A general formula by $LDL^{T}$ decomposition for the type-I seesaw mechanism

TL;DR

This paper introduces a basis-independent formula for the type-I seesaw mechanism using an $LDL^{T}$ decomposition of the inverse right-handed neutrino mass matrix, facilitating analysis of symmetries and fine-tunings.

Contribution

It provides a novel, mathematically elegant $LDL^{T}$ decomposition-based formula for the inverse mass matrix in the type-I seesaw mechanism, avoiding cubic equations.

Findings

The formula is basis-independent and simplifies analysis.

It enables investigation of flavor and $CP$ symmetries.

It aids in understanding fine-tuning issues.

Abstract

By performing an approximate spectral decomposition to the inverse mass matrix of the right-handed neutrinos $M^{-1}$, we obtain a basis-independent formula for the type-I seesaw mechanism. Mathematically, it is based on the generalized Cholesky (or $LDL^{T}$) decomposition of the symmetric matrix $M^{-1} = L D L^{T}$, with a diagonal matrix $D$ and a lower unitriangular matrix $L$. Since the diagonalization of $L$ can be inverted without solving cubic equations, the formula will be useful to investigate general properties of the mechanism, such as flavor symmetries, generalized $CP$ symmetries, and fine-tunings.