Flavor Mixing and the Permutation Symmetry among Generations

TL;DR

This paper explores how the permutation symmetry among fermion generations in the standard model can be restored by treating mass matrix parameters as physical variables, leading to covariant tensor equations for neutrino oscillations.

Contribution

It introduces a framework where permutation symmetry is preserved by including mass parameters as variables, unifying known relations as covariant tensor equations.

Findings

Permutation symmetry can be restored with mass parameters as variables.

Neutrino oscillation formulas are covariant tensor equations under the symmetry.

Renormalization group equations are compatible with the symmetry.

Abstract

In the standard model, the permutation symmetry among the three generations of fundamental fermions is usually regarded to be broken by the Higgs couplings. It is found that the symmetry is restored if we include the mass matrix parameters as physical variables which transform appropriately under the symmetry operation. Known relations between these variables, such as the renormalization group equations, as well as formulas for neutrino oscillations (in vacuum and in matter), are shown to be covariant tensor equations under the permutation symmetry group.