Photon-electron scattering taking vacuum energy into account

We deal with photon-electron scattering between the two uncharged conducting parallel plates. The electromagnetic vacuum field between the two plates is defined by the configuration of space and also interacts with the electrons. We first deal with the relevant operators for the electron and photon fields and with the Feynman propagator. We compute theS-matrix for photon-electron scattering, taking into account the influence of the vacuum field. The computed photon-electron scattering cross section also manifests the influence of the vacuum field. We give an example for low-energy scattering of the influence of the vacuum field upon the scattering cross section.     

Scattering with vacuum field taken into account

We present the calculation of the photon-electron scattering by taking into account both the static and the dynamic vacuum fields. The vacuum field is represented by the configuration of the space between two uncharged conducting parallel plates which can be at rest (static vacuum field) or moving from each other at a constant velocity (dynamic vacuum field). Correction terms in the computedS-matrix and scattering cross section manifest the influence of the static and dynamic vacuum field.     

Large-angle scattering of heavy ions: (I). General structure of elastic cross section

Closed-form expressions are derived for the differential cross section of elastic heavy-ion scattering at large angles. The derivation is based on the general form of the elastic partial-wave S-matrix in real l-space. By a generalization of analytic techniques developed in earlier work, it is shown that the large-angle scattering cross section has a universal structure involving combinations of Bessel functions and the Fourier transforms of the rapidly varying parts of the S-matrix, irrespective of their dynamical origin. Anomalous large-angle scattering is attributed to deviations of the S-matrix from its “normal strong-absorption profile”, and general conditions for backward-angle enhancement are given. Our model-independent formulation provides the framework for an “inductive” method of analyzing experimental angular distributions and excitation functions aimed at identifying, as uniquely as possible, the dynamical mechanisms that operate in large-angle heavy-ion scattering. Extensions of the formalism to inelastic scattering and transfer reactions, and applications of the analytic method, will be described in subsequent papers.     

Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half-spaces

Abstract

The small-slope approximation (SSA) for wave scattering at the rough interface of two homogeneous half-spaces is developed. This method bridges the gap between two classical approaches to the problem: the method of small perturbations and the Kirchhoff (or quasi-classical) approximation. In contrast to these theories, the SSA is applicable irrespective of the wavelength of radiation, provided that the slopes of roughness are small compared with the angles of incidence and scattering.

The resulting expressions for the SSA are given for the entries of an S-matrix that represents the scattering amplitudes of plane waves of different polarizations interacting with the rough boundary. These formulae are quite general and are valid, in fact, for waves of different origins. Apart from the shape of the boundary, some functions in these formulae are coefficients of the expansion of the S-matrix into a power series in terms of elevations. These roughness independent functions are determined by a specific scattering problem. In this paper they are calculated for the case of electromagnetic scattering at the interface of two dielectric half-spaces. In contrast to an earlier paper by the author, where only the formulae for the reflected field were presented, in this paper both reflected and transmitted fields are considered in detail.

The a priori symmetry relations that this scattering problem should obey (reciprocity and energy conservation) are formulated in terms of the S-matrix.

The statistical moments of scattering amplitudes are directly related to the mean-reflection coefficient and scattering cross sections, which are usually determined experimentally. The corresponding formulae are given here for the case of Gaussian space-homogeneous statistics of roughness.     

Interference between diffractive,refractive, and Coulomb effects in the cross sections for the elastic scattering of light nuclei by nuclei at intermediate energies

Patterns typical of Fraunhofer and Fresnel diffractive scattering and of rainbow scattering are considered for elastic nucleus-nucleus scattering in the region of intermediate energies. The interference contributions to the respective differential cross section that are responsible for the formation of these patterns are singled out on the basis of the S-matrix approach.     

Topics in the S-matrix theory of massless particles

We discuss how massless particle reactions may be incorporated into standard S-matrix theory. The crucial element for doing so is a low-energy zero. Examples of reactions where such zeros occur are weak interaction processes involving neutrinos, chirally symmetric massless pion scattering, and two-photon exchange between neutral systems. These zeros make two-body unitarity a good approximation for sufficiently low energy despite the coalescence of multiparticle thresholds. Through two-body unitarity, these zeros produce lines of zeros in the absorptive parts and double spectral functions. These lines of zeros are the S-matrix analog of the requirement of an infrared finite field theory. Not only do they produce finite total cross sections at finite energies, but they also allow both upper and lower bounds to be derived for these cross sections at high energies. This upper bound is our main result. If a plausible smoothness assumption is made, we find σ_tot <s^? (where ? is arbitrarily small). In particular, the experimentally observed linear rise of the neutrino proton cross section cannot continue indefinitely.     

