Wigner Formula Research Articles

Branching ratios and CP asymmetries of the quasi-two-body decays Bc→K0∗(1430,1950)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow \\ K^{*}_0(1430,1950) D_{(s)} \\rightarrow K \\pi D_{(s)} $$\\end{document} in the PQCD approach

In this paper, we investigate the quasi-two-body decays Bc→K0∗(1430,1950)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1430,1950) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document} within the perturbative QCD (PQCD) framework. The S-wave two-meson distribution amplitudes (DAs) are introduced to describe the final state interactions of the Kπ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\pi $$\\end{document} pair, which involve the time-like form factors and the Gegenbauer polynomials. In the calculations, we adopt two kinds of parameterization schemes to describe the time-like form factors: one is the relativistic Breit–Wigner (RBW) formula, which is usually more suitable for the narrow resonances, and the other is the LASS line shape proposed by the LASS Collaboration, which includes both the resonant and nonresonant components. We find that the branching ratios and the direct CP violations for the decays Bc→K0∗(1430)D(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1430) D_{(s)}$$\\end{document} obtained from those of the quasi-two-body decays Bc→K0∗(1430)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1430) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document} under the narrow width approximation (NWA) can be consistent well with the previous PQCD results calculated in the two-body framework by assuming that K0∗(1430)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^*_0(1430)$$\\end{document} is the lowest lying q¯s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{q} s$$\\end{document} state, which is the so-called scenario II (SII). We conclude that the LASS parameterization is more suitable to describe the K0∗(1430)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_0^{*}(1430)$$\\end{document} than the RBW formula, and the nonresonant components play an important role in the branching ratios of the decays Bc→K0∗(1430)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1430) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document}. In view of the large difference between the decay width measurements for the K0∗(1950)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_0^{*}(1950)$$\\end{document} given by BaBar and LASS collaborations, we calculate the branching ratios and the CP violations for the quasi-two-body decays Bc→K0∗(1950)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1950) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document} by using two values, ΓK0∗(1950)=0.100±0.04\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _{K^*_0(1950)}=0.100\\pm 0.04$$\\end{document} GeV and ΓK0∗(1950)=0.201±0.034\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _{K^*_0(1950)}=0.201\\pm 0.034$$\\end{document} GeV, besides the two kinds of parameterizations for the resonance K0∗(1950)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^*_0(1950)$$\\end{document}. We find that the branching ratios and the direct CP violations for the decays Bc→K0∗(1950)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1950) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document} have not as large difference between the two parameterizations as the case of decays involving the K0∗(1430)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^*_0(1430)$$\\end{document}, especially for the results with ΓK0∗(1950)=0.201±0.034\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _{K^*_0(1950)}=0.201\\pm 0.034$$\\end{document} GeV. The effect of the nonresonant component in the K0∗(1950)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^*_0(1950)$$\\end{document} may be not so serious as that in the K0∗(1430)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^*_0(1430)$$\\end{document}. The quasi-two-body decays Bc+→K0∗+(1430)D0→K0π+D0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^+_c \\rightarrow K^{*+}_0(1430) D^{0} \\rightarrow K^0 \\pi ^+ D^{0}$$\\end{document} and Bc+→K0∗0(1430)D+→K+π-D+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^+_c \\rightarrow K^{*0}_0(1430) D^{+} \\rightarrow K^+ \\pi ^- D^{+}$$\\end{document} have large branching ratios, which can reach to the order of 10-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-4}$$\\end{document} and are most likely to be observed in the future LHCb experiments. Furthermore, the branching ratios of the quasi-two-body decays Bc→K0∗(1950)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1950) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document} are about one order smaller than those of the corresponding decays Bc→K0∗(1430)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1430) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document}.

