A Generic Diagonalization of the 3X3 Neutrino Mass Matrix and Its Implications on the mu-tau Flavor Symmetry and Maximal CP Violation

Abstract

In the flavor basis where the mass eigenstates of three charged leptons are identified with their flavor eigenstates, one may diagonalize a 3 × 3 Majorana neutrino mass matrix M ν by means of the standard parametrization of the 3×3 neutrino mixing matrix V . In this treatment the unphysical phases of M ν have to be carefully factored out, unless a special phase convention for neutrino fields is chosen so as to simplify M ν to M ′ ν without any unphysical phases. We choose this special flavor basis and establish some exact analytical relations between the matrix elements of M ′ ν M ′ ν † and seven physical parameters -three neutrino masses (m 1 , m 2 , m 3 ), three flavor mixing angles (θ 12 , θ 13 , θ 23 ) and the Dirac CP-violating phase (δ). Such results allow us to derive the conditions for the µ-τ flavor symmetry with θ 23 = π/4 and maximal CP violation with δ = ±π/2, which should be useful for discussing specific neutrino mass models.In particular, we show that θ 23 = π/4 and δ = ±π/2 keep unchanged when constant matter effects are taken into account for a long-baseline neutrino oscillation experiment.

Flavor structures of charged fermions and massive neutrinos

“…If θ 23 = π/4 and δ ν = ±π/2 hold in vacuum, Eq. ( 136) implies that θ 23 = π/4 and δ ν = ±π/2 must simultaneously hold in matter [404,405,480]. In other words, matter effects respect the µ-τ reflection symmetry of three massive neutrinos, which naturally leads us to…”

“…For example, the well-known Fritzsch texture is definitely disfavored for M ν in the inverted neutrino mass ordering, simply because m ee = 0 has been ruled out in this case. The 3 × 3 Majorana neutrino mass matrix M ν with an exact µ-τ reflection symmetry is of the following form [404,433,480,705,815,816]:…”

On Q6 flavor symmetry and the breaking of μ ↔ τ symmetry

“…The first one is ruled out by the current experimental data on neutrino oscillations, but not even the second condition is allowed at the 1σ or 3σ level. At present, the Long-baseline energy experiment NOνA has disfavored the exact µ − τ symmetry, therefore some possible breakings of the µ ↔ τ symmetry have been explored [13][14][15][16][17][18][19][20][21][22][23][24] (for generic models see [11,[25][26][27][28][29][30][31][32]). If this symmetry is broken, it is vital to investigate the source of the breaking and the framework where this is realized.…”

Almost exact diagonal reflection symmetries and three-zero texture

“…where sin(θ ντ − θ τ ) = sin θ l . Moreover, we assume s e ≃ ± m e /m µ ≃ ± 0.07, (32) that is suggested from d -e unification and a zero texture (M d,e ) 11 = 0. For the four-zero texture, the sign of s e corresponds to that of the mass eigenvalues m ei as follows [50];…”