Expansion of the Potential of a Homogeneous Circular Torus in Terms of Geometrical Parameter

The spatial potential of a homogeneous gravitating (or statically charged) circular torus is expanded in terms of geometrical parameter q. The first term of the expansion (coefficient of zero-power parameter q) coincides with the potential of a thin ring the mass of which is equal to the mass of torus and the radius of which is equal to the radius of the axial line. It is shown that the coefficients of terms with odd powers are zeros. Even terms of the expansion of the potential are analytically represented in terms of standard complete elliptic integrals. The corresponding series describes the potential of torus in entire space including internal region. The proposed method for representation of the potential makes it possible to obtain gravitational energy of a homogeneous circular torus. The method is used to calculate masses of two rings of the Chariklo asteroid. The mass of the internal ring is M_r1 ≈ 9.8 × 10¹⁸ g and the ratio of such a mass to the mass of asteroid is \(\frac{{{M_{r1}}}}{{{M_0}}}\)≈ 0.001. The results for the external ring are M_r2 ≈ 10¹⁸ g and \(\frac{{{M_{r1}}}}{{{M_0}}}\)≈ 10_–4.     

Resolvent Expansion and Time Decay of the Wave Functions for Two-Dimensional Magnetic Schrödinger Operators

We consider two-dimensional Schrödinger operators H(B, V) given by Eq. (1.1) below. We prove that, under certain regularity and decay assumptions on B and V, the character of the expansion for the resolvent (H(B, V) ? λ)^?1 as λ → 0 is determined by the flux of the magnetic field B through \({\mathbb{R}^2}\) . Subsequently, we derive the leading term of the asymptotic expansion of the unitary group e ^{?i t H(B, V)} as t → ∞ and show how the magnetic field improves its decay in t with respect to the decay of the unitary group e ^{?i t H(0, V)}.     

Pseudoscalar Higgs boson in experiments at <Emphasis Type="Italic">e</Emphasis><Superscript>+</Superscript><Emphasis Type="Italic">e</Emphasis><Superscript>−</Superscript> colliders

The potential of experiments at e ⁺ e ^? colliders to search for a signal of a new pseudoscalar Higgs boson and to study its CP properties is analyzed. A new Higgs boson coupling to the b quark and τ lepton, which is parameterized in a model-independent way as m _b /v(a + iγ₅ b), is chosen to be an indicator of its CP nature. We study the e ⁺ e ^? → τ⁺τ^?vv, e ⁺ e ^? → b \(\bar b\)vv, and e ⁺ e ^? → e ⁺ e ^? b \(\bar b\) processes highly sensitive to anomalous Higgs boson couplings due to the dominant contributions from the vector boson fusion processes W*W* → H and Z*Z* → H in the central region. It is shown that the study of polarization of particles in the final state could help to separate the contributions from the scalar and pseudoscalar Higgs bosons. For instance, the consideration of the cascade decays of τ leptons in the e ⁺ e ^? → τ⁺τ^?vv process allows one to determine reliably the CP state of the Higgs boson as well as the value and the sign of the b parameter. As a result of our analysis, we develop a search strategy for a signal of a new pseudoscalar Higgs boson in future experiments at linear e ⁺ e ^? colliders, which will allow one to understand the CP nature of the Higgs boson and set the bounds on its couplings to fermions.     

Analytic calculation of feynman amplitudes

A new method is proposed for calculating amplitudes of processes involving spin-1/2 fermions. The method is based on the use of the isotropic-vierbein vectors in Minkowski space and the basis spinors associated with them. The amplitudes for the electron-positron interaction processes e^?e⁺ → ff and e^?e⁺ → W^?W⁺ and the amplitude for one of the possible diagrams for the process e⁺e^? → e⁺e^?e⁺e^?e⁺e^? are calculated in order to test and illustrate the potential of the method.     

Quark mixing and exponential form of the Cabibbo-Kobayashi-Maskawa matrix

Various forms of representation of the mixing matrix are discussed. An exponential parametrization e ^ of the Cabibbo-Kobayashi-Maskawa matrix is considered in the context of the unitarity requirement, this parametrization being the most general form of the mixing matrix. An explicit representation for the exponential mixing matrix in terms of the first and second degrees of the matrix  exclusively is obtained. This representation makes it possible to calculate this exponential mixing matrix readily in any order of the expansion in the small parameter λ. The generation of new unitary parametric representations of the mixing matrix with the aid of the exponential matrix is demonstrated.     

