Jarlskog Invariant of the Neutrino Mapping Matrix

Understand

The Jarlskog Invariant $J_{\nu-map}$ of the neutrino mapping matrix is calculated based on a phenomenological model which relates the smallness of light lepton masses $m_e$ and $m_1$ (of $\nu_1$) with the smallness of $T$ violation.

• For small $T$ violating phase $\chi_l$ in the lepton sector, $J_{\nu-map}$ is proportional to $\chi_l$, but $m_e$ and $m_1$ are proportional to $\chi_l^2$.
• This leads to $ J_{\nu-map} \cong {1/6}\sqrt{\frac{m_e}{m_\mu}}+O \bigg(\sqrt{\frac{m_em_\mu}{m_\tau^2}}\bigg)+O \bigg(\sqrt{\frac{m_1m_2}{m_3^2}}\bigg)$.
• Assuming $\sqrt{\frac{m_1m_2}{m_3^2}}<<\sqrt{\frac{m_e}{m_\mu}}$, we find $J_{\nu-map}\cong 1.16\times 10^{-2}$, consistent with the present experimental data.