Probabilities of forbidden magnetic-dipole transitions in the hydrogen atom and hydrogen-like ions

Explicit formulasfor strongly forbidden magnetic-dipole transitiions between states njl and n′jl in the hydrogen atom and light hydrogen-like ions are derived. The expressions for transition probabilities are presented in the form W _{n′jl; njl} ^(M1) = D _n′n ^lj αm _e (αZ)¹⁰ (in relativistic units), where m _e is the electron mass, α is the fine-structure constant, and Z is the nuclear charge; the constants D _n′n ^lj are presented in an analytical form. Before now, only the D ₂₁ ^01/2 coefficient corresponding to the 1s _1/2–2s _1/2 transition was known in explicit form. The results obtained can be used in designing an experiment on parity violation in the hydrogen atom.     

Coulomb Problem for <Emphasis Type="Italic">Z</Emphasis> &gt; Z<Subscript>cr</Subscript> in Doped Graphene

The dynamics of charge carriers in doped graphene, i.e., graphene with a gap in the energy spectrum depending on the substrate, in the presence of a Coulomb impurity with charge Z is considered within the effective two-dimensional Dirac equation. The wave functions of carriers with conserved angular momentum J = M + 1/2 are determined for a Coulomb potential modified at small distances. This case, just as any two-dimensional physical system, admits both integer and half-integer quantization of the orbital angular momentum in plane, M = 0, ±1, ±2, …. For J = 0, ±1/2, ±1, critical values of the effective charge Z_cr(J, n) are calculated for which a level with angular momentum J and radial quantum numbers n = 0 and n = 1 reaches the upper boundary of the valence band. For Z < Z_cr (J, n = 0), the energy of a level is presented as a function of charge Z for the lowest values of orbital angular momentum M, the level with J = 0 being the first to descend to the band edge. For Z>Z_cr (J, n = 0), scattering phases are calculated as a function of hole energy for several values of supercriticality, as well as the positions ε₀ and widths γ of quasistationary states as a function of supercriticality. The values of ε₀* and width γ* are pointed out for which quasidiscrete levels may show up as Breit–Wigner resonances in the scattering of holes by a supercritical impurity. Since the phases are real, the partial scattering matrix is unitary, so that the radial Dirac equation is consistent even for Z > Z_cr. In this single-particle approximation, there is no spontaneous creation of electron–hole pairs, and the impurity charge cannot be screened by this mechanism.     

Probing of <Emphasis Type="Italic">Z</Emphasis>γγ production sensitivity to anomalous quartic gauge couplings at LHC experiments with √ <Emphasis Type="Italic">s</Emphasis> =13 TeV

Sensitivity to anomalous ZZγγ and Zγγγ couplings in Zγγ production was probed for the ATLAS experiment at Large Hadron Collider. Zγγ process with anomalous couplings simulation in ppcollisions with √ s = 13 TeV was performed using VBFNLO MC generator. The expected limits on the Effective Field Theory parameters f _T0/Λ4, f _T5/Λ4, f _T9/Λ4, f _M2/Λ4, f _M3/Λ4 were extracted for 5 fb^?1 integral luminosity using the distribution on the invariant mass of Zγγ from the combination of charged leptonic decay channels of Z boson (Zγγ → μ⁺μ^?γγ and Zγγ → e⁺e^?γγ).     

The 3–3–1 Model with RH Neutrinos and Associated ZH Production at High-Energy ${\bf \textit{e}}^{+}{\bf \textit{e}}^{-}$ Collider

In the context of SU(3) _C ?? SU(3) _L ?? U(1) _X (3–3–1) model with right-handed neutrinos, we study the Higgsstrahlung process e ^?+? e ^???→ZH and calculate the cross section of this process at leading order. Our numerical results showed that the production cross sections for this process can be significantly large as \(M_{Z'}\approx \sqrt{s}\). With reasonable values of the Z′ mass M _Z′, Z′ exchange can generate large corrections to the cross sections of this process, which might be detected in the future high-energy linear e ^?+? e ^??? collider experiments.     

