Minimal A4 Type-II Seesaw Realization of Testable Neutrino Mass Sum Rules

TL;DR

The paper builds an $A_4$ flavor-symmetric model with a type-II seesaw using an $SU(2)_L$ scalar triplet $\boldsymbol{\triangle}$ and an $A_4$-triplet Higgs, yielding a residual lepton triality in the charged-lepton sector. A neutrino mass sum rule, derived from basis invariants of $\boldsymbol{\text{M}}_\nu$, fixes the absolute neutrino mass scale once the measured $\Delta m_{ij}^2$ are input, and robustly favors inverted ordering with distinctive predictions for $|m_{ee}|$, $m_{\text{eff}}$, and CP phases. The model predicts a tight correlation between $\theta_{23}$ and the Dirac phase $\delta_{CP}$, near maximal $\theta_{23}$, and constrains the Majorana phases, while charged-lepton flavor violation is suppressed by the approximate triality, allowing only specific three-body $\tau$ decays. These predictions render the framework highly testable in upcoming oscillation experiments (e.g., DUNE, Hyper-K), direct-mass probes (KATRIN, Project-8), and $0\nu\beta\beta$ experiments (KamLAND-Zen2).

Abstract

We propose a flavour model based on an $A_4$ symmetry combined with a type-II seesaw mechanism for neutrino mass generation. The resulting neutrino mass matrix obeys a sum rule that, together with the measured mass-squared differences, fully determines the absolute neutrino mass spectrum. The constrained flavour structure yields correlated predictions for lepton mixing parameters, leads to inverted ordering after imposing mixing constraints, restricts the Majorana phases and implies a neutrinoless double beta decay rate close to its maximal value for inverted ordering. In the charged lepton sector an approximate triality symmetry arises in the seesaw limit, suppressing muon flavour-violating processes and allowing only specific $τ$ decay channels. The model provides a tightly constrained and experimentally testable framework linking neutrino masses, lepton mixing and lepton-number-violating observables.

Figures (5)

• Figure 1: Region in the $(\theta,\phi)$ plane satisfying the neutrino mass constraints. $(\theta,\phi)$ are the $\Delta$ vev alignment angles, see Eq. \ref{['eq:alignment']}. The black contour corresponds to the best-fit values of the neutrino mass-squared differences deSalas:2020pgw, while the pink band shows the corresponding $3\sigma$ allowed range. The blue segments, artificially thickened for visibility, indicate the subset of solutions that are also compatible with the observed lepton mixing angles. All viable solutions lie close to the limiting vev alignments $(1,1,0)$, $(1,0,1)$, and $(0,1,1)$.
• Figure 2: Correlation of the atmospheric mixing angle $\theta_{23}$, left panel: with the Dirac CP-violating phase $\delta_{\rm CP}$ and right panel: Majorana phase $\phi_{13}$.
• Figure 3: Correlations between CP-violating phases. Left panel: $\delta_{CP}$ vs $\phi_{13}$. Right panel: $\phi_{12}$ vs $\phi_{13}$. Note that the Majorana phase $\phi_{12}$ is forced to be close to the values maximizing $|m_{ee}|$, see Eqs. \ref{['eq:phi12']} and \ref{['eq:mee']}.
• Figure 4: Effective Majorana mass $|m_{ee}|$ as a function of the lightest neutrino mass $m_{\text{lightest}}$. The blue and red bands correspond to the allowed regions for normal and inverted mass ordering, respectively, obtained by varying the oscillation parameters within their current $3\sigma$ ranges. The purple band shows the prediction of the model, which enforces inverted ordering sum rule and restricts the Majorana phase $\phi_{12}$ (see Eq. \ref{['eq:phi12']}), leading to a highly localized region for $|m_{ee}|$. This results in a sharp and testable prediction for neutrinoless double beta decay experiments.
• Figure 5: Schematic Feynman diagrams of the allowed $\tau^-$ three-body decay channels mediated by the scalars $\phi^0_1$ and $\phi^0_2$.