Fermion Dark Matter and Radiative Neutrino Masses from Spontaneous Lepton Number Breaking

Cesar Bonilla; Eduardo Peinado; J. M. Lamprea; Leon M.G. de la Vega; Roberto A. Lineros

REVIEW 2 major objections 6 minor 1 cited by

Fermion Dark Matter and Radiative Neutrino Masses from Spontaneous Lepton Number Breaking

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A fermion dark matter candidate in a radiative neutrino mass model can account for all of the observed relic abundance in three mass windows below 1 TeV.

desk verdict A competent numerical scan of a known scotogenic model whose three DM mass regions are real but cut-dependent; worth refereeing as a phenomenological update, not as a discovery. read the letter →

arxiv 1908.04276 v2 pith:HEQ2KFVO submitted 2019-08-12 hep-ph

Cesar Bonilla , Leon M.G. de la Vega , J. M. Lamprea , Roberto A. Lineros , Eduardo Peinado This is my paper · ORCID

classification hep-ph PACS 14.60.Pq12.60.Fr14.80.-j

keywords neutrinomassdarkmatterscotogenicmodelMajoronspontaneousleptonnumberbreakingZ2symmetrydirectdetectionXENON1T

topics Dark Matter

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 a single fermionic dark matter candidate can solve two puzzles at once: it is the lightest $\mathbb{Z}_2$-odd Majorana fermion in a model where neutrino masses are generated at one loop, and it can make up all of the observed dark matter. The key move is to break lepton number spontaneously with a scalar singlet whose CP-even component mixes with the Higgs doublet, opening a new annihilation portal. A numerical scan restricted to theoretically and experimentally allowed parameters finds three dark matter mass windows below 1 TeV, around 8 to 20 GeV, just below half the Higgs mass, and above 80 GeV, each giving the right relic abundance with a spin-independent cross section below the current XENON1T limit. If correct, this gives direct detection experiments concrete mass ranges to confirm or exclude.

What carries the argument

The paper works in the scotogenic model, a one-loop radiative neutrino mass framework with an unbroken $\mathbb{Z}_2$ symmetry that stabilizes dark matter. The central object that carries the argument is the lepton-number-carrying scalar singlet $\sigma$, whose vacuum expectation value $v_\sigma$ breaks $U(1)_L$, gives mass to the Majorana fermions $N_i$, and produces the Majoron $J$, the massless Nambu-Goldstone boson of the broken symmetry. What rescues the fermionic dark matter candidate is the mixing angle $\alpha$ between the CP-even component of $\sigma$ and the SM Higgs doublet: this mixing creates the $s$-channel annihilation diagrams $N_1N_1\to h_{1,2}\to$ SM particles that supply the correct relic density without requiring large neutrino Yukawa couplings. It is also what keeps the model testable, because it yields a spin-independent scattering cross section near current direct detection sensitivities.

What would settle it

Two checks would settle the central claim: re-run the numerical scan with the Majoron-annihilation cap and the no-co-annihilation condition removed, and see whether the three mass regions persist; and watch the next-generation direct detection experiments XENONnT and LZ, which will either observe signals in the predicted windows or push upper limits below the model's predicted $\sigma_{SI}$ and exclude those windows.

Watch

Extended reading notes

Core claim

The central claim is that the spontaneously broken lepton number is not just the origin of neutrino masses but also the source of a new annihilation channel that makes the fermion dark matter candidate work. In the minimal scotogenic model, lepton-flavor-violation bounds force the neutrino Yukawa couplings to be tiny, so the usual $t$-channel annihilation is suppressed and the calculated relic density is too large. Here, after $\langle\sigma\rangle\neq 0$ breaks lepton number, the CP-even singlet mixes with the SM Higgs doublet, and dark matter annihilates through the $s$-channel into quarks, gauge bosons, and Higgs bosons. The paper's scan identifies three mass regions below 1 TeV, $8\lesssim m_{N_1}\lesssim 20$ GeV with $b\bar b$ dominance, $m_{N_1}\lesssim m_h/2$ with resonant annihilation, and $m_{N_1}\gtrsim 80$ GeV with $WW$ and $ZZ$ final states, where the relic density matches $\Omega_c h^2 = 0.120\pm 0.001$ and $\sigma_{SI}$ stays below the XENON1T bound, under the scan requirement that annihilation into Majorons is subdominant at freeze-out.

