3N scattering in a three-dimensional operator formulation

A recently developed formulation for a direct treatment of the equations for two- and three-nucleon bound states as set of coupled equations of scalar functions depending only on vector momenta is extended to three-nucleon scattering. Starting from the spin-momentum dependence occurring as scalar products in two- and three-nucleon forces together with other scalar functions, we present the Faddeev multiple scattering series in which order by order the spin degrees can be treated analytically leading to 3D integrations over scalar functions depending on momentum vectors only. Such formulation is especially important in view of awaiting extension of 3N Faddeev calculations to projectile energies above the pion production threshold and applications of chiral perturbation theory 3N forces, which are to be most efficiently treated directly in such three-dimensional formulation without having to expand these forces into a partial-wave basis.     

A Three-Dimensional Treatment of the Three-Nucleon Bound State

Recently a formalism for a direct treatment of the Faddeev equation for the three-nucleon bound state in three dimensions has been proposed. It relies on an operator representation of the Faddeev component in the momentum space and leads to a finite set of coupled equations for scalar functions which depend only on three variables. In this paper we provide further elements of this formalism and show the first numerical results for chiral NNLO nuclear forces.     

Momentum space 3N Faddeev calculations of hadronic and electromagnetic reactions with proton-proton Coulomb and three-nucleon forces included

We extend our approach to incorporate the proton-proton (pp) Coulomb force into the three-nucleon (3N) momentum space Faddeev calculations of elastic proton-deuteron (pd) scattering and breakup to the case when also a three-nucleon force (3NF) is acting. In addition, we formulate that approach in the application to electron- and g \gamma -induced reactions on ³He . The main new ingredient is a 3-dimensional screened pp Coulomb t -matrix obtained by a numerical solution of a 3-dimensional Lippmann-Schwinger equation (LSE). The resulting equations have the same structure as the Faddeev equations which describe pd scattering without 3NF acting. That shows the practical feasibility of both presented formulations.     

A bridge between hyperspherical and integro-differential approaches to the many-body bound states

The solution of the Schrödinger equation can be obtained from the one of a system of coupled differential equations generated from the potential harmonic expansion of the bound-state wave function of a system of identical particles governed by two-body central interactions. It is shown that the system of coupled equations can be transformed into an equivalent integro-differential equation. For three bosons inS states this equation is identical to the Faddeev equation as written by Noyes. The integro-differential equations describing the triton for non-central realisticN-N forces are explicitly given.Laboratoire associé au C.N.R.S.     

Slow neutron scattering parameters for stable nuclei with nucleon numbers betweenA=79 andA=96

The relation between the method of coupled channels for rearrangement reactions (CRC) and the bound state approximation to the channel coupling array formalism (BSCCA), advocated in recent years, is investigated in detail for a simple 3-body system expressed in terms of truncated component wavefunctions of Faddeev type. The system is described by coupled differential or integral equations that are truncated into a model space of strongly-coupled channels. It is shown that CRC can be derived from the truncated coupled equations, either in differential or integral form, provided care is taken to use the entire model space. The corresponding BSCCA in this model space can be obtained from a restrictive condition on the integral from of the coupled equations, while it cannot be obtained consistently from the differential form of the coupled equations. The boundary conditions for the component functions are discussed in detail.     

On the possibility of an analytic solution of the three-body problem

Three-body Faddeev equations in the Noyes-Fiedeldey form are rewritten as a matrix analog of a one-dimensional nonrelativistic Schrödinger equation. Unlike the method of K-harmonics, where a similar equation was obtained by expansion of a three-body Schrödinger equation wavefunction into the orthogonal set of functions of two variables (K-harmonics), the use of the Noyes-Fiedeldey form of Faddeev equations allows us to limit ourselves to the expansion in functions of one variable only. The solutions of the above mentioned matrix equation are obtained. These solutions converge uniformly within every interval of continuity of the matrix, which corresponds to the potential of that equation. Their asymptotic behavior for large interparticle distances is discussed. The solutions for the harmonic oscillator, inverse-square, and Coulomb-Kepler potentials are found. It is shown that energy levels in the last case may be calculated from a simple formula which is very similar to the corresponding formula for the two-body Coulomb-Kepler problem. This formula can be easily generalized to the case of n particles interacting with inverse distance potentials.     

