D. Hüber

The Three nucleon continuum: Achievements, challenges and applications

Abstract After a brief historic overview the basic equations for three-nucleon (3N) scattering based on general two-nucleon and 3N forces are reviewed and the main steps for their derivation are given. Also the expressions for the various observables, elastic and breakup cross sections, as well as the great variety of spin observables are displayed and derived. The treatment of the 3N Faddeev equations in momentum space and in a partial wave decomposition is outlaid in some detail, the handling of the singularities in the integral kernel described and the algorithms and techniques used to solve the large set of equations in the discretized form are presented. Accuracy tests in form of benchmark calculations, where our results are compared to the ones of other techniques, are given. The bulk part of this review, however, is devoted to the comparison of very many observables in elastic nucleon-deuteron (Nd) scattering and the breakup process to the predictions based on the most modern nucleon-nucleon (NN) forces AV18, Nijmegen93, Nijmegen I and II and a recently updated OBE-potential CD Bonn. Overall the agreement with the data is excellent and there is little room left for the action of a three-nucleon force (3NF). The effects of the π-π, π-ϱ and ϱ-ϱ exchange 3NFs of the Tucson-Melbourne model are studied. They are in general small and in the few cases where discrepancies to data occur using NN forces only, they go into the wrong direction. We propose quite a few measurements, which should help to get more information on the potential energy of three nucleons. Several special topics are discussed: Do certain 3N scattering observables scale with the triton binding energy? Which of the 3N breakup cross sections are totally insensitive to the choice of the NN force and which are very sensitive? How well can one extract the nn scattering length from the 3N breakup? We discuss the outsticking discrepancy of the 3N analyzing power Ay in low energy elastic Nd scattering; the eigen phase shifts and mixing parameters in elastic nd scattering; the simplifications of 3N scattering at high energies and the formulation of the optical potential for elastic nd scattering and its limiting form at high energies. Alternative approaches to solve 3N scattering: in configuration space, using finite rank expansions of NN forces, variational techniques and the hybrid Sendai method are briefly described as well as the proton proton Coulomb force problem in the pd system and the question how to incorporate relativity. Finally some applications are sketched where a 3N final state occurs and where the interaction among the nucleons requires a correct treatment. We mention inelastic electron scattering as well as π-absorption on 3He (3H) and nonmesonic decay of the hypertriton.

A New look into the partial wave decomposition of three nucleon forces

Abstract. We demonstrate that the partial-wave decomposition of three-nucleon forces used up to now in momentum space has to be necessarily unstable for high partial waves. This does not affect the applications performed up to now, which were restricted to low partial waves. We present a new way to perform the partial-wave decomposition free of that defect. This is exemplified for the most common two-pion-exchange Tucson-Melbourne three-nucleon force. For the lower partial waves the results of the old method are reproduced.

Momentum-space calculations for three-nucleon scattering including a three-nucleon force

The formalism to include a three-nucleon force into three-nucleon continuum calculations is presented. First numerical results, obtained in momentum space, are shown. The two- and three-nucleon forces have been restricted to act only in the1S0 and3S1-3D1 partial-wave states. As two-nucleon interaction the Bonn-B potential and as three-nucleon interaction the Tucson-Melbourne two-pion exchange model has been used.

Inclusive scattering of polarized electrons on polarized {sup 3}He: Effects of final state interaction and the magnetic form factor of the neutron

Effects of final state interaction on asymmetries in inclusive scattering of polarized electrons on polarized 3He are investigated using consistent 3He bound state wave function and 3N continuum scattering states. Significant effects are found, which influence the extraction of the magnetic neutron form factor from AT ′ . The enhancement found experimentally for ATL′ near the 3N breakup threshold, which could not be explained in calculations carried through in plane wave impulse approximation up to now, occurs now also in ∗present address: Institut für Kernphysik, Fachbereich 5 der Technischen Hochschule Darmstadt, D-64289 Darmstadt, Germany †present address: Los Alamos National Laboratory, Theoretical Division, Los Alamos, NM 87545

The c-Term of the Tucson-Melbourne Three-Body Force: To Be or Not to Be

Abstract. By Faddeev calculations of 3H we study the dependence of the binding energy on the three-nucleon force. We adopt the 2π-exchange Tucson-Melbourne three-nucleon force and investigate phenomenologically the dependence on the strength of the individual three-body force operators (the a-, b-, c-, and d-terms). While it is well known that the a-term is not as important as the b- and d-terms to gain the experimental binding energy, we find two solutions for the c-term, one around the value used in the original Tucson-Melbourne model and a new one close to zero. A tensor-analyzing power T20 of the pd elastic scattering using the modified Tucson-Melbourne model, which follows the recommendation by chiral perturbation theory that the short-range c-term should be dropped, describes the data well.

Algorithm for solving the Faddeev equations in the three-body continuum under avoidance of moving logarithmic singularities

A new method is presented for solving the Faddeev equations in the three-body continuum, which avoids the moving logarithmic singularities present in momentum space methods used up to now. The new algorithm leads to a simple structure of the Faddeev integral kernel, what simplifies significantly the numerical realization. Its application in nuclear physics is, however, still plagued by the presence of the virtual-state pole in the nucleon-nucleon1S0 channel. Omitting that channel in calculations with the Bonn-B potential we demonstrate excellent agreement between three-nucleon observables obtained with the new and a former method. Since the codes are quite different, this can be considered as a convincing test.

Effects of the Tucson-Melbourne three-nucleon force in the proton-deuteron breakup process at Ep = 65 MeV

Abstract We present the calculated cross sections and vector analyzing powers using the Bonn B nucleon-nucleon potential and the Tucson-Melbourne three-nucleon force ( 3NF ) for six collinearity and quasi-free scattering breakup configurations. These calculations are compared to the results of the recent kinematically complete pd experiments at E p = 65 MeV. The Tucson-Melbourne 3NF , adjusted together with the Bonn B potential to reproduce the triton binding energy, leads to small effects both in cross sections and analyzing powers in all six studied configurations.

Low energy proton-deuteron versus neutron-deuteron breakup in four configurations: implications for Coulomb-force effects

Abstract We have measured cross sections of the pd breakup reaction at a laboratory proton energy of 10.5 MeV in four kinematically complete arrangements comprising space-star, collinearity, final-state interaction and quasi-free scattering conditions. We present our results and compare them to the predictions of rigorous three-nucleon Faddeev calculations using different realistic nucleon-nucleon potentials. We find cases of good agreement but also cases of clear discrepancies. The inclusion of the 2π-exchange Tucson-Melbourne three-nucleon force does not remove these discrepancies. Considering also pd data at 13 and 19 MeV and nd data at 10.5 and 13 MeV, we draw tentative conclusions about possible Coulomb-force effects in these configurations. In the quasi-free scattering, space-star, and collinear configurations, the Coulomb-force effects might possibly be responsible for at least part of the observed discrepancies. For the np FSI configuration the Coulomb-force effects are likely to be small.

Achievements and Challenges in 3N- and 4N-Systems

Recent results on rigorous solutions of the 3N Faddeev equations but also open problems and suggestions for new experiments in 3N scattering are presented. For 4 He binding energies and properties of the ground state wavefunction for various realistic NN forces are given based on precise solutions of the 4N Faddeev-Yakubovsky equations.

Nuclear Three- and Four-Body Systems

Recent insight in 3- and 4-nucleon bound state properties, the hypertriton and 3N scattering are reviewed. New results for inclusive electron scattering on 3He (3H) including exactly the final state interaction are given.