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Few-Body Systems in Low Energy Effective Theory. 鎌田裕之(九州工業大学) E. Epel b aum ( Juelich 研究所+Bonn大学) W. Glöckle ( Bochum 大学) Ulf-G. Meissner ( Bonn 大学). KEK研究会 『 原子核・ハドロン物理:横断研究会 』 高エネルギー加速器研究機構、素粒子原子核研究所 2007 年 11 月 19 日 ( 月 ) ~ 11 月 21 日 ( 木 ) KEK 4 号館 1 階セミナーホール.

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Few body systems in low energy effective theory

Few-Body Systems in Low Energy Effective Theory

鎌田裕之(九州工業大学)

E. Epelbaum(Juelich研究所+Bonn大学)

W. Glöckle(Bochum大学)

Ulf-G. Meissner(Bonn大学)

KEK研究会

『原子核・ハドロン物理:横断研究会』

高エネルギー加速器研究機構、素粒子原子核研究所2007年11月19日(月)~11月21日(木)KEK4号館1階セミナーホール


New generation formalism

Basic Yukawa formalism

  • 2NF

  • 1.Bonn Potential

  • 2.Argonne Potential

  • 3.Nijmegen Potential

  • 3NF

  • Fujita-Miyazawa

  • Urbana IX

  • Tucson-Melbourn

New generation formalism

Meson theoretical realistic formalism

★Consistence with QCD

★Unification of 2NF and 3NF

★Applicability

Chiral Perturbation theoretical formalim


Line up
Line up

  • Feshbach-Bloch-Horowitzの方法

  • Low-Momentum NN Interaction

  • Okubo方程式を解く

  • Okubo理論を場の量子論に適用する

  • カイラル摂動理論

  • 3体力&4体力

  • Summary & Outlook


Interaction
Low-Momentum NN Interaction

EffectiveTheory

- Tutorial introduction -

Feshbach-Bloch-Horowitzの方法

B&H,NP8,(1958)91.

Okubo 理論:S.Okubo,PTP12,(1954)603


Feshbach bloch horowitz
Feshbach-Bloch-Horowitzの方法

B&H,NP8,(1958)91.

Q

P

λ

λ

Veff=Veff(E)


Okubo理論と散乱振幅

Low-Momentum NN Interaction


Aについての非線形方程式


Kubo i
kubo方程式を解く (I)


Aについての線形方程式


Binding energies of and
Binding energies of 3H and 4He

Λ→

Λ→

S. Fujii, E. Epelbaum, H. Kamada, R. Okamoto, K. Suzuki,  W. Glöckle,

Physical Review C 70, 024003 (2004)


Binding energy of h

Binding Energy of 3H

Eb [MeV]


Binding energy of he

Binding Energy of 3He

Eb [MeV]



Kubo ii
kubo方程式を解く (II)

ポイント:ベキ展開によって漸化式を求める。



Okubo理論を場の量子論に適用する

  • フォック空間 

    φ:π中間子が現れない(on-mass-shell)→P

    ψ:π中間子の現れる(1個、2個、3個・・・)

    ψ=ψ(1)+ ψ(2) + ψ(3)・・・       →Q

0π:NNのみ

1π

2π

φ

ψ(1)

ψ(2)

ψ(*)

。。。


Okubo理論を場の量子論に適用する

  • フォック空間 

    φ:π中間子が現れない(on-mass-shell)

    ψ:π中間子の現れる(1個、2個、3個・・・)

    ψ=ψ(1)+ ψ(2) + ψ(3)・・・

  • Full Hamiltonian

    Η =H0+HI

    H0=HN0 + Hπ0HN0=-N†(∇2/2m)N

    Hπ0=(1/2)π2+(1/2)(∇π)2+(1/2)mπ2π2

    N(π):核子(π中間子)の場の演算子

  • 相互作用HIは、例えばカイラル・ラグランジアンを用いる


Chiral perturbation theory
Chiral Perturbation Theory

Chirality:

Symmetry of massless QCD Lagrangian:

SU(Nf)L× SU(Nf)R×U(1)V×U(1)A

Nambu-Goldstone-WeinbergRealization:

Mechanism of the sponteneous breaking symmetry:

SU(2)L× SU(2)R ~  SU(2)A× SU(2)V ⇒  SU(2)V

SU(2)A× SU(2)V~SO(4)  Dim[SO(4)]=4>3πfields

Nonlinear realization :π→π’=f(π;g)


