EEffective interactions between nuclear clusters
Yoshiko Kanada-En’yo
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Dean Lee
Facility for Rare Isotope Beams and Department of Physics and Astronomy,Michigan State University, East Lansing, MI 48824, USA
The effective interactions between two nuclear clusters, d + d , t + t , and α + α , are investigatedwithin a cluster model using local nucleon-nucleon ( NN ) forces. It is shown that the interaction inthe spin-aligned d + d system is repulsive for all inter-cluster distances, whereas the α + α and spin-aligned t + t systems are attractive at intermediate distances. The Pauli blocking between identical-nucleon pairs is responsible for the cluster-cluster repulsion and becomes dominant in the shallowbinding limit. We demonstrate that two d -clusters could be bound if the NN force has nonzerorange and is strong enough to form a deeply bound d -cluster, or if the NN force has both even-parity and odd-parity attraction. Effective dimer-dimer interactions for general quantum systemsof two-component fermions are also discussed in heavy-light mass limit, where one component ismuch heavier than the other, and their relation to inter-cluster interactions in nuclear systems arediscussed. Our findings provide a conceptual foundation for conclusions obtained numerically in theliterature, that increasing the range or strength of the local part of the attractive nucleon-nucleoninteraction results in a more attractive cluster-cluster interaction. I. INTRODUCTION
Nuclear clustering is a fascinating and important fea-ture of many nuclear systems. Developed cluster struc-tures appear in excited states of several nuclei and alsoin the ground states of systems such as 2 α clustering in Be(0 +1 ) and O+ α clustering in Ne [1, 2]. While α clusters are the most common type of cluster structure,deuteron and triton clusters have also been suggested inlight p -shell nuclei and at the surface of closed shell corenuclei. In highly excited states, cluster states containingmore than two clusters such as 3 α structures in C and4 α structures in O have been attracting great interestin theoretical and experimental studies [1–4].The formation of clusters has been also investigated atthe nuclear surface of sd - and heavier nuclei where spa-tial cluster correlations beyond mean-field may emerge[5, 6]. Concerning a two-nucleon pair with a strong spa-tial correlation, deuteron-like pn and dineutron nn cor-relations are also recent hot topics. For the latter, twoneutrons are not bound in a free space, but the nn cor-relation is rather strong in loosely-bound neutron-richsystems such as He and Li and can be regarded as a( nn )-cluster [7–11]. The possibility of an α + nn + nn structure has been proposed for an excited state of He[12]. Another candidate for multi-dineutron systems is nn + nn clustering in a four-neutron system called thetetraneutron. But this remain a controversial issue: ex-perimental signals of a tetraneutron resonance have beenrecent reported [13, 14] while several theoretical studiesare not able to accommodate such a resonance [15–21].The effective interactions between clusters play an im-portant role in cluster phenomena in nuclear systems.For example, the ground state of Be is a quasi-bound 2 α state formed by a short-range repulsion and a medium-range attraction of the effective α - α interaction, which has been experimentally observed from the α - α scatter-ing phase shifts. This α - α interaction also describes the3 α structure of the Hoyle state, C(0 +2 ). The short-range repulsion and medium-range attraction, which areexperimentally known from the scattering phase shifts,are essential to describe the developed 3 α structure in C(0 +2 ). In a microscopic α + α cluster model with theresonating group method (RGM), the repulsive effect ofthe α - α interaction was described by a nodal structure ofthe inter-cluster wave function caused by the Pauli repul-sion between identical nucleons in different clusters [23].A similar Pauli effect contributes to the effective inter-action between two dineutrons and produces significantrepulsion in the tetraneutron system [15, 16].In Ref. [24] it was observed that the α - α interactiondetermines whether nuclear matter forms a nuclear liq-uid or a Bose-Einstein condensate (BEC) of alpha parti-cles. First principles calculations showed that the rangeand strength of the local part of the nucleon-nucleon in-teraction were essential for overcoming the Pauli block-ing repulsion between the α particles [24, 25]. Here theterm “local interaction” refers to an interaction kernelthat is diagonal in the particle positions. These resultsshow that cluster-cluster interactions are important notonly for understanding specific nuclear states with well-defined cluster substructures, but also important for un-derstanding the balance of attractive and repulsive forcesin nuclear matter.Nuclear clustering is characterized by spatial correla-tions of the nucleons, and there are clear analogies to uni-versal phenomena in other quantum degenerate fermionicsystems. Dineutron correlations can be understoodin terms of the universal properties of two-componentfermionic superfluids at large scattering length [26–28],and α condensation in nuclear matter can be related tothe general theory of fermionic quartet condensation [29– a r X i v : . [ nu c l - t h ] A ug M and m with M > m . We find that this ap-proach is useful for understanding the competition be-tween attractive and repulsive forces analytically in thelimit M (cid:29) m , and we will refer to it as the heavy-lightansatz or Born-Oppenheimer approximation [36]. Ques-tions to be answered are whether the nuclear force be-haves as a short-range force, thus producing universalrepulsion between two dimers, and, if so, how the attrac-tive α - α interaction forms as the number and binding ofthe constituent nucleons within the clusters increase.In this work, we start with a general discussion of theeffective dimer-dimer interactions using the heavy-lightansatz and consider the relation to the effective inter-cluster interaction for the spin-aligned d + d system, whichcan be viewed as a two-dimer system composed of two-component fermions with components corresponding toisospin. We then investigate the effective inter-clusterinteractions of d + d , t + t , and α + α systems with amicroscopic cluster model using Brink-Bloch two-clusterwave functions [37] with effective nucleon-nucleon ( N N )forces. We find a repulsive interaction in the spin-aligned d + d system, attractive interactions in the spin-aligned t + t and α + α systems, and strong attractive interactionsin the spin-opposed d + d and t + t systems. By analyz-ing single-particle orbitals in the two-cluster systems, theimpact of antisymmetrization between identical nucleonson the cluster-cluster interaction is illuminated. Energiesof the lowest states of two-cluster systems are calculatedwith the generator coordinate method (GCM) [38, 39].The paper is organized as follows. In the next sec-tion, two-dimer systems with the heavy-light ansatz are described and effective dimer-dimer interactions are dis-cussed. In Sec. III, effective interactions between twoclusters in nuclear systems are investigated. A summaryis given in Sec. IV. Appendix A gives solutions of thetwo-delta potential problem in one dimension, and Ap-pendix B describes parametrization of the effective N N force. Inter-cluster wave functions in two-cluster systemsare described in Appendix C.
