Majorana CP Violation in Approximately  -  Symmetric Models with det(M ) = 0

Abstract

We discuss effects of Majorana CP violation in a model-independent way for a given phase structure of flavor neutrino masses. To be more predictive, we confine ourselves to models with det(M ν ) = 0, where M ν is a flavor neutrino mass matrix, and to be consistent with observed results of the neutrino oscillation, the models are subject to an approximate μ-τ symmetry. There are two categories of approximately μ-τ symmetric models classified as (C1) yielding sin 2 2θ 23 ≈ 1 and sin 2 θ 13 1 and (C2) yielding sin 2 2θ 23 ≈ 1 and Δm 2 /|Δm 2 atm | 1, where θ 23(13) stands for the mixing of massive neutrinos ν 2 and ν 3 (ν 1 and ν 3 ) and Δm 2 (Δm 2 atm ) stands for the mass squared difference for atmospheric (solar) neutrinos. The Majorana phase can be large for the normal mass hierarchy and for the inverted mass hierarchy with m 1 ≈ −m 2 only realized in (C1) while they are generically small for the inverted mass hierarchy with m 1 ≈ m 2 in both (C1) and (C2). These results do not depend on a specific choice of phases in M ν but hold true in any models with det(M ν ) = 0 because of the rephasing invariance.Subject Index: 152, 154 §1. IntroductionNeutrinos are oscillating and mixed with each other among three flavor neutrinos. Such oscillations have been confirmed to occur for the atmospheric neutrinos, 1) the solar neutrinos, 2), 3) the reactor neutrinos 4) and the accelerator neutrinos. 5) Three massive neutrinos have masses m 1,2,3 measured as mass squared differences defined by Δm 2 = m 2 2 −m 2 1 and Δm 2 atm = m 2 3 −m 2 1 . Three flavor neutrinos ν e,μ,τ are mixed into three massive neutrinos ν 1,2,3 during their flight and the mixing can be described by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix 6) parameterized by three mixing angles θ 12,23,13 , one Dirac CP violating phase δ CP and three Majorana phase φ 1,2,3 , 7) where Majorana CP violating phases are given by two combinations of φ 1,2,3 .It is CP property of neutrinos that has currently received much attention since the similar CP property of quarks has been observed and successfully described by the Kobayashi-Maskawa mixing matrix. 8) If neutrinos exhibit CP violation, there is a new seed to produce the baryon number in the Universe by the Fukugida-Yanagida mechanism of the leptogenesis, 9) which favors the seesaw mechanism 10) of creating tiny neutrino masses. However, there is no direct linkage between CP violation of three flavor neutrinos and that of the leptogenesis since the CP violating phases are associated with heavy neutrinos but not with three flavor neutrinos. If the number * )

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

“…Switching off the unphysical phases and taking account of the constraint equations, we find that it is possible to reach an agreement between the results obtained in Ref. [13] and ours. Finally, we point out an immediate and interesting application of Eq.…”

“…Such an exercise has been done in Ref. [13]. Switching off the unphysical phases and taking account of the constraint equations, we find that it is possible to reach an agreement between the results obtained in Ref.…”

Flavor structures of charged fermions and massive neutrinos

“…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]:…”

Approximately $μ$-$τ$ Symmetric Minimal Seesaw Mechanism and Leptonic CP Violation

“…In this case, the larger Majorana CP violation is expected because the role of X and Y is almost interchanged and we can find that X becomes complex, yielding larger ρ, thereby, giving larger ϕ, and Y becomes almost real. The detailed discussions based on this expectation will be presented elsewhere [11].…”

“…( 10) and Eq. (11). To perform numerical calculation, we have assumed ω = 0 and the tri-bi maximal mixing for M (+) ν [9] which gives sin 2 θ 12 = 1/3, sin 2 θ 23 = 1/2, sin 2 θ 13 = 0.…”