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Searching for True Muonium in Relativistic Heavy Ion Collisions

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Pith's one-line read Heavy-ion collisions could make true muonium detectable for the first time.

desk verdict A careful production calculation whose central numbers rest on an overturned survival assumption: true muonium with a 512-fm Bohr radius will not simply sail through the QGP. read the letter →

arxiv 2505.10070 v1 pith:PSX2346G submitted 2025-05-15 nucl-th hep-ph

Zuo-Tang Liang , Qian Yang , Hong Zhang , Jiaxing Zhao , Pengfei Zhuang This is my paper · ORCID
classification nucl-thhep-ph PACS 25.75.-q36.10.-k12.20.-m
keywords truemuoniumquark-gluonplasmaheavy-ioncollisionsBoltzmanntransportviscoushydrodynamicsQEDboundstatesthermalproductionRHICLHC
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 hydrogen-like atom made of a muon and an antimuon—has never been observed. This paper argues that the quark-gluon plasma created in relativistic heavy-ion collisions should produce it in detectable numbers, through quark-antiquark and quark-gluon fusion with an extra gluon in the final state. Using thermal dressed quark masses and a viscous hydrodynamic description of the expanding fireball, the authors obtain effective production cross sections of 1.23 $\mu$b at RHIC and 14.2 $\mu$b at the LHC, with yields of roughly $10^4$ and $10^5$ per billion central collisions. That is more than a thousand times higher than an earlier estimate, and it would make heavy-ion collisions a viable discovery channel for the first observation of true muonium. The same signal would also act as a sensitive probe of the early-time temperature of the quark-gluon plasma.

What carries the argument

The object doing the work is the $n^3S_1$ true-muonium bound state, a nonrelativistic QED atom with binding energy $-1.4/n^2$ keV, mass $2m_\mu=211$ MeV, and Bohr radius roughly $512n$ fm. Its production is encoded in a Coulomb-resummed vertex $iZ_n=i(\alpha^2/4\pi)\sqrt{m_\mu^3 m_n/n^3}$, equivalent to evaluating the bound-state wavefunction at zero separation. The two NLO amplitudes (Eq. (4) and its crossing) are integrated with thermal distributions of dressed quarks, antiquarks, and gluons, and the resulting collision terms enter an analytic solution of the relativistic Boltzmann equation (Eq. (7)) over a viscous hydrodynamic background; the sum over principal quantum numbers factorizes into $\zeta_3=\sum_n n^{-3}$. The step function $\Theta(T-T_c)$ restricts production to the deconfined phase.

What would settle it

Compute the thermally averaged Coulomb-dissociation cross section of the $1.4$ keV-bound state by scattering thermal gluons and quarks at $T\sim300$–$500$ MeV and add it as a loss term in Eq. (5); if the surviving yield falls far below the quoted $\mathcal{O}(10^4)$–$\mathcal{O}(10^5)$, the discovery claim fails. A targeted experimental check would be a high-statistics search for a narrow peak near $211$ MeV in the dimuon or dielectron invariant-mass spectrum of central Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV.

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Extended reading notes

Core claim

The paper's central claim is that the quark-gluon plasma acts as a thermal source of the $n^3S_1$ state of true muonium, with production rates large enough for discovery at RHIC and the LHC. The calculation includes two next-to-leading-order processes, $q\bar q\to(\mu^+\mu^-)g$ and $qg\to(\mu^+\mu^-)q$, with the virtual photon coupled to the bound state through a Coulomb-resummed bubble amplitude, and it follows the produced states through an expanding viscous fluid using an analytic Boltzmann transport solution. The result is an effective cross section of $1.23\,\mu$b for central Au+Au at $\sqrt{s_{NN}}=200$ GeV and $14.2\,\mu$b for Pb+Pb at $\sqrt{s_{NN}}=5.02$ TeV, corresponding to $\mathcal{O}(10^4)$ and $\mathcal{O}(10^5)$ true muonium per billion central collisions. The authors take these values, over $10^3$ times the earlier estimate, as sufficient to allow a discovery of $(\mu^+\mu^-)$ and to make true muonium a new probe of the early quark-gluon plasma.

Load-bearing premise

The calculation assumes true muonium, once formed in the plasma, survives long enough to be counted; the Boltzmann equation drops dissociation and only the annihilation channel is ruled out.

Editorial extensions

If this is right

  • A narrow resonance near $211$ MeV should appear in the $\mu^+\mu^-$ (or $e^+e^-$) invariant-mass spectrum of central heavy-ion collisions at RHIC and the LHC.
  • The predicted di-lepton yield from true muonium lies between that of $J/\psi$ and $\Upsilon$, within reach of existing heavy-ion detectors.
  • Because production is concentrated at early times and the yield changes by a factor of two for a 30 MeV change in temperature, the signal is a sensitive early-QGP thermometer.
  • QGP production exceeds ultra-peripheral production by about two orders of magnitude, making heavy-ion collisions the most promising route to first observation.
  • The effective cross sections, $1.23\,\mu$b at RHIC and $14.2\,\mu$b at the LHC, are comparable to $\Upsilon$ production.

