1. Structure of 10Be and 10B hypernuclei studied with four-body cluster modelΛ Λ
E. Hiyama (RIKEN)
Submitted in PRC last August and waiting for referee’s comment
Today, I will give a talk about structure of these A=10 hypernuclei. I submitted this work to PRC last month and I am waiting for referee comments.

2. Hypernuclear g-ray data since 1998 (figure by H.Tamura)
This is hypernuclear gamma-ray data sicne So far, these gamma-ray data have been obtained by Tamura san and Kishimoto san. And by combining these experimental data and theoretical calculations by shell-model and few-body technique, we could succeed in extracting spin-spin, spin-orbit, and tensor terms in Lambda N interaction. Today, among these data, let’s focus on this B10L.
・Millener (p-shell model), 　・ Hiyama (few-body)

3. Four-body structure of these hypernuclei is important for
the study of ΛN spin-dependent force and 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.

4. To obtain information about ΛN spin-dependent force Λ p
10B(K-,π-) 10B α α Λ
BNL-E930
10B Λ
3/2-
α+α+p+Λ
In the case of B10L, this hypernuclei is important to obtain information about Lambda N spin-dependent force. So far, in BNL-E930, high-resolution gamma-ray experiment was performed to measure the splitting of this doublet state. However, they did not observe gamma-ray transition between the ground state doublet. This suggests as follows: this energy splitting is less than 100 keV or 2- state is ground state.
α+α+p p 2-
ΔE<100keV?
or 2- is ground state? α α γ 9B 1-
We could not obtain observed γ-ray.
Important issue: which part is supported theoretically?

5. To obtain information about ΛN spin-spin force Λ p
D.J. Millener
tried to calculate spin-doublet
splitting by shell model.
The calculated result was
more than 100 keV. α α
BNL-E930
10B Λ
3/2-
α+α+p+Λ
In order to explain this data, Millener..
α+α+p p 2-
ΔE<100keV?
or 2- is ground state? α α γ 9B 1-
We could not obtain observed γ-ray.
What value in my calculation?

6. ΛN interaction: Nijmegen ’97fp
ΛN interaction: Nijmegen ’97f α α
Not original one but simulated one
The ΛN-ΣN coupling interaction can be
renomalized into the ΛN-ΛN interaction
effectively.
10B Λ
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

7. Λ p α α 10B J=2- =>S=1,0 J=1- =>S=0 40 keV ΔE<100keV?
This is our calculational results, This is the ground state of core nucleus, 9Be. And this is the calculated ground-state doublet of B10L.These are results using even- and odd-state spin-spin Lambda N force. When we use spin-spin force, splitting energy is about 0.15 MeV, which is similar for the calculation by Millener. And when we include spin-orbit force, the energy splitting was reduced and energy is 40 keV. This result support this part among these two suggestions. But this is theoretical calculation.
10B Λ
-8.34
40 keV
ΔE<100keV?
or 2- is ground state?

8. Λ p α α 10B BNL-E930 2- γ 1- Experimentally, we need to conclude
Which is final conclusion?, namely
ΔE<100keV?
or 2- is ground state?
For this study, it might be good way to
observe weak decay from 10B and to measure
πangular distribution to determine spin and parity. Λ p α α Λ
10B Λ
3/2-
α+α+p+Λ
α+α+p
BNL-E930 2-
ΔE<100keV?
or 2- is ground state? γ 1-

9. Λ p α α 10B BNL-E930 2- γ 1- Problem: if the ground state is 1- state,
We have weak-decay both from 1- and 2- state.
Because splitting energy is less than 100 keV,
Transition probability T(M1,2-1-) is small.
Then, the weak decay occurs from 2-.
So, it might be difficult to identify which is the
Ground state by observing weak decay. Λ p α α
10B Λ
3/2-
α+α+p+Λ
α+α+p
BNL-E930 2-
ΔE<100keV?
or 2- is ground state? γ M1 1-

10. Λ n α α 10Be 2- γ 1- Then, in order to solve this issue,
I propose to measure weak decay from
mirror hypernucleus, 10Be. Λ Λ n α α
α+α+n+Λ
10Be Λ 2-
ΔE<100keV?
or 2- is ground state? γ M1 1-

11. Decay Pion Spectroscopy for Light and Exotic -Hypernuclei
Fragmentation Process p e’ e
12C 
12B* K+
Example:
Access to variety of light and exotic hypernuclei, some of which cannot be produced or measured precisely by other means
At JLab and Mainz, they are planning to perform decay pion spectroscopy for light Lambda hypernuclei using electron beam. We will get only ground state from this experiment. Then, I think that we can get spin and parity of Be10L from this experiment. If the spin and parity of Be10L is determined, then we can determine spin and parity of mirror hypernucleus 10BL. So, I hope that this experiment will be done in the future.  
4H
Fragmentation (<10-16s)
4He
 -
Weak mesonic two body decay (~10-10s)

12. Decay pion spectroscopy at Mainz Λ nα α
10Be Λ
I think it is possible to get spin and parity of 10Be from
decay pion spectroscopy.
If the spin and parity of 10Be is determined,
then we can determine spin and parity of mirror hypernucleus
10B. Because spin-structure of 10Be is the same as 10B.
So, I hope that this experiment will be done in the future. Λ Λ Λ Λ Λ

13. Furthermore, four-body structure of these hypernuclei
is important for the study of CSB interaction. 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.

