A Novel One-loop Model for Majorana Neutrino Mass and Dark Matter

Mohamed Amin Loualidi; Mohamed Belfkir; Salah Nasri

REVIEW 4 major objections 4 minor 57 references

A Novel One-loop Model for Majorana Neutrino Mass and Dark Matter

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper constructs the first complete field-theoretic realization of the one-loop T4-3-i topology for Majorana neutrino mass, with a Dirac mediator that eliminates the tree-level seesaw and an exact Z2 symmetry that stabilizes dark…

desk verdict A solid, careful T4-3-i model-building paper whose central 'first complete realization' claim is currently under-argued relative to Ref. [18]; worth refereeing, not desk-rejecting. read the letter →

arxiv 2608.12646 v1 pith:M5A5YXNF submitted 2026-08-12 hep-ph

Mohamed Belfkir , Mohamed Amin Loualidi , Salah Nasri This is my paper · ORCID

classification hep-ph

keywords MajorananeutrinomassradiativeseesawT4-3-itopologyWeinbergoperatordarkmatterinertscalarschargedleptonflavorviolationrank-twomatrix

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

This paper constructs the first complete field-theoretic model in which neutrino mass comes from the finite one-loop topology T4-3-i, a loop realization of the dimension-five Weinberg operator that had been classified but never fully realized with all symmetries intact. The construction removes the usual tree-level type-I or type-III seesaw by making the fermion attached to the lepton-Higgs pairs a Dirac field, and puts lepton-number violation on a separate Majorana fermion inside the loop; an exact Z2 symmetry keeps the new scalars inert and stabilizes the lightest odd particle as dark matter. The minimal version, T4-3-i-B1, contains one Dirac fermion, one Majorana fermion, one inert scalar doublet, and one inert scalar singlet, and yields a rank-two neutrino mass matrix, so one neutrino is massless at leading order. The authors show, by scanning the parameter space against oscillation, flavor-violation, electroweak, Higgs, relic-density, and direct-detection data, that both normal and inverted ordering and both fermionic and scalar dark-matter candidates remain viable. If the model is right, it closes a long-standing gap in radiative neutrino mass models and ties the neutrino mass matrix, dark-matter stability, and charged-lepton flavor violation to one small set of couplings.

What carries the argument

The load-bearing object is the T4-3-i topology, a one-loop realization of the Weinberg operator $LLHH$ in which a single fermion connects the two external lepton-Higgs pairs, together with the Dirac-mediator trick that prevents that fermion from generating a tree-level type-I seesaw. The second load-bearing element is the compact mass identity $M_\nu = \Lambda(Y y^T + y Y^T)$, where $\Lambda$ collects the trilinear scalar coupling, the Higgs VEV, the Majorana mass, and the loop function; this rank-two structure is what forces one neutrino mass to zero. The loop function, the $Z_2$ charges, and the field content (one Dirac singlet $N$, one Majorana singlet $\psi$, one inert doublet $\phi$, one inert singlet $\phi'$) are what carry the entire phenomenological analysis.

What would settle it

Computing the full renormalization of the dimension-five Weinberg operator in this model would settle the central claim: if a counterterm must be introduced to cancel a divergence, the one-loop diagram is not the leading source of neutrino mass, and separately a measurement of three nonzero neutrino masses would rule out the minimal rank-two prediction.

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Extended reading notes

Core claim

The central claim is that the T4-3-i topology can be realized without a competing lower-order seesaw: with a Dirac singlet mediator N, a Majorana singlet psi inside the loop, inert scalar doublet and singlet, and an exact Z2 symmetry, the Weinberg operator is generated only at one loop. The resulting mass matrix takes the form $M_\nu = \Lambda (Y y^T + y Y^T)$; since it is a sum of two outer products it has rank at most two, so $\det M_\nu = 0$ and one neutrino is massless, with $m_1 = 0$ in normal ordering and $m_3 = 0$ in inverted ordering. The paper classifies the electroweak charge assignments of the loop fields into four model families, A, B, C, and D, and analyzes the minimal singlet-doublet case B1. In the numerical fit, both fermionic and scalar dark matter are viable: the fermion has only a loop-induced Higgs coupling and a spin-independent scattering rate below current sensitivity, while the scalar has a tree-level Higgs portal and can be as heavy as roughly 900 GeV while remaining below current direct-detection bounds. The same couplings that fix the neutrino mass matrix also control charged-lepton flavor violation, so oscillations, $\mu \to e \gamma$, $\mu \to 3e$, and $\mu$--$e$ conversion in nuclei are correlated predictions.

