Binding of antikaons and Lambda(1405) clusters in light kaonic nuclei
BBinding of antikaons and
Λ(1405) clusters in light kaonic nuclei
Yoshiko Kanada-En’yo
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
The energy spectra of light-mass kaonic nuclei were investigated using the theoretical frameworkof the 0 s -orbital model with zero-range ¯ KN and ¯ K ¯ K interactions of effective single-channel realpotentials. The energies of the ¯ KNN , ¯
KNNN , ¯
KNNNN , ¯ K ¯ KN , and ¯ K ¯ KNN systems werecalculated in the cases of weak- and deep-binding of the ¯ KN interaction, which was adjusted to fitthe Λ(1405) mass with the energy of the ¯ KN bound state. The results qualitatively reproduced theenergy systematics of kaonic nuclei calculated via other theoretical approaches. In the energy spectraof the ¯ KNN and ¯ K ¯ KNN systems, the lowest states ¯
KNN ( J π , T = 0 − , /
2) and ¯ K ¯ KNN (0 + , KN (1 / − ,
0) state, respectively. Higher ( J π , T ) states including ¯ KNN (1 − , / K ¯ KNN (0 + , K ¯ KNN (1 + ,
1) were predicted at energies of 9–25 MeV below the antikaon-decay threshold.The effective Λ(1405)-Λ(1405) interaction in the ¯ K ¯ KNN system was also investigated via a ¯ KN +¯ KN -cluster model. Strong and weak Λ(1405)-Λ(1405) attractions were obtained in the S π = 0 + and S π = 1 − channels, respectively. The Λ(1405)-Λ(1405) interaction in the ¯ K ¯ KNN system wascompared with the effective d - d interaction in the NNNN system, and the properties of dimer-dimerinteractions in hadron and nuclear systems were discussed.
I. INTRODUCTION
Kaonic nuclei have recently become a hot topic inhadron physics. In particular, light-mass kaonic nu-clei have been intensively investigated to understand thestructures of exotic hadrons that are interpreted as mul-tihadron systems called hadron molecular states. Onecandidate for the simplest such system is the Λ(1405)state (denoted as Λ ∗ ), which is the lowest negative-parityΛ state with ( J π , T ) = (1 / − , π Σ spectra at anenergy slightly below the ¯ KN threshold, and is consid-ered to be a ¯ KN quasibound state produced by a strong¯ KN interaction. This picture of the ¯ KN molecular statefor the Λ(1405) resonance led to the concept of kaonicstates with an antikaon deeply bound via the ¯ KN in-teraction in light-mass nuclei such as K − pp and K − ppn [1–5], for which various few-body calculations have beendeveloped [6–19]. Several experiments have been per-formed in searching of the ¯ KN N state [20–23], but theevidence has not yet been confirmed [24, 25]. A similarchallenging is investigating double-kaonic nuclei with twoantikaons [26–28].For kaonic nuclei of mass number A = 2, intensivestudies of the ¯ KN N system have been performed bymany groups. To clarify the properties of the strangenessdibaryon, the ¯ K ¯ KN N system is also a key issue. It hasalso attracted interest in properties of the ¯ K ¯ K interac-tion concerning the kaon condensation in dense nuclearmatter. To experimentally search for the quasibound¯ K ¯ KN N state, a formation mechanism via a Λ ∗ + Λ ∗ doorway was proposed [29]. Furthermore, the effectiveΛ ∗ -Λ ∗ interaction in the ¯ K ¯ KN N system may draw gen-eral interest in dimer-dimer interactions in hadron sys-tems. For such a system consisting of isospin SU(2)bosons and fermions, the question of what role is playedby the nucleon Fermi statistics and antikaon Bose statis-tics in the effective dimer-dimer interaction between two Λ ∗ particles arises.The original idea for kaonic nuclei was based on thephenomenological ¯ KN interaction called the Akaishi-Yamazaki (AY) interaction [1, 5]. The AY interactionis characterized by an extremely strong ¯ KN attractionin the T = 0 channel, which reproduces an Λ(1405) massat an energy 27 MeV below the ¯ KN threshold reportedin Particle Data Group (PDG) table [30]. On the otherhand, a weaker ¯ KN interaction was proposed in detailedanalyses of π Σ scattering based on the chiral SU(3) ef-fective field theory, from which the Λ ∗ resonance-poleposition was obtained at only 8–12 MeV below the ¯ KN threshold [31]. In three-body calculations of the ¯ KN N ,the deep-type AY interaction predicted a deeply bound¯
KN N state, whereas the weak-type chiral interactionobtained a smaller binding energy (B.E.) for the ¯
KN N state. Moreover, for other kaonic nuclei, theoretical pre-dictions of the binding energies are spread over a rangebecause of the uncertainty of ¯ KN interactions and modelambiguities in theoretical treatments, such as the energydependence of the interaction as well as channel coupling.In this paper, my aim is to investigate the energy spec-tra of single- and double-kaonic nuclei, particularly, the¯ K ¯ KN N system. I do not intend to predict precise val-ues of the energy spectra, which may depend upon thedetails of hadron-hadron interactions as well as modeltreatments. Instead, I investigate the energy systematicsof kaonic nuclei to understand their global features and toextract universal properties independently from the un-certainty in hadron-hadron interactions. In this paper,I apply a simple model of the 0 s -orbital configurationto kaonic nuclei and calculate their energy spectra byassuming zero-range ¯ KN and ¯ K ¯ K interactions of effec-tive single-channel real potentials. The ¯ KN interactionis tuned to fit the Λ ∗ mass with the ¯ KN bound state intwo cases of weak- and deep-binding. As for the N N in-teraction, I adopt a finite-range effective
N N interactionadjusted to reproduce S -wave N N -scattering lengths. I a r X i v : . [ nu c l - t h ] A ug discuss the important roles of isospin symmetry in theenergy spectra of kaonic nuclei. I also investigate the ef-fective Λ ∗ -Λ ∗ interaction with a ¯ KN + ¯ KN -cluster model.For comparison with the effective d - d interaction in the N N N N system, I discuss the properties of dimer-dimerinteractions and binding mechanisms in hadron and nu-clear systems.This paper is organized as follows. In Sec. II, the theo-retical frameworks for the 0 s -orbital and ¯ KN + ¯ KN clus-ter models are explained. In Sec. III, the results of the 0 s -orbital model for kaonic nuclei are presented. In Sec. IV,the ¯ K ¯ KN N system is investigated via the ¯ KN + ¯ KN -cluster model, and the effective Λ ∗ -Λ ∗ interactions arediscussed. Finally, a summary is given in Sec. V. II. FORMULATION OF SINGLE- ANDDOUBLE-KAONIC NUCLEIA. s -orbital model for single- and double-kaonicnuclei For single and double-kaonic nuclei with A K antikaonsand A nucleons, I assume the 0 s -orbital configurationwith one-range Gaussian wave functions and express thewave functions for the ( J π , T ) states with angular mo-mentum ( J ), parity ( π ), and isospin ( T ) asΨ ( J π ,T )¯ K AK N A (1 (cid:48) , . . . , A (cid:48) K , , . . . , A )= φ K ( r (cid:48) ) · · · φ K ( r A (cid:48) K ) φ N ( r ) · · · φ N ( r A ) ⊗ [ s · · · s A ] S ⊗ [ t (cid:48) · · · t A (cid:48) K t · · · t A ] T , (1) φ N ( r ) = (cid:18) ν N π (cid:19) / e − ν N r , (2) φ K ( r ) = (cid:18) ν K π (cid:19) / e − ν K r (3)with J = S and π = ( − A K . Here, nucleon spins s i arecoupled to the total nuclear spin S , and antikaon isospins t i (cid:48) and nucleon isospins t i are coupled to the total isospin T . The spatial configuration of identical particles in the0 s -orbit is symmetric. The nucleon Fermi statistics aretaken into account in the nucleon-spin and -isospin con-figuration. For double-kaonic nuclei, the isospins of twoantikaons in the 0 s -orbit are coupled to form an isovector( τ K = 1) as [ t (cid:48) t (cid:48) ] τ K =1 to satisfy the Bose statistics.The Gaussian width parameter ν K for φ K is chosen tobe ν K ≡ m K m N ν N (4)with a ratio m K /m N ≈ / m K to the nucleon mass m N , such that the center-of-mass(cm) motion can be exactly removed from the total wavefunction. Hence, the antikaon orbit φ K has a broader dis-tribution than the nucleon orbit φ N . The internal wave functions of the N N , ¯ K ¯ K , and ¯ KN pairs can be writtenas Φ ¯ K ¯ K ( r i (cid:48) j (cid:48) ) = (cid:16) ν K π (cid:17) / e − νK r i (cid:48) j (cid:48) , (5)Φ NN ( r ij ) = (cid:16) ν N π (cid:17) / e − νN r ij , (6)Φ ¯ KN ( r i (cid:48) j ) = (cid:18) λπ (cid:19) / e − λ r i (cid:48) j , (7) λ = 2 m ¯ K m N + m ¯ K ν N , (8)where r kl = r l − r k . The root-mean-square (rms) dis-tances (cid:112) (cid:104) r kl (cid:105) of the N N , ¯ K ¯ K , and ¯ KN pairs are givenas R NN = 1 / √ ν N , R ¯ K ¯ K = 1 / √ ν K , and R ¯ KN =1 / √ λ , respectively.
