few-body systems in low energy effective theory
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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Few-Body Systems in Low Few-Body Systems in Low Energy Effective Theory Energy Effective Theory
Few-Body Systems in Low Few-Body Systems in Low Energy Effective Theory Energy Effective Theory
鎌田裕之(九州工業大学)鎌田裕之(九州工業大学)E. EpelE. Epel bb aumaum (( JuelichJuelich 研究所+Bonn大研究所+Bonn大
学)学)W. GlöckleW. Glöckle (( BochumBochum 大学)大学)
Ulf-G. MeissnerUlf-G. Meissner (( BonnBonn 大学)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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Basic Yukawa formalism
Meson theoretical realistic formalism
★Consistence with QCD★Unification of 2NF and 3NF
★Applicability
2NF1. Bonn Potential2. Argonne Potential3. Nijmegen Potential 3NF1. Fujita-Miyazawa2. Urbana IX3. Tucson-Melbourn
New generation formalism
Chiral Perturbation theoretical formalim
Chiral Perturbation theoretical formalim
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www.kyutech.ac.jp Line upLine upLine upLine 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
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Low-Momentum NN Low-Momentum NN InteractionInteractionLow-Momentum NN Low-Momentum NN InteractionInteraction
- Tutorial introduction -
EffectiveEffective TheoryTheoryEffectiveEffective TheoryTheory
Feshbach-Bloch-HorowitzFeshbach-Bloch-Horowitz の方法の方法
Okubo 理論:S.Okubo,PTP12,(1954)603
B&H, NP8,(1958)91.
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www.kyutech.ac.jp Feshbach-Bloch-HorowitzFeshbach-Bloch-Horowitz の方法の方法Feshbach-Bloch-HorowitzFeshbach-Bloch-Horowitz の方法の方法
B&H, NP8,(1958)91.
V eff= V eff (E)
QP
λ λ
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www.kyutech.ac.jp Okubo理論と散乱振幅Okubo理論と散乱振幅Okubo理論と散乱振幅Okubo理論と散乱振幅Low-Momentum NN Low-Momentum NN InteractionInteractionLow-Momentum NN Low-Momentum NN InteractionInteraction
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Aについての非線形方程式
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www.kyutech.ac.jp OO kubokubo 方程式を解く 方程式を解く (I)(I)OO kubokubo 方程式を解く 方程式を解く (I)(I)
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Aについての線形方程式
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Binding energies of Binding energies of 33H H and and 44HeHeBinding energies of Binding energies of 33H H and and 44HeHe
Λ→Λ→
S. Fujii, E. Epelbaum, H. Kamada, R. Okamoto, K. Suzuki, W. Glöckle ,Physical Review C 70, 024003 (2004)
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Binding Energy of Binding Energy of 33 H H Binding Energy of Binding Energy of 33 H H
Potential No 3NF With 3NF
CDBONNNijmegen INjimegen IINijmegen 93AV-18
8.013 7.741 7.659 7.668 7.628
8.48 8.48 8.48 8.48 8.48
Exp. 8.48
Eb [MeV]
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Binding Energy of Binding Energy of 33 He He Binding Energy of Binding Energy of 33 He He
Potential No 3NF
CDBONNNijmegen INjimegen IINijmegen 93AV-18
7.288 7.083 7.008 7.014 6.917
Exp. 7.718
Eb [MeV]
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Binding Energy of αparticle Binding Energy of αparticle Binding Energy of αparticle Binding Energy of αparticle
Eb [MeV]Potential No 3NF With
3NF
CDBONNNijmegen INjimegen IINijmegen 93AV-18
26.26 24.98 24.56 24.53 24.25
28.40 28.60 28.56 28.36
Exp. 28.30
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www.kyutech.ac.jp OO kubokubo 方程式を解く 方程式を解く (II)(II)OO kubokubo 方程式を解く 方程式を解く (II)(II)
ポイント:ベキ展開によって漸化式を求める。
*
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核子間相互作用核子間相互作用核子間相互作用核子間相互作用
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www.kyutech.ac.jp Okubo理論を場の量子論に適用するOkubo理論を場の量子論に適用するOkubo理論を場の量子論に適用する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)→Pψ : π 中間子の現れる ( 1個、2個、3
個・・・)ψ = ψ (1)+ ψ (2) + ψ (3)・・・
→ Q
・
0 π: NNのみ
。。。
1 π 2 π
ψ (1) ψ ( 2 ) ψ (*)φ
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www.kyutech.ac.jp Okubo理論を場の量子論に適用するOkubo理論を場の量子論に適用するOkubo理論を場の量子論に適用する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)ψ : π 中間子の現れる ( 1個、2個、3個・・・)ψ = ψ (1)+ ψ (2) + ψ (3)・・・ Full H amiltonianΗ =H0+HI
H0=HN0 + Hπ0 HN0=-N† (∇2 /2m)N Hπ0= ( 1/2)π 2+( 1/2) (∇ π ) 2 +( 1/2)mπ
2 π 2
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
・
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Chiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation Theory
Chirality :Symmetry of massless QCD Lagrangian: SU(N f )L× SU(N f )R×U(1)V×U(1)A
Nambu - Goldstone - Weinberg Realization :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>3 πfieldsNonlinear realization :π→π’ =f( π ;g)→→→
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www.kyutech.ac.jp 相互作用H相互作用HII相互作用H相互作用HII
Low Energy Coefficient: CT,CS,C1,C2,C4
