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     

The Gell-Mann-Goldberger formula for long-range potential scattering

A derivation of the Gell-Mann-Goldberger (GG) formula and cut-off versions of this formula for the T-matrix involving long-range potentials is given. The derivation is based on the time-dependent and recently developed stationary formalisms for scattering via long-range potentials. A stationary S-operator expression for two-body Coulomb-like scattering is derived. Using the well-known expression for the off-energy-shell “T-matrix” for a pure Coulomb potential the energy-shell limit of this stationary expression is shown to yield the pure Coulomb scattering amplitude. A proof of the convergence of the perturbation series corresponding to the Gell-Mann-Goldberger formula for the two-body Coulomb-like T-matrix is given.     

A spin-isospin-dependent 3N scattering formalism in a 3D Faddeev scheme

We introduce a spin-isospin-dependent three-dimensional approach for the formulation of the three-nucleon scattering. The Faddeev equation is expressed in terms of vector Jacobi momenta and spin-isospin quantum numbers of each nucleon. Our formalism is based on connecting the transition amplitude T to momentum-helicity representations of the two-body t -matrix and the deuteron wave function. Finally, the expressions for nucleon-deuteron elastic scattering and full breakup process amplitudes are presented.     

Theory of the Trojan-Horse method

The Trojan-Horse method is an indirect approach to determine the energy dependence of S factors of astrophysically relevant two-body reactions. This is accomplished by studying closely related three-body reactions under quasi-free scattering conditions. The basic theory of the Trojan-Horse method is developed starting from a post-form distorted wave Born approximation of the T-matrix element. In the surface approximation the cross-section of the three-body reaction can be related to the S-matrix elements of the two-body reaction. The essential feature of the Trojan-Horse method is the effective suppression of the Coulomb barrier at low energies for the astrophysical reaction leading to finite cross-sections at the threshold of the two-body reaction. In a modified plane wave approximation the relation between the two- and three-body cross-sections becomes very transparent. The appearing Trojan-Horse integrals are studied in detail.     

The Bethe-Goldstone equation with singular interactions: A fully off-shell solution for the boundary condition model

The mutual interaction of a pair of fermions imbedded in a many-body system of identical particles when they are excited out of the filled Fermi sea, is studied via the T-matrix or transition amplitude specified by the Bethe-Goldstone (BG) equation. The role of the bare two-body interaction is emphasised, and in particular the consequences are elucidated of whether the potential is “well-behaved” (nonsingular) or not. The properties of the BG T-matrix, including generalized orthonormality and completeness relations, are derived both for nonsingular potentials and for singular potentials containing an infinite hard core. General analytic properties are exploited to derive relations that express the fully off-shell BG T-matrix purely in terms of the half-shell amplitude (and the properties of any possible bound states in the medium). The general formalism is illustrated by deriving exact analytic expressions for the fully off-shell BG T-matrices for a pair of particles with equal and opposite momenta interacting via either of two singular model interactions; namely, the pure hard-core interaction and the boundary condition model. Results for both models are expressed in terms of the solution to a simple one-dimensional Fredholm integral equation. The analytic properties of the solutions are discussed and exploited to prove both their uniqueness and that they satisfy the various general relations derived. To our knowledge, these results represent the first exact nontrivial solution to the fully off-shell BG equation for any local potential, or singular limiting case thereof.     

Low-energy elastic scattemng of pions by the deuteron

The problem of the elastic scattering of pions by a deuteron is considered using the separable representation of the two-body t-matrix. The Faddeev equations are reduced to a set of one-dimensional integral equations by separating the angular variables. The dependence of the π-d scattering length on the form of two-body interaction and on the values of the π-N scattering lengths is studied in the case of a one-term nonlocal potential with separable variables. The π-d scattering length proves to be practically independent of the two-body interaction form, and is essentially dependent on the values of the π-N scattering lengths.     

