International Journal of Modern Physics A

Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. Here, we investigate a theory that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of space-time. Next, we suggest replacing in the string holonomy group, the Lorentz group by a discrete subgroup, which turns space-time into a 4-dimensional crystal with defects.

The CMS experiment was completely assembled in the fall of 2008 after a decade of design, construction and installation. During the last two years, cosmic ray data were taken on a regular basis. These data have enabled CMS to align the detector components, both spatially and temporally. Initial use of muons has also established the relative alignment of the CMS tracking and muon systems. In addition, the CMS calorimetry has been crosschecked with test beam data, thus providing an initial energy calibration of CMS calorimetry to about 5%. The CMS magnet has been powered and field mapped. The trigger and data acquisition systems have been installed and run at full speed. The tiered data analysis system has been exercised at full design bandwidth for Tier0, Tier1 and Tier2 sites. Monte Carlo simulation of the CMS detector has been constructed at a detailed geometric level and has been tuned to test beam and other production data to provide a realistic model of the CMS detector prior to first collisions.

Experiments at the Relativistic Heavy Ion Collider (RHIC) have yielded an abundance of data which suggest that a new state of matter has been produced in ultra-relativistic heavy ion collisions. This new state of matter is a strongly interacting Quark Gluon Plasma that behaves like a nearly perfect fluid with very low viscosity. I will review the experimental observations that point to the existence of the sQGP, and in particular I will emphasize the particle spectra and the flow data.

Data from the Large Electron Positron collider (LEP) at CERN on hadron production in gamma-gamma interactions exceed the predictions of the standard model by an order of magnitude at the highest observed transverse momenta in three channels. The amplitude for the process is asymptotically proportional to the sum of the squares of the charges of quarks. The data are suggestive of models where quarks have unit charges, or larger, and where partons have substructure. A previously proposed model of electro-strong interactions includes both these features. Definitive measurements could be made with either of the linear electron-positron colliders that have been proposed, viz. the International Linear Collider (ILC) or the Compact Linear Collider (CLIC). However, an electron-electron collider employing the recently developed "plasma wakefield" acceleration technique could provide the most affordable option. An independent check of the multi-muon events that were recently reported at Fermilab could also be made with this type of collider.

I argue that TeV neutrino physics might become an exciting frontier of particle physics in the era of the Large Hadron Collider (LHC). The origin of non-zero but tiny masses of three known neutrinos is probably related to the existence of some heavy degrees of freedom, such as heavy Majorana neutrinos or heavy Higgs bosons, via a TeV-scale seesaw mechanism. I take a few examples to illustrate how to get a balance between theoretical naturalness and experimental testability of TeV seesaws. Besides possible collider signatures at the LHC, new and non-unitary CP-violating effects are also expected to show up in neutrino oscillations for type-I, type-(I+II) and type-III seesaws at the TeV scale.

We review briefly the history of the ν_e mass since 1933, and point out how the KATRIN experiment can resolve the mystery of the tachyonic mass indicated by the Mainz-Trotsky experiments.

Some of the developments related to quantum anomalies and path integrals during the past 10 years are briefly discussed. The covered subjects include the issues related to the local counter term in the context of 2-dimensional path integral bosonization and the treatment of chiral anomaly and index theorem on the lattice. We also briefly comment on a recent analysis of the connection between the two-dimensional chiral anomalies and the four-dimensional black hole radiation.

The paper presents a summary of some recent work on a SU(5) grand unification scheme for effective 3 + 1 dimensional fields dynamically localized on a domain-wall brane. This is achieved through the confluence of the clash-of-symmetries mechanism for symmetry breaking through domain-wall formation and the Dvali-Shifman gauge boson localization idea. It is shown that it requires an E₆ invariant action, yielding a domain-wall solution that has E₆ broken to differently embedded SO(10) ⊗ U(1) subgroups in the two bulk regions on the opposites of the wall.

We present an Abelian decomposition of Einstein's general relativity, viewing Einstein's theory as a gauge theory of Lorentz group and identifying the gravitational connection as the gauge potential of Lorentz group. The decomposition confirms the existence of the restricted gravity which is much simpler than Einstein's theory but which has the full local Lorentz invariance (and thus the full general invariance). Moreover, it tells that Einstein's theory can be viewed as the restricted gravity which has the Lorentz covariant valence connection as the gravitational source. With the Abelian decomposition we show how to construct all possible vacuum gravitational connections, which can be classified by the knot topology π₃(S³) = π₃(S²). We discuss the physical implications of our result in quantum gravity.