II. EFFECTIVE INTERACTION BETWEENTWO DIMERSA. Heavy-light ansatz M (cid:29) m We consider a mass imbalanced system of two-component fermions, where the two fermion componentshave masses M and m with M > m . We assume an at-tractive and local
M m potential that produces a bound
M m dimer and no interaction between identical parti-cles. We consider the limit M (cid:29) m , and we call theresulting simplifications the heavy-light ansatz. The dis-cussion will begin with the one-dimensional case, but willthen move to the three-dimensional case soon afterwards.The heavy particles are stationary at coordinates at { R , R , . . . } , and the light particles are feel the poten-tials produced by the heavy particles. The Hamiltonianis H = (cid:88) i h ( i ) , h ( i ) ≡ t ( i ) + U ( i ) , (1) t ( i ) = − (cid:126) m ∂ ∂x i , U ( i ) = (cid:88) j v ( | x i − R j | ) , (2)where U is the one-body potential. The ground state isa Slater determinant of single particle states,Ψ(1 , . . . , A m ) = A { ψ · · · ψ A m } = 1 √ A m ! det { ψ · · · ψ A m } . (3) A m is the total number of light m -particles and A is theantisymmetrizer, and the single-particle states are h ( i ) ψ n ( i ) = e n ψ n ( i ) . (4)We here use the notation for the one-dimensional (1D)system, but it can be readily applied to the three dimen-sional (3D) problem by replacing x → x and R → R . Itis also straightforward to extend the model to a nonlocal M m interaction.For the single-dimer system (
M m ), the Hamiltonianand wave function are given as h (0) = t + v ( x ) , (5) h (0) φ (0) ( x ) = (cid:15) (0) φ (0) ( x ) , (6)where x , (cid:15) (0) , and φ (0) are the relative coordinate, dimerenergy, and dimer wave function respectively. For sim-plicity, the phase of φ (0) is chosen to be real.To discuss the effective interaction between two dimers,we consider a two-dimer system M m + M m with twoheavy M -particles placed at x = − R/ x = R/ R . The Hamiltonian for two light m -particles is writtenas H = h (1) + h (1 (cid:48) ) , (7) h ( i ) = t ( i ) + U ( x i ) , U ( x ) = v L ( x ) + v R ( x ) , (8) v L ( x ) = v ( x + R/ , v R ( x ) = v ( x − R/ , (9)where the first and second m particles are labeled as 1and 1 (cid:48) . The energy E ( R ) and the two-body wave functionΨ(1 , (cid:48) ) of the lowest state are given as E ( R ) = (cid:15) + (cid:15) , (10)Ψ(1 , (cid:48) ) = A{ ψ (1) ψ (1 (cid:48) ) } , (11)where (cid:15) i and ψ i are the i th single-particle energy andstate obtained by solving the one-body problem ofthe single-particle Hamiltonian, h ( i ) ψ n ( i ) = (cid:15) n ψ n ( i ).Because of the symmetry of the one-body potential U ( x ) = U ( − x ), ψ n ( i ) are parity eigenstates with ψ ( x ) = ψ ( − x ), ψ ( x ) = − ψ ( − x ). The effective dimer-dimerinteraction is given by the relative energy E ( R ) − (cid:15) (0) measured from the two-dimer threshold energy. This ex-pression is exact for the heavy-mass limit, whereas it cor-responds to the Born-Oppenheimer approximation for fi-nite mass ratio. B. A cluster model for two-dimer system
1. Frozen dimer ansatz
For a general discussion of the effective dimer-dimerinteraction, we apply a cluster model to the two-dimersystem with a frozen dimer ansatz to approximately eval-uate the energy E ( R ). In this model, the system is ex-pressed as the antisymmetrized product of “atomic or-bitals” given by the isolated dimer wave functions aroundthe left and right M -particles asΦ(1 , (cid:48) ) = N A{ φ (0)L (1) φ (0)R (1 (cid:48) ) } , (12) φ (0)L ( i ) = φ (0) ( x i + R/ , φ (0)R ( i ) = φ (0) ( x i − R/ , (13)where N is the normalization factor. We introduce thefollowing notation for the matrix elements of one-bodyoperators O with respect to φ (0)L and φ (0)R as (cid:104) φ (0)L |O| φ (0)L (cid:105) = (cid:104)O(cid:105) LL , (cid:104) φ (0)R |O| φ (0)R (cid:105) = (cid:104)O(cid:105) RR , (14) (cid:104) φ (0)L |O| φ (0)R (cid:105) = (cid:104)O(cid:105) LR , (cid:104) φ (0)R |O| φ (0)L (cid:105) = (cid:104)O(cid:105) RL . (15)Here, the single-particle wave functions φ (0)L and φ (0)R arenot orthogonal but has a nonzero norm overlap (cid:104) (cid:105) LR = (cid:104) (cid:105) RL (cid:54) = 0, which vanishes in the limit of large R . Nev-ertheless, the total wave function Φ(1 , (cid:48) ) satisfies thePauli principle (Fermi statistics) because of the antisym-metrizer, and N = 1 / (cid:112) − (cid:104) (cid:105) is obtained from thenormalization condition (cid:104) Φ(1 , (cid:48) ) | Φ(1 , (cid:48) ) (cid:105) = 1.
2. Orthonormal bases sets: Molecular orbitals andorthonormal atomic orbitals
The atomic orbitals φ (0)L and φ (0)R with small separa-tion distance ( R ) overlap considerably with each otherand venture far into the Pauli forbidden region. In thiscase it is more natural to view the total wave functionΦ(1 , (cid:48) ) rewritten using a new orthonormal basis set, tak-ing into account the invariance of the normalized Slaterdeterminant under any linear transformation of the ba-sis vectors. One choice is the basis set of “molecularorbitals” asΦ(1 , (cid:48) ) = A{ ϕ + (1) ϕ − (1 (cid:48) ) } , (16) ϕ ± ( i ) = 1 (cid:112) ± (cid:104) (cid:105) RL ) { φ (0)L ( i ) ± φ (0)R ( i ) } , (17)where ϕ + and ϕ − are positive- and negative-parity or-bitals around whole system in analogy to covariant bondsof homonuclear diatomic molecules. This expression withthe molecular orbitals respects the parity symmetry ofthe one-body potential and is useful to discuss the two-dimer system in the overlapping region. However, at longdistances, the atomic orbital picture is more natural forthe probability of an m -particle on the left or right. Asyet another alternative basis set, “orthonormal atomicorbitals” can be also be defined asΦ(1 , (cid:48) ) = A{ ϕ L (1) ϕ R (1 (cid:48) ) } , (18) ϕ L ( i ) = 1 √ ϕ + ( i ) + ϕ − ( i )) , (19) ϕ R ( i ) = 1 √ ϕ + ( i ) − ϕ − ( i )) . (20)It should be commented that the former set { ϕ + , ϕ − } are obtained by solving the generalized eigenvalue prob-lem for the 2 × { φ (0)L , φ (0)R } . In contrast,the latter set of { ϕ L , ϕ R } is obtained by solving the gen-eralized eigenvalue problem for the norm and the positionoperator x .As a demonstration, we show the molecular orbitals { ϕ + , ϕ − } and the orthonormal atomic orbitals { ϕ L , ϕ R } for a Gaussian wave function φ (0) ( x ) = (cid:0) νπ (cid:1) / e − νx with ν = 0 .
25 fm − in Fig. 1. Fig. 1(a)-(d) compare themolecular orbitals with the original atomic orbitals fordistances R → R = 2, 4, and 8 fm. As the two dimerscome close to each other, the positive-parity orbital isformed by merging the left and right atomic orbitals,while the negative-parity molecular orbital has an extranode at the origin. In Fig. 1(e)–(h), the orthonormal -0.4 0 0.4 0.8 -8 -6 -4 -2 0 2 4 6 8 (h)atom. R=8 w a v e f n . (f m - / ) x (fm) φ L(0) , φ R(0) ϕ L ϕ R -0.4 0 0.4 0.8 -8 -6 -4 -2 0 2 4 6 8 (b)mole. R=2 w a v e f n . (f m - / ) x (fm) φ L(0) , φ R(0) ϕ + ϕ - -0.4 0 0.4 0.8 -8 -6 -4 -2 0 2 4 6 8 (c)mole. R=4 w a v e f n . (f m - / ) x (fm) φ L(0) , φ R(0) ϕ + ϕ - -0.4 0 0.4 0.8 -8 -6 -4 -2 0 2 4 6 8 (d)mole. R=8 w a v e f n . (f m - / ) x (fm) φ L(0) , φ R(0) ϕ + ϕ - -0.4 0 0.4 0.8 -8 -6 -4 -2 0 2 4 6 8 (e)atom. R → w a v e f n . (f m - / ) x (fm) φ L(0) , φ R(0) ϕ L ϕ R -0.4 0 0.4 0.8 -8 -6 -4 -2 0 2 4 6 8 (f)atom. R=2 w a v e f n . (f m - / ) x (fm) φ L(0) , φ R(0) ϕ L ϕ R -0.4 0 0.4 0.8 -8 -6 -4 -2 0 2 4 6 8 (g)atom. R=4 w a v e f n . (f m - / ) x (fm) φ L(0) , φ R(0) ϕ L ϕ R -0.4 0 0.4 0.8 -8 -6 -4 -2 0 2 4 6 8 (a)mole. R → w a v e f n . (f m - / ) x (fm) φ L(0) , φ R(0) ϕ + ϕ - FIG. 1: (a)-(d) Molecular orbitals { ϕ + , ϕ − } and (e)-(h) orthonormal atomic orbitals { ϕ L , ϕ R } of the two-dimer system for aGaussian wave function of a single dimer φ (0) ( x ) = (cid:0) νπ (cid:1) / e − νx with ν = 0 .
25 fm − . The separation distances are chosen tobe R → R = 2 ,
4, and 8 fm. The original atomic orbitals { φ (0)L , φ (0)R } on the left and right are also shown. atomic orbitals { ϕ L , ϕ R } are compared with the originalatomic orbitals { φ (0)L , φ (0)R } . At short distances R ≤ ϕ L and ϕ R are significantly distorted from the original or-bitals because of antisymmetrization, while at long dis-tances the effect of antisymmetrization vanishes and theyapproach the original orbitals φ (0)L and φ (0)R . C. Effective interaction between two dimers
Two dimers can not exist at the same position be-cause of the Pauli principle between identical fermions.This effect gives a repulsive contribution to the effectivedimer-dimer interaction at short distance. As shown inFig. 1(e)-(h), significant distortion occurs in the “physi-cal” atomic orbitals, ϕ L and ϕ R , at short distances be-cause of the antisymmetrization effect. As a result ofthe distortion, each dimer loses some internal energy. Onthe other hand, the M m potential term between differentdimers can give an attractive contribution. This showsthat the effective dimer-dimer interaction is determinedby the competition between the internal energy loss and the energy gain from the inter-cluster potential term. Inthis section, we investigate the two-dimer energy E ( R )with the frozen dimer ansatz and discuss the effectivedimer-dimer interaction.
