November 3, 2003 13:21 WSPC/Trim Size: 9in x 6in for Proceedings pre-hiyama1 WHY INTERESTING TO STUDY STRUCTURE OF LIGHT HYPERNUCLEI? E.HIYAMA High Energy Accelerator Research Organization(KEK), Tsukuba 305-0801, Japan M. KAMIMURA Kyushu University, Fukuoka 812-8581 T. MOTOBA Osaka Electro-Communication University, Neyagawa 572-8530 T. YAMADA Kanto-Gakuin University, Yokohama 236-8501 Y. YAMAMOTO Tsuru University, Tsuru 402-8555 The structure studied of 9Λ Be and 13 Λ C related to Y N spin-orbit force are discussed. In the microscopic α + x(= 0, n, p, d, t,3 He, α) model for 6ΛΛ He, 7ΛΛ He, 7ΛΛ Li, 8ΛΛ Li, 9 Li, 9 Be and 10 Be, we predict the binding energies and the excited-state enΛΛ ΛΛ ΛΛ ergies in bound region. Also, the importance of ΛΛ − ΞN coupling in 4ΛΛ H is discussed. 1. Introduction Recently in hypernuclear physics, we have obtained some of epoch-making experimental data by KEK-E373 1 , BNL-E906 16 , -E929 3 and -E930 4 . In near future, at TJLAB, BNL, DAΦNE, J-PARC and GSI facilities, it is planned to produce many single Λ hypernuclei and double Λ hypernuclei. Experimental data from these facilities will continuously provide us with many interesting and important few-body problems. Why is it interesting and important to study structure of light hypernuclei? Why do the facilities intend to produce many hypernuclei? 1 November 3, 2003 13:21 WSPC/Trim Size: 9in x 6in for Proceedings pre-hiyama1 2 One of the major purpose of the strangeness nuclear physics is to understand the baryon-baryon interaction. The baryon-baryon interaction is fundamental and important for the study of the nuclear physics. In general, it is the most useful for determination of any interaction to perform scattering experiments. However, in hypernuclear physics, since hyperon(Y )nucleon(N) scattering experiments are difficult to perform, the existing data are very limited. Furthermore, there is no hyperon(Y )-hyperon(Y ) scattering data. The Y N and Y Y interactions so far proposed have a large degree of ambiguity. In order to obtain useful information about Y N and Y Y interactions, we need many Y Y and Y N scattering data. In near future, some of Y N scattering experiments will be performed at J-PARC facility. However, it is very difficult to perform Y Y scattering experiments. Therefore, it is absolutely important to constrain ambiguity of Y N and Y Y interactions from spectroscopic study of many kinds of single Λ and double Λ hypernuclei. We obtain information about the Y N and Y Y interactions by combing theory and experiment in the following: (1) Firstly, we have candidate Y N and Y Y interactions, which are based on the one-boson-exchange model and the consistent quark model. (2) Secondly, we have hypernuclear γray spectroscopy experiments performed in order to provide information about the Y N and Y Y interactions. (3)However, state-of-art structure calculations using the models of the Y N and Y Y interactions are indispensable because the γ-ray data do not give information on the Y N and Y Y interactions directly. This theoretical component is our calculational contribution from few-body systems. As an example, we shall discuss γ-ray spectroscopy experimental and theoretical structure calculations related to the Y N spin-orbit interaction. Secondly, the ΛΛ − ΞN coupling and structure study of 4ΛΛ H will be discussed. It is important to obtain information on particle-conversion interactions such as ΛN −ΣN and ΛΛ−ΞN couplings in hypernuclear physics, since the mass difference between Λ and Σ, and ΛΛ and ΞN