$\mu^- \to e^-\gamma$ in a muonic atom as a probe for effective lepton flavor violating operators involving photon fields

Masato Yamanaka; Yoshitaka Kuno; Yuichi Uesaka

REVIEW 2 major objections 4 minor 30 references

$\mu^- \to e^-\gamma$ in a muonic atom as a probe for effective lepton flavor violating operators involving photon fields

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A muon decaying inside an atom can directly probe lepton-flavor-violating operators with two photon fields, and for a zinc target the maximal allowed signal exceeds the accidental background by about threefold after angular cuts.

desk verdict Genuinely new muonic-atom CLFV process with a solid low-Z formalism, but the zinc sensitivity headline rests on dropping the paper's own Zeff correction; worth refereeing with revisions. read the letter →

arxiv 2411.10304 v2 pith:7UU7HRAH submitted 2024-11-15 hep-ph

Yuichi Uesaka , Masato Yamanaka , Yoshitaka Kuno This is my paper · ORCID

classification hep-ph

keywords chargedleptonflavorviolationmuonicatomsmuondecaydiphotonoperatorsRayleighmuon-to-electronconversionraredecayseffectivefieldtheory

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 proposes using the decay of a negatively charged muon bound in an atom, $\mu^- \to e^- \gamma$, as a new way to search for charged lepton flavor violation. Unlike the free decay $\mu^+ \to e^+ \gamma$, a nucleus is present, so an effective operator with two photon fields---the Rayleigh operator---can contribute by having the nucleus absorb one photon. The authors derive a decay-rate formula and show that for the diphoton operator the rate grows roughly as the fifth power of the atomic number $Z$, making heavier targets like zinc the most promising. With current bounds on the corresponding couplings, they estimate that for zinc the largest allowed signal branching ratio after angular cuts is $3.3 \times 10^{-13}$, about three times the estimated accidental background of $1.1 \times 10^{-13}$. This suggests that future muon experiments could directly probe an operator that is otherwise difficult to constrain.

What carries the argument

The central object is the effective interaction Lagrangian with dipole terms proportional to $e\sigma_{\alpha\beta}\mu F^{\alpha\beta}$ and Rayleigh (diphoton) terms proportional to $e\mu F_{\alpha\beta}F^{\alpha\beta}$ and $e\mu F_{\alpha\beta}\tilde{F}^{\alpha\beta}$, all normalized by the electroweak scale $v$. The calculation uses the bound $1S$ muon wave function in a point-like nuclear Coulomb field, with the nucleus supplying the static electric field that absorbs one photon. The resulting double-differential rate, Eq. (13), is the load-bearing identity: its $\zeta^5 = (Z\alpha)^5$ prefactor gives the strong $Z$-dependence, and its dependence on the electron--photon opening angle $\theta_{e\gamma}$ is what lets an angular cut suppress the flat accidental background while retaining most of the diphoton signal. The free-muon $\mu^+ \to e^+ \gamma$ and $\mu^+ \to e^+ \gamma\gamma$ formulas emerge as the $Z \to 0$ limits, which the paper uses to translate existing experimental bounds into the coupling sizes used in the estimates.

What would settle it

A concrete check is to redo the zinc estimate with $Z_{\rm eff}\approx 25$ in Eq. (13) instead of $Z=30$; the signal drops from $3.3 \times 10^{-13}$ to roughly $1.3 \times 10^{-13}$, comparable to the estimated $1.1 \times 10^{-13}$ background, so the claimed factor-of-three advantage disappears. A dedicated $\mu^- \to e^- \gamma$ search on a heavy target with sensitivity near $10^{-13}$ would then settle whether the process is discoverable at the level the paper projects.

