Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Conditions of general $Z_{2}$ symmetry in an arbitrary basis and
  TM$_{1,2}$ mixing for the minimal type-I seesaw mechanism   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 $(M_{22} a_{i}^{+} - M_{12} b_{i}^{+}) \,
( M_{22} a_{j}^{-} - M_{12} b_{j}^{-}) = - \det M \, b_{i}^{+} \, b_{j}^{-}$
for the $Z_{2}$-symmetric and -antisymmetric part of a Yukawa matrix
$Y_{ij}^{\pm} \equiv (Y \pm T Y )_{ij} /2 \equiv (a_{j}^{\pm}, b_{j}^{\pm})$
and 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 $m m^{\dagger}$ for a
$Z_{2}$-symmetric $m$. Two eigenvectors $u_{1,2}$ of $m m^{\dagger}$ coincide
with any of those of $T$, and the remaining one is a vector $(u_{1} \times
u_{2})^{*}$ 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 $\mu-\tau$
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 $S_{2} Y = \pm Y$ is phenomenologically
rejected because it predicts $m_{2}=0$ or $m_{1}, m_{3} = 0$.

2
The categories of this paper are: math
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
3
The categories of this paper are: physics