Reading between the lines

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

  • If Coulomb dissociation of the loosely bound state were included as a loss term, the net yield could drop by orders of magnitude; a quantitative breakup calculation would directly test the paper's central assumption.
  • The same thermal-production machinery could be applied to other shallow QED bound states that might form inside a quark-gluon plasma, such as pionium.
  • Because the predicted peak sits near $211$ MeV with little background, existing high-statistics dimuon data sets from RHIC and the LHC could be reanalyzed for the signal without new runs.
  • The steep temperature dependence suggests true muonium could be combined with direct-photon spectra to separate early-time and late-time temperature measurements of the plasma.
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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 / 5 minor

Summary. The manuscript proposes that true muonium (μ⁺μ⁻) can be produced thermally in the quark-gluon plasma formed in relativistic heavy-ion collisions. The authors compute the NLO scattering amplitudes for q q̄ → (μ⁺μ⁻)g and qg → (μ⁺μ⁻)q using DQPM thermal masses, and couple a gain-only Boltzmann equation to viscous hydrodynamic simulations of the QGP. They report effective production cross sections of 1.23 μb at RHIC and 14.2 μb at the LHC, corresponding to yields of O(10⁴) and O(10⁵) per billion central collisions, and conclude that heavy-ion collisions are a promising discovery channel for true muonium.

Significance. If the yield claim were correct, the paper would establish a new production and detection channel for an as-yet-unobserved QED bound state and would introduce a new early-time thermometer for the QGP. The transport framework, the analytic solution of Eq. (7), and the coupling to realistic hydrodynamic evolution are useful features, and the scattering amplitudes follow standard Feynman rules. However, the central quantitative claim rests on the unsupported neglect of dissociation in Eq. (5). Since true muonium is an extremely fragile object with a 512 fm Bohr radius and 1.4 keV binding energy, the absence of any estimate of Coulomb breakup by thermal partons is a load-bearing gap that undermines the reported yields and cross sections.

major comments (2)
  1. [Production in heavy ion collisions, Eq. (5)] The gain-only Boltzmann equation is not justified. The text asserts that dissociation can be safely ignored because of the small phase-space distribution and relatively long lifetime, but neither property controls the breakup rate in a hot medium. For a state with Bohr radius r_b ≈ 512 fm, the geometric ionization cross section by a scattering thermal parton is σ ≈ π r_b² ≈ 8×10⁵ fm², and with typical QGP parton densities the resulting dissociation rate is many orders of magnitude larger than the QGP expansion rate. Ruling out only the annihilation channel (μ⁺μ⁻) → q q̄ via thermal quark masses does not address Coulomb breakup. Without a loss term in Eq. (5), the yields computed from Eqs. (7)–(9) are unsupported and could be wrong by orders of magnitude.
  2. [Characteristic Scales] The time-scale argument has the wrong implication. From the paper's own numbers, τ_b ≈ 1.7×10⁻²¹ s, r_b ≈ 512 fm, τ_QGP ≈ 10⁻²³ s, and r_D ≈ 2.11/T ≈ 1 fm. If τ_b ≫ τ_QGP and r_b ≫ r_QGP, then the pair cannot form a bound state inside the QGP; if it binds only after freeze-out, the in-medium collision term in Eq. (6) with the vacuum Coulomb-resummed vertex of Eq. (3) is not the correct production amplitude. Moreover, a Yukawa potential with r_D/r_b ≈ 0.002 has no 1s bound state, so the vertex of Eq. (3), obtained from free-photon resummation in Fig. 2, is not the in-medium formation amplitude. The quoted 1.5 eV modification of the binding energy is not the relevant criterion: the medium modifies the wave function and the production vertex itself.
minor comments (5)
  1. [Abstract] The abstract says 'per billion AA collisions' while the text says 'per billion central collisions'; the centrality condition should be stated consistently.
  2. [Introduction] 'Six decades smaller' should read 'six orders of magnitude'.
  3. [Fig. 3 caption] The caption assigns T = 500, 300, 200 MeV to the orange, green, and blue curves, but the legend in the figure itself does not show this color–temperature correspondence; please add the temperatures to the legend.
  4. [Notation] The notation n³S₁ should be explicitly defined as the n-th ³S₁ state of the μ⁺μ⁻ system; the symbols n, m_n, and ζ₃ are used later without a clear definition of the principal quantum number summation range.
  5. [References] Reference [37] is an arXiv preprint; if a published journal version exists, it should be cited instead.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the true muonium yield is a genuine model output; the few self-citations are technical tools, not load-bearing inputs.

