Jarlskog Invariant of the Neutrino Mapping Matrix

Abstract

The Jarlskog Invariant $J_{ν-map}$ 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 $ν_1$) with the smallness of $T$ violation. For small $T$ violating phase $χ_l$ in the lepton sector, $J_{ν-map}$ is proportional to $χ_l$, but $m_e$ and $m_1$ are proportional to $χ_l^2$. This leads to $ J_{ν-map} \cong {1/6}\sqrt{\frac{m_e}{m_μ}}+O \bigg(\sqrt{\frac{m_em_μ}{m_τ^2}}\bigg)+O \bigg(\sqrt{\frac{m_1m_2}{m_3^2}}\bigg)$. Assuming $\sqrt{\frac{m_1m_2}{m_3^2}}<<\sqrt{\frac{m_e}{m_μ}}$, we find $J_{ν-map}\cong 1.16\times 10^{-2}$, consistent with the present experimental data.