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明　孝之　　大阪工業大学 　　　　　　阪大 RCNP

Tensor optimized shell model using bare interaction for light nuclei. 明　孝之　　大阪工業大学 　　　　　　阪大 RCNP. 共同研究者 土岐　博 　 阪大 RCNP 池田　清美　　理研. 1. RCNP 研究会「少数粒子系物理の現状と今後の展望」 @RCNP 2008.12.23-25. Outline. Tensor Optimized Shell Model (TOSM) Unitary Correlation Operator Method (UCOM)

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明　孝之　　大阪工業大学 　　　　　　阪大 RCNP

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  1. Tensor optimized shell model using bare interaction for light nuclei 明　孝之　　大阪工業大学 　　　　　　阪大RCNP 共同研究者 土岐　博　 阪大RCNP 池田　清美　　理研 1 RCNP研究会「少数粒子系物理の現状と今後の展望」 @RCNP 2008.12.23-25

  2. Outline • Tensor Optimized Shell Model (TOSM) • Unitary Correlation Operator Method (UCOM) • TOSM + UCOM with bare interaction • Application of TOSM to Li isotopes • Halo formation of 11Li • TM, K.Kato, H.Toki, K.Ikeda, PRC76(2007)024305 • TM, K.Kato, K.Ikeda, PRC76(2007)054309 • TM, Sugimoto, Kato, Toki, Ikeda, PTP117(2007)257 • TM. Y.Kikuchi, K.Kato, H.Toki, K.Ikeda, PTP119(2008)561 • TM, H. Toki, K. Ikeda, Submited to PTP

  3. Motivation for tensor force • Tensor force (Vtensor) plays a significant role in the nuclear structure. • In 4He, • ~ 80% (GFMC) R.B. Wiringa, S.C. Pieper, J. Carlson, V.R. Pandharipande, PRC62(2001) • We would like to understand the role of Vtensor in the nuclear structure by describing tensor correlation explicitly. • model wave function (shell model and cluster model) • He, Li isotopes (LS splitting, halo formation, level inversion) • Structures of light nuclei with bare interaction • tensor correlation + short-range correlation 3

  4. Tensor & Short-range correlations • Tensor correlation in TOSM (long and intermediate) • 2p2h mixing optimizing the particle states (radial & high-L) • Short-range correlation • Short-range repulsionin the bare NN force • Unitary Correlation Operator Method (UCOM) S D H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61 T. Neff, H. Feldmeier, NPA713(2003)311 4 4

  5. Property of the tensor force • Centrifugal potential (1GeV@0.5fm) pushes away the L=2 wave function. Long and intermediate ranges 5

  6. Tensor-optimized shell model (TOSM) TM, Sugimoto, Kato, Toki, Ikeda PTP117(2007)257 • Tensor correlation in the shell model type approach. • Configuration mixingwithin 2p2h excitationswith high-L orbitTM et al., PTP113(2005) TM et al., PTP117(2007) T.Terasawa, PTP22(’59)) • Length parameters such as are determined independently and variationally. • Describe high momentum component from VtensorCPP-HF by Sugimoto et al,(NPA740) / Akaishi (NPA738)CPP-RMF by Ogawa et al.(PRC73), CPP-AMD by Dote et al.(PTP115) 4He 6 6

  7. Hamiltonian and variational equations in TOSM TM, Sugimoto, Kato, Toki, Ikeda, PTP117(’07)257 • Effective interaction : Akaishi force (NPA738) • G-matrix from AV8’ with kQ=2.8 fm-1 • Long and intermediate ranges of Vtensor survive. • Adjust Vcentral to reproduce B.E. and radius of 4He 7

  8. 4He in TOSM vnn: G-matrix Shrink Length parameters 0 1 2 3 4 5 6 Lmax Higher shell effect good convergence 8 8 Cf. K. Shimizu, M. Ichimura and A. Arima, NPA226(1973)282.

