anti-K anti-K N molecule state in three-body calculation
aa r X i v : . [ nu c l - t h ] M a y ¯ K ¯ KN molecule state in three-body calculation Yoshiko Kanada-En’yo and Daisuke Jido
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
A ¯ K ¯ KN system with I = 1 / J P = 1 / + is investigated with non-relativistic three-bodycalculations by using effective ¯ KN and ¯ K ¯ K interactions. The present investigation suggests thata weakly bound state for the ¯ K ¯ KN system can be formed below the two-body threshold of ¯ K and quasibound ¯ KN with a 40 ∼
60 MeV decay width of ¯ K ¯ KN → πY ¯ K . This corresponds to anexcited Ξ baryon with J P = 1 / + located around 1.9 GeV. Studying the wave function of the ¯ K ¯ KN system obtained in this formulation, we find that the three-body bound system has a characteristicstructure of ¯ KN + ¯ K cluster with spatial extent. PACS numbers:
I. INTRODUCTION
The study of hadron structure is one of the most impor-tant issues in hadron physics. Recent interest in this lineis developed in exploring quasi-bound systems of mesonsand baryons governed by strong interactions among thehadrons. One of the long-standing candidates is theΛ(1405) resonance considered as a quasi-bound state of¯ KN system [1]. It has been also suggested that the f (980) scalar meson is a molecular state of ¯ KK [2]. Fornuclear systems, bound states of an eta meson in nu-clei and an anti-kaon in light nuclei were predicted inRefs. [3, 4, 5]. Recently a multi-hadron state was pro-posed for the Θ + baryon resonance to explain its narrowwidth [6].In such multi-hadron systems, anti-kaon plays a uniqueand important role due to its heavy mass and Nambu-Goldstone boson nature. The heavier kaon mass indi-cates stronger s -wave interactions around the thresholdthan those for pion according to the chiral effective the-ory. In addition, since typical kaon kinetic energy inthe system estimated by range of hadronic interaction issmall in comparison with the kaon mass, we may treatkaons in multi-hadron systems in many-body formula-tions.The strong attraction in the ¯ KN system led to theidea of deeply bound kaonic states in light nuclei, suchas K − pp and K − ppn , by Akaishi and Yamazaki [5, 7,8, 9, 10]. Later, many theoretical studies on the struc-ture of the K − pp system have been done, for example inFaddeev calculations [11, 12] and in variational calcula-tions [13, 14], having turned out that the K − pp system isbound with a large width. Experimental search for thesestates has been reported [15, 16, 17, 18], whereas clearevidences are not obtained yet and interpretations of theexperimental data are controversial [19, 20]. Motivatedby the strong ¯ KN ( I = 0) attractions, quests of multi- ¯ K nuclei are challenging issues [8, 21, 23, 24, 25]. This isalso one of the key subjects related to kaon condensationin dense nuclear matter [26, 27, 28].The key issue for the study of the ¯ KN interaction isthe resonance position of the Λ(1405) in the ¯ KN scat-tering amplitude. The Λ(1405) is observed around 1405 MeV in the π Σ final state interaction, as summarized inthe particle data group [29]. Based on this fact, a phe-nomenological effective ¯ KN potential (AY potential) wasderived in Refs. [5, 13], having relatively strong attrac-tion in the I = 0 channel to provide the K − p bound stateat 1405 MeV. Recent theoretical studies of the Λ(1405)based on chiral unitary approach have indicated that theΛ(1405) is described as a superposition of two pole statesand one of the state is considered to be a ¯ KN quasiboundstate embedded in the strongly interacting π Σ contin-uum [30, 31, 32, 33, 34, 35]. This double-pole conjecturesuggests that the resonance position in the ¯ KN scatter-ing amplitudes with I = 0 is around 1420 MeV, which ishigher than the energy position of the nominal Λ(1405)resonance. Based on this chiral SU(3) coupled-channeldynamics, Hyodo and Weise have derived another effec-tive ¯ KN potential (HW potential) [36]. The HW po-tential provides a ¯ KN quasibound state at ∼ s -wave ¯ KN effective potential is that the ¯ KN interac-tion with I = 0 is strongly attractive and describes theΛ(1405) resonance as a ¯ KN quasibound state. Basingon these strong ¯ KN interactions, we examine possiblebound states of a lightest two-anti-kaon nuclear system,namely ¯ K ¯ KN with I = 1 / J P = 1 / + , in thehadronic molecule picture. We expect that the three-body ¯ K ¯ KN system will form a bound state due to thestrong attraction in the two-body ¯ KN subsystem pro-ducing the 10 ∼
30 MeV binding energy. The questionsraised here are whether the strong attraction is enoughfor a three-body bound state below the threshold of the¯ K and the ¯ KN quasibound state, and what structure thebound state has, if it is formed.The ¯ K ¯ KN molecule state may have characteristic de-cay patterns depending on the binding energy. If thethree-body state is above the threshold of the ¯ K and the¯ KN quasibound state, it can decay into ¯ K and Λ(1405),and, consequently, the bound state has a large width.If the ¯ K ¯ KN system is bound below the threshold, thebound state has a comparable width to the Λ(1405),and the main decay mode is a three-body π Σ ¯ K andtwo-hadron decays are strongly suppressed in contrastto usual excited baryons. For deeply bound ¯ K ¯ KN sys-tems, the molecular picture may be broken down andtwo-hadron decays may be favored. Presently experi-mental data for S = − K − , K + ) processes atJ-PARC.In this paper, we investigate the ¯ K ¯ KN system with I = 1 / J P = 1 / + with non-relativistic three-bodycalculations by using the HW and AY potential as effec-tive ¯ KN interactions. In Sec. II, we describe the frame-work of the present calculations. We apply a variationalapproach with a Gaussian expansion method [37] to solvethe Schr¨odinger equation of the three-body system. Bytreating the imaginary potentials perturbatively, we findthe ¯ K ¯ KN quasibound state. In Sec. III, we present ourresults of the three-body calculation. In analysis of thewave functions, we discuss the structure and the bindingmechanism of the ¯ K ¯ KN state. The effects of ¯ K ¯ K in-teractions on the ¯ K ¯ KN system are also discussed. Sec-tion IV is devoted to summary and concluding remarks. II. FRAMEWORK
We consider a non-relativistic three-body potentialmodel for the ¯ K ¯ KN system. We calculate the ¯ K ¯ KN wave function by using effective two-body interac-tions [13, 36] in local potential forms. We apply a varia-tional approach with a Gaussian expansion method [37]in solving the Schr¨odinger equation for the three-bodysystem. In this section we explain the details of theframework and interactions used in this work. A. Hamiltonian
