We propose a novel approach to solve the three-nucleon (3N) Faddeev equation which avoids the complicated singularity pattern going with the moving logarithmic singularities of the standard approach. In this new approach the treatment of the 3N Faddeev equation becomes essentially as simple as the treatment of the two-body Lippmann-Schwinger equation. Very good agreement of the new and old approaches in the application to nucleon-deuteron elastic scattering and the breakup reaction is found.
The continuum Faddeev equations for the neutron-neutron-alpha (n-n-$α$) system are formulated for a general interaction as well as for finite rank forces. In addition, the capture process n+n+$α\rightarrow ^6$He+$γ$ is derived.
A new, more efficient approach to include a three nucleon force into three-nucleon continuum calculations is presented. Results obtained in the new and our old approach are compared both for elastic nucleon-deuteron scattering as well as for the breakup process. The advantages of the new scheme are discussed.
We demonstrate a computational scheme which drastically decreases the required time to get theoretical predictions based on chiral two- and three-nucleon forces for observables in three-nucleon continuum. For a three-nucleon force containing N short-range terms all workload is reduced to solving N+1 Faddeev-type integral equations. That done, computation of observables for any combination of strengths of the contact terms is done in a flash. We demonstrate on example of the elastic nucleon-deuteron scattering observables the high precision of the proposed emulator and its capability to reprodu…
The Faddeev equations for the three-body bound state are solved directly as thre e-dimensional integral equations without employing partial wave decomposition. Two-body forces of the Malfliet-Tjon type and simple spin independent genuine three-body forces are considered for the calculation of the three-body binding energy.
The Complex Energy Method [Prog. Theor. Phys. {\bf 109}, 869L (2003)] is applied to the 4-body Faddeev-Yakubovsky equations in the 4-nucleon system. We obtain a well converged solution in all energy regions below and above the 4-nucleon break-up threshold.
The paper is devoted to the $\bar{K}NNN$ system, consisting of an antikaon and three nucleons. Four-body Faddeev-type AGS equations are being solved in order to find possible quasi-bound state in the system.
The paper is devoted to the $\bar{K}NNN$ system, which is an exotic system consisting of an antikaon and three nucleons. Four-body Faddeev-type AGS equations, which are being used for evaluation of the possible quasi-bound state in the system are described.
We present a practical method to solve Faddeev three-body equations at energies above three-body breakup threshold as integral equations in coordinate space. This is an extension of previously used method for bound states and scattering states below three-body breakup threshold energy. We show that breakup components in three-body reactions produce long-range effects on Faddeev integral kernels in coordinate space, and propose numerical procedures to treat these effects. Using these techniques, we solve Faddeev equations for neutron-deuteron scattering to compare with benchmark solutions.
The method of screening and renormalization is used to include the Coulomb interaction between the charged particles in the description of few-body nuclear reactions. Calculations are done in the framework of Faddeev-type equations in momentum-space. The reliability of the method is demonstrated. The Coulomb effect on observables is discussed.
Relativistic Faddeev equations for three-body scattering at arbitrary energies are solved in first order in the two-body transition operator in terms of momentum vectors without employing a partial wave decomposition. Relativistic invariance is incorporated within the framework of Poincar{é} invariant quantum mechanics. Based on a Malfliet-Tjon type interaction, observables for elastic and breakup scattering are calculated and compared to the nonrelativistic ones.
The halo nucleus $^{11}$Li is treated as a three-body system consisting of an inert core of $^{9}$Li plus two valence neutrons. The Faddeev equations are solved using separable potentials to describe the two-body interactions, corresponding in the n-$^{9}$Li subsystem to a p$_{1/2}$ resonance plus a virtual s-wave state. The experimental $^{11}$Li energy is taken as input and the $^{9}$Li transverse momentum distribution in $^{11}$Li is studied.
Momentum space three-body Faddeev-like equations are used to calculate elastic, transfer and charge exchange reactions resulting from the scattering of deuterons on 12C and 16O or protons on 13C and 17O; 12C and 16O are treated as inert cores. All possible reactions are calculated in the framework of the same model space. Comparison with previous calculations based on approximate methods used in nuclear reaction theory is discussed.
