Scotogenic Dirac Model

• Scotogenic Dirac Model is a framework where neutrino masses arise radiatively via loop diagrams, linking Dirac neutrino mass generation to dark matter stability.
• It forbids tree-level Dirac Yukawa couplings through discrete or gauged symmetries, ensuring neutrinos remain Dirac while stabilizing the lightest dark-sector particle.
• The model offers versatile one-loop and two-loop realizations, embedding in U(1) extensions and predicting testable signatures in LFV, dark matter, and collider experiments.

The Scotogenic Dirac Model is a class of extensions of the Standard Model in which neutrinos acquire naturally small Dirac masses through loop diagrams involving fields from the dark sector, so that neutrino mass generation and dark-matter stability are tied to the same symmetry structure. In the minimal construction of Farzan and Ma, the Standard Model gauge group is supplemented by an additional global or gauged $U(1)_{B-L}$, a softly broken $Z_2^A$ that forbids the tree-level Dirac Yukawa coupling $L H \nu^c$, and an exact $Z_2^B$ under which the new particles are odd; the lightest odd state is then stable and can play the role of dark matter (Farzan et al., 2012). Subsequent work broadened the term to encompass generalized one-loop and two-loop realizations, anomaly-free Abelian extensions, residual $Z_2$, $Z_3$, or $Z_6$ symmetries, $Z'$-portal dark matter, and leptoquark-embedded constructions that attempt to address flavor anomalies together with neutrino masses and dark matter (Wang et al., 2017, Li et al., 2022, Chuliá et al., 2024).

1. Origins and defining idea

The adjective scotogenic was introduced for models in which neutrino masses arise radiatively from couplings to dark matter, “from the Greek ‘scotos’ meaning darkness.” The 2012 Dirac version was formulated as an “analogous mechanism for Dirac neutrino masses” relative to the earlier one-loop Majorana setup, and one of its distinctive observations was that “the lightest Dirac fermion which appears in the loop diagram generating neutrino mass can be a viable dark matter candidate,” a possibility that “does not exist for the Majorana case” (Farzan et al., 2012).

Within the later literature, “Dirac scotogenic model” no longer denotes a single unique Lagrangian. It denotes a family of constructions in which three ingredients recur. First, the tree-level operator $\overline{L}\tilde H \nu_R$ is forbidden by a discrete or gauge symmetry. Second, the same symmetry forbids Majorana masses, so neutrinos remain Dirac. Third, the lightest nontrivially charged field under the residual dark symmetry is stable and can serve as dark matter. This architecture appears in generalized inert-doublet models with a global $U(1)_n$ (Hagedorn et al., 2018), in $Z_2^A$0 models with residual $Z_2^A$1 or $Z_2^A$2 symmetries (Wang et al., 2017), in $Z_2^A$3 models descending from $Z_2^A$4 (Ma, 2019), in Stueckelberg realizations of unbroken gauged $Z_2^A$5 (Leite et al., 2020), and in residual $Z_2^A$6 constructions where a single Abelian discrete symmetry protects both “Diracness” and dark-matter stability (Chuliá et al., 2022, Chuliá et al., 2024).

A common misconception is that scotogenic models are intrinsically Majorana. The Dirac scotogenic literature directly contradicts this: exact or residual symmetries are engineered specifically so that “Majorana masses for $Z_2^A$7” are forbidden while the neutrino mass is generated only radiatively (Farzan et al., 2012, Wang et al., 2017).

2. Minimal one-loop realization

In the minimal model, the gauge group is $Z_2^A$8 with an additional global or gauged $Z_2^A$9. The discrete symmetries are $L H \nu^c$0, under which only the right-handed neutrinos $L H \nu^c$1 are odd and which forbids $L H \nu^c$2, and an exact $L H \nu^c$3 (“dark parity”), under which the new fields $L H \nu^c$4 are odd while all Standard Model fields are even (Farzan et al., 2012).

The extra field content consists of an inert scalar doublet $L H \nu^c$5 with hypercharge $L H \nu^c$6, a real scalar singlet $L H \nu^c$7, three copies of vector-like gauge-singlet Dirac fermions $L H \nu^c$8, and three right-handed neutrinos $L H \nu^c$9. The $Z_2^B$0 charge assignment is such that $Z_2^B$1 for $Z_2^B$2, $Z_2^B$3 for $Z_2^B$4, and $Z_2^B$5 for $Z_2^B$6. In this setup, $Z_2^B$7 “forbids Majorana masses for $Z_2^B$8 and for $Z_2^B$9” (Farzan et al., 2012).