In the present formulation, we use the Hamiltonian forthe ¯ K ¯ KN system given by H = T + V ¯ KN ( r ) + V ¯ KN ( r ) + V ¯ K ¯ K ( r ) , (1)with the kinetic energy T , the effective ¯ KN interaction V ¯ KN and the ¯ K ¯ K interaction V ¯ K ¯ K . These interactionsare given by local potentials as functions of ¯ K - N dis-tances, r , r , and the ¯ K - ¯ K distance, r , which are de-fined by r = | x − x | , r = | x − x | and r = | x − x | .The vectors x , x , x denote the spatial coordinates ofthe first anti-kaon ( ¯ K ), the second anti-kaon ( ¯ K ) andthe nucleon, respectively. For convenience, we use Ja-cobian coordinates, r c and R c , in three rearrangementchannels c = 1 , , K − and ¯ K and between proton and neutron byusing the averaged masses. We do not consider three-body forces nor transitions to two-hadron decays. K r R N K K N K R c=1 c=2 c=3r R FIG. 1: Three Jacobian coordinates of the ¯ K ¯ KN system. The kinetic energy T is simply expressed by the Jaco-bian coordinates for a rearrangement channel as T ≡ − µ r c ∇ r c + − µ R c ∇ R c , (2)with the reduced masses µ r c and µ R c for the correspond-ing configuration. For instance, µ r = M K M N / ( M K + M N ) and µ R = M K ( M K + M N ) / (2 M K + M N ) for therearrangement channel c = 1. Here M N and M K denotethe averaged nucleon and kaon masses, respectively, as M N = 938 . M K = 495 . V ¯ KN and V ¯ K ¯ K , we uselocal potentials obtained by s -wave two-body scatteringamplitudes with isospin symmetry. In the present three-body calculations, we take l -independent potentials forsimplicity. The details of the local potentials will be dis-cussed in Sec. II B. The coupled-channel effects of the¯ KN to other relevant channels have been already im-plemented to the effective single-channel ¯ KN interaction V ¯ KN . Consequently V ¯ KN has an imaginary part owingto scattering states below the ¯ KN threshold, and thepresent Hamiltonian is not hermitian. In the calcula-tions of ¯ K ¯ KN wave functions, we first use only the realpart of V ¯ KN in a variational approach, and then we cal-culate the energy E with the expectation value of thetotal Hamiltonian (1) with respect to the obtained wavefunctions. The width of the bound state is evaluated bythe imaginary part of the energy as Γ = − E . Thedetails of the calculational procedure will be describedlater. B. Effective interactions
In this subsection, we explain the details of the ef-fective interactions of the ¯ KN and ¯ K ¯ K two-body sub-systems in our formulation. We use two effective ¯ KN interactions which were derived in different ways, andcompare the results obtained by these interactions to es-timate theoretical uncertainties. Both ¯ KN interactionshave so strong attraction as to provide the Λ(1405) asa quasibound state of the ¯ KN system. The importantdifference is the binding energy of the ¯ KN system, asalready mentioned in introduction.One of the ¯ KN interactions which we use here is givenby Hyodo and Weise in Ref. [36]. This effective interac-tion was derived based on the chiral unitary approach [38]for s -wave scattering amplitude with strangeness S = − KN amplitude can be obtained by reduc-ing the four (five)-channel problem for I = 0 ( I = 1) toa ¯ KN single-channel problem including dynamics of therest channels. The local ¯ KN potential was constructedin coordinate space so as to reproduce the single-channel¯ KN interaction as a solution of the Schr¨odinger equationfor the ¯ KN system with the local potential.The potential is written in a one-range Gaussian formas V ¯ KN = U I =0¯ KN exp (cid:2) − ( r/b ) (cid:3) P ¯ KN ( I = 0)+ U I =1¯ KN exp (cid:2) − ( r/b ) (cid:3) P ¯ KN ( I = 1) , (3)with the isospin projection operator P ¯ KN ( I = 0 ,
1) andthe range parameter b = 0 .
47 fm. The range param-eter was optimized for the parametrization referred asHNJH [39] in Ref. [36]. We refer this potential as “HW-HNJH potential”. The strength U ¯ KN ( I = 0 ,
1) has en-ergy dependence and is parametrized in terms of a thirdorder polynomial in the energy ω : U I =0 , KN ( ω ) = K I =0 , + K I =0 , ω + K I =0 , ω + K I =0 , ω , (4)for 1300 MeV ≤ ω ≤ K I =0 , i for I = 0 and I = 1 are given in Table IV and V ofRef. [36], respectively. Here we use the corrected versionof the local potentials [36].In chiral unitary approaches for the meson-baryon in-teractions, two poles are generated at 1.4 GeV region,and the Λ(1405) resonance is described as a ¯ KN qua-sibound state in the strongly interacting π Σ contin-uum [30, 32]. In this case, the peak position of the ¯ KN scattering amplitudes in the I = 0 channel appears at ω ∼ π Σ channel. The HW potential re-produces this feature of ¯ KN scattering amplitudes, andtherefore the position of the ¯ KN quasibound state is lo-cated at ω ∼ KN quasibound state bysolving the ¯ KN two-body Schr¨odinger equation in the s -wave I = 0 channel. We get a ¯ KN quasibound stateat 1423 MeV with a width Γ = 44 MeV, treating theimaginary part of the potentials perturbatively. This isa weakly bound ¯ KN state located at 11 MeV below the¯ KN threshold ( M K + M N = 1434 MeV). The root-mean-square of the ¯ K - N distance ( d ¯ KN ) is also calculated as1.9 fm.The HW-HNJH potential V I =0 , KN ( ω ) depends on theenergy of the subsystem ¯ KN . We regard, however, ω asan interaction parameter and use fixed values of ω , sincethe energy dependence is small in the region of interest, ω = 1400 ∼ ω = M N + M K − δω with δω =0 and 11 MeV. The choice of δω = 0MeV corresponds to ¯ KN at the threshold, while δω = 11MeV is for the binding energy B ( ¯ KN ) = 11 MeV of the¯ KN bound state obtained by the HW-HNJH potential.In each of the fixed HW potential, we calculate the energy E Re = − B ( ¯ K ¯ KN ) of the ¯ K ¯ KN system and also theenergy − B ( ¯ KN ) of the ¯ KN system.The other ¯ KN interaction which we use is givenby Akaishi and Yamazaki (AY) in Refs. [5, 13]. TheAY potential was derived in a phenomenological wayto start with the ansatz that the Λ(1405) resonanceis a K − p bound state at 1405 MeV as reported byPDG [29]. The AY potential is energy independent andwas parametrized so as to reproduce the ¯ KN quasiboundstate at the PDG values of the Λ(1405): V I =0¯ KN ( r ) = ( − − i ) exp (cid:2) − ( r/ . (cid:3) , (5) V I =1¯ KN ( r ) = ( − − i ) exp (cid:2) − ( r/ . (cid:3) . (6)The AY potential has a stronger attraction in the I = 0channel because it was adjusted to generate a ¯ KN ( I = 0)quasibound state at ∼ KN binding energy B ( ¯ KN ) ∼
30 MeV which is muchlarger than B ( ¯ KN ) = 11 MeV for the HW-HNJH poten-tial. By solving the ¯ KN two-body Schr¨odinger with aperturbative treatment of the imaginary potential, weget a ¯ KN ( I = 0) state at 1403 MeV with a widthof 40 MeV, and the root-mean-square ¯ K - N distance is d ¯ KN = 1 . I = 0 channel of the AY potential, the ¯ KN ( I = 0) qua-sibound state has the deeper binding and smaller radiusthan those given by the HW-HNJH potential. We com-ment that the size of the ¯ K - N state is sensitive to thebinding energy [40].For the ¯ K ¯ K interactions, there are few experimentaldata. The low energy theorem based on the current alge-bra suggests a repulsive interaction. For the s -wave inter-action of ¯ K ¯ K , the I = 0 is forbidden due to Einstein-Bosestatistics. Thus we assume V I =0¯ K ¯ K = 0. We introduce theeffective interaction of the subsystem ¯ K ¯ K with I = 1, V I =1¯ K ¯ K , in a Gaussian form V I =1¯ K ¯ K ( r ) = U I =1¯ K ¯ K exp (cid:2) − ( r/b ) (cid:3) P ¯ K ¯ K ( I = 1) , (7)where the range parameter b is chosen to be the samevalue as that of the ¯ KN interaction. The interaction V I =1¯ K ¯ K is a real function, since there are no decay channelsopen for the ¯ K ¯ K system.The strength U I =1¯ K ¯ K is estimated by theoretical calcula-tions of the scattering length of K + K + , which is equiv-alent to that of ¯ K ¯ K with I = 1. Recently, the K + K + scattering length has been obtained in lattice QCD cal-culation as a K + K + = − . ± .