14. Charge Symmetry breaking
Exp.
0 MeV
3He+Λ
0 MeV
3H+Λ
-1.00
-1.24 1+ 1+
0.24 MeV
In S=-1 sector, there is experimental data of 4HeL and H4L. This is experimental data. Lambda separation energy is measured with respect to 3He plus Lambda and H plus Lambda. The energy difference between the ground states of these hypernuclei is 0.35 MeV. Also, the energy difference between the excited state is 0.24 MeV. Historically it is considered that these energy difference come from the charge symmetry breaking effect between nLambda and pLambda interactions.
-2.04
-2.39 0+
0.35 MeV 0+ Λ p n n p n Λ p Λ p Λ n 4H
4He Λ Λ

15. in p-shell Λ hypernuclei as well as s-shell Λ hypernuclei.
Furthermore, 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 and 10 Λ hypernuclei is suited.
Because, core nuclei with A=6 are iso-triplet states and core nuclei with A=9
are iso-doublet states. n p n n p p α α α
6He
6Li(T=1)
6Be
9Be 9B n p α α α α

16. Λ Λ n p α α α α 10Be 10B Λ Λ Λ 7He 7Be 7Li(T=1)
Then, A=7　Λ hypernuclei are also iso-triplet states.
And A=10 Λ
hypernuclei are
iso-doublet states. Λ Λ n p α α α α
10Be
10B Λ Λ
It is possible that CSB interaction between Λ and valence nucleons
contribute to the Λ-binding energies in these hypernuclei.

17. 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:E experiment
Preliminary data: 5.68±0.03±0.22
-3.79
7Be Λ
7Li
(T=1) Λ
7He Λ

18. Exp. Λ Λ p n α α α α 10B 10Be 9Be+Λ 0 MeV 0 MeV 9B+Λ
Recently, 10B(e,e’K+) 10Be has been done
at JLab. Analysis is in progress. Λ
Exp: ± 0.12 MeV
These are experimental data of A=10 hypernuclei. The energies are measured from these threshold. Energy difference is 0.22 MeV. However, in these hypernuclei, we have large error bar. Among them,recently this experiment has been..
Exp: ± 0.22 MeV
Exp: 0.22 ±0.25 MeV Λ Λ p n α α α α
10B
10Be Λ Λ

19. Λ Λ n p α α α α 10Be 10B Important issue:
Can we describe the Λ binding energies of A=7 and 10 Λ hypernuclei?
To study the effect of CSB in iso-triplet A=7 and 10 hypernuclei. p n n n p p Λ Λ Λ α α α
7Be Λ
7He
For this purpose, we study structure of A=7 hypernuclei and A=10 hyprenuclei with these 4-body cluster model.. Λ
7Li(T=1) Λ
E. Hiyama, Y. Yamamoto, T. Motoba and
M. Kamimura,PRC80, (2009) Λ Λ n p α α α α
10Be
10B Λ Λ

20. Now, it is interesting to see as follows:
What is the level structure of A=7 and 10 hypernuclei
without CSB interaction?
(2) What is the level structure of A=7 and 10 hypernuclei
with CSB interaction?

21. 6Be 6Li 6He 7Be 7Li 7He Without CSB BΛ：CAL= 5.21 EXP= 5.16
BΛ： EXP= 5.68±0.03±022
JLAB:E 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

22. Without CSB interaction
9Be+Λ
0 MeV
0 MeV
9B+Λ
Exp: ± 0.12 MeV
Exp: ± 0.22 MeV
Cal: 8.67 MeV
Cal: 8.84 MeV
Exp: 0.22 ±0.25 MeV
Cal: 0.17 MeV
In good agreement
with the data Λ Λ p n α α α α
10B
10Be Λ Λ

23. 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

24. 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

25. With CSB interaction Λ n Λ p α α α α 10Be 10B 9Be+Λ 0 MeV 0 MeV 9B+Λ
No plan
Propose to measure binding energy
at J-PARC
Recently, 10B(e,e’K+) 10Be has been done
at JLab. Analysis is in progress. Λ
Exp: ± 0.12 MeV
Cal: 8.67 MeV (without CSB)
Exp: ± 0.22 MeV
Cal: 8.77 MeV (with CSB)
Cal: 8.84 MeV (without CSB)
For this purpose, recently,…
And furthermore, we propose to measure B10L in the future at J-PARC facility.
Cal: 8.76 MeV (with CSB)
Exp: 0.22 ±0.25 MeV
Cal: 0.17 MeV (without CSB)
Cal: 0.01 MeV (With CSB) Λ n
Error bar is
large to discuss
CSB interaction. Λ p α α
We can extract information on
CSB interaction, if they provide
new data of A=10 hypernuclei
within 100 keV accuracy. α α
10Be Λ
10B Λ

26. Summary
We calculated spin-doublet states of in 　10B.
The calculated splitting energy is 0.04 MeV which is consistent with the
experimental fact of no observed γ-ray.
For the detailed investigation of spin doublet state in 10B,
I propose to measure π weak decay from 10B as well as 10Be. Λ Λ Λ Λ
(2)The CSB effects in A=10 hypernuclei are investigated quantitatively on
the basis of the phenomenological CSB interaction,
which describe the experimental energy difference between 4H and
4He.
However, the error bars of energies in A=10 hypernuclei are too large
to discuss the CSB effect in these hypernuclei.
Then, we propose to measure 10B by (π,K) or (K,π) at J-PARC in the
future. Λ Λ Λ