A novel approach to solve the Faddeev equation for three-body scattering at arbitrary energies is proposed. This approach disentangles the complicated singularity structure of the free three-nucleon propagator leading to the moving and logarithmic singularities in standard treatments. The Faddeev equation is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. In its simplest form the Faddeev equation for identical bosons, which we are using, is an integral equation in five variables, magnitudes of relative momenta and an…
A novel approach is developed to find the three-body breakup amplitudes and cross sections within the modified Faddeev equation framework. The method is based on the lattice-like discretization of the three-body continuum with a three-body stationary wave-packet basis in momentum space. The approach makes it possible to simplify drastically all the three- and few-body breakup calculations due to discrete wave-packet representations for the few-body continuum and simultaneous lattice representation for all the scattering operators entering the integral equation kernels. As a result, the few-bod…
FaCE is a self contained programme, with namelist input, that solves the three body Faddeev equations. It enables the inclusion of excitation of one of the three bodies, whilst the other two remain inert. It is particularly useful for obtaining the binding energies and bound state structure compositions of light exotic nuclei treated as three-body systems, given the three effective two body interactions. A large variety of forms for these interactions may be defined, and supersymmetric transformations of these potentials may be calculated whenever two body states need to be removed due to Paul…
Algorithm, based on explicit representations for analytic continuation of the T-matrix Faddeev components on unphysical sheets, is worked out for calculations of resonances in the three-body quantum problem. According to the representations, poles of T-matrix, scattering matrix and Green function on unphysical sheets, interpreted as resonances, coincide with those complex energy values where appropriate truncations of the scattering matrix have zero as eigenvalue. Scattering amplitudes on the physical sheet, necessary to construct scattering matrix, are calculated on the basis of the Faddeev d…
We demonstrate the feasibility and efficiency of the Coulomb-Sturmian separable expansion method for generating accurate solutions of the Faddeev equations. Results obtained with this method are reported for several benchmark cases of bosonic and fermionic three-body systems. Correct bound-state results in agreement with the ones established in the literature are achieved for short-range interactions. We outline the formalism for the treatment of three-body Coulomb systems and present a bound-state calculation for a three-boson system interacting via Coulomb plus short-range forces. The corres…
The polarization transfer coefficient $K_y^{y'}$ for the $^2$H$(\vec{d}, \vec{p})^3$H reaction at energies $E_{\rm lab.} = 30$, 50, 70, and 90keV is calculated using the four-body Faddeev-Yakubovsky equation. We employ a separable type of the Paris potential. The experimental data [Phys. Rev. {\bf C64}, 047601 (2001)] agrees with our solution at the scattering angle $θ_{\rm c.m.}=0^\circ$, and there is no discrepancy pointed out in that paper. We found that the $K_y^{y'}$ is sensitive to the incident deuteron energy at this angle.
Algorithm, based on explicit representations for analytic continuation of the T-matrix Faddeev components on unphysical sheets, is worked out for calculations of resonances in the three-body quantum problem. According to the representations, poles of the T--matrix, scattering matrix and resolvent on unphysical sheets, interpreted as resonances, coincide with those complex energy values where appropriate truncations of the scattering matrix have zero as eigenvalue. Scattering amplitudes on the physical sheet, necessary to construct scattering matrix, are calculated on the basis of the Faddeev d…
Nucleon-knockout reactions on proton targets (p, pN ) have experienced a renewed interest due to the availability of inverse-kinematics experiment with exotic nuclei. Various theoretical descriptions have been used to describe these reactions, such as the Distorted-Wave Impulse Approximation (DWIA), the Faddeev-type formalism and the Transfer to the Continuum method. Our goal is to benchmark the observables computed with the Faddeev and Transfer to the Continuum formalisms in the intermediate energy regime relevant for the experimental (p, pn) and (p, 2p) studies. In this paper, we analyze the…
