Three- and four-body structure of hypernuclei It is my great pleasure to invite this nice opportunity to give a talk in this seminar. Today, I will give a talk about structure of hypernuclei from view point of few-body problem. E. Hiyama (RIKEN)

Introduction

Major goals of hypernuclear physics 1) To understand baryon-baryon interactions Fundamental and important for the study of nuclear physics 2) To study the structure of multi-strange systems To understand the baryon-baryon interaction, two-body scattering experiment is most useful. Total number of Nucleon (N) -Nucleon (N) data: 4,000 YN and YY potential models so far proposed (ex. Nijmegen, Julich, Kyoto-Niigata) have large ambiguity. ・ Total number of differential cross section Hyperon (Y) -Nucleon (N) data: 40 ・ NO YY scattering data

Therefore, as a substitute for the 2-body limited YN and non-existent YY scattering data, the systematic investigation of the structure of light hypernuclei is essential.

X Strategy to determine YN and YY interactions from the studies of light hypernuclear structure YN and YY interactions based on meson theory: Nijmegen, Ehime, Julich・・ 　 based on constituent quark model: Kyoto-Niigata,・・ using a few-body method My role ① ③ Use Suggest to improve Accurate calculation of hypernuclear structure Few-body, cluster model, shell model, ….. Compare theoretical results with experimental data X ② No direct information Spectroscopy experiments 　　・ High-resolution γ-ray spectroscopy experiment 　 by Tamura and his collaborators 　　・ Emulsion experiment 　 by Nakazawa and his collaborators

High-precision calculations of various 3- and 4-body systems: Our few-body caluclational method Gaussian Expansion Method (GEM) , since 1987 , ・A variational method using Gaussian basis functions ・Take all the sets of Jacobi coordinates Developed by Kyushu Univ. Group, Kamimura and his collaborators. Review article : E. Hiyama, M. Kamimura and Y. Kino, Prog. Part. Nucl. Phys. 51 (2003), 223. High-precision calculations of various 3- and 4-body systems: Exotic atoms / molecules , 3- and 4-nucleon systems, Multi-cluster structure of light nuclei, Light hypernuclei, 3-quark systems, ……….

Sec. 2. S=-1 hypernuclei and YN interaction Strategy to determine YN and YY interactions from the studies of light hypernuclear structure YN and YY interactions based on meson theory: Nijmegen, Ehime, Julich・・ 　 based on constituent quark model: Kyoto-Niigata,・・ using a few-body method My role ① ③ Use Suggest to improve Accurate calculation of hypernuclear structure Few-body, cluster model, shell model, ….. Sec. 2. S=-1 hypernuclei and YN interaction Compare theoretical results with experimental data X ② No direct information Spectroscopy experiments 　　・ High-resolution γ-ray spectroscopy experiment 　 by Tamura and his collaborators 　　・ Emulsion experiment 　 by Nakazawa and his collaborators

Section 2. S= -1 hypernuclei and YN interaction

ΛN interaction (effectively including ΛN -ΣN coupling) One of the important issue Almost determined since 1998 ----- SLS (Symmetric LS) ----- Symmetric LS (SLS) ----- ALS (Anti symmetric LS)

12C YN LS force and energy-splitting in 9Be and 13C 8Be 9Be 13C Λ Λ Λ Λ ----- SLS (Symmetric LS) 12C 8Be 9Be ----- ALS (Antisymmetric LS) 13C Λ Λ [vanishes in S=0 nuclei, Pauli] [breaks charge symmetry] In the ALS part : 0 < VALS (meson theory) VALS (constituent quark model) << Kyoto-Niijata FSS potential Nijgemen model D, F , soft core ’97a-f It is important to extract information about these LS force from the study of the structure of Λ hypernuclei.

α α α α α 12C 13C 8Be 9Be 13C 9Be BNL-E930 BNL-E929 (0s) ΔE 1/2- (0p) Λ Λ (0s) ΔE 3/2- 3/2+ ΔE　 LS splitting γ γ 2+ 5/2+ γ γ 0+ 1/2+ 0+ 1/2+ 12C 13C 8Be 9Be Λ Λ 3- and 4-body calculations: E. Hiyama, M. Kamimura, T. Motoba, T. Yamada and Y. Yamamoto Phys. Rev. Lett. 85 (2000) 270. Microscopic α-cluster model + Λ Λ α Λ 13C 9Be α α Λ Λ α α YN LS force 　　　1) Meson theory : Nijmegen Model D, F, soft core’97 a – f. 　　　2) Qurak model : Kyoto-Niigata, FSS　　　　

