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REVIEW 3 major objections 5 minor 31 references

Neutrino models with a zero mass eigenvalue

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A zero neutrino mass eigenvalue survives adding a third right-handed neutrino, and the extension makes thermal leptogenesis viable at the TeV scale in the scotogenic model.

desk verdict A clever zero-mass construction and a careful leptogenesis analysis, undermined by a PMNS ansatz the solar angle already rules out. read the letter →

arxiv 2412.05774 v2 pith:WLGFV3WJ submitted 2024-12-08 hep-ph

Daijiro Suematsu This is my paper · ORCID
classification hep-ph
keywords neutrinomasszeroeigenvalueseesawmechanismscotogenicmodeltribimaximalmixingleptogenesisnormalorderingneutrinolessdoublebetadecay
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

The paper argues that cosmological hints favoring a zero lightest neutrino mass can be realized in seesaw-type models with three right-handed neutrinos, not just the minimal two. It starts from the known fact that two right-handed neutrinos produce one zero mass eigenvalue, then adds a third neutrino while keeping that eigenvalue zero through special Yukawa flavor patterns. The extension is not cosmetic: it is needed for thermal leptogenesis, because with only two right-handed neutrinos the CP asymmetry vanishes under the assumed flavor structure. With normal mass ordering, the model fixes the absolute masses at $m_2 = 0.0087$ eV and $m_3 = 0.0503$ eV, giving $\sum_j m_j = 0.059$ eV, and predicts effective masses of $m_{\beta\beta} = 0.0035$ eV for neutrinoless double $\beta$ decay and $m_\beta = 0.0086$ eV for $\beta$ decay; the inverted-ordering version predicts $m_{\beta\beta} = 0.049$ eV and may already conflict with the current bound. In the scotogenic version, the same construction allows successful leptogenesis with the lightest right-handed neutrino as light as about $10^4$ GeV, far below the usual $3\times10^9$ GeV seesaw bound.

What carries the argument

The central object is the seesaw-type neutrino mass matrix $(M_\nu)_{\alpha\beta} = \sum_k h^\nu_{\alpha k} h^\nu_{\beta k} e^{i\gamma}\Lambda_k$, where $\Lambda_k = \langle\phi\rangle^2/M_{N_k}$ in the type-I seesaw model and is the one-loop expression in eq. (5) in the scotogenic model (a radiative neutrino mass model with an inert doublet and a dark matter candidate). A tribimaximal mixing matrix $U_\nu$ diagonalizes $M_\nu$, and the PMNS matrix is constructed as $V = V_{\rm CKM}^T U_{\rm TBM}$, motivated by the SU(5) relation $M_e = M_d^T$ between charged-lepton and down-quark mass matrices. The special Yukawa flavor patterns (10) and (17) keep one eigenvalue zero after the third right-handed neutrino is added, while the complex relative phase in the extended mass term provides CP violation. The decay-asymmetry formula (26), with its flavor-structure coefficients, carries the leptogenesis calculation.

What would settle it

A decisive observation would be a precise measurement of the lightest neutrino mass: with normal ordering the model requires $m_1 = 0$ and $\sum_j m_j = 0.059$ eV, so any reliable determination with $m_1 > 0$ (or a mass sum clearly above $0.07$ eV while normal ordering is favored) would falsify this branch. For the inverted-ordering version, the model predicts $m_{\beta\beta} = 0.049$ eV, so next-generation neutrinoless double $\beta$ decay searches that push the limit below roughly $0.01$ eV with no signal would exclude it.

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Extended reading notes

Core claim

The central claim is that a model built on the PMNS ansatz $V = V_{\rm CKM}^T U_{\rm TBM}$ -- tribimaximal mixing in the neutrino sector combined with charged-lepton mixing inherited from the CKM matrix -- can support a zero neutrino mass eigenvalue even after three right-handed neutrinos are introduced. The Yukawa couplings must obey the flavor relations in eqs. (10) and (17), and the third right-handed neutrino receives its mass from $M_j + \Delta M$ with $\Delta M = M_0 e^{i\xi}$, whose phase relative to the singlet VEV generates the CP asymmetry in $N_1$ decay. This extension is argued to be necessary for leptogenesis to work in this setup. Using neutrino oscillation data, the normal-ordering absolute masses come out as $m_2 = 0.0087$ eV and $m_3 = 0.0503$ eV; the inverted-ordering version predicts $m_{\beta\beta} = 0.049$ eV, close enough to the current neutrinoless double $\beta$ decay bound that the paper treats the normal ordering as the viable branch.

