Faddeev Yakubovsky Research Articles

Description of Four- and Five-Nucleon Systems by Solving Faddeev-Yakubovsky Equations in Configuration Space

The Faddeev Yakubovsky equations constitute a rigorous formulation of the quantum mechanical N body problem in the framework of non relativistic dynamics. They allow the exact solutions of the Schrodinger equation for bound and scattering states to be obtained. In this review, we will present the general formalism as well as the numerical tools we use to solve Faddeev Yakubovsky equations in configuration space. We will consider in detail the description of the four and five nucleon systems based on modern realistic nuclear Hamiltonians. Recent achievements in this domain will be summarized. Some of the still controversial issues related with the nuclear Hamiltonians as well as the numerical methods traditionally employed to solve few nucleon problems will be highlighted.

The Faddeev–Yakubovsky Symphony

We briefly summarize the main steps leading to the Faddeev-Yakubovsky equations in configuration space for N=3, 4 and 5 interacting particles.

Four-Body Scattering Equations Including a Three-Body Force in the Faddeev–Yakubovsky Theory

Faddeev–Yakubovsky four-nucleon scattering equations are derived including a three-body force. From the asymptotic form of the Yakubovsky-components we obtain scattering amplitudes of all outgoing channels in a direct manner. We present these equations in the momentum space representation.

Notice of Retraction: Three-dimensional study of the six-body bound-state for the case of effective three-body configuration model

A recently developed three-Dimensional (3D) approach, without Partial-Wave (PW) decomposition is considered to investigate solutions of Faddeev–Yakubovsky (F–Y) integral equations in momentum space representation. The F–Y formulation for six-nucleon bound-state problem leads to five coupled equations in terms of five independent structures, which can be reduced to two coupled ones for considering the effective interactions of the two loosely bound nucleons with respect to the α-core nucleus. In order to study, the effective structure of six-nucleon bound-state in 3D representation, the coupled equations are represented in momentum space as a two coupled 3D integral equations. This is the first step to approach the calculations, of six-nucleon bound-state problem in 3D approach. In our calculations, we considered simple potential models, namely spin-independent two-body (2B) interactions. Our numerical results are fair compatible with results of other methods.

Resonances in Electron and Atom Scattering by Molecules. II. Results of Calculations

We describe the main features of the resonant scattering of electrons by molecules and calculate the parameters of the resonances within the framework of two different approaches: (1) the theory of collisions in a two-body system (where it is applicable); (2) the quantum theory of scattering in a fewbody system based on the Faddeev–Yakubovsky equations. We present the results of calculations of the resonant cross sections of electronic and atomic collisions with molecules and compare the obtained results with the available experimental data, as well as with the results of calculations based on other approximations. In addition, we discuss some biological applications (e.g., the properties of biopolymer molecules).

Resonances in Scattering. I. Basic Equations and Main Approximations

We describe the main features of resonances in scattering, determining the resonances in view of the theory of collisions in a two-body system, as well as the resonances emerging as a result of collisions in a few-body system. We analyze regularities in the emergence of such resonances and their characteristics. We discuss the results of calculations of the resonant processes occurring during collisions of electrons with diatomic molecules, in view of the quantum theory of scattering in a few-body system based on the Faddeev–Yakubovsky equations.

Four-body calculation of 12C(α, γ)16O radiative capture reaction at stellar energies

On the basis of the four-alphamodel, the 12C(α, γ)16Oradiative capture process is investigated by using the four-body Faddeev–Yakubovsky equations as well as the two- and three-body electromagnetic currents. The present calculation is an application of our current conservation realistic potentials method for the 12C(α, γ)16Oradiative capture process. This work clears the way formore refinedmodels of radiative capture based on two- and three-body realistic potentials and current conservation. The calculation is carried out by considering the 4He + 12C (1 + 3) and the 8Be + 8Be (2 + 2) subamplitudes, respectively. Radiative capture 12C(α, γ)16Oreaction is one of the most important reactions in nuclear astrophysics. For this reaction, the electric dipole transitions between states with the same isospin are forbidden in the first order. Because the state 1+ and 0+ ground state nuclei 16O have zero isospin, thus the electric dipole radiations are not at the first order between two levels and electric dipole radiation will be the second order and electric dipole radiation is the same order as the electric quadrupole radiation. Therefore, we must consider the effects of both radiations. In comparison with other theoretical methods and available experimental data, good agreement is achieved for the E 1 and E 2 contribution to the cross section and the astrophysical S factor for this process.