Load-bearing premise

The paper assumes, on the authority of the topology classification, that the one-loop diagram is genuinely finite and needs no Weinberg-operator counterterm, which is what makes the loop the leading source of neutrino mass.

Editorial extensions

If this is right

The loop is the leading source of neutrino mass, so the smallness of neutrino masses is explained by the one-loop suppression plus the small Yukawa couplings, with no tree-level seesaw hidden in the model.
The mass matrix has rank two, so the minimal model predicts exactly one massless neutrino: $m_1=0$ in normal ordering and $m_3=0$ in inverted ordering.
The fermionic dark-matter candidate scatters off nuclei only through a loop-induced Higgs coupling, placing its spin-independent cross section below about $10^{-49}$ cm$^2$ in the accepted samples.
The scalar dark-matter candidate annihilates efficiently in the Higgs-resonance region near $m_{H_1^0}\simeq m_h/2$ and through coannihilation with nearby inert scalars, with cross sections up to about $10^{-48}$ cm$^2$ that next-generation xenon experiments can probe.
In inverted ordering the mass sum is near 0.1 eV and the neutrinoless-double-beta mass $m_{\beta\beta}$ lies in the range targeted by next-generation experiments, so lepton-number-violating searches are the sharpest test of that scenario.

Reading between the lines

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

If the model is extended with additional Dirac or Majorana generations, the rank of the neutrino mass matrix can rise to three; this would be the natural way to make all three neutrino masses nonzero while keeping the same one-loop mechanism, at the cost of losing the massless-neutrino prediction.
Because the fermionic dark-matter candidate is essentially invisible to current and planned direct-detection experiments, its discovery would have to come through indirect detection or collider production of the coannihilating inert scalars, a route the paper leaves for future work.
The surviving charge-assignment families A, C, and D listed but not scanned could produce models with richer scalar spectra and different coannihilation channels; their cLFV rates and collider signatures are a direct testing ground for whether the T4-3-i idea generalizes.

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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

Short version: this is a genuine contribution to the radiative-neutrino-mass catalog. The construction is clean: a Dirac mediator N removes the tree-level type-I seesaw, a separate Majorana fermion psi carries lepton-number violation inside the loop, and an exact Z2 keeps the scalars inert and stabilizes DM. The hypercharge classification of the singlet-fermion T4-3-i variants is useful, and the minimal singlet-doublet B1 model is economical. The rank-two mass matrix M_nu = Lambda(Y y^T + y Y^T) is derived transparently, and the resulting prediction of one massless neutrino is structural, not a fitting artifact. The phenomenological scan is broad and honestly reported: both neutrino mass orderings, both DM candidates, and the main constraints (oscillations, cLFV, EWPO, h->gamma gamma, relic density, direct detection) are covered, with benchmark points given. The paper earns its place in the T4-3-i lineage.

The soft spots are real but not fatal. The biggest one is the novelty claim. The paper says Ref. [18] 'reduces to this topology after U(1)_B-L breaking' and then asserts that no fully symmetry-invariant realization has been constructed. That is the load-bearing sentence, and it is doing too much work. A spontaneously broken gauge symmetry is still a complete renormalizable field-theoretic realization; if the low-energy content of Wang--Han contains the same one-loop diagram and no dominant tree-level mass, the 'first complete' claim collapses. The authors need to explain precisely why that model does not qualify. I suspect they can defend it, perhaps by showing that Ref. [18] contains additional tree-level terms or lacks the exact stabilizing symmetry, but as written the comparison is one sentence and it has to be a paragraph.

Second, no code or data are shipped. The scans use 10^6 proposed points in roughly fifteen dimensions and accept only 157--1641 points. That is not damning, given how many hard constraints are applied, but without code or data the reader cannot assess scan coverage, and the small accepted samples make the viability claim more provisional than the prose suggests. A reproducibility appendix or public code would materially improve the paper.

Third, the finiteness of the T4-3-i loop is inherited from the topology classification of Ref. [1] rather than demonstrated for this specific field content. This is probably fine, but a one-line finiteness check or an explicit reference to the calculation would close the gap. The citation pattern is otherwise solid; the relevant one-loop classification literature is there, and the authors engage with prior scotogenic and T4-2-i work.

Who is this for: practitioners in radiative seesaw and DM model building. It is a serious candidate for the topology list and will be cited. My recommendation: send it to peer review. The referee should push on the Ref. [18] comparison, ask for code or a much more detailed scan description, and the authors should fix those points in revision. This is a revise-and-resubmit, not a desk-reject.