is very small compared with between N and ∆. Especially, ΛΛ − ΞN coupling is very important in binding mechanism of 4ΛΛ H. Recently at BNL-E906 which was designed to observe the production of double Λ hypernuclei, they obtained the results which indicated the production of the double Λ hypernucleus, 4 4 ΛΛ H. However, they could not determine the binding energy of ΛΛ H. Here, we emphasized that ΛΛ − ΞN coupling is one of the most important components to make 4ΛΛ H bound. November 3, 2003 13:21 WSPC/Trim Size: 9in x 6in for Proceedings pre-hiyama1 3 2. Model and Method In order to solve three- and four-body problems precisely, we have employed the Gaussian Expansion Method 5 . This has been successfully applied to the bound states of various three- and four-body systems 6,7,8 . The basis function work excellently in describing both short-range correlations and long-range tail behavior. In Ref. [5], we proposed infinitesimally-Gaussian basis functions, which are mathematically equivalent to the previous ones 6,7 , but the different in expression, so that the calculation of the interaction matrix elements can be performed much more easily. Owing to the use of these new basis functions, applicability of our calculational method is much extended even to four-body systems with complicated interactions. 3. Y N spin-orbit force and 9 Λ Be and 13 Λ C One of the characteristic phenomena in non-strange nuclear physics is that there is a strong N N spin-orbit force which leads to magic number nuclei. How large is the spin-orbit force in comparison with the N N spin-orbit force? However, because the there is no Y N spin-polarized scattering data, we have no information on the strength of Y N spin-orbit force experimentally. Two dedicated experiments with excellent energy resolution have been performed to obtain information about the ΛN spin-orbit interaction: One + 9 (E930 4 ) measures γ-rays from decays of the 5/2+ 1 and 3/21 states in Λ Be, − − and the other (E929 3 ) measures those from the 3/21 and 1/21 states in 13 Λ C. With the prospect of new data from high-resolution γ-ray measurements, we calculated the splittings of the spin-orbit double states in 9Λ Be 8 and 13 Λ C . For this purpose we employed all available versions of the Nijmegen OBE models 9,10 , and also took quark-model interaction properties 11 into account in the separation calculations. To perform careful structure calculations, we employ the microscopic three-body model (2α + Λ) for 9Λ Be and the four-body model (3α + Λ) for 13 6,7 . Λ C. The details of the calculational framework are described elsewhere + + 9 The calculated energy splitting of the 5/21 -3/21 doublet in Λ Be was − 13 predicted to be 80-200 keV and that the of 3/2− 1 -1/2 1 doublet in Λ C was predicted to be 390 - 960 keV for the NSC97a-f, ND, and NF potentials. On the other hand, in the case of the quark-model based spin-orbit force, we predicted significantly small energy splitting, namely 35-40 keV in 9Λ Be and 150-200 MeV in 13 Λ C. November 3, 2003 13:21 WSPC/Trim Size: 9in x 6in for Proceedings pre-hiyama1 4 Recently data of E930 and E929 have been reported to be 43.0 ± 5.0 3 keV in 9Λ Be 4 and 152 ± 54 ± 36 keV in 13 Λ C . We see that the predicted energy splittings using the quark-model-based spin-orbit force can explain both data consistently. On the other hand, predictions using any of the Nijmegen models are larger than both data. From the comparison between the experimental data and our theoretical calculations, we find that the ΛN spin-orbit force is 20-30 times smaller than the N N spin-orbit force. 