Quasi-Dirac Neutrino Mass Models

• Quasi-Dirac neutrinos are defined by dominant Dirac mass terms with tiny Majorana corrections that split each neutrino state into two almost-degenerate Majorana partners.
• Various models, including seesaw mechanisms, radiative constructions, and symmetry-based frameworks, illustrate how small lepton-number-violating effects produce characteristic mass splittings.
• These models have practical implications across neutrino oscillation experiments, collider searches, and dark matter studies, offering distinct signatures such as slow oscillations and modified dilepton ratios.

Searching arXiv for recent and foundational papers on quasi-Dirac neutrino mass models. Quasi-Dirac neutrino mass models are frameworks in which Dirac mass terms dominate the neutrino sector while lepton-number-violating Majorana terms are sufficiently small that each Standard-Model-like neutrino mass eigenstate splits into two almost-degenerate Majorana states. Equivalently, the scenario can be characterized by pairs of neutrinos with almost degenerate masses, with splittings such as $\delta m_k^2 \equiv (m_k^+)^2-(m_k^-)^2 \ll m_k^2$, and it interpolates between the exact Dirac limit and generic Majorana neutrinos (Anamiati et al., 2017, Carloni et al., 25 Mar 2025, Anamiati et al., 2019). Across the recent literature, quasi-Dirac constructions appear in low-scale and high-scale seesaw settings, radiative models, $B-L$ gauge extensions with family symmetry, and frameworks that connect neutrino mass to dark matter or leptogenesis [(Arbeláez et al., 2021); (Machado et al., 2011); (Nga et al., 30 Nov 2025); (Fong et al., 2020)].

1. Formal definition and mass-matrix structure

In the simplest one-generation formulation, one introduces both a Dirac mass term $m_D$ and two small Majorana masses $m_L,m_R$. In the two-component basis $(\nu_L,N^c)$, the mass part of the Lagrangian is

$$L_{\rm mass} = -\frac12 \Bigl(\overline{\nu_L^c},\overline{N}\Bigr) \begin{pmatrix} m_L & m_D\ m_D & m_R \end{pmatrix} \begin{pmatrix} \nu_L\ N^c \end{pmatrix} +\text{h.c.}\,.$$

In the limit $m_L,m_R\to0$, lepton number is exact and the two states form a Dirac fermion of mass $m_D$ (Anamiati et al., 2019).

Defining

$$\varepsilon\equiv\frac{m_L+m_R}{2m_D},\qquad \theta\equiv\frac{m_L-m_R}{4m_D},$$

with $\varepsilon,\theta\ll1$, the mass eigenvalues are

$B-L$0

and the small splitting is $B-L$1. Here $B-L$2 measures the departure from maximal active-sterile mixing, while $B-L$3 controls the quasi-Dirac splitting (Anamiati et al., 2019).

For three active and three sterile Weyl fields, the generic quasi-Dirac structure is encoded in the $B-L$4 symmetric mass matrix

$B-L$5

so that the six Weyl spinors pair up into three Dirac fermions in the exact Dirac limit, and small nonzero $B-L$6 split each mass-degenerate pair into two Majorana states (Anamiati et al., 2017). In oscillation language, the $B-L$7 charged-current mixing matrix can be parametrized by 12 real angles and 12 phases; in the exactly degenerate limit there is a $B-L$8 redundancy among each degenerate pair, so only 13 real parameters survive (Anamiati et al., 2017).

This formalism makes clear that “quasi-Dirac” is not a single model but a regime of parameter space. The physical content depends on how the small Majorana terms arise, how the quasi-Dirac pairs couple to charged leptons and gauge bosons, and whether the dominant observables are oscillatory, collider-based, cosmological, or connected to dark sectors.

2. Symmetry realizations and model architectures

Several explicit constructions realize quasi-Dirac neutrinos through distinct symmetry patterns and field contents.