Referee Report

4 major / 4 minor

Summary. This paper proposes a new radiative neutrino mass model based on the one-loop T4-3-i topology. The construction promotes the mediator fermion to a Dirac singlet N, places lepton-number violation in a separate Majorana singlet psi, and imposes an exact Z2 symmetry under which the new scalars and psi are odd. The resulting minimal model T4-3-i-B1 contains one Dirac fermion, one Majorana fermion, an inert doublet, and an inert singlet, and yields a neutrino mass matrix of the form Lambda(Y y^T + y Y^T), which has rank two and hence predicts one massless neutrino. The authors classify the allowed charge assignments, derive charged-lepton flavor violation, oblique parameters, Higgs diphoton, relic density, and direct detection observables, and perform numerical scans over the parameter space for fermionic and scalar dark matter in both normal and inverted neutrino mass orderings.

Significance. If the construction is correct and genuinely novel, the paper would provide the first complete symmetry-invariant realization of the finite T4-3-i topology, with a robust structural prediction of one massless neutrino and a broad phenomenological survey connecting neutrino mass, dark matter, and cLFV. The derivation of the rank-two mass matrix is clean, and the paper includes a number of machine-verifiable analytic formulas and a detailed set of constraints. However, the novelty claim relative to Ref. [18] is not convincingly argued, and the numerical evidence for global model viability rests on very sparse accepted samples. The paper is therefore of interest to the radiative neutrino mass community, but the central claims require further support before publication.

major comments (4)

[Sec. I, Ref. [18]] The claim that this is the first complete field-theoretic realization of T4-3-i is not supported by the discussion of Ref. [18]. The paper states that the radiative linear-seesaw model with U(1)_B-L 'reduces to this topology after U(1)_B-L breaking,' but does not explain why a spontaneously broken Abelian symmetry is not a complete field-theoretic realization. If the low-energy content of Ref. [18] contains the same one-loop diagram without a tree-level Weinberg term, the central novelty claim collapses. A direct Lagrangian-level comparison of the two models, including the fate of the tree-level seesaw after symmetry breaking, is required.
[Sec. III.A] The paper relies on the topology classification of Ref. [1] for the finiteness of the T4-3-i diagram, but does not demonstrate for the specific field content of T4-3-i-B1 that no Weinberg-operator counterterm is generated at one loop. Since the entire radiative mechanism depends on the loop being the leading source of neutrino mass, the authors should provide an explicit power-counting or counterterm analysis, or state clearly that the absence of a counterterm follows directly from the classification and verify that the specific couplings and quantum numbers do not alter this conclusion.
[Sec. VI] The numerical evidence for model viability is based on very small accepted samples: 157, 156, 1641, and 1636 points out of 10^6 proposed points per scan in a parameter space with roughly 15-20 free parameters. No convergence diagnostics, coverage tests, or independent scan repetitions are provided, and no code or data are released. This is insufficient to establish that the reported best-fit points are global or representative. The authors should provide acceptance rates, a discussion of the effective number of independent parameters, and either multiple independent scans or a Markov-chain-based exploration to confirm the global character of the claimed best-fit regions.
[Sec. VI and Table VII] The reported total chi-squared values (3.65, 4.69, 4.55, 3.80) are surprisingly small given the number of Gaussian observables included in the likelihood, such as the Higgs diphoton rate, oblique parameters, and relic abundance. The small values suggest either a high degree of fine-tuning or that the quoted chi-squared is not the full negative log-likelihood over all included observables. The authors should specify the number of effective data points, the degrees of freedom, and how the hard cuts and Gaussian terms are combined in the reported chi-squared.

minor comments (4)

[Table III] The free-parameter table lists lambda_2, lambda_3 and kappa_1 twice, and also repeats Rey_alpha, Imy_alpha; the duplicate rows should be removed or merged into a single set of ranges for each scan type.
[Sec. II.A] In the classification of models, the text lists T4-3-i-D3 with alpha=-3, but later states that 'the T4-3-i-D model with alpha=+3 is not included in this list' without having mentioned alpha=+3. Please clarify the quantum numbers and electric charges of the alpha=+3 case.
[Eq. (V.63)] The loop-induced effective Higgs coupling for fermionic dark matter is given only for M_psi < m_phi^pm, but the numerical scan may explore points with M_psi > m_phi^pm. The domain of validity and the appropriate analytic continuation of the logarithm should be specified.
[References] Several references are dated 2026 and may still be preprints; if the manuscript is intended for a journal, the authors should verify that all cited works have been accepted or are otherwise stable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rank-two mass structure and one-loop observables follow from the stated Lagrangian, and the numerical fit uses external data as constraints rather than renaming fitted inputs as predictions.

