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REVIEW 2 major objections 4 minor 78 references

Taming flavour violation in the Inverse Seesaw

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read In the inverse seesaw, the absence of muon-to-electron gamma decay together with sizeable muon-to-3e or conversion rates would be a sign of a non-degenerate, mixed heavy sterile spectrum.

desk verdict Useful new parametrisations of the ISS(3,3), but the headline 'no μ→eγ with sizeable μ→3e' rests on an unquantified residual-dipole approximation that needs a numerical check. read the letter →

arxiv 2412.13696 v2 pith:3SW2YZLT submitted 2024-12-18 hep-ph hep-ex

Jonathan Kriewald , Ana M. Teixeira This is my paper · ORCID
classification hep-phhep-ex
keywords inverseseesawleptonflavourviolationnon-unitarityheavyneutralleptonsZ-penguinradiativedecaysmuontoelectronconversionneutrinomassmodel
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 establishes that, within the inverse seesaw with three generations of sterile fermions, the sources of charged lepton flavour violation can be disentangled. It develops new parametrisations of the Yukawa couplings that encode the non-unitarity of the effective PMNS matrix (the $\eta$-matrix) directly, so that low-energy bounds can be imposed by construction. Using these, it shows that radiative decays $\ell_\alpha\to\ell_\beta\gamma$ are, to a very good approximation, controlled only by the off-diagonal $\eta_{\alpha\beta}$, while Z-penguin observables such as $\mu\to 3e$ and $\mu$--$e$ conversion also depend on the mixing $V_2$ among the heavy states and on the heavy mass splittings. A future experimental pattern with no $\mu\to e\gamma$ but sizeable $\mu\to 3e$ or conversion would therefore indicate a non-degenerate, non-trivially mixed heavy sector.

What carries the argument

The key machinery is a set of three equivalent parametrisations of the Yukawa matrix $Y_D$ (polar decomposition, Cholesky/QR decomposition, and singular value decomposition) in which the active-sterile mixing is encoded in the $\eta$-matrix, the deviation from unitarity of the effective PMNS matrix, and the mixing among heavy states is isolated in a unitary factor $V_2$ (Eqs. (25)--(27)). This algebraic separation, combined with the asymptotic loop functions $G_\gamma(x)\to 1/2$ and $F_Z(x)\to 5/2-\frac{5}{2}\log x$ for $x\gg1$, is what splits observables into those that only feel $\eta_{\alpha\beta}$ (radiative decays) and those that additionally feel $V_2$ and the heavy mass splittings (Z-penguin processes). The split is exact in the degenerate limit and approximate for a non-degenerate spectrum.

What would settle it

Compute the exact one-loop photon-penguin amplitude for the benchmark heavy spectrum $M_R=\mathrm{diag}(0.9,1,1.1)M_0$ with a non-trivial $V_2$; if the resulting $\mathrm{BR}(\mu\to e\gamma)$ is not suppressed by several orders of magnitude relative to $\mathrm{BR}(\mu\to3e)$ and the $\mu$--$e$ conversion rate, the clean separation fails. Experimentally, a measurement of $\mu\to e\gamma$ at future sensitivity in a parameter region the model predicts to be silent would falsify the disentanglement claim.

Watch

Extended reading notes

Core claim

The central claim is that the inverse seesaw with three generations of sterile fermions, the ISS(3,3), can generate sizeable rates for Z-penguin dominated processes $\mu\to3e$, neutrinoless $\mu$--$e$ conversion, and some collider signals even when the off-diagonal entries of the $\eta$-matrix are set to zero, provided the heavy pseudo-Dirac pairs are non-degenerate and their mixing $V_2$ is non-trivial. In that configuration the radiative decays $\ell_\alpha\to\ell_\beta\gamma$ remain negligible, because the photon-dipole amplitudes depend essentially only on $\eta_{\alpha\beta}$. This is demonstrated with a benchmark $M_R=\mathrm{diag}(0.9,1,1.1)M_0$ and one or two non-zero angles in $V_2$, where $\mu\to3e$ and $\mu$--$e$ conversion lie within future reach while $\mu\to e\gamma$ is switched off by construction. The paper concludes that a future null result for $\mu\to e\gamma$ combined with a positive signal in the Z-penguin observables would hint at a non-trivially mixed and non-degenerate heavy spectrum.

