Renormalizable minimal SO(10) GUT in 4D and 5D

This report is a review of the present status of GUT, especially renormalizable minimal SO(10) GUT, and its future prospect. It consists of two parts. In part I, I review how the minimal renormalizable supersymmetric SO(10) GUT, an SO(10) framework with only one ${\bf 10}$ and one $\bar{\bf 126}$ Higgs multiplets in the Yukawa sector, is attractive because of its high predictivity. Indeed it not only gave a consistent predictions on neutrino oscillation data but also did reasonable and interesting values for Leptogenesis, LFV, muon g-2, neutrinoless double beta decay etc. However, this model suffers from problems, apart from the small deviations from the observed values, related to running of gauge couplings and proton decay. The gauge coupling unification may be spoiled due to the presence of intermediate scales much lighter than the grand unification (GUT) scale. In addition, the gauge couplings blow up around the GUT scale because of the presence of Higgs multiplets of large representations. In order to remedy these pathologies, in part II, we extend GUT into 5D. We propose two approaches: one is to consider the warped extra dimension, using the bulk Higgs profile to explain the intermediate energy scales. Another is to use the orbifold GUT. Both approaches are complementary to each other.Comment: A talk in the workshop on GUT held at Ritsumeikan Univ. on Dec.17-19 200

An a posteriori verification method for generalized real-symmetric eigenvalue problems in large-scale electronic state calculations

An a posteriori verification method is proposed for the generalized real-symmetric eigenvalue problem and is applied to densely clustered eigenvalue problems in large-scale electronic state calculations. The proposed method is realized by a two-stage process in which the approximate solution is computed by existing numerical libraries and is then verified in a moderate computational time. The procedure returns intervals containing one exact eigenvalue in each interval. Test calculations were carried out for organic device materials, and the verification method confirms that all exact eigenvalues are well separated in the obtained intervals. This verification method will be integrated into EigenKernel (https://github.com/eigenkernel/), which is middleware for various parallel solvers for the generalized eigenvalue problem. Such an a posteriori verification method will be important in future computational science.Comment: 15 pages, 7 figure