1. Wave functions of single-kaonic nuclei
The ¯ KN bound state with ( J π , T ) = ( − ,
0) corre-sponding to Λ ∗ is expressed asΦ ( − , KN (1 (cid:48) ,
1) = φ K ( r (cid:48) ) φ N ( r ) ⊗ s ⊗ [ t (cid:48) t ] T =0 , (9)where the nucleon spin s = {↑ , ↓} specifies the intrinsicspin of the Λ ∗ system.The ¯ KN N states with ( J π , T ) = (0 − , ), (0 − , ), and(1 − , ) are written asΨ (0 − ,T )¯ KNN (1 (cid:48) , ,
2) = φ K ( r (cid:48) ) φ N ( r ) φ N ( r ) ⊗ [ s s ] S =0 ⊗ (cid:104) t (cid:48) [ t t ] τ N =1 (cid:105) T , (10)Ψ (1 − , )¯ KNN (1 (cid:48) , ,
2) = φ K ( r (cid:48) ) φ N ( r ) φ N ( r ) ⊗ [ s s ] S =1 ⊗ (cid:104) t (cid:48) [ t t ] τ N =0 (cid:105) T = , (11)where τ N indicates the total nucleon isospin. The J = 0states contain an isovector N N pair, and the J = 1 statecontains a deuteron-like isoscalar N N pair because ofthe nucleon Fermi statistics for the total nucleon spin, S = J . The (0 − , ) state is the lowest ¯ KN N state, whichhas been intensively studied by three-body calculations,whereas the (1 − , ) state was predicted to be a higher¯ KN N state [15, 16].For kaonic nuclei with mass numbers A = 3 and A = 4,I consider the ( J π , T ) = ( − ,
0) and (0 − , ) states with H and He cores, respectively, asΨ ( − , KNNN (1 (cid:48) , , ,
3) = φ K ( r (cid:48) ) φ N ( r ) φ N ( r ) φ N ( r ) ⊗ [ s s s ] S = ⊗ (cid:104) t (cid:48) [ t t t ] τ N = (cid:105) T =0 , (12)Ψ (0 − , )¯ KNNNN (1 (cid:48) , , , , φ K ( r (cid:48) ) φ N ( r ) φ N ( r ) φ N ( r ) φ N ( r ) ⊗ [ s s s s ] S =0 ⊗ (cid:104) t (cid:48) [ t t t t ] τ N =0 (cid:105) T = . (13)
2. Wave functions of double-kaonic nuclei
The 0 s -orbital states of double-kaonic nuclei containan isovector ¯ K ¯ K pair because of the Bose statistics. The¯ K ¯ KN system with ( J π , T ) = (
12 + ,
0) is given byΨ (
12 + , )¯ K ¯ KN (1 (cid:48) , (cid:48) ,
1) = φ K ( r (cid:48) ) φ K ( r (cid:48) ) φ N ( r ) ⊗ s ⊗ (cid:104) [ t (cid:48) t (cid:48) ] τ K =1 t (cid:105) T = . (14)For the ¯ K ¯ KN N system with the 0 s -orbital configura-tion, the ( J π , T ) = (0 + , T ) states consist of isovector N N and ¯ K ¯ K pairs, which are coupled to the total isospins T = 0, 1, and 2, and the ( J π , T ) = (1 + , T ) state containsan isoscalar N N pair and an isovector ¯ K ¯ K pair asΨ (0 + ,T )¯ K ¯ KNN (1 (cid:48) , (cid:48) , ,
2) = φ K ( r (cid:48) ) φ K ( r (cid:48) ) φ N ( r ) φ N ( r ) ⊗ [ s s ] S =0 ⊗ (cid:104) [ t (cid:48) t (cid:48) ] τ K =1 [ t t ] τ N =1 (cid:105) T , (15)Ψ (1 + , K ¯ KNN (1 (cid:48) , (cid:48) , ,
2) = φ K ( r (cid:48) ) φ K ( r (cid:48) ) φ N ( r ) φ N ( r ) ⊗ [ s s ] S =1 ⊗ (cid:104) [ t (cid:48) t (cid:48) ] τ K =1 [ t t ] τ N =0 (cid:105) T =1 . (16) B. ¯ KN + ¯ KN cluster model for ¯ K ¯ KNN
To discuss the effective Λ ∗ -Λ ∗ interaction, I apply a¯ KN + ¯ KN -cluster model to the ¯ K ¯ KN N system. I con-sider two ( ¯ KN )-clusters with the 0 s -orbital configurationlocated at − R / R / R = | R | asΨ ( S π T )¯ KN + ¯ KN ( R ; 1 (cid:48) , (cid:48) , , n A S (cid:48) (cid:48) (cid:110) φ K − R ( r (cid:48) ) φ N − R ( r ) φ K R ( r (cid:48) ) φ N R ( r ) ⊗ [ s s ] S ⊗ (cid:104) [ t (cid:48) t (cid:48) ] τ K [ t t ] τ N (cid:105) T (cid:111) , (17) φ K X ( r ) = (cid:18) ν K π (cid:19) / e − ν K ( r − X ) , (18) φ N X ( r ) = (cid:18) ν N π (cid:19) / e − ν N ( r − X ) , (19)where n is the normalization factor. The operators A and S (cid:48) (cid:48) are the antisymmetrized and symmetrized op-erators for nucleons and kaons, respectively, A = 1 − P √ , S (cid:48) (cid:48) = 1 + P (cid:48) (cid:48) √ , (20)which are equivalent to the internal-parity-projection op-erators of the N N and ¯ K ¯ K pairs.Hereafter, I consider the T = 0 states with τ K = τ N ≡ τ leading to the asymptotic Λ ∗ +Λ ∗ state at R → ∞ , andtake the isospin τ = 0 and τ = 1 components, which Idenoted as Ψ ( S π KN + ¯ KN ( R , τ ; 1 (cid:48) , (cid:48) , , τ -mixing) of τ = 0 and τ = 1 is given byΨ ( S π KN + ¯ KN ( R ; 1 (cid:48) , (cid:48) , , (cid:88) τ =0 , c τ Ψ ( S π KN + ¯ KN ( R , τ ; 1 (cid:48) , (cid:48) , , , (21)where the coefficients c τ are determined by diagonaliza-tion of the norm and Hamiltonian matrices for τ = { , } at each distance R . Note that the parity π of the¯ KN + ¯ KN system is related to the total nucleon spin S as S π = 0 + and 1 − because of the nucleon Fermi andantikaon Bose statistics. These correspond to the selec-tion rule of S π = 0 + and 1 − for two Λ ∗ particles in Fermistatistics.For the ¯ KN + ¯ KN -cluster state in the S π = 0 + chan-nel, the τ = 1 component has a spatial-even ¯ K ¯ K pairand a singlet-even N N pair, while the τ = 0 compo-nent consists of a spatial-odd ¯ K ¯ K pair and a singlet-odd N N pair. The former, the τ = 1 component, becomesequivalent to the lowest 0 s -orbital ¯ K ¯ KN N state with( J π , T ) = (0 + ,
0) at R = 0. The latter is forbidden inthe 0 s -orbital model space and therefore goes to an ex-cited configuration with two 0 p -orbital particles in the R → KN + ¯ KN -cluster state in the S π = 1 − chan-nel leads to a spin-aligned Λ ∗ + Λ ∗ state having negativeparity. The τ = 1 component of the S π = 1 − statehas a spatial-even ¯ K ¯ K pair and a triplet-odd N N pair,whereas the τ = 0 component is composed of a spatial-odd ¯ K ¯ K pair and a triplet-even N N pair. Neither com-ponent is not allowed in the 0 s -orbital configuration. In-stead, the τ = 1 component becomes a one-antikaon ex-citation and the τ = 0 component goes to a one-nucleonexcitation in the R → KN +¯ KN -cluster wave functions before A and S (cid:48) (cid:48) can berewritten in the separable form of the cm, inter-cluster,and ¯ KN -cluster internal wave functions with Jacobi co-ordinates as φ K − R ( r (cid:48) ) φ N − R ( r ) φ K R ( r (cid:48) ) φ N R ( r )= φ cm ( r cm ) ⊗ φ rel ( R , r rel ) ⊗ Φ ¯ KN ( r (cid:48) ) ⊗ Φ ¯ KN ( r (cid:48) ) , (22) φ cm ( r cm ) = (cid:18) · γπ (cid:19) e − γ r , (23) φ rel ( R , r rel ) = (cid:18) γπ (cid:19) / e − γ ( r rel − R ) , (24) γ = m + m m ν N , (25) r cm = 12 (cid:110) m N r (cid:48) + m K r m N + m K + m N r (cid:48) + m K r m N + m K (cid:111) , (26) r rel = m N r (cid:48) + m K r m N + m K − m N r (cid:48) + m K r m N + m K . (27) C. Hamiltonian and two-body interactions