  9. Configuration of 4He in TOSM 4 Gaussians instead of HO c.m. excitation = 0.6 MeV • 0 of pion nature. • deuteron correlation with (J,T)=(1,0) Cf. R.Schiavilla et al. (GFMC) PRL98(’07)132501 9

  10. Tensor & Short-range correlations • Tensor correlation in TOSM (long and intermediate) • 2p2h mixing optimizing the particle states (radial & high-L) • Short-range correlation • Short-range repulsionin the bare NN force • Unitary Correlation Operator Method (UCOM) S D TOSM+UCOM H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61 T. Neff, H. Feldmeier NPA713(2003)311 10 10

  11. Unitary Correlation Operator Method TOSM short-range correlator Bare Hamiltonian Shift operator depending on the relative distance r 2-body cluster expansion of Hamiltonian H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61 11 11

  12. Short-range correlator : C (or Cr) 1E Originalr2 3GeV repulsion 3E Vc 1O C 3O AV8’ : Central+LS+Tensor 12

  13. 4He in UCOM (Afnan-Tang, Vc only) C 13 13

  14. 4He with AV8’in TOSM+UCOM AV8’ : Central+ LS+Tensor exact Kamada et al. PRC64 (Jacobi) • Gaussian expansion for particle states (6 Gaussians) • Two-body cluster expansion of Hamiltonian 14

  15. Extension of UCOM : S-wave UCOM for only relative S-wave wave function – minimal effect of UCOM SD coupling 5 MeV gain 15 15

  16. Different effects of correlation function • S-wave S No Centrifugal Barrier Short-range repulsion D • D-wave due to Centrifugal Barrier 16 16

  17. Saturation of 4He in UCOM UCOM: short-range + tensor T. Neff, H. Feldmeier NPA713(2003)311 Short tensor Energy TOSM+UCOM Long tensor Benchmark cal. Kamada et al. PRC64 17

  18. 4He in TOSM + S-wave UCOM T (exact) Kamada et al. PRC64 (Jacobi) Remaining effect : 3-body cluster term in UCOM VLS E VC VT 18

  19. Summary • Tensor and short-range correlations • Tensor-optimized shell model (TOSM) • He & Li isotopes (LS splitting, Halo formation) • Unitary Correlation Operator Method (UCOM) • Extended UCOM : S-wave UCOM • In TOSM+UCOM, we can study the nuclear structure starting from the bare interaction. • Spectroscopy of light nuclei (p-shell, sd-shell) 19 19

  20. Pion exchange interaction vs. Vtensor Involve large momentum Delta interaction Tensor operator Yukawa interaction - Vtensor produces the high momentum component. 20

  21. Characteristics of Li-isotopes Halostructure • Breaking of magicity N=8 • 10-11Li, 11-12Be • 11Li … (1s)2 ~ 50%.(Expt by Simon et al.,PRL83) • Mechanism is unclear 11Li 21

  22. Pairing-blocking : K.Kato,T.Yamada,K.Ikeda,PTP101(‘99)119,Masui,S.Aoyama,TM,K.Kato,K.Ikeda,NPA673('00)207. TM,S.Aoyama,K.Kato,K.Ikeda,PTP108('02)133, H.Sagawa,B.A.Brown,H.Esbensen,PLB309('93)1. 22

  23. 11Li in coupled 9Li+n+n model • System is solved based on RGM TOSM • Orthogonality Condition Model (OCM) is applied. 23

  24. 11Li G.S. properties (S2n=0.31 MeV) Rm Simon et al. P(s2) Tensor +Pairing Pairing correlation couples (0p)2 and (1s)2 for last 2n

  25. 2n correlation density in 11Li s2=4% s2=47% 9Li 9Li n n Di-neutron type config. Cigar type config. H.Esbensen and G.F.Bertsch, NPA542(1992)310 25 K. Hagino and H. Sagawa, PRC72(2005)044321

  26. Short-range correlator : C (or Cr) Hamiltonian in UCOM 2-body approximation in the cluster expansion of operator

  27. LS splitting in 5He with tensor corr. • T. Terasawa,PTP22(’59) • S. Nagata, T. Sasakawa, T. Sawada R. Tamagaki, PTP22(’59) • K. Ando, H. Bando PTP66(’81) • TM, K.Kato, K.Ikeda PTP113(’05) • Orthogonarity Condition Model (OCM) is applied. 27

  28. Phase shifts of 4He-n scattering 28

  29. 6He in coupled 4He+n+n model System is solved based on RGM TOSM • Orthogonality Condition Model (OCM) is applied. 29

  30. Tensor correlation in 6He Ground state Excited state TM, K. Kato, K. Ikeda, J. Phys. G31 (2005) S1681 30 30

  31. 6He results in coupled 4He+n+n model Theory With Tensor complex scaling for resonances • (0p3/2)2 can be described in Naive 4He+n+n model • (0p1/2)2 loses the energy Tensor suppression in 0+2 31 31

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