1. Expression for general potentials
The present model with the frozen dimer ansatz cor-responds to an approximation of the single-particle wavefunctions ψ in the two-dimer system with linear combi-nation of the left and right atomic orbitals as ψ ≈ φ = c L φ (0)L + c R φ (0)R , which is equivalent to a two-level problemgiven as (cid:104) H (cid:105) αβ = (cid:18) (cid:15) (0) + (cid:104) v R (cid:105) LL (cid:15) (0) (cid:104) (cid:105) LR + (cid:104) v L (cid:105) LR (cid:15) (0) (cid:104) (cid:105) RL + (cid:104) v R (cid:105) RL (cid:15) (0) + (cid:104) v L (cid:105) RR (cid:19) , (21) (cid:104) (cid:105) αβ = (cid:18) (cid:104) (cid:105) LR (cid:104) (cid:105) RL (cid:19) , (22)with ( α, β ) = ( φ (0)L , φ (0)R ). By solving the generalizedeigenvalue problem for these 2 × ϕ + and ϕ − as the eigensolutionswith eigenvalues (cid:15) + and (cid:15) − as (cid:15) ± = (cid:15) (0) + (cid:104) v R (cid:105) LL ± (cid:104) (cid:105) RL ± (cid:104) v R (cid:105) RL ± (cid:104) (cid:105) RL . (23)The total energy of the two-dimer system and the relativeenergy measured from the threshold energy are obtainedas E = (cid:15) + + (cid:15) − = 2 (cid:15) (0) + 21 − (cid:104) (cid:105) ( (cid:104) v R (cid:105) LL − (cid:104) (cid:105) RL (cid:104) v R (cid:105) RL ) , (24)∆ E ( R ) ≡ E ( R ) − (cid:15) (0) = 21 − (cid:104) (cid:105) ( (cid:104) v R (cid:105) LL − (cid:104) (cid:105) RL (cid:104) v R (cid:105) RL ) . (25)Note that the kinetic energy contribution does not explic-itly appear in the present expression of ∆ E ( R ), thoughit is implicitly contained in the exchange potential termwith the relation (cid:104) v R (cid:105) RL = (cid:15) (cid:104) (cid:105) RL − (cid:104) t (cid:105) RL .For the general case, we consider an attractive poten-tial v ( x ) ≤ r (0) and a dimer wavefunction φ (0) ( x ) ≥ b (0) . Let us con-sider two terms in the expression (cid:104) v R (cid:105) LL − (cid:104) (cid:105) RL (cid:104) v R (cid:105) RL .The first term, (cid:104) v R (cid:105) LL = (cid:90) v R ( x ) | φ (0)L ( x ) | dx ≤ , (26)gives a negative (attractive) contribution and is the directpotential term obtained by folding the right-side poten-tial with the density ρ (0)L ( x ) ≡ | φ (0)L ( x ) | of the left-sideatomic orbital. Roughly speaking, this term gives a fi-nite contribution in the R < r (0) + b (0) region, where thedimer density has overlap with the closest edge of theexternal potential.The second term, −(cid:104) (cid:105) RL (cid:104) v R (cid:105) RL = −(cid:104) (cid:105) RL (cid:90) φ (0)L ( x ) v R ( x ) φ (0)R ( x ) dx ≥ , (27)gives a positive (repulsive) contribution corresponding toan exchange potential term. This term becomes signifi-cant in the R < b (0) + min( b (0) , r (0) ) region for the over-lapping region of the two atomic orbitals and the right-side potential.As an alternative expression, the sum of the direct andexchange potential terms can be rewritten as (cid:104) v R (cid:105) LL − (cid:104) (cid:105) RL (cid:104) v R (cid:105) RL = (cid:104) φ (0)L | P ⊥ R v R | φ (0)L (cid:105) , (28)where P ⊥ R ≡ − | φ (0)R (cid:105)(cid:104) φ (0)R | is the projection operatoronto the space orthogonal to φ (0)R . It means that the sumis the transition from φ (0)L to the orthogonal component P ⊥ R | φ (0)L (cid:105) of the left-side particle by the external potential v R on the right, and the exchange potential term arisesfrom the orthogonal condition.For the two-dimer energy ∆ E ( R ) measured from thethreshold in Eq. (25), the overall factor −(cid:104) (cid:105) is positivebecause |(cid:104) (cid:105) RL | ≤
1. Therefore, the sign of the effectivedimer-dimer interaction is determined by the competitionbetween the attraction from the direct potential termof Eq. (26) and the repulsive effect from the exchangepotential term of Eq. (27).For the case where the local potential v ( x ) has a rangelonger than the dimer size, r (0) > b (0) , the effectivedimer-dimer interaction can be attractive in the inter-mediate distance region of 2 b (0) < R < r (0) + b (0) . Inthis region the two atomic orbitals have almost no over-lap φ (0)L ( x ) φ (0)R ( x ) ∼
0, and the exchange potential termis small compared with the direct potential term. In theopposite case that v ( x ) is a has a range shorter than thedimer size, r (0) < b (0) , the effective dimer-dimer interac-tion can be repulsive because of the strong contributionfrom the exchange potential term. Also in the case oflong-range but nonlocal potential v ( x, x (cid:48) ), the effectivedimer-dimer interaction may again be repulsive, becausethe nonlocality generally suppresses the matrix element (cid:104) v R (cid:105) LL in the direct potential term but enhances (cid:104) v R (cid:105) RL in the exchange potential term.All of the expressions derived in this section can beapplied to dimer-dimer systems in three dimensions, justby replacing the one-dimensional integrals in the expec-tation values with three-dimensional integrals. We notethat these findings provide a conceptual foundation forthe conclusions obtained numerically in Ref. [24, 35], thatincreasing the range or strength of the local part of theparticle-particle interaction produces a more attractivecluster-cluster interaction. D. Effective dimer-dimer interaction withzero-range potential
As an example of short-range potentials, we show thatthe effective dimer-dimer interactions with M (cid:29) m in 1Dand 3D for a zero-range potential are always repulsive forany R .
1. Frozen cluster ansatz
Firstly, we discuss the dimer-dimer interaction in 1Dwith the frozen cluster ansatz. For the delta potential v ( x ) = − (cid:126) κ m δ ( x ) , (29)the energy and wave function of a single dimer are givenas (cid:15) (0) = − (cid:126) m κ , φ (0) ( x ) = √ κ e − κ | x | , (30)where 1 /κ is roughly regarded as the dimer size b (0) .For the two-dimer system with the distance R , one cancalculate matrix elements as (cid:104) (cid:105) RL = (1 + κ R )e − κ R , (cid:104) v R (cid:105) RL = 2 (cid:15) (0) e − κ R , (cid:104) v R (cid:105) LL = 2 (cid:15) (0) e − κ R , (31)and obtain energies for the positive- and negative-paritymolecular orbitals (cid:15) ± = (cid:15) (0) + 2 (cid:15) (0) e − κ R ± e − κ R ± e − κ R (1 + κ R ) , (32)and the two-dimer energy from the threshold is∆ E ( R ) = | (cid:15) (0) | κ R e − κ R − (1 + κ R ) e − κ R > . (33)This shows that the two dimers feel a repulsive dimer-dimer interaction for any R .Next, we show the result for the dimer-dimer interac-tion in 3D obtained with the frozen cluster ansatz. Forthe renormalization of the single-delta potential in 3D,we assume that we have dimer with energy (cid:15) (0) ( < φ (0) ( r ) = (cid:114) κ π e − κ | r | | r | , (34)with the definition κ ≡ (cid:112) m | (cid:15) (0) | / (cid:126) . For the two-dimersystem in 3D, we consider the single-particle energies fortwo delta potentials at − R / R / R = | R | . Using the frozendimer ansatz, the matrix elements are obtained as (cid:104) (cid:105) RL = e − κ R , (cid:104) v R (cid:105) RL = 2 (cid:15) (0) e − κ R κ R , (cid:104) v R (cid:105) LL = 0 , (35)and the energies for the positive- and negative-paritymolecular orbitals are (cid:15) ± = (cid:15) (0) ± (cid:15) (0) e − κ R κ R (1 ± e − κ R ) , (36)and the two-dimer energy measured from threshold is∆ E ( R ) = | (cid:15) (0) | − κ R κ R (1 − e − κ R ) > , (37)indicating again a repulsive dimer-dimer interaction.Our results for the zero-range potential in 1D and 3Dusing the frozen dimer ansatz clearly show that the repul-sive dimer-dimer interaction originates from the exchangepotential term (cid:104) v R (cid:105) RL , i.e. , the antisymmetrization orPauli blocking effect.
