Abstract
Singular values are used to construct physically admissible 3-dimensional mixing matrices characterized as contractions. Depending on the number of singular values strictly less than one, the space of the 3-dimensional mixing matrices can be split into four disjoint subsets, which accordingly corresponds to the minimal number of additional, non-standard neutrinos. We show in numerical analysis that taking into account present experimental precision and fits to different neutrino mass splitting schemes, it is not possible to distinguish, on the level of 3-dimensional mixing matrices, between two and three extra neutrino states. It means that in 3+2 and 3+3 neutrino mixing scenarios, using the so-called α parametrization, ranges of non-unitary mixings are the same. However, on the level of a complete unitary 3+1 neutrino mixing matrix, using the dilation procedure and the Cosine-Sine decomposition, we were able to shrink bounds for the “light-heavy” mixing matrix elements. For instance, in the so-called seesaw mass scheme, a new upper limit on |Ue4| is about two times stringent than before and equals 0.021. For all considered mass schemes the lowest bounds are also obtained for all mixings, i.e. |Ue4|, |Uμ4|, |Uτ4|. New results obtained in this work are based on analysis of neutrino mixing matrices obtained from the global fits at the 95% CL.
Highlights
Connection to Dark Matter, Baryogenesis via Leptogenesis, and feebly interaction dark sectors, referred to as ‘Neutrino Portal’ [23,24,25,26]
We show in numerical analysis that taking into account present experimental precision and fits to different neutrino mass splitting schemes, it is not possible to distinguish, on the level of 3-dimensional mixing matrices, between two and three extra neutrino states
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]
Summary
In [32] a region Ω of physically admissible mixing matrices was defined as the convex hull spanned on 3 × 3 unitary UPMNS mixing matrices with parameters restricted by experiments. It has been shown that singular values control the minimal dimension of possible extensions of the 3 × 3 matrix U PMNS to a complete unitary matrix of some BSM models, which means that the minimal number of additional neutrinos is not arbitrary but depends on singular values. A distinction between the minimal dimension of the unitary extension of matrices from the region Ω is encoded in the number of singular values strictly smaller than one. This fact allows to divide Ω into four disjoint subsets: Ω1, Ω2, Ω3 and Ω4 characterized as. Such an internal structure of Ω provides motivation for analysis of the neutrino mixing matrices with respect to the minimal number of additional neutrino states
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