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arxiv: 2606.26752 · v1 · pith:24Z7HP2Pnew · submitted 2026-06-25 · ✦ hep-ph

Global analysis of a minimally extended scotogenic model

Huchan Lee , Sin Kyu Kang This is my paper

Pith reviewed 2026-06-26 04:22 UTC · model grok-4.3

classification ✦ hep-ph
keywords scotogenic modeldark matterneutrino massesvacuum stabilityoblique parametersZ invisible decayneutrino hierarchyelectroweak precision
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The pith

Global analysis of minimally extended scotogenic model constrains fermionic dark matter to 120-350 GeV and CP-odd scalar to 350-600 GeV.

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

The paper performs a global numerical scan of a minimally extended scotogenic model that generates neutrino masses radiatively, supplies a dark matter candidate, and stabilizes the Standard Model vacuum at high scales. It enforces bounded-from-below conditions, vacuum stability, and perturbativity under renormalization-group flow together with flavor observables, the muon anomalous magnetic moment, oblique parameters, and leptonic Z and Higgs decays. The scan identifies four concrete outcomes: the DESI baryon acoustic oscillation bound would exclude the inverted neutrino hierarchy if confirmed, the oblique parameters lie in a range reachable by future precision measurements, viable fermionic dark matter masses occupy 120-350 GeV while the CP-odd scalar occupies 350-600 GeV, and the predicted Z to invisible width agrees with the world average at 3 sigma and with recent ATLAS data at 3 sigma.

Core claim

A comprehensive numerical scan of the minimally extended scotogenic model yields viable parameter space that simultaneously satisfies all theoretical and experimental constraints, with a fermionic dark matter candidate mass between 120 and 350 GeV, a CP-odd scalar mass between 350 and 600 GeV, a clear preference for the normal neutrino mass hierarchy once the DESI bound is imposed, and a Z to invisible branching ratio compatible with existing measurements at the 3 sigma level.

What carries the argument

The minimally extended scalar sector that restores high-scale vacuum stability while retaining the scotogenic radiative mechanism for neutrino masses and the stability of the dark matter candidate.

If this is right

  • The DESI BAO bound excludes the inverted neutrino hierarchy if upheld by other experiments.
  • Oblique parameters fall inside the projected sensitivity of future precision measurements.
  • Fermionic dark matter masses are restricted to the interval 120-350 GeV.
  • The CP-odd scalar mass is restricted to the interval 350-600 GeV.
  • The Z to invisible decay width remains compatible with the world average and recent ATLAS data at 3 sigma.

Where Pith is reading between the lines

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

  • The reported mass windows for dark matter and the CP-odd scalar point to concrete search channels at current and future colliders.
  • The preference for normal hierarchy under the DESI bound can be cross-checked by independent neutrino oscillation and cosmology experiments.
  • The projected oblique-parameter shifts offer a near-term target for electroweak precision programs at proposed lepton colliders.

Load-bearing premise

The chosen minimal addition to the scalar sector is enough to restore vacuum stability and the numerical scan samples the viable parameter space without large gaps or biases.

What would settle it

Independent experimental confirmation that the neutrino mass hierarchy is inverted would exclude all viable regions found by the scan.

Figures

Figures reproduced from arXiv: 2606.26752 by Huchan Lee, Sin Kyu Kang.

