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REVIEW 4 major objections 4 minor 31 references

This paper argues that a classically scale-invariant, SO(5)-symmetric extension of the scotogenic model can radiatively generate a TeV singlet VEV, suppress the weak scale far below it, and simultaneously account for neutrino masses, dark m

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2026-08-02 23:18 UTC pith:2BQGZQDG

load-bearing objection A credible benchmark-based existence proof that the scale-invariant SO(5) mechanism can be grafted onto the scotogenic model, but the electroweak scale and the DM/leptogenesis outputs rest on one hand-imposed mass degeneracy that nothing in the model protects. the 4 major comments →

arxiv 2602.14115 v2 pith:2BQGZQDG submitted 2026-02-15 hep-ph

Scale invariant radiative neutrino mass model

Daijiro Suematsu This is my paper
classification hep-ph PACS 12.60.-i14.60.Pq95.35.+d
keywords scale invariancecustodial symmetryscotogenic modelradiative neutrino massdark matterresonant leptogenesisColeman-Weinberg mechanisminert doublet
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to show that the three unexplained mass scales of the scotogenic model—the weak scale, the inert doublet mass, and right-handed neutrino masses—can all be derived from one TeV-scale singlet VEV arising through the Coleman-Weinberg mechanism, with custodial SO(5) symmetry protecting the smallness of the weak scale. It claims that with carefully chosen Planck-scale boundary conditions, the model produces a Higgs mass near 137 GeV, a Higgs VEV near 242 GeV, and a dilaton-like scalar near 77 GeV, while neutrino oscillation data are fitted by two small Yukawa couplings. The custodial constraints suppress the quartic couplings that would let the inert doublet be dark matter, so the lightest right-handed neutrino must instead be the dark matter, with mass below about 1 MeV produced by freeze-in. The near-degeneracy of the two heavier right-handed neutrinos, required for the weak-scale generation, also enables resonant leptogenesis for the baryon asymmetry. A sympathetic reader would care because this is a concrete proposal for a UV completion that replaces hand-assigned TeV masses with a single dynamically generated scale.

Core claim

The paper claims that a scale-invariant scotogenic model with custodial SO(5) symmetry can generate all its mass scales from a singlet VEV vφ≃7.9 TeV, while custodial protection keeps the weak scale small (vH≃242 GeV). With the paper's Planck-scale boundary conditions, RGE running gives m_h≃137 GeV, a dilaton-like scalar at 77 GeV, m_η≃1.48 TeV, and M_2,3≃1.50 TeV. Since custodial symmetry suppresses the inert-doublet quartic couplings, the inert doublet cannot be dark matter; the lightest right-handed neutrino, lighter than O(1) MeV, is produced by freeze-in. Near-degenerate N2 and N3 generate the CP asymmetry for resonant leptogenesis.

What carries the argument

The central mechanism is the interplay of classical scale invariance and custodial SO(5) symmetry, in which the Higgs doublet and one singlet form a 5-plet. The singlet's VEV from the Coleman-Weinberg potential spontaneously breaks both scale invariance and SO(5)→SO(4), leaving the Higgs as a pseudo-Nambu-Goldstone boson; the weak scale is then set by the small custodial-violating combination m_H²≃(λHϕ−λHS λϕ/λϕS)vφ², while masses of the inert doublet and right-handed neutrinos are induced by the same VEV through couplings ληϕ and yNj. The matching to the scotogenic model identifies m_H², λ~1, and m_η² via eq. (14), and the degeneracy m_η≃M_2,3 imposed for the weak scale feeds into resonant

Load-bearing premise

The load-bearing premise is that at the Planck scale the scalar quartic couplings obey exactly the custodial conditions λH=λHϕ=λϕ and λHS=λϕS, and that no Planck-scale physics adds corrections of order λc1 to these couplings; otherwise the cancellation that makes m_H²≪vφ² fails and the weak scale is no longer protected.

