Calculation of the transfer integrals in Wannier representation for a planar trans-polyacetylene chain

In this paper,the electronic transfer integrals,the energy gap,and the bandwidth of a planar trans-polyacetylene chain are calculated in Wannier representation,in which a combination of the wave function of hydrogen-like atoms is used to stand for the Wannier function.When the effective nuclear charge number Z=2.125 and the distortion amplitude of the carbon sites u=0.0038 nm,the nearest,next,and third neighborhopping energies obtained are -3.224 78 eV,-2.388 61 eV,0.148 14 eV,0.006 65 eV,and 0.00650 eV,respectively.The energy bandwidth and gap corresponding to these values are Wd=11.19 eV and Eg=1.70 eV,respectively.These results coincide with the experimental values.     

Computationally efficient recurrence relations for one-dimensional Franck–Condon overlap integrals

Calculations based on analytical expressions for the harmonic oscillator Franck–Condon factors often yield numerically unstable and erroneous results for large values of the oscillator quantum numbers. This instability arises from inherent machine precision limits and large number round-off associated with the products and ratios of factorial and gamma functions in these expressions; the analytical expressions themselves are exact. This paper presents, first, efficient, exact recurrence relations to evaluate Franck–Condon factors for the harmonic oscillator model. The recurrence relations, which are similar to those originally found by Manneback, Wagner and Ansbacher avoid the direct use of the factorial and gamma functions. Second, a variational strategy for the evaluation of Franck–Condon factors for the Morse oscillator is proposed. The Schrödinger equation for the Morse model is solved variationally with a large enough basis set of one-dimensional harmonic oscillator functions to get good agreement with the analytic eigenvalues of the Morse potential itself. The eigenvectors of this analysis are then used together with the associated harmonic oscillator Franck–Condon overlap matrix elements to evaluate the overlap for the Morse potential. This approach allows one, in principle, to estimate Franck–Condon overlap up to states near to the dissociation limit of the Morse oscillator.     

Differential cross section and related integrals for the molière potential

The Molière potential is widely used in radiation damage simulation studies. It is not much used in analytical transport theory calculations because of the awkward expression for the differential cross section corresponding to the potential. We follow a two step process to obtain a useful cross section: adopting the Lindhard, Nielsen and Scharff (LNS) approximations in order to generate a simpler form of the Molière cross section and then creating a simple, easy-to-use, fit to that approximate form. Within the framework of the LNS treatment of atomic cross sections, our fit is accurate to 6%. Simple forms for the total cross section and several related quantities are presented.     

Numerical evaluation of tensor Feynman integrals in Euclidean kinematics

For the investigation of higher order Feynman integrals, potentially with tensor structure, it is highly desirable to have numerical methods and automated tools for dedicated, but sufficiently ‘simple’ numerical approaches. We elaborate two algorithms for this purpose which may be applied in the Euclidean kinematical region and in d=4?2ε dimensions. One method uses Mellin–Barnes representations for the Feynman parameter representation of multi-loop Feynman integrals with arbitrary tensor rank. Our Mathematica package AMBRE has been extended for that purpose, and together with the packages MB (M. Czakon) or MBresolve (A.V. Smirnov and V.A. Smirnov) one may perform automatically a numerical evaluation of planar tensor Feynman integrals. Alternatively, one may apply sector decomposition to planar and non-planar multi-loop ε-expanded Feynman integrals with arbitrary tensor rank. We automatized the preparations of Feynman integrals for an immediate application of the package sector_decomposition (C. Bogner and S. Weinzierl) so that one has to give only a proper definition of propagators and numerators. The efficiency of the two implementations, based on Mellin–Barnes representations and sector decompositions, is compared. The computational packages are publicly available.     

