Few-body aspect of hypernuclear physics E. Hiyama (RIKEN)

In hypernuclear physics, there are many interesting subjects from view points of few-body problem. Then, so far, I have been studying these single Lambda hypernuclei within the framework of 3- and 4-body problem.

ΛΛ hypernuclei What is interesting and important from view points of And I also have been studying these double Lambda hypernuclei. Here, I focus on structure of double Lambda hypernuclei. I discuss about.. What is interesting and important from view points of few-body problem.

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-strangeness systems Due to the difficulty of YN and YY 2-body scattering experiment for the study of baryon-baryon interaction, the systematic investigation of the structure of light hypernuclei is essential.

Hypernuclear g-ray data since 1998 (figure by H.Tamura) ・Millener (p-shell model), 　・ Hiyama (few-body)

What is the structure when one or more Λs are added to a nucleus? + + + + ・・・・ Λ Λ Λ Λ Λ It is conjectured that extreme limit, which includes many Λs in nuclear matter, is the core of a neutron star. nucleus In this meaning, the sector of S=-2 nuclei , double Λ hypernuclei and Ξ hypernuclei is just the entrance to the multi-strangeness world. 　However, we have hardly any knowledge of the YY interaction 　because there exist no YY scattering data. Then, in order to understand the YY interaction, it is crucial to study the structure of double Λ hypernuclei and Ξ hypernuclei.

In 2001, the epoch-making data 　　has been reported by the　 　　KEK-E373 experiment. Observation of 　6He ΛΛ Uniquely 　identified 　without 　ambiguity for　the first time Λ Λ α+Λ+Λ α ６．９1±0.16 MeV 0+

Strategy of how to determine YY interaction from the study of light hypernuclear structure Nijmegen model D 6He Suggest reducing the strength of spin-independent force by half ① use ③ ΛΛ Λ Λ Accurate structure calculation α compare between the theoretical result and the experimental data of the biding energy of 6He ② prediction of new double Λ hypernuclei ④ ΛΛ Spectroscopic experiments Emulsion experiment (KEK-E373) by Nakazawa and his collaborators

Approved proposal at J-PARC KEK-E373 experiment analysis is still in progress. Approved proposal at J-PARC ・E07 “Systematic Study of double strangness systems at J-PARC” by Nakazawa and his collaborators It is difficult to determine (1) spin-parity (2) whether the observed state is the ground state or an excited state My theoretical contribution using few-body calculation comparison Emulsion experiment Theoretical calculation input: ΛΛ interaction to reproduce the observed binding energy of 　6He ΛΛ the identification of the state

10Be 10Be Successful example to determine spin-parity of double Λ hypernucleus --- Demachi-Yanagi event for 10Be ΛΛ Observation of 　10Be --- KEK-E373 experiment ΛΛ Λ Λ 8Be+Λ+Λ α α 11.90±0.13 MeV 10Be ground state ? excited state ? ΛΛ 10Be ΛΛ Demachi-Yanagi event

Successful interpretation of spin-parity of Λ Λ α α E. Hiyama, M. Kamimura,T.Motoba, T. Yamada and Y. Yamamoto Phys. Rev. 66 (2002) , 024007 11.83 11.90 α+Λ+Λ 6.91 ±0.16 MeV Λ Λ Demachi-Yanagi event -14.70 α

Hoping to observe new　double Λ hypernuclei in future experiments, I predicted level structures of these　double Λ hypernuclei within the framework of the α+x+Λ+Λ 4-body model. E. Hiyama, M. Kamimura, T. Motoba, T.Yamada and Y. Yamamoto Phys. Rev. C66, 024007 (2002) Λ Λ 3He x = t n p d = = = = = α x 7He 7Li 8Li 8Li 9Be ΛΛ ΛΛ ΛΛ ΛΛ ΛΛ

Spectroscopy of ΛΛ-hypernuclei E. Hiyama, M. Kamimura,T.Motoba, T. Yamada and Y. Yamamoto Phys. Rev. 66 (2002) , 024007 A 11 ΛΛ hypernuclei > new data (2009) I have been looking forward to having new data in this mass-number region.

