Flavor-violating dark matter at MEG-II and Mu3e

Daniele Barducci; Giulio Marino; Paolo Panci; Robert Ziegler

REVIEW 3 major objections 6 minor 94 references

Flavor-violating dark matter at MEG-II and Mu3e

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Rare muon decays may expose dark matter at TeV scales.

desk verdict Solid systematic EFT study of μ→eχχ at MEG-II/Mu3e, but it underplays an existing total-muon-lifetime/EW constraint that could cut into its TeV-scale reach and freeze-in region. read the letter →

arxiv 2608.02740 v1 pith:V2E3DREV submitted 2026-08-03 hep-ph hep-ex

Daniele Barducci , Giulio Marino , Paolo Panci , Robert Ziegler This is my paper · ORCID

classification hep-phhep-ex

keywords leptonflavorviolationdarkmattermuondecayMichelspectrumfreeze-ineffectivefieldtheoryMEG-IIMu3e

topics Dark Matter

open problems Dark Matter

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

This paper argues that high-intensity muon experiments can detect dark matter through the lepton-flavor-violating decay $\mu^+\to e^+\chi\bar\chi$, even though charged lepton flavor is conserved in the Standard Model except for tiny neutrino-mass effects. The key move is to ignore the total event rate and compare only the shape of the positron energy and angular spectrum, the Michel spectrum, with the Standard Model prediction. The authors show that this shape-only analysis reaches new-physics scales of order TeV for dark matter masses above roughly 1 MeV, and that the radiative decay $\mu^+\to e^+\chi\bar\chi\gamma$ at MEG-II extends sensitivity down to massless dark matter. If correct, the same interactions could also generate the observed dark matter abundance by freeze-in when the reheating temperature is below about 10 MeV, making these experiments probes of both particle physics and cosmology.

What carries the argument

The machinery is an effective field theory of dimension-six four-fermion operators $(\bar e \Gamma_X \mu)(\bar\chi \Gamma_X \chi)$ with $\Gamma_X\in\{1,\gamma^5,\gamma^\mu,\gamma^\mu\gamma^5,\sigma^{\mu\nu}\}$, normalized by an effective scale $\Lambda^{\mu e}_X$. The analysis builds a shape-only $\chi^2$ from binned positron energy and $\cos\theta_e$ distributions, fixing the total muon count to the Standard Model expectation from a calibration phase, which isolates spectral distortion from any overall rate change. For light dark matter, the radiative decay adds the photon energy and opening angles as extra kinematic variables, and the freeze-in calculation uses the Boltzmann equation with decay and annihilation collision terms to connect the same operator to the relic abundance.

What would settle it

A measurement of the total muon lifetime at the level of $10^{-6}$ that sets $\mathrm{BR}(\mu^+\to e^+\chi\bar\chi)\lesssim 10^{-6}$ would falsify the claimed O(TeV) reach for vector operators, since at $\Lambda\sim 1$\text{--}$2$ TeV the vector branching ratio is roughly $10^{-3}$ to $10^{-4}$.

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Extended reading notes

Core claim

The central claim is that the decay $\mu^+\to e^+\chi\bar\chi$, mediated by dimension-six four-fermion operators of all Lorentz structures, can be distinguished from the Standard Model Michel decay by shape alone, provided the dark matter mass exceeds about 1 MeV and the experiment records enough muons. For pure right-handed or vector/axial couplings, the positron angular distribution differs from the Standard Model and yields effective scales $\Lambda^{\mu e}_X$ of order TeV; for pure left-handed couplings the signal becomes degenerate with the Standard Model as $m_\chi\to 0$, but the radiative decay $\mu^+\to e^+\chi\bar\chi\gamma$ removes that degeneracy and restores sensitivity down to the massless limit. In addition, for reheating temperatures between the big bang nucleosynthesis bound and about 10 MeV, muon decay and $\mu\bar e$ annihilation through the same operators can produce the observed dark matter relic abundance via freeze-in, and the parameter region overlaps with the projected reach of MEG-II and Mu3e.