A detailed comparison of Faddeev and variational wave functions for $^3$H, calculated with realistic nuclear forces, has been made to study the form of three-body correlations in few-body nuclei. Three new three-body correlations for use in variational wave functions have been identified, which substantially reduce the difference with the Faddeev wave function. The difference between the variational upper bound and the Faddeev binding energy is reduced by half, to typically $<2\%$. These three-body correlations also produce a significant lowering of the variational binding energy for $^4$He an…
Four-body Faddeev-Yakubovsky calculations for Lambda-Lambda-p-n do not produce a bound state for Lambda-Lambda hydrogen 4, although suitably defined three-body Faddeev calculations for Lambda-Lambda-d produce a 1+ bound state for Lambda-Lambda interactions fitted to the Lambda-Lambda helium 6 emulsion event from KEK E373. This is opposite to the normal situation where a four-body Faddeev-Yakubovsky calculation yields stronger binding than that due to a suitably defined three-body Faddeev calculation. For stranger systems, Faddeev calculations using Lambda-Xi interactions which simulate the Nij…
We solve the Faddeev equations for the two meson-one baryon system $ππN$ and coupled channels using the experimental two-body $t$-matrices for the $πN$ interaction as input and unitary chiral dynamics to describe the interaction between the rest of coupled channels. In addition to the $N^*(1710)$ obtained before with the $ππN$ channel, we obtain, for $J^π=1/2^+$ and total isospin of the three-body system $I=1/2$, a resonance peak whose mass is around 2080 MeV and width of 54 MeV, while for $I=3/2$ we find a peak around 2126 MeV and 42 MeV of width. These two resonances can be identified with t…
The equations which relate three-body and two-body symmetry violating scattering amplitudes are derived in the first order of symmetry violating interactions. They can be used to obtain three-body symmetry violating scattering amplitudes from two-body symmetry violating scattering amplitudes calculated in low energy effective field theory.
We discuss the appearance of spurious solutions of few-body equations for Faddeev amplitudes. The identification of spurious states, i.e., states that lack the symmetry required for solutions of the Schroedinger equation, as well as the symmetrization of the Faddeev equations is investigated. As an example, systems of three and four electrons, bound in a harmonic-oscillator potential and interacting by the Coulomb potential, are presented.
Evidence for particle stability of Lambda-Lambda hydrogen-4 (4LLH) has been suggested by the BNL-AGS E906 experiment. We report on Faddeev-Yakubovsky calculations for the four-body Lambda-Lambda-p-n system using Lambda-N interactions which reproduce the observed binding energy of Lambda hydrogen-3 (3LH) within a Faddeev calculation for the Lambda-p-n subsystem. No 4LLH bound state is found over a wide range of Lambda-Lambda interaction strengths, although the Faddeev equations for a three-body Lambda-Lambda-d model of 4LLH admit a 1+ bound state for as weak a Lambda-Lambda interaction strength…
\noindent{\bf Background:} Deuteron-induced nuclear reactions are an essential tool for probing the structure of nuclei as well as astrophysical information such as $(n,γ)$ cross sections. The deuteron-nucleus system is typically described within a Faddeev three-body model consisting of a neutron ($n$), a proton ($p$), and the target nucleus ($A$) interacting through pairwise phenomenological potentials. While Faddeev techniques enable the exact description of the three-body dynamics, their predictive power is limited in part by the omission of irreducible neutron-proton-nucleus three-body for…
The formal scattering theory is developed for the three-particle differential Faddeev equations. The theory is realised along the same line as in the standard two-body case. The solution of the scattering problem is expressed in terms of the matrix T-operator constructed from the matrix resolvent of the differential Faddeev equations. The relationships of the matrix T-operator with elements of transition operators and Faddeev T-matrix components have been established.
We implement complex scaling of Faddeev equations using hyper-spheric coordinates and adiabatic expansion. Complex scaling of coordinates allows convenient calculations of three-body resonances. We derive the necessary equations and investigate the adiabatic spectrum at large distances. We illustrate the viability of the implementation by calculations of several three-body resonances: a $0^+$ resonance in a model benchmark system of three identical bosons; the $2^+$ resonance in the $^6$He nucleus within the $α+n+n$ model; and the two $0^+$ resonances in $^{12}$C within the three-$α$ model.
The set of Faddeev and Lippmann--Schwinger integral equations for three-body systems involving Coulomb interactions deduced from a ``three-potential'' picture are shown to be compact for all energies and a method of solution is given.