2. Exact solution and asymptotic expansion
We can also obtain exact solutions for the two-dimerenergy in 1D and 3D by solving the two-delta poten-tials and see again the universal repulsion of the effectivedimer-dimer interaction in the M (cid:29) m limit.We express the exact energies (cid:15) exact ± in terms of bindingmomenta κ ± defined as (cid:15) exact ± = − (cid:126) m κ ± . (38)For the exact solutions of the positive- and negative-parity bound states of the 1D two-delta potential, κ + and κ − are given as κ + = κ (cid:110) κ R W ( κ R e − κ R ) (cid:111) , (39) κ − = κ (cid:110) κ R W − ( κ R e − κ R ) (cid:111) , (40)where W and W − are branches of the Lambert W func-tion. With these solutions for κ ± , the two-dimer energymeasured from the threshold is expressed as∆ E ( R ) = (cid:15) (0) (cid:16) κ κ + κ − κ − (cid:17) . (41)For large κ R we have the asymptotic forms κ + → κ (cid:16) − κ R − κ R e − κ R + · · · (cid:17) , (42) κ − → κ (cid:16) − e − κ R − κ R e − κ R + · · · (cid:17) , (43)and hence∆ E ( R ) = | (cid:15) (0) | (cid:104) κ R e − κ R − − κ R + · · · (cid:105) . (44)One can see that the leading term | (cid:15) (0) | κ R e − κ R isconsistent with that of the approximate result in Eq. (33)of the frozen dimer ansatz.Similarly, the bound-state solutions for the 3D two-delta potential have binding momenta κ + and κ − of theform κ + = κ (cid:110) κ R W (e − κ R ) (cid:111) , (45) κ − = κ (cid:110) κ R W − (e − κ R ) (cid:111) . (46)The asymptotic forms for large κ R are κ + → κ (cid:16) − κ R κ R − e − κ R κ R + · · · (cid:17) , (47) κ − → κ (cid:16) − e − κ R κ R − e − κ R κ R + · · · (cid:17) , (48)and hence∆ E ( R ) = | (cid:15) (0) | (cid:104) − κ R κ R − − κ R ( κ R ) + · · · (cid:105) . (49)One can see again that the leading term is consistent withthat in Eq. (37) of the frozen dimer ansatz.In Fig. 2, we compare the single-particle energiesand the two-dimer energy measured from the thresh-old energy for exact solutions and approximate ones ofthe frozen cluster ansatz. In the 1D results shown inFigs. 2(a) and (b), one can see that the frozen clusteransatz is a good approximation for κ R (cid:38)
2, but getsworse for R (cid:46) /κ , where two dimers are closer thantwice of the dimer size b (0) ∼ /κ . For the 3D case,it is a good approximation for κ R (cid:38) . III. NUCLEAR SYSTEMS OF TWO CLUSTERS: d + d , t + t , AND α + α A. Cluster model wave functions
We now discuss the effective interactions between twonuclear clusters by applying the Brink-Bloch clustermodel [37]. We consider d + d , t + t , and α + α systemswith d , t , and α clusters consisting of two, three and fournucleons, respectively. We denote the mass number of acluster as c ( c = 2 , ,
4) and use a label “ c + c ” for thetwo-cluster systems.
1. Single-cluster wave function
In the cluster model, a single cluster is assumed to bea c -nucleon state with the harmonic oscillator 0 s -orbitconfiguration noted as (0 s ) c . The wave function for thecluster placed at R is written as a product of single-particle Gaussian wave functions asΦ c R (1 , . . . , c )= A (cid:110) φ (0) R ( r ) · · · φ (0) R ( r c ) ⊗ χ c ( s , . . . , s c ) (cid:111) , (50) φ (0) R ( r ) = (cid:18) νπ (cid:19) / exp (cid:2) − ν ( r − R ) (cid:3) , (51)where s i indicates the nucleon spin and isospin degrees offreedom of the i th nucleon, and χ c is the spin and isospinfunction of the ( S = 1 , T = 0), ( S = 1 / , T = 1 / S = 0 , T = 0) states for the deuteron, triton, and α clusters, respectively.
2. Two-cluster wave function
For the d -cluster with S = 1, we consider the spin-aligned [ d + d ] S =2 and spin-opposed [ d + d ] S =0 states.Similarly, for the t -cluster with S = 1 /
2, the spin-aligned[ t + t ] S =1 and spin-opposed [ t + t ] S =0 states are considered. -2-1.5-1-0.5 0 0 1 2 3 4 5 6 (a) ε + ε − ε ± ⁄ ε ( ) κ R exact in 1Dapprox. in 1D 0 0.5 1 1.5 2 0 1 2 3 4 5 6 (b) ∆ E ⁄ ε ( ) κ R exact in 1Dapprox. in 1D-2-1.5-1-0.5 0 0 1 2 3 4 5 6 (c) ε + ε − ε ± ⁄ ε ( ) κ R exact in 3Dapprox. in 3D 0 0.5 1 1.5 2 0 1 2 3 4 5 6 (d) ∆ E ⁄ ε ( ) κ R exact in 3Dapprox. in 3D FIG. 2: Energies of the two-dimer system for delta potentialin the heavy-light ansatz in one dimension (1D) and threedimensions (3D). The approximate values with the frozendimer ansatz and exact values are compared. (a) Single-particle energies (cid:15) ± in 1D. (b) The two-dimer energy fromthe threshold energy, ∆ E = (cid:15) + + (cid:15) − − (cid:15) (0) , in 1D. (c) Single-particle energies (cid:15) ± in 3D. (d) The two-dimer energy fromthe threshold energy in 3D. Energies are plotted in units of1 / | (cid:15) (0) | = 2 m/ ( (cid:126) κ ). The wave function of a two-cluster system with sepa-ration distance R is given asΦ c + c ( R ; 1 , . . . , c, (cid:48) , . . . , c (cid:48) )= ( N ) n id A (cid:110) Φ c − R (1 , . . . , c )Φ c R (1 (cid:48) , . . . , c (cid:48) ) (cid:111) = ( N ) n id A (cid:110) φ (0) − R ( r ) · · · φ (0) − R ( r c ) φ (0) R ( r (cid:48) ) · · · φ (0) R ( r c (cid:48) ) ⊗ (cid:2) χ c ( s , . . . , s c ) χ c ( s (cid:48) , . . . , s c (cid:48) ) (cid:3) S (cid:111) , (52)where spins of two clusters are coupled to S in total,and R is chosen to be (0 , , R ) on the z axis. n id is thenumber of pairs of identical nucleons. n id = 2 , , d + d ] S =2 , [ t + t ] S =1 , [ α + α ] S =0 respectively, and n id = 0 , d + d ] S =0 , [ t + t ] S =0 respectively.The nuclear matter densities of two-cluster wave func-tions are shown in Fig. 3. The [ d + d ] S =2 , [ t + t ] S =1 ,and [ α + α ] S =0 systems are composed of d = ( p ↑ n ↑ ), t = ( p ↑ n ↑ n ↓ ), and α = ( p ↑ p ↓ n ↑ n ↓ ), respectively, andthey show a dumbbell-like drop in the density in the R (cid:46) d + d ] S =0 system with no identical-nucleon pairs,the two clusters can penetrate each other without anyPauli blocking and merge into a He state with an (0 s ) configuration in the R → t + t ] S =0 state con-taining two identical-nucleon pairs shows a weaker Pauliblocking effect than the [ t + t ] S =1 state.For the [ d + d ] S =2 , [ t + t ] S =1 , and [ α + α ] S =0 sys-tems, the total wave function Φ c + c ( R ) is expressed bya Slater determinant of non-orthonormal atomic orbitals { φ (0) − R ( i ) , φ (0) R ( i (cid:48) ) } , which can be transformed into themolecular orbitals set { ϕ + ( i ) , ϕ − ( i (cid:48) ) } or the orthonor-mal atomic orbitals set { ϕ L ( i ) , ϕ R ( i (cid:48) ) } under invarianceof the total wave function as described previously in Sec-tion II.
