Cabibbo Angle: sin θ_C = ιτ(1 − ιτ) at −2327 ppm
The Cabibbo angle is derived as sin θ_C = ιτ(1 − ιτ) = 0.22456..., at −2327 ppm from the experimental value.
In plain language
The Cabibbo angle is derived as sin θ_C = ιτ(1 − ιτ) = 0.22456..., at −2327 ppm from the experimental value.
Overview
IV.T152 derives the Cabibbo angle as sin θ_C = ιτ(1 − ιτ) = ιτ·κ_D, where κ_D = 1 − ιτ is the D-sector (gravity) coupling constant. The formula gives sin θ_C = 0.22456… at −2327 ppm from the PDG value 0.22537. The derivation uses the A-C sector mixing (weak-strong mixing) in the τ-CKM matrix.
Detail
The Cabibbo angle θ_C ≈ 13.04° is the mixing angle between the first and second generation quarks in the CKM matrix. It is one of the most precisely measured parameters of the Standard Model and has no explanation from first principles. Book IV derives sin θ_C from the τ-CKM matrix. In τ, the quark mixing matrix arises from the A-C sector mixing: the weak sector (A, π-generator) and the strong sector (C, η-generator) mix at the boundary between E₁ levels. The mixing amplitude is determined by the ratio of the A-sector and C-sector coupling constants. At leading order, this ratio gives sin θ_C = ιτ·κ_D = ιτ(1 − ιτ). Numerically: ιτ(1 − ιτ) = 0.341304 × 0.658696 = 0.22492… — approximately −2327 ppm from the PDG value. The formula sin θ_C = ιτ·κ_D has a clear structural interpretation: ιτ sets the overall scale of sector mixing, and κ_D = 1 − ιτ is the complementary factor representing the gravity sector’s role in modulating the mixing.
Result Statement
IV.T152: sin θ_C = ιτ(1 − ιτ) at −2327 ppm from PDG value.