Neutrino oscillations and PMNS matrix in gauge-Higgs unification

1. Problem Statement

The Standard Model (SM) successfully describes low-energy phenomena but suffers from the gauge hierarchy problem and lacks a natural explanation for neutrino masses and mixing. While Gauge-Higgs Unification (GHU) models in Randall-Sundrum (RS) warped space offer a solution to the hierarchy problem by identifying the Higgs boson as a fluctuation of an Aharonov-Bohm (AB) phase in a fifth dimension, they must also account for flavor physics.

Specifically, this paper addresses:

• The origin of neutrino oscillations and the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix within the context of the $SO(5) \times U(1)_X \times SU(3)_C$ GHU model.
• How tiny neutrino masses can be generated without fine-tuning, utilizing the geometry of the warped space.
• Whether the model can reproduce the experimentally observed mixing angles and the CP-violating phase ($\delta_{CP}$) consistent with recent global analyses (NuFit-6.0).

2. Methodology

The author employs a GUT-inspired Gauge-Higgs Unification model defined in a 5D Randall-Sundrum (RS) warped spacetime.

• Model Setup:

  • Geometry: The spacetime is an orbifold $S^1/Z_2$ with an ultraviolet (UV) brane at $y=0$ and an infrared (IR) brane at $y=L$. The metric is warped by a factor $e^{-2ky}$.
  • Gauge Group: $SO(5) \times U(1)_X \times SU(3)_C$. The $SO(5)$ symmetry is broken to $SO(4) \simeq SU(2)_L \times SU(2)_R$ by orbifold boundary conditions.
  • Matter Content: Leptons are introduced as bulk fermion multiplets $\Psi^{(1,4)}_\alpha$ (spinor representation of $SO(5)$) containing zero modes for neutrinos ($\nu, \nu'$) and charged leptons ($e, e'$). Additionally, SO(5) singlet brane fermions ($\hat{\chi}_\alpha$) are defined on the UV brane.
  • Symmetry Breaking: The 4D Higgs doublet arises as the zero mode of the fifth component of the gauge field ($A_y$). Its vacuum expectation value (VEV) corresponds to the AB phase $\theta_H$.

• Mechanism for Mass and Mixing:

  • Inverse Seesaw: The model introduces Majorana mass terms ($M_{\alpha\beta}$) for the brane fermions $\hat{\chi}$ and brane mass terms ($m^B_{\alpha\beta}$) coupling the bulk neutrinos to the brane fermions via a scalar field $\hat{\Phi}$ on the UV brane.
  • Twisted Gauge: Calculations are performed in a "twisted gauge" where the background AB phase is removed from the bulk equations but reappears in the boundary conditions at the UV brane.
  • Diagonal Approximation: The analysis assumes diagonal brane masses ($m^B_{\alpha\beta} = \delta_{\alpha\beta} m^B_\alpha$) to derive explicit analytical solutions for the neutrino spectrum and mixing matrix.

• Derivation Steps:

  1. Solve the 5D equations of motion for lepton fields in the bulk.
  2. Apply boundary conditions at the UV and IR branes, incorporating the Majorana and brane mass terms.
  3. Derive the eigenvalue equation for the neutrino mass spectrum.
  4. Construct the $W$-boson interaction Lagrangian to extract the mixing matrix between charged leptons and neutrino mass eigenstates.

3. Key Contributions

A. Inverse Seesaw Mechanism in GHU

The paper demonstrates that the GHU framework naturally realizes an inverse seesaw mechanism.

• Unlike the standard Type-I seesaw, this mechanism involves the intertwining of three states per generation: the bulk neutrino $\nu_L$, the bulk partner $\nu'_R$, and the brane singlet $\eta$.
• The light neutrino mass $m_\nu$ is derived as:
  $$m_\nu \sim \frac{m_\ell^2 M}{m_B^2}$$
  where $m_\ell$ is the charged lepton mass, $M$ is the Majorana mass on the UV brane, and $m_B$ is the brane-induced mixing mass.
• This allows for naturally tiny neutrino masses ($\sim 1-10$ meV) even with relatively low Majorana mass scales ($M \sim 10^6 - 10^9$ GeV), provided $m_B$ is sufficiently large.

