Minimal Scotogenic Model: Neutrinos & Dark Matter

• The minimal scotogenic model is a radiative neutrino-mass framework that extends the Standard Model by adding an inert scalar doublet and Z2-odd Majorana singlets, enabling one-loop mass generation.
• It offers two formulations—the original three-singlet version and a reduced two-singlet variant—where the Z2 symmetry forbids tree-level neutrino masses while stabilizing a dark matter candidate.
• The model’s phenomenology spans neutrino oscillations, dark matter relic abundance, electric dipole moments, leptogenesis, and renormalization-group evolution, with implications for both collider tests and cosmology.

Searching arXiv for core and papers on the minimal scotogenic model, including the supplied EDM paper and closely related variants. arxiv_search(query="minimal scotogenic model neutrino masses dark matter electric dipole moments leptogenesis review", max_results=10, sort_by="relevance") Gathering a focused set of arXiv references on the minimal scotogenic model and its main phenomenological directions. Looking up arXiv records relevant to the minimal scotogenic model. The minimal scotogenic model is a radiative neutrino-mass framework in which the Standard Model is supplemented by an inert scalar doublet and $\mathbb{Z}_2$-odd Majorana singlet fermions, with the exact discrete symmetry simultaneously forbidding tree-level neutrino masses and stabilizing a dark-matter candidate. In the literature, the expression “minimal scotogenic model” is used in two closely related senses: the original Ma setup with three singlet fermions and one inert doublet, and a reduced two-singlet version that still reproduces oscillation data but leaves one active neutrino massless [(Bouchand et al., 2012); (Escribano, 2021); (Abada et al., 2018)]. Across both usages, the defining structure is the same: neutrino mass is generated at one loop by dark-sector fields, and the lightest $\mathbb{Z}_2$-odd state is stable.

1. Terminology and field-content conventions

A recurrent source of ambiguity is that “minimal” does not denote a unique fermion multiplicity. In papers on renormalization-group evolution and generalized field multiplicities, the minimal scotogenic model is identified with Ma’s original $(n_N,n_\eta)=(3,1)$ construction: three Majorana singlets $N_k\sim(1,1,0)$, one inert scalar doublet $\eta\sim(1,2,1/2)$, and an exact $\mathbb{Z}_2$ under which all new fields are odd and all Standard Model fields are even [(Bouchand et al., 2012); (Escribano, 2021)]. In papers emphasizing electric dipole moments or reduced-parameter leptogenesis, the same label is also applied to a two-singlet reduction, again with one inert doublet and the same $\mathbb{Z}_2$, but with only two $N_i$ fields (Abada et al., 2018, Hugle et al., 2018).

Usage in the literature	New singlet fermions	Immediate consequence
Original Ma setup	3	Full-rank light-neutrino mass matrix is possible
Reduced “minimal version”	2	One active neutrino is massless

In both cases, the $\mathbb{Z}_2$ symmetry has two central consequences. First, the standard Yukawa term $\overline{L}\tilde H N$ is forbidden, so there is no tree-level type-I seesaw mass. Second, the inert doublet does not acquire a vacuum expectation value, $\mathbb{Z}_2$0, and the lightest $\mathbb{Z}_2$1-odd particle is stable [(Bouchand et al., 2012); (Abada et al., 2018)].

2. Lagrangian structure and scalar sector

The characteristic Yukawa interaction is of the form

$\mathbb{Z}_2$2

or, in the two-singlet notation,

$\mathbb{Z}_2$3

with $\mathbb{Z}_2$4 and $\mathbb{Z}_2$5 [(Bouchand et al., 2012); (Abada et al., 2018)]. The scalar potential is the inert-doublet one,

$\mathbb{Z}_2$6

or equivalently with $\mathbb{Z}_2$7 replacing $\mathbb{Z}_2$8 [(Bouchand et al., 2012); (Abada et al., 2018)].

The neutral inert component is decomposed as

$\mathbb{Z}_2$9

and electroweak symmetry breaking induces the inert spectrum

$(n_N,n_\eta)=(3,1)$0

together with the charged-scalar mass

$(n_N,n_\eta)=(3,1)$1

in the notation of the RGE analysis (Bouchand et al., 2012). In the two-singlet formulation the same mass splitting is written as

$(n_N,n_\eta)=(3,1)$2

with the same physical content: $(n_N,n_\eta)=(3,1)$3 controls the CP-even/CP-odd inert splitting (Abada et al., 2018).

The central structural role of $(n_N,n_\eta)=(3,1)$4 is common to all formulations. It controls the neutral-scalar splitting, enters linearly in the loop-induced neutrino mass, and parametrizes lepton-number violation. In the limit $(n_N,n_\eta)=(3,1)$5, $(n_N,n_\eta)=(3,1)$6, neutrino masses vanish, and lepton number is restored; this is the standard ’t Hooft naturalness argument for small $(n_N,n_\eta)=(3,1)$7 [(Bouchand et al., 2012); (Abada et al., 2018)].