Load-bearing premise

The argument assumes that the photon-loop piece of radiative decays is controlled by the active-heavy mixing alone, so setting that mixing to zero silences those decays while Z-penguin processes stay sensitive to the heavy states' internal mixing and mass differences.

Editorial extensions

If this is right

  • If the central claim is correct, a future observation of $\mu\to3e$ or $\mu$--$e$ conversion without $\mu\to e\gamma$ would be a smoking gun for a non-degenerate, mixed heavy sterile spectrum in the ISS(3,3).
  • The new $\eta$-based parametrisations allow a controlled exploration of the ISS(3,3) parameter space that automatically respects low-energy universality bounds, avoiding the numerical instability of the standard seesaw parametrisation.
  • A high-luminosity $\mu^+e^-$ collider would provide the strongest future constraints on $\tau$--$e$ and $\tau$--$\mu$ flavour violation, outperforming Z-pole searches, while dedicated muon facilities remain the best probes of the $\mu$--$e$ sector.
  • Simultaneous presence of two heavy-mixing angles in $V_2$ can induce new flavour-violating directions (for example, $\theta_{13}$ together with $\theta_{23}$ generates $\mu\to3e$), so the pattern of future cLFV signals encodes information about the heavy mixing matrix.

Reading between the lines

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

  • A direct extension of the disentangling logic suggests that a signal in $\tau\to3\ell$ without $\tau\to\ell\gamma$ at a future tau-factory or high-luminosity Z-factory would similarly point to heavy-sector mixing, giving a flavour-universal diagnostic.
  • The approximate nature of the 'switched-off' radiative decay invites a dedicated higher-order computation of the mass-splitting corrections to the dipole amplitude; if those corrections are comparable to the leading Z-penguin rates for splittings around 10%, the clean experimental discriminant would be blurred.
  • Model builders aiming to evade $\mu\to e\gamma$ bounds while predicting observable $\mu\to3e$ could use this parametrisation as a target: arrange for a non-degenerate, mixed heavy sector with small off-diagonal $\eta$.
  • The ratio $\mathrm{BR}(\mu\to3e)/\mathrm{BR}(\mu\to e\gamma)$ measured in a single future run could serve as a quantitative probe of the heavy mass splitting in the ISS(3,3), should the model be realised.
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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 / 4 minor

Summary. This paper studies the charged-lepton flavour phenomenology of the Inverse Seesaw with three generations of sterile fermions (ISS(3,3)). Its technical core is a set of new parametrisations of the Dirac Yukawa matrix Y_D based on polar (Eq. 12), QR (Eq. 18) and singular-value (Eq. 21) decompositions, in which the deviation from unitarity of the effective PMNS matrix (the η-matrix of Eq. 5) is imposed directly as input, while the mixing within the heavy sterile sector is confined to a 'hidden' unitary matrix V_2 (Eqs. 25-27). After deriving the algebraic identities connecting η to the Yukawa couplings, the authors use the new parametrisations to scan the ISS(3,3) parameter space and compare the reach of low-energy cLFV searches (μ→eγ, μ→3e, μ-e conversion, τ decays), Z-pole LFUV observables, and the proposed μTRISTAN collider. The central result is a proposed disentanglement: with off-diagonal η_ij set to zero, radiative decays ℓα→ℓβγ are controlled solely by η_αβ via BR(ℓα→ℓβγ) ≈ (3α/2π)|η_αβ|² (Eq. 33), while Z-penguin dominated processes (μ→3e, μ-e conversion in nuclei) can still receive sizeable contributions from V_2 mixing combined with a non-degenerate heavy spectrum (benchmark M_R = diag(0.9, 1, 1.1) M_0, Figure 4). The paper concludes that in the absence of radiative decays, sizeable rates for Z-penguin dominated observables could hint at a non-trivially mixed and non-degenerate heavy spectrum (Abstract and Section 5).

