Non-Group Scotogenic Model: Neutrino Mass & Dark Matter

• The non-group Scotogenic Model is a framework where neutrino masses are generated via a one-loop mechanism with inert scalars and singlet fermions.
• This approach circumvents traditional flavor-group extensions, using either a minimal Z2 symmetry, emergent parity from U(1) breaking, or non‐invertible symmetries to maintain stability.
• The model tightly connects neutrino physics with dark-matter and collider phenomenology, offering insights into lepton flavor violation and leptogenesis through controlled radiative effects.

The non-group Scotogenic Model denotes a class of scotogenic neutrino-mass constructions in which the defining radiative mechanism is retained, but the organizing symmetry principle differs from the conventional pattern of an ad hoc discrete stabilizing symmetry supplemented by non-Abelian flavor-group model building. In the current literature, the label is used in more than one technically distinct sense: for minimal $Z_2$-only setups with no extra nontrivial flavor or gauge groups, for ultraviolet completions in which the low-energy dark parity emerges as a residual subgroup of a broken continuous symmetry such as $U(1)_{B-L}$ or $U(1)_L$, and for recent models based on non-invertible $Z_M$ symmetries. Across these variants, the common core is the one-loop generation of Majorana neutrino masses through inert electroweak multiplets and singlet fermions, together with a stable dark-sector state (Garbrecht et al., 2024, Chuliá et al., 2019, Nomura et al., 14 Jul 2025).

1. Nomenclature and conceptual scope

The literature does not use the term “non-group” uniformly. Instead, three recurrent usages can be identified.

Usage	Defining feature	Representative papers
Minimal $Z_2$-only usage	No nontrivial flavor or gauge groups beyond the Standard Model and the dark $Z_2$	(Garbrecht et al., 2024, Escribano, 2021)
Emergent-parity usage	Dark symmetry appears as a residual subgroup of broken $U(1)_{B-L}$ or $U(1)_L$	(Chuliá et al., 2019, Portillo-Sánchez et al., 2023)
Non-invertible usage	Operator selection is controlled by non-invertible $Z_M$ fusion rules	(Nomura et al., 14 Jul 2025)

In the first usage, “non-group” emphasizes the absence of extra non-Abelian flavor or gauge structures. A particularly explicit example is the minimal setup with two inert scalar doublets and one heavy Majorana singlet, where purely flavored leptogenesis is driven by the interference of the two scalar multiplets and not by an enlarged flavor sector (Garbrecht et al., 2024). Closely related generalized formulations allow arbitrary numbers of inert doublets and singlet fermions while assuming only the minimal dark parity and no additional flavor symmetry (Escribano, 2021, Escribano et al., 2020).

In the second usage, the point is not the absence of symmetry, but the absence of an externally imposed discrete group. The stabilizing dark symmetry is obtained instead as a residual subgroup of a continuous symmetry already present or postulated in the ultraviolet. In this formulation, $U(1)_{B-L}\to Z_{2n}$ or $U(1)_{B-L}$0 simultaneously controls Majorana neutrino mass generation and dark-matter stability (Chuliá et al., 2019, Portillo-Sánchez et al., 2023).

The third usage is newer and more specialized. The “No-group Scotogenic Model” based on non-invertible $U(1)_{B-L}$1 symmetry replaces ordinary charge addition by fusion rules of conjugacy classes. In that construction, the symmetry enforces the Yukawa texture, forbids tree-level seesaw terms, and yields a one-zero neutrino-mass texture while an accidental $U(1)_{B-L}$2 stabilizes the inert state (Nomura et al., 14 Jul 2025).

2. Scotogenic kernel and field-theoretic structure

All scotogenic realizations share the Ma-type loop mechanism. In its canonical form, the Standard Model is extended by an inert scalar doublet $U(1)_{B-L}$3 and singlet Majorana fermions $U(1)_{B-L}$4, odd under a dark parity. The exact $U(1)_{B-L}$5 forbids the tree-level Yukawa $U(1)_{B-L}$6, keeps $U(1)_{B-L}$7, and stabilizes the lightest odd state (Bouchand et al., 2012).