ΛN LS force and 9Be and 13C 9Be 13C 43±5 Exp. 5/2+ 3/2+ keV BNL-E930 H. Akikawa et al. Phys. Rev. Lett. 88 (2002) 082501; H. Tamura et al. Nucl. Phys. A754 (2005) 58c 152 1/2- 3/2- BNL-E929 ± 54 36 keV S.Ajimura et al. Phys. Rev. Lett. 86,(2001) 4255 43±5 ΛN LS force and 9Be and 13C Λ Λ 9Be Λ 3/2+ 5/2+ 3/2+ 35 40 keV 2) Quark ～ 150 200 1/2- 3/2- 80 200 keV ～ 5/2+ 1) Meson Nijmegen model D,F Soft core ’97a-f 13C Λ 1/2- 360 960 keV ～ 3/2- 1) Meson

LS splitting in 9Be α α 9Be 43±5 Meson Theory SLS SLS + ALS 5/2+ 5/2+ We suggested there are 2 paths to improve the Meson models : reduce the SLS strength or enhance the ALS strength so as to reproduce the observed LS splittings in 9Be and 13C. LS splitting in 9Be Λ Meson Theory (Large) (Small) SLS SLS +　ALS 5/2+ 5/2+ Λ 80～200 keV Λ 140～250 keV 3/2+ 3/2+ 3/2+ 5/2+ keV 43±5 (Large) - (Large) Exp. SLS + ALS 5/2+ Λ 35～40keV 3/2+ α α Quark-based 9Be Λ

LS splitting in 9Be Λ Recently, a new YN interaction based on meson theory, extended soft core potential 06 (ESC06) by Th. A Rijken 9Be Λ BNL-E930 (small) SLS + ALS 4-body calculation (2007) 3/2+ 3/2+ 39 keV 5/2+ 5/2+ keV 43±5 Exp. ESC06 Good agreement Hiyama (2007) H. Akikawa et al. Phys. Rev. Lett. 88,(2002)82501; H. Tamura et al. Nucl. Phys. A754,58c(2005)

④ ⑤ Strategy to determine YN and YY interactions from the studies of light hypernuclear structure Meson theory :Nijmegen Quark model :Kyoto-Niigata YN SLS+ALS potentials ① Use ③ ④ new version potential (ESC06) Suggest to improve Accurate calculation of energy splitting (9Be and 13C) using the YN SLS+ALS potentials Λ Λ comparison again: good agreement ② comparison ⑤ Spectroscopy experiments 　　 High-resolution γ-ray spectroscopy experiment in 9Be and 13C 　 by Tamura and his collaborators by Kishimoto and his collaborators Λ Λ

Hypernuclear g-ray data since 1998 Taken by Tamura Λ N This is hypernuclear gamma-ray data since 1998. So far, we succeeded in getting so many data like this. Then, combining with these experimental data and theoretical calculation such as shell model and few-body technique, we succeeded in extracting information about these Lambda N interaction. ・Millener (p-shell model), 　・ Hiyama (few-body)

spin-orbit force of ΣN interaction ΣN interaction-> Σ hypernuclei In S= -1 sector, what are the open questions in　YN interaction? 　spin-orbit force of ΣN interaction ΣN interaction-> Σ hypernuclei Now, we know that spin-orbit force in ΛN interaction is so small. Then, how about spin-orbit force in ΣN interaction? Next, we want to know about it. Please perform experiment to get information about ΣN interaction.

(2) ΣN interaction-> Σ hypernuclei In S= -1 sector, what are the open questions in　YN interaction? (2) ΣN interaction-> Σ hypernuclei Σ N N N 4He Σ First observation of Σ hypernucleus

Possibility of another Σ hypernucleus Possible existence of bound 7Li state α Σ T. Yamada and K. Ikeda, PRC46, 1315 (1992). p p n n n p α α α 6Be 6Li(T=1,T=0) 6He A=6 nuclear system is iso-triplet system. Let’s add Σ　particle into these nuclei. According to Yamada et al, 7Li has possibility to have bound state. Σ Σ separation energy is 1.6 MeV. Γ=6.0 MeV. 7Li (K-(in-flight), π-) ??