We review recent works discussing the effects of variation of fundamental "constants" on a variety of physical systems. These are motivated by theories unifying gravity with other interactions that suggest the possibility of temporal and spatial variation of the fundamental constants in an expanding Universe. The effects of any potential variation of the fine-structure constant and fundamental masses could be seen in phenomena covering the lifespan of the Universe, from Big Bang nucleosynthesis to quasar absorption spectra to modern atomic clocks. We review recent attempts to find such variations and discuss some of the most promising new systems where huge enhancements of the effects may occur.

We study a model for the mass matrices of the quarks and leptons. Two of the three flavor mixing angles of the quarks can be calculated in terms of the quark mass ratios. They agree very well with the experiments. We are able to relate the mass eigen values of the charged leptons and of the neutrinos to the mixing angles and can predict the masses of the neutrinos. We find a normal hierarchy -the masses are 0.004 eV, 0.01 eV and 0.05 eV. The atmospheric mixing angle is given by the mass ratios of the charged leptons and the neutrinos. We find 38 degrees, consistent with the experiments. The mixing element, connecting the first neutrino with the electron, is found to be 0.05.

The use of the binary tetrahedral group (T′) as flavor symmetry is discussed. I emphasize the CKM quark and PMNS neutrino mixings. I present a novel formula for the Cabibbo angle.

These days we learn that, in our Universe, the dark matter occupies about 25% of the content, compared to only 5% of the "visible" ordinary matter. We propose that the description of the dark matter would be an extension of the Standard Model - a gauge theory. We all know that in the Standard Model we have three generations but still don't know why - the so-called "family problem". On other hand, in view of the masses and oscillations, the neutrinos now present some basic difficulty in the Standard Model. In this note, I propose that on top of the SU_c(3)×, SU(2) × U(1) standard model there is an SU_f(3) extension - a simple SU_c(3) × SU(2) × U(1) × SU_f(3) extended standard model. The family gauge bosons (familons) are massive through the so-called "colored" Higgs mechanism while the remaining Higgs particles are also massive. The three neutrinos, the electron-like, muon-like, and tao-like neutrinos, form the basic family triplets. Hopefully all the couplings to the "visible" matter are through the neutrinos, explaining why the dark matter is more than the visible matter in our Universe.

We compare the conventional description of the interaction of matter with the four known forces in the standard model with an alternative Weyl description in which the chiral coupling is extended to include gravity. The two are indistinguishable at the low energy classical level of equations of motion, but there are subtle differences at the quantum level when nonvanishing torsion and the Adler-Bell-Jackiw anomaly are taken into account. The spin current and energy-momentum of the chiral theory then contain non-Hermitian terms which are not present in the conventional theory. In the chiral alternative, CPT invariance is not automatic because chirality supersedes Hermiticity but full Lorentz invariance holds. New fermion loop processes associated with the theory are discussed together with a perturbative regularization which explicitly maintains the chiral nature and local symmetries of the theory.

Electroweak precision physics is reviewed from the LEP era with implications for Higgs bosons and supersymmetry, together with expectations for the LHC and a future Linear Collider ILC.

We employ two simple and robust results to constrain the mixing matrix of the isosinglet scalar mesons f₀(1710), f₀(1500), f₀(1370): one is the approximate SU(3) symmetry empirically observed in the scalar sector above 1 GeV and confirmed by lattice QCD, and the other is the scalar glueball mass at 1710 MeV in the quenched approximation. In the SU(3) symmetry limit, f₀(1500) becomes a pure SU(3) octet and is degenerate with a₀(1450), while f₀(1370) is mainly an SU(3) singlet with a slight mixing with the scalar glueball which is the primary component of f₀(1710). These features remain essentially unchanged even when SU(3) breaking is taken into account. The observed enhancement of ωf₀(1710) production over ɸf₀(1710) in hadronic J/ψ decays and the copious f₀(1710) production in radiative J/ψ decays lend further support to the prominent glueball nature of f₀(1710). We deduce the mass of the pseudoscalar glueball G from an η-η′-G mixing formalism based on the anomalous Ward identity for transition matrix elements. With the inputs from the recent KLOE experiment, we find a solution for the pseudoscalar glueball mass around (1.4±0.1) GeV, which is fairly insensitive to a range of inputs with or without Okubo-Zweig-Iizuka-rule violating effects. This affirms that η(1405), having a large production rate in the radiative J/ψ decay and not seen in γγ reactions, is indeed a leading candidate for the pseudoscalar glueball. It is much lower than the results from quenched lattice QCD (> 2.0 GeV) due to the dynamic fermion effect. It is thus urgent to have a full QCD lattice calculation of pseudoscalar glueball masses.

