We find that the presence of a global Le − Lµ − Lτ (≡ L ′ ) symmetry and an S2 permutation symmetry for the µ-and τ -families supplemented by a discrete Z4 symmetry naturally leads to almost maximal atmospheric neutrino mixing and large solar neutrino mixing, which arise, respectively, from type II seesaw mechanism initiated by an S2-symmetric triplet Higgs scalar s with L ′ = 2 and from radiative mechanism of the Zee type initiated by two singly charged scalars, an S2-symmetric h + with L ′ = 0 and an S2-antisymmetric h ′+ with L ′ = 2. The almost maximal mixing for atmospheric neutrinos is explained by the appearance of the democratic coupling of s to neutrinos ensured by S2 and Z4 while the large mixing for solar neutrinos is explained by the similarity of h +and h ′+ -couplings described by f hand µ+ (µ−) stand for h + (h ′+ )-couplings, respectively, to leptons and to Higgs scalars.

We have constructed an SU (3)L × U (1)N gauge model utilizing an U (1) L ′ symmetry, where L ′ = Le − Lµ − Lτ , which accommodates tiny neutrino masses generated by L ′ -conserving one-loop and L ′ -breaking two-loop radiative mechanisms. The generic smallness of two-loop radiative effects compared with one-loop radiative effects describes the observed hierarchy of ∆m T , where i (= 1, 2, 3) denotes three families. It is found that our model is relevant to yield quasi-vacuum oscillations for solar neutrinos.

We argue the possibility that a real part of a flavor neutrino mass matrix only respects a µ-τ symmetry. This possibility is shown to be extended to more general case with a phase parameter θ, where the µ-τ symmetric part has a phase of θ/2. This texture shows maximal CP violation and no Majorana CP violation.The present experimental data on neutrino oscillations [1,2] indicate the mixing angles [3] satisfying 0.70 < sin 2 2θ ⊙ < 0.95, 0.92 < sin 2 2θ atm , sin 2 θ CHOOZ < 0.05,where θ ⊙ is the solar neutrino mixing angle, θ atm is the atmospheric neutrino mixing angle and θ CHOOZ is for the mixing angle between ν e and ν τ . These mixing angles are identified with the mixings among three flavor neutrinos, ν e , ν µ and ν τ , yielding three massive neutrinos, ν 1,2,3 : θ 12 = θ ⊙ , θ 23 = θ atm and θ 13 = θ CHOOZ . These data seem to be consistent with the presence of a µ-τ symmetry [4,5,6,7] in the neutrino sector, which provides maximal atmospheric neutrino mixing with sin 2 2θ 23 = 1 as well as sin θ 13 = 0. Although neutrinos gradually reveal their properties in various experiments since the historical Super-Kamiokande confirmation of neutrino oscillations [1], we expect to find yet unknown property related to CP violation [8]. The effect of the presence of a leptonic CP violation can be described by four phases in the PMNS neutrino mixing matrix, U P MN S [9], to be denoted by one Dirac phase of δ and three Majorana phases of β 1,2,3 as U P MN S = U ν K [10] with U ν =   c 12 c 13 s 12 c 13 s 13 e −iδ −c 23 s 12 − s 23 c 12 s 13 e iδ c 23 c 12 − s 23 s 12 s 13 e iδ s 23 c 13 s 23 s 12 − c 23 c 12 s 13 e iδ −s 23 c 12 − c 23 s 12 s 13 e iδ c 23 c 13 ,   ,where c ij = cos θ ij and s ij = sin θ ij (i, j=1,2,3) and Majorana CP violation is specified by two combinations made of β 1,2,3 such as β i − β 3 in place of β i in K. To examine such effects of CP violation, there have been various discussions [11] including those on the possible textures of flavor neutrino masses [12,13,14,15]. In this note, we would like to focus on the role of the µ-τ symmetry in models with CP violation [14,15], which can be implemented by introducing complex flavor neutrino masses. The µ-τ symmetric texture gives sin θ 13 = 0 as well as maximal atmospheric neutrino mixing characterized by c 23 = σs 23 = 1/ √ 2 (σ = ±1). Because of sin θ 13 = 0, Dirac CP violation is absent in Eq.(2) and CP violation becomes of the Majorana type. Since the µ-τ symmetry is expected to be approximately realized, its breakdown is signaled by sin θ 13 = 0. To have sin θ 13 = 0, we discuss another implementation of the µ-τ symmetry such that the symmetry is only respected by the real part of M ν . The discussion is based on more general case, where M ν is controlled by one phase to be denoted by θ and the specific value of θ = 0 yields the µ-τ symmetric real part. It turns out that Majorana CP violation is absent because all three Majorana phases are calculated to be −θ/4 while Dirac CP violation becomes maximal.Our complex flavor neutrino mass matrix o...

