Faddeev-Yakubovsky Equations Research Articles

Hidden-flavor four-quark states in the charm and bottom region

We discuss the spectrum and the internal composition of ground and excited four-quark states in the charm and bottom energy region. To this end we extend previous calculations within the framework of the relativistic four-body Faddeev-Yakubovsky equation to include quantum numbers with JPC=0++,0−+,1−−,1+− and 1++ and study their internal composition in terms of heavy-light meson pairs, hadroquarkonia and diquark-antidiquark clusters. We observe similar patterns in the charm and bottom energy region with different compositions of the four-quark states depending on JPC quantum numbers. Most notably, we find that all states with C·P=+1 are dominated by heavy-light meson contributions, whereas for axial-vector states with JPC=1+− including the Zc(3900) we find a much more complicated picture depending on the flavor content. We systematically compare our results for the spectrum with existing experimental results and provide predictions for future analyses. Published by the American Physical Society 2024

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.

Emergence of 4H Jπ = 1− resonance in contact theories

We obtain the s- and p-wave low-energy scattering parameters for n3H elastic scattering and the position of the 4H Jπ=1− resonance using the pionless effective field theory at leading order. Results are extracted with three numerical techniques: confining the system in a harmonic oscillator trap, solving the Faddeev-Yakubovsky equations in configuration space, and using an effective two-body cluster approach. The renormalization of the theory for the relevant amplitudes is assessed in a cutoff-regulator range between 1fm−1 and 10fm−1.Most remarkably, we find a cutoff-stable/RG-invariant resonance in the 4H Jπ=1− system. This p-wave resonance is a universal consequence of a shallow two-body state and the introduction of a three-body s-wave scale set by the triton binding energy. The stabilization of a resonant state in a few-fermion system through pure contact interactions has a significant consequence for the powercounting of the pionless theory. Specifically, it suggests the appearance of similar resonant states also in larger nuclei, like 16-oxygen, in which the theory's leading order does not predict stable states. Those resonances would provide a starting state to be moved to the correct physical position by the perturbative insertion of sub-leading orders, possibly resolving the discrepancy between data and contact EFT.

$$\Lambda -\alpha $$ Potential by Folding $$\Lambda $$-Nucleon Interaction with Realistic $$\alpha $$ Wave Function

We construct a folding potential between the \(\alpha \) and \(\Lambda \) particles based on underlying nucleon-nucleon and hyperon-nucleon interactions. Starting from a phenomenological \(\Lambda \)-N potential and a Gaussian form of the \(\alpha \)-particle wave function we obtain the \(\alpha \)-\(\Lambda \) potential and with this potential the binding energy of \(^5_\Lambda \)He is (3.10 MeV), which is consistent with recent experimental data \(3.12 \pm 0.02\) MeV. When in turn an exact solution of the four-body Faddeev-Yakubovsky equation for the \(\alpha \)-particle calculated with the CDBonn, Nijmegen or Argonne V18 realistic nucleon-nucleon potential is used and the phenomenological Gaussian \(\Lambda \)-N potential is replaced by the realistic (e.g. Nijmegen NSC97f) potential approximated by a rank-1 separable form, then \(^5_\Lambda \)He is overbound. In particular, its binding energy given by the folding potential generated with the \(\alpha \) particle wave function based on the CDBonn potential is 7.47 MeV. Although the rank-1 separable \(\Lambda \)-N potential reproduces the exact scattering length and the effective range of the original \(\Lambda \)-N potential, the \(\Lambda \)-\(\alpha \) folding potential results from these \(\Lambda \)-N potential give a large binding energy of \(^5_\Lambda \)He .

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.

Integral equations and binding energies for Λ − NN and Λ − NNN systems

A systematic treatment of the procedure used to construct and solve the Faddeev-Yakubovsky integral equations for three and four different particles with nonzero spin is introduced. The four-body T-matrices are written in terms of solutions of the three-body Faddeev integral equations in total angular momenta representation. General spin-angular structure of the four-body Faddeev-Yakubovsky equations is written down and reduced to a spin-angular structure of the three-body system by elimination of one body. Partial-wave expansion performed for the system of four-body integral equations gives an infinite system of one-dimensional coupled equations with integral kernels containing four types of partial components of the three-body Faddeev integral equation solutions. The given method is tested on Λ−NN and Λ−NNN systems with two-body potential written down in only separable form. Both separable NN potential and ΛN potential rewritten in separable form for s-wave are used to determine the Λ-hyperon binding energies in Λ−NN and Λ−NNN systems. Calculation also includes the hyperon conversion in ΛN→ΣN process. Λ-hyperon separation energies were calculated from binding energies and turned out to be 0.147 MeV in Λ−NN system and 2.04 MeV in Λ−NNN system, respectively, that is in close coincidence with experimental ones.