006 fm [41]. This valueis consistent with the leading order calculation of the chi-ral perturbation theory, a K + K + = − .
147 fm, which isobtained by a K + K + = − m K πf K , (8)with f K = 115 MeV. In the present calculation, thestrength U I =1¯ K ¯ K is adjusted to reproduce the scatteringlength a I =1 KK = − .
14 fm. For the HW-HNJH poten-tial, with the interaction range b = 0 .
47 fm, we obtain U I =1¯ K ¯ K = 313 MeV, and we find U I =1¯ K ¯ K = 104 MeV with b = 0 .
66 fm for the AY potential. We will also try aweaker repulsion with a I =1 KK = − .
10 using U I =1¯ K ¯ K = 205MeV and 70 MeV for the HW-HNJH and the AY po-tentials, respectively. In order to examine how strongthe ¯ KN interaction for the three body system, at firstwe neglect the ¯ K ¯ K repulsion, then we see the effect ofthe repulsive interaction with including the effective ¯ K ¯ K interaction given above. C. Three-body wave function
The three-body ¯ K ¯ KN wave function Ψ is described asa linear combination of amplitudes Φ ( c ) I KK ( r c , R c ) of tworearrangement channels c = 1 , c = 2 is included by symmetrization of the two anti-kaon in the wave function. In the present calculation, wetake the model space limited to l c = 0 and L c = 0 of theorbital-angular momenta for the Jacobian coordinates r c and R c in the channel c . Then the wave function of the¯ K ¯ KN system with I = 1 / J P = 1 / + is written asΨ = 1 + P √ , (9)Φ = Φ ( c =1) I KK =0 ( r , R ) (cid:2) [ ¯ K ¯ K ] I KK =0 N (cid:3) I =1 / + Φ ( c =1) I KK =1 ( r , R ) (cid:2) [ ¯ K ¯ K ] I KK =1 N (cid:3) I =1 / + Φ ( c =3) I KK =1 ( r , R ) (cid:2) [ ¯ K ¯ K ] I KK =1 N (cid:3) I =1 / , (10)where P is the exchange operator between thetwo anti-kaons, ¯ K and ¯ K for two bosons. The (cid:2) [ ¯ K ¯ K ] I KK N (cid:3) I =1 / specifies the isospin configuration ofthe wave function Φ ( c ) I KK ( r c , R c ), meaning that the to-tal isospin 1 / K ¯ KN system is given by combi-nation of total isospin I KK for the ¯ K ¯ K subsystem andisospin 1 / (cid:2) [ ¯ K ¯ K ] I KK =0 N (cid:3) I =1 / @in the c = 3 is not necessary, be-cause it vanishes after the symmetrization in the case of l = 0.As mentioned above, we omit basis wave functions with l c ≥ L c ≥ s -wave two-body dynamics. We commentthat components with non-zero angular momenta of two-body subsystems are contained in the model wave func-tion through rearrangement of three-body configurations,although the s -wave component is expected to be domi-nant.In solving the Schr¨odinger equation for the ¯ K ¯ KN sys-tem, we adopt the Gaussian expansion method for three- body systems given in Ref. [37]. The spatial wave func-tions Φ ( c ) I KK ( r c , R c ) of the subcomponent of Eq. (10) areexpanded in terms of the Gaussian basis functions, φ Gn ( r )and ψ Gn ( R ):Φ ( c ) I KK ( r c , R c ) = n max ,N max X n c ,N c A c,I KK n c ,N c φ Gn c ( r c ) ψ GN c ( R c ) . (11)The coefficients A c,I KK n c ,N c are determined by variationalprinciple when we solve the Schr¨odinger equation. InEq. (11), n max and N max are the numbers of the Gaus-sian basis, and the basis functions are defined by φ Gn ( r ) = N n e − ν n r , (12) ψ Gn ( R ) = N N e − λ N R , (13)where the normalization constants are given by N n =2(2 ν n ) / π − / and N N = 2(2 λ N ) / π − / , and theGaussian ranges, ν n and λ N , are given by ν n = 1 /r n , r n = r min (cid:18) r max r min (cid:19) n − n max − , (14) λ N = 1 /R N , R N = R min (cid:18) R max R min (cid:19) N − N max − . (15)We take enough bases for the present system by usingthe values given in Table I for the basis numbers andrange parameters in the channel c = 1 and c = 3. Wefind that the mixing effect of the rearrangement channel c = 3 is very small in the present results of the ¯ K ¯ KN system. This is because the ¯ K ¯ K interactions are notattractive and therefore the ¯ K - ¯ K correlation is not strongin the ¯ K ¯ KN system. This is different from the case ofthe K − pp system where the p - p correlation is significantbecause of the attractive N N interaction.
