Poincaré Invariant Three-Body Scattering (original) (raw)

Poincaré invariant three-body scattering at intermediate energies

Physical Review C, 2008

The relativistic Faddeev equation for three-nucleon scattering is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. The equation is solved through Padé summation, and the numerical feasibility and stability of the solution is demonstrated. Relativistic invariance is achieved by constructing a dynamical unitary representation of the Poincaré group on the three-nucleon Hilbert space. Based on a Malfliet-Tjon type interaction, observables for elastic and break-up scattering are calculated for projectile energies in the intermediate energy range up to 2 GeV, and compared to their nonrelativistic counterparts. The convergence of the multiple scattering series is investigated as a function of the projectile energy in different scattering observables and configurations. Approximations to the two-body interaction embedded in the three-particle space are compared to the exact treatment.

First order relativistic three-body scattering

Physical Review C, 2007

Relativistic Faddeev equations for three-body scattering at arbitrary energies are formulated in momentum space and in first order in the two-body transition-operator directly solved in terms of momentum vectors without employing a partial wave decomposition. Relativistic invariance is incorporated within the framework of Poincaré invariant quantum mechanics, and presented in some detail. Based on a Malfliet-Tjon type interaction, observables for elastic and break-up scattering are calculated up to projectile energies of 1 GeV. The influence of kinematic and dynamic relativistic effects on those observables is systematically studied. Approximations to the two-body interaction embedded in the three-particle space are compared to the exact treatment.

Three-body scattering at intermediate energies

Physical Review C, 2005

The Faddeev equation for three-body scattering at arbitrary energies is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. In its simplest form the Faddeev equation for identical bosons, which we are using, is a three-dimensional integral equation in five variables, magnitudes of relative momenta and angles. This equation is solved through Padé summation. Based on a Malfliet-Tjon-type potential, the numerical feasibility and stability of the algorithm for solving the Faddeev equation is demonstrated. Special attention is given to the selection of independent variables and the treatment of three-body break-up singularities with a spline based method. The elastic differential cross section, semi-exclusive d(N,N ′) cross sections and total cross sections of both elastic and breakup processes in the intermediate energy range up to about 1 GeV are calculated and the convergence of the multiple scattering series is investigated in every case. In general a truncation in the first or second order in the two-body t-matrix is quite insufficient.

Three-body elastic and inelastic scattering at intermediate energies

Nuclear Physics A, 2007

The Faddeev equation for three-body scattering at arbitrary energies is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. For identical bosons this results in a threedimensional integral equation in five variables, magnitudes of relative momenta and angles. The cross sections for both elastic and breakup processes in the intermediate energy range up to about 1 GeV are calculated based on a Malfliet-Tjon type potential, and the convergence of the multiple scattering series is investigated.

Relativistic three-particle scattering equations

Physical Review C, 1993

We derive a set of relativistic three-particle scattering equations in the three-particle c.m. frame employing a relativistic three-particle propagator suggested long ago by Ahmadzadeh and Tjon in the c.m. frame of a two-particle subsystem. We make the coordinate transformation of this propagator from the c.m. frame of the two-particle subsystem to the three-particle c.m. frame. We also point out that some numerical applications of the Ahmadzadeh and Tjon propagator to the three-nucleon problem use unnecessary nonrelativistic approximations which do not simplify the computational task, but violate constraints of relativistic unitarity and/or covariance.

A New Approach to the 3D Faddeev Equation for Three-body Scattering

Few-Body Systems, 2009

A novel approach to solve the Faddeev equation for three-body scattering at arbitrary energies is proposed. This approach disentangles the complicated singularity structure of the free threenucleon propagator leading to the moving and logarithmic singularities in standard treatments. The Faddeev equation is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. In its simplest form the Faddeev equation for identical bosons, which we are using , is an integral equation in five variables, magnitudes of relative momenta and angles. The singularities of the free propagator and the deuteron propagator are now both simple poles in two different momentum variables, and thus can both be integrated with standard techniques.

Theory of non-relativistic three-particle scattering

Physica, 1967

A new method, using asymptotically stationary states, is developed to calculate the S-matrix for the scattering of a non-relativistic particle by the bound state of two other particles. For the scattering with breakup of this bound state, we obtain a simplified form of the Faddeev integral equations. It is then shown that all scattering processes without breakup of the bound state can be simply connected to the process where possible excitation or deexcitation of the bound state but no particle exchange occurs. The scattering amplitude for these processes satisfies a two-particle Lippmann-Schwinger equation with a compact kernel. It is in a form where all quantities have a clear physical meaning, so that it is well suited as a starting point for the application of approximation methods.

Relativistic Description of Two-body Scattering Reactions

African Institute for Mathematical Sciences, UCT, South Africa, 2007

In this essay, two-body elastic scattering is treated in the framework of relativistic quantum mechanics. By using lowest-order Feynman diagrams, detailed derivations of the invariant matrix element and differential cross-section for unpolarized electron-proton scattering are made. First, the proton is approximated as a spin-half point particle which allows a consistent quantum electrodynamical description of the scattering process. The Feynman rules and trace algebra have been employed in constructing the relativistic expression of invariant amplitude. Moreover, by using a suitable basis for the second rank tensor, the hadronic tensor for point proton is generalized to include the electromagnetic form factors which lead us to treat the proton as an extended object within the finite volume. The calculated differential cross section for a point proton is compared to the Rutherford and Mott predictions at laboratory angles between 0 and 180 degrees and initial electron energy between 1 MeV and 1 GeV by using numerical simulations. The results are plotted against scattering angle in the laboratory frame. In the simulation it is shown that the calculated differential cross-section agrees with the Mott prediction. It also shows the expected deviation from the Rutherford prediction.

Relativistic Three-Particle Dynamical Equations I. Theoretical Development

Annals of Physics, 1994

Starting from the two-particle Bethe-Salpeter equation in the ladder approximation and integrating over the time component of momentum, we rederive three dimensional scattering integral equations satisfying constraints of relativistic unitarity and covariance, first derived by Weinberg and by Blankenbecler and Sugar.

Relativistic effects in exclusive pd breakup scattering at intermediate energies

Physics Letters B, 2008

The relativistic Faddeev equation for three-nucleon scattering is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. Relativistic invariance is achieved by constructing a dynamical unitary representation of the Poincaré group on the three-nucleon Hilbert space. The exclusive breakup reaction at 508 MeV is calculated based on a Malfliet-Tjon type of two-body interaction and the cross sections are compared to measured cross sections at this energy. We find that the magnitude of the relativistic effects can be quite large and depends on the configurations considered. In spite of the simple nature of the model interaction, the experimental cross sections are in surprisingly good agreement with the predictions of the relativistic calculations. We also find that although for specific configurations the multiple scattering series converges rapidly, this is in general not the case.