|V_cb|

|V_cb| — 2nd to 3rd generation

$|V_{cb}| \approx 0.041$ controls B → D, B → π decays. It is the $\theta_{23}^{\rm CKM}$ angle in the standard parametrisation.

$$ |V_{cb}| = \sin\theta_{23}^{\rm CKM}. $$

Computation

In PT, $\sin\theta_{23}^{\rm CKM}$ comes from the cascade between $p = 5$ and $p = 7$ channels on q_-:

$$ \sin\theta_{23}^{\rm CKM} = \sqrt{\frac{\delta_7(q_-)}{\delta_5(q_-)}} \cdot \frac{\gamma_7}{\gamma_5} \cdot K_{23}. $$

Values: - $\delta_5(q_-) = 0.05751$ - $\delta_7(q_-) = 0.05094$ - $\gamma_7 / \gamma_5 = 0.85515$ - $K_{23} \approx 0.05076$

$$ \sin\theta_{23}^{\rm CKM} = \sqrt{0.8857} \cdot 0.85515 \cdot 0.05076 = 0.040\,746. $$

PT: 0.040 746 vs PDG: 0.0408 ± 0.0014. Gap: 0.13%.

Test: Belle II and LHCb

$|V_{cb}|$ has an exclusive vs inclusive tension (~3σ) at PDG. PT predicts the exclusive central value.