![]() ![]() Another method introduces a generalized CP transform on the basis of a discrete flavor symmetry, i.e., a finite flavor group and a CP symmetry are combined. A popular approach is adding a model-independent perturbation to the leading TBM matrix. Hence, diverse methods to generate a nonzero reactor mixing angle have been proposed. The discovery of a relatively large reactor mixing angle and the remarkable progress of measurements of neutrino oscillation parameters, however, strictly constrain or exclude the above patterns. These patterns, especially the TBM, derived from flavor groups such as and, are compatible with previous published data of atmospheric neutrino and solar neutrino. Before the determination of the reactor mixing angle, the following are various candidates of the mixing patterns: bimaximal mixing (BM), tri-bi-maximal mixing (TBM), golden ratio mixing (GRM), hexagonal mixing (HM), etc. However, the origin of the flavor mixing is elusive for theorists. The discovery of neutrino flavor oscillation is a tremendous achievement in particle physics during the last two decades, which manifests that neutrinos have masses, and the leptonic mixing matrix is nontrivial. ![]() The implications of the correlations are shown through the predicted flavor ratio of high-energy astrophysical neutrinos (HANs) at Earth. Using the data in these regions, correlations of the leptonic mixing angles and the Dirac CP violating phase are obtained. We find that the scattered points are concentrated on several special regions. At the 3 level of the recent global fit data of neutrino oscillations, the distribution of the scattered points of the angles between the vectors, which are constructed by the element of the leptonic mixing matrix, is analysed. In this paper, we introduce an alternative geometric method to study the correlations between the leptonic mixing parameters. Leptonic mixing patterns are usually extracted on the basis of groups or algebraic structures.
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