To extract sum rules of CKM matrix elements, eigenvalue problems for quasi-democratic mass matrices are solved in the first order perturbation approximation with respect to small deviations from the democratic limit. Mass spectra of up and down quark sectors and the CKM matrix are shown to have clear and distinctive hierarchical structures. Numerical analysis shows that the absolute values of calculated CKM matrix elements fit the experimental data quite well. The order of the magnitude of the Jarlskog parameter is estimated by the relation IJI : : : : : : ,;'i(mc/mt + ms/mb)lVu.12IVcbl/4. §1. Introduction 281 Fundamental fermions, quarks and leptons, exist with a broad mass spectrum, ranging from zero or almost zero masses of neutrinos to 180 GeV of the top quark. Magnitude of the weak Cabbibo-Kobayashi-Maskawa (CKM) matrix elements 1), 2) tends to decrease rapidly from the diagonal to the off-diagonal direction. It is impossible to explain such hierarchical structures of fundamental fermions within the framework of the standard model. To do so it is necessary to postulate some working hypothesis on the mass matrices from outside of the model.Among various forms of mass matrices, 3) -6) democratic mass matrices with small correction terms explain such hierarchical structures in a simple and systematic way. 7) -19) The calculability of the CKM matrix elements in terms of quark mass ratios is examined under the hypothesis of the universal strength of Yukawa couplings in which all Yukawa coupling constants are assumed to have equal moduli. 15) -18) It was shown that it is possible to find the CKM matrix elements from quasi-democratic mass matrices which are consistent with present experimental data to high accuracy. 19) In this article, sum rules of the CKM matrix elements are extracted from Hermitian quasi-democratic mass matrices which have small deviation terms and phases in off-diagonal elements. For this purpose we solve mass eigenvalue problems in a first order perturbation approximation with respect to small deviations around the democratic limit.We postulate here that mass matrices for up and down quark sectors are quasidemocratic and represented generically in the forms *)
Renormalization group analysis is made on the relation mH ~ v'2mt for masses of the top quark and the Higgs boson,. which is predicted by the standard model based on generalized covariant derivatives with gauge and Higgs fields. This relation is a low energy manifestation of a tree level constraint which holds among the quartic Higgs self-coupling constant and the Yukawa coupling constants at a certain high energy scale J-to. With the renormalization group equation at one-loop level, the evolution of the constraint is calculated from J-to down to the low energy region around the observed top quark mass. The result of analysis shows that the Higgs boson mass satisfies mt ;; mH ;; v'2mt for a wide range of the energy scale J-to ~ mt and it approaches 177 Ge V (:::::l mt) for large values of J-to.Confirmation of the existence of the top quark by the Collider Detector Fermilab (CDF and DO) groups 1), 2) has shown that the standard model is a consistent and correct theory of fundamental interactions. To enrich the model further, however, we must solve many basic problems remained concerning its scalar sector. In particular, it is expected to predict the Higgs boson mass for an experiment to observe it in near future.Recently, one of the authors 3), 4) has reformulated the standard model by using the concept of generalized covariant derivatives with gauge and Higgs fields which act on a multi-spinor field consisting of all the chiral fermion fields. In the new model, the bosonic part of the Lagrangian is determined by field strengths of the gauge and Higgs fields which are constructed from the commutators of the generalized covariant derivatives. As one of interesting outcomes of this scheme, the strength of quartic self-interaction in the Higgs potential is fixed exclusively in terms of the Yukawa coupling constants and, as a result, the approximate relation mH :::::i v2mt holds for the masses of the top quark and the Higgs boson at the tree level.Unified description of the gauge and Higgs fields has been pioneered by Connes. 5) Using a noncommutative geometry, he introduced the Higgs field as a connection along the discrete direction in a doublely sheeted Minkowski spacetime. His theory predicts the tree level relation mtop = 2 m wand mHiggs = 3.14 m w . Alvarez et al. 6), 7) investigated the evolution of these relations under the one-loop renormalization group equations. 8), 9) Following their method, we analyze quantum effects on the restriction predicted by the standard model with the generalized covariant derivatives in this article.Collecting three generations of the electroweak doublets o/lj and singlets o/ej of lepton fields (the doublets o/qj, and singlets o/uj and o/dj of quark fields) into a at
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