The renormalizable interactions relevant for neutrino mass are

$Z_2$0

together with the Dirac mass term $Z_2$1 and the softly $Z_2$2-breaking trilinear scalar interaction

$Z_2$3

After electroweak symmetry breaking, $Z_2$4, the $Z_2$5-term mixes $Z_2$6 and $Z_2$7, and the two neutral states are

$Z_2$8

This structure is the canonical one-loop Scotogenic Dirac Model (Farzan et al., 2012).

Later papers often repackage the same mechanism in different notation. A widely used modern formulation employs three right-handed neutrinos $Z_2$9, three vector-like singlet fermions $Z_3$0, one inert scalar doublet $Z_3$1, one real scalar singlet $Z_3$2, and a $Z_3$3 symmetry under which $Z_3$4 forbids tree-level Dirac and Majorana masses while the residual $Z_3$5 stabilizes the lightest odd state (Guo et al., 2020, Guo et al., 22 Aug 2025). The field content differs in notation, but the operational principle is the same: forbidden tree-level mass, mixed neutral scalars, and a loop connecting $Z_3$6 to $Z_3$7 through dark-sector mediators.

3. Radiative Dirac mass generation

In the minimal realization, the one-loop neutrino mass arises from the chain

$Z_3$8

The resulting Dirac mass matrix is

$Z_3$9

Three structural properties are emphasized in the original analysis. Each matrix element is proportional to the soft parameter $Z_6$0 through $Z_6$1; $Z_6$2 remains exact so neutrinos are purely Dirac; and small $Z_6$3 arises from loop suppression, large $Z_6$4, and small $Z_6$5 (Farzan et al., 2012).

The same loop structure reappears across later realizations. In the generalized model with two inert scalar doublets and one Dirac singlet fermion $Z_6$6, neutrino masses are generated at one loop by $Z_6$7, with neutral-scalar mixing governed by $Z_6$8 (Hagedorn et al., 2018). In the $Z_6$9 formulation, the loop involves $Z'$0 and yields

$Z'$1

with $Z'$2 (Guo et al., 22 Aug 2025, Guo et al., 2020).

Not all Dirac scotogenic models are one-loop. A two-loop realization with two $Z'$3 triality symmetries and a global spontaneously broken $Z'$4 generates Dirac neutrino masses only at two-loop order through the dark sector. In that construction, strict lepton-number conservation is enforced at higher orders, a physical Diracon emerges from spontaneous $Z'$5 breaking, and invisible Higgs decays receive contributions both from $Z'$6 and from the Higgs-to-dark-matter mode (Bonilla et al., 2016).

A second misconception is that every Scotogenic Dirac Model predicts three nonzero neutrino masses. Some realizations are explicitly rank-2. In the FIMP version, the couplings to $Z'$7 are taken extremely small, so “rank $Z'$8 and one neutrino is exactly massless (to very good approximation)” (Guo et al., 22 Aug 2025). In the residual-$Z'$9 model of Centelles Chuliá et al., the use of only two heavy Dirac fermions likewise leaves one neutrino massless (Chuliá et al., 2024).

4. Dark matter sector and flavor structure

The original model already exhibits the two characteristic dark-matter options of the Dirac scotogenic framework. The lightest scalar can be dark matter, typically a mostly $\overline{L}\tilde H \nu_R$0 or $\overline{L}\tilde H \nu_R$1 state. For a mostly singlet scalar, Higgs-portal scattering can lie “easily below current XENON bounds,” and “typical parameter ranges” are $\overline{L}\tilde H \nu_R$2–$\overline{L}\tilde H \nu_R$3 with small $\overline{L}\tilde H \nu_R$4–$\overline{L}\tilde H \nu_R$5 mixing, $\overline{L}\tilde H \nu_R$6, to evade $\overline{L}\tilde H \nu_R$7-mediated bounds (Farzan et al., 2012).

The alternative is fermion dark matter. In the minimal model, the lightest Dirac fermion $\overline{L}\tilde H \nu_R$8 is protected by $\overline{L}\tilde H \nu_R$9 and annihilates dominantly through the $U(1)_n$0 couplings into $U(1)_n$1, or through $U(1)_n$2 gauge channels if the symmetry is gauged. The relic-density estimate quoted in the original analysis requires $U(1)_n$3, implying that for $U(1)_n$4 and $U(1)_n$5, one finds $U(1)_n$6. If $U(1)_n$7 is gauged with $U(1)_n$8 and $U(1)_n$9, then $Z_2^A$00-mediated annihilation can fix $Z_2^A$01 (Farzan et al., 2012).

