Dimuonium: True Muonium Bound State

• Dimuonium is a pure QED bound state of a muon and an antimuon with hydrogen-like characteristics, a Bohr radius of about 512 fm, and a binding energy of 1.4 keV.
• It serves as a precision-QED system and a muon-sector new-physics probe, with distinct para (γγ decay) and ortho (e⁺e⁻ decay) signatures that frame development of discovery strategies.
• Multiple production methods—including fixed-target interactions, e⁺e⁻ collisions, heavy quarkonium decays, and ultraperipheral heavy-ion collisions—highlight its experimental challenges and promise.

Dimuonium, also called true muonium, is the QED bound state $(\mu^+\mu^-)$, one of the pure QED bound systems of leptons together with positronium and ditauonium. It is a hydrogen-like system with a Bohr radius of about $512$ fm and a ground-state binding energy of $1.4$ keV, and it remains unobserved experimentally despite sustained theoretical attention (Feng et al., 20 Aug 2025, Chen et al., 2012). Contemporary work treats dimuonium simultaneously as a precision-QED system, a muon-sector new-physics probe, and a challenging experimental target whose production may be accessible in fixed-target reactions, low-energy $e^+e^-$ machines, heavy quarkonium decays, photon-photon collisions, and ultraperipheral heavy-ion collisions (Gninenko et al., 3 Mar 2025, Fox et al., 2022).

1. Bound-state structure and quantum numbers

Dimuonium is the bound state of a muon and an antimuon held together by electromagnetic interaction. Because its constituents are leptons, it is free of nuclear-structure ambiguities that complicate atoms with composite nuclei, and its dynamics are commonly modeled with Coulombic wavefunctions and NRQED matching (Feng et al., 20 Aug 2025, Uskov et al., 2022).

For the ground state, the wavefunction at the origin enters essentially every production and annihilation calculation: $|\Psi(0)|^2 = \frac{(\alpha m_\mu)^3}{8\pi}.$ This quantity appears explicitly in photon-photon production, collider production, and heavy-quarkonium decay calculations (Feng et al., 20 Aug 2025).

The lowest-lying spectroscopic states are the spin-singlet para-dimuonium and spin-triplet ortho-dimuonium. In the notation used across the literature, these are the $^1S_0$ and $^3S_1$ states, respectively. The $^3S_1$ state has $J^{PC}=1^{--}$ and the $^1S_0$ state has $512$0 (Gninenko et al., 3 Mar 2025). Selection rules strongly affect accessible production modes: even-spin, $512$1-even states can be formed in $512$2 fusion, while the neutral vector $512$3 state requires three photons in UPC triphoton production or other non-$512$4 mechanisms because of the Landau–Yang constraint (d'Enterria et al., 13 Mar 2025, Feng et al., 23 Apr 2026).

A compact summary of the lowest states is given below.

State	Lifetime	Dominant decay
Para-dimuonium $512$5	$512$6 fs	$512$7
Ortho-dimuonium $512$8	$512$9 ps	$1.4$0

The para-state lifetime is also quoted as $1.4$1 ps in production studies, while the triplet lifetime is consistently quoted near $1.4$2 ps; the difference reflects the usual precision conventions across papers rather than a physics discrepancy (Gninenko et al., 3 Mar 2025, d'Enterria et al., 13 Mar 2025).

2. Decays, spectroscopy, and precision observables

Dimuonium decays are state-dependent and experimentally consequential. Para-dimuonium decays dominantly to $1.4$3, with branching ratio $1.4$4 in the NLO photon-photon production study and $1.4$5 in UPC phenomenology (Feng et al., 20 Aug 2025, d'Enterria et al., 13 Mar 2025). For para-dimuonium in the UPC treatment, the diphoton width is

$1.4$6

numerically yielding approximately $1.4$7 meV (d'Enterria et al., 13 Mar 2025).

Ortho-dimuonium is longer-lived and phenomenologically more favorable in channels with displaced charged tracks. Its dominant decay is to $1.4$8, with branching $1.4$9 in the photon-photon NLO analysis, and the annihilation width is written in the rare-decay study as

$e^+e^-$0

This is the basis of several discovery strategies centered on low-mass displaced $e^+e^-$1 vertices (Feng et al., 20 Aug 2025, Czarnecki et al., 2017).