TABLE I: The numbers and range parameters of the basisfunctions for the rearrangement channels, c = 1 and c = 3.channel n max r min r max N max R min R max (fm) (fm) (fm) (fm) c = 1 15 0.2 20 25 0.2 200 c = 3 15 0.2 20 15 0.2 20 D. Procedure of calculations
The wave function of the ¯ K ¯ KN system is obtained bysolving the Schr¨odinger equation:[ T + V ¯ KN ( r ) + V ¯ KN ( r ) + V ¯ K ¯ K ( r ) − E ] Ψ = 0 . (16)The effective interaction V ¯ KN is complex due to the pres-ence of the decay channels below the threshold, while V ¯ K ¯ K is expressed by real numbers.In order to solve this equation with variational princi-ple, we treat the imaginary part of the potentials pertur-batively. Separating the real part of the Hamiltonian, wewrite H Re = T + Re V ¯ KN ( r ) + Re V ¯ KN ( r ) + V ¯ K ¯ K ( r ) . (17)We first calculate the wave function for the real part of the Hamiltonian, H Re , with variational principle inthe model space of the Gaussian expansion described inSec. II C. This is equivalent to determine the eigenenergy E Re and the coefficients A c,I KK n c ,N c of the Gaussian wavefunctions φ Gn c ( r c ) ψ GN c ( r c ) (cid:2) [ ¯ K ¯ K ] I KK N (cid:3) I =1 / by diagonal-izing the norm matrix and Hamiltonian matrix D φ Gn ′ c ( r c ) ψ GN ′ c ( r c ) (cid:2) [ ¯ K ¯ K ] I ′ KK N (cid:3) I =1 / (cid:12)(cid:12) H Re (cid:12)(cid:12) φ Gn c ( r c ) ψ GN c ( r c ) (cid:2) [ ¯ K ¯ K ] I KK N (cid:3) I =1 / E . (18)After this variational calculation, we take the lowest-energy solution for H Re . The binding energy B ( ¯ K ¯ KN )of the three-body system is given as B ( ¯ K ¯ KN ) = − E Re .It should be checked if the bound state of the ¯ K ¯ KN islower than the threshold of the ¯ K and the quasiboundstate of ¯ KN , because the ¯ K ¯ KN solution obtained abovethe ¯ KN + ¯ K threshold is not a three-body bound statebut a two-body continuum state with ¯ K and ¯ KN quasi-bound state.Next we estimate the imaginary part of the energy E for the total Hamiltonian H by calculating the expec-tation value with the wave function Ψ obtained by theHamiltonian H Re : E Im = h Ψ | Im V ¯ KN | Ψ i . (19)The total decay width for ¯ K ¯ KN is estimated as Γ = − E Im . In the present calculation, we have only three-body decays to π Σ ¯ K and π Λ ¯ K by the model setting.We also calculate several quantities characterizing thestructure of the three-body system, such as spatial con-figurations of the constituent particles and probabilitiesto have specific isospin configurations. These values arecalculated as expectation values of the wave functions.The root-mean-square (r.m.s.) radius of the ¯ K distribu-tion is defined as the average of the distribution of eachanti-kaon by r ¯ K ≡ q(cid:10) Ψ (cid:12)(cid:12) ( x + x ) (cid:12)(cid:12) Ψ (cid:11) , (20)which is measured from the center of mass of the three-body system. For the two-body ¯ KN system, r ¯ K is givenby the spatial coordinate of ¯ K , x ¯ K , measured from thecenter of mass of the two-body system as r ¯ K = q(cid:10) x K (cid:11) .We also calculate the r.m.s. value of the relative ¯ K - ¯ K distance defined by d ¯ K ¯ K ≡ q h Ψ | r | Ψ i . (21)The probabilities for the three-body system to have theisospin I ¯ K ¯ K states are introduced byΠ (cid:16)(cid:2) ¯ K ¯ K (cid:3) I KK (cid:17) ≡ h Ψ | P ¯ K ¯ K ( I KK ) | Ψ i , (22) where P ¯ K ¯ K ( I KK ) is the projection operator for theisospin configuration (cid:2) [ ¯ K ¯ K ] I KK N (cid:3) I =1 / , as given before.We also calculate the r.m.s. radius of the ¯ K distributionfor the each isospin state r ¯ K | I KK ≡ s (cid:10) Ψ (cid:12)(cid:12) ( x + x ) (cid:12)(cid:12) P ¯ K ¯ K ( I KK )Ψ (cid:11) h Ψ | P ¯ K ¯ K ( I KK ) | Ψ i , (23)which is normalized by Eq. (22).In order to investigate the structure of the ¯ K ¯ KN sys-tem further, we calculate the expectation values with theunsymmetrized wave function Φ given in Eq. (10). Al-though these expectation values are not observable inthe real ¯ K ¯ KN system having two identical bosons, theyare helpful to understand the structure of the three-bodysystem and to investigate the symmetrization effect. Wecalculate norms of the wave functions for the isospin (cid:2) [ ¯ K N ] I KN ¯ K (cid:3) I =1 / states, where the total isospin 1 / K ¯ KN system is given by combination of totalisospin I KN for the ¯ K N subsystem and isospin 1 / K ,Π (cid:16)(cid:2) ¯ K N (cid:3) I KN (cid:17) Φ ≡ (cid:10) Φ (cid:12)(cid:12) P ¯ K N ( I KN ) (cid:12)(cid:12) Φ (cid:11) , (24)where P ¯ K N ( I KN ) is again the isospin projection opera-tor. Note that this cannot be interpreted as a probabilityfor the (cid:2) [ ¯ K N ] I KN ¯ K (cid:3) I =1 / states in the physical ¯ K ¯ KN system, since the norm is not normalized to be unitywithout the symmetrization of the anti-kaon wave func-tions. The r.m.s. values of the ¯ K - N distance and the¯ K N - ¯ K distance for Φ are evaluated as d ¯ K N ≡ s h Φ | r | Φ ih Φ | Φ i , (25a) d ( ¯ K N ) − ¯ K ≡ s h Φ | R | Φ ih Φ | Φ i , (25b)respectively.The perturbative treatment performed above is jus-tified qualitatively in the case of |h Ψ | Im V | Ψ i| ≪ |h Ψ | Re V | Ψ i| . For the two-body system ¯ KN , wefind the perturbative treatment good, observing that |h Im V ¯ KN i| = 22 MeV is much smaller than |h Re V ¯ KN i| ∼
100 MeV in the HW-HNJH potential case, for instance.This is responsible for that we get the reasonable energy E = 1423 − i MeV for the ¯ KN with the perturba-tive treatment. (In a full calculation with the ¯ KN effec-tive interaction, the scattering amplitude reproduces theΛ(1405) resonance at 1420 MeV with 40 MeV width) Alsoin the case of the ¯ K ¯ KN system, it is found that the abso-lute values of the perturbative energy |h Ψ | Im V ¯ KN | Ψ i| =20 ∼
30 MeV is much smaller than the real potential en-ergy |h Ψ | Re V ¯ KN + Re V ¯ K ¯ K | Ψ i| = 100 ∼
200 MeV in thepresent calculations. This is because the dominant con-tribution of h V ¯ KN i comes from I = 0 channel which hasthe strong attractive potential with the weak imaginarypart compared with the real part. III. RESULTS
In this section, we show the results of calculation ofthe ¯ K ¯ KN system with I = 1 / J P = 1 / + . For the¯ KN interactions, we compare two potential, the HW-HNJH potential and the AY potential, as discussed in theprevious sections. We first show the calculation withoutthe ¯ K ¯ K interactions V ¯ K ¯ K , in order to see if the ¯ KN interaction is strong enough for binding the three-bodysystem. Later in subsection III B, we discuss the effect of V ¯ K ¯ K by introducing possible repulsive ¯ K ¯ K interactionsin the I = 1 channel. A. Bound ¯ K ¯ KN state without ¯ K ¯ K interaction In this subsection, we present the results of the ¯ K ¯ KN state calculated without the V ¯ K ¯ K interaction.
