J_PMNS

J_PMNS — Jarlskog invariant for neutrinos

The Jarlskog invariant measures CP violation "volume". For PMNS:

$$ J_{\rm PMNS} = \frac{1}{8}\sin 2\theta_{12}\sin 2\theta_{23}\sin 2\theta_{13}\cos\theta_{13}\sin\delta_{\rm CP}. $$

Computation

PT values: - $\sin^2\theta_{12} = 0.303\,684 \Rightarrow \sin 2\theta_{12} = 0.9201$ - $\sin^2\theta_{23} = 0.573\,252 \Rightarrow \sin 2\theta_{23} = 0.9893$ - $\sin^2\theta_{13} = 0.022\,216 \Rightarrow \sin 2\theta_{13} = 0.2949$, $\cos\theta_{13} = 0.9888$ - $\delta_{\rm CP} = 197.358° \Rightarrow \sin\delta_{\rm CP} = -0.2978$

$$ J_{\rm PMNS} = \frac{1}{8} \cdot 0.9201 \cdot 0.9893 \cdot 0.2949 \cdot 0.9888 \cdot (-0.2978) = -0.009\,889. $$

PT gives $|J_{\rm PMNS}| = 0.009\,889$. Sign is convention-dependent.

PT: 0.009 889 vs PDG: ~0.0099. Gap: 0.11%.

Matter/antimatter asymmetry

$J_{\rm PMNS}$ is ~30× larger than $J_{\rm CKM}$. This PMNS amplification makes leptogenesis plausible as the origin of cosmic matter/antimatter asymmetry.