Conditions of general $Z_{2}$ symmetry and TM$_{1,2}$ mixing for the minimal type-I seesaw mechanism in an arbitrary basis

Abstract

In this paper, using a formula for the minimal type-I seesaw mechanism by LDL T decomposition, conditions of general Z 2 -invariance of the neutrino mass matrix m is obtained for Lagrangian parameters in an arbitrary basis. The conditions are found to be (Mand the right-handed neutrino mass matrix M ij . In other words, the symmetric and antisymmetric part of b i must be proportional to those of the quantity (M 22 a i − M 12 b i ).Since the eigenvectors of the generator T are orthogonal, we can analyze the eigenvectors and mass eigenvalues of mm † for a Z 2 -symmetric m. Two eigenvectors u 1,2 of mm † coincide with any of those of T , and the remaining one is a vector (u 1 × u 2 ) * orthogonal to them. Furthermore, if the Yukawa matrix does not have the Z 2 symmetry, two nonzero neutrino masses are concisely represented.These results are applied to three Z 2 symmetries, the µ − τ symmetry, the TM 1 mixing, and the magic symmetry which predicts the TM 2 mixing. In particular, for the TM 2 mixing, the magic (anti-)symmetric Yukawa matrix with S 2 Y = ±Y is phenomenologically rejected because it predicts m 2 = 0 or m 1 , m 3 = 0. I. INTRODUCTIONThe structure of the lepton mixing matrix often involves a certain Z 2 symmetry of the neutrino mass matrix m. The bi-maximal mixing [1] accompanies the µ − τ symmetry [2-22], and the trimaximal mixing [23-30] does the magic symmetry [31]. The tri-bi-maximal (TBM) mixing [32] realized by the combination of these two mixings has the Klein symmetry K 4 ≃ Z 2 × Z 2 [33-35]. And for CP phases, µ − τ reflection symmetry [36-60] predicts the

Conditions of general Z2 symmetry and TM1,2 mixing for the minimal type-I seesaw mechanism in an arbitrary basis

“…Since they are orthogonality conditions for two 2-dimensional vectors (without Hermitian conjugation), all solutions can be classified into two types [30].…”

“…However, the discussion has been limited to a certain basis and/or specific structures. In a previous paper [30], we considered the conditions of TM 1,2 mixing in a general basis of M R for the minimal type-I seesaw model . Since the conditions in the previous paper were rather complicated, in this paper, we organize the conditions for the TM 1,2 mixing into a more concise form and determine all the constraints that come from phenomenological parameters.…”

“…In Ref [30],. the solution for B1 = 0 or B2,3 = 0 is classified in another class that is merged to this class.…”