1. Structure of light hypernuclei
E. Hiyama (RIKEN)

2. Nuclear chart with strangeness
Multi-strangeness system
such as Neutron star Λ Λ
What is interesting to study hypernuclei?
This is three dimensional nuclear chart. This is nonstrangeness sector. Many participants in this symposium are interested in this sector. When a hyperon such as a Lambda particle is added to this sector, we have S=-1 sector such as
Single Lambda and singel Sigma hypernuclei. One can add one more Lambda to this sector, then we have S=-2 sector.
Furthermore, we can add many Lambdas, then the extreme limit to have many Lambda particle is core of neutron star.
Hypernuclear physists are quite interested in these strangeness sector.
What is purpose to study these sectors in hypernuclear physics?

3. n n 6 H t 3n Outline (1) Introduction (2) 7He Λ Λ n n n n Λ ４ 4He Λ Λ

4. Sec.1 Introduction
Let’s begin.

5. Major goals of hypernuclear physics
1) To understand baryon-baryon interactions
2) To study the structure of multi-strangeness systems
In order to understand the baryon-baryon interaction, two-body scattering
experiment is most useful.
Total number of
Nucleon (N) -Nucleon (N) data: 4,000
Study of NN intereaction
has been developed.
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
since it is difficult to perform YN scattering experiment even at J-PARC.

6. 2) To study the structure of multi-strangeness systems
Hypernucleus =
Many-body system of
neutrons,
protons,
hyperons nn n p p n Y n n n p p Y
Once the Hamiltonian is determined,
it is possible , using few-body calculation method (Gaussian
Expansion Method), to precisely calculate
the structure of many-body systems consisting of
neutrons, protons and hyperons.
As a result, we can predict new phenomena
such as we have never imagined before.

7. No Pauli principle
Between N and Λ
Λ particle can reach deep inside, and attract the surrounding nucleons towards the interior
of the nucleus. N Λ
Due to the attraction of
Λ N interaction, the
resultant hypernucleus will
become more stable against
the neutron decay.
Hypernucleus Λ
neutron decay threshold γ
nucleus
hypernucleus

8. Nuclear chart with strangeness
Multi-strangeness system such as Neutron star
Extending drip-line! Λ
Interesting phenomena concerning
the neutron halo have　been
observed near the neutron drip line
of light nuclei.
Outline
How is structure change when a Λ　particle is injected
into neutron-rich nuclei ? 8

9. Question : How is structure change when a Λ particle
is injected into neutron-rich nuclei? n n n n
6 H
7He Λ t Λ Λ α Λ
Observed at JLAB, Phys. Rev. Lett. 110, (2013).
Observed by FINUDA group,
Phys. Rev. Lett. 108, (2012). n n Λ
C. Rappold et al., HypHI collaboration
Phys. Rev. C 88, (R) (2013) 3n Λ

10. ・A variational method using Gaussian basis functions
In order to solve few-body problem accurately,
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,
4He-atom tetramer

11. 3n three-body calculation of Section 2 E. Hiyama, S. Ohnishi,Λ
E. Hiyama, S. Ohnishi,
B.F. Gibson, and T. A. Rijken,
PRC89, (R) (2014). n n 　Λ

12. What is interesting to study nnΛ system?　Λ
S=0
The lightest nucleus to have a bound state is deuteron.
n+p threshold n p
J=1+ d
-2.22 MeV
Exp.
S=-1 (Λ　hypernucear sector)
n+p+Λ
Lightest hypernucleus to have a bound state n p
d+Λ
3H (hyper-triton)
0.13 MeV Λ
J=1/2+ 　Λ
Exp.

13. Observation of nnΛ system (2013)
-23.7fm
nnΛ　breakup threshold n n ?
They did not report the
binding energy. 　Λ
scattering length:-2.68fm
Observation of nnΛ system (2013)
Lightest hypernucleus to have a bound state
Any two-body systems are unbound.=>nnΛ system is bound.
Lightest Borromean system.

14. Theoretical important issue: Do we have bound state for nnΛ system?
If we have a bound state for this system, how much is
binding energy?
nnΛ　breakup threshold n n ?
They did not report the
binding energy. 　Λ
NN interaction : to reproduce the observed binding energies
of 3H and 3He
NN: AV8 potential
We do not include 3-body force for nuclear sector.
How about YN interaction?

