Low-energy parity restoration and unification mass scale within maximal symmetries

We investigate the hierarchy of gauge boson masses in the maximal grand unified theory by studying the renormalization group equations for the running coupling constants associated with the symmetry breaking of SU(16)viaSU(12) _q×SU(4) _l×U(1) _|B|?|L| chain. Particular attention is given to the contribution of Higgs scalars to these equations. It is found that the intermediate mass scale M_L, associated with right-handed gauge bosons could be as low as 10 ³ GeV only for sin ²θ _w(M _L) as high as 0.265 with α _s(M _L)=0.13. In this chain of symmetry breaking, we have also examined the lowest unification mass that is allowed by the low-energy data for sin ²θ _w(M _L) and the assumed gauge hierarchy. This has been done in two cases; first for the case where SU(3) ^c is vectorial, second, for the case where SU(3) ^c is axial. In both cases the lowest unification mass scales were found to be 10 ¹³, 10 ¹¹, 10 ⁸ and 10 ⁷ GeV for sin ²θ _w(M _L) = 0.22, 0.24, 0.26,and0.265 respectively with α _s(M _L) = 0.13. The implication of these low unification masses on baryon non-conserving processes is also discussed.     

P-odd one-meson exchanges revisited: a non-linear chiral model approach

Coupling constants of theP-odd interaction of nucleon with the π-, ρ-, ω-mesons are calculated in the non-linear chiral model. The hierarchy of these constants governed by small parameters sin² θ _w and 1/N _c is established. According to this hierarchy the ratio of the weak πNN and ρNN coupling constants is small, being ?sin² θ _w /N _c . The absolute values of all considered constants prove to be essentially larger than those in theSU(6) _W model, thus pointing that theP-odd effects due to the overlap of the nucleons' wave functions are of importance.     

Dynamically broken supergravity and the hierarchy problem

We propose a scenario for grand unified models based on local supersymmetry. We give arguments that condensates of strongly interacting gauge theories might break local supersymmetry. The gravitino mass induces mass splittings in the low energy theory and allows us to understand a hierarchy of M_p = 10¹⁹ GeV to M_w = 10² GeV naturally.     

The effect of Higgses and “technions” on the calculation of sin2θw and Mx

The effect of the scalar sector on the calculation of sin²θ_w and M_x in grand unified theories is studied. We consider first elementary Higgs scalars and assume the “big desert” hypothesis. It is argued that Higgses other than the usual doublet can also be light (～ M_w). One can obtain bigger values for sin²θ_w by having light (～ M_w) scalar colour sextets which can give rise to interesting phenomenology. As an example, one can build an SU(5) model giving sin²θ_w ? 0.23 at the one-loop level. We also calculate the uncertainty in sin²θ_w and M_x due to the lack of knowledge of the specific masses of superheavy Higgses. We find that this uncertainty is small for reasonable SU(5) models but large in all the SO(10) versions except the minimal. Finally (and alternatively) we consider the effect of a technicolour interaction. The pseudo-Goldstone bosons (technions) increase the calculated sin²θ_w by as much as ～0.01 but M_x remains nearly unchanged. Second-order contributions due to the technifermions tend to cancel the increase on sin²θ_w and in turn increase M_x.     

Fourth-order invariant parameters for F2 isolated fundamental states of tetrahedral molecules: The study of the ν4 band of 12CH4

The ambiguity of effective Hamiltonians for triply degenerate F₂ states of tetrahedral molecules is studied on the basis of a set of 70 fits of experimental data for ν₄ of ¹²CH₄. The formalism of irreducible tensors of the paper by J. P. Champion and G. Pierre [J. Mol. Spectrosc.79, 255–280 (1980)] is used. It is shown that, in agreement with recent theoretical propositions [V. I. Perevalov, Vl. G. Tyuterev, and B. I. Zhilinskii, Dokl. Acad. Nauk SSSR263, 868–872 (1982)], any of q²J⁴ diagnonal coupling parameters may be changed significantly without noticeable variation in the quality of fits; reduced Hamiltonians having fewer numbers of adjusted parameters are first applied to describe the ν₄ energy levels of methane. With good accuracy, the behavior of fitted q²J⁴ parameters obey linear equations derived in the referred paper. A new set of q²J⁴-type diagonal coupling parameters [w₁, w₂, w₃; ?] for triply degenerate F₂ fundamental states is suggested. Parameters w_i are invariant under unitary transformations and can be used to refine molecular force field: their values are determined for ν₄ of ¹²CH₄.     

