{"id":"f0d3c6f6-6dad-4330-a27b-b111f1423bad","arxiv_id":"2505.02676","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"J/psi decays can produce ortho-dimuonium at a branching fraction of about 1.5e-12, while Upsilon decays allow tauonium production, with several channels potentially observable at planned facilities.","lead":"This paper calculates how often the quarkonium particles J/psi and Upsilon decay into leptonium, the QED bound states of lepton pairs. It finds ortho-dimuonium may be produced at a rate observable at the future Super Tau-Charm Facility, offering a path to the first detection of dimuonium.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed ortho-dimuonium inclusive rate is an exclusive sum over X = e+e- and mu+mu- only; no argument is given that hadronic or multi-photon X are negligible, so Br(J/psi -> (mu+mu-)[3S1] + X) = 1.5e-12 is not a demonstrated inclusive branching fraction.","rationale":"The reader's weakest_assumption identifies exactly the point I consider load-bearing: Eqs. (47)-(50) label an exclusive sum as an inclusive branching fraction, with no quantitative argument that the omitted final states are negligible. I agree with the reader's verdict CONDITIONAL and with the positive assessment of the standard NRQED calculation. The paper has genuine independent support: the J/psi -> l+l- cross-check against PDG (Eqs. 7-8), the consistent comparison with Ref. [27] including the zeta(3) excited-state factor, and explicit analytic expressions for the new four-lepton channels. None of that, however, addresses the missing-X question. I considered whether the concern could be dismissed by NRQCD scaling: the (mu+mu-)[3S1] is a color-singlet electromagnetic final state, so one might expect hadronic X to be suppressed by alpha_s^2 relative to the purely leptonic X. But that scaling is not presented in the paper, and one-loop QCD corrections to an electromagnetic exclusive decay are typically 10-30% effects, not 10^-3, so the burden is on the authors to show the hadronic X channels are subdominant. The fact that the triplet has JPC = 1--, identical to the J/psi, makes this a genuine gap rather than a formality. The proposed test would settle the question: if a leading-order NRQCD computation of J/psi -> (mu+mu-)[3S1] + 2 gluons (or + pi0) gives a rate within a factor of a few of 1.5e-12, the inclusive label and the STCF projection are not justified; if it is orders of magnitude smaller, the abstract's claim survives. Until that check is done, the correct statement is that the sum of the considered exclusive channels is 1.5e-12, not the inclusive branching fraction.","tokens_in":22648,"tokens_out":2431,"duration_ms":29555,"concrete_test":"Estimate the hadronic contribution J/psi -> (mu+mu-)[3S1] + X_had at leading order by computing the same Feynman diagrams with the off-shell photon replaced by a virtual photon that couples to a hadronic system, e.g. model X = pi0 via gamma* -> 2 gluons -> pi0 or compute J/psi -> (mu+mu-)[3S1] + 2 gluons in NRQCD factorization, and compare with the leptonic channels (38) and (44). A simpler decisive check: compute the continuum J/psi -> mu+mu- + (anything) rate with the mu+mu- pair projected onto the 3S1 bound-state window and compare the integrated yield against the two exclusive leptonic channels. If the hadronic X contribution is below 10% of 1.5e-12, the inclusive claim is essentially correct; if it is comparable or larger, the headline branching fraction and the 'discoverable at STCF' projection should be revised.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The headline number Br(J/psi -> (mu+mu-)[3S1] + X) = 1.5e-12 is obtained by adding Eqs. (38) and (44), i.e. X = e+e- and X = mu+mu- only, with the radiative channel vanishing for the triplet. The text explicitly defines X as gamma or charged leptons, so this is an exclusive sum, not an inclusive branching fraction. No operator-product expansion, NRQCD scaling argument, or phase-space estimate is given for X = hadrons, X = gamma gamma, X = extra lepton pairs, or X = pi0, etc. The (mu+mu-)[3S1] has JPC = 1--, the same quantum numbers as the J/psi itself, so C-parity and the available couplings do not forbid production with a hadronic system; the same virtual-photon and multi-gluon mechanisms that drive ordinary J/psi hadronic decays can in principle contribute. The paper provides no estimate of such channels, so the quoted inclusive number could be an underestimate, and the STCF discovery projection of roughly five raw events per year inherits this uncertainty. A related but secondary discrepancy is that the abstract promises unlike-flavor radiative channels (l1+ l2-)+gamma with l1 != l2 that do not appear in the body; however, the central claim rests on the ortho-dimuonium inclusive rate, so the missing-X question is the load-bearing concern.