Commutator representations of differential calculi on the quantum group SUq(2)

Let (Γ, d) be the 3D-calculus or the 4D_±-calculus on the quantum group SU_q (2). We describe all pairs (π, F) of a ^*-representation π of (SU_q(2)) and of a symmetric operator F on the representation space satisfying a technical condition concerning its domain such that there exist a homomorphism of first order differential calculi which maps dx into the commutator [iF, π(x)] for x ε (SU_q (2)). As an application commutator representations of the two-dimensional left-covariant calculus on Podles quantum 2-sphere S_qc² with c = 0 are given.     

Single and double dissociation in πp collisions at 11 and 16 GeV/c

The reactions π⁻p → 2π⁻π⁺p, π⁻p → 2π⁻π⁺π^op and π⁻p → 2π⁻2π⁺n are analysed at 11 and 16 GeV/c using longitudinal phase space (LPS) plots. The weighted LPS distributions for π⁻p → 2π⁻π⁺p is dominated by two well separated structures corresponding to single diffraction dissociation of the pion, π⁻p → (2π⁻π⁺)p, and of the proton, π⁻p → π⁻(π⁻π⁺p). The former is more abundant than the latter, and both are approximately constant with energy. In contrast, processes of type π⁻p → (2π)(πp) decrease with increasing energy.

In the five-body reactions the weighted LPS distribution reveals especially at 16 GeV/c a maximum for single dissociation of the proton into 3πp, namely π⁻p → π⁻(π⁻π⁺π^op); this process is likely to be diffractive. The neutron channel has a corresponding maximum displaced toward a multiperipheral configuration π⁻p → π⁻(π⁻2π⁺)n. Another strong maximum corresponds to the pion dissociation π⁻p → (2π⁻π⁺π^o)p. This is interpreted to be an ω-exchange process because no analogous structure occurs in π⁻p → (2π⁻2π⁺)n. Finally, a broad structure reveals double dissociation of both incident particles; it occurs in the two channels π⁻p → (2π⁻π⁺)(π^op) and π⁻p → (2π⁻π⁺)(π⁺n), being stronger in the latter. Further analysis of this process in terms of isospin exchange suggests that it is partially diffractive. Factorization is also discussed.

An appendix gives general aspects of the LPS analysis for the asymptotic study of n-body collisions at very high energy.     

Anisotropic profile decay on perturbed Au(111) vicinal surfaces

The anisotropy of the surface free energy has a large influence on the decay rate of periodic surface profiles at temperatures below the roughening temperature. This has been demonstrated by a study of the periodic surface profiles that were etched into vicinal Au(111) crystals in two orientations: with their wavevector parallel (ψ = 0) and perpendicular (ψ = π/2) to the intrinsic step direction of the vicinal surface. Due to this modulation, steps become either oscillatory in shape or remain straight, with their average separation modulated. Accordingly, profile decay is driven primarily by step self-energy (ψ = 0) or step interaction energy (ψ = π/2). The rate of decay was measured under ultra-high vacuum conditions at 750°C using a scanning tunneling microscope for imaging. A large anisotropy of decay was observed on two vicinal crystals of 1.5 and 5° miscut, with the ψ = 0 decay being much faster than with ψ = π/2). The results are evaluated and discussed in the framework of recent theory.     

On the reentrant behaviour of the two-dimensional Ising model

The phase diagram of the two-dimensional Ising model with binary disorder is evaluated (in the region -1 < J₁/J₀ < 0) using a pertubative approach.     

Chiral symmetry amplitudes in the S-wave isoscalar and isovector channels and the σ, f0(980), a0(980) scalar mesons

We use a non-perturbative approach which combines coupled channel Lippmann-Schwinger equations with meson-meson potentials provided by the lowest order chiral Lagrangian. By means of one parameter, a cut off in the momentum of the loop integrals, which results of the order of 1 GeV, we obtain singularities in the S-wave amplitudes corresponding to the σ, f₀ and a₀ resonances. The ππ → ππ, phase shifts and inelasticities in the T = 0 scalar channel are well reproduced as well as the π⁰η and mass distributions in the T = 1 channel. Furthermore, the total and partial strong decay widths of the f₀ and a₀ resonances are properly reproduced. The results seem to indicate that chiral symmetry constraints at low energy and unitarity in coupled channels is the basic information contained in the meson-meson interaction below GeV.     

Hadronic production by ee collisions at the energy 990 MeV with the Orsay storage ring

Data were taken at the energy 2E = 990 MeV to search for multibody events, with the same large solid angle detector which has been used for the measurement of the , ω andφ production by e⁺e⁻ annilations. Assuming a π⁺π⁻π⁰π⁰ production by the quasi two-body process e⁺e⁻ → → ωπ⁰ we give the correspondi ng cross section σ(e⁺e⁻ → π⁺π⁻π⁰π⁰) = (1.1 ± 0.5) 10⁻³² cm². Since no events with 3 and 4 charged pions have been observed σ(e⁺e⁻ → π⁺π⁻π⁰π⁻) 1.5 × 10⁻³³ cm².     