A standard Lagrangian sector is

$U(1)_{B-L}$8

together with the scalar potential

$U(1)_{B-L}$9

After electroweak symmetry breaking, $U(1)_L$0, and the CP-even/odd neutral masses split through $U(1)_L$1. Because conventions differ across papers, the splitting is written either as $U(1)_L$2 or as $U(1)_L$3, depending on the normalization of the scalar potential (Bouchand et al., 2012, Portillo-Sánchez et al., 2023).

The one-loop neutrino mass matrix in the single-inert-doublet formulation is

$U(1)_L$4

This exhibits the defining scotogenic feature: neutrino mass vanishes when the CP-even/odd inert states are degenerate, so the lepton-number-violating parameter $U(1)_L$5 controls both the loop amplitude and the direct-detection-safe mass splitting of the inert neutral scalars (Bouchand et al., 2012).

Generalizations promote $U(1)_L$6 to a family $U(1)_L$7 and the quartic to a symmetric matrix $U(1)_L$8. In that case,

$U(1)_L$9

in the small-$Z_M$0, small-mixing limit, with $Z_M$1 encoding both diagonal splittings and inter-doublet mixings (Escribano, 2021).

3. Realizations without conventional flavor-group model building

In minimal $Z_M$2-only realizations, the non-group character lies in the absence of any nontrivial flavor or gauge extension beyond the dark parity. The prototype with two inert scalar doublets $Z_M$3 and a single heavy Majorana fermion $Z_M$4 is especially notable because flavored CP violation arises from interference between the two scalar multiplets and their mixings, so a second heavy singlet is not necessary for leptogenesis (Garbrecht et al., 2024). More general multi-doublet/multi-singlet formulations preserve this philosophy: the radiative mechanism is retained while the flavor structure is carried by the Yukawas $Z_M$5 and the scalar-sector tensors $Z_M$6, not by extra flavor symmetries (Escribano, 2021).

A second branch of non-group constructions replaces the imposed dark parity by an emergent remnant. In the $Z_M$7 framework, spontaneous breaking by a scalar of even charge yields $Z_M$8, with Standard Model fields transforming as even elements and dark fields as odd elements. This simultaneously allows Majorana neutrino masses and stabilizes the lightest odd state (Chuliá et al., 2019). Closely related ultraviolet completions based on global $Z_M$9 classify all models in which $Z_2$0 at low energies and the effective $Z_2$1 operator is generated only after integrating out heavy scalar mediators. Under the assumptions of global $Z_2$2, tree-level completions, and at most two singlet scalars, $Z_2$3 ultraviolet extensions were identified, organized into topologies I–V (Portillo-Sánchez et al., 2023).

These emergent-parity models also address the origin of the small lepton-number-violating parameter. In the one-singlet completion IV$Z_2$4,

$Z_2$5

while in the two-singlet completion II$Z_2$6,

$Z_2$7

In both cases, the smallness of $Z_2$8 is linked to symmetry breaking and heavy-mediator suppression rather than inserted by hand (Portillo-Sánchez et al., 2023).

A third structurally different realization is the non-invertible $Z_2$9 model. There, the lepton doublets, right-handed charged leptons, singlet neutrinos, Higgs, and inert doublet are assigned to conjugacy classes $Z_2$0, and local operators are allowed only if iterated fusion contains the identity class. The minimal assignment with $Z_2$1 forbids $Z_2$2, forbids $Z_2$3, enforces diagonal charged leptons, and permits a highly constrained $Z_2$4 texture (Nomura et al., 14 Jul 2025).

4. Neutrino-mass textures, rank, and flavor organization

The non-group scotogenic literature is characterized less by a single preferred texture than by a set of mechanisms for obtaining flavor structure without conventional non-Abelian model building. In generalized multi-$Z_2$5 models, the matrix structure $Z_2$6 implies that the light-neutrino mass matrix is no longer a simple sum of outer products. As a consequence, even a single Majorana singlet can generate more than one non-zero light-neutrino mass eigenvalue if $Z_2$7, and with $Z_2$8 one can obtain three massive neutrinos and large mixing even for diagonal Yukawas in the singlet sector (Escribano, 2021, Escribano et al., 2020).

