Implementation of the 8-Nucleon Yakubovsky Formalism for Halo Nucleus

In order to study the bound-state structure of the Helium halo nuclei, the 8-nucleon Yakubovsky formalism has been implemented for 8He in a 5-body sub-cluster model, i.e. α + n + n + n + n. In this case, the 8nucleon Yakubovsky equations has been obtained in the form of two coupled equations, based on the two independent components. In addition, by removing the contribution interactions of the 8 and 7’s bound nucleons in the formalism, the obtained equations explicitly reduce to the 6-nucleon Yakubovsky equations for 6He, in the case of effective 3-body model, i.e. α + n + n. In view of the expectation for the dominant structure of 8He, namely an inert α-core and four loosely-bound neutrons, Jacobi configurations of the two components in momentum space have been represented to provide technicalities which were considered useful for a numerical performance, such as bound-state calculations and momentum density distributions for halo-bound neutrons. PACS. No: 21.45.+v Few-body systems; 21.30.Fe Forces in hadronic systems and effective interactions; 21.30.-x Two-nucleon interactions; 21.10.Dr Binding energies and masses.


Introduction
Studies of rich-neutron light nuclei present stringent tests for their calculations as they probe aspects of the interactions that are less prevalent in nuclei closer to stability. Among them, the physics of strong interactions gives rise to new structures in rich-neutron light nuclei. One of the best prominent cases are the helium halo nuclei, namely 6 He and 8 He with two and four loosely-bound neutrons, forming an extended halo around the 4 He core. In this regard, the rich-neutron helium isotopes, 6 He and 8 He, are the topic of this study. Indeed, both nuclei are Borromean halos [1,2], they have no bound excited states, and they exhibit the binding energy. Both helium halo nuclei are radioactive and decay quickly that complicates efforts to measure their properties. 8 He has a time-life of only a tenth of a second and in this little time, can be considered as a stable nucleus. Practically, detection of the rare 8 He is a major step for particle physicists. But, it is very important and interesting to understand how helium configures itself with strong interactions after being produced from a particle accelerator.
In recent decades, combination of nuclear and atomic physics methods has allowed new courses of precision measurements of the ground-state energies and charge radii of 6 He [3,4] and 8 He [5,6]. Dealing with the theoretical study of these nuclei is very challenging, because one needs to describe the small separation energy of the halo neutrons and the large radius of the whole system, in addition to the challenges of binding energy calculations. Because they are light-mass nuclei, one can use traditional methods to study them, such as Green's Function Monte Carlo (GFMC) method [7] and the No-Core Shell Model (NCSM) [8]. In addition, some of the standard methods in nuclear theory has been the progress of Effective Field Theory (EFT) and the Renormalization Group (RG) to nuclear forces. While such helium halo nuclei have been investigated using the NCSM [9], but still, no sense has been obtained for the halo structure properties in such theoretical studies.
In this paper, to approach a simple physical structure of the observed effects, namely 8 He and 6 He halo-bound nuclei, and to understand the physics that governs the neutron distributions in helium halos, it was felt that by using current/modern computers with high computational speed, beyond previous works [10][11][12][13][14], a further step to study the light halo nuclei, namely 8 He as an 8-nucleon bound system problem, within the reliable Yakubovsky method [15] is now desirable. It is worth mentioning that our obtained results are more difficult to implement bound-state problem in few-body branch, due to its halo nature and high-dimensional calculations. However, we provide technicalities for numerical performances.
This paper is written as follows: In sect. I, the Yakubovsky formalism to the 8-nucleon bound system problem using the standard notation [16] is explicitly derived. In this case, the Lippmann-Schwinger scheme twice step-bystep has been applied on the integral form of Schrödinger equation, and then the identity of the nucleons are implemented. Finally, the 8-nucleon Yakubovsky formalism for effective -core structure in a 5-body sub-cluster model of 8 He leads to a set of two coupled equations related to 2 different components. In sect. II, useful technicalities for numerical applications have been provided. To this aim, corresponding Jacobi configurations of each Yakubovsky component is schematically represented and then the integral form of coupled equations are represented by introducing basis states based on Jacobi momenta in partial-wave (PW) analysis. Finally, a summary in sect. III is given.

