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Few-body aspects of strangeness nuclear physics

Few-body aspects of strangeness nuclear physics. Submitted to Phys. Rev. C. E. Hiyama (Nara Women’s Univ.). n. Λ. p. 7. Li. Λ. α. One of the major purpose of hypernuclear physics is to understand the baryon-baryon interaction in unified way.

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Few-body aspects of strangeness nuclear physics

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  1. Few-body aspects of strangeness nuclear physics Submitted to Phys. Rev. C E. Hiyama (Nara Women’s Univ.) n Λ p 7 Li Λ α

  2. One ofthe major purpose of hypernuclear physics is to understand the baryon-baryon interaction in unified way. Since the hyperon-hyperon scattering data is extremely limited, then hyperon(Y)-nucleon(N) interaction so far proposed have a large degree of ambiguity. Therefore, it is very important to obtain information about YN interaction from the spectroscopy of many single Λ hypernuclei.

  3. For this purpose, so far several high-resolution γ-ray spectroscopy experiment such as 9Be, 13C and 7Li have been performed at KEK and BNL. Λ Λ Λ These experiments have been done for the study of YN spin-orbit force and spin-spin force. 7 13 9 Li C Be Λ Λ Λ n Λ Λ α Λ p α α α α α

  4. Spin-orbit force 13 9 C Be Λ Λ Λ Λ α α α α α BNL-E929 BNL-E930

  5. BNL-E930 Λ 3/2+ γ ΔE Spin-orbit splitting 2+ 5/2+ γ 0+ 1/2+ 8Be 9Be Λ 1/2- BNL-E929 ΔE Λ 3/2- E. Hiyama et al. Phys. Rev. Lett. 85, 270 (2000) γ γ 0+ 1/2+ 13C 12C Λ

  6. E. Hiyama et al. Phys. Rev. Lett. 85, 270 (2000) Λ α α α α 8Be = 9Be= Λ α 12C = α α α 13C= Λ Λ α α

  7. YN spin-orbit force ・Nijmegen model ・quark-based model (made by Fujiwara et al.) Since explicit form of the quark-based LS and ALS interaction was not available, we tried to use the form of Nijmegen model for the LS force and ALS parts. So, we tried to enlarge the strength of ALS part to be 85% of the SLS with the opposite sigh.

  8. ΛN spin orbit force and 9Be and 13C Λ BNL-E930 Λ 3/2+ 5/2+ +2.5 31.4 keV Exp. -3.6 H. Akikawa et al. Phys. Rev. Lett. 88,(2002) 82501. BNL-E929 1/2- 3/2- 152 54 36 keV Exp. ± ± S.Ajimura et al. Phys. Rev. Lett. 86,(2001) 4255 9Be Λ 35 40 keV 3/2+ 80 200 keV 3/2+ ～ ～ 5/2+ 5/2+ Quark Meson 13C Λ 150 200 keV 1/2- 360 960 keV 1/2- ～ ～ 3/2- 3/2- Quark Meson

  9. Therefore, by comparing our theoretical calculation and the experiments of γ-ray spectroscopy, we can understand that the desirable strength of YN spin-orbit force should be very small.

  10. For the study of spin-spin force 7 Li Λ n Λ p α

  11. Tamura et al. α+n+p done by BNL-E930 3+ 7/2+ 0.47 MeV 1+ 5/2+ done by KEK-E419 6Li 3/2+ ΛN spin-spin splitting energy σΛ・σN n p 1/2+ 0.69 MeV Λ α 7Li Λ

  12. By comparing with these high-resolution γ-ray experimental data and shell model calculation with the restricted configuration of (0s)4(0p)n0sΛ by Millner, we succeeded in obtaining useful information about ΛN spin-dependent force partially. M. Ukai et al., Phys. Rev. C 73, 012501(R) (2006) D.J. Millener, Nucl. Phys. A754, 48c (2005)

  13. We have 2 important issues: • Can we explain consistently two spin-doublets of 3/2+-1/2+ and 7/2+-5/2+ states using ΛN spin-orbit force and spin-spin force based on the experimental data for 9Be, 7Li and 4H? (2) How is the level structure of the other A=7 hypernuclei, namely, 7He, 7Li (T=1) and 7Be using the above using ΛN interaction? Λ Λ Λ Λ Λ Λ

  14. My contribution To understand the hypernuclear structure by performing our these four-body calculations and To use this structure information to understand the ΛN spin-spin force and spin-orbit force.

