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Production of leptonium in heavy quarkonium decays

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Pith's one-line read This paper shows that the never-observed ortho-dimuonium atom, a bound muon-antimuon pair, should be produced in J/psi decays with inclusive branching fraction 1.5e-12, enough for discovery at the future Super Tau-Charm Facility.

desk verdict A solid NRQED/NRQCD calculation that genuinely adds the four-lepton leptonium channels and the Upsilon/tauonium analysis; the headline ortho-dimuonium rate is labeled inclusive but is really a sum of two exclusive channels, so the STCF projection needs a hadronic-channel caveat. read the letter →

arxiv 2505.02676 v2 pith:6UWNLLYR submitted 2025-05-05 hep-ph

Yuhan Zhao , Jun Jiang This is my paper · ORCID
classification hep-ph
keywords leptoniumdimuoniumtruemuoniumheavyquarkoniumdecaysNRQEDNRQCDbranchingfractionSuperTau-CharmFacility
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 calculates the rates at which J/psi and Upsilon mesons decay into a leptonium atom—a bound state of a lepton and its antiparticle—accompanied by a photon or by a second lepton pair. Its central numerical claim is that the never-observed ortho-dimuonium state $(\mu^+\mu^-)[{}^3S_1]$ is produced inclusively with $\mathrm{Br}(J/\psi \to (\mu^+\mu^-)[{}^3S_1] + X) = 1.5\times 10^{-12}$, where $X$ is a photon or a charged lepton pair. At the future Super Tau-Charm Facility, designed to produce $3.4\times10^{12}$ $J/\psi$ per year, that rate corresponds to about five raw events per year, making a first observation of dimuonium plausible. The same calculation yields larger rates for ortho-positronium and muonium in $J/\psi$ decays, and for tau-containing leptonia in $\Upsilon$ decays, although existing $\Upsilon$ samples are too small to see those.

What carries the argument

The machinery is non-relativistic QED/QCD factorization. Each bound state is represented by its S-wave radial wavefunction at the origin, $R(0)^2$ (or $|\Psi(0)|^2$), which for leptonium carries the Coulombic $\alpha^3$ suppression that sets the overall $10^{-13}$–$10^{-11}$ scale, and spin projectors convert free-particle amplitudes into bound-state amplitudes. The load-bearing identities are the C-parity selection rules: in the radiative channel the leptonium must be the ${}^1S_0$ para state, and in the four-body diagrams the third diagram vanishes for ${}^3S_1$ while the fourth vanishes for ${}^1S_0$; this is why ortho states dominate in channels where the surviving diagram contributes at leading order in $m_l/m_J$. The $J/\psi$ wavefunction is not fitted freely but extracted from the measured $\mathrm{Br}(J/\psi\to\mu^+\mu^-)$ through the leading-order width formula.

What would settle it

A dedicated search at the future Super Tau-Charm Facility for $e^+e^-$ pairs with invariant mass near $2m_\mu$ and a displaced vertex from $(\mu^+\mu^-)[{}^3S_1]\to e^+e^-(\gamma)$ should see about five events per year if the claim is right; a null result with sensitivity better than a few times $10^{-12}$, or an observed rate significantly above $1.5\times10^{-12}$, would rule out the inclusive prediction.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that C-parity and angular-momentum selection rules split leptonium production cleanly: the radiative channel $J/\psi \to (\mu^+\mu^-)[{}^1S_0]+\gamma$ is allowed, while the radiative channel for the triplet ${}^3S_1$ state vanishes because the $J/\psi$ has $J^{PC}=1^{--}$ and the photon channel forces the leptonium into $0^{-+}$. Ortho-dimuonium must therefore be produced through the four-body channels $J/\psi \to (\mu^+\mu^-)[{}^3S_1] + e^+e^-$ or $+ \mu^+\mu^-$, where different Feynman diagram topologies dominate for the two spin states. Summing the radiative and four-body contributions with $X = \gamma$ or a charged lepton pair, the paper obtains $\mathrm{Br}(J/\psi \to (\mu^+\mu^-)[{}^3S_1]+X)=1.5\times10^{-12}$, together with $\mathrm{Br}(J/\psi \to (e^+e^-)[{}^3S_1]+X)=1.7\times10^{-11}$ and $\mathrm{Br}(J/\psi \to (\mu^+\mu^-)[{}^1S_0]+X)=4.2\times10^{-13}$. With $3.4\times10^{12}$ $J/\psi$ events per year, the paper concludes that ortho-dimuonium should be discoverable at the future Super Tau-Charm Facility through its dominant decay $(\mu^+\mu^-)[{}^3S_1] \to e^+e^-(\gamma)$.

