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REVIEW 3 major objections 4 minor 97 references

Lepton flavor violation in the Majorana and Dirac scotogenic models

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Radiative neutrino models differ in their τ→3μ decay rates, offering a way to distinguish them in future experiments.

desk verdict A solid, useful extension of LFV calculations in the scotogenic models; the main caveat is that the numerical maxima are computed on a restricted slice of parameter space, not over the full Casas-Ibarra freedom. read the letter →

arxiv 2502.04733 v2 pith:73V3NV6U submitted 2025-02-07 hep-ph

Raghavendra Srikanth Hundi This is my paper · ORCID
classification hep-ph
keywords leptonflavorviolationscotogenicmodelMajorananeutrinoDiracτ→3μradiativemassmuondecayboxdiagram
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 analyzes lepton flavor violating processes in two minimal radiative neutrino mass models: the Majorana scotogenic model (MSM), where neutrinos are Majorana particles, and the Dirac scotogenic model (DSM), where neutrinos are Dirac. After imposing current experimental constraints, the author finds that the three-body decay τ→3μ can reach a branching ratio of about $10^{-10}$ in the MSM and $10^{-11}$ in the DSM. This difference arises because the MSM has an additional box diagram contribution to three-body decays, making such decays a promising way to distinguish the two models in future searches.

What carries the argument

The key objects are two related one-loop neutrino mass models: the MSM with a Z2 symmetry and the DSM with an exact U(1)_{B-L} and softly broken Z2(A). The LFV amplitudes are computed from penguin diagrams (γ, Z, H) and box diagrams. The crucial difference is the right-hand box diagram in Fig. 2, which is possible only in the MSM because it relies on the Majorana nature of the internal fermions; this diagram adds to the branching ratios of all 3-body decays in the MSM, while the DSM has only the left-hand box diagram.

What would settle it

A future experiment that probes τ→3μ to a branching ratio of $10^{-11}$ and observes no events would cast doubt on the MSM's predicted $10^{-10}$ rate, while leaving the DSM's $10^{-11}$ prediction on the edge; conversely, an observation around $10^{-10}$ with no accompanying 2-body signals would support the MSM's extra box diagram.

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Extended reading notes

Core claim

The central claim is that the 3-body LFV decay τ→3μ can reach branching ratios as high as $10^{-10}$ in the Majorana scotogenic model and $10^{-11}$ in the Dirac scotogenic model, after satisfying bounds on μ→eγ, μ→3e, μ→e conversion, and perturbativity of Yukawa couplings. The MSM receives an extra box diagram contribution not present in the DSM, giving it a systematically larger τ→3μ rate. In contrast, 2-body decays like τ→eγ and τ→μγ have essentially the same maximal rates in both models because they come only from penguin diagrams with the same topology.

Load-bearing premise

The scan sets the unphysical matrices R and S to the identity and sets Majorana phases to zero, so the quoted maximal branching ratios hold only for that slice of parameter space, not for the full model.

Editorial extensions

If this is right

  • The τ→3μ decay channel can discriminate between Majorana and Dirac scotogenic models, provided future experiments reach sensitivities of O(10^-10) to O(10^-11).
  • The decay μ→3e can impose constraints on the parameter space of both models beyond those from μ→eγ, especially in the inverted neutrino mass ordering and for large Yukawa couplings.
  • The μ→e conversion rate in nuclei is unlikely to give additional constraints once the bound on μ→eγ is satisfied.
  • Normal neutrino mass ordering yields larger maximal branching ratios for most LFV tau decays than inverted ordering in both models.

Reading between the lines

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

  • The paper fixes the Casas-Ibarra matrices R and S to the identity, so the quoted maximal branching ratios may not be true global maxima; exploring non-trivial R and S could either enhance or suppress the rates.
  • The predicted τ→3μ rates sit at the boundary of future experimental sensitivity, so a null result in upcoming searches would not decisively rule out either model, but an observation would strongly favor the Majorana version, assuming this parameter slice is representative.
  • The same box-diagram distinction could be tested in other observables, such as the lepton flavor violating decay of a Z boson or Higgs boson, where similar Majorana vs Dirac differences might appear.
  • The DSM's Dirac nature and exact B-L symmetry mean neutrinoless double beta decay is absent, whereas the MSM predicts it; combining LFV rates with 0νββ searches could further separate the two models.
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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

3 major / 4 minor

Summary. The paper performs a comparative analysis of charged-lepton flavor violation in the Majorana scotogenic model (MSM) and the Dirac scotogenic model (DSM). It derives analytic expressions for the two-body radiative decays, the three-body decays, and muon-to-electron conversion in both models, emphasizing that the MSM has an additional box-diagram contribution to three-body final states. After scanning the parameter space and imposing the current bounds on μ→eγ, μ→3e, μ→e conversion, and perturbativity of the Yukawa couplings, the author reports that τ→3μ can reach branching ratios of about 10^-10 in the MSM and 10^-11 in the DSM, with the 3-body tau rates in the DSM suppressed by roughly an order of magnitude relative to the MSM.