Photoejection from Various Systems and Radiative-Rate Coefficients

Photoionization or photodetachment is an important process. It has applications in solar- and astrophysics. In addition to accurate wave function of the target, accurate continuum functions are required. There are various approaches, like exchange approximation, method of polarized orbitals, close-coupling approximation, R-matrix formulation, exterior complex scaling, the recent hybrid theory, etc., to calculate scattering functions. We describe some of them used in calculations of photodetachment or photoabsorption cross sections of ions and atoms. Comparisons of cross sections obtained using different approaches for the ejected electron are given. Furthermore, recombination rate coefficients are also important in solar- and astrophysics and they have been calculated at various electron temperatures using the Maxwell velocity distribution function. Approaches based on the method of polarized orbitals do not provide any resonance structure of photoabsorption cross sections, in spite of the fact that accurate results have been obtained away from the resonance region and in the resonance region by calculating continuum functions to calculate resonance widths using phase shifts in the Breit–Wigner formula for calculating resonance parameters. Accurate resonance parameters in the elastic cross sections have been obtained using the hybrid theory and they compare well with those obtained using the Feshbach formulation. We conclude that accurate results for photoabsorption cross sections can be obtained using the hybrid theory.

Real-time quantum calculations of phase shifts using wave packet time delays

We present a method to extract the phase shift of a scattering process using the real-time evolution in the early and intermediate stages of the collision in order to estimate the time delay of a wave packet. This procedure is convenient when using noisy quantum computers for which the asymptotic out-state behavior is unreachable. We demonstrate that the challenging Fourier transforms involved in the state preparation and measurements can be implemented in $1+1$ dimensions with current trapped ion devices and IBM quantum computers. We compare quantum computation of the time delays obtained in the one-particle quantum mechanics limit and the scalable quantum field theory formulation with accurate numerical results. We discuss the finite volume effects in the Wigner formula connecting time delays to phase shifts. The results reported involve two- and four-qubit calculations, and we discuss the possibility of larger scale computations in the near future.

Using the image processing in digital system to determine the nuclear parameters of LR-115 detector

The relative resolution of the nuclear track detector to distinguish between the different energies of alpha particles has been investigated. This done by modeling the nuclear track of LR-115 detector, the revealed tracks have been classified and their diameters were measured with numerical densities replication depending on the fitting by resonance Breit–Wigner formula, the optimum etching time was chose with the least standard deviation of the spectral distribution of tracks diameter according to the average track diameter (D). The spectral distribution width (ΔD) was determined at Full Width at Half Maxima (FWHM). The results and comparisons showed the resonance Breit–Wigner formula it had a preference in improving the relative resolution of the detector as compared with the Gaussian and Wblfit formulas. It turns out the application of image analysis software by MATLAB technique leads to give fast and accurate nuclear track parameters for LR-115. The digitally counting system is high accurate and infinite speed in completing the procedure. The results obtained with respect to the measurement of 238U concentration in the standard samples, showed the superiority of digitally system in the determination of these concentration in comparison with manually system, with average error rate no exceeding of 4.175%.

Generalized Model for the Electrical Conductance of a One-Atom Metallic Constriction with Electrochemical Potential Control

Starting from the Breit–Wigner model, we present a generalized approach for calculating the electrical conductance of a one-atom metallic constriction with electrochemical-potential control. In addition, the corresponding current intensity is determined. On the one hand, the nanoconstriction is conceived as a dilute (degenerate) Fermi gas (relatively low electron concentrations are considered) and, on the other hand, non-dilute gas is considered. Under these conditions, the Fermi energy-level into the Breit–Wigner formula is replaced by the corresponding chemical-potential energy. We emphasize the role of the optical potential as significant ingredient of the chemical potential.

A hybrid model description of 13C(p,γ)14N capture reaction

The radiative capture reaction [Formula: see text] is analyzed using a hybrid model approach where the non-resonant component has been constructed employing the potential model with a folded M3Y potential. The one-level Breit–Wigner formula has been used to estimate the cross-sections of the resonant decays of dominant ([Formula: see text]) state of [Formula: see text]N at 8.06[Formula: see text]MeV. The contribution of the broad resonance at 8.77[Formula: see text]MeV ([Formula: see text]) has been dealt with differently. While Breit–Wigner formula has been used where the excitation function data exist, the [Formula: see text]-matrix prediction for the cross-section of decay to a bound state of [Formula: see text]N from the broad resonance has been used where excitation function data are not available. The single particle spectroscopic factors for ground and six excited states of [Formula: see text]N have been obtained from the fits. The resulting astrophysical [Formula: see text]-factor at zero relative energy is [Formula: see text][Formula: see text]keV b. The value is in good agreement with the previously reported [Formula: see text]-matrix result and also consistent within error bars with the published values.

