A geometrical approach to neutrino oscillation parameters

TL;DR

The paper proposes a phenomenological geometric constraint on PMNS mixing: $2(\\theta_{12}+\\theta_{13}+\\theta_{23})=180^\\circ$, casting the doubled mixing angles as the interior angles of a Euclidean triangle. Using current global-fit NO values, it derives a predictive $\\theta_{23}^{pred}= (48.05\\pm0.74)^\\circ$ and finds consistency with the data within $1\\sigma$, while the triangle also reproduces the mass-squared ratio structure to within a few percent but underestimates the absolute splittings by about a factor of $\sim10$, suggesting the triangle encodes dimensionless hierarchy rather than the overall scale. The framework is explicitly phenomenological and falsifiable, offering clear tests via upcoming precision measurements (e.g., JUNO, DUNE, Hyper-K) that will either confirm the geometric relation or rule it out. If validated, the approach could point toward deeper structural or symmetry-based origins for lepton flavor, whereas even a refutation would provide a valuable benchmark for understanding why neutrino mixing deviates from simple symmetric patterns.

Abstract

We propose a geometric hypothesis for neutrino mixing: twice the sum of the three mixing angles equals $180^\circ$, forming a Euclidean triangle. This condition leads to a predictive relation among the mixing angles and, through trigonometric constraints, enables reconstruction of the mass-squared splittings. The hypothesis offers a phenomenological resolution to the $θ_{23}$ octant ambiguity, reproduces the known mass hierarchy patterns, and suggests a normalized geometric structure underlying the PMNS mixing. We show that while an order-of-magnitude scale mismatch remains (the absolute splittings are underestimated by $\sim 10\times$), the triangle reproduces mixing ratios with notable accuracy, hinting at deeper structural or symmetry-based origins. We emphasize that the triangle relation is advanced as an empirical, phenomenological organizing principle rather than a result derived from a specific underlying symmetry or dynamics. It is testable and falsifiable: current global-fit values already lie close to satisfying the condition, and improved precision will confirm or refute it. We also outline and implement a simple $χ^2$ consistency check against global-fit inputs to quantify agreement within present uncertainties.

Figures (2)

• Figure 1: Mixing triangle constructed with angles $\phi_{12}, \phi_{13}, \phi_{23}$ and sides labeled $a, b, c$ corresponding to $\Delta m^2_{32}, \Delta m^2_{21}, \Delta m^2_{31}$ respectively. The vertex labels reflect mass-squared eigenvalues $m_1^2$, $m_2^2$, and $m_3^2$.
• Figure 2: Variation of the triangle-predicted quantities with $\theta_{23}$. Left: Scaled value of $\Delta m^2_{32}$ as a function of $\theta_{23}$, assuming the triangle condition and fixing $\Delta m^2_{21}$, $\Delta m^2_{31}$ and $\theta_{13}$. Right: The corresponding value of $\theta_{12}$ obtained via the triangle constraint $\theta_{12} = 90^\circ - \theta_{13} - \theta_{23}$. The shaded region shows the global-fit uncertainty range for $\theta_{12}$, which is visibly exceeded when $\theta_{23}$ is tuned to match $\Delta m^2_{32}$.