In the present calculation, I omit the charge-symmetrybreaking and the Coulomb force. The Hamiltonian forsingle-kaonic nuclei ¯ KN A with mass number A is givenby H ¯ KN A = t kin1 (cid:48) + A (cid:88) i t kin i − T kincm + (cid:88) T =0 , A (cid:88) i v T ¯ KN (1 (cid:48) , i ) + (cid:88) ( ST ) (cid:88) i N N force [32], which isoften used with cluster models for nuclear systems. Inthe present calcualtion, the N N spin-orbit and tensorinteractions are omitted. The Volkov central N N forceis given in two-range Gaussian form as v ( ST ) NN ( i, j ) = u ( ST ) NN ( r ij ) P ( ST ) ij , (31) u ( ST ) NN ( r ) = f ( ST ) NN (cid:88) k =1 , V k e − r η k , (32)where P ( ST ) ij is the projection operator to the ( ST ) stateof the N N pair. The range parameters η k and global-strength parameters V k are given in the Volkov parame-terization, whereas the strength ratios f ( ST ) NN of four com-ponents ( ST ) = (10) , (01) , (11), and (00) are adjustable parameters, which I tune to fit the S -wave N N -scatteringlengths and the α + α -scattering phase shifts. For the spa-tial parts of the expectation values of the N N interactionfor the 0 s -orbital N N pair, I use the notation V ( ST ) NN ≡ (cid:104) φ N φ N | u ( ST ) NN ( r ) | φ N φ N (cid:105) . (33)For the ¯ KN and ¯ K ¯ K interactions, I consider the S -wave interactions of the effective single-channel real po-tentials and assume zero-range (delta function) forces forsimplicity. The imaginary part of the ¯ KN interaction,which corresponds to the π Σ decays via the Σ- ¯ KN cou-pling, is omitted. The ¯ KN interaction in the T = 0 and T = 1 channels is written as v T ¯ KN ( i (cid:48) , j ) = u T ¯ KN ( r i (cid:48) j ) P Ti (cid:48) j (34)with the delta function u T ¯ KN ( r ) = U T ¯ KN δ ( r ). Here, theisospin-projection operators can be expressed as P T =0 kl = − τ k · τ l and P T =1 kl = τ k · τ l . For the ¯ K ¯ K interaction,the spatial-even term exists only in the T = 1 channeland is given by v T =1¯ K ¯ K ( i (cid:48) , j (cid:48) ) = u T =1¯ K ¯ K ( r i (cid:48) j (cid:48) ) P T =1 i (cid:48) j (cid:48) (35)with u T =1¯ K ¯ K ( r ) = U T =1¯ K ¯ K δ ( r ). For these zero-range interac-tions, the spatial parts of the expectation values for the¯ KN and ¯ K ¯ K pairs in the 0 s -orbit are obtained as V T ¯ KN ≡ (cid:104) φ ¯ K φ N | u T ¯ KN ( r ) | φ ¯ K φ N (cid:105) = U T ¯ KN (cid:16) λπ (cid:17) , (36) V T ¯ K ¯ K ≡ (cid:104) φ ¯ K φ ¯ K | u T ¯ K ¯ K ( r ) | φ ¯ K φ ¯ K (cid:105) = U T ¯ K ¯ K (cid:16) ν K π (cid:17) . (37)The strengths U T =0¯ KN , U T =1¯ KN , and U T =1¯ K ¯ K of the interactionsare tuned as follows. I first adjust the strength U T =0¯ KN ofthe ¯ KN interaction in the T = 0 channel to make the Λ ∗ energy fit with the energy T kin0 + V T =0¯ KN of the ¯ KN state.The strengths U T =1¯ KN and U T =1¯ K ¯ K are adjusted to repro-duce the strength ratios F T =1¯ KN ≡ u T =1¯ KN ( r ) /u T =0¯ KN ( r ) and F T =1¯ K ¯ K ≡ u T =1¯ K ¯ K ( r ) /u T =0¯ KN ( r ) of the ¯ KN and ¯ K ¯ K interac-tions used in other theoretical works with kaonic nuclei.The adopted values of these parameters are explainedlater. D. Parameter settings For the N N interaction, I use the values of V = − . 65 MeV, V = 61 . 14 MeV, η = 1 . 80 fm, and η = 1 . 01 fm of the Volkov No.2 parametrization [32].I tune the ratio parameters f ( ST ) NN to fit the experimen-tal data of the S -wave N N -scattering lengths in thespin-triplet and -singlet channels and the α + α -scatteringphase shifts and set values of f (10) NN = 1 . f (01) NN = 0 . f (11) NN = − . 2, and f (00) NN = − . 2. This parametrizationdescribes a stronger triplet-even N N interaction to forma bound deuteron state and a weaker singlet-even N N interaction describing an unbound nn state. The odd-channel N N interactions are weak repulsions.In the present calculation, I adopt two sets of param-eters of the 0 s -orbit width ( ν N ) and the strengths of the¯ KN and ¯ K ¯ K interactions. One is the set-I parametriza-tion for the weak-binding case, and the other is the set-II parametrization for the deep-binding case. In eachparametrization, I use fixed ν N and ν K values consis-tently for all kaonic and normal nuclei. In the set-I (weak-binding) case, I use ν N = 0 . 16 fm − , which was optimizedfor the deuteron energy in the 0 s -orbital model with thetuned N N interaction. In the set-II (deep-binding) case,I choose ν N = 0 . 25 fm − which reproduces the bindingenergy and nuclear size of the He system. To deter-mine the strengths U T =0¯ KN , U T =0¯ K ¯ K , and U T =1¯ K ¯ K of the ¯ KN and ¯ K ¯ K interactions, I adopt the Λ ∗ energy ( (cid:15) Λ ∗ ) andstrength ratios F T =1¯ KN and F T =1¯ K ¯ K that are given by a weak-type chiral interaction for the set-I (weak-binding) case,and those that are given by the deep-type phenomeno-logical AY interaction for the set-II (deep-binding) case.For the set-I (weak binding) case with ν N = 0 . 16 fm − ,I adjust U T =0¯ KN to fit (cid:15) Λ ∗ = − 10 MeV, which correspondsto the Λ ∗ resonance-pole position of the chiral SU(3)analysis [31]. For the strength U T =1¯ KN , I adopt the value F T =1¯ KN = 0 . 457 of the effective single-channel ¯ KN poten-tials derived from the chiral SU(3) coupled-channel anal-ysis. The original ¯ KN potential in Ref. [31] is energy-dependent and contains imaginary terms, but I omit theenergy dependence and use only the real part of the in-teraction at the Λ ∗ resonance-pole position (1,421 MeVof the Λ ∗ mass). For the strength U T =1¯ K ¯ K , I take the value F T =1¯ K ¯ K = − . 345 of a ¯ K ¯ K interaction from Ref. [28].For the set-II (deep-binding) case with ν N = 0 . − , U T =0¯ KN is adjusted to fit (cid:15) Λ ∗ = − 27 MeV fromthe PDG value of Λ ∗ [30]. To determine U T =1¯ KN and U T =1¯ K ¯ K , I employ the value F T =1¯ KN = 0 . 294 of the deep-type single-channel AY interaction [1, 5], and the value F T =1¯ K ¯ K = − . 175 from Ref. [28].The expectation values of the single-particle kinetic en-ergy and the spatial parts of the N N , ¯ KN , and ¯ K ¯ K -interaction terms of the two-particle pairs are listed inTable I. In both the set-I and set-II cases, the ¯ KN at-traction is stronger in the T = 0 channel than in the T = 1 channel with a factor of 2–3, and the T = 1¯ K ¯ K interaction is the weak repulsion. Note that the¯ KN and ¯ K ¯ K interactions adopted here are delta forcesrenormalized to reproduce energy expectation values inthe present 0 s -orbital model space with a given ν N value.Such renormalized delta forces cannot be applied to vari-ational calculations beyond the assumed model setting. III. RESULTS OF KAONIC NUCLEI WITH THE s -ORBITAL MODELA. Energy counting With the present 0 s -orbital model, I calculate the en-ergies E ( J π ,T )¯ KN A and E ( J π ,T )¯ K ¯ KN A of the ( J π .T ) states of thekaonic nuclei, ¯ KN A and ¯ K ¯ KN A . These are obtainedby counting the spin and isospin components of the N N , ¯ KN , and ¯ K ¯ K pairs and can be expressed sim-ply with the expectation value terms, T kin0 , V ( ST ) NN , V T ¯ KN ,and V T ¯ K ¯ K . Hence, I obtain the lowest ( J π , T ) statesof each system of the ¯ KN (1 / − , KN N (0 − , / K ¯ KN (1 / + , / K ¯ KN N (0 + , KN N N (1 / − , KN N N N (0 − , / KN (1 / − , 0) corresponding to the Λ ∗ state is E (1 / − , KN = T kin0 + V T =0¯ KN = (cid:15) Λ ∗ , (38)which is used as an input to determine the interactionstrengths in the present framework. For ¯ KN N (0 − , / K ¯ KN (1 / + , / K ¯ KN N (0 + , E (0 − , / KNN = 2 T kin0 + V (01) NN + 2 (cid:16) V T =0¯ KN (cid:17) + 2 (cid:16) V T =1¯ KN (cid:17) = (cid:15) Λ ∗ + (cid:15) nn + 12 V T =0¯ KN + 12 V T =1¯ KN , (39) E (1 / + , / K ¯ KN = 2 T kin0 + 2 (cid:16) V T =0¯ KN (cid:17) + 2 (cid:16) V T =1¯ KN (cid:17) + V T =1¯ K ¯ K = (cid:15) Λ ∗ + T kin0 + 12 V T =0¯ KN + 12 V T =1¯ KN + V T =1¯ K ¯ K , (40) E (0 + , K ¯ KNN = 3 T kin0 + V (01) NN + 4 (cid:16) V T =0¯ KN (cid:17) + 4 (cid:16) V T =1¯ KN (cid:17) + V T =1¯ K ¯ K = 2 (cid:15) Λ ∗ + (cid:15) nn + V T =0¯ KN + V T =1¯ KN + V T =1¯ K ¯ K , (41)where (cid:15) nn = T kin0 + V (01) NN is the energy of a two-neutron state with the 0 