Observable #24 — |V_tb| — The Theory of Persistence

Formula

$$|V_{tb}| = \cos\theta_{23}^{\rm CKM} \cdot \cos\theta_{13}^{\rm CKM}$$

Input theorems

This derivation uses the following theorems from the PT chain:

Derivation

|V_tb| — last diagonal

$|V_{tb}| \approx 0.999$, nearly exactly 1. This quasi-unitarity reflects the geometric isolation of the top in PT — it saturates amplitudes at the fixed point.

$$ |V_{tb}| = \cos\theta_{23}^{\rm CKM} \cdot \cos\theta_{13}^{\rm CKM}. $$

Computation

PT values: - $\cos\theta_{23} = \sqrt{1 - (0.040\,746)^2} = 0.999\,170$ - $\cos\theta_{13} = \sqrt{1 - (0.003\,814)^2} = 0.999\,993$

$$ |V_{tb}| = 0.999\,170 \cdot 0.999\,993 = 0.999\,215. $$

PT: 0.999 215 vs PDG: 0.9991 ± 0.0001. Gap: 0.012%.

Top coupling universality

$|V_{tb}| \approx 1$ ensures top decays almost exclusively to W + b. PT predicts this near-universality as consequence of unitary Yukawa $y_t = 1$ (cf. m_t).