Neutrino Masses & Mixing
Deriving the neutrino mass scale, mass squared splittings, generation count, PMNS mixing angles, and wave sizes — all from me, mp, and Nc = 3. Zero free parameters.
§1 — Neutrino Mass Scale
Neutrinos are “colorless” — they do not couple to the strong force. Their mass scale is set by a seesaw-like ratio between the lepton and baryon sectors:
where d = 3 (the number of spatial dimensions). The leading factor me3/(d × mp2) is a seesaw-like ratio: neutrinos are third-order standing wave modes (e → p → e), averaged over d spatial axes. The correction factor (1 + 1/(d × 4π)) is the Wyler per-axis correction on the transverse sphere S2. Neutrinos are purely transverse Weyl spinors with no longitudinal polarization, so the relevant manifold is Sd−1 = S2 (Vol = 4π), not the full Sd = S3 (Vol = 2π2) used for massive Dirac particles. Same formula, sphere dimension matches the transverse-only geometry.
me = 0.5046 MeV (from m(16,32))
me3 = (0.5046)3 = 0.1285 MeV3
mp = 6π5 × me = 926.5 MeV
mp2 = (926.5)2 = 8.58 × 105 MeV2
M0 = me3 / (d × mp2) = 0.1285 / (3 × 8.58 × 105)
= 4.989 × 10−8 MeV = 49.89 meV
Neutrinos are Weyl spinors (purely transverse, d−1 = 2 polarizations).
Correction uses Vol(Sd−1) = Vol(S2) = 4π, not Vol(S3) = 2π2.
Meff = M0 × (1 + 1/(3 × 4π)) = 49.89 × 1.02653
= 51.21 meV = 0.05121 eV
Observed: ∼0.050 eV (from oscillation data)
Physical Meaning
Neutrinos are “hadron-blind” — their mass scale comes from a seesaw-like ratio of the lepton and baryon sectors. The cubic dependence on me reflects the fact that neutrinos are third-order standing wave modes: they arise from the residual oscillation left over after the dominant electromagnetic (me) and strong (mp) modes have been accounted for. The factor d = 3 in the denominator enforces the lattice’s three-dimensional structure — neutrino mass is divided equally among the three spatial directions. The transverse Wyler correction uses Vol(S2) = 4π because neutrinos have no longitudinal mode — the same Wyler formula used for massive particles, but with the sphere dimension matching the transverse-only geometry of Weyl spinors.
§2 — Mass Squared Splitting Δm231 (Atmospheric)
The atmospheric mass squared splitting is the dominant oscillation frequency, measured by atmospheric and accelerator neutrino experiments. GWT predicts:
where Neff = Ntop × (1 + 1/Vol(S3)) = 25 × (1 + 1/(2π2)) = 26.27. Here Ntop = d × 2d + 1 = 25 counts the topological modes (24 breathers + 1 kink) of the lattice, and the Wyler correction uses Vol(S3) = 2π2 because this is a topological mode count, not a polarization correction.
Meff2 = (0.05121 eV)2 = 2.623 × 10−3 eV2
1 − 1/Neff = 1 − 1/26.27 = 0.9619
Δm231 = 0.9619 × 2.623 × 10−3
= 2.523 × 10−3 eV2
Observed: 2.534 × 10−3 eV2 (NuFIT 6.0, 2024)
§3 — Mass Squared Splitting Δm221 (Solar)
The solar mass squared splitting drives the slower oscillation seen in reactor and solar neutrino experiments. GWT predicts:
The prefactor d/(4 Neff) = 3/(4 × 26.27) = 0.02855 encodes the spatial fraction d/(d+1) = 3/4 of the lattice participating in solar oscillations, divided by the effective mode count Neff.
Δm221 = d/(4 Neff) × Meff2
= 0.02855 × 2.623 × 10−3 eV2
= 7.49 × 10−5 eV2
Observed: 7.53 × 10−5 eV2 (NuFIT 6.0)
§4 — Number of Neutrino Generations
The Standard Model treats Nν = 3 as an empirical input. GWT derives it from the lattice:
Each neutrino generation corresponds to one independent oscillation direction in the 3D lattice. A standing wave can oscillate along the x, y, or z axis — giving exactly three independent neutrino species. There cannot be a fourth, because the lattice has no fourth spatial dimension.
Observed: Nν = 2.984 ± 0.008 (LEP, Z-boson invisible width)
Why Exactly Three
The LEP measurement is not exactly 3 because it measures the Z-boson’s invisible decay width and divides by the predicted single-neutrino rate. The 0.5% gap is a radiative correction in the Standard Model’s own calculation. GWT predicts the integer 3 exactly: one generation per spatial dimension. No more, no less. A fourth generation would require a fourth spatial dimension, which the lattice does not possess.
§5 — PMNS Mixing Angles
The PMNS matrix describes how the three neutrino mass eigenstates (ν1, ν2, ν3) mix to form the three flavor eigenstates (νe, νμ, ντ). In GWT, the PMNS matrix is derived from a single rotation applied to the tribimaximal (TBM) leading order:
where θ = arcsin((σμ/σe)) = arcsin((me/mμ)1/d) and axisi = μ-directioni × max(1, σp/σi). The axis is the muon direction in the TBM flavour triangle, rescaled by the proton wrapping factor. The angle is the standing-wave size ratio of the muon at the electron’s scale. Zero free parameters. See §27 of the Hamiltonian page for the full geometric derivation.