full rationale

The paper's central output—effective cross sections of 1.23 μb at RHIC and 14.2 μb at LHC, and yields O(10^4) and O(10^5) per billion central collisions—is obtained by integrating NLO QED/QCD scattering amplitudes (Eq. (4)) with DQPM thermal masses and couplings that are calibrated to lattice entropy from Ref. [38], and by transporting those rates through a MUSIC hydrodynamic medium whose initial conditions are fixed by measured charged-particle multiplicities. These inputs are external to the present work and do not encode the target yield. The only places where prior work by the same authors enters are the analytic Boltzmann solution in Eq. (7), cited to Refs. [37,42], and the comparison baseline Ref. [37]; these are mathematical or reference tools, not assumptions that define the predicted yield. The decision to drop dissociation in Eq. (5), while physically questionable for a 512-fm-radius state in a Debye-screened plasma, is a modeling approximation rather than a circular reduction: no dissociation parameter is fitted and then relabeled as a prediction, and omitting a loss term cannot make the gain-only result equal to any input by construction. No equation reduces to another by definition, no fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' own prior work. The self-citations are ancillary to the main derivation, so the score is 1 rather than 0 only to reflect that the transport solution is reused from the authors' earlier framework without independent re-derivation; it remains standard and checkable.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The calculation rests on several model assumptions not proven here: DQPM quasiparticle masses and coupling fitted to lattice entropy, vacuum Coulomb wavefunction for the bound-state vertex, a gain-only Boltzmann equation with no dissociation, and hydrodynamic initial conditions tied to final multiplicity. No new particles or entities are introduced.

free parameters (6)
  • lambda in g^2(T) = 2.42
    Fitted parameter in the piecewise strong-coupling formula Eq. (2) from Ref. [38]; controls the parton masses and production rates.
  • T* in g^2(T) = 1.19 T_c
    Boundary between the two pieces of the strong-coupling fit in Eq. (2), taken from the DQPM calibration to lattice entropy.
  • low-T power in g^2(T) = 3.1
    The exponent in Eq. (2) for T < T*, fitted to lattice QCD entropy density in Ref. [38].
  • strange quark mass offset = 0.045 GeV
    An additive shift in Eq. (1c) that sets the strange quark thermal mass in the DQPM.
  • shear viscosity to entropy ratio = 0.08
    Chosen value for the viscous hydrodynamic simulation; the paper does not scan over this parameter.
  • initial proper time and temperature profile = tau0=0.6 fm/c, Tmax=320 MeV (RHIC); tau0=0.4 fm/c, Tmax=450 MeV (LHC)
    Initial conditions for MUSIC are fixed by final charged-hadron multiplicity, and the yield depends strongly on this early-time temperature.
assumptions (5)
  • domain assumption The QGP does not modify the muon-antimuon bound-state wavefunction, so the vacuum Coulomb-resummed vertex iZ_n is valid.
    Invoked in the derivation of Eq. (3) and in the Characteristic Scales argument that Debye screening changes binding by only 1.5 eV; this is a load-bearing physical assumption.
  • domain assumption Thermal quark masses from DQPM are large enough that the leading-order q qbar to muon pair process is kinematically forbidden, leaving only NLO channels.
    Used to select the two NLO processes q qbar to (mu+ mu-) g and q g to (mu+ mu-) q; if the masses are different, the rates change substantially.
  • ad hoc to paper Dissociation of true muonium inside the QGP is negligible, so the Boltzmann equation has only gain terms.
    Asserted before Eq. (5) with the phrase 'one can safely ignore the dissociation'; no quantitative breakup estimate is given.
  • domain assumption Quarks, antiquarks, and gluons in the QGP follow thermal distributions with DQPM dressed masses, and the emitted gluon has only transverse polarizations.
    Used in the collision terms in Eq. (6) and in the squared amplitude Eq. (4).
  • domain assumption The QGP spacetime evolution is described by MUSIC with eta/s=0.08, zero bulk viscosity, and the Bernhard lattice EoS, with initial conditions tuned to final charged-hadron multiplicity.
    The predicted yield depends on the early-time temperature profile through Eq. (7); this is a standard but unverified input set.

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Pith. "Pith review of Searching for True Muonium in Relativistic Heavy Ion Collisions." pith.science (2026). https://pith.science/paper/PSX2346G

@misc{pith2026250510070,
  author       = {Pith},
  title        = {Pith review of: Searching for True Muonium in Relativistic Heavy Ion Collisions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PSX2346G}},
  note         = {Machine review of arXiv:2505.10070}
}
abstract

We investigate the production of the as-yet-undetected true muonium within the quark-gluon plasma formed in relativistic heavy-ion collisions, employing a relativistic Boltzmann transport framework coupled to viscous hydrodynamic simulations. The obtained effective cross sections for central collisions are 1.23~$\mu b$ in AuAu collisions with $\sqrt{s_{\rm NN}}=200$~GeV and 14.2~$\mu b$ in PbPb collisions with $\sqrt{s_{\rm NN}}=5.02$~TeV, resulting in a yield of $\mathcal{O}(10^4)$ and $\mathcal{O}(10^5)$ true muonium per billion $AA$ collisions at RHIC and the LHC, respectively. This establishes heavy-ion collisions as a promising process for detecting true muonium.

Figures

Figures reproduced from arXiv: 2505.10070 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic Feynman diagrams of NLO processes. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Feynman diagrams for the bubble amplitude with [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The collision terms as functions of true muo [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. The true muonium yield at each proper time [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4. The true muonium yield as a function of the number [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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