Faddeev Approach Research Articles

Space-like electromagnetic form factors of lambda- and sigma-baryons from quark-diquark Faddeev equations

An important goal of ongoing and future experiments is to explore spectra and transition form factors of baryons with non-zero strangeness. Of particular interest is the transition form factorγ(∗)Σ0→Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma ^{(*)} \\Sigma ^0 \\rightarrow \\Lambda $$\\end{document} in the time-like momentum region that can be extracted from Dalitz decays. On the road towards a theoretical description of these form factors we extend a covariant dynamical quark-diquark model for the baryon Faddeev equation to the strange-quark sector. Based on an excellent description of the mass spectrum of selected baryon octet and decuplet states and reasonable results for the nucleon form factors we determine the elastic electromagnetic form factors of Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} and Σ+,Σ0,Σ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma ^+, \\Sigma ^0, \\Sigma ^-$$\\end{document} hyperons in the space-like region as well as the ones for the octet transition γ(∗)Σ0→Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma ^{(*)} \\Sigma ^0 \\rightarrow \\Lambda $$\\end{document}. We discuss qualitative and quantitative features of the diquark-quark picture and compare systematically with previous results from a three-body Faddeev approach and lattice data where available.

The nucleon-induced deuteron breakup process as a laboratory for chiral dynamics

The nucleon-induced deuteron breakup reaction is studied within the Faddeev approach at incoming nucleon laboratory energies of 135 and 200 MeV. The chiral semilocal momentum-space (SMS) potential developed up to N4LO+, supplemented by the N2LO three-nucleon interaction, is used. Our investigation is focused on the determination of theoretical uncertainties in a predicted cross section related to its dependence on the value of the cutoff parameter of the regulator. We also compare predictions based on the complete N2LO potential with those based on the two-nucleon force upgraded to the N4LO+ order and augmented with the N2LO three-nucleon force. In addition, we study the three-nucleon force effects predicted by this model of interaction. Our systematic study covers the entire kinematically allowed phase space; however, our main results are obtained when additional restrictions on energies and cross section values are imposed. In such a case, we observe that the dependence of the differential cross sections on the regulator cutoff is moderate at 135 MeV and much stronger at 200 MeV. For the latter energy, it can amount to up to 45% in specific kinematic configurations. Taking into account terms beyond, N2LO in a two-body interaction changes the cross section up to 20% (27%) at E = 135(200) MeV. The inclusion of the three-nucleon force leads to effects of approximately 27% at both energies. We illustrate these dependencies with a few examples of the exclusive cross section as a function of the arc length of the S-curve.

Simulation Chemical Reactions in Faddeev Approach

The main features of resonance in scattering are described and resonances are determined on the basis of the theory of collisions in a two-body system, as well as resonances emerging as a result of collisions in a few-body system. Regularities in the emergence of such resonances and their characteristics are analyzed. The results of calculations of these resonant processes occurring during collisions of electrons with diatomic molecules, made on the basis of the quantum theory of scattering in a few body system, based on Faddeev-Yakubovsky equations are discussed. The results of calculations of the resonant cross sections of electron and atom collisions with molecules are presented. Obtained results are compared with the available experimental data and with the results of calculations based on other approximations. In addition some biological applications (e.g. properties biopolymer molecules) are presented.

Quantifying uncertainties due to irreducible three-body forces in deuteron-nucleus reactions