Feynman formulas and path integrals for some evolution semigroups related to τ-quantization

This note is devoted to Feynman formulas (i.e., representations of semigroups by limits of n-fold iterated integrals as n → ∞) and their connections with phase space Feynman path integrals. Some pseudodifferential operators corresponding to different types of quantization of a quadratic Hamiltonian function are considered. Lagrangian and Hamiltonian Feynman formulas for semigroups generated by these operators are obtained. Further, a construction of Hamiltonian (phase space) Feynman path integrals is introduced. Due to this construction, the Hamiltonian Feynman formulas obtained here and in our previous papers do coincide with Hamiltonian Feynman path integrals. This connects phase space Feynman path integrals with some integrals with respect to probability measures. These connections enable us to make a contribution to the theory of phase space Feynman path integrals, to prove the existence of some of these integrals, and to study their properties by means of stochastic analysis. The Feynman path integrals thus obtained are different for different types of quantization. This makes it possible to distinguish the process of quantization in the language of Feynman path integrals.     

Inclusive Single- and Dijet Rates in Next-to-Leading Order QCD for γ*p and γ*γ Collisions

We present one- and two-jet inclusive cross sections for γ*γ scattering and virtual photoproduction in ep collisions. The hard cross sections are calculated in next-to-leading order QCD. Soft and collinear singularities are extracted using the phase-space-slicing method. The initial state singularity of the virtual photon depends logarithmically its’ virtuality. This logarithm is large and has to be absorbed into the parton distribution function of the virtual photon. We define for this purpose an factorization scheme similar to the real photon case. We numerically study the dependence of the inclusive cross sections on the transverse energies and rapidities of the outgoing jets and on the photon virtuality. The ratio of the resolved to the direct cross section in ep collisions is compared to ZEUS data.     

Feynman path integrals and the trace formula for the Schrödinger operators

We study Schrödinger operators of the form on ^d , whereA ² is a strictly positive symmetricd×d matrix andV(x) is a continuous real function which is the Fourier transform of a bounded measure. If _n are the eigenvalues ofH we show that the theta function is explicitly expressible in terms of infinite dimensional oscillatory integrals (Feynman path integrals) over the Hilbert space of closed trajectories. We use these explicit expressions to give the asymptotic behaviour of (t) for smallh in terms of classical periodic orbits, thus obtaining a trace formula for the Schrödinger operators. This then yields an asymptotic expansion of the spectrum ofH in terms of the periodic orbits of the corresponding classical mechanical system. These results extend to the physical case the recent work on Poisson and trace formulae for compact manifolds.Partially supported by the USP-Mathematisierung, University of Bielefeld (Forschungsprojekt Unendlich dimensionale Analysis)     

Exact solutions of Dyson–Schwinger equations for iterated one-loop integrals and propagator-coupling duality

The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar φ³ theory, from all nestings and chainings of a primitive self-energy subdivergence. Here we formulate the nonperturbative problems which these resummations approximate. For Yukawa theory, at spacetime dimension d=4, we obtain an integrodifferential Dyson–Schwinger equation and solve it parametrically in terms of the complementary error function. For the scalar theory, at d=6, the nonperturbative problem is more severe; we transform it to a nonlinear fourth-order differential equation. After intensive use of symbolic computation we find an algorithm that extends both perturbation series to 500 loops in 7 minutes. Finally, we establish the propagator–coupling duality underlying these achievements making use of the Hopf structure of Feynman diagrams.     

Novel semiconducting iron–quinizarin metal–organic framework for application in supercapacitors*

We present conductivity data for a newly synthesised metal–organic framework FeQ, Fe(C₁₄H₆O₄).H₂O demonstrating significant electronic transport. The electrical conductivity of the material is expected to be through the π–π interaction of the ligand-quinizarin and is measured to be 1.73?×?10^?2?Scm^?1_. Its potential role as supercapacitor electrodes is discussed.     

On the nonexistence of Liouvillian first integrals for generalized Liénard polynomial differential systems

We consider generalized Liénard polynomial differential systems of the form ? = y, ? = -g(x) - f (x) y, with f (x) and g(x) two polynomials satisfying deg(g) ≤ deg(f). In their work, Llibre and Valls have shown that, except in some particular cases, such systems have no Liouvillian first integral. In this letter, we give a direct and shorter proof of this result.     