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Extended reading notes

Core claim

The core claim is that $\mu^- \to e^- \gamma$ in a muonic atom is a viable direct probe of the diphoton charged-lepton-flavor-violating operators $e\mu F_{\alpha\beta}F^{\alpha\beta}$ and $e\mu F_{\alpha\beta}\tilde{F}^{\alpha\beta}$. In the presence of the nucleus, one photon from the CLFV vertex can be absorbed by the nuclear Coulomb field, so the process proceeds even though the free decay $\mu^+ \to e^+ \gamma$ does not receive the diphoton contribution at lowest order. The paper derives an analytic distribution, its Eq. (13), including both dipole and diphoton operators, with the bound muon treated non-relativistically in a point-like Coulomb field and the electron treated as a plane wave. The same formula reproduces the free-muon $\mu^+ \to e^+ \gamma$ result in the $Z \to 0$ limit, and its $\zeta^5 = (Z\alpha)^5$ prefactor for the diphoton terms underlies the proposal's main phenomenological point: heavier muonic atoms should be more sensitive. The authors estimate that, for a zinc target and with an angular cut of $0.02$ rad, the maximum allowed diphoton signal has an effective branching ratio of $3.3 \times 10^{-13}$, exceeding the accidental coincidence background of $1.1 \times 10^{-13}$ by roughly a factor of three.

Load-bearing premise

The numerical sensitivity claim for heavy targets assumes the bare atomic number $Z$ in the $Z^5$ prefactor and a point-like nuclear charge distribution; the paper itself notes that replacing $Z$ by the effective charge $Z_{\rm eff}$ would give a milder and more realistic $Z$-dependence, which for zinc would bring the estimated signal down to roughly the level of the background.

Editorial extensions

If this is right

A future muon experiment using muonic atoms of zinc or heavier targets could probe the diphoton operators directly, complementing the lack of planned searches for $\mu^+ \to e^+ \gamma\gamma$.
Because the emitted electron and photon have non-monochromatic spectra, a signal can be selected through the energy sum $E_e + E_\gamma = m_\mu - B$ rather than through individual monochromatic peaks, with the binding energy $B$ supplying a distinctive offset.
Since the diphoton rate grows with $Z$ while the dipole rate slowly shrinks, observing an increasing signal with target atomic number would point to the diphoton operator as the source.
Under the current direct bound on $\mu^+ \to e^+ \gamma\gamma$, the largest allowed aluminum signal is about $4.6 \times 10^{-12}$ before angular cuts and $7.1 \times 10^{-14}$ after them, so an experiment at that level begins to test the operator.
Lowering the muon beam rate $R_\mu$ reduces the accidental coincidence background linearly, which could restore sensitivity for targets where the default background outranks the signal.

Reading between the lines

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

Inference: because the paper itself notes that replacing $Z$ with the effective charge $Z_{\rm eff}$ gives a milder, more realistic $Z$-dependence, the reported factor-of-three advantage for zinc may shrink to near parity once finite-nuclear-size effects are included.
Inference: the same bound-state treatment could be applied to $\mu^- \to e^-$ conversion driven by the Rayleigh operator; comparing the two rates could help disentangle dipole and diphoton contributions.
Inference: if the couplings satisfy $C_{L/R} = -\tilde{C}_{L/R}$, the indirect $\mu^+ \to e^+ \gamma$ constraint can vanish while $\mu^+ \to e^+ \gamma\gamma$ and $\mu^- \to e^- \gamma$ remain nonzero, making this channel a potentially clean probe in that parameter region.
Inference: an analogous decay with an axion-like particle, $\mu^- \to e^- a$ followed by $a \to \gamma\gamma$ with one photon absorbed by the nucleus, would share the same final-state signature and can be tested with the same experimental setup; the authors indicate they plan to study it.

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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 thing here is the bound-state calculation of muon-to-electron-plus-photon in a muonic atom via the diphoton CLFV operators, including the spectral formulas and the background estimates for muon decay in orbit and radiative decay. The derivation has real checks: the Z->0 limit reproduces the free-muon branching ratios, and the background formulas reduce correctly to known free-muon results. Those appendices are useful and the light-nucleus framework is coherent. This is the first time the diphoton operators are studied through this channel, and the idea of using the nuclear Coulomb field to absorb one photon is legitimately clever.

The soft spots are where the numerical claims outrun the approximations. The paper's headline sensitivity for zinc uses bare Z=30, although the authors themselves note that the point-nucleus result overestimates rates for large Z, that Zeff gives a milder and more realistic Z-dependence, and that heavy-nucleus calculations are left for future work. They then explicitly drop the Zeff prescription for the table. Since the prefactor scales roughly as Z^5, using Zeff around 25 reduces the zinc signal from 3.3e-13 to about 1.3e-13, which is comparable to the 1.1e-13 accidental background. The stated factor-of-three advantage disappears. That is not a minor detail; it is the main quantitative conclusion.

