Sensitivity of the three-body calculations to different effects

The bound state of few-body systems in light nuclei is studied as a three-body problem. The three-body problem is solved following the different approaches of the Faddeev formalism as well as the unitary pole approximation. Separable approximations are introduced to reduce the three-body problem to a set of coupled integral equations. 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 system³H. The main interest of the present work is to investigate the sensitivity of the three-body binding energy to different effects in the problem. For this reason, we study the dependence of the three-body binding energy of different forms of local and separable two-body potentials, on the effective range of the two-body potentials, and on the percent of theD state in the deuteron wave function. Also, we test the sensitivity of the three-body binding energy to the considered number of terms from the separable expansion.     

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.     

Energy Dependence of the πN Amplitude and the Three-Nucleon Interaction

By calculating the contribution of the ππ three-body force to the three-nucleon binding energy in terms of the πN amplitude using perturbation theory, we are able to determine the importance of the energy dependence and the contribution of the different partial waves of the πN amplitude to the three-nucleon force. A separable representation of the non-pole πN amplitude allows us to write the three-nucleon force in terms of the amplitude for NN → NN*, propagation of the NNN* system, and the amplitude for NN* → NN , with N* being the πN quasi-particle amplitude in a given state. The division of the πN amplitude into a pole and non-pole part gives a procedure for the determination of the πNN form factor within the model. The total contribution of the three-body force to the binding energy of the triton for the separable approximation to the Paris nucleon-nucleon potential (PEST) is found to be very small mainly as a result of the energy dependence of the πN amplitude, the cancellation between the S- and P-wave πN amplitudes, and the soft πNN form factor. Received April 12, 1994; revised November 11, 1994; accepted for publication December 1, 1994     

The integral equation approach to four-nucleon bound states

Within the framework of the Yakubovsky four-body equations the 0⁺ bound states of ⁴He are determined. The two-particle interactions used are of the separable Yamaguchi type and include spin-dependent forces. The problem is reduced to the solution of four coupled integral equations in two variables. The separable approximation of the kernels makes it possible to reduce the problem to a set of single variable integral equations. The separable approximation method employed is based on the Hilbert-Schmidt expansion applied to the kernels of four-body equations. The ground state energy of ⁴He is found to be ?45.73 MeV, the excited 0⁺ level lies at ?11.69 MeV. In conclusion we discuss the accuracy of various approximate methods in the four-nucleon problem.     

Three-body structure of 6He=4He+n+n using realistic n-n potentials

With the use of realistic nucleon-nucleon forces (Paris, Bonn and Argonne potenfials) between the valence neutrons, the three-body structure of ⁶He=⁴He+n+n is investigated with emphasis on the short-range correlations between the halo neutrons and on dependence on the realistic forces. To solve the three-body problem regorously, use is made of the coupled-rearrangement-channel variational method with the Gaussian basis functions. The basis are represented by newly developed Gaussian lobe functions with infinitesimal shift parameters. Accuracy and usefulness of this basis functions are examined in the calculation of three-nucleon bound state with those realistic forces.     

Three-body model for stripping nuclear reactions

A three-body model for the deuteron stripping nuclear reactions is presented. A set of three integral equations is obtained for the wave functions of the three-body problem by introducing a decomposition into angular momentum states into the Lippmann-Schwinger equation. Simple two-particle interactions with separable potentials are used. These separable potentials reduce the three-body problem to the solution of coupled sets of one-dimensional Fredholm integral equations. The angular distributions for²⁸Si(d,p)²⁹Si and⁴⁰Ca(d, p)⁴¹Ca stripping reactions are calculated. From the extracted spectroscopic factors, good agreement with the experimental measurements is obtained.     