4. Double Λ hypernuclei 4.1. ΛN − ΣN coupling and 4 Λ He In non-strangeness nuclei, there is particle conversion such as N N → N ∆. But, since the mass difference is large, it is considered that the contribution of ∆ particle is very small. On the other hand, in single Λ hypernuclei, the mass difference between Λ and Σ, which is 80 MeV, is much smaller than that between N and ∆. For this study, so far, many authors 12,8,13,14 pointed out that the Σ mixing is significantly important in making 4Λ H and 4Λ He bound. Jacobian coordinates of 4Λ He are illustrated in Fig.1 of Ref. [8]. The four-body wavefunction are written as ΨJ M (4Λ H,4Λ He) = 4 Y =Λ,Σ c=1 I,j l,L,λ s0 ,s,S,t0,t0 n,N,ν ×AN (c) AY,nlN LνλIjs 0sSt0 t (c) (c) (c) [φnl (rc )ψN L (Rc )]I ξνλ (ρc ) j (σ) (σ) (σ) (σ) × [χ1/2 (N1 )χ1/2 (N2 )]s0 χ1/2 (N3 ) s χ1/2 (Y ) (τ) (τ) (τ) (τ) × [χ1/2 (N1 )χ1/2 (N2 )]t0 χ1/2 (N3 ) t χtY (Y ) S JM T =1/2 , (1) where AN are the three-nucleon antisymmetrization operator and the isospin tY = 0 for Y = Λ and tY = 1 for Y = Σ. Here, n, N and ν denote the size of the Gaussian basis functions for four-body systems. The eigenenergies of Hamiltonian and coefficients C are determined by Rayleigh-Ritz variational method. As for the Y N interaction with ΛN − ΣN coupling terms, we consider those of Nijmegen soft core ’97f, but we simulate them by using a soft-core potential with central, spin-orbit and tensor terms 15 which reproduce the scattering phase shifts obtained by the original interaction. In the following, November 3, 2003 13:21 WSPC/Trim Size: 9in x 6in for Proceedings pre-hiyama1 5 we refer to this phase-shift equivalent soft-core potential as NSC97f. We employ the AV8 potential as the N N interaction. All the calculations shown below have been done both for 4Λ H and 4Λ He, but we present only the case of the former since the results very similar to each other. Calculated result of 4Λ He is shown in Fig.1. In the case of only (i): (NNN) Λ –BΛ (MeV) (ii): (NNN)Σ + 1 3 0 (unbound) + 0 He+ Λ (unbound) –0.54 –1 –1.15 –2 –2.39 1 + 0 + Exp. –2.28 (i) 1 + 0 + (i)+(ii) 4 Λ He Figure 1. Calculated energy levels of 4 He Λ the 3N +Λ channel adopted, both of the two states are unbound. When the N N N Σ channel is included, 0+ and 1+ states become bound with respect to 3 He + Λ threshold. Therefore, by solving the coupled four-body problem of 4Λ He, we found that the Σ channel component plays an essential role in the binding mechanism of the 4Λ He. November 3, 2003 13:21 WSPC/Trim Size: 9in x 6in for Proceedings pre-hiyama1 6 4.2. ΛΛ − ΞN coupling and 4 ΛΛ H In S = −2 nuclei, since the mass difference between ΛΛ and ΞN is only 28 MeV, then ΛΛ − ΞN particle conversion must be more important. For this study, we obtained three epoch-making experimental data on double Λ hypenuclei at KEK-E373 and BNL-E906. First one is observation of 6ΛΛ He, which is called the NAGARA event 1 . This event has been uniquely identified on the basis of observing the sequential weak decay, and the precise experimental value of the ΛΛ binding energy BΛΛ = 7.25 ± 0.19+0.18 −0.11MeV 10 has been obtained. Second one is the observation of ΛΛ Be, which is called exp , obtained is the Demachi-Yanagi event 16,17 . The binding energy ,BΛΛ +0.35 12.33 −0.21 MeV. It has not been, however, determined whether this event was the ground state or the excited state. Thirdly, at BNL-E906 which was designed to observe the production of the double Λ hypernuclei, they obtained the results which indicated the production