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The smallest non-Abelian groups, S3 and D4, yield three-Higgs-doublet models that generate both neutrino mass scales and a residual Z2 that stabilizes dark matter.

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2026-07-10 16:40 UTC pith:SUKVUFAX

load-bearing objection Solid reduction of discrete-dark-matter + scoto-seesaw to the two smallest non-Abelian groups and a 3HDM; existence of CP-conserving fits is shown, but vacuum alignment is assumed rather than derived. the 2 major comments →

arxiv 2607.07853 v1 pith:SUKVUFAX submitted 2026-07-08 hep-ph

Neutrino Masses and Dark Matter Stability in 3HDMs with Minimal Non-Abelian Discrete Symmetries

Cesar Bonilla , Andres Layana-Ramirez This is my paper
classification hep-ph
keywords neutrino massesdark matter stability3HDMS3D4discrete dark matterscoto-seesawresidual Z2
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.

Two long-standing gaps in the Standard Model—tiny neutrino masses and a stable dark-matter particle—can be closed together by adding three Higgs doublets and a discrete flavor symmetry that is spontaneously broken. The paper shows that the two smallest non-Abelian groups with a two-dimensional representation, S3 and D4, are already enough. After the vacuum alignment leaves a residual Z2 parity, one right-handed neutrino generates the atmospheric mass scale at tree level while the dark-sector fields running in a loop generate the solar scale. With one extra singlet right-handed neutrino the mass matrix reaches rank 2 (S3) or rank 3 (D4) and fits all five oscillation observables inside 1σ of current global fits. Both constructions keep CP conserved in the Yukawa and scalar sectors and require fewer scalars than earlier A4 realizations.

Core claim

Next-to-minimal three-Higgs-doublet models based on S3 or D4, each containing one extra singlet right-handed neutrino NS, produce a neutrino mass matrix of rank 2 or 3 that simultaneously accommodates all measured oscillation parameters while a residual Z2 from spontaneous breaking of the discrete group stabilizes a dark-matter candidate that participates in the one-loop mass contribution.

What carries the argument

The discrete-dark-matter residual Z2 that remains after the flavor-doublet scalar Φ acquires the vacuum expectation value alignment (⟨H2⟩,0); this parity both protects the lightest Z2-odd particle and forces the dark fields to run in the one-loop diagram that supplies the solar mass scale.

Load-bearing premise

The vacuum of the scalar potential must settle into the exact alignment that leaves an unbroken residual Z2; if that alignment is not a stable minimum, both the dark-matter protection and the radiative mass contribution disappear.

What would settle it

A dedicated minimization of the full three-doublet potential that shows no region of parameter space realizes the required vacuum alignment (⟨H2⟩,0) while remaining consistent with the measured Higgs mass and the quoted collider bounds on the new scalars.

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

If this is right

  • S3 forces the lightest neutrino to be exactly massless, fixing the neutrinoless double-beta-decay effective mass to the narrow window 1.45–3.70 meV.
  • D4 yields three massive neutrinos and predicts an effective Majorana mass of order 1–8 meV for mν1 ≲ 5 meV, still well below present limits but within reach of next-generation 0νββ experiments.
  • Both constructions reduce the scalar sector from four doublets (needed by A4) to three while conserving CP, removing the need for complex phases in the potential.
  • The same residual Z2 that stabilizes dark matter also forbids flavor-changing neutral currents involving the inert doublet, automatically satisfying an important phenomenological constraint.

Where Pith is reading between the lines

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

  • Because the models keep every Yukawa coupling real, any future measurement of a non-zero Dirac CP phase in neutrino oscillations would require either complex phases beyond the present minimal assignment or an additional source of CP violation not considered here.
  • The upper inert-doublet mass window used for the scalar dark-matter benchmarks is already known to be under-abundant once extra portal couplings are open; a full relic-density scan may therefore force the dark-matter candidate into the fermionic N2 state, tightening the Yukawa bounds further.
  • The same residual-Z2 construction can be asked of the next-smallest groups that contain two-dimensional representations, offering a systematic way to enlarge the list of viable discrete dark-matter flavor models without increasing the number of Higgs doublets.

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

2 major / 4 minor

Summary. The paper constructs next-to-minimal three-Higgs-doublet models realizing the discrete dark matter (DDM) mechanism with the smallest non-Abelian groups admitting a 2-dimensional irrep, S3 and D4. A residual Z2 from the assumed vev alignment of the flavor-doublet scalar Φ stabilizes a dark-sector particle that mediates a one-loop (scotogenic) contribution to neutrino masses, while a tree-level type-I seesaw from the active RH neutrino supplies the atmospheric scale. An extra singlet RH neutrino NS is required to raise the neutrino mass matrix to rank 2 (S3) or 3 (D4). Explicit real-Yukawa benchmarks are given that place all five oscillation observables inside 1σ of NuFit-6.0 under normal ordering, with CP conserved in both Yukawa and scalar sectors and scalar masses satisfying the listed collider and perturbativity cuts.

