There are physical Majorana phases in the lepton flavor mixing matrix when neutrinos are Majorana fermions. In the case of two degenerate neutrinos, the physical Majorana phase plays a crucial role in the stability of the maximal flavor mixing between the second and the third generations with respect to quantum corrections. The physical Majorana phase of π guarantees that the maximal mixing is stable with respect to quantum corrections, while the Majorana phase of zero causes the maximal mixing destroyed by quantum corrections when neutrino masses are of O(1 eV). The continuous change of the Majorana phase from π to 0 makes the maximal mixing destroyed by quantum corrections with O(1 eV) degenerate neutrino masses. On the other hand, when there is a large mass hierarchy among neutrinos, the maximal flavor mixing is not destroyed by quantum corrections independently of the Majorana phase. * )
The Superkamiokande experiment suggests large flavor mixing between ν µ and ν τ . We show that both the second and third generation neutrino masses are larger than O(0.1 eV) when the mixing angle receives significant corrections from the renormalization group equation (RGE). This implies that this mixing angle must be small at the right-handed neutrinos' decoupling scale when the results of the LSND experiments are correct and when the atmospheric neutrino anomaly is accounted for by ν µ -ν τ oscillation. * ) †) The confirmation of LSND results still awaits future experiments. Recent measurements in the KARMEN detector exclude part of the LSND allowed region. 13) We use m t = 174.5 GeV, m c = 0.657 GeV, m b = 3.02 GeV, m s = 9.935 × 10 −2 GeV, m τ = 1.746 GeV, and m µ = 1.0273 × 10 −3 GeV at µ = M Z . 14) * * ) Equation (2 . 15) was first derived by Babu, Leung and Pantaleone in Ref.3). * * * ) Since we use a diagonal base of the charged lepton, sin 2 2θ 23 in this paper is an observable quantity.†) As shown below, we do not need to calculate Eq. (2 . 16) in our analysis.
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 κ ij s which are coefficients of these effective operators under the approximation to neglect the corrections of O(κ 2 ). As a consequence, we find that all phases in κ 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 energyscale.
The effect of correlation on Peierls transition, which is accompanied by a dimerization, t d , of a bond alternation for transfer energy, has been examined for a half-filled one-dimensional electron system with on-site repulsive interaction (U ). By applying the renormalization group method to the interaction of the bosonized Hamiltonian, the dimerization has been calculated variationally and self-consistently with a fixed electron-phonon coupling constant (λ) and it is shown that t d takes a maximum as a function of U . The result is examined in terms of charge gap and spin gap and is compared with that of the numerical simulation by Hirsch [Phys. Rev. Lett 51 (1983) 296]. Relevance to the spin Peierls transition in organic conductors is discussed. KEYWORDS: Peierls transition, Hubbard model, dimerization, renormalization, charge gap, spin gapPeierls transition for a one-dimensional half-filled electron system coupled with phonon has been studied extensively since the Su-Schrieffer-Heeger model was proposed for the quasi-one-dimensional conductor polyacetylene with a bond alternation.1) The role of correlation in such a model has been examined by introducing an onsite repulsive interaction, U . The calculation using the Hartree-Fock approximation leads to the Peierls state, which exists only for U smaller than a critical value of the order of the band width.2, 3) This study has been further developed by taking into account a one-dimensional quantum fluctuation. The numerical simulation exhibits an enhancement of dimerization in the presence of repulsive interaction. 4,5,6,7,8,9) A notable finding is that the dimerization takes a maximum at a value of U being nearly the band width 5,8,9) and that the charge gap becomes much larger than the dimerization gap.5) Regarding the case of weak coupling, the analytical method of a renormalization group (RG) has also exhibited the enhancement of t d as a function of U .10) The effect of finite phonon frequency has been explored using the RG method based on the bosonization.11, 12, 13) It has been shown that the half-filled case leads to a competition between the state with both spin and charge gaps and the state with only a charge gap on the plane of the phonon frequency and the electron-phonon coupling constant.13) In contrast, the case of strong coupling with large U has been examined in terms of the opposite approach, i.e., the expansion of 1/U ) and mapping the electron system into a spin 1/2 chain system, which leads to the spin Peierls transition. The successful treatment of the quantum fluctuation leads to the occurrence of the spin Peierls transition for an arbitrary magnitude of the electron-phonon coupling constant.
The effect of the charge ordering on the spin-Peierls (SP) state has been examined by using a Peierls-Hubbard model at quarter-filling with dimerization, on-site and nearest-neighbor repulsive interactions. By taking account of the presence of dimerization, a bond distortion is calculated variationally with the renormalization group method based on bosonization. When the charge ordering appears at V = Vc with increasing the nearest-neighbor interaction (V ), the distortion exhibits a maximum due to competition between the dimerization and the charge ordering. It is shown that the second-order phase transition occurs from the SP state with the bond alternation to a mixed state with an additional component of the site alternation at V = Vc.
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