Flavor invariants and renormalization-group equations in the leptonic sector with massive Majorana neutrinos

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.

Hilbert series for leptonic flavor invariants in the minimal seesaw model

“…In ref. [18], implementing the mathematical tool of Hilbert series (HS) and plethystic logarithm (PL) from the invariant theory [19,20], we have studied the algebraic structures of the invariant ring in the low-energy effective theory with three massive Majorana neutrinos and explicitly constructed all the 34 basic flavor invariants. All the flavor invariants, built upon the matrix polynomials of M l and M ν in eq.…”

“…• Since the basic invariants in the low-energy effective theory have been found in ref. [18], it is interesting to examine the basic invariants in the high-energy full theory and to establish a connection between these two sets of basic invariants. For this purpose, we start with the construction of flavor invariants directly from M l , M D and M R in eq.…”

“…where the non-negative integers {a, b, c, d, e, f } stand for the power indices of the corresponding matrices and the ellipses " • • • " denote the additional possible matrices in the set A or B. As one can see, the structures of the building blocks in the seesaw model are much more complicated than those in the low-energy effective theory, where all the building blocks transform with the unique unitary matrix U L [18]. This reflects the fact that richer leptonic flavor structures and a more complicated invariant ring exist in the full seesaw model at the high-energy scale.…”

“…where 3 L (or N R ) and 3 * L (or N * R ) denote respectively the fundamental and anti-fundamental representation of U(3) (or U(N )) and the subscript "s" refers to the symmetric part. Recalling that the character functions of the fundamental and anti-fundamental representation of U(N ) group are N i=1 z i and N i=1 z −1 i , respectively, with z i being the coordinates on the maximum torus of U(N ) [18], one can calculate the character functions of any representations via the tensor product decomposition. In the subsequent two subsections, we will consider two concrete models, namely, the toy model with N = 1 and the realistic MSM with N = 2.…”

“…A systematic approach to decomposing an arbitrary invariant into the polynomial of the basic invariants and to finding out all the syzygies at a certain degree is presented in appendix C of ref. [18].…”