Faddeev Equations Research Articles

Particle representation for the kaonic NNK¯ system

The kaonic system $NN\overline{K}$ is studied based on the configuration space Faddeev equations. We consider two models associated with isospin ``natural'' basis and isospin ``charge'' basis. One basis is related to another by a unitary transformation. We show that the ``particle representation'' for $NN\overline{K}({s}_{NN}=0)$ system motivated by the charge basis does not describe the system in terms of coupled particle channels $pp{K}^{\ensuremath{-}}/pn{\overline{K}}^{0}$. The coupling is associated with the nondiagonal elements of the matrix representation for the $N\overline{K}$ potential in the charge basis. The matrix can be diagonalized by a simple unitary transformation. With this relation, the Kyoto potential is discussed. The particle configurations of the $NN\overline{K}$ system may be classified by the presence or absence of the Coulomb interaction according to an analogy with the $NNN$ system. The $NN\overline{K}({s}_{NN}=0)$ system is represented by four particle configurations: $pp{K}^{\ensuremath{-}}$, $np{K}^{\ensuremath{-}}$, $np{\overline{K}}^{0}$, and $nn{\overline{K}}^{0}$.

Prediction of possible exotic states in the system * * Partly supported by the National Natural Science Foundation of China (11735003, 1191101015, 11475227), and by the Youth Innovation Promotion Association CAS (2016367)

We investigate the three-body system in order to look for possible exotic states in the framework of the fixed-center approximation of the Faddeev equation. We assume the scattering of on a clusterized system , which is known to generate , or a in a clusterized system , which is shown to generate . In the case of - scattering, we find evidence of a bound state below the threshold with a mass of around 1700 MeV and a width of about 180 MeV. Considering - scattering, we obtain a bound state just below the threshold with a mass of around 1680 MeV and a width of about 160 MeV.

Impact of dynamical chiral symmetry breaking and dynamical diquark correlations on proton generalized parton distributions

We calculate the leading-twist, helicity-independent generalized parton distributions (GPDs) of the proton, at finite skewness, in the Nambu--Jona-Lasinio (NJL) model of quantum chomodynamics (QCD). The NJL model reproduces low-energy characteristics of QCD, including dynamical chiral symmetry breaking (DCSB). The proton bound-state amplitude is solved for using the Faddeev equation in a quark-diquark approximation, including both dynamical scalar and axial vector diquarks. GPDs are calculated using a dressed non-local correlator, consistent with DCSB, which is obtained by solving a Bethe-Salpeter equation. The model and approximations used observe Lorentz covariance, and as a consequence the GPDs obey polynomiality sum rules. Extractions of electromagnetic and gravitational form factors are performed. We find a D-term of $-1.08$ when the non-local correlator is properly dressed, and $0.85$ when the bare correlator is used instead, suggesting that within this framework proton stability requires the constituent quarks to be dressed consistently with DCSB. We also find that the anomalous gravitomagnetic vanishes, as required by Poincar\'{e} symmetry.

Particle representation for $NN{\bar K}$ system

In the framework of the Faddeev equations in configuration space we perform an analysis of quasi-bound state of the NN{\bar K}NNK‾ system within a particle model. In our approach, the system NN{\bar K}(s_{NN}=0)NNK‾(sNN=0) (NN{\bar K}(s_{NN}=1))NNK‾(sNN=1)) is described as a superposition of ppK^{-}ppK− and pn{\bar K}^0pnK‾0 (nn{\bar K}^{0}nnK‾0 and pn{ K}^-pnK−) states, which is possible due to a particle transition. The relation of the particle model to the theory of a two-state quantum system is addressed and discussed taking into account the possibilities of deep and shallow NN{\bar K}(s_{NN}=0)NNK‾(sNN=0) quasi-bound states.