1. Energy, width and decay modes of ¯ K ¯ KN state First of all, it is very interesting that the ¯ K ¯ KN boundstate is obtained below the threshold of the ¯ K and thequasibound ¯ KN state in both calculations with the HW-HNJH and with the AY potentials, as seen in Fig. 2,where we show the level structure of the ¯ K ¯ KN sys-tem. In Table II, we show the results for the energiesand radius of the ¯ K ¯ KN state as well as those for the¯ KN state. For the energy-dependent HW-HNJH po-tential, we take two energies δω = 0 and 11 MeV with ω = M N + M K − δω . In the table, B ( ¯ KN ) and B ( ¯ K ¯ KN )denote the binding energies for the ¯ KN and ¯ K ¯ KN sys-tems measured by the energies of the two-body and three-body break-up states, respectively.The ¯ K ¯ KN bound state appears as small as 1 MeVbelow the threshold of the ¯ K and quasibound ¯ KN state( − B ( ¯ KN )) in both cases of the HW-HNJH potential.The calculated value for r ¯ K is about 4 fm, which is much larger than that of the two-body ¯ KN bound state. It in-dicates that the ¯ K ¯ KN state is loosely bound with a largeradius. In the case of the AY potential, the ¯ K ¯ KN energyis −
36 MeV, which is 5 MeV below the ¯ KN + ¯ K thresh-old ( −
31 MeV). The AY potential gives a deeper bindingand a smaller radius, r ¯ K = 2 fm than the HW-HNJHpotential, reflecting the stronger ¯ KN ( I = 0) attractionin the AY potential.The ¯ K ¯ KN → πY ¯ K decay width of the ¯ K ¯ KN state isevaluated by the imaginary ¯ KN potentials as Γ( ¯ K ¯ KN → πY ¯ K ) = − E Im . In the present results, we obtain thewidth in the range of 51 MeV to 57 MeV. This impliesthat the ¯ K ¯ KN state has a comparable width to thatof the ¯ KN state for Λ(1405). It is interesting to notethat the dominant contribution in the decay width of the¯ K ¯ KN state comes from the I = 0 channel for the ¯ KN subsystem, while the I = 1 channel gives only a few MeVcontribution. This means that the ¯ K ¯ KN state has domi-nant π Σ ¯ K decay and relatively small π Λ ¯ K decay modes.This characteristic decay pattern comes from the strongattraction in the ¯ KN channel with I = 0 reproducingthe Λ(1405) as a quasibound state. -40-30-20-10 0 * Λ KKN (a)HW(b)AY threshold E n e r gy ( M e V ) K+K+N +K
FIG. 2: Energies of the ¯ K ¯ KN calculated with (a) the HW-HNJH ( δω = 11 MeV) potential and (b) the AY potentialwithout the ¯ K ¯ K interaction. The calculated thresholds of ¯ K and the quasibound ¯ KN state are also shown. The ¯ K + ¯ K + N threshold is located at 1930 MeV.
2. Structure of ¯ K ¯ KN state Let us discuss the binding mechanism and the struc-ture of the ¯ K ¯ KN system with I = 1 / K ¯ KN system,since the potential energy is determined by both of theisospin and spatial structure.We first discuss the binding mechanism based on theisospin configurations. For convenience, we introduce thefollowing notations; [ ¯ K ¯ K ] I denotes the isospin configu-ration (cid:2) [ ¯ K ¯ K ] I KK = I N (cid:3) I =1 / for the two-body ¯ K ¯ K sub-system with I = 0 or 1 in the three-body system, and[ ¯ KN ] stands for (cid:2) [ ¯ KN ] I KN =0 ¯ K (cid:3) I =1 / for the two-body¯ KN subsystem with I = 0, which is Λ(1405) + ¯ K -likeisospin configuration.According to the following argument, it is easy to un-derstand that the [ ¯ K ¯ K ] and [ ¯ KN ] configurations areenergetically favoured to gain potential energy of the¯ KN subsystem. The effective ¯ KN potential V I ¯ KN hasstrong attraction in the I = 0 and weak attraction in the I = 1 channel. From group-theoretical arguments it iseasily found that, on one hand, the former configuration[ ¯ K ¯ K ] consists of [ ¯ KN ] and [ ¯ KN ] components with aratio 3:1. As a result, both of two anti-kaons effectivelyfeel the potential V I =0¯ KN + V I =1¯ KN , which is moderate at-traction. On the other hand, in the latter configuration[ ¯ KN ] , one of the anti-kaons couples with the nucleon in I KN = 0 and gains the strong attraction of V I =0¯ KN , and,at the same time, the other anti-kaon feels much weakerattraction as V I =0¯ KN + V I =1¯ KN .In the present calculation, it is found that the prob-ability Π([ KK ] ) for the three-body system to have the[ ¯ K ¯ K ] configuration is dominant in the ¯ K ¯ KN wave func-tions as shown in Table III, in which Π([ ¯ K ¯ K ] ) is 0.87for the result of the HW-HNJH ( δω = 11 MeV) poten-tial, and it is 0.91 in the case of the AY potential. Thesevalues are in between two limits, [ ¯ KN ] and [ ¯ K ¯ K ] ; Inthe [ ¯ KN ] limit, the probability Π([ ¯ K ¯ K ] ) should be0.75 when the symmetrization of two anti-kaons is ig-nored, while Π([ ¯ K ¯ K ] ) = 1 for the pure [ ¯ K ¯ K ] state.The present calculation implies that the ¯ K ¯ KN state isregarded as an admixture of the isospin configurations[ ¯ K ¯ K ] and [ ¯ KN ] . Therefore, the rearrangement of theisospin configurations is essential in the ¯ K ¯ KN boundstate.Next we discuss the spatial structure of the ¯ K ¯ KN state. Since two anti-kaons are identical bosonic parti-cles, ¯ K and ¯ K cannot be identified in the symmetrizedwave function Ψ. For an intuitive understanding, it ishelpful to analyze the wave function Φ obtained beforethe symmetrization. For the wave function Φ, we candefinitely calculate the expectation values for the ¯ K - N and ( ¯ K N )- ¯ K distances in the rearrangement channel c = 1 as given by Eq. (25). In the calculated results,it is found that d ¯ K N is almost the same as the ¯ K - N distance in the ¯ KN quasibound state, while d ( ¯ K N ) − ¯ K is remarkably large. Actually, d ( ¯ K N ) − ¯ K is more thanthree times larger than d ¯ K N in all choices of the inter-actions as