Analysis of (π ± ,K ＋ ) and (K － ,K ＋ ) spectra in DWIA

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

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, Σand Ξ hypernuclear spectrum on several targets with our modification. Σ Repulsive Λ nucleon nucleon -30MeV -50MeV -50MeV

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. Double differential cross section Kinematical factor Strength function Elementary cross section Strength function Meson distorted waves Include the hyperon potential in Green function Distortion factor “Green function”

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

7. Λ hypernuclear production spectra on 28Si target dΛ pΛ Local Optimal Fermi Averaging 27Si+Λ sΛ Woods-Saxon parameters V0=-28[MeV],VLS=2[MeV],W0=-0.5[MeV],R=r0(A-1)1/3,r0=1.080+0.395A-2/3[fm] Ref.D. J. Millener,et.al. PRC38(1988)2700

8. Σ hypernucler production spectrum on 28Si target ・ 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 -10MeV -30MeV -50MeV 0MeV W0Σ= 20MeV +10MeV +50MeV +90MeV ⇒QF spectrum can be reproduced by small repulsive potential.

9. Σ hypernucler production spectrum on 28Si 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) Σ－27Al:UΣ WΣ Batty-DD SCL-RMF2 SCL-RMF1 Is the Quasi-Free data reproduced ??

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

11. Study of Ξ-nucleus potential by (K－,K+) reaction 12C(K－,K+) PK=1.80GeV/c 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 V0 of about 14 MeV within the Woods-Saxon prescription (DWIA calculation). Theoretical curve: INC(Y. Nara et al.,NPA614(1997)433.) DWIA(Tadokoro et al,PRC51(1995)2656.)

12. Ξ－hypernuclear production spectra on several targets Woods-Saxon Potential: V0Ξ=-15MeV Calculation in Green function method Q.F. Q.F. Q.F. Q.F. Exp.Data:E176

13. Ξ－hypernuclear production spectra on 12C target Quasi-Free p3/2-1 s1/2-1 Det. Res. :2MeV Woods-Saxon Potential V0Ξ=-15MeV W0Ξ= 1MeV pΞ 11B+Ξ- sΞ p3/2-1 s1/2-1

14. Ξ－hypernuclear production spectra on Al target 26Mg+Ξ－ dΞ Det. Res. :2MeV Woods-Saxon Potential V0Ξ=-15MeV W0Ξ= 1MeV pΞ 6(deg.) sΞ 0(deg.) 12(deg.)

15. Ξ－hypernuclear production spectra on Ni target Quasi-Free Det. Res. :2MeV Woods-Saxon Potential V0Ξ=-15MeV W0Ξ= 1MeV 57Co+Ξ－

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 V0～－30ＭｅＶ • 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 V0～－15ＭｅＶ, QF spectra on various targets are reproduced.

17. Ξ－hypernuclear production spectra on 12C target Quasi-Free Woods-Saxon Potential V0Ξ=-15MeV W0Ξ= 1MeV p3/2-1 Exp.Data:E176 11B+Ξ- s1/2-1

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

19. Σ hypernucler production spectrum on 28Si target ・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 -50MeV 0MeV W0Σ= 20MeV Batty’s DD -10MeV +10MeV +90MeV

20. Recent analysis of hypernuclear Quasi-Free spectrum Experimental side: Σnucleus potential ～90MeV Noumi et al.(E438) Theoretical side: Kohno et al.(SCDW) Harada and Hirabayashi(DWIA) T. Harada, Y. Hirabayashi, Nucl. Phys. A 744 (2004) 323 M. Kohno, Y. Fujiwara, M. Kawai et al., PTP112 (2004)895 • 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. 1999 M.T.Lopez-Arias 1995

23. Λ hypernuclear production spectra on 51V fΛ dΛ sΛ pΛ

24. Ξ－hypernuclear production spectra(Bound region) Woods-Saxon potential V0Ξ=-15MeV 26Mg+Ξ－ Det. Res. :2MeV 5MeV 3MeV dΞ pΞ sΞ 0.5MeV 1MeV

25. Ξ－hypernuclear production spectra(Bound region) Woods-Saxon potential V0Ξ=-15MeV 59Co+Ξ－ Det. Res. :2MeV fΞ 5MeV 3MeV dΞ pΞ 0.5MeV sΞ 1MeV