  15. Applied to Gaussian Expansion Method Developed by Kyushu Univ. group Kamimura (1) 3-cluster structure of light nuclei (2) Coulomb 3-body muonic molecular ions appearing in the muon-catalyzed fusion cycles (1987～) (3) 3-nucleon bound states with realistic NN and 3N forces (1988) (4)Metastable antiprotonic helium atom (He++p+e)(1995～) E. Hiyama, M. Kamimura and Y. Kino, Prog. Part. Nucl. Phys. 51 (2003), 223.

  16. Now, I have been applying our method to hypernuclear structure. (1) Can we explain consistently two spin-doublets of 3/2+-1/2+ and 7/2+-5/2+ states using ΛN spin-orbit force and spin-spin force based on the experimental data for 9Be, 7Li and 4H? (2) How is the level structure of the other A=7 hypernuclei, namely, 7He, 7Li (T=1) and 7Be using the above using ΛN interaction? Λ Λ Λ Λ Λ Λ

  17. 9 ΨJM( 7Li)=∑ΦJM(rc,Rc,ρC) Λ C=1

  18. (spatial)=φnl(c)(rc)ψνλ(c)(ρc)χNL(c)(Rc) ^ 2 φnlm(c)=rle-(r/r ) Ylm(rc), rn=r1an-1(n=1～nmax) n 2 Ψνλμ(ρc)=ρλe-(ρ/ρ) Yλμ(ρc), ρμ=ρ1αμ-1 (μ=1～μmax) ^ μ χNL(c)(Rc)=RLe-(R/R )YLM(Rc), RN=R1AN-1(N=1～Nmax) Geometric progression (H-E)ΨJM=0 The Schödinger equation is solved with Rayleigh-Ritz variational method. For the angular-momentum component of the wavefunction, the approximation with l,L,λ≤2 was found to be sufficient to obtain in getting satisfactory convergence of the binding energies. But, no truncation of the interaction is made in the angular- momentum space.

  19. α-Ninteraction: potential which reproduce reasonably well the low-lying states and low-energy scattering phase shifts of the αN systems 7 Li Λ n p Λ α α-Λ interaction: Nijgemen soft core ’97f YNG folded into the density of the α cluster

  20. 7 Li Λ ΛN interaction: Nijmegen ’97f n Not original one but simulated one p Λ The ΛN-ΣN coupling interaction can be renomalized into the ΛN-ΛN interaction effectively. α 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

  21. V0+σΛ・σNVs 4H Λ NSC97f 0MeV 3H+Λ N N 1+ -1.04 -2.05 MeV 1+ Λ 0+ N -2.00 0+ -2.43 MeV Exp. Adjusted so as to reproduce the observed data of 4H Λ

  22. (σΛ+σN)/2・VSLS+(σΛ-σN)/2・VALS Adjusted so as to reproduce the data of 9Be Λ BNL-E930 Λ 3/2+ +2.5 31.4 keV γ -3.6 2+ 5/2+ γ 0+ 1/2+ 8Be 9Be Λ

  23. 1/2- BNL-E929 152 54 Λ Exp. ± 36 keV ± 3/2- γ γ 0+ 1/2+ 13C 12C Λ α Λ Calculated energy splitting 0.2 MeV → consistent with the data within the error 13C Λ α α

  24. Here, in the study of A=7 hypernuclei based on the α+Λ+N+N 4-body model, before 4-body calculation, it is absolutely necessary to examine whether the model with the interaction adopted is able to reproduce reasonably well the following observed quantities: • Energies of the low-lying states and scattering phase shifts • of the α+N, NN and αNN nuclear systems • (ii) BΛ of hypernuclei composed of α+Λ, α+N+Λ

  25. In our model, the observed low-energy properties of the α+N Nuclei and the existing Λ-binding energies of the α+Λ and α+Λ+N hypernuclei have been reproduced accurately. n n p p 7 Li Λ Λ Λ α α n n p p p Λ n Λ Λ α α α

  26. This encourages us to perform the 4-body calculation with NO adjustable parameter at this stage, expecting high reliability of the results.

  27. 0 MeV α+Λ+n+p threshold ~ ~ n Λ p 7 Li 7/2+ Λ 4H α Λ N 5/2+ N Λ N 3/2+ 0MeV 3H+Λ 1+ 1+ -1.04 -1.04 1/2+ similar 0+ 0+ -2.00 -2.00 Exp. Cal.

