Patterns in the Fermion Mixing Matrix, a bottom-up approach

TL;DR

This paper introduces a new parametrization of 3x3 hermitian matrices to analyze fermion mass matrices, revealing how phase differences influence observable mixing angles, thus providing insights into fermion mixing patterns.

Contribution

A novel, compact parametrization of unitary matrices diagonalizing hermitian matrices, enabling analysis of fermion mixing patterns and phase influence on mixing angles.

Findings

Phase differences control the size of mixing angles.

New parametrization simplifies pattern extraction.

Predicts mixing angles from phase differences.

Abstract

We first obtain the most general and compact parametrization of the unitary transformation diagonalizing any 3 by 3 hermitian matrix H, as a function of its elements and eigenvalues. We then study a special class of fermion mass matrices, defined by the requirement that all of the diagonalizing unitary matrices (in the up, down, charged lepton and neutrino sectors) contain at least one mixing angle much smaller than the other two. Our new parametrization allows us to quickly extract information on the patterns and predictions emerging from this scheme. In particular we find that the phase difference between two elements of the two mass matrices (of the sector in question) controls the generic size of one of the observable fermion mixing angles: i.e. just fixing that particular phase difference will "predict" the generic value of one of the mixing angles, irrespective of the value of anything else.