s -orbital configuration and hasa positive value. The energies of ¯ KN N N (1 / − , 0) and¯ KN N N N (0 − , / 2) are written as E (1 / − , KNNN = 3 T kin0 + 3 (cid:16) V (01) NN (cid:17) + 3 (cid:16) V (01) NN (cid:17) + 3 (cid:16) V T =0¯ KN (cid:17) + 3 (cid:16) V T =1¯ KN (cid:17) = (cid:15) Λ ∗ + (cid:15) H + 12 V T =0¯ KN + 32 V T =1¯ KN , (42) E (0 − , / KNNNN = 4 T kin0 + 6 (cid:16) V (01) NN (cid:17) + 6 (cid:16) V (01) NN (cid:17) + 4 (cid:16) V T =0¯ KN (cid:17) + 4 (cid:16) V T =1¯ KN (cid:17) = (cid:15) Λ ∗ + (cid:15) He + 3 V T =1¯ KN , (43)where (cid:15) H = 2 T kin0 + V (10) NN + V (01) NN and (cid:15) He = 3 T kin0 +3 V (10) NN + 3 V (01) NN are the energies of the H and He nucleiwith the 0 s -orbital configuration, respectively. TABLE I: Expectation values and factors for energies of kaonic and normal nuclei in the 0 s -orbital model. Upper: the single-particle kinetic energy and the spatial part of the expectation values of the interaction terms for the NN , ¯ KN , and ¯ K ¯ K pairsfor set-I and set-II parametrization in units of MeV. Lower: factors of each term as A tot − T kin0 V (10) NN V (01) NN V T =0¯ KN V T =1¯ KN V T =0¯ K ¯ K set-I ( ν N = 0 . 16 fm − ) 9.95 − . − . − . − . 11 4.59set-II ( ν N = 0 . 25 fm − ) 15.55 − . − . − . − . 51 4.97kaonic nuclei( J π , T ) (cid:104) T kin (cid:105) (cid:104) v (10) NN (cid:105) (cid:104) v (01) NN (cid:105) (cid:104) v T =0¯ KN (cid:105) (cid:104) v T =1¯ KN (cid:105) (cid:104) v T =1¯ K ¯ K (cid:105) ¯ KN (1 / − , 0) 1 0 0 1 0 0¯ KNN (0 − , / 2) 2 0 1 2( ) 2( ) 0¯ KNN (0 − , / 2) 2 0 1 0 2 0¯ KNN (1 − , / 2) 2 1 0 2( ) 2( ) 0¯ K ¯ KN (1 / + , / 2) 2 0 0 2( ) 2( ) 1¯ K ¯ KNN (0 + , 0) 3 0 1 4( ) 4( ) 1¯ K ¯ KNN (0 + , 1) 3 0 1 4( ) 4( ) 1¯ K ¯ KNN (0 + , 2) 3 0 1 0 4 1¯ K ¯ KNN (1 + , 1) 3 1 0 4( ) 4( ) 1¯ KNNN (1 / − , 0) 3 3( ) 3( ) 3( ) 3( ) 0¯ KNNNN (0 − , / 2) 4 6( ) 6( ) 4( ) 4( ) 0nuclei( J π , T ) (cid:104) T kin (cid:105) (cid:104) v (10) NN (cid:105) (cid:104) v (01) NN (cid:105) NN (0 + , 1) 1 0 1 NN (1 + , 0) 1 1 0 NNN (1 / + , / 2) 2 3( ) 3( ) NNNN (0 + , 0) 3 6( ) 6( ) The energy counting for the lowest and other ( J π , T )states is summarized in Table I. The factor for each in-teraction term is given by the product of the numberof pairs and the spin-isospin component per pair. Thestrong ¯ KN interaction in the T = 0 channel generallyinduces isoscalar ¯ KN correlation. On the other hand,for N N pairs, an isoscalar N N correlation is favoredbecause the triplet-even ( ST ) = (10) term is strongerthan the singlet-even ( ST ) = (01) term in the effective N N interaction. Furthermore, nuclear systems in the0 s -orbit favor spin and/or isospin saturation as in the He system because of the Pauli principle of nucleons.In kaonic nuclei, the isoscalar ¯ KN and N N correlationscompete against each other. In A = 2 kaonic nuclei,the ¯ KN N (0 − , / 2) and ¯ K ¯ KN N (0 + , 0) states containingan isovector ( ST ) = (01) N N pair are energetically fa-vored over the ¯ KN N (1 − , / 2) and ¯ K ¯ KN N (1 + , 1) stateswith a dueteron-like ( ST ) = (10) N N pair, indicatingthat isoscalar ¯ KN correlation is superior to isoscalar( ST ) = (10) N N correlation. In kaonic nuclei with A ≥ 3, the isospin saturation occurs in the nuclear part;consequently, the isoscalar ¯ KN correlation gradually de-creases with the increase of A , as can be seen in thereduction of the T = 0 component of the ¯ KN pairs. Thefraction of the T = 0 component is 1 in the ¯ KN (1 / − , in the ¯ KN N (0 − , / K ¯ KN (1 / + , / K ¯ KN N (0 + , 0) states, in the ¯ KN N N (1 / − , 0) state,and in the ¯ KN N N N (0 − , / 2) state. B. Energy spectra of kaonic nuclei The calculated energies obtained using set-I (weak-binding) and set-II (deep-binding) are listed in Tables IIand III, respectively. For kaonic nuclei, the total ener-gies ( E ), ¯ K -separation energies ( S ¯ K ), and Λ ∗ -separationenergies ( S Λ ∗ ) are shown. For normal nuclei, the total en-ergies, nucleon-separation energies ( S N ), and deuteron-separation energies ( S d ) are shown. Moreover, the con-tributions of the kinetic energy and N N , ¯ KN , and ¯ K ¯ K -interaction terms are listed in the table.In the lowest states, i.e., ¯ KN (1 / − , KN N (0 − , / KN N N (1 / − , KN N N N (0 − , / K ¯ KN N (0 + , ∗ are deeplybound due to the remarkable contribution of the ¯ KN interaction with S Λ ∗ (cid:38) 10 MeV in the set-I result and S Λ ∗ (cid:38) 20 MeV in the set-II result. An exception isthe ¯ K ¯ KN (1 / + , / K + Λ ∗ -threshold energy in the set-I result andis weakly bound with S Λ ∗ = 7 . J π , T ) states of the kaonic nuclei are TABLE II: The energies of the ( J π , T ) states of kaonic and normal nuclei calculated by the 0 s -orbital model with the set-I(weak-binding) parametrization. The total energy ( E = − B.E.) and the contributions of kinetic ( T kin ), NN ( v NN ), ¯ KN ( v ¯ KN ),and ¯ K ¯ K ( v NN ) interactions are listed. Separation energies S ¯ K and S Λ ∗ for kaonic nuclei and S N and S d for normal nuclei arealso shown. Energies are in units of MeV.set-I ( (cid:15) Λ ∗ = − 10 MeV, ν N = 0 . 16 fm − ) (cid:104) T kin (cid:105) (cid:104) v NN (cid:105) (cid:104) v ¯ KN (cid:105) (cid:104) v ¯ K ¯ K (cid:105) E S ¯ K S Λ ∗ ¯ KN (1 / − , 0) 10.0 0 . − . . − . . − ¯ KNN (0 − , / 2) 19.9 − . − . . − . . 8) 10 . KNN (0 − , / 2) 19.9 − . − . . − . . − ¯ KNN (1 − , / 2) 19.9 − . − . . − . . . K ¯ KN (1 / + , / 2) 19.9 0 . − . . − . − . − . K ¯ KNN (0 + , 0) 29.9 − . − . . − . . . K ¯ KNN (0 + , 1) 29.9 − . − . . − . . − ¯ K ¯ KNN (0 + , 2) 29.9 − . − . . − . . − ¯ K ¯ KNN (1 + , 1) 29.9 − . − . . − . . − ¯ KNNN (1 / − , 0) 29.9 − . − . . − . . . KNNNN (0 − , / 2) 39.8 − . − . . − . . . (cid:104) T kin (cid:105) (cid:104) v NN (cid:105) E S N S d NN (0 + , 1) 10.0 − . . − − NN (1 + , 0) 10.0 − . − . . − NNN (1 / + , / 2) 19.9 − . − . . − NNNN (0 + , 0) 29.9 − . − . . . (cid:15) Λ ∗ = − 27 MeV, ν N = 0 . 25 fm − ) T kin v NN v ¯ KN v ¯ K ¯ K E S ¯ K S Λ ∗ ¯ KN (1 / − , 0) 15.6 0 . − . . − . . − ¯ KNN (0 − , / 2) 31.1 − . − . . − . . 8) 20 . KNN (0 − , / 2) 31.1 − . − . . − . . − ¯ KNN (1 − , / 2) 31.1 − . − . . − . . − . K ¯ KN (1 / + , / 2) 31.1 0 . − . . − . . . K ¯ KNN (0 + , 0) 46.7 − . − . . − . . . K ¯ KNN (0 + , 1) 46.7 − . − . . − . . − ¯ K ¯ KNN (0 + , 2) 46.7 − . − . . − . . − ¯ K ¯ KNN (1 + , 1) 46.7 − . − . . − . . − ¯ KNNN (1 / − , 0) 46.7 − . − . . − . . . KNNNN (0 − , / 2) 62.2 − . − . . − . . . T kin v NN E S N S d NN (0 + , 1) 15.6 − . . − − NN (1 + , 0) 15.6 − . − . . − NNN (1 / + , / 2) 31.1 − . − . . − NNNN (0 + , 0) 46.7 − . − . . . relatively unfavored because they have weaker isoscalar¯ KN correlations than the lowest states.I compare the present results with other theoretical results in Table IV. The binding energies of the low-est states are compared with the theoretical values fromRefs. [17–19] of the few-body calculations using weak-type and deep-type ¯ KN interactions. The energy spectraof the set-I result agrees reasonably well with the resultsof other calculations with weak-type chiral interactions,and the set-II result corresponds well with other theoret-ical results with deep-type AY interactions. In Table IV,I