Model realization	Core ingredients	Characteristic relation
General oscillation formalism (Anamiati et al., 2017)	Active $B-L$9 and sterile $m_D$0 with $m_D$1	Three quasi-Dirac pairs; 12 angles and 12 phases
Minimal linear seesaw (Arbeláez et al., 2021)	Add singlets $m_D$2 and $m_D$3; $m_D$4	$m_D$5
$m_D$6 model (Machado et al., 2011)	Exotic $m_D$7 charges, $m_D$8 family symmetry, two Higgs doublets and singlets	Tree-level tribimaximal mixing and one quasi-Dirac pair
Radiative model with inert scalar (Nga et al., 30 Nov 2025)	Vectorlike singlet $m_D$9, inert doublet $m_L,m_R$0, $m_L,m_R$1, approximate $m_L,m_R$2	$m_L,m_R$3 and radiative neutrino mass

In the $m_L,m_R$4 model with local symmetry and exotic right-handed-neutrino charges, anomaly cancellation forces the three right-handed neutrinos to carry non-standard $m_L,m_R$5: two with $m_L,m_R$6 and one with $m_L,m_R$7. The two $m_L,m_R$8 fields are placed in an $m_L,m_R$9 doublet, while the third is in an $(\nu_L,N^c)$0 singlet. After integrating out two heavy right-handed neutrinos, the effective neutrino Yukawa sector contains one Dirac-type term and two effective Majorana terms, producing a tree-level $(\nu_L,N^c)$1 spectrum with two non-degenerate Majorana states and one quasi-Dirac pair (Machado et al., 2011).

In the minimal linear seesaw model, one adds two kinds of gauge-singlet neutrinos $(\nu_L,N^c)$2 and $(\nu_L,N^c)$3 per family. In the basis $(\nu_L,N^c)$4 the neutrino mass matrix is

$(\nu_L,N^c)$5

with $(\nu_L,N^c)$6. The block-diagonalized light mass matrix is

$(\nu_L,N^c)$7

while each heavy pair is quasi-Dirac, with the exact relation $(\nu_L,N^c)$8 (Arbeláez et al., 2021).

In the radiative model of a neutral vectorlike fermion and inert scalar doublet, the gauge symmetry is the Standard Model one, supplemented by a $(\nu_L,N^c)$9 under which the new fields are odd and all Standard Model fields are even. There is also an accidental lepton-like $$L_{\rm mass} = -\frac12 \Bigl(\overline{\nu_L^c},\overline{N}\Bigr) \begin{pmatrix} m_L & m_D\ m_D & m_R \end{pmatrix} \begin{pmatrix} \nu_L\ N^c \end{pmatrix} +\text{h.c.}\,.$$0 broken softly by small Majorana masses $$L_{\rm mass} = -\frac12 \Bigl(\overline{\nu_L^c},\overline{N}\Bigr) \begin{pmatrix} m_L & m_D\ m_D & m_R \end{pmatrix} \begin{pmatrix} \nu_L\ N^c \end{pmatrix} +\text{h.c.}\,.$$1 and by the small Yukawa coupling $$L_{\rm mass} = -\frac12 \Bigl(\overline{\nu_L^c},\overline{N}\Bigr) \begin{pmatrix} m_L & m_D\ m_D & m_R \end{pmatrix} \begin{pmatrix} \nu_L\ N^c \end{pmatrix} +\text{h.c.}\,.$$2. In the basis $$L_{\rm mass} = -\frac12 \Bigl(\overline{\nu_L^c},\overline{N}\Bigr) \begin{pmatrix} m_L & m_D\ m_D & m_R \end{pmatrix} \begin{pmatrix} \nu_L\ N^c \end{pmatrix} +\text{h.c.}\,.$$3, the neutral-fermion mass matrix is

$$L_{\rm mass} = -\frac12 \Bigl(\overline{\nu_L^c},\overline{N}\Bigr) \begin{pmatrix} m_L & m_D\ m_D & m_R \end{pmatrix} \begin{pmatrix} \nu_L\ N^c \end{pmatrix} +\text{h.c.}\,.$$4