BFKL resummation effects in γ*γ* → ρρ

We calculate the leading order BFKL amplitude for the exclusive diffractive process in the forward direction, which can be studied in future high energy e ⁺ e ^- linear colliders. The resummation effects are very large compared to the fixed-order calculation. We also estimate the next-to-leading logarithmic corrections to the amplitude by using a specific resummation of higher order effects and find a substantial growth with energy, but smaller than in the leading logarithmic approximation. CPHT Unité mixte 8627 du CNRS LPT Unité mixte 8627 du CNRS An erratum to this article is available at .     

Trace Anomalies and Chiral Ward Identities

In a simple Abelian spinor field theory, the canonical trace identities for certain axial-vector and axial-scalar operators are re-examined in dimensional regularization, some disagreements with previous results are found and an interesting new phenomenon is observed and briefly discussed。     

Ω-Expansion of van Kampen through functional integrals

We give a functional integral version of the van Kampen-expansion method for the master equation. Explicit expressions are given for the generating functional of correlation functions.     

First integrals of the discrete nonconservative and nonholonomic systems

In this paper we show that the first integrals of the discrete equation of motion for nonconservative and nonholonomic mechanical systems can be determined explicitly by investigating the invariance properties of the discrete Lagrangian. The result obtained is a discrete analogue of the generalized theorem of Noether in the Calculus of variations.     

Bilinear Identities and Hirota’s Bilinear Forms for the (γn, σk)-KP Hierarchy

In this paper, we discuss how to construct the bilinear identities for the wave functions of the (γ_n, σ_k)-KP hierarchy and its Hirota’s bilinear forms. First, based on the corresponding squared eigenfunction symmetry of the KP hierarchy, we prove that the wave functions of the (γ_n, σ_k)-KP hierarchy are equal to the bilinear identities given in Sec.3 by introducing N auxiliary parameters z_i, i = 1, 2,?…?, N. Next, we derived the bilinear equations for the tau-function of the (γ_n, σ_k)-KP hierarchy. Then, we obtain the bilinear equations for the taufunction of the mixed type of KP equation with self-consistent sources (KPESCS), which includes both the first and the second type of KPESCS as special cases by setting n = 2 and k = 3. Finally, using the relation between the Hirota bilinear derivatives and the usual partial derivatives, we show the procedure of translating the Hirota’s bilinear equations into the mixed type of KPESCS.     

D~*Dρ and B~*Bρ strong couplings in light-cone sum rules

We present an improved calculation of the strong coupling constants g_(D~*Dp) and g_(B~*Bp) in light-cone sum rules, including one-loop QCD corrections of leading power with meson distribution amplitudes. We further compute subleading-power corrections from two-particle and three-particle higher-twist contributions at leading order up to twist-4 accuracy. The next-to-leading order corrections to the leading power contribution numerically offset the subleading-power corrections to a certain extent, and our numerical results are consistent with those of previous studies on sum rules. A comparison between our results and existing model-dependent estimations is also made.     

Electron-hole transitions in multiply charged ions for precision laser spectroscopy and searching for variations in α

We consider transitions of electron holes (vacancies in otherwise filled shells of atomic systems) in multiply charged ions that, due to level crossing of the holes, have frequencies within the range of optical atomic clocks. Strong E1 transitions provide options for laser cooling and trapping, while narrow transitions can be used for high-precision spectroscopy and tests of fundamental physics. We show that hole transitions can have extremely high sensitivity to α variation and propose candidate transitions that have much larger α sensitivities than any previously seen in atomic systems.     

Is d* a candidate for a hexaquark-dominated exotic state?

We confirm our previous prediction of a d* state with I(JP) = 0(3+) [Phys. Rev. C 60, 045203(1999)] and report for the first time based on a microscopic calculation that d* has about 2/3 hidden color(CC)configurations and thus is a hexaquark-dominated exotic state. By performing a more elaborate dynamical coupledchannels investigation of the △△-CC system within the framework of the resonating group method(RGM) in a chiral quark model, we find that the d* state has a mass of about 2.38–2.42 Ge V, a root-mean-square radius(RMS) of0.76–0.88 fm, and a CC fraction of 66%–68%. The last may cause a rather narrow width for the d* which, together with the quantum numbers and our calculated mass, is consistent with the newly observed resonance-like structure(M ≈2380 Me V, Γ≈70 Me V) in double-pionic fusion reactions reported by the WASA-at-COSY Collaboration.