also compare the present results for rms distances R NN , R ¯ KN , and R ¯ K ¯ K for N N , ¯ KN , and ¯ K ¯ K pairs in kaonicnuclei with other theoretical results. The rms distancesare constant for a fixed ν N value in the present 0 s -orbitalmodel, whereas they are dependent on the system inother calculations with few-body approaches that includedynamical effects. Nevertheless, the present calculationsusing set-I and set-II yield reasonable results for R NN , R ¯ KN , and R ¯ K ¯ K in kaonic nuclei that are comparableto other calculations of weak-type and deep-type inter-actions, respectively. Hence, the present choices of ν N adopted for sets-I and II are reasonable for global de-scriptions of the system sizes of kaonic nuclei.In Fig. 1, the energy spectra of kaonic nuclei are showntogether with other theoretical results. Figure 1(a) showsthe set-I (weak-binding) result in comparison with othertheoretical results for weak-type chiral interactions fromRefs. [12, 15–19]. Figure 1(b) shows the set-II (deep-binding) result compared with other calculations for thedeep-type AY interaction. In each group of weak- anddeep-type calculations, the present calculation describesthe energy systematics of other theoretical results. Thismeans that the binding energies of kaonic nuclei are notvery sensitive to the details of the ¯ KN interaction butessentially depend upon the energy of the ¯ KN boundstate corresponding to Λ ∗ . Moreover, the leading part ofthe binding energies may be understood by simple energycounting in the present 0 s -orbital model, in which thespin and isospin symmetries play essential roles in thebinding mechanism of light-mass kaonic nuclei. C. ¯ KNN system In the ¯ KN N system, the ( J π , T ) = (0 − , / 2) state isthe lowest and has been investigated by many groups asa deeply bound K − pp system. For this state, I obtaina binding energy that approximately two times largerthan the Λ ∗ -binding energy (see Table II and Fig. 1(a)for the set-I result, and Table III and Fig. 1(b) for theset-II result). This energy relation E (0 − , / KNN ≈ (cid:15) Λ ∗ isnaively understood by the energies for two ¯ KN pairs inthe ¯ KN N (0 − , / 2) state. Quantitatively, it can be de-scribed by the present model with energy counting as E (0 − , / KNN = 2 (cid:15) Λ ∗ + 12 ( V T =1¯ KN − V T =0¯ KN ) + V (01) NN , (44)meaning that the N N attraction compensates for theenergy loss by reducing the isoscalar ¯ KN correlation in¯ KN N (0 − , / KN N (0 − , / 2) and ¯ K ¯ KN (1 / + , / KN interaction. As shown in Eqs. (39) and (40), theenergy difference is just the last term, V (01) NN of the N N attraction in ¯ KN N (0 − , / 2) and the V T =1¯ K ¯ K term of the¯ K ¯ K repulsion in ¯ K ¯ KN (1 / + , / ∗ -separation energies oftwo systems indicating the important role of the singlet-even N N attraction in the binding mechanism of the¯ KN N (0 − , / 2) state.The ¯ KN N (1 − , / 2) state with a deuteron-like ( ST ) =(10) N N pair has a higher energy than the lowest¯ KN N (0 − , / 2) state having a ( ST ) = (01) N N pair,despite the triplet-even N N interaction being strongerthan the singlet-even N N interaction. This is becausethere is no isoscalar ¯ KN correlation in ¯ KN N (1 − , / / T = 0 component. In theset-I result, the ¯ KN N (1 − , / 2) state is weakly boundwith S Λ ∗ = 5 . S Λ ∗ = 9 MeV for a coupled-channel Fad-deev calculation [16]. In the set-II result, I obtained E (1 − , / KNN = − . ∗ -decay threshold at − . D. ¯ K ¯ KNN system The ( J π , T ) = (0 + , 0) state is the lowest of the¯ K ¯ KN N system, and is deeply bound because of thestrong isoscalar ¯ KN correlation with a fraction of 3/4 forthe T = 0 component. In both the set-I and II results,the energy of ¯ K ¯ KN N (0 + , 0) is approximately twice thatof ¯ KN N (0 − , / K -separation en-ergy is almost constant between the two systems. Thisenergy relation, E (0 + , K ¯ KNN ≈ E (0 − , / KNN , (45)is also roughly satisfied in other theoretical results forRefs. [17, 18]. In the present 0 s -orbital model, it is easyto see the energy relation from Eqs. (39) and (41) as E (0 + , K ¯ KNN = 2 E (0 − , / KNN − (cid:15) nn + V T =1¯ K ¯ K , (46)where the last two terms yield minor contributions as − (cid:15) nn = − . V T =1¯ K ¯ K = 4 . − (cid:15) nn = − . V T =1¯ K ¯ K = 5 . N N pair in the ¯ KN N and ¯ K ¯ KN N systems in a Born-Oppenheimer picture of light-mass antikaons aroundheavy-mass nucleons. Two nucleons in the singlet-evenchannel are unbound without antikaons, but they aredeeply bound by a surrounding antikaon in the ¯ KN N system and further deeply bound by two antikaons in the¯ K ¯ KN N system. The mechanism for binding the two nu-cleons by an antikaon in the ¯ K − pp system was originallyinterpreted as a super-strong nuclear force caused by amigrating ¯ K meson by Yamazaki and Akaishi [5, 34]. In TABLE IV: Binding energies (B.E.) and rms distances of NN ( R NN ), ¯ KN ( R ¯ KN ), and ¯ K ¯ K ( R ¯ K ¯ K ) pairs in the lowest statesof kaonic and normal nuclei. Calculated values obtained with the set-I (weak-binding) and set-II (deep-binding) cases arecompared with other theoretical results obtained with weak-type chiral and deep-type AY interactions by Maeda et al. [18],Ohnishi et al. [19], and Barnea et al. [17]. For values of weak-type chiral interactions, the weak-chiral-regime result of Ref. [18],the Kyoto type-I result of Ref. [19], and the BGL result of Ref. [17] are listed. For normal nuclei, the AV4’ result obtained byOhnishi et al. from Ref. [19] is also shown. present Maeda [18] Ohnishi [19] Barnea [17]set-I set-II Chiral AY Chiral AY Chiral ν N (fm − ) 0.16 0.25kaonic nuclei( J π , T )¯ KN (1 / − , 0) B.E. (MeV) 10 27 8.3 26.6 11.4 R ¯ KN (fm) 2.17 1.73 2.25 1.41 1.87¯ KNN (0 − , / 2) B.E. (MeV) 20.8 47.8 23.8 51.5 27.9 48.7 15.7 R NN (fm) 1.77 1.41 1.93 1.62 2.16 1.84 R ¯ KN (fm) 2.17 1.73 1.80 1.55¯ K ¯ KNN ; (0 + , 0) B.E. (MeV) 40.7 97.3 43 93 32.1 R NN (fm) 1.77 1.41 1.57 1.35 1.84 R ¯ K ¯ K (fm) 2.50 2.00 2.31¯ KNNN (1 / − , 0) B.E. (MeV) 40.4 73.6 42 69 45.3 72.6 R NN (fm) 1.77 1.41 1.89 1.75 1.99 1.87 R ¯ KN (fm) 2.17 1.73 1.79 1.63¯ KNNNN (0 − , / 2) B.E. (MeV) 60.8 93.2 67.9 85.2 R NN (fm) 1.77 1.41 1.98 2.07 R ¯ KN (fm) 2.17 1.73 1.83 1.81nuclei( J π , T ) NN (1 + , 0) B.E. (MeV) 1.6 0.8 2.24 2.24 R NN (fm) 1.77 1.41 4.04 4.04 NNN (1 / + , / 2) B.E. (MeV) 6.7 6.6 8.99 8.99 NNNN (0 + , 0) B.E. (MeV) 23.5 28.7 32.1 32.1 the perturbative picture, the constant S ¯ K in the ¯ KN N and ¯ K ¯ KN N systems can be described by the conden-sation of two antikaons in the same orbit around twonucleons. if the ¯ K ¯ K interaction is minor. It should benoted that, when three antikaons around two nucleonsare considered, the additional(third) antikaon no longerexhibits isoscalar ¯ KN correlation because the isospin isalready saturated in the ¯ K ¯ KN N system.The higher states ¯ K ¯ KN N (0 + , K ¯ KN N (0 + , K ¯ KN N (1 + , KN correla-tions because these state have lower symmetry in theisospin coupling between antikaons and nucleons than¯ K ¯ KN N (0 + , 0) does. In particular, the ¯ K ¯ KN N (0 + , K ¯ KN N (0 + , 2) states are composed of isovector N N and ¯ KN pairs coupled to T = 1 