Effect of isospin averaging for $ppK^-$ kaonic cluster

The kaonic cluster ppK^-ppK− is described by isospin-dependent N{\bar K}NK‾ potentials with significant difference between singlet and triplet components. The quasi-bound state energy of the system is calculated based on the configuration space Faddeev equations within isospin and averaged potential models. The isospin averaging of N{\bar K}NK‾ potentials is used to simplify the isospin model to isospinless one. We show that three-body bound state energy E_{3}E3 has a lower bound within the isospin formalism due to relation \left\vert E_{3}(V_{NN}=0)\right\vert<2\left\vert E_{2}\right\vert|E3(VNN=0)|<2|E2|, where E_{2}E2 is the binding energy of isospin singlet state of the N{\bar K}NK‾ subsystem. The averaged potential model demonstrates opposite relation between |E_{2}||E2| and |E_{3}(V_{NN}=0)||E3(VNN=0)|.

Exploring the unknown Lambda-neutron interaction

No published \LambdaΛn scattering data exist. A relativistic heavy-ion experiment has suggested that a \LambdaΛnn bound state was seen. However, several theoretical analyses have cast serious doubt on the bound-state assertion. Nevertheless, there could exist a three-body \LambdaΛnn resonance. Such a resonance could be used to constrain the \LambdaΛn interaction. We discuss \LambdaΛnn calculations using nn and \LambdaΛn pairwise interactions of rank-one, separable form that fit effective range parameters of the nn system and those hypothesized for the as yet unobserved \LambdaΛn system based upon four different \LambdaΛN potentials. The use of rank-one separable potentials allows one to analytically continue the \LambdaΛnn Faddeev equations onto the second complex energy plane in search of resonance poles, by examining the eigenvalue spectrum of the kernel of the Faddeev equations. Although each of the potential models predicts a \LambdaΛnn sub-threshold resonance pole, scaling of the \LambdaΛn interaction by as little as \sim∼5% does produce a physical resonance. This suggests that one may use photo-(electro-)production of the \LambdaΛnn system from tritium as a tool to examine the strength of the \LambdaΛn interaction.

Investigation of the interaction of circularly and linearly polarized photon beams with a polarized He3 target

Photodisintegration of polarized 3He by linearly or circularily polarized photons offers a rich choice of observables which can be calculated with high precision using a rigorous scheme of three-nucleon Faddeev equations. Using the (semi)phenomenological AV18 nucleon-nucleon potential combined with the Urbana IX three-nucleon force we investigate sensitivity of 3He photodisintegration observables to underlying currents taken in the form of a single-nucleon current supplemented by two-body contributions for $\pi$- and $\rho$-meson exchanges or incorporated by the Siegert theorem. Promising observables to be measured for two- and three-body fragmentation of 3He are identified. These observables form a challenging test ground for consistent forces and currents being under derivation within the framework of chiral perturbation theory. For thre-body 3He photodisintegration several kinematicaly complete configurations, including SST and FSI, are also discussed.

A three-dimensional momentum-space calculation of three-body bound state in a relativistic Faddeev scheme

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 integral equation using an iterative scheme. The relativistic effects on three-body binding energy are calculated for the Malfliet-Tjon potential. Our calculations show that the relativistic effects lead to a roughly 2% reduction in the three-body binding energy. The contribution of different Faddeev components in the normalization of the relativistic three-body wave function is studied in detail. The accuracy of our numerical solutions is tested by calculation of the expectation value of the three-body mass operator, which shows an excellent agreement with the relativistic energy eigenvalue.

Numerical solution of time-dependent three-particle Faddeev equations: Calculation of rearrangement S matrices

Numerical solution of time-dependent three-particle Faddeev equations: Calculation of rearrangement <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> matrices

Implications of an increased Λ-separation energy of the hypertriton

Stimulated by recent indications that the binding energy of the hypertriton could be significantly larger than so far assumed, requirements of a more strongly bound HΛ3 state for the hyperon-nucleon interaction and consequences for the binding energies of A=4,5 and 7 hypernuclei are investigated. As basis, a YN potential derived at next-to-leading order in chiral effective field theory is employed. Faddeev and Yakubovsky equations are solved to obtain the corresponding 3- and 4-body binding energies, respectively, and the Jacobi no-core shell model is used for Λ5He and Λ7Li. It is found that the spin-singlet Λp interaction would have to be much more attractive which can be, however, accommodated within the bounds set by the available Λp scattering data. The binding energies of the HeΛ4 hypernucleus are predicted to be closer to the empirical values than for YN interactions that produce a more weakly bound HΛ3. The quality of the description of the separation energy and excitation spectrum for Λ7Li remains essentially unchanged.

