1. Analysis of (π ±,K ＋ ) and (K －,K ＋ ) spectra in DWIA HYP06, Friday, Oct. 13, 2006, Mainz, Germany H. Maekawa, K. Tsubakihara, A. Ohnishi Division of Physics, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan 1.Introduction and our purpose 2.Model (DWIA with Green function method and Local optimal Fermi averaging) 3.Results( Λ 、 Σ 、 Ξ Quasi-Free spectra) 4.Summary

2. Do we understand hypernuclear Quasi-Free spectrum ? Previous DWIA calculation of (K,π), (π,K) and (K,K) reactions Bound state region Successful expression of the hypernuclear production spectra QF(continuum) region It is not possible to reproduce QF spectrum well though there are a lot of attempts.(S.W.Hong et al. 1999, M.T.Lopez-Arias 1995) Auerbach et al., Annals of Physics 148(1983)381. Traditional Fermi averaging

3. Recent analysis of hypernuclear Quasi-Free spectrum Theoretical Cal. Distorted wave impulse approximation: T. Harada, Y. Hirabayashi, Nucl. Phys. A 744 (2004) 323. Semi Classical Distorted Wave model: M. Kohno, Y. Fujiwara, M. Kawai et al., PTP112 (2004)895. Cascade model: Y. Nara, A. Ohnishi, T. Harada and A. Engel, NPA614(1997)433. T. Harada, Y. Hirabayashi, Nucl. Phys. A 744 (2004) 323 The key in this problem→Fermi averaging with on-shell classical kinematics of t-matrix(Harada and Hirabayashi)

4. Purpose of our study ・ In optimal Fermi averaging, the t-matrix is averaged under the on-shell kinematics in the free space(no potential effects) ・ We would like to include potential effects with the on-shell condition into the Fermi averaging procedure. ・ To confirm the validity our extension of Fermi averaging with potential effects, we attempt to calculate Λ, Σ and Ξ hypernuclear spectrum on several targets with our modification. ・ In optimal Fermi averaging, the t-matrix is averaged under the on-shell kinematics in the free space(no potential effects) ・ We would like to include potential effects with the on-shell condition into the Fermi averaging procedure. ・ To confirm the validity our extension of Fermi averaging with potential effects, we attempt to calculate Λ, Σ and Ξ hypernuclear spectrum on several targets with our modification. Λ Σ Repulsive -30MeV -50MeV-50MeV nucleon

5. Model: Green function method by Morimatsu and Yazaki Ref) O.Morimatsu and K.Yazaki, Nucl. Phys. A483(1988)493. S.Tadokoro,Y.Akaishi,H.Kobayashi. Phys.Rev.C51(1995)2656. M.T.Lopez-Arias, Nucl. Phys. A582(1995)440. Elementary cross section Kinematical factor Meson distorted waves Include the hyperon potential in Green function Strength function Distortion factor Double differential cross section “Green function” Strength function

6. Local Optimal Fermi Averaging of t-matrix (LOFAt) π Ｎ Ｋ Ｙ We’d like to include the potential effects in the production points. Local Optimal Fermi Averaging of t-matrix (LOFAt) Energy conservation equation “potential” →Include the potential effects into Fermi-averaging

7. sΛsΛ pΛpΛ dΛdΛ Λ hypernuclear production spectra on 28 Si target Local Optimal Fermi Averaging 27 Si+Λ V 0 =-28[MeV],V LS =2[MeV],W 0 =-0.5[MeV],R=r 0 (A-1) 1/3,r 0 =1.080+0.395A -2/3 [fm] Ref. D. J. Millener,et.al. PRC38(1988)2700 Woods-Saxon parameters

8. Σ hypernucler production spectrum on 28 Si target -50MeV 0MeV +90MeV -10MeV +10MeV ・ Σ Quasi-Free analysis(Noumi et al., Harada and Hirabayashi, Kohno et al.): Σ - nucleus pot.:Repulsive (Woods-Saxon),V=+30MeV ～ +90MeV With potential effect W 0 Σ = 20MeV ⇒ QF spectrum can be reproduced by small repulsive potential. -30MeV +50MeV

9. Σ hypernucler production spectrum on 28 Si target We consider the two type potentials derived from the Σ atomic data. 1.Batty density dependent potential 2.SCL-RMF model by Tsubakihara, Maekawa, Ohnishi(talk in previous session) Batty-DD SCL-RMF1 SCL-RMF2 Is the Quasi-Free data reproduced ?? Σ － 27 Al:U Σ WΣWΣ

10. Σ hypernucler production spectrum on 28 Si target Batty’s DD SCL-RMF1 Derived from Σ － X-ray data potential ⇒ QF spectrum can be reproduced well using density dependent potentials derived from atomic data (rather than the case of simple Woods-Saxon type potentials) ⇒ QF spectrum can be reproduced well using density dependent potentials derived from atomic data (rather than the case of simple Woods-Saxon type potentials) SCL-RMF2 ⇒ Σ-nucleus potential is … Attractive pocket and Repulsive core Structure of Attractive pocket and Repulsive core is favored. ⇒ Σ-nucleus potential is … Attractive pocket and Repulsive core Structure of Attractive pocket and Repulsive core is favored.

