels as direct predictions.
  • The larger J/ψ rate raises the expected number of events at a future Z factory.
  • The same framework supplies consistent rates for the radial excitations n=2 and n=3.
  • Comparison with data on multiple channels can test the model uniformly across states.

Reading between the lines

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

  • The result implies that relativistic effects retained in the Bethe-Salpeter approach matter for electroweak decay rates of charmonium.
  • Similar calculations could be performed for radiative decays of the Higgs boson into the same states.
  • Tension with other quarkonium models would isolate the role of the instantaneous kernel choice.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript employs the Bethe-Salpeter model in the instantaneous approximation to compute branching fractions for the radiative decays Z → charmonium + γ. It treats S-wave states (ψ(nS), η_c(nS) for n=1,2,3) and P-wave states (χ_c0(mP), χ_c1(mP), h_c(mP) for m=1,2) via leading-order Feynman diagrams and concludes that the J/ψ branching fraction is larger than prior theoretical estimates, making it more promising for experiment.

Significance. If the numerical results hold under the stated approximations, the work supplies updated predictions for rare Z decays that could guide searches at future e+e- colliders. The relativistic bound-state treatment via the BS equation is a standard tool in this area and the explicit listing of multiple channels provides a systematic survey.

major comments (2)
  1. [Abstract and results section] Abstract and results section: the central claim that prior J/ψ results were underestimated rests on decay amplitudes obtained in the instantaneous BS approximation with only LO diagrams. Because the Z mass sets a hard scale and the photon carries O(M_Z) momentum, the neglect of retardation and higher-order corrections can shift the overlap integrals by an amount comparable to the reported difference; no sensitivity study or comparison to a covariant kernel is provided to demonstrate control.
  2. [Method of amplitude construction] Method of amplitude construction: the instantaneous approximation sets the relative energy component to zero by construction, which directly affects the momentum-space wave-function overlap that enters the Z → charmonium + γ matrix element. Without an explicit check (e.g., variation of the kernel or comparison to a non-instantaneous treatment) it is unclear whether the enhancement relative to earlier work is physical or an artifact of the truncation.
minor comments (2)
  1. [Notation] Notation for the P-wave states should be clarified (e.g., explicit definition of the polarization tensors or radial wave functions used for χ_c1 and h_c).
  2. [Results] A table comparing the new branching fractions directly with the numerical values from the cited prior works would make the “underestimated” statement quantitative rather than qualitative.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, clarifying our use of the instantaneous Bethe-Salpeter framework while acknowledging its limitations.

read point-by-point responses

  1. Referee: [Abstract and results section] Abstract and results section: the central claim that prior J/ψ results were underestimated rests on decay amplitudes obtained in the instantaneous BS approximation with only LO diagrams. Because the Z mass sets a hard scale and the photon carries O(M_Z) momentum, the neglect of retardation and higher-order corrections can shift the overlap integrals by an amount comparable to the reported difference; no sensitivity study or comparison to a covariant kernel is provided to demonstrate control.

    Authors: The instantaneous approximation is the standard reduction of the BS equation employed in the majority of charmonium decay calculations in the literature. Our enhancement for the J/ψ channel originates from the relativistic momentum-space wave functions solved with the chosen kernel, which differ systematically from the non-relativistic or quark-model inputs used in earlier estimates. We agree that a dedicated sensitivity study or covariant comparison would strengthen the robustness claim; we will therefore add an explicit paragraph in the revised manuscript discussing the expected size of retardation effects and the model dependence of the result. revision: partial

  2. Referee: [Method of amplitude construction] Method of amplitude construction: the instantaneous approximation sets the relative energy component to zero by construction, which directly affects the momentum-space wave-function overlap that enters the Z → charmonium + γ matrix element. Without an explicit check (e.g., variation of the kernel or comparison to a non-instantaneous treatment) it is unclear whether the enhancement relative to earlier work is physical or an artifact of the truncation.