Load-bearing premise

The load-bearing premise is that the charged-lepton diagonalization matrix is essentially the CKM matrix, so the PMNS matrix equals $V = V_{\rm CKM}^T U_{\rm TBM}$; the paper itself notes that only more than half of the elements of this $V$ lie inside the 3$\sigma$ range of the global fit, so if the real charged-lepton mixing differs, the predicted mixing angles, the zero eigenvalue, and the fixed mass values lose their anchor.

Editorial extensions

If this is right

  • In the normal ordering, the zero lightest mass fixes the mass sum at $\sum_j m_j = 0.059$ eV, giving future cosmological probes of the neutrino mass sum a narrow target to confirm or exclude.
  • The inverted-ordering version predicts $m_{\beta\beta} = 0.049$ eV, so continued improvement of neutrinoless double beta decay bounds without a signal would eliminate that branch.
  • In the scotogenic model, $N_1$ can be as light as about $10^4$ GeV and still generate the observed baryon asymmetry, placing the right-handed neutrinos closer to accelerator reach than in high-scale seesaw.
  • The singlet-scalar scattering $N_k N_k \to N_1 N_1$ can thermally produce the lightest right-handed neutrino even when its Yukawa coupling $h_1$ is very small, and the small $h_1$ delays $N_1$ decay until washout has frozen out.
  • In the normal-ordering branch, the predicted effective masses $m_{\beta\beta} = 0.0035$ eV and $m_\beta = 0.0086$ eV are below current experimental sensitivity, so this version of the model would show up first in mass-sum cosmology rather than in decay searches.

Reading between the lines

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

  • If the same zero-eigenvalue construction is carried over to other radiative mass models, the low-scale leptogenesis mechanism should survive as long as $\Lambda_k$ can be small at TeV-scale $N_k$ masses; the scotogenic model is the paper's concrete example rather than the only possible host.
  • The ansatz $V = V_{\rm CKM}^T U_{\rm TBM}$ directly ties quark mixing to lepton mixing, so a precise determination of the Dirac CP phase that is incompatible with the predicted Jarlskog invariant $J \simeq 4\times10^{-4}$ (or the appendix's $J = -0.009$) would cast doubt on the whole construction.
  • The small allowed range of $h_1$ implies a long-lived lightest right-handed neutrino, so searches for displaced or late decays of TeV-scale neutral fermions could probe this scenario even if direct pair production is beyond reach.
  • The late $N_1$ decay also makes $N_1$ a natural common source for the baryon asymmetry and for the dark matter abundance, giving the scotogenic branch a phenomenological link that the type-I seesaw branch lacks.
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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 / 5 minor

Summary. The paper studies seesaw and scotogenic neutrino-mass models with a vanishing light neutrino mass eigenvalue, extending the usual two-right-handed-neutrino construction to three right-handed neutrinos while keeping the zero eigenvalue. The neutrino-sector mixing is taken tribimaximal, and the charged-lepton rotation is assumed to be the CKM matrix, giving the PMNS ansatz in Eq. (7). From the zero-mass assumption and the measured squared-mass differences the authors derive the absolute masses m2 = 0.0087 eV and m3 = 0.0503 eV for normal ordering, compute the effective masses for neutrinoless double beta decay and beta decay, and analyze thermal leptogenesis. The main phenomenological result is that successful leptogenesis can occur in the scotogenic model for MN1 as low as about 10^4 GeV, in contrast to the type-I seesaw lower bound around 3 x 10^9 GeV.