Significance. The disentanglement proposal is the paper's most valuable contribution. If it survives quantitative scrutiny, it provides a falsifiable diagnostic—measurable by MEG II, Mu3e, COMET and Mu2e—that separates active-sterile flavour mixing (η) from mixing within the heavy sterile sector (V_2). The algebraic identities of Section 2 are standard matrix decompositions that can be verified by hand, and the SVD parametrisation is a practical improvement over the modified Casas-Ibarra form: the hidden parameters enter through bounded angles rather than hyperbolic functions, and the global η bounds of Ref. [40] can be imposed directly. The phenomenological analysis treats a broad set of observables with consistent current and projected sensitivities, and the V_2-induced predictions of Section 4.2 are concrete and testable. The main caveats are that no numerical code is provided, the residual V_2-induced photon-dipole contribution is asserted rather than computed (Major Comment 1), and the normalisation of the parametrisations relative to Eq. (6) is ambiguous as printed (Major Comment 2).

major comments (2)
  1. [Section 4.2, Figure 4] The claim that setting off-diagonal η_ij = 0 switches off the photon dipole is exact only in the degenerate-mass limit and is left unquantified for the benchmark M_R = diag(0.9, 1, 1.1) M_0, where the paper states that 'the rate of the radiative decay μ → eγ is negligible by construction'. With V_1 = 1 and a 12-rotation in V_2, the GIM-violating dipole amplitude is (y_e y_μ/2) sinθ12 cosθ12 [Gγ(x2) − Gγ(x1)], with y_i = √(2η_ii) as stated after Eq. (24) and Gγ given by Eq. (28). Using Gγ(x) ≈ 1/2 − (3/2) log(x)/x and the maximal diagonal η values of Eq. (32), I estimate the residual BR(μ→eγ) ≈ (3α/2π)|(y_e y_μ/2) sinθ12 cosθ12 [Gγ(x2) − Gγ(x1)]|² to be of order 10^-14 at M_0 = 1 TeV, rising to order 10^-13 at M_0 ≈ 300 GeV for sinθ12 ≈ 0.3, i.e., at or above the MEG II projection of 6 × 10^-14 in the lower part of the mass range of Figure 4. For the 9-11 TeV masses of Figure 5 the residual is instead negligible (order 10^-17), so the concern is specific to the low-mass end of Figure 4, where the displayed μ→3e and μ-e conversion constraints are nevertheless sizable. The authors should add the μ→eγ contours, or an analytic estimate of the residual, over the full M_0 range of Figure 4, and should restrict the disentanglement claim in the Abstract, in the Introduction ('only observables that depend on Z-penguin transitions can be affected'), and in Section 5 to the region where the residual is below the quoted sensitivity. The logarithmic enhancement of the Z-penguin relative to the asymptotically flat dipole makes the proposed pattern plausible for M_0 ≳ 1 TeV, so I regard this as a missing calculation rather than a disproof of the claim.
  2. [Section 2, Eqs. (6), (24)-(27)] As printed, the parametrisations are not consistent with Eq. (6) under the stated relation m_D = v Y_D/√2. Inserting Eq. (27) into Eq. (6) gives η = (1/4) V_1^* diag(y²) V_1^T, a factor of two below Eq. (24); likewise, Eqs. (25) and (26) yield η = η_input/2 when evaluated through Eq. (6). The identities in Eqs. (13), (14), (19) and (24) are internally consistent; the discrepancy appears only when Eq. (6) is applied to the explicit Y_D displayed in Eqs. (25)-(27). The stated translation y_i = √(2η_ii) is compatible with Eq. (24), but requires either doubling the coefficients in Eqs. (25)-(26) (√2/v → 2/v in Eq. (25); 1/v → √2/v in Eqs. (26) and (27)) or dropping the 1/√2 factor in the m_D definition for these parametrisations. Because y_i enters the perturbativity bounds in Figures 1-5 and the V_2-induced amplitudes in Figures 4-5, and because the plotted η values must correspond to the physical η of Eq. (6) rather than half of it, the convention should be stated unambiguously and the numerical implementation checked against it.
minor comments (4)
  1. [Section 4.1, Eq. (33), Figure 3] Since BR(ℓα→ℓβγ) ≈ (3α/2π)|η_αβ|² is the defining relation of the scan variable, the μ→eγ exclusion contour in the left panel of Figure 3 is a direct translation of the experimental bound onto |η_eμ| and carries no model information beyond Eq. (33); the informative content of the figure lies in the comparison with the Z-penguin and box observables. I suggest stating this explicitly in the text and drawing the μ→eγ curve as a reference rather than as an independent constraint.
  2. [Section 2, Eq. (15)] The inequality in Eq. (15) and the discussion around it are attributed to 'Schwartz'; the correct spelling is 'Schwarz' (Hermann Schwarz). The same typo appears near Eq. (17).
  3. [Appendix A] The algorithm in Appendix A is attributed to 'Cholesky-Banachiwiecz'; the correct spelling is 'Cholesky-Banachiewicz'.
  4. [Figures 4-5] The phases δij appearing in the parametrisation of V_2 (Eq. (22)) are not specified for the scans of Figures 4 and 5. Please state whether they are all set to zero, since the interference effects discussed in Figure 5 (e.g., the mutual suppression of the μ-e rates for large sinθ13) depend on these phases.