Background: Deuteron-induced nuclear reactions are an essential tool for probing the structure of nuclei as well as astrophysical information such as $(n,\ensuremath{\gamma})$ 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 force ($n\text{\ensuremath{-}}p\text{\ensuremath{-}}A$ 3BF).Purpose: Our goal is to quantify systematic uncertainties stemming from the reduction of deuteron-nucleus ($d+A$) dynamics to a picture of three pointlike nuclear clusters interacting via pairwise nucleon-nucleus forces, using as testing grounds $d+\ensuremath{\alpha}$ scattering and the $^{6}\mathrm{Li}$ ground state. We particularly focus on quantifying uncertainties arising from the full antisymmetrization of the ($A+2$)-body system with the target nucleus fixed in its ground state.Methods: We adopt the ab initio no-core shell model coupled with the resonating group method (NCSM/RGM) to compute microscopic $n\text{\ensuremath{-}}\ensuremath{\alpha}$ and $p\text{\ensuremath{-}}\ensuremath{\alpha}$ interactions, and use them in a three-body description of the $d+\ensuremath{\alpha}$ system by means of momentum-space Faddeev-type equations. Simultaneously, we also carry out ab initio calculations of $d+\ensuremath{\alpha}$ scattering and $^{6}\mathrm{Li}$ ground state by means of six-body NCSM/RGM calculations to serve as a benchmark for the three-body model predictions given by the Faddeev calculations.Results: By comparing the Faddeev and NCSM/RGM results, we show that the irreducible $n\text{\ensuremath{-}}p\text{\ensuremath{-}}\ensuremath{\alpha}$ 3BF has a non-negligible effect on bound state and scattering observables alike. Specifically, the Faddeev approach yields a $^{6}\mathrm{Li}$ ground state that is approximately 600 keV shallower than the one obtained with the NCSM/RGM. Additionally, the Faddeev calculations for $d+\ensuremath{\alpha}$ scattering yield a ${3}^{+}$ resonance that is located approximately 400 keV higher in energy compared to the NCSM/RGM result. The shape of the $d+\ensuremath{\alpha}$ angular distributions computed using the two approaches also differ, owing to the discrepancy in the predictions of the ${3}^{+}$ resonance energy.Conclusions: The Faddeev three-body model predictions for $d+\ensuremath{\alpha}$ scattering and $^{6}\mathrm{Li}$ using microscopic $n\text{\ensuremath{-}}\ensuremath{\alpha}$ and $p\text{\ensuremath{-}}\ensuremath{\alpha}$ potentials differ from those computed microscopically with the NCSM/RGM. These discrepancies are due to the $n\text{\ensuremath{-}}p\text{\ensuremath{-}}\ensuremath{\alpha}$ 3BF, which arises from two-nucleon exchange terms in the microscopic $d\text{\ensuremath{-}}\ensuremath{\alpha}$ interaction and are not accounted for in the three-body model Faddeev calculations. This study lays the foundation for future parametrizations of the 3BF due to Pauli exclusion principle effects in improved three-body calculations of deuteron-induced reactions.

Resonances in Chemical Reactions in Faddeev Approach

The main features of resonance in scattering are described and resonances are determined on the basis of the theory of collisions in a two-body system, as well as resonances emerging as a result of collisions in a few-body system. Regularities in the emergence of such resonances and their characteristics are analyzed. The results of calculations of these resonant processes occurring during collisions of electrons with diatomic molecules, made on the basis of the quantum theory of scattering in a few-body system, based on Faddeev-Yakubovsky equations are discussed. The results of calculations of the resonant cross sections of electron and atom collisions with molecules are presented. Obtained results are compared with the available experimental data and with the results of calculations based on other approximations. In addition, some biological applications (e.g. properties biopolymer molecules) are presented.

Electron-Phonon Model in the Faddeev-Jackiw Quantization Formalism

The Faddeev and Jackiw procedure for the quantization of constrained gauge systems is used on the analysis of an interacting electron-phonon model including Chern-Simons term. The generalized FJ brackets of this system with first-class constraints are found by removing the singularity in the symplectic matrix. This was done by adding the gauge-fixing term in the symplectic potential. Compared to Dirac’s method, this Faddeev and Jackiw approach is quite straightforward for these systems.

Various types of theoretical uncertainties by example of the elastic nucleon-deuteron scattering spin correlation coefficient Cz,x

The elastic nucleon-deuteron scattering process can be used to study three-nucleon dynamics, but in order to draw reliable conclusions based on the comparison of theoretical predictions and data, an estimation of theoretical uncertainties is necessary. We consider the statistical errors arising from a propagation of uncertainties of the two-nucleon interaction parameters to the three-nucleon system, truncation errors, regulator dependencies, and uncertainties arising from using the formalism of partial waves. We give an example of the nucleon-deuteron spin correlation coefficient Cz,x, obtained within the Faddeev approach, to discuss magnitudes of the above mentioned theoretical uncertainties. We find that using various models of the nuclear forces results in the dominant theoretical uncertainty for the nucleon-deuteron elastic scattering observables.