full rationale

The central derivation is self-contained. Eq. (III.29) gives M_nu = Lambda(Y y^T + y Y^T) directly from the Yukawa and mass terms of Eq. (III.26), so rank(M_nu) <= 2 and one massless neutrino is a mathematical consequence of the field content, not an input fitted to neutrino data. The oscillation parameters are used as likelihood inputs to constrain the complex Yukawa couplings; the resulting m_beta, m_betabeta, and sum m_i are then computed from the diagonalized matrix and are not themselves fitted, which is standard parameter estimation rather than circular "prediction". cLFV rates, h->gamma gamma, oblique parameters, relic density, and direct-detection rates are evaluated from the same fitted couplings and masses but are imposed as independent hard constraints or separate likelihood terms; no observable is used both to determine and to validate itself. The one-loop finiteness of the T4-3-i topology is imported from Ref. [1], an external classification, and is a stated premise rather than a circular derivation. The novelty claim that no fully symmetry-invariant realization existed before this work is an unsupported completeness assertion, especially relative to Ref. [18], but lack of proof of novelty is not logical circularity. Self-citations (Refs. [17], [78-80]) are present but merely support collider-phenomenology and previous T4-2-i constructions; they are not load-bearing for the neutrino-mass or dark-matter claims. Therefore no circular step is exhibited.

Assumptions & free parameters 6 free parameters · 5 assumptions · 3 invented entities

The model introduces a small number of new fields (one Dirac fermion, one Majorana fermion, two inert scalars) and many continuous parameters, including Yukawa couplings and scalar potential coefficients. The neutrino mass mechanism and dark matter candidates depend on these parameters, which are scanned and fitted to data. The paper does not provide a parameter-free prediction that could be checked independently, aside from the structural rank-two mass matrix and the resulting massless neutrino.

free parameters (6)

Yukawa vector Y_alpha (alpha = e, mu, tau) = complex; benchmarks in Table VI
Complex Yukawa couplings fitted to neutrino oscillation data and cLFV constraints.
Yukawa vector y_alpha (alpha = e, mu, tau) = complex; benchmarks in Table VI
Complex Yukawa couplings entering the one-loop mass matrix and cLFV.
y'_11 = 5.36e-6 (psiNO best fit)
Coupling between psi, N, and phi', sets the size of neutrino mass.
Scalar masses and mixings (m_H1, m_H2, m_A1, m_A2, m_phi+-, theta_H) = see Table V
Physical scalar spectrum inputs scanned to fit relic density, direct detection, and electroweak constraints.
Quartic couplings kappa1, kappa4, kappa5, lambda2, lambda3 = see Table V
Quartic scalar couplings scanned; kappa1 controls h to gamma gamma and the h psi psi loop.
Dirac fermion mass M_N = 1169 GeV (psiNO best fit)
Mass of the Dirac mediator, scanned as a free parameter.

assumptions (5)

domain assumption The T4-3-i one-loop diagram is finite and requires no Weinberg-operator counterterm.
Inherited from the topology classification in Ref. [1]; this makes the loop the leading source of neutrino mass.
domain assumption Lepton number is conserved by all couplings except the Majorana mass term of psi.
Assumed to forbid the tree-level type-I seesaw and to localize lepton-number violation inside the loop.
domain assumption The universe underwent standard thermal freeze-out in a radiation-dominated cosmology.
Used in the relic density calculation; non-thermal histories would change the viable parameter space.
domain assumption All scalar potential parameters are real, so there is no explicit CP violation in the scalar sector.
Assumed in Sec. II.B to simplify the scalar mass matrices and mixing angles.
standard math The standard three-flavor neutrino oscillation framework with unitarity of the PMNS matrix.
Used to diagonalize the rank-two mass matrix and to compare with NuFIT oscillation profiles.

invented entities (3)

Dirac fermion N, an SU(2)L singlet independent evidence
purpose: Mediator connecting the external lepton-Higgs pairs; its Dirac nature removes the tree-level type-I seesaw.
Has a TeV-scale mass and Yukawa couplings; would appear in lepton flavor violation and collider searches, although its collider phenomenology is not studied here.
Majorana fermion psi, an SM singlet independent evidence
purpose: Provides lepton-number violation inside the loop and can be a dark matter candidate.
Can be fermionic dark matter; its loop-induced Higgs coupling suppresses direct detection, but its Yukawa couplings are constrained by cLFV.
Inert scalar doublet phi and inert scalar singlet phi' independent evidence
purpose: Run in the loop and provide a scalar dark matter candidate, H_1^0.
The lightest neutral scalar can be dark matter with a tree-level Higgs portal; mass splittings are constrained by electroweak precision and collider searches.