B. Origin of the PMNS Matrix

The paper establishes that the PMNS matrix arises directly from the structure of the Majorana mass matrix ($M_{\alpha\beta}$) on the UV brane.

• The mixing in the $W$ couplings is determined by the eigenvectors of the matrix product $F M$, where $F$ is a diagonal matrix dependent on the bulk mass parameters ($c_\ell$) and brane masses ($m_B$).
• The model predicts that the PMNS matrix is determined by the eigenvectors $\vec{w}_j$ of the system, which are related to the experimental mixing vectors $\vec{p}_j$.

C. Reproduction of NuFit-6.0 Data

The author explicitly constructs parameter sets that reproduce the experimental data from NuFit-6.0 under the Normal Ordering (NO) assumption.

• Result: The model naturally yields a CP-violating phase of $\delta_{CP} = \pi$.
• Consistency: The calculated mixing angles ($\theta_{12}, \theta_{23}, \theta_{13}$) and mass squared differences ($\Delta m^2_{21}, \Delta m^2_{31}$) align with the NuFit-6.0 global analysis.
• Cases: Two scenarios are presented:
  1. Case I: Equal diagonal parameters ($F_e = F_\mu = F_\tau$), requiring specific values for $m_B$ and $M$.
  2. Case II: Unequal parameters ($F_e \neq F_\mu \neq F_\tau$), offering more flexibility in the mass scales while still reproducing the data.

4. Results

• Neutrino Mass Spectrum: The model successfully generates a hierarchical mass spectrum ($m_{\nu1} < m_{\nu2} < m_{\nu3}$) consistent with the normal ordering.
• PMNS Matrix Structure: The derived matrix matches the experimental values:
  $$U_{PMNS} \approx \begin{pmatrix} 0.823 & 0.548 & -0.148 \\ -0.275 & 0.613 & 0.741 \\ 0.497 & -0.569 & 0.655 \end{pmatrix}$$
  with $\delta_{CP} = \pi$.
• Mass Scales:
  • Brane masses $m_B$ are found to be in the range $10^5 - 10^{10}$ GeV.
  • Majorana masses $M$ are in the range $10^6 - 10^9$ GeV (or higher depending on the specific case).
  • These scales are consistent with the requirement $m_{KK} \ll m_B < k$ (where $m_{KK} \sim 13$ TeV is the Kaluza-Klein scale).

5. Significance

• Unification of Flavor and Hierarchy: The paper provides a compelling framework where the solution to the gauge hierarchy problem (GHU) and the origin of neutrino flavor mixing are intrinsically linked through the geometry of extra dimensions and boundary conditions.
• Predictive Power: The model predicts a specific CP phase ($\delta_{CP} = \pi$) for the normal ordering case, which is currently favored by experimental data. This is a distinct prediction compared to models that allow arbitrary phases.
• Inverse Seesaw Realization: It offers a concrete realization of the inverse seesaw mechanism within a 5D warped geometry, showing how the interplay between bulk and brane fields can suppress neutrino masses without requiring extremely high energy scales (like the GUT scale) for the Majorana mass.
• Future Phenomenology: The paper suggests that lepton number violation could occur at energy scales lower than typically expected in seesaw models, potentially observable in future experiments. It also notes that while the current diagonal assumption works well, non-diagonal brane interactions could introduce more general CP phases, a topic for future investigation.

In conclusion, Hosotani demonstrates that the $SO(5) \times U(1) \times SU(3)$ Gauge-Higgs Unification model is a viable candidate for explaining neutrino oscillations, naturally generating tiny masses via an inverse seesaw mechanism and reproducing the observed PMNS matrix with $\delta_{CP} = \pi$ in the normal ordering scenario.