3. One-loop neutrino-mass mechanism

Neutrino mass is generated by the one-loop diagram with $(n_N,n_\eta)=(3,1)$8 and $(n_N,n_\eta)=(3,1)$9 running in the loop. In the three-singlet Ma model, the mass matrix is

$N_k\sim(1,1,0)$0

with the sum over $N_k\sim(1,1,0)$1 implied (Bouchand et al., 2012). In the two-singlet version the same structure appears as

$N_k\sim(1,1,0)$2

and reduces, in the small-splitting limit, to a form explicitly proportional to $N_k\sim(1,1,0)$3 (Abada et al., 2018).

This loop-induced mass can be written compactly as $N_k\sim(1,1,0)$4 or $N_k\sim(1,1,0)$5, depending on convention, and is routinely matched to oscillation data with a Casas–Ibarra parametrization adapted to the scotogenic loop structure (Abada et al., 2018, Borah et al., 2020). The rank of the light-neutrino mass matrix is bounded by the number of singlets. With two singlets, one light neutrino is massless; with three singlets, full rank is possible (Abada et al., 2018, Escribano, 2021).

A notable technical consequence is that the model admits TeV-to-tens-of-TeV singlet masses with Yukawas of order $N_k\sim(1,1,0)$6–$N_k\sim(1,1,0)$7 and tiny $N_k\sim(1,1,0)$8, while still reproducing realistic neutrino masses (Bouchand et al., 2012). This feature underlies much of the model’s phenomenology: the Yukawas can be large enough to matter for flavor, dark matter, and CP violation, yet the neutrino mass remains radiatively suppressed.

4. Dark matter sectors and mass regimes

Because the exact $N_k\sim(1,1,0)$9 stabilizes the lightest odd state, the minimal scotogenic model admits both scalar and fermionic dark matter. In the original three-singlet formulation, viable candidates are a neutral inert component of $\eta\sim(1,2,1/2)$0 or the lightest singlet fermion $\eta\sim(1,2,1/2)$1 [(Bouchand et al., 2012); (Escribano, 2021)]. The two-singlet EDM analysis likewise identifies two possibilities: fermionic dark matter if $\eta\sim(1,2,1/2)$2 is lightest, and scalar dark matter if the lighter neutral inert state is lightest (Abada et al., 2018).

For scalar dark matter, two standard mass windows recur. One is a low-mass region below the $\eta\sim(1,2,1/2)$3 threshold and near the Higgs resonance. The other is a high-mass inert-doublet regime beginning around $\eta\sim(1,2,1/2)$4–$\eta\sim(1,2,1/2)$5 and extending to multi-TeV masses (Abada et al., 2018, Borah et al., 2020). In the intermediate interval $\eta\sim(1,2,1/2)$6, the relic density is often underabundant because annihilation into gauge bosons and co-annihilation with $\eta\sim(1,2,1/2)$7 and $\eta\sim(1,2,1/2)$8 are too efficient; this is the inert-doublet “desert” emphasized in scalar-DM studies (Bernardini et al., 2020).

For fermionic dark matter, the relic abundance is controlled primarily by Yukawa-mediated annihilation and co-annihilation. In the EDM study, obtaining the observed relic abundance for $\eta\sim(1,2,1/2)$9 above the electroweak scale typically requires $\mathbb{Z}_2$0, while direct detection remains negligible because the singlet fermion couples only to leptons at tree level (Abada et al., 2018). The same basic distinction reappears in phase-transition studies: scalar dark matter is strongly constrained by XENON1T when a strong first-order electroweak transition is imposed, whereas the leptophilic fermion-dark-matter scenario remains largely unconstrained by direct detection (Borah et al., 2020).

The dark sector also exhibits a cosmological bifurcation between thermal and non-thermal production. In the fermion-dark-matter $\mathbb{Z}_2$1-leptogenesis analysis, the lightest singlet $\mathbb{Z}_2$2 can be either a thermal WIMP, assisted by co-annihilation with the inert doublet, or a freeze-in relic when its Yukawa couplings are sufficiently small (Mahanta et al., 2019). Scalar dark matter can likewise receive a non-thermal contribution from late decays of $\mathbb{Z}_2$3, opening otherwise underabundant regions of the inert-doublet desert (Bernardini et al., 2020).

5. Flavor, CP violation, EDMs, and baryogenesis

The same Yukawa sector that generates neutrino mass also drives charged-lepton flavor violation, CP violation, and leptogenesis. In the two-singlet minimal model, charged-lepton electric dipole moments vanish at one loop and first arise at two loops. The general charged-lepton EDM takes the form

$\mathbb{Z}_2$4

with antisymmetric CP invariants $\mathbb{Z}_2$5 constructed from the Yukawas; in this model the Dirac-type loop function $\mathbb{Z}_2$6 vanishes identically, so only the Majorana-type contribution survives (Abada et al., 2018). The resulting phenomenology is sharply asymmetric between dark-matter realizations: sizeable electron EDMs, up to and beyond the planned ACME sensitivity $\mathbb{Z}_2$7, occur only in the fermionic dark-matter scenario, while scalar dark matter keeps $\mathbb{Z}_2$8 below that reach (Abada et al., 2018).

