Embedding Generalized CP Symmetry in One Zero Texture Neutrino Mass Models

1. Problem Statement

The Standard Model (SM) fails to explain the non-zero masses of neutrinos, a fact confirmed by oscillation experiments. While the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix parameters (mixing angles and mass squared differences) are now measured with high precision, fundamental mysteries remain:

• The neutrino mass hierarchy (Normal vs. Inverted).
• The nature of neutrinos (Dirac vs. Majorana).
• The origin of CP violation in the lepton sector.
• The specific values of the Majorana phases.

Traditional flavor symmetry models (like Tribimaximal Mixing, TBM) predict a zero reactor angle ($\theta_{13}=0$), which is experimentally excluded. Furthermore, simple texture zeros (vanishing elements in the mass matrix) alone often lack sufficient predictive power. The authors aim to bridge this gap by combining Generalized CP (GCP) symmetries with one-zero texture constraints to create a highly predictive framework that can be tested against current and future cosmological and oscillation data.

2. Methodology

Theoretical Framework

The study utilizes a framework where the neutrino mass matrix ($m_\nu$) is constrained by:

1. Residual Flavor Symmetries ($G_i$): Symmetries that leave $m_\nu$ invariant ($G_i^T m_\nu G_i = m_\nu$).
2. Generalized CP Symmetries ($X_i$): Transformations that convert $m_\nu$ into its complex conjugate ($X_i^T m_\nu X_i = m_\nu^*$).
3. Complex Tribimaximal (cTBM) Matrix: The authors use a cTBM mixing matrix ($U_{cTBM}$) which incorporates non-zero Majorana phases ($\rho, \sigma$) while retaining the TBM mixing angle structure. This distinguishes their work from models where $G_i = X_i$; here, they focus on the case $G_i \neq X_i$.

Construction of Models

• Symmetry Imposition: The authors impose two specific CP symmetries ($X_1$ and $X_4$ for Case I; $X_2$ and $X_4$ for Case II) on the mass matrix. This leads to specific residual flavor symmetries ($G_1$ and $G_2$).
• Diagonalization: The resulting mass matrices in the cTBM basis are diagonalized by a single real rotation angle ($\theta$) in the 2-3 or 1-3 plane, depending on the case.
• Texture Zeros: Six distinct one-zero texture patterns ($m_I$ to $m_{VI}$) are imposed on the mass matrix. This introduces a complex constraint equation: $m_1 A + m_2 B + m_3 C = 0$, where $A, B, C$ are products of mixing matrix elements.
• Parameter Space Scan: The authors perform a numerical scan of $10^8$ points, varying:
  • Mass squared differences ($\Delta m^2_{solar}, \Delta m^2_{atm}$) within $3\sigma$ ranges.
  • Rotation angle $\theta$ ($0$ to $2\pi$).
  • Majorana phases $\rho, \sigma$ ($0$ to $2\pi$).
  • The ratio of mass squared differences $R_\nu = \Delta m^2_{21} / |\Delta m^2_{31}|$ within NuFIT 6.0 $3\sigma$ limits.

Experimental Constraints

The models are tested against:

• Oscillation Data: NuFIT 6.0 ($3\sigma$ ranges for angles and mass differences).
• Neutrinoless Double Beta Decay ($0\nu\beta\beta$): Limits from KamlandZen, LEGEND, CUORE, and nEXO on the effective Majorana mass ($m_{ee}$).
• Cosmological Data:
  • Planck data: $\sum m_i < 0.12$ eV.
  • DESI/SDSS+Pantheon+DES-SN (95% CL): $\sum m_i < 0.17$ eV.
  • Future DESI projection: $\sum m_i < 0.072$ eV.

3. Key Contributions

1. Novel Symmetry Combination: The paper systematically explores the intersection of Generalized CP symmetries (specifically $X_1$ and $X_2$ cases with $G_i \neq X_i$) and one-zero textures. This reduces the parameter space significantly compared to models using only flavor symmetries.
2. Predictive Correlations: The framework establishes strict correlations between:
   • The rotation angle $\theta$ and the mixing angles ($\theta_{13}, \theta_{23}$).
   • The Dirac CP phase ($\delta$) and the atmospheric mixing angle ($\theta_{23}$).
   • The Majorana phases ($\rho, \sigma$).
3. Hierarchy Discrimination: The analysis provides a definitive test for the mass hierarchy based on cosmological bounds, effectively ruling out the Inverted Hierarchy (IH) for most viable textures under current and future constraints.
4. Texture Viability: It identifies which of the six one-zero texture patterns are physically viable under GCP symmetries and which are ruled out by experimental data.

