Neutrino Flavors on the Helix: Neutrino oscillation and the Dirac CP phase in the Euler lens

Abstract

Neutrino flavour oscillation is written in the language of the Euler circle. Each mass eigenstate is a unit-modulus phasor whose time derivative is the phasor turned through a right angle, ψ′ = −iEψ; lifted along a time axis the phasor becomes a helix, and the three eigenstates become three helices of slightly unequal pitch. Their beat is the oscillation, with wavelength fixed by the pitch difference, 4πE/Δm². The Dirac phase δ enters as a relative azimuthal twist of the helix bundle; complex conjugation, which carries neutrinos to antineutrinos, reflects the bundle and sends δ to −δ, so a nonzero δ makes the two beat patterns differ. That difference is CP violation, the ingredient leptogenesis needs to account for the matter excess of the universe. Amplitude is conserved along every helix: oscillation redistributes flavour without loss, the CP asymmetry sums to zero across channels, and the only true attrition is the slow loss of coherence as the eigenstate packets separate. The construction is developed in vacuum; matter effects enter as a tilt of the helix pitches that mimics a CP asymmetry and must be separated from the genuine phase. The present experimental position, from the T2K and NOvA beams and the first JUNO oscillation results of 2026, which do not yet establish a nonzero δ, to the DUNE and Hyper-Kamiokande programmes built to reach discovery significance, is set against the geometric picture.