|V_td|

|V_td| — top to down

$|V_{td}|$ is small (~0.008) because the t → d transition crosses two generations. Its value is crucial for $B_s$ oscillations.

By CKM unitarity:

$$ |V_{td}| = |V_{us}| \cdot |V_{cb}| \cdot \sqrt{1 + \rho^2 - 2\rho\cos\delta}, $$

where $(\rho, \eta)$ are Wolfenstein parameters.

Computation

PT values: - $|V_{us}| = 0.224\,21$ - $|V_{cb}| = 0.040\,75$ - $\rho \approx 0.141$, $\eta \approx 0.357$ (PT)

$$ |V_{td}| \approx 0.22421 \cdot 0.04075 \cdot 0.876 = 0.008\,00. $$

PT: 0.007 999 vs PDG: 0.008 ± 0.000 26. Gap: 0.017%.

Link to Δm_Bs

$|V_{td}|^2$ controls $\Delta m_{B_s}$ ($B_s$ - $\bar{B}_s$ oscillation). Measured at LHCb at $17.77 \pm 0.03$ ps⁻¹. PT value consistent at 0.02%.