Leptogenesis studies likewise show a strong dependence on fermion multiplicity and flavor structure. In the three-singlet minimal scotogenic model, hierarchical thermal leptogenesis with three right-handed neutrinos can lower the $\mathbb{Z}_2$9 mass scale to around $\mathbb{Z}_2$0 without any degeneracy in the heavy spectrum, provided the lightest active neutrino mass is around $\mathbb{Z}_2$1 and the $\mathbb{Z}_2$2 Yukawas are suppressed (Hugle et al., 2018). By contrast, in the two-right-handed-neutrino case the lower bound remains of order $\mathbb{Z}_2$3, essentially identical to standard thermal leptogenesis (Hugle et al., 2018).

When $\mathbb{Z}_2$4 is stable fermion dark matter and the asymmetry is instead generated by $\mathbb{Z}_2$5 decays, the scale depends strongly on the neutrino ordering. For normal ordering, successful leptogenesis is pushed several orders of magnitude above the TeV scale; for inverted ordering, $\mathbb{Z}_2$6 leptogenesis can occur at a scale of a few tens of TeV, and inclusion of lepton-flavor effects lowers the scale by around an order of magnitude in both cases (Mahanta et al., 2019). Scalar-dark-matter studies of $\mathbb{Z}_2$7 leptogenesis similarly identify viable TeV-scale baryogenesis, including non-thermal production of the lightest stable state in the inert-doublet desert (Bernardini et al., 2020).

6. Renormalization-group structure and theoretical consistency

The model’s high-energy behavior is unusually structured for a radiative neutrino-mass theory. A full set of one-loop renormalization-group equations for Ma’s scotogenic model was derived in the dedicated RGE study, including the gauge couplings, Yukawas, Majorana masses, scalar couplings, scalar masses, and the effective Weinberg-like operators that appear after integrating out singlets (Bouchand et al., 2012). A central result is that $\mathbb{Z}_2$8 runs multiplicatively: $\mathbb{Z}_2$9 If $N_i$0 at some scale, it remains zero under the RG flow; this matches the restoration of lepton number in that limit (Bouchand et al., 2012).

The effective theory below singlet thresholds contains two relevant dimension-five operators, $N_i$1 and $N_i$2, which mix through $N_i$3. This operator mixing distinguishes the scotogenic model from standard seesaw EFTs and modifies the running of neutrino parameters (Bouchand et al., 2012). In the numerical example of that paper, a bimaximal leptonic mixing pattern at the GUT scale evolves into a realistic low-energy spectrum with a large $N_i$4, suggesting nontrivial interplay between flavor symmetries and scotogenic RG flow (Bouchand et al., 2012).

The generalized-multiplicity analysis adds a cautionary point: the same Yukawas that help neutrino masses and LFV can destabilize the inert vacuum at high scales. In the beta function for the inert mass parameter $N_i$5, the negative trace term

$N_i$6

can drive $N_i$7 negative, spontaneously breaking the $N_i$8 and spoiling both dark-matter stability and the radiative origin of neutrino mass (Escribano, 2021). This “parity problem” already identified in the standard scotogenic model is therefore intrinsic to the minimal setup and becomes sharper when Yukawas are large (Escribano, 2021).

7. Generalizations and the historiography of “minimality”

A substantial part of the recent literature uses the minimal scotogenic model as a benchmark from which controlled extensions are built. Singlet–triplet scotogenic models add a fermion triplet $N_i$9 and a real scalar triplet $\mathbb{Z}_2$0, retaining the inert-doublet mechanism while enabling low-scale leptogenesis and a tree-level shift in the $\mathbb{Z}_2$1-boson mass through $\mathbb{Z}_2$2 (Singh et al., 2023). A one-singlet, two-inert-doublet construction shows that purely flavored leptogenesis can be realized with only one heavy Majorana fermion, provided there are two gauge multiplets that can interfere (Garbrecht et al., 2024). Modular $\mathbb{Z}_2$3 models keep the Ma field content but replace the ad hoc $\mathbb{Z}_2$4 with modular invariance, drastically reducing flavor freedom and predicting narrow ranges for $\mathbb{Z}_2$5, $\mathbb{Z}_2$6, and CP phases (Nomura et al., 2019). Non-invertible-symmetry realizations preserve the original field content while enforcing one-zero textures in the neutrino mass matrix and correlated charged-lepton-flavor-violation patterns (Nomura et al., 14 Jul 2025). Gauged $\mathbb{Z}_2$7 embeddings use the scotogenic sector as the neutrino-mass and dark-matter backbone while adding new states to address $\mathbb{Z}_2$8, the CDF-II $\mathbb{Z}_2$9-mass anomaly, and a $\overline{L}\tilde H N$0 scalar excess (Borah et al., 2023).

This broader literature makes the meaning of “minimal scotogenic model” precise by contrast. Minimality can refer to minimal field content for the original one-loop radiative mechanism, minimal singlet multiplicity compatible with oscillation data, or minimal parameterization after imposing additional flavor structure. What remains invariant is the defining scotogenic triad: an exact dark symmetry, an inert electroweak multiplet, and radiative Majorana neutrino mass generated by dark-sector fields.