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

Renormalization Group Equation of Quark-Lepton Mass Matrices in the SO(10) Model with Two Higgs Scalars

The renormalization group equations (RGEs) of the mass matrices of quarks and leptons in a SO(10) model with two Higgs scalars in the Yukawa coupling are studied. This model is the minimal model of SUSY and non-SUSY SO(10) GUT which can reproduce all the experimental data. Non-SUSY SO(10) GUT model has the intermediate energy phase, Pati-Salam phase, and passes through the symmetry breaking pattern, $SO(10) \to SU(2)_L \times SU(2)_R \times SU(4)_C \to SU(2)_L \times U(1)_Y \times SU(3)_C$. Though minimal, it has, after the Pati-Salam phase, four Higgs doublets in Yukawa interactions. We consider the RGE's of the Yukawa coupling constants of quarks and charged leptons and of the coupling constants of the dimension five operators of neutrinos corresponding to the above symmetry breaking pattern. The scalar quartic interactions are also incorporated.Comment: 16 pages, 1 figure; A discussion of scalar self couplings contributions to the RGE of neutrino mass matrix and some references added, a figure replace

Consistent Digital Curved Rays and Pseudoline Arrangements

Representing a family of geometric objects in the digital world where each object is represented by a set of pixels is a basic problem in graphics and computational geometry. One important criterion is the consistency, where the intersection pattern of the objects should be consistent with axioms of the Euclidean geometry, e.g., the intersection of two lines should be a single connected component. Previously, the set of linear rays and segments has been considered. In this paper, we extended this theory to families of curved rays going through the origin. We further consider some psudoline arrangements obtained as unions of such families of rays

Looking up at seesaw and GUT scales from TeV

In this talk, I discuss how we may approach physics at the seesaw- and GUT-scales using data from the TeV scale. Even though we cannot hope to directly reach these energy scales using particle accelerators, we may get convinced of grand unification and seesaw mechanism based on experimental data if Nature is kind to us. In addition, we may find compelling reason to believe in leptogenesis based on experimental data. This cannot be achieved by a single experiment, but rather a collection of them, based on LHC, ILC, neutrino oscillation, neutrinoless double beta decay, direct dark matter detection, CMB power spectrum and its tensor mode.Comment: Proceedings of "Grand Unified Theories: Current Status and Future Prospects: An International Workshop", Kusatsu, Shiga, Japan, 17-19 December 200

SO(10) Group Theory for the Unified Model Building

The complete tables of Clebsh-Gordan (CG) coefficients for a wide class of SO(10) SUSY grand unified theories (GUTs) are given. Explicit expression of all states and corresponding multiplets under standard model gauge group G_{321} = SU(3)_C x SU(2)_L x U(1)_Y, necessary for evaluation of the CG coefficients are presented. The SUSY SO(10) GUT model considered here incudes most of the Higgs irreducible representations usually used in the literature: 10, 45, 54, 120, 126 126-bar and 210. Mass matrices of all G_{321} multiplets are found for the most general superpotential. These results are indispensible for the precision calculations of the gauge coupling unification and proton decay, etc.Comment: 62 pages, LaTeX, we comment on results of hep-ph/0402122 and hep-ph/0405074; typos corrected;references updated, small improvements of the tex

Commuting difference operators arising from the elliptic C_2^{(1)}-face model

We study a pair of commuting difference operators arising from the elliptic C_2^{(1)}-face model. The operators, whose coefficients are expressed in terms of the Jacobi's elliptic theta function, act on the space of meromorphic functions on the weight space of the C_2 type simple Lie algebra. We show that the space of functions spanned by the level one characters of the affine Lie algebra sp(4,C) is invariant under the action of the difference operators.Comment: latex2e file, 19 pages, no figures; added reference