4. Key Results

Mass Hierarchy and Cosmology

• Inverted Hierarchy (IH) Ruled Out: For both $X_1$ and $X_2$ cases, the Inverted Hierarchy is inconsistent with Planck data ($\sum m_i < 0.12$ eV) and the DESI/SDSS+Pantheon+DES-SN dataset ($\sum m_i < 0.17$ eV).
  • Exception: The texture $m_I$ (zero at 1,1) is ruled out in IH because it predicts an unphysically large reactor angle $\theta_{13}$ ($\sin^2 2\theta_{13} \gtrsim 0.5$) to satisfy the texture zero condition.
• Normal Hierarchy (NH) Viability:
  • $X_1$ Case: Only textures $m_{II}$ and $m_{III}$ are compatible with both Planck ($\sum m_i < 0.12$ eV) and the looser DESI bound. Textures $m_{IV}, m_V, m_{VI}$ are compatible with the DESI bound but violate the stricter Planck bound.
  • $X_2$ Case: Textures $m_{I}, m_{II}, m_{III}$ are compatible with the Planck bound. Textures $m_{IV}, m_V, m_{VI}$ violate the Planck bound but satisfy the DESI bound.
• Future Constraints: The projected DESI bound ($\sum m_i < 0.072$ eV) is extremely stringent.
  • In the $X_1$ scenario, this would rule out all matrices except $m_I$ (which is already problematic in IH, but the text implies a specific exclusion logic for NH as well, though the abstract notes $m_I$ is the only survivor in X1 for this bound).
  • In the $X_2$ scenario, this would rule out all matrices except $m_I, m_{II}, m_{III}$.

Effective Majorana Mass ($m_{ee}$)

• The predicted $m_{ee}$ for viable textures falls within the sensitivity range of current ($KamlandZen$) and future ($LEGEND, nEXO$) experiments.
• For $m_{II}$ and $m_{III}$ (the most robust models), $m_{ee}$ ranges from $\approx 0.005$ eV to $\approx 0.30$ eV, with lower bounds often near the sensitivity threshold of next-generation experiments.

Correlations and Parameters

• Rotation Angle ($\theta$): The analysis tightly constrains the rotation angle $\theta$:
  • $X_1$ Case: $14.3^\circ - 15.6^\circ$.
  • $X_2$ Case: $10.1^\circ - 11.1^\circ$.
• CP Violation: Strong correlations exist between the Dirac phase $\delta$ and the atmospheric angle $\theta_{23}$. As $\theta_{23}$ approaches maximal mixing, $\delta$ tends toward $\pi/2$ or $3\pi/2$, suggesting maximal CP violation.
• Jarlskog Invariant ($J$): The models predict specific values for $J$, providing a testable signature for future long-baseline experiments (DUNE, Hyper-Kamiokande).

5. Significance

This work significantly advances the phenomenology of neutrino mass models by demonstrating that Generalized CP symmetries combined with texture zeros offer a powerful, predictive framework.

• Testability: The model makes sharp predictions for the atmospheric mixing angle ($\theta_{23}$) and the Dirac CP phase ($\delta$), which can be tested by upcoming experiments like NOνA, DUNE, and Hyper-Kamiokande.
• Cosmological Impact: The study highlights the critical role of cosmological mass sum limits ($\sum m_i$) in distinguishing between theoretical models. The tightening of these bounds (e.g., by DESI) could effectively rule out large classes of texture-zero models, leaving only a few viable candidates ($m_{II}, m_{III}$).
• Experimental Guidance: By identifying specific textures ($m_{II}, m_{III}$) that survive all current constraints, the paper guides experimentalists on where to look for signals in neutrinoless double beta decay and oscillation data.

In conclusion, the paper argues that the combination of GCP symmetry and one-zero textures is a viable path to understanding neutrino properties, provided the mass hierarchy is Normal and the sum of masses satisfies the tightest cosmological bounds.