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Dynamics of Macroscopic and Microscopic 　　 　　　　　　　 Three-Body Systems

Outline

Three-body systems of composite particles (clusters)Macroscopic = Use of fewer degrees of freedom　　　 20C+n+n ： 　 20C: shell-model inert core 3α ： 　　　　 α: (0s)4 nucleon cluster 　　　 3-nucleon ： 　 N: (0s)3 quark clusterPauli principle, nonlocality, energy-dependence 　　　　

Y. Suzuki (Niigata)

Collaborators: Y. Fujiwara (Kyoto), H. Matsumura (Niigata), M. Orabi (Niigata)

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Unexplored Three-body System

Pauli constraint acts only between core-n

Giant two-neutron haloS-wave dominance

W.Horiuchi and Y.S. PRC, in press

Reaction cross sections

A~ 60

Borromean, n-dripline

SVM on CG

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x1=x2=x x ＝ 5 fm θ=17○

Two-neutron Correlation Function

22C

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12C as 3α System

Ali-Bodmer potential: shallow, L-dependent, no bound states Buck-Friedrich-Wheatley potential: deep, L-independent, redundant states 0s, 1s, 0d bound states

These 2αpotentials produce poor results for 3α and 4α systems

Supersymmetric transform

ααlocal potential in macroscopic approach

D.Baye, PRL58(1987)

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Solution with Removal of Redundant States

Orthogonalizing pseudo potential

Allowed states

(for any pairwise redundant states)

Kukulin and Pomerantsev, Ann. Phys. 111 (1978)

Solution is to be found in allowed state space

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Comparison of 3α Allowed States

Matsumura,Orabi,Suzuki,Fujiwara, NPA, in press

important in shell model

0+

Q=30

Ns=174

(NA=129,

NF=43)

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Energy of 12C from 3αThreshold

BFW potential

Expt.

HOFS

Tursunov,Baye,Descouvemont. NPA723(2003)Matsumura,Orabi,Suzuki,Fujiwara, NPA, in press

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energy-dependent, nonlocal potential

Intercluster potential (RGM)

Note ：

A

BB

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(self-consistency)

Fujiwara et al., Prog.Theor.Phys.107(2002)

Use of 2-cluster RGM kernel

A B

C

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Summary of 3α Calculations

Interaction States eliminated ground state energy (MeV)

BFW Bound states of -0.22 the potential BFW HOWF -19.3

2αRGM HOWF -9.6 Kernel NN potential (HOWF) -11.3 (microscopic) Expt. -7.27

Matsumura,Orabi,Suzuki,Fujiwara, NPA, in pressFujiwara et al., Few-Body Systems 34(2004),PRC70(2004)

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Meson Theory Short-Ranged Interaction Compositeness of Baryons (0s)3 quark cluster Baryon-Baryon Interaction with SU(6) quark model OGEP+EMEP at quark level FSS: Pseudo Scalar, Scalar PRC54 (1996) fss2: Pseudo Scalar, Scalar, Vector PRC65 (2002)

Application to Triton and Hypertriton PRC66 (2002), PRC70 (2004)

Fujiwara,Suzuki,Nakamoto, PPNP, in press

Three-Nucleon System with Quark-Model Potential

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np Phase Shifts (S,P,D)

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Deuteron properties

np effective range parameters

Prediction with Quark Model Potential Isospin basis, NoCSB

T

t

t

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Triton Binding Energy vs Deuteron D-state Probability

8.519 MeV

8.394 MeV

8.48 MeVPRC66(2002)

Ｓａｌａｍａｎｃａ PRC65(2002)７ . ７２ ＭｅＶ （５ｃｈ ) ＰＤ＝４ . ８５％　　　Ｔａｋｅｕｃｈｉ et al. NPA508(1990) ８ . ０１ MeV （５ｃｈ） PD ＝５ . ５８％

no charge dependenceexcept CD-Bonn

(34 ch)

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Two-nucleon system; Tensor force and Central force are counterbalanced Tensor force more (less) attractive (D-state probability larger (weaker)) Central force less (more) attractive More-nucleon system; Effects of Tensor force are reduced D-state probability larger (Central force less attractive) Weaker binding

Role of Tensor force in many-nucleon system

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　　　　　　 3α-system 　 Triton--system 　　　　　　

Local pot. 　　 BFW Realistic ForceNonlocal pot. 2αRGM NNRGM Kernel (fss2, …)

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Summary

1. Macroscopic three-body systems with clusters are useful, with the following reservations 2. A significant difference appears in 3αsystem 　 depending on the choice of redundant states (Pauli principle effects) Its reason is now clear. 3. The quark model potential gives larger binding for triton in spite of large D-state probability (energy-dependent, nonlocal potential)

Use of 2-cluster RGM kernels in three-cluster system is appealing, though further study remains to clarify roles of off-shell property, E-dependence, etc

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model model parametersparameters 　　　　　　

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Decomposition of triton energy:

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Three-body problems: advantage: accurate solutions for bound states possible　　　　　　　　　　　　　　　　　 　 Faddeev, Variational (CBF,SVM,…) interest: interplay between interaction and structure

Three-body systems with composite particles (clusters) 　 micoscopic macroscopic mapping interaction between clusters role of Pauli principle

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Three-body System

Pauli constraint acts only between core-n

Giant two-neutron halo

Density of n-n relative motion

W.Horiuchi and Y.S. PRC, in press

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αα 　 Phase Shifts 　　　

Redundant states: 0s, 1s, 0d

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np Phase Shifts

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Hypertriton (pnΛ)

Potential BΛ(keV) PΣ(%)

fss2 289 0.805 FSS 878 1.361

Exp. 130(50)

Nogga,Kamada,Glockle,PRL88(2002)

Miyagawa,Kamada,Glockle,Stoks,PRC51(1995)

1S0/3S1 ΛN interaction

PRC70(2004)

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NN and YN total cross sections(fss2)

recent KEK exp’t Y. Kondo et al. Nucl. Phys. A676 (2000) 371

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αα RGM Phase Shifts 　

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T

t

t

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Triton Binding Energy vs Deuteron D-state Probability

8.519 MeV

8.394 MeV

8.48 MeVPRC66(2002)

Ｓａｌａｍａｎｃａ PRC65(2002)７ . ７２ ＭｅＶ （５ｃｈ ) ＰＤ＝４ . ８５％　　　Ｔａｋｅｕｃｈｉ et al. NPA508(1990) ８ . ０１ MeV （５ｃｈ） PD ＝５ . ５８％

no charge dependenceexcept CD-Bonn

(34 ch)