shown in Table. III. For example, we obtain d ¯ K N = 1 . d ( ¯ K N ) − ¯ K = 6 . δω =11 MeV) potential. It indicates that oneof the kaons widely distributes around the nucleon withvery loosely binding and the other kaon is moving in thevicinity of the nucleon. In addition, the wave functionΦ contains dominantly the [ ¯ K N ] component and littlethe [ ¯ K N ] configuration, which are shown as the norms of the corresponding wave functions in TableIII. Thismeans that the ¯ K N subsystem has more likely the I = 0component and, thus the Φ has a Λ(1405) + ¯ K clusterstructure [44]. It is worth noting that, even though thegroup theoretical argument suggests that the probabilityΠ([ ¯ K ¯ K ] ) for the normalized wave function should be0.75 without the symmetrization, the [ ¯ K ¯ K ] componentis obtained as Π([ KK ] ) ∼ . K ¯ KN bound state can beinterpreted as a hybrid of the two configurations: the[ ¯ K ¯ K ] in the inner region (I) and the Λ(1405)+ ¯ K clusterin the asymptotic region (II) as shown in the schematicfigure (Fig. 3). As mentioned above, one of the anti-kaons distributes in a spatially wide region and the otheranti-kaon distributes near the nucleon. In the inner re-gion (I), two anti-kaons are coupled to isospin symmetricwith the isospin configuration [ ¯ K ¯ K ] because they oc-cupy the same orbit and are spatially symmetric. In theouter region (II), the nucleon and one of the anti-kaonsform a Λ(1405) state and the other anti-kaon is mov-ing around the Λ(1405). This kaon-halo like structureis reminiscent of the neutron-halo observed in unstablenuclei. B. effect of repulsive ¯ K ¯ K interaction In the previous subsection, we presented the resultscalculated without ¯ K ¯ K interactions. We discuss pos-sible effects of the ¯ K ¯ K interactions. As already men-tioned, the I = 0 is forbidden for s -wave ¯ K ¯ K states dueto Einstein-Bose statistics. We here investigate how therepulsive ¯ K ¯ K interactions may affect the ¯ K ¯ KN state byintroducing V I ¯ K ¯ K for the I = 1 channel.We use the V I =1¯ K ¯ K interactions adjusted to reproducethe scattering length a I =1 KK = − .
14 fm of lattice QCDcalculation. We also use a weaker repulsion with a I =1 KK = − .
10 fm. The calculated results with V I =1¯ K ¯ K are shownin Table IV. It is found that the ¯ K ¯ KN bound stateis obtained even with the possible repulsive ¯ K ¯ K inter-actions. Because of the repulsive ¯ K ¯ K interactions, theΛ(1405) + ¯ K cluster develops more and the anti-kaon isfurther loosely bound. As a result, the absolute valueof the imaginary energy Im E decreases as shown in Ta-ble IV. The value of d ( ¯ K N ) − ¯ K is extremely large as d ( ¯ K N ) − ¯ K ≥
16 fm in the HW-HNJH potential, and theΛ(1405) + ¯ K cluster feature becomes more remarkable.There still remains ambiguity of the strengths of the¯ K ¯ K interactions due to few experimental data. Here weestimate possible boundaries of the repulsive interactionfor the formation of the bound ¯ K ¯ KN state. For theAY potential, the ¯ K ¯ KN state is still bound even with afurther strong ¯ K ¯ K interaction as a I =1 KK = − .
20 fm. Inthe case of the HW-HNJH potential, the ¯ K ¯ KN energy is TABLE II: Energies and root-mean-square (r.m.s.) radii and distances of the ¯ KN and the ¯ K ¯ KN states calculated withoutthe ¯ K ¯ K interaction. For the ¯ KN interactions, the AY potential and the HW-HNJH potential with the energy parameter ω = M N + M K − δω (MeV) are used. For the ¯ KN state, the real energy E = − B ( ¯ KN ), the imaginary energy E Im , ther.m.s. ¯ K - N distances( d ¯ KN ), r ¯ K are shown. For the ¯ K ¯ KN state, the real energy E = − B ( ¯ K ¯ KN ), the energy relative to the¯ KN + ¯ K threshold, the imaginary energy E Im , r ¯ K are shown. We also list the expectation values of Im V I =0¯ KN and Im V I =1¯ KN separately. AY HW-HNJH δω = 0 δω = 11¯ KN ( I = 0) state − B ( ¯ KN ) (MeV) − − − E Im (MeV) − − − d ¯ KN (fm) 1.4 2.0 1.9 r ¯ K (fm) 0.9 1.3 1.2¯ K ¯ KN ( I = 1 /
2) state − B ( ¯ K ¯ KN ) (MeV) − − − − B ( ¯ K ¯ KN ) + B ( ¯ KN ) (MeV) − − − E Im (MeV) − − − h Im V I =0 i (MeV) − − − h Im V I =1 i (MeV) − − − r ¯ K (fm) 2.0 4.2 3.8TABLE III: Properties such as the isospin configurations and radii in the three-body ¯ K ¯ KN system calculated without the¯ K ¯ K interaction. In the upper part, we show the values in the total wave function Ψ obtained after the symmetrization. Theprobabilities P ([ ¯ K ¯ K ] , ) to have the [ ¯ K ¯ K ] , configurations are listed. The r ¯ K in the [ ¯ K ¯ K ] and [ ¯ K ¯ K ] components are shownseparately. The calculated values for d ¯ K ¯ K are also shown. We also present the expectation values for the unsymmetrizedwave function Φ. The norms of the wave functions for the isospin configurations [ ¯ K N ] and [ ¯ K N ] , and distances d ¯ K N and d ( ¯ K N ) − ¯ K calculated with the wave function Φ are shown.AY HW-HNJH δω = 0 δω = 11expectation values for ΨΠ([ ¯ K ¯ K ] ) 0.09 0.13 0.13Π([ ¯ K ¯ K ] ) 0.91 0.87 0.87 r ¯ K | I KK =0 (fm) 2.7 5.2 4.8 r ¯ K | I KK =1 (fm) 1.9 4.0 3.7 d ¯ K ¯ K (fm) 3.1 6.4 5.9expectation values for ΦΠ([ ¯ K N ] ) Φ K N ] ) Φ d ¯ K N (fm) 1.2 1.7 1.6 d ( ¯ K N ) − ¯ K (fm) 3.2 6.8 6.2 found above the ¯ KN + ¯ K threshold with more repulsive¯ K ¯ K interactions than a I =1 KK = − .
15 fm. In such a case,a resonance state or a virtual state may appear near thethreshold. This might have a larger width than that ofthe three-body bound state. For the detailed structureof such states, we need to use formulations beyond thepresent framework, because the continuum states are nottaken into account in the present calculations.