of the double Λ hypernucleus, 4ΛΛ H. However, they could not determine the binding energy of 4 ΛΛ H. Analysis of KEK-E373 experiment is still in progress. Furthermore, it is planned to produce many double Λ hyperuclei at J-PARC, GSI and BNL. Here, the important issues are: (1) Does the Y Y interaction which is designed to reproduce the binding energy of 6ΛΛ He make 4ΛΛ H bound? If so, how large is binding energy? And how important is the effect of ΛΛ − ΞN coupling in making 6ΛΛ He and 4ΛΛ H bound? (2) Does the Y Y interaction which is designed to reproduce the binding energy of 6ΛΛ He, also reproduce the Demachi-Yanagi event consistently? (3) If this Y Y interaction is used for the other double Λ hypernuclei, how is the theoretical prediction of the level structure? At first, we discuss subject (1). Recently, Filikhin and Gal 18 calculated N N ΛΛ four-body system. The ΛΛ interaction was improved so as to reproduce the binding energy of 6ΛΛ He which was observed recently as NAGARA event. They concluded that the ground state of 4ΛΛ H was not bound. Recently, Nemura and Akaishi 19 performed more precise four-body calculation of 4ΛΛ H and they had the same conclusion as Filikhin and Gal as long as the ΛN interaction is adjusted so as to reproduce the observed splitting energy of 4Λ H. Here, it should be noted that they did not included ΛΛ − ΞN coupling in this system. This mass difference is much smaller than the mass difference between Λ and Σ. Therefore, it is expected that the effect is not so important in 6ΛΛ He. On the other hand, it is expected that the effect is not so important in 6ΛΛ He. November 3, 2003 13:21 WSPC/Trim Size: 9in x 6in for Proceedings pre-hiyama1 7 The reasons is as follows: In the case of 6ΛΛ He, two protons, two neutrons and two Λs occupy in the lowest s-shell and this s-shell is closed. When two Λs in 6ΛΛ He are converted into ΞN by ΛΛ − ΞN coupling, a additional nucleon is forbidden to occupy the s-shell due to Pauli blocking. Therefore, the effect of ΛΛ − ΞN coupling is small in 6ΛΛ He and p-shell double Λ hypernuclei. This Pauli suppression effect in p-shell double Λ hypernuclei was already pointed out in Ref. [20,21,22]. On the other hand, in the s-shell double Λ hypernucleus, 4ΛΛ H, the Ξ and the additional nucleon converted by the ΛΛ − ΞN coupling are allowed to occupy the lowest s-shell. Then, since there is no Pauli blocking, the ΛΛ − ΞN effect can be large. This is also pointed out in Ref. [23]. For this study, it is highly desirable to perform 4-body calculation taking N N ΛΛ and N N N Ξ channel explicitly. In the present calculation, all the interactions are taken to be of central type. As for the N N interaction, Minnesota potential which reproduce the binding energies of 3 H, 3 He and 4 He was used. As for the ΛN interaction, simulated Nijmegen model D modified so as to reproduce the binding energies of hypertriton and 4Λ H. In the case of Y Y interaction, ΛΛ, ΛΛ − ΞN coupling and ΞN − ΞN potentials are used. Here, Y Y interactions are improved so as to reproduce the binding energy of 6ΛΛ He within the framework of α + Λ + Λ and α + Ξ + N coupled threebody model. In this three-body calculation of 6ΛΛ He, Pauli blocking effect is included by the orthogonal condition model. Here four types of ΛΛ − ΞN potentials are employed. The strength of case (1) is the smallest of four potentials. This strength is corresponding to that of original Nijmegen model D. The strength of case (2) is lager than case (1). This strength is corresponding to that of Nijmegen soft