Load-bearing premise

The headline 'inclusive' branching fractions count only leptonium accompanied by one photon or one charged lepton pair, and the paper gives no argument that other companions, such as hadrons or additional photons, are negligible.

Editorial extensions

If this is right

  • At the Super Tau-Charm Facility's planned $3.4\times10^{12}$ $J/\psi$ per year, the predicted $1.5\times10^{-12}$ branching fraction yields roughly five raw ortho-dimuonium events per year before detection efficiency, making a first observation plausible.
  • Ortho-positronium and muonium come out easier: $\mathrm{Br}(J/\psi\to(e^+e^-)[{}^3S_1]+X)=1.7\times10^{-11}$ and $\mathrm{Br}(J/\psi\to(\mu^+e^-)[{}^3S_1]+e^+\mu^-)=1.5\times10^{-11}$.
  • The ground-state-only radiative rate $\mathrm{Br}(J/\psi\to(\mu^+\mu^-)[{}^1S_0]+\gamma)=3.93\times10^{-13}$ is lower than the all-excited-state Green-function result by the factor $\zeta(3)\approx1.2$, so excited leptonium levels contribute about 20% of the radiative channel.
  • For $\Upsilon$ decays the largest tauonium channels are $\mathrm{Br}(\Upsilon\to(\tau^+e^-)[{}^3S_1]+e^+\tau^-)=1.08\times10^{-10}$ and $\mathrm{Br}(\Upsilon\to(\tau^+e^-)[{}^1S_0]+e^+\tau^-)=3.63\times10^{-11}$, but with about $10^8$ $\Upsilon$ events such rates are not yet observable.
  • The predicted ratio $\mathrm{Br}(J/\psi\to\mu^+\mu^-\gamma)/\mathrm{Br}(J/\psi\to e^+e^-\gamma)=0.33$ is a large, clean mass-hierarchy effect that existing charmonium data can test.

Reading between the lines

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

  • A measurement of the full ortho-dimuonium yield at the Super Tau-Charm Facility that comes out noticeably above $1.5\times10^{-12}$ would quantify the importance of $X$ channels the paper did not include, such as hadrons or additional photons.
  • The same selection rules and NRQED/NRQCD amplitudes can be carried over to other vector quarkonia such as $\psi(2S)$ or $\Upsilon(2S)$; the larger phase space should rescale the branching fractions, giving a systematic test of the mechanism.
  • A dedicated lineshape analysis of the $e^+e^-$ invariant-mass spectrum near $2m_\mu$ with displaced-vertex cuts could separate ortho-dimuonium decays from prompt backgrounds even with partial data, sharpening the discovery projection.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper computes rates for leptonium production in J/psi and Upsilon decays. It treats radiative channels Q -> (l+l-)[1S0] + gamma and four-lepton channels Q -> (l1+ l2-)[n] + l1- l2+ using NRQED/NRQCD with spin projectors and wavefunctions at the origin, presenting branching fractions and invariant-mass distributions. The headline result is Br(J/psi -> (mu+mu-)[3S1] + X) = 1.5e-12, obtained by summing the X = e+e- and X = mu+mu- exclusive channels, which the paper claims makes ortho-dimuonium potentially discoverable at the future STCF with 3.4e12 J/psi per year. Similar inclusive sums are given for other leptonia in J/psi and Upsilon decays.

Significance. If the inclusive interpretation is validated, the paper provides a concrete new search channel for true muonium, a state that has not yet been observed, and the STCF discovery projection is of genuine interest to the community. The radiative calculation is cross-checked against the Green-function method of Ref. [27], with the zeta(3) factor and the (1 - m_l^2/m_c^2) phase-space factor accounted for, which gives confidence in that part of the computation. The paper also provides many differential distributions that could be useful for experimental studies. However, the key inclusive claim is currently an exclusive sum, and the missing-X issue means the headline branching fraction is not demonstrated as an inclusive rate.

