Harmonic Analysis of Linear Fields on the Nilgeometric Cosmological Model

To analyze linear field equations on a locally homogeneous spacetime by means of separation of variables, it is necessary to set up appropriate harmonics according to its symmetry group. In this paper, the harmonics are presented for a spatially compactified Bianchi II cosmological model -- the nilgeometric model. Based on the group structure of the Bianchi II group (also known as the Heisenberg group) and the compactified spatial topology, the irreducible differential regular representations and the multiplicity of each irreducible representation, as well as the explicit form of the harmonics are all completely determined. They are also extended to vector harmonics. It is demonstrated that the Klein-Gordon and Maxwell equations actually reduce to systems of ODEs, with an asymptotic solution for a special case.Comment: 28 pages, no figures, revised version to appear in JM

Scalar fields on SL(2,R) and H^2 x R geometric spacetimes and linear perturbations

Using appropriate harmonics, we study the future asymptotic behavior of massless scalar fields on a class of cosmological vacuum spacetimes. The spatial manifold is assumed to be a circle bundle over a higher genus surface with a locally homogeneous metric. Such a manifold corresponds to the SL(2,R)-geometry (Bianchi VIII type) or the H^2 x R-geometry (Bianchi III type). After a technical preparation including an introduction of suitable harmonics for the circle-fibered Bianchi VIII to separate variables, we derive systems of ordinary differential equations for the scalar field. We present future asymptotic solutions for these equations in a special case, and find that there is a close similarity with those on the circle-fibered Bianchi III spacetime. We discuss implications of this similarity, especially to (gravitational) linear perturbations. We also point out that this similarity can be explained by the "fiber term dominated behavior" of the two models.Comment: 23 pages, no figures, to be published in Class. Quant. Gravi

Cosmological Family Asymmetry and CP violation

We discuss how the cosmological baryon asymmetry can be achieved by the lepton family asymmetries of heavy Majorana neutrino decays and they are related to CP violation in neutrino oscillation, in the minimal seesaw model with two heavy Majorana neutrinos. We derive the most general formula for CP violation in neutrino oscillation in terms of the heavy Majorana masses and Yukawa mass term. It is shown that the formula is very useful to classify several models in which $e-$, $\mu-$ and $\tau-$leptogenesis can be separately realized and to see how they are connected with low energy CP violaton. To make the models predictive, we take texture with two zeros in the Dirac neutrino Yukawa matrix. In particular, we find some interesting cases in which CP violation in neutrino oscillation can happen while lepton family asymmetries do not exist at all. On the contrary, we can find $e-$, $\mu-$ and $\tau-$leptogenesis scenarios in which the cosmological CP violation and low energy CP violation measurable via neutrino oscillations are very closely related to each other. By determining the allowed ranges of the parameters in the models, we predict the sizes of CP violation in neutrino oscillation and $|V_{e3}^{MNS}|$. Finally, the leptonic unitarity triangles are reconstructed.Comment: 22 pages, 9 figures A figure caption correcte

Leptogenesis and Low energy CP violation, a link

How is CP violation of low energy related to CP violation required from baryon number asymmetry ? We give an example which shows a direct link between CP violation of neutrino oscillation and baryogenesis through leptogenesis.Comment: 3 pages and 2 figures, Talk presented at 4th Nufac02, July 1-6, 200

A4A_4 Modular Flavour Model of Quark Mass Hierarchies close to the Fixed Point τ=ω\tau = \omega

We investigate the possibility to describe the quark mass hierarchies as well as the CKM quark mixing matrix without fine-tuning in a quark flavour model with modular $A_4$ symmetry. The quark mass hierarchies are considered in the vicinity of the fixed point $\tau = \omega \equiv \exp({i\,2\pi/3})$ (the left cusp of the fundamental domain of the modular group), $\tau$ being the VEV of the modulus. The model involves modular forms of level 3 and weights 6, 4 and 2, and contains eight constants, only two of which, $g_u$ and $g_d$, can be a source of CP violation in addition to the VEV of the modulus, $\tau = \omega + \epsilon$, $(\epsilon)^* \neq \epsilon$, $|\epsilon|\ll 1$. We find that in the case of real (CP-conserving) $g_u$ and $g_d$ and common $\tau$ ($\epsilon$) in the down-quark and up-quark sectors, the down-type quark mass hierarchies can be reproduced without fine tuning with $|\epsilon| \cong 0.03$, all other constants being of ${\cal O}(1)$, and correspond approximately to $1 : |\epsilon| : |\epsilon|^2$. The up-type quark mass hierarchies can be achieved with the same $|\epsilon| \cong 0.03$ but allowing $g_u\sim {\cal O}(10)$ and correspond to $1 : |\epsilon|/|g_u| : |\epsilon|^2/|g_u|^2$. In this setting, we discuss the CKM quark mixing and CP violation.Comment: 32 pages, no figure. arXiv admin note: text overlap with arXiv:2102.0748

A4A_4 Modular Flavour Model of Quark Mass Hierarchies close to the Fixed Point τ=i∞\tau = i\infty

We study the possibility to generate the quark mass hierarchies as well as the CKM quark mixing and CP violation without fine-tuning in a quark flavour model with modular $A_4$ symmetry. The quark mass hierarchies are considered in the vicinity of the fixed point $\tau = i\infty$, $\tau$ being the vacuum expectation value of the modulus. We consider first a model in which the up-type and down-type quark mass matrices $M_u$ and $M_d$ involve modular forms of level 3 and weights 6, 4 and 2 and each depends on four constant parameters. We also consider the case of $M_u$ and $M_d$ depending on the same $\tau$ and involving modular forms of weights 8, 4, 2 and 6, 4, 2, respectively, with $M_u$ receiving a tiny SUSY breaking or higher dimensional operator contribution. Both the mass hierarchies of up-type and down-type quarks as well and the CKM mixing angles and CP violating phase are reproduced successfully with one complex parameter and all parameters being in magnitude of the same order. The relatively large value of ${\rm Im}\,\tau$, needed for describing the down-type quark mass hierarchies, is crucial for obtaining the correct up-type quark mass hierarchies.Comment: 37 pages. arXiv admin note: substantial text overlap with arXiv:2212.1333

Democratic Mass Matrices from Broken O(3)L×O(3)RO(3)_L\times O(3)_R Flavor Symmetry

We impose $O(3)_L\times O(3)_R$ flavor symmetry in the supersymmetric standard model. Three lepton doublets $\ell_i$ transform as an $O(3)_L$ triplet and three charged leptons $\bar e_i$ as an $O(3)_R$ triplet, while Higgs doublets $H$ and $\bar H$ are $O(3)_L\times O(3)_R$ singlets. We discuss a flavor $O(3)_L\times O(3)_R$ breaking mechanism that leads to "successful" phenomenological mass matrices, so-called "democratic" ones, in which the large \n_\mu-\n_\tau mixing is naturally obtained. Three neutrinos have nearly degenerate masses of order 0.1\eV which may be accesible to future double \b-decay experiments. We extend our approach to the quark sector and show that it is well consistent with the observed quark mass hierarchies and the CKM matrix elements. However, the large mass of the top quark requires a relatively large coupling. constant.Comment: 12 pages, LaTex file, No figures, minor correction