A remark on the mathematics of the seesaw mechanism

TL;DR

This paper proves that matrices of the seesaw type inherently produce a large hierarchy in neutrino masses without approximation, using the Courant-Fisher-Weyl theorem, and discusses related inequalities for their singular values.

Contribution

It introduces a direct proof of the neutrino mass hierarchy property for seesaw matrices using a classical theorem, avoiding approximate methods.

Findings

Matrices of seesaw type have a large gap in their singular spectrum.

The Courant-Fisher-Weyl theorem can be used to prove the hierarchy property directly.

Additional inequalities for singular values of seesaw matrices are sketched.

Abstract

To demonstrate that matrices of seesaw type lead to a hieararchy in the neutrino masses, i.e. that there is a large gap in the singular spectrum of these matrices, one generally uses an approximate block-diagonalization procedure. In this note we show that no approximation is required to prove this gap property if the Courant-Fisher-Weyl theorem is used instead. This simple observation might not be original, however it does not seem to show up in the literature. We also sketch the proof of additional inequalities for the singular values of matrices of seesaw type.