---
title: 'Minimal Scotogenic Model: Neutrinos & Dark Matter'
url: https://www.emergentmind.com/topics/minimal-scotogenic-model
type: topic
---

# Minimal Scotogenic Model: Neutrinos & Dark Matter

Searching arXiv for core and recent 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 [1205.0008; 2110.07890; 1802.00007]. 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 [1205.0008; 2110.07890]. 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 [1802.00007; 1804.09660].

| 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, \(\langle\eta^0\rangle=0\), and the lightest \(\mathbb{Z}_2\)-odd particle is stable [1205.0008; 1802.00007].

## 2. Lagrangian structure and scalar sector

The characteristic Yukawa interaction is of the form
\[
\mathcal{L}\supset -\,\overline{L}\,h_\nu\,\tilde\eta\,N_R-\frac12\,\overline{N^c}M_NN+\text{h.c.},
\]
or, in the two-singlet notation,
\[
\mathcal{L}\supset \frac12\overline{N_i}(i\!\not\!\partial-m_i)N_i-y_{i\alpha}\,\eta\,\overline{N_i}P_LL_\alpha+\text{H.c.},
\]
with \(N_i\sim(1,1,0)\) and \(\eta\sim(1,2,1/2)\) [1205.0008; 1802.00007]. The scalar potential is the inert-doublet one,
\[
V = m_1^2 \Phi^\dagger \Phi + m_2^2 \eta^\dagger \eta + \frac12\lambda_1(\Phi^\dagger\Phi)^2 + \frac12\lambda_2(\eta^\dagger\eta)^2 + \lambda_3(\Phi^\dagger\Phi)(\eta^\dagger\eta)+\lambda_4(\Phi^\dagger\eta)(\eta^\dagger\Phi)+\frac12\lambda_5\left[(\Phi^\dagger\eta)^2+\text{h.c.}\right],
\]
or equivalently with \(H\) replacing \(\Phi\) [1205.0008; 1802.00007].

The neutral inert component is decomposed as
\[
\eta^0=\frac{1}{\sqrt2}(\eta_R+i\eta_I),
\]
and electroweak symmetry breaking induces the inert spectrum
\[
m_R^2=m_2^2+(\lambda_3+\lambda_4+\lambda_5)v^2,\qquad
m_I^2=m_2^2+(\lambda_3+\lambda_4-\lambda_5)v^2,
\]
together with the charged-scalar mass
\[
m_{\eta^\pm}^2=m_2^2+\lambda_3v^2
\]
in the notation of the RGE analysis [1205.0008]. In the two-singlet formulation the same mass splitting is written as
\[
m_R^2-m_I^2=2\lambda_5 v^2,
\]
with the same physical content: \(\lambda_5\) controls the CP-even/CP-odd inert splitting [1802.00007].

The central structural role of \(\lambda_5\) 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 \(\lambda_5\to0\), \(m_R\to m_I\), neutrino masses vanish, and lepton number is restored; this is the standard ’t Hooft naturalness argument for small \(\lambda_5\) [1205.0008; 1802.00007].

## 3. One-loop neutrino-mass mechanism

Neutrino mass is generated by the one-loop diagram with \(N_k\) and \(\eta_{R,I}\) running in the loop. In the three-singlet Ma model, the mass matrix is
\[
\mathcal{M}_{\nu,ij} =
\frac{h_{ik}\, h_{jk}}{16\pi^2} \, M_k
\left[
\frac{m_R^2}{m_R^2-M_k^2}\ln \frac{m_R^2}{M_k^2}
-\frac{m_I^2}{m_I^2-M_k^2}\ln \frac{m_I^2}{M_k^2}
\right],
\]
with the sum over \(k\) implied [1205.0008]. In the two-singlet version the same structure appears as
\[
(m_\nu)_{\alpha\beta}
=
\sum_{i=1}^2
\frac{y_{i\alpha}y_{i\beta}\,m_i}{2(4\pi)^2}
\left[
\frac{m_R^2}{m_R^2-m_i^2}\ln\frac{m_R^2}{m_i^2}
-
\frac{m_I^2}{m_I^2-m_i^2}\ln\frac{m_I^2}{m_i^2}
\right],
\]
and reduces, in the small-splitting limit, to a form explicitly proportional to \(\lambda_5\) [1802.00007].

This loop-induced mass can be written compactly as \(m_\nu = y^T\Lambda y\) or \(M_\nu = Y\Lambda^{-1}Y^T\), depending on convention, and is routinely matched to oscillation data with a Casas–Ibarra parametrization adapted to the scotogenic loop structure [1802.00007; 2003.02276]. 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 [1802.00007; 2110.07890].

A notable technical consequence is that the model admits TeV-to-tens-of-TeV singlet masses with Yukawas of order \(0.1\)–\(0.4\) and tiny \(\lambda_5\), while still reproducing realistic neutrino masses [1205.0008]. 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 \(\mathbb{Z}_2\) 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 \(\eta\) or the lightest singlet fermion \(N_1\) [1205.0008; 2110.07890]. The two-singlet EDM analysis likewise identifies two possibilities: fermionic dark matter if \(N_1\) is lightest, and scalar dark matter if the lighter neutral inert state is lightest [1802.00007].