Load-bearing premise

The three mass regions and the claim that the cross sections stay 'naturally' below XENON1T come from a scan that discards parameter points in which dark matter annihilates mostly into the massless Goldstone boson (the Majoron) or co-annihilates with heavier partners; if those selection cuts were relaxed, the same model would still give the right relic density but through channels that direct detection experiments would not see.

Editorial extensions

If this is right

The 8 to 20 GeV window announces itself through $b\bar b$ final states and lies within reach of upcoming liquid-xenon experiments.
The resonant window near $m_{N_1}\simeq m_h/2$ produces invisible Higgs decays into $N_1N_1$ and $JJ$, so measuring the Higgs invisible width tests it.
Above 80 GeV, annihilation into $W^+W^-$ and $ZZ$ dominates, and gamma-ray telescopes can already exclude some points that explain only part of the relic density.
Since $\sigma_{SI}$ ranges from near the XENON1T bound down to the neutrino floor, null results from next-generation experiments will shrink the surviving parameter space.

Reading between the lines

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

Relaxing the scan's selection cuts would still yield the right relic abundance through the Majoron channel, but with $\sigma_{SI}$ far below the neutrino floor, which would make the model much harder to verify.
The same $s$-channel portal could be added to other radiative neutrino mass models with a spontaneously broken global symmetry and a CP-even scalar mixing with the Higgs, generalizing the mechanism beyond this concrete setup.
A further test could come from the ratio of invisible Higgs decay channels: the model predicts a specific relationship between $h\to N_1N_1$ and $h\to JJ$ that a future Higgs factory could compare with the dark matter mass region inferred from direct detection.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This is a solid, workmanlike phenomenological paper. It does not invent a new model—the spontaneous-lepton-number scotogenic setup is Babu–Ma, and the s-channel portal was introduced by two of the same authors in Ref. [25]. What is new is the systematic numerical scan and the three benchmark points: a 10 GeV b-bbar region, a 59 GeV resonant region, and a 707 GeV VV region. The loop formula has the corrected 1/2 factor, and the benchmarks satisfy the stated constraints: relic abundance near 0.12, XENON1T, Fermi-LAT, LFV, and S/T. That is real work, honestly presented, and the tables are complete enough that a patient reader could reproduce the key numbers.

The soft spots are real but not fatal. The three mass regions are not intrinsic to the model; they emerge after two selection cuts in Sec. IV: BR(N1N1→JJ)<10% and mN2,3>1.1 mN1. The first cut is explicitly justified as keeping direct detection above the neutrino floor. That is a sensible phenomenological choice—nobody can discover a WIMP that scatters below the neutrino floor—but it means the 'predictions' are tailored. If the Majoron channel dominates freeze-out, the same Lagrangian gives the right relic density with arbitrarily small direct detection rate. The co-annihilation cut is milder but still removes a physically possible corner. I would not call the paper circular: relic density and sigma_SI are computed from parameters, not fitted to data. But the central claim is conditional on cuts that are imposed by hand. The second issue is reproducibility: no scan code or data is provided, so the exact boundaries of the three regions in Fig. 3 cannot be independently checked. That is a reproducibility gap, not a correctness gap.

Citation pattern is fine. The paper leans on earlier work by the same group where appropriate; that is legitimate. Who is this for? Model builders working on scotogenic DM and Majoron physics, and experimentalists who want benchmark WIMP points for XENONnT/LZ. It deserves a serious referee. I would tell the referee to focus on whether the cuts are adequately disclosed and to ask for a sensitivity appendix showing how the regions change when the cuts are relaxed. My own verdict: conditional accept.

Referee Report

2 major / 6 minor

Summary. The paper studies an extension of the scotogenic model in which a scalar singlet σ carrying lepton number acquires a vev, spontaneously breaking a global U(1)_L and generating the Majorana masses of three Z2-odd fermions N_i as well as, at one loop, the light neutrino masses. The lightest Z2-odd fermion N1 is the dark matter candidate, and its annihilations proceed through a t-channel mediated by the inert scalars and an s-channel mediated by the CP-even scalars h1,h2 whose mixing originates from the singlet-doublet potential. Using a MicrOMEGAS-based scan, the authors impose perturbativity, boundedness, LEP/LHC Higgs and electroweak constraints, neutrino oscillation data, and DM relic and direct-detection constraints. They identify three mass regions for N1 below 1 TeV (low mass around 8–20 GeV, resonant near mh/2, and high mass above 80 GeV) where N1 can account for 100% of the observed relic abundance while the predicted spin-independent cross section lies below the XENON1T bound. Three benchmark points with full parameter values and derived observables are provided.

