Structure of 10Be and 10B hypernuclei studied with four-body cluster model Λ 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.

Hypernuclear g-ray data since 1998 (figure by H.Tamura) This is hypernuclear gamma-ray data sicne 1998. 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)

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.

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?

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?

ΛN interaction: Nijmegen ’97f p 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

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?

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-

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-

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-

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)

Decay pion spectroscopy at Mainz Λ n α 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

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.

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

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

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.

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:E01-011 experiment Preliminary data: 5.68±0.03±0.22 -3.79 7Be Λ 7Li (T=1) Λ 7He Λ

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: -8.89 ± 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: -9.11 ± 0.22 MeV Exp: 0.22 ±0.25 MeV Λ Λ p n α α α α 10B 10Be Λ Λ

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, 054321 (2009) Λ Λ n p α α α α 10Be 10B Λ Λ

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?

6Be 6Li 6He 7Be 7Li 7He Without CSB BΛ：CAL= 5.21 EXP= 5.16 BΛ： EXP= 5.68±0.03±022 JLAB:E01-011 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

Without CSB interaction 9Be+Λ 0 MeV 0 MeV 9B+Λ Exp: -8.89 ± 0.12 MeV Exp: -9.11 ± 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

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

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

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: -8.89 ± 0.12 MeV Cal: 8.67 MeV (without CSB) Exp: -9.11 ± 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

Summary We calculated spin-doublet states of 1-- 2- 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