Faddeev Equations Research Articles

Antiproton-deuteron hydrogenic states in optical models

By solving the Faddeev equations for the p̄pn system, we compute the antiproton-deuteron level shifts and widths for the lowest hydrogenic states as well as the corresponding p̄d scattering lengths and volumes. The p̄d annihilation densities are obtained and compared to the nuclear density of deuterium. The validity of the Trueman relation for composite particles is studied. The strong part of N̄N interaction is described by two different optical models, including the p̄p-n̄n coupling and n-p mass difference, while for NN several realistic interactions are used.

Possible Extension of the Three-body Force by the Unitary Clothing Transformations Method in the Faddeev Equations

Possible Extension of the Three-body Force by the Unitary Clothing Transformations Method in the Faddeev Equations

Computational approaches for three-nucleon systems

We revisit the three-nucleon system with two kinds of nucleon–nucleon interactions: separable potentials and chiral forces, showing the computational aspects in detail. We start with S-wave separable potentials for which there are simplified forms for the Faddeev equations describing the scattering and binding of three nucleons. We then discuss the partial-wave projected case with a chiral potential considering only one channel so that the computational details can be clearly shown.

A Realistic Approach to the $$\varXi NN$$ Bound-State Problem Based on Faddeev Equation

The Faddeev equations for the \(\varXi NN\) bound-state problem are solved where the three \(S=-2\) baryon–baryon interactions of Jülich-Bonn-München chiral EFT, HAL QCD and Nijmegen ESC08c are used. The T-matrix \(T_{\varXi N, \varXi N}\) obtained within the original \(\varLambda \varLambda -\varXi N-\varSigma \varSigma /\varXi N-\varLambda \varSigma -\varSigma \varSigma \) coupled-channel framework is employed as an input to the equations. We found no bound state for Jülich-Bonn-München chiral EFT and HAL QCD but ESC08c generates a bound state with the total isospin and spin-parity \((T,J^{\pi })=(1/2, 3/2^+)\) where the decays into \(\varLambda \varLambda N\) are suppressed.

Comprehensive investigation of the symmetric space-star configuration in the nucleon-deuteron breakup

We examine a description of available cross section data for symmetric space star (SST) configurations in the neutron-deuteron (nd) and proton-deuteron (pd) breakup reaction using numerically exact solutions of the three-nucleon (3N) Faddeev equation based on two- and three-nucleon (semi)phenomenological and chiral forces. The predicted SST cross sections are very stable with respect to the underlying dynamics for incoming nucleon laboratory energies below $\approx 25$ MeV. We discuss possible origins of the surprising discrepancies between theory and data found in low-energy nd and pd SST breakup measurements.

Efficient emulator for solving three-nucleon continuum Faddeev equations with chiral three-nucleon force comprising any number of contact terms

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 reproduce exact results.

Trinucleon form factors with relativistic multirank separable kernels

This paper studies elastic electron-trinucleon scattering in the relativistic impulse approximation. The amplitudes for a trinucleon have been obtained by solving the relativistic generalization of the Faddeev equations with a multirank separable kernel of the nucleon-nucleon interactions. The static approximation and additional relativistic corrections for the trinucleon electromagnetic form factors have been calculated for the momentum transfer squared up to 50 fm−2.

Theoretical Study of Weakly-Bound Triatomic Systems with Faddeev Equations in the Total Orbital Momentum Representation

Theoretical Study of Weakly-Bound Triatomic Systems with Faddeev Equations in the Total Orbital Momentum Representation

Perturbative Treatment of Three-Nucleon Force Contact Terms in Three-Nucleon Faddeev Equations

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 the three-nucleon Hamiltonian parameters.

Form factors of the nucleon axial current

A symmetry-preserving Poincaré-covariant quark+diquark Faddeev equation treatment of the nucleon is used to deliver parameter-free predictions for the nucleon's axial and induced pseudoscalar form factors, GA and GP, respectively. The result for GA can reliably be represented by a dipole form factor characterised by an axial charge gA=GA(0)=1.25(3) and a mass-scale MA=1.23(3)mN, where mN is the nucleon mass; and regarding GP, the induced pseudoscalar charge gp⁎=8.80(23), the ratio gp⁎/gA=7.04(22), and the pion pole dominance Ansatz is found to provide a reliable estimate of the directly computed result. The ratio of flavour-separated quark axial charges is also calculated: gAd/gAu=−0.16(2). This value expresses a marked suppression of the size of the d-quark component relative to that found in nonrelativistic quark models and owes to the presence of strong diquark correlations in the nucleon Faddeev wave function – both scalar and axial-vector, with the scalar diquark being dominant. The predicted form for GA should provide a sound foundation for analyses of the neutrino-nucleus and antineutrino-nucleus cross-sections that are relevant to modern accelerator neutrino experiments.