Inelastic electron scattering from He

The spectra of electrons scattered inelastically from ⁴He have been measured at incident energies from 150 MeV to 400 MeV for scattering angles from 38° to 90°. Through the use of a liquid ⁴He target, a high momentum resolution (≈0.25%) was obtained, and the 20 MeV 0⁺ state of the -particle was observed for the first time in electron scattering. The excitation energy and the total width of this state were determined and are in good agreement with the results from other experimental methods. It was found that the total disintegration cross section appears be smaller than 2(σ_p+σ_n) by more than an order of magnitude at the lowest q²(≈ 0.33 fm⁻²). In earlier work the assumption of a total cross section of 2(σ_p+σ_n) was found to be quite accurate for higher q². As a new result, the partial radiative width for the 0⁺ state is determined to be Γ_rad = 1.1±0.3 meV.     

Upper bound to the charge-transfer energy on a 2D Wigner lattice in high-Tc cuprates

Assuming that the energy of a 2D Wigner lattice, which has been frequently observed on the CuO₂ plane in layered high-T_c cuprates, is composed of Coulomb energy, kinetic energy, and charge-transfer energy which is necessary to transfer electrons or holes onto the CuO₂ plane (the magnitude of the energy is expected to be of the order of the work function of the compound constituting the plane from which the electrons are transferred, if the CuO₂ plane easily accepts the electrons or holes, as if the CuO₂ plane were crystal surface), we find in the mean-field approximation, that the total energy for a unit lattice, E₀, can be given by a function of the charge-transfer energy E_φ as E₀= (E_φ−E_φ⁰+O((E_φ−E_φ⁰)₂)(>0), where the value of E_φ⁰(1.5 in the ground state, for example) is independent of the lattice configuration. This relation implies that the upper bound to the charge-transfer energy is given by E_φ⁰. The smallness of E_φ⁰, compared with the work function of ordinary metals and alloys (with the work function around 4 eV), is necessary for the realization of the 2D Wigner lattice in many layered high-T_c cuprates.     

Magnetic moment of the Δ (1232) resonance

The reaction γp → π^°γ′p has been measured with the TAPS BaF₂ calorimeter at the Mainz Microtron accelerator facility MAMI for energies between √2 = 1221–1331 MeV. Cross sections differential in angle and energy have been determined for the photon γ′ in three bins of the excitation energy. This reaction channel provides access to the magnetic dipole moment of the Δ⁺(1232) resonance and, for the first time, a value of μ_Δ+ = (2.7^+1.0_−1.3(stat) ± 1.5(syst) ± (theor)) π_N has been extracted.     

Electro-pion production on O to bou nd states of the residual nuclei

Differential cross sections for the reactions 16O(γ, π−π+−)16F16F to the sum of the four lowest lying states in 16F and 16N have been measured as a function of angle for pions with a kinetic energy of 30 MeV. The extracted ratios R = σ(γ, π−)/σ(γ, π+), the first ones to discrete final states as a function of angle, are in fair agreement with results obtained for the nucleon. For positive pions the energy dependence of the cross section has been measured at the angles of 45° and 90°. Distorted wave impulse approximation calculations fail to describe the energy dependence.     

A study of the Li(π, pp)He reaction within the three-body problem

The differential cross section for the ⁶Li(π⁺, pp)⁴He reaction for symmetrical coplanar kinematics has been calculated at the incident pion energy E_π = 70 MeV within the three-body problem + 2N taking into account the Pauli exclusion principle in each N subsystem. Three-body distortions in the final state have been taken into account in the eikonal approximation. The treatment is based on the superposition of the single-particle and two-particle mechanisms of absorption leading to the rescattering of the π- and ρ-mesons through the Δ-isobar. The analytical character of the adopted wave function for the ⁶Li nucleus (the multidimensional gaussian basis) makes it possible to ascertain clearly that the reaction amplitude of the type under study is insensitive to short-range NN correlations both in the initial and final states. The calculated momentum distribution of recoil -particles is in good agreement with the experimental data available, but the experimental accuracy should be improved in order to see the contribution of a specific three-body part of the ⁶Li wave function orthogonal to the ad channel. Some problems to be solved in future are discussed.     

Regge-pole description of γN→πN backward-scattering data

Baryon Regge-pole contributions to u-channel helicity amplitudes for the γN→πN processes are derived, with attention to kinematic singularities and threshold conditions. An N, Nγ and Δδ Regge-exchange model is proposed that describes the backward scattering data on γp→π^op and γp→π⁺n at high energy. The N and Nγ trajectories are found to be nearly degenerate, with residues in the ratio β(Nγ)/β(N)≈0.6. Structure in the differential cross sections is explained as dominance at small u giving way to dominance at large u. An isoscaar-isovector admixture for the γ-coupling to is required by the fits. The solution extrapolates through the mean γp→π^op 180^o differential cross section at intermediate energies, as required by duality.     