Circularity Check

1 steps flagged · score 6.0 of 10

The ‘absence of radiative decays’ in the claimed smoking-gun pattern is imposed by setting η_ij = 0, so the no-μ→eγ leg is an input, not a prediction; the residual V2-induced dipole for non-degenerate masses is asserted but not computed.

  1. self definitional [Section 4.2 (Figure 4 benchmark), with Eq. (33); restated in the Abstract and Section 5]
    "As previously argued, the γ dipoles are to a very good approximation only proportional to one of the off-diagonal ηij. Consequently, contributions from γ-dipoles can be “switched off” by construction – and thus it is possible to have sizeable rates for µ → 3e and neutrinoless µ − e conversion in muonic atoms (induced by sizeable ηii and non-trivial mixing in the heavy sector) while having negligible rates for µ → eγ."

    The paper’s own Eq. (33) gives BR(ℓα → ℓβγ) ≃ (3α/2π)|ηαβ|^2. In the benchmark of Figure 4, the authors set all off-diagonal η_ij = 0. Therefore the “negligible by construction” μ→eγ rate is not a prediction but a direct consequence of setting the parameter that controls the dipole to zero. The headline pattern — no radiative decay but sizeable Z-penguin rates — is thus a description of the chosen input point, not an independent derivation. The remaining independent content, that V2 mixing with non-degenerate masses generates Z-penguin rates while leaving the dipole off, is asserted “to a very good approximation” without computing the residual dipole for MR = diag(0.9,1,1.1)M0, so the disentanglement is partly an input assumption.

full rationale

The paper is largely a reparametrisation study: the polar/QR/SVD decompositions of YD are mathematical identities, and encoding low-energy deviations from unitarity in the η matrix is a legitimate tool. The main phenomenological claim, however, reduces in part to construction. Eq. (33) makes radiative decays proportional to |η_αβ|^2, and the authors then set η_ij = 0 in the non-degenerate benchmark, declaring μ→eγ “negligible by construction.” Consequently, the abstract’s statement that “in the absence of radiative decays … sizeable rates for Z-penguin dominated observables could hint at a non-trivially mixed and non-degenerate heavy spectrum” is not a prediction of the absence; it is a conditional built on an input choice. The non-trivial part — that V2-driven Z-penguins survive — is physically plausible and is computed for μ→3e and μ-e conversion, but the dipole leg is not independently verified for the non-degenerate case; the paper’s own caution that the mixing dependence is “in general more complicated” for non-degenerate spectra underscores this gap. Self-citations to [23] and [55] appear for Z-pole and μTRISTAN projections, but they are not load-bearing for the central disentanglement argument and are normal scholarly references. Thus the paper has meaningful independent content, but one of the two legs of the advertised smoking-gun pattern is imposed by definition, giving a partial-circularity score of 6 rather than a higher one.

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

The central claim rests on the ISS(3,3) model structure, the leading-order perturbative expression for eta, one-loop loop-function results imported from the literature, and external fits for neutrino oscillations and eta bounds. The paper introduces no new fundamental free parameters beyond scan inputs; the heavy mixing angles V2 and mass scale M0 are the physical unknowns the analysis is designed to probe, and the eta entries are bounded inputs from [40] rather than fitted outputs. The 'invented entities' list is empty: no new particles, forces, or conserved quantities are postulated.

