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.
Highlights
In a class of seesaw models [1], massive neutrinos turn out to be Majorana particles, namely, they are their own antiparticles [3, 4]
We have performed a systematic investigation on the flavor invariants in the leptonic sector with massive Majorana neutrinos based on the invariant theory
The physical observables should not depend on the choice of the basis, it is useful to study quantities that are invariant under the flavor basis transformations
Summary
We explain what flavor invariants are and how to construct them based on the lepton mass matrices. As proved in the seminal works by Processi [21] and Formanek [22], the first fundamental theorem for the invariants of N × N matrices Ai (for i = 1, 2, · · · , k with k being a positive integer) under the U(N ) group action Ai → ULAiUL† states that the polynomial invariant of Ai (for i = 1, 2, · · · , k) is a polynomial of Tr Ai1Ai2 · · · Aij , where Ai1Ai2 · · · Aij run over all possible non-commutative monomials. In the subsequent two sections, we concentrate on the flavor invariants and their RGEs in the cases of two- and three-generation leptons, respectively
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