Neutrino mixing parameters and masses from $Δ(96)\rtimes H_{CP}$ in the tri-direct CP approach

TL;DR

This work conducts a comprehensive, model‑independent analysis of lepton mixing and neutrino masses in a minimal two‑right‑hand‑neutrino seesaw framework governed by the Δ(96)×H_{CP} symmetry under the tri-direct CP approach. By scanning all remnant symmetry patterns in the charged-lepton, atmospheric, and solar sectors, it identifies 54 viable breaking patterns, with 42 NO and 12 IO cases when φ_sol transforms as $m{ar{3}_{1}}$; five NO and one IO patterns are explored in detail, revealing TM1 and other predictive mixing patterns with a small set of input parameters ($|m_a|$, $r$, $ar{ ext{eta}}$, and optionally $x$). The results yield concrete correlations among the lepton mixing angles, Dirac and Majorana CP phases, and the absolute neutrino masses, many of which are testable in upcoming experiments such as JUNO, DUNE, and T2HK. The framework demonstrates high predictive power, including sum rules and fixed first-column (or third-column) structures for several breaking patterns, while RG effects remain subdominant. This approach provides a transparent route to discriminate among flavor-symmetry breaking patterns via future precision neutrino measurements and neutrinoless double beta decay experiments.

Abstract

We present a comprehensive model independent analysis of all breaking patterns resulting from $Δ(96)\rtimes H_{CP}$ in the tri-direct CP approach of the minimal seesaw model with two right-handed neutrinos. The three generations of left-handed lepton doublets are assumed to transform as the irreducible triplet $\bm{3_{0}}$ of $Δ(96)$, and the two right-handed neutrinos are assigned to singlets. In the case that both flavon fields $φ_{\text{atm}}$ and $φ_{\text{sol}}$ transform as triplet $\bm{\bar{3}_{0}}$, only one phenomenologically viable lepton mixing pattern is obtained for normal ordering neutrino masses. The lepton mixing matrix is predicted to be the TM1 pattern, with neutrino masses, mixing angles, and CP violation phases depending on only three real input parameters. When $φ_{\text{sol}}$ is assigned to the $\bm{\bar{3}_{1}}$ representation, an additional real parameter $x$ must be included. Then we find 42 (12) independent phenomenologically interesting mixing patterns for normal (inverted) ordering neutrino masses, and the corresponding predictions for lepton mixing parameters and neutrino masses are obtained. Furthermore, we perform a detailed numerical analysis for five (one) example breaking patterns with some benchmark values of $x$ for normal (inverted) ordering. For the five normal examples, the absolute values of the first columns of the PMNS matrix are fixed to be $\left(\sqrt{\frac{2}{3}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}\right)^{T}$, $\frac{1}{5}\left(\sqrt{17},2,2\right)^{T}$, $\frac{1}{\sqrt{38}}\left(5,2,3\right)^{T}$, $\frac{1}{\sqrt{57}}\left(\sqrt{37},\sqrt{10},\sqrt{10}\right)^{T}$ and $\frac{1}{3}\left(\sqrt{6},1,\sqrt{2}\right)^{T}$, respectively. For the inverted example, the absolute value of the third column of the PMNS matrix is $\frac{1}{2\sqrt{11}}\left(1,5,3\sqrt{2}\right)^{T}$.

Figures (11)

• Figure 1: Contour plots of $\delta_{CP}/\pi$ within $\sin^2\theta_{13}-\sin^2\theta_{23}$ plane, and the dimensionless observable quantities $\sin^{2}\theta_{23}$, $\sin^{2}\theta_{13}$, $\delta_{CP}$ and $m_{2}^2/m_{3}^2$ in the $\eta$/$\pi-r$ plane. On the left panel, the results arise from applying the sum rule outlined in Eq. \ref{['eq:delta_CP_sumrule1']}. The red regions show allowed ranges of $\sin^2\theta_{13}$ and $\sin^2\theta_{23}$, calculated from random variations of input parameters $r$ and $\eta$ while ensuring agreement with experimental $3\sigma$ ranges for mixing angles, the Dirac CP phase, and $\Delta m^2_{21}/\Delta m^2_{31}$ in table \ref{['tab:bf_13sigma_data']}. On the right panel, the orange, blue, claret and green regions represent the 3$\sigma$ ranges of $\sin^{2}\theta_{23}$, $m_{2}^2/m_{3}^2$, $\sin^{2}\theta_{13}$ and $\delta_{CP}$ respectively. The dashed lines denote the best fit values of them.
• Figure 2: The predicted values of our model with $\eta=3\pi/5$ for the mixing angles $\sin^2\theta_{12}$, $\sin^2\theta_{23}$, the Dirac CP phase $\delta_{CP}$, the Majorana CP phase $\beta$ and mass ratio $m_2/m_3$ as a function of $\sin\theta_{13}$. Horizontal and vertical bands show the experimentally determined $1\sigma$ and $3\sigma$ ranges Esteban:2024eli for each parameter.
• Figure 3: Best fit results for 42 viable breaking patterns showing $\chi^2$ minima, lepton mixing angles, and CP violation phases for the NO case. Red dashed lines indicate best fit values, light blue bands show the $1\sigma$ and $3\sigma$ ranges from NuFIT 6.0 with Super-Kamiokande atmospheric data Esteban:2024eli. The pale green band shows the predicted $3\sigma$ range for $\sin^{2}\theta_{12}$ after 6 years of JUNO data JUNO:2022mxj. The faint green areas indicate the resolution (in degrees) for $\sin^{2}\theta_{23}$ and $\delta_{CP}$ after 15 years of DUNE operation DUNE:2020ypp.
• Figure 4: The the best fit $\chi^2$ for all 12 viable breaking patterns with IO neutrino mass spectrum, covering the three lepton mixing angles and CP violation phases.
• Figure 5: Contour plots of $\sin^{2}\theta_{23}$, $\sin^{2}\theta_{13}$, $\delta_{CP}$ and $m_{2}^2/m_{3}^2$ in the $\eta$/$\pi-r$ plane for breaking pattern $\mathcal{F}_{6}$ with $x=3$. On the right panel, we show the predictions for mixing parameters and mass ratio as functions of $\sin\theta_{13}$ for $\mathcal{F}_{6}$ with $x=3$ and $\eta=5\pi/3$.