Dirac Seesaw Mechanism for Neutrino Masses

• Dirac seesaw models are mechanisms that generate naturally small Dirac neutrino masses by coupling SM lepton doublets with heavy fermions and extra scalar fields under conserved lepton number.
• They employ anomaly-free U(1)(B–L) charges and discrete flavor symmetries to forbid Majorana terms and establish predictive mass mixing patterns.
• These models predict distinct experimental signatures, such as the absence of neutrinoless double beta decay and potential TeV-scale collider signals, enabling practical tests of beyond-SM physics.

A Dirac seesaw for neutrino masses is a class of mechanisms that generate naturally small Dirac neutrino masses by leveraging the interplay between Standard Model (SM) lepton doublets, heavy singlet or multiplet fermions, and additional scalar fields, with lepton number conservation engineered at either the operator or symmetry level. Unlike the classic Majorana seesaw, where lepton number is violated by a large Majorana mass, Dirac seesaw frameworks maintain lepton number (L) as a symmetry, so the light neutrinos are Dirac fermions distinct from their antiparticles. These models typically embed the seesaw suppression either through the Dirac-analog of the type-I/III seesaws or via more exotic scalar and flavor symmetry structures.

1. Structural Foundation and Generic Mechanism

Dirac seesaw models extend the SM by adding heavy fermions (singlets, triplets, or other SU(2) multiplets) and at least one right-handed neutrino (ν_R), while enforcing lepton number conservation either as a global or (more naturally) as a local, anomaly-free symmetry. A prominent realization appears in anomaly-free U(1)₍B–L₎ models with unconventional charge assignments for ν_R, forbidding the standard tree-level Yukawa coupling (ℓ_L φ ν_R) and thus suppressing unwanted Majorana mass terms (Ma et al., 2014, Ma et al., 2015).

The general structure of the minimal Dirac seesaw is as follows: left-handed neutrino doublets (ν_L) are coupled to heavy vector-like fermions N and, via at least one scalar S, to ν_R. The effective Lagrangian contains terms such as: $$-\mathcal{L} \supset y \, \overline{ℓ}_L \phi N_R + M_N\, \overline{N}_L N_R + λ\, \overline{N}_L S\, ν_R + \text{h.c.}$$ where y and λ are Yukawa couplings, M_N is a large Dirac mass, and S is an additional scalar with an appropriately chosen charge to allow the Yukawa couplings but forbid (ν_R ν_R) Majorana terms.

Upon electroweak symmetry breaking (and S acquiring a vacuum expectation value), the light neutrino Dirac mass is generated via a seesaw-suppressed effective operator: $$m_ν \sim \frac{y λ\, \langle φ \rangle\, \langle S\rangle}{M_N}$$ where the suppression by M_N ensures naturally small neutrino masses even for order-one couplings and vacuum expectation values (VEVs) at the electroweak or TeV scale.

2. Symmetry Realizations and Model Building

To maintain lepton number conservation and ensure the absence of Majorana mass terms, specific symmetry assignments are required:

• Exotic U(1)₍B–L₎ charges: Assigning right-handed neutrinos charges such as (+5, –4, –4), with heavy mediators carrying B–L = –1, cancels gauge anomalies and prevents Majorana mass generation (Ma et al., 2014, Ma et al., 2015).
• Additional scalars with non-standard charges: Scalars such as X₃ (B–L = +3), as opposed to the canonical X₂ (B–L = +2) in standard Majorana seesaws, mediate the required couplings while precluding lepton-number–violating terms (Ma et al., 2014).
• Flavor symmetries: Discrete non-Abelian groups such as S₃ enable texture zeros and enforce maximal or hierarchical mixing structures in both Dirac and charged lepton sectors. Assignments of SM lepton doublets and heavy mediators to S₃ multiplets (singlets or doublets) determine the allowed Yukawa matrices and lead, e.g., to maximal atmospheric mixing (θ₂₃ ≈ π/4) (Ma et al., 2014).
• Generalized inverse seesaw: Supersymmetric scenarios or extended left-right symmetric models naturally realize Dirac seesaw mechanisms using new vector-like fields and impose discrete symmetries (e.g., ℤ₃ subgroups) to forbid unwanted terms (Chuliá et al., 2020).

3. Representative Model Structures

A prototypical Dirac seesaw neutrino mass matrix takes the form (for a single generation and canonical S₃/B–L assignment) (Ma et al., 2014): $$\mathcal{M}_{(ν,N)} = \begin{pmatrix} 0 & m_0 \ m_3 & M \end{pmatrix}$$ with:

• $m_0 \sim y \langle φ \rangle$ from the SM Higgs coupling to N,
• $m_3 \sim f \langle X_3 \rangle$ from the new scalar coupling,
• $M$ the heavy singlet Dirac mass.