3. Parity and orbital-angular-momentum projections
We consider the parity ( π ) and orbital-angular-momentum ( L ) projection of the two-cluster wave func-tions as Φ πc + c ( R ) = P π Φ c + c ( R ) , (53)Φ Lπc + c ( R ) = P L P π Φ c + c ( R ) , (54)with the L and π projection operators P L and P π .The intrinsic energy E int ( R ) at a distance R is calcu-lated using the π -projected wave function without the L -projection as E int c + c ( R ) = (cid:104) Φ πc + c ( R ) | H | Φ πc + c ( R ) (cid:105)(cid:104) Φ πc + c ( R ) | Φ πc + c ( R ) (cid:105) . (55)Similarly the L π -projected energy is calculated with the L π -projected wave function as E Lπc + c ( R ) = (cid:104) Φ Lπc + c ( R ) | H | Φ Lπc + c ( R ) (cid:105)(cid:104) Φ Lπc + c ( R ) | Φ Lπc + c ( R ) (cid:105) . (56) (a) R → ρ / A (f m - ) z (fm) [d+d] S=2 ,[t+t]
S=1 ,[ α + α ] S=0 [d+d]
S=0 [t+t]
S=0 (b) R=2 (fm) ρ / A (f m - ) z (fm) [d+d] S=2 ,[t+t]
S=1 ,[ α + α ] S=0 [d+d]
S=0 [t+t]
S=0 (c) R=4 (fm) ρ / A (f m - ) z (fm) [d+d] S=2 ,[t+t]
S=1 ,[ α + α ] S=0 [d+d]
S=0 [t+t]
S=0 (d) R=8 (fm) ρ / A (f m - ) z (fm) [d+d] S=2 ,[t+t]
S=1 ,[ α + α ] S=0 [d+d]
S=0 [t+t]
S=0
FIG. 3: Nuclear matter density of the two-cluster wave func-tions for d + d , t + t , and α + α systems with distances R → R = 2 ,
4, and 8 fm. The densities are integrated over x and y and normalized with the mass number A = 2 c as ρ ( z ) /A .The normalized densities of the [ d + d ] S =2 , [ t + t ] S =1 , and[ α + α ] S =0 states, which are consistent with each other, areplotted with dashed lines, and those of [ d + d ] S =0 and [ t + t ] S =0 states are shown by dotted and solid lines, respectively. We take π = − and L = 1 ( P -wave) for the [ t + t ] S =1 system as required by antisymmetry, and π = + and L = 0 ( S -wave) for the other systems.The total angular momentum and parity are J π =0 − , − , − for the [ t + t ] S =1 system, and J π = S π for theother systems. Strictly speaking, the J π = 2 + and 0 + states are coupled in the d + d system and the J π = 0 − , − and 2 − states are coupled in the [ t + t ] S =1 system becauseof the N N spin-orbit and tensor interactions, but we omitsuch the channel couplings due to our assumption of ef-fective
N N central forces for simplicity.
4. GCM calculation of two-cluster systems
We calculate the energy E c + c of the ground states oftwo-cluster systems with the GCM [38, 39] by superpos-ing L π -projected wave functionsΨ GCM c + c = (cid:88) k c k Φ Lπc + c ( R k ) , (57)where coefficients c k are determined by solving the dis-cretized Hill-Wheeler equation [38], i.e. , solving the gen-eralized eigenvalue problem for norm and Hamiltonianmatrices with respect to k . This GCM calculation corre-sponds to optimization of the inter-cluster wave functionas described in Appendix C.We also perform one-dimensional GCM calculations(1d-GCM) by superposing the π -projected wave func-tions instead of the L π projected ones asΨ c + c = (cid:88) k c k Φ ± c + c ( R k ) . (58)In the 1d-GCM calculation, all nucleons ( i =1 , . . . , c, (cid:48) . . . , c (cid:48) ) are confined for the x and y directionsin the same Gaussian orbit (cid:0) νπ (cid:1) / exp (cid:2) − ν ( x i + y i )],whereas the inter-cluster motion in the z direction is op-timized by the superposition. After diagonalization ofthe norm and Hamiltonian matrices, one obtains the 1d-GCM energy E for the lowest solution of the one-dimensional motion. B. Hamiltonian and effective nuclear force
The Hamiltonian of nuclear systems is given as H = (cid:88) i t i − T cm + (cid:88) i 2) = V N ( r ) × (cid:2) f E P ( E ) + f E P ( E )+ f O P ( O ) + f O P ( O ) (cid:3) , (61) r ≡ | r − r | , (62)where the radial function V N ( r ) is given by a two-range Gaussian form. In the original expression, the Volkov force is given by the Wigner, Bertlett, Heisen-berg, and Majorana terms. Details of parametrization ofthe Volkov No.2 force and its relation to ratios f E , f E , f O , and f O in Eq. (61) are explained in Appendix B.The parameter sets { f E , f E , f O , f O } adopted in thepresent calculation are summarized in Table I. The firstset is a purely even-parity force with SU4 symmetry as v SU4 N = V N ( r ) (cid:2) P ( E ) + P ( E ) (cid:3) , (63)which we call the SU4-even force. This force acts on spa-tial even components of otherwise-nucleon pairs, ( p ↑ p ↓ ),( p ↑ n ↑ ), ( p ↑ n ↓ ), and ( n ↑ n ↓ ) with the same strength. Notethat the [ d + d ] S =2 state is equivalent to a four-neutronsystem of two dineutrons ( nn ) + ( nn ) in the case of SU4-symmetric forces. The second set is a tuned force v tuned N = V N ( r ) (cid:2) . P ( E ) + 0 . P ( E ) − . P ( O ) − . P ( O ) (cid:3) , (64)adjusted to fit the experimental data of S -wave N N scat-tering lengths in the spin-triplet and spin-singlet channelsand α - α scattering phase shifts. This tuned force con-tains a stronger E force and a weaker E force with theratio of 1.3/0.7 to describe a bound deuteron state andan unbound nn state.In addition, we consider two optional sets to make the[ d + d ] S =2 system to be bound, which do not describephysical nuclear systems. One is a strong-even N N force3 v SU4 N = V N ( r ) (cid:2) P ( E ) + 3 P ( E ) (cid:3) , (65)which is three times as strong as the SU4-even N N force.The other is a state-independent N N force containing E , O , and O attraction with the same strength as the E component of v tuned N v st-ind N = V N ( r ) (cid:2) . P ( E ) + 1 . P ( E )+ 1 . P ( O ) + 1 . P ( O ) (cid:3) . (66) C. Energy of single-cluster systems In Table II, we show the total, kinetic and potentialenergies for a single-cluster system of d , t , and α calcu-lated with the (0 s ) c configurations. Values of the widthparameter ν used in the present calculation are also listedin the table. For the v SU4 N force, ν is fixed to be a com-mon value ν = 0 . 25 fm − , which reproduces the root-mean-square (rms) radius of an α particle. For otherthree forces, v tuned N , v st-ind N , and 3 v SU4 N , we use the values ν = 0 . 16, 0 . 16, and 0 . 35 fm − , respectively, which areoptimized to minimize the d -cluster energy.Let us compare the energies of the d , t , and α clus-ters obtained with the SU4-even force ( v SU4 N ). As theconstituent nucleons increases, the single-cluster systemobtains a deeper binding because the kinetic energy lossis proportional to c − c ( c − / TABLE I: Parameter sets of the Volkov No.2 force for four-types of the NN forces, v SU4 N (SU4-even), v tuned N (tuned), v st-ind N (state-independent), and 3 v SU4 N (strong-even) forces. Details of the strength parameters ( f E , f E , f O , and f O ) for the E , E , O ,and O terms and the parameters ( W B , H , and M ) for the Wigner, Bartlett, Heisenberg, and Majorana terms are describedin Appendix B. f E f E f O f O W B H Mv SU4 N SU4-even 1 1 0 0 0.5 0 0 0.5 v tuned N tuned 1.3 0.7 − . − . v st-ind N state-independent 1.3 1.3 1.3 1.3 1 0.3 0 03 v SU4 N strong-even 3 3 0 0 1.5 0 0 1.5 -80-60-40-20 0 20 40 60 0 1 2 3 4 5 6 ε d = − ε d = − 24 MeV v N (r) ( M e V ) r (fm) v Ntuned NSU4 FIG. 4: Radial dependence of the E component of thetuned ( v tuned N ) and strong-even (3 v SU4 N ) forces. The internalwave function Φ d ( r ) ∝ exp[ − νr ] of a d -cluster with the(0 s ) configuration with ν = 0 . 16 fm − for the tuned forceand that with ν = 0 . 35 fm − for the strong-even force areshown by pink and light-blue colored areas, respectively, inan arbitrary unit. The tuned N N force ( v tuned N ) gives a bound d state atthe energy (cid:15) d = − . ν = 0 . 