CKM & PMNS Mixing Angles
Quark and neutrino mixing matrices derived from mass ratios and Nc = 3 tetrahedral geometry. Both CP-violating phases are dihedral angles of the regular tetrahedron — the fundamental symmetric object in three dimensions.
§1 — CKM Matrix: Quark Mixing
The Cabibbo–Kobayashi–Maskawa (CKM) matrix describes how the three quark generations mix under the weak interaction. In GWT, each quark is a standing wave with frequency proportional to its mass. The CKM elements are frequency ratios of these modes: two standing waves couple at the proton’s 2D nodal boundary, and the coupling amplitude scales as the square root of their frequency ratio. The ½ power (surface geometry, d−1 = 2) distinguishes quark mixing from lepton mixing (which uses the ⅓ power from 3D bulk geometry). Zero free parameters.
Vus — The Cabibbo Angle
The Cabibbo angle receives contributions from both the down-type and up-type sectors. These contributions are orthogonal in flavour space (they rotate in perpendicular planes), so they add in quadrature:
md = 4.783 MeV, ms = 98.56 MeV (down-type)
mu = 2.214 MeV, mc = 1271 MeV (up-type)
md/ms = 4.783 / 98.56 = 0.04853
mu/mc = 2.214 / 1271 = 0.001742
λ² = 0.04853 + 0.001742 = 0.05027
λ = √(0.05027) = 0.2242
Observed: Vus = 0.22500 ± 0.00067 (PDG 2024)
The down-type sector contributes √(md/ms) = 0.2203 and the up-type contributes √(mu/mc) = 0.0417. Because they rotate in orthogonal flavour planes, their squares add: λ² = (md/ms) + (mu/mc). The ½ power comes from the (d−1) = 2 dimensional proton boundary where quarks couple — this is the surface geometry analog of the ⅓ power used in PMNS bulk geometry.
Vcb — Second-Generation Coupling
The 2–3 generation coupling is the up-type Cabibbo angle. Just as Vus receives its leading contribution from √(md/ms), Vcb comes from the corresponding up-type mass ratio:
mu = 2.214 MeV (n = 13, p = 31)
mc = 1271 MeV (n = 11, p = 27)
mu/mc = 2.214 / 1271 = 0.001742
Vcb = √(0.001742) = 0.04173
Observed: Vcb = 0.04182 ± 0.00085 (PDG 2024)
This replaces the previous Wolfenstein form Vcb = Aλ² which required the ad hoc amplitude A = √(2/d). The unified formula Vcb = √(mu/mc) has zero free parameters and is more accurate: 0.1σ vs the previous 0.2σ. The ½ power (surface geometry) is the same as in Vus and Vub.
Vub — Third-Generation Coupling
The 1–3 generation coupling spans the full up-type mass hierarchy. The surface-geometry formula gives the direct frequency ratio:
mu = 2.214 MeV (n = 13, p = 31)
mt = 176,547 MeV (n = 12, p = 24)
mu / mt = 2.214 / 176,547 = 1.254 × 10−5
Vub = √(1.254 × 10−5) = 0.003541
Observed: Vub = 0.00369 ± 0.00011
Using the GWT §24 sine-Gordon masses (mu = 2.214 MeV, mt = 176,547 MeV) gives 4% accuracy. This is the weakest CKM prediction, as Vub connects generation 1 to generation 3 in a single step. The ½ power again reflects 2D surface geometry.
CKM CP Phase δCKM
The CP-violating phase of the CKM matrix is the dihedral angle of a regular tetrahedron — the fundamental 3D symmetric object uniquely determined by Nc = 3:
In bond physics, the antibonding enhancement is fanti = 2d/(2d−1) = 6/5. This measures the asymmetry between constructive and destructive interference channels in d = 3 dimensions. The same asymmetry governs the CP-violating phase of the quark mixing matrix.
The cosine of the CP phase is the reciprocal of twice the antibonding factor:
cos(δ) = 1/(2fanti) = (2d−1)/(4d) = 5/12 = 0.41667
This connects CP violation directly to the bonding/antibonding asymmetry of 3D standing waves.
δCKM = arccos(5/12) = 65.38°
Consistency check: for d = 2, cos(δ) = 3/8 → δ = 67.98°. For d = 1, cos(δ) = 1/4 → δ = 75.52°. The phase decreases with increasing d, converging to arccos(1/2) = 60° as d → ∞.