Dynamical effects of a one-dimensional multibarrier potential of finite range

We discuss the properties of a large number N of one-dimensional (bounded) locally periodic potential barriers in a finite interval. We show that the transmission coefficient, the scattering cross section σ, and the resonances of σ depend sensitively upon the ratio of the total spacing to the total barrier width. We also show that a time dependent wave packet passing through the system of potential barriers rapidly spreads and deforms, a criterion suggested by Zaslavsky for chaotic behaviour. Computing the spectrum by imposing (large) periodic boundary conditions we find a Wigner type distribution. We investigate also the S-matrix poles; many resonances occur for certain values of the relative spacing between the barriers in the potential. Received 1st August 2001 and Received in final form 18 November 2001     

Complete Analysis of the ηN S-Wave Scattering-Length Values and Its Natural Limitations in Any Single-Resonance Model

We show that the S-wave ηN scattering length can be extracted in a model-independent way only if the restricting assumption is adopted that one resonance describes the behaviour of the S-wave entirely. To obtain the ηN S-wave scattering length in a model-independent way one needs only two wellknown quantities; the πN elastic S-wave T-matrix at the η-production threshold, and the near threshold π ⁻ p →ηn total cross section slope. The results are independent of the particular parametrization of the elastic πN S-wave T-matrix and of the number of channels used. These assumptions are more general than the assumptions of the existing single-resonance models employed up to now for extracting the ηN S-wave scattering length. We show that the ηN S-wave scattering length value of other single-resonance models agrees with the model-independent estimate if the input data agree with the commonly accepted values. The existence of the upper limit of the real part and the almost fixed value of the imaginary part of the ηN S-wave scattering length are demonstrated, and the nature of limitations originating directly from first principles is explained as the reduction of the full model to the single resonance in the S-wave. The expected differences from the single-resonance estimate for models that are more general is shown. A simple criterion for recognizing the importance of parts of the model that are added in addition to the single resonance in the S-wave is given as well. Received May 15, 1995; revised September 14, 1995; accepted for publication October 11, 1995     

Extension of Haag’s theorem in the case of the lorentz invariant noncommunitative quantum field theory in a space with arbitrary dimension

The generalized Haag theorem was proven in SO(1, k) invariant quantum field theory. Apart from the k + 1 variables, an arbitrary number of additional coordinates, including noncommutative ones, can occur in the theory. In SO(1, k) invariant theory new corollaries of the generalized Haag theorem are obtained. It has been proven that the equality of four-point Wightman functions in the two theories leads to the equality of elastic scattering amplitudes and thus to the equality of the total cross sections in these theories. It was also shown that at k > 3 the equality of (k + 1) point Wightman functions in the two theories leads to the equality of the scattering amplitudes of some inelastic processes. In the SO(1, 1) invariant theory it was proven that if in one of the theories under consideration the S-matrix is equal to unity, then in another theory the S-matrix equals unity as well.     

Evanescent modes in a multiple scattering factorization

We discuss differences between the exactS-matrix for scattering on serial structures and a known factorized expression constructed of single-elementS-matrices. As an illustration, we use an exactly solvable model of a quantum wire with two point impurities. The work has been partially supported by the Grants AS No. 148409 and GA CR No. 202-93-1314.     

Extraction of Astrophysical Cross Sectionsin the Trojan-Horse Method

 The Trojan-horse method has been proposed to extract S-matrix elements of a two-body reaction at astrophysical energies from a related reaction with three particles in the final state. This should be useful in cases where the direct measurement of the two-body reaction at the necessary low energies is experimentally difficult. The formalism of the Trojan-horse method for nuclear reactions is developed in detail from basic scattering theory including spin degrees of freedom of the nuclei and we specify the necessary approximations. The energy dependence of the three-body reaction is determined by characteristic functions that represent the theoretical ingredients for the method. In a plane-wave Born approximation of the T-matrix the differential cross section assumes a simple structure. Received August 31, 1999; revised June 14, 2000; accepted for publication June 30, 2000     

Correlation between zeroes and poles ofS matrix for complex potentials

Using several illustrative examples, the nature of resonance poles and the corresponding zeroes of the s-waveS matrix is examined for several potentials having an absorptive pocket followed by a barrier. It is shown that even though the presence of absorption practically suppresses the manifestation of resonance in the elastic scattering cross section, the effect of the resonances generated by the absorptive pocket is more clearly manifested in the absorption cross section provided the barrier width is not too large. We further find that the signature of barrier top resonances are also more clearly manifested in the absorption cross section rather than in the elastic scattering cross section. These results have been interpreted in terms of complex resonance poles and corresponding zeroes of theS matrix. This implies that in complex potential scattering like heavy ion collisions, the reaction channel cross section peak is a more reliable signature of resonance phenomenon than the variation of the elastic channel cross section with energy.     

Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models

The general properties of the factorized S-matrix in two-dimensional space-time are considered. The relation between the factorization property of the scattering theory and the infinite number of conservation laws of the underlying field theory is discussed. The factorization of the total S-matrix is shown to impose hard restrictions on two-particle matrix elements: they should satisfy special identities, the so-called factorization equations. The general solution of the unitarity, crossing and factorization equations is found for the S-matrices having isotopic O(N)-symmetry. The solution turns out to have different properties for the cases N = 2 and N 3. For N = 2 the general solution depends on one parameter (of coupling constant type), whereas the solution for N 3 has no parameters but depends analytically on N. The solution for N = 2 is shown to be an exact soliton S-matrix of the sine-Gordon model (equivalently the massive Thirring model). The total S-matrix of the model is constructed. In the case of N 3 there are two “minimum” solutions, i.e., those having a minimum set of singularities. One of them is shown to be an exact S matrix of the quantum O(N)-symmetric nonlinear σ-model, the other is argued to describe the scattering of elementary particles of the Gross-Neveu model.     