I. The 8-nucleon Yakubovsky Formalism
In branch of few-body systems, each N-body bound system has ( − 1)/2 different two-body interactions. Therefore, in the 8-nucleon bound system there are 28 different two-nucleon interactions or 28 different cluster decompositions ( 7 ) having 7 clusters. They are indexed by the only pairing forces in cluster 7 they contain, e.g. 7 = 12 ≡ 12 + 3 + 4 + 5 + 6 + 7 + 8. In order to study the 8-nucleon bound system in the framework of the Yakubovsky method using the sub-cluster notation [16], the method is to first sum up the pair forces in each 7-body fragment ( 7 ), in a second step among all 6-body fragments ( 6 ), and then in a third step among all 5-body fragments ( 5 ). We cut off that formalism ending with 5-body sub-clusters in the spirit of the usually used approximating effective 5-body model that is + + + + . In the standard formalism, for an individual -particle, such two-body interactions, namely − and -clusters have not been used, because in this framework -core interacts as a four-nucleon subsystem in the 8-nucleon system. We start with the non-relativistic Schrödinger equation for the 8-nucleon bound system as where 0 refers to the free Hamiltonian operator of the 8-nucleon system that will be introduced in the next section, and ∑ briefly, we work out that method ending with ( 5 ) in the spirit of the applied though approximate effective + + + + in a 5-body model. Further details could be found in Refs. [12,13]. In the following, the single nucleons in sub-clusters, namely 7-, 6-and 5-body fragments, i.e. ( 7 ), ( 6 ) and ( 5 ) respectively, will no longer be indexed.
In the next step, the identity of the nucleons in sub-cluster components are implemented. We start from Eq. (1.4) choosing the case 7-body fragments 7 ≡ 12 with 6-body fragments 6  ) and 12,12+34 12+34+56 coupled in the equations. According to the 5-body sub-cluster distributions of above independent components, it is well-known that two first components are related to 4 + + + + that refer to approximately effective 5-body model ( + + + + ). Respectively, third component refers to 3 + 2 + + + , which is not present in the effective 5-body -core model. Fourth component is a linear combinations that refer again to the third component. This is also beyond the effective 5-body -core model. The last one is related to 2 + 2 + 2 + 2 sub-cluster which is not contained in the effective 5-body -core model, either.

1234
, are related to approximately effective configuration of + + + + , where the -core and four loosely bound neutrons approximation are valid. Now, here we convinced enough why we choose two specific components, and those components are related to the effective -core structure. By considering the distinct Jacobi configurations related to remaining Yakubovsky components, Fig.1 , and obviously other components will not be taken into account.
Hint: after removing the contribution interactions of the seventh and eighth nucleons in Eq. (1.34), i.e. removing their permutation operators, the 8-nucleon obtained equations leads to the 6-nucleon ones for halo nucleus 6 He (See Eq. (67) in Ref. [12]). Such reduction, confirms that our implementing the Yakubovsky formalism for the 8nucleon bound system for 8 He, is appropriate approximation to describe the effective -core structure for Helium halo-bound nuclei, similar to the approximation of the six-nucleon for 6 He in the effective 3-body model [12,13].
In the above explanations we clearly have discussed why only the two specific components, 12;123 1234 and 12;12+34 1234 , are relevant to the effective -core structure of 8 He, and the other Yakubovsky components in Eq. (1.34), will not be taken into account. Therefore, Eq. (1.34) leads to It is worth mentioning that switching off some Yakubovsky independent components changes the Hamiltonian of the comprehensive 8-nucleon system, so the Hamiltonian does not contain all interactions, but we show that in the remaining components the effective interactions of -core as the stable 4-nucleon sub-system is governed in the remaining components (Refer to Fig.1). It is obvious that after removing the contribution interactions  Consequently, the formalism can be simplified and the numerical solution of integral would be much faster in comparison with the case where was need to solve first sub-cluster Faddeev-like equation to obtain 123 and 12+34 by using Pade′ approximation [10]. In addition to Eq. (1.36), the 4-nucleon Yakubovsky coupled equations has been obtained in [10] as In the next step we will describe characteristic numerical techniques such as introducing Jacobi configuration of each independent component in momentum space representation to provide corresponding basis states and to understand the halo structure model of the 8-nucleon system with respect to the regarded relevant remaining components; integral representation of the coupled equations; eigenvalue equation form and iteration method for solution of it, and discussions about the halo structure configurations that describe the 8-nucleon halo-bound system in the case of effective -core structure.

