Abstract
To demonstrate that matrices of seesaw type lead to a hierarchy 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 CourantâFisherâWeyl 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.
Highlights
The terms in the Standard Model Lagrangian1 giving mass to the neutrinos can be gathered in a matrix of the general form Mν = mL mD tmD MR (1)which has to be symmetric and where each entry is a complex 3 à 3 matrix acting on the generation space
The neutrino masses are the singular values of Mν, that is to say the eigenvalues of the positive definite matrix MνâMν, where the star means matrix adjoint
To explain the smallness of the observed neutrino masses it is generally argued that if mD is small with respect to MR, the singular values of Mν split into two families: one very small, and one large
Summary
The terms in the Standard Model Lagrangian giving mass to the neutrinos can be gathered in a matrix of the general form (see for instance [3]). To explain the smallness of the observed neutrino masses it is generally argued that if mD is small with respect to MR, the singular values of Mν split into two families: one very small, and one large (of the order of MR). Our purpose here is just to make the simple observation that if one uses the Courant-Fischer-Weyl theorem no approximation is needed to prove the gap property for the singular values for an arbitrary number of generations, in the form of exact inequalities like (3).
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