相互作用H

Low Energy Coefficient: CT,CS,C1,C2,C4



Okubo
Okubo 方程式


Yukawaforce:1πon exchange

Contact force


Chiral perturbation theory1
Chiral Perturbation Theory

2NF

3NF

4NF

ν=0

&

ν=2

π+N

Δ+heavy meson

expansion

ν=3

ν=4


2NF

3NF

4NF

Chiral Perturbation Theory

Nonrelativistic

limit

ν=0

&

ν=2

π+N

Δ+heavy meson

expansion

ν=3

ν=4


Chiral perturbation theory2
Chiral Perturbation Theory

2NF

3NF

4NF

ν=0

&

FM3NF

ν=2

π+N

Δ+heavy meson

expansion

ν=3

ν=4


NLO

NNLO

Cross section

Ay

T20

T21

T22

3MeV

NLO

NNLO


NLO

NNLO

Cross section

Ay

T20

T21

T22

10 MeV

NLO

NNLO


NLO

NNLO

Cross section

Ay

T20

T21

T22

65 MeV

NLO

NNLO


Three-body break-up reaction

FSI configration

QFS configuration

Space Star configuration

13 MeV

NLO

NNLO



Chiral perturbation theory3
Chiral Perturbation Theory

2NF

3NF

4NF

ν=0

&

FM3NF

ν=2

π+N

Δ+heavy meson

expansion

ν=3

ν=4



Tucson melbourn 3nf
Tucson Melbourn 3NF

g  (σ・q)

4π (m2+q2)

(σ・q’)

m2+q’2 

W=

F(q,q’)

F(q,q’)=a +b (q ・q’)+c(q2+q’2)+d σ・(q×q’)


Relation to TM – 3NF parameters

F(q,q’)=a +b (q ・q’)+c(q2+q’2)+d σ・(q×q’)

c1,c2 and c3 are

parameter free.

The condition c=0 makes

the 3NF new as called

TM’-3NF.


Faddeev three-body calculation

for the proton-deuteron elastic

scattering with the

realistic NN potential and

the three-nucleon force

65

Differential Cross Section

Elab[MeV]

2NF only

190

135

3NF included

TM’ 3NF

Urbana IX 3NF

Sagara Discrepancy

Phys. Rev. C 63, 024007 (2001)


Faddeev three-body calculation

for the proton-deuteron elastic

scattering with the

realistic NN potential and

the three-nucleon force

65

Tensor Polarization T20

Elab[MeV]

2NF only

3NF (original TM)

190

135

TM’ 3NF

Urbana IX 3NF




Chiral perturbation theory4
Chiral Perturbation Theory

2NF

3NF

4NF

ν=0

&

ν=2

π+N

Δ+heavy meson

expansion

ν=3

ν=4


2006.11.17

35 Diagram

21 Diagram

8 Diagram


Fujita-Miyazawa 3NF

b-term,d-term

Urbana 3NF

Tucson-Melbourne 3NF

Brazil 3NF

(1957)

a-term,(c-term)

Scalar Short range

U0

πρ exchange:

F(IΔ+), Kroll-Ruderman term

Illinoi Model

Chiral perturbation Theoretical 3NF (NNNLO)

・3π exchange

・・・・・・・{

・2π-1π term

・2π exchange between all threenucleons

・contact 1πexchange

・contact 2πexchnge



Chiral perturbation theory5
Chiral Perturbation Theory

2NF

3NF

4NF

ν=0

&

ν=2

π+N

Δ+heavy meson

expansion

ν=3

ν=4



α粒子(4核子系)における4体力の寄与

  • CT=0 の場合

  • Gaussian:            - 270 keV

  • (Λ,Λ)=(400,500):    - 386 keV

  •      =(550,500):   - 219 keV

〔MeV/c〕

Acta Physica Polonica B37, 2889-2903 (2006)




[MeV]

V5 < 6kV


Summary
Summary

  • Okubo理論は、カイラル摂動理論に用いることによって、多核子間のポテンシャルをコンシステントに導く.(2NF,3NF,4NF,・・・)

  • TM3体力のc項の不必要性を予言し、それによって、陽子・重陽子散乱におけるT20などの偏極量を改善している.

  • 3体力は重要視されつつあり,中重核への適用が期待される.


Outlook

LEC(Cn, Dn, En,・・・)は,QCDラグランジアンから求めるべき“観測量”である.

Outlook

QCD

Quark, Gluon

・・・ confinement ?

χPT

N, π


QCD

χPT

N, π: confinement

P

Q

VNN

VNNN

VNNNN

VπN

Vππ


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