Der Grundzustand der Urfermionen im Gitterraum

The energy of the Dirac sea of interacting urfermions in a lattice space withZ ³ points is calculated using Heisenberg's Hamiltonian and a two-particle approximation which is a variational calculation with the test function ¦?〉=e ⁱη¦D ₀〉; ¦D ₀〉 is the Dirac sea without interaction,η=(ψ ^° ψ) a bilinear expression of the urfermion creation and annihilation operators. The same result is obtained by a BCS-calculation. Beyond that, we derive simple lower and upper bounds for the energy. Excited states are considered consisting of a particle-antiparticle pair with the energyE=2√ω ²+M ². The massM and the interaction constantW are connected by the equation (4W)^?1=Z ^?3∑(ω ²+M ²)^?1/2. For usual masses 4W～√Z/1 (1 a nuclear length). Methods are discussed to improve the results.     

Differential-geometric structures associated with Lagrangians corresponding to scalar physical fields

The paper is devoted to the investigation, using the method of Cartan–Laptev, of the differential-geometric structure associated with a Lagrangian L, depending on a function z of the variables t, x ¹,...,x ⁿ and its partial derivatives. Lagrangians of this kind are considered in theoretical physics (in field theory). Here t is interpreted as time, and x ¹,...,x ⁿ as spatial variables. The state of the field is characterized by a function z(t, x ¹,..., x ⁿ ) (a field function) satisfying the Euler equation, which corresponds to the variational problem for the action integral. In the present paper, the variables z(t, x ¹,..., x ⁿ are regarded as adapted local coordinates of a bundle of general type M with n-dimensional fibers and 1-dimensional base (here the variable t is simultaneously a local coordinate on the base). If we agree to call t time, and a typical fiber an n-dimensional space, then M can be called the spatiotemporal bundle manifold. We consider the variables t, x ¹,...,x ⁿ , z (i.e., the variables t, x ¹,...,x ⁿ with the added variable z) as adapted local coordinates in the bundle H over the fibered base M. The Lagrangian L, which is a coefficient in the differential form of the variational action integral in the integrand, is a relative invariant given on the manifold J ¹ _H (the manifold of 1-jets of the bundle H). In the present paper, we construct a tensor with components Λ⁰⁰, Λ⁰ⁱ , Λ ^ij (Λ ^ij = Λ ^ji ) which is generated by the fundamental object of the structure associated with the Lagrangian. This tensor is an invariant (with respect to admissible transformations the variables t, x ¹,...,x ⁿ , z) analog of the energy-momentum tensor of the classical theory of physical fields. We construct an invariant I, a vector G ⁱ , and a bivalent tensor G ^jk generated by the Lagrangian. We also construct a relative invariant of E (in the paper, we call it the Euler relative invariant) such that the equation E = 0 is an invariant form of the Euler equation for the variational action integral. For this reason, a nonvariational interpretation of the Euler equation becomes possible. Moreover, we construct a connection in the principal bundle with base J ² H (the variety of 2-jets of the bundle H) and with the structure group GL(n) generated by the structure associated with the Lagrangian.     

Electron gas induced in SrTiO<Subscript>3</Subscript>

This mini-review is dedicated to the 85th birthday of Prof. L.V. Keldysh, from whom we have learned so much. In this paper, we study the potential and electron density depth profiles in surface accumulation layers in crystals with a large and nonlinear dielectric response such as SrTiO₃ (STO) in the cases of planar, spherical, and cylindrical geometries. The electron gas can be created by applying an induction D₀ to the STO surface. We describe the lattice dielectric response of STO using the Landau–Ginzburg free energy expansion and employ the Thomas–Fermi (TF) approximation for the electron gas. For the planar geometry, we arrive at the electron density profile n(x) ∝ (x + d)^–12/7, where d ∝ D₀^–12/7. We extend our results to overlapping electron gases in GTO/STO/GTO heterojunctions and electron gases created by spill-out from NSTO (heavily n-type doped STO) layers into STO. Generalization of our approach to a spherical donor cluster creating a big TF atom with electrons in STO brings us to the problem of supercharged nuclei. It is known that for an atom with a nuclear charge Ze where Z > 170, electrons collapse onto the nucleus, resulting in a net charge Zn < Z. Here, instead of relativistic physics, the collapse is caused by the nonlinear dielectric response. Electrons collapse into the charged spherical donor cluster with radius R when its total charge number Z exceeds the critical value Z_c ≈ R/a, where a is the lattice constant. The net charge eZ_n grows with Z until Z exceeds Z* ≈ (R/a)^9/7. After this point, the charge number of the compact core Z_n remains ≈ Z*, with the rest Z* electrons forming a sparse TF atom with it. We extend our studies of collapse to the case of long cylindrical clusters as well.     

