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少数体系アプローチの研究と今後の課題Few-Body Approach and Future Problems

・ NN interaction is characterized by strong short-range repulsion 　 and long-range tensor force ・ Accurate solution is possible for FBS・ The interplay between BB interaction and dynamics of strongly interacting few-body quantum systems is revealed ・ The effect of three-body forces is one of current issues

Y. Suzuki (Niigata)

1. Ab initio calculation in FBS2. Towards more-particle systems3. Continuum problems4. Breakup reactions

Present status and future direction on

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NN potential

Even partial waves

Odd partial waves

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Benchmark calculation for the ground state of 4He FY CRCGV, SVM, HH (Variational) GFMC NCSM, EIHH (P-space effective int.)

1.1 Various accurate methods for bound states

H.Kamada et al. PRC64 (2001)

AV8’

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Correlation functions for s-shell nuclei

Triplet even

Singlet even

Y. Suzuki, W. Horiuchi, arXiv (2008)

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Correlation functions (continued)

Triplet odd

Coulomb

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Density

Hiyama et al. PRC70 (2004)

1.2 First excited state of 4He

3N+N cluster state

Inelastic electron scatt. form factor

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±

Horiuchi,Ikeda: PTP 40(1968)

Quartets:

J: 1/2 + 1/2 + 0 = 0, 1 T: 1/2 + 1/2 = 0, 1 3N + N structure

Asymmetric clusters Parity inverted state E.g. Ammonia molecule of NH3

Inversion doublets: J: 1/2+ 1/2 + 1 = 0, 1, 2

Questions arising from 3N+N clusters with spins

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Quartets, Negative parity partners, 0-0 and 0-1 level spacing

W.Horiuchi et al. PRC78 (2008)

1.3 Energy levels of 4He

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Only 02+0 has a peak near 3N surface, indicating a resonance

Spectroscopic amplitude (SA)

W.Horiuchi et al. PRC78 (2008)

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Width of 0- : 0.61 MeV (Cal)　 0.84 MeV

(Exp)

Negative parity partners

Peak position Centrifugal barrier

3N+N cluster structure Inversion doublet

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・ Binding energies・ The ground state of 10B (1+ or 3+) S.C.Pieper et al. PRC66 (2002) E.Caurier et al., PRC66 (2002)・ Scattering observables Nd scattering

1.4 Three-body forces

See Proceedings of FM 50 (2007)

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Effects of three-body forces: Correct spin-parity of 10B～ 20MeV contribution for 12C

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--- sensitive to short-range and tensor correlations--- 1.5 Momentum distribution

Dueteron: D-wave fills the dip of S-wave Effects of short-range repulsion

6He: nn (pp) pair 6Li: np pair

W. Horiuchi et al. PRC76 (2007) T. Suda et al. 6He(p,dn)4He

理研 .08R. Schiavilla et al. PRL98 (2007)

Ｑ =0 ：　 Back to back geometry 　　 pn (lines) pp (symbols)

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Dependence on Q

R.B. Wiringa et al. PRC78 (2008)

pn (lines) 4 pp (symbols) 1

Q=p1+p2

q=(p1-p2)/2

R. Subedi et al. Science 320 (2008) Exp. for 12C

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H.Nemura et al. PRL94 (2005)

1.6 Accurate calculations needed to explore YN and YY interactions in Hypernuclei

Interactions are poorly known experimentally

ΛN-ΣN coupling,

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・ GFMC (A ～ 12)

・ NCSM, UMOA (P-space effective interaction) Intruder states (e.g. Excited 0+ states of 12C and 16O) Slow convergence

・ Transformation to milder interaction (indep. of P and Q) UCOM (Unitary transf., cluster exp.) Transcorrelated method (Similarity transf.)

