Conditions of general Z2 symmetry and TM1,2 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 LDLT (or generalized Cholesky) decomposition, conditions of general Z2-invariance for the neutrino mass matrix m are obtained in an arbitrary basis. The conditions are found to be (M22ai+−M12bi+)(M22aj−−M12bj−)=−det⁡Mbi+bj− for the Z2-symmetric and -antisymmetric part of a Yukawa matrix Yij±≡(Y±TY)ij/2≡(aj±,bj±) and the right-handed neutrino mass matrix Mij. In other words, the symmetric and antisymmetric part of bi must be proportional to those of the quantity a˜i≡ai−M12M22bi. They are equivalent to the condition that m is block diagonalized by eigenvectors of the generator T.These results are applied to three Z2 symmetries, the μ−τ symmetry, the TM1 mixing, and the magic symmetry which predicts the TM2 mixing. For the case of TM1,2, the symmetry conditions become M222a˜1TBMa˜2TBM=−det⁡Mb1TBMb2TBM and M222a˜1,2TBMa˜3TBM=−det⁡Mb1,2TBMb3TBM with components a˜iTBM and biTBM in the TBM basis v1,2,3. In particular, for the TM2 mixing, the magic (anti-)symmetric Yukawa matrix with S2Y=±Y is phenomenologically excluded because it predicts m2=0 or m1,m3=0. In the case where Yukawa is not (anti-)symmetric, the mass singular values are displayed without a root sign.