Energy scan of cross sections for <Emphasis Type="Italic">e</Emphasis><Superscript>+</Superscript><Emphasis Type="Italic">e</Emphasis><Superscript>–</Superscript> annihilation into quarkonia and light hadrons in the Belle experiment

The cross sections of the reactions e⁺e^– → ?(nS)π⁺π^? (n = 1, 2,3) and e⁺e^– → h _b (nP)π⁺π^? (n = 1, 2) are measured as a function of the cms collision energy from their thresholds up to 11.02 GeV using the data of the Belle experiment operating at the KEKB e⁺e^– collider. The peaks of the ?(10 860) and ?(11020) resonances are observed in the cross sections with an insignificant contribution of the continuum. The decay ?(11020) → h _b (nP)π⁺π^? is found to fully proceed through intermediate isovector states Z _b (10610) and Z _b (10650).     

Screening of strongly charged macroparticles in liquid electrolyte solutions

A modified Poisson-Boltzmann model has been proposed which makes it possible to describe the screening of strongly charged macroparticles in liquid electrolyte Z: Z solutions in the case when parameter B= ZeQ₀/εRT?1(Q₀ is the surface electric charge, T is the temperature, ε is the solution permittivity, and Z is the valence of ions) provided that the solution is dilute: κR ≡ (8πZ²e²n_i0/εT)^1/2R?1 (n_i0 is the equilibrium number density of ions). It is assumed that the charge Q₀ of a macroparticle appears as a result of adsorption of ions of a certain polarity on its surface. Quantitative criteria of division of dissolved ions into capable and incapable of adsorption are formulated. For aqueous solutions, the adsorption mechanism always leads to values of B ? 1. It is shown that the charge inversion effect predicted by other authors on the basis of different models must be observed for such solutions for all Z ≥ 1. The effect of Brownian movement of macroparticles on their screening is considered. It is shown that viscous forces emerging during such movement lead to peripheral destruction (“washing out”) of the screening ionic shell of macroparticles and, as a result, to violation of their electroneutrality. This results in the emergence of two types of oppositely charged compound particles with small radii close to R and with radii much larger than R, the charge polarity of the latter being opposite to the polarity of Q₀. It is found that both types of ions of compound particles obey the “law of distribution” of the mean energy of their electric field, expressed by formula (29). The problem of ionic screening of gas bubbles accompanied by the formation of bubstons (bubbles stabilized by ions) is considered separately. It is shown that the bubston radius R in pure water and in aqueous solutions of electrolytes is equal to 14 nm irrespective of the ion number density n_i0. The value of n_i0 determines the number density n _b of bubstons themselves, which are formed spontaneously under equilibrium conditions.     

Effective refractive index of a quasi-two-dimensional polydomain film of a conjugated polymer

The relation between the effective ordinary refractive index n* = (n _o n _e )^1/2 of a quasi-two-dimensional polydomain uniaxial film of a conjugated polymer F8BT in the visible transparency region and the refractive indices (n _o,e ) of uniaxial domains with the optical axes randomly oriented in the plane of the film has been confirmed experimentally. The permissible interval of variations in n* has been established and a strong spectral dispersion of this interval near the long-wavelength electronic absorption band of the film has been demonstrated.     

