Jarlskog Invariant of the Neutrino Mapping Matrix

Abstract

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 $J_{\nu -map}\cong 1/6\sqrt{\frac{m_e}{m_{\mu }}}+O\left(\sqrt{\frac{m_e{m}_{\mu }}{m_{\tau }^2}}\right)+O\left(\sqrt{\frac{m_1{m}_2}{m_3^2}}\right)$. Assuming $\sqrt{\frac{m_1{m}_2}{m_3^2}}<<\sqrt{\frac{m_e}{m_{\mu }}}$, we find $J_{\nu -map}\cong 1.16\times 10^{-2}$, consistent with the present experimental data.

Comments: 19 pages