A formula by $LD{L}^T$ decomposition for the minimal type-I seesaw mechanism and conditions of $CP$ symmetry in an arbitrary basis

Abstract

In this paper, defining a formula by $LD{L}^T$ decomposition for the minimal type-I seesaw mechanism, we obtain conditions of $CP$ symmetry for the neutrino mass matrix $m$ in an arbitrary basis. The conditions are found to be $\mathrm{Re}(M_{22}{a}_i-M_{12}{b}_i)\mathrm{Im}(M_{22}{a}_j-M_{12}{b}_j)=-\det M\mathrm{Re}{b}_i\mathrm{Im}{b}_j$ or $=-\det M\mathrm{Im}{b}_i\mathrm{Re}{b}_j$ for the Yukawa matrix $Y_{ij}=(a_j,b_j)$ and the right-handed neutrino mass matrix $M_{ij}$. In other words, the real or imaginary part of $b_i$ must be proportional to the real or imaginary part of the quantity $(M_{22}{a}_i-M_{12}{b}_i)$.

13 pages