Probing Non-Holomorphic Modular $A_4$ Double Seesaw: Signatures in Neutrino Oscillation Experiments and Implications for Leptogenesis

Technical Summary: Probing Non-Holomorphic Modular $A_4$ Double Seesaw

Problem Statement
The origin of tiny neutrino masses and the baryon asymmetry of the Universe (BAU) remain unexplained within the Standard Model (SM). While the Type-I seesaw mechanism offers a natural explanation for neutrino masses and a framework for thermal leptogenesis, it typically treats Right-Handed Neutrino (RHN) Majorana masses as arbitrary input parameters, lacking a dynamical origin. Furthermore, in minimal Type-I leptogenesis, there is often no direct correlation between high-energy leptogenesis parameters and low-energy neutrino oscillation observables, rendering the mechanism difficult to probe indirectly. Extended frameworks, such as the Double Seesaw (DSS) mechanism, introduce additional sterile fermions to generate RHN masses dynamically, but many realizations (e.g., in Left-Right Symmetric Models) introduce additional gauge bosons that complicate the leptogenesis dynamics with extra scattering and washout processes. Additionally, conventional discrete flavor models relying on flavon fields often suffer from complex vacuum alignment problems and increased model-building complexity.

Methodology
The authors propose a non-holomorphic modular $A_4$ framework to realize the DSS mechanism. The model extends the SM with:

1. Three generations of Right-Handed Neutrinos ($N_R$).
2. Three generations of left-handed sterile neutrino fields ($S_L$).
3. An $A_4$-singlet scalar flavon $\rho$.

Key methodological features include:

• Modular Symmetry: Instead of spontaneous symmetry breaking via flavon vacuum expectation values (VEVs) in flavor space, flavor structures are governed by the vacuum expectation value of a complex modulus $\tau$. The Yukawa couplings are described by polyharmonic Maaß forms, allowing for non-holomorphic modular weights (positive, zero, and negative).
• Field Assignments: $N_R$ and $S_L$ transform as triplets under $A_4$, while the scalar $\rho$ is a trivial singlet. Modular weight assignments are chosen such that a bare Majorana mass term for $N_R$ is forbidden, while the necessary Dirac and Majorana mass terms for the DSS structure are allowed.
• Mass Generation: The light active neutrino mass matrix ($m_\nu$) and the effective RHN Majorana mass matrix ($M_N$) are derived by integrating out the heavy sterile states ($S_L$). The hierarchy $M_S \gg M_{RS} \gg M_D$ is realized, where $M_S$ is the sterile Majorana scale, $M_{RS}$ is the RHN-sterile Dirac scale, and $M_D$ is the active-RHN Dirac scale.
• Numerical Analysis: The authors perform a comprehensive scan of the modular parameter space (complex modulus $\tau$ and dimensionless coupling ratios). They constrain the model using global fit data from NuFIT 6.0 (neutrino oscillation parameters).
• Experimental Projections: The allowed parameter space is tested against projected sensitivities of DUNE, T2HK, and JUNO using the GLoBES framework.
• Leptogenesis: The study analyzes thermal leptogenesis in two regimes:
  1. Unflavored regime: For $M_{N_1} > 10^{12}$ GeV, solving Boltzmann equations for decays and inverse decays.
  2. Two-flavor regime: For $10^9 < M_{N_1} < 10^{12}$ GeV, where the $\tau$ flavor is resolved but $e$ and $\mu$ remain coherent.
  3. Resonance diagnostics: A scan-level check for quasi-degenerate RHN pairs ($N_2, N_3$) that could enter the resonant regime.

Key Contributions and Results

1. Dynamical Origin of RHN Masses: The framework successfully forbids bare RHN Majorana masses via modular weight selection rules. The RHN masses are induced entirely through the double seesaw structure involving the sterile sector, linking the high-energy mass scale to the low-energy neutrino sector.
2. Neutrino Oscillation Phenomenology:
   • The model yields viable solutions exclusively for Normal Ordering (NO) within the explored parameter space.
   • The best-fit point is found at $\text{Re}(\tau) = -0.018$ and $\text{Im}(\tau) = 1.65$.
   • The model predicts $\sin^2\theta_{13}$ within the narrow interval $[0.020, 0.024]$ and allows the Dirac CP phase $\delta_{CP}$ to span its entire physical range $[-180^\circ, 180^\circ]$.
   • Experimental Constraints: The projected sensitivities of DUNE and T2HK are shown to significantly constrain the atmospheric mixing angle $\theta_{23}$ and mass splitting $\Delta m^2_{31}$, potentially resolving the octant degeneracy and excluding higher-octant solutions allowed by the model if the lower octant is realized in nature. JUNO provides complementary high-precision probes for solar parameters ($\theta_{12}, \Delta m^2_{21}$) and $\Delta m^2_{31}$.
3. Leptogenesis Success:
   • Unflavored Benchmark: For a representative point with $M_{N_1} \approx 1.34 \times 10^{13}$ GeV and $m_1 \approx 1.82 \times 10^{-2}$ eV, the system operates in the strong-washout regime ($K_1 \approx 25.73$). The numerical solution of Boltzmann equations yields a baryon asymmetry $Y_{\Delta B} \simeq 8.11 \times 10^{-11}$, which is $\approx 0.93$ times the observed value ($Y_{\Delta B}^{obs} \simeq 8.7 \times 10^{-11}$). This result is achieved with minimal decay and inverse-decay dynamics, without requiring additional scattering terms.
   • Two-Flavor Benchmark: For a point with $M_{N_1} \approx 1.82 \times 10^{11}$ GeV, a semi-analytic treatment of the two-flavor regime (resolved $\tau$, coherent $e+\mu$) yields $|Y_{\Delta B}| \simeq 5.34 \times 10^{-11}$ ($\approx 0.61 Y_{\Delta B}^{obs}$), demonstrating the viability of flavored leptogenesis in this framework.
   • Resonance Diagnostics: The scan reveals that the parameter space admits configurations where the heavier pair $(N_2, N_3)$ is quasi-degenerate ($\Delta M_{23} \sim \Gamma_{avg}$) while $N_1$ remains isolated. However, the authors note that successful BAU reproduction does not strictly require these resonant configurations, as hierarchical benchmarks also succeed.

Significance and Claims
The paper establishes the non-holomorphic modular $A_4$ double seesaw as a predictive setting that unifies low-energy neutrino phenomenology with high-energy thermal leptogenesis. Its primary significance lies in:

• Predictivity: It links the RHN mass spectrum and Yukawa textures directly to low-energy oscillation parameters through the modular symmetry, removing the arbitrariness of RHN mass inputs found in minimal Type-I seesaws.
• Testability: The model's predictions for oscillation parameters are directly testable by forthcoming precision experiments (DUNE, T2HK, JUNO). The ability of these experiments to constrain the model's parameter space (particularly regarding $\theta_{23}$ octants) is a central claim.
• Minimalism: By utilizing modular symmetry and a singlet scalar, the framework avoids the need for non-trivial flavon alignments and additional gauge sectors (unlike LRSM realizations of DSS), resulting in a technically simpler leptogenesis scenario closer to the minimal Type-I case.
• Interconnection: The work demonstrates that the observed BAU, neutrino masses, and mixing patterns can emerge as interconnected consequences of a single underlying construction, with the oscillation-constrained parameter points simultaneously determining the ingredients required for successful thermal leptogenesis.

The authors maintain a modest stance regarding the resonant regime, clarifying that while quasi-degenerate spectra exist within the allowed space, they are not a necessary ingredient for explaining the BAU in this framework, as hierarchical non-resonant realizations are already successful.