Figure 1
Figure 1. Figure 1: Neutrino mass generation diagram at the one-loop level. Here, [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: RG evolution of the SM (upper panels) and the scotogenic model under study (lower [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Diagram contributing to the muon anomalous magnetic moment and the charged lepton [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: One-loop Feynman diagrams for the Z → ℓαℓβ vertex corrections within this BSM model. Here, the symbol ℓ denotes the SM charged leptons (e, µ, τ ), while νi represent the massless SM Dirac neutrinos. hm and Am are the CP-even and -odd scalars, respectively. The indices α, β, i serve as generation indices ranging from 1 to 3, whereas the indices m, n label the generations of the new scalars from 1 to 2. Prel… view at source ↗
Figure 5
Figure 5. Figure 5: Flavor-violating Z → ℓαℓβ CT topologies. Here, η + is the NP charged scalar and χk (k = 1, 2, 3) is the RH neutrinos. follows: Aexp = ASM + ANP (34) If a BSM model does not extend the SM gauge symmetry by additional gauge symmetries, and does not modify the vacuum structure of the SM scalar potential, it is sufficient to identify the NP amplitude ANP solely with the contributions from the new particles [49… view at source ↗
Figure 6
Figure 6. Figure 6: One-loop Feynman diagrams contributing to the [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: CT topologies for the flavor-violating Z → νανβ process. Analogous to the vertex correction diagrams in [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: One-loop Feynman diagrams contributing to the [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The SM contributions to h → ℓαℓα are obtained in analogy with the leptonic Z decay by decoupling the χ-related fields and taking the appropriate limits of the extended scalar sector to recover the SM scalar sector. The one-loop amplitude for the H → ℓαℓβ can be decomposed in terms of the form factors: A [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: One-loop Feynman diagrams for the h1 → ℓαℓβ vertex corrections within this BSM model. Here, the symbol ℓ denotes the SM charged leptons (e, µ, τ ), while νi represent the massless SM Dirac neutrinos. hm and Am are the CP-even and -odd scalars, respectively. The indices α, β, i serve as generation indices ranging from 1 to 3, whereas the indices m, n label the generations of the new scalars from 1 to 2. Equ… view at source ↗
Figure 10
Figure 10. Figure 10: One-loop Feynman diagrams contributing to the [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Scanned DM mass versus relic density (left panel) and spin-independent proton cross [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Scanned results of the oblique parameters. The left panel takes the absolute values of [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Scanned results of the lepton flavor universality. The dashed line with ”SM” [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Scanned results of H → Invisible versus Z → Invisible. In both panels, the dashed line denotes the SM prediction for the Z → Invisible branching ratio given in Table V. The left panel is based on the world avergae experimental bound, whereas the right panel is based on the recent ATLAS experimental bound [90] . VI. CONCLUSION The SM has explained many phenomena with remarkable precision, yet several key o… view at source ↗
Figure 15
Figure 15. Figure 15: Feynman diagrams contributing to the photon SE in the scotogenic model. For the [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Feynman diagrams contributing to the photon SE in the scotogenic model. For the [PITH_FULL_IMAGE:figures/full_fig_p032_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Feynman diagrams contributing to the Z boson tadpole in the scotogenic model, where i, j = 1, 2. In the SM limit, hi and ai reduce to the SM Higgs boson h and the would-be Goldstone boson a of the Z boson, respectively, while the contributions involving the χ, ηR,I and H+ fields are absent. Z Z hi , ai , ηR,I Z Z G+, H+ Z Z W+ Z Z νi νi Z Z ui , di , ei ui , di , ei Z Z aj (ηI ) hi (ηR) Z Z G+, H+ G+, H+ … view at source ↗
Figure 18
Figure 18. Figure 18: Feynman diagrams contributing to the Z boson tadpole in the scotogenic model, where i, j = 1, 2. In the SM limit, hi and ai reduce to the SM Higgs boson h and the would-be Goldstone boson a of the Z boson, respectively, while the contributions involving the χ and H+ fields are absent. × (MηR − MηI + MZ)(MηR + MηI + MZ)B0  M2 Z, M2 ηR , M2 ηI  − 1 64c 2 w(−1 + D)π 2s 2 w  4MH+ − M2 Z e − 2es2 w 2 B0 … view at source ↗
Figure 19
Figure 19. Figure 19: Feynman diagrams contributing to the Z boson tadpole in the scotogenic model, where i, j = 1, 2. In the SM limit, hi and ai reduce to the SM Higgs boson h and the would-be Goldstone boson a of the Z boson, respectively, while the contributions involving the χ, ηR,I and H+ fields are absent. − 2DB0  M2 Z, m2 uk , m2 uk (−2 + D)M2 Z + 4m2 uk  − 4M2 Zs 4 w  9(−23 + 19D)M2 Z × DB0  M2 Z, M2 W , M2 W … view at source ↗
Figure 20
Figure 20. Figure 20: Feynman diagrams contributing to the Z boson tadpole in the scotogenic model, where i, j = 1, 2. In the SM limit, hi and ai reduce to the SM Higgs boson h and the would-be Goldstone boson a of the Z boson, respectively, while the contributions involving the χ, ηR,I and H+ fields are absent. × B0  M2 W , M2 W , M2 Z  + X 3 k=1 1 32c 2 w(−1 + D)M2 Z π 2s 2 w e 2B0  M2 W , 0, m2 ek M2 W − m2 ek (−2 + … view at source ↗
Figure 21
Figure 21. Figure 21: Feynman diagrams contributing to the scalar tadpole in the scotogenic model, where [PITH_FULL_IMAGE:figures/full_fig_p043_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Feynman diagrams contributing to the scalar SE in the scotogenic model, where [PITH_FULL_IMAGE:figures/full_fig_p043_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Feynman diagrams contributing to the charged lepton tadpole in the scotogenic [PITH_FULL_IMAGE:figures/full_fig_p044_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Following [75], the charged lepton propagator - with incoming and outgoing leptons [PITH_FULL_IMAGE:figures/full_fig_p044_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Feynman diagrams contributing to the neutrino SE in the scotogenic model, where [PITH_FULL_IMAGE:figures/full_fig_p045_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Feynman diagrams contributing to the neutrino SE in the SM, where [PITH_FULL_IMAGE:figures/full_fig_p045_26.png] view at source ↗
read the original abstract