IV. SUMMARY AND CONCLUDINGREMARKS
We have investigated the ¯ K ¯ KN system with I = 1 / J P = 1 / + as an example of multi-hadron systemswith anti-kaons. We have performed a non-relativisticthree-body calculation by using the effective ¯ KN inter-actions proposed by Hyodo-Weise and Akaishi-Yamazaki.With the ¯ KN potentials and no ¯ K ¯ K interaction, thepresent calculation suggests that a weakly bound ¯ K ¯ KN state can be formed below the ¯ KN + ¯ K threshold energy. TABLE IV: Theoretical results of the ¯ K ¯ KN state calculated with the ¯ K ¯ K interactions. The range parameters of the ¯ K ¯ K interactions are taken to be b = 0 .
47 fm and b = 0 .
66 fm for the HW-HNJH and the AY potentials, respectively. Energies androot-mean-square (r.m.s.) radii and distances are listed. The expectation values for the various isospin configurations, and thosefor the unsymmetrized wave function Φ are also shown. The detailed descriptions are given in captions of Tables II and III. Inthe calculations for HW-HNJH( δω = 0 MeV) with U I =1 KK = 70 MeV, we take the extended basis space as ( n max , N max ) = (15 , r min , r max , R min , R max ) = (0 . , , . , c = 1 to get the convergent solution.AY+ V ¯ K ¯ K HW-HNJH + V ¯ K ¯ K δω = 0 δω = 11 a I =1 KK (fm) 1.0 1.4 1.0 1.4 1.0 1.4 U I =1 KK (MeV) 70 104 205 313 205 313¯ K ¯ KN ( I = 1 /
2) state − B ( ¯ K ¯ KN ) (MeV) − − − − − − − B ( ¯ K ¯ KN ) + B ( ¯ KN ) (MeV) − − − − − − E Im (MeV) − − − − − − h Im V I =0 i (MeV) − − − − − − h Im V I =1 i (MeV) − − − − − r ¯ K (fm) 2.7 3.2 12 49 10 32expectation values for ΨΠ([ ¯ K ¯ K ] ) 0.13 0.14 0.20 0.24 0.20 0.23Π([ ¯ K ¯ K ] ) 0.87 0.86 0.80 0.76 0.80 0.77 r ¯ K | I KK =0 (fm) 3.4 3.9 13 50 11 33 r ¯ K | I KK =1 (fm) 2.6 3.0 12 48 9.8 31 d ¯ K ¯ K (fm) 4.1 4.9 18 76 16 49expectation values for ΦΠ([ ¯ K N ] ) Φ K N ] ) Φ d ¯ K N (fm) 1.2 1.3 1.9 2.0 1.8 1.9 d ( ¯ K N ) − ¯ K (fm) 4.2 5.0 19 76 16 49 The AY potential for the ¯ KN interactions provides adeeper bound state of the ¯ K ¯ KN system because of thestronger attraction in the ¯ KN channel with I = 0 thanthe case of the HW potential.Investigating the wave function obtained by the three-body calculation, we have found that the ¯ K ¯ KN boundstate can be interpreted as a hybrid of two configura-tions; In the inner region, two kaons are spatially sym-metric and couples to I KK = 1, while in the asymptoticregion where one kaon far from the nucleon, the systemis regarded as the Λ(1405)+ ¯ K like cluster. The root-mean-square radius of ¯ K distribution is found to be alarge value due to the Λ(1405)+ ¯ K like component with aloosely bound kaon around the Λ(1405). This kaon-halolike structure is reminiscent of the neutron-halo observedin unstable nuclei.We have also evaluated the decay width of the ¯ K ¯ KN system to πY ¯ K , obtaining Γ = 40 ∼
60 MeV, whichis comparable to the width of Λ(1405). The dominantmode is found to be ¯ K ¯ KN → ¯ K ( π Σ) I =0 , reflecting the¯ K +Λ(1405) cluster structure. In this estimation, we havenot considered three-body forces nor transitions to two-hadron decays. Nevertheless, such effects are expected tobe suppressed because the overlap of the wave functionsof three particles in a compact region is very small in the present case that one of the ¯ K s is loosely bound anddistributes very widely around the ¯ KN subsystem.To estimate the effect of unknown ¯ K ¯ K interactions, wehave introduced a repulsive interaction with I = 1 whichreproduces the K + K + scattering length obtained by lat-tice QCD calculation. It is interesting that a bound stateof the ¯ K ¯ KN system is possible even with the repulsive¯ K ¯ K interactions. The repulsive nature of the ¯ K ¯ K in-teraction suggests that the Λ(1405)+ ¯ K cluster developsmore and the anti-kaon is further loosely bound.The peculiar structure that the wave function of one ofthe anti-kaon spreads for long distance is a consequencethat the ¯ KN interaction is strong enough to form a qua-sibound ¯ KN and ¯ K ¯ KN states, but is not so strong forthe deeply bound ¯ K ¯ KN state. This is very significantfor the multi-kaon system. In such systems, some of theanti-kaon could be bound very loosely, so that many-body absorptions of anti-kaons are suppressed and theanti-kaons keep their identity.So far the experimental data for the S = − K ¯ KN molecular state suggested in thepresent calculation is one of the excited Ξ baryon with0 KN Λ (1405) KKN K (r) K (r) [KK] I=1 I=0 [KN] K r (I) (II) (3.2 fm)6.2 fm(1.2 fm) FIG. 3: Schematic figure for the ¯ K ¯ KN state. K ( r ) and K ( r ) indicate the anti-kaon wave functions. One of the anti-kaons distributes in a wide region and the other anti-kaondistributes near the nucleon. The values shown in the figurewithout the parenthesis are d ¯ K N and d ( ¯ K N ) − ¯ K calculatedwith the HW-HNJH potential ( δω = 11 MeV), while the val-ues in the parentheses are those for the AY potential. In theinner region (I), two anti-kaons are coupled to isospin sym-metric with the isospin configuration [[ ¯ K ¯ K ] I KK =1 N ] I =1 / be-cause they occupy the same orbit and are spatial symmetric.In the outer region (II), the nucleon and one of the anti-kaonsform a Λ(1405) state and the other anti-kaon is moving aroundthe Λ(1405). J P = 1 / + sitting around 1.9 GeV and having charac-teristic properties. The main decay mode is the three-body decay of the ¯ Kπ Σ with I = 0 for the final π Σ,since the Λ(1405) component in the three-body systemis a doorway for the decay. In addition, productions ofsuch a multi-hadron system may be strongly dependenton transfered momentum, since the loosely bound statehas a