A novel treatment of the proton-proton Coulomb force in proton-deuteron Faddeev calculations: Breakup

We extend our approach to incorporate the proton-proton (ppCoulomb force into the three-nucleon (3NFaddeev calculations from elastic proton-deuteron (pdscattering to the breakup process. The main new ingredient is a 3-dimensional screened pp Coulomb t -matrix obtained by a numerical solution of the 3-dimensional Lippmann-Schwinger (LS) equation. We demonstrate numerically that the proton-deuteron (pdbreakup observables can be determined from the resulting on-shell 3N amplitudes increasing the screening radius. However, contrary to the pd elastic scattering, the screening limit exists only after renormalisation of the pp t -matrices.     

A novel treatment of the proton-proton Coulomb force in proton-deuteron Faddeev calculations: Elastic scattering

We propose a novel approach to incorporate the proton-proton (pp) Coulomb force into the three-nucleon (3N) Faddeev calculations. The main new ingredient is a 3-dimensional screened pp Coulomb t -matrix obtained by a numerical solution of the 3-dimensional Lippmann-Schwinger (LS) equation. We demonstrate numerically and provide analytical insight that the elastic proton-deuteron (pd ) observables can be determined directly from the resulting on-shell 3N amplitude increasing the screening radius. The screening limit exists without the need of renormalisation not only for observables but for the elastic pd amplitude itself.     

Polarization-Transfer Coefficients in and Reactions at MeV: Experiment Compared to Calculations with Recent Nuclear-Interaction Models

The polarization-transfer coefficients K , K and K , K have been measured in the elastic scattering reactions D(, )p and D(, )d at MeV, respectively. They are compared to solutions of the three-nucleon Faddeev equations obtained with the recent nucleon-nucleon interactions AV18, CD Bonn, NijmI and II. Effects of the Tucson-Melbourne three-nucleon force, adjusted separately to reproduce the triton binding energy for each of these potentials, are studied. Both and exhibit a scaling behaviour with the triton binding energy. For and the various predictions with two-nucleon forces only agree practically with each other but spread after inclusion of the three-nucleon force. The agreement of theory and data is fair but the neglect of the proton-proton Coulomb force precludes a final conclusion. Received July 28, 1997; revised February 3, 1998; accepted for publication March 11, 1998     

Proton-proton scattering without Coulomb force renormalization

We demonstrate numerically that proton-proton (pp scattering observables can be determined directly by standard short-range methods using a screened pp Coulomb force without renormalization. In the examples the appropriate screening radii are given. We also numerically investigate solutions of the 3-dimensional Lippmann-Schwinger (LS) equation for a screened Coulomb potential alone in the limit of large screening radii and confirm analytically predicted properties for off-shell, half-shell and on-shell Coulomb t -matrices. These 3-dimensional solutions will form a basis for a novel approach to include the pp Coulomb interaction into the 3N Faddeev framework.     

Measurements of the Cross Sections and A y for D(p, n) Inclusive Breakup Reaction at 170 MeV

The effects of three-nucleon force (3NF) has been actively studied by using the nucleon–deuteron (Nd) scattering states. The differential cross sections of the elastic Nd scattering at the energy below 150 MeV can be well reproduced by incorporating 3NF in the Faddeev calculation based on modern nucleon–nucleon (NN) interactions. On the other hand, the differential cross sections of Nd elastic and inelastic scatterings at 250 MeV show large discrepancies between the data and the Faddeev calculations with 3NF. It indicates the presence of the missing features of the three nucleon system at this energy region. For the systematic study about the energy dependence of this large discrepancies, we measured the differential cross sections and the vector analyzing power A _y for the ²H(p, n) inclusive breakup reaction at 170 MeV. The experiment was carried out at RCNP by detecting scattered neutrons by using the neutron detector NPOL3. The data was compared with the results of the Faddeev calculations with and without the 3NF.     

Studies of the three-nucleon system dynamics: Cross sections of the deuteron-proton breakup at 130 MeV

The three-nucleon system is the simplest non-trivial testing ground in which the quality of modern nucleon-nucleon interaction models, as well as additional dynamical ingredients referred to as three-nucleon forces, can be probed quantitatively by means of a rigorous technique of solving the Faddeev equations. A large set of high precision, exclusive cross-section data for the $^1{\rm H}({\vec d},pp)n$ breakup reaction at 130?MeV was obtained at KVI, Groningen. It allowed to establish for the first time a clear evidence of the three-nucleon force contributions to the cross sections of the breakup process and to confirm recent predictions of sizable influences of the Coulomb force in this reaction.     

pd scattering at 250 MeV and three-nucleon forces

We have measured angular distributions of the differential cross-section, the analyzingpower and all of the spin transfer coefficients K _y ^y, K _x ^x, K _z ^x, K _x ^z, and K _z ^z for the pd elastic scattering at 250 MeV. These are the first measurements of a complete set of proton polarization observables for pd elastic scattering at intermediate energies. The present data are compared with theoretical predictions based on exact solutions of the three-nucleon Faddeev equations and modern realistic nucleon-nucleon potentials combined with three-nucleon forces (3NF), namely the Tucson-Melbourne (TM) 2 -exchange model, a modification thereof (TM) closer to chiral symmetry, and the Urbana IX model.Received: 30 September 2002, Published online: 22 October 2003PACS: 21.45. + v Few-body systems - 21.30.-x Nuclear forces - 24.70. + s Polarization phenomena in reactions     

Two-body scattering without angular-momentum decomposition: Fully off-shell T-matrices

Two-body scattering is studied by solving the Lippmann-Schwinger equation in momentum space without angular-momentum decomposition for a local short-range interaction plus Coulomb. The screening and renormalization approach is employed to treat the Coulomb interaction. Benchmark calculations are performed by comparing our procedure with a configuration space calculation, using the standard partial-wave decomposition, for ¹²C - ¹⁰Be elastic scattering. The fully off-shell T -matrices are also calculated for the final goal of studying the three-body scattering by solving Faddeev/AGS equations.     