(3)Charge symmetry breaking (4) ΛN－ΣN coupling In S= -1 sector, what are the open questions in　YN interaction? (3)Charge symmetry breaking (4) ΛN－ΣN coupling JLAB J-PARC : Day-1 experiment ・E13 “γ-ray spectroscopy of light hypernuclei” by Tamura and his collaborators 11B 4He Λ Λ ・E10 “Study on Λ-hypernuclei with the doubleCharge-Exchange reaction” by Sakaguchi , Fukuda and his collaboratiors 9He 6H Λ Λ

(3) Charge Symmetry breaking Energy difference comes from dominantly Coulomb force between 2 protons. Charge symmetry breaking effect is small. In S=0 sector Exp. N+N+N 0 MeV - 7.72 MeV 0.76 MeV 1/2+ - 8.48 MeV 3He 1/2+ 3H n p p n n p

Exp. 4H 4He In S= -1 sector 3He+Λ 3H+Λ 0 MeV 0 MeV -1.00 -1.24 1+ 1+ -2.04 -2.39 0+ 0.35 MeV 0+ n n p n Λ p Λ p 4H 4He Λ Λ

Σ In order to explain the energy difference, 0.35 MeV, N N N N + N Λ N ・E. Hiyama, M. Kamimura, T. Motoba, T. Yamada and Y. Yamamoto, Phys. Rev. C65, 011301(R) (2001). ・A. Nogga, H. Kamada and W. Gloeckle, Phys. Rev. Lett. 88, 172501 (2002) ・H. Nemura. Y. Akaishi and Y. Suzuki, Phys. Rev. Lett.89, 142504 (2002). Coulomb potentials between charged particles (p, Σ±) are included.

4H 4He 3He+Λ 3H+Λ + 0 MeV 0 MeV -1.00 -1.24 1+ 1+ -2.04 0+ -2.39 0+ n (Exp: 0.24 MeV) (cal: -0.01 MeV(NSC97e)) -2.04 0+ -2.39 0+ (Exp: 0.35 MeV) (cal. 0.07 MeV(NSC97e)) Among these calculation, Nogga and his collaborators investigated the charge symmetry breaking effect by sophisticated 4-body calculation using modern realistic YN and NN interactions. These are results using Nijmegen soft core ’97e model. The calculated energy difference in the ground state is 0.07 MeV. And this value in the excited state is -0.01 MeV. Both of energy difference in the ground state and the excited state are inconsistent with the data. At the present, there exist no YN interaction to reproduce the charge symmetry breaking effect. n n p n Λ p Λ p 4H ・A. Nogga, H. Kamada and W. Gloeckle, Phys. Rev. Lett. 88, 172501 (2002) 4He Λ Λ ・E. Hiyama, M. Kamimura, T. Motoba, T. Yamada and Y. Yamamoto, Phys. Rev. C65, 011301(R) (2001). ・H. Nemura. Y. Akaishi and Y. Suzuki, Phys. Rev. Lett.89, 142504 (2002). N N N N + Σ N Λ N

4H 4He 3H+Λ 3He+Λ 0 MeV 0 MeV -1.00 -1.24 1+ 1+ -2.04 0+ -2.39 0+ n n (Exp: 0.24 MeV) (cal: -0.01 MeV(NSC97e)) -2.04 0+ -2.39 0+ (Exp: 0.35 MeV) (cal. 0.07 MeV(NSC97e)) n n p n Λ p Λ p 4H 4He Λ Λ There exist NO YN interaction to reproduce the data. why can not we reproduce the data? Let’s discuss experimental data again.

We get binding energy by decay π spectroscopy. π-+1H+3He →2.42 ±0.05 MeV 4He Λ π-+1H+1H+2H → 2.44 ±0.09 MeV Total: 2.42 ±0.04 MeV Then, binding energy of 4He is reliable. Λ decay π-+1H+3H →2.14 ±0.07 MeV Two different modes give 0.22 MeV 4H Λ π-+2H+2H → 1.92 ±0.12 MeV Total: 2.08 ±0.06 MeV This value is so large to discuss CSB effect. Then, for the detailed CSB study, we should perform experiment to confirm the Λ separation energy of 4H. Λ For this purpose, at JLAB, it is planned to to perform ・・・ Key experiment to get information about CSB. 　4He (e, e’K+) 4H Λ

Possible new understanding JLAB experiment 　4He (e, e’K+) 4H Λ Possible new understanding If the binding energy of 4H is the same as that of 4He, we find that we have no charge symmetry breaking effect in S=-1 sector. Λ (2) If the binding energy of 4H is quite different from that of 4He, we find that we have charge symmetry breaking (CSB) effect between Λn and Λp interaction. (It should be noted that CSB interaction in S=0 is very small.) We need the JLAB experiment. Λ

Furthermore, we need more Λ hypernuclear data to get information on CSB.

