Creating true muonium via charmonium radiative decay

Hai-Bo Li; Jian-Ping Dai; Shuai Zhao; Zong-Ying Zheng

REVIEW 2 major objections 6 minor 50 references

Creating true muonium via charmonium radiative decay

T0 review · 2 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper predicts that the radiative decay J/psi -> gamma + (mu+mu-)_bound produces true para-muonium at a branching fraction of about 7 x 10^-13, and argues that a future super tau-charm facility could detect a few events per year.

desk verdict A clean parameter-free QED prediction for J/psi -> gamma + true muonium at ~7e-13; worth refereeing despite a mislabeled selection rule and a speculative reach section. read the letter →

arxiv 2412.12592 v1 pith:LR7P6M3F submitted 2024-12-17 hep-ph hep-ex

Jian-Ping Dai , Hai-Bo Li , Shuai Zhao , Zong-Ying Zheng This is my paper · ORCID

classification hep-phhep-ex

keywords truemuoniumJ/psiradiativedecaycharmoniumCoulombresummationSommerfeldenhancementQEDboundstatessupertau-charmfacilitybranchingfraction

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

True muonium, the bound state of a muon and an antimuon, is one of the simplest QED atoms and has never been observed. The paper shows that the radiative decay of the charmonium vector meson J/psi into a photon and a para-muonium ground state is a viable production channel, with a predicted branching fraction of about 7 x $10^{-13}$. That rate is too small for current BESIII data, but a proposed super tau-charm facility producing 3.4 x $10^{12}$ J/psi per year would yield 2-3 signal events per year. The calculation is built from final-state radiation amplitudes and a Coulomb Green function resummation, and it cancels the nonperturbative charmonium wave function by normalizing to the measured J/psi -> mu+mu- branching fraction.

What carries the argument

The central object is the imaginary part of the nonrelativistic Coulomb Green function for the muon-antimuon pair, evaluated at zero separation. Its bound-state poles give the squared wave function at the origin, |psi_n(0)|^2 = $alpha^{3}$ $m_mu^{3}$/(8 pi $n^{3}$), and its branch cut gives the Sommerfeld-Schwinger-Sakharov enhancement factor for the continuum. The paper's key move is to express the desired decay width as the free-pair width times this Green function's imaginary part, then take the ratio with the J/psi -> mu+mu- width so the charmonium wave function R(0) cancels; the remaining input is the measured J/psi leptonic branching fraction.

What would settle it

Measure the J/psi -> gamma mu+mu- differential rate near threshold at a super tau-charm facility with 3.4 x $10^{12}$ J/psi events, and look for the bound-state peak at E = -$alpha^{2}$ m_mu/4 with the predicted height corresponding to a 7 x $10^{-13}$ branching fraction; alternatively, perform an explicit one-loop calculation of the two 'vanishing' diagrams to test the parity argument.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that J/psi -> gamma + (mu+mu-)_bound is a quantitatively predicted and experimentally meaningful channel for producing true para-muonium. Using a nonrelativistic spin-triplet projector for the J/psi, the final-state-radiation amplitude, and the imaginary part of the Coulomb Green function for the muon pair, the authors obtain R|E<0 ~ 1.18 x $10^{-11}$ for the ratio of the bound-state decay width to the J/psi -> mu+mu- width, which translates into Br(J/psi -> gamma (mu+mu-)) ~ 7.03 x $10^{-13}$ when multiplied by the measured 5.961% leptonic branching fraction. The same machinery gives the above-threshold continuum rate, controlled by the Sommerfeld factor; in a threshold window as narrow as the ground-state binding energy it is comparable to the bound-state contribution (R|E>0 ~ 4.91 x $10^{-12}$), and it grows quickly with the window size. The paper then confronts this rate with detector reality: with 3.4 x $10^{12}$ J/psi per year at a future super tau-charm facility, one expects 2-3 bound-state events per year, and the 1.3 mm lab-frame decay length of the boosted true muonium could be separated with vertex resolution at the 0.4 mm level.

Load-bearing premise

The calculation assumes, without derivation, that parity makes the two Feynman diagrams with the photon emitted from the J/psi itself vanish, so only final-state muon radiation contributes; if that cancellation fails, the predicted branching fraction changes.