so that the mass splitting satisfies $$L_{\rm mass} = -\frac12 \Bigl(\overline{\nu_L^c},\overline{N}\Bigr) \begin{pmatrix} m_L & m_D\ m_D & m_R \end{pmatrix} \begin{pmatrix} \nu_L\ N^c \end{pmatrix} +\text{h.c.}\,.$$5, yielding a quasi-Dirac pair with $$L_{\rm mass} = -\frac12 \Bigl(\overline{\nu_L^c},\overline{N}\Bigr) \begin{pmatrix} m_L & m_D\ m_D & m_R \end{pmatrix} \begin{pmatrix} \nu_L\ N^c \end{pmatrix} +\text{h.c.}\,.$$6 (Nga et al., 30 Nov 2025).

These realizations show that quasi-Diracness can emerge from approximate global symmetry, local gauge symmetry with exotic charge assignments, or seesaw textures with suppressed lepton-number violation.

3. Neutrino-mass generation mechanisms

Quasi-Dirac neutrino models are often organized by the mechanism that turns small lepton-number violation into observable light-neutrino mass splittings.

In the radiative inert-doublet construction, neutrino masses arise at one loop through $$L_{\rm mass} = -\frac12 \Bigl(\overline{\nu_L^c},\overline{N}\Bigr) \begin{pmatrix} m_L & m_D\ m_D & m_R \end{pmatrix} \begin{pmatrix} \nu_L\ N^c \end{pmatrix} +\text{h.c.}\,.$$7 with $$L_{\rm mass} = -\frac12 \Bigl(\overline{\nu_L^c},\overline{N}\Bigr) \begin{pmatrix} m_L & m_D\ m_D & m_R \end{pmatrix} \begin{pmatrix} \nu_L\ N^c \end{pmatrix} +\text{h.c.}\,.$$8 in the loop. In the quasi-Dirac limit $$L_{\rm mass} = -\frac12 \Bigl(\overline{\nu_L^c},\overline{N}\Bigr) \begin{pmatrix} m_L & m_D\ m_D & m_R \end{pmatrix} \begin{pmatrix} \nu_L\ N^c \end{pmatrix} +\text{h.c.}\,.$$9, the small splitting of the quasi-Dirac masses suitably suppresses neutrino mass while preserving viable dark matter annihilation, direct detection, and charged lepton flavor violation. Numerically, taking $m_L,m_R\to0$0, $m_L,m_R\to0$1, $m_L,m_R\to0$2, $m_L,m_R\to0$3 reproduces $m_L,m_R\to0$4 (Nga et al., 30 Nov 2025).

In the linear seesaw realization, the same matrix structure that gives the light mass matrix also fixes the heavy-pair splitting. The heavy masses are

$m_L,m_R\to0$5

and therefore

$m_L,m_R\to0$6

The heavy quasi-Dirac splitting is thus exactly the light-neutrino mass (Arbeláez et al., 2021).

In the quasi-Dirac leptogenesis framework, one considers one generation of Standard Model doublet $m_L,m_R\to0$7, mirror doublet $m_L,m_R\to0$8, and two heavy singlets $m_L,m_R\to0$9. The heavy-sector mass matrix

$m_D$0

gives

$m_D$1

so the heavy states form a quasi-Dirac pair. At low energy, the light sector has

$m_D$2

and hence

$m_D$3

In this setup, the same quasi-Diracness that splits the heavy pair also splits the light pair (Fong et al., 2020).

Radiative corrections can also generate quasi-Dirac splittings in otherwise more constrained textures. In the $m_D$4 solar-neutrino model with tribimaximal mixing at tree level, loop-induced Majorana mass insertions are parametrized by small dimensionless parameters $m_D$5. In the CASE A approximation, one may take $m_D$6, and the quasi-Dirac pair splits as

$m_D$7

with $m_D$8 (Rossi-Torres et al., 2013).

4. Oscillation phenomenology across solar, terrestrial, and astrophysical baselines

The defining experimental signature of quasi-Dirac neutrinos is the appearance of very slow oscillations driven by tiny mass splittings and, in the general case, by additional mixing angles and phases.