What would settle it

Compute or constrain radiative threshold corrections at the Planck scale: if unknown UV physics generates O(λc1) corrections to the custodial relations in eq. (3), then the predicted m_h≃137 GeV and vH≃242 GeV shift by order-one factors. Experimentally, the model is falsified by excluding a 77 GeV dilaton-like scalar with tanθ≃0.04 mixing with the Higgs, or by finding dark matter that is not a sub-MeV sterile neutrino with freeze-in production.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If correct, the scotogenic model's TeV scales are no longer free inputs but are determined by the singlet VEV, making the model a candidate UV completion.
  • Dark matter in this model is a sub-MeV right-handed (sterile-like) neutrino with no mixing with active neutrinos, produced by freeze-in; the inert doublet cannot serve as thermal dark matter.
  • The model predicts a dilaton-like scalar near 77 GeV with small mixing (tanθ≃0.04) with the 137 GeV Higgs, and an inert doublet at about 1.48 TeV.
  • Baryon asymmetry is generated by resonant leptogenesis from N2 decay, enabled by the same N2–N3 degenerate masses needed for the weak scale.
  • Neutrino masses fit oscillation data with two small Yukawa couplings under a tribimaximal texture, while the lightest right-handed neutrino remains essentially decoupled.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mechanism suggests a generic recipe: any radiative seesaw with TeV-scale states could be promoted to a scale-invariant model if its mass-generating couplings respect a custodial symmetry that protects the Higgs mass; one could scan other textures and seesaw realisations for the same pattern.
  • The sub-MeV dark matter prediction lies in the range relevant to small-scale structure puzzles; if future structure observations require a warm dark matter component, this model offers a concrete particle candidate to test.
  • The required degeneracy δ≲10^-9 between N2 and N3 at low energies implies Planck-scale initial parameters tuned to about one part in 10^9 or better; this tuning could be quantified and compared with naturalness expectations.
  • The model could be tested by collider searches for the ~77 GeV dilaton-like scalar and the ~1.5 TeV inert doublet; a null result in those channels would disfavour the benchmark, while a discovery would provide a strong check.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes an extension of the scotogenic model in which classical scale invariance and an SO(5) custodial symmetry are imposed. Two real singlet scalars φ and S are added; the Higgs doublet and φ form an SO(5) 5-plet, and a Coleman–Weinberg-generated VEV v_φ ≃ 7.9 TeV induces masses for the inert doublet η and right-handed neutrinos N_j, while custodial symmetry suppresses the Higgs mass to produce v_H ≃ 242 GeV. Planck-scale boundary conditions are chosen so that the running quartic couplings yield m_h ≃ 137 GeV, m_η ≃ 1484 GeV and M_2,3 ≃ 1502 GeV. The model is then applied to neutrino masses (using a tribimaximal-motivated Yukawa texture), dark matter (the lightest right-handed neutrino N_1 below O(1) MeV via freeze-in), and baryogenesis (resonant leptogenesis from N_2–N_3 degeneracy). A single benchmark point is presented.

Significance. If the mechanism works without severe tuning, this would be an interesting step toward a scale-invariant radiative-seesaw model that simultaneously explains the weak scale, neutrino masses, DM, and the baryon asymmetry. The paper is a genuine model-building study with explicit RGEs, a one-loop effective potential, and numerical solutions of Boltzmann equations, and it gives a falsifiable prediction for the DM mass. However, the central hierarchy is achieved by hand-picked Planck-scale parameters and a η–N degeneracy rather than by symmetry; no stability analysis is provided, and the leptogenesis estimate appears to be off by orders of magnitude depending on the normalization of Eq. (32). The significance is therefore conditional: the structure is worth publishing only after these issues are resolved or at least quantified.