Developments regarding three-body reaction channels within the R-matrix formalism

At low incident energies of nucleon-induced reaction cross sections exhibit a striking resonance structure which cannot properly be described by (semi-) microscopic models. Usually R-matrix theory is applied which provides a sufficiently accurate but phenomenological description of the resonance region. However, standard R-matrix theory is only suited for two-particle channels. Three- and many-particle channels which may occur at rather low incident energies and are usually treated in approximative or effective way. In this contribution an extension to unequal masses of the R-matrix formulation of Glockle based on the Faddeev equation is performed and proper expressions for numerical implementation are given.

Исследование трехатомных кластеров в рамках уравнений Фаддеева

The binding energies of 4He3 and 4He26Li three-atomic clusters were calculated within the differential Faddeev equations. Obtained results show that different methods provide similar results for different potential models. It is shown that the excited state energy of each system is close to it’s respective two-body threshold, which could indicate on the Efimov nature of the excited state of 4He26Li.

Momentum-Space Probability Density of $${}^6$$He in Halo Effective Field Theory

We compute the momentum-space probability density of ${}^6$He at leading order in Halo EFT. In this framework, the ${}^6$He nucleus is treated as a three-body problem with a ${}^4$He core ($c$) and two valence neutrons ($n$). This requires the $nn$ and $nc$ t-matrices as well as a $cnn$ force as input in the Faddeev equations. Since the $nc$ t-matrix corresponds to an energy-dependent potential, we consider the consequent modifications to the standard normalization and orthogonality conditions. We find that these are small for momenta within the domain of validity of Halo EFT. In this regime, the ${}^6$He probability density is regulator independent, provided the cutoff is significantly above the EFT breakdown scale.

Nd breakup within isospinless AAB model

The three-nucleon system (ppn or nnp) may be considered as a system of identical particles within a framework of the isospin formalism (AAA model). Here we propose to treat the system as a system with only two identical particles as a natural way to take into account the charge dependence of the nucleon–nucleon (NN) interaction. We apply the ‘given charge formalism’ to formulate the ‘isospinless’ AAB model. The charge independence breaking effect for Nd scattering is modeled by using a phenomenological adjustment for s-wave NN potential. The modeling is based on the configuration space Faddeev equations. Calculations of the phase shifts, inelasticities, and breakup amplitudes are performed for nd and pd scattering at different energies for both AAA and AAB models. It has found that the numerical differences between the results obtained with these two models are relatively small (about 1%), however, noticeable. In this regard, we consider the numerical accuracy of the calculations and in particular the effect of oscillations of the phase shifts and inelasticities that depend on the asymptotic cutoff parameter. The choice of the cutoff parameter is performed to minimize the impact of oscillations on the accuracy for the phenomenological (MT I–III) and realistic (AV14) potentials.

Equivalence of three-particle scattering formalisms

In recent years, different on-shell $\mathbf{3}\to\mathbf{3}$ scattering formalisms have been proposed to be applied to both lattice QCD and infinite volume scattering processes. We prove that the formulation in the infinite volume presented by Hansen and Sharpe in Phys.~Rev.~D92, 114509 (2015) and subsequently Brice\~no, Hansen, and Sharpe in Phys.~Rev.~D95, 074510 (2017) can be recovered from the $B$-matrix representation, derived on the basis of $S$-matrix unitarity, presented by Mai {\em et al.} in Eur.~Phys.~J.~A53, 177 (2017) and Jackura {\em et al.} in Eur.~Phys.~J.~C79, 56 (2019). Therefore, both formalisms in the infinite volume are equivalent and the physical content is identical. Additionally, the Faddeev equations are recovered in the non-relativistic limit of both representations.