16 fm − , while itgives the same α energy (cid:15) α = − . v st-ind N obtains (cid:15) d = − . N N force becausethe N N force in the E channel is unchanged.The strong-even N N force (3 v SU4 N ) gives a deeplybound d state with ν = 0 . 35 fm − at (cid:15) d = − . E component and thedeuteron wave function for the tuned and strong-even N N forces are shown in Fig. 4. Compared with thetuned force, the d -cluster for the strong-even force is moredeeply bound and the cluster size is much smaller. TABLE II: Energies of single-cluster and two-cluster sys-tems calculated with the cluster model using four types ofthe NN force. For single-cluster systems, the total ( (cid:15) c ), ki-netic ( T ), and potential ( V ) energies are shown together withthe adopted ν values (fm − ). For two-cluster systems, GCMenergies measured from the c + c threshold energy (2 (cid:15) c ) and1d-GCM energies relative to the one-dimensional c + c decaythreshold energy (2 (cid:15) c + (cid:126) ω/ 2) are shown. For the [ t + t ] S =1 system, the GCM result for the L π = 1 − state and the 1d-GCM result for the π = − state are shown. For other systems,the GCM result for the L π = 0 + state and the 1d-GCM re-sult for the π = + state are shown. For unbound systems,positive energies are obtained in the present framework of abound state approximation with R ≤ 10 fm. The energy unitis MeV. v SU4 N : SU4 ν (cid:15) c T V ∆ E c + c ∆ E c + c d . − . d + d ] S =2 unbd.(1.34) unbd.(0.98)[ d + d ] S =0 − . − . t − . − . t + t ] S =1 unbd.(1.14) − . t + t ] S =0 − . − . α − . − . α + α ] S =0 − . − . v tuned N : tuned ν (cid:15) c T V ∆ E c + c ∆ E c + c d − . − . d + d ] S =2 unbd.(2.9) unbd.(0.97) α − . − . t + t ] S =1 − . − . v SU4 N : strong even ν (cid:15) c T V ∆ E c + c ∆ E c + c d − . − . d + d ] S =2 − . − . v st-ind N : state-independent ν (cid:15) c T V ∆ E c + c ∆ E c + c d − . − . d + d ] S =2 − . − . D. Two-cluster systems 1. GCM and 1d-GCM resuts To obtain the lowest states of two-cluster systems, weperform the GCM calculations using the two-cluster wavefunctions with R k = 0 . , , . . . , 10 fm. The calculationscorrespond to a bound state approximation in a finitebox boundary R k ≤ 10 fm. We also perform the 1d-GCMcalculations to check whether two clusters effectively feelan attraction forming a one-dimension bound state ornot.In Table II, the GCM and 1d-GCM energies of two-cluster systems are listed. For the GCM result, theenergy is measured from the c + c threshold energy as∆ E c + c ≡ E c + c − (cid:15) c . For the 1d-GCM result, the en-ergy is measured from the one-dimensional c + c decaythreshold ∆ E c + c = E c + c − (2 (cid:15) c + 12 (cid:126) ω ) (67)are shown. Here the one-dimensional decay thresholdcontains an extra kinetic energy cost 2( (cid:126) ω/ 4) for local-ization in two directions on the xy plane.For the [ d + d ] S =2 system with the v tuned N (tuned) and v SU4 N (SU4-even) forces, no bound state is obtained inboth the GCM and 1d-GCM calculations, indicatingthat the effective interaction between to d -clusters in the S = 2 channel is repulsive. For the [ α + α ] S =0 systemwith the v tuned N (tuned) force, the GCM calculation ob-tains a weakly bound state without the Coulomb force,but an unbound state with the Coulomb force, consis-tent with the observed quasi-bound 2 α state of Be(0 + ).The [ t + t ] S =1 system with the v SU4 N (SU4-even) force, isnot bound in the GCM calculation but bound in the 1d-GCM calculation meaning that the effective interactionbetween two t -clusters in the S = 1 channel is a weakattraction.In the [ d + d ] S =0 system for the spin-opposed d + d inthe S = 0 channel, two d -clusters are deeply bound andform an α particle because there is no Pauli blocking inthis system. Also the [ t + t ] S =0 system forms a boundstate because of a weaker Pauli effect than the [ t + t ] S =1 system. 2. Energy curves of two-cluster systems To discuss effective inter-cluster interactions, we an-alyze the R dependence of the L π -projected energies E Lπc + c ( R ) for the two-cluster wave functions Φ Lπc + c ( R ) withthe distance R . In Fig. 5, we show the total, kinetic, andpotential energy contributions of d + d , t + t , and α + α systems calculated with the SU4-even ( v SU4 N ) force. Eachenergy contribution is shifted by subtracting the “asymp-totic” value at R → ∞ . In this plot, the two-clusterdecay threshold energy 2 (cid:15) c is located at (cid:126) ω/ R = ∞ , which contains the kinetic energy cost for localization of the inter-clusterwave function in the R direction. In Fig. 6(a), we showthe intrinsic energy E int c + c ( R ) for Φ πc + c ( R ) measured fromthe one-dimensional decay threshold energy (2 (cid:15) c + (cid:126) ω ).Since Φ πc + c ( R ) contains a kinetic energy cost 3( (cid:126) ω/ 4) forthe localization in three directions, the offset energy at R → ∞ is (cid:126) ω/ d + d ] S =2 , [ t + t ] S =1 , and[ α + α ] S =0 , the effective inter-cluster interaction in the[ d + d ] S =2 system is found to be repulsive for all R ,whereas those in the [ t + t ] S =1 and [ α + α ] S =0 systemsare attractive in the medium distance region. The ki-netic energy term gives a repulsive contribution in theshort-distance range because of the Pauli effect, whereasthe potential energy term gives an attractive contribu-tion in a slightly longer range than the kinetic repulsion.As the mass number c of clusters increases, the potentialenergy attraction rapidly increases, and finally producesthe medium-range attraction of the effective interactionin [ α + α ] S =0 .In the [ d + d ] S =0 and [ t + t ] S =0 systems, the effectiveinter-cluster interactions are attractive because two clus-ters feel either no or a weaker Pauli effect. In particular,two d -clusters in the S = 0 channel feel a rather strongattraction at short distances and come close to each otherwithout the Pauli repulsion. In these spin-opposed states,two clusters merge into bound α and He states losingtheir identity.In both the spin-aligned and spin-opposed cases, thecompetition between kinetic an potential energy termsplays an important role in the effective inter-cluster in-teractions. The relatively short-range repulsion of theinter-cluster interactions comes from the Pauli effect be-tween identical nucleons mainly through the kinetic en-ergy term. 3. Energy contributions in the atomic- andmolecular-orbital pictures For further discussion of the effective inter-cluster in-teraction in the [ d + d ] S =2 , [ t + t ] S =1 , and [ α + α ] S =0 sys-tems, we count kinetic and potential energy contributionswith the atomic- and molecular-orbitals, which are de-scribed in the previous section. For a general discussion,we here choose f E = f E ≡ f even and f O = f O ≡ f odd and consider a SU4-symmetric N N force as v N = V N ( r )( f even P even + f odd P odd ) , (68)with P even ≡ P ( E )+ P ( E ) and P odd ≡ P ( O )+ P ( O ).The energy for a single cluster is given as (cid:15) c = ( c − 1) ¯ T + c ( c − V E0 , (69)¯ T ≡ (cid:104) φ (0)0 | t | φ (0)0 (cid:105) = 34 (cid:126) ω, (70)¯ V E0 ≡ f even (cid:104) φ (0)0 φ (0)0 | V N | φ (0)0 φ (0)0 (cid:105) , (71)2 -60-40-20 0 20 40 60 0 1 2 3 4 5 6 7 8 H+ H S=2,L=0m50-bh0- ν E n e r gy ( M e V ) R (fm) H)T − T ∞ V − V ∞ E − E ∞ -60-40-20 0 20 40 60 0 1 2 3 4 5 6 7 8 H+ H S=1,L=1m50-bh0- ν E n e r gy ( M e V ) R (fm) H)T − T ∞ V − V ∞ E − E ∞ -60-40-20 0 20 40 60 0 1 2 3 4 5 6 7 8 He+ He S=0,L=0m50-bh0- ν E n e r gy ( M e V ) R (fm) He)T − T ∞ V − V ∞ E − E ∞ -60-40-20 0 20 40 60 0 1 2 3 4 5 6 7 8 H+ H S=0,L=0m50-bh0- ν E n e r gy ( M e V ) R (fm) H)T − T ∞ V − V ∞ E − E ∞ -60-40-20 0 20 40 60 0 1 2 3 4 5 6 7 8 H+ H S=1,L=1m50-bh0- ν E n e r gy ( M e V ) R (fm) H)T − T ∞ V − V ∞ E − E ∞ FIG. 5: L π -projected energies E Lπc + c ( R ) of two-cluster systems with separation distance R calculated with the SU4-even NN force ( v SU4 N ). Total ( E ), kinetic ( T ), and potential ( V ) energy contributions of (a) [ d + d ] S =2 , (b) [ α + α ] S =0 , (d) [ d + d ] S =0 ,and (e) [ t + t ] S =0 for the L π = 0 + states and those of (c) [ t + t ] S =1 for the L π = 1 − state are shown. The asymptotic valuesat R → ∞ are subtracted from each energy contribution. Black dashed lines show the two-cluster decay threshold relative tothe asymptotic total energy at R → ∞ , given as 2 (cid:15) (0) c − E Lπc + c ( ∞ ) = − (cid:126) ω/ where φ (0)0 = φ (0) R =0 . Note that the odd component givesno contribution to single-cluster systems.For the [ d + d ] S =2 , [ t + t ] S =1 , and [ α + α ] S =0 systems, theintrinsic energy can be expressed with the orthonormalatomic orbitals { ϕ L , ϕ R } as E int c + c − (cid:15) c = ¯ T + 2∆ (cid:15) c + ( c − c ) ¯ V ELR + c ¯ V OLR , (72)∆ (cid:15) c = c (cid:0) ¯ T LL − ¯ T (cid:1) + c ( c − (cid:0) ¯ V ELL − ¯ V E0 (cid:1) , (73)¯ T LL ≡ (cid:104) ϕ L | t | ϕ L (cid:105) , ¯ V ELL ≡ f even (cid:104) ϕ L ϕ L | V N | ϕ L ϕ L (cid:105) , ¯ V ELR ≡ f even (cid:104) ϕ L ϕ R | V N P even | ϕ L ϕ R (cid:105) , ¯ V OLR ≡ f odd (cid:104) ϕ L ϕ R | V N P odd | ϕ L ϕ R (cid:105) . (74)The first term ¯ T is the kinetic energy cost to localizetwo clusters with the distance R . ∆ (cid:15) c in the second termstands for the internal energy loss of a cluster by thecluster distortion from { φ (0)L , φ (0)R } to { ϕ L , ϕ R } because ofthe Pauli effect. The fourth term is the potential energycontribution of the odd part, which vanishes for the SU4-even N N force. The third term for the even part ofthe potential energy contribution is proportional to thefactor c − c counting the number of different-nucleonpairs. In the [ α + α ] S =0 system, this factor is c − c =12 and this third term gives a significant contributionto produce the medium-range attraction of the effectiveinter-cluster interaction, whereas the [ d + d ] S =2 system contains only two different-nucleon pairs, which is notenough to compensate the repulsion from the first andsecond terms.The energy can be expressed also by molecular or-bitals { ϕ + , ϕ − } . In the present cluster model, ϕ + and ϕ − become the harmonic-oscillator 0 s and 0 p orbits inthe R → { ϕ + , ϕ − } → { ϕ s , ϕ p } . In this limit, theintrinsic energy of the two-cluster systems are written as E int c + c − (cid:15) c = ¯ T + 2∆ (cid:15) c + ( c − c ) ¯ V E sp + c ¯ V O sp , (75)∆ (cid:15) c = c (cid:126) ω + c ( ¯ V E pp − ¯ V E0 ) , (76)¯ V E pp ≡ f even (cid:104) ϕ p ϕ p | V N | ϕ p ϕ p (cid:105) , ¯ V E sp ≡ f even (cid:104) ϕ s ϕ p | V N P even | ϕ s ϕ p (cid:105) , ¯ V O sp ≡ f odd (cid:104) ϕ s ϕ p | V N P odd | ϕ s ϕ p (cid:105) . (77)Here ϕ s = φ (0)0 and (cid:104) ϕ p | t | ϕ p (cid:105) = 5 (cid:126) ω/ (cid:15) c contains the significant re-pulsive effect from the kinetic energy cost for raising halfof A = 2 c nucleons from the 0 s orbit to the 0 p orbit toavoid Pauli blocking. 4. Bound states of [ d + d ] S =2 with unrealistic NN forces As shown in Eq. (72), two d -clusters in the S = 2channel can be bound if the third term 2 ¯ V ELR and/orthe fourth term 4 ¯ V OLR could give attractive contributions3 -15-10-5 0 5 10 15 20 0 1 2 3 4 5 6 7 8 (a) SU4 even force E n e r gy ( M e V ) R (fm) [d+d] S=2 [t+t] S=1 [ α + α ] S=0 -15-10-5 0 5 10 15 20 0 1 2 3 4 5 6 7 8 (b) [d+d] S=2 E n e r gy ( M e V ) R (fm) tunedstrong evenstate independent-15-10-5 0 5 10 15 0 1 2 3 4 5 6 7 8 (c) [d+d] S=2 E n e r gy ( M e V ) R,z (fm) tunedstrong even| φ L(0) | , | φ R(0) | | ϕ L | , | ϕ R | FIG. 6: Intrinsic energies E int c + c ( R ) of (a) [ d + d ] S =2 , [ t + t ] S =1 , and [ α + α ] S =0 with the SU4-even ( v SU4 N ) force, (b) [ d + d ] S =2 calculated with the tuned ( v tuned N ), state-independent( v st-ind N ), and strong-even (3 v SU4 N ) forces, and those of (c) [ d + d ] S =2 calculated with the tuned and strong-even forces. Theenergies measured from the one-dimensional decay thresholdenergy (2 (cid:15) c + (cid:126) ω ) are plotted. In the panel (c), the densitiesof the orthonormal atomic orbitals ϕ L ( z ) and ϕ R ( z ) in the[ d + d ] S =2 system with ν = 0 . 35 fm − at the distance R =2 . φ (0)L ( z ) and φ (0)R ( z ) are shownby cyan colored areas. strong enough to compensate the kinetic energy increase¯ T and the reduced binding energy of the clusters, 2∆ (cid:15) c .We consider two choices corresponding to artificial N N forces which produce a bound [ d + d ] S =2 state. One isthe strong-even force 3 v SU4 N , and the other is the state-independent force v st-ind N . Although these forces do notdescribe physical nuclear systems, it is worth consideringthese examples in order to better understand the under-lying physics involved.The energies of [ d + d ] S =2 obtained with the GCMand 1d-GCM calculations for the 3 v SU4 N and v st-ind N forcesare shown in Table II together with the deuteron energy (cid:15) d , and the R -plot of the intrinsic energies is shown inFig. 6(b).The strong-even force (3 v SU4 N ) gives the deeply bounddeuteron cluster with the cluster size smaller than therange of the N N force, as shown in Fig. 4. This is incontrast to the tuned N N force v tune N , which provides aloosely bound deuteron state with a larger size. More-over, for the deeply bound “deuteron” state, the poten-tial energy contribution becomes twice the kinetic energycontribution (see Table II). As seen in the energy curve ofthe [ d + d ] S =2 system in Fig. 6(b), a medium-range attrac-tion of the effective inter-cluster interaction is obtainedwith the strong-even force. In Fig. 6(c), we show single-particle densities of the orthonormal atomic orbitals inthe [ d + d ] S =2 system with the distance R = 2 . V ELR gives significant contribution to the binding ofthe two d -clusters. In other words, the deeply bound d -clusters effectively feel a longer-range N N force than theweakly bound d -clusters. This binding mechanism of the[ d + d ] S =2 system is similar to that of a two-dimer system( M m + M m ) with a long-range M m potential previouslydiscussed with the heavy-light ansatz.The second case is the state-independent force v st-ind N ,which contains even-parity and odd-parity componentswith the same strength. It should be commented thatthis N N force is an exactly local potential, whereas other N N forces are not but state-dependent forces having noodd-parity component or a weakly repulsive odd-paritycomponent. The odd component in v st-ind N force gives nocontribution to the internal energy of clusters but pro-vides an additional attraction to the inter-cluster poten-tial. In Fig. 6(b), the intrinsic energy of the [ d + d ] S =2 system obtained with the v st-ind N force is shown by adash-dotted line. The energy curve shows an attractivecluster-cluster interaction over a wide range, R (cid:46) R → s ) (0 p ) configu-ration.4 IV. SUMMARY We began with a discussion of effective dimer-dimerinteractions for general two-component fermion systemsusing the heavy-light ansatz. In our analysis we wereable to give a conceptual understanding of why increasingthe range or strength of the local part