Although dimuonium is a pure QED atom in its binding dynamics, rare hadronic channels are not absent. The decay

$e^+e^-$2

has branching ratio

$e^+e^-$3

so approximately one in $e^+e^-$4 ortho-dimuonia decays into this channel (Czarnecki et al., 2017). The emitted photon has energy

$e^+e^-$5

providing a distinctive monochromatic signature unavailable in positronium because of the lower mass threshold (Czarnecki et al., 2017).

The same virtual $e^+e^-$6 dynamics induces a calculable hyperfine-splitting correction. The real part of the shift is quoted as about $e^+e^-$7 MHz, corresponding to a $e^+e^-$8 part-per-million effect, while the loop also reproduces the leading $e^+e^-$9 contribution to the muon anomalous magnetic moment,

$|\Psi(0)|^2 = \frac{(\alpha m_\mu)^3}{8\pi}.$0

as a consistency check (Czarnecki et al., 2017). This establishes that “pure QED” should not be misread as “completely isolated from hadronic vacuum-polarization effects”; rather, hadronic contributions are small, structured, and calculable.

3. Production in lepton colliders and heavy quarkonium decays

A major production axis is $|\Psi(0)|^2 = \frac{(\alpha m_\mu)^3}{8\pi}.$1 fusion at high-luminosity $|\Psi(0)|^2 = \frac{(\alpha m_\mu)^3}{8\pi}.$2 facilities. The relevant EPA process is

$|\Psi(0)|^2 = \frac{(\alpha m_\mu)^3}{8\pi}.$3

with total cross section obtained by convolution with the Weizsäcker–Williams photon flux (Feng et al., 20 Aug 2025). The first full NLO QED calculation for this channel finds that the corrections are negative for both para and ortho states. Quantitatively, the para correction is

$|\Psi(0)|^2 = \frac{(\alpha m_\mu)^3}{8\pi}.$4

corresponding to $|\Psi(0)|^2 = \frac{(\alpha m_\mu)^3}{8\pi}.$5, while the ortho correction ranges from $|\Psi(0)|^2 = \frac{(\alpha m_\mu)^3}{8\pi}.$6 at BEPCII to $|\Psi(0)|^2 = \frac{(\alpha m_\mu)^3}{8\pi}.$7 at CEPC (Feng et al., 20 Aug 2025). The same study quotes order $|\Psi(0)|^2 = \frac{(\alpha m_\mu)^3}{8\pi}.$8 para-dimuonium events/year at Belle II/SuperKEKB and about $|\Psi(0)|^2 = \frac{(\alpha m_\mu)^3}{8\pi}.$9 para-dimuonium plus $^1S_0$0 ortho-dimuonium events/year at STCF, indicating that LO-based discovery projections remain essentially intact after NLO refinement (Feng et al., 20 Aug 2025).

Heavy quarkonium decays provide an orthogonal route. In the NRQED/NRQCD analysis of $^1S_0$1 and $^1S_0$2 decays, the most favorable dimuonium channel is inclusive ortho-dimuonium production in $^1S_0$3 decay: $^1S_0$4 The corresponding inclusive para-dimuonium branching fraction is

$^1S_0$5

and the radiative para channel is

$^1S_0$6

With the annual production of $^1S_0$7 $^1S_0$8 events at STCF, this translates to approximately $^1S_0$9 ortho-dimuonium events/year and about $^3S_1$0 para-dimuonium events/year under the stated assumptions (Zhao et al., 5 May 2025).

Dedicated threshold machines have also been proposed. The DIMUS concept uses collisions of $^3S_1$1 MeV electrons and positrons at a $^3S_1$2 angle in a compact $^3S_1$3 m circumference collider, targeting peak luminosity $^3S_1$4 and $^3S_1$5 million dimuons per year (Fox et al., 2022). The large crossing angle boosts the produced atom so that the decay point is shifted from the beam-collision region; a related collider design quotes a decay length

$^3S_1$6

with $^3S_1$7 mm at $^3S_1$8, enabling strong rejection of prompt Bhabha backgrounds (Bogomyagkov et al., 2017). This suggests that direct threshold production is not only a spectroscopy strategy but also a background-engineering strategy.