Treatment of Confinement in Three-Quark Calculations

 The treatment of confining interactions in non-relativistic three-quark systems is addressed. Usually in the Faddeev equations the Faddeev components are coupled by the total potential. In the new treatment proposed here the Faddeev components are coupled only by the non-confining short-range part of the potential, thus allowing its channel-by-channel investigation. The convergence in angular-momentum channels is much faster. Received August 28, 1998; accepted for publication October 16, 1998     

Elastic pion-deuteron scattering in the resonance region as a three-body problem

We study elastic pion-deuteron scattering in the Δ(1236) energy region by means of the three-body Faddeev equations. We present a compact angular momentum reduction of the Faddeev integral equation for separable amplitudes, neglecting the nucleon spin, and solve the resulting coupled integral equations. We examine the dependence of the elastic scattering amplitude on the deuteron structure, on the pion-nucleon scattering amplitude, and on the various orders of multiple scattering. The differential cross section is very sensitive to multiple scattering effects at backward angles. We find that a number of conventional approximations do not well reproduce these multiple scattering effects in the resonance region.     

A iterative method of bound state energy calculation

An iterative method of solution of the Lippmann-Schwinger-type equation is proposed. This method is applicable to the bound state energy calculations of strongly coupled particles and can also be applied to the Faddeev equations. The convergence of this method is tested on the Lippmann-Schwinger equation.     

Effective Action for Scalar Fields in Two-Dimensional Gravity

We consider a general two-dimensional gravity model minimally or nonminimally coupled to a scalar field. The canonical form of the model is elucidated, and a general solution of the equations of motion in the massless case is reviewed. In the presence of a scalar field all geometric fields (zweibein and Lorentz connection) are excluded from the model by solving exactly their Hamiltonian equations of motion. In this way the effective equations of motion and the corresponding effective action for a scalar field are obtained. It is written in a Minkowskian space-time and does not include any geometric variables. The effective action arises as a boundary term and is nontrivial both for open and closed universes. The reason is that unphysical degrees of freedom cannot be compactly supported because they must satisfy the constraint equation. As an example we consider spherically reduced gravity minimally coupled to a massless scalar field. The effective action is used to reproduce the Fisher and Roberts solutions.     

Faddeev approach to the three-body problem in total-angular-momentum representation

For a system of three charged particles the Faddeev equations are derived in the total-angular-momentum representation. They have the form of coupled sets of partial differential equations in three-dimensional space and can be used to develop new efficient numerical procedures to tackle the three-body Coulomb problem. The asymptotic conditions at large distances corresponding both to binary scattering and bound-state problems are presented. The behaviour of the Faddeev components near the triple and double collision points is studied.     

Asymptotic method for determining the amplitude for three-particle breakup: Neutron-deuteron scattering

The process of neutron-deuteron scattering at energies above the deuteron-breakup threshold is described within the three-body formalism of Faddeev equations. Use is made of the method of solving Faddeev equations in configuration space on the basis of expanding wave-function components in the asymptotic region in bases of eigenfunctions of specially chosen operators. Asymptotically, wave-function components are represented in the form of an expansion in an orthonormalized basis of functions depending on the hyperangle. This basis makes it possible to orthogonalize the contributions of elastic-scattering and breakup channels. The proposed method permits determining scattering and breakup parameters from the asymptotic representation of the wave function without reconstructing it over the entire configuration space. The scattering and breakup amplitudes for states of total spin S = 1/2 and 3/2 were obtained for the s-wave Faddeev equation.     

Nucleon-nucleon interaction in the three-nucleon system

The three-nucleon system is reconsidered. The Faddeev equations are given leading to a set of integral equations. Solving these integral equations, suitable forms are considered for the nucleon-nucleon interaction. In the bound state of three-nucleon system, the form of the nuclear forces from the nucleon-nucleon interaction is important. In the present calculations, we consider the nuclear forces resulting from the nucleon-nucleon interaction by the exchange of a scalar meson, a pseudoscalar meson, and a massless vector meson. With this different meson exchange nucleon-nucleon interaction, the binding energy of the three-nucleon system is calculated by solving the Faddeev integral equations giving a value of 8.452 MeV.     

