The objective is to carry out few-nucleon calculations without the traditionally employed angular momentum decomposition. It is well known that nuclear scattering at intermediate energies of a few hundred MeV requires quite a few angular momentum states in order to achieve convergence of e.g. scattering observables. Presently employed computational methods for three nucleon (3N) scattering at higher energies using conventional partial wave expansions have intrinsic limitations, since with increasing energy the amount of channel quantum numbers strongly proliferates, leading to increasing numerical difficulties with respect to accuracy as well as storage requirements.
Carrying out 3N scattering calculations in three dimensions allows to take Faddeev calculations into the so-called intermediate energy regime which means projectile energies from about 0.5 to 3 GeV, where measurements of proton-deuteron (pd) reactions are carried out at the Cooler Synchrotron (COSY) at the FZ-Juelich and the Research Center for Nuclear Physics (RCNP). Theoretical analyses of pd breakup processes at these energies often rely on models based on tree-level diagrams. Our work will be able to thoroughly investigate the importance of multiple scattering within a pd reaction and check if commonly used assumptions are valid.
The discretized Faddeev equation for the scattering of 3 identical particles (neglecting spin and isospin degrees of freedom for our first calculations) is a three dimensional integral equation in 5 variables. These 5 variables are the magnitudes of the the Jacobi momenta p and q, and angles between the momentum vectors and a fixed projectile momentum q0. Typical grids for the momentum variables are 50 and 50, and for the angular variables are 23 x 23 x 23. The integral equation is solved iteratively, first the Neumann series is generated, then the series is summed by a Pade method. A converged result needs typically 15-18 terms in the sum.
For the integration itself, Gaussian quadrature is used. For the kernel of the integral equation, a two-body t-matrix (with the two-nucleon interaction as driving term) is obtained by solving a system of linear equations of the form A*x=b, where A is typically a 2000*2000 matrix. This system is solved for about 80 different vectors b. Since the momentum region which contributes to a solution of the two-body t-matrix is quite different from the region of importance in a 3-body calculation, the solutions of t are mapped onto a momentum grid relevant for the Faddeev equations by repeated application of the Lippmann-Schwinger equation. For treating the deuteron pole, the residue of the t-matrix at the pole has to be extracted with high precision. The moving singularities inherent in any 3-body calculation above the 3-body break-up, are treated by subtraction, and the logarithmic singularities of the subtraction term are integrated by a spline method, i.e. it is semi-analytic. The latter proved to be important for the accuracy and simplicity of the calculation. The interpolations needed to solve the 3-body transition amplitude are based on cubic Hermite splines. The number of required interpolations is typically 109.
The Faddeev equations as function of vector variables for scattering below and above the break-up threshold were solved and scattering observables (elastic and inelastic differential and total cross sections) are obtained. Calculations are carried out for projectile energies up to 1 GeV.
A first test of the accuracy and convergence of our numerical calculations is the insertion of our converged solution for the integral equation for the transition amplitude one further time and a recalculation of the observables with this new solution. The agreement of the two solutions is 0.001\% or better, which means that our calculations are properly converged. A second and more stringent test is a comparison of values for the total cross section obtained from integrating the differential cross section with the ones obtained via the optical theorem. For a specially tuned grid we could achieve an agreement of 0.5% between the two calculations. From which we conclude that our algorithm is correct and stable. We have a very good understanding of how the choice of our grid parameters influences the accuracy of our calculations as function of laboratory energy. For general calculations fulfilling the optical theorem within 5-10% is sufficient.
The solution of the Faddeev equation gives the three-body transition amplitude T, from which then scattering observables are obtained. Since we neglect spin and iso-spin degrees of freedom, we have elastic and inelastic differential and total cross sections, which correspond to the spin-averaged cross section in experiments. Though our currently employed model for the NN interaction is relatively simple, we nevertheless can study general properties of these cross sections as function of increasing projectile energy. Of specific interest are the convergence properties of the multiple scattering series as function of the projectile energy, i.e. how many rescattering terms contribute at moderately high projectile energies. As example the full Faddeev solution at 1 GeV projectile energy is shown the elastic cross section at backward angle (corresponding to the reaction pd -> dp) is shown in the left panel of Fig. 1. Our full Faddeev calculation is compared to results obtained from the first four orders of the multiple scattering series when added successively. The right panel of Fig. 1 shows the semi-exclusive cross section for a reaction where two particles are scattering into forward direction with a relative energy between 0 and 3 MeV and the remaining one in backward direction. In both cases it is clear that even at 1 GeV at least two (three) rescattering terms are need to be summed to get close to the cross sections obtained from a full Faddeev calculation. This finding is relevant, since it indicates that projectile energies of about 1 GeV are not yet high enough to allow the omission of rescattering terms.