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Low Energy Coefficient: D1,D2,C1~ C7
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Okubo 方程式Okubo 方程式
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Yukawaforce: 1 πon exchangeContact force
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Chiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation Theory
&
π + N Δ + heavy meson
expansion Q
ν =0
ν =2
ν =3
ν =4
3NF2NF 4NF
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&
π + N Δ + heavy meson
expansionQ
Nonrelativisticlimit
0
ν =0
ν =2
ν =3
ν =4
2NF 3NF 4NF
Chiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation Theory
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Chiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation Theory
&
π + N Δ + heavy meson
expansionQ
ν =0
ν =2
ν =3
ν =4
FM 3 NF
2NF 3NF 4NF
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3MeV
Cross section
Ay
T20
T21
T22
NLO
NNLO
NLO NNLO
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Cross section
Ay
T20
T21
T22
10 MeV
NLO
NNLO
NLO NNLO
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Cross section
Ay
T20
T21
T22
65 MeV
NLO
NNLO
NLO NNLO
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FSI configration
QFS configuration
Space Star configuration
13 MeV
Three-body break-up reaction
NLO
NNLO
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65 MeV
NLO
NNLO
Three-body break-up reaction
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Chiral Perturbation Theory
&
π + N Δ + heavy meson
expansionQ
ν =0
ν =2
ν =3
ν =4
FM 3 NF
2NF 3NF 4NF
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NNLO 3NF
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www.kyutech.ac.jp Tucson Melbourn 3NFTucson Melbourn 3NFTucson Melbourn 3NFTucson Melbourn 3NF
g ( σ ・ q )
4 π ( m 2+ q 2)W=
( σ ・ q’ )
m 2+ q’ 2 F( q,q’ )
F
F ( q,q’ ) =a +b (q ・ q’)+c(q2+q’2)+d σ ・( q×q’ )
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F ( q,q’ ) =a +b (q ・ q’)+c(q2+q’2)+d σ ・( q×q’ )
21 3
2 2
42
4 2, ,
, .0
m c c
f f
c
f
b
dc
a
The condition c=0 makes the 3NF new as called TM’-3NF.
c1,c2 and c3 are parameter free.
Relation to TM – 3NF parameters
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Faddeev three-body calculation for the proton-deuteron elastic
scattering with the realistic NN potential and the three-nucleon force
Differential Cross Section
Elab [MeV ]
Phys. Rev. C 63, 024007 (2001)
3
135
65
1902 NF only
3 NF included
TM’ 3NF
Urbana IX 3NF
Sagara Discrepancy
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Tensor Polarization T20
Elab [MeV ]
3 65
190135
Faddeev three-body calculation for the proton-deuteron elastic
scattering with the realistic NN potential and the three-nucleon force
2 NF only
3 NF ( original TM)
TM’ 3NF
Urbana IX 3NF
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NNLO 3NF
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Low Energy Constant
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Chiral Perturbation Theory
&
π + N Δ + heavy meson
expansion Q
ν =0
ν =2
ν =3
ν =4
3NF2NF 4NF
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2006.11.17
21 Diagram
35 Diagram
8 Diagram
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Fujita-Miyazawa 3NF
b-term, d-term
Urbana 3NF Tucson-Melbourne 3NFBrazil 3NF
Scalar Short rangeU0
a-term,(c-term)
πρ exchange :F(IΔ+) , Kroll-Ruderman term
Chiral perturbation Theoretical 3 NF (NNNLO)
・2 π-1π term・2 π exchange between all three nucleons・ contact 1πexchange・ contact 2πexchnge
Illinoi Model
・3 π exchange・・・・・・・{
(1957)
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4NF4NF
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Chiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation Theory
&
π + N Δ + heavy meson
expansion Q
ν =0
ν =2
ν =3
ν =4
3NF2NF 4NF
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Possible Diagrams (NNNLO)Possible Diagrams (NNNLO)
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CT=0 の場合Gaussian : - 270 k eV(Λ,Λ )=(400 , 500): - 386 keV =(550 , 500): - 219 keV
~
Acta Physica Polonica B37, 2889-2903 (2006)
〔 MeV/c 〕
α粒子(4核子系)における4体力の寄与
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5NF5NF
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[MeV
]
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[MeV
]
V5 < 6kV
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SummarySummary
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.
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OutlookOutlookLEC ( 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 .
Quark, Gluon
N, π
QCD
χPT・・・ confinement ?
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QCD
χPT N, π: confinement
VNN VNNN VNNNN
VπN Vππ…
P Q