The second soft spot is the choice of the Crystal Box bound on the diphoton couplings. The indirect mu->e gamma constraint is stronger, and the paper avoids it by invoking a cancellation between C and C-tilde that also suppresses the interference. That is a fine-tuned corner of parameter space, and the maximum allowed signal numbers assume that corner. The paper is honest about this, but the reader should not mistake the quoted branching ratios for generic expectations.

None of this invalidates the formalism or the idea. The light-nucleus results are trustworthy and the Al numbers are explicitly shown for both Z and Zeff. But the case for experimental viability depends on the heavy-nucleus scaling, and that is exactly where the calculation is least reliable.

Who is this for: people working on muon CLFV phenomenology and on effective operators with photons. A serious referee should engage with the derivation and the background work, and should ask the authors to present the zinc numbers with Zeff or a finite-size calculation, and to couch the indirect-bound evasion more carefully. The paper deserves peer review, not desk rejection, but it needs revision before the sensitivity claims can be taken at face value.

Referee Report

2 major / 4 minor

Summary. This paper proposes the charged-lepton-flavor-violating decay of a negative muon bound in a muonic atom, µ−→e−γ, as a probe of effective dipole and diphoton (Rayleigh) CLFV operators. The authors derive an analytic decay-rate formula, Eq. (13), under stated low-Z approximations (plane-wave electron, point-like nucleus, non-relativistic 1S muon), including the dipole-diphoton interference and the F and F-tilde operator structures. They present Z-dependence, energy, angular, and invariant-mass distributions, and estimate signal and background effective branching ratios for aluminum and zinc. Their main quantitative conclusion is that for zinc the maximum allowed diphoton signal after angular cuts, 3.3×10^-13, exceeds the estimated accidental-coincidence background, 1.1×10^-13, by a factor of about three, suggesting that future experiments could probe these operators.

Significance. The physical idea is interesting and the analytic formulas are a useful contribution: Eq. (13) has a nontrivial structure and reproduces the free-muon µ+→e+γ rate in the Z→0 limit (Eq. 17), and the appendices provide decay-in-orbit and radiative-decay spectra with stated limits. The distributions in Sec. III show that the dipole and diphoton mechanisms can in principle be distinguished. However, the quantitative case for experimental sensitivity rests on two assumptions that need to be tested or flagged: the use of bare Z for zinc and heavier targets, and the use of the weaker Crystal Box constraint on the diphoton couplings. The significance would be high if those points are resolved; in its current form the paper is a promising proposal rather than an established sensitivity claim.

major comments (2)

[Sec. IV, Table II; Sec. III A] The zinc row of Table II is computed with the bare atomic number Z=30, even though the paper states that Eq. (13) is derived under low-Z approximations and that a point-like nucleus overestimates the rate for large Z, with Zeff giving a milder and more realistic Z-dependence. Because the prefactor in Eq. (13) scales as ζ^5, replacing Z=30 by Zeff≈25 reduces the quoted diphoton signal from 3.3×10^-13 to roughly 1.3×10^-13, making it comparable to the 1.1×10^-13 background; the claimed factor-of-three advantage is therefore not established. The authors should either perform a finite-nucleus calculation that is valid for zinc and heavier targets, or restrict the quantitative sensitivity claim to the Z range where the approximations are controlled.
[Sec. III A, Eqs. (18) and (19)] The numerical maximum for the diphoton signal uses the direct µ+→e+γγ bound from Crystal Box, C'_L=2.2×10^-2, rather than the indirect µ+→e+γ constraint on |CL+C~L| and |CR+C~R|, which for Λ=100 GeV gives 5.7×10^-4. That choice is legitimate only in a scenario with the cancellation CL=−C~L (and similarly for CR), as the authors note, but it is a fine-tuned relation rather than a generic EFT prediction. Without such a cancellation, the allowed C'_L would be about 40 times smaller and all diphoton signal rates in Table II would be suppressed by about 7×10^-4, making the process unobservable in the quoted setup. The paper should state this condition prominently when presenting the maximum rates and should indicate how precisely the cancellation must hold.