    Authors: Setting the relative energy to zero is the defining step of the instantaneous approximation and is applied uniformly to both the bound-state wave functions and the decay amplitude. This truncation is the same one used in our prior BS studies of charmonium transitions, where it reproduces known decay constants and widths to within typical hadronic uncertainties. The reported enhancement is therefore a direct consequence of the relativistic overlap integrals obtained inside this consistent framework rather than an uncontrolled artifact. A full non-instantaneous calculation lies outside the scope of the present work. revision: no

standing simulated objections not resolved
  • A quantitative comparison to a covariant (non-instantaneous) Bethe-Salpeter kernel or an explicit retardation study would require a separate, substantially larger computational project.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; decay rates computed from independently solved wave functions

full rationale

The paper solves the instantaneous Bethe-Salpeter equation for charmonium wave functions (typically with parameters fixed by the meson spectrum) and then evaluates the Z → charmonium + γ matrix elements at leading order. No quoted step shows a branching fraction being fitted to itself or renamed as a prediction; the central numerical claim (higher J/ψ rate than prior work) is an output of the overlap integrals, not an input. The derivation therefore remains self-contained against external benchmarks such as measured masses and other decay channels.

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

No full manuscript text is available, so free parameters, axioms, and invented entities cannot be extracted; the abstract alone does not list any.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Rare Radiative Decays of Z boson into S- and P-wave Charmonium within the Bethe-Salpeter Model." pith.science (2026). https://pith.science/paper/TM2ONPID

@misc{pith2026260624189,
  author       = {Pith},
  title        = {Pith review of: Rare Radiative Decays of Z boson into S- and P-wave Charmonium within the Bethe-Salpeter Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TM2ONPID}},
  note         = {Machine review of arXiv:2606.24189}
}
abstract

We employ the Bethe-Salpeter (BS) model to investigate radiative decays of the Z boson into charmonium. Within the instantaneous approximation of the BS equation, we analyze the relevant decay channels for S-wave charmonium states ($\psi(nS)$, $\eta_{c}(nS)$, $n=1,2,3$) and P-wave charmonium states ($\chi_{c0}(mP)$, $\chi_{c1}(mP)$, $h_{c}(mP)$, $m=1,2$). We compute all branching fractions using leading-order Feynman diagrams. For the $J/\psi$ channel, our BS calculations suggest that prior theoretical results were likely underestimated, making our prediction more promising for future experimental measurements.

Figures

Figures reproduced from arXiv: 2606.24189 by the authors.

Figure 1
Figure 1. Leading-order Feynman diagrams for the Z → HQQ¯γ process, where HQQ¯ denotes a charmo￾nium state. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Coefficients C1 (left) and Ci , C′ i (right) multiplied by M2 Z versus |~q| in the range 0 < |~q| < 4 GeV for the ηc state. It follows from the expressions of Ci given in Eqs. 23 and 28 that all Ci , C′ i carry dimensions of 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Coefficients C1 (left) and Ci , C′ i (right) multiplied by M2 Z versus |~q| in the range 0 < |~q| < 4 GeV for the J/ψ state. We can obtain ψ ++ v (q⊥) and ψ −− v (q⊥) using Eq. 7. Substituting them into Eq. 40 and Eq. 41, taking 12 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Coefficients C1 (left) and Ci , C′ i (right) multiplied by M2 Z versus |~q| in the range 0 < |~q| < 4 GeV for the χc0(1P) state. The corresponding decay width reads as ΓZ→χc0γ = 1 2 8π  eeQggQ v cos θW 2 (M2 Z − M2 ) M5 Z h 4M2 ZX 2 2 + (X1 − X3) 2 (M2 Z − M2 ) 2 i (…
Figure 5
Figure 5. Figure 5: Coefficients C1 (left) and Ci , C′ i (right) multiplied by M2 Z versus |~q| in the range 0 < |~q| < 4 GeV for the χc1(1P) state. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: , with hc(1P) taken as an example. C1 0 1 2 3 4 -20 -10 0 10 20 |q| (GeV) C1×MZ 2 C2 C3 C1 ' C2 ' C3 ' 0 1 2 3 4 -3 -2 -1 0 1 2 |q| (GeV) Coefficient values×MZ 2 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references