free parameters (5)
  • eta_ee, eta_mu_mu, eta_tau_tau (diagonal non-unitarity entries) = 1.4e-3, 1.4e-4, 8.9e-4 (upper bounds from [40])
    Diagonal eta set to the maximal values allowed by the global fit [40]; they control the overall size of flavour-violating rates, and the new parametrisations (Eqs. 25-27) are built on them.
  • a, b, c (off-diagonal eta magnitudes, with phases delta_12, delta_13, delta_23) = Varied 0 to 1 in scans; set to 0 in Figures 4-5
    The hierarchy of active-sterile flavour mixing; varied up to the Schwarz bound Eq. (15). In the central disentanglement plots they are set to zero.
  • sin theta_12, sin theta_13, sin theta_23 (V2 mixing angles) = Varied between 10^-4 and about 0.5; benchmark MR = diag(0.9,1,1.1) M0
    The heavy-sector mixing angles, the key hidden parameters whose impact on Z-penguin observables is the paper's main result.
  • M0 (heavy mass scale) = 10^2 to 10^4 GeV in scans; 9, 10, 11 TeV in Figure 5
    Overall heavy sterile mass scale, treated as a free scan parameter; controls perturbativity and the logarithmic Z-penguin mass dependence.
  • m_lightest (lightest active neutrino mass) = 10^-5 eV
    Chosen input for the light spectrum (normal ordering, NuFit central values [80]); affects mu_X via Eq. (8).
assumptions (5)
  • domain assumption ISS(3,3) Lagrangian and mass matrix structure (Eqs. 1-2): 3+3 sterile fermions with small lepton-number-violating mu_X, mu_R, with block form m_D, M_R, mu_X.
    The entire analysis lives in this model; the paper cites [1-3] for the mechanism and does not derive it.
  • domain assumption Perturbative block-diagonalisation of the seesaw mass matrix, yielding Eq. (3) for m_nu and Eq. (6) for eta at leading order in mu_X, mu_R much smaller than m_D much smaller than M_R.
    All parametrisations of Section 2 rest on the leading-order relation eta approximately 1/2 m_D* (M_R^-1)+ (M_R^-1) m_D^T; higher-order corrections are neglected.
  • standard math Validity of the one-loop loop-function results for dipole and Z-penguin amplitudes (Eqs. 28-31 from [41]) and their asymptotic behaviour.
    The disentanglement claim relies on G_gamma approaching 1/2 and F_Z approaching (5/2) log(x) asymptotics; these are imported from prior literature.
  • domain assumption Neutrino oscillation data as input: normal mass ordering, NuFit central values [80], lightest mass 10^-5 eV.
    mu_X is fixed from oscillation data via Eq. (8); the analysis does not scan over the neutrino parameters.
  • domain assumption The eta-matrix bounds of [40] at 95% C.L. (Eq. 32) correctly encode low-energy universality and cLFV constraints.
    Diagonal eta are pinned to these fitted bounds; all contour plots inherit them.

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Cite this review

Pith. "Pith review of Taming flavour violation in the Inverse Seesaw." pith.science (2026). https://pith.science/paper/3SW2YZLT

@misc{pith2026241213696,
  author       = {Pith},
  title        = {Pith review of: Taming flavour violation in the Inverse Seesaw},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SW2YZLT}},
  note         = {Machine review of arXiv:2412.13696}
}
abstract

The Inverse Seesaw mechanism remains one of the most attractive explanations for the lightness of neutrino masses, allowing for natural low-scale realisations. We consider the prospects of a simple extension via 3 generations of sterile fermions - the so called ISS(3,3) - in what concerns numerous lepton flavour observables. In order to facilitate a connection between the Lagrangian parameters and low-energy data, we systematically develop new parametrisations of the Yukawa couplings. Relying on these new parametrisations to explore the parameter space, we discuss the complementary role of charged lepton flavour violation searches in dedicated facilities, as well as in lepton colliders (FCC-ee and $\mu$TRISTAN). Our results reveal the strong synergy of the different indirect searches in probing the distinct flavour sectors of the model. In particular, we show that in the absence of radiative decays $\ell_\alpha\to\ell_\beta\gamma$, sizeable rates for $Z$-penguin dominated observables could hint at a non-trivially mixed and non-degenerate heavy spectrum.

Figures

Figures reproduced from arXiv: 2412.13696 by the authors.

Figure 1
Figure 1. Casas-Ibarra parametrisation: constraints on the ISS(3,3) parameter space from current [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Constraints from current experimental bounds and future sensitivities from [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Constraints from current experimental bounds (filled contours) and future sensitivities [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Constraints from current experimental bounds (filled contours) and future sensitivities [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Constraints from current experimental bounds (filled contours) and future sensitivities [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

Discussion (0). Continue with ORCID to comment.

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