4. Fixed-target, heavy-ion, and ultraperipheral production channels

A recent fixed-target proposal introduces a new mechanism for forming true muonium in Drell–Yan dimuon production off nuclei in a thin foil target. In this picture, a fraction of the $^3S_1$9 pairs produced in proton–nucleus Drell–Yan reactions bind via the static Coulomb interaction when produced with small relative momentum (Gninenko et al., 3 Mar 2025). The discovery signature is an excess of $^3S_1$0 pairs with invariant mass near $^3S_1$1 MeV and a displaced vertex downstream of the foil, with decay time consistent with the ortho-dimuonium lifetime (Gninenko et al., 3 Mar 2025). For $^3S_1$2 POT at $^3S_1$3 GeV and $^3S_1$4 GeV with an optimal $^3S_1$5Be foil thickness of $^3S_1$6 mm, the surviving $^3S_1$7 yield is quoted as $^3S_1$8 and $^3S_1$9, respectively; after including passage, excitation, and dissociation, the number of $J^{PC}=1^{--}$0 atoms with $J^{PC}=1^{--}$1 GeV surviving to vacuum is $J^{PC}=1^{--}$2 and $J^{PC}=1^{--}$3 (Gninenko et al., 3 Mar 2025). Low-$J^{PC}=1^{--}$4, low-density targets such as Be are identified as optimal.

UPCs furnish both para- and ortho-production channels. In exclusive photon fusion for even-spin resonances, para-dimuonium is treated as an even-spin QED bound state producible in $J^{PC}=1^{--}$5 fusion. For PbPb at $J^{PC}=1^{--}$6 TeV, one study quotes a para-dimuonium cross section of $J^{PC}=1^{--}$7b and an expected yield of $J^{PC}=1^{--}$8 events, while FCC PbPb at $J^{PC}=1^{--}$9 TeV yields $^1S_0$0b and $^1S_0$1 events (d'Enterria et al., 13 Mar 2025). A separate LHC study using a direct bound-state projection formalism gives

$^1S_0$2

for Pb+Pb at $^1S_0$3 TeV, and finds that EPA overestimates production at low $^1S_0$4 by $^1S_0$5 (Bertulani et al., 2023). These two predictions are model-dependent but numerically compatible at the order-of-magnitude level.

For the vector state, triphoton fusion has emerged as a distinct UPC mechanism: $^1S_0$6 In Pb+Pb UPCs at $^1S_0$7 TeV, the predicted dimuonium cross section via $^1S_0$8 fusion is $^1S_0$9 for $512$00, compared with $512$01 for the radiative two-photon channel, so triphoton production is dominant by orders of magnitude in that setup (Feng et al., 23 Apr 2026). The same paper quotes, for ortho-dimuonium, LHC values of $512$02--$512$03 and $512$04--$512$05 events/year, depending on $512$06 and detector-threshold assumptions (Feng et al., 23 Apr 2026). The preferred signature is an extremely low-$512$07 displaced $512$08 pair with invariant mass near $512$09 MeV (Feng et al., 23 Apr 2026).

Dimuonium production has also been studied in the QGP created in relativistic heavy-ion collisions. There the dominant process is identified as

$512$10

with transport evolution governed by a Boltzmann-type equation because dimuonium does not thermalize with the medium (Chen et al., 2012). The overall yield is small, about $512$11 for central RHIC collisions and three times larger at LHC, but the low-$512$12 spectrum can be fitted with effective temperatures of $512$13 MeV at RHIC and $512$14 MeV at LHC, implying that the transverse-energy distribution encodes QGP thermodynamics (Chen et al., 2012). This suggests that, even where total rates are marginal, dimuonium can function as a QED probe of hot QCD matter.