Variational approach to N -body interactions in finite volume

We explore variational approach to the finite-volume $N$-body problem. The general formalism for N non-relativistic spinless particles interacting with periodic pair-wise potentials yields N-body secular equations. The solutions depend on the infinite-volume N-body wave functions. Given that the infinite-volume N-body dynamics may be solved by the standard Faddeev approach, the variational N-body formalism can provide a convenient numerical framework for finding discrete energy spectra in periodic lattice structures.

Faddeev approach to the reaction K−d→πΣn at pK=1GeV/c

The reaction $K^-d \to \pi \Sigma n$ is studied within a Faddeev-type approach, with emphasis on the specific kinematics of the E31 experiment at J-PARC, i.e. $K^-$ beam momentum of $p_K = 1$ GeV/c and neutron angle of $\theta_n=0^\circ$. The employed Faddeev approach requires as main input amplitudes for the two-body subsystems $\bar KN \to \bar KN$ and $\bar KN \to \pi\Sigma$. For the latter results from recently published chiral unitary models of the $\bar KN$ interaction are utilized. The $\bar KN \to \bar KN$ amplitude itself, however, is taken from a recent partial-wave analysis. Due to the large incoming momentum of the $K^-$, the $\bar KN$ interaction is probed in a kinematical regime where those chiral potentials are no longer applicable. A comparison of the predicted spectrum for various $\pi \Sigma$ charge channels with preliminary data is made and reveals a remarkable agreement as far as the magnitude and the line shape in general is concerned. Noticeable differences observed in the $\pi\Sigma$ spectrum around the $\bar KN$ threshold, i.e. in the region of the $\Lambda$(1405) resonance, indicate a sensitivity to the details of the employed $\bar KN \to \pi\Sigma$ amplitudes and suggest that pertinent high-precision data could indeed provide substantial constraints on the structure of the $\Lambda$(1405).

Electromagnetic transition form factors of baryons in a relativistic Faddeev approach

The covariant Faddeev approach which describes baryons as relativistic three-quark bound states and is based on the Dyson-Schwinger and Bethe-Salpeter equations of QCD is briefly reviewed. All elements, including especially the baryons’ three-body-wave-functions, the quark propagators and the dressed quark-photon vertex, are calculated from a well-established approximation for the quark-gluon interaction. Selected previous results of this approach for the spectrum and elastic electromagnetic form factors of ground-state baryons and resonances are reported. The main focus of this talk is a presentation and discussion of results from a recent investigation of the electromagnetic transition form factors between ground-state octet and decuplet baryons as well as the octet-only Σ0 to Λ transition.

Two-body and three-body rare-gas clusters

Properties of a helium triatomic system are studied using the Faddeev approach. Spectra and radii of all possible homogeneous and heterogeneous rare-gas dimers are calculated. It is shown that all rare-gas dimers have excited states with the energy close to the breakup threshold.

International Symposium on New Developments in Methods and Applications of Few-body Physics: In Memory of Professor SA Sofianos

PrefaceThe symposium was held in memory of Professor S A Sofianos, a distinguished physicist, who dedicated all his academic life to serving the University of South Africa and the Few-body physics community. Fourteen researchers from Brazil, Russia, Japan and South Africa presented their work at the symposium, while many others who were interested could not participate because of financial and visa challenges. Almost all the participants have collaborated with Professor S A Sofianos in their research.A wide range of few-body physics topics were discussed during the symposium. Nuclear interactions as well as bound and scattering processes formed the bases of all the discussions of the symposium. The long-standing Coulomb problem and the almost forgotten Marchenko inversion resurfaced. New developments in few-body methods such as the Artificial Neural Networks, Integrodifferential Equations Approach, Gaussian Expansion Method and the Faddeev Approach, which are applicable to stable nuclear systems, were discussed. Various three- and four-body nuclear systems were considered for illustrations of these methods. Developments in the Jost function and Jost matrix methods and their application to unstable exotic nuclear systems were also discussed. Professor S A Sofianos worked on all, and many more, of these methods and few-body systems.List of Committee, Sponsors and Participants are available in this pdf

Scattering length calculations via Faddeev approach

The results of calculation of the scattering length for the collision of a 4He atom with a 4He7Li dimer are presented. These investigations are based on the hard-core version of the Faddeev differential equations.