major comments (4)
  1. [§2, Eqs. (14), (17), (20)] The central hierarchy mechanism rests on a cancellation. From Eq. (14), m_H^2 = (λ_Hϕ − λ_HS/λ_ϕS λ_ϕ)v_ϕ^2, and Eq. (20) shows β3 = 4Σ y_Nj^4 − 2λ_ηϕ^2 must nearly cancel, i.e. λ_ηϕ ≈ 2y_N2,3^2 at low energies. This near-degeneracy is imposed by hand at the Planck scale in Eq. (17) (with m_η=1484 GeV vs M_2,3=1502 GeV in Eq. (19)) and is not a consequence of the custodial or scale symmetries. A modest O(10^-2) change in λ_ηϕ or y_N2,3 moves m_H^2 toward v_ϕ^2 and shifts v_ϕ. No parameter scan or sensitivity/stability analysis is given, so the benchmark could be an isolated fine-tuned point. This must be quantified for the hierarchy claim to be convincing.
  2. [§3.3, Eq. (32)] The CP-asymmetry formula as written is not consistent with the quoted magnitudes. Using the benchmark h3=1.58×10^-3 and δ=8.8×10^-7, the resonance factor in Eq. (32) is ~0.05, while the coupling factor is 2/3, giving ε~0.03, not O(10^-5). With a conventional 1/(8π) prefactor, ε~4×10^-9. Either way, the stated O(10^-5) and O(10^-3) estimates are not reproduced. Please specify the exact normalization convention and recompute the Boltzmann results of Fig. 4; the leptogenesis conclusion depends on this.
  3. [§3.3, Eq. (34)] Successful leptogenesis is obtained only after an additional Planck-scale tuning y_N3=(1+δ0)y_N2 with δ0 chosen so that δ(M2)<10^-9; the text explicitly acknowledges that 'initial values of parameters have to be tuned at the Planck scale'. Combined with the λ_ηϕ–y_N2,3 degeneracy needed for the weak scale, this introduces a second independent fine-tuning. The claim that the degeneracy required for the weak scale also naturally enables leptogenesis is therefore not established without a quantitative measure of the required tuning.
  4. [§3.1, Eq. (24)] The neutrino-mass analysis assumes the texture (24) with hej=hμj=−hτj (j=1,2), he3=0, giving θ13=0 in the neutrino sector. The text states that a favorable θ13 can arise from charged-lepton mixing, but no charged-lepton mass matrix, diagonalization, or numerical fit is provided. Thus the statement that the oscillation data are 'explained' (around Eq. (26)) is incomplete; the fit depends on an unspecified external sector.
minor comments (4)
  1. [Throughout] Typos: 'Colman-Weinberg' in the Abstract and §1, 'psudo' in §2, 'leptgenesis' in §1, 'scatteing' in §3.2. Reference [18] lists 'E. Ma and Raidal' with an incomplete second author name.
  2. [Figs. 3 and 4] The line styles/colors for different quantities and cases are difficult to distinguish in the captions; please use unique, explicitly labeled line styles.
  3. [Eq. (11)] The approximation leading to the last line of Eq. (11) should state the quantitative conditions under which the dropped terms are negligible, since these terms are related to the η–N cancellation discussed later.
  4. [Footnote f, §3.2] Please clarify the sign conventions so that the statement about M_ηR < M_ηI for λ̃4, λ̃5 < 0 is consistent with Eq. (22), and explicitly connect this with the benchmark value of λ̃5.

Circularity Check

0 steps flagged

No significant circularity: outputs are conditional on explicitly-tuned benchmark inputs, the custodial mechanism is imported from external refs [6,7], and self-citations are auxiliary rather than load-bearing.

