Zee-Babu model in a non-holomorphic modular $A_4$ symmetry and modular stabilization

1. Problem Statement

The paper addresses two primary challenges in neutrino physics and model building:

1. Neutrino Mass Origin: Explaining the tiny active neutrino masses without introducing excessive free parameters. The Zee-Babu (ZB) model is a promising radiative seesaw mechanism (generating masses at the two-loop level) but typically requires many free parameters to fit oscillation data.
2. Flavor Symmetry and Modulus Stabilization: While modular flavor symmetries (specifically $A_4$) have been successful in supersymmetric (SUSY) models using holomorphic modular forms, their application to non-supersymmetric theories has been limited. Furthermore, in modular models, the vacuum expectation value (VEV) of the modulus field $\tau$ is a free parameter that must be stabilized. The authors aim to construct a minimal ZB model using non-holomorphic modular forms (which allow for negative weights and are applicable to non-SUSY theories) and investigate the stabilization of $\tau$ near the fixed point $\omega = e^{2\pi i/3}$.

2. Methodology

The authors employ a theoretical framework combining the Zee-Babu mechanism with non-holomorphic modular $A_4$ symmetry.

• Model Construction:

  • Symmetry: They utilize the non-holomorphic modular $A_4$ symmetry, formulated recently by Qu and Ding, which allows for modular forms with negative weights.
  • Field Assignments:
    • Left-handed leptons ($L_L$) and right-handed charged leptons ($\ell_R$) are assigned to $A_4$ triplets with modular weights $-1$ and $+1$, respectively.
    • The singly-charged scalar $S^-$ and doubly-charged scalar $k^{++}$ are assigned weights $+2$ and $0$.
    • All scalars are trivial singlets under $A_4$.
  • Lagrangian: The charged-lepton mass matrix and the neutrino sector are constructed using modular forms $Y^{(k)}_r(\tau, \bar{\tau})$. The neutrino mass is generated via the Zee-Babu two-loop mechanism involving $S^-$ and $k^{++}$.
  • Non-Holomorphic Forms: Unlike holomorphic models, the modular forms depend on both $\tau$ and $\bar{\tau}$, satisfying the Laplacian condition for polyharmonic Maaß forms. This introduces negative weight forms (e.g., $Y^{(-2)}$) which are crucial for the model's structure.

• Numerical Analysis:

  • The authors perform a $\chi^2$ analysis using neutrino oscillation data from NuFit 6.0 (5$\sigma$ intervals).
  • They treat the modulus $\tau$ and the parameter $\tilde{b}_k$ (related to the coupling of the doubly-charged boson) as free parameters.
  • They compare the results of the full numerical scan against the specific fixed point $\tau = \omega$.

• Modulus Stabilization:

  • They propose a mechanism to stabilize the modulus $\tau$ near $\omega$ in a non-SUSY framework.
  • They construct a modular-invariant potential $V(\tau, \bar{\tau})$ using the expansion of modular forms around the fixed point. They demonstrate that a potential of the form $V \propto (2\text{Im}\tau)^4 |Y^{(4)}_1(\tau)|^2$ has a minimum at $\tau = \omega$.
  • They discuss how small deviations from $\omega$ can be induced by adding perturbative terms or radiative corrections to the potential.

3. Key Contributions

1. Minimal Parameter Zee-Babu Model: The authors construct a ZB model with the minimum number of free parameters (only two complex parameters for the neutrino sector: $\tau$ and $\tilde{b}_k$). This is achieved by leveraging the constraints of the non-holomorphic modular $A_4$ symmetry.
2. Non-Holomorphic Application: They successfully apply non-holomorphic modular forms to a non-supersymmetric Zee-Babu model, demonstrating that richer flavor structures and negative modular weights can be utilized without SUSY.
3. Localization at $\tau \approx \omega$: They discover that the model only yields phenomenologically viable solutions (fitting neutrino data) when the modulus $\tau$ is localized very close to the fixed point $\omega = e^{2\pi i/3}$.
4. Stabilization Mechanism: They provide a concrete potential and analytical expansion for stabilizing the modulus at $\tau = \omega$ in a non-SUSY context, addressing a common gap in modular model building.

4. Results

• Neutrino Hierarchy: The model exclusively allows for the Normal Hierarchy (NH) of neutrino masses. The Inverted Hierarchy (IH) is ruled out because it fails to satisfy the experimental constraints within the 5$\sigma$ range of the model's parameter space.
• Modulus Value: The allowed region for the modulus is highly localized:
  • $\tau \approx \omega$ with a small absolute deviation of approximately 0.006.
  • This small deviation is critical; at the exact fixed point $\tau = \omega$, the model predicts two massless neutrinos (rank-1 mass matrix), which is experimentally excluded. The deviation allows for the correct mass splitting.
• Mixing Angles and Phases:
  • $s^2_{12}$: The model predicts a value slightly deviated from the upper bound of the experimental 3$\sigma$ range.
  • $s^2_{23}$: The allowed region is localized near the lower experimental bound.
  • CP Phases:
    • The Dirac CP phase ($\delta_{CP}$) is predicted to be in the range $[-35^\circ, -2.5^\circ]$.
    • The Majorana phase ($\alpha$) is predicted to be in the range $[0^\circ, 65^\circ]$.
  • These predictions are sharp and testable.
• Neutrinoless Double Beta Decay ($0\nu\beta\beta$):
  • The effective Majorana mass $\langle m_{ee} \rangle$ is predicted to be very small, approximately 4–4.4 meV.
  • This is well below the current upper bounds from KamLAND-Zen ($< 28-122$ meV), making it a distinct prediction for future experiments.
• Modulus Stabilization: The potential analysis confirms that $\tau = \omega$ is a stable minimum. The required small deviation ($\delta \tau \approx 0.006$) can be naturally explained by small perturbations (e.g., radiative corrections or higher-order potential terms) in the non-SUSY framework.

5. Significance

• Phenomenological Success: The paper demonstrates that a highly constrained model (only 2 complex parameters) can successfully describe all current neutrino oscillation data, provided the modulus is stabilized near a specific fixed point.
• Theoretical Advancement: It bridges the gap between modular flavor symmetries and non-supersymmetric theories. By utilizing non-holomorphic forms, it opens new avenues for model building where SUSY is not assumed, avoiding potential issues with SUSY breaking effects on flavor structures.
• Predictive Power: The model makes specific, testable predictions for the Dirac CP phase, Majorana phases, and the effective mass for neutrinoless double beta decay. The prediction of a very small $\langle m_{ee} \rangle$ and a specific $\delta_{CP}$ range offers clear targets for next-generation experiments like DUNE, Hyper-Kamiokande, and future $0\nu\beta\beta$ searches.
• Analytical Tools: The paper provides explicit expansions of modular forms and their derivatives near $\tau = \omega$ (Appendix B), offering a valuable toolkit for other researchers constructing models near modular fixed points.

In conclusion, this work presents a robust, minimal, and predictive extension of the Zee-Babu model that relies on non-holomorphic modular symmetry, successfully explaining neutrino data while offering a mechanism for modulus stabilization in non-supersymmetric theories.