In the six-state formalism, the flavor-transition amplitude is

$m_D$9

with $$\varepsilon\equiv\frac{m_L+m_R}{2m_D},\qquad \theta\equiv\frac{m_L-m_R}{4m_D},$$0. One-parameter fits with all other new angles set to zero give at 95% CL

$$\varepsilon\equiv\frac{m_L+m_R}{2m_D},\qquad \theta\equiv\frac{m_L-m_R}{4m_D},$$1

from solar data, and

$$\varepsilon\equiv\frac{m_L+m_R}{2m_D},\qquad \theta\equiv\frac{m_L-m_R}{4m_D},$$2

from atmospheric and long-baseline accelerator/reactor data. However, with suitable changes to the lepton mixing matrix, limits on such mass splittings are much weaker, or even completely absent. In particular, for special “blind” directions with $$\varepsilon\equiv\frac{m_L+m_R}{2m_D},\qquad \theta\equiv\frac{m_L-m_R}{4m_D},$$3 or $$\varepsilon\equiv\frac{m_L+m_R}{2m_D},\qquad \theta\equiv\frac{m_L-m_R}{4m_D},$$4, $$\varepsilon\equiv\frac{m_L+m_R}{2m_D},\qquad \theta\equiv\frac{m_L-m_R}{4m_D},$$5 and $$\varepsilon\equiv\frac{m_L+m_R}{2m_D},\qquad \theta\equiv\frac{m_L-m_R}{4m_D},$$6 lose any $$\varepsilon\equiv\frac{m_L+m_R}{2m_D},\qquad \theta\equiv\frac{m_L-m_R}{4m_D},$$7 dependence (Anamiati et al., 2017).

Solar neutrino data provide a particularly direct test when a quasi-Dirac pair induces $$\varepsilon\equiv\frac{m_L+m_R}{2m_D},\qquad \theta\equiv\frac{m_L-m_R}{4m_D},$$8. In the four-flavor basis, the effective Hamiltonian in matter is

$$\varepsilon\equiv\frac{m_L+m_R}{2m_D},\qquad \theta\equiv\frac{m_L-m_R}{4m_D},$$9

and to leading order the quasi-Dirac pair gives

$\varepsilon,\theta\ll1$0

A scan over $\varepsilon,\theta\ll1$1 using Homestake, SAGE/GALLEX, Super-Kamiokande, SNO, and Borexino data yields the 2$\varepsilon,\theta\ll1$2 region

$\varepsilon,\theta\ll1$3

(Rossi-Torres et al., 2013).

When the splittings are too small to be directly resolved, the quasi-Dirac signal is encoded in nonstandard mixing combinations. In the DUNE and JUNO analysis, the relevant reparametrization-invariant quantities are seven combinations $\varepsilon,\theta\ll1$4 built from $\varepsilon,\theta\ll1$5. In the exact Dirac limit one has

$\varepsilon,\theta\ll1$6

and a corresponding relation for $\varepsilon,\theta\ll1$7; violations of these equalities signal quasi-Diracness (Anamiati et al., 2019).

Diffuse high-energy astrophysical neutrinos probe an entirely different splitting range. Assuming the source redshift distribution follows the star-formation-rate density, one obtains

$\varepsilon,\theta\ll1$8

with

$\varepsilon,\theta\ll1$9

and $B-L$00. Using IceCube all-sky flux measurements from TeV to PeV energies, values of $B-L$01 in the range $B-L$02 are disfavored at $B-L$03, while there is a mildly significant preference at $B-L$04 driven by the low-energy tension between cascade and track samples (Carloni et al., 25 Mar 2025).

5. Collider, flavor, dark-matter, and leptogenesis connections

A distinctive feature of quasi-Dirac model building is that the same small lepton-number-violating parameters that control oscillations can also govern collider observables, flavor violation, dark matter, and baryogenesis.

