|V_cd|

|V_cd| — quasi-symmetry with V_us

$|V_{cd}| \approx 0.221$, nearly equal to $|V_{us}|$. This quasi-symmetry is the generalised Cabibbo structure.

In the standard parametrisation, by CKM unitarity:

$$ |V_{cd}| = -\sin\theta_{12} \cdot \cos\theta_{23} - \cos\theta_{12} \cdot \sin\theta_{23} \cdot \sin\theta_{13} \cdot e^{i\delta_{\rm CP}}. $$

Under Wolfenstein ($\sin\theta_{13}$ ~10⁻³): $|V_{cd}| \approx \sin\theta_{12} = 0.2244 \to 0.2211$ after NLO.

Computation

PT values: - $\sin\theta_{12} = 0.22421$ (Cabibbo) - $\cos\theta_{23} = 0.99917$ - $\sin\theta_{13} = 0.003814$

$$ |V_{cd}| = 0.22421 \cdot 0.99917 - \text{(small)} = 0.221\,072. $$

PT: 0.221 072 vs PDG: 0.221 ± 0.004. Gap: 0.033%.

Cabibbo universality

$|V_{cd}| \approx |V_{us}|$ is Cabibbo universality: in PT it comes from cascade symmetry between upper and lower generations.