Flavor Invariants and Renormalization-group Equations in the Leptonic Sector with Massive Majorana Neutrinos

TL;DR

This paper systematically studies flavor invariants and their renormalization-group equations in the leptonic sector with Majorana neutrinos, revealing the algebraic structure and providing tools to analyze leptonic flavor physics.

Contribution

It identifies 34 basic flavor invariants, constructs their RGEs, and demonstrates how to extract physical observables, offering a comprehensive framework for leptonic flavor analysis.

Findings

34 basic flavor invariants identified

Explicit RGEs derived for all invariants

Method to extract physical observables from invariants

Abstract

In the present paper, we carry out a systematic study of the flavor invariants and their renormalization-group equations (RGEs) in the leptonic sector with three generations of charged leptons and massive Majorana neutrinos. First, following the approach of the Hilbert series from the invariant theory, we show that there are 34 basic flavor invariants in the generating set, among which 19 invariants are CP-even and the others are CP-odd. Any flavor invariants can be expressed as the polynomials of those 34 basic invariants in the generating set. Second, we explicitly construct all the basic invariants and derive their RGEs, which form a closed system of differential equations as they should. The numerical solutions to the RGEs of the basic flavor invariants have also been found. Furthermore, we demonstrate how to extract physical observables from the basic invariants. Our study is helpful for understanding the algebraic structure of flavor invariants in the leptonic sector, and also provides a novel way to explore leptonic flavor structures.