Diagnosing holographic type dark energy models with the Statefinder hierarchy,composite null diagnostic and <Emphasis Type="Italic">w</Emphasis>-<Emphasis Type="Italic">w′</Emphasis> pair

The main purpose of this work is to distinguish various holographic type dark energy (DE) models, including the ΛHDE, HDE, NADE, and RDE model, by using various diagnostic tools. The first diagnostic tool is the Statefinder hierarchy, in which the evolution of Statefinder hierarchy parmeter S ⁽¹⁾ ₃(z) and S ⁽¹⁾ ₄(z) are studied. The second is composite null diagnostic (CND), in which the trajectories of {S ⁽¹⁾ ₃, ?} and {S ⁽¹⁾ ₄, ?} are investigated, where ? is the fractional growth parameter. The last is w-w′ analysis, where w is the equation of state for DE and the prime denotes derivative with respect to lna. In the analysis we consider two cases: varying current fractional DE density Ω _de0 and varying DE model parameter C. We find that: (1) both the Statefinder hierarchy and the CND have qualitative impact on ΛHDE, but only have quantitative impact on HDE. (2) S ⁽¹⁾ ₄ can lead to larger differences than S ⁽¹⁾ ₃, while the CND pair has a stronger ability to distinguish different models than the Statefinder hierarchy. (3) For the case of varying C, the {w,w′} pair has qualitative impact on ΛHDE; for the case of varying Ω _de0, the {w, w′} pair only has quantitative impact; these results are different from the cases of HDE, RDE, and NADE, in which the {w,w′} pair only has quantitative impact on these models. In conclusion, compared with HDE, RDE, and NADE, the ΛHDE model can be easily distinguished by using these diagnostic tools.     

Two remarks on the painlevé test

Using the example of two previously discussed autonomous differential equations it is pointed out that the Painlevé test with an ansatz w = Σ_n=p^∞w_n(z − z₀)ⁿ, p < 0, integer, will not reveal movable branch points if all leading order contribution arise from terms with equal degree of nonlinearity. Just contrary to this case, for the reduced damped and driven φ⁴ equation with p = −1 two resonance conditions arise at order n = 3 which cannot be satisfied simultaneously.     

High-Speed Cylindrical Collapse of Type-I Matter

In this paper, the cylindrical symmetric gravitational collapse with anisotropic pressure has been investigated using high-speed approximation scheme. The collapsing speed of the fluid is assumed to be very large. To see the effects of pressure, we have used the equations $ \sqrt{p_{R}/\rho}=k$ and $\sqrt{p_{T}/\rho}=w$ of states for radial pressure and tangential pressure, respectively. It is observed that if the ratios of both pressures, that is, tangential and radial pressures, to energy density are bounded from below by some positive value, there arise two possibilities depending on whether 1?+?k ²???2 w ²?>?0 or 1?+?k ²???2w ²?<?0. For 1?+?k ²???2 w ²?>?0, the high-speed approximation scheme fails, while for 1?+?k ²???2 w ²?<?0, the high-speed approximation works. For vanishing w and k, the high-speed scheme does not break down, and, as a result, a naked singularity forms in this case. For p _T ?=?p _R ?=?p, all the results reduce to the perfect fluid case obtained by Nakao and Morisawa (Prog Theor Phys 113:73, 2005).     

Computing the Scaling Exponents in Fluid Turbulence from First Principles: Demonstration of Multiscaling

We develop a consistent closure procedure for the calculation of the scaling exponents ζ _n of the nth-order correlation functions in fully developed hydro-dynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents ζ _n . This hierarchy was discussed in detail in a recent publication by V. S. L'vov and I. Procaccia. The scaling exponents in this set of equations cannot be found from power counting. In this paper we present in detail the lowest non-trivial closure of this infinite set of equations, and prove that this closure leads to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integro-differential equations, reflecting the nonlinearity of the original Navier–Stokes equations. Nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L. We demonstrate that the solvability condition of our equations leads to non-Kolmogorov values of the scaling exponents ζ _n . Finally, we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence.     

Horizontal interactions and their symmetries in GUTs

Using very general assumptions we find and discuss a large class of unified models with horizontal symmetries. We classify them and show on the basis of renormalisation group equations that the typical horizontal mass scale must be at least 10⁹–10¹³ GeV, depending on the model. A class of nonsupersymmetric theories with horizontal symmetries is discovered which predicts a proton lifetimeτ _p ≧10³³ and sin² θ _w ?0.23. It is also argued that supersymmetric unified models involving horizontal symmetries are unlikely to meet present experimental and theoretical requirements—contrary to ordinary supersymmetric theories without horizontal sector.     

Transformation groups for soliton equations: IV. A new hierarchy of soliton equations of KP-type

A new approach to soliton equations, based on τ functions (or Hirota's dependent variables), vertex operators and the Clifford algebra of free fermions, is applied to study a new hierarchy of Kadomtsev-Petviashvili type equations (the BKP hierarchy). The infinite-dimensional orthogonal group acts on the space of BKP τ-functions. The Sawada-Kotera equation is obtained as a reduction of BKP. Its infinitesimal transformations constitute the Euclidean Lie Algebra A₂⁽²⁾.     

