Role of tensor force in He and Li isotopes with tensor optimized shell model

1. Role of tensor force in He and Li isotopes with tensor optimized shell model Takayuki MYO Osaka Institute of Technology Hiroshi TOKI RCNP, Osaka Univ. Kiyomi IKEDA RIKEN Atsushi UMEYA RIKEN The Fifth Asia-Pacific Conference on Few-Body Problems in Physics @ Seoul, Korea, 2011.8.22-26

2. Purpose & Outline • We would like to understand role of Vtensorin 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+UCOMto He & Li isotopes with Vbare TM, K. Kato, H. Toki, K. Ikeda, PRC76(2007)024305 TM, H. Toki, K. Ikeda, PTP121(2009)511 TM, A. Umeya, H. Toki, K. Ikeda, PRC(2011) in press.

3. Deuteron properties& tensor force AV8’ S D Vcentral AV8’ Rm(s)=2.00 fm Rm(d)=1.22 fm r d-wave is “spatially compact” (high momentum) Vtensor

4. Tensor-optimized shell model (TOSM) TM, Sugimoto, Kato, Toki, Ikeda PTP117(2007)257 4He • Configuration mixingwithin 2p2h excitationswith high-Lorbits.TM et al., PTP113(2005) TM et al., PTP117(2007) T.Terasawa, PTP22(’59)) • Length parameters such as b0s, b0p, … are optimized independently (or superposed by many Gaussian bases). • Describe high momentum component from Vtensor • Spatial shrinkage of relative D-wave component as seen in deuteronHF by Sugimoto et al,(NPA740) / Akaishi (NPA738) RMF by Ogawa et al.(PRC73), AMD by Dote et al.(PTP115) 4 4

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

6. Configurations of 5He in TOSM Gaussian expansion particle states C1 C0 C3 C2 nlj proton neutron hole states (harmonic oscillator basis) c.m. excitation is excluded by Lawson’s method Application to Hypernuclei by Umeya Tomorrow

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

8. 4He in TOSM + S-wave UCOM T (exact) Kamada et al. PRC64 (Jacobi) TM, H. Toki, K. Ikeda PTP121(2009)511 • variational calculation VLS E • Gaussian expansion • with 9Gaussians VC VT 8 good convergence

9. 4-8He with TOSM+UCOM • Difference from 4He in MeV ~6 MeV in 8He using GFMC • No VNNN • No continuum 4p4h in TOSM ~7 MeV in 8He using Cluster model (PLB691(2010)150 , TM et al.)

10. 4-8He with TOSM+UCOM • Excitation energies in MeV • No VNNN • No continuum • Excitation energy spectra are reproduced well

11. 5-9Li with TOSM+UCOM Preliminary results • Excitation energies in MeV • No VNNN • No continuum • Excitation energy spectra are reproduced well

12. Matter radius of Heisotopes TM, R. Ando, K. Kato PLB691(2010)150 Cluster model Expt TOSM Halo Skin 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)252501 12

13. Configurations of 4He • 0 of pion nature. • deuteron correlation with (J,T)=(1,0) Cf. R.Schiavilla et al. (VMC) PRL98(’07)132501 • 4He contains p1/2 of “pn-pair”. 13 13

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

15. 6He : Hamiltonian component • Difference from 4He in MeV bhole=1.5 fm =18.4 MeV (hole) same trend in 5,7,8He

16. Summary TOSM+UCOM with bare nuclear force. 4He contains “pn-pair” of p1/2 than p3/2. He isotopes with p3/2has large contributionsof Vtensor& Kinetic energy than those with p1/2. Vtensorenhances 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 11Li K. Ikeda, T. Myo, K. Kato and H. Toki Springer, Lecture Notes in Physics 818 (2010) “Clusters in Nuclei” Vol.1, 165-221. 16

17. 4-8He in TOSM+UCOM Argonne V8’ w/o Coulomb force. Convergence for particle states. Lmax =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.

18. 5He : Hamiltonian component • Difference from 4He in MeV bhole=1.5 fm =18.4 MeV (hole)

19. LS splitting in 5He & tensor correlation val.-n Pauli-Blocking • 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(a+n OCM) • 30% of the observed splitting from Pauli-blocking • d-wave splitting is weaker than p-wave splitting

20. Tensor correlation & Splitting in 5He Vtensor~exact in 4He Enhancement of Vtensor

21. 4-8He with TOSM Minnesota force (Central+LS) • Difference from 4He in MeV • No VNNN • No continuum

22. Short-range correlator : C (or Cr) [fm] Shift function Transformed VNN (AV8’) 1E Originalr2 s(r) 3GeV repulsion 3E Vc 1O C 3O We further introduce partial-wave dependencein “s(r)” of UCOM S-wave UCOM 22

23. Importance of tensor force • Tensor force (Vtensor) plays a significant role in the nuclear structure. • In 4He, Vtensor ~  Vcentral (AV18, GFMC) • Main origin : p-exchange in pn-pair • We would like to understand the role of Vtensor 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) 23

24. 7He : Hamiltonian component bhole=1.5 fm • Difference from 4He in MeV

25. 8He : Hamiltonian component bhole=1.5 fm • Difference from 4He in MeV