A remark on asymptotic completeness of local fields

Assuming the existence of an asymptotically complete Wightman field with non-trivialS-matrix, we construct a local field such that the Haag-Ruelle scattering theory applied to this field leads to.     

On the Connection between Particles and Poles of the S-Matrix

In axiomatic S-matrix theory it is usually assumed that stable particles give rise to simple poles of the S-matrix for real negative energies while unstable particles give rise to poles close to the real axis on an unphysical sheet of the energy Riemann surface. The stable particle — pole association has been known for a long time not to be always true. For example in potential scattering what is relevant in this case in fact is not the S-matrix but the Jost function. The zeroes of this function for real negative energies are in fact in one-to-one correspondence with the bound states, while the correspondence may break down for the poles of the S-matrix. On the other hand it has recently been pointed out that there also is in general no connection between unstable particles and poles of the S-matrix.     

Construction ofS-matrix from scattering data

A method of construction of the scatteringS-matrix from scattering data is proposed. The scattering matrix is expressed in the form of a rational fraction and takes fully into account the analytic properties of theS-matrix. The method generates a unique and stable analytic continuation of theS-matrix into the complex energy plane. The method is applied to calculation of energy and widths of several resonances in nuclear and atomic physics. Its efficiency is compared with some recently proposed methods.     

Virtual black holes in generalized dilaton theories

The virtual black hole phenomenon, which has been observed previously in specific models, is established for generic 2D dilaton gravity theories with scalar matter. The ensuing effective line element can become asymptotically flat only for two classes of models; among them spherically reduced theories and the string inspired dilaton black hole. We present simple expressions for the lowest order scalar field vertices of the effective theory which one obtains after integrating out geometry exactly. Treating the boundary in a natural and simple way, asymptotic states, tree-level vertices and the tree-level S-matrix are conformally invariant. Examples are provided pinpointing the physical consequences of virtual black holes on the (CPT-invariant) S-matrix for gravitational scattering of scalar particles. For minimally coupled scalars the evaluation of the S-matrix in closed form is straightforward. For a class of theories including the string inspired dilation black hole all tree-graph vertices vanish, which explains the particular simplicity of that model and at the same time shows yet another essential difference to the Schwarzschild case.Received: 7 August 2002, Revised: 2 June 2003, Published online: 11 July 2003D.V. Vassilevich: On leave from V. Fock Institute of Physics, St. Petersburg University, 198904 St. Petersburg, Russia     

Exact two-particle S-matrix of quantum sine-Gordon solitons

The exact and explicit formulas for the quantumS-matrix elements of the soliton-antisoliton scattering which satisfy unitarity and crossing conditions and have correct analytical properties are constructed. ThisS-matrix is in agreement with the massive Thirring model perturbation theory and with the semiclassical sine-Gordon results.     

On the solution of a “2D Coulomb + Aharonov-Bohm” problem: oscillator strengths in the discrete spectrum and scattering

In this paper we present an exact analytic solution of the Schrödinger equation both inthe discrete and continuous spectra for the combination of a 2D Coulomb potential and theAharonov-Bohm flux. We analyze the influence of the Aharonov-Bohm flux on the energyspectrum of such a system and show that its presence leads to the broadening of theelectron density in the bound states with the given value of the principal quantum number.We have shown that the scattering phase shift, which determines theS-matrix, can be represented as a sum of the Aharonov-Bohm scatteringphase, first obtained by Henneberger, and a “modified” 2D Coulomb phase. We have noticed,that the Aharonov-Bohm scattering phase has a full analogy with the “quantum defect” forsuch a system. We have shown also, that the presence of the Aharonov-Bohm flux affects theradiation spectrum of the electron in this case, and this fact is demonstrated bycalculations of the corresponding oscillator strengths. The explicit analytic expressionfor the scattering cross section on such a system is found in the frame of the eikonalapproach. Obtained formula contains the two exact limiting cases, namely, the “pure” 2DCoulomb scattering as well as the “pure” Aharonov-Bohm effect. The mutual influence of a2D Coulomb potential and the Aharonov-Bohm flux is also discussed.     

Quantum interference and statistical fluctuations in deep-inelastic heavy-ion reactions

Starting from the general multi-channel formulation of the scattering matrix we perform a coarse-grain average of the cross section appropriate for the observed quantities in heavy-ion reactions. We keep full track of the geometrical factors for the coupling of partial waves and channel spin. Relating the diagonal part of the S-matrix product semiclassically to the statistical treatment in terms of phase-space trajectories we are dealing with quantum fluctuations and statistical fluctuations on an even basis. Quantum corrections play a minor role if the underlying statistical model takes effects of the nonthermalized relative motion properly into account. This is true even for small energy losses.