After integrating out the heavy mediator, the effective light neutrino Dirac mass is: $m_ν \sim \frac{m_0\, m_3}{M}$ A key feature is that both $m_0$ and $m_3$ are induced by scalar VEVs, and the absence of Majorana terms is a consequence of symmetry assignments, not fine-tuning.

4. Phenomenological and Oscillation Consequences

Dirac seesaw models make several testable predictions:

• Absence of neutrinoless double beta decay: As L is conserved, these models preclude the $0\nu\beta\beta$ process at tree level, which distinguishes them sharply from Majorana seesaw scenarios (Ma et al., 2014, Ma et al., 2015).
• Hierarchical or maximal mixing patterns: By suitable choices of flavor symmetry representations and VEV alignments, the mass matrices can accommodate normal or inverted hierarchy, degenerate or hierarchical masses, and realistic PMNS mixing matrices. Maximal atmospheric mixing, for instance, emerges from μ–τ symmetric structures realized in specific S₃ assignments (Ma et al., 2014).
• Parameter correlations: The imposition of S₃ or similar symmetry reduces the number of independent Yukawa couplings, resulting in tight correlations among mixing angles and mass differences, yielding predictive textures subject to direct experimental test.
• Lepton number conservation: The conserved global (or gauged) lepton number implies lepton-number–violating signatures are strongly suppressed or absent.

5. Theoretical Implications and Extensions

Dirac seesaw constructions pave the way for additional model building and phenomenology:

• Connecting to dark matter stability: The introduction of right-handed neutrinos with non-standard quantum numbers under U(1) extensions naturally ties their stability and interactions to those of candidate dark sectors (Chen et al., 2022).
• Accommodating baryogenesis: Some realizations link the smallness of Dirac neutrino masses to mechanisms for baryogenesis via leptogenesis that exploit heavy mediator decays with CP violation, even in the absence of Majorana mass terms, by tracking lepton and anti-lepton sectors using asymmetries in heavy fermion decays (Chen et al., 2022).
• Collider signatures: Heavy singlet Dirac fermions and new scalar states can yield distinctive signals at the LHC or future colliders, particularly if some mediators have masses in the TeV range. Non-unitarity in the lepton mixing matrix and lepton flavor violation are also possible when the mixing of active–sterile states is sizable (Chen et al., 2022, Chuliá et al., 2020).
• GUT and left-right symmetric embedding: The Dirac seesaw mechanism is compatible with embeddings within Pati–Salam, SU(5), or SO(10) unified frameworks, as well as with left–right symmetric extensions where discrete or gauge symmetries are appropriately arranged to protect lepton number (Chen et al., 2022).

6. Comparison with Majorana Seesaw and Model Classification

In contrast to the familiar Majorana seesaws (type I/II/III) where the Majorana mass term breaks lepton number by two units, Dirac seesaw models achieve suppression of neutrino mass entirely through Dirac mass entries and their interplay with heavy mediator masses and scalar VEVs, avoiding direct L violation. The table below outlines representative distinctions:

Feature	Majorana Seesaw	Dirac Seesaw
Lepton number (L)	Broken (ΔL=2), L not conserved	Conserved globally or locally
Neutrino nature	Majorana (ν ≡ ν̄)	Dirac (ν ≠ ν̄)
Mass generation	Type I: $m_\nu \sim y^2v^2/M$	$m_\nu \sim y_1 y_2 v v'/M$
0νββ decay	Typically allowed	Forbidden at tree level
Scalar sector	Often minimal, or triplets	Extended: new scalars, S₃/A₄, etc.
Flavor symmetry	Optional, parameter reduction	Often critical for predictive power

Model classification is determined by the choice of fermion and scalar field content, the symmetry assignments (including gauge, global, and flavor symmetries), and the structure of the vacuum alignment.

7. Future Prospects and Experimental Tests

Dirac seesaw models are subject to ongoing and forthcoming tests:

• Neutrino oscillation experiments: Searches for deviations from Maximal |U_e3| or specific mixing angle correlations provide direct tests of flavor-symmetric Dirac seesaw models.
• Searches for lepton-number violation: Non-observation of neutrinoless double beta decay at increased sensitivity would bolster Dirac frameworks.
• Collider and flavor physics: The discovery of heavy singlets, non-standard scalars, lepton-flavor violation, or non-unitary PMNS matrix elements would help distinguish models with new Dirac seesaw states accessible at the TeV scale.

Overall, the Dirac seesaw mechanism represents a robust and versatile approach to understanding the smallness and structure of neutrino masses while offering a pathway to link lepton number conservation, neutrino mixing, baryogenesis, and dark matter via well-motivated extensions of SM symmetries and field content (Ma et al., 2014, Ma et al., 2015, Chen et al., 2022, Chuliá et al., 2020).