Cross section of neutrons from the H2(n,2n) reaction at En=15 MeV

The double-differential cross section of the deuteron breakup reaction $^{2}\mathrm{H}(n,2n$) has been studied experimentally with a neutron beam energy of 15 MeV. Special attention has been devoted to estimation of background condition and multiple scattering effect in the scattering sample. Experimental data have been compared with models based on phase-space approximation used in the ENDF/B-VIII.0 data library and in the MCNP neutron transport code, as well as with rigorous model based on Faddeev equations used for cross section evaluations in JENDL data library. It was found that experimental data are better reproduced by Faddeev model, however, the model overestimates data in the low-energy region of the neutron spectrum ($<4$ MeV).

Three-body model for K(1460) resonance

The three-body $KK\bar K$ model for the $K(1460)$ resonance is developed on the basis of the Faddeev equations in configuration space. A single-channel approach is using with taking into account the difference of masses of neutral and charged kaons. It is demonstrated that a splitting the mass of the $K(1460)$ resonance takes a place around 1460 MeV according to $K^0K^0{\bar K}^0$, $K^0K^+K^-$ and $K^+K^0{\bar K}^0$, $ K^+K^+K^-$ neutral and charged particle configurations, respectively. The calculations are performed with two sets of $KK$ and $K\bar K$ phenomenological potentials, where the latter interaction is considered the same for the isospin singlet and triplet states. The effect of repulsion of the $KK$ interaction on the mass of the $KK\bar K$ system is studied and the effect of the mass polarization is evaluated. The first time the Coulomb interaction for description of the $K(1460)$ resonance is considered. The mass splitting in the $K$(1460) resonances is evaluated to be in range of 10 MeV with taking into account the Coulomb force. The three-body model with the $K\bar K$ potential, which has the different strength of the isospin singlet and triplet parts that are related by the condition of obtaining a quasi-bound three-body state is also considered. Our results are in reasonable agreement with the experimental mass of the $K(1460)$ resonance.

Relativistic rank-one separable kernel for helium-3 charge form factor

Elastic electron-helium-3 scattering is studied in the relativistic impulse approximation. The amplitudes for the three-nucleon system are obtained by solving the relativistic generalization of the Faddeev equations with a separable rank-one kernel of the nucleon-nucleon interaction. The contribution of partial-wave states with nonzero orbital moment of a nucleon pair is investigated. The static approximation and additional relativistic corrections for the He3 charge form factor are calculated for the momentum transfer squared up to 100 fm−2. The contribution of the D13 partial-wave state and relativistic corrections are found to be sizable at high energies.

Faddeev calculations on lambda hypertriton with potentials from Gel’fand–Levitan–Marchenko theory

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 (). In coordinate space, the dynamics of the system is described using a set of coupled hyperradial equations obtained from the differential Faddeev equations. By solving the eigenvalue problem derived from this set of coupled hyperradial equations, the binding energy and root mean square matter radius computed are found to be −2.462 MeV and 7.00 fm, respectively. The potentials are also shown to display a satisfactory convergence behaviour.

Few-Body Systems Consisting of Mesons

We present a work which is meant to inspire the few-body practitioners to venture into the study of new, more exotic, systems and to hadron physicists, working mostly on two-body problems, to move in the direction of studying related few-body systems. For this purpose we devote the discussions in the introduction to show how the input two-body amplitudes can be easily obtained using techniques of the chiral unitary theory, or its extensions to the heavy quark sector. We then briefly explain how these amplitudes can be used to solve the Faddeev equations or a simpler version obtained by treating the three-body scattering as that of a particle on a fixed center. Further, we give some examples of the results obtained by studying systems involving mesons. We have also addressed the field of many meson systems, which is currently almost unexplored, but for which we envisage a bright future. Finally, we give a complete list of works dealing with unconventional few-body systems involving one or several mesons, summarizing in this way the findings on the topic, and providing a motivation for those willing to investigate such systems.