Significance. If the reported maxima are robust, the paper gives a concrete way to distinguish the Majorana and Dirac variants of the minimal scotogenic model in upcoming τ→3μ searches, and it provides the first systematic LFV analysis of the DSM including tau decays. The structural observation that the MSM receives an extra box diagram while the two-body penguin rates coincide is well argued and is the paper's most valuable contribution. The paper also usefully emphasizes the role of Yukawa perturbativity in determining which of μ→eγ, μ→3e, and μ→e conversion actually constrain the parameter space. However, the numerical 'maximal' claims are obtained on a restricted Casas-Ibarra slice, and the paper itself flags unresolved discrepancies between its Eqs. (24) and (27) and the corresponding expressions in the literature; these issues must be addressed before the quantitative headline results can be taken as robust.

major comments (3)
  1. [Sec. 2.1, Eq. (8); Sec. 2.2, Eq. (15)]
  2. [Sec. 4.1, Eq. (24) and Eq. (27)]
  3. [Sec. 5.1, Figs. 6 and 7; Sec. 5.2, Figs. 9 and 10]
minor comments (4)
  1. [Sec. 6, Conclusions]
  2. [Sec. 5.2, first paragraph]
  3. [Fig. 3 and Fig. 5]
  4. [Sec. 5, neutrino input]

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the LFV predictions are computed from external neutrino-oscillation data and scanned model parameters, not fitted to the LFV observables.

full rationale

The paper's derivation chain is self-contained with respect to the LFV predictions. Neutrino masses, mixing angles, and delta_CP are taken from external global fits [75]; the Yukawa couplings are fixed through the Casas-Ibarra parametrization (Eqs. (6) and (14)) with the stated simplifying choices R=I and R=S=I. These choices restrict the scanned parameter slice and therefore affect whether the quoted values are true model maxima, but they do not make the predicted branching ratios equal to an input: no LFV observable is fitted, and the tau->3mu maxima are obtained by scanning m_eta, M_k, lambda_5 or A/v subject to mu->e gamma, mu->3e, mu->e conversion, and perturbativity constraints. The MSM-versus-DSM comparison follows from an extra box diagram (Fig. 2), a structural difference, not from an input. The self-citations [50,51,71] are contextual and not load-bearing. The paper explicitly flags unresolved discrepancies with [83] in Eq. (24) and in the |A2|^2 term of Eq. (27); that is a correctness risk, not a circular construction. No equation is equivalent to another by construction, and no fitted parameter is renamed as a prediction.

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

The models introduce no new particles beyond the known scotogenic field content. The main discretionary inputs are the small λ5 and A/v parameters, which are tuned to maximize LFV rates, and the simplifying choice R=S=I.

free parameters (5)
  • λ5 (MSM quartic) = ~10^-11 to 10^-10
    Scanned near minimal values to suppress ηR-ηI splitting, keeping Yukawa couplings O(1) and maximizing LFV rates.
  • A/v (DSM soft term) = ~10^-10 to 10^-9
    Scanned near minimal values to enhance f^D couplings and maximize branching ratios.
  • lightest neutrino mass = varied in [0, √(Δm²_s)]
    Undetermined by oscillation data; varied over the allowed range for both orderings.
  • M1, mη±, mχ, δM = scan ranges, e.g. M1=5-50 TeV, mη±=0.1-1 TeV, δM=1 TeV
    Mass parameters scanned to produce maximal LFV while satisfying μ→eγ, μ→3e, and perturbativity constraints.
  • R, S, Majorana phases = R=S=I; phases=0
    Model parameters fixed by hand; they are free in the Casas-Ibarra parametrization and affect LFV rates.
assumptions (5)
  • domain assumption Radiative neutrino mass formulas (Eqs. 5 and 13)
    Taken from prior literature (Ma 2006; Farzan-Ma 2012) and assumed valid for all scanned parameters.
  • ad hoc to paper Casas-Ibarra parametrization with R=S=I
    The simplification to unit R and S and zero Majorana phases is not without loss of generality for LFV rates.
  • domain assumption Perturbativity bound |f| ≤ √(4π)
    Standard requirement that Yukawa couplings remain in the perturbative regime, used to define PER.
  • domain assumption Neglect of Z and Higgs penguin contributions
    Argued to be subdominant due to heavy propagator masses and small lepton Yukawa couplings.
  • domain assumption Mass lower bounds mR, mI, mζ ≥ 5 GeV
    Imposed to avoid collider exclusions, but the specific 5 GeV choice is not derived.