and T = 2, respectively,and the ¯ K ¯ KN N (1 + , 1) state contains an isoscalar N N pair.Comparing ¯ K ¯ KN N (0 + , 1) and ¯ K ¯ KN N (1 + , KN and N N correlations. ¯ K ¯ KN N (0 + , 1) has a moderateisoscalar ¯ KN component with a fraction of 1 / N N component, whereas the ¯ K ¯ KN N (1 + , N N component but noisoscalar ¯ KN correlation with a fraction 1 / T = 0component. For these two states, S ¯ K is constant at9 . . K ¯ KN N (0 + , K ¯ KN N (1 + , 1) is the same value as the energy dif-ference between ¯ KN N (0 − , / 2) and ¯ KN N (1 − , / ≈ 20 MeVin the set-II result.In future experimental searches for double-kaonic nu-clei, the ¯ K ¯ KN N states might be observed as the quasi-bound resonances in the invariant mass spectra of suchmodes as the ΛΛ, ΛΣ ± π ∓ , and Ξ − p decays. The ΛΛmode for the T = 0 spectrum shows the ¯ K ¯ KN N (0 + , -100-80-60-40-20 0 20 E n e r gy [ M e V ] KN − J π =1/2 − T=0 KNN − J π =0 − T=1/2 KNN − J π =1 − T=1/2 KKN − − J π =1/2 + T=1/2 KKNN − − J π =0 + T=0 KNNN − J π =1/2 − T=0 KNNNN − J π =0 − T=1/2 set-I Dote-chr Barnea-chrMaeda-chr Ohnishi-chrBayer-chr Oset-chr -100-80-60-40-20 0 20 E n e r gy [ M e V ] KN − J π =1/2 − T=0 KNN − J π =0 − T=1/2 KNN − J π =1 − T=1/2 KKN − − J π =1/2 + T=1/2 KKNN − − J π =0 + T=0 KNNN − J π =1/2 − T=0 KNNNN − J π =0 − T=1/2 set-II Maeda-AYOhnishi-AY (b) set−II(a) set−Ideep−typeweak−type FIG. 1: Energies E = − B.E. of the kaonic nuclei. (a) Upper: the set-I (weak-binding) results are shown together with othertheoretical results obtained by few-body approaches using weak-type chiral interactions by Dot´e et al. (ORB type-I case) [12],Barnea et al. (BGL case) [17], Maeda et al. (weak-chiral-regime case) [18], Ohnishi et al. (Kyoto type-I case) [19], Bayar etal. (normal-radius case) [15], and Oset et al. (normal-radius case) [16]. (b) Lower: the set -II (deep-binding) results are showntogether with other theoretical results for the deep-type AY interaction by Maeda et al. [18] and Ohnishi et al. [19]. contribution, whereas the ΛΣ ± π ∓ and Ξ − p modes probeboth the T = 0 and T = 1 components and may containthe ¯ K ¯ KN N (0 + , 1) and ¯ K ¯ KN N (1+ , 1) contributions athigher energies than the ¯ K ¯ KN N (0 + , 0) contribution. E. Antiknock binding in single-kaonic nuclei In Fig. 2(a), I plot the ¯ K -separation energies ( S ¯ K ) forthe lowest states of single-kaonic nuclei calculated withset-I and II. For comparison, I also show theoretical re-sults from Refs. [18, 19]. In each group of weak- anddeep-type calculations, the A dependence of S ¯ K exhibitsa similar trend, in that S ¯ K increases gradually up to A = 3 and becomes saturated at A = 4 because theisoscalar ¯ KN correlation vanishes in the nuclear-isospin-saturated system. In Fig. 2(b), I compare the separationenergies, S ¯ K for single-kaonic nuclei and S N for normalnuclei of set-I, which are plotted as functions of A tot − A = 2 and 3 systems, a nucleon in a normal nu-cleus is rather weakly bound because of the relatively weak N N interaction compared with an antikaon that isdeeply bound by the strong ¯ KN attraction in a kaonicnucleus. However, at A = 4 for He, S N increases dras-tically. This is in contrast with the gradual change of S ¯ K with the increase of A in kaonic nuclei. IV. RESULTS OF ¯ KN + ¯ KN -CLUSTER MODELA. Effective Λ ∗ - Λ ∗ interaction To investigate the effective Λ ∗ -Λ ∗ interaction, I applythe ¯ KN + ¯ KN -cluster model to the ¯ K ¯ KN N system withtotal isospin T = 0. As described in Sec. II B, I assume0 s -orbital configuration for each ¯ KN cluster and considerthe two-cluster wave function with a distance R . In thecluster limit at a large distance R , each ¯ KN cluster formsan isoscalar ¯ KN bound state that corresponds to the Λ ∗ state. In this asymptotic Λ ∗ +Λ ∗ state, two S π = 0 + and1 − channels are allowed because of Fermi statistics of Λ ∗ particles, and their energies are degenerate at R → ∞ .1 S e p a r a ti on e n e r gy [ M e V ] A set-I Maeda-chr Ohnishi-chrset-II Maeda-AYOhnishi-AY 0 20 40 60 0 1 2 3 4 S e p a r a ti on e n e r gy [ M e V ] A tot -1 S K : set-I S K :Ohnishi-chr S N : presentS N :OhnishiS N :expt. (b)(a) FIG. 2: ¯ K -separation energies ( S ¯ K ) of kaonic nuclei andnucleon-separation energies ( S N ) of normal nuclei. (a) Theset-I (weak-binding) and set-II (deep-binding) results of S ¯ K for the lowest states of single-kaonic nuclei. (b) The set-I (weak-binding) results of S ¯ K for single-kaonic nuclei andof S N for normal nuclei. For comparison, other theoreticalresults for S ¯ K that were calculated with weak-type chiral in-teractions by Maeda et al. (weak-chiral-regime case) [18] andOhnishi et al. (Kyoto type-I case) [19] are also presented,together with those with deep-type AY interaction calculatedby Maeda et al. [18] and Ohnishi et al. [19]. For the S N ofnormal nuclei, the experimental values, and the AV4’ resultby Ohnishi et al. [19] are also shown. The mixing of τ = 0 and τ = 1 components withrespect to the isospin τ K = τ N = τ of N N and ¯ K ¯ K pairs, which are coupled to total isospin T = 0, is takeninto account in each S π channel.The ¯ KN + ¯ KN -cluster wave function with τ mixingcan smoothly connect two limits; the shell model state at R → ∗ + Λ ∗ state at R → ∞ . Asthe two Λ ∗ -clusters approach each other, the isospin rear-rangement occurs through the isospin exchange betweentwo clusters via the ¯ KN and N N interactions.Because of the Bose and Fermi statistics, there are se-lection rules in the spatial symmetry of the ¯ K ¯ K and N N pairs as follows. In the ( S π T ) = (0 + 0) channel,the ¯ KN + ¯ KN -cluster system is described by a linearcombination of two isospin components: the τ = 1 com-ponent with spatial-even N N and ¯ K ¯ K pairs, and the τ = 0 component with spatial-odd N N and ¯ K ¯ K pairs.In the shell-model limit at R → 0, the former compo-nent goes to the 0 s -orbital ¯ K ¯ KN N (0 + , 0) state, whichis the lowest state in the 0 s -orbital model. The latter τ = 0 component is forbidden in the 0 s -orbital modelspace and instead goes to a (0 s ) (0 p ) configuration withan antikaon and a nucleon excited into 0 p -orbits. On theother hand, in the ( S π T ) = (1 − 0) channel with negativeparity, either one of N N and ¯ K ¯ K pairs is a spatial-oddstate. The τ = 0 component contains a spatial-odd ¯ K ¯ K pair, whereas the τ = 1 component has a spatial-odd N N pair. In the shell-model limit at R → 0, they be-come excited (0 s ) (0 p ) states. In the τ = 0 component,there’s an antikaon excitation and in the τ = 1 compo-nent, there(s a nucleon excitation.Such selection rules for ¯ K ¯ K and N N pairs play impor-tant roles in the effective Λ ∗ -Λ ∗ interaction, particularlyat short distances. In the asymptotic Λ ∗ + Λ ∗ state at alarge R , each channel of S π = 0 + and 1 − contains τ = 1and τ = 0 components with a ratio of 3:1. On the otherhand, in the shell-model limit at R → 0, the τ -mixingin the S π = 0 + channel is equivalent to the mixing ofthe (0 s ) and (0 s ) (0 p ) configurations, and that in the S π = 1 − channel corresponds to the configuration mixingof the antikaon and nucleon excitations in the (0 s ) (0 p )configuration.I