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Cite this review

Pith. "Pith review of A Novel One-loop Model for Majorana Neutrino Mass and Dark Matter." pith.science (2026). https://pith.science/paper/M5A5YXNF

@misc{pith2026260812646,
  author       = {Pith},
  title        = {Pith review of: A Novel One-loop Model for Majorana Neutrino Mass and Dark Matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5A5YXNF}},
  note         = {Machine review of arXiv:2608.12646}
}

abstract

We present the first complete field-theoretic realization of the finite one-loop T4-3-i topology for Majorana neutrino mass. Here, T4-3-i denotes a one-loop realization of the Weinberg operator in which a fermion links the two external lepton--Higgs pairs. If this fermion is a Majorana singlet or triplet, the same interactions generate a tree-level type-I or type-III seesaw contribution, respectively, so that the loop is not the leading source of neutrino mass. This lower-order contribution is removed by taking the mediator to be a Dirac fermion $N$, placing lepton-number violation in a separate Majorana fermion $\psi$ inside the loop, and imposing an exact $Z_2$ symmetry that keeps the new scalars inert and stabilizes the lightest odd state. We classify the allowed electroweak charge assignments and study the minimal singlet-doublet realization, denoted T4-3-i-B1, which contains one Dirac fermion, one Majorana fermion, an inert scalar doublet, and an inert scalar singlet. The resulting rank-two neutrino mass matrix predicts one massless neutrino. We confront both normal and inverted neutrino-mass orderings with neutrino-oscillation and cosmological data, charged-lepton flavor violation, including $\mu-e$ conversion, electroweak precision observables, $h\to\gamma\gamma$, theoretical consistency conditions, the relic abundance, and direct-detection limits. Both fermionic and scalar dark matter are viable. The fermionic candidate has only a loop-induced Higgs coupling and consequently a strongly suppressed spin-independent scattering rate, whereas the scalar candidate couples through a tree-level Higgs portal and can lie above the neutrino floor while remaining compatible with current limits. In both cases, coannihilation with inert scalars is essential for reproducing the observed relic abundance.

Figures

Figures reproduced from arXiv: 2608.12646 by the authors.

**Figure 1.** FIG. 1. Left: Topology T4-3-i accompanied by the tree level type-I or type-III seesaw contributions when Ψ is a Majorana [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** FIG. 2. Results for fermionic DM with NO. Left panel: relic abundance as a function of [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** FIG. 3. Neutrino-sector predictions for fermionic DM with NO. Top-left panel: solar projection in the (∆ [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: FIG. 4. cLFV predictions for fermionic DM with NO. Left panel: radiative decays [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: FIG. 5. Same as Fig. 2 but for IO [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

**Figure 6.** Figure 6: FIG. 6. Same as Fig. 3 but for IO [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: FIG. 7. Same as Fig. 4 but for IO [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

**Figure 8.** Figure 8: FIG. 8. Scalar DM with NO. Left panel: relic density as a function of [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: FIG. 9. Neutrino sector predictions for scalar DM with NO. The panel order, color code and external references are the same [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

**Figure 10.** Figure 10: FIG. 10. cLFV predictions for scalar DM with NO. The panel order, color convention and experimental lines are the same as [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

**Figure 11.** Figure 11: FIG. 11. Same as Fig. 8 but for IO [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]

**Figure 12.** Figure 12: FIG. 12. Same as Fig. 9 but for IO [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]

**Figure 13.** Figure 13: FIG. 13. Same as Fig. 10 but for IO [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]

**Figure 14.** Figure 14: FIG. 14. Higgs diphoton constraint in the four scans. Top-left panel: fermionic DM with NO. Top-right panel: fermionic DM [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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Un- like the fermionic case, its coupling to the Higgs is already present at the renormalizable level. The diagonal trilinear interaction controlling elastic Higgs-mediated scattering is ghH 0 1H 0 1 =− √ 2c HsHµϕ−υ c2 H(κ1 +κ 2 +κ 3)−υ s 2 Hκ4.(V.66) The scalar potential also contains the verticeshH 0 2H0 2 andhH 0 2H0

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