Neutrino mixing, interval matrices and singular values

Abstract

We study the properties of singular values of mixing matrices embedded within an experimentally determined interval matrix. We argue that any physically admissible mixing matrix needs to have the property of being a contraction. This condition constrains the interval matrix, by imposing correlations on its elements and leaving behind only physical mixings that may unveil signs of new physics in terms of extra neutrino species. We propose a description of the admissible threedimensional mixing space as a convex hull over experimentally determined unitary mixing matrices parametrized by Euler angles, which allows us to select either unitary or nonunitary mixing matrices. The unitarity-breaking cases are found through singular values and we construct unitary extensions, yielding a complete theory of minimal dimensionality larger than three through the theory of unitary matrix dilations. We discuss further applications to the quark sector.

New limits on neutrino non-unitary mixings based on prescribed singular values

“…In this paper we investigate in detail our original idea [32] on how the notions of singular values and contractions which are coming from the matrix theory, influence limits on neutrino mixing parameters within the standard PMNS mixing matrix framework [33][34][35], when additional neutrino states are added [32,36].…”

“…We approach the problem of non-unitarity of the neutrino mixing matrix differently and explore mathematical properties of matrices, and their consequences for neutrino mixing analysis. To describe the neutrino mixing and experimental data in a uniform way, singular values are perfect quantities, giving a possibility to identify regions of physically admissible 3×3 mixing matrices [32]. Singular values restrict neutrino mixing ranges to a special class of matrices known as contractions which have the largest singular values σ 1 smaller than one,…”

Studies of Non-standard Particle Mixings Through Singular Values

“…For matrices representing physical mixings, the contraction property is necessary but it is not sufficient due to just mentioned restrictions on mixing elements imposed by experimental data. However, a workaround of the problem can be in constructing mixing matrices as the convex combination of unitary matrices which allows to define the space of physically admissible matrices in the following way [5]:…”

“…However, in particle physics, they have not been receiving too much attention so far. We present the survey of a recently developed approach [5] to study properties of interval matrices that emerge in the particle mixing phenomenon with the help of singular values. However, the methodology may equally well be applicable to the quark sector.…”

Matrix norms and search for sterile neutrinos

“…Moreover, they posses important properties which help with theoretical calculations and manipulation of matrices. Most of the background for such analysis was established in [30] where a notion of singular values, matrix norms and contraction was introduced. This machinery allows us to determine when a nonunitary mixing matrix can describe physical phenomenon.…”