Editorial extensions

If this is right

With 10^10 accumulated J/psi events, BESIII would expect well below one bound-state event, so current data cannot test this rate.
At the proposed super tau-charm facility, 3.4 x 10^12 J/psi per year translates to 2-3 bound-state events per year, making the search statistically plausible but far from easy.
The lab-frame decay length of the boosted true muonium is about 1.3 mm, so a vertex resolution near 0.4 mm would separate the signal from prompt QED backgrounds.
Widening the muon-pair energy window to the MeV scale boosts the continuum contribution toward R ~ 1e-8, meaning thousands of events per year at STCF, though those are not bound-state signals.
Because the prediction is normalized to the measured J/psi -> mu+mu- rate, the QED part is free of hadronic uncertainties, so a future measurement would test the Coulomb-resummation formalism directly.

Reading between the lines

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

The same ratio method should transfer directly to Upsilon decays into gamma plus true tauonium, with the tau decay width kept finite; the paper mentions tauonium only in passing, so a dedicated calculation would be a natural next step.
The parity argument that eliminates the charmonium-radiation diagrams is asserted without derivation; an explicit one-loop check would either confirm the 7e-13 prediction or reveal an additional amplitude.
A realistic search might target the threshold-enhanced continuum rather than the bound-state peak, since the paper's Eq. (35) shows the continuum rate grows rapidly with the experimental energy resolution.
The ratio R cancels the J/psi wave function, so this channel is a rare case where a hadronic decay's QED part is computable almost parameter-free; measuring it would probe Coulomb resummation in an entirely new regime.

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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

Bottom line: this is a clean, internally consistent QED calculation that gives a new prediction Br(J/psi -> gamma + true muonium) ~ 7e-13, normalized to the measured Br(J/psi -> mu+mu-). The bound-state number is parameter-free, and the Coulomb resummation machinery is standard but applied to a new initial state. I think the paper deserves a serious referee.

The genuine new result is Eq. (39), and the authors do two useful things: they calculate the low-invariant-mass FSR amplitude explicitly, and they cross-check the bound-state width via an independent spin-projector/wavefunction calculation. The phase-space integral and the ratio R check out. The citation to their own earlier work [13] is contextual, not load-bearing.

The soft spots are mostly presentation and experimental extrapolation, not the physics. The paper says the photon-from-J/psi diagrams vanish by parity. That's the wrong name: it's charge-conjugation invariance (J/psi has C=-1; gamma and the virtual photon each have C=-1, so the J/psi -> gamma gamma* amplitude is C-forbidden). The conclusion is right, so the central branching fraction is not threatened. The m_c choice is a real but modest uncertainty: using M_J/psi/2 shifts the rate down by about a third, not by an order of magnitude. The experimental section is optimistic in the way this kind of paper usually is: 0.4 mm neutral-vertex resolution is not demonstrated, and 2-3 events per year at STCF is a challenge, not a discovery. The authors do state the resolution condition, so it's not hidden.

One thing I'd want the referee to push on: the Lambda-dependence of the E>0 contribution. The bound-state rate is clean, but the continuum contribution depends strongly on the energy window, and the claim that a few-MeV window could give 1e-8 is extrapolating the small-s1 approximation beyond its validity. That section could use a clearer caveat.

My verdict: send it to review. It's a solid order-of-magnitude prediction for an unobserved system, the derivation is checkable, and the main numerical result is robust at the factor-2 level. The paper's usefulness to BESIII/STCF planning is real, and the experimental discussion, while speculative, is clearly flagged as such.

Referee Report

2 major / 6 minor

Summary. The paper computes the branching fraction for J/ψ decaying into a real photon plus a bound para-muonium state, J/ψ → γ + (μ⁺μ⁻)_{bound}. Using a nonrelativistic spin projector for the charmonium and keeping only final-state-radiation diagrams, the authors derive the differential width near the μ⁺μ⁻ threshold, resum Coulomb ladder exchanges via the Green-function formalism, and integrate over the bound-state poles. The ratio R = Γ(J/ψ→γ(μ⁺μ⁻)_{bound}) / Γ(J/ψ→μ⁺μ⁻) is found to be R|E<0 ≈ 1.18 × 10⁻¹¹, which, multiplied by the measured Br(J/ψ→μ⁺μ⁻) = 5.961%, gives Br(J/ψ→γ(μ⁺μ⁻)_{bound}) ≈ 7.03 × 10⁻¹³. The paper also estimates the above-threshold continuum contribution in a chosen energy window and discusses detection prospects at BESIII and the proposed Super Tau-Charm Facility.

