Unified Parametrization for Quark and Lepton Mixing Angles

TL;DR

This paper introduces a unified parametrization of quark and lepton mixing matrices using angles based on simple algebraic numbers related to the golden ratio, providing a new perspective on their structure.

Contribution

It proposes a novel parametrization with zeroth order angles at pi/12 and pi/5, linking mixing angles to algebraic numbers and the golden ratio, and introduces a perturbation scheme for current data.

Findings

The zeroth order angles are close to observed values.

Corrections are small and similarly signed for quarks and leptons.

The parametrization simplifies the structure of mixing matrices.

Abstract

We propose a new parametrization for the quark and lepton mixing matrices: the two 12-mixing angles (the Cabibbo angle and the angle responsible for solar neutrino oscillations) are at zeroth order pi/12 and pi/5, respectively. The resulting 12-elements in the CKM and PMNS matrices, V_{us} and U_{e2}, are in this order irrational but simple algebraic numbers. We note that the cosine of pi/5 is the golden ratio divided by two. The difference between pi/5 and the observed best-fit value of solar neutrino mixing is of the same order as the difference between the observed value and the one for tri-bimaximal mixing. In order to reproduce the central values of current fits, corrections to the zeroth order expressions are necessary. They are small and of the same order and sign for quarks and leptons. We parametrize the perturbations to the CKM and PMNS matrices in a 'triminimal' way, i.e., with three small rotations in an order corresponding to the order of the rotations in the PDG-description of mixing matrices.