Significance. If the claimed regions are robust, the paper is a useful phenomenological demonstration that spontaneous lepton number breaking in the scotogenic framework provides an s-channel scalar portal that can make sub-TeV fermionic dark matter viable, in contrast to the simplest scotogenic model where the t-channel is too suppressed by lepton-flavor constraints. The corrected one-loop neutrino mass formula, the explicit use of multiple constraints, and the three detailed benchmarks are strengths. The main caveat is that the central 'natural' direct-detection statement is obtained after imposing selection cuts on the predicted observables, so the significance is conditional on those detectability priors; with appropriate qualification and a robustness check, the paper would be a solid phenomenological contribution.

major comments (2)

[Section IV, last paragraph before IV.A; footnote 3; Section IV.A] The paper's central claim that the model 'naturally' predicts σSI below XENON1T in three dark matter mass regions is conditional on two output-dependent cuts: BR(N1N1→JJ)<10%, introduced to 'guarantee detectability ... and to avoid direct detection cross sections in regions far below the neutrino floor,' and mN2,3>1.1 mN1, which discards co-annihilation. These are selections on the predicted observables, not symmetry or collider constraints from Section III. Parameter points with dominant annihilation into Majorons can reproduce the relic abundance while giving σSI far below the neutrino floor, and near-degenerate N2,N3 spectra are physically allowed. The abstract and conclusions should therefore qualify the claim as conditional on these detectability priors, remove or redefine the word 'naturally,' and the analysis should quantify how the allowed regions and the σSI distribution change when the cuts are relaxed, for example by showing the rejected points in Fig. 3 or reporting a scan without the BR(JJ) cut. This is load-bearing because it determines the direct-detection prediction attributed to the model.
[Section IV.A and Fig. 3] The boundaries of the three mass regions are presented as approximate ranges, but the paper does not report the number of scanned points, the total accepted/rejected statistics, or any coverage checks. The scan ranges (mN1∈[8,1000] GeV, mηR∈[110,5000] GeV, vσ∈[500,10000] GeV) are chosen without stated physical motivation, and the three regions are inferred from point clouds rather than from an exhaustive or statistically characterized scan. This makes it difficult for the reader to judge whether the three regions are robust features of the model or artifacts of the selected scan volume and sampling density. Please report scan statistics and point densities, and state whether the region boundaries are stable under variation of the scan ranges.

minor comments (6)

[Section III.B] The sentence 'there are 6 physical scalars: three CP-even hi (i=1,2) and ηR...' contains a typo: there are only two CP-even eigenstates h1 and h2, and counting ηR together with them gives three CP-even scalars.
[Eq. (13)] The displayed neutrino mass formula contains a stray closing parenthesis after 32π²; please correct the typography.
[Section III.D and footnote 4] The main text says the charged-lepton mass matrix is not assumed flavor-diagonal, while footnote 4 states that Yν and YN are taken real and diagonal and that the charged-lepton Yukawa matrix is non-diagonal. Please clarify in the main text that the PMNS mixing is generated through the charged-lepton rotation U_l, since the LFV calculation and the interpretation of the neutrino Yukawa couplings depend on this basis choice.
[Section IV.A and Fig. 4] Please specify whether the plotted gamma-ray cross sections for the purple and pink points, which correspond to subdominant dark matter, are rescaled by (Ωh²/Ωc h²)²; otherwise the statement that Fermi-LAT excludes some of these points is ambiguous.
[Table III] The column header BR(h2→h1h1) is confusing for BM1 and BM2, where this decay is kinematically forbidden; using an explicit '—' with a footnote stating 'kinematically forbidden' would be clearer.
[Throughout] There are numerous typographical errors, including 'chaged leptons', 'diﬁned', 'In additon', 'dectection', and inconsistent spacing in Section IV; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: relic abundance and sigma_SI are computed outputs of the scanned Lagrangian parameters, not fitted inputs.