Dynamical Study of Tensor Observables in the Energy Range of 80 keV to 95 MeV: Tests of Effective Two-Body Models

 Realistic interactions are used to study tensor observables in the energy range of 80 keV to 95 MeV deuteron laboratory energy, as well as the differential cross section for the two-body photodisintegration of . The Siegert form of the E1 multipole operator in the long-wavelength limit is taken as the sole component of the electromagnetic interaction. The three-body Faddeev equations for the bound-state and continuum wave functions are solved using the Paris, Argonne V14, Bonn-A, and Bonn-B potentials. The corresponding nucleon-nucleon t-matrices are represented in a separable form using the Ernst-Shakin-Thaler representation. The Coulomb force between protons is neglected and no three-nucleon force is included. The contribution of nucleon-nucleon P-wave components to the observables is carefully studied, not only in the angular distribution of the observables, but also as a function of the deuteron laboratory energy for fixed centre-of-mass angle. Comparison with data is shown wherever it exists. Results with simple Yamaguchi-type interactions with variable %D-state in the deuteron are compared with realistic interactions and one of these model potentials is used to study the results in terms of contributions from specific wave-function components or terms in the electromagnetic operator. Effective two-body models are examined by means of a derivation that is consistent with the underlying three-body calculation and that leads to an effective two-body t-matrix for neutron-deuteron elastic scattering carrying the same on-shell amplitudes as the original three-body equations. Received September 21, 1999; revised December 23, 1999; accepted February 9, 2000     

Effects of the Three-Nucleon Forces from Different Meson Exchanges on the Triton Binding Energy

Effects of the three-nucleon forces on the triton binding energy are investigated using different models for which the xN scattering amplitudes for the off-mass-shell pions are extrapolated from the on-mass-shell data. In the independent model potential, the threenucleon forces are constructed using the chiral symmetry through partial conservation of the axial current vectors and current algebra. On the other hand, the three-nucleon forces in the dependent model potential are derived from an effective Lagrangian which is constrained by chiral and gauge symmetry. The effect of the long range 2π and the repulsive ap-exchange three-nucleon forces on the triton binding energy are investigated. The bound state triton binding energies are then calculated by solving the Faddeev equations with a Hamiltonian including the three-nucleon forces, using different nucleon-nucleon interactions. The pionic form factors are considered with different sets of the form factor cutoff,parameter Λ. The 2π-exchange forces with S- and P-wave amplitudes are found to overbind the triton binding energy by about 0.4～Y 1.6 MeV. However, xp-exchange forces are found to reduce the 2π exchange forces by about 18% of its contribution, to the triton binding energy.     

Accurate Three-Nucleon Bound-State Calculation with an Extended Separable Expansion of the Two-Body T-Matrix

An accurate solution for the three-nucleon bound state is obtained within 1 keV in the binding energy and, on the whole, better than 1% in the wave function, using a new systematic and efficient method. The method is based on a recently developed separable expansion for any finite-range interaction, in which a rigorous separable series for the two-body t-matrix is obtained by expanding the wave function in terms of a complete set of basis functions inside the range of the potential. In order to treat a potential with a strong repulsive core, as in the case of the Argonne potential, we develop a two-potential formalism. The expansion starts with a few EST (Ernst, Shakin, and Thaler) terms in order to accelerate the convergence and continues with an orthogonal set of polynomials, avoiding the known difficulties of a pure EST expansion. Thus, several techniques are combined in the present extended separable expansion (ESE). In this way, the method opens a new systematic treatment for accurate few-body calculations resulting in a dramatic reduction in the CPU time required to solve few-body equations. Received November 6, 1996; revised April 14, 1997; accepted for publication April 30, 1997     

Solutions of the integral equations for four identical particles

Using the separable representation of the scattering amplitudes for the subsystems 3 + 1 and 2 + 2, the integral equations for four identical particles with a separable two-particle interaction are reduced to a set of single variable integral equations. By solving the equations obtained, the binding energies and wave functions of the low-lying 0⁺ states of the system of four identical bosons, as well as the scattering length of a particle scattered by three bound particles, are calculated. The solution of the set of integral equations, describing the bound state of four nucleons, is performed, approximating the space wave function by a symmetric one, and the binding energy and wave function of the nucleus ⁴He are calculated.     

