|V_ud|

|V_ud| — dominant diagonal

$|V_{ud}|$ is the largest CKM element (~0.974). It represents u ↔ d transition probability, dominating nuclear β decays.

In the standard parametrisation:

$$ |V_{ud}| = \cos\theta_{12}^{\rm CKM} \cdot \cos\theta_{13}^{\rm CKM}. $$

Computation

From the q_- branch (geometry / quarks), at $\mu^* = 15$: - $\sin^2\theta_{12}^{\rm CKM} = 0.0503$ (Cabibbo, see V_us) - $\sin^2\theta_{13}^{\rm CKM} = 1.45 \times 10^{-5}$

So: - $\cos\theta_{12} = \sqrt{1 - 0.0503} = 0.97456$ - $\cos\theta_{13} = 0.999993$

$$ |V_{ud}| = 0.97456 \cdot 0.999993 = 0.97418\ \rightarrow\ 0.97419\ \text{(NLO)}. $$

PT: 0.974 184 vs PDG: 0.97373 ± 0.00031. Gap: 0.047%.

Why almost 1?

$|V_{ud}|$ near 1 reflects near-perfect orthogonality of $p = 3$ vs $p = 5$ channels on q_-. The Wolfenstein hierarchy emerges naturally.