Although in the framework of the Standard Model (SM) of particles, Lepton Flavor Violating (LFV) processes such as µ → eγ, µ → eee and µN → eN are strictly forbidden, in various beyond SM scenarios such as Minimal Supersymmetric Standard Model (MSSM), new sources of LFV appear that can lead to sizeable rates for these processes. There are on-going experiments that search for the LFV processes. If the rates of these processes are close to the present bound, these experiments will collect abundant amount of data which makes precision measurement of LFV parameters a close possibility. We show that by studying the polarization of the final particles, information on the CP-violating phases of the underlying theory can be also derived. We discuss the possibility to reduce the degeneracy of parameters by combining information from various experiments.

The issues of holography and possible links with gauge theories in spacetime physics is discussed, in an approach quite distinct from the more restricted AdS-CFT correspondence. A particular notion of holography in the context of black hole thermodynamics is derived (rather than conjectured) from rather elementary considerations, which also leads to a criterion of thermal stability of radiant black holes, without resorting to specific classical metrics. For black holes that obey this criterion, the canonical entropy is expressed in terms of the microcanonical entropy of an Isolated Horizon which is essentially a local generalization of the very global event horizon and is a null inner boundary of spacetime, with marginal outer trapping. It is argued why degrees of freedom on this horizon must be described by a topological gauge theory. Quantizing this boundary theory leads to the microcanonical entropy of the horizon expressed in terms of an infinite series asymptotic in the cross-sectional area, with the leading 'area-law' term followed by finite, unambiguously calculable corrections arising from quantum spacetime fluctuations.

Over the last decade, astronomical observations show that the acceleration of the expansion of the universe is greater than expected from our understanding of conventional general relativity, the mass density of the visible universe, the size of the visible universe and other astronomical measurements. The additional expansion has been attributed to a variety of phenomenon that have been given the general name of dark energy. Dark energy in the universe seems to comprise a majority of the energy in the visible universe amounting to about three times the total mass energy. But locally the dark energy density is very small. However it is not zero. In this paper I describe the work of others and myself on the question of whether dark energy density can be directly detected. This is a work-in-progress and I have no answer at present.

Neutrino oscillation was discovered about 10 years ago. Since then, the knowledge on neutrino masses and mixing angles have been improving substantially. This article describes neutrino oscillation experiments; the discovery, the present status and the future prospect.

We study the moments of multiplicty distribution and its relation to the Lee-Yang zeros of the generating function in electron-positron and hadron-hadron high energy collision. Our work shows that GMD moments can reproduce the oscillatory behaviour as shown in the experimental data and predicted by quantum chromodynamics at preasymptotic energy, while it can also be used to distinguish electron-positron (e⁺e^-) multiplicity data from hadron-hadron (pp and ) multiplicity. Furthermore, there seems to be a link between the development of shoulder structure in the multiplicity distribution and the development of ear structure in the Lee-Yang zeros. We predict that these structures is going to be very obvious at 14 TeV. We argue that the development of these structures indicates an ongoing transition from quark-dominated soft scattering events to gluon-dominated semihard scattering events.

In this paper, we study the zero and finite temperature Casimir force acting on a perfectly conducting rectangular piston inside a closed rectangular cavity with infinitely permeable walls. Explicit formulas for the Casimir force are derived. We show that at any temperature, the Casimir force always tends to move the piston away from the walls and towards its equilibrium position. In the high temperature regime, we show that the leading term of the Casimir force is linear in temperature and therefore the Casimir force has a classical limit.

Implications of texture specific mass matrices have been investigated for both quarks and neutrinos. Interestingly, for the case of quarks Fritzsch-like texture 4 zero mass matrices have been found to be compatible with the present precisely known sin 2β as well as other precise CKM matrix elements. In the case of leptonic mass matrices, for both Majorana and Dirac neutrinos we find that for texture 4, 5, 6 zero mass matrices the inverted hierarchy and degenerate scenarios of neutrino masses are ruled out by the present data.