VEPP-2000 project: Collider,detectors, and physics program

The new VEPP-2000 e⁺e^? collider of maximum energy 2000 MeV, which is under construction at the Budker Institute of Nuclear Physics (Siberian Division, Russian Academy of Sciences, Novosibirsk), is briefly described. Experiments at VEPP-2000 will be performed with two upgraded detectors, CMD-2M and SND. A precise measurement of the total cross section for the process e⁺e^? → hadrons and of the partial cross sections for its exclusive hadronic channels is the main point of the physics program for this machine. These measurements will be aimed at testing QCD and the VMD and CVC models, as well as at refining the hadron contribution to fundamental constants such as the muon anomalous magnetic moment \(a_\mu = \frac{{g - 2}}{2}\) and the fine-structure constant α_em(M _Z ² ). Measurements of the nucleon form factors in the reactions e⁺e^? → p\(\bar p\), n\(\bar n\) at their threshold will also be of great importance.     

Quantifying Contextuality of Empirical Models in Terms of Trace-Distance

In order to quantify contextuality of empirical models, the quantity of contextuality (QoC) of empirical models is introduced in terms of the trace-distance. Let Q C(e) denote the QoC of an empirical model e. The following conclusions are proved. (i) An empirical model e is non-contextual if and only if Q C(e)=0, and then it is contextual if and only if Q C(e)>0; (ii) the QoC function QC is convex, contractive and continuous. Finally, the QoC of some famous models is computed, including PM-isotropic boxes P M ^α , M-isotropic boxes M ^α , C H _n -isotropic boxes \(CH_{n}^{\alpha }\) as well as K box, where α∈[0,1]. Moreover, P M ^α is non-contextual if and only if \(\alpha \in [\frac {1}{6},\frac {5}{6}]\); M ^α is non-contextual if and only if \(\alpha \in [0,\frac {4}{5}]\); when n is even, \(CH_{n}^{\alpha }\) is non-contextual if and only if \(\alpha \in [\frac {1}{n},\frac {n-1}{n}]\), and when n is odd, \(CH_{n}^{\alpha }\) is non-contextual if and only if \(\alpha \in [0,\frac {n-1}{n}]\). The most important thing is that it is very easy to compare the QoC of any two isotropic boxes discussed in the above.     

Analysis of the production of Higgs boson pairs at the one-loop level in the minimal supersymmetric standard model

Within theminimal supersymmetric standardmodel, the amplitudes and total cross sections for the processes e ⁺ e ^? → hh, e ⁺ e ^? → hH, e ⁺ e ^? → HH, and e ⁺ e ^? → AA are calculated in the first order of perturbation theory with allowance for a complete set of one-loop diagrams in the m _e → 0 approximation. Analytic expressions are obtained for the quantities under consideration; numerical results are presented in a graphical form. It is shown that the cross section for the process e ⁺ e ^? → hh is larger than those for the other processes (and is on the same order of magnitude as the cross section for the corresponding processes in the Standard Model). In the case of the collision energy equal to √s = 500 GeV, an integrated luminosity in the region ∫ ? ≥ 500 fb^?1, and a longitudinal polarization of the e ⁺ e ? beams used, 520, 320, and 300 production events are possible in the processes e ⁺ e ^? → hh (at M _h = 115 GeV), e ⁺ e ^? → HH, and e ⁺ e ^? → AA (at M _H,A = 120 GeV), respectively. Even at M _H,A ≈ 500 GeV and √s = 1.5 TeV, not less than 200 events for each of the processes can be accumulated. The cross section for the process e ⁺ e ^? → hH is small (about 10^?2 fb), which complicates the detection of the sought signal significantly.     

Calculation of the diffusion coefficient in acoustic turbulence

The first (Born) approximation commonly used to calculate the diffusion coefficient D_T of a passive scalar in acoustic turbulence is shown to be insufficient. Even for a small main parameter—the Mach number, M?1—the next approximation gives a larger contribution to D_T than does the first approximation, but negative in sign. We present a procedure for correctly calculating D_T based on the solution of a nonlinear DIA (direct interaction approximation) equation for the mean Green’s function of the problem. We include an additional term in the general formula for D_T that directly describes the compressibility of acoustic turbulence. This term has not been known previously and has been disregarded even in the Born approximation. A positive value was obtained for D_T=CM³u₀/p₀. The spectrum E(x) was assumed to be smooth at distances Δ x～M²?1.     