It is interesting to investigate the charge symmetry breaking effect For this purpose, It is interesting to investigate the charge symmetry breaking effect in p-shell Λ hypernuclei as well as s-shell Λ hypernuclei. For this purpose, to study structure of A=7 Λ hypernuclei is suited. Because, core nuclei with A=6 are iso-triplet states. p p n n n p α α α 6Be 6Li(T=1) 6He

Then, A=7 Λ hypernuclei are also iso-triplet states. α α α 7He 7Be 7Li(T=1) Λ Λ Λ Then, A=7　Λ hypernuclei are also iso-triplet states. It is possible that CSB interaction between Λ and valence nucleons contribute to the Λ-binding energies in these hypernuclei.

Exp. 6He 6Be 6Li 7Be 7Li 7He Emulsion data Emulsion data BΛ=5.16 MeV 1.54 Emulsion data Emulsion data 6He 6Be 6Li (T=1) BΛ=5.16 MeV These are experimental data of A=7 hypernuclei. These two data of Be7L and Li7L are taken by emulsion data. As mentioned by the previous speaker, Hashimoto san, we got new data of He7L. The value is like this. We see that as the number of neutron increase, Lambda separation energy increase. BΛ=5.26 MeV JLAB:E01-011 experiment Preliminary data: 5.68±0.03±0.22 -3.79 7Be Λ 7Li (T=1) Λ 7He Λ

Can we describe the Λ binding energy of 7He observed at JLAB Important issue: Can we describe the Λ binding energy of 7He observed at JLAB using　ΛN　interaction to reproduce the Λ binding energies of 7Li (T=1) and 7Be ? To study the effect of CSB in iso-triplet A=7 hypernuclei. Λ Λ Λ p n n n p p The detailed study has been done in this paper. Λ Λ Λ α α α 7He 7Be 7Li(T=1) Λ Λ Λ For this purpose, we study structure of A=7 hypernuclei within the framework of α+Λ+N+N 4-body model. E. Hiyama, Y. Yamamoto, T. Motoba and M. Kamimura,PRC80, 054321 (2009)

Li ΛN interaction: Nijmegen ’97f Not original one but simulated one p Λ The ΛN-ΣN coupling interaction can be renomalized into the ΛN-ΛN interaction effectively. α VΛN=V0+σΛ・σNVs+(σΛ+σN)/2・VSLS+(σΛ-σN)/2・VALS Made by Yamamoto so as to reproduce the phase shifts given by the original one Strengths of Vs,VSLS,VALS are adjusted so as to reproduce of the observed data of 4H, 7Li(T=0), 9Be and 13C. Λ Λ Λ Λ

Now, it is interesting to see as follows: What is the level structure of A=7 hypernuclei without CSB interaction? (2) What is the level structure of A=7 hypernuclei with

6Be 6Li 6He 7Be 7Li 7He Without CSB BΛ：CAL= 5.21 EXP= 5.16 BΛ： EXP= 5.68±0.03±022 JLAB:E01-011 experiment These are results of A=7 hypernuclei without CSB interaction. We see that our Λ binding energies of 7BeL and Li7L are in good agreement with the data. Let see the Λ separation energy of He7L. Our result is not inconsistent with data. CAL= 5.36 preliminary 7Be Λ 7Li (T=1) Λ 7He Λ

Exp. 4H 4He Next we introduce a phenomenological CSB potential with the central force component only. Strength, range are determined ao as to reproduce the data. 0 MeV 3He+Λ 0 MeV 3H+Λ -1.00 -1.24 1+ 1+ 0.24 MeV -2.04 -2.39 0+ 0.35 MeV 0+ n n p n Λ p Λ p Exp. 4H 4He Λ Λ

With CSB 5.29 MeV (With CSB) 5.21 (without CSB) 5.44(with CSB) 5.28 MeV( withourt CSB) 5.44(with CSB) 5.21 (without CSB) 5.29 MeV (With CSB) 5.36(without CSB) This is our results with a phenomenological CSB interaction. The binding of Li7 with and without CSB is almost the same. Because, there is cancellation between ｎΛ and pΛ CSB interaction. On the other hand, in the case of Be7L, the energy of Be7L with CSB make deeper bound by 0.2 MeV comparing with this value. And in the case of He7L, the energy with CSB interaction make less bound by 0.2 MeV comparing with this. Then, we found that binding energies with CSB interaction of He7L and Be7L became inconsistent with the data. 5.16(with CSB) BΛ： EXP= 5.68±0.03±0.22 Inconsistent with the data p n α

Comparing the data of A=4 and those of A=7, Let me compare with the experimental data of A=4 hypernuclei and data of A=7 hypernuclei. Comparing the data of A=4 and those of A=7, tendency of BΛ is opposite. How do we understand these difference?