calculate the energy of the ¯ KN + ¯ KN -cluster statewith and without τ -mixing at each distance R . In Fig. 3,I show the R dependence of the total energy of the ¯ KN +¯ KN state in the S π = 0 + and 1 − channels. Note that theasymptotic Λ ∗ +Λ ∗ state at a large distance R contains anadditional energy cost T kin0 for localization of the relativemotion, and the energy calculated with τ -mixing equals2 (cid:15) Λ ∗ + T kin0 at sufficiently large R . In both the S π = 0 + and 1 − channels, I obtain the energy minimum at R → τ -mixing, which indicates anattractive Λ ∗ -Λ ∗ interaction. The attraction of the Λ ∗ -Λ ∗ interaction in the S π = 0 + channel is strong enoughto form a deeply bound ¯ K ¯ KN N (0 + , 0) state in the shell-model limit, whereas that in the S π = 1 − channel isweaker.In Table V, I list values at R → P ( τ = 1) for the τ = 1 compo-nent. I also show the ¯ K - and Λ ∗ -separation energies ofthe ¯ K ¯ KN N states, which are evaluated by the energy inthe shell-model limit, as measured from the correspond-ing decay-threshold energies. For the S π = 0 + state inthe shell-model limit, the τ = 1 configuration is dom-inant and the τ = 0 mixing effect is negligibly small,meaning that the 0 s -orbital model used in the presentwork well approximates the deeply bound Λ ∗ + Λ ∗ statein the S π = 0 + channel.For the S π = 1 − channel, I obtain a value of S Λ ∗ =2 . τ -mixing (see TableV), indicating that the Λ ∗ -Λ ∗ attraction forms a (quasi)bound S π = 1 − state at a slightly lower energy thanthe two-cluster-threshold energy, 2 (cid:15) Λ ∗ . In the set-I re-sult with τ -mixing, a negative value S Λ ∗ = − . ∗ -separation energy. This suggeststhat the Λ ∗ -Λ ∗ attraction in the S π = 1 − channel isinsufficient to form a bound state, but may produce a L π = 1 − resonance near the 2Λ ∗ -threshold energy. Theweaker Λ ∗ +Λ ∗ attraction in the S π = 1 − channel than inthe S π = 0 + channel is described by a much weaker ¯ KN attraction and a somewhat weaker N N attraction as wellas a larger kinetic-energy loss for the 0 p -orbit excitation.Let me discuss the S π = 1 − state in ore detail. Com-paring the energies of the τ = 1 and τ = 0 configurationswithout τ -mixing, the τ = 1 component is favored at all R because of the stronger isoscalar ¯ KN correlation thanthe τ = 0 component. In the result with τ -mixing, the τ = 1 configuration dominates the S π = 1 − state in theshell-model limit with probability P ( τ = 1) = 0 . 84 forset-I and P ( τ = 1) = 0 . 92 for set-II. In Table VI, I showthe energy contributions of the ¯ KN + ¯ KN state at R → ∗ cluster. From the table, one can see that the τ = 1 component with a single-nucleon excitation gainsenergy in the ¯ KN attraction but loses energy due to ¯ K ¯ K repulsion, whereas the τ = 0 component with a single-antikaon excitation gains energy through N N attractionbut somewhat loses energy through the ¯ KN attraction.In principle, in the shell-model limit of ¯ KN + ¯ KN with S π = 1 − , the τ = 1 and τ = 0 configurations cancompete and the mixing ratio depends upon details ofthe interactions. In the present calculation, the nucleonexcitation in the τ = 1 component is favored over the an-tikaon excitation in the τ = 0 component; this can be ex-plained by the nucleon feeling a broader ¯ KN mean-field,allowing it to more easily excite into the 0 p -orbit thanan antikaon, because light-mass antikaons have broaderdensity distributions than nucleons in the present model.The τ -mixing ratio may change if the antikaon mass isheavier than the physical antikaon mass. For example, ifI assume equal kaon and nucleon masses m ¯ K = m N andkeep the other parameters unchanged, I obtain a lowerenergy for the τ = 0 component with an antikaon ex-citation than for the τ = 1 component with a nucleonexcitation in the small- R region, resulting in significant τ -mixing in the ¯ KN + ¯ KN state with S π = 1 − . B. Comparison of Λ ∗ - Λ ∗ and d - d interactions The Λ ∗ -Λ ∗ interaction discussed previously is regardedas an effective dimer-dimer interaction in the kaonic nu-clei. I here compare its properties with those of the d - d interaction in nuclear systems. A deuteron is a weaklybound ( ST ) = (10) N N state. I describe the N N + N N system in the S π = 0 + , 1 − , and 2 + channels with a d + d cluster model called the Brink-Bloch model [33] as donein Ref. [35]. The detailed properties of the effective d - d interaction have been investigated in the previous pa-per [35]. In the present paper, I show the energy of the N N + N N system at R → S π states of two systems based onthe number of spatial-odd ¯ K ¯ K and N N pairs. Becauseof the nucleon Fermi statistics, the S π = 0 + , S π = 1 − ,and S π = 2 + states of the N N + N N system containzero, one, and two spatial-odd N N pairs, respectively.Similarly, in the ¯ KN + ¯ KN system, the τ = 1 and τ = 0components of the S π = 0 + state contain zero and twospatial-odd pairs, respectively, while the S π = 1 − statehas one spatial-odd pair. In the shell-model limit, thesestates having no, one, and two spatial-odd pairs corre-spond to the (0 s ) , (0 s ) (0 p ), (0 s ) (0 p ) configurations,which have kinetic energy of 3 T kin0 , (3 + ) T kin0 , and(3 + ) T kin0 , respectively.In Fig. 4 and Table VI, I show the results of the¯ KN + ¯ KN and N N + N N systems in the shell-modellimit. The energy contributions measured from twice ofthe internal energies of a single cluster are shown. Fromthe energy spectra of Fig. 4, the two clusters in the lowest S π = 0 + channel for the (0 s ) configuration are deeplybound in both systems. The binding energy for the twoΛ ∗ clusters from the threshold is approximately 20 MeV,coinciding with that for two deuteron clusters. However,the detailed contributions of the four pairs between thetwo clusters differ. According to the energy counting inthe present model, the Λ ∗ +Λ ∗ and d + d binding energiesare given as∆ E (0 + , K ¯ KNN ≡ E (0 + , K ¯ KNN − (cid:15) Λ ∗ = T kin0 + V (01) NN + V T =0¯ KN + V T =1¯ KN + V T =1¯ K ¯ K , (47)∆ E (0 + , NNNN ≡ E (0 + , NNNN − (cid:15) d = T kin0 + 3 V (01) NN + V (10) NN . (48)In the N N N N system, the potential energy contributionis always attractive for all four N N pairs between thetwo clusters, while in the ¯ K ¯ KN N system, the strongattraction in the isocalar ¯ KN pair compensates for therepulsion in the isovector ¯ K ¯ K pair.In the S π = 1 − channel of the N N N N and ¯ K ¯ KN N systems at R → 0, two clusters gain some amount ofpotential energy but lose kinetic energy for one 0 p -orbitexcitation in the (0 s ) (0 p ) configuration. In the ¯ K ¯ KN N system, the Λ ∗ -Λ ∗ interaction in the S π = 1 − channelis the weak attraction and almost forms a bound stateat an energy close to the 2Λ ∗ threshold. To gain the¯ KN attraction efficiently from the Λ ∗ + Λ ∗ state to theshell-model-limit state, isospin rearrangement plays anessential role. This is a unique characteristic of kaonicnuclei, but cannot be seen in the N N + N N system be-cause such isospin rearrangement is not allowed in the( S π T ) = (1 − 0) state. Hence, there is no attraction ofthe d - d interaction in the S π = 1 − channel.The N