Based on a new approach (the so-called Yukawaon model) to the mass spectra and mixings, a neutrino mass matrix which is described in terms of the up-quark masses and CKM matrix parameters is proposed. The mass matrix successfully leads to a nearly tribimaximal mixing without assuming any discrete symmetry.

We first review a method to generate nonlinear Dirac equations. The method demands the nonlinear extensions preserve several physical properties like locality, Hermiticity, Poincaré invariance and separability. The last constraint results in nonlinear extensions of non-polynomial type. A class of nonlinear extensions that simultaneously violate Lorentz invariance is also constructed. We then review, using the classes of nonlinear extensions with or without violation of Lorentz symmetry, the sub-leading modifications to the neutrino oscillation probabilities in the ν_µ-ν_τ sector. The parameters in our models are bounded using the current experimental data. These are then used to estimate corrections to the oscillation probabilities and the corresponding energies at which the corrections will be sizeable. Thus one may test quantum nonlinearities in future higher energy experiments.

It is found that a term normally discarded in the standard treatment of the MSW effect might be relevant in the case of non-adiabatic varying matter density, leading to a second order field equation, instead of the usual first order "Schrodinger equation". This leads to dispersion relation that gives rise to the possibility of neutrino trapping in a dense medium as well as the coupling of neutrino oscillation to neutral current interaction. This is found to be in agreement with previous results¹.

The corresponding conserved probability current is derived for this second order equation, and applied to the case of 2-flavor neutrino oscillation in a dense medium. The results in this work might be applicable to the oscillation of neutrinos in dense astrophysical medium.

CMB (Cosmic Microwave Background) polarization observations test many aspects of cosmological models. Effective pseudoscalar-photon interaction(s) would induce a rotation of linear polarization of electromagnetic wave propagating with cosmological distance in various cosmological models. CMB polarization observations are superb tests of these models and have the potential to discover new fundamental physics. Pseudoscalar-photon interaction is proportional to the gradient of the pseudoscalar field. From phenomenological point of view, this gradient could be neutrino number asymmetry, other density current, or a constant vector. In these situations, Lorentz invariance or CPT may effectively be violated. In this paper, we review these results and anticipate what more precise observations can tell us about fundamental physics, inflation, etc. Better accuracy in CMB polarization observation is expected from PLANCK mission to be launched this year. Dedicated CMB polarization observers like B-Pol mission, CMBpol mission and LiteBIRD mission would probe this fundamental issue more deeply in the future. With these sensitivities, cosmic polarization rotations from effective pseudoscalar-photon interaction, Faraday polarization rotations from primordial and large-scale magnetic field, and tensor modes effects would have chances to be detected and distinguished. The subtracted tensor-mode effects are likely due to primordial gravitational waves. We discuss the direct detectability of these primordial gravitational waves using space GW detectors.

We find that we can account for the possible descrepancy of the primordial abundance of ⁷Li between the observation and the prediction of the Big-Bang Nucleosynthesis in a scenario of the Big-Bang Nucleosynthesis with the Minimal Supersymmetric Standard Model. This scenario is consistent with a stau-neutralino coannihilation scenario to explain the relic abundance of dark matter. The solution to the discrepancy is given by taking the values of parameters; the mass of the neutralino as 300 GeV and the mass difference between the stau and the neutralino as (100 – 120) MeV.

Phenomenology of unparticle physics at linear as well as hadron colliders was briefly reviewed in this talk, including real emissions and virtual exchanges of unparticle. Existing constraints from collider as well as astrophysics experiments were briefly discussed.

We calculate the production rates of the second Kaluza-Klein (KK) photon γ⁽²⁾ and Z boson Z⁽²⁾ at the LHC including all significant processes in the minimal universal extra dimension (MUED) model. For discrimination of the MUED model from other TeV scale models in hadron collider experiments, γ⁽²⁾ and Z⁽²⁾ play a crucial role. In order to discuss the discrimination and calculate their production rates, we derive effective Lagrangian containing KK number violating operators. We show that KK number violating processes are extremely important for the compactification scale larger than 800 GeV. We find that, with an integrated luminosity of 100 fb^-1, γ⁽²⁾ and Z⁽²⁾ are produced 10⁶ - 10² for the compactification scale between 400 GeV and 2000 GeV.