Why CSB interaction which reproduce the energy difference of A=4 hypernuclei, do not reproduce the energy difference in p-shell hypernuclei such as A=7 system? In my calculation, ΛN-ΣN coupling effect is not included explicitly and mass difference of Σ. (2) The binding energy of 4H is incorrect. Then, we should measure the binding energy of this hypernucleus again. If the experiment will be done successful, It might the energies of 4H and 4He are the same. Here, let me discuss with this part. Λ Λ Λ (3) odd-state CSB interaction whose contribution is negligible in A=4 hypernuclei, contribute to p-shell Λ hypernuclei with opposite sign of the even state of CSB interaction.

CSB interaction contribute to those hypernuclei. Λ Λ Λ α α α 7He 7Be 7Li(T=1) Λ Λ Λ A=7　Λ hypernuclei are p-shell nuclei. Then, it is possible that odd state CSB interaction contribute to those hypernuclei. Now, let me introduce a phenomenological odd state CSB interaction. Parameters are adjusted so as to reproduce the observed binding energy of 7He. Λ

In order to check the validity of the odd CSB interaction, It is suited for study the structure of A=10 Λ hypernuclei such as 10B and 10Be. Λ Λ These are p-shell Λ hypernuclei. Λ Λ n p α α α α These are experimental data of these hypernuclei taken by emulsion. However, the number of emulsion data are a few. Then, the error bar is large. Among these experiment, recently, to produce this experiment has been done at JLAB and the analysis is in progress. 10B 10Be Λ Λ Exp. BΛ=9.11±0.22 MeV Number of event (emulsion data): 3 Exp. BΛ=8.89±0.12 MeV Number of event (emulsion data): 10 At JLAB, the analysis is in progress.

Λ Λ n p α α α α 10B 10Be Exp. BΛ=9.11±0.22 MeV Number of event (emulsion data): 3 Exp. BΛ=8.89±0.12 MeV Number of event (emulsion data): 10 At JLAB, the analysis is in progress. If the odd CSB interaction to reproduce the binding energy of 7He reproduce the binding energy of 10Be which will be reported soon, we can check the validity of odd state CSB interaction. Λ Λ

6Be 6Li 6He 7Be 7Li 7He BΛ：CAL= 5.21 EXP= 5.16 JLAB:E01-011 experiment BΛ： EXP= 5.68±0.03±022 JLAB:E01-011 experiment We tune the odd state CSB interaction. And we get this energy value in He7L. CAL= 5.18 →5.66 preliminary 7Be Λ 7Li (T=1) Λ 7He Λ

Now, I shall show you the Λ separation energy of 10B and 10Be. Results without CSB interaction Results with CSB interaction Λ Λ n p α α α α 10B 10Be Λ Λ Exp. BΛ=9.11±0.22 MeV Number of event (emulsion data): 3 Exp. BΛ=8.89±0.12 MeV Number of event (emulsion data): 10 At JLAB, the analysis is in progress.

Λ Λ p n α α α α 10B 10Be Without CSB interaction CAL:BΛ=8.76 MeV Exp. BΛ=9.11±0.22 MeV Number of event (emulsion data): 3 Exp. BΛ=8.89±0.12 MeV Number of event (emulsion data): 10 At JLAB, the analysis is in progress.

Λ Λ p n α α α α Where? Jlab 10B 10Be With CSB interaction which reproduce the observed binding energy of 7He Λ Λ Λ p n α α α α Where? Please measure BΛ precisely. Jlab 10B I will wait for the future observed data from JALB. 10Be Λ Λ BΛ=8.56 MeV (without CSB) BΛ=8.76 MeV(without CSB) BΛ=8.35 MeV (with CSB) BΛ= 8.97 MeV (with CSB) Exp. BΛ=8.89±0.12 MeV Number of event (emulsion data): 10 Exp. BΛ=9.11±0.22 MeV Number of event (emulsion data): 3 If the calculated results are consistent with the observed data at JLAB in the future, we can extract information on the odd state of CSB interaction.