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We construct an SU(3) L ϫU(1) N gauge model based on an S 2L permutation symmetry for left-handed and families, which provides the almost maximal atmospheric neutrino mixing and the large solar neutrino mixing of the large mixing angle type. Neutrinos acquire one-loop radiative masses induced by the radiative mechanism of the Zee type as well as tree level masses induced by the type II seesaw mechanism utilizing interactions of lepton triplets with an SU(3)-sextet scalar. The atmospheric neutrino mixing controlled by the tree-level and radiative masses turns out to be almost maximal owing to the presence of S 2L supplemented by a Z 4 discrete symmetry. These symmetries ensure the near equality between the e -and e -radiative masses dominated by contributions from heavy leptons contained in the third members of lepton triplets, whose Yukawa interactions conserve S 2L even after the spontaneous breaking. The solar neutrino mixing controlled by radiative masses, including a -mass, which are taken to be of similar order, turns out to be described by large solar neutrino mixing angles.

We demonstrate the usefulness of flavour neutrino masses expressed in terms of Mee, Meµ and Meτ . The analytical expressions for the flavour neutrino masses, mass eigenstates and physical CP-violating Majorana phases for texture two zeros are obtained exactly.PACS numbers: 14.60.Pq I. INTRODUCTIONThe atmospheric neutrino oscillations have been experimentally confirmed by the Super-Kamiokande collaboration [1]. A similar oscillation phenomenon, the solar neutrino oscillations, has been long suggested [2] and have been finally confirmed by various collaborations [3].Theoretically, the neutrino oscillations are realised if neutrinos have different masses and can be explained by mixings of three flavour neutrinos (ν e , ν µ , ν τ ). These mixings are well described by a unitary matrix U [4] involving three mixing angles θ 12 , θ 23 , θ 13 , which converts three neutrino mass eigenstates (ν 1 , ν 2 , ν 3 ) with masses (m 1 , m 2 , m 3 ) into (ν e , ν µ , ν τ ). Furthermore, leptonic CP violation is induced if U contains phases which are given by one CP-violating Dirac phase δ and two CP-violating Majorana phases α 2 , α 3 [5].There have been various discussions of flavour neutrino mass matrix to ensure the appearance of the observed neutrino mixings and masses [6]. The flavour neutrino mass matrix M is parameterized bywhere diag(m 1 , m 2 , m 3 ) = U T M U . The elements of the mass matrix, M ij (i, j = e, µ, τ ), are functions of the mixing angles, mass eigenstates and CP phases :M ij = f (θ 12 , θ 23 , θ 13 , m 1 , m 2 , m 3 , δ, α 2 , α 3 ).Our recent discussions [7] have found that M µµ , M τ τ , M µτ as well as m 1 , m 2 , m 3 are expressed in terms of M ee , M eµ , M eτ as follows: M µµ,τ τ,µτ = f (M ee , M eµ , M eτ , θ 12 , θ 23 , θ 13 , δ), (3) and m 1,2,3 = f (M ee , M eµ , M eτ , θ 12 , θ 23 , θ 13 , δ, α 2 , α 3 ). (4) Various specific textures of flavour neutrino mass matrices for the maximal CP violation and the maximal atmospheric neutrino mixing have been obtained by using the relation of Eq.(3) [7].

To discuss the possible contribution of parafermions to the dark matter abundance, we extend the Boltzmann equation for fermionic dark matter to include parafermions. Parafermions can accommodate r particles per quantum state ð2 ≤ r < ∞Þ, in which the parafermion of order r ¼ 1 is identical to the ordinary fermion. It is found that the parafermionic dark matter can be more abundant than the fermionic dark matter in the present Universe.

A new type of neutrino mixing named bi-pair neutrino mixing is proposed to describe the current neutrino mixing pattern with a vanishing reactor mixing angle and is determined by a mixing matrix with two pairs of identical magnitudes of matrix elements. As a result, we predict sin 2 θ12 = 1 − 1/ √ 2 (≈ 0.293) for the solar neutrino mixing and either sin 2 θ23 = tan 2 θ12 or cos 2 θ23 = tan 2 θ12 for the atmospheric neutrino mixing. We determine flavor structure of a mass matrix M , leading to diagonal masses of m1,2,3, and find that Mµµ − Mee/t 2 12 : |Mµτ | : Mττ − Mee/t 2 12 =t 2 23 : |t23| : 1 for the normal mass hierarchy if m1 = 0, where tij=tan θij (i, j=1, 2, 3) and Mij (i, j=e, µ, τ ) stand for flavor neutrino masses. For the inverted mass hierarchy, the bi-pair mixing scheme turns out to satisfy the strong scaling ansatz requiring that |Mµµ| : |Mµτ | : |Mττ | = 1 : |t23| : t 2 23 if m3 = 0.