--- title: Quasi-Dirac Neutrino Mass Models url: https://www.emergentmind.com/topics/quasi-dirac-neutrino-mass-models type: topic --- # Quasi-Dirac Neutrino Mass Models 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 [1710.06249; 2503.19960; 1907.00980]. 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 [2104.08023; 1105.6064; 2512.00854; 2007.09158]. ## 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 $m_D$ [1907.00980]. 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 \[ m_{1,2}=m_D(1\pm\varepsilon), \] and the small splitting is $\Delta m^2\approx 4m_D^2\varepsilon$. Here $\theta$ measures the departure from maximal active-sterile mixing, while $\varepsilon$ controls the quasi-Dirac splitting [1907.00980]. For three active and three sterile Weyl fields, the generic quasi-Dirac structure is encoded in the $6\times6$ symmetric mass matrix \[ m_\nu= \begin{pmatrix} m_L & M_D\\ M_D^T & m_R \end{pmatrix}, \qquad m_L,m_R\ll M_D, \] so that the six Weyl spinors pair up into three Dirac fermions in the exact Dirac limit, and small nonzero $m_L,m_R$ split each mass-degenerate pair into two Majorana states [1710.06249]. In oscillation language, the $3\times6$ charged-current mixing matrix can be parametrized by 12 real angles and 12 phases; in the exactly degenerate limit there is a $U(2)^3$ redundancy among each degenerate pair, so only 13 real parameters survive [1710.06249]. 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 [1710.06249] | Active $\nu$ and sterile $N^c$ with $m_L,m_R\ll M_D$ | Three quasi-Dirac pairs; 12 angles and 12 phases | | Minimal linear seesaw [2104.08023] | Add singlets $N$ and $S$; $M_\varepsilon\ll m_D\ll M_R$ | $\Delta M=m_\nu$ | | $B-L\otimes S_3$ model [1105.6064] | Exotic $B-L$ charges, $S_3$ family symmetry, two Higgs doublets and singlets | Tree-level tribimaximal mixing and one quasi-Dirac pair | | Radiative model with inert scalar [2512.00854] | Vectorlike singlet $N_L,N_R$, inert doublet $\eta$, $Z_2$, approximate $U(1)_L$ | $\Delta M/M\ll1$ and radiative neutrino mass | In the $B-L$ model with local symmetry and exotic right-handed-neutrino charges, anomaly cancellation forces the three right-handed neutrinos to carry non-standard $B-L$: two with $B-L=-4$ and one with $B-L=+5$. The two $B-L=-4$ fields are placed in an $S_3$ doublet, while the third is in an $S_3$ 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 $4\times4$ spectrum with two non-degenerate Majorana states and one quasi-Dirac pair [1105.6064]. In the minimal linear seesaw model, one adds two kinds of gauge-singlet neutrinos $N$ and $S$ per family. In the basis $\{\nu_L^c,N,S\}$ the neutrino mass matrix is \[ M_\nu= \begin{pmatrix} 0 & m_D & M_\varepsilon\\ m_D^T & 0 & M_R\\ M_\varepsilon^T & M_R^T & 0 \end{pmatrix}, \] with $M_\varepsilon\ll m_D\ll M_R$. The block-diagonalized light mass matrix is \[ m_\nu = m_D M_R^{-1}M_\varepsilon^T + \bigl(m_D M_R^{-1}M_\varepsilon^T\bigr)^T, \] while each heavy pair is quasi-Dirac, with the exact relation $\Delta M = m_\nu$ [2104.08023]. In the radiative model of a neutral vectorlike fermion and inert scalar doublet, the gauge symmetry is the Standard Model one, supplemented by a $Z_2$ under which the new fields are odd and all Standard Model fields are even. There is also an accidental lepton-like $U(1)_L$ broken softly by small Majorana masses $\mu_L,\mu_R$ and by the small Yukawa coupling $h'$. In the basis $(N_L,N_R^c)$, the neutral-fermion mass matrix is \[ {\cal M}_N= \begin{pmatrix} \mu_L & M\\ M & \mu_R \end{pmatrix}, \qquad \mu_{L,R}\ll M, \] so that the mass splitting satisfies $\Delta M\simeq \mu_L+\mu_R\ll M$, yielding a quasi-Dirac pair with $\Delta M/M\ll1$ [2512.00854]. 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 $l_L\to N_i\to l_L$ with $\eta^0=(S+iA)/\sqrt2$ in the loop. In the quasi-Dirac limit $(\mu,h'\ll M)$, 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 $\lambda_5\sim0.01$, $h\sim0.05$, $\mu_{\rm eff}/M\sim10^{-5}$, $M\sim1\,{\rm TeV}$ reproduces $m_\nu\sim0.05\,{\rm eV}$ [2512.00854]. 