Exact Phenomenology of the Neutrino Cuboid: Normal Mass Ordering, the Tribimaximal Limit, and Cosmological Constraints

Abstract

We investigate the exact phenomenology of a geometric neutrino cuboid defined by $m_1=m_0\sin \xi$, $m_2=m_0\cos \xi \sin \zeta$, and $m_3=m_0\cos \xi \cos \zeta$, whose cubic point corresponds to mass degeneracy and the tribimaximal values of the solar and atmospheric mixing angles. Imposing the mass-mixing alignment $\xi ={\theta }_{12}$ and $\zeta ={\theta }_{23}$, we derive closed-form consistency relations without relying on an expansion about the cubic limit. These relations show that the observed condition ${\sin }^2{\theta }_{12}<1/3$ selects normal mass ordering and, in the resulting normal-ordering regime, the physical condition $0<\Delta m_{21}^2/\Delta m_{31}^2<1$ requires ${\theta }_{23}$ to lie in the lower octant. Representative JUNO-based inputs give ${\sin }^2{\theta }_{23}\simeq 0.440$, $(m_1,m_2,m_3)\simeq (0.0938,0.0942,0.1063)\mathrm{eV}$, and ${\sum }_i{m}_i\simeq 0.294\mathrm{eV}$. We demonstrate that the first-order expansion around the tribimaximal point is numerically unstable for the solar-to-atmospheric mass-splitting ratio because of a leading-order cancellation. Present atmospheric-angle data yield only mild tension with exact alignment, whereas stringent neutrino-mass bounds in baseline $\Lambda$CDM cosmology strongly disfavor it; its viability under extended cosmological models remains model dependent. We finally formulate geometric alignment residuals as general null-test observables, providing a systematic framework for testing the neutrino cuboid with future oscillation and absolute-mass measurements.

Comments: 41 pages