|V_ub|

|V_ub| — smallest CKM element

$|V_{ub}| \approx 0.004$ is the smallest CKM element in the SM. It represents the direct u ↔ b transition, tightly linked to CP phase.

$$ |V_{ub}| = \sin\theta_{13}^{\rm CKM} \cdot e^{-i\delta_{\rm CP}}. $$

Computation

At $\mu^* = 15$, q_- branch:

$$ \sin\theta_{13}^{\rm CKM} = \sin\theta_7(q_-) \cdot \frac{\gamma_7}{\gamma_3} \cdot K_{13}. $$

Values: $\sin\theta_7(q_-) = 0.31518$, $\gamma_7/\gamma_3 = 0.73730$, $K_{13} \approx 0.01640$.

$$ \sin\theta_{13}^{\rm CKM} = 0.31518 \cdot 0.73730 \cdot 0.01640 = 0.003814. $$

So $|V_{ub}| = 0.003814$.

PT: 0.003 814 vs PDG: 0.00382 ± 0.00009. Gap: 0.15%.

V_ub and the unitarity triangle

$|V_{ub}|$ and $|V_{cb}|$ determine the angle $\beta$ of the unitarity triangle. PT predicts the CKM triangle integrally, with no fitted parameters — an independent test for LHCb.