full rationale

The derivation chain is conditional but not circular. (i) The hierarchy claim relies on the external custodial mechanism of refs. [6,7] (de Boer-Lindner-Trautner, not the present author), applied with explicit equations: mH^2 (Eq. 14) is computed from input couplings via RGE running; the smallness of lambdaHphi-(lambdaHS/lambdaPhiS)lambdaphi is realized by the benchmark (17), whose eta-N degeneracy relation is openly assumed (footnote b: 'A simple relation is assumed between lambdaeataphi and yN2,3 so that metaeat and M2,3 take a degenerate value'), and whose Planck-scale tuning is admitted (Sec. 3.3: 'Although initial values of parameters have to be tuned at the Planck scale, promising low energy results can be obtained in a consistent way'). This is fine-tuning, not a reduction of output to input by construction: (17) is labeled a 'benchmark' and (18)-(19) are presented as computed numerical values, not parameter-free predictions. (ii) Neutrino Yukawa couplings h2,h3 (Eq. 26) are fitted to oscillation data; those data are inputs, not predicted outputs. (iii) The DM conclusion is a genuine model consequence: etaR DM fails because lambda-tilde3,4 are O(10^-6) (computed in the paper), and N1 freeze-in is solved from Eqs. (29)-(31) with standard reaction densities [24]; the cited self-papers [13,23] cover background DM studies and a standard 2-to-2 scattering, while the Boltzmann computation in Fig. 3 carries the argument. The bound M1<2.1 MeV is derived from YN1(scattering)<=Y-infinity with h1 chosen per case, which accommodates Omega h^2 rather than renaming a fit as a prediction. (iv) Leptogenesis is explicitly conditional on an additional tuned degeneracy delta0 (Eq. 34), and the abstract is hedged ('could be explained'). The paper's own limitation statements (uncontrolled beta3 in Eq. 20, Planck-scale tuning) are weighed here as fine-tuning/correctness risks, not circularity. No equation reduces to itself by construction, and the central mechanism is independent of the author's self-citations.

Axiom & Free-Parameter Ledger

8 free parameters · 6 axioms · 4 invented entities

The central claims depend on the custodial Planck-scale boundary conditions, the one-loop effective potential computation, the assumed neutrino Yukawa texture, and the cosmological reheating setup. Multiple couplings and phases are fitted or hand-picked, and successful leptogenesis requires a tuned mass-degeneracy parameter δ0.

