CP-conserving SO(3) parameterization of the neutrino mixing matrix

TL;DR

The paper tackles the order-dependence of the standard PMNS parameterization by introducing an SO(3) based, CP-conserving neutrino mixing framework, $U_{ m SO3}$, realized as a single rotation with $\delta_{\mathrm{CP}} = 180^ ^\circ$. Using NuFIT-6.0, it obtains a set of democratic rotation angles $\theta_x \approx 43.8^\circ$, $\theta_y \approx 21.7^\circ$, $\theta_z \approx 28.3^\circ$ (total $\theta \approx 56.5^\circ$), and demonstrates that this parameterization can reproduce observed oscillation data without relying on the conventional order of Euler rotations. The approach yields tighter predictions for the absolute neutrino masses, $m_\beta$ and $m_{\beta\beta}$, under CP conservation and provides experimentally testable targets for next-generation neutrinoless double beta decay and tritium beta decay experiments, while offering a new perspective for flavor symmetry and leptogenesis model-building. Overall, the work proposes a falsifiable CP-conserving alternative to the PMNS description and a framework for CP-conserving neutrino oscillation analyses with potential implications for fundamental flavor structure.

Abstract

The pattern of neutrino mixing, usually parameterized by the Pontecorvo-Maki-Nakagawa-Sakata $U_{\rm PMNS}$ matrix, still remains a striking puzzle in particle physics. $U_{\rm PMNS}$ is one of six possible products of multiplying three Euler matrices. Here we discuss the neutrino mixing matrix parameterization for three known flavours of neutrinos based on the SO(3) group represented by one three-dimensional rotation matrix $U_{\rm SO3}$. The mixing matrix $U_{\rm SO3}$ with cyclic order of the Lie group generators implies $δ_{\rm{CP}}=180^\circ$ for clockwise rotation in three dimensions, a viable scenario for normal mass ordering. We determine a range of rotation angles for $U_{\rm SO3}$ which deviate substantially from the almost maximal mixings in the standard $U_{\rm PMNS}$ scenario, yielding `democratic' values for the mixing angles. With the fixed $δ_{\rm CP}$ value, the $U_{\rm SO3}$ parameterization can be validated or falsified by the next generation of neutrinoless double beta decay experiments and puts a stringent constraint on the absolute neutrino mass. The proposed $U_{\rm SO3}$ neutrino mixing parameterization is suited for independent CP-conserving neutrino oscillation experimental analysis.

Figures (4)

• Figure 1: The $U$ matrix constructed from Euler angles written in (\ref{['eul']}) for 6 orderings. The framed subplot represents the standard PMNS matrix order $R_x R_y R_z$. The red-green-blue arc plots with $10^\circ$ marks visualise angles of performed rotations.
• Figure 2: The proposed direct rotation by $\theta=\sqrt{\theta_{x}^2+\theta_{y}^2+\theta_{z}^2}\simeq 56.5^{\circ}$. The arc plot visualises values of the projected rotations for the determined neutrino mixing angles in Eq. (\ref{['angles2']}).
• Figure 3: Probability densities (PDF) based on NuFIT-6.0 data esteban2025nufit for the original PMNS parameterization with any values of $\delta_{CP}$ ($\theta_{23,13,12}$, top plots) and the proposed order-independent SO(3) parameterization, so with $\delta_{CP}=180^\circ$ ($\theta_{x,y,z}$, bottom plots). Values inside each plot are angles for PDF maxima.
• Figure 4: Prediction for $m_{\beta}$ (left) and $m_{\beta\beta}$ (right) against the lightest neutrino mass ($m_1$) with $\theta_{x}=43.8^{\circ}\pm 2.1 ^{\circ}, \theta_{y}=21.7^{\circ}\pm 0.6^{\circ}, \theta_{z}=28.3^{\circ}\pm 0.8^{\circ}$ for NO. In both panels, the green patch represents the SO(3) prediction, whereas the yellow and gray shaded regions represent the 1$\sigma$ and 3$\sigma$ allowed region esteban2025nufit using $U_{\rm PMNS}$ parameterization. The dark blue patch in the right panel shows the 1$\sigma$$U_{\rm PMNS}$ allowed region with $\delta_{\rm CP}=180^{\circ}$ and $\alpha_{1,2}=0$.