μ–τ symmetry and maximal CP violation

Abstract

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...

Implications of μ-τ flavored CP symmetry of leptons

“…Among the very first symmetries studied for neutrinos is the µ-τ exchange symmetry [11][12][13][14][15][16][17][18][19][20][21][22][23], which not only predicted maximal θ 23 but also that θ 13 = 0. Many other symmetries such as S 4 , A 4 , ∆(3n 2 ), etc., were considered later on.…”

“…The Lagrangian for N , 19) gives rise to M R in the CP µτ symmetric form (2.4) after η k acquire generic vevs. GCP…”

Pseudo-Dirac neutrinos from flavour dependent CP symmetry

“…The latter depends upon perturbation if p is not purely imaginary. One obtains the µ-τ reflection symmetry [37][38][39][40][41][42][43][44] (see also [45] for a recent review) results ±δ CP = 2θ 23 = π 2 , sin α 21 = sin α 31 = 0 for special class of residual symmetries for which p is purely imaginary. While this symmetry is close to the observed values there already exists hints on its small breaking since the best fit values of θ 23 is found to be different from the maximal values for both the orderings in neutrino masses [46].…”

What doesμ−τsymmetry imply about neutrino mixings?

“…We divide M ν into the µ-τ symmetric part M sym and its breaking part M b [24,25,26] expressed in terms of…”