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Cite this review

Pith. "Pith review of Lepton flavor violation in the Majorana and Dirac scotogenic models." pith.science (2026). https://pith.science/paper/73V3NV6U

@misc{pith2026250204733,
  author       = {Pith},
  title        = {Pith review of: Lepton flavor violation in the Majorana and Dirac scotogenic models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/73V3NV6U}},
  note         = {Machine review of arXiv:2502.04733}
}
abstract

In this work we have considered two minimal versions of scotogenic models, where neutrinos acquire masses through a radiative mechanism. We call these two models as Majorana and Dirac scotogenic models. In the former model, neutrinos have Majorana nature, and in the later one, neutrinos are Dirac particles. These two models are related to each other in terms of additional fields and symmetries of the model. Hence, to compare these two models in future experiments, we have analyzed lepton flavor violating (LFV) processes in both of them, in the charged lepton sector. We have found that the 3-body LFV decays in both these models can get different contributions. Among all the LFV decays and after satisfying relevant constraints, we have found that $\tau\to3\mu$ can have a branching ratio as high as $10^{-10}(10^{-11})$ in the Majorana(Dirac) scotogenic model.

Figures

Figures reproduced from arXiv: 2502.04733 by the authors.

Figure 1
Figure 1. Feynman diagrams of the sub-processes which lead to the decay [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Box diagrams which lead to the decay ℓα → ℓβℓρℓδ. The left-hand side diagram is possible in both the MSM and DSM, whereas, the right-hand side diagram is possible only in the MSM. Interchanging ℓβ with ℓρ in these diagrams give additional diagrams for the above decay. hand side diagram of this figure. Hence, this diagram is possible in both the MSM and DSM. Also notice that, the amplitude of this diagram is the same… view at source ↗
Figure 3
Figure 3. Br(µ → 3e) and CR(µ → e, Au) versus PERM, after satisfying the experimental bound on Br(µ → eγ). The plots are for the case of NO and mη ± M < M1. The color coding in the left-hand side plot is as follows: green points violate the perturbativity bound on the Yukawa couplings, red points violate the experimental bound on Br(µ → 3e), blue points are allowed by the above mentioned bounds. See text, for more details. M1… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: , after satisfying the experimental bound on Br(µ → eγ). In this figure, the values 0.5 1 2 10-13 10-12 10-11 10-10 10-9 10-8 10-7 PERM Br(μ → 3e) 0.5 1 2 10-17 10-16 10-15 10-14 10-13 PERM CR(μ → e, Au) [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]
Figure 5
Figure 5. Figure 5: Br(µ → 3e) and CR(µ → e, Au) versus PERM, after satisfying the experimental bound on Br(µ → eγ). The plots are for the case of IO and mη ± M > M1. The color coding in the left-hand side plot is same as that in [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]
Figure 6
Figure 6. Figure 6: Branching ratios of various LFV tau decays in the MSM and for the case [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]
Figure 7
Figure 7. Figure 7: Branching ratios of various LFV tau decays in the MSM and for the case of [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]
Figure 8
Figure 8. Figure 8: Br(µ → 3e) versus PERD, after satisfying the experimental bound on Br(µ → eγ). The plots are for the case of IO. Top-left and -right plots are for the cases mη ± D < M′ 1 and mη ± D > M′ 1 , respectively. The bottom plot is for the case mη ± D ∼ M′ 1 . The color coding…
Figure 9
Figure 9. Figure 9: Branching ratios of various LFV tau decays in the DSM and for the case of NO, [PITH_FULL_IMAGE:figures/full_fig_p033_9.png]
Figure 10
Figure 10. Figure 10: Branching ratios of various LFV tau decays in the DSM and for the case of IO, [PITH_FULL_IMAGE:figures/full_fig_p035_10.png]

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