11 Role of tensor force in light nuclei based on the tensor optimized shell model Hiroshi TOKI RCNP, Osaka Univ. Manuel Valverde RCNP, Osaka Univ. Atsushi UMEYA RIKEN Kiyomi IKEDA RIKEN Takayuki MYO 明 孝之 Osaka Institute of Technology International conference on the structure of baryons Osaka Univ.,

Outline 2 Tensor correlation in light nuclei Tensor Optimized Shell Model (TOSM) to describe tensor correlation Unitary Correlation Operator Method (UCOM) for short-range correlation TOSM+UCOM with bare nuclear force Application of TOSM to Li isotopes halo formation in 11 Li

Tensor force (V tensor ) plays a significant role in the nuclear structure. –In 4 He,  V tensor  ~  V central  – ~ 80% (GFMC) 3 Importance of tensor force R.B. Wiringa, S.C. Pieper, J. Carlson, V.R. Pandharipande, PRC62(2001) We would like to understand role of V tensor in the nuclear structure by describing tensor correlation explicitly. tensor & short range correlations from V NN (bare) He, Li isotopes (LS splitting, halo formation, level inversion)

S D Energy MeV Kinetic19.88 Central Tensor LS P(L=2) 5.77% Radius 1.96 fm V central V tensor AV8’ Deuteron & tensor force R m (s)=2.00 fm R m (d)=1.22 fm d-wave is “spatially compact” r AV8’

TM, Sugimoto, Kato, Toki, Ikeda PTP117(2007)257 5 Tensor-optimized shell model (TOSM) 5 4 He Tensor correlation in the shell model type approach. Configuration mixing within 2p2h excitations with high- L orbits. TM et al., PTP113(2005) TM et al., PTP117(2007) T.Terasawa, PTP22(’59)) Length parameters {b  } such as b 0s, b 0p, … are optimized independently (or superposed by many Gaussian bases). –Describe high momentum component from V tensor (Shrinkage) HF by Sugimoto et al,(NPA740) / Akaishi (NPA738) RMF by Ogawa et al.(PRC73), AMD by Dote et al.(PTP115)

Configurations in TOSM protonneutron Gaussian expansion nlj particle states hole states (harmonic oscillator basis) c.m. excitation is excluded by Lawson’s method Application to Hypernuclei by Umeya  coupling C0C0 C1C1 C2C2 C3C3

7 4 He in TOSM Orbitb particle /b hole 0p 1/ p 3/ s 1/ d 3/ d 5/ f 5/ f 7/ Length parameters good convergence Higher shell effect L max v nn : G -matrix Cf. K. Shimizu, M. Ichimura and A. Arima, NPA226(1973)282. shrink

8 Unitary Correlation Operator Method 8 H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61 short-range correlator Bare Hamiltonian Shift operator depending on the relative distance r TOSM 2-body cluster expansion

9 Short-range correlator : C (or C r ) 3GeV repulsion Original  r 2 CC VcVc 1E1E 3E3E 1O1O 3O3O s(r) [fm] We further introduce partial-wave dependence in “s(r)” of UCOM S-wave UCOM Shift functionTransformed V NN (AV8’)

10 T VTVT V LS VCVC E (exact) Kamada et al. PRC64 (Jacobi) Gaussian expansion with 9 Gaussians variational calculation TM, H. Toki, K. Ikeda PTP121(2009)511 4 He in TOSM + S-wave UCOM good convergence

11 Configurations of 4 He in TOSM + short-range UCOM (0s 1/2 ) % (0s 1/2 ) −2 JT (p 1/2 ) 2 JT JT= JT= (0s 1/2 ) −2 10 (1s 1/2 )(d 3/2 ) (0s 1/2 ) −2 10 (p 3/2 )(f 5/2 ) Radius [fm]  of pion nature. deuteron correlation with (J,T)=(1,0) Cf. R.Schiavilla et al. (GFMC) PRL98(’07) He contains p 1/2 of “pn”-pair.

5,6 He in TOSM+UCOM Argonne V8’ with no Coulomb force. Convergence for particle states. –L max ~10 –6~8 Gaussian for radial component Lawson method to eliminate CM excitation. Bound state approximation for resonances. Cf) GFMC : K.M. Nollett et al., PRL99 (2007) NCSM : S. Quaglioni, P. Navrátil, PRL101 (2008) SVM : Y.Suzuki, W.Horiuchi, K. Arai, NPA823 (2009) 1

4,5,6 He with TOSM+UCOM Difference from 4 He in MeV AV8’ Preliminary

5 He : Hamiltonian component Difference from 4 He in MeV 5 He3/2  1/2  TT  Central  9.0  7.0  Tensor  5.6  1.1  LS  EE Diff.=3.9 [MeV] =18.4 MeV (hole) b hole =1.5 fm for 4,5 He

Tensor correlation & Splitting in 5 He  V tensor  ~exact in 4 He Enhancement of V tensor

LS splitting in 5 He with tensor correlation T. Terasawa, A. Arima, PTP23 (’60) 87, 115. S. Nagata, T.Sasakawa, T.Sawada, R.Tamagaki, PTP22(’59) K. Ando, H. Bando PTP66 (’81) 227 TM, K.Kato, K.Ikeda PTP113 (’05) 763 (  +n OCM) 30% of the observed splitting from Pauli-blocking d-wave spliiting is weaker than p-wave splitting Pauli-Blocking

Phase shifts of 4 He-n scattering

18 Characteristics of Li-isotopes Breaking of magicity N= Li, Be 11 Li … (1s) 2 ~ 50%. (Expt by Simon et al.,PRL83) Mechanism is unclear 11 Li Tanihata et al., PRL55(1985)2676. PLB206(1998)592. Halo structure

19 9 Li in TOSM Tensor-optimized shell model TM et al., PTP121(2009), PTP117(2007). 0s+0p+1s0d within 2p2h excitations, G-matrix (Akaishi) Length parameters b 0s, b 0p, … are determined independently and variationally (or Gaussian expansion). –Describe high momentum component from V tensor cf. CPP-HF by Sugimoto et al.,NPA740 / Akaishi NPA738

Energy surface for b-parameter in 9 Li V tensor is optimized with shrunk HO basis cf. 1 st order (residual interaction): T. Otsuka et al. PRL95(2005) TOSM pairing correlation nn Dominant part of the tensor correlation pnpn

21 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 11 Li in coupled 9 Li+n+n model System is solved based on RGM Orthogonality Condition Model (OCM) is applied. TOSM

23 11 Li G.S. properties (S 2n =0.31 MeV) Tensor +Pairing Simon et al. P(s 2 ) RmRm E(s 2 )-E(p 2 )  0.1 [MeV] Pairing correlation couples (0p) 2 and (1s) 2 for last 2n (pn)(pn)(nn)

24 Coulomb breakup strength of 11 Li E1 strength by using the Green’s function method +Complex scaling method +Equivalent photon method (TM et al., PRC63(’01)) Expt: T. Nakamura et al., PRL96,252502(2006) Energy resolution with =0.17 MeV. No three-body resonance T.Myo, K.Kato, H.Toki, K.Ikeda PRC76(2007)024305

25 Summary TOSM+UCOM with bare nuclear force. V tensor enhances LS splitting energy. Coexistence of tensor and pairing correlations, explicitly Halo formation in 11 Li Review Di-neutron clustering and deuteron-like tensor correlation in nuclear structure focusing on 11 Li K. Ikeda, T. Myo, K. Kato and H. Toki Springer, Lecture Notes in Physics 818 (2010) "Clusters in Nuclei" Vol.1,