S-wave state and D-wave state of the vector meson

The S-wave state ³ S ₁ and D-wave state ³ D ₁ are investigated by applying the Salpeter equation. The coupled equations of the S-wave component and the D-wave component are obtained. In nonrelativistic limit, the coupled equations are decoupled and reduced to two Schrödinger equations describing the S-wave state and the D-wave state, respectively. It is shown that the S-D coupling will be of order v ⁴ or of higher order. For vector mesons 1^??, the contribution to the decay constant comes only from the S-wave state in the nonrelativistic limit. Even when only the simple potential, the scalar and the zero component of the vector potential are considered and the orbit-spin term and tensor term are neglected, the D-wave contribution to the decay constant should also be considered in higher order.     

Effective carrier lifetime and base-widening in shortp + nn +-diodes

The paper shows the charge storage behavior of short psn-silicon diodes under the condition of quasi-stationary equilibrium. Recombination in such diodes in the region of high current injection is characterized by an effective lifetime τ _F which may be determined simply by measuring the chargeQ _s stored in the forward biased diode. For short base widthsw _B the recombination in the weaklyn-doped middle region is negligibly small in relation to that in the high-doped contacts. The effective lifetime of the charge carriers injected into the middle region is, except for an additive quantity, identical with the mean transit timeT $$\tau _F = T + T_0 .$$ The injected charge carriers travel without recombination during a transit timeT through the base widthw, given by the doping profile, and then recombine in the high-doped contacts. A parameterw ₀ interpreted as ‘base-widening’ leads in a simplified concept to an effective storage regionw _eff =w+w ₀. The parameterw ₀ can be associated in the experiment with an additional storage effect in then ⁺-substrate. The storage effects in all three layers of the psn-structure are discussed theoretically.     

Theory of light scattering from a system of interacting Brownian particles

Starting from a N-particle diffusion equation for a system of N interacting spherical Brownian particles, a non-linear transport equation for concentration fluctuations δc(r, t) of the particles is derived. This dynamic equation is transformed into a hierarchy of equations for retarded propagators of increasing numbers of concentration fluctuations. A cluster expansion to lowest order in the average concentration results in a set of two coupled equations. The spectrum of light scattered by the interacting particles is in general not a Lorentzian, due to the non-linear term in the transport equation. For small scattering wave vectors k the width is D(ω)k², where ω is the transferred frequency. It is shown that D(0) = D_e, the effective diffusion coefficient. For a hardcore interaction potential the spectrum is Lorentzian and it is found that D_e = D₀(1 + φ), where D₀ is the diffusion constant for independent particles and φ the volume concentration of Brownian particles.     

A theoretical study of alkaline-earth cations in crystalline sodium chloride,potassium chloride and potassium bromide

Theoretical methods, based on Mott-Littleton techniques are used to investigate the defect structures of alkali-halide crystals doped with divalent ions. The systems studied are those having Mg²⁺, Ca²⁺, Sr²⁺ and Ba²⁺ as impurity ions in NaCl, KCl and KBr crystals. Our calculations find comparable stability of the (110) nearest-neighbour and (200) next-nearest-neighbour complexes whereas the (211) complex is found to be less stable. Significant trends in the variation of binding energy with dopant ion radius are predicted. We also consider the activation energies for the w₁, w₂, w₃ and w₄ type jumps which occur in the vicinity of the impurity ion. The results are, where possible, compared with experimentally determined defect energies and their implications in diffusion processes of M²⁺ impurities in the alkali halides are discussed.     

Equilibrium and stability of the charged particles’ fluxes drifting crosswise to the strong magnetic field at the expense of the Hall charge separation

The current equilibrium is investigated, where the generation of the Hall electric field on the magnetic Debye radius r _B = B ₀/(4πen _e) is considered by the drifting of the relativistic electrons crosswise to the strong magnetic field. In this case, the electron propagation is possible at the distance d that is essentially larger than the electron radius of the backward reflection in the magnetic field r ₀ ? m _e v _z c/(eB ₀). The instability of the joint drift motion of ions and electrons is investigated for the frequency oscillation w much higher than the ion cyclotron frequency w _Bi and by 4π n _i m _i c ² ? B ₀ ² and (k · B ₀) = 0. It is shown that the resonance effects by the ion beam’s plasma frequency w ? kv ₀ = w _pi leads to the generation of the nonpotential perturbations with the characteristic increment Imw ～ 10^?1 ÷ 10^{? 2} w _pi. Estimates show that the instability, associated with the propagation of the high-energy ion beam through the strong magnetic field, can essentially be like the edge-localized mode in tokamaks.     

A determination of sin2θ w and the lepton and quark electroweak couplings frome + e − data

Published data on lepton pair and quark pair final states ine ⁺ e ^? annihilation have been analysed in a self consistent way to yield values for the lepton and quark weak vector and axial vector couplings. Generation universality has been tested for the leptons and under the assumption of the standard model of the weak interaction, the parameter sin²θ _w has been determined separately for the lepton and quark sector. In the renormalisation scheme α,G _μ and sin²θ _w , the result for the lepton final states is sin²θ _w = 0.212±0.014 and for the hadron final states, sin²θ _w = 0.236±0.015. The combined result for this single parameter in the model is sin²θ _w = 0.223±0.011±0.007, corresponding tom _Z =93.0 _?1.8 ^+2.0 GeV.