Significance. The algebraic construction is transparent, and the leptogenesis analysis has a genuinely useful feature: the CP asymmetry in Eq. (26) is invariant under unitary redefinitions of the neutrino Yukawa couplings, so that part of the paper is not entangled with the PMNS fit. If the mixing ansatz Eq. (7) were compatible with data, the paper would offer a concrete zero-eigenvalue framework with definite predictions for m_beta_beta, m_beta, and the sum of neutrino masses, as well as an interesting new production mechanism for the lightest right-handed neutrino through scattering mediated by the singlet scalar S. However, the assumed PMNS matrix is not a mild approximation but is excluded by the measured solar angle, and the exclusion of the inverted ordering rests on an unsupported reading of the neutrinoless double beta decay bound. These are load-bearing problems for the paper's phenomenological claims. The paper is also clearly written and the derivations are mostly easy to follow, but the central data-compatibility claim fails.

major comments (3)
  1. [§2, Eq. (7)] The PMNS ansatz V = V_CKM^T U_TBM is not merely in mild tension with neutrino oscillation data; it is excluded by the solar angle. Using the absolute values in Eq. (7), sin^2(theta_12) = |V_e2|^2 / (1 - |V_e3|^2) ≈ 0.43^2 / 0.978 ≈ 0.19, far below the 3-sigma lower bound of about 0.27 in the global fit cited in the paper. Since theta_12 is independent of the Dirac CP phase, no choice of phases can repair this mismatch, and the appendix's vector-like lepton extension, Eq. (37), still has |V_e2| = 0.43. The statement that 'more than half of the V components is included in the 3 sigma range' is not an adequate acceptance criterion for a unitary matrix, because individual elements are correlated; the mismatch in |V_e2| alone corresponds to a many-sigma failure. All subsequent phenomenological outputs, including the effective masses in Eqs. (24)-(25), depend on the assumed charged-lepton sector, so the model as presented has not been shown to reproduce the measured neutrino mixing matrix.
  2. [§2, after Eq. (24), and §4] The inverted ordering is dismissed because 'the present bound for the neutrinoless double beta decay may have excluded the model already.' Equation (24) gives m_beta_beta = 0.049 eV for IO, whereas the KamLAND-Zen bound quoted in the paper is in the range 36-156 meV, or 28-122 meV depending on the nuclear matrix elements. The IO prediction is therefore not excluded by this bound. The authors may wish to argue that IO is disfavored by cosmological data, but the paper itself stresses that the cosmological bound is model-dependent and that it is premature to base strong conclusions on it. The unsupported exclusion of IO should be removed or replaced by a quantitative discussion of the actual constraint.
  3. [§3.1, Eq. (26) and text after Eq. (29)] The numerical leptogenesis analysis uses |epsilon| ~ 10^-7 and the estimate Y_L = epsilon Y_N1^eq as if these conditions guarantee the observed baryon asymmetry, but the final asymmetry depends on the full Boltzmann dynamics, including the delay of the N1 decay controlled by h1. The rate estimates and Fig. 4 support this for the chosen benchmark, but the region of parameter space in which the no-washout estimate is quantitatively accurate is not mapped. This should be presented as an order-of-magnitude estimate rather than as a derivation, or supplemented by a scan over h1 and the right-handed neutrino mass ratios.
minor comments (5)
  1. [Throughout] There are numerous typos and malformed references, including 'on the oher hand', 'scotgenic', 'chrged', 'exteriment', 'Collabolation', 'α=e,µ.τ' in Eq. (1), and 'arXiv:2407.18047 [astro-phCO]' in Ref. [7]. These should be corrected in a revised version.
  2. [§3.3, Eq. (34)] The washout cross section in Eq. (34) contains terms involving M_eta, but it is then applied to both the scotogenic model and the type-I seesaw model in Fig. 3. For the type-I case the limit M_eta -> infinity, or a separate formula, should be given explicitly.
  3. [§3.1, Eqs. (29)-(31)] The parameter n is used in the exponents before its definition through MN3 = 10^n MN1 is recalled in Fig. 1; defining n once near Eq. (29) would make the bounds on h1 much easier to parse.
  4. [Appendix, Eq. (35)] The notation in Eq. (35), in particular the inequality F F^dagger > M_E^2, is dimensionally ambiguous because F is a coupling while M_E is a mass. Please specify the mass dimension of F and state the condition in dimensionless form.
  5. [§3.1, footnote f and Introduction] The statement in the introduction that the three-right-handed-neutrino extension 'is found to be supported by imposing that the baryon number asymmetry is generated through thermal leptogenesis' is stronger than what is shown: the two-right-handed-neutrino model gives epsilon = 0 only for the specific TBM-aligned flavor structure used here, as footnote f itself acknowledges. The claim should be made conditional on that flavor structure.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the zero-mass eigenvalue and TBM ansatz are explicit inputs, and the quoted mass values are read off from measured squared-mass differences rather than derived from the model.