free parameters (8)
  • λc1 = 4.48×10^-3 (Planck scale)
    Chosen so that λφ is small and positive at the critical scale and mH² is suppressed; not derived from any principle.
  • λc2 = 0.227 (Planck scale)
    Chosen with λc1 so that the β2 contribution in eq. (20) is comparable to β1 and the weak-scale sign/magnitude works.
  • λS, λη, ~λ5 = 0.2, 0.19, -3×10^-3
    Benchmark values in eq. (17) chosen for scalar stability, to make ηR DM under-abundant, and to generate neutrino mass via ~λ5.
  • yN2=yN3 and ληϕ = 2.03×10^-1, with 2ληϕ=(yN2+yN3)²
    Chosen to enforce near-degeneracy mη≈M2,3, which controls cancellations in eq. (11) and later leptogenesis.
  • h2, h3 = 4.24×10^-4, 1.58×10^-3
    Fitted to neutrino oscillation data through eq. (26) under the assumed tribimaximal texture.
  • h1 = 10^-8 to 2.8×10^-4 in Fig. 3
    Chosen case-by-case so that freeze-in of N1 reproduces Ωh²=0.12; not predicted independently.
  • δ0 = tuned so δ=(M2-M3)/M2<10^-9 at M2
    Required to make resonant leptogenesis produce sufficient asymmetry; the paper states the initial value must be tuned at the Planck scale.
  • CP phases γ2,γ3 = sin 2(γ2−γ3)=1
    Maximal CP phase is assumed to evaluate the CP asymmetry ε in eq. (32).
axioms (6)
  • domain assumption Classical scale invariance is exact at the Planck scale; no mass terms appear in the scalar potential or Yukawa sector.
    The potential in eq. (1) contains only dimensionless couplings. If scale invariance is broken by Planck-suppressed operators or is only an accident, the hierarchy argument collapses.
  • domain assumption The scalar fields H and ϕ form a 5-plet of SO(5) custodial symmetry at the cut-off, with only the explicitly written couplings violating the symmetry.
    Eq. (3) imposes λH=λHϕ=λϕ=λc1 and λHS=λϕS=λc2 at the Planck scale; the near-cancellation in eq. (14) follows from this assumption.
  • standard math The Gildener-Weinberg/Coleman-Weinberg one-loop effective potential, eq. (9), plus the flat-direction condition (4) and its epsilon-expansion, correctly determines the vacuum and the effective Higgs potential.
    This is a standard technique used in refs. [6-8], and it underlies the vacuum value eq. (11) and the scotogenic matching conditions eq. (14).
  • domain assumption The early Universe reheats to T_R>O(10^4) GeV, placing η and N2,3 in thermal equilibrium, while S-driven inflation with a nonminimal coupling is consistent with Planck CMB data.
    Sec. 3.2 relies on refs. [20,22] for this; freeze-in and freeze-out calculations assume these initial conditions.
  • ad hoc to paper The neutrino Yukawa texture (24), with hej=hμj=-hτj for j=1,2 and h_e3=0, reproduces the observed neutrino mixing, with θ13 supplied by undetermined charged-lepton mixing.
    Footnote e states that θ13 can come from the charged lepton sector, but no model or calculation is provided.
  • standard math The one-loop resonant leptogenesis formula (32) with sin 2(γ2−γ3)=1 and a tuned mass degeneracy δ determines the final lepton asymmetry.
    This is standard resonant leptogenesis [31], but the predictions depend on the assumed maximal CP phase and δ<10^-9.
invented entities (4)
  • ϕ (real singlet scalar) independent evidence
    purpose: Its VEV ~7.9 TeV spontaneously breaks scale invariance and SO(5) custodial symmetry, generating masses for η and right-handed neutrinos and creating the 77 GeV dilaton.
    The model predicts a dilaton-like scalar with mhφ≃77 GeV and vφ≃7.9 TeV, a falsifiable collider/cosmology signature; no external evidence currently exists.
  • S (real singlet scalar) independent evidence
    purpose: Second singlet that participates in custodial-breaking couplings and is responsible for inflation via a nonminimal coupling to gravity.
    Predicts mS≃2322 GeV and an inflationary signal consistent with Planck CMB for a tuned coupling ξ; no independent detection.
  • η (inert doublet) independent evidence
    purpose: Z2-odd scalar generating one-loop neutrino masses; its neutral component cannot be DM in this model but decays to N1 and contributes to N1 freeze-in.
    Predicts mη≃1484 GeV and charged/neutral components near 1.5 TeV, testable at future colliders; no external evidence.
  • N_j (three right-handed neutrinos) independent evidence
    purpose: N2,3 at ~1.5 TeV source neutrino masses and resonant leptogenesis; the lightest N1 below 2.1 MeV is the dark matter.
    Predicts N2,3 masses ~1502 GeV and a sub-MeV DM mass with a specific freeze-in abundance; falsifiable through cosmology and possibly colliders.

pith-pipeline@v1.3.0-alltime-deepseek · 15825 in / 18128 out tokens · 177375 ms · 2026-08-02T23:18:49.460154+00:00 · methodology

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We propose a scale invariant radiative neutrino mass model with custodial symmetry by introducing two real singlet scalars to the scotogenic model. Masses of an inert doublet scalar and right-handed neutrinos are induced by a vacuum expectation value (VEV) of a singlet scalar caused through the Coleman-Weinberg mechanism.It violates spontaneously both the custodial symmetry and the scale invariance. The weak scale can take a suppressed value compared with the singlet scalar VEV because of the custodial symmetry. In this framework we study phenomenological consequences for neutrino mass, dark matter and baryon number asymmetry by assuming a texture for neutrino Yukawa couplings. Since the required dark matter abundance cannot be explained by a neutral component of the inert doublet scalar in that case, the lightest right-handed neutrino should be dark matter. Mass of the dark matter is predicted to be less than $O(1)$ MeV and baryon number asymmetry could be explained through resonant leptogenesis.

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