The two-inert-doublet, one-singlet prototype provides a particularly explicit example. Summing over scalar flavors $Z_2$9, the neutrino mass matrix is

$U(1)_{B-L}$0

In this setup, the neutrino mass matrix has rank two, and the $U(1)_{B-L}$1 matrix can be reconstructed from neutrino data and the Yukawas through an explicit inversion formula (Garbrecht et al., 2024).

Predictive flavor organization without non-Abelian groups also appears in generalized-CP constructions. In the scotogenic model for co-bimaximal mixing, the relevant symmetry content consists of softly broken lepton numbers $U(1)_{B-L}$2, a non-standard CP symmetry of $U(1)_{B-L}$3–$U(1)_{B-L}$4 reflection type, and simple $U(1)_{B-L}$5 parities. The loop-generated mass matrix satisfies

$U(1)_{B-L}$6

which implies

$U(1)_{B-L}$7

while leaving $U(1)_{B-L}$8 arbitrary. The construction is explicitly described as one that does not invoke non-Abelian flavor groups (Ferreira et al., 2016).

In the non-invertible $U(1)_{B-L}$9 realization, the symmetry-enforced Yukawa texture

$U(1)_L$0

leads to the one-zero texture

$U(1)_L$1

Permuting the class assignments of $U(1)_L$2 permutes the vanishing entry, so the framework naturally realizes the family of one-zero textures discussed in the neutrino-texture literature (Nomura et al., 14 Jul 2025).

5. Dark matter, leptogenesis, and cosmological roles

The dark-matter candidate in non-group scotogenic models is always the lightest state protected by the surviving dark-sector selection rule, but its identity varies substantially. In standard and generalized inert-doublet realizations, the candidate can be either the lightest inert scalar or the lightest singlet fermion. In the ultraviolet-emergent $U(1)_L$3 models, the presence of additional singlet-like scalars and a massless Majoron enriches annihilation channels, and the viable relic density can be achieved in regions consistent with $U(1)_L$4 without relying solely on coannihilations (Portillo-Sánchez et al., 2023).

In the minimal two-doublet, one-singlet prototype, the lightest neutral scalar from the lighter inert doublet is the typical dark-matter candidate. Direct detection through $U(1)_L$5-exchange is suppressed by requiring a neutral-scalar mass splitting $U(1)_L$6, while Higgs-mediated elastic scattering is governed by

$U(1)_L$7

A representative scan with $U(1)_L$8 and $U(1)_L$9 yields viable dark matter for $Z_M$0 (Garbrecht et al., 2024).

A more elaborate cosmological role appears in the asymmetric-mediator extension. There the model contains the usual inert doublet $Z_M$1, right-handed Majorana neutrinos $Z_M$2, and an additional real $Z_M$3-odd singlet scalar $Z_M$4 that constitutes the dark matter. CP-violating decays of $Z_M$5 generate equal and opposite asymmetries in the lepton sector and in the $Z_M$6-odd doublet sector; the latter is relayed to $Z_M$7 through the decay induced by $Z_M$8. Within the quoted viable window,

$Z_M$9

together with

$U(1)_{B-L}\to Z_{2n}$0

and

$U(1)_{B-L}\to Z_{2n}$1

the model yields $U(1)_{B-L}\to Z_{2n}$2, and $U(1)_{B-L}\to Z_{2n}$3 few GeV naturally gives $U(1)_{B-L}\to Z_{2n}$4 (Asai et al., 2022).

Leptogenesis is likewise broadened beyond the standard heavy-singlet picture. In the two-inert-doublet, one-singlet setup, leptogenesis is purely flavored: the sum of flavor asymmetries vanishes, but flavor-dependent washouts produce a non-zero baryon asymmetry. The closed-time-path treatment shows that both wavefunction and vertex source terms contribute, and that large widths limit resonant enhancement so that vertex sources can dominate when the mixed-scalar widths are large (Garbrecht et al., 2024).