Zur Chronologie des alten Ägypten

The velocityv of the propagation of discharge along the anode of a self-quenchingG—M-counter is a function of total pressureP, pressure of the quenching gasP _D, radius of the cathoder _a and of the anoder _i andV _ü the difference between working- and starting-potential. For the mixtures argon-methylal, argon-alcohol and helium-alcohol isv=v ₀·exp[k·(V _ü/V _e)^1/2] withv ₀ the velocity at the starting potentialV _e v ₀=(a+b·P _D/P)·V _n ^1/2 ·exp [(c?d·P_D/P·V _n ^?1/2 ] andV _n=V _e·(lnr _a/r _i)^?1.k, a, b, c andd are characteristical constants of the filling gas.     

Brownian motion on a manifold as a limit of Brownian motions with drift

Let ?ⁿ be n-dimensional Euclidean space and let M ? ?ⁿ be a smooth compact m-dimensional Riemannian manifold (without boundary) embedded in ?ⁿ. By a Brownian motion on M we mean a Markovian process whose transition semigroup is defined by the generator ?½Δ_M, where Δ_M stands for the Laplace-Beltrami operator on M (see, e.g., [2]). This note extends a series of papers in which a measure generated by a Brownian motion on M on the space of trajectories (with values in M) can be represented as the weak limit of measures on the space of trajectories in the ambient space ?ⁿ (see [7–10]). Namely, we claim that a sequence of diffusion processes on ?n which are Brownian motions with drift (in the direction of the manifold) with infinitely increasing modulus converges in distribution to a Brownian motion on the manifold.     

A strongly asymmetric single-electron transistor operating as a zero-biased electrometer

We have studied a strongly asymmetric Al single-electron transistor with R₁ ? R₂ and C₁ ? C₂, where R_{1, 2} and C_{1, 2} are the tunnel resistances and capacitances of the first and second junction respectively. Due to the asymmetry in its electric parameters leading to strong asymmetry of the nonlinear I–V curve at zero bias (V = 0), the transistor demonstrated a remarkable current response to an AC signal at the values of the gate charge Q₀ close to (n + 1/2)e, where n is integer. A rather delicate regime of the transistor operation (V ? e/C_Σ) being important for unperturbed measurements was examined. The measured curves are in good agreement with a model based on the orthodox theory of single electron tunneling. This specific zero bias regime of an asymmetric transistor opens new opportunities for a single-electron transistor as an ultrasensitive charge/field sensor.     

Effect of the nuclear charge of a fast structural ion on its internal effective stopping in collisions with atoms

The energy losses of fast structural ions in collisions with atoms have been considered in the eikonal approximation. The structural ions are ions consisting of a nucleus and a certain number of electrons bound to it. The effect of nuclear charge Z of the ion on its effective deceleration κ^(p) (energy losses associated with excitation of only intrinsic ion shells) has been analyzed. It is shown that the allowance for the interaction of an atom with the ion nucleus for Z _a Z/v > 1, where Z _a is the charge of the atomic nucleus and v is the velocity of collisions in atomic units, considerably affects the value of κ^(p), which generally necessitates taking into account nonperturbatively the effect of both charges Z _a and Z on κ^(p).     

Analysis of the angular distribution of Auger electrons in a Xe atom

The angular distribution of Auger electrons, and the numerically calculated anisotropy parameters of the angular distribution α for (M₃ → N₂N₃), (M₃ → N₃N₃), (M₄ → N₁N₃), (M₄ → N₄N₅), (M₄ → N₅N₅), and (M_4,5 → O_{2, 3}O_{2, 3}) transitions in a Xe atom are given. The matrix elements are calculated by using the nonrelativistic Hartree-Fock method in LS coupling and the relativistic Hartree-Fock-Dirac method in both jj coupling (the single-configuration approximation) and intermediate coupling (the multiconfiguration approximation).