Significance. If the constructions hold, the work supplies the most economical non-Abelian DDM realizations of the scoto-seesaw framework, reducing the scalar content from the four doublets of the earlier A4 model to three while still fitting current oscillation data. The models yield concrete, CP-conserving predictions for the effective Majorana mass ⟨metaeta⟩ in the few-meV range and a fully specified scalar spectrum that can be confronted at colliders. The appendices provide complete group-theory Clebsch-Gordan rules, the full scalar potentials with tadpole and mass formulae, and the finite one-loop Passarino-Veltman coefficients, making the results reproducible. These are genuine strengths of a well-executed model-building paper.

major comments (2)
  1. [Section 2, Eq. (1); Appendices B.1–B.2] Section 2, Eq. (1) and the paragraph that follows: the residual Z2 that stabilizes the dark-matter candidate (and thereby legitimizes the one-loop scoto contribution) is obtained only after imposing by hand the alignment ⟨Φ⟩=(⟨H2⟩,0). Appendices B.1–B.2 give the tadpole equations, physical masses and copositivity conditions under that alignment, but never demonstrate that the chosen vacuum is a local minimum of the full S3 or D4 potential, that soft-breaking or radiative corrections do not reintroduce a vev for η, or that the residual generator remains unbroken once all allowed operators are retained. If the true minimum is misaligned, both the residual parity and the dark-matter interpretation of the loop particle collapse, undermining the central claim.
  2. [Section 2.1; Conclusions] Section 2.1 and Conclusions: the benchmarks place the CP-even dark scalar η0 in the high-mass inert-doublet window (mDM ≃ 540–550 GeV), yet the text explicitly states that the additional portal couplings open further annihilation channels that generically drive the relic abundance below the observed value. Presenting these points as dark-matter candidates while acknowledging under-abundance, and deferring a dedicated scan, leaves the phenomenological viability of the residual-Z2 dark sector incomplete for the very parameter points used to fit neutrino data.
minor comments (4)
  1. [Appendix B] Appendix B, just before Eq. (38): the phrase “In this analysis we got use of tan β” is ungrammatical; replace with “make use of” or “use”.
  2. [Section 5] Conclusions, first paragraph of the DM discussion: several words are concatenated without spaces (“presentedinSections3and4”, “theneutralcomponentoftheinertdoublet”). These should be corrected for readability.
  3. [Tables 4 and 7] Table 4 and Table 7: the extremely high-precision floating-point entries for the charged-lepton Yukawas (e.g., 1.876449333333 imes10−2) give a false impression of fine-tuning; rounding to a few significant figures consistent with the scan precision would improve clarity.
  4. [Section 3] Section 3, paragraph after Eq. (16): the KamLAND-Zen bound is quoted as ⟨metaeta⟩<28–122 meV; a brief note that the range reflects nuclear-matrix-element uncertainty would help non-specialist readers.

Circularity Check

1 steps flagged

No significant circularity: free-parameter scans demonstrate existence of fits to external oscillation data under a chosen residual-Z2 alignment; metaeta ranges follow from the fitted benchmarks and rank structure rather than a self-definitional loop.

specific steps
  1. fitted input called prediction [Section 3, Eqs. (15)–(16) and Table 5 (analogous for D4 in Sec. 4)]
    "Using the benchmark values of Table 5, this yields the prediction ⟨mββ⟩S3≃1.45–3.70meV, where the lower (upper) value corresponds to a destructive (constructive) interference between the two contributions"

    The benchmark of Table 5 is obtained by scanning free Yukawas yi, heavy masses mN,mS and scalar couplings until sin2θij and Δm2ij all fall inside the 3σ ranges of the external global fit (Table 2). ⟨mββ⟩ is then evaluated from those same fitted Uej and mνj (with mν1=0 by the rank-2 structure and discrete Majorana phases). The numerical interval is therefore statistically forced by the fit to the input oscillation data rather than an independent, parameter-free prediction.