Phenomenological use of the Callan-Symanzik equation

We find that the Callan-Symanzik equation with the intrinsic breaking term omitted suggests ansatz for the differential cross section in the high energy wide angle limit. The agreement with experimental data for the reactions ep → ep, π⁻p → π⁻p, pp → pp is very satisfactory.     

πN scattering and electromagnetic corrections in the perturbative chiral quark model

We apply the perturbative chiral quark model to give predictions for the electromagnetic O(p²) low-energy couplings of the ChPT effective Lagrangian that define the electromagnetic mass shifts of nucleons and first-order (e²) radiative corrections to the πN scattering amplitude. We estimate the leading isospin-breaking correction to the strong energy shift of the π⁻p atom in the 1s state, which is relevant for the experiment “Pionic Hydrogen” at PSI.     

Electron coincidence spectroscopy of O2: Valence electron momentum distributions and binding energies

The binding energy spectra and momentum distributions for the valence orbital transitions of molecular oxygen have been obtained using the (e,2e) reaction at total electron energies of 400 and 1200 eV. The outer π_g orbital is found to have a wave-function which is significantly more extended in momentum space than that of the more-tightly bound π_u orbitals. This is interpreted as a consequence of the single occupancy of the anti-bonding π_g orbitals. Peaks at 39 and 47 eV are assigned to ^4,2Σ_g⁻ ion states on the basis of the observed momentum distributions. The momentum distribution and observed intensity for the 32.5 eV transition supports the assignment of 2 Π_u for this ion state. The measured relative strengths for the various Π_u and Σ_u transitions are in qualitative agreement with CI and Green's function calculations.     

Self energies of the pion and the Δ isobar from the He(e,e′π)H reaction

In a kinematically complete experiment at the Mainz microtron MAMI, pion angular distributions of the ³He(e,e′π⁺)³H reaction have been measured in the excitation region of the Δ resonance to determine the longitudinal (L), transverse (T), and the LT interference part of the differential cross section. The data are described only after introducing self-energy modifications of the pion and Δ-isobar propagators. Using Chiral Perturbation Theory (ChPT) to extrapolate the pion self energy as inferred from the measurement on the mass shell, we deduce a reduction of the π⁺ mass of MeV/c² in the neutron-rich nuclear medium at a density of fm⁻³. Our data are consistent with the Δ self energy determined from measurements of π⁰ photoproduction from ⁴He and heavier nuclei.     

Incommensurability in an exactly-soluble quantal and classical model for a kicked rotator

The Hamiltonian H = 2πp + V(θ) Σ^∞_−∞ δ(t − n)(0 θ < 2π) is solved exactly, classically and quantally; the so lutions depend strongly on . There is no classical chaos and the phase cylinder p, θ is filled with invariant curves, which are finite loops around the cylinder if is sufficiently irrational and are translates of the infinitely long p axis if is rational. Quantal quasi-energy states correspond exactly to these invariant curves: localized in p and extended in θ if is sufficiently irrational, and extended in p and localized in θ if is rational. For a classical or quantal initial pure-momentum state, the energy at time t = n grows as n² if is rational (resonance) and remains bounded if is sufficiently irrational (non-resonance). If is very nearly rational (marginal resonance), the energy may grow as n^λ where λ is expressed in terms of exponents describing the irrationality of and the continuity class of V(θ). If the value of is uncertain, ensemble-averaging over shows that the energy grows ultimately as n, i.e. diffusively, as though under random impulses.     

Whole-field digital measurement of principal stress directions in photoelasticity

A method that uses three light-field isoclinic images and the associated light-field unloaded model (or white) images for whole-field digital determination of the principal stress directions in photoelasticity is presented. Relevant theory is derived and explicit conditions for directly determining the directions to the range 0–π/2 are given. Tests of this method on a directly loaded two-dimensional disc and a stress frozen photoelastic slice are demonstrated. The results agree well with the values obtained from the manual method and/or the theory.     

Influence of electromagnetic fluctuations on the resonant tunneling of electrons

We have considered the influence of electromagnetic fluctuations on electron tunneling via one non-degenerate resonant level, the problem that is relevant for electron transport through quantum dots in the Coulomb blockade regime. We show that the overall effect of the fluctuations depends on whether the electron bands in external electrodes are empty or filled. In the empty band case, depending on the relation between the tunneling rate Γ and characteristic frequency Ω of the fluctuations, the field either simply shifts the conductance peak (for rapid tunneling, Γ Ω) or broadens it (for Γ Ω). In the latter case, the system can be in three different regimes for different values of the coupling g between electrons and the field. Increasing interaction strength in the region g < 1 leads to gradual suppression of the conductance peak at the bare energy of the resonant level ε₀, while at g 1 it leads to the formation of a new peak of width at the energy ε₀ + E_cis a charging energy. For intermediate values of g the conductance is non-vanishing in the entire energy range from ε₀ to ε₀ + E_c. For filled bands the problem is essentially multi-electron in character. One consequence of this is that, in contrast to the situation with the empty band, the fluctuations of the resonant level do not suppress conductance at resonance for g < 1. At g> 1 a Coulomb gap appears in the position of the resonant level as a function of its bare energy which leads to suppression of conductance.