The standard model of particle physics is generalized with a horizontal symmetry so that all its results are successfully reproduced without inducing any unphysical modes of existence. Two stage breakdowns of symmetries result in mass matrices of Majorana and Dirac types for fundamental fermions and predict rich physical modes of boson fields, some of which might be observed by the LHC experiment.

A simple inhomogeneous cosmological model with a local void is constrained with the latest Union supernova compilation. To fit the supernova data, a large local void on the scales of 1 Gpc is found, contrary to the small scales of 200 Mpc in the previous finding. A more realistic inhomogeneous cosmological model may be required to fit the supernova data. Alternatively, a clumpy universe with α < 1 can fit the supernova data with reduced local void scales.

A quasilocal framework of stationary and dynamical untrapped hypersurfaces is introduced to generalize the notions of energy and angular momentum of isolated and dynamical trapping horizons to general strong gravitating systems.

In the framework of a proposed space-time symmetric model of elementary particles, a simplified semi-classical geometrical dynamics leads to the quantum mechanical equations, which may shed light on the physical reality of an individual micro-particle. As a result, the precise P-odd effects would be made explicit in those equations.

A Chou-Yang Type multiplicity distribution which consists of a stochastic (binomial) component for z = n_f - n_b and a nonstochastic (Generalized Multiplicity Distribution) component for n = n_f + n_b, is used to analyze the forward-backward multipicity distributions for hadron hadron collision over a wide range of cms energy. The results are compared with the multiplicity distribution using the Negative Binomial Distribution. Finally, cluster size r = 3.30±0.14 and the correlation parameter b = 0.98 are predicted for 14 TeV.

Nonlinear corrections are proposed to the discrete equispaced area spectrum of quantum black holes obtained previously in some quantisation schemes. It is speculated that such a modified spectrum might be related to the fine structure found using the loop quantum gravity approach.

We give a brief summary of the recently constructed superconformal Chern-Simons-matter theories and and how they describe the dynamics of M2-branes at orbifolds.

Inspired by the space-time noncommutativity on a D₅-brane world, in a type IIB string theory, we explore the possibility of an emergent 4D ordinary space-time in the formalism. In particular, a curved D₃-brane dynamics is worked out to obtain an axially symmetric and a spherically symmetric AdS and dS black holes. Extremal geometries are analyzed, using the noncommutative scaling. The emerging two dimensional semi-classical black holes are investigated to yield evidence for extra dimensions in the curved brane-world. Interestingly, a tunneling between dS to AdS vacua in the formalism is briefly discussed by incorporating the Hagedorn transitions in string theory.

Quantum black holes are the smallest and heaviest conceivable elementary particles. They have a microscopic size but a macroscopic mass. Several fundamental types have been constructed with some remarkable properties. Quantum black holes in the neighborhood of the Galaxy could resolve the paradox of ultra-high energy cosmic rays detected in Earth's atmosphere. They may also play a role as dark matter in cosmology.

The gauge theory - gravity duality has provided us a way of studying QCD at high energies or short distances from straightforward calculations in classical general relativity. Among numerous results obtained so far, one of the most striking is the universality of the ratio of the shear viscosity to the entropy density. For all gauge theories with Einstein gravity dual this ratio has been found to be η/s = 1/4π. In this note, we consider higher curvature-corrected black hole solutions for which η/s can be smaller than 1/4π, thus violating the conjecture bound. Here we shall argue that the Gauss-Bonnet gravity and (Riemann)² gravity theories, in particular, provide concrete examples in which inconsistency of a theory, such as a violation of microcausality at short distances, and a classical limit on black hole entropy are correlated.

Electromagnetic duality has been utilized to study the isocolor charge-dyon interactions in Restricted Quantum Chromodynamics (RCD),in terms of current-current correlation (in magnetic gauge)using dielectric and permeability parameters of the associated vacuum. In the state of dyonic superconductivity, it has been shown that the dual propagators behave as 1/k⁴ (for small k²), which in analogy with superconductivity (dual superconductivity) leads to the confinement of colored fluxes associated with dyonic quarks vide generalized Meissner effect. Based on semi-quantitative analysis of vortex solutions of RCD and by calculating the masses for the massive collective modes of the condensed vacuum, the expressions for the London penetration depth, coherence length and the associated flux energy functions for the type I and type II superconducting media have been obtained. It has further been demonstrated that in the type I medium, vortices tend to coalesce and hence are attractive, while the energy function supports repulsive forces between vortices in the type II superconducting medium. The RCD has been supersymmetrized in N=1 limit and the supersymmetric dyonic solutions have been obtained. In the dyonic background gauge one-loop quantum corrections to the dyonic mass have been calculated and it has been shown that the one-loop quantum corrections lead no change in classical mass of the dyon.