of the attractiveparticle-particle interaction results in a more attractivedimer-dimer interaction.We then considered the effective cluster-cluster inter-actions of the d + d , t + t , and α + α systems using amicroscopic cluster model with Brink-Bloch two-clusterwave functions. As the effective N N force, we use theVolkov central force with two sets of the parametriza-tion, the SU4-even and tuned N N forces. The latteris adjusted to fit the data of the S -wave N N scatter-ing lengths and the α - α scattering phase shifts. It wasshown that the effective inter-cluster interaction in the[ d + d ] S =2 system is repulsive for all R , whereas those inthe [ t + t ] S =1 and [ α + α ] S =0 systems are attractive atintermediate distances.In these systems, the kinetic energy term gives a re-pulsion to the effective inter-cluster interaction becauseof Pauli blocking of identical-nucleon pairs. Meanwhile,the potential energy term gives an attractive contributionwith a slightly longer range than the kinetic energy re-pulsion. As the mass number increases, the potential en-ergy contribution increases rapidly and produces enoughmedium-range attraction to form a bound 2 α state inthe absence of Coulomb effects. For the [ d + d ] S =0 and[ t + t ] S =0 systems, the effective inter-cluster interactionsare attractive since the two clusters feel a weaker Paulirepulsion or none at all. They then merge to form an α or He respectively, giving up their initial two-clusterstructures.Since the [ d + d ] S =2 system is a two-dimer system oftwo-component fermions in the isospin sector, the effec-tive inter-cluster interaction in this system can help toconnect with our analysis of the dimer-dimer interactionsfor general fermionic systems. We extended our analysisof the effective inter-cluster interaction of the [ d + d ] S =2 system by artificially changing the N N forces. It wasfound that two d -clusters could be bound if two nucle-ons are deeply bound to form a compact d -cluster with astrong even-parity N N force, or if the N N force containsboth even-parity and odd-parity attraction. Acknowledgments The computational calculations of this work were per-formed using the supercomputer at the Yukawa Institutefor Theoretical Physics at Kyoto University. The workwas supported by Grants-in-Aid of the Japan Societyfor the Promotion of Science (Grant Nos. JP18K03617and 18H05407), the U.S. Department of Energy (de-sc0018638), and the Nuclear Computational Low-EnergyInitiative (NUCLEI) SciDAC project. Appendix A: Two-dimer system with deltapotentials in 1D As explained in Sec. II, the solution for the two-dimersystem M m + M m with a delta M m potential in theheavy-light ansatz ( M (cid:29) m ) is obtained by solving thesingle-particle problem in the two-delta potential U ( x ) = v ( x + R v ( x − R , (A1) v ( x ) = − (cid:126) κ m δ ( x ) . (A2)In the frozen dimer ansatz, single-particle energies andwave functions are approximately expressed as (cid:15) ± = (cid:15) (0) + 2 (cid:15) (0) e − κ R ± e − κ R ± e − κ R (1 + κ R ) , (A3) ϕ ± ( x ) = 1 (cid:112) ± e − κ R (1 + κ R )] × (cid:104) ˜ φ ( κ ; x + R ± ˜ φ ( κ ; x − R (cid:105) , (A4)˜ φ ( κ ; x ) = √ κ e − κ | x | , (A5)where (cid:15) (0) = − (cid:126) m κ and ˜ φ ( κ ; x ) = φ (0) ( x ) are thesingle-particle energy and wave function for the bound-state solution in the single-delta potential U ( x ) = v ( x ). ϕ + ( x ) and ϕ − ( x ) are the molecular orbitals with positiveand negative parities.For the exact energies (cid:15) exact ± , we define valuables κ ± = (cid:112) − m(cid:15) exact ± (cid:126) . (A6) κ ± satisfy equations κ ± κ = 1 ± e − κ ± R , (A7)and the solutions are given as κ + = κ (cid:110) κ R W ( κ R e − κ R ) (cid:111) , (A8) κ − = κ (cid:110) κ R W − ( − κ R e − κ R ) (cid:111) . (A9)The exact single-particle energies and wave functions arewritten with κ ± as (cid:15) exact ± = − (cid:126) m κ ± , (A10) ψ ± ( x ) = 1 (cid:112) ± e − κ ± R (1 + κ ± R )] × (cid:104) φ ( κ ± ; x + R ± φ ( κ ± ; x − R (cid:105) . (A11)Note that the negative-parity state is not bound for κ R < 1, meaning that the two-delta potential is notenough to bind two fermions.5By comparing Eqs. (A4) and (A11), one can see thatthe approximate single-particle wave functions ϕ ± ( x ) areexpressed in a similar form to ψ ± ( x ), but κ for the un-perturbed energy (cid:15) (0) is used in ϕ ± ( x ) instead of κ ± forthe exact solutions.Also for the two-delta potential in 3D, exact single-particle energies ( (cid:15) exact ± = − (cid:126) m κ ± ) for the positive-andnegative-parity bound states can be expressed in similarforms with κ ± given in Eq. (45), which satisfy equations κ ± κ = 1 ± e − κ ± R κ R . (A12)The full details for the 3D case can obtained from theauthors upon request. Appendix B: Effective NN interaction The effective N N force v N ( i, j ) used in the presentcalculations of two-cluster systems is the Volkov centralforce [40], which is a finite-range two-body nuclear forcewith a Gaussian form as v N (1 , V N ( r )( W + BP σ − HP τ − M P σ P τ ) , (B1) V N ( r ) = (cid:88) k =1 , V k e − r η k , r ≡ √ r − r , (B2)where P σ and P τ are the exchange operators of nucleon-spins and isospins, respectively. For the strength andrange parameters, we use the Volkov No.2 parametriza-tion given as V = − . 65 MeV, V = 61 . 14 MeV, η = 1 . 80 fm, and η = 1 . 01 fm.The Volkov N N force can be rewritten as, v N (1 , 2) = V N ( r ) (cid:2) f E P ( E ) + f E P ( E )+ f O P ( O ) + f O P ( O ) (cid:3) , (B3)with f E = W + B + H + M,f E = W − B − H + M,f O = W − B + H − M,f O = W + B − H − M. (B4)It means that the strengths of the E , E , O , and O terms can be adjusted by W B , H , and M for the Wigner,Bartlett, Heisenberg, and Majorana terms, respectively,in the Volkov force.The parameter sets of W , B , H , and M for the SU4-even, tuned, strong-even, and state-independent forcesused in the present calculation and corresponding valuesof f E , f E , f O , and f O are summarized in Table I.The v tuned N force is adjusted to fit the experimental dataof the S -wave N N scattering lengths a t in the spin-tripletand a s in the spin-singlet, and also the α - α scatteringphase shifts. The theoretical values obtained with the v tuned N force are a t = 5 . a s = − . pn scattering are a t = 5 . 42 fm and a s = − . 75 fm [41]. Appendix C: Relative wave function betweenclusters in two-cluster wave functions The spacial part of the two-cluster wave functionΦ c + c ( R ) in Eq. (52) can be rewritten in a separable formof the cm, inter-cluster, and intrinsic coordinates asΦ c − R ( r , . . . , r c )Φ c R ( r (cid:48) , . . . , r c (cid:48) )= φ cm ( r cm ) ⊗ φ rel ( R , r rel ) ⊗ Φ c ( ξ ) ⊗ Φ c ( ξ (cid:48) ) ⊗ (cid:2) χ c ( s , . . . , s c ) χ c ( s (cid:48) , . . . , s c (cid:48) ) (cid:3) S , (C1) φ cm = (cid:18) cνπ (cid:19) e − cν r , (C2) φ rel ( R , r rel ) = (cid:18) γπ (cid:19) / e − γ ( r rel − R ) = (cid:88) L Γ L ( r rel , R ) (cid:88) m Y LM (ˆ r rel ) Y ∗ LM ( ˆ R ) , (C3)Γ L ( r rel , R ) ≡ π ( 2 γπ ) i L (2 γRr rel ) e − γ ( r + R ) , (C4) r cm ≡ c c (cid:88) i =1 ( r i + r i (cid:48) ) , r rel ≡ c c (cid:88) i =1 ( r i − r i (cid:48) ) , (C5)with γ ≡ c ν . Here r cm , r rel , ξ , and ξ (cid:48) indicate thecm coordinate, the inter-cluster coordinate, and intrinsiccoordinates of the first and second clusters, respectively.The GCM calculation is performed by superposing L π -projected wave functions as given in Eq. (57). 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