Total S-matrix and adiabatic amplitudes for the three-body problem

The three-body quantum scattering problem reduced by the expansion of the wavefunction over the specially constructed basis to a two-body problem is considered. The asymptotics of this basis, as well as the solutions of the effective two-body equations are derived. A total S-matrix for 2 (2, 3) processes is expressed in terms of adiabatic amplitudes and vice versa.     

Localized excitation transport on substitutionally disordered lattices

Average-T-matrix and coherent medium theories are used to study the motion of localized excitations on Substitutionally disordered lattices. We derive equations which relate coherent medium results for bond and site averaging and show how these reduce to the two-body solution results of Gochanour, Andersen, and Fayer. Numerical results forP ₀(t), the probability of remaining at the origin for two-dimensional nearest-neighbor lattices are presented.Supported in part by the National Science Foundation. (CHE81-00407).     

Light front approach to correlations in hot quark matter

We investigate two-quark correlations in hot and dense quark matter. To this end we use the light front field theory extended to finite temperature T and chemical potential μ. Therefore it is necessary to develop quantum statistics formulated on the light front plane. As a test case for light front quantization at finite T and μ we consider the NJL model. The solution of the in-medium gap equation leads to a constituent quark mass which depends on T and μ. Two-quark systems are considered in the pionic and diquark channel. We compute the masses of the two-body system using a T-matrix approach.     

Derivation of extended boundary condition method from general definition of -matrix elements and Lippman–Schwinger equation for transition operator

Waterman's surface-integral expressions for the T-matrix elements are derived on the basis of the quantum-mechanical potential scattering approach in electromagnetic scattering problem. We use general definition of the elements of the T-matrix as the matrix elements of the dyadic transition operator and Lippman–Schwinger volume integral equation for the dyadic transition operator. The interrelations of the Q- and Waterman's T-matrix with the transition operator are shown.     

Thermodynamic consistency for nuclear matter calculations

We investigate the relation between the binding energy and the Fermi energy and between different expressions for the pressure in cold nuclear matter. For a self-consistent calculation based on a Φ derivable T-matrix approximation with off-shell propagators, the thermodynamic relations are well satisfied unlike for a G-matrix or a T-matrix approach using quasi-particle propagators in the ladder diagrams. Received: 8 February 2001 / Accepted: 11 June 2001     

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.     

T-matrix analysis of biexcitonic correlations in the nonlinear optical response of semiconductor quantum wells

A detailed numerical analysis of exciton-exciton interactions in semiconductor quantum wells is presented. The theory is based on the dynamics-controlled truncation formalism and evaluated for the case of resonant excitation of 1s-heavy-hole excitons. It is formulated in terms of standard concepts of scattering theory, such as the forward-scattering amplitude (or T-matrix). The numerical diagonalization of the exciton-exciton interaction matrix in the 1s-approximation yields the excitonic T-matrix. We discuss the role of the direct and exchange interaction in the effective two-exciton Hamiltonian, which determines the T-matrix, evaluated within the 1s-subspace, and also analyze the effects of the excitonic wave function overlap matrix. Inclusion of the latter is shown to effectively prevent the 1s-approximation from making the Hamiltonian non-hermitian, but a critical discussion shows that other artefacts may be avoided by not including the overlap matrix. We also present a detailed analysis of the correspondence between the excitonic T-matrix in the 1s-approximation and the well-known T-matrix governing two-particle interactions in two dimensional systems via short-range potentials. Received 3 August 2001 and Received in final form 26 December 2001     

Mean-field T-matrix approach to elastic wave scattering by small and point-like objects

The T-matrix of a small inclusion embedded in a homogeneous matrix is calculated for vector elastic waves. The theory relies on an approximation of the local stress and momentum inside the inclusion by volumetric averages, which allows the summation of the full multiple-scattering series. The limiting case of a point scatterer is discussed. The mean-field T-matrix and its point limit both satisfy the elastodynamic version of the optical theorem. The accuracy of the results is discussed by comparison with some exact solutions for spheres. In particular, the mean-field T-matrix allows the approximate modeling of low-frequency resonances by small, high-contrast objects. The present theory could be applied to the multiple scattering of elastic waves by a collection of small or fuzzy resonant scatterers.     

