A<Subscript>4</Subscript> symmetry and lepton masses and mixing

Stimulated by Ma’s idea, which explains the tribimaximal neutrino mixing by assuming an A₄ flavor symmetry, a lepton mass matrix model is investigated. A Frogatt–Nielsen-type model is assumed, and the flavor structures of the masses and mixing are caused by the VEVs of SU(2)_L singlet scalars φ_i ^u and φ_i ^d (i=1,2,3), which are assigned to 3 and (1 ,1 ’,1 ”) of A₄, respectively. Possible charged lepton and neutrino mass spectra and mixing are investigated.     

Hierarchical neutrino masses and large mixing angles from the Fritzsch texture of lepton mass matrices

We show that the Fritzsch texture of lepton mass matrices can naturally lead to the bi-large flavor mixing pattern, if three neutrinos have a normal but weak mass hierarchy (typically, m₁:m₂:m₃1:3:10). The effective mass of the tritium beta decay and that of the neutrinoless double beta decay are too small to be observable in this ansatz, but CP violation at the percent level is allowed and could be measured in long-baseline neutrino oscillations.     

Quark and lepton masses from Gaussian landscapes

The flavor structure of the standard model (SM) might arise from random selection on a landscape. We propose a class of simple models, "Gaussian landscapes," where Yukawa couplings derive from overlap integrals of Gaussian wave functions on extra-dimensions. Statistics of vacua are generated by scanning the peak positions of these zero-modes, giving probability distributions for all flavor observables. Gaussian landscapes can account for all observed flavor patterns with few free parameters. Although they give broad probability distributions, the predictions are correlated and accounting for measured parameters sharpens the distributions of future neutrino measurements.     

Baryogenesis from mixing of lepton doublets

It is shown that the mixing of lepton doublets of the Standard Model can yield sizable contributions to the lepton asymmetry, that is generated through the decays of right-handed neutrinos at finite temperature in the early Universe. When calculating the flavour-mixing correlations, we account for the effects of Yukawa as well as of gauge interactions. We compare the freeze-out asymmetry from lepton-doublet mixing to the standard contributions from the mixing and direct decays of right-handed neutrinos. The asymmetry from lepton mixing is considerably large when the mass ratio between the right-handed neutrinos is of order of a few, while it becomes Maxwell-suppressed for larger hierarchies. For an intermediate range between the case of degenerate right-handed neutrinos (resonant leptogenesis) and the hierarchical case, lepton mixing can yield the main contribution to the lepton asymmetry.     

Diophantine collinearity and lepton masses

The combined requirements of Diophantine quantization (integral or possibly half-integral solutions of mass equations) and collinearity (with respect to a rational quantum number which need not be specified explicitly) lead to limitations on heavy lepton masses, with four allowed values in the 13.3–16.9 GeV range.     

Unifying CP violations of quark and lepton sectors

A preliminary determination of the Dirac phase in the PMNS matrix is \(\delta _\mathrm{PMNS}\approx -\frac{\pi }{2}\). A rather accurately determined Jarlskog invariant J in the CKM matrix is close to the maximum. Since the phases in the CKM and PMNS matrices will be accurately determined in the future, it is an interesting problem to relate these two phases. This can be achieved in a families-unified grand unification if the weak CP violation is introduced spontaneously à la Froggatt and Nielsen at a high energy scale, where only one meaningful Dirac CP phase appears.     

Finite modular groups and lepton mixing

We study lepton mixing patterns which are derived from finite modular groups ΓN, requiring subgroups Gν and Ge to be preserved in the neutrino and charged lepton sectors, respectively. We show that only six groups ΓN with N=3, 4, 5, 7, 8, 16 are relevant. A comprehensive analysis is presented for Ge arbitrary and Gν=Z₂×Z₂, as demanded if neutrinos are Majorana particles. We discuss interesting patterns arising from both groups Ge and Gν being arbitrary. Several of the most promising patterns are specific deviations from tri-bimaximal mixing, all predicting θ₁₃ non-zero as favoured by the latest experimental data. We also comment on prospects to extend this idea to the quark sector.     

A maximal atmospheric mixing from a maximal CP violating phase

We point out an elegant mechanism to predict a maximal atmospheric angle, which is based on a maximal CP violating phase difference between second and third lepton families in the flavour symmetry basis. In this framework, a discussion of the general formulas for θ₁₂${\theta }_{12}$, |U_e3|$|U_{e3}|$, δ   and their possible correlations in some limiting cases is provided. We also present an explicit realisation in terms of an SO(3)$\mathit{SO}(3)$ flavour symmetry model.     

A Chiang-type lagrangian in $$\mathbb {CP}^2$$

We analyse a monotone lagrangian in \(\mathbb {CP}^2\) that is hamiltonian isotopic to the standard lagrangian \(\mathbb {RP}^2\), yet exhibits a distinguishing behaviour under reduction by one of the toric circle actions, namely it intersects transversally the reduction level set and it projects one-to-one onto a great circle in \(\mathbb {CP}^1\). This lagrangian thus provides an example of embedded composition fitting work of Wehrheim–Woodward and Weinstein.     

Parametrizing the lepton mixing matrix in terms of charged lepton corrections

We consider a parametrization of the lepton mixing matrix in which the deviations from maximal atmospheric mixing and vanishing reactor mixing are obtained in terms of small corrections from the charged lepton sector. Relatively large deviations for the reactor mixing angle from zero as indicated by T2K experiment can be obtained in this parametrization. We are able to further reduce the number of complex phases, thus, simplifying the analysis. In addition, we have obtained the sides of unitarity triangles and the vacuum oscillation probabilities in this parametrization. The Jarlskog rephasing invariant measure of CP violation at the leading order has a single phase difference which can be identified as Dirac-type CP violating phase in this parametrization.     

Unitarity boomerangs of quark and lepton mixing matrices

The most popular way to present mixing matrices of quarks (CKM) and leptons (PMNS) is the parametrization with three mixing angles and one CP-violating phase. There are two major options in this kind of parametrizations, one is the original Kobayashi–Maskawa (KM) matrix, and the other is the Chau–Keung (CK) matrix. In a new proposal by Frampton and He, a unitarity boomerang is introduced to combine two unitarity triangles, and this new presentation displays all four independent parameters of the KM parametrization in the quark sector simultaneously. In this Letter, we study the relations between KM and CK parametrizations, and also consider the quark–lepton complementarity (QLC) in the KM parametrization. The unitarity boomerang is discussed in the situation of the CK parametrization for comparison with that in the KM parametrization in the quark sector. Then we extend the idea of unitarity boomerang to the lepton sector, and check the corresponding unitarity boomerangs in the two cases of parametrizations.