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22980,"tokens_out":10814,"duration_ms":125355,"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":[{"comment":"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).","section":"Section II, Eqs. (47)-(50)"},{"comment":"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.","section":"Section II, Eqs. (37)-(46)"}],"minor_comments":[{"comment":"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.","section":"Abstract and Introduction"},{"comment":"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.","section":"Section II.E"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Section II.D, Eq. (24)"}],"recommendation":"major_revision","confidential_remarks":"The referee report is based on the manuscript text as provided. The central technical issue is that the paper's headline 'inclusive' branching fraction is an exclusive sum over X = gamma and X = charged lepton pair, with no estimate of hadronic or multi-photon X. I believe this is fixable within the manuscript's scope, either by renaming the quoted quantities or by adding an estimate of the missing channels, but the claim as it stands is not supported. The abstract also overstates the coverage of radiative channels for unlike-flavor leptonium. The underlying NRQED/NRQCD calculations appear standard and the radiative cross-check is a strength."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a straightforward NRQED/NRQCD calculation that does one genuinely useful thing: it gives the four-lepton production channels for all six leptonia in J/psi and Upsilon decays, including the ortho states, and it extends the analysis to tauonium. The radiative para-dimuonium channel repeats Dai et al., but the authors check against it and explain the factor of zeta(3) from summing excited states, which is a nice cross-check rather than a claim of novelty.\n\nThe new numbers look credible. The amplitudes are standard, the wavefunctions at the origin are extracted from measured leptonic widths (a normal external input), and the differential distributions are given for each mode, which is exactly what an experimental collaborator needs for a search. The Upsilon section is largely a kinematic extension, but it is done systematically, and the tauonium results are new.\n\nThe soft spots are real but not fatal. The biggest one: the paper calls Eqs. (47)-(50) inclusive branching fractions, but they are sums over X = gamma or one charged lepton pair. For the headline ortho-dimuonium channel, X is just e+e- and mu+mu-. No estimate is given for hadronic X, additional photons, or extra lepton pairs. Since (mu+mu-)[3S1] has the same JPC as the J/psi, there is no symmetry forbidding production with pions or kaons, and the C-parity argument that eliminates them in the radiative channel does not apply to strong or electromagnetic multi-particle final states. The authors need either an order-of-magnitude bound on such channels or a clear statement that 'inclusive' means 'within the computed exclusive set'. The STCF discovery estimate of five raw events per year inherits this uncertainty.\n\nTwo smaller issues. Many four-lepton branching fractions are quoted without uncertainties; the 2% error on the wavefunction is an easy propagation. And the abstract promises radiative channels for all six flavor combinations, but the body only computes the like-flavor ones and (tau+tau-) for Upsilon. The unlike-flavor radiative modes are absent.