Modified T-matrix formulation for multiple scattering of acoustic waves

Abstract

In studying the multiple scattering of acoustic waves by distributed discrete scatterers, the ‘two-exterior’ T-matrix or the modified T-matrix is needed. In this paper, the modified T-matrix formulae for a scatterer of arbitrary shape are derived, based on Huygen's principle and the method of optimal truncation (MOOT), respectively. Analytical expressions are given for the modified T-matrix elements of a spheroid in the low-frequency limit. The agreement with the existing results is shown to be exact to the given order of ka. It is also shown that the results based on Huygen's principle are merely the special cases of those based on MOOT.     

Multiple scattering theory in condensed materials

Second-order elliptic differential equations (such as the time-independent single particle Schrödinger equation) may be solved in a finite closed disjoint region of space independently of the rest of space. The solution in all space may then be determined by solving the equations in the exterior region together with boundary conditions at the junction of the two regions. These boundary conditions are determined by the previously found interior solution. This means that such regions may be taken as ‘black boxes’ whose exact details do not matter. The simplest example of this is phase-shift scattering theory from a single scatterer where all the scattering properties are described by the phase shifts, and the exact details of the scattering potential are unimportant. In a macroscopic condensed system, however, there are many core regions and one is really concerned with the multiple scattering which takes place between these different scattering centres. Much of this article is devoted to investigating the formal properties of scattering theory when there are many non-overlapping spherical regions of radius R _M, each of which is described by its own scattering matrix, or, equivalently for a spherically symmetric potential, by its phase shifts. Non-spherically symmetric and spin-dependent potentials are permitted, but for simplicity we assume initially that the interstitial region between each disjoint scattering region has zero potential. The generalization of the multiple scattering formalism for non-zero interstitial potential is also given at a later stage.

It is shown that in such a system a generalized T-matrix may be defined which describes the radiation from one of the core regions when another one has been excited. It is then a many channel T-matrix in which the channels are the different disjoint scattering regions. It is shown that the formal properties of this T matrix are the same as for a normal T matrix. In § 2 we review the properties of ordinary scattering theory, and then in § 3 we show that analogous properties for the generalized T matrix hold. An exact expression for the density of particle eigenstates is derived in terms of the positions and scattering matrices of the individual scattering centres. This expression reduces to the standard KKR band structure equation for the infinite regular lattice. We also consider how to construct the density of eigenstates and the charge density for such a system. These latter quantities may be approached in two different ways: the usual way is to consider the scattering material to occupy all space, but from a multiple scattering viewpoint one must consider the total volume of condensed material to be small compared with all space, even if both limit to infinity. It is not obvious that the latter method leads to the same results as the former (formally the density of eigenvalues is identical to the free electron density of eigenvalues in the latter method) and it is shown how the differences in the two approaches are resolved. We also discuss the expansion of some of these results for a perfect lattice. While the usual expansions are pseudo-potential expansions, a manifestly ‘on-energy shell’ expansion is derived which does not contain the arbitrary parameters of the pseudo-potential expansions. Finally, in § 4, we review the most significant contributions of other authors to the theory of multiple scattering.     

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.     

Ground state pressure and energy density of an interacting homogeneous Bose gas in two dimensions

We consider an interacting homogeneous Bose gas at zero temperature in two spatial dimensions. The properties of the system can be calculated as an expansion in powers of g, where g is the coupling constant. We calculate the ground state pressure and the ground state energy density to second order in the quantum loop expansion. The renormalization group is used to sum up leading and subleading logarithms from all orders in perturbation theory. In the dilute limit, the renormalization group improved pressure and energy density are expansions in powers of the T ^2B and T ^2Bln(T ^2B), respectively, where T ^2B is the two-body T-matrix. Received 19 April 2002 Published online 13 August 2002