|V_cs|

|V_cs| — the other dominant diagonal

$|V_{cs}| \approx 0.975$ is the second CKM diagonal. Like $V_{ud}$, it reflects PT channel orthogonality.

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

Computation

Values: - $\cos\theta_{12} = 0.97456$ - $\cos\theta_{23} = 0.99917$ - $\sin\theta_{12} = 0.22421$ - $\sin\theta_{23} = 0.04075$ - $\sin\theta_{13} = 0.003814$

$$ |V_{cs}| \approx 0.97456 \cdot 0.99917 - 0.22421 \cdot 0.04075 \cdot 0.003814 \cdot \cos(\delta_{\rm CP}) = 0.974\,406. $$

PT: 0.974 406 vs PDG: 0.975 ± 0.006. Gap: 0.061%.

Consequence for D-mesons

$|V_{cs}|$ controls charm → strange decays (D⁰ → K⁻π⁺, etc.). PT precision 0.06% compares with BaBar/Belle measurements.