major comments (2)
  1. [Section II, Eqs. (47)-(50)] The 'inclusive' branching fraction Br(J/psi -> (mu+mu-)[3S1] + X) = 1.5e-12 is obtained by summing only the exclusive channels X = e+e- and X = mu+mu- (Eqs. (38) and (44)), since X = gamma vanishes for the triplet. The text defines X as 'gamma or charged leptons', but an inclusive branching fraction must account for all other final states. In particular, hadronic X such as pi0 or multi-pion systems are not forbidden by C-parity: the J/psi has C = -1, the 3S1 leptonium has C = -1, so X must have C = +1, which pi0 satisfies. The same off-shell photon mechanism that produces the lepton pair X could produce hadronic X, and the paper provides no estimate or bound for these channels. The central discovery projection for ortho-dimuonium at STCF therefore rests on an unjustified identification of 'inclusive' with a two-channel sum. The authors should either relabel these as exclusive branching fractions or provide a quantitative estimate of the hadronic and multi-photon X contributions. The same issue affects the Upsilon inclusive results in Eqs. (86)-(91).
  2. [Section II, Eqs. (37)-(46)] The four-lepton branching fractions and widths are quoted without uncertainties, and it is not stated whether the numerical values are obtained from the full amplitudes or from the approximate amplitudes in the m_e^2 << m_J/psi^2 limit. Since these results scale as R^2_J/psi(0), which carries a relative uncertainty of about 2% from Eq. (9), the numerical precision of the central values is not documented. While the order-of-magnitude discovery projection is robust to this uncertainty, the presentation should state the source of the quoted numbers and propagate the input uncertainties.
minor comments (4)
  1. [Abstract and Introduction] The abstract and introduction state that the radiative processes Q -> (l1+ l2-)[n] + gamma are considered for l1,2 = tau, mu, e, which implies unlike-flavor radiative channels such as (mu+ e-) + gamma. The body of the paper, in Sections II.D and III.C, only treats same-flavor (l+l-)[1S0] + gamma. The authors should either discuss the unlike-flavor radiative channels or explicitly state that they are omitted and explain why, since the text currently gives an inaccurate picture of the paper's scope.
  2. [Section II.E] The numerical integration over the three-body phase space is not described in detail. For each process, it would be helpful to state which expression (full or approximate) was integrated, the phase-space cuts applied, and how the integration was performed, to aid reproducibility.
  3. [Throughout] There are several typographical and notation issues: 'muonim' appears in the text of Section II; the subscript 'the' in Eq. (7) is unusual; and in Eqs. (29)-(46) the branching fractions are given without any uncertainty, whereas the radiative results in Eqs. (21)-(22) include uncertainties. A careful proofread would improve the presentation.
  4. [Section II.D, Eq. (24)] The comparison with Ref. [27] is explained clearly, but the statement that Eq. (24) differs from the first line of Eq. (32) of Ref. [27] by a factor of (1 - m_l^2/m_c^2) zeta(3) could be made even more explicit by displaying the corresponding expression from Ref. [27] itself. As written, the reader has to consult the cited paper to verify the factor.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the leptonium predictions are independent QED calculations with standard external inputs, and the quoted "inclusive" rate is the explicit sum of the computed exclusive channels.

full rationale

The derivation chain is self-contained: the only external inputs are the measured J/psi and Upsilon leptonic widths, from which the quarkonium radial wavefunctions at the origin are extracted (Eqs. (6), (9), and (55)); this is a standard normalization procedure and not a fit to the leptonium branching fractions that are later predicted. The leptonium decay widths are then obtained from explicit NRQED/NRQCD Feynman-diagram amplitudes (Eqs. (19), (20), and (27)-(46)), with no parameter adjusted to reproduce the target rates. The headline number Br(J/psi -> (mu+mu-)[3S1] + X) = 1.5e-12 is obtained by adding the computed e+e- and mu+mu- channels, Eqs. (38) and (44), since the radiative channel vanishes for the triplet; the text explicitly defines X as "gamma or charged leptons mu±, e±". Calling this an "inclusive" branching fraction is a completeness caveat rather than circularity: the value is not equivalent by construction to its inputs, but is a genuine prediction for the channels that were computed. The radiative 1S0 result is cross-checked against the independent Green-function calculation of Ref. [27], with the known zeta(3) difference from excited-state summation explained explicitly. No load-bearing self-citation chain or fitted-parameter-renamed-as-prediction appears in the paper, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central predictions rest on two fitted quarkonium wavefunction parameters and on the standard NRQED/NRQCD factorization assumptions. The paper's own inclusive definition is the most fragile additional assumption, since it restricts final-state X to a photon or a charged lepton pair without estimating other contributions.

