Five-body Cluster Structure of the double Λ hypernucleus 11 Be Emiko Hiyama (RIKEN) ΛΛ

・ Four-body structure of 7 He, 7 Li, 8 Li, 9 Li, 9 Be, 10 Be ΛΛ ・ Five-body structure of 11 Be ΛΛ α ｎ Λ Λ α p Λ Λ α d Λ Λ α t Λ Λ α 3He Λ Λ αα ΛΛ n Outline of my talk ・ Introduction

Introduction

Λ + ・・・・ It is conjectured that extreme limit, which includes many Λs in nuclear matter, is the core of a neutron star. 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. What is the structure when one or more Λs are added to a nucleus? nucleus ΛΛΛ Λ +++

Uniquely identified without ambiguity for the first time α+Λ+Λ 0+0+ In 2001, the epoch-making data has been reported by the KEK-E373 experiment. Observation of 6 He ΛΛ α ΛΛ ６．９ 1±0.16 MeV

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

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

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

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

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

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

α x Λ Λ x = n pd t 3 He = = = = = 7 He 7 Li 8 Li ΛΛ 8 Li 9 Be ΛΛ 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, (2002)

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

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

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

α ΛΛ n 11 Be ΛΛ α 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. (γ ～ MeV is sufficient.) 5-body calculation of 11 Be ΛΛ

5-body calculation of 11 Be (Hida event) ΛΛ V ΛΛ : the same one used in reproducing the observed binding energy of 6 He V Λα :the same as the one used in (obtained by folding Nijgemen soft core ’97f into the α-cluster density; local plus non-local potentials) α ΛΛ n 11 Be ΛΛ α The core nucleus 9 Be has been extensively studied with the α+ α+ n microscopic 3-body cluster model. V Λn :Nijgemen soft core ’97f ΛΛ Λn α Λ ΛΛ （ V αα, V nα ) 6 He ΛΛ

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

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

Form of each basis function 5-body spatial function Gaussian Expansion Method (GEM) (review paper) E. H., Y. Kino and M. Kamimura, Prog. Part. Nucl. Phys., 51 (2003) 223. 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

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

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

α α Λ Λ n CAL : MeV EXP : MeV 5 He (1/2 - ) 6 He ( 1 - ) Λ CAL : MeV EXP : MeV α α Λ Λ n Λ α α Λ Λ n 9 Be (1/2 + ) Λ CAL : MeV EXP : MeV (The energy is measured from the full-breakup threshold of each subsystem)

ΛΛ αα ΛΛ n 10 Be (0 +, 2 + ) Λ CAL (2 + ): MeV EXP (2 + ): MeV CAL (0 + ): MeV EXP (0 + ): MeV ΛΛ α α ΛΛ n 6 He (0 + ) Λ CAL (0 + ): MeV EXP (0 + ): MeV ΛΛ α α Λ Λ n 10 Be (1 - ) CAL : MeV EXP : MeV Λ 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. 11 Be Λ adjustedpredicted

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

To be published in Phys. Rev. Lett.

What is structure of 11 Be ? ΛΛ Λ Hypernucleus Λ particle can reach deep inside, and attract the surrounding nucleons towards the interior of the nucleus. No Pauli principle Between N and Λ Λ particle plays a ‘glue like role’ to produce a dynamical contraction of the core nucleus.

By reduction of B(E2) due to the addition of Λ particle to the core nucleus, we can find the contraction of nucleus by glue-like role of Λ particle. Theoretical calculation E. Hiyama et al. Phys. Rev. C59 (1999), KEK-E419 α n p R α-np 6 Li α n p Λ Λ 7 Li Λ R α-np ( 6 Li) > R α-np ( 7 Li) Reduced by about 20 % B(E2: 3 + →1 + : 6 Li)=9.3 ±0.5e 2 fm 4 →B(E2:5/2 + →1/2 + : 7 Li)= 3.6 ±2.1 e 2 fm 4

20% reduction 8% reduction α n α α Λ Λ n α α ΛΛ n α 9 Be 10 Be Λ 11 Be ΛΛ Λ

α α 11 Be ΛΛ ΛΛ n

How large is the ΛΛ interaction?

Estimation of strength of ΛΛ interaction ⊿ B ΛΛ ≡ B ΛΛ - ２ B Λ ２ Λ separation energy １ Λ separation energy α＋Λ＋Λα＋Λ＋Λ 6.91±0.16 MeV 0+0+ α Λ Λ 6 He ΛΛ ⊿ B ΛΛ ( 6 He) ≡ B ΛΛ ( 6 He)-2B Λ ( 5 He) ΛΛ Λ 6.91 MeV 3.12 MeV =0.67 MeV ΛΛ bond energy BΛBΛ BΛBΛ Calculated ΔB ΛΛ ( 11 Be)=0.29 MeV: smaller than 6 He ΛΛ Why so small?

答え： mass-number-dependence は、 A=7 ～１０のダブル Λ ハイパー 核の ΛN 相互作用のスピンスピン部分から来る。 ΔB ΛΛ （ A Z ）＝ B ΛΛ （ A Z ）－２ B Λ （ A-1 Z) ΛΛΛ α x S=0 σΛ・σxσΛ・σx σΛ・σxσΛ・σx cancel x Λ α σx・σΛσx・σΛ No conribution of σ x ・ σ Λ There is contribution of σ x ・ σ Λ 6 Li+Λ 3/2 + 1/2 + 7 Li Λ 0.69 MeV M. Danysz et al., Nucl. Phys. 49, 121 (1963) “This definition is of simple meaning only When the nuclear core is spinless.

Core nucleus J0J0 BΛBΛ BΛBΛ J2J2 J1J1 Spin-averaged Λ ハイパー核 B Λ =J 0 /(2J 0 +1)B Λ (J 1 =J 0 -1/2) +(J 0 +1)/(2J 0 +1)B Λ (J 2 =J 0 +1/2) ΔB ΛΛ （ A Z ）＝ B ΛΛ （ A Z ）－２ B Λ （ A-1 Z) ΛΛΛ ΔB ΛΛ （ A Z ）＝ B ΛΛ （ A Z ）－２ B Λ （ A-1 Z) ΛΛΛ

ΔB ΛΛ include rearrangement effects in nuclear cores due to participation of Λ particle. We can see shrinkage effect of 9 Be due to the addition of two Λ particles. Then, we cannot use ΔB ΛΛ to extract information on ΛΛ interaction. α n α α Λ Λ n α α Λ Λ n α 9 Be 10 Be Λ 11 Be ΛΛ Λ

V bond ( A Z) ≡B ΛΛ （ A Z ）－ B ΛΛ （ A Z;V ΛΛ =0) ΛΛ V ΛΛ Then, we propose the following equation to extract information on ΛΛ interaction. The calculated ΛΛbond energies are reasonable.

As mentioned before, Hida event has another possibility, namely, observation of 12 Be. ΛΛ αα Λ Λ nn 12 Be ΛΛ B ΛΛ = 22.06±1.15 MeV 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 methodfor 6-body problem. Fortunately, we will have much more powerful supercomputer (HITACHI SR16000) at KEK in June in This supercomputer enable me to make six-body calculation. For the confirmation of Hida event, we expect to have more precise data at J-PARC. ΛΛ

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

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

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

Thank you!