Observation of Hida event KEK-E373 experiment Λ Λ Λ Λ n n n α α α α 11Be From the 12Be ΛΛ ΛΛ BΛΛ= 20.49±1.15 MeV BΛΛ= 22.06±1.15 MeV Important issue: Is the Hida event the observation of a ground state or an excited state? It is neccesary to perform 5-body calculation of this system. Why 5-body?

Core nucleus, 9Be is well described as α+α+ n three-cluster model. ΛΛ Λ Λ Then, 11Be is considered to be suited for studying with α+α+ n +Λ+Λ 5-body model. ΛΛ n α α Difficult 5-body calculation: 1) 3 kinds of particles (α, Λ, n) 2) 5 different kinds of interactions Λ Λ Λ n 3) Pauli principle between α and α, and between α and n α Λ α But, I have succeeded in performing this calculation. n α α

5-body calculation of 11Be ΛΛ 11Be ΛΛ Λ Λ n α α (γ～10000 MeV is sufficient.) rules out the Pauli-forbidden states from the 5-body wave unction. The Pauli-forbidden states (f ) are the 0S, 1S and 0D states of the α α relative motion, and the 0S states of the α n relative motion. This method for the Pauli principle is often employed in the study of light nuclei using microscopic cluster models.

5-body calculation of 11Be ΛΛ ΛΛ Λ Λ n α α A variational method: Gaussian Expansion Method (GEM) (review paper) E. H., Y. Kino and M. Kamimura, Prog. Part. Nucl. Phys., 51 (2003) 223. expansion coefficient specifies 5-body basis functions of each Jacobi-coordinate set specifies many sets of Jacobi coordinates

An example of 5-body basis function: 5-body spatial function spin function (a Jacobi coordinate set) spin specify radial dependence (shown below) specify angular momenta

Form of each basis function 5-body spatial function Gaussian for radial part : geometric progression for Gaussian ranges : Similarly for the other basis : Use of this type gaussian basis is known to be very suitable for describing simultaneously both the short-range correlations and long-range tail behaviour of few-body systems; This is precisely shown in Gaussian Expansion Method (GEM) (review paper) E. H., Y. Kino and M. Kamimura, Prog. Part. Nucl. Phys., 51 (2003) 223.

Some of important Jacobi corrdinates of the α+ α+ n + Λ+ Λ system. Two αparticles are symmetrized. Two Λparticles are antisymmetrized. 120 sets of Jacobi corrdinates are employed.

5He (3/2-) 8Be (0+) 9Be (3/2-) Before doing full 5-body calculation, it is important and necessary to reproduce the observed binding energies of all the sets of subsystems in 11Be. In our calculation, this was successfully done using the same interactions for all subsystems: ΛΛ Λ Λ Λ Λ Λ Λ n n n α α α α α α 8Be (0+) 9Be (3/2-) 5He (3/2-) CAL : +0.80 MeV EXP : +0.80 MeV CAL : +0.09 MeV EXP : +0.09 MeV CAL : -1.57 MeV EXP : -1.57 MeV

10Be (0+, 2+ ) 6He (0+ ) 10Be (1-) adjusted predicted n Λ n Λ Λ Λ Λ Λ α α α α α α ΛΛ ΛΛ ΛΛ 10Be (0+, 2+ ) 6He (0+ ) 10Be (1-) Λ Λ Λ Λ Λ CAL (0+): -6.93 MeV EXP (0+): -6.93 MeV All the potential parameters have been fixed in the 2- and 3-body subsystems. Therefore, let me emphasize that the energies of these 4-body susbsystems and the 5-body system are predicted with no adjustable parameters. CAL (2+): -10.96 MeV EXP (2+): -10.98 MeV CAL : -10.64 MeV EXP : -10.64 MeV CAL (0+): -14.74 MeV EXP (0+): -14.69 MeV All the potential parameters have been adjusted in the 2- and 3-body subsystems. Therefore, energies of these 4-body susbsystems and the 5-body system are predicted with no adjustable parameters. 11Be Λ Λ