・ Semi-microscopic model Assuming a core nucleus or a cluster

・ DFT

2 Extension to more-particle systems

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2.1 GFMC

S.C.Pieper et al. PRC66 (2002)

GFMC propagation requires huge storage of memory ～ 3A-1 2A 2A 12C(A=12) ～ 3×1012 (3 兆）

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2.2 NCSM

Convergence for intruder states is slowHuge size of memory is required

12C Nmax=8 M=0 states in m-scheme Basis dimension 594,496,743 (6 億）　　　 No. of nonzero matrix elements for 2B potentials 539,731,979,351 (5400 億）

01 02

P. Maris et al. arXiv (2008)

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2.3 Transcorrelated Method

E-indep. effective interaction eliminating short-range repulsion

Separation of short-range repulsionChoosing f(r) to eliminate W

HTC is indep. of P and Q, non-Hermitean.Energy minimization is not applicable.

Y.Suzuki et al. PTP113 (2005)

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Assuming clusters Phenomenological interaction is used Pauli-forbidden states

2.4 Semi-microscopic model

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Ambiguity in cluster potentials

Y. Suzuki et al. PLB659 (2008)

Energy-indep. nonlocal potential

Exp. -7.27 0.38 MeV

12C=3αmodel

Dep. of E on phase-equivalent α-α potentials

Different off-shell behavior

---RGM formalism---

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2.5 Density Functional Theory

P. Hohenberg, W. Kohn, PR136 (1964)W. Kohn, L.J. Sham, PR140 (1965)

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Critical difference between electron gasses and nuclei Self-bound system with no external (s.p.) potential

Is the DFT justifiable for nuclei?

Y. Suzuki, W. Horiuchi, arXiv (2008)

Correlation functions are basic variables

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・ Strength function CSM, LITM ・ Scattering phase shifts

3.1 Application of discretized states

to continuum problems

K.M. Nollett et al. PRL99 (2007)

Effects of three-body forces in α+n scattering phase shifts

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3.2 Complex Scaling Method

T.Myo et al. PRC63 (2001)

4He+n+n model for 6He

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3.3 Lorentz Integral Transform method

Invert Lorentz integral transform to obtain R or σ

V.D.Efros et al. PLB338 (1994)

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4He photo-absorption cross section

Proc. of FM 50

S.Quaglioni et al. PLB652 (2007)

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3.4 Scattering phase shift

α+n scatteringeffective force(central+LS)

R-matrix (lines)SAGF (symbols)

---correcting spectroscopic amplitude with Green’s function (SAGF)---

Study with realistic interactions is in progress

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4.1 Breakup reactions of halo nuclei

Elastic scattering of 6He on 12C

Breakup effects of fragile nucleus

α+ n + n three-body model for 6He

Continuum-discretized states Coupled-channel calculation 　(cdcc)

T. Matsumoto et al. PRC70 (2004)

VMC wave function for 6He

Glauber model: 3α microscopic cluster model w.f. for 12CNN profile function

Eikonal approx.: N-12C optical potential

Folding

Full

40 MeV/nucleon

B. Abu-Ibrahim et al. NPA 728 (2003)

Breakup effects are taken into account by Glauber- and Eikonal-model calculations

Description of the elastic breakup reaction of two-neutron halo nucleus

Challenging four-body problem including continuum final states ・ How to solve ・ Final-state interaction ・ Extraction of E1 strength function or effects of other multipoles

6He breakup on 208Pb at 240 MeV/A

Coulomb-corrected eikonal model J. Margueron et al. NPA703 (2002): P. Capel et al. PRC78 (2008)

D. Baye et al. submitted T. Aumann et al. PRC59 (1999)

Hoping for

1. Fundamental and Breakthrough Works 2. Center for Discussions and Facilities 3. Positions for Young Promising Physicists

Example: α ＋ α S-wave scattering phase shifts with realistic potentials ～ 4000 (Nα)2 times Time(α+n) Time(α+n)=0.1 day on a PC Nα=10 at least 40,000 days on a single processor Demand for a number of parallel processors