About the Breit–Wigner Formula for Identical Particles

It is noted that in a number of authoritative publications (including reviews of the Particle Data Group) the Breit–Wigner formula for a resonance production cross section is erroneously given without the combinatorial factor of 2! that must be inserted in the case of collisions of identical particles—for example, when γγ collisions are considered.

Different manifestations of S-matrix poles

Making use of the analytical properties of the S-matrix and a theorem of Mittag-Leffler, model independent non-relativistic expressions for cross sections in single channel elastic scattering, scattering phase shifts and survival probabilities of resonances are derived. Provided certain conditions are satisfied by the poles, the residues can also be estimated analytically. Considerations of the low energy behaviour of the S-matrix and cross sections reveal additional conditions on the residues of the poles appearing in the Mittag-Leffler expansions. The exact expressions for the resonant cross section and phase shift are shown to reduce to the commonly used Breit–Wigner formula plus corrections. The latter is shown to approach the exact result with the example of a meson and a baryon resonance. Finally, a comparison of the exact expressions with some realistic examples is presented. The calculations of survival probabilities in particular reveal the reason behind the non-observability of non-exponential decay.

S-wave resonance contributions to in the perturbative QCD factorization approach**Supported by National Natural Science Foundation of China (11235005, 11547038)

By employing the perturbative QCD (PQCD) factorization approach, we study the quasi-two-body decays, where the pion pair comes from the S-wave resonance f0(X). The Breit–Wigner formula for the f0(500) and f0(1500) resonances and the Flatté model for the f0(980) resonance are adopted to parameterize the time-like scalar form factors in the two-pion distribution amplitudes. As a comparison, Bugg's model is also used for the wide f0(500) in this work. For decay rates, we found the following PQCD predictions: (a) when the contributions from f0(980) and f0(1500) are all taken into account; (b) in the Breit-Wigner model and in Bugg's model.

The three-body decays B(s) → ρf0(X)→ρπ−π+ in pQCD factorization approach

In this paper, we study three-body decays B(s)→ρf0(X)→ρπ−π+ in the perturbative QCD (pQCD) factorization approach. By using two-pion distribution amplitudes (DAs), these decays can proceed mainly via quasi-two-body channels. The Flatté model for the f0(980) resonance and the Breit–Wigner formula for the f0(500), f0(1500), and f0(1790) resonances are adopted to parameterize the time-like scalar form factors. We also use Bugg's model for the wide f0(500) as a comparison. We evaluate S-wave resonance contributions, and show the contributions to the B(s)→ρπ+π− decay spectrums with respect to the two-pion invariant mass M(π+π−). The pQCD prediction of B→ρf0(500) is B(B0→ρ0f0(500))=4.43×10−8, B(B+→ρ+f0(500))=3.84×10−8 for Breit–Wigner model, and B(B0→ρ0f0(500))=4.91×10−8, B(B+→ρ+f0(500))=4.07×10−8 for Bugg's model. We also predict other resonance contributions B(Bs→ρ0f0(980))=8.91×10−8, B(Bs→ρ0f0(1500))=1.43×10−8 and B(Bs→ρ0f0(1790))=2.78×10−9. This study in pQCD approach can provide a ready reference to the running and forthcoming experiments. Further improvements in theories and experiments are expected so that we can understand the inner structure of the scalar mesons f0.

High-precision evaluation of Wigner's d matrix by exact diagonalization.