5. Matter interactions, foil transport, and computational infrastructure

Passage through matter is central both to detector concepts and to dedicated foil-based spectroscopy proposals. The interaction is modeled as scattering in the external screened Coulomb field of atomic nuclei, typically in the non-relativistic Born approximation (Uskov et al., 2022, Alizzi et al., 2024). In MuMuPy, the core observable is the atomic form factor

$512$15

from which transition and breakup cross sections are computed (Uskov et al., 2022). The screened target potential is modeled with Thomas-Fermi-Molière screening: $512$16 MuMuPy implements three independent calculation strategies—Dewangan’s method, Afanasev and Tarasov’s method, and a new hypergeometric-based method—and supports arbitrary hydrogenic quantum numbers, target $512$17 from $512$18 to $512$19, and states up to $512$20 (Uskov et al., 2022).

A dedicated transport study for dimuonium traversing Be, Al, and Pb foils formulates the state populations through the kinetic system

$512$21

Using $512$22 different approximations of the atomic potential, that work finds that the atomic-potential-model-dependent error in the yields of the low lying states is quite small: absolute cross sections can vary by as much as $512$23, but ratios of cross sections vary by less than $512$24, so the impact on the final low-lying yields is negligible within the applied Born approximation (Alizzi et al., 2024). The same analysis reports $512$25 yields up to $512$26 and $512$27 yields up to $512$28 of the initial $512$29 population for a representative setup, with good convergence already by $512$30 for low-lying states (Alizzi et al., 2024). A methodological correction is emphasized: the quantization axis should be chosen along the dimuonium beam, not along the transferred momentum (Alizzi et al., 2024).

Matter effects also enter fixed-target discovery proposals through dissociation. In the Drell–Yan foil mechanism, the breakup cross section is parameterized as

$512$31

with survival probability

$512$32

This makes target optimization a balance between production, transmission, excitation, and breakup (Gninenko et al., 3 Mar 2025).

6. Experimental status, theoretical caveats, and outlook

The experimental status remains unchanged: positronium was observed in 1951, whereas the search for dimuonium is still in vain (Feng et al., 20 Aug 2025). What has changed is the breadth of viable production strategies and the maturity of the corresponding theory. Recent calculations argue that observation is tenable at Belle II and especially at STCF in $512$33 production (Feng et al., 20 Aug 2025), promising in $512$34 decays at STCF (Zhao et al., 5 May 2025), plausible in dedicated threshold machines such as DIMUS (Fox et al., 2022), and potentially accessible in fixed-target Drell–Yan-on-foil searches (Gninenko et al., 3 Mar 2025) and in UPCs at the LHC or FCC (Bertulani et al., 2023, Feng et al., 23 Apr 2026).

Several recurring misconceptions are corrected by the current literature. First, the statement that Coulomb corrections are negligible is only partially true. In high-energy $512$35 electroproduction and paradimuonium production by a relativistic electron in an atomic field, Coulomb corrections to cross sections integrated over final-electron momentum are small, but Coulomb corrections to the differential cross sections are large: in the peak region near $512$36, the exact cross section is up to $512$37 lower than the Born result, whereas for $512$38 it is about $512$39 higher (Krachkov et al., 2017). Second, EPA-based estimates are not uniformly adequate across phase space: the LHC para-dimuonium study finds a $512$40 overestimate at low $512$41 (Bertulani et al., 2023). Third, large production rates do not automatically imply straightforward reconstruction. In UPC para-dimuonium searches, the decay photons have energies close to $512$42 MeV each, and detection is described as extremely challenging because of low energies, small opening angles, and continuum/instrumental backgrounds (d'Enterria et al., 13 Mar 2025).

The convergence of theory and experimental design is strongest around displaced-vertex and exclusive low-mass signatures. Ortho-dimuonium searches repeatedly target displaced $512$43 decays near $512$44 MeV (Gninenko et al., 3 Mar 2025, Feng et al., 23 Apr 2026), while para-dimuonium programs emphasize narrow diphoton structure at the same mass scale (Bertulani et al., 2023, d'Enterria et al., 13 Mar 2025). A plausible implication is that the first observation will be driven less by raw production cross section than by the channel in which kinematic cleanliness, vertexing, and trigger thresholds can be made commensurate with the picosecond-scale lifetimes and sub-GeV visible final states.