LeptonicCPviolation in a two parameter model

We further study the “complementary” ansatz, $\mathrm{Tr}(M_{\nu })=0$, for a prediagonal light Majorana type neutrino mass matrix. Previously, this was studied for the $CP$ conserving case and the case where the two Majorana type $CP$ violating phases were present but the Dirac type $CP$ violating phase was neglected. Here we employ a simple geometric algorithm which enables us to “solve” the ansatz including all three $CP$ violating phases. Specifically, given the known neutrino oscillation data and an assumed two parameter (the third neutrino mass $m_3$ and the Dirac $CP$ phase $\delta$) family of inputs we predict the neutrino masses and Majorana $CP$ phases. Despite the two parameter ambiguity, interesting statements emerge. There is a characteristic pattern of interconnected masses and $CP$ phases. For large $m_3$ the three neutrinos are approximately degenerate. The only possibility for a mass hierarchy is to have $m_3$ smaller than the other two. A hierarchy with $m_3$ largest is not allowed. Small $CP$ violation is possible only near two special values of $m_3$. Also, the neutrinoless double beta decay parameter is approximately bounded as $0.020\mathrm{eV}<|m_{ee}|<0.185\mathrm{eV}$. As a by-product of looking at physical amplitudes we discuss an alternative parametrization of the lepton mixing matrix which results in simpler formulas. The physical meaning of this parametrization is explained.