full rationale

The paper is transparent about its assumptions. Eq. (8) imposes a zero eigenvalue and tribimaximal diagonalization by hand, and Eq. (7) adopts V=V_CKM^T U_TBM as a phenomenological ansatz; neither is presented as a prediction. The absolute masses in Eq. (18), m2=0.0087 eV and m3=0.0503 eV, are obtained by converting the measured squared-mass differences [1] under m1=0, and the paper says so (by using the experimental values for the NO given in [1]). The effective masses in Eqs. (24)-(25) are then standard algebraic consequences of those inputs plus the assumed PMNS matrix; they are not fitted parameters renamed as predictions. The leptogenesis section is a consistency check: h2 and h3 are fixed by the mass splittings and the seesaw/scotogenic formulas, then Eq. (26) is evaluated and compared with the requirement eps about 10^-7; the comparison could have failed and does fail in parts of parameter space (Fig. 1). The extension to a third right-handed neutrino is motivated by the vanishing of the two-RH asymmetry under the chosen alignment, but this is an in-model argument, not a claim that general leptogenesis requires three right-handed neutrinos (footnote f concedes the opposite). Several structural inputs are cited to the author's previous work ([15], [17], [20], [21], [22]), but the key algebraic relations are displayed and checked in the text, and no uniqueness theorem or unverified self-citation is used to force the result. The weak agreement of Eq. (7) with global-fit ranges (More than half of the V components is included in the 3 sigma range) is a phenomenological viability problem, not circularity, since the matrix is not fitted to the observables later labeled predictions. The derivation chain is therefore self-contained relative to its stated assumptions, with only the usual amount of author self-citation.

Assumptions & free parameters 7 free parameters · 6 assumptions · 3 invented entities

The model rests on a number of hand-picked parameters and structural assumptions. The zero eigenvalue is imposed by the Yukawa texture; the PMNS matrix is an ansatz; the right-handed neutrino mass origin via the singlet scalar S is assumed; and the leptogenesis benchmarks choose masses and couplings to make the mechanism work. No independent experimental evidence is provided for the new particles.

