Quasi-Dirac Neutrinos: A Hybrid Mass Mechanism

• Quasi-Dirac neutrinos are mass eigenstates formed by combining nearly degenerate Majorana neutrinos with opposite CP parity to effectively mimic Dirac fermions.
• They arise from a small symmetry-breaking perturbation in models with both Dirac and Majorana mass terms, as seen in inverse seesaw, B-L, and radiative mass generation frameworks.
• Their tiny mass splittings lead to distinctive oscillation phenomena and experimental signatures across solar, atmospheric, astrophysical, and collider settings.

Quasi-Dirac neutrinos are mass eigenstates formed when two Majorana neutrinos of nearly equal mass but opposite CP parity combine to behave almost as a single Dirac fermion, up to a tiny mass splitting induced by weakly broken lepton number. In contrast to pure Dirac neutrinos (exact lepton number) or pure Majorana neutrinos (maximal lepton-number violation), quasi-Dirac states interpolate between these extremes and possess unique phenomenological signatures. The quasi-Dirac paradigm arises generically from a small symmetry-breaking perturbation in models containing both Dirac and Majorana mass terms, and is realized in various frameworks such as inverse and linear seesaw mechanisms, B–L models, and radiative mass generation schemes (Rossi-Torres et al., 2013, Nga et al., 30 Nov 2025, Machado et al., 2011, Arbeláez et al., 2021).

1. Theoretical Foundations: Mass Matrices and Mixing

Quasi-Dirac neutrinos require the coexistence of Dirac and (small) Majorana masses for each generation. In the minimal scenario, the $2 \times 2$ mass matrix takes the form: $$\mathcal{M}_\nu = \begin{pmatrix} m_L & m_D \ m_D & m_R \end{pmatrix}$$ where $m_D$ is a “large” Dirac mass and $m_L, m_R$ are “small” Majorana terms. If $m_L = m_R = 0$, the neutrino is Dirac; for $m_L,m_R \ll m_D$, the eigenstates are nearly degenerate Majorana neutrinos with small mass splitting $\Delta m \sim (m_L + m_R)$.

The diagonalization yields mass eigenstates: $$m_{1,2} \simeq m_D (1 \pm \varepsilon),\;\; \varepsilon = \frac{m_L + m_R}{2 m_D}$$ and mixing angle $\theta \sim \frac{m_L - m_R}{4 m_D}$. The eigenstates are

$$\nu_1 \simeq \tfrac{1}{\sqrt{2}}\left[(1+\theta)\nu_L + (1-\theta)N_R^c \right],\;\; \nu_2 \simeq \tfrac{i}{\sqrt{2}} \left[(-1+\theta)\nu_L + (1+\theta)N_R^c \right]$$

For three generations and added sterile states, the general $6\times6$ mass matrices require diagonalization by a unitary $U$ with up to 12 angles and 12 phases, reflecting the expanded parameter space in oscillation physics (Anamiati et al., 2017, Anamiati et al., 2019).

2. Realizations in Non-Minimal Models

Quasi-Dirac spectra naturally emerge from mechanisms that softly break lepton number, such as:

• Inverse and Linear Seesaws: These involve pairs of heavy singlet fermions with small lepton-number breaking terms $\mu$, leading to quasi-Dirac heavy neutrinos. The mass splitting is controlled via $\Delta M \simeq \mu$, with the light neutrino mass proportional to $\mu$ times (mixing)$^2$ (Anamiati et al., 2016, Arbeláez et al., 2021).
• $B-L$ and flavor symmetry models: Nonstandard $B-L$ charge assignments and discrete symmetries like $S_3$ or $S_4$ can protect or orchestrate the quasi-Dirac pattern at tree level, with the Dirac structure enforced and Majorana splittings induced only radiatively or by explicit symmetry breaking (Machado et al., 2011, Morisi et al., 2011, Rossi-Torres et al., 2013).
• Radiative inverse-seesaw and dark matter: Models with quasi-Dirac TeV-scale vectorlike fermions can generate neutrino masses radiatively and simultaneously stabilize scalar dark matter; active neutrino masses are doubly suppressed by both heavy mass scale and quasi-Dirac splitting, $m_\nu \sim (v^2/M)(\Delta M / M)$ (Nga et al., 30 Nov 2025).

3. Oscillation Phenomenology

Quasi-Dirac neutrinos introduce new oscillation modes driven by tiny mass splittings, yielding observable effects only over very long baselines or in the presence of high sensitivity:

• For each quasi-Dirac pair, oscillation probabilities include an extra term: $$P_{\alpha\beta} \sim \sin^2 2\theta \sin^2\left( \frac{\Delta m^2 L}{4E} \right )$$ where $\Delta m^2 = 2 m_D \Delta m$ is typically much smaller than standard oscillation scales (Rossi-Torres et al., 2013, Sen, 2022, Carloni et al., 25 Mar 2025). The splitting of Dirac pairs enables active-sterile (or left-right handed) oscillations with frequencies set by $\Delta m^2$.
• In the exact Dirac limit ($\Delta m^2 \rightarrow 0$), these long-wavelength oscillations vanish, while the presence of nonzero splittings in the range $10^{-12}$--$10^{-18}$ eV$^2$ leads to distinctive modulations in solar, reactor, atmospheric, supernova, or even astrophysical neutrino fluxes (Anamiati et al., 2019, Anamiati et al., 2017).
• In multi-generation scenarios, the full $6\times6$ mixing matrix introduces new parameters; even if splittings are too small to resolve, precision oscillation data can constrain nonstandard mixing angles and test “Diracness” through relationships among observable $X_i$ parameters (Anamiati et al., 2019).

