General solutions for TM${}_{1,2}$ mixing in the minimal type-I seesaw mechanism and phenomenological constraints to Yukawa matrix of neutrinos

Abstract

In this paper, we concisely represent the general solution for the TM${}_{1,2}$ mixing in the type-I minimal seesaw mechanism and show phenomenological constraints of the Yukawa matrix of neutrinos $Y$. First, conditions of $Z_2$ symmetry associated with TM${}_{1,2}$ restrict two complex parameters. In addition, three complex parameters are constrained from five phenomenological parameters (two neutrino masses $m_i$, the mixing angle ${\theta }_{13}$, the Dirac phase $\delta$, and the Majorana phase $\alpha$) and the overall phase. As a result, there is only one free complex parameter for $Y$, except for the right-handed neutrino masses $M_{1,2}$. For the TM${}_1$ mixing with normal hierarchy, the symmetry condition is imposed only on $Y$, and the physical parameters depend on $M_R$. In other situations, the dimensionless parameters $({\theta }_{13},\delta$ and $\alpha )$ do not depend on $M_R$ but only on certain combinations of the components of $Y$.

Comments: 14 pages