J_CKM

J_CKM — quark Jarlskog invariant

The CKM analogue of J_PMNS:

$$ J_{\rm CKM} = \Im(V_{us} V_{cb} V_{ub}^* V_{cs}^*). $$

This invariant controls CP violation in the quark sector.

Computation

PT values: - $V_{us} = 0.224\,21$ - $V_{cb} = 0.040\,75$ - $|V_{ub}| = 0.003\,814$, phase $\delta_{\rm CP}^{\rm CKM} = 66.912°$ - $V_{cs} = 0.974\,406$

$$ J_{\rm CKM} = 0.22421 \cdot 0.04075 \cdot 0.003814 \cdot 0.97441 \cdot \sin(66.912°) = 3.093 \times 10^{-5}. $$

PT: 3.093 × 10⁻⁵ vs PDG: 3.08 × 10⁻⁵. Gap: 0.44%.

J_CKM ≪ J_PMNS

$J_{\rm CKM} \approx 3 \times 10^{-5}$, ~300× smaller than $J_{\rm PMNS} \approx 10^{-2}$. This difference makes leptogenesis (vs pure baryogenesis) plausible as origin of matter-antimatter asymmetry.