full rationale

The paper's derivation chain is a Lagrangian-level model (eqs. 2 and 4), one-loop neutrino masses (eq. 13), and a parameter scan using MicrOMEGAS that computes the thermal relic abundance and spin-independent cross section from the same Lagrangian parameters. The observed relic abundance and XENON1T bound are used as external constraints, not as fitted inputs: no parameter is tuned to the dark matter observables, and the reported sigma_SI values are outputs of the scanned points, as shown in Tables II-IV. The selection cuts in Section IV, namely BR(N1N1 -> JJ) < 10% 'to guarantee detectability in DM direct detection experiments and to avoid direct detection cross sections in regions far below the neutrino floor,' and the footnote discarding co-annihilation with mN2,3 <= 1.1 mN1, narrow the definition of 'viable' and therefore shape the boundaries of the three mass regions. However, these are scope conditions, not equations that identify the predicted quantity with the input; the paper does not claim a prediction from the model alone independent of these stated selection criteria. The self-citation to Ref. [25] introduces the s-channel annihilation portal as a model component, but the numerical calculation here is independent and externally checkable, so that citation is not load-bearing circularity. Overall, no step in the derivation reduces by construction to its own inputs.

Assumptions & free parameters 8 free parameters · 5 assumptions · 3 invented entities

The paper demonstrates existence of viable parameter points by scanning a high-dimensional Lagrangian. That is honest but weak: every phenomenological output such as relic density and sigma_SI is a function of many hand-chosen parameters, so the predictions are not sharp. The ledger shows seven groups of free parameters and several imposed ordering and cut assumptions. The model also introduces three new beyond-SM fields, so the economy of the construction is moderate: it connects dark matter and neutrino mass, but at the cost of a new global symmetry, a scalar singlet, an inert doublet, and three Majorana fermions.

free parameters (8)

Scalar quartic couplings lambda2 to lambda8 = Scanned in [10^-6, 1]; benchmark values in Table II
They set scalar masses, mixing, and dark matter annihilation rates. No symmetry or measurement fixes them; they are scanned.
Singlet vev v_sigma = 1.08, 5.80, 10.4 TeV for BM1 to BM3
Sets the lepton-number-breaking scale and gives Majorana masses to N_i through YN. Scanned in [500, 10000] GeV.
Heavy Majorana masses mN1, mN2, mN3 = BM1: 10, 119, 316 GeV; BM2: 59.1, 184, 410 GeV; BM3: 707, 924, 940 GeV
mN1 is the dark matter mass; the ordering mN1 below mN2 and mN3 below m_eta is imposed by hand.
Inert scalar masses m_etaR and m_eta+/- = BM1: 520, 486 GeV; BM2: 666, 645 GeV; BM3: 1132, 1119 GeV
m_etaI is derived from lambda5 and m_etaR. These masses control the loop suppression of neutrino masses.
Neutrino Yukawas Y_nu_i = |Y_nu| around 3 x 10^-5 to 2 x 10^-4 in benchmarks
Chosen small to suppress lepton flavor violation and to fit neutrino masses; no independent prediction.
Majorana Yukawas YN_i = 1 x 10^-3 to 0.11 in benchmarks
Together with v_sigma they determine mN_i. Scanned freely.
Doublet-singlet mixing sin(alpha) = 0.13, 0.16, 0.16 for BM1 to BM3
The key portal for s-channel annihilation and direct detection. Scanning selects these values; they are not predicted.
Charged-lepton Yukawa matrix = Not tabulated; chosen to produce the PMNS angles
Neutrino oscillation angles are inserted by hand through the charged-lepton sector rather than derived from the model.