Significance. If correct, the calculation provides a concrete, QED-dominated production mechanism for true muonium with a clean normalization to the measured J/ψ→μ⁺μ⁻ branching fraction: the ratio R is, up to the charm-mass choice, parameter-free, since the nonperturbative wave function R(0) cancels. The phase-space integrals and the Coulomb Green-function treatment are explicit and checkable, and the numerical factor is internally consistent. The C-parity selection rule that eliminates the charmonium-radiation diagrams is physically important and correctly identifies FSR as the only leading contribution, although the paper mislabels it as parity. The result is of direct interest to the true-muonium program and to the planning of future tau-charm facilities. The main weakness is the sensitivity of the numerical prediction to the charm-quark mass, and the experimental feasibility section rests on assumptions about neutral-vertex resolution that are not demonstrated.

major comments (2)

[Sec. V, Eqs. (36)-(39)] The numerical prediction uses m_c = 1.27 GeV, but the nonrelativistic derivation in Sec. II relies on M_J/ψ ≈ 2m_c, which with M_J/ψ = 3.097 GeV gives m_c ≈ 1.55 GeV. The PDG value 1.27 GeV is an MS-bar running mass, not the pole mass appropriate for the nonrelativistic bound-state expansion used in the spin projector. Since R|E<0 ∝ 1/m_c², replacing m_c by M_J/ψ/2 ≈ 1.55 GeV reduces Eq. (39) from 7.03 × 10⁻¹³ to about 4.7 × 10⁻¹³. The authors should either justify the scale choice explicitly or present the result as a function of m_c with a corresponding uncertainty.
[Sec. II, paragraph after Fig. 1] The statement that diagrams (a) and (b) vanish "due to the conservation of parity" is not the correct selection rule. Parity does not forbid J/ψ → γγ*; the correct rule is charge-conjugation invariance: C(J/ψ) = -1 while a two-photon state (real plus virtual) has C = +1, so the amplitude J/ψ → γγ* is exactly zero. Because this step is the basis for keeping only the final-state-radiation diagrams, the explanation should be corrected and the asserted explicit verification should either be shown or replaced by a precise reference.

minor comments (6)

[Sec. IV, after Eq. (32)] The sentence claiming that Eq. (32) was verified "in another approach" is not substantiated by any calculation; please include the cross-check or explicitly state that the derivation is omitted.
[Sec. V, event yield estimate] The estimate of 2-3 events per year at STCF assumes 100% detection efficiency and does not include the reconstruction efficiency for the γγ decay mode or the associated acceptance; the event yield should be presented as a raw rate with these efficiencies clearly separated.
[Eq. (2)] The reduction from the trace over the spin projector to the second line of Eq. (2) is not shown; a brief derivation or a specific reference would make the amplitude easier to verify.
[Sec. IV, Eq. (24)] The symbol Γ is overloaded: it denotes the muon decay width in the Coulomb Green function and also the J/ψ decay width in the ratio R; please use distinct symbols such as Γ_μ.
[Sec. IV, paragraph on muon lifetime] The statement that the lifetime of true muonium is "on the order of several picoseconds" is imprecise for the 1¹S₀ ground state, whose lifetime is about 0.6 ps; please specify the state and distinguish para from ortho.
[Reference [11]] Reference [11] appears malformed; the correct citation is D. B. Cassidy and A. P. Mills, Jr., Nature 449, 193 (2007).