The precise calculations of Wigner's d matrix are important in various research fields. Due to the presence of large numbers, direct calculations of the matrix using Wigner's formula suffer from a loss of precision. We present a simple method to avoid this problem by expanding the d matrix into a complex Fourier series and calculate the Fourier coefficients by exactly diagonalizing the angular momentum operator J(y) in the eigenbasis of J(z). This method allows us to compute the d matrix and its various derivatives for spins up to a few thousand. The precision of the d matrix from our method is about 10(-14) for spins up to 100.

Electron attachment to the SF6 molecule

Various models for transition between electron and nuclear subsystems are compared in the case of electron attachment to the SF6 molecule. Experimental data, including the cross section of electron attachment to this molecule as a function of the electron energy and vibrational temperature, the rate constants of this process in swarm experiments, and the rates of the chemionization process involving Rydberg atoms and the SF6 molecule, are collected and treated. Based on the data and on the resonant character of electron capture into an autodetachment ion state in accordance with the Breit–Wigner formula, we find that intersection of the molecule and negative ion electron terms proceeds above the potential well bottom of the molecule with the barrier height 0.05–0.1 eV, and the transition between these electron terms has both the tunnel and abovebarrier character. The limit of small electron energies e for the electron attachment cross section at room vibrational temperature takes place at e ≪ 2 meV, while in the range 2 meV ≪ e ≪ 80 meV, the cross section is inversely proportional to e. In considering the attachment process as a result of the interaction between the electron and vibrational degrees of freedom, we find the coupling factor f between them to be f = aT at low vibrational temperatures T with a ≈ 3 × 10−4 K−1. The coupling factor is independent of the temperature at T > 400 K.

S-wave resonance contributions to theB(s)0→J/ψπ+π−andBs→π+π−μ+μ−decays

We study $S$-wave resonance contributions to the ${B}_{(s)}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${B}_{s}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ decays in the perturbative QCD framework by introducing two-hadron distribution amplitudes for final states. The Breit--Wigner formula for the ${f}_{0}(500)$, ${f}_{0}(1500)$, and ${f}_{0}(1790)$ resonances and the Flatt\'e model for the ${f}_{0}(980)$ resonance are adopted to parametrize the timelike scalar form factors in the two-pion distribution amplitudes, which include both resonant and nonresonant contributions. The resultant branching fraction and differential branching fraction in the pion-pair invariant mass for each resonance channel are consistent with experimental data. The determined $S$-wave two-pion distribution amplitudes, containing the information of both resonant and nonresonant rescattering phases, can be employed to predict direct $CP$ asymmetries of other three-body hadronic $B$ meson decays in various localized regions of two-pion phase space.

Analytical formula for numerical evaluations of the Wigner rotation matrices at high spins

The Wigner d function, which is the essential part of an irreducible representation of SU(2) and SO(3) parameterized with Euler angles, has been know to suffer from a serious numerical errors at high spins, if it is calculated by means of the Wigner formula as a polynomial of cos and sin of half of the second Euler angle. This paper shows a way to avoid this problem by expressing the d functions as the Fourier series of the half angle. A precise numerical table of the coefficients of the series is obtainable from a web site.

Phonon tunneling through a double barrier system

The tunneling of optical and acoustic phonons at normal incidence on a double-barrier is studied in this paper. Transmission coefficients and resonance conditions are derived theoretically under the assumption that the long-wavelength approximation is valid. It is shown that the behavior of the transmission coefficients for the symmetric double barrier has a Lorentzian form close to resonant frequencies and that Breit–Wigner's formula have a general validity in one-dimensional phonon tunneling. Authors also study the so-called generalized Hartman effect in the tunneling of long-wavelength phonons and show that this effect is a numerical artifact resulting from taking the opaque limit before exploring the variation with a finite barrier width. This study could be useful for the design of acoustic devices.