Major goals of hypernuclear physics 1) To understand baryon-baryon interactions Fundamental and important for the study of nuclear physics 2) To study the structure of multi-strange systems To understand the baryon-baryon interaction, two-body scattering experiment is most useful. Total number of Nucleon (N) -Nucleon (N) data: 4,000 YN and YY potential models so far proposed (ex. Nijmegen, Julich, Kyoto-Niigata) have large ambiguity. ・ Total number of differential cross section Hyperon (Y) -Nucleon (N) data: 40 ・ NO YY scattering data

Hypernuclear physics Λ Λ Λ particle can reach deep inside, and attract the surrounding nucleons towards the interior of the nucleus. There is no Pauli Pricliple between N and Λ. Nucleus How about hypernuclear physics? Let’s add one hyperon such as Lambda particle to this nucleus. Since there is no Pauli Priciple between N and Lambda, then Lambda particle can reeach deep inside, and attract the surrounding nucleons towards the interior of the nucleus. As a results,… We call it glue-like role of Lambda particle. Hypernucleus Λ particle plays a ‘glue like role’ to produce a dynamical contraction of the core nucleus. How do we observe nuclear shrinkage effect by experiment?

Rα-np(6Li) > Rα-np(7Li) Reduced by about 19 % Theoretical calculation by Hiyama et al. B(E2: 5/2+ → 1/2+) =2.85 e2fm4 reduced by 22% KEK-E419 Λ n Λ n α α 7Li Rα-np Λ p p 6Li Rα-np(6Li) > Rα-np(7Li) Reduced by about 19 % Λ B(E2: 3+→1+:6Li)=9.3 ±0.5e2fm4 →B(E2:5/2+→1/2+:7Li)= 3.6 ±2.1 e2fm4 Λ The shrinkage effect on the nuclear size included by the Λ particle was confirmed for the first time.

Are all nuclei compressed by the injection of a Λ? or not? Ground states of stable nuclei with A ≥ 11 :Not compressed Some excited states of stable nuclei with A ≥ 11 :shrunk by as much as 30 % E. Hiyama et al., Prog. Theor. Phys. 97, 881 (1997). α α 13C Λ α Λ

Example :13C Λ Λ α α α 12C Λ +0.86 Loosely coupled α clustering state 0+2 0 MeV 3α threshold Λ 0+1 Shell-like compact state -7.27 How is the structure change when a Λ particle is injected into 2 kinds of 0+ states in 12C ?

α α α O O C This difference comes from the state dependence of nucleon The density of α―α relative motion as a function of α―α distance. α excitate-state C α α C O Drastic shrinkage ground-state O C C C No change This difference comes from the state dependence of nucleon density distribution in core nucleus.

12C 13C B(E2):Reduced B(E2) B(E2):Enhanced B(E2):No change 2+2 cluster-like states 0+2 3αthreshold 2+2 2+1 B(E2):Reduced shell-like states B(E2) 0+2 0+1 This is energy level of C12 and this is that of C13L. These two are cluster-like levels and shell-like level. If Lambda particle is added to these levels, due to shrinkage effect this BE2 value should be reduced. On the other hand, B(E2) between shell-like states should be almost the same as the B(E2) in core nucleus due to no shrinkage. B(E2):Enhanced 12C 2+1 α B(E2):No change 0+1 α 13C α α α Λ α Λ

In this way, by the measurement of B(E2) in the hypernuclei, we can recognize which states are cluster-like states and which states are shell-like states.

By the measurement of level energy and determination of the spin parity of the excited states of A ≥ 11 hypernuclei, we can give an important contribution to the study of light nuclei with A ≥ 10.

Schematic illustration shell-like states α-clustering states Does energy gain go in parallel way for all the states? No ! 2+2 0+2 2+1 2+2 0+2 01+ 2+1 A≥10 core nucleus 01+ Λ A≥11 Λ hypernucleus

Energy gain by Λ-particle addition ΔE(shell-like) > ΔE(clustering) state clustering state shell-like state Level crossing A≥10 core nucleus A≥11 Λ hypernucleus

For example of level crossing : 12C and 13C Λ α α α α α α Λ 13C 12C Λ

Level crossing between shell-like state and clustering state shell-like states ← clustering states shell-like states B =7.83MeV Level crossing B (EXP,CAL)=11.69MeV

12C

By measurement of energy gain by Λ particle addition, we can recognize which states are cluster-like states and which states are shell-like states.

Concluding remark GSI JLAB J-PARC J-PARC Multi-strangeness system such as Neutron star GSI JLAB J-PARC J-PARC

Thank you!