  1. [1]

    Bodwin, Eric Braaten, and G

    Geoffrey T. Bodwin, Eric Braaten, and G. Peter Lepage. Err atum: Rigorous qcd analysis of inclusive annihilation and production of heavy quarkonium [phys. rev. d51, 1125 (1995)]. Physical Review D , 55(9):5853–5854, May 1997. 26

    1995

  2. [2]

    Measurement of double charmonium p roduction in e+e− annihilations at√ s = 10

    Bernard Aubert et al. Measurement of double charmonium p roduction in e+e− annihilations at√ s = 10. 6 GeV. Phys. Rev. D , 72:031101, 2005

    2005

  3. [3]

    Study of double charmonium production in e+ e- annihilation at s**(1/2) ~ 10.6-GeV

    Kazuo Abe et al. Study of double charmonium production in e+ e- annihilation at s**(1/2) ~ 10.6-GeV. Phys. Rev. D , 70:071102, 2004

    2004

  4. [4]

    Problems of double charm production in e+ e- annihilation at s**(1/2) = 10.6-GeV

    Kui-Yong Liu, Zhi-Guo He, and Kuang-Ta Chao. Problems of double charm production in e+ e- annihilation at s**(1/2) = 10.6-GeV. Phys. Lett. B , 557:45–54, 2003

    2003

  5. [5]

    E. E. Salpeter. Mass corrections to the fine structure of h ydrogen-like atoms. Physical Review , 87(2):328–343, 1952

    1952

  6. [6]

    Relativistic effects in the semileptonic bc decays to charmonium with the b ethe-salpeter method

    Zi-Kan Geng, Tianhong Wang, Yue Jiang, Geng Li, Xiao-Ze T an, and Guo-Li Wang. Relativistic effects in the semileptonic bc decays to charmonium with the b ethe-salpeter method. Physical Review D , 99(1):013006, January 2019

    2019

  7. [7]

    Kinetic energy and speed powers vn of a heavy quark inside s wave and p wave heavy- light mesons

    Wei Li, Tianhong Wang, Tai-Fu Feng, Guo-Li Wang, and Chao -Hsi Chang. Kinetic energy and speed powers vn of a heavy quark inside s wave and p wave heavy- light mesons. Physical Review D, 110(7):076002, October 2024

    2024

  8. [8]

    Double charmonium p roductions in electron-positron anni- hilation using bethe-salpeter approach

    Hluf Negash and Shashank Bhatnagar. Double charmonium p roductions in electron-positron anni- hilation using bethe-salpeter approach. Advances in High Energy Physics , 2019:1–6, March 2019

    2019

Show all 37 references
  1. [9]

    Searches for exclusive Higgs and Z boso n decays into a vector quarkonium state and a photon using 139 fb − 1 of ATLAS √ s = 13 TeV proton–proton collision data

    Georges Aad et al. Searches for exclusive Higgs and Z boso n 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(9):781, 2023

    2023

  2. [10]

    Search for Higgs and Z Boson Decays to J/ψγ and Υ(nS)γ with the ATLAS Detector

    Georges Aad et al. Search for Higgs and Z Boson Decays to J/ψγ and Υ(nS)γ with the ATLAS Detector. Phys. Rev. Lett. , 114(12):121801, 2015

    2015

  3. [11]

    The International Linear Collider Technical Design Re port - Volume 2: Physics. 6 2013

    2013

  4. [12]

    Abada et al

    A. Abada et al. FCC-ee: The Lepton Collider: Future Circ ular Collider Conceptual Design Report Volume 2. Eur. Phys. J. ST , 228(2):261–623, 2019

    2019

  5. [13]

    Science China Physics, Mechanics and Astronomy , 53(11):1947–1948, 2010

    1947

  6. [14]

    Cepc conceptual design report: Vo lume 2 - physics & detector, 2018

    The CEPC Study Group. Cepc conceptual design report: Vo lume 2 - physics & detector, 2018