Lower Bound for ppK– Quasi-Bound State Energy

The kaonic cluster $$pp{{K}^{ - }}$$ is described as $$NN\bar {K}({{s}_{{NN}}} = 0)$$ system within the isospin formalism based on the configuration space Faddeev equations. In the model, the isospin singlet $$N\bar {K}$$ state is a deeply bound state associated with $$\Lambda (1405)$$ hyperon. The phenomenological isospin-dependent $$N\bar {K}$$ potentials are used for the calculation of $$pp{{K}^{ - }}$$ quasi-bound state energy. The existence of a lower bound for the energy is shown by theoretical and numerical analysis. In contrast to the isospin formalism based model, the isospinless model obtained by isospin averaging $$N\bar {K}$$ potential is also considered. The results of both models are compared.

Nucleon elastic form factors at accessible large spacelike momenta

A Poincar\'e-covariant quark+diquark Faddeev equation is used to compute nucleon elastic form factors on $0\leq Q^2\leq 18 \,m_N^2$ ($m_N$ is the nucleon mass) and elucidate their role as probes of emergent hadronic mass in the Standard Model. The calculations expose features of the form factors that can be tested in new generation experiments at existing facilities, e.g. a zero in $G_E^p/G_M^p$; a maximum in $G_E^n/G_M^n$; and a zero in the proton's $d$-quark Dirac form factor, $F_1^d$. Additionally, examination of the associated light-front-transverse number and anomalous magnetisation densities reveals, inter alia: a marked excess of valence $u$-quarks in the neighbourhood of the proton's centre of transverse momentum; and that the valence $d$-quark is markedly more active magnetically than either of the valence $u$-quarks. The calculations and analysis also reveal other aspects of nucleon structure that could be tested with a high-luminosity accelerator capable of delivering higher beam energies than are currently available.

Signature of the N* resonance in the mass spectrum of the KK¯N decay channels

Three-body calculations of $I=\frac{1}{2}$, $J^{\pi}=\frac{1}{2}^{+}$ state of $K\bar{K}N$ system were performed. Using separable potentials for two-body interactions in the Faddeev equation, different reaction process for $K\bar{K}N$ three-body system were studied. Within this method, the $\pi\Sigma{K}$ and $\pi\eta{N}$ mass spectra were extracted. Different types of $\bar{K}N-\pi\Sigma$ potentials based on phenomenological and chiral SU(3) approach were used and the dependence of the mass spectra to the $\bar{K}N$ model of interaction were studied. It was shown that the $\pi\Sigma{K}$ and $\pi\eta{N}$ mass spectra could be a useful tool to study the properties of the $N^{\star}$ resonance.

Three-body losses of a polarized Fermi gas near a p -wave Feshbach resonance in effective field theory

We study three-body recombination of fully spin-polarized ${}^6$Li atoms that are interacting resonantly in relative p-waves. Motivated by a recent experiment, we focus on negative scattering volumes where three atoms recombine into a deep dimer and another atom. We calculate the three-body recombination rate using a Faddeev equation derived from effective field theory. In particular, we study the magnetic field and temperature dependences of the loss rate and use the recombination data to determine the effective range of the p-wave atom-atom interaction. We also predict the existence of a shallow three-body bound state that manifests itself as a prominent feature in the energy-dependent three-body recombination rate.

Three-nucleon force effects in inclusive spectra of the neutron-deuteron breakup reaction

We investigate the sensitivity of the non-exclusive nucleon induced deuteron breakup reaction to the three-nucleon interaction and distributions of three-nucleon force effects in inclusive spectra. To this end we solve the three-nucleon Faddeev equation at a number of incoming nucleon laboratory energies using the CD Bonn nucleon-nucleon interaction alone or combined with the 2{\pi}-exchange Tucson-Melbourne three-nucleon force. Based on these solutions energy spectra of an outgoing nucleon, at a specified detection angle as well as spectra integrated over that angle, are calculated. By integrating the spectra at a given angle over the energy of the outgoing nucleon the angular distributions of three-nucleon force effects in the breakup process are additionally obtained. Contrary to elastic nucleon-deuteron scattering, where at higher energies significant three-nucleon force effects were encountered for scattering angles around the minimum of the cross section, for the breakup process only moderate effects are found and they are restricted to forward angles. Results of the present investigation show that the large three-nucleon force effects found for some specific complete breakup configurations are reduced substantially in the incomplete spectra when averaging over contributing complete geometries is performed.