large spatial distribution, which leads a softer formfactor. These would be good indications for identifyingthe molecular state in experiments. In the present work, we have discussed the bound¯ K ¯ KN state obtained by the the single-channel three-body calculation, where effects of coupled-channelmeson-baryon interactions are taken into account as ef-fective single-channel ¯ KN interactions. Such coupled-channel effects could debase clear resonance shape for thethree-body quasi-bound state in the spectrum, or couldpush the quasi-bound state up to the two-body contin-uum and the quasi-bound state could be a virtual state.Nevertheless, in principle, the spectra which will be ob-served in experiments have information of resonances andvirtual states. For complete understanding of the ¯ K ¯ KN molecule state, further detailed studies involving dynam-ics of three-body resonances are necessary. For instance,coupled-channel calculations of the three-body system in-cluding ¯ KN ↔ π Σ [42], such as Faddeev type calcula-tions done for the ppK − system in Refs. [11, 12] andfor two-meson and one baryon system with S = − K ¯ KN quasibound state impliesa possible existence of an excited baryon with a moleculestructure of two mesons surrounding a baryon. Thepresent investigation may be an important step to leadfundamental information on the physics of multi strangesystems, such as anti-kaons in nuclear medium. Acknowledgments
The authors would like to thank Dr. Hyodo for valu-able discussions. They are also thankful to membersof Yukawa Institute for Theoretical Physics (YITP) andDepartment of Physics in Kyoto University, especiallyfor fruitful discussions. This work is supported in partby the Grant for Scientific Research (No. 18540263 andNo. 20028004) from Japan Society for the Promotion ofScience (JSPS) and from the Ministry of Education, Cul-ture, Sports, Science and Technology (MEXT) of JapanA part of this work is done in the Yukawa InternationalProject for Quark-Hadron Sciences (YIPQS). The com-putational calculations of the present work were done byusing the supercomputer at YITP. [1] R. H. Dalitz and S. F. Tuan, Phys. Rev. Lett. , 425(1959); Annals Phys. , 307 (1960)[2] J. D. Weinstein and N. Isgur, Phys. Rev. Lett. , 659(1982); Phys. Rev. D , 2236 (1990).[3] Q. Haider and L. C. Liu, Phys. Lett. B , 257 (1986).[4] T. Kishimoto, Phys. Rev. Lett. , 4701 (1999).[5] Y. Akaishi and T. Yamazaki, Phys. Rev. C , 044005(2002).[6] T. Kishimoto and T. Sato, Prog. Theor. Phys. , 241(2006)[7] T. Yamazaki and Y. Akaishi, Phys. Lett. B , 70(2002). [8] T. Yamazaki, A. Dote and Y. Akaishi, Phys. Lett. B ,167 (2004).[9] A. Dote, H. Horiuchi, Y. Akaishi and T. Yamazaki, Phys.Rev. C , 044313 (2004).[10] Y. Akaishi, A. Dote and T. Yamazaki, Phys. Lett. B ,140 (2005).[11] N. V. Shevchenko, A. Gal, and J. Mares, Phys. Rev. Lett. , 082301 (2007).[12] Y. Ikeda and T. Sato, Phys. Rev. C , 035203 (2007).[13] T. Yamazaki and Y. Akaishi, Phys. Rev. C76 ,045201(2007).[14] A. Dote, T. Hyodo and W. Weise, Nuclear Physics A , 197-206 (2008).[15] M. Agnello et al. [FINUDA Collaboration], Phys. Rev.Lett. , 212303 (2005).[16] T. Suzuki et al. , Phys. Lett. B , 263 (2004).[17] M. Sato et al. , Phys. Lett. B , 107 (2008).[18] T. Suzuki et al. [KEK-PS E549 Collaboration], Phys.Rev. C , 068202 (2007).[19] V. K. Magas, E. Oset, A. Ramos and H. Toki, Phys. Rev.C , 025206 (2006).[20] T. Yamazaki and Y. Akaishi, Nucl. Phys. A , 229(2007).[21] A. Dote, Y. Akaishi and T. Yamazaki, Nucl. Phys. A , 391 (2005).[22] T. Yamazaki and Y. Akaishi, Proc. Japan Acad. B ,144 (2007).[23] D. Gazda, E. Friedman, A. Gal and J. Mares, Phys. Rev.C , 055204 (2007); Phys. Rev. C 77, 019904(E) (2008).[24] D. Gazda, E. Friedman, A. Gal and J. Mares,arXiv:0801.3335 [nucl-th].[25] T. Muto, Prog. Theor. Phys. Suppl. , 623 (2007).[26] D. B. Kaplan and A. E. Nelson, Phys. Lett. B , 57(1986).[27] T. Muto, R. Tamagaki and T. Tatsumi, Prog. Theor.Phys. Suppl. , 159 (1993).[28] T. Muto, T. Takatsuka, R. Tamagaki and T. Tatsumi,Prog. Theor. Phys. Suppl. , 221 (1993).[29] W. M. Yao et al. [Particle Data Group], J. Phys. G ,1 (2006).[30] J. A. Oller and U. G. Meissner, Phys. Lett. B B 500 ,263 (2001).[31] D. Jido, A. Hosaka, J. C. Nacher, E. Oset and A. Ramos,Phys. Rev. C , 025203 (2002). [32] D. Jido, J. A. Oller, E. Oset, A. Ramos, and U. G. Meiss-ner, Nucl. Phys. A , 181 (2003).[33] T. Hyodo, D. Jido and A. Hosaka, Prog. Theor. Phys.Suppl. 168 (2007) 32.[34] T. Hyodo, D. Jido and A. Hosaka, arXiv:0803.2550 [nucl-th].[35] In a different context, the double pole structure ofthe Λ(1405) was discussed in P. J. Fink, Jr., G. He,R. H. Landau and J. W. Schnick, Phys. Rev. C , 2720(1990).[36] T. Hyodo and W. Weise, Phys. Rev. C , 035204 (2008).[37] E. Hiyama , Y. Kino and M. Kamimura, Prog. Part. Nucl.Phys. , 223 (2003).[38] N. Kaiser, P. B. Siegel, and W. Weise, Nucl. Phys. A
325 (1995); E. Oset and A. Ramos, Nucl. Phys. A
99 (1998); J. A. Oller and U. G. Meissner, Phys. Lett. B , 263 (2001); M. F. M. Lutz and E. E. Kolomeitsev,Nucl. Phys. A , 193 (2002).[39] T. Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev.
C68 , 018201 (2003); Prog. Theor. Phys. , 73 (2004).[40] T. Sekihara, T. Hyodo and D. Jido, arXiv:0803.4068[nucl-th].[41] S. R. Beane, T. C. Luu, K. Orginos, A. Parreno,M. J. Savage, A. Torok and A. Walker-Loud [NPLQCDCollaboration], arXiv:0709.1169 [hep-lat].[42] Y. Ikeda and T. Sato, talk in JPS meeting, September2007, Hokkaido, Japan.[43] A. Martinez Torres, K. P. Khemchandani and E. Oset,Phys. Rev. C , 042203(R) (2008).[44] The idea of Λ(1405) cluster in kaonic nuclei was proposedin the K − pppp