core ’89. The strength of case (3) is larger than case (2). This strength is corresponding to that of Nijmegen model F. The strength of case (4) is the largest of four potentials. This strength is corresponding to that of the extended soft core potential which was recently proposed by Nijmegen group. The calculated binding energies of 6ΛΛ He using four types of ΛΛ − ΞN coupling are well reproduced the observed binding energy. The calculated Ξ mixing probabilities in 6ΛΛ He are 0.1 %, 0.6 %, 1.1 % and 1.6 % in the case of (1), (2), (3) and (4), respectively. The calculated binding energies of 4ΛΛ H are as follows: The case (1) ∼ (3) do not make 4ΛΛ H bound, while case (4) makes this hypernucleus weakly bound. At present, we have no experimental information about the strength of ΛΛ− ΞN coupling. If the binding energy of 4ΛΛ H is determined experimentally in the future, we can obtain useful information on the ΛΛ − ΞN coupling potential. November 3, 2003 13:21 WSPC/Trim Size: 9in x 6in for Proceedings pre-hiyama1 8 4.3. p-shell double Λ hypernuclei 8 9 Be Λ Be (MeV) 2 + 0 + 10 ΛΛBe 3.00 2.0 0.09 α+ α 5 Λ He + α + –4.0 α + α+ Λ+Λ α+ α + Λ BΛΛ(2 ) = 12.28 (exp. 12.33) –2.0 BΛ = 6.73 (exp. 6.71) 0.0 + 3/2 + 5/2 2.83 (3.05) –6.0 + 1/2 –8.0 0.00 (0.00) –6.64 cal.(exp.) Ex B ΛΛ (0 ) = 15.14 –10.0 + –12.0 –14.0 –16.0 Figure 2. Calculated energy levels of 8 Be, 9 Be Λ 6 ΛΛ He+ α 2 + 0 + and 2.86 –15.05 0.00 Ex 10 Be. ΛΛ In order to answer second issue(2), we made structure calculation of based on an α + α + Λ + Λ four-body model. The calculated BΛΛ for the 2+ excited state in 10 ΛΛ Be, 12.28 MeV, agree exp +0.35 with the experimental value BΛΛ = 12.33 −0.21MeV in the Demachi-Yanagi event as shown in Fig.1. We therefore interpret this event as the observation of the 2+ excited state of 10 ΛΛ Be, which is consistent with NAGARA event 10 ΛΛ Be November 3, 2003 13:21 WSPC/Trim Size: 9in x 6in for Proceedings pre-hiyama1 9 (MeV) 2.0 7 ΛΛHe 6 ΛΛHe 0.0 α +Λ+Λ α +n+ Λ + Λ 7 ΛΛLi 8 ΛΛLi 9 ΛΛLi α +p+Λ + Λ α+d+Λ+Λ α+ t+ Λ+Λ 9 ΛΛBe 10 ΛΛBe α+ 3 He+Λ+Λ α+α+Λ+Λ –2.0 –4.0 –6.0 6 + 0 –8.0 6 ΛΛHe+n –7.25 ΛΛHe+p – 3/2 –7.48 6 + 2 ΛΛHe+d ΛBe+ Λ 8 ΛLi +Λ –8.47 – –10.0 5/2 – + –12.0 3 1.36 5.96 5/2 – + –14.0 1 –12.91 0.00 7/2 5.92 – 4.54 7/2 + 2 4.55 – 0.73 1/2 – 3/2 1/2 – 3/2 0.71 –16.00 0.00 –17.05 0.00 + 0 –15.05 0.00 Ex Ex Ex Figure 3. Calculated energy levels of the double Λ hypernuclei, 8 Li, 9 Li, 9 Be and 10 Be. ΛΛ ΛΛ ΛΛ ΛΛ of 2.86 Ex – –16.0 6 ΛΛ He +α 3 ΛΛHe+ He 8 – 3/2 6 6 ΛΛHe + t 5.63 6 He, 7 He, 7 Li, ΛΛ ΛΛ ΛΛ 6 ΛΛ He. Then since our four-body calculations are predictive, hoping to have the data of many double Λ hypernuclei in the future experiments at JPARC, GSI, we have predicted level structure for double Λ hypernuclei in Fig. 2 with A = 7 ∼ 10 within the framework of an α + x + Λ + Λ fourbody model, where x = n, p, d, t,3He and α. This is corresponding to the third issue. It is our intention that these extensive four-body cluster model calculations should contribute to stimulating comprehensive spectroscopic studies of double Λ hypernuclei. 5. Conclusion We have shown that there are a lot of interesting and important few-body problems in hypernuclear physics. As examples, we reported three- and 4 7 four-body structure studies of single Λ hypernuclei of 9Λ Be, 13 Λ C, ΛΛ H, ΛΛ He, 7 8 9 9 10 ΛΛ Li, ΛΛ Li, ΛΛ Li, ΛΛ Be and ΛΛ Be. In the future, at TJLAB, BNL, DAΦNE, J-PARC and GSI facilities, they are planning to produce many single Λ and double Λ hypernuclei. 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