A recently developed three-dimensional approach (without partial-wave decomposition) is considered to investigate solutions of Faddeev-Yakubovsky integral equations in momentum space for three- and four-body bound states, with the inclusion of three-body forces. In the calculations of the binding energies, spin-dependent nucleon-nucleon (NN) potential models (named, S$_{3}$, MT-I/III, YS-type and P$_{5.5}$GL) are considered along with the scalar two-meson exchange three-body potential. Good agreement of the presently reported results with the ones obtained by other techniques are obtained, dem…
Three-body Faddeev equations are considered as a spectral problem for nonsymmetrical matrix operator. Invariant subspaces related to physical and spurious solutions to Faddeev equations and its adjoint are described. Respective eigenvectors corresponding to invariant subspaces are constructed for Faddeev operator and its adjoint. Extensions of the formalism on CCA equations are discussed.
A novel method for calculating resonances in three-body Coulombic systems is presented. The Faddeev-Merkuriev integral equations are solved by applying the Coulomb-Sturmian separable expansion method. To show the power of the method we calculate resonances of the three-$α$ and the $H^-$ systems.
Obtaining cross sections for nuclear reactions at intermediate energies based on the Glauber formulation has a long tradition. Only recently the energy regime of a few hundred MeV has become accessible to ab-initio Faddeev calculations of three-body scattering. In order to go to higher energies, the Faddeev equation for three-body scattering is formulated and directly solved without employing a partial wave decomposition. In the simplest form the Faddeev equation for interacting scalar particles is a three-dimensional integral equation in five variables, from which the total cross section, the…
Using separable $\bar{K}N-πΣ$ potentials in the Faddeev equations, we calculated the binding energies and widths of the $K^{-}pp$, $K^{-}ppn$ and $K^{-}ppp$ quasi-bound states on the basis of three- and four-body Alt-Grassberger-Sandhas equations in the momentum representation. One- and two-pole version of $\bar{K}N-πΣ$ interaction are considered and the dependence of the resulting few-body energy on the two-body $\bar{K}N-πΣ$ potential was investigated. The $s$-wave [3+1] and [2+2] sub-amplitudes are obtained by using the Hilbert-Schmidt expansion procedure for the integral kernels. As a resu…
In this paper, we study the relativistic effects in a three-body bound state. For this purpose, the relativistic form of the Faddeev equations is solved in momentum space as a function of the Jacobi momentum vectors without using a partial wave decomposition. The inputs for the three-dimensional Faddeev integral equation are the off-shell boost two-body $t-$matrices, which are calculated directly from the boost two-body interactions by solving the Lippmann-Schwinger equation. The matrix elements of the boost interactions are obtained from the nonrelativistic interactions by solving a nonlinear…
The first ever numerical solution of five-body Faddeev-Yakubovsky equations is presented in this work. Modern realistic Nucleon-Nucleon Hamiltonians have been tested when describing low energy elastic neutron scattering on $^4$He nucleus. Obtained results have been compared with those available in the literature and based on solution of the Schrödinger equation.
We use equations for Faddeev amplitudes to solve the shell-model problem for four nucleons in the model space that includes up to 14 hbar Omega harmonic-oscillator excitations above the unperturbed ground state. Two- and three-body effective interactions derived from the Reid93 and Argonne V8' nucleon-nucleon potentials are used in the calculations. Binding energies, excitations energies, point-nucleon radii and electromagnetic and strangeness charge form factors for 4He are studied. The structure of the Faddeev-like equations is discussed and a formula for matrix elements of the permutation o…
Coupled-channel three-body calculations of an $I=1/2$, $J^π=0^-$ $\bar{K}NN$ quasi-bound state in the $\bar{K}NN - πΣN$ system were performed and the dependence of the resulting three-body energy on the two-body $\bar{K}N - πΣ$ interaction was investigated. Earlier results of binding energy $B_{K^-pp} \sim 50 -70$ MeV and width $Γ_{K^-pp} \sim 100$ MeV are confirmed [N.V. Shevchenko {\it et al.}, Phys. Rev. Lett. {\bf 98}, 082301 (2007)]. It is shown that a suitably constructed energy-independent complex $\bar{K}N$ potential gives a considerably shallower and narrower three-body quasi-bound st…
We propose a method that allows for the efficient solution of the three-body Faddeev equations in the presence of infinitely rising confinement interactions. Such a method is useful in calculations of nonrelativistic and especially semirelativistic constituent quark models. The convergence of the partial wave series is accelerated and possible spurious contributions in the Faddeev components are avoided. We demonstrate how the method works with the example of the Goldstone-boson-exchange chiral quark model for baryons.