For heavy quasi-Dirac neutrinos at colliders, a central discriminator is the same-sign to opposite-sign dilepton ratio

$B-L$05

The Majorana limit is $B-L$06, the Dirac limit is $B-L$07, and the quasi-Dirac regime is $B-L$08. In the linear seesaw scenario, because $B-L$09 is at most $B-L$10, the condition $B-L$11 requires extremely small heavy-light mixing. Numerical scans show that for a lightest neutrino mass $B-L$12 in the range $B-L$13, the quasi-Dirac window occurs for $B-L$14 when $B-L$15 and for $B-L$16 when $B-L$17, with mixings $B-L$18. For $B-L$19, displaced vertices with decay lengths $B-L$20 become relevant (Arbeláez et al., 2021).

In the radiative inert-doublet model, the lightest $B-L$21-odd scalar, taken to be $B-L$22, is stable and serves as the dark matter candidate. The small splitting $B-L$23 forbids $B-L$24-exchange in direct detection. For $B-L$25, the leading annihilation channels are $B-L$26 via gauge quartic terms and $B-L$27 via the $B-L$28 portal, with

$B-L$29

yielding $B-L$30. Benchmark solutions include $B-L$31 for gauge-portal domination, $B-L$32 for Higgs-portal domination, and $B-L$33 for the mixed case. Direct detection gives $B-L$34 for $B-L$35 and $B-L$36, in agreement with LUX/XENON limits. The same model predicts $B-L$37, and for $B-L$38 with $B-L$39 one saturates the current MEG bound (Nga et al., 30 Nov 2025).

In the leptogenesis construction, quasi-Diracness enhances the self-energy contribution to the CP asymmetry of heavy-singlet decays. In the resonant limit $B-L$40, the total CP asymmetry can reach $B-L$41, and in the $B-L$42-symmetric benchmarks the upper bound is

$B-L$43

Successful resonant leptogenesis requires $B-L$44 for weak-scale $B-L$45–TeV, which translates into

$B-L$46

In the same model, charged-lepton flavor violation is unobservably small, $B-L$47, and neutrinoless double-$B-L$48 decay is suppressed, $B-L$49 (Fong et al., 2020).

6. Constraints, blind directions, and future tests

The phenomenology of quasi-Dirac neutrino mass models is shaped by a recurrent tension: one-parameter perturbations of the Dirac limit can be tightly constrained, yet more general deformations of the mixing matrix can hide the same splittings. The six-state oscillation analysis explicitly demonstrates that very stringent bounds on mass splittings follow in restricted parameter scans, whereas suitable changes to the lepton mixing matrix can weaken or eliminate those bounds (Anamiati et al., 2017).

For the “small-splitting, angle-dominated” regime, DUNE and JUNO improve the sensitivity to the reparametrization-invariant combinations $B-L$50. In the combined DUNE + JUNO analysis with a 3% Daya Bay prior on the effective reactor angle, one finds at $B-L$51

$B-L$52

and at $B-L$53

$B-L$54

Defining the “Diracness” test

$B-L$55

the exact Dirac limit is $B-L$56, the combined sensitivity is $B-L$57 at $B-L$58, and if true $B-L$59 one would claim discovery of quasi-Dirac neutrinos at $B-L$60 (Anamiati et al., 2019).

Solar and atmospheric oscillation experiments remain directly relevant to model building. In the leptogenesis scenario, future precision could cover most of the band $B-L$61 required by successful resonant leptogenesis (Fong et al., 2020). In the astrophysical domain, more years of IceCube data will shrink statistical errors and probe down to $B-L$62, while KM3NeT and IceCube-Gen2 can test the $B-L$63 hint at $B-L$64 or exclude it decisively (Carloni et al., 25 Mar 2025).

Taken together, these results indicate that quasi-Dirac neutrino mass models are best regarded as a broad class of softly lepton-number-violating constructions rather than a single mechanism. Their common signature is the emergence of almost-degenerate Majorana pairs with characteristic oscillation, collider, flavor, and cosmological consequences; their principal model-building challenge is to explain why the Majorana terms are small enough to preserve quasi-Dirac behavior while still generating observable effects (Anamiati et al., 2017, Anamiati et al., 2019, Nga et al., 30 Nov 2025).