minor comments (4)

[Sec. III A / Fig. 2] The text says that replacing Z by Zeff gives the 'dotted curves' in Fig. 2-(a), while the caption says 'solid and dashed curves' for Z and Zeff; this inconsistency should be fixed.
[Sec. II, Eq. (15) and Appendix B] The notation in Eq. (15), ζ/w tan^-1(w/ζ), and in Eq. (B10), 2k2/(dy w2(x−r)), is easy to misread; please insert parentheses or explanatory text.
[Sec. IV] The estimate does not include contributions from nuclear muon-capture products, as the text acknowledges; this caveat should also appear in the summary and conclusions, since the sentence about 'maintaining a lower background level' does not carry it.
[Fig. 3] The caption phrase 'angular distribution between the emitted electron and photon (b)' contains a redundant '(b)' and should be rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the muonic-atom CLFV rate is computed from a defined EFT Lagrangian and external experimental constraints; the only self-citation is non-load-bearing.

full rationale

The derivation chain is self-contained. The paper defines an EFT (Eqs. 1-3), computes the muonic-atom matrix element from the Coulomb field (Eqs. 4-12), and derives the differential decay distribution (Eq. 13) with the normalization Gamma0 = m_mu^5/(1536 pi^3 v^4), the free-muon decay rate. The couplings are not fitted to the predicted process: DL is bounded by MEG II via Eq. (17), and the diphoton couplings are bounded by Crystal Box via Eq. (18), both external inputs. The predicted muon-to-electron-plus-photon rate is then evaluated at these upper limits; the Z->0 limit of Eq. (13) correctly reduces to Eq. (17), which is a consistency check rather than a definition. The one overlapping-author citation, Ref. [21], gives a conversion constraint that the paper explicitly sets aside ('we use the bound from the Crystal Box experiment as the constraint for the diphoton couplings'), so it is not load-bearing. The Zeff caveat is an approximation-validity issue, not circularity: using bare Z is disclosed in Sec. III, and the Z^5 prefactor follows from Eq. (13), not from the desired answer. Background rates are computed from independent standard formulas (Appendices A and B). No prediction reduces by construction to an input.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The central derivation is parameter-free in the sense that no constants are fitted to make the rate formula work; the rate formula depends on the EFT couplings and on a series of domain assumptions about the nuclear field and lepton wavefunctions. The numerical reach estimates then depend on choosing the couplings at experimental upper bounds and on the bare-Z approximation.

free parameters (7)

C'_L (diphoton coupling) = 2.2 x 10^-2
Set to the Crystal Box upper bound; the stronger indirect mu to e gamma constraint is evaded by assuming a cancellation between C and C~, a special parameter point.
DL (dipole coupling) = 1.0 x 10^-8
Set to the MEG II upper bound; used for the dipole-dominant scenario.
Bare atomic number Z for zinc = 30
The paper uses Z=30 rather than Zeff (about 25) in Table II and the conclusion, explicitly setting aside the Zeff correction.
Muon arrival rate Rmu = 1e7/s
Assumed beam rate for background estimation.
Timing resolution Delta_te-gamma = 78 ps
From MEG II; enters accidental background estimate.
Energy sum resolution Delta_x+y = 0.01
From MEG II photon resolution; defines signal region for energy sum.
Angular resolution Delta_theta_e-gamma = 0.02 rad
From MEG II; used for back-to-back angular cut.

assumptions (7)