A generalized Scotogenic Dirac Model with fermionic dark matter sharpened this picture by showing a strong complementarity between dark-matter direct detection and charged lepton flavor violation. There, “due to the strong limits from the latter, dark matter annihilations are suppressed and the relic abundance is set by coannihilations with (and annihilations of) the new scalars if the latter and the Dirac fermion are sufficiently degenerate in mass.” The quoted viable region is

$Z_2^A$02

with $Z_2^A$03 and $Z_2^A$04 (Hagedorn et al., 2018).

The flavor sector of the minimal model was also developed explicitly. With an $Z_2^A$05 symmetry, the loop-induced neutrino mass matrix can take the texture

$Z_2^A$06

and in the tribimaximal basis this becomes

$Z_2^A$07

In that setup, small nonzero $Z_2^A$08 induce deviations from tri-bimaximal mixing, in particular $Z_2^A$09, allowing $Z_2^A$10 while keeping $Z_2^A$11 and $Z_2^A$12 within experimental ranges (Farzan et al., 2012).

5. Gauge extensions, residual symmetries, and model-building variants

A major line of development embeds the Scotogenic Dirac Model into gauged Abelian symmetries. In $Z_2^A$13 constructions, anomaly cancellation strongly constrains the spectrum. For one-loop models, the anomaly-free conditions were shown to imply the unique solution $Z_2^A$14, $Z_2^A$15, and $Z_2^A$16, while spontaneous breaking of $Z_2^A$17 leaves a residual $Z_2^A$18 in one-loop realizations or a residual $Z_2^A$19 in two-loop realizations. The residual discrete symmetry both forbids the tree-level Dirac Yukawa coupling and stabilizes the lightest inert state (Wang et al., 2017).

The $Z_2^A$20 framework provides another route. There the extra gauge symmetry arises from $Z_2^A$21, and the literature discusses two distinct dark-matter scenarios: one with light Dirac fermion dark matter and another with self-interacting scalar dark matter with a light scalar mediator that decays only to two neutrinos (Ma, 2019). A related $Z_2^A$22 study focusing on the $Z_2^A$23 portal found that after combining dilepton searches at the LHC, $Z_2^A$24, relic abundance, direct detection, and indirect detection, “the resonance region $Z_2^A$25 is the viable parameter space” (Han et al., 2018).

An especially economical realization keeps gauged $Z_2^A$26 unbroken and gives the associated gauge boson a mass through the Stueckelberg mechanism. In that construction, $Z_2^A$27 acquires a Proca-type mass without any Higgs vacuum expectation value, matter parity $Z_2^A$28 remains exact, and the lightest $Z_2^A$29-odd state is absolutely stable. Both scalar and fermion dark matter are possible, and the one-loop mass formula again takes the standard mixed-scalar form proportional to $Z_2^A$30 (Leite et al., 2020).

Residual $Z_2^A$31 symmetries have become particularly prominent. One model starts from a global $Z_2^A$32 softly broken to a residual $Z_2^A$33, “lepton quarticity,” which simultaneously forbids Majorana masses for $Z_2^A$34 and $Z_2^A$35 and ensures that the lightest dark-sector particle is absolutely stable (Chuliá et al., 2022). Another describes $Z_2^A$36 as the unbroken subgroup of the so-called 445 $Z_2^A$37 symmetry and emphasizes that the exact $Z_2^A$38 protects both the Dirac nature of the neutrinos and the stability of the dark-matter candidate (Chuliá et al., 2024). A gauged-lepton-number realization goes further and proposes “the first scotogenic neutrino mass model with gauged lepton number $Z_2^A$39, which is spontaneously broken by three units $Z_2^A$40 down to a residual discrete gauge symmetry $Z_2^A$41” (Hernández et al., 23 Dec 2025).

Systematic classification has also been carried out at the level of anomaly-free Abelian symmetries. A comprehensive scan of active $Z_2^A$42 and dark $Z_2^A$43 charge assignments identified large sets of anomaly-free chiral solutions that realize one-loop Dirac scotogenic neutrino masses via dimension-5 or dimension-6 operators, including models with no massless chiral fermions (Bernal et al., 2021).

Finally, the framework has been embedded into still more elaborate structures. One example is the supersymmetric extension of the original model, in which inert doublet and singlet superfields reproduce the one-loop Dirac mass mechanism while enlarging the dark sector (Farzan et al., 2012). Another is the embedding with leptoquarks, where “plenty of diagrams associated with the two-loop realizations of $Z_2^A$44 can support the coexistence of leptoquarks and dark matter candidates,” leading to models that could address $Z_2^A$45, $Z_2^A$46, $Z_2^A$47, neutrino masses, and dark matter in a unified picture (Li et al., 2022).