Load-bearing premise

The whole reach rests on the assumption that the total muon lifetime does not already provide a stronger bound than the spectral shape, because the analysis deliberately ignores the overall event rate in favor of shape only.

Editorial extensions

If this is right

MEG-II and Mu3e can constrain $\Lambda^{\mu e}_X$ to $\mathcal O(1\text{--}2)$ TeV for dark matter masses above about 1 MeV, a reach comparable to supernova cooling bounds.
The radiative decay $\mu\to e\chi\bar\chi\gamma$ at MEG-II remains sensitive down to massless dark matter and is competitive with, or stronger than, the non-radiative channel in several regions.
For a purely left-handed vector interaction the non-radiative search loses sensitivity as $m_\chi\to 0$, but the radiative channel removes this degeneracy.
For scalar interactions the non-radiative channel gives significantly stronger bounds than the radiative one because the scalar radiative signal is suppressed by the small invariant mass of the invisible pair.
If the reheating temperature lies between roughly 4 and 10 MeV, freeze-in through the same lepton-flavor-violating operator can produce the observed dark matter abundance in regions accessible to MEG-II and Mu3e.

Reading between the lines

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

A precision measurement of the total muon lifetime at the $10^{-6}$ level would independently test the claimed reach: for vector operators the lepton-flavor-violating branching ratio is roughly $1/(2G_F^2\Lambda^4)$, so a TeV-scale $\Lambda$ corresponds to a branching ratio of order $10^{-3}$ to $10^{-4}$, far above the current lifetime precision.
Because the shape-only analysis uses the Michel spectrum as its own background, the same procedure could be recycled for other invisible final states, such as $\mu\to e$ plus a light scalar, and for constraining the Lorentz structure of any future signal by comparing energy and angular bins.
A detected signal would make the freeze-in link testable: for reheating temperatures above about 10 MeV the same operator would overproduce dark matter, so a positive observation at MEG-II or Mu3e would either pin the reheating temperature below that scale or require additional suppression of the electron-flavor diagonal coupling.

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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

The genuinely new content here is the complete Lorentz-structure scan of μ→eχχ, the radiative μ→eχχγ channel as a handle for light DM, and the freeze-in connection with T_R below ~10 MeV. The rate calculations, benchmark definitions, and the shape-based statistical procedure are clean and presented in enough detail that a reader can reproduce them. The authors also flag the calibration region near the Michel endpoint and the left-handed degeneracy honestly. That is real, credit-worthy work.

The soft spot is the total muon lifetime. The paper mentions in Sec. 5.2 that LFV operators are constrained by the EW fit, but it never quantifies that bound or shows it in the sensitivity plots. For the vector operator, Γ_NP/Γ_SM ≈ 1/(2 G_F^2 Λ^4), so Λ = 1 TeV gives BR_NP ≈ 3.7×10⁻³ and Λ = 2 TeV gives ≈ 2.3×10⁻⁴. The measured muon lifetime and global EW fit plausibly constrain such an additional width at the 10⁻⁴–10⁻³ level. If that is right, the shape-only projections in Figs. 4 and 7 are over-optimistic for the low-Λ side of the bands, and the freeze-in curves for mχ ~ 0.1–1 MeV at T_R = 6–10 MeV—which sit at Λ ~ 2–4 TeV—may partially lie in an already-disfavored region. The paper's own rough SN1987A bound (Λ_μeV ≳ 2.8 TeV) is also not drawn in the freeze-in figures, even though it is estimated in Sec. 5.2. This is a specific, addressable gap rather than a fatal flaw: the radiative channel for mχ < 1 MeV and the high-mass reach above ~2 TeV can still survive, but the claimed sensitivity and the cosmological overlap need to be re-plotted with the lifetime/EW and SN bounds included.

Minor: the radiative background estimate relies on a simple random-coincidence model; that is fine as a first pass, but the 4% systematic is a guess. The scalar and tensor results in the appendix are a useful completeness check.