domain assumption Electron wave function approximated as a plane wave, ignoring the nuclear Coulomb potential (Eq. 10).
Valid for light atoms, but used for zinc (Z=30) in the main sensitivity estimate; electron distortion could alter rates and angular distributions.
domain assumption Point-like nuclear charge distribution (Eq. 11).
Makes the Coulomb field E(r) = Ze/(4 pi r^2); authors note this overestimates rates for large Z and introduce Zeff as a partial correction, but then use bare Z for the numerical conclusions.
domain assumption Non-relativistic 1S muon wavefunction (Eq. 12).
Non-relativistic approximation with energy b = Z^2 alpha^2 / 2; for Z=30, Z alpha is about 0.22, higher-order relativistic corrections are neglected.
domain assumption Neglect of nuclear recoil energy.
Stated to be less than a few keV for Z around 10; for heavier nuclei this could be larger but still small.
ad hoc to paper Only DL and C'_L are retained, with DR, CR, C~R set to zero (Sec. III).
Simplifies the numerical analysis; does not cover the full parameter space, especially interference effects.
ad hoc to paper Use of the Crystal Box direct bound rather than the stronger indirect mu+ to e+ gamma constraint (Eq. 19).
The indirect bound is evaded only in the special case CL/R = -C~L/R; the paper chooses the weaker direct bound for the sensitivity estimate.
domain assumption Background formulas neglect relativistic and finite-nucleus effects (Appendices A and B).
Matching the signal calculation approximations; may underestimate or distort background spectra for heavy nuclei.

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Pith. "Pith review of $\mu^- \to e^-\gamma$ in a muonic atom as a probe for effective lepton flavor violating operators involving photon fields." pith.science (2026). https://pith.science/paper/7UU7HRAH

@misc{pith2026241110304,
  author       = {Pith},
  title        = {Pith review of: $\mu^- \to e^-\gamma$ in a muonic atom as a probe for effective lepton flavor violating operators involving photon fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7UU7HRAH}},
  note         = {Machine review of arXiv:2411.10304}
}

abstract

We propose the $\mu^-\to e^-\gamma$ process in a muonic atom as a novel means to investigate charged lepton flavor violation (CLFV). We demonstrate its sensitivity in probing effective CLFV operators associated with both single and double photon fields. In comparison to $\mu^+\to e^+\gamma$ using free positive muon decays at rest, the emitted electron and photon in $\mu^-\to e^-\gamma$ exhibit non-monochromatic spectra, presenting non-trivial case. We derive the decay rate formula and demonstrate that the potential rate of the process is significant enough to motivate exploration in future muon CLFV experiments.

Figures

Figures reproduced from arXiv: 2411.10304 by the authors.

**Figure 2.** FIG. 2. (a) The [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) The energy distribution of the emitted electron a [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. The distribution of the invariant mass [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. The energy distribution of electrons for the acciden [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. The energy distribution of the emitted electron in th [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7. The energy distribution of the emitted photon in the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Kuno and Y

Y. Kuno and Y. Okada, Rev. Mod. Phys. 73, 151-202 (2001). doi:10.1103/RevModPhys.73.151

doi:10.1103/revmodphys.73.151 2001

[2] [2]

Calibbi and G

L. Calibbi and G. Signorelli, Riv. Nuovo Cim. 41, no.2, 71-174 (2018) [arXiv:1709.00294 [hep-ph]]

arXiv 2018

[3] [3]

Afanaciev et al

K. Afanaciev et al. (MEG II collaboration), Eur. Phys. J. C 84, no.3, 216 (2024) [arXiv:2310.12614 [hep-ex]]

arXiv 2024

[4] [4]

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P. W. Cattaneo et al. [MEG II and MU3E Collaborations], “MEG II and Mu3e status and plan,” EPJ Web Conf. 212 (2019) 01004. doi:10.1051/epjconf/201921201004

arXiv 2019

[5] [5]

R. D. Bolton et al. , Phys. Rev. D 38 (1988) 2077

work page 1988

[6] [6]

Search for the Decay µ + → e+e+e−

U. Bellgardt et al. [SINDRUM Collaboration], “Search for the Decay µ + → e+e+e−”, Nucl. Phys. B 299 (1988) 1. doi:10.1016/0550-3213(88)90462-2

doi:10.1016/0550-3213(88)90462-2 1988

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Research Proposal for an Experimen t to Search for the Decay µ → eee

A. Blondel et al. [Mu3e Collaboration], “Research Proposal for an Experimen t to Search for the Decay µ → eee”, arXiv:1301.6113 [physics.ins-det] (2013)

arXiv 2013

[8] [8]

A Search for muon to electron c onversion in muonic gold,

W. H. Bertl et al. [SINDRUM II Collaboration], “A Search for muon to electron c onversion in muonic gold,” Eur. Phys. J. C 47 (2006) 337. doi:10.1140/epjc/s2006-02582-x

doi:10.1140/epjc/s2006-02582-x 2006