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We have used the SmallGroups library of groups, together with the computer algebra systems GAP and Mathematica, to search for groups with a three-dimensional irreducible representation in which one of the group generators has a twice-degenerate eigenvalue while another generator has non-degenerate eigenvalues. By assuming one of these group generators to commute with the charged-lepton mass matrix and the other one to commute with the neutrino (Dirac) mass matrix, one derives group-theoretical predictions for the moduli of the matrix elements of either a row or a column of the lepton mixing matrix. Our search has produced several realistic predictions for either the second row, or the third row, or for any of the columns of that matrix.
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Advantages of the original symmetrical form of the parametrization of the lepton mixing matrix are discussed. It provides a conceptually more transparent description of neutrino oscillations and lepton number violating processes like neutrinoless double beta decay, clarifying the significance of Dirac and Majorana phases. It is also ideal for parametrizing scenarios with light sterile neutrinos.
We prove that the classic falling-block video game Tetris (both survival and board clearing) remains NP-complete even when restricted to 8 columns, or to 4 rows, settling open problems posed over 15 years ago [BDH+04]. Our reduction is from 3-Partition, similar to the previous reduction for unrestricted board sizes, but with a better packing of buckets. On the positive side, we prove that 2-column Tetris (and 1-row Tetris) is polynomial. We also prove that the generalization of Tetris to larger $k$-omino pieces is NP-complete even when the board starts empty, even when restricted to 3 columns or 2 rows or constant-size pieces. Finally, we present an animated Tetris font.
We study an energy-scale dependence of the lepton-flavor-mixing matrix in the minimal supersymmetric standard model with the effective dimension-five operators which give the masses of neutrinos. We analyze the renormalization group equations of kappa_{ij}s which are coefficients of these effective operators under the approximation to neglect the corrections of O(kappa^2). As a consequence, we find that all phases in $kappa$ do not depend on the energy-scale, and that only n_g-1 (n_g: generation number) real independent parameters in the lepton-flavor-mixing matrix depend on the energy-scale.
Inspired by a new relation $theta_{13}^{rm PMNS}={theta_C}/{sqrt{2}}$ observed from the relatively large $theta_{13}^{rm PMNS}$, we find that the combination of this relation with the quark-lepton complementarity and the self-complementarity results in correlations of the lepton mixing angles with the quark mixing angles. We find that the three mixing angles in the PMNS matrix are all related to the Wolfenstein parameter $lambda$ in the quark mixing, so they are also correlated. Consequently, the PMNS matrix can be parameterized by $lambda$, A, and a Dirac CP-violating phase $delta$. Such parametrizations for the PMNS matrix have the same explicitly hierarchical structure as the Wolfenstein parametrization for the CKM matrix in the quark mixing, and the bimaximal mixing pattern is deduced at the leading order. We also discuss implications of these phenomenological relations in parametrizations.
The Superkamiokande experiment suggests the large flavor mixing between nu_mu and nu_tau. We show that the mixing angle receives significant corrections from the renormalization group equation (RGE) when both the second and the third generation neutrino masses are larger than O(0.1eV). This means that the mixing angle must be small at the decoupling scale of right-handed neutrinos in the model containing a sterile neutrino nu_s with the mass spectrum of m_nu_s = m_nu_e << m_nu_mu = m_nu_tau.
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