Convergence of the ground-state energy of the α+α+ n +Λ+Λ 5-body system ( ) 11Be ΛΛ J=3/2-

Phys. Rev. Lett. 104,212502 (2010)

For this study, it is necessary to calculate 6-body problem. As mentioned before, Hida event has another possibility, namely, observation of 12Be. ΛΛ Λ Λ BΛΛ= 22.06±1.15 MeV n n 12Be α α ΛΛ For this study, it is necessary to calculate 6-body problem. At present, it is difficult for me to perform 6-body calculation. However, I think, it is good chance to develop my method for 6-body problem. I started to make program for this calculation this month. If you are interested in this calculation, let’s work together with me! For the confirmation　of Hida event, we expect to have more precise data at J-PARC.

Spectroscopy of ΛΛ-hypernuclei At J-PARC 11Be , A=12, 13, …… ΛΛ For the study of this mass region, we need to perform more 5-body cluster-model calculations.

Therefore, we intend to calculate the following 5-body systems. Λ Λ Λ Λ Λ Λ Λ Λ p t 3He d α α α α α α α α 11B 13B 13C 12B ΛΛ ΛΛ ΛΛ ΛΛ Λ Λ To study 5-body structure of these hypernuclei is interesting and important as few-body problem. α α α 14C ΛΛ

Ξhypernuclei

For the study of ΞN interaction, it is important to study Ξ- core nucleus For the study of ΞN interaction, it is important to study the structure of Ξ hypernuclei. However, so far there is no observed Ξ hypernucleus. Then, it is important to predict theoretically what kinds of Ξ hypernuclei will exist as bound states.

t α Ξ- Ξ- Ξ- α α α α α Ξ- (9Be+Ξ-) (6He+Ξ-) p n n p n p Ξ- Ξ- (d+Ξ-) For this purpose, recently, I studied these Ξ hypernuclei. p n n p n p α Ξ- Ξ- Ξ- (５Li+Ξ-) (d+Ξ-) (t +Ξ-)- ( Ξ- t n n - Ξ- n α α α α α Ξ- (9Be+Ξ-) (6He+Ξ-) (11B+Ξ-) (ESC04) ΞN interaction to reproduce the experimental data of 12C(K-,K+) reaction 　　・T. Fukuda et al. Phys. Rev. C58, 1306, (1998); 　　 ・P.Khaustov et al. Phys. Rev. C61, 054603 (2000).

α α Spectroscopy of Ξ hypernuclei at J-PARC t α α α α n n p n n n n n Ξ- n Ξ- t n p n n n n n α p Ξ- Ξ- α α α Ξ- Ξ- α α MeV (pn+Ξ ) (pnn+Ξ ) (5He+Ξ) (6He+Ξ) ( ９B e+Ξ) (11B+Ξ) -2 d+Ξ 1/2 + αn+Ξ -5 (α+Ξ)+nn 1- 1/2+ t +Ξ (8Be+Ξ)+n -10 2- 11B+Ξ 1+ 0+ 2- 1- -20 1- 2-

p K+ K- Ξ- 11B 11B 12C Ξ hypernucleus Approved proposal at J-PARC ・E05 “Spectroscopic study of Ξ-Hypernucleus, 12Be, via the 12C(K-,K+) Reaction” by Nagae　and his collaborators Day-1 experiment K+ K- p Ξ- 11B 11B 12C Ξ hypernucleus First observation of Ξ hypernucleus This observation will give information about ΞN　interaction.

α α Spectroscopy of Ξ hypernuclei at J-PARC t α α α α n n p n n n n n Ξ- n Ξ- t n p n n n n n α p Ξ- Ξ- α α α Ξ- Ξ- α α MeV (pn +Ξ) (pnn+Ξ) (5He+Ξ) (6He+Ξ) ( ９Be+Ξ) (11B+Ξ ) -2 d+Ξ 1/2 + αn+Ξ -5 (α+Ξ)+nn 1- 1/2+ t +Ξ (8Be+Ξ) +n -10 2- 11B+ Ξ 1+ 0+ 2- 1- -20 1- 2-

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

Thank you!