We perform a global analysis of a minimally extended scotogenic model motivated by observed non-zero neutrino masses, viable dark matter (DM) candidates, and the instability of the Standard Model (SM) vacuum at high-energies. We examine the bounded-from-below conditions, vacuum stability, and RG-driven perturbativity bounds arising from the extended scalar sector, alongside a comprehensive set of flavor and electroweak (EW) precision observables - including the muon anomalous magnetic moment $\Delta a_{\mu}$, the radiative decays $\ell_{\alpha} \rightarrow \ell_{\beta} \gamma$ and $\ell_{\alpha} \rightarrow 3\ell_{\beta}$, and the $\mu \rightarrow e$ conversion rate, the oblique parameters, and leptonic decays of $Z$ and $H$ bosons. A numerical scan reveals four notable features: the DESI BAO bound would rule out the inverted hierarchy if confirmed by other experiments; the oblique parameters are projected to be within the reach of future precision measurements; the viable fermionic DM candidate mass lies in the range $120-350 \func{GeV}$, while the CP-odd scalar is constrained to $350-600 \func{GeV}$; and our result on $Z \rightarrow \func{Invisible}$ is compatible with the world average at the $3\sigma$ level and is favored by the recent ATLAS measurement at the $3\sigma$ level.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript performs a global analysis of a minimally extended scotogenic model to simultaneously address non-zero neutrino masses, viable dark matter candidates, and SM vacuum instability. It incorporates bounded-from-below conditions, vacuum stability, RG-driven perturbativity bounds from the extended scalar sector, and a broad set of flavor and electroweak precision observables (including Δa_μ, ℓ_α → ℓ_β γ, ℓ_α → 3ℓ_β, μ → e conversion, oblique parameters, and leptonic Z/H decays). A numerical scan over the parameter space yields four main results: the DESI BAO bound would rule out the inverted neutrino hierarchy if confirmed; oblique parameters lie within reach of future precision measurements; viable fermionic DM masses are 120-350 GeV and the CP-odd scalar is 350-600 GeV; and the Z → invisible width is compatible with the world average at 3σ and favored by recent ATLAS data at 3σ.