Predictions for charmed nuclei based on Y_c N forces inferred from lattice QCD simulations

Charmed nuclei are investigated utilizing varLambda _c N and varSigma _c N interactions that have been extrapolated from lattice QCD simulations at unphysical masses of m_pi = 410–570 MeV to the physical point using chiral effective field theory as guideline. Calculations of the energies of varLambda _c single-particle bound states for various charmed nuclei from ^{ 5}_{varLambda _c}Li to ^{209}_{varLambda _c}Bi are performed using a perturbative many-body approach. This approach allows one to determine the finite nuclei varLambda _c self-energy from which the energies of the different bound states can be obtained. Though the varLambda _c N interaction inferred from the lattice results is only moderately attractive, it supports the existence of charmed nuclei. Already the lightest nucleus considered is found to be bound. The spin-orbit splitting of the p- and d-wave states turns out to be small, as in the case of single varLambda hypernuclei. Additional calculations based on the Faddeev-Yakubovsky equations suggest that also A=4 systems involving a varLambda _c baryon are likely to be bound, but exclude a bound ^{, 3}_{varLambda _c}hbox {He} state.

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.

Ab initio calculations of 5H resonant states

By solving the 5-body Faddeev-Yakubovsky equations in configuration space with realistic nuclear Hamiltonians we have studied the resonant states of 5H isotope. Two different methods, allowing to bypass the exponentially diverging boundary conditions, have been employed providing consistent results. The existence of 5H broad Jπ=1/2+,3/2+,5/2+ states as S-matrix poles has been confirmed and compared with the, also calculated, resonant states in 4H isotope. We have established that the positions of these resonances only mildly depend on the nuclear interaction model.

Few-nucleon and many-nucleon systems with semilocal coordinate-space regularized chiral nucleon-nucleon forces

We employ a variety of ab initio methods including Faddeev-Yakubovsky equations, No-Core Configuration Interaction Approach, Coupled-Cluster Theory and In-Medium Similarity Renormalization Group to perform a comprehensive analysis of the nucleon-deuteron elastic and breakup reactions and selected properties of light and medium-mass nuclei up to 48Ca using the recently constructed semilocal coordinate-space regularized chiral nucleon-nucleon potentials. We compare the results with those based on selected phenomenological and chiral EFT two-nucleon potentials, discuss the convergence pattern of the chiral expansion and estimate the achievable theoretical accuracy at various chiral orders using the novel approach to quantify truncation errors of the chiral expansion without relying on cutoff variation. We also address the robustness of this method and explore alternative ways to estimate the theoretical uncertainty from the truncation of the chiral expansion.

Solution of the n−He4 elastic scattering problem using the Faddeev-Yakubovsky equations

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 Schr\"odinger equation.

Description of Five-Nucleon Systems Using Faddeev-Yakubovsky Equations

A brief overview of Faddeev-Yakubovsky equations is presented before deriving 5-body ones. Numerical formalism, enabling to solve these equations in configuration space for a system of five nucleons is described. Microscopic calculations are realized to determine phaseshifts of low energy neutron scattering on $$^4$$ He and $$1/2^+$$ resonance position of $$^5$$ H, employing phenomenological MT I-III potential.

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.

Description of the four-nucleon collisions by including breakup

Four-nucleon reactions above the breakup threshold are described by solving Faddeev-Yakubovsky equations for the realistic nuclear Hamiltonians. Complex-scaling method is applied in order to simplify the boundary conditions.

Modern nuclear force predictions forn−H3scattering above the three- and four-nucleon breakup thresholds

Background: Description of the collision process, which includes breakup, is one of the most challenging problems of the quantum mechanics. Recently I have developed a formalism based on the complex-scaling method, which describes accurately nuclear collisions in three- and four-body systems.Purpose: To provide accurate calculations for $n\ensuremath{-}^{3}\mathrm{H}$ scattering above the three- and four-nucleon breakup thresholds.Method: A four-nucleon system is described in configuration space employing Faddeev-Yakubovsky equations. The complex-scaling method is applied to overcome the difficulties related with the complicated boundary conditions.Results: Elastic observables as well as total breakup cross sections are calculated for neutron scattering on tritium at 14.1, 18, and 22.1 MeV using realistic $\mathit{NN}$ interactions. Excellent agreement is found with the pioneering calculations of this process reported by A. Deltuva et al. [Phys. Rev. C 86, 011001 (2012)]. Strong correlation of the calculated cross sections is established with model-predicted trinucleon binding energy. The forementioned observables reveal little sensitivity to the short-range details of $\mathit{NN}$ interaction.Conclusion: Reliable and accurate methods are now available to study four-nucleon scattering including the breakup.

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.