Photoproduction of Triplets on Free Electrons and the Search for the Dark Photon

We have investigated the photoproduction of triplets on free electrons, γe^– → e⁺e^–e^–, in which dark photon A′ can be formed as an intermediate state with subsequent decay into an e⁺e^– pair. This effect appears as a result of the so-called kinetic mixing and is characterized by the small parameter describing the intensity of the interaction of the dark photon with charged Standard Model (SM) particles in terms of electric charge e. The search for a manifestation of A′ in this process is advantageous since the background to the A′ signal is purely electrodynamic and, hence, can be calculated with the required accuracy. We calculate this background taking into account the identity of final electrons. As regards A′, its contribution is taken into account only in Compton-type diagrams (of virtual Compton scattering) in which a virtual dark photon is a time-like particle and its propagator has the form of a Breit–Wigner resonance. It is only in the vicinity of resonance that A′ can be manifested. We calculate the invariant mass distribution of both e⁺e^– pairs formed in the process and analyze the kinematic region in which relatively small squares of momenta transferred from the target electron to the formed electrons are excluded. In these conditions, the contribution to the differential cross section from Compton-type diagrams is not suppressed relative to the contribution of the remaining (Borsellino) diagrams. A number of limitations on parameter depending on the dark photon mass and the statistics (number) of events are obtained for a special method of gathering events in which the invariant mass of one e⁺e^– pair remains fixed and of the other pair is scanned.     

Effects of four-fermion contact interactions in the Bhabha process at linear electron-positron colliders

For the case of longitudinally polarized primary beams, effects are investigated that are induced by four-fermion contact interactions in the process e⁺e^? → e⁺e^? occurring at linear electron-positron colliders of energy in the range \(\sqrt s = 0.5 - 1 TeV\). Model-independent constraints on the parameters of four-fermion contact interactions are obtained. It is shown that the sensitivity of the Bhabha process to the parameter Λ_LR is considerably higher than the sensitivity of the leptonic process e⁺e^? → μ⁺μ^?; at the same time, the sensitivities of these processes to the parameters Λ_RR and Λ_LL are commensurate.     

Nuclear composition of 1-to 20-PeV primary cosmic rays according to lateral features of the electron-photon component of all extensive air showers and of extensive air showers accompanying high-energy gamma rays and hadrons in x-ray emulsion chambers at the Tien Shan level

Data from the Tien Shan array Adron on the dependence of the lateral distributions of the electron-photon component (age parameter S) in extensive air showers of cosmic rays on the number of electrons, N _e , which is a quantity that characterizes the primary-nucleus energy E₀, are subjected to a comparative analysis. The distributions in question are given both for all showers and for showers accompanying high-energy gamma rays and hadrons in x-ray emulsion chambers. According to calculations, events associated with the latter are generated predominantly by primary protons, and this makes it possible to assess their role at various values of E₀. The distributions with respect to S suggest a significant fraction of light nuclei, predominantly protons, in the region after the knee in the spectrum for N _e >10⁶, at least up to N _e =5.6×10⁶ (E₀ ～ 10 PeV).     

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.     

Exact asymptotic form for the β function in quantum electrodynamics

It is shown that the asymptotic form of the Gell-Mann-Low function in quantum electrodynamics can be determined exactly: β(g) = g for g → ∞, where g = e ² is the running fine-structure constant. This solves the problem of electrodynamics at small distances L (for which dependence g ∞ L ^?2 holds) and completely eliminates the problem of “zero charge.”     

Effect of FCNC mediated <Emphasis Type="Italic">Z</Emphasis> boson on lepton flavor violating decays

We study the three body lepton flavor violating (LFV) decays μ ^?→e ^? e ⁺ e ^?, \(\tau^{-} \to l_{i}^{-} l_{j}^{+} l_{j}^{-}\) and the semileptonic decay τ→μφ in the flavor changing neutral current (FCNC) mediated Z boson model. We also calculate the branching ratios for LFV leptonic B decays, B _d,s →μe, B _d,s →τe, B _d,s →τμ and the conversion of muon to electron in Ti nucleus. The new physics parameter space is constrained by using the experimental limits on μ ^?→e ^? e ⁺ e ^? and τ ^?→μ ^? μ ⁺ μ ^?. We find that the branching ratios for τ→eee and τ→μφ processes could be as large as \({\sim}{\mathcal{O}}(10^{-8})\) and \(\mathrm{Br}(B_{d,s} \to \tau \mu,~ \tau e) \sim {\mathcal{O}}(10^{-10})\). For other LFV B decays the branching ratios are found to be too small to be observed in the near future.     

Separation of electrons and pions in the SND detector calorimeter

A method is developed for discriminating between e⁺e^? → e⁺e^? and e⁺e^? → π⁺π^? events in the center-of-mass (cms) energy range of 0.5 to 1.0 GeV by energy deposition in the electromagnetic calorimeter of the SND detector using a machine learning algorithm. Identification efficiencies for e⁺e^? → e⁺e^? and e⁺e^? → π⁺π^? events are 99.6 and 99.8%, respectively.     