Significance. If the numerical scan is shown to be robust and unbiased, the work supplies concrete, testable constraints on an extension of the scotogenic framework that unifies several BSM motivations. The reported mass windows for the fermionic DM and CP-odd scalar, together with the hierarchy implication from DESI, constitute falsifiable predictions for upcoming experiments. The breadth of included observables (flavor, precision EW, and stability) is a methodological strength.

major comments (2)
  1. [numerical scan results] The section presenting the numerical scan results does not specify the sampling algorithm, prior ranges on the Yukawa couplings and scalar masses, treatment of parameter correlations, or convergence criteria used to generate the quoted mass ranges (120-350 GeV for fermionic DM and 350-600 GeV for the CP-odd scalar) and hierarchy statements. These omissions are load-bearing because all four notable features are direct outputs of the scan rather than independent predictions.
  2. [model definition and constraints] The claim that the chosen minimal scalar extension restores vacuum stability and perturbativity while preserving all other model features lacks explicit verification of the RG evolution and bounded-from-below conditions for the added parameters; incomplete enforcement could introduce bias into the viable regions sampled for the neutrino hierarchy and DM mass ranges.
minor comments (1)
  1. [abstract] The abstract states compatibility of Z → invisible 'at the 3σ level' with both the world average and ATLAS but does not quote the numerical prediction or the exact pull values.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and agree that additional methodological details and explicit verifications will strengthen the presentation of our results.

read point-by-point responses

  1. Referee: The section presenting the numerical scan results does not specify the sampling algorithm, prior ranges on the Yukawa couplings and scalar masses, treatment of parameter correlations, or convergence criteria used to generate the quoted mass ranges (120-350 GeV for fermionic DM and 350-600 GeV for the CP-odd scalar) and hierarchy statements. These omissions are load-bearing because all four notable features are direct outputs of the scan rather than independent predictions.

    Authors: We agree that a more detailed description of the scan is required. In the revised manuscript we will insert a new subsection (likely Section 4.1) that specifies the sampling algorithm employed, the prior ranges adopted for the Yukawa couplings and scalar masses, the treatment of parameter correlations, and the convergence criteria used to obtain the reported mass windows and hierarchy statements. This addition will make the robustness of the quoted ranges fully transparent. revision: yes

  2. Referee: The claim that the chosen minimal scalar extension restores vacuum stability and perturbativity while preserving all other model features lacks explicit verification of the RG evolution and bounded-from-below conditions for the added parameters; incomplete enforcement could introduce bias into the viable regions sampled for the neutrino hierarchy and DM mass ranges.

    Authors: We acknowledge that the manuscript would benefit from more explicit documentation of the RG evolution and bounded-from-below conditions for the newly introduced scalar parameters. In the revision we will add a dedicated paragraph (or short appendix) presenting the relevant RG trajectories and confirming that the bounded-from-below conditions remain satisfied throughout the scanned parameter space, thereby removing any potential ambiguity regarding bias in the viable regions. revision: yes

Circularity Check

0 steps flagged

Numerical scan outputs are direct constraint results with no circular reduction

full rationale

The paper conducts a global numerical scan over the parameter space of the minimally extended scotogenic model, enforcing bounded-from-below conditions, vacuum stability, RG perturbativity, and experimental observables including oblique parameters, flavor processes, and Z/H decays. The four notable features (DESI impact on hierarchy, future reach of oblique parameters, viable DM and scalar mass windows, and Z invisible compatibility) are explicitly the surviving regions after these constraints are applied. No step claims a first-principles derivation that reduces by construction to the scan inputs, no self-citation is load-bearing for any uniqueness theorem, and no fitted quantity is relabeled as an independent prediction. The chain is a standard computational exploration of viable parameter space and remains self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claims rest on the standard scotogenic radiative mechanism plus an unspecified minimal scalar extension, with a large number of free parameters scanned numerically against data and theoretical bounds.

free parameters (1)
  • Yukawa couplings and scalar masses in the extension
    Scanned over to satisfy all constraints and produce the reported mass ranges
axioms (2)
  • domain assumption Radiative neutrino mass generation via scotogenic loops remains valid after minimal extension
    Core assumption of the model class invoked throughout the abstract
  • ad hoc to paper Numerical scan covers all viable regions without missing solutions
    Implicit in reporting specific mass intervals from the scan
invented entities (1)
  • Minimally extended scalar sector no independent evidence
    purpose: Restore vacuum stability at high scales
    New scalars added to address SM instability; no independent evidence provided beyond the scan

pith-pipeline@v0.9.1-grok · 5777 in / 1472 out tokens · 51549 ms · 2026-06-26T04:22:04.442279+00:00 · methodology

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Reference graph

Works this paper leans on

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