N N N (2 + 0) state and the τ = 0 componentof the ¯ K ¯ KN N (0 + 0) state correspond to the (0 s ) (0 p ) N N N N (2 + 0) state gains potential energybecause of the attractive N N interaction, whereas the τ = 0 component of the ¯ K ¯ KN N (0 + 0) state containingisoscalar ¯ K ¯ K and N N pairs looses the potential energyof the ¯ KN and ¯ K ¯ K interactions. V. SUMMARY I investigated the energy systematics of single- anddouble-kaonic nuclei in the mass number A ≤ s -orbital model using zero-range ¯ KN and ¯ K ¯ K interactions. The ¯ KN interaction was tuned to fit theΛ(1405) mass with the energy of the ¯ KN bound state.For the N N interaction, I adopted the Volkov finite-range central interaction with a tuned parametrizationadjusted to reproduce the S -wave N N -scattering lengths.I calculated the energy spectra of the ¯ KN N , ¯ KN N N N ,¯ KN N N , ¯ K ¯ KN , and ¯ K ¯ KN N systems in the cases ofweak- and deep-binding and compared the results withother theoretical calculations with weak-type chiral anddeep-type AY interactions. The present results quali-tatively reproduce the energy systematics of kaonic nu-clei calculated via other theoretical approaches. In thepresent 0 s -orbital model, the energy spectra of kaonicnuclei were given by simple energy counting of isospincomponents of N N , ¯ KN , ¯ K ¯ K pairs. The approximateenergy relations for the lowest states of the ¯ KN , ¯ KN N ,and ¯ K ¯ KN N systems were obtained as E (0 − , / KNN ≈ (cid:15) Λ ∗ and E (0 + , K ¯ KNN ≈ E (0 − , / KNN , which are universal featuresthat are independent of the Λ(1405) mass.For the ¯ KN N and ¯ K ¯ KN N systems, I discussed theimportant roles of the isospin symmetry in the energyspectra of the ( J π , T ) states. In addition to the low-est ¯ KN N (0 − , / 2) and ¯ K ¯ KN N (0 + , 0) states contain-ing the isovector ( ST ) = (01) N N pair, I also obtainthe ¯ K ¯ KN N (0 + , KN N (1 − , / K ¯ KN N (1 + , ST ) = (10) N N pairs like deuterons. The predicted ¯ K -separation ener-gies for these states are S ¯ K =9–25 MeV. In future exper-imental searches for ¯ K ¯ KN N states, the ¯ K ¯ KN N (0 + , K ¯ KN N (1 + , 1) states may contribute to the T = 1components of invariant mass spectra.I also investigated the effective Λ ∗ -Λ ∗ interaction withthe ¯ KN + ¯ KN -cluster model and obtained a strong at-traction in the S π = 0 + channel and a weak attractionin the S π = 1 − channel. In comparing the Λ ∗ -Λ ∗ inter-action in the ¯ K ¯ KN N system with the d - d interaction inthe N N N N system, I discussed the properties of dimer-dimer interactions in hadron and nuclear systems.In the present calculation, zero-range real potentialswere used for the ¯ KN and ¯ K ¯ K interactions. Moreover, -60-40-20 0 20 40 60 0 1 2 3 4 5 6 7 8 ε Λ∗ S π =0 + (a) set-Iweak E ( KKNN ) [ M e V ] R [fm] τ -mix τ =1 τ =0-60-40-20 0 20 40 60 0 1 2 3 4 5 6 7 8 ε Λ∗ S π =1 − (b) set-Iweak E ( KKNN ) [ M e V ] R [fm] τ -mix τ =1 τ =0-60-40-20 0 20 40 60 0 1 2 3 4 5 6 7 8 ε Λ∗ S π =1 − (c) set-IIdeep E ( KKNN ) [ M e V ] R [fm] τ -mix τ =1 τ =0 FIG. 3: Energies of the ¯ KN + ¯ KN -cluster system with inter-cluster distances R for (a) the S π = 0 + and (b) S π = 1 − statesof the set-I result. (c) Those for the S π = 1 − state of the set-II result. The energies calculated with and without τ -mixingare shown. Arrows show the Λ ∗ + Λ ∗ threshold energy, whichis ¯ T below the asymptotic energy at R → ∞ obtained with τ -mixing. kaonic nuclei were simply described using the 0 s -orbitaland cluster models. Despite such simple theoretical treat-ments of interactions and wave functions, the present re-sults succeeded in globally describing the energy system-atics of kaonic nuclei obtained with precise few-body cal-culations. The energy-counting rule in the present modelis useful for understanding the leading properties of en-ergy spectra in kaonic nuclei; it also enables one to ex-4 TABLE V: Energies of the ¯ KN + ¯ KN ( S π T ) states in the shell-model ( R → 0) limit, as calculated with and without τ -mixing.The set-I and II results are shown in the upper and lower parts, respectively.set-I ( (cid:15) Λ ∗ = − 10 MeV, ν N = 0 . 16 fm − ) (cid:104) T kin (cid:105) (cid:104) v NN (cid:105) (cid:104) v ¯ KN (cid:105) (cid:104) v ¯ K ¯ K (cid:105) E S ¯ K S Λ ∗ P ( τ = 1)¯ KN + ¯ KN ( S π T ) = (0 + τ -mixing 30.0 − . − . . − . . . τ = 1 29.9 − . − . . − . . . τ = 0 43.1 0 . − . . . − . − . KN + ¯ KN ( S π T ) = (1 − τ -mixing 36.5 − . − . . − . . − . τ = 1 36.5 0 . − . . − . − . − . τ = 0 36.5 − . − . . − . − . − . (cid:15) Λ ∗ = − 27 MeV, ν N = 0 . 25 fm − ) (cid:104) T kin (cid:105) (cid:104) v NN (cid:105) (cid:104) v ¯ KN (cid:105) (cid:104) v ¯ K ¯ K (cid:105) E S ¯ K S Λ ∗ P ( τ = 1)¯ KN + ¯ KN ( S π T ) = (0 + τ -mixing 46.9 − . − . . − . τ = 1 46.7 − . − . . − . . . τ = 0 67.4 1 . − . . . KN + ¯ KN ( S π T ) = (1 − τ -mixing 57.0 0 . − . . − . . . τ = 1 57.0 1 . − . . − . . − . τ = 0 57.0 − . − . . − . − . − . KN + ¯ KN ( S π T ) and NN + NN ( S π T ) states in the shell-model limit, as measured from thetwo-cluster threshold energies. The set-I result of the total energy (∆ E ), kinetic (∆ T kin ), NN (∆ v NN ), ¯ KN (∆ v ¯ KN ), and¯ K ¯ K (∆ v ¯ K ¯ K ) interaction-energy contributions measured from twice of the internal energies of a single cluster are listed. Forthe ¯ KN + ¯ KN states, the results obtained with and without τ -mixing are shown. All energies are in units of MeV.set-I ( (cid:15) Λ ∗ = − 10 MeV, ν N = 0 . 16 fm − )configuration ∆ T kin ∆ v NN ∆ v ¯ KN ∆ v ¯ K ¯ K ∆ E ¯ KN + ¯ KN (0 + τ -mixing (0 s ) + (0 s ) (0 p ) − . − . . − . τ = 1 (0 s ) − . − . . − . τ = 0 (0 s ) (0 p ) . . . . KN + ¯ KN (1 − τ -mixing (0 s ) (0 p ) − . − . . . τ = 1 (0 s ) (0 p ) . − . . . τ = 0 (0 s ) (0 p ) − . . . . T kin ∆ v NN ∆ ENN + NN (0 + s ) − . − . NN + NN (1 − s ) (0 p ) − . . NN + NN (2 + s ) (0 p ) − . . tract universal features independently from the details of the hadron-hadron interactions. For precise predictions5 -30-20-10 0 10 20 30 40 KKNN − −∆ T kin ε Λ * + ( τ =1)1 − ( τ -mix)1 − ( τ =1)1 − ( τ =0)0 + ( τ =0) NNNN ∆ T kin ε d + − + ∆ E [ M e V ] FIG. 4: Energy spectra of the ¯ K ¯ KNN and NNNN systemscalculated with the ¯ KN + ¯ KN and NN + NN -cluster modelsin the shell-model limit. The energies are measured from thetwo-cluster threshold energies, 2 (cid:15) Λ ∗ for the ¯ K ¯ KNN systemand 2 (cid:15) d for the NNNN system. For the ¯ K ¯ KNN system, theenergies without τ -mixing and the S π = 1 − energy with τ -mixing are shown. Energy levels above the threshold are notbound states, but energies obtained for the lowest shell-modelconfigurations are plotted. of the energies and widths of the quasibound states ofkaonic nuclei, it is necessary to perform further investiga-tions with sophisticated calculations beyond the presentframework. Higher-order effects in hadron interactionssuch as the ¯ KN - π Σ coupling or imaginary part, the en-ergy dependences of the ¯ KN interaction, and non-central N N forces should be also taken into account. Acknowledgments This work was inspired by discussions with Prof. Leefor dimer-dimer interactions. 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