The work is devoted to comparison of two different approaches to calculation of three-body resonances on the basis of the Faddeev differential equations. The first one is the well known complex scaling approach. The second method is based on an immediate calculation of the zeros of the scattering matrix continued to the physical sheet.
A modified version of the Faddeev three-body equation to accommodate the Coulomb interaction, which was used in the study of three-nucleon bound states, is applied to the proton-deuteron scattering problem at energies below the three-body breakup threshold. A formal derivation of the equation in a time-independent scattering theory is given. Numerical results for phase shift parameters are presented to be compared with those of another methods and results of the phase shift analysis. Differential cross sections and nucleon analyzing powers are calculated with the effects of three-nucleon force…
In this paper, I prove that the sub-Coulomb $A(d,p)B$ reaction amplitude, which is a solution of the three-body Faddeev equations in the Alt-Grassberger-Sandhas (AGS) form, is peripheral if a peripheral is the corresponding DWBA amplitude. Hence the Faddeev's reaction amplitude for the sub-Coulomb $A(d,p)B$ reactions can also be parametrized in terms of the ANC of the $(n\,A)$ bound state. First, I consider the original AGS equations with separable potentials and prove that such equations are peripheral at sub-Coulomb energies. After that, the two-particle AGS equations are derived for the gen…
We present a perturbative approach to solving the three-nucleon continuum Faddeev equation. This approach is particularly well suited to dealing with variable strengths of contact terms in a chiral three-nucleon force. We use examples of observables in the elastic nucleon-deuteron scattering as well as in the deuteron breakup reaction to demonstrate high precision of the proposed procedure and its capability to reproduce exact results. A significant reduction of computer time achieved by the perturbative approach in comparison to exact treatment makes this approach valuable for fine-tuning of …
Given the phenomenological success of the Nambu-Jona-Lasinio model in describing the meson physics in the low energy limit, it is tempting to find the fully relativistically structured nucleon solution in the same model under the similar approximation employed in the mesonic sector. To achieve this goal we need to solve a relativistic Faddeev equation. The factorizability of the two-body T-matrix reduces the three-body Faddeev equation to a tractable two-body Bethe-Salpeter equation. The reduced equation is then solved numerically. Our result indicates that the nucleon consists of three loosel…
pd capture processes at various energies have been analyzed based on solutions of 3N-Faddeev equations and using modern NN forces. The application of the Siegert theorem is compared to the explicit use of $π$- and $ρ$-like exchange currents connected to the AV18 NN interaction. Overall good agreement with cross sections and spin observables has been obtained but leaving room for improvement in some cases. Feasibility studies for 3NF's consistently included in the 3N continuum and the 3N bound state have been performed as well.
Background: The numerical solution of few-body scattering problems with realistic interactions is a difficult problem that normally must be solved on powerful supercomputers, taking a lot of computer time. This strongly limits the possibility of accurate treatments for many important few-particle problems in different branches of quantum physics. Purpose: To develop a new general highly effective approach for the practical solution of few-body scattering equations that can be implemented on a graphics processing unit. Methods: The general approach is realized in three steps: (i) the reformulat…
Low-energy muon-transfer cross sections and rates in collisions of muonic atoms with hydrogen isotopes are calculated using a six-state close-coupling approximation to coordinate-space Faddeev-Hahn-type equations. In the muonic case satisfactory results are obtained for all hydrogen isotopes and the experimentaly observed strong isotopic dependence of transfer rates is also reproduced. A comparison with results of other theoretical and available experimental works is presented. The present model also leads to good transfer cross sections in the well-understood problem of antihydrogen formation…
Effective lambda-proton and lambda-neutron potentials, restored from theoretical scattering phases through Gel'fand-Levitan-Marchenko theory, are tested on a lambda hypertriton through three-body calculations. The lambda hypertriton is treated as a three-body system consisting of lambda-proton, lambda-neutron and proton-neutron subsystems. Binding energy and root-mean-square radius are computed for the ground state of lambda hypertriton ($J^π=1/2^+$). In coordinate space, the dynamics of the system is described using a set of coupled hyperradial equations obtained from the Differential Faddeev…
We have introduced a spin-isospin dependent three-dimensional approach for formulation of the three-nucleon scattering. 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.