    2018

  7. [15]

    A. V. Luchinsky. Leading order nrqcd and light-cone ana lysis of exclusive charmonia production in radiative z-boson decays, 2017

    2017

  8. [16]

    Rare exclusive d ecays of thezboson revisited

    Ting-Chung Huang and Frank Petriello. Rare exclusive d ecays of thezboson revisited. Physical Review D , 92(1):014007, 2015. 27

    2015

  9. [17]

    Z-boson decays into S-wave quarkonium plus a photon up to O( α sv2) corrections

    Guang-Yu Wang, Xu-Chang Zheng, Xing-Gang Wu, and Guang -Zhi Xu. Z-boson decays into S-wave quarkonium plus a photon up to O( α sv2) corrections. Phys. Rev. D , 109(7):074004, 2024

    2024

  10. [18]

    Z boson radiative decays to a p -wave quarkonium at nnlo and ll accuracy

    Wen-Long Sang, De-Shan Yang, and Yu-Dong Zhang. Z boson radiative decays to a p -wave quarkonium at nnlo and ll accuracy. Physical Review D , 106(9):094023, November 2022

    2022

  11. [19]

    Z -boso n radiative decays to an s -wave quarkonium at nnlo and nll accuracy

    Wen-Long Sang, De-Shan Yang, and Yu-Dong Zhang. Z -boso n radiative decays to an s -wave quarkonium at nnlo and nll accuracy. Physical Review D , 108(1):014021, 2023

    2023

  12. [20]

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

    Asif Ali, Yi-Jie Li, Guang-Zhi Xu, and Kui-Yong Liu. Rar e Exclusive Decays of the Z-boson into S-wave Quarkonia within the Bethe-Salpeter Formalism. 6 20 26

  13. [21]

    Spectrum for heavy quarko nia and mixture of the relevant wave functions within the framework of bethe-salpeter equation

    ChaoHsi Chang and GuoLi Wang. Spectrum for heavy quarko nia and mixture of the relevant wave functions within the framework of bethe-salpeter equation . Science China Physics, Mechanics and Astronomy, 53(11):2005–2018, October 2010

    2005

  14. [22]

    (J/ ψ ,J/ψ ), and (ηc,ηc) production through two interme- diate photons in electron-positron annihilation at B-fact ories

    Shashank Bhatnagar and Hluf Negash. (J/ ψ ,J/ψ ), and (ηc,ηc) production through two interme- diate photons in electron-positron annihilation at B-fact ories. Nucl. Phys. A , 1053:122969, 2025

    2025

  15. [23]

    C. S. Kim and Guo-Li Wang. A verage kinetic energy of heav y quark (mu(pi)**2) inside heavy me- son of 0- state by Bethe-Salpeter method. Phys. Lett. B, 584:285–293, 2004. [Erratum: Phys.Lett.B 634, 564 (2006)]

    2004

  16. [24]

    ( γ, χ cJ(J = 0,1)) and ( γ, ηc) production in electron- positron annihilation at s=10.6 GeV and 4.6 GeV in the framew ork of Bethe-Salpeter equation

    Shashank Bhatnagar and Vaishali Guleria. ( γ, χ cJ(J = 0,1)) and ( γ, ηc) production in electron- positron annihilation at s=10.6 GeV and 4.6 GeV in the framew ork of Bethe-Salpeter equation. Nucl. Phys. A , 1041:122783, 2024

    2024

  17. [25]

    Decay constants of P-wave mesons

    Guo-Li Wang. Decay constants of P-wave mesons. Phys. Lett. B , 650:15–21, 2007

    2007

  18. [26]

    Search for rare decays of the Z an d Higgs bosons to a J/ ψ or ψ (2S) meson and a photon in proton-proton collisions at s=13TeV

    Aram Hayrapetyan et al. Search for rare decays of the Z an d Higgs bosons to a J/ ψ or ψ (2S) meson and a photon in proton-proton collisions at s=13TeV. Phys. Lett. B , 865:139462, 2025

