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arxiv: 2606.26752 · v2 · 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 2026-07-01 07:15 UTC · model grok-4.3

classification ✦ hep-ph
keywords scotogenic modelneutrino massesdark mattervacuum stabilityelectroweak precision observableslepton flavor violationoblique parametersZ invisible decay
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The pith

A numerical scan of the minimally extended scotogenic model identifies viable fermionic dark matter masses between 120 and 350 GeV while the DESI BAO bound excludes the inverted neutrino hierarchy.

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

The paper conducts a global analysis of a minimally extended scotogenic model to determine whether it can simultaneously account for nonzero neutrino masses, furnish a dark matter candidate, and resolve the Standard Model vacuum instability at high scales. Constraints are imposed from bounded-from-below conditions, renormalization-group evolution ensuring perturbativity, and an extensive list of flavor and electroweak observables that includes the muon anomalous magnetic moment, lepton-flavor-violating processes, oblique parameters, and Z and Higgs leptonic decays. The scan produces four concrete results: the inverted neutrino hierarchy is ruled out once the DESI baryon-acoustic-oscillation bound is adopted, future precision measurements can reach the predicted oblique-parameter shifts, the viable fermionic dark-matter mass window is 120-350 GeV while the CP-odd scalar lies between 350 and 600 GeV, and the computed Z to invisible width lies within 3 sigma of both the world average and the recent ATLAS datum.

Core claim

A numerical scan of the minimally extended scotogenic model that incorporates vacuum-stability and perturbativity requirements together with flavor and electroweak precision observables yields viable parameter space in which the fermionic dark-matter candidate has mass 120-350 GeV, the CP-odd scalar has mass 350-600 GeV, the oblique parameters lie within reach of forthcoming measurements, the Z to invisible branching ratio is compatible with the world average at the 3 sigma level and favored by recent ATLAS data at the 3 sigma level, and the inverted neutrino mass hierarchy is excluded if the DESI BAO bound is confirmed.

What carries the argument

The minimally extended scotogenic model, an extension of the Standard Model by additional scalar and fermionic fields that generates neutrino masses radiatively at one loop while supplying a stable dark-matter candidate and addressing vacuum instability.

If this is right

  • The inverted neutrino hierarchy is ruled out once the DESI BAO bound is adopted.
  • Projected values of the oblique parameters fall inside the sensitivity of next-generation precision electroweak measurements.
  • Direct searches for fermionic dark matter should target the 120-350 GeV interval.
  • The CP-odd scalar is restricted to the 350-600 GeV window by the combined constraints.
  • The predicted Z to invisible width remains compatible with existing data at the 3 sigma level.

Where Pith is reading between the lines

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

  • The narrow mass windows narrow the target range for upcoming direct-detection and collider searches for both the dark-matter fermion and the additional scalar.
  • Preference for the normal hierarchy under DESI data aligns the model with independent cosmological indications that already favor normal ordering.
  • If future measurements tighten the oblique-parameter bounds without finding the predicted shifts, the viable parameter space would shrink further.
  • The compatibility with the ATLAS Z invisible result suggests that modest improvements in invisible-width precision could begin to discriminate among scotogenic variants.

Load-bearing premise

The chosen set of flavor, electroweak, vacuum-stability, and perturbativity constraints is assumed to be sufficient to capture all relevant restrictions on the model's parameter space.

What would settle it

A confirmed measurement of the inverted neutrino hierarchy that remains consistent with the DESI BAO bound, or a dark-matter particle whose mass lies outside the 120-350 GeV window while satisfying all other listed observables, would falsify the scan results.

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 $120-350 \operatorname{GeV}$, while the CP-odd scalar is constrained to $350-600 \operatorname{GeV}$; and our result on $Z \rightarrow \operatorname{Invisible}$ 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

0 major / 3 minor

Summary. The manuscript performs a global analysis of a minimally extended scotogenic model, incorporating additional scalars and fermions to generate neutrino masses, provide dark matter candidates, and address SM vacuum instability. Constraints from bounded-from-below conditions, vacuum stability, RG perturbativity, flavor observables (including Δa_μ, radiative decays, μ→e conversion), oblique parameters, and leptonic Z/H decays are applied via a numerical scan over scalar potential couplings, masses, and Yukawa couplings. The scan yields four reported features: DESI BAO data would exclude inverted neutrino hierarchy; oblique parameters lie within future experimental reach; viable fermionic DM mass in 120-350 GeV and CP-odd scalar in 350-600 GeV; and Z→invisible width compatible with world average at 3σ and favored by recent ATLAS data at 3σ.

Significance. If the scan methodology and constraint implementation are robust, the results supply concrete, testable mass windows for DM and scalar states, a potential hierarchy discriminator via BAO, and indications that precision EW observables can probe the model, strengthening the phenomenological case for minimally extended scotogenic scenarios beyond existing literature.

minor comments (3)
  1. The abstract states that a numerical scan was performed but provides no information on methodology, prior ranges, convergence checks, or post-hoc cuts. The full manuscript should include a dedicated subsection (e.g., §3 or §4) detailing the scan procedure, parameter ranges, and validation to allow reproduction of the quoted mass windows and 3σ statements.
  2. The four notable features are presented as direct outputs of the scan; clarify in the results section whether any (e.g., Z invisible width) constitute genuine predictions or are by construction within the fitted observables.
  3. Minor notation inconsistency: the abstract uses “Z → Invisible” while the text likely employs standard Γ(Z→inv); ensure uniform notation throughout.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment, including the recommendation for minor revision. The referee summary accurately captures the scope and main findings of our global analysis. No major comments were raised that require point-by-point rebuttal.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper performs a standard global numerical scan of a scotogenic model extension, imposing external constraints from vacuum stability, perturbativity, flavor observables, EW precision data, and DM requirements. The reported features (mass ranges, hierarchy exclusion under DESI, oblique parameter projections, Z invisible width compatibility) are direct outputs of this scan applied to the chosen observables; none reduce by construction to the inputs via self-definition, renaming, or self-citation chains. The derivation remains self-contained against the listed external benchmarks with no load-bearing internal reduction exhibited.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model extension itself introduces additional scalar fields and couplings whose values are not fixed by the Standard Model. The global analysis then fits or scans over these new parameters subject to theoretical and experimental constraints. No machine-checked proofs or parameter-free derivations are referenced.

free parameters (2)
  • scalar potential couplings and masses
    The minimally extended scalar sector adds multiple new quartic couplings and mass parameters that are scanned numerically to satisfy vacuum stability and perturbativity.
  • Yukawa couplings to new fermions
    These control neutrino mass generation and dark matter interactions and are varied within the scan.
axioms (2)
  • domain assumption The Standard Model vacuum instability is cured by the new scalars without introducing new instabilities at higher scales.
    Invoked when imposing bounded-from-below and RG perturbativity conditions on the extended potential.
  • domain assumption All relevant constraints are captured by the listed flavor, EW precision, and vacuum-stability observables.
    The global analysis treats the chosen set of measurements as exhaustive.

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    Generate an initial point in the parameter space yielding a finite log likelihood value, defined as lnL= X i lnL i =− X i Opred i − Oexp i 2 2σ2 i ,(49) whereO pred i andO exp i denote the predicted and experimental values of thei-th observable, respectively, andσ i is the corresponding 1σuncertainty

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