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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 * )

The μ - τ symmetry can reproduce the consistent results with experimental data of θ13, and θ23 (θ13, and θ23 respectively denote the νe - ντ, and νμ - ντ, mixing angles). However, we can not address the issue of the leptonic CP violation in μ - τ symmetric models. So we add the μ - τ symmetry breaking part to include the CP violation. We characterize leptonic CP violation in terms of three phases, where one is conventional phase δ and others are additional phases ρ and γ. These δ, ρ and γ are, respectively, the phases of νe - ντ, νe - νμ and νμ - ντ mixings. The ρ and γ are redundant but the effect of ρ remains in the leptonic CP violation which is characterized by δ + ρ. The δ arises from the μ - τ symmetry breaking part of the Meμ and Meτ while ρ arises from of μ - τ symmetric part of the Meμ and Meτ, where Mij stands for ij (i,j = e,μ,τ) element of M(= [Formula: see text] for Mν being a flavor neutrino mass matrix). Moreover, θ23 can be exactly estimated to be: [Formula: see text] ( sin θ ∝ sin θ13 cos (δ + ρ)[Formula: see text], sin ϕ ∝ Mμμ - Mττ, where [Formula: see text] is the solar neutrino mass difference squared). The conditions of maximal atmospheric neutrino mixing are given by [Formula: see text] and Mμμ = Mττ,which indicate maximal Dirac CP violation.