A Gauge Model for Quasi-Dirac Neutrinos

Technical Summary: A Gauge Model for Quasi-Dirac Neutrinos

Problem Statement
The nature of neutrino mass—whether Dirac or Majorana—remains an open question in particle physics. While neutrino oscillations confirm non-zero masses, the mechanism generating them is unknown. A pure Dirac scenario requires right-handed neutrinos ($N_R$) to be massless and lepton number to be strictly conserved. However, theoretical models often struggle with the "naturalness problem," where tiny Dirac masses imply unnaturally small Yukawa couplings. Conversely, if neutrinos are Majorana, lepton number is violated. The paper addresses the "quasi-Dirac" scenario, where neutrinos possess dominant Dirac masses but tiny Majorana masses, implying approximate lepton number conservation. The authors aim to construct a gauge-theoretic model that naturally generates small Dirac masses without fine-tuned couplings, while ensuring anomaly cancellation and consistency with current experimental bounds.

Methodology
The authors propose an extension of the Standard Model (SM) by introducing a new Abelian gauge symmetry, $U(1)_{L_\mu - L_\tau}$. The methodology involves:

1. Field Content and Charge Assignment:

   • SM Sector: Quarks and the SM Higgs are neutral under the new symmetry. Lepton doublets ($L_i$) and singlets ($e^C_{Ri}$) carry charges $z_i$, chosen such that $z_1=0$ and $z_2 = -z_3$ (where $z_2 \equiv z$). This ensures the first generation is trivial while the second and third are vector-like under the new symmetry, canceling chiral anomalies within the SM sector.
   • Right-Handed Neutrinos: Six SM-singlet Weyl fermions ($N_1$ through $N_6$) are introduced with specific charges ($2z, 2z, 2z, -8z, -8z, 10z$) to ensure total anomaly cancellation.
   • Scalar Sector: Four dark Brout–Englert–Higgs fields ($\phi_1$ to $\phi_4$) with charges $(-z, -2z, -3z, 16z)$ are introduced to break the $U(1)_{L_\mu - L_\tau}$ symmetry.

2. Lagrangian Construction:

   • Renormalizable Terms ($d \leq 4$): The model includes kinetic terms, gauge interactions, and scalar potentials. Crucially, the $U(1)_{L_\mu - L_\tau}$ symmetry forbids renormalizable Majorana mass terms for the right-handed neutrinos $N_{1,2,3}$ and trilinear Dirac mass terms.
   • Higher-Dimensional Operators ($d=5$): Dirac neutrino masses are generated via dimension-5 operators suppressed by a high energy scale $M \sim 10^{14}$ GeV. These operators involve the scalar fields $\phi_i$ and the SM Higgs $H$.
   • Symmetry Breaking: The model assumes spontaneous symmetry breaking (SSB) where the scalars acquire vacuum expectation values (VEVs). A hierarchical pattern is assumed: $v_1, v_2, v_3 \ll v_4 \sim \text{TeV}$.

3. Mass Matrix Analysis:

   • The authors construct a $9 \times 9$ neutrino mass matrix involving active neutrinos ($\nu$) and the six singlets ($N$).
   • They analyze the sub-matrices to demonstrate that the active neutrinos acquire Dirac masses ($D_{3\times3}$) of order $10^{-2}$ eV, perturbed by small Majorana masses ($R_{3\times3}$) of order $10^{-3}$ eV or smaller.
   • The heavy singlets ($N_{4,5,6}$) acquire large Majorana masses ($\sim$ TeV), while $N_6$ remains light ($\sim 0.1$ GeV).

Key Contributions and Results

• Natural Quasi-Dirac Scenario: The model successfully realizes a quasi-Dirac neutrino scenario where the Dirac mass is dominant. The smallness of the neutrino mass is explained naturally via the suppression by the high scale $M$ and the VEVs of the scalar fields, analogous to the Froggatt–Nielsen mechanism, rather than requiring tiny Yukawa couplings.
• Anomaly-Free Chiral Fermions: The specific charge assignment of the six singlet fermions ensures the model is free of gauge anomalies without introducing global symmetries.
• Neutrino Mixing and Masses: The Dirac mass matrix is shown to be flexible enough to reproduce realistic neutrino oscillation data (mass squared differences and mixing angles) by tuning the Yukawa couplings and VEVs. The model predicts that the right-handed neutrino Majorana masses are at least an order of magnitude smaller than the Dirac masses, preserving approximate lepton number conservation.
• Gauge Boson Phenomenology: The model predicts a new massive gauge boson $X_\mu$ (associated with $U(1)_{L_\mu - L_\tau}$) with a mass $m_X \sim 5\text{--}10$ TeV (or higher to satisfy cosmological bounds). The mixing between the SM $Z$ boson and $X_\mu$ is negligible ($\alpha \ll \epsilon$).
• Cosmological Constraints: The authors address the production of right-handed neutrinos in the early universe. To satisfy Big Bang Nucleosynthesis (BBN) constraints on the number of relativistic degrees of freedom, the new gauge boson mass is required to be $m_X \gtrsim 43$ TeV. This pushes the symmetry breaking scale $v_4$ to the tens of TeV range.
• Dark Matter Candidate: The lightest singlet mass eigenstate, $N^m_6$ (mass $\sim 0.1$ GeV), is identified as a potential sub-GeV dark matter candidate. Its decay is heavily suppressed (lifetime $\gg$ age of the universe) due to double seesaw suppression of mixing angles.

Significance and Claims
The paper claims to provide a theoretically motivated gauge model for quasi-Dirac neutrinos that avoids the naturalness problems often associated with Dirac mass generation. By utilizing the $U(1)_{L_\mu - L_\tau}$ symmetry, the model naturally forbids large Majorana masses for the active neutrinos while allowing for small, non-zero Majorana terms.

The authors emphasize that:

1. The model does not rely on global or discrete symmetries, which can be unstable under quantum gravity effects.
2. It offers a testable prediction: the existence of a heavy $X$ boson that could be directly produced at future muon colliders with center-of-mass energies $\sqrt{s} \geq m_X > 5$ TeV.
3. The model is consistent with current LHC limits and cosmological constraints, provided the new physics scale is sufficiently high (PeV scale in extreme cases).
4. The Weinberg operator, typically associated with integrating out heavy right-handed neutrinos, is absent in the Lagrangian because the relevant right-handed neutrinos in this model are light, distinguishing this mechanism from standard Type-I seesaw scenarios.

The work concludes that while the parameters can be relaxed, the primary achievement is constructing a self-consistent framework where quasi-Dirac neutrinos emerge naturally from gauge principles and anomaly cancellation.