4. Experimental Constraints and Signatures

A wide range of experiments have set bounds or revealed phenomenological windows for quasi-Dirac neutrinos:

• Solar and atmospheric oscillation data: Bounds on quasi-Dirac mass splittings are $\lesssim 10^{-12}$ eV$^2$ (solar sector) and $\lesssim 10^{-5}$ eV$^2$ (atmospheric sector), excluding substantial regions of parameter space (Rossi-Torres et al., 2013, Anamiati et al., 2017, Anamiati et al., 2019).
• Supernova neutrinos: SN1987A data constrain $\delta m^2 \sim 10^{-20}$ eV$^2$, and future core-collapse supernovae observed by DUNE or Hyper-K could reach $\delta m^2 \sim 10^{-22}$--$10^{-23}$ eV$^2$ (Sen, 2022).
• Astrophysical neutrinos: IceCube diffuse-flux data have recently excluded $$2\times 10^{-19} \lesssim \delta m^2 \lesssim 3 \times 10^{-18}$$ eV$^2$ at more than $3\sigma$ and observe a mild preference for $\delta m^2 \sim 2\times 10^{-19}$ eV$^2$, opening a new regime for experimental tests (Carloni et al., 25 Mar 2025).
• Laboratory searches: DUNE and JUNO will further tighten constraints on quasi-Dirac mixing angles and relations by precision oscillation fits; colliders search for the lepton-number violation ratio $R_{\ell\ell}$ in same-sign/opp-sign dilepton events as a probe of quasi-Dirac nature, with $$R_{\ell\ell} = \Delta M^2 / (2\Gamma^2 + \Delta M^2)$$, interpolating between 0 (Dirac) and 1 (Majorana) (Anamiati et al., 2016, Arbeláez et al., 2021).

Observable	Quasi-Dirac Signature	Sensitivity/bounds
Solar neutrino $P_{ee}$	Reduced survival, slow active-sterile beats	$\varepsilon^2 \lesssim 10^{-12}$ eV$^2$ (Rossi-Torres et al., 2013)
Supernova $\bar{\nu}_e$	Energy-dependent dips in spectrum	$\delta m^2 \sim 10^{-20}$ eV$^2$ (Sen, 2022)
IceCube flux	Suppressed low-energy cascade events	$\delta m^2 \sim 10^{-19}$ eV$^2$ (Carloni et al., 25 Mar 2025)
Same-sign dileptons (LHC)	$R_{\ell\ell} \in [0,1]$	$R_{\ell\ell}$ tracks $\Delta M/\Gamma$ (Anamiati et al., 2016)

5. Neutrinoless Double Beta Decay and Lepton Number Violation

In standard light-neutrino exchange, quasi-Dirac neutrinos suppress the effective Majorana mass $m_{ee}$ since the two nearly-degenerate Majorana components contribute with opposite CP phases, causing cancellation: $$m_{ee} \simeq \sum_i (\cos^2 \theta_i m_{+,i} - \sin^2 \theta_i m_{-,i}) \sim \delta m_i$$ The contribution is typically far below experimental sensitivity (Gu, 2011, Morisi et al., 2011). However, models can generate significant $0\nu\beta\beta$ signals via short-range tree-level heavy-scalar exchange, independently of the tiny neutrino splitting, allowing observable decay rates with quasi-Dirac spectra (Gu, 2011). The lower bound on $m_{ee}$ is increased by a factor of $\sim2$ in scenarios where one state is quasi-Dirac and the others are Majorana (Morisi et al., 2011).

6. Cosmological and Dark Matter Implications

Quasi-Dirac neutrinos are consistent with cosmological bounds on $\sum m_\nu$ due to sub-eV mass scales accessible in pure Dirac or quasi-Dirac limits (Rossi-Torres et al., 2013). In radiative scotogenic models at the TeV scale, quasi-Dirac heavy fermions can generate viable dark matter candidates—namely, inert scalar components (e.g., $A$)—with relic density and direct-detection cross sections compatible with current experimental bounds. The suppression of lepton-number violating processes by the tiny splitting ($\Delta M/M \ll 1$) simultaneously enables small neutrino masses and dark matter stability (Nga et al., 30 Nov 2025).

7. Outlook and Prospective Tests

Future terrestrial and astrophysical experiments will extend the search for quasi-Dirac physics:

• JUNO, DUNE: precision measurement of oscillation parameters and $X_i$ combinations can test Diracness and quasi-Dirac signatures well below the current percent level (Anamiati et al., 2019).
• IceCube-Gen2, KM3NeT: improved high-energy neutrino flux data, enabling deeper exclusion or discovery of $10^{-19}$ eV$^2$ splittings (Carloni et al., 25 Mar 2025).
• LHC, intensity frontier: displaced vertex and lepton-number violating searches probe the same-sign/opp-sign dilepton ratio across $0 < R_{\ell\ell} < 1$, discriminating between Dirac, Majorana, and quasi-Dirac scenarios (Anamiati et al., 2016, Arbeláez et al., 2021).
• Neutrinoless double-beta decay: nonstandard mechanisms may yield observable $0\nu\beta\beta$ rates even if $m_{ee}$ from light-quasi-Dirac neutrino exchange is suppressed (Gu, 2011, Morisi et al., 2011).
• Astrophysical and supernova neutrinos: Multi-messenger and long-baseline measurements reaching $\delta m^2 \sim 10^{-22}$ eV$^2$ test beyond-current parameter windows (Sen, 2022).

Collectively, the quasi-Dirac paradigm provides a testable, theoretically motivated bridge between Dirac and Majorana neutrino physics, impacting classic observables in oscillation experiments, collider signatures, and cosmological measurements. It remains a key target for experimental scrutiny in the next decade.