This paper is for particle phenomenologists working on dark matter, lepton flavor, or the MEG-II/Mu3e physics case. It deserves a serious referee: the central method is sound, and the missing bound can be added by the authors in revision. I would not desk-reject it.

Referee Report

3 major / 6 minor

Summary. The paper investigates the sensitivity of MEG-II and Mu3e to lepton-flavor-violating muon decays into a positron and a dark matter pair, μ+→e+χχ̅, described by dimension-six four-fermion operators. The authors compute the differential decay rates for all Lorentz structures, perform a shape-only analysis of the Michel spectrum (Sec. 3), and show that scales Λ_μe_X of order TeV can be probed for m_χ above about 1 MeV, with the pure right-handed vector case giving the strongest reach. For lighter DM, the radiative decay μ+→e+χχ̅γ at MEG-II is proposed as a complementary channel that restores sensitivity down to the massless limit and breaks the left-handed degeneracy (Sec. 4). The paper also computes freeze-in production of χ through the same LFV operators and finds an overlap between the experimental reach and the parameter region reproducing the observed relic abundance for reheating temperatures below about 10 MeV (Sec. 6). Appendices collect results for scalar and tensor interactions and a flavor-scheme example. The central claims are the TeV-scale laboratory reach and the freeze-in connection.

Significance. If the claimed sensitivities hold, this work identifies a new, phenomenologically motivated use of high-intensity muon facilities: probing LFV interactions with a dark sector at effective scales comparable to, or exceeding, those from direct searches. The paper's systematic enumeration of Lorentz structures, full m_χ-dependent differential rates, and explicit treatment of detector effects are strengths, as is the proposal of the radiative channel to cover the calibration-dominated region. The freeze-in connection is an interesting addition that links rare-decay searches to early-universe DM production. However, the significance is substantially tempered by two issues that are not adequately treated in the manuscript: the unquantified total muon-lifetime/EW-fit constraint on the same operators, and the paper's own SN1987A estimate (Eq. 5.6) which is stronger than the projected reach and would exclude the claimed freeze-in overlap. These must be resolved before the main conclusions can be accepted.

major comments (3)

[Sec. 5.2] The paper mentions 'constraints on the total muon lifetime as part of the EW fit' without quantifying them. Because the shape-only analysis of Sec. 3.1 explicitly discards total-rate information by fixing the total number of muons to the SM expectation, the existing muon-lifetime bound is essential for assessing the reach. For the vector benchmark, Eq. (2.6) gives Γ_NP/Γ_SM = 1/(2 G_F² Λ⁴), so BR_NP ≈ 3.7×10⁻³ at Λ = 1 TeV and ≈ 2.3×10⁻⁴ at Λ = 2 TeV. The muon lifetime determines G_F with relative precision around 10⁻⁶, and a shift in G_F is constrained by the electroweak fit (e.g., via M_W). A conservative bound of BR_NP ≲ 10⁻⁴–10⁻³ would exclude Λ below roughly 1.3–2.4 TeV, covering most of the projected MEG-II/Mu3e reach in Figs. 4, 5, and 7 and removing the claimed freeze-in overlap. The authors should compute this bound explicitly and show it in the sensitivity plots.
[Sec. 5.2, Eq. (5.6)] The authors' own SN1987A estimate gives Λ_μe_V ≳ 2.8 TeV for the vector benchmark. This lower bound is stronger than the projected sensitivities shown in Figs. 4, 5, and 7, and it lies above the freeze-in curves in Figs. 6 and 7 once Eq. (6.8) is corrected as discussed in the next major comment. If this estimate is taken at face value, the central claims that MEG-II and Mu3e 'can probe' this parameter space and that freeze-in through the same operators is testable are internally inconsistent with the complementary bound presented in the same paper. The authors must either provide a more reliable SN calculation that relaxes the bound or reframe the conclusions to state that the experiments probe scales below an existing exclusion. Showing the SN bound in Figs. 6 and 7 is mandatory.
[Sec. 6, Eq. (6.8)] The displayed analytical freeze-in formulas contain (10 MeV/T_R)^{5/2} e^{-10 MeV/T_R}. The Boltzmann suppression of the thermal muon abundance should be e^{-m_μ/T_R} with the corresponding power of (m_μ/T_R), since x_R = m_μ/T_R and F_{1→3}(x_R) ≃ (1/3)√(2/π) x_R^{5/2} e^{-x_R} as stated in the text. As written, the exponential suppression is underestimated by a factor e^{-(m_μ−10 MeV)/T_R}, which is about e^{-95.7 MeV/10 MeV} ≈ 10⁻⁴ at T_R = 10 MeV. This is a quantitative error in a central equation. The numerical coefficient (4.73 TeV) should be re-derived with the correct factor, and the claim that the analytical expressions 'provide an excellent approximation to the full numerical solution' should be rechecked after the correction.