6. Phenomenology, constraints, and experimental status

Charged lepton flavor violation is among the most important probes. In the $Z_2^A$48 formulation, all LFV is mediated by the charged scalar $Z_2^A$49 and the heavy fermions $Z_2^A$50, with the experimental bounds “especially given by decays $Z_2^A$51 and $Z_2^A$52” placing severe constraints on the Yukawa coupling $Z_2^A$53 and on the masses $Z_2^A$54 and $Z_2^A$55 (Guo et al., 2020). In the generalized inert-doublet setup, the explicit bound $Z_2^A$56 implies $Z_2^A$57 for $Z_2^A$58 (Hagedorn et al., 2018). Comparative analyses of Majorana and Dirac scotogenic models further indicate that $Z_2^A$59 can reach branching ratios as high as $Z_2^A$60 in the Dirac case, which is within the reach of future planned experiments (Hundi, 7 Feb 2025).

Dark-matter detection is highly model dependent. In the fermion-dark-matter realization of the minimal $Z_2^A$61 model, the relic abundance is “basically by annihilating through another Yukawa $Z_2^A$62,” while direct detection is loop suppressed and the current bounds are “relatively loose and can barely exclude more parameter region beyond the LFV” (Guo et al., 2020). In the $Z_2^A$63 analysis of the same model class, the spin-independent direct-detection rate is typically $Z_2^A$64, below current LUX-ZEPLIN sensitivity, while the same parameter space can still be probed by LFV and collider searches (Borah et al., 2022).

Cosmological radiation density introduces an additional handle that is specific to Dirac neutrino models. Because right-handed neutrinos can thermalize through dark-sector interactions, the Dirac scotogenic model can generate an observable $Z_2^A$65. One study found that the parameter space consistent with dark-matter phenomenology and neutrino mass bounds “can also be probed at future cosmic microwave background experiments like CMB-S4 via precision measurements of $Z_2^A$66,” with a target region $Z_2^A$67 (Borah et al., 2022). In a FIMP realization, $Z_2^A$68 receives both thermal and non-thermal contributions, and viable parameter regions survive after imposing relic-density, LFV, BBN, CMB, and $Z_2^A$69 constraints for all next-to-lightest odd particles considered (Guo et al., 22 Aug 2025).

Collider signatures are likewise diverse. In inert-doublet realizations, $Z_2^A$70 followed by $Z_2^A$71 yields the characteristic $Z_2^A$72 signature, and the exclusion limits from collider searches provide “a complementary detecting capability compared to the LFV and dark matter detections” (Guo et al., 2020). In the generalized model, sufficiently small mass splittings and Yukawas can instead generate long-lived charged states, producing “charged-track or disappearing-track signatures” (Hagedorn et al., 2018). Gauged versions add dilepton $Z_2^A$73 resonances as the “golden channel” (Wang et al., 2017, Han et al., 2018).

Electroweak precision tests have also been studied. In the residual-$Z_2^A$74 Dirac Scotogenic Model, the CDF-II $Z_2^A$75-boson mass result can be accommodated if the dark matter is mainly a singlet scalar, whereas a dark matter candidate mainly composed of an $Z_2^A$76 scalar doublet “cannot concurrently satisfy: (a) the dark matter relic density (b) the $Z_2^A$77 anomaly and (c) the direct detection constraints” (Chuliá et al., 2022).

The most recent broad phenomenological survey emphasizes that the Dirac scotogenic framework allows “novel low-mass scalar and fermionic dark matter, a feature not shared by its canonical Majorana counterpart.” After imposing neutrino masses, electroweak vacuum stability, charged-lepton-flavor violation, and dark-matter constraints, the surviving parameter space includes low-mass singlet-scalar and fermion dark matter made viable by coannihilation, alongside the more familiar inert-doublet and Higgs-portal regimes (Chuliá et al., 2024).

A final misconception is that the model is experimentally equivalent to its Majorana analogue except for the absence of neutrinoless double beta decay. The literature indicates a broader distinction. Dirac scotogenic constructions can feature viable Dirac fermion dark matter already in the minimal setup (Farzan et al., 2012), residual $Z_2^A$78 symmetries that tie dark-matter stability to “Diracness” (Chuliá et al., 2022), observable $Z_2^A$79 from thermalized $Z_2^A$80 (Borah et al., 2022), and low-mass scalar or fermionic dark matter regions “not shared by its canonical Majorana counterpart” (Chuliá et al., 2024).