  28. 0 MeV α+Λ+n+p threshold ~ ~ -6.23 Agree with the binding energy of of the ground state of 1/2+ Due to the repulsive nature of NSC97f 0.86 Exp. (1/2+)9.28 MeV -7.13 -8.41 The odd-state interaction of the other Nijmegen model are attractive. 0.97 0.97 -9.38 overbound by 0.5MeV

  29. The important role of the repulsive odd-state interaction does not necessarily mean that the odd-state part in NSC97f is more realistic than the other interactions. The detailed reason is discussed later.

  30. 0 MeV α+Λ+n+p threshold ~ ~ -6.23 0.86 Splitting energy is larger than the experimental data -7.13 The odd-state interaction is adjusted so as to reproduce the observed splitting energy. -8.41 0.97 0.97 Exp.0.69 MeV -9.38

  31. 0 MeV α+Λ+n+p threshold ~ ~ 7/2+ 0.54 5/2+ 3/2+ 1/2+

  32. Now, we come to the important stage of looking at the role of the SLS and ALS interactions for splitting energies. It should be noted here that SLS and ALS interactions work differently for two doublets states in 7Li. Λ The spin-orbit contribution to the ground-state doublets(1/2+-3/2+) very small The spin-orbit contribution to the excited-state doublets(5/2+-7/2+) large 7 Li Λ n p n p Λ L=2 Λ L=0 α α

  33. 0 MeV α+Λ+n+p threshold ~ ~ 7 Li Λ 7/2+ S=3/2 0.54 5/2+ n p Λ α 3/2+ 1/2+

  34. 0 MeV α+Λ+n+p threshold ~ ~ S=1/2 n p 0.54 Λ α S=3/2 n p Λ α

  35. 0 MeV α+Λ+n+p threshold ~ ~ 0.54 In this way, owing to the combined effect of SLS and ALS, our final result reproduces nicely the observed energies of the spin-doublet states of 7Li. Λ

  36. It is interesting to see the level structure of the other A=7 hypernuclei such as 7He, 7Li (T=1) and 7Be. Λ Λ Λ p n n n Λ Λ p Λ p α α α 7 He 7 Li 7 Be Λ Λ Λ

  37. n n p p n p 5He 5He 5He Λ Λ Λ 7He 7 Be 7Li Λ Λ Λ 5He α Λ = Λ E. Hiyama, M. Kamimura, T. Motoba, T. Yamada and Y. Yamamoto, Phys. Rev. C53 (1996), 2075

  38. E. Hiyama, M. Kamimura, T. Motoba, T. Yamada and Y. Yamamoto, Phys. Rev. C53 (1996), 2075

  39. How is the level structure based on α+Λ+N+N 4-body model using ΛN interaction which was applied to the level structure of 7Li(T=0)? Λ

  40. Charge symmetry breaking effect Cal.BΛ=5.56MeV

  41. In this way, here in this talk, I discussed about the level structure of A=7 hypernuclei. Comment on the role of ΛN-ΣN coupling Our basis assumption in this work: The ΛN-ΣN coupling interaction can be renomalized into the ΛN-ΛNinteraction. In this sprit, the even-state part of our ΛN interaction were adjusted so as to reproduce the 0+ and 1+ state of 4H. Λ

  42. However, the role of the ΛN-ΣN coupling may be important for 4H and 7Li. Λ Λ

  43. 0 MeV α+Λ+n+p threshold ~ ~ The repulsive odd-state interaction such as NSC97f reproduce the observed binding energy of the ground state of 7Li. 7/2+ 0.54 5/2+ Exp. (1/2+)9.28 MeV Λ 3/2+ 1/2+

  44. But, it might be reasonable to consider that the ΛN-ΣN coupling works more repulsively in 7Li. Λ At the present, it is likely that the role of the odd-state repulsion in our treatment is a substitute for this effect.

  45. ・Y. Akaishi et al. Phys. Rev. Lett. 84, 3539 (2000) ・B. F. Gibson et al. Phys. Rev. C6, 741 (1972) N1 N2 N3 Λ The extra contribution to the 0+-1+ splitting of 4H from the 3-body correlated ΛN-ΣN mixing. Λ Σ N1 N2 N3 Λ E. Hiyama et al., Phys. Rev. C65, 011301(R) (2001) Obtained the value of 0.3 MeV for the three-body contribution of ΛN-ΣN coupling in 4H Λ In the shell model calculation, Millner calculated the spin-doublet states in 7Li including ΛN-ΣN coupling and he concluded that this contribution was small in these splitting. Λ

  46. However, it is an open problem to study ΛN-ΣN coupling effect consistently for 4H and 7Li. Λ Λ n p n p Σ Λ + α α Future my work

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