Three-atomic clusters of rare gases within Faddeev approach

The properties of the rare gas clusters of helium and neon are studied. The calculations of the bound state energies of helium and neon trimers are performed within Faddeev differential equation in total angular momentum representation.

Towards a Microscopic Understanding of Nucleon Polarizabilities

We outline a microscopic framework to calculate nucleon Compton scattering from the level of quarks and gluons within the covariant Faddeev approach. We explain the connection with hadronic expansions of the Compton scattering amplitude and discuss the obstacles in maintaining electromagnetic gauge invariance. Finally we give preliminary results for the nucleon polarizabilities.

Spectra for theA= 6 reactions calculated from a three-body resonance model

We develop a resonance model of the transition matrix for three-body breakup reactions of the A = 6 system and present calculations for the nucleon observed spectra, which are important for inertial confinement fusion and Big Bang nucleosynthesis (BBN). The model is motivated by the Faddeev approach where the form of the T matrix is written as a sum of the distinct Jacobi coordinate systems corresponding to particle configurations (α , n -n ) and (n ; n -α) to describe the final state. The structure in the spectra comes from the resonances of the two-body subsystems of the three-body final state, namely the singlet (T = 1) nucleon-nucleon (NN ) anti-bound resonance, and the N α resonances designated the ground state (Jπ = ) and first excited state (Jπ = ) of the A = 5 systems 5 He and 5 Li. These resonances are described in terms of single-level, single-channel R -matrix parameters that are taken from analyses of NN and Nα scattering data. While the resonance parameters are approximately charge symmetric, external charge-dependent effects are included in the penetrabilities, shifts, and hard-sphere phases, and in the level energies to account for internal Coulomb differences. The shapes of the resonance contributions to the spectrum are fixed by other, two-body data and the only adjustable parameters in the model are the combinatorial amplitudes for the compound system. These are adjusted to reproduce the observed nucleon spectra from measurements at the Omega and NIF facilities. We perform a simultaneous, least-squares fit of the tt neutron spectra and the 3 He3 He proton spectra. Using these amplitudes we make a prediction of the α spectra for both reactions at low energies. Significant differences in the tt and 3 He3 He spectra are due to Coulomb effects.

Resonance model for the three-body states of the A=6 reactions

We present an R -matrix-based model for three-body final states that has aspects of the Faddeev approach to three-particle scattering. The model is applied to describing the nucleon spectra for breakup reactions in the A = 6 systems. Calculations using a charge-symmetric parametrization agree fairly well with the experimental data, although they indicate larger contributions from the 5 He or 5 Li ground state are necessary.

Three-Body Calculation of 4He(αα,γ)12C Reaction at Stellar Energies**Supported by the University of Birjand.

The triple-alpha reaction is the key to our understanding about the nucleosynthesis and the observed abundance of 12C in stars. On the basis of our model, the triple-alpha radiative capture process is studied by using the three-body Faddeev approach as well as two- and three-body electromagnetic currents. The bound and resonance states of 12C are calculated via an inverse process: three-α photodisintegration of a 12C nucleus.

Electromagnetic baryon form factors in the Poincaré-covariant Faddeev approach

Baryons are treated as three-quark systems using QCD degrees of freedom in Poincaré-covariant bound-state equations. The quark self-energy as well as the interaction between quarks are approximated by a vector-vector interaction via a single dressed-gluon exchange (rainbow-ladder truncation), thereby allowing a unified study of quark, meson and baryon properties. Here we will focus on the calculation of electromagnetic properties of spin-1/2 and spin-3/2 ground state baryons.

Octet and decuplet masses: A covariant three-body Faddeev calculation

We determine the baryon octet and decuplet masses as well as their wave functions in a covariant three-body Faddeev approach. We work in an approximation where irreducible three-body forces are neglected. In the two-body interactions we take into account a well explored rainbow-ladder kernel as well as flavor dependent meson-exchange terms motivated from the underlying quark-gluon interaction. Without flavor dependent forces we find agreement with the experimental spectrum on the 5-10 percent level. Including the flavor dependent terms on an exploratory level delivers a Sigma/Lambda splitting with the correct sign although the magnitude is still too small.