Theoretical description of reactions in general, and the theory for $(d,p)$ reactions, in particular, needs to advance into the new century. Here deuteron stripping processes off a target nucleus consisting of ${A}$ nucleons are treated within the framework of the few-body integral equations theory. The generalized Faddeev equations in the AGS form, which take into account the target excitations, with realistic optical potentials provide the most advanced and complete description of the deuteron stripping. The main problem in practical application of such equations is the screening of the Coul…
An interaction of a photon with $^3H$ is invstigated based on a three dimensional Faddeev approach. In this approach the three-nucleon Faddeev equations with two-nucleon interactions are formulated with consideration of the magnitude of the vector Jacobi momenta and the angle between them with the inclusion of the spin-isospin quantum numbers, without employing a partial wave decomposition. In this formulation the two body t-matrices and triton wave function are calculated in the three dimensional approach using AV18 potential. In the first step we use the standard single nucleon current in th…
$\Kbar N$ interactions are investigated {\it via} an effective non-linear chiral meson-baryon Lagrangian. The adjustable parameters are determined by a fitting procedure on the $K^-p$ threshold branching ratios and total cross-section data for $p^{lab}_K\le$ 250 MeV/c. We produce predictions for the $Σπ$ mass spectrum, and scattering lenghts $a_{K^-p}$, $a_n(K^-n \to K^-n)$, $a_n0(\Kbar0 n \to \Kbar0 n)$, and $a_{ex}(K^-p \to \Kbar0 n)$. The $\Kbar N$ amplitudes thus obtained, as well as those for other two-body channels ($πN$, $NN$, and $YN$) are used as input to predict the scattering length…
The Faddeev equation for three-body scattering at arbitrary energies is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. In its simplest form the Faddeev equation for identical bosons, which we are using, is a three-dimensional integral equation in five variables, magnitudes of relative momenta and angles. This equation is solved through Pade summation. Based on a Malfliet-Tjon-type potential, the numerical feasibility and stability of the algorithm for solving the Faddeev equation is demonstrated. Special attention i…
We propose a novel method for calculating resonances in three-body Coulombic systems. The method is based on the solution of the set of Faddeev and Lippmann-Schwinger integral equations, which are designed for solving the three-body Coulomb problem. The resonances of the three-body system are defined as the complex-energy solutions of the homogeneous Faddeev integral equations. We show how the kernels of the integral equations should be continued analytically in order that we get resonances. As a numerical illustration a toy model for the three-$α$ system is solved.
The Continuum Discretized Coupled Channels (CDCC) method is a well established theory for direct nuclear reactions which includes breakup to all orders. Alternatively, the 3-body problem can be solved exactly within the Faddeev formalism which explicitly includes breakup and transfer channels to all orders. With the aim to understand how CDCC compares with the exact 3-body Faddeev formulation, we study deuteron induced reactions on: i) $^{10}$Be at $E_{\rm d}= 21.4, 40.9 \; {\rm and} \; 71$ MeV; ii) $^{12}$C at $E_{\rm d} = 12 \; {\rm and} \; 56$ MeV; and iii) $^{48}$Ca at $E_{\rm d} = 56$ MeV…
The continuum discretized coupled channels (CDCC) method is compared to the exact solution of the three-body Faddeev equations in momentum space. We present results for: i) elastic and breakup observables of d-12C at E_d=56 MeV, ii) elastic scattering of d-58Ni at E_d=80 MeV, and iii) elastic, breakup and transfer observables for 11Be+p at E_{11Be}/A=38.4 MeV. Our comparative studies show that, in the first two cases, the CDCC method is a good approximation to the full three-body Faddeev solution, but for the 11Be exotic nucleus, depending on the observable or the kinematic regime, it may miss…