Four-Body Faddeev–Yakubovsky Equations Using Two-Cluster RGM Kernels: Applications to 4N and 4α Systems

Four-cluster Faddeev–Yakubovsky calculations using two-cluster RGM kernels are carried out for identical clusters. A precise ground-state energy of the α-particle, predicted by the quark-model nucleon–nucleon (NN) interaction fss2, is Eα = −26.61 MeV, including approximate effects of the Coulomb force and the charge dependence of the 2N force. The missing −1.7 MeV in the experimental value −28.3 MeV is about half of 3–4 MeV, predicted by modern meson-exchange 2N potentials, implying that almost half of 3–4 MeV is attributed to the off-shell effect of our nonlocal NN interaction fss2. As to the applications to four-α system, a method to eliminate the Faddeev redundant components from the basic Faddeev–Yakubovsky equations is proposed.

Trimer–Tetramer Interwoven States in the Scaling Limit

We report here some results we have obtained on the scale dependence of tetramer energies at the unitary limit, considering the number of tetramer energy levels appearing between the ground and the excited Efimov trimers. Our numerical investigation is done by solving a renormalized set of Faddeev–Yakubovsky equations for identical bosons with zero-range interaction, requiring a four-body scale, which in principle can be independent of the trimer properties. The ratio between the three- and four-body scales is introduced by considering the two lower trimer states and corresponding associated tetramers. We conclude that at least three tetramers are possible to exist between two Efimov states by varying the relation between such scales, and considering the relation between three-body Efimov states in the unitary limit. The results for the trimer–tetramer interwoven states are shown through a correlation between tetramers attached to consecutive Efimov trimer states.

Range Corrections to Universal Tetramer Properties

The universal properties of weakly-bound tetramers close to the scaling limit are investigated by solving a subtracted set of Faddeev–Yakubovsky (FY) equations for identical bosons with a zero-range interaction. The solution demands a four-body scale independent of the trimer properties. Furthermore, the effect of a finite effective range is introduced in the FY equations, which we show produces results that are distinct from the scale variation. In particular range effects to two universal scaling functions for the tetramers are investigated. The correlation between successive tetramer energies corresponding to states within two Efimov trimer energies, proposed before and studied close to the unitary limit; and the correlation between the position of the four-atom recombination peaks. In this case, we found a shift in the scaling function due to the range, which can be associated to the shift of the data found for caesium atoms, with respect to zero-range calculations, due to a nonvanishing range in the actual experimental setups.

Four-Boson Systems Close to a Universal Regime

Universal properties of weakly-bound four-boson systems near the scaling limit are discussed by considering recent results obtained from the solution of Faddeev-Yakubovsky (FY) equations, which confirm a previous conjecture on a four-body scale dependence. In the present contribution, within a discussion on our numerical results obtained for the binding energies of two consecutive tetramer states, we are analyzing the relative relevance of the two possible configurations of the four-body system.

Recent Progress in Ab-initio Four-Body Scattering Calculations

In the first part of the contribution, we discuss the results of a recent benchmark calculation of n − 3H and p − 3He phase-shifts below the trinucleon disintegration thresholds. Three different methods—Alt, Grassberger and Sandhas, Hyperspherical Harmonics, and Faddeev–Yakubovsky—have been used and their results are compared. For both n − 3H and p − 3He we observe a rather good agreement between the three different theoretical methods. In the second part of the contribution, we study the longitudinal asymmetry \({A_z^{n^3{\rm He}}}\) in the 3He(n, p)3H reaction in order to obtain information about the parity-violating components of the nucleon–nucleon interaction.

Binding and structure of tetramers in the scaling limit

The momentum-space structure of the Faddeev-Yakubovsky (FY)components of weakly-bound tetramers is investigated at the unitary limit using a renormalized zero-range two-body interaction. The results, obtained by considering a given trimer level with binding energy $B_3$, provide further support to a universal scaling function relating the binding energies of two successive tetramer states. The correlated scaling between the tetramer energies comes from the sensitivity of the four-boson system to a short-range four-body scale. Each excited $N-$th tetramer energy $B_4^{(N)}$ moves as the short-range four-body scale changes, while the trimer properties are kept fixed, with the next excited tetramer $B_4^{(N+1)}$ emerging from the atom-trimer threshold for a universal ratio $B_4^{(N)}/B_3 = B_4^ {(N)}/B_4^{(N+1)} \simeq 4.6$, which does not depend on $N$. We show that both channels of the FY decomposition [atom-trimer ($K-$type) and dimer-dimer ($H-$type)] present high momentum tails, which reflect the short-range four-body scale. We also found that the $H-$channel is favored over $K-$channel at low momentum when the four-body momentum scale largely overcomes the three-body one.