θ12 (Solar Angle)
The TBM leading order gives sin²θ12 = 1/d = 1/3 (35.26°). The rotation correction R(θ, axis) × UTBM brings this down to 33.49°, within +0.1σ of observation.
θ23 (Atmospheric Angle)
The TBM leading order gives θ23 = 45° (maximal mixing). The rotation correction shifts this to 49.28°, within +0.1σ of observation. The deviation from 45° arises from the proton wrapping factor in the rotation axis.
θ13 (Reactor Angle)
The TBM leading order gives θ13 = 0. The rotation axis has a nonzero (1,1,1) component (12% of the axis norm) due to the tau wrapping correction, which mixes the TBM eigenvector with the degenerate subspace and generates θ13 ≠ 0. The predicted 8.63° is within +0.5σ of observation.
δCP (CP-Violating Phase)
The CP-violating phase in the neutrino sector is the supplement of the tetrahedral angle: arccos(−1/3) = 109.47°, appearing as −109.5° because of the opposite handedness relative to the CKM matrix (quark sector). This angle is unique to three-dimensional geometry — it is the angle between any two bonds in a tetrahedron, the most symmetric arrangement of four points in 3D.
The Geometric Pattern
All four PMNS parameters are derived from UPMNS = R(θ, axis) × UTBM with geometrically determined rotation parameters. The axis is the muon direction in the TBM flavour triangle, rescaled by the proton wrapping factor max(1, σp/σi). The angle is the standing-wave size ratio θ = arcsin((me/mμ)1/d). All three mixing angles are predicted within 1σ of observation with zero free parameters. The CP phase is the tetrahedral dihedral angle δ = arccos(−1/d). The Standard Model requires 7 free parameters for the neutrino sector; GWT derives all of them from d = 3, me, mμ, mτ, and mp.
§6 — Neutrino Wave Sizes
In GWT, every particle is a standing wave with a definite spatial extent. Neutrinos are 1D weak standing waves — their wave function extends along one lattice direction, and their characteristic size is the Compton wavelength:
Because neutrino masses are tiny, their wave sizes are enormous by particle physics standards — comparable to biological cells.
ν3 (heaviest)
m3 = Meff = 0.05121 eV = 0.05121 × 10−6 MeV
λC = ℏc / m3 = 197.3 MeV·fm / (0.05121 × 10−6 MeV)
= 197.3 / (5.121 × 10−8) fm
= 3.853 × 109 fm = 3.853 μm
ν2 (middle)
From the mass splittings, m2 is determined by Δm221 and the hierarchy. The resulting Compton wavelength:
ν1 (lightest)
Why Neutrinos Are Ghostly
The “ghostliness” of neutrinos — their incredibly tiny interaction cross sections — is explained by the ratio of their wave size to the weak interaction range:
rweak = ℏc / (MZc2) ≈ 2.16 × 10−18 m = 2.16 × 10−3 fm
λC(ν3) / rweak = 3.853 × 10−6 m / 2.16 × 10−18 m
= 1.78 × 1012
The neutrino wave is nearly two trillion times larger than the range over which it can interact. This enormous mismatch explains why neutrinos have cross sections of order σ ∼ 10−44 cm2 — the wave simply does not “fit” into the interaction region.
Physical Picture
A neutrino is a standing wave stretching across microns of space, but it can only exchange energy through the weak force, which operates at scales of 10−18 m. Imagine trying to thread a 20-meter rope through a keyhole — the probability is proportional to the ratio of keyhole size to rope length, squared. This is why neutrinos pass through light-years of lead: they are enormous waves trying to interact through a tiny channel.
§7 — Summary
All neutrino predictions from GWT, derived from me, mp, and d = 3 — which are themselves derived from the three lattice constants {k, a, η}. Zero free parameters.
| Quantity | GWT Prediction | Observed | Accuracy |
|---|---|---|---|
| Mass scale Meff | 0.05121 eV | ∼0.050 eV | ∼2% |
| Δm231 | 2.523 × 10−3 eV2 | 2.534 × 10−3 eV2 | 0.4% |
| Δm221 | 7.49 × 10−5 eV2 | 7.53 × 10−5 eV2 | 0.6% |
| Nν | 3 | 2.984 ± 0.008 | 0.5% |
| θ12 (solar) | 33.49° | 33.41° | +0.1σ |
| θ23 (atm.) | 49.28° | 49.20° | +0.1σ |
| θ13 (reactor) | 8.63° | 8.57° | +0.5σ |
| δCP | −109.5° | −90° to −135° | consistent |
| ν3 wave size | 3.853 μm | — | prediction |
| ν2 wave size | 14.93 μm | — | prediction |
| ν1 wave size | 19.75 μm | — | prediction |
What This Means
The Standard Model requires at least 7 free parameters to describe the neutrino sector (3 masses, 3 mixing angles, 1 CP phase) and leaves the mass scale itself unexplained. GWT derives all of them from three lattice constants and d = 3.
Both mass squared splittings are predicted within 1% of observation: Δm231 to 0.4% and Δm221 to 0.6%. All three PMNS mixing angles are predicted within 1σ: θ12 = 33.49° (+0.1σ), θ23 = 49.28° (+0.1σ), θ13 = 8.63° (+0.5σ). The transverse Wyler correction (S2 instead of S3) reflects the purely transverse nature of neutrinos as Weyl spinors. The three wave size predictions (ν1, ν2, ν3) are unique to GWT and currently untestable — but they explain, for the first time, why neutrinos are ghostly.
The neutrino sector is not mysterious. It is geometry.