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 \[ M_{N_{a,b}} \simeq \frac{M_R}{2}+\frac{m_D^2}{4M_R} \mp \frac12\Bigl(m_D M_R^{-1}M_\varepsilon^T + (m_D M_R^{-1}M_\varepsilon^T)^\dagger\Bigr), \] and therefore \[ \Delta M = m_D M_R^{-1}M_\varepsilon^T + (m_D M_R^{-1}M_\varepsilon^T)^\dagger = m_\nu. \] The heavy quasi-Dirac splitting is thus exactly the light-neutrino mass [2104.08023]. In the quasi-Dirac leptogenesis framework, one considers one generation of Standard Model doublet $\nu_L$, mirror doublet $\nu'_L$, and two heavy singlets $N_R,N'_R$. The heavy-sector mass matrix \[ {\cal M}_{\rm heavy}= \begin{pmatrix} m & M\\ M & m' \end{pmatrix} \] gives \[ M_{1,2}\simeq M\mp\frac{m+m'}{2},\qquad \Delta M\simeq m+m', \] so the heavy states form a quasi-Dirac pair. At low energy, the light sector has \[ m_\nu=\frac{y\,y'\,v\,f}{M},\qquad \delta m\simeq\frac{|w\,v^2+w' f^2|}{M}, \] and hence \[ \Delta m^2\simeq 4m_\nu\delta m. \] In this setup, the same quasi-Diracness that splits the heavy pair also splits the light pair [2007.09158]. Radiative corrections can also generate quasi-Dirac splittings in otherwise more constrained textures. In the $3+1$ solar-neutrino model with tribimaximal mixing at tree level, loop-induced Majorana mass insertions are parametrized by small dimensionless parameters $\epsilon_l,\epsilon'_l$. In the CASE A approximation, one may take $\epsilon_e=\epsilon_\mu=\epsilon_\tau\equiv\epsilon$, and the quasi-Dirac pair splits as \[ m_{2,4}\simeq m_D(1\mp\epsilon),\qquad \Delta m_{42}^2\simeq 4m_D^2\epsilon, \] with $\epsilon\lesssim10^{-6}$ [1302.5590]. ## 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 \[ A(\nu_\alpha\to\nu_\beta)=\sum_{j=1}^6 U_{\beta j}U_{\alpha j}^* \exp[-i\,m_j^2L/(2E)], \] with $m_{i\pm}^2=m_i^2\pm(\delta\epsilon_i)^2/2$. One-parameter fits with all other new angles set to zero give at 95% CL \[ \delta\epsilon_1^2\lesssim1.7\times10^{-11}\,{\rm eV}^2,\qquad \delta\epsilon_2^2\lesssim1.7\times10^{-11}\,{\rm eV}^2, \] from solar data, and \[ \delta\epsilon_3^2\lesssim1\times10^{-5}\,{\rm eV}^2 \] 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 $\phi_i=\pi/4$ or $3\pi/4$, $P_{ee}$ and $P_{\mu\mu}$ lose any $\delta\epsilon_i^2$ dependence [1710.06249]. Solar neutrino data provide a particularly direct test when a quasi-Dirac pair induces $\nu_e\to\nu_s$. In the four-flavor basis, the effective Hamiltonian in matter is \[ H_{\rm eff}(r)=\frac{1}{2E}M^\nu M^{\nu\dagger}+V(r), \qquad V(r)=\sqrt2\,G_F\bigl[n_e(r)-\tfrac12 n_n(r)\bigr], \] and to leading order the quasi-Dirac pair gives \[ P_{e\to s}(E)\simeq \sin^2(2\theta_{es}) \sin^2\!\Bigl[\frac{\Delta m_{42}^2L}{4E}\Bigr], \qquad \theta_{es}\simeq \epsilon/\sqrt6. \] A scan over $(\epsilon,m_1)$ using Homestake, SAGE/GALLEX, Super-Kamiokande, SNO, and Borexino data yields the 2$\sigma$ region \[ 5\times10^{-9}\lesssim \epsilon \lesssim 10^{-6}, \qquad m_4\in[0.01,0.2]\ {\rm eV} \] [1302.5590]. 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 $X_1,\ldots,X_7$ built from $U_{e2},U_{e3},U_{e5},U_{e6},U_{\mu2},U_{\mu3},U_{\mu5},U_{\mu6}$. In the exact Dirac limit one has \[ X_5=X_1X_3,\qquad X_6=X_2X_4, \] and a corresponding relation for $\Re X_7$; violations of these equalities signal quasi-Diracness [1907.00980]. Diffuse high-energy astrophysical neutrinos probe an entirely different splitting range. Assuming the source redshift distribution follows the star-formation-rate density, one obtains \[ P_{\alpha\beta}(E)=\frac12\sum_{k=1}^3 |U_{\alpha k}|^2|U_{\beta k}|^2 \left[1+\cos\!\left(\frac{\delta m_k^2L_{\rm eff}}{2E}\right)\right], \] with \[ L_{\rm eff}=\int_0^{z_{\max}}\frac{dz'}{H(z')(1+z')^2}, \] and $z_{\max}\approx6$. Using IceCube all-sky flux measurements from TeV to PeV energies, values of $\delta m^2$ in the range $5\times10^{-19}\,{\rm eV}^2\le\delta m^2\le3\times10^{-18}\,{\rm eV}^2$ are disfavored at $>3\sigma$, while there is a mildly significant preference at $\delta m^2\simeq1.9\times10^{-19}\,{\rm eV}^2$ driven by the low-energy tension between cascade and track samples [2503.19960]. ## 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 \[ R_{\ell\ell}=\frac{\# {\rm SS}}{\# {\rm OS}} =\frac{\Delta M^2}{2\Gamma^2+\Delta M^2}. \] The Majorana limit is $\Gamma\ll\Delta M\implies R_{\ell\ell}\to1$, the Dirac limit is $\Delta M\ll\Gamma\implies R_{\ell\ell}\to0$, and the quasi-Dirac regime is $\Delta M\sim\Gamma\implies 03\sigma$ or exclude it decisively [2503.19960]. 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 [1710.06249; 1907.00980; 2512.00854]. Source: https://www.emergentmind.com/topics/quasi-dirac-neutrino-mass-models