A study of the ground-state energy of 4He with nucleon–nucleon potentials

Ground-state properties are evaluated for the finite nucleus 4He starting from realistic nucleon–nucleon interactions within the framework of the Green’s function approach. For the sake of comparison, the same calculations are performed using the Brueckner–Hartree–Fock approximation. For that purpose four high-quality modern nucleon–nucleon interactions represented in momentum space are employed: the Argonne V18, CD-Bonn, Bonn A and N 3LO potentials. In these potentials, the effects of charge dependence are taken into account. Additional binding energy is obtained from the inclusion of the hole–hole scattering term within the framework of the Green function approach. It has been shown that the Green function results agree well with the results obtained by accurate methods for few-nucleon systems such as the Faddeev–Yakubovsky calculation. In this study, a comparison of the calculated ground-state energies, obtained by using the Green function approach and different nucleon–nucleon potentials, with experimental values is carried out. The results show good agreement between the calculated values and the experimental ones.

Some applications of the Faddeev–Yakubovsky equations to the cold-atom physics

We present some recent applications of the Faddeev--Yakubovsky equations in describing atomic bound and scattering problems. We consider the scattering of a charged particle $X$ by atomic hydrogen with special interest in $X=p,e^{\pm}$, systems of cold bosonic molecules and the bound and scattering properties of N=3 and N=4 atomic $^4$He multimers.

Stability of the s- and p-shell α cluster hypernuclei with strangeness S = −2 to −4

The energies of the s- and p-shell hypernuclei in the α cluster model with strangeness content S = −1 to −4 are calculated in a variational Monte Carlo framework. Two types of Λα and Ξ0α potentials and a variety of three-range Gaussian YY potentials simulated from models of the Nijmegen group are used. The Woods–Saxon Ξ0α potential with central depths −24.0 and −14.0 MeV is considered. Unlike the earlier Faddeev–Yakubovsky results, the system He is as strongly bound for the Isle Ξ0α potential as for the Woods–Saxon shape. The hypernuclei of the 8Be core with strange baryons Ξ0 and ΛΛΞ0 are also found to be stable, but the latter one is overbound without dispersive Λαα force. The strong conversion makes S = −3 hypernucleus Be unstable and no definitive conclusion about the stability of He can be drawn. The attractive three-body ααα force is required to fit the ground-state energy and root-mean-square radius of 12C for a αα potential, similar to earlier works. The BΛ of 13ΛC is not consistent with the data without Λαα dispersive force. The system C has been found to be stable.

Neutron–Triton Elastic Scattering

The Kohn variational principle and the hyperspherical harmonics technique are applied to study the n − 3H elastic scattering at low energies. In this contribution the first results obtained using a non-local realistic interaction derived from the chiral perturbation theory are reported. They are found to be in good agreement with those obtained solving the Faddeev–Yakubovsky equations. The calculated total and differential cross sections are compared with the available experimental data. The effect of including a three-nucleon interaction is also discussed.

Bound state of the αΛΛΞ0 system

The Faddeev–Yakubovsky equations in configuration space are applied to the study of the four-body system αΛΛΞ comprising three distinct particles. We use the OBE-simulating potential of the NSC97 model for the ΛΞ and ΛΛ interactions. For the Ξα and Λα interactions we use phenomenological potentials. The Faddeev–Yakubovsky equations for the system and its subsystems are numerically solved in s-wave approach by the cluster reduction method. We evaluate the binding energy of the hypothetical multi-strangeness nucleus . We find that the existence of the ground state of this nucleus depends strongly on the behavior of the Ξα potential at small distances. In particular, for a Woods–Saxon type potential having no repulsive core, the system can be bound.

WIDTH OF θ-PENTAQUARKS IN RELATIVISTIC QUARK MODEL

The relativistic generalization of Faddeev–Yakubovsky approach is constructed in the form of the dispersion relations. The five-quark amplitudes for the low-lying pentaquarks including the u-, d-, s-quarks are calculated. The poles of these amplitudes determine the masses and widths of θ-pentaquarks.

Faddeev–Yakubovsky calculations for light ΛΛ hypernuclei

New Faddeev–Yakubovsky calculations are reported for 6 ΛΛ He and 10 ΛΛ Be in terms of clusters of α's and Λ's, using ΛΛ s-wave potentials motivated by several of the Nijmegen model interactions. The self consistency of the ΛΛ hypernuclear world data for these species is discussed. The newly reported 6 ΛΛ He event is found to be compatible with ΛΛ interaction strengths provided by the Nijmegen soft-core one-boson-exchange model NSC97. Faddeev calculations for 5 ΛΛ H and 5 ΛΛ He suggest that these ΛΛ hypernuclei are stable against emitting Λ's for any (essentially attractive) ΛΛ interaction, whereas calcualtions for 4 ΛΛ H do not allow a clear-cut conclusion whether or not it is particle stable.