Observed: δCKM ≈ 65.5° ± 2.0° (PDG 2024)
Jarlskog Invariant J
The Jarlskog invariant is a rephasing-invariant measure of CP violation in the quark sector. It combines all mixing angles and the CP phase into a single number:
s12 = λ = 0.2242
s23 = Vcb = 0.04173
s13 = Vub = 0.003541
δ = arccos(5/12) = 65.38°
c12 = √(1 − 0.2242²) = 0.9745
c23 = √(1 − 0.04173²) = 0.9991
c13 = √(1 − 0.003541²) = 0.99999
J = 0.9745 × 0.9991 × (0.99999)² × 0.2242 × 0.04173 × 0.003541 × sin(65.38°)
J = 0.9745 × 0.9991 × 1.0000 × 0.2242 × 0.04173 × 0.003541 × 0.9091
J ≈ 2.93 × 10−5
Observed: J = 3.08 × 10−5 ± 0.15 × 10−5
The Jarlskog invariant is a derived quantity — it inherits the deviation in Vub. The −1.0σ accuracy reflects the consistent use of the four geometric inputs.
§2 — PMNS Matrix: Neutrino Mixing
The Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix describes lepton-sector mixing. Neutrinos mix far more strongly than quarks — their angles are large, not small. GWT derives the full PMNS matrix from a single rotation applied to the tribimaximal leading order: UPMNS = R(θ, axis) × UTBM, where θ = arcsin((me/mμ)1/d) and axisi = μ-diri × max(1, σp/σi). All three angles within 1σ, zero free parameters. See §27 of the Hamiltonian page for the full geometric derivation.
θ12 — Solar Angle
sin²θ12 = 1/d = 1/3 gives the tetrahedral angle θ12 = 35.26°. This comes from the democratic coupling of the d = 3 lattice.
UPMNS = R(θ, axis) × UTBM corrects the TBM leading order. The rotation brings θ12 from 35.26° down to 33.49°, within +0.1σ of observation.
Observed: θ12 = 33.41° ± 0.78° (NuFIT 6.0)
θ23 — Atmospheric Angle
TBM gives θ23 = 45° (maximal mixing). The rotation correction shifts this to 49.28°. The deviation from 45° arises from the proton wrapping factor in the rotation axis, which breaks the exact μ–τ symmetry.
Observed: θ23 = 49.20° ± 1.05° (NuFIT 6.0)
θ13 — Reactor Angle
TBM gives θ13 = 0. The rotation axis has a nonzero (1,1,1) component (12% of the axis norm) due to the tau wrapping correction σp/στ = 1.242. This component mixes the TBM eigenvector with the degenerate subspace, generating θ13 ≠ 0. The tau correction nearly doubles this component (from 6.9% to 12.4%), which is why it has such a large effect on θ13.
Observed: θ13 = 8.57° ± 0.12° (NuFIT 6.0)
PMNS CP Phase δCP
The leptonic CP phase is the supplementary dihedral angle of the tetrahedron — the opposite handedness from the quark sector:
CKM: δ = arccos(1/(2fanti)) = arccos(5/12) = 65.38° (antibonding geometry)
PMNS: δ = arccos(−1/3) = 109.47° (tetrahedral dihedral, opposite handedness)
Both phases arise from d = 3 wave geometry: CKM from the bonding/antibonding asymmetry, PMNS from the tetrahedral symmetry of democratic mixing.
In the standard PMNS parameterization, this is written as δCP = −109.5° (or equivalently +250.5°), reflecting the opposite-handedness convention.
Observed: between −90° and −135° (best fit ~−100° to −120°)
The PMNS CP phase is still poorly measured. Current data are fully consistent with the GWT prediction of −109.5°. Future long-baseline experiments (DUNE, Hyper-Kamiokande) will test this to high precision.
§3 — The Geometric Pattern
Both CP-violating phases come from the same geometric object — the regular tetrahedron — with opposite signs:
| Sector | Formula | Value | Handedness |
|---|---|---|---|
| CKM (quarks) | arccos(5/12) | 65.38° | Positive |
| PMNS (leptons) | arccos(−1/3) | 109.47° | Negative |
| Sum | — | 174.85° | Near-supplementary |
The PMNS phase is the bare tetrahedral dihedral angle arccos(−1/3). The CKM phase comes from the antibonding geometry of 3D standing waves: cos(δCKM) = 1/(2fanti) = (2d−1)/(4d) = 5/12. The CP phase encodes the asymmetry between constructive and destructive interference channels — the same fanti = 6/5 that governs antibonding in molecules.
Why These Phases?
The PMNS phase arccos(−1/3) is the tetrahedral dihedral angle — the unique angle at which four equidistant planes meet in 3D space. The CKM phase arccos(5/12) = arccos(1/(2fanti)) comes from the antibonding geometry: fanti = 2d/(2d−1) = 6/5 measures the asymmetry between constructive and destructive wave interference, and the CP phase encodes this asymmetry as a complex rotation.
CP violation is not a mysterious accident. It is the geometric inevitability of three generations living in three dimensions, with phases set by the wave interference geometry of d = 3.