minor comments (6)

[Sec. 3.1, Eq. (3.7)] The sum in the χ² definition is written with a 'P_i' that should be a summation symbol ∑_i.
[Sec. 3.3] The sentence 'we will fix keep fixed the Michel spectrum normalization' contains a typo; it should read 'we will keep fixed'.
[Sec. 6, after Eq. (6.2)] The phrase 'since the decay channel considered here is open only for m_χ < m_μ/2' appears in the discussion of the scattering contribution; this should be clarified to avoid confusion.
[Fig. 4 and Fig. 7] The y-axis labels '102 103' are ambiguous; please use superscripts or explicit units (GeV) in the axis title.
[Sec. 5.2, Tab. 1] The statement that the trapping regions at low Λ are 'essentially excluded by direct experimental searches' is too strong for the muon operator; the NA64 limit Λ_μμ ≳ O(10 GeV) only excludes a small part of the trapping interval, which extends up to several TeV.
[Introduction and Abstract] The phrase 'comparable to astrophysical constraints from supernova cooling' is imprecise given Eq. (5.6), which gives a bound stronger than the projected reach; the comparison should be quantified.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the MEG-II/Mu3e reach is derived from a shape-only chi-square fit to alpha with an independent EFT mapping, and the freeze-in curves impose Omega h^2=0.12; self-citations enter only complementary SN bounds.

full rationale

The central derivation is self-contained and not circular. The experimental sensitivity is obtained by constructing a shape-only chi-square (Eq. 3.7) over the Michel spectrum, fitting the parameter alpha defined in Eq. (3.3), and then converting alpha into a bound on Lambda_mu_e_X through the closed-form kinematic relation Eq. (3.8) together with the total rates Eq. (2.6). The mapping from alpha to Lambda is a theoretical prediction of the EFT, not a fitted quantity renamed as a prediction; alpha is an independently fitted shape parameter, and no target observable is used as an input to itself. The freeze-in analysis solves the Boltzmann equation (Eq. 6.1) with collision terms computed from the same LFV operators and imposes the observed relic abundance Omega h^2 = 0.12 as an external constraint; the resulting curves are matching conditions, not derivations of the experimental reach from the DM abundance. The paper does not import a uniqueness theorem from the authors' prior work, and it does not smuggle in the LFV ansatz by citing unpublished assumptions. The only author-overlapping citations are Refs. [83,100-102], which enter the complementary supernova-cooling estimates in Sec. 5; those bounds are not used to derive the MEG-II/Mu3e reach in Figs. 4, 5, or 7, nor do they feed into the freeze-in curves. The detector response and efficiencies are taken from external experimental references and Ref. [19], which are used as input data, not as a substitute for the physics result. The unquantified muon-lifetime/EW-fit constraint discussed in the reader's take is a plausible correctness concern about the total-rate normalization, but it is not a circularity: the analysis deliberately discards total-rate information, and a stronger external bound would shrink the claimed region without making the derivation circular. Accordingly, no step reduces by construction to its own input, and the paper earns a low score reflecting only the presence of minor, non-load-bearing self-citations.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The central EFT and freeze-in derivations are mostly self-contained and use external experimental inputs as anchors. The main extra assumptions are the operator basis, the Z2-stable Dirac DM state, and the freeze-in approximations. The reach figures additionally depend on assumed systematic uncertainties, on a free Michel-event normalization for MEG-II, and on an implicit neglect of the total muon-lifetime constraint, the last of which is flagged as a red flag.