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the bound-state branching fraction is a parameter-free QED ratio normalized to measured J/psi->mu+mu-, with only contextual self-citation.

full rationale

The central prediction Br(J/psi -> gamma(mu+mu-)_bound) ~ 7e-13 is obtained as R|E<0 times the measured Br(J/psi -> mu+mu-), where R is computed from QED amplitudes and Coulomb resummation. The J/psi radial wave function R(0) appears in both the numerator and denominator of R and cancels (Eqs. (28)-(32)); no fitted parameter is introduced. The normalization by the measured leptonic branching fraction is a standard external input, not a circular reuse of the quantity being predicted. The only self-citation (Ref. [13]) states that true muonium has not yet been observed experimentally and plays no role in the derivation. The parity/C-parity suppression of the photon-from-charmonium diagrams is a selection rule external to the model, and the m_c dependence is an ordinary parametric uncertainty (R is proportional to 1/m_c^2), not an ansatz fitted to the target rate. The derivation is self-contained QED plus standard NRQCD projector inputs, with the numerical result benchmarked against measured Br(J/psi -> mu+mu-) only as a normalization.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central prediction rests on standard QED plus the nonrelativistic charmonium approximation. The only nonstandard input is the assertion that direct photon emission from the charmonium vanishes, which is not proven. True muonium is a known predicted state, so no entities are invented.

free parameters (2)

m_c (charm quark mass) = 1.27 GeV
Taken from PDG, but scheme-dependent. The nonrelativistic relation M_J/psi ≈ 2 m_c would give about 1.55 GeV; the chosen value changes the central branching fraction by roughly 30-40%.
Lambda (energy window for the E>0 continuum) = alpha^2 m_mu/4 ≈ 1.4 keV, later MeV-scale
Chosen by hand in Sec. V to match detector resolution; it affects the continuum discussion but not the bound-state branching fraction.

assumptions (3)

domain assumption J/psi is a nonrelativistic ccbar system with M_J/psi ≈ 2 m_c
Used in the amplitude calculation in Sec. II; standard in NRQCD but subject to ~v^2 corrections.
ad hoc to paper The diagrams with the photon emitted from the charmonium vanish by parity
Asserted in Sec. II without a shown derivation; load-bearing because the amplitude omits these diagrams.
standard math Coulomb rescattering of the muon pair is resummed with the Fadin-Khoze Green function formula
Standard QED result cited to Refs [40-43] and used in Sec. IV.

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Cite this review

Pith. "Pith review of Creating true muonium via charmonium radiative decay." pith.science (2026). https://pith.science/paper/LR7P6M3F

@misc{pith2026241212592,
  author       = {Pith},
  title        = {Pith review of: Creating true muonium via charmonium radiative decay},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LR7P6M3F}},
  note         = {Machine review of arXiv:2412.12592}
}

abstract

True muonium, the bound state of a muon and an antimuon ($\mu^+\mu^-$), has long been theoretically predicted but remains experimentally elusive. We investigate the production of true para-muonium in the radiative decay of $J/\psi$ meson,and analyze the prospects for detecting true muonium in current and future high-energy $e^+e^-$ experiments, particularly focusing on the BESIII experiment and the proposed Super Tau-Charm Facility. Although the events are rare at the super tau-charm facility, the detection of true para-muonium via $J/\psi$ radiative decays could become feasible at its future updates.

Figures

Figures reproduced from arXiv: 2412.12592 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

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(37) Using the branching fraction of J/ψ → µ+µ− [44] as follows: Br(J/ψ →µ+µ−) = 5.961%, (38) we can then calculate the branching fraction of J/ψ → γ(µ+µ−)

(36) With these inputs, we get the ratio R for the bound state contributions R|E<0 ≈ 1.18 × 10−11. (37) Using the branching fraction of J/ψ → µ+µ− [44] as follows: Br(J/ψ →µ+µ−) = 5.961%, (38) we can then calculate the branching fraction of J/ψ → γ(µ+µ−). The result reads Br(J...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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M. A. J. Cassidy DB, Nature 7159, 195 (2007)

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In the process, the photon can be radi- ated either from the J/ψ meson or from outgoing muons (µ+ and µ−). However, due to the conservation of par- ity, the ﬁrst two diagrams, where the photon is radiated directly from the J/ψ meson, yield zero contribution, a result that can be rigorously veriﬁed through explicit cal- culations. Consequently, the nonzero...

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