IIAS-2

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Tensor optimized shell model using bare
interaction for light nuclei
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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

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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

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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 repulsion in 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
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Property of the tensor force
Long and intermediate ranges

• Centrifugal potential (1GeV_at_0.5fm) pushes away
  the L2 wave function.

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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 orbit TM 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 Vtensor
  CPP-HF by Sugimoto et al,(NPA740) / Akaishi
  (NPA738)CPP-RMF by Ogawa et al.(PRC73), CPP-AMD
  by Dote et al.(PTP115)

4He
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Hamiltonian and variational equations in TOSM
TM, Sugimoto, Kato, Toki, Ikeda, PTP117(07)257

• Effective interaction Akaishi force (NPA738)
• G-matrix from AV8 with kQ2.8 fm-1
• Long and intermediate ranges of Vtensor survive.
• Adjust Vcentral to reproduce B.E. and radius of
  4He

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4He in TOSM
vnn G-matrix
Shrink
Length parameters
Orbit bparticle/bhole
0p1/2 0.65
0p3/2 0.58
1s1/2 0.63
0d3/2 0.58
0d5/2 0.53
0f5/2 0.66
0f7/2 0.55
0 1 2 3 4 5 6
Lmax
Higher shell effect
good convergence
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Cf. K. Shimizu, M. Ichimura and A. Arima,
NPA226(1973)282.
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Configuration of 4He in TOSM
Energy (MeV) ? 28.0
? 51.0
4 Gaussians instead of HO
(0s1/2)4 85.0
(0s1/2)2JT(0p1/2)2JT JT10 5.0
JT01 0.3
(0s1/2)210(1s1/2)(0d3/2)10 2.4
(0s1/2)210(0p3/2)(0f5/2)10 2.0
PD 9.6
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
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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 repulsion in the bare NN force
• Unitary Correlation Operator Method (UCOM)

S
D
TOSMUCOM
H. Feldmeier, T. Neff, R. Roth, J. Schnack,
NPA632(1998)61 T. Neff, H. Feldmeier
NPA713(2003)311
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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
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Short-range correlator C (or Cr)
1E
Original?r2
3GeV repulsion
3E
Vc
1O
?C
3O
AV8 CentralLSTensor
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4He in UCOM (Afnan-Tang, Vc only)
?C
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4He with AV8 in TOSMUCOM
AV8 Central LSTensor
exact
Kamada et al. PRC64 (Jacobi)

• Gaussian expansion for particle states (6
  Gaussians)
• Two-body cluster expansion of Hamiltonian

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Extension of UCOM S-wave UCOM
for only relative S-wave wave function
minimal effect of UCOM
SD coupling
5 MeV gain
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Different effects of correlation function

• S-wave

S
No Centrifugal Barrier
Short-range repulsion
D

• D-wave

due to Centrifugal Barrier
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Saturation of 4He in UCOM
UCOM short-range tensor
T. Neff, H. Feldmeier NPA713(2003)311
Short tensor
Energy
TOSMUCOM
Long tensor
Benchmark cal. Kamada et al. PRC64
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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
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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 TOSMUCOM, we can study the nuclear structure
  starting from the bare interaction.
• Spectroscopy of light nuclei (p-shell, sd-shell)

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Pion exchange interaction vs. Vtensor
Involve large momentum
Delta interaction
Tensor operator
Yukawa interaction
- Vtensor produces the high momentum component.
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Characteristics of Li-isotopes
Halo structure

• Breaking of magicity N8
• 10-11Li, 11-12Be
• 11Li (1s)2 50. (Expt by Simon et
  al.,PRL83)
• Mechanism is unclear

11Li
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Pairing-blocking K.Kato,T.Yamada,K.Ikeda,PTP1
01(99)119, Masui,S.Aoyama,TM,K.Kato,K.Ikeda,NPA
673('00)207. TM,S.Aoyama,K.Kato,K.Ikeda,PTP10
8('02)133, H.Sagawa,B.A.Brown,H.Esbensen,PLB3
09('93)1.
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11Li in coupled 9Linn model

• System is solved based on RGM

TOSM

• Orthogonality Condition Model (OCM) is applied.

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11Li G.S. properties (S2n0.31 MeV)
Rm
Simon et al.
P(s2)
Tensor Pairing
E(s2)-E(p2) 2.1 1.4 0.5 -0.1 MeV
Pairing correlation couples (0p)2 and (1s)2 for
last 2n
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2n correlation density in 11Li
s24
s247
9Li
9Li
n
n
Di-neutron type config.
Cigar type config.
H.Esbensen and G.F.Bertsch, NPA542(1992)310
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K. Hagino and H. Sagawa, PRC72(2005)044321
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Short-range correlator C (or Cr)
Hamiltonian in UCOM
2-body approximation in the cluster expansion of
operator
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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.

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Phase shifts of 4He-n scattering
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6He in coupled 4Henn model

• System is solved based on RGM

TOSM

• Orthogonality Condition Model (OCM) is applied.

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Tensor correlation in 6He
Ground state
Excited state
TM, K. Kato, K. Ikeda, J. Phys. G31 (2005) S1681
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6He results in coupled 4Henn model
Theory With Tensor
complex scaling for resonances

• (0p3/2)2 can be described in Naive 4Henn model
• (0p1/2)2 loses the energy

Tensor suppression in 02
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