free parameters (2)

Effective Michel-event normalization N_mu+ at MEG-II = 10^5 to 10^7
Chosen as a free range in Sec. 3.2 because MEG-II only records e+gamma coincidences; the derived Lambda bounds scale roughly as N_mu+ to the 1/4, so this assumption sets the headline reach.
Systematic uncertainty epsilon_syst = 10^-3 to 10^-2; 4x10^-2 for radiative photon channel
Assumed rather than measured. At Mu3e's large dataset the limits are systematics-dominated, and the 4% photon-trigger uncertainty drives the radiative analysis in Sec. 4.

assumptions (5)

domain assumption The dimension-six four-fermion EFT with Lambda much larger than m_mu, X=Y, and no pseudotensor structure is a complete low-energy description.
Sec. 2 introduces Eq. (2.2) and the X=Y restriction; all sensitivity numbers depend on this operator basis.
domain assumption chi is a Dirac fermion stabilized by a Z2 symmetry and is the only dark-sector state.
Sec. 2; the decay final state and freeze-in calculation would change for bosonic or unstable dark matter.
domain assumption Initial DM abundance at reheating is zero, and Maxwell-Boltzmann statistics with m_e=0 describe the freeze-in collision terms.
Sec. 6, around Eqs. (6.1) and (6.2); these approximations set the thermal curves in Figs. 6 and 7.
ad hoc to paper The existing total muon-lifetime constraint is not stronger than the shape-only sensitivity.
Sec. 5.2 mentions the lifetime only in passing; the paper's O(TeV) reach and freeze-in overlap implicitly require this premise, but a 10^-6 lifetime measurement corresponds to Lambda above about 8 TeV for the vector benchmark.
domain assumption The positron spectrum of MEG-II random-coincidence e+gamma events is equivalent to the SM Michel spectrum for shape analysis.
Sec. 3.2 and Sec. 4; this allows reusing MEG-II data, but the trigger and acceptance distortions are only approximately modeled.

invented entities (1)

Light Dirac fermion dark matter chi with LFV four-fermion couplings to electrons and muons independent evidence
purpose: Final state in mu to e chi chi and mu to e chi chi gamma, and the freeze-in relic population
chi is assumed and not observed; the paper computes falsifiable decay spectra and relic-density targets, so the model can be tested, but no independent observation currently supports it.

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Cite this review

Pith. "Pith review of Flavor-violating dark matter at MEG-II and Mu3e." pith.science (2026). https://pith.science/paper/V2E3DREV

@misc{pith2026260802740,
  author       = {Pith},
  title        = {Pith review of: Flavor-violating dark matter at MEG-II and Mu3e},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V2E3DREV}},
  note         = {Machine review of arXiv:2608.02740}
}

abstract

We investigate the potential of high-intensity muon experiments such as MEG~II and Mu3e to uncover dark matter (DM) through the lepton-flavor-violating decay $\mu^+\to e^+\chi\bar\chi$. We describe the underlying interactions in terms of dimension-six four-fermion operators and systematically explore all allowed Lorentz structures. We show that precision measurements of the Michel spectrum can probe new-physics scales of ${\cal O}({\rm TeV})$ for DM masses above approximately $1~{\rm MeV}$. For lighter DM, whose signal is confined close to the Michel endpoint, the radiative decay $\mu^+\to e^+\chi\bar\chi\gamma$ at MEG~II opens a complementary window, with sensitivity comparable to and, in some regions, stronger than that of the non-radiative channel. Remarkably, for reheating temperatures below the tens-of-MeV scale but above the BBN bound, MEG~II and Mu3e can probe regions in which freeze-in through the very same LFV interactions accounts for the observed DM abundance.

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