15. Non-strangeness nucleiΔ
Nucleon can be converted into Δ.
However, since mass difference between
nucleon and Δ is large, then probability
of Δ in nucleus is not large. N N
On the other hand, the mass difference
between Λ and Σ is much smaller, then
Λ　can be converted into Σ particle
easily. Δ
In non-strangeness nuclei, nucleon can be converted into Delta. However, since mass difference between nucleon and Delta is large, then, probability of Delta in nucleus is not large. On the other hand, the mass difference between Lambda and Sigma is smaller, then, there is significant probability of Sigma in Lambda hypernuclei.
300MeV N Λ Σ N
80MeV Σ Λ N Λ

16. To take into account of Λ particle to be converted into
Σ particle, we should perform below calculation using
realistic hyperon(Y)-nucleon(N) interaction. n n n n + 　Λ Σ
YN interaction: Nijmegen soft core ‘97f potential (NSC97f)
proposed by Nijmegen group
reproduce the observed binding energies of 3H, 4H and 4He Λ Λ Λ

17. -BΛ
0 MeV
d+Λ p n 3H Λ Λ
1/2+
1/2+
-0.19 MeV
-0.13 ±0.05 MeV
Cal.
Exp.

18. What is binding energy of nnΛ?p n n
4He n Λ Λ 4H Λ
-BΛ p Λ p
4He
-BΛ Λ 4H Λ Λ
3He+Λ
0 MeV
0 MeV
3H+Λ 1+ 1+ 1+ 1+
-0.57
-0.54
-1.24
-1.00 0+ 0+ 0+
-2.39
-2.04 0+
-2.28
-2.33
Exp.
Cal.
Cal.
Exp.
What is binding energy of nnΛ?

19. Λ VT ΛN-ΣN -BΛ 0 MeV We have no bound state in nnΛ system.
1/2+
nnΛ threshold
We have no bound state in nnΛ system.
This is inconsistent with the data. n n
0 MeV 　Λ
Now, we have a question.
Do we have a possibility to have a bound state in nnΛ system tuning
strength of YN potential ?
It should be noted to maintain consistency with the binding energies
of 3H and 4H and 4He. Λ Λ Λ
VT ΛN-ΣN
X1.1, 1.2

20. How to tune YN interaction
J=1/2+ Λ
T=0, S=1 n p 　Λ
S=0 state dominates
between Λ and nucleons.
We multiply tensor part of ΛN-ΣN　potential (VTΛN-ΣN)
by factor 1.1 and 1.2.
nnΛ　　J=1/2+ n n
T=1, S=0
Both S=0 and S=1
Components
of YN interaction contribute. 　Λ
S=1
S=0

21. n n 　Λ
VT ΛN-ΣN
VT ΛN-ΣN
X1.1
X1.2
When we have a bound state in nnΛ system, what are binding energies of 3H and A=4 hypernuclei? Λ

22. Λ n Λ n p We have no possibility to have a bound state in nnΛ system.　Λ
We have no possibility
to have a bound state
in nnΛ system. n Λ n p

23. Question: If we tune 1S0 state of nn interaction,
Do we have a possibility to have a bound state in nnΛ?
In this case, the binding energies of 3H and 3He reproduce
the observed data?
Some authors pointed out to have dineutron bound state in
nn system. Ex. H. Witala and W. Gloeckle, Phys. Rev. C85,
(2012).
T=1, 1S0 state
I multiply component of 1S0 state by 1.13 and
1.35. What is the binding energies of nnΛ？ n n 　Λ

24. Λ n n unbound nn 0 MeV -0.066MeV -1.269 MeV 1S0X1.13 1S0X1.35 n n　Λ
1/2+
MeV
We do not find any possibility to have a bound state
in nnΛ.
N+N+N
3H (3He)
-7.77(-7.12)
-8.48 (-7.72)
-9.75 (-9.05)
1/2+
(-13.23)MeV
Exp.
Cal.
Cal.
1/2+
Cal.

25. 3n 3H Summary of nnΛ system:
Motivated by the reported observation of data suggesting
a bound state nnΛ, we have calculated the binding energy
of this hyperucleus taking into account ΛN-ΣN explicitly.
We did not find any possibility to have a bound state in this
system. However, the experimentally they reported
evidence for a bound state. As long as we believe the data,
we should consider additional
missing elements in the present calculation. But, I have no idea.
Unfortunately, they did not report binding energy.
It might be good idea to
perform search experiment of nnΛ system at Jlab to conclude
whether or not the system exists as bound state experimentally. n n n n 3H 3n Λ Λ n p
(e,e’K+)

26. 6H n n t Four-body calculation of Section 3 Λ
E. H, S. Ohnishi, M. Kamimura, Y. Yamamoto, NPA 908 (2013) 29.

