Outline Tensor Optimized Shell Model TOSM Unitary Correlation

Outline ü Tensor Optimized Shell Model (TOSM) ü Unitary Correlation Operator Method (UCOM) ü TOSM + UCOM with bare interaction • Application of TOSM to Li isotopes • • • Halo formation of 11 Li TM, K. Kato, H. Toki, K. Ikeda, 　　　　　　　PRC 76(2007)024305 TM, K. Kato, K. Ikeda, PRC 76(2007)054309 TM, Sugimoto, Kato, Toki, Ikeda, PTP 117(2007)257 TM. Y. Kikuchi, K. Kato, H. Toki, K. Ikeda, PTP 119(2008)561 TM, H. Toki, K. Ikeda, Submited to PTP 2

Motivation for tensor force • Tensor force (Vtensor) plays a significant role in the nuclear structure. – In 4 He, – ~ 80% (GFMC) R. B. Wiringa, S. C. Pieper, J. Carlson, V. R. Pandharipande, PRC 62(2001) • We would like to understand the role of Vtensor in the nuclear structure by describing tensor correlation explicitly. ü model wave function (shell model and cluster model) ü He, Li isotopes (LS splitting, halo formation, level inversion) • Structures of light nuclei with bare interaction ü tensor correlation + short-range correlation 3

Tensor & Short-range correlations • Tensor correlation in TOSM (long and intermediate) – – 2 p 2 h mixing optimizing the particle states (radial & high-L) • Short-range correlation – Short-range repulsion in the bare NN force – Unitary Correlation Operator Method (UCOM) S D H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA 632(1998)61 T. Neff, H. Feldmeier, NPA 713(2003)311 4

Property of the tensor force Long and intermediate ranges 5 • Centrifugal potential (1 Ge. V@0. 5 fm) pushes away the L=2 wave function.

Tensor-optimized shell model (TOSM) TM, Sugimoto, Kato, Toki, Ikeda • Tensor correlation PTP 117(2007)257 in the shell model type approach. 4 He • Configuration mixing within 2 p 2 h excitations with high-L orbit 　TM et al. , PTP 113(2005) TM et al. , PTP 117(2007) T. Terasawa, PTP 22(’ 59)) • Length parameters such as determined independently and variationally. are – Describe high momentum component from Vtensor CPP-HF by Sugimoto et al, (NPA 740) / Akaishi (NPA 738) CPP-RMF by Ogawa et al. (PRC 73), CPP-AMD by Dote et al. (PTP 115) 6

Hamiltonian and variational equations in TOSM TM, Sugimoto, Kato, Toki, Ikeda, PTP 117(’ 07)257 • Effective interaction : Akaishi force (NPA 738) – G-matrix from AV 8’ with k. Q=2. 8 fm-1 – Long and intermediate ranges of Vtensor survive. – Adjust Vcentral to reproduce B. E. and radius of 4 He 7

4 He in TOSM vnn: G-matrix Shrink Length parameters Orbit bparticle/bhole 0 1 2 good convergence 3 Lmax 4 5 6 0 p 1/2 0. 65 0 p 3/2 0. 58 1 s 1/2 0. 63 0 d 3/2 0. 58 0 d 5/2 0. 53 0 f 5/2 0. 66 0 f 7/2 0. 55 Higher shell effect Cf. K. Shimizu, M. Ichimura and A. Arima, NPA 226(1973)282. 8

Configuration of 4 He in TOSM 28. 0 Energy (Me. V) 4 Gaussians instead of HO 51. 0 (0 s 1/2)4 85. 0 % (0 s 1/2)2 JT(0 p 1/2)2 JT JT=10 5. 0 JT=01 0. 3 (0 s 1/2)210(1 s 1/2)(0 d 3/2)10 2. 4 (0 s 1/2)210(0 p 3/2)(0 f 5/2)10 2. 0 P[D] 9. 6 c. m. excitation = 0. 6 Me. V • 0 of pion nature. • deuteron correlation with (J, T)=(1, 0) Cf. R. Schiavilla et al. (GFMC) PRL 98(’ 07)132501 9

Tensor & Short-range correlations • Tensor correlation in TOSM (long and intermediate) – – 2 p 2 h mixing optimizing the particle states (radial & high-L) • Short-range correlation – Short-range repulsion in the bare NN force – Unitary Correlation Operator Method (UCOM) S D TOSM+UCOM H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA 632(1998)61 T. Neff, H. Feldmeier NPA 713(2003)311 10

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, NPA 632(1998)61 11

Short-range correlator : C (or Cr) 1 E Original r 2 3 Ge. V repulsion 3 E Vc 1 O 3 O AV 8’ : Central+LS+Tensor C 12

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

4 He with AV 8’ in TOSM+UCOM AV 8’ : Central+ LS+Tensor exact Kamada et al. PRC 64 (Jacobi) • Gaussian expansion for particle states (6 Gaussians) • Two-body cluster expansion of Hamiltonian 14

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

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

Saturation of 4 He in UCOM Energy UCOM: short-range + tensor Short tensor T. Neff, H. Feldmeier NPA 713(2003)311 TOSM+UCOM Long tensor Benchmark cal. Kamada et al. PRC 64 17

4 He in TOSM + S-wave UCOM T (exact) Kamada et al. PRC 64 (Jacobi) Remaining effect : 3 -body cluster term in UCOM VLS E VC VT 18

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

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

Characteristics of Li-isotopes Halo structure Breaking of magicity N=8 • 10 -11 Li, 11 -12 Be • 11 Li … (1 s)2 ~ 50%. (Expt by Simon et al. , PRL 83) • Mechanism is unclear 11 Li 21

Pairing-blocking : 22 　　K. Kato, T. Yamada, K. Ikeda, PTP 101(‘ 99)119, 　 Masui, S. Aoyama, TM, K. Kato, K. Ikeda, NPA 673('00)207. TM, S. Aoyama, K. Kato, K. Ikeda, PTP 108('02)133, H. Sagawa, B. A. Brown, H. Esbensen, PLB 309('93)1.

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

11 Li G. S. properties (S 2 n=0. 31 Me. V) Simon et al. Rm P(s 2) Tensor +Pairing E(s 2)-E(p 2) 2. 1 1. 4 　0. 5 -0. 1 [Me. V] Pairing correlation couples (0 p)2 and (1 s)2 for last 2 n 24

2 n correlation density in 11 Li s 2=4% 9 Li Cigar type config. 9 Li s 2=47% n n Di-neutron type config. H. Esbensen and G. F. Bertsch, NPA 542(1992)310 K. Hagino and H. Sagawa, PRC 72(2005)044321 25

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

LS splitting in 5 He with tensor corr. • T. Terasawa, PTP 22(’ 59) • S. Nagata, T. Sasakawa, T. Sawada R. Tamagaki, 　　PTP 22(’ 59) • K. Ando, H. Bando PTP 66(’ 81) • TM, K. Kato, K. Ikeda PTP 113(’ 05) • Orthogonarity Condition Model (OCM) is applied. 27

Phase shifts of 4 He-n scattering 28

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

Tensor correlation in 6 He Ground state Excited state TM, K. Kato, K. Ikeda, J. Phys. G 31 (2005) S 1681 30

6 He results in coupled 4 He+n+n model complex scaling for resonances Theory With Tensor • (0 p 3/2)2 can be described in Naive 4 He+n+n model • (0 p 1/2)2 loses the energy Tensor suppression in 0+31 2