Geometry of the neutrino mixing space

TL;DR

This paper explores the geometric structure of neutrino mixing matrices within the spectral norm unit ball, revealing how these matrices can be represented as convex combinations of unitary matrices and analyzing their volume and structural properties.

Contribution

It introduces a geometric framework for neutrino mixing matrices, determines the Carathéodory's number for their convex representations, and compares volumes across different neutrino scenarios.

Findings

Maximum of four unitary matrices needed for any 3x3 neutrino mixing matrix.

Carathéodory's number is two for one additional neutrino, three for two.

Volume comparisons between mathematical structures and physical mixing regions.

Abstract

We study a geometric structure of a physical region of neutrino mixing matrices as part of the unit ball of the spectral norm. Each matrix from the geometric region is a convex combination of unitary PMNS matrices. The disjoint subsets corresponding to a different minimal number of additional neutrinos are described as relative interiors of faces of the unit ball. We determined the Carath\'eodory's number showing that, at most, four unitary matrices of dimension three are necessary to represent any matrix from the neutrino geometric region. For matrices which correspond to scenarios with one and two additional neutrino states, the Carath\'eodory's number is two and three, respectively. Further, we discuss the volume associated with different mathematical structures, particularly with unitary and orthogonal groups, and the unit ball of the spectral norm. We compare the obtained volumes to the volume of the region of physically admissible mixing matrices for both the CP-conserving and CP-violating cases in the present scenario with three neutrino families and scenarios with the neutrino mixing matrix of dimension higher than three.