Leptonic CP violation: zero, maximal or between the two extremes

Abstract

Discovery of the CP-violation in the lepton sector is one of the challenges of the particle physics. We search for possible principles, symmetries and phenomenological relations that can lead to particular values of the CP-violating Dirac phase, δ. In this connection we discuss two extreme cases: the zero phase, δ = 0, and the maximal CP-violation, δ = ±π/2, and relate them to the peculiar pattern of the neutrino mixing. The maximal CP-violation can be related to the ν µ − ν τ reflection symmetry. We study various aspects of this symmetry and introduce a generalized reflection symmetry that can lead to an arbitrary phase that depends on the parameter of the symmetry transformation. The generalized reflection symmetry predicts a simple relation between the Dirac and Majorana phases. We also consider the possibility of certain relations between the CP-violating phases in the quark and lepton sectors.

Mu-tau reflection symmetry with a texture-zero

“…This invariant is clearly nonzero for physical values and corresponds to one of the invariants in refs. [54,55] adapted to the CP µτ symmetry case. We can also note that if we perform a change of basis of M ν only by the first matrix of (3.5), we obtain a real symmetric matrix which can be diagonalized by a real orthogonal matrix.…”

Deviation from bimaximal mixing and leptonic CP phases in S4 family symmetry and generalized CP

“…The simplest example is the so-called µ − τ reflection symmetry which is a combination of the canonical CP transformation and the µ − τ exchange symmetry. The invariance of the light neutrino mass matrix under µ − τ reflection in the charged lepton diagonal basis leads to maximal atmospheric mixing angle θ 23 and maximal Dirac CP phase δ CP with δ CP = ±π/2 [33][34][35][36][37][38][39]. The phenomenological implications of the generalized CP symmetry has been analyzed within the context of popular A 4 [40], S 4 [30,[41][42][43][44] and T [45] family symmetries.…”

Generalised CP and Δ(6n 2) family symmetry in semi-direct models of leptons

“…The idea was first explored in the context of CP symmetry combined with µ − τ reflection symmetry, resulting in the atmospheric mixing angle θ 23 and the Dirac CP phase δ CP both being determined to be maximal, as discussed in [21][22][23][24][25][26][27]. More recently, the phenomenological consequences of imposing both an S 4 family symmetry and a generalised CP symmetry have been investigated in a model-independent way [28][29][30][31][32][33].…”

“…It is interesting that the mixing angles θ 13 and θ 23 are related as follows 2 cos 2 θ 13 sin 2 θ 23 = 1, or 2 cos 2 θ 13 sin 2 θ 23 = 1 − 2 sin 2 θ 13 , 27) where the second relation is for the PMNS matrix obtained by exchanging the second and the third rows of U V P M N S . Moreover, θ 12 and θ 13 are related by cos 2 θ 13 cos 2θ 12 = ±2 sin θ 13 cos 2θ 13 cos ϕ 8 , Hence ϕ 8 is constrained to lie in the region…”