11. P. Khaustov et al., Phys. Rev. C61(2000) 054603-1. Reasonable agreement between the data and theory is achieved by assuming a Ξ-nucleus potential well depth V 0 of about 14 MeV within the Woods-Saxon prescription (DWIA calculation). 12 C(K －,K + ) P K =1.80GeV/ c Study of Ξ-nucleus potential by (K －,K + ) reaction Theoretical curve: DWIA DWIA (Tadokoro et al,PRC51(1995)2656.) INC INC ( Y. Nara et al.,NPA614(1997)433. )

12. Ξ － hypernuclear production spectra on several targets Woods-Saxon Potential: V 0 Ξ =- 15MeV Exp.Data:E17 6 Calculation in Green function method Q.F.

13. Ξ － hypernuclear production spectra on 12 C target sΞsΞ pΞpΞ 11 B+Ξ - Quasi-Free Det. Res. :2MeV Woods-Saxon Potential V 0 Ξ =-15MeV W 0 Ξ = 1MeV p 3/2 -1 s 1/2 -1 p 3/2 - 1

14. Ξ － hypernuclear production spectra on Al target Det. Res. :2MeV sΞsΞ pΞpΞ dΞdΞ 12(deg. ) 6(deg.) 0(deg.) Woods-Saxon Potential V 0 Ξ =-15MeV W 0 Ξ = 1MeV 26 Mg+Ξ －

15. Ξ － hypernuclear production spectra on Ni target Det. Res. :2MeV Woods-Saxon Potential V 0 Ξ =-15MeV W 0 Ξ = 1MeV 57 Co+Ξ － Quasi-Free

16. Summary DWIA with Quantum mechanical treatment of QF region(Green function method) Fermi averaging (In ordinary DWIA) On-shell classical kinematics (Optimal Fermi average, SCDW,INC) Potential effects at reaction points (Local optimal Fermi average; Ours) are found to explain various hyperon production QF spectrum. We propose the “Local optimal Fermi averaging” of t-matrix To include the potential effects into optimal Fermi averaging We calculate the hypernucler Quasi-Free spectrum. Λ:With V 0 ～－ 30 ＭｅＶ Both of QF and Bound state spectrum are reproduced very well We confirm the validity of our extension of F.A. Σ:QF spectra is compatible with atomic data Batty’s DD pot.,SCL-RMF→works well Ξ: With V 0 ～－ 15 ＭｅＶ, QF spectra on various targets are reproduced.

17. Ξ － hypernuclear production spectra on 12 C target Quasi-Free 11 B+Ξ - Exp.Data:E17 6 p 3/2 -1 s 1/2 -1 Woods-Saxon Potential V 0 Ξ =-15MeV W 0 Ξ = 1MeV

18. Σ hypernucler production spectrum on 28 Si target Batty’s DD

19. Σ hypernucler production spectrum on 28 Si target -50MeV 0MeV +90MeV Batty’s DD -10MeV +10MeV ・ Σ atomic data analysis(Batty et al., Mares et al.):Σ-nucleus pot.:Repulsive core + attractive pocket ・ Σ Quasi-Free analysis(Noumi et al., Harada and Hirabayashi, Kohno et al.):Σ- nucleus pot.:Repulsive (Woods-Saxon),V=+30MeV ～ +90MeV Optimal Fermi averaging W 0 Σ = 20MeV

20. Recent analysis of hypernuclear Quasi-Free spectrum Experimental side: Theoretical side: Noumi et al.(E438) Harada and Hirabayashi(DWIA) Kohno et al.(SCDW) T. Harada, Y. Hirabayashi, Nucl. Phys. A 744 (2004) 323 M. Kohno, Y. Fujiwara, M. Kawai et al., PTP112 (2004)895 Σnucleus potential ～ 90MeV The key in this problem→Fermi averaging of t-matrix

21. Several Fermi averagings for t-matrix in DWIA Previous procedure Auerbach et al. Annals of Physics 148(1983)381. Recent extension of Fermi averaging(Optimal Fermi averaging) T. Harada, Y. Hirabayashi, Nucl. Phys. A 744 (2004) 323

22. Do we understand hypernuclear Quasi-Free spectrum ? Previous DWIA calculation of (K,π), (π,K) and (K,K) reactions Bound state region Successful expression of the hypernuclear production spectra QF(continuum) region It is not possible to reproduce QF well though there are a lot of attempts. S.W.Hong et al. 1999M.T.Lopez-Arias 1995

23. Λ hypernuclear production spectra on 51 V sΛsΛ pΛpΛ dΛdΛ fΛfΛ

24. Ξ － hypernuclear production spectra(Bound region) Det. Res. :2MeV Woods-Saxon potential V 0 Ξ =-15MeV 26 Mg+Ξ － 0.5MeV 1MeV 3MeV5MeV sΞsΞ pΞpΞ dΞdΞ

25. Ξ － hypernuclear production spectra(Bound region) Det. Res. :2MeV Woods-Saxon potential V 0 Ξ =-15MeV 59 Co+Ξ － 0.5MeV 1MeV 3MeV5MeV sΞsΞ pΞpΞ dΞdΞ fΞfΞ