Role of tensor force in He and Li isotopes with tensor optimized shell model Hiroshi TOKI RCNP, Osaka Univ. Kiyomi IKEDA RIKEN Atsushi UMEYA RIKEN Takayuki MYO Osaka Institute of Technology The Fifth Asia-Pacific Conference on Few-Body Problems in Seoul, Korea,

Purpose & Outline 2 We would like to understand role of V tensor in the nuclear structure by describing strong tensor correlation explicitly. Tensor Optimized Shell Model (TOSM) to describe tensor correlation Unitary Correlation Operator Method (UCOM) to describe short-range correlation TOSM+UCOM to He & Li isotopes with V bare TM, K. Kato, H. Toki, K. Ikeda, PRC76(2007) TM, H. Toki, K. Ikeda, PTP121(2009)511 TM, A. Umeya, H. Toki, K. Ikeda, PRC(2011) in press.

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

TM, Sugimoto, Kato, Toki, Ikeda PTP117(2007)257 4 Tensor-optimized shell model (TOSM) 4 4 He 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 such as b 0s, b 0p, … are optimized independently (or superposed by many Gaussian bases). –Describe high momentum component from V tensor –Spatial shrinkage of relative D-wave component as seen in deuteron HF by Sugimoto et al,(NPA740) / Akaishi (NPA738) RMF by Ogawa et al.(PRC73), AMD by Dote et al.(PTP115)

Hamiltonian and variational equations in TOSM c.m. excitation is excluded by Lawson’s method (0p0h+1p1h+2p2h) TOSM code : p-shell region Particle state : Gaussian expansion for each orbit Gaussian basis function

Configurations of 5 He 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 C0C0 C1C1 C2C2 C3C3 Tomorrow

7 Unitary Correlation Operator Method 7 H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61 short-range correlator Bare Hamiltonian Shift operator depending on the relative distance TOSM 2-body cluster expansion Amount of shift, variationally determined. (short-range part)

8 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

4-8 He with TOSM+UCOM Difference from 4 He in MeV No V NNN No continuum ~6 MeV in 8 He using GFMC ~7 MeV in 8 He using Cluster model (PLB691(2010)150, TM et al.) 4p4h in TOSM

4-8 He with TOSM+UCOM Excitation energies in MeV No V NNN No continuum Excitation energy spectra are reproduced well

5-9 Li with TOSM+UCOM Excitation energies in MeV No V NNN No continuum Preliminary results Excitation energy spectra are reproduced well

12 Matter radius of He isotopes I. Tanihata et al., PLB289(‘92)261 G. D. Alkhazov et al., PRL78(‘97)2313 O. A. Kiselev et al., EPJA 25, Suppl. 1(‘05)215. P. Mueller et al., PRL99(2007) TOSM Expt TM, R. Ando, K. Kato PLB691(2010)150 Halo Skin Cluster model

13 Configurations of 4 He (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] Cf. R.Schiavilla et al. (VMC) PRL98(’07) He contains p 1/2 of “pn-pair”. 0  of pion nature. deuteron correlation with (J,T)=(1,0)

14 Tensor correlation in 6 He 14 Ground state Excited state TM, K. Kato, K. Ikeda, J. Phys. G31 (2005) S1681 Tensor correlation is suppressed due to Pauli-Blocking val.-n halo state (0 + )

6 He : Hamiltonian component Difference from 4 He in MeV 6 He n 2 config(p 3/2 ) 2 (p 1/2 ) 2  Kin  Central  27.8  22.9  14.1  Tensor  12.0  12.8  0.2  LS  4.0  =18.4 MeV (hole) b hole =1.5 fm same trend in 5,7,8 He

16 Summary TOSM+UCOM with bare nuclear force. 4 He contains “pn-pair” of p 1/2 than p 3/2. He isotopes with p 3/2 has large contributions of V tensor & Kinetic energy than those with p 1/2. V tensor enhances LS splitting energy. –TM, A. Umeya, H. Toki, K. Ikeda, PRC(2011) in press. 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,

4-8 He in TOSM+UCOM Argonne V8’ w/o 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. TM, H. Toki, K. Ikeda, PTP121(2009)511 TM, A. Umeya, H. Toki, K. Ikeda, PRC, in press.

5 He : Hamiltonian component Difference from 4 He in MeV 5 He 3/2  1/2  n config.p 3/2 p 1/2  Kin  Central  9.0  7.0  Tensor  5.6  1.1  LS  =18.4 MeV (hole) b hole =1.5 fm

LS splitting in 5 He & 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 splitting is weaker than p-wave splitting Pauli-Blocking val.-n

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

4-8 He with TOSM No V NNN No continuum Difference from 4 He in MeV Minnesota force (Central+LS)

22 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’)

Tensor force (V tensor ) plays a significant role in the nuclear structure. –In 4 He,  V tensor  ~  V central  (AV18, GFMC) –Main origin :  -exchange in pn-pair 23 Importance of tensor force We would like to understand the role of V tensor in the nuclear structure by describing tensor correlation explicitly. tensor correlation + short range correlation model wave functions : shell model, cluster model (TOSM: Tensor Optimized Shell Model) He, Li isotopes (LS splitting, halo formation, level inversion)

7 He : Hamiltonian component Difference from 4 He in MeV 7 He 3/2  1 3/2  2 3/2  3 n 3 config(p 3/2 ) 3 (p 3/2 ) 2 (p 1/2 )(p 3/2 )(p 1/2 ) 2  Kin  Central  39.1  33.6  27.0  Tensor  19.9  16.0  10.0  LS  8.6  2.8  1.0 b hole =1.5 fm

8 He : Hamiltonian component Difference from 4 He in MeV 8 He n 4 config(p 3/2 ) 4 (p 3/2 ) 2 (p 1/2 ) 2 (p 3/2 ) 3 (p 3/2 )  Kin  Central  59.2  51.6  56.2  Tensor  25.2  17.1  23.4  LS  11.6  2.55  7.1 b hole =1.5 fm