Quasi two-zero texture in Type-II seesaw at fixed points from modular $A_4$ symmetry

We study a quasi two-zero neutrino texture based on a type-II seesaw model with modular $A_4$ symmetry to evade the cosmological bound on the sum of neutrino masses while keeping some predictability in the neutrino sector. Working on three fixed points for modulus, we discuss predictions of the model and show the allowed points satisfying the cosmological bound on neutrino mass from both CMB and CMB+BAO data.

💡 Research Summary

The paper investigates a “quasi two‑zero” neutrino mass texture within a type‑II seesaw framework that is endowed with a modular A₄ flavor symmetry. Traditional two‑zero textures, which assume a diagonal charged‑lepton mass matrix and enforce two exact zeros in the neutrino mass matrix, are attractive because they reduce the number of free parameters and lead to sharp predictions for mixing angles, CP phases and neutrinoless double‑beta decay. However, such textures inevitably predict a sum of neutrino masses Σ m_ν that exceeds the most recent cosmological limits (Σ m_ν < 120 meV from Planck CMB alone, and Σ m_ν < 72 meV when CMB is combined with BAO data).

To overcome this tension, the authors relax the assumption of a diagonal charged‑lepton sector. They assign the three left‑handed lepton doublets L to three distinct A₄ singlets {1, 1′′, 1′} with modular weight –5, while the right‑handed charged leptons ℓ̄ form an A₄ triplet with weight –1. Two scalar triplets Δ_u (Y = +1) and Δ_d (Y = –1) are introduced to implement the type‑II seesaw mechanism in a supersymmetric setting, ensuring gauge anomaly cancellation. The modular forms of weight 6 (Y^{(6)}) and weight 10 (Y^{(10)}) provide the Yukawa structures; the former enters the charged‑lepton sector, the latter the neutrino sector.

The charged‑lepton mass matrix m_ℓ depends on three real parameters a_e, b_e, c_e and three complex parameters p, q, r. Its diagonalisation yields a non‑trivial left‑handed mixing matrix V_L, which, unlike the usual diagonal case, contributes to the observable PMNS matrix. The authors focus on three “fixed points” of the modulus τ: τ = i, τ = ω = e^{2πi/3}, and τ → i∞. At these points the modular forms simplify dramatically:

• τ = i preserves a Z₂ subgroup, giving Y^{(6)} a set of non‑zero real components and Y^{(10)} a specific pattern that leads to one massless charged lepton at leading order (the electron mass arises at sub‑leading order).
• τ = ω preserves Z₃, causing Y^{(6)} to vanish at leading order, so V_L ≈ I and the lepton mixing is essentially dictated by the neutrino sector.
• τ → i∞ preserves another Z₂, yielding Y^{(6)} ∝ (1,0,0) and again V_L ≈ I.

In the neutrino sector, the type‑II seesaw generates a mass matrix m_ν = (v_{Δ_u}/√2) · a_ν Y^{(10)}. Because only the A₄ singlets 1 and 1′ appear in Y^{(10)}, the matrix automatically satisfies a “two‑zero” condition: two distinct entries (ab) and (cd) vanish simultaneously. By introducing λ₁ = D₁, λ₂ = D₂ e^{–iα}, λ₃ = D₃ e^{–iβ} (with D_i the real mass eigenvalues) and defining U′ = V_L^* V^*, the authors derive compact relations (Λ₁₂, Λ₁₃) that connect the mass ratios D₁/D₂, D₁/D₃ and the Majorana phases α, β directly to the elements of U′. Consequently, the solar and atmospheric mass‑squared differences, the effective 0νββ mass ⟨m_{ee}⟩, and the total mass Σ m_ν can all be expressed analytically in terms of a small set of parameters (a_e, b_e, c_e, p, q, r) and the deviation parameters ε_{ij} that quantify how far τ is from the exact fixed point.

The authors perform a comprehensive numerical scan for both normal hierarchy (NH) and inverted hierarchy (IH). For each τ‑fixed point they vary the free parameters while imposing: (i) the latest global fit values of the three mixing angles and the two mass‑splittings, (ii) the cosmological bounds Σ m_ν < 120 meV (CMB only) and Σ m_ν < 72 meV (CMB + BAO), and (iii) the current KamLAND‑Zen limit on ⟨m_{ee}⟩ (28–122 meV). The results show:

• For τ = ω and τ → i∞, the charged‑lepton mixing is negligible, and a sizable region of parameter space satisfies both the neutrino oscillation data and the stringent Σ m_ν < 72 meV bound, especially in NH. The predicted Dirac CP phase clusters around ±π/2, while the Majorana phases obey a correlated pattern (α ≈ β). The effective 0νββ mass lies in the 10–30 meV range, well within the reach of upcoming experiments.
• For τ = i, the electron mass is generated only at sub‑leading order, requiring hierarchical choices such as |p|, |q| ≪ |r| (or analogous permutations). Within these hierarchies, viable points exist that respect Σ m_ν < 120 meV, but the Σ m_ν < 72 meV constraint is more restrictive, allowing only a narrow band of solutions.
• In the IH case, allowed regions are considerably smaller. Nevertheless, for τ = ω a few points survive the Σ m_ν < 72 meV bound, predicting ⟨m_{ee}⟩ in the 30–70 meV window.

Overall, the study demonstrates that a modular A₄‑symmetric type‑II seesaw model with a quasi‑two‑zero texture can reconcile the predictive power of texture‑zero approaches with the stringent cosmological limits on the sum of neutrino masses. The model yields concrete predictions for the Dirac CP phase, the Majorana phases, and the neutrinoless double‑beta decay rate, all of which can be tested by next‑generation long‑baseline neutrino experiments, cosmological surveys, and 0νββ searches. Moreover, the reliance on fixed points of the modulus τ links the phenomenology to the geometry of extra dimensions in string theory, providing an appealing theoretical motivation for the chosen flavor structure.