The hyperspherical-harmonics expansion method and the integral-equation approach to solving the few-body problem in momentum space

The Faddeev and Faddeev-Yakubovsky equations for three- and four-body systems are solved by applying the hyperspherical-harmonics expansion to them in momentum space. This coupling of two popular approaches to the few-body problem together with the use of the so-called Raynal-Revai transformation, which relates hyperspherical functions, allows the few-body equations to be written as one-dimensional coupled integral equations. Numerical solutions for these are achieved through standard matrix methods; these are made straightforward, because a second transformation renders potential multipoles easily calculable. For sample potentials and a restricted size of matrix in each case, the binding energies extracted match those previously obtained in solving the Schrödinger equation through the hyperspherical-harmonics expansion in coordinate space.Work supported in part by the National Science Foundation through grant No. PHY83-06584 and grant No. PHY87-12229     

Coupled channel T-operator equations with connected kernels: (I). Three-body problem with pairwise interactions

Previous work on T-operator coupled equations for two-channel systems is generalized and applied to the problem of three bodies interacting via pair potentials. Sets of coupled, integral equations for the two-body arrangement channel T-operators are derived using a channel coupling array W, and the connectedness properties of the kernels of these equations are discussed. It is shown that either disconnected or connected (iterated) kernels can be obtained by various choices of W. One particular realization of the coupled equations is seen to be similar but not identical to the Lovelace form of the Faddeev equations. Since the matrix form of the coupled equations is similar to the one-body Lippmann-Schwinger equation, the introduction of Møller wave operators is straightforward, and these are used to derive coupled integral equations for the channel state vectors.     

Axisymmetric wave propagation of thermoelastic transversely isotropic half-space under buried loading using potential functions

By virtue of a new scalar potential function and Hankel integral transforms, the wave propagation analysis of a thermoelastic transversely isotropic half-space is presented under buried loading and heat flux. The governing equations of the problem are the differential equations of motion and the energy equation of the coupled thermoelasticity theory. Using a scalar potential function, these coupled equations have been uncoupled and a six-order partial differential equation governing the potential function is received. The displacements, temperature, and stress components are obtained in terms of this potential function in cylindrical coordinate system. Applying the Hankel integral transform to suppress the radial variable, the governing equation for potential function is reduced to a six-order ordinary differential equation with respect to z. Solving that equation, the potential function and therefore displacements, temperature, and stresses are derived in the Hankel transformed domain for two regions. Using inversion of Hankel transform, these functions can be obtained in the real domain. The integrals of inversion Hankel transform are calculated numerically via Mathematica software. Our numerical results for displacement and temperature are calculated for surface excitations and compared with the results reported in the literature and a very good agreement is achieved.     

Bound-State Solutions of the Klein-Gordon Equation for the Generalized <Emphasis Type="Italic">PT</Emphasis>-Symmetric Hulthén Potential

The one-dimensional Klein-Gordon equation is solved for the PT-symmetric generalized Hulthén potential in the scalar coupling scheme. The relativistic bound-state energy spectrum and the corresponding wave functions are obtained by using the Nikiforov-Uvarov method which is based on solving the second-order linear differential equations by reduction to a generalized equation of hypergeometric type. PACS numbers: 03.65.Fd, 03.65.Ge     

The Hadamard Construction of Green's Functions on a Curved Space-Time with Symmetries

The Hadamard constituents of Green's functions for a ζ-parametrized generalization of the massless scalar d'Alembert equation to a curved space-time including the conformally invariant wave equation: the world function of space-time, the transport scalar, and the tail-term coefficients, being simultaneously coefficients in the Schwinger-DeWitt expansion of the Feynman propagator for the corresponding invariant Klein-Gordon equation, are considered on a general static spherically symmetric and (2,2)-decomposable metric. The construction equations determining the Hadamard building elements are cast into a symmetry-adapted form and used to obtain, on a specific model metric, exact explicit solutions.     

Simplified Bethe-Salpeter equation for positronium

The Bethe-Salpeter equation for a fermion-antifermion system, coupled by photons, is considered in the Feynman gauge. The kernel is that resulting from exchange of a single photon. The usual reduction of the sixteen B-S spinor amplitudes in terms of tensors leads to 16 coupled integro-differential equations. By straightforward application of charge conjugation-, parity-, and Lorentz-invariance, the system of coupled equations is reduced to ones involving no more than eight and as few as three scalar structure functions for the various parity, charge conjugation, and total angular momentum states. The results hold for arbitrary coupling strength. As a check of the equations obtained, a perturbation theory is carried out for the Coulomb interaction. It leads to effective potentials in agreement with those obtained previously to order mα⁴ for positronium.