II. Technicalities for Numerical Applications
In this section, first, Jacobi configurations in momentum space are represented to define corresponding momentum basis states for evaluations of the integral kernels. Next, details for a typical algorithm are described to solve coupled integral equations based on PW representation. In order to describe the corresponding momentum basis states, for two independent components of the dominant -core structure of 8 He, namely 12;123 1234 and 12;12+34 1234 the standard Jacobi momenta are described in Fig. 1, respectively. (1) (2) Fig. 1: Schematic representation of the two independent components of 8-nucleon bound system for 8 He (four loosely bound neutrons with an inert -core) in Jacobi configurations. In the bag the -core tightly interacts in a 4-nucleon subsystem.
For the first component 12;123 1234 in terms of the first configuration in Fig. 1 sets corresponding to -set chains where is an individual particle momentum in the center of mass situation (under the condition ∑ = 0) that is described by relative Jacobi momenta ; ( = 1, 2, 3,4,5,6,7,8). Similarly, to 12;12+34 1234 in terms of the second configuration in Fig. 1 belongs here the first part in the above-introduced basis states are magnitude of momentum vectors in PW representation. In the second part the orbital angular momentum go with the \ , which are two-body spins for nucleons , are total 1-and 2-body angular momenta coupled out of orbital and spin angular momenta, are total -body angular momenta except 5 and 7 . These are total angular momenta of 5, 6 and 7, 8 respectively. Evidently, the third part refers to isospin. In this step, in order to implement the numerical techniques, the two coupled equations, where is the energy eigenvalue at which auxiliary Yakubovsky kernel eigenvalue is ( ) = 1. Because the energy is varied such that one reaches eigenvalue to be one. The Yakubovsky kernel of the equations ( ) is just depend to energy E, and ( ) is eigenvalue with ( 12;123 1234 , 12;12+34 1234 ) as the corresponding eigenvector. In order to solve the eigenvalue equation, Eq. (2.9), it can be used the Gaussian quadrature grid points. The coupled equations represent a set of homogenous integral equations, which turn into a high-dimension matrix eigenvalue equation after discretization. Starting from an arbitrary initial ≡ 0 one generates by consecutive applications of ( ) a sequence of amplitudes , which after orthogonalization form a basis into which is expanded. Then the energy is varied such that one reaches ( ) = 1. In order to investigate the loosely interactions of the halo bound neutrons in 8 He, calculations of the momentum probability density as ( )[ ] in terms of [ −1 ], correspondingly ( ) can be done. Why we just select 4 and 6 Jacobi momenta to calculate the momentum probability density? Because they are the only ones that make up contributions of the halo neutrons (refer to Eqs. (2.1) and (2.3)). By calculating momentum probability densities, we can represent the situation of the halo bound neutrons in momentum space. It is expected that the representation of momentum probability densities approximately are wide and short in comparison with tightly bound neutrons, namely neutrons inside the -core subsystem. Such those momentum distributions confirms that the 8-nucleon in the case of effective -core model is a halo structure system, exactly like halo nucleus 8 He.

III. Summary
The applications of Faddeev and Yakubovsky equations are traditionally for 3-and 4-nucleon bound and scattering states. Today, after the experiences with 4-, 5-and 6-nucleon bound-state problems within the Yakubovsky approach [13,14] that the technical expertise has developed in the recent years and the very strong increase of computer power just recently achieved, it is possible to approach the 8-nucleon bound-state problem in that formalism and then numerical solution. For a full solution of the 8-nucleon Yakubovsky equations for 8 He, modern supercomputers with parallel algorithms is required. But in order to approach the calculations of the 8-nucleon bound system, for the first time, we have implemented the 8-nucleon Yakubovsky equations for the halo nucleus 8 He in the case of simple effective -core structure. To this end, we stop the sequential sub-clustering with 5 fragments, namely + + + + , though the additional step with 4 fragments could be easily performed, but is not applicable for a physical system. In this method, we restricted the formulation to two-nucleon forces only because loosely bound neutrons with the center of an inert -core tend to have much stronger pairing interactions for large radii neutrons [5]. Therefore, as a simplification in the formalism, and above fact in such a model, three-nucleon forces was ignored. As a result, the implementation of the 8-nucleon bound system in the halo structure model within the Yakubovsky approach leads to two coupled equations in terms of two independent Yakubovsky components. Subsequently, in order to approach the halo structures of 8 He, in addition to the analytical implementations, we have introduced the Jacobi configurations of those components in momentum space representation. Eventually, we provide technicalities which were considered useful for a numerical performance, such as bound-state calculations and proving the halo structure of bound neutrons that are outside of the -core.