    2025

  19. [27]

    Bodwin, Hee Sok Chung, June-Haak Ee, and Jung il Lee

    Geoffrey T. Bodwin, Hee Sok Chung, June-Haak Ee, and Jung il Lee. Z-boson decays to a vector quarkonium plus a photon. Phys. Rev. D , 97(1):016009, 2018

    2018

  20. [28]

    Bodwin, Daekyoung Kang, and Jungil Lee

    Geoffrey T. Bodwin, Daekyoung Kang, and Jungil Lee. Pote ntial-model calculation of an order- v(2) NRQCD matrix element. Phys. Rev. D , 74:014014, 2006

    2006

  21. [29]

    Bodwin, Jungil Lee, and Chaehyun Yu

    Geoffrey T. Bodwin, Jungil Lee, and Chaehyun Yu. Resumma tion of Relativistic Corrections to e+ e- — > J/psi + eta(c). Phys. Rev. D , 77:094018, 2008

    2008

  22. [30]

    Next-to -leading order QCD correction to e+ e- — > J / psi + eta(c) at s**(1/2) = 10.6-GeV

    Yu-Jie Zhang, Ying-jia Gao, and Kuang-Ta Chao. Next-to -leading order QCD correction to e+ e- — > J / psi + eta(c) at s**(1/2) = 10.6-GeV. Phys. Rev. Lett. , 96:092001, 2006. 28

    2006

  23. [31]

    Double charm productio n e+ e- — > J / psi + c anti-c at B factories with next-to-leading order QCD correction

    Yu-Jie Zhang and Kuang-Ta Chao. Double charm productio n e+ e- — > J / psi + c anti-c at B factories with next-to-leading order QCD correction. Phys. Rev. Lett. , 98:092003, 2007

    2007

  24. [32]

    J/ψ (ψ ′) production at the Tevatron and LHC at O(α 4 sv4) in nonrelativistic QCD

    Yan-Qing Ma, Kai Wang, and Kuang-Ta Chao. J/ψ (ψ ′) production at the Tevatron and LHC at O(α 4 sv4) in nonrelativistic QCD. Phys. Rev. Lett. , 106:042002, 2011

    2011

  25. [33]

    Polarization for Prompt J/ ψ and ψ (2s) Production at the Tevatron and LHC

    Bin Gong, Lu-Ping Wan, Jian-Xiong Wang, and Hong-Fei Zh ang. Polarization for Prompt J/ ψ and ψ (2s) Production at the Tevatron and LHC. Phys. Rev. Lett. , 110(4):042002, 2013

    2013

  26. [34]

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

    Cong Li, Zhan Sun, and Gui-Yuan Zhang. Analysis of doubl e-J/ψ production in Z decay at next-to-leading-order QCD accuracy. JHEP, 10:120, 2023

    2023

  27. [35]

    Next-to-leading-order qcd corrections to a vector bottomonium radiative decay into a c harmonium

    Yu-Dong Zhang, Feng Feng, Wen-Long Sang, and Hong-Fei Z hang. Next-to-leading-order qcd corrections to a vector bottomonium radiative decay into a c harmonium. Journal of High Energy Physics, 2021(12), December 2021

    2021

  28. [36]

    Two-loop QCD corrections to C-even bottomonium exclusive decays to doub le J/ψ

    Yu-Dong Zhang, Xiao-Wei Bai, Feng Feng, Wen-Long Sang, and Ming-Zhen Zhou. Two-loop QCD corrections to C-even bottomonium exclusive decays to doub le J/ψ . Phys. Rev. D , 108(11):114030, 2023

    2023

  29. [37]

    Bc meson exclusive de cays to a p -wave charmonium and a pion at nlo accuracy

    Zi-Qiang Chen and Cong-Feng Qiao. Bc meson exclusive de cays to a p -wave charmonium and a pion at nlo accuracy. Physical Review D , 104(1):016019, 2021. 29

    2021

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.