Ab-Initio Calculations of Four-Nucleon Elastic Scattering

We present microscopic calculations of low-energy scattering observables in all possible four-nucleon systems: n-³H, p-³He, and p-³H. Results were obtained by solving Faddeev-Yakubovski equations in configuration space, appropriately modified to include Coulomb and three-nucleon forces.     

Variational calculations of resonant states in 4He

We present a modified R-matrix method which allows microscopic calculations of nucleon-nucleus scattering at low energies. This method may be applied in conjunction with any of the commonly used methods for the ground states of few-body systems, i.e. the Green function Monte Carlo, Faddeev and variational techniques. We then report results of variational calculations of low-energy tp scattering in the region of 0⁺, 0^? and 2^? resonances in ⁴He. The energies and widths of these resonances are calculated using realistic two- and three-nucleon interactions; but, the Coulomb interaction is neglected. The three-nucleon interaction is found to have a much smaller effect on the energies of the resonances than on the ground-state energy.     

Triton photodisintegration in three-dimensional approach

Two- and three-particles photodisintegration of the triton is investigated in a three-dimensional (3D) Faddeev approach. For this purpose the Jacobi momentum vectors for three-particles system and spin-isospin quantum numbers of the individual nucleons are considered. Based on this picture the three-nucleon Faddeev integral equations with the two-nucleon interaction are formulated without employing the partial-wave decomposition. The single-nucleon current as well as π- and ρ-like exchange currents are used in an appropriate form to be employed in the 3D approach. The exchange currents are derived from AV18 NN force. The two-body t-matrix, deuteron and triton wave functions are calculated in the 3D approach by using the AV18 potential. Benchmarks are presented to compare the total cross-section for the two- and three-particles photodisintegration in the range of E _γ < 30 MeV. The 3D Faddeev approach shows promising results.     

Calculations of Three-Nucleon Reactions

Faddeev calculations using the chiral three-nucleon force in next-to-next-to-next-to-leading-order show that this force is too weak to provide an explanation for the low-energy A _y puzzle. The large discrepancy between data and theory for the neutron–neutron quasi-free-scattering cross section in low energy neutron–deuteron breakup requires a modification of the ${^{1}S_0}$ neutron–neutron force. We discuss the consequences that a bound ${^{1}S_0}$ state of two neutrons has on neutron–deuteron scattering observables. At higher energies we compare the solutions of the non-relativistic three-nucleon Faddeev equations with three-nucleon force included to the solutions of its Poincaré invariant version.     

Extraction of the Neutron-Neutron Scattering Length a nn from Kinematically Complete Neutron-Deuteron Breakup Experiments

Data for the neutron-neutron final-state-interaction cross section obtained recently in a kinematically complete neutron-deuteron breakup experiment have been reanalyzed using rigorous solutions of the three-nucleon Faddeev equations with realistic nucleon-nucleon interactions. A discrepancy was found with respect to a recent analysis based on the W-matrix approximation to the Paris potential. We also estimate theoretical uncertainties in extracting the neutron-neutron scattering length resulting from the use of different nucleon-nucleon interactions and the possible action of the two-pion-exchange three-nucleon force. We find that there exists a certain production angle for the interacting neutron-neutron pair where the uncertainties become minimal. Received August 27, 1994; revised February 13, 1995; accepted for publication March 21, 1995     

Dependence of the Three-Nucleon Binding Energy on Some Different Effects

The bound state of three-nucleon system is studied as a three-body problem which is solved following the different approaches of the Faddeev formalism as well as the unitary pole approximation. The three-body problem is reduced to a set of coupled integral equations by using separable approximations. Numerical calculations are carried out for the resulting integral equations and the separable expansion. In the present work, we calculate the ground-state binding energy of the bound three-nucleon system ³H. The main interest of the present work is to investigate the sensitivity of the three-nucleon binding energy to different effects. For this reason, we study the dependence of this energy on different forms of local and separable nucleon-nucleon potentials, the effective range of the nucleon-nucleon interaction, and on the percent of the D state in the deuteron wave function. Also we test the sensitivity of the three-nucleon binding energy to the considered number of terms from the separable expansion.     

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

 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 T ₂₀ 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. Received April 22, 1999; revised June 6, 2000; accepted for publication June 16, 2000     

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.     

Faddeev equations and the method of hyperspherical harmonics in the problem of three-nucleon continuum

A method for solving the Faddeev equations in configuration space is developed for a three-nucleon system in the continuum by using the decomposition over a hyperspherical basis. The wave functions of Nd-system, phase shifts, and cross sections of Nd-scattering at subthreshold energies are calculated. Also, within the framework of this method, one-dimensional integral equations are formulated for the problem of infinite motion of all three strongly interacting particles, and the Faddeev equations for a system of three hadrons with Coulomb interaction in the continuum are modified. Similar methods of investigation of three-particle systems are reviewed.     

3H at Next-to-Next-to-Next-to Leading Order of the Chiral Expansion

The chiral three-nucleon force (3NF) at next-to-next-to-next-to leading order (N³LO) is used to calculate the triton wave function and the doublet nucleon–deuteron scattering length. This allows us to fix the values of the low-energy constants which are free parameters of the theory. The obtained values of these parameters, the expectation values of the kinetic energy, two- and three-body potentials and individual contributions of different parts of 3NF are given.