free parameters (7)
  • h1 (N1 Yukawa coupling) = not fixed by oscillation data; constrained by eqs. (30)-(31), e.g. 4.2e-9 to 4.2e-7 for MN1 = 1e4 GeV
    The zero mass eigenvalue leaves h1 unconstrained by neutrino data; it controls N1 decay temperature and thermal production.
  • MN1, MN2, MN3 (right-handed neutrino masses) = type-I: MN1 = 3e9 GeV, MN3 = 10^n MN1; scotogenic: MN1 = 1e4 GeV, MN3 = 10^n MN1, MN2 = 10^(n/2) MN1
    The absolute mass scale of the right-handed neutrinos is chosen to satisfy the leptogenesis condition; the ratios define the benchmark scans.
  • u (VEV of singlet scalar S) = 1.5e12 GeV (type-I), 5e6 GeV (scotogenic)
    Sets the right-handed neutrino mass scale through yk = MNk/u; value chosen so that yk takes plausible values.
  • M0, xi (parameters in Delta M for third RH neutrino mass) = M1 = M0 and xi = -pi/2 in the benchmark
    These parameters, with gamma = pi/4, give sin(gamma - tilde gamma) = 0.92, providing the CP violation needed for leptogenesis.
  • lambda5 (quartic coupling in scotogenic model) = |lambda5| = 1e-5
    Chosen to bring the right-handed neutrino mass scale down to 1e4-1e6 GeV in the scotogenic model.
  • M_eta (inert doublet mass) = 1 TeV
    A benchmark value consistent with dark matter and loop suppression.
  • gamma (CP phase of <S>) = pi/4
    Chosen in the benchmark to give large sin(gamma - tilde gamma).
assumptions (6)
  • domain assumption The neutrino mass matrix is exactly diagonalized by the tribimaximal mixing matrix U_nu of eq. (9).
    Used to derive the Yukawa coupling textures in eqs. (10) and (17); this is a classic ansatz, not derived from a symmetry in this paper.
  • ad hoc to paper The PMNS matrix is given by V = V_CKM^T U_nu, with Ue = V_CKM, motivated by SU(5) and Me = Md^T.
    Phenomenological ansatz introduced around eq. (7); the paper admits only about half of its elements lie within 3 sigma of the global fit.
  • ad hoc to paper Right-handed neutrino masses originate from the VEV of a singlet scalar S through yk S Nk Nk^c, with spontaneous CP violation from the potential.
    This is the 'certain assumption for origin of right-handed neutrino mass' stated in the abstract; it is carried over from the author's earlier models and adds the S-mediated production channel.
  • standard math The effective neutrino mass formula of the type-I seesaw (eq. 4) or the scotogenic model (eqs. 5-6) applies.
    Standard results; the scotogenic formula is a one-loop correction from the inert doublet.
  • ad hoc to paper h1 << h2,3, so that the lightest mass eigenvalue remains zero in the three-right-handed-neutrino extension.
    Required for eq. (18) and for the mass eigenvalues to be essentially independent of h1; quantified in footnote d.
  • domain assumption Thermal leptogenesis follows the standard out-of-equilibrium decay of N1 with the washout processes of Section 3.3.
    Standard leptogenesis framework, including the Boltzmann evolution in Fig. 4.
invented entities (3)
  • Singlet scalar S
    purpose: Generates the right-handed neutrino masses via its VEV, breaks lepton number and CP, and mediates NkNk -> N1N1 scattering that thermally produces N1.
    The paper's own estimates show no observable signal from S; the only indirect handle is the assumed contribution to N1 production in leptogenesis, which is internal to the model.
  • Inert doublet eta
    purpose: Provides radiative neutrino mass in the scotogenic model and a dark matter candidate.
    The dark matter relic abundance and LFV rates are not computed in this paper; the claimed DM role is standard from refs. [10,11] but no new evidence is given here.
  • Third right-handed neutrino (N3 in NO, N1 in IO)
    purpose: Introduces a nonzero CP asymmetry for leptogenesis, which is zero in the two-right-handed-neutrino version with the assumed texture.
    No direct observable is predicted; its existence is inferred from the requirement that leptogenesis work.

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Cite this review

Pith. "Pith review of Neutrino models with a zero mass eigenvalue." pith.science (2026). https://pith.science/paper/WLGFV3WJ

@misc{pith2026241205774,
  author       = {Pith},
  title        = {Pith review of: Neutrino models with a zero mass eigenvalue},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WLGFV3WJ}},
  note         = {Machine review of arXiv:2412.05774}
}
read the original abstract

Absolute values of the neutrino mass are not known still now although their upper bounds are constrained through several experiments and observations. Recent analyses of cosmological observations present severe constraint on the sum of neutrino masses. It might suggest an interesting possibility for the absolute values of neutrino mass and their ordering. In this paper, taking it as a useful hint, we study possible neutrino models with a zero mass eigenvalue from a view point of neutrino oscillation data and baryon number asymmetry in the Universe. We focus our study on the seesaw type mass generation by making a certain assumption for origin of right-handed neutrino mass.

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Reviewed August 11, 2026 · model on record in the stance chip above.