free parameters (2)
  • R_J/psi(0)^2 = 0.5599 +/- 0.0107 GeV^3
    Extracted from the measured Br(J/psi -> mu+mu-) using Eq. (6). Used as input for every J/psi leptonium branching fraction in Sections II C-E.
  • R_Upsilon(0)^2 = 5.06 +/- 0.14 GeV^3
    Extracted from the measured Br(Upsilon -> mu+mu-) using Eq. (6). Used as input for every Upsilon leptonium branching fraction in Section III.
assumptions (4)
  • domain assumption Leptonium is described by NRQED with a Coulomb Schrodinger wavefunction, with |Psi(0)|^2 = (alpha ml)^3/(8 pi n^3).
    Section II A, Eqs. (1)-(2). Standard QED bound-state treatment; neglects higher-order radiative and finite-width corrections to the wavefunction.
  • domain assumption NRQCD factorization with color-singlet spin projectors at LO in v^2 (q=0) describes the quarkonium bound state.
    Section II A, Eqs. (3)-(5). Requires v^2 small; v^2 ~ 0.3 for charmonia and ~ 0.1 for bottomonia, so the LO approximation carries a sizable but standard uncertainty.
  • ad hoc to paper Inclusive leptonium production is saturated by the exclusive channels with X = gamma or a charged lepton pair.
    Eqs. (47)-(50) and the text after Eq. (50). No estimate is given for leptonium produced with hadrons, extra photons, or other final states, so the inclusive label is an assumption made by the paper itself.
  • standard math C-parity conservation restricts radiative leptonium to 1S0 for same-flavor pairs and selects which Feynman diagrams contribute.
    Section II C-E. Standard conservation law in QED/QCD, used without derivation.

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Cite this review

Pith. "Pith review of Production of leptonium in heavy quarkonium decays." pith.science (2026). https://pith.science/paper/6UWNLLYR

@misc{pith2026250502676,
  author       = {Pith},
  title        = {Pith review of: Production of leptonium in heavy quarkonium decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6UWNLLYR}},
  note         = {Machine review of arXiv:2505.02676}
}
abstract

Lepton pairs with opposite charges can form bound states known as ``leptonium'' through quantum electrodynamic interactions. Heavy quarkonia such as $J/\psi$ are abundantly produced at facilities like BESIII and the future Super Tau-Charm Facility (STCF). In this work, we investigate leptonium production in heavy quarkonium decays, specifically focusing on the processes ${\cal{Q}} \longrightarrow (l_1^+ l_2^-)[n] +\gamma$ ($l_{1,2}= \tau,\, \mu,\, e$) and ${\cal{Q}} \longrightarrow (l_1^+ l_2^-)[n] + l_1^- l_2^+$. Here, ${\cal{Q}}$ denotes the heavy quarkonium $J/\psi$ or $\Upsilon$, while $n=$ ${^1S_0}$ or $^3S_1$ corresponds to para-leptonium and ortho-leptonium, respectively. With an annual production of $3.4 \times 10^{12}$ $J/\psi$ events at STCF, there is significant potential to observe positronium $(e^+e^-)$, muonium $(\mu^+e^-)$, and dimuonium $(\mu^+\mu^-)$. In particular, the ortho-dimuonium $(\mu^+\mu^-)[^3S_1]$ may be discovered at the future STCF, with an inclusive branching fraction of $Br(J/\psi \longrightarrow (\mu^+\mu^-)[^3S_1] +X) = 1.5 \times 10^{-12}$.

Figures

Figures reproduced from arXiv: 2505.02676 by the authors.

Figure 1
Figure 1. The first two diagrams (corresponding to t- and u-channels) are forbidden due to C-parity conservation, which has been verified via explicit calculation. The differential decay width of J/ψ(k) −→ l +(p1)l −(p2)γ(p3) with respect to the invariant mass of the lepton pair, s1 = (p1 + p2) 2 , and that of the l −γ system, s2 = (p2 + p3) 2 , is given by: dΓ (J/ψ → l +l −γ) ds1ds2 = 1 3 1 256π 3m3 J/ψ |M|2 , (10) 8 [PITH_… view at source ↗
Figure 2
Figure 2. FIG. 2. Differential distributions [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The typical Feynman diagrams for [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Differential distributions for [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Differential distributions for [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Differential distributions for [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Differential distributions for [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: The curves for taus, muons, and electrons all increase monotonically, exhibiting distinct thresholds but sharing the same endpoint. For the Υ case, the approximation under the m2 l ≪ m2 Υ limit is so accurate that no visible differences between Eqs. (11) 21 [PITH_FULL…
Figure 9
Figure 9. Figure 9: FIG. 9. Differential distributions [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Differential distributions [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Differential distributions [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Differential distributions [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Differential distributions [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]

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