6. Constraints, renormalization-group structure, and experimental tests

Non-group scotogenic models are constrained by the same broad phenomenological triad as the canonical Ma model—lepton flavor violation, collider signatures, and dark-matter searches—but the detailed pattern is more model-dependent. A canonical radiative-decay formula is

$U(1)_{B-L}\to Z_{2n}$5

and current bounds drive strong restrictions on the Yukawa combinations relevant for neutrino mass and fermion dark matter (Portillo-Sánchez et al., 2023). In the non-invertible $U(1)_{B-L}\to Z_{2n}$6 realization, the symmetry-imposed one-zero texture simultaneously suppresses one radiative LFV channel at one loop; for the displayed assignment, $U(1)_{B-L}\to Z_{2n}$7 is absent while $U(1)_{B-L}\to Z_{2n}$8 and $U(1)_{B-L}\to Z_{2n}$9 remain potentially sizeable (Nomura et al., 14 Jul 2025).

Ultraviolet-emergent models add Majoron phenomenology. Because the spontaneous breaking of $U(1)_{B-L}$00 leaves a massless Goldstone boson $U(1)_{B-L}$01, one obtains invisible-Higgs and lepton-Majoron signatures absent from the canonical model. Representative quoted limits include $U(1)_{B-L}$02, implying $U(1)_{B-L}$03 when $U(1)_{B-L}$04, and $U(1)_{B-L}$05, implying $U(1)_{B-L}$06 in a related completion (Portillo-Sánchez et al., 2023, Escribano et al., 2021). Loop-induced couplings $U(1)_{B-L}$07 then connect Higgs physics, astrophysical cooling bounds, and searches such as $U(1)_{B-L}$08.

Renormalization-group behavior is one of the most distinctive theoretical differences between standard and generalized scotogenic models. In the original model, $U(1)_{B-L}$09 is multiplicatively renormalized: $U(1)_{B-L}$10 Hence, if $U(1)_{B-L}$11 at some scale, it remains zero at one loop; this expresses the protective lepton-number limit and the technical naturalness of small $U(1)_{B-L}$12 (Bouchand et al., 2012). In generalized multi-$U(1)_{B-L}$13 models, however, the RGE for $U(1)_{B-L}$14 contains a negative trace term,

$U(1)_{B-L}$15

which can drive inert-scalar mass parameters negative in the ultraviolet and induce spontaneous $U(1)_{B-L}$16 breaking. Large quartics, especially $U(1)_{B-L}$17, can counteract this flow and in some cases develop Landau poles before the would-be breaking scale (Escribano, 2021, Escribano et al., 2020).

Collider phenomenology reflects the enlarged inert sector. The standard signatures are Drell–Yan production of $U(1)_{B-L}$18-odd scalars, leading to leptons or jets plus missing energy. Two-inert-doublet models add cascade structures not present in the minimal model, including the final states $U(1)_{B-L}$19, $U(1)_{B-L}$20, and $U(1)_{B-L}$21 (Ahriche, 2022). A plausible implication is that the most incisive experimental discrimination among non-group variants will come not from a single observable, but from correlated information spanning LFV, invisible Higgs widths, inert-scalar spectroscopy, and dark-matter scattering.

Overall, the non-group Scotogenic Model is best regarded not as a unique Lagrangian but as a technically coherent family of radiative neutrino-mass frameworks that avoid conventional flavor-group model building in different ways. Some versions minimize symmetry content to a bare $U(1)_{B-L}$22; others derive the dark parity as a residual subgroup of a broken continuous symmetry; the most recent versions use non-invertible symmetries to enforce Yukawa textures. What unifies them is the persistence of the scotogenic loop, the intimate relation between neutrino mass and dark-sector structure, and a phenomenology in which flavor, Higgs, collider, and dark-matter observables remain tightly entangled (Garbrecht et al., 2024, Portillo-Sánchez et al., 2023, Nomura et al., 14 Jul 2025).