full rationale

The paper’s central results are (i) construction of next-to-minimal S3 and D4 3HDMs whose residual Z2 (from a chosen vev alignment) stabilizes a dark-sector particle that participates in a one-loop neutrino-mass contribution, and (ii) numerical demonstration that free Yukawa couplings, heavy-fermion masses and scalar quartics can be chosen so that the five oscillation observables lie inside the 3σ ranges of external global fits. This is a standard existence proof via random scan (Eq. (2) and Sec. 2.1), not a claim that the mixing angles or mass-squared differences are derived from first principles without free parameters. The residual Z2 is obtained by imposing the alignment ⟨Φ⟩=(⟨H2⟩,0) by hand (Sec. 2); the potential is then analyzed under that assumption (App. B), which is an explicit modeling choice rather than a circular reduction of a claimed prediction to its own input. Self-citations to the authors’ prior A4 work supply historical context for the DDM+scoto-seesaw framework but are not invoked as load-bearing uniqueness theorems that force the present S3/D4 constructions. The only mild circularity is that the numerical ⟨mββ⟩ “predictions” are computed from the same fitted benchmarks that already reproduce the oscillation data; because the models have enough parametric freedom to accommodate essentially any angles (subject only to the rank of Mν), the quoted meV ranges are largely forced by the external data plus the discrete CP phases. This is ordinary model-building practice, not a self-definitional or uniqueness-imported loop, and does not elevate the score above 2.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 2 invented entities

The central existence claim rests on standard group theory and QFT, on the domain assumption that a residual Z2 from a chosen vev alignment stabilizes DM, and on a large set of free Yukawa and scalar parameters that are fitted to oscillation data. The new fields (Φ, ND, NS) are invented entities whose only independent handle is the usual collider/DM phenomenology of inert doublets and heavy Majorana fermions.

free parameters (3)
  • Yukawa couplings yD_i, yS_i and charged-lepton yij
    Scanned over 10^{-10} to √(4π) and fitted so that the resulting Mν and Mℓ reproduce the five oscillation observables and charged-lepton masses.
  • Heavy Majorana masses mN, mS
    Scanned over 10^2–10^5 GeV; their values enter both the tree-level seesaw and the loop functions C1–C3 and are chosen to fit Δm².
  • Scalar quartics λ1…λ8 and tan β
    Varied subject to bounded-from-below, perturbativity and |θH|≲0.317; they set the physical scalar spectrum and the loop functions that contribute to Mν.
axioms (3)
  • domain assumption Spontaneous breaking of GD by ⟨Φ⟩=(vH2,0) leaves a residual Z2 that stabilizes the lightest odd particle.
    Stated in Section 2, Eq. (1); required for the discrete-dark-matter interpretation.
  • ad hoc to paper All Yukawa couplings may be taken real (CP conserved in Yukawa and scalar sectors).
    Imposed throughout the scan (Section 2.1); sufficient for normal ordering but excludes inverted-ordering solutions that might require complex phases.
  • standard math Standard type-I seesaw and one-loop scotogenic mass formulae hold after the discrete symmetry is imposed.
    Used to write Eqs. (11)–(14) and (21); derived from the Lagrangian terms given in the text.
invented entities (2)
  • Flavor-doublet scalar Φ=(H2,η) and Majorana doublet ND=(N1,N2) no independent evidence
    purpose: Provide the residual Z2, the tree-level seesaw partner N1 and the dark-sector loop mediators η,N2.
    Postulated as the minimal GD-doublet content; independent evidence is only the usual inert-doublet and heavy-neutrino phenomenology.
  • Extra singlet RH neutrino NS no independent evidence
    purpose: Lift the neutrino mass matrix from rank 1 to rank 2 (S3) or 3 (D4) so that both mass-squared differences can be accommodated.
    Required once the purely minimal content is shown to be insufficient; no independent evidence beyond the oscillation fit itself.

pith-pipeline@v1.1.0-grok45 · 23094 in / 3130 out tokens · 39761 ms · 2026-07-10T16:40:14.828940+00:00 · methodology

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read the original abstract

We present minimal three-Higgs-doublet models (3HDMs) based on global non-Abelian discrete symmetries that simultaneously explain neutrino masses and dark matter stability. A residual $Z_2$ parity from the spontaneous breaking of the new symmetry stabilizes the dark matter candidate, which runs in the loop generating neutrino masses at one loop alongside a tree-level type-I seesaw contribution. We identify $S_3$ and $D_4$ as the smallest non-Abelian groups realizing this framework and determine the minimal models in both cases that are consistent with current neutrino oscillation data. The resulting models conserve CP in both the Yukawa and scalar sectors.

Figures

Figures reproduced from arXiv: 2607.07853 by Andres Layana-Ramirez, Cesar Bonilla.

Figure 1
Figure 1. Figure 1: νL νL = vev vev νL νL ⟨H1⟩ ⟨H1⟩ + N1 νL νL η η ⟨H1⟩ ⟨H1⟩ N2 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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