27. The detail analysis will be reported by the next speaker.Λ Λ t

28. 5H:super heavy hydrogen
Γ=1.9±0.4 MeV
Phys. Rev. Lett. 108, (2012).
1/2+
FINUDA experiment
1.7±0.3 MeV
t+n+n+Λ
t+n+n
EXP:BΛ=4.0±1.1 MeV
4H+n+n Λ n n
0.3 MeV t n n 6H Λ Λ t
5H:super heavy hydrogen

29. Before experiment, the following authors calculated the
binding energies by shell model picture and G-matrix theory.
(1) R. H. Dalitz and R. Kevi-Setti, Nuovo Cimento 30, 489 (1963).
(2) L. Majling, Nucl. Phys. A585, 211c (1995).
(3) Y. Akaishi and T. Yamazaki, Frascati Physics Series Vol. 16 (1999).
Motivating the experimental data, I calculated the binding energy of
6H and I shall show you my result. Λ

30. 6H 6H 6H Before doing full 4-body calculation,
it is important and necessary to reproduce the observed binding energies of all the sets of subsystems in 6H. Λ
Namely, All the potential parameters are needed to
adjust in the 2- and 3-body subsystems. 6H 6H Λ Λ 6H n n Λ n n n n Λ Λ Λ t t t
Among the subsystems, it is extremely important to
adjust the energy of 5H core nucleus.

31. To calculate the binding energy of 6H, it is very important
Framework:
To calculate the binding energy of 6H, it is very important
to reproduce the binding energy of the core nucleus 5H. Λ
transfer reaction p(6He, 2He)5H
A. Korcheninnikov, et al. Phys. Rev. Lett.
87 (2001)
Γ=1.9±0.4 MeV
1/2+
1.7±0.3 MeV
To reproduce the data, for example,
R. De Diego et al, Nucl. Phys. A786 (2007), 71.
calculated the energy and width of 5H with t+n+n
three-body model using complex scaling method.
The calculated binding energy for the ground state
of 5H is 1.6 MeV with respect to t+n+n threshold and
width has 1.5 MeV.
t+n+n

32. On the contrary, if we tune the potentials to have a bound state
Exp: 1.7 ±0.3 MeV
Even if the potential parameters were tuned
so as to reproduce the lowest value of the
Exp. , E=1.4 MeV, Γ=1.5 MeV,
we do not obtain any bound state of 6H.
Γ=1.9 ±0.4 MeV Λ
Γ= 2.44 MeV ½+
Γ= 0.91 MeV
1.69 MeV
1.17 MeV
0 MeV
Γ=0.23 MeV
t+n+n+Λ
t+n+n
E=-0.87 MeV 0+
4H+n+n Λ
-　2.0
4H+n+n 0+
-2.07 MeV Λ
On the contrary, if we tune the
potentials to have a bound state
in 6H, then what is the
energy and width of 5H? Λ

33. n n 5 H n t 6 H n t 5H:super heavy hydrogen FINUDA experiment t+n+n+Λ
Γ= 1.9±0.4 MeV
Phys. Rev. Lett. 108, (2012).
1/2+
FINUDA experiment
1.7±0.3 MeV
t+n+n+Λ
t+n+n
5 H
EXP:BΛ=4.0±1.1 MeV
4H+n+n Λ
t+n+n+Λ n n
0.3 MeV t
6 H Λ n n
5H:super heavy hydrogen t Λ
But, FINUDA group provided the bound state of 6H. Λ

34. How should I understand the inconsistency between our results
and the observed data?
We need more precise data of 5H.
A. Korcheninnikov, et al. Phys. Rev. Lett.
87 (2001)
Γ=1.9±0.4 MeV
To get bound state of 6H, the energy should
be lower than the present data.
It is planned to measure the energy and width
of 5H more precisely　at RCNP . Λ
1/2+
1.7±0.3 MeV
t+n+n

35. [3] A.A. Korosheninnikov et al., PRL87 (2001) 092501
We cited this experiment.
However, you have many
different decay widths.
Width is strongly related to
the size of wavefunction.
Then, I hope that
The decay width will be
determined at RCNP this year.
[3] A.A. Korosheninnikov et al., PRL87 (2001)
[8] S.I. Sidorchuk et al., NPA719 (2003) 13
[4] M.S. Golovkov et al. PRC 72 (2005)
[5] G. M. Ter-Akopian et al., Eur. Phys. J A25 (2005) 315.