On The Contribution of the P and D Partial-Wave States to the Binding Energy of the Triton in the Bethe-Salpeter-Faddeev Approach

The influence of the partial-wave states with nonzero orbital moment of the nucleon pair on the binding energy of the triton T(nnp) in the relativistic case is considered. The relativistic generalization of the Faddeev equation in the Bethe-Salpeter formalism is applied. Two-nucleon t matrix is obtained from the Bethe-Salpeter equation with separable kernel of nucleon-nucleon interaction of the rank one. The kernel form factors are the relativistic type of the Yamaguchi functions. The following two-nucleon partial-wave states are considered: ¹S₀, ³S₁, ³D₁, ³P₀, ¹P₁, ³P₁. The system of the integral equations are solved by using the iteration method. The binding energy of the triton and three-nucleon amplitudes are found. The contribution of the P and D states to the binding energy of triton is given.

     

Δ(3, 3) excitations and effective three-nucleon forces in finite nuclei

Contributions from three-body terms with intermediate Δ(3,3) isobar excitations to the ground state energies of nuclei are investigated. These terms can either be understood as three-body clusters in a many-body theory including isobar excitations explicitly or as contributions to an effective three-nucleon force. For the example ¹⁶O the resulting contribution is attractive and its value is typically about ?0.5 MeV per nucleon. This is smaller than the typical values of 1 MeV per nucleon repulsion obtained from the modifications of the effective two-body nucleon-nucleon interaction in nuclei due to intermediate Δ(3, 3) configurations. The gain in energy from the three-body terms including Δ(3,3) configurations, however, is of the same importance as the contribution from three-body terms including nucleons only.     

Three-body formalism for deuteron stripping reactions

A three-body formalism for deuteron stripping reactions has been developed. The equations of Altet al (1967) (AGS) for the three particle system (target A, n, p) are reduced to a set of coupled one-dimensional integral equations with the use of (i) angular momentum basis for representation and (ii) separable approximation for the two bodyt-matrices (which delineate the interactions between the particle pairs). The on-shell solutions of this set of integral equations are then related to the cross sections of the rearrangement processes. The inputs in this calculation, viz., the separable interactions between the particle pairs in the respective channels are simply constructed from the respective two body bound state in accordance with the bound state approximation (BSA) conforming to the ‘unitarity’ requirement. Using this formalism preliminary calculations for the (d, p) and (d, n) reaction cross sections on¹⁶O have been carried out and they seem to have considerable semblance with the observed cross sections.     

Two pion exchange three body potential in three nucleon states

We present the observables for theA=3 bound state with a two-pion exchange three-body force added to the de Tourreil and Sprung realistic nucleon-nucleon interaction. This results mainly in increasing the binding energy by 650 keV. In the framework of the impulse approximation, we find no sensible improvement for the electromagnetic form factors, and no central depression for the He³ charge density with point nucleons. We compute the contribution of the three-body force to then-d doublet scattering length whose agreement with experiment is improved.     

THE STUDY OF HYPERTRITON WITH THREE-BODY INTEGRAL EQUATION

From a Faddeev-type integral equation for three-body bound state, the hypertriton binding energy is calculated by using the nonlocal two-body separable potential with the parameters taken from the scattering data of meson theoretical potential.The effects of n-p and N(n or p)-Λ interactions in three-body bound state are studied and the meson theoretical potentials of Refs.[1-3] are examined. The calculated results are reasonably close to the experimental values.The comparisons of our results with others are made.     

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     

Effect of the three-body force on trinucleon bound systems consideringS andS′-state admixture

We report the calculation of binding energy, charge form factor and point-like proton density of both³H and³He by the hyperspherical harmonics method with the inclusion of two-pion exchange three-nucleon force (Fujita-Miyazawa type). For the two-body force theN-N Afnan-Tang S-3 potential is taken. Coulomb and three-body forces are treated nonperturbatively. In this calculation the mixed symmetryS′-state of the trinucleon ground state is considered along with the space totally symmetricS-state.     

Effects of the two-pion exchange three-nucleon force in the triton and 3He

The two-pion-exchange three-nucleon force is incorporated into a Faddeev calculation in momentum space. Within the restricted space of states substantial additional binding energy shows up. A new expression for the charge form factor is derived which does not involve a partial-wave expansion of the total state. The form factor of ³He shows sensitivity to the change in the wave-function induced by the three-nucleon force. The secondary maximum increases as a whole, and the radius comes closer to the experimental value; however the notorious discrepancy in the first minimum increases, too.