\n\nOn balance, the paper is worth serious refereeing. The central computation is sound, and the missing-X question is answerable rather than a contradiction. If the authors add a conservative estimate of the omitted channels, or appropriately demote the 'inclusive' label, the paper is publishable. I would send it to a good referee.","headline":"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.","tokens_in":23520,"tokens_out":3423,"would_cite":true,"duration_ms":39627,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["leptonium","dimuonium","true muonium","heavy quarkonium decays","NRQED","NRQCD","branching fraction","Super Tau-Charm Facility"],"falsifier":"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.","tokens_in":22428,"feed_emoji":"⚛️","tokens_out":16816,"duration_ms":179301,"temperature":0.7,"pith_summary":"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.","feed_headline":"Dimuonium should appear in J/psi decays at STCF","feed_subtitle":"Predicted branching fraction 1.5e-12 yields about five raw events per year of the never-seen muon atom.","key_machinery":"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.","core_discovery":"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 $(\\mu^+\\mu^-)[{}^3S_1] \\to e^+e^-(\\gamma)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the accumulated $10^{10}$ $J/\\psi$ events at the current charmonium experiment, the baseline that the radiative dimuonium channel cannot reach.","marker":"[6]"},{"why":"Sets the annual $3.4\\times10^{12}$ $J/\\psi$ yield of the future tau-charm facility, the exposure that makes the predicted $1.5\\times10^{-12}$ branching fraction observable.","marker":"[7]"},{"why":"Provides the leptonium binding energies and wavefunctions, including higher excited states, used for the Coulombic bound-state description.","marker":"[5]"},{"why":"The prior Green-function calculation of dimuonium production in radiative $J/\\psi$ decays; the paper compares its ratio with this result and isolates the $\\zeta(3)$ excited-state factor.","marker":"[27]"},{"why":"Introduces non-relativistic QED, the effective theory used to treat leptonium as a Coulombic bound state.","marker":"[34]"},{"why":"Establishes the NRQCD factorization and spin-projector technique used for the quarkonium side of the amplitudes.","marker":"[40]"},{"why":"Supplies the experimental inputs: the $J/\\psi$ total width, lepton masses, and the measured $\\mathrm{Br}(J/\\psi\\to\\mu^+\\mu^-)$ from which the $J/\\psi$ radial wavefunction at the origin is extracted.","marker":"[44]"}],"fun_headline_variants":["Ortho-dimuonium predicted in J/psi decays at STCF","J/psi decays may reveal ortho-dimuonium at STCF","Ortho-dimuonium could be first seen at STCF","Selection rules point to ortho-dimuonium at STCF"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Ortho-dimuonium predicted in J/psi decays at STCF","J/psi decays may reveal ortho-dimuonium at STCF","Ortho-dimuonium could be first seen at STCF","Selection rules point to ortho-dimuonium at STCF"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000446,"raw_usage":{"total_tokens":2393,"prompt_tokens":1225,"completion_tokens":1168,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":841,"completion_tokens_details":{"reasoning_tokens":1093}},"tokens_in":841,"tokens_out":1168,"duration_ms":12745,"temperature":1.0,"reasoning_tokens":1093,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:44:58.999775+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Deutsch, Evidence for the Formation of Positronium in Gases, Phys","cited_arxiv_id":null,"evidence_quote":"Supplies the accumulated $10^{10}$ $J/\\psi$ events at the current charmonium experiment, the baseline that the radiative dimuonium channel cannot reach."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Sets the annual $3.4\\times10^{12}$ $J/\\psi$ yield of the future tau-charm facility, the exposure that makes the predicted $1.5\\times10^{-12}$ branching fraction observable."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the leptonium binding energies and wavefunctions, including higher excited states, used for the Coulombic bound-state description."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The prior Green-function calculation of dimuonium production in radiative $J/\\psi$ decays; the paper compares its ratio with this result and isolates the $\\zeta(3)$ excited-state factor."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces non-relativistic QED, the effective theory used to treat leptonium as a Coulombic bound state."},{"cited_title":"Sommerfeld, ¨Uber die Beugung und Bremsung der Elektronen, Annalen Phys","cited_arxiv_id":null,"evidence_quote":"Establishes the NRQCD factorization and spin-projector technique used for the quarkonium side of the amplitudes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the experimental inputs: the $J/\\psi$ total width, lepton masses, and the measured $\\mathrm{Br}(J/\\psi\\to\\mu^+\\mu^-)$ from which the $J/\\psi$ radial wavefunction at the origin is extracted."}],"review_version":1}