36. 6 H t+n+n+Λ No peak?! 4H+n+n 0.3 MeV FINUDA data
Theoretically, we might understand by the following reason.
If the state is resonant state, the reaction cross section would be much smaller than
that we expect. => I should calculate reaction cross section 6Li (π,K) 6H. Λ

37. 7He Λ n n α Λ
7He Λ 37 37 37

38. n n n n 6He α 7He α 6He : One of the lightest n-rich nucleiΛ n n
7He: One of the lightest
n-rich hypernuclei Λ
7He Λ α Λ
Observed at JLAB,
Phys. Rev. Lett. 110, (2013).

39. 6He 7He Λ -4.57 α+Λ+n+n 5He+n+n 5/2+ 3/2+ 1/2+
CAL: E. Hiyama et al., PRC 53, 2075 (1996), PRC 80, (2009)
6He
7He Λ 2+
α+Λ+n+n
0 MeV
0 MeV
α+n+n
One of the excited
state was observed at Jlab.
5He+n+n 0+ Λ Λ
Exp:-0.98
-1.03 MeV
5/2+
Neutron
halo states
3/2+
-4.57
BΛ: CAL= 5.36 MeV
BΛ： EXP= 5.68±0.03±0.25
1/2+
Observed at J-Lab experiment
Phys. Rev. Lett.110,
(2013).
-6.19 MeV

40. 7Li(e,e’K+)7ΛHe Third peak? (FWHM = 1.3 MeV) Fitting results
At present, due to poor statics,
It is difficult to have the third peak.
Theoretically, is it possible to
have new state?
Let’s consider it.
Third peak?
(FWHM = 1.3 MeV)
Fitting results

41. theory Exp. Core nucleus 6He (2+,1-,0+)?
Γ＝12.1 ±1.1 MeV
(2+,1-,0+)?
2.5 MeV Γ=
Γ=113 ±20 keV Γ= 2+
1.797 MeV
1.79 MeV
0 MeV
α+n+n
-0.98 0+
-0.97 MeV
6He
theory
Exp.
Myo et al., PRC 84, (2011).
Core nucleus

42. n n 7He α 7He Λ Λ -4.57 α+Λ+n+n 5He+n+n 5/2+ 3/2+ 1/2+ 0 MeV Neutron
New prediction
5/2+
-1.88 Γ=0.7 MeV
3/2+
-2.04, Γ=0.5 MeV
5He+n+n Λ
-3.12
5/2+
Neutron
halo states
3/2+
-4.57
1/2+
-6.19 MeV

43. Exp. 7He 6He Λ (2+,1-,0+)? -4.57 α+Λ+n+n 5He+n+n 5/2+ 3/2+ 1/2+
Γ＝12.1 ±1.1 MeV
7He Λ
(2+,1-,0+)?
If we find these two excited states at Jlab,
in 6He, existence of the second 2+ state
is promising. Please search the second 2+
state in 7He at Jlab. 2+ 2+
1.797 MeV Λ
Γ=113 ±20 keV
α+Λ+n+n
0 MeV
0 MeV
α+n+n
5/2+
-1.88 Γ=0.7 MeV
-0.98
3/2+
-2.04, Γ=0.5 MeV 0+
5He+n+n Λ
Neutron
halo states
-3.12
5/2+
6He
3/2+
-4.57
Exp.
1/2+
If we find these two states at Jlab, these existence
contribute to unstable nuclear physics.
-6.19 MeV

44. n n 6 H t 3n Neutron-rich Λ hypernuclei Summary 7He Λ Λ n n n n Λ ４

45. Nuclear chart with strangeness
Multi-strangeness system
such as Neutron star Λ Λ
J-PARC
Jlab
GSI
This is three dimensional nuclear chart. This is nonstrangeness sector. Many participants in this symposium are interested in this sector. When a hyperon such as a Lambda particle is added to this sector, we have S=-1 sector such as
Single Lambda and singel Sigma hypernuclei. One can add one more Lambda to this sector, then we have S=-2 sector.
Furthermore, we can add many Lambdas, then the extreme limit to have many Lambda particle is core of neutron star.
Hypernuclear physists are quite interested in these strangeness sector.
What is purpose to study these sectors in hypernuclear physics?

46. Thank you!