§4 — Three Generations = Three Dimensions
The number of fermion generations equals the number of spatial dimensions:
Each generation corresponds to a standing-wave mode polarized along one spatial axis:
| Generation | Polarization | Leptons | Quarks |
|---|---|---|---|
| 1st | x-axis | e, νe | u, d |
| 2nd | y-axis | μ, νμ | c, s |
| 3rd | z-axis | τ, ντ | t, b |
The mass hierarchy between generations arises from coupling complexity along each axis. The first-generation mode has the simplest boundary conditions (lowest energy), while higher generations require progressively more complex waveforms (higher energy = higher mass).
Mixing occurs because the mass eigenstates (standing-wave modes along lattice axes) are not perfectly aligned with the weak-interaction eigenstates (modes that couple to the transverse W-boson waves). The misalignment angles are the CKM and PMNS matrices.
§5 — Unified CKM Matrix Construction
All four CKM parameters (θ12, θ23, θ13, δ) are now derived from quark mass ratios + tetrahedral geometry. The full 3×3 matrix is built in the standard PDG parametrization:
θ12 = arcsin(√(md/ms + mu/mc)) = 12.957° (quadrature Cabibbo)
θ23 = arcsin(√(mu/mc)) = 2.392° (up-type Cabibbo)
θ13 = arcsin(√(mu/mt)) = 0.203° (1–3 surface overlap)
δ = arccos(5/12) = 65.38° (tetrahedral boundary geometry)
|Vud| = 0.97453 |Vus| = 0.22422 |Vub| = 0.00354
|Vcd| = 0.22408 |Vcs| = 0.97368 |Vcb| = 0.04173
|Vtd| = 0.00852 |Vts| = 0.04101 |Vtb| = 0.99912
J = Im(VudVcsV*usV*cd) = 2.93 × 10−5
Observed: J = 3.08 × 10−5 (−4.8%, −1.0σ)
Mean error 0.64% across all 9 elements. Unitarity exact by construction. The previous Wolfenstein A = √(2/d) is no longer needed — Vcb = √(mu/mc) is simpler and more accurate.
Why Surface Geometry?
All CKM angles use the ½ power (square root of mass ratios). This is the (d−1) = 2 dimensional surface geometry, because all six quarks are confined inside the proton. The proton’s 2D boundary is where the weak interaction couples generation modes. In contrast, the PMNS matrix uses the ⅓ power (bulk, d = 3) because leptons are free particles that extend through the full 3D lattice.
§6 — Summary: All Mixing Predictions
Every mixing angle and CP phase, compared against experiment:
| Parameter | GWT | Observed | Accuracy |
|---|---|---|---|
| CKM (Quark Sector) — Unified Construction | |||
| Vus | 0.22422 | 0.22500 ± 0.00067 | −1.2σ |
| Vcb | 0.04173 | 0.04182 ± 0.00085 | −0.1σ |
| Vub | 0.003541 | 0.00369 ± 0.00011 | −1.4σ |
| δCKM | 65.38° | ≈ 65.5° ± 2.0° | −0.1σ |
| J (Jarlskog) | 2.93 × 10−5 | 3.08 × 10−5 | −1.0σ |
| Vtd | 0.00852 | 0.00857 ± 0.00020 | −0.3σ |
| Vts | 0.04101 | 0.04110 ± 0.00085 | −0.1σ |
| PMNS (Lepton Sector) | |||
| θ12 (solar) | 33.49° | 33.41° | +0.1σ |
| θ23 (atmospheric) | 49.28° | 49.20° | +0.1σ |
| θ13 (reactor) | 8.63° | 8.57° | +0.5σ |
| δPMNS | −109.5° | ≈ −100° to −120° | consistent |
| Full Matrix (mean |error| = 0.64%) | |||
| Vud | 0.97453 | 0.97435 ± 0.00016 | +1.1σ |
| Vcs | 0.97368 | 0.97349 ± 0.00016 | +1.2σ |
| Vtb | 0.99912 | 0.99912 ± 0.00004 | +0.1σ |
The Pattern
All mixing parameters from zero free parameters. The CKM matrix is fully determined by four mass ratios and one geometric angle: θ12 = arcsin(√(md/ms + mu/mc)), θ23 = arcsin(√(mu/mc)), θ13 = arcsin(√(mu/mt)), δ = arccos(5/12). All 9 elements within 1.4σ, mean error 0.64%.
The quark sector (CKM) has small mixing angles because quark mass ratios are small — the three generation modes are nearly orthogonal. The lepton sector (PMNS) has large mixing angles because neutrino mass ratios are close to unity — the three modes are nearly democratic.
Both sectors derive from d = 3 wave geometry: CKM uses the ½ power (surface, d−1 = 2) with CP phase from antibonding asymmetry arccos(1/(2fanti)) = arccos(5/12); PMNS uses the ⅓ power (bulk, d = 3) with CP phase from tetrahedral symmetry arccos(−1/3). One geometry, two boundary conditions.