Zur Theorie von Systemen aus drei Teilchen mit Coulomb-Wechselwirkung

The wave functions of muon molecules as well as of the corresponding electron molecules and of the systeme ⁺ e ^? e ⁺ have been approximated by simple analytical functions, the best choice of parameter values being obtained by straightforward variational procedure. The results are discussed in some detail and binding energies tabulated.     

Two-photon production of vector resonance in <Emphasis Type="Italic">e</Emphasis><Superscript>+</Superscript><Emphasis Type="Italic">e</Emphasis><Superscript>−</Superscript> collisions

A parameterization of the amplitude of the two-photon production of 1⁺⁺ and 1^?+ vector resonances V in the e ⁺ e ^? → e ⁺ e ^? V reaction has been obtained. The differential (in momentum transfer) scattering cross section dσ/dQ ² has been found for this reaction in the equivalent-photon approximation. The result obtained is model-independent.     

Proc ess es e+ e− → cccc and e+ e− → J/ψ + gg at $$\sqrt s = 10.59 G eV$$

Experimental data obtained by the BELLE Collaboration for inclusive J/ψ production in the processes e⁺e^? → J/ψ + gg and e⁺e^? → J/ψ + cc are discussed. These data are compared with the predictions of perturbative QCD that were obtained by two methods, that which employs information about the J/ψ wave function and that which relies on the hypothesis of quark-hadron duality exclusively. Both computational methods yield results that disagree with the experimental data considerably. The dependence of the cross section for the process e⁺e^? → J/ψ + gg on the effective gluon mass is studied. The cross section for the production of doubly charmed baryons Ξ _cc ^* is estimated.     

Determination of quark-antiquark component of the photon wave function for <Emphasis Type="Italic">u,d, s</Emphasis> quarks

Based on the data for the transitions π⁰, η, η′ → γγ*(Q²) and reactions of the e⁺e^? annihilations e⁺e^? → ρ⁰, ω, ? and e⁺e^? → hadrons at 1<E _e+e? <3.7 GeV, we determine the light-quark components of the photon wave function \(\gamma * (Q^2 ) \to q\bar q(q = u,d,s)\) for the region 0 ? Q² ? 1 (GeV/c)².     

Measurement of cross sections for processes involving <Emphasis Type="Italic">e</Emphasis><Superscript>+</Superscript><Emphasis Type="Italic">e</Emphasis><Superscript>−</Superscript> annihilation to hadrons at the VEPP-2M collider

Results obtained by measuring, in experiments at the VEPP-2M e ⁺ e ^? collider with the CMD-2 and SND detectors at the Novosibirsk Institute of Nuclear Physics, cross sections for e ⁺ e ^? → hadrons annihilation processes at energies in the range 360 ≤ √s ≤ 1380 MeV are surveyed. The measured cross sections and the vector-meson parameters deduced from an analysis of these cross sections are presented. The cross section for the process e ⁺ e ^? → π ⁺ π ^? is compared with the spectral function for the decay τ ^? → π ^? π ⁰ ν _τ.     

A Note on Holographic Dark Energy with Varying <Emphasis Type="Italic">c</Emphasis><Superscript>2</Superscript> Term

We reconsider the holographic dark energy (HDE) model with a slowly time varying c ²(z) parameter in the energy density, namely \(\rho _{D}=3{M_{p}^{2}} c^{2}(z)/L^{2}\), where L is the IR cutoff and z is the redshift parameter. As the system’s IR cutoff we choose the Hubble radius and the Granda-Oliveros (GO) cutoffs. The latter inspired by the Ricci scalar curvature. We derive the evolution of the cosmological parameters such as the equation of state and the deceleration parameters as the explicit functions of the redshift parameter z. Then, we plot the evolutions of these cosmological parameters in terms of the redshift parameter during the history of the universe. Interestingly enough, we observe that by choosing L = H ^?1 as the IR cutoff for the HDE with time varying c ²(z) term, the present acceleration of the universe expansion can be achieved, even in the absence of interaction between dark energy and dark matter. This is in contrast to the usual HDE model with constant c ² term, which leads to a wrong equation of state, namely that for dust w _D =0, when the IR cutoff is chosen the Hubble radius.     

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).     

CP violation at colliders

The prospects of experimental detection ofCP violation ate ⁺ e ^? and\(pp/p\bar p\) colliders are reviewed. After a general discussion on the quantities which can measureCP violation and on the implications of theCPT theorem, various possibilities of measuringCP violation arising outside the standard model are taken up.CP violation in leptonic processes, especially polarization effects ine ⁺ e ^?→l ⁺ l ^? are discussed next.CP violation in\(t\overline t \) andW ⁺ W ^? production and decay is also described.