In Fig. 2 the cross section in the center-of-mass frame for selected exclusive break-up configurations are shown. The cross section in the left panel describes a collinear configuration, where all three scattered particles leave along a line defined by the beam direction. Here the first order calculation grossly overestimates the final result, and at least 3 rescattering terms are needed to come close to the full solution. In the right panel the so-called space-star configuration is shown, where the three scattered particles leave in a plane perpendicular to the beam directions. Here there is no contribution from a first order term, and it is clear from the figure that low order rescattering terms are not converged to the full solution of the Faddeev equation.
Nuclear scattering at intermediate energies of a few hundred MeV requires quite a few angular momentum states in order to achieve convergence of e.g. scattering observables. Presently employed computational methods for 3N scattering at higher energies using conventional partial wave expansions have intrinsic limitations, since with increasing energy the number of channel quantum numbers strongly proliferates, leading to increasing numerical difficulties with respect to accuracy and as well as storage requirements.
This work represents an alternative computational approach by solving the Faddeev equations directly in a three-dimensional (3D) form in momentum space. The incorporation of the boundary conditions for three-body scattering does not change for 3D solutions of the Faddeev equations. In the integral from in momentum space, which we are using, they are automatically included. This approach will allow to extend the investigations concerning the importance of three-nucleon forces in a computationally sound way to the energy regime up to 300 MeV, a regime where currently experimental efforts the Kernfysisch Versneller Instituut (KVI), the Cooler Synchrotron (COSY), as well as the Research Center for Nuclear Physics (RCNP) are under way.
H. Liu, Ch. Elster, W. löckle, Three-Body Scattering at Intermediate Energies, nucl-th/0410051 and submitted to Phys. Rev. C.
H. Liu, Ch. Elster, W. löckle, Three-Body Scattering without Partial Waves, 19th European Conference on Few-Body Problems in Physics, AIP Conference Proceedings, Vol. 768, p.430, NY, 2005
H. Liu, Ch. Elster, W. Glöckle, Model Study of Three-Body Forces in the Three-Body Bound State, Few-Body Systems 33, 241 (2003).URL http://www.phy.ohiou.edu/~elster/nersc05/nersc05ann.html
Fig. 1: Left Panel: The elastic differential cross section for three-body scattering at backward angle as function of projectile laboratory energy. The result of the full full Faddeev equation is given by the solid black line and compared with calculations based on first order, 2nd order, 3rd order, and 4th order in the two-body t-matrix as indicated in the legend. Right Panel: The center-of-mass cross section for the semi-exclusive break-up reaction in which two particles emerge in forward direction with a relative energy between 0 and 3 MeV, and one particle in backward direction as function of the projectile laboratory energy The result of the full full Faddeev equation is given by the solid black line and compared with calculations based on first order, 2nd order, 3rd order, and 4th order in the two-body t-matrix as indicated in the legend.
Fig. 2: Left Panel: The exclusive differential break-up cross section in the center-of-mass frame for the laboratory projectile energy 1 GeV. The break-up configuration is collinear with all three particles leaving along a line defined by the beam direction. The result of the full full Faddeev equation is given by the solid black line. The contributions of the lowest orders of the multiple scattering series added up successively are given by the colored curves as indicated in the legend. Right Panel:The exclusive differential break-up cross section in the center-of-mass frame for the laboratory projectile energy 1 GeV for the space-star configuration, where all three particles emerge in a plane perpendicular to the beam direction with an angle of 120 degrees. The result of the full full Faddeev equation is given by the solid black line. The contributions of the lowest orders of the multiple scattering series added up successively are given by the colored curves as indicated in the legend.