N-N-N∗ model calculations for experimental η-mesic 3He searches

The possibility for the existence of the exotic [Formula: see text] states is explored with the objective of calculating the [Formula: see text] momentum distribution inside such nuclei. Even though the latter is an essential ingredient for the analysis of the experimental data on the [Formula: see text], [Formula: see text] and [Formula: see text] reactions aimed at finding an [Formula: see text]-mesic 3He, the data analysis is usually performed by approximating the [Formula: see text] momentum distribution by that of a nucleon. Here, we present calculations performed by solving the three-body Faddeev equations to obtain the momentum distribution of the [Formula: see text] inside possible ([Formula: see text])[Formula: see text], ([Formula: see text])[Formula: see text] and ([Formula: see text])[Formula: see text]-[Formula: see text] states. The [Formula: see text] momentum distributions are found to be much narrower than those of the nucleons and influence the data selection criteria.

Can a $$\varLambda nn$$ Resonance Constrain the $$\varLambda n$$ Amplitude?

We address the question of constraints on the $$\varLambda n$$ amplitude as a result of observing a $$\varLambda nn$$ resonance above the $$\varLambda nn$$ three-body threshold. To examine this question, we will first demonstrate that by starting with a $$\varLambda n$$ interaction based upon the experimental $$\varLambda p$$ data and then scaling the $$\varLambda n$$ potential by $$\approx 8\%$$ one can generate a $$\varLambda nn$$ resonance, and with a sufficiently large scaling one can generate a $$\varLambda nn$$ bound state. This is achieved using the three-body Faddeev equations with rank one separable potentials for all pairwise interactions. The use of separable potentials is motivated by two factors: (i) the Faddeev equations reduce to a set of one dimensional integral equations that can be analytically continued into the complex energy plane where resonances reside. (ii) The $$\varLambda n$$ amplitude can be defined in terms of the effective range parameters. As a result we can explore the question: What constraint can be placed on the $$\varLambda n$$ effective range parameters by the $$\varLambda nn$$ resonance parameters?

Nd-Scattering within MGL Approach for Configuration-Space Faddeev Equations

We study the neutron-deuteron elastic and breakup scattering on the basis of the configuration-space Faddeev equations. The Merkuriev–Gignoux–Laverne approach has been generalized for arbitrary nucleon-nucleon potentials and with an arbitrary number of partial waves. Neutron-deuteron observables at the incident nucleon energy 14.1 MeV are calculated using the realistic AV14 nucleon-nucleon potential. Results are compared with those of other authors and with experimental neutron-deuteron scattering data. The computational procedure is presented in details.

Model with Coupled Internal and External Channels for 2N and 3N Systems

A Faddeev-type formalism for a three-body problem with two-body interactions containing internal degrees of freedom is introduced. The Faddeev equations for the basic objects such as total resolvent, scattering wave function and transition operators are derived explicitly. As a particular example, the dibaryon model for NN interaction with internal channel corresponding to a six-quark bag is considered. It is shown that this interaction model allows to reproduce both real and imaginary parts of some NN partial phase shifts at energies up to 600 MeV. The developed formalism can be also applied to other quantum systems.

Possible bound states with hidden bottom from K¯(*)B(*)B¯(*) systems

We study the three-body systems of $\bar{K}^{(*)}B^{(*)}\bar{B}^{(*)}$ by solving the Faddeev equations in the fixed-center approximation, where the light particle $\bar{K}^{(*)}$ interacts with the heavy bound states of $B\bar{B}$ ($B^*\bar{B}^*$) forming the clusters. In terms of the very attractive $\bar{K}^*B$ and $\bar{K}^*B^*$ subsystems, which are constrained by the observed $B_{s1}(5830)$ and $B_{s2}^*(5840)$ states in experiment, we find two deep bound states, containing the hidden-bottom components, with masses $11002\pm 63$ MeV and $11078\pm 57$ MeV in the $\bar{K}^*B\bar{B}$ and $\bar{K}^*B^*\bar{B}^*$ systems, respectively. The two corresponding states with higher masses of the above systems are also predicted. In addition, using the constrained two-body amplitudes of $\bar{K}B^{(*)}$ and $\bar{K}\bar{B}^{(*)}$ via the hidden gauge symmetry in the heavy-quark sector, we also find two three-body $\bar{K}B\bar{B}$ and $\bar{K}B^{*}\bar{B}^*$ bound states.