The λφ⁴ model is conventionally used to explain the origin of mass of elementary particles through the Spontaneous Symmetry Breaking (SSB) phenomena. The triviality status of the λφ⁴ model in 4-dimensional spacetime remains an open question despite attempts by several authors. This study establishes a new approach to determine the triviality status of the λφ⁴ model based on an unpublished note by Professor Bryce DeWitt. We adopted the DeWitt's Ansatz for the 2-point connected correlation function on the lattice where α is a parameter that measures the departure from triviality. Calling α's continuum counterpart as β, then a non-zero value of β signifies non-triviality of the λφ⁴ model. The 2-point connected correlation function, given in terms of an Euclidean functional integral, is computed numerically via Monte Carlo methods. Our analysis, based on β, is different from the traditional analysis based on the renormalized coupling constant λ_R. To test the new approach, we performed the simulation in 2 dimensions and obtained results that are consistent with previous findings: 2-dimensional λφ⁴ model is non-trivial. Finally, for the case in 4 dimensions, our results show that the model is non-trivial.

We investigate the gauge-Higgs unification models within the scheme of the coset space dimensional reduction, beginning with two types of set up; fourteen-dimensional gauge theory with simple gauge groups and ten-dimensional gauge theory with direct product gauge groups. We found some phenomenologically acceptable models through an exhaustive search for the candidates of the coset spaces, the gauge group in higher dimension, and fermion representation.

Casimir energy is calculated for the 5D electromagnetism in the warped geometry. It is compared with the flat case(arXiv:0801.3064). A new regularization, called sphere lattice regularization, is taken. It is based on the minimal area principle and is a direct realization of the geometrical approach to the renormalization group. The properly regularized form of Casimir energy, is expressed in a closed form. We numerically evaluate Λ(4D UV-cutoff), ω(5D bulk curvature, warp parameter) and T(extra space IR parameter) dependence of the Casimir energy. The warp parameter ω suffers from the renormalization effect. We examine the meaning of the weight function and finally reach a new definition of the Casimir energy where the 4D momenta( or coordinates) are quantized with the extra coordinate as the Euclidean time. We comment on the cosmological term at the end.

The magnetofluid unification is constructed using lagrangian approach by imposing a non-Abelian gauge symmetry to the matter inside the fluid. The model provides a general description for relativistic fluid interacting with either Abelian or non-Abelian gauge field. The differences with the hybrid magnetofluid model are discussed, and some physical consequences of this formalism are briefly worked out.

We study the charmful three-body baryonic B decays with D^(*) or J/Ψ in the final state. We explain the measured rates of , , and . In particular, the branching fraction of predicted to be of order 2.3 × 10^-6 is in accordance with the Belle measurement, .

We calculate the two-boson-exchange (TBE) corrections to the parity-violating asymmetry of the elastic electron-proton scattering in a parton model using the formalism of generalized parton distributions (GPD's). We find that these effects can reach a few percent and are comparable in size with the current experimental measurements of strange-quark effects in the proton neutral weak current. Our result is also compared with the result of the hadronic model.

This paper aims at the development of 't Hooft's quantization proposal to describe composite quantum mechanical systems. In particular, we show how 't Hooft's method can be utilized to obtain from two classical Bateman oscillators a composite quantum system corresponding to a quantum isotonic oscillator. For a suitable range of parameters, the composite system can be also interpreted as a particle in an effective magnetic field interacting through a spin-orbital interaction term. In the limit of a large separation from the interaction region we can identify the irreducible subsystems with two independent quantum oscillators.

We consider an exact µ-τ reflection symmetry in neutrino sector realized at the GUT scale in the context of the seesaw model with and without supersymmetry. It is shown that the renormalization group (RG) evolution from the GUT scale to the seesaw scale gives rise to breaking of the µ-τ symmetry which is essential to achieve a successful leptogenesis. We show that CP violation responsible for the generation of baryon asymmetry of our universe can be directly linked with CP violation measurable through neutrino oscillation as well as neutrino mixing angles θ₁₂ and θ₁₃.