assumptions (5)

domain assumption An exact unbroken Z2 symmetry is imposed, making the lightest Z2-odd particle absolutely stable.
Assumed in Section II; no Z2-breaking term is added to the Lagrangian.
domain assumption The global U(1)L is spontaneously broken by the vev of sigma, producing an exactly massless Majoron J.
Section II.A. The paper does not assess constraints on a massless Majoron from N_eff or stellar energy loss.
domain assumption Perturbativity and bounded-from-below conditions restrict couplings to |lambda_i| and |Y|^2 <= 4 pi and to eq. (15).
Section III.A invokes these as permissible parameter restrictions.
domain assumption Neutrino oscillation data, normal ordering, best-fit angles, and the cosmological bound on the sum of neutrino masses are taken as external constraints.
Section III.D uses the global fit values as fixed inputs.
ad hoc to paper Co-annihilation and Majoron-dominated annihilation are excluded by fiat before defining viable regions.
Section IV footnotes and the final paragraph impose mN2 and mN3 above 1.1 times mN1 and BR(N1N1 to JJ) below 10%.

invented entities (3)

Scalar singlet sigma with lepton number 2 independent evidence
purpose: Breaks global lepton number spontaneously, generates Majorana masses for N_i, produces the Majoron, and mixes with the SM Higgs to provide the s-channel dark matter annihilation portal.
Predicts invisible Higgs decay h to JJ and a second scalar h2; these are testable but not yet observed.
Inert doublet eta, Z2-odd independent evidence
purpose: Mediates the one-loop neutrino mass diagram and can host scalar dark matter in other mass orderings.
Predicts charged and neutral scalars near the weak scale with LEP and LHC signatures; no direct evidence to date.
Three Majorana fermions N_i, Z2-odd independent evidence
purpose: Run in the neutrino mass loop; N1 is the dark matter candidate.
Predicts dark matter signals including sigma_SI and gamma rays, plus LFV and collider signatures; no direct evidence so far.

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Cite this review

Pith. "Pith review of Fermion Dark Matter and Radiative Neutrino Masses from Spontaneous Lepton Number Breaking." pith.science (2026). https://pith.science/paper/HEQ2KFVO

@misc{pith2026190804276,
  author       = {Pith},
  title        = {Pith review of: Fermion Dark Matter and Radiative Neutrino Masses from Spontaneous Lepton Number Breaking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HEQ2KFVO}},
  note         = {Machine review of arXiv:1908.04276}
}

abstract

In this paper, we study the viability of having a fermion Dark Matter particle below the TeV mass scale in connection to the neutrino mass generation mechanism. The simplest realization is achieved within the scotogenic model where neutrino masses are generated at the 1-loop level. Hence, we consider the case where the dark matter particle is the lightest $\mathbb{Z}_2$-odd Majorana fermion running in the neutrino mass loop. We assume that lepton number is broken dynamically due to a lepton number carrier scalar singlet which acquires a non-zero vacuum expectation value. In the present scenario the Dark Matter particles can annihilate via $t$- and $s$-channels. The latter arises from the mixing between the new scalar singlet and the Higgs doublet. We identify three different Dark Matter mass regions below 1 TeV that can account for the right amount of dark matter abundance in agreement with current experimental constraints. We compute the Dark Matter-nucleon spin-independent scattering cross-section and find that the model predicts spin-independent cross-sections ``naturally'' dwelling below the current limit on direct detection searches of Dark Matter particles reported by XENON1T.

Figures

Figures reproduced from arXiv: 1908.04276 by the authors.

**Figure 2.** Feynman diagrams for the annihilation channels of the fermion dark matter in the model. On [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Dark matter mass − σSI plane showing the solutions in the model that satisfy all theoretical and experimental constraints given in Section III. The latest bound on direct dark matter detection is set by the XENON1T experiment [34] (top shaded area). The dashed lines represent the expected sensitivities in forthcoming experimental searches such as XENONnT [49], LUX-ZEPLIN (LZ) [50], DarkSide 20k [51], DARWIN [52] and… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Predictions for the velocity averaged cross section of dark matter annihilation into gamma rays [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Flavour and precision probes of a class of scotogenic models
hep-ph 2025-06 conditional novelty 5.0 of 10

Full next-to-leading-order corrections to leptonic Z and Higgs decays are derived for the T1-2-A scotogenic model, and the resulting cLFV and electroweak observables are scanned to identify future collider probes.

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Reviewed August 14, 2026 · model on record in the stance chip above.

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The above considerations are made in such a way that they all satisfy the theoretical and experimental constraints described in Sec. III. We computed the value of theS and T parameters using the expressions given in [46, 47], takingU = 0 [30] and keeping those solutions that are in agreement with the bounds given in eq. (19). It is worth to mention that w...

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