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

TL;DR

This paper introduces a new formula based on LDL^T decomposition to analyze the minimal type-I seesaw mechanism, deriving conditions for CP symmetry in neutrino mass matrices across arbitrary bases.

Contribution

It provides a novel LDL^T decomposition-based formula to determine CP symmetry conditions in the minimal type-I seesaw mechanism, applicable in any basis.

Findings

Derived explicit CP symmetry conditions for neutrino mass matrices.

Established proportionality relations between real and imaginary parts of Yukawa matrix components.

Presented a generalized approach for analyzing CP symmetry in neutrino models.

Abstract

In this paper, defining a formula by $LDL^{T}$ 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 ${\rm Re\,} (M_{22} a_{i} - M_{12} b_{i}) \, {\rm Im \,} ( M_{22} a_{j} - M_{12} b_{j}) = - \det M \, {\rm Re\,} b_{i} \, {\rm Im \,} b_{j}$ or $ = - \det M \, {\rm Im \,} b_{i} \, {\rm 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 $(M_{22} a_{i} - M_{12} b_{i})$.