For scalar dark matter, two standard mass windows recur. One is a low-mass region below the \(W^+W^-\) threshold and near the Higgs resonance. The other is a high-mass inert-doublet regime beginning around \(535\)–\(550\;\mathrm{GeV}\) and extending to multi-TeV masses [1802.00007; 2003.02276]. In the intermediate interval \(M_W < M_{DM}\le 550\;\mathrm{GeV}\), the relic density is often underabundant because annihilation into gauge bosons and co-annihilation with \(\eta_I^0\) and \(\eta^\pm\) are too efficient; this is the inert-doublet “desert” emphasized in scalar-DM studies [2004.02276].

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 \(m_{N_1}\) above the electroweak scale typically requires \(y_{i\alpha}\gtrsim \mathcal{O}(0.1)\), while direct detection remains negligible because the singlet fermion couples only to leptons at tree level [1802.00007]. 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 [2003.02276].

The dark sector also exhibits a cosmological bifurcation between thermal and non-thermal production. In the fermion-dark-matter \(N_2\)-leptogenesis analysis, the lightest singlet \(N_1\) 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 [1906.03577]. Scalar dark matter can likewise receive a non-thermal contribution from late decays of \(N_1\), opening otherwise underabundant regions of the inert-doublet desert [2004.02276].

## 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
\[
d_{\alpha} = -\frac{e\, m_\alpha}{(4\pi)^4 m_{\eta^+}^2}
\sum_{\beta}\sum_{i,j=1}^2\left[
J^M_{ij\alpha\beta}\sqrt{x_i x_j}\,I_M(x_i,x_j)
+
J^D_{ij\alpha\beta}I_D(x_i,x_j)
\right],
\]
with antisymmetric CP invariants \(J^{M,D}\) constructed from the Yukawas; in this model the Dirac-type loop function \(I_D\) vanishes identically, so only the Majorana-type contribution survives [1802.00007]. The resulting phenomenology is sharply asymmetric between dark-matter realizations: sizeable electron EDMs, up to and beyond the planned ACME sensitivity \(|d_e|/e\sim10^{-30}\,\mathrm{cm}\), occur only in the fermionic dark-matter scenario, while scalar dark matter keeps \(|d_e|\) below that reach [1802.00007].

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 \(N_1\) mass scale to around \(10\;\mathrm{TeV}\) without any degeneracy in the heavy spectrum, provided the lightest active neutrino mass is around \(10^{-12}\;\mathrm{eV}\) and the \(N_1\) Yukawas are suppressed [1804.09660]. By contrast, in the two-right-handed-neutrino case the lower bound remains of order \(10^{10}\;\mathrm{GeV}\), essentially identical to standard thermal leptogenesis [1804.09660].

When \(N_1\) is stable fermion dark matter and the asymmetry is instead generated by \(N_2\) 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, \(N_2\) 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 [1906.03577]. Scalar-dark-matter studies of \(N_2\) leptogenesis similarly identify viable TeV-scale baryogenesis, including non-thermal production of the lightest stable state in the inert-doublet desert [2004.02276].

## 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 [1205.0008]. A central result is that \(\lambda_5\) runs multiplicatively:
\[
\mathcal{D}\lambda_5 \propto \lambda_5.
\]
If \(\lambda_5=0\) at some scale, it remains zero under the RG flow; this matches the restoration of lepton number in that limit [1205.0008].

The effective theory below singlet thresholds contains two relevant dimension-five operators, \(\kappa^{(11)}\) and \(\kappa^{(22)}\), which mix through \(\lambda_5\). This operator mixing distinguishes the scotogenic model from standard seesaw EFTs and modifies the running of neutrino parameters [1205.0008]. 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 \(\theta_{13}\), suggesting nontrivial interplay between flavor symmetries and scotogenic RG flow [1205.0008].

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 \(m_\eta^2\), the negative trace term
\[
-4\,\mathrm{Tr}\!\left[y_a^\dagger M_N^\ast M_N y_b\right]
\]
can drive \(m_\eta^2\) negative, spontaneously breaking the \(\mathbb{Z}_2\) and spoiling both dark-matter stability and the radiative origin of neutrino mass [2110.07890]. This “parity problem” already identified in the standard scotogenic model is therefore intrinsic to the minimal setup and becomes sharper when Yukawas are large [2110.07890].

## 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 \(\Sigma\) and a real scalar triplet \(\Omega\), retaining the inert-doublet mechanism while enabling low-scale leptogenesis and a tree-level shift in the \(W\)-boson mass through \(v_\Omega\) [2309.12755]. 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 [2404.02207]. Modular \(A_4\) models keep the Ma field content but replace the ad hoc \(\mathbb{Z}_2\) with modular invariance, drastically reducing flavor freedom and predicting narrow ranges for \(\sum m_\nu\), \(m_{ee}\), and CP phases [1908.07457]. 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 [2507.10299]. Gauged \(U(1)_{L_\mu-L_\tau}\) embeddings use the scotogenic sector as the neutrino-mass and dark-matter backbone while adding new states to address \((g-2)_\mu\), the CDF-II \(W\)-mass anomaly, and a \(95\;\mathrm{GeV}\) scalar excess [2310.11953].

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.

Source: https://www.emergentmind.com/topics/minimal-scotogenic-model