CP Violation and Strong Pion-Pion Interactions in the Weak $${B^\pm \to \pi^\pm \pi^\mp \pi^\pm}$$ Decays

An analysis of $${B^\pm \to \pi^\pm \pi^\mp \pi^\pm}$$ decay data in a quasi two-body B ± → π ± [π + π −] S, P, D QCD factorization framework is performed. The short distance amplitudes are calculated including next-to-leading order in α s vertex and penguin corrections. The long distance amplitudes due to the [π + π −] S, P, D final state interactions are described by the pion non-strange scalar and vector form factors for the [π + π −] S S-wave state and [π + π −] P P-wave state, respectively and by a relativistic Breit–Wigner formula for the[π + π −] D D-wave state. We achieve a good fit of the data with only three parameters for the S wave, one for the P wave and one for the D wave. Our model gives a unified unitary description of the three scalar resonances, f 0(600), f 0(980) and f 0(1400) in terms of the pion scalar form factor. We predict for the B to f 2(1270) transition form factor, $${F^{Bf_2}(m_\pi^2)=0.098\pm0.007}$$ .

Switch of Conducting Orbital by Bias-Induced Electronic Contact Asymmetry in a Bipyrimidinyl-biphenyl Diblock Molecule: Mechanism to Achieve a pn Directional Molecular Diode

We have both experimentally and theoretically studied low voltage rectification phenomena in an asymmetric single diblock molecular junction and have analyzed its mechanism. First-principles nonequilibrium Green’s function (NEGF) theory was applied to electric current calculations. Then we identified a conducting molecular orbital based on the Breit–Wigner formula combined with the effective molecular projected state Hamiltonian (MPSH) analysis, where real and complex energy terms are renormalized explicitly into the standard MPSH by a Feshbach projection operator technique. The analysis discusses voltage dependences of conducting molecular orbitals as well as of site energy alignment defined in terms of molecular fragments. We found quantitative results of (i) electronic switching of the conducting molecular orbital caused by bias-dependent intrinsic electronic couplings with electrodes and (ii) change of site energy alignment driven by bias voltage sweep without significant additional charge injection. ...

Quantum mechanics of successive measurements with arbitrary meter coupling

We study successive measurements of two observables using von Neumann's measurement model. The two-pointer correlation for arbitrary coupling strength allows retrieving the initial system state. We recover Lüders rule, the Wigner formula and the Kirkwood–Dirac distribution in the appropriate limits of the coupling strength.

Wigner formula of rotation matrices and quantum walks

Quantization of a random-walk model is performed by giving a qudit (a multicomponent wave function) to a walker at site and by introducing a quantum coin, which is a matrix representation of a unitary transformation. In quantum walks, the qudit of the walker is mixed according to the quantum coin at each time step, when the walker hops to other sites. As special cases of the quantum walks driven by high-dimensional quantum coins generally studied by Brun, Carteret, and Ambainis, we study the models obtained by choosing rotation as the unitary transformation, whose matrix representations determine quantum coins. We show that Wigner's $(2j+1)$-dimensional unitary representations of rotations with half-integers $j$'s are useful to analyze the probability laws of quantum walks. For any value of half-integer $j$, convergence of all moments of walker's pseudovelocity in the long-time limit is proved. It is generally shown for the present models that, if $(2j+1)$ is even, the probability measure of limit distribution is given by a superposition of $(2j+1)∕2$ terms of scaled Konno's density functions, and if $(2j+1)$ is odd, it is a superposition of $j$ terms of scaled Konno's density functions and a Dirac's delta function at the origin. For the two-, three-, and four-component models, the probability densities of limit distributions are explicitly calculated and their dependence on the parameters of quantum coins and on the initial qudit of walker is completely determined. Comparison with computer simulation results is also shown.

Prediction of size disorder induced resonant tunneling in large scale nanodot arrays

AbstractWe have successfully investigated the effect of size fluctuation on the resonant tunneling characteristics of nanodots arrays. The approach was realized with taking into account the induced electron energy level disorder under the framework of the generalized Breit–Wigner formula. It is found that the dot size fluctuation has played an important role in both the tunneling current profile and number of the resonant peaks. By the aid of the statistical concept to characterize the size fluctuation in large scale nanodots systems, we have shown a clear picture for the dependence of the resonant peak number and peak to valley ratio on the degree of disorder and spatial configuration in the nanodots. The whole investigation will give a general prediction of the origin of resonant tunneling phenomena observed in large scale nanodot systems. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)