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

2. 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.

3. 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.

4. Introduction

5. 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.

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

7. 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.

8. 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+

9. 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

10. 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

11. 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

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

13. 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) Λ Λ
3He x = t n p d = = = = = α x
7He
7Li
8Li
8Li
9Be ΛΛ ΛΛ ΛΛ ΛΛ ΛΛ

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

15. 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?

16. 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

17. 5-body calculation of 11BeΛΛ
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.

18. 5-body calculation of 11BeΛΛ 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

19. 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

20. 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.

21. 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.

22. 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 : MeV
EXP : MeV
CAL : MeV
EXP : MeV
CAL : MeV
EXP : MeV

23. 10Be (0+, 2+ ) 6He (0+ ) 10Be (1-) adjusted predicted n Λ n Λ Λ Λ Λ Λα α α α α α ΛΛ ΛΛ ΛΛ
10Be (0+, 2+ )
6He (0+ )
10Be (1-) Λ
Λ Λ
Λ Λ
CAL (0+): MeV
EXP (0+): 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+): MeV
EXP (2+): MeV
CAL : MeV
EXP : MeV
CAL (0+): MeV
EXP (0+): 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

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

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

26. 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.

27. 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.

28. 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

29. Ξhypernuclei

30. 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.

31. 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, (2000).

32. α α Spectroscopy of Ξ hypernuclei at J-PARC t α α α α n n p n n n n nΞ- 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-

33. 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.

34. α α Spectroscopy of Ξ hypernuclei at J-PARC t α α α α n n p n n n n nΞ- 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-

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

36. Thank you!