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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. 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

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  1. 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

  2. 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

  3. 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

  4. 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

  5. Feshbach-Bloch-Horowitzの方法 B&H,NP8,(1958)91. Q P λ λ Veff=Veff(E)

  6. Okubo理論と散乱振幅 Low-Momentum NN Interaction

  7. Aについての非線形方程式

  8. Okubo方程式を解く (I)

  9. Aについての線形方程式

  10. 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)

  11. Binding Energy of 3H Eb [MeV]

  12. Binding Energy of 3He Eb [MeV]

  13. Binding Energy of αparticle Eb [MeV]

  14. Okubo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。

  15. 核子間相互作用

  16. 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(*) 。。。 ・

  17. 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 ・

  18. 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) → → →

  19. 相互作用HI Low Energy Coefficient: CT,CS,C1,C2,C4

  20. Low Energy Coefficient: D1,D2,C1~C7

  21. Okubo 方程式

  22. Yukawaforce:1πon exchange Contact force

  23. Chiral Perturbation Theory 2NF 3NF 4NF ν=0 & ν=2 π+N Δ+heavy meson expansion ν=3 ν=4

  24. 2NF 3NF 4NF Chiral Perturbation Theory Nonrelativistic limit 0 ν=0 & ν=2 π+N Δ+heavy meson expansion ν=3 ν=4

  25. Chiral Perturbation Theory 2NF 3NF 4NF ν=0 & FM3NF ν=2 π+N Δ+heavy meson expansion ν=3 ν=4

  26. NLO NNLO Cross section Ay T20 T21 T22 3MeV NLO NNLO

  27. NLO NNLO Cross section Ay T20 T21 T22 10 MeV NLO NNLO

  28. NLO NNLO Cross section Ay T20 T21 T22 65 MeV NLO NNLO

  29. Three-body break-up reaction FSI configration QFS configuration Space Star configuration 13 MeV NLO NNLO

  30. Three-body break-up reaction 65 MeV NLO NNLO

  31. Chiral Perturbation Theory 2NF 3NF 4NF ν=0 & FM3NF ν=2 π+N Δ+heavy meson expansion ν=3 ν=4

  32. NNLO 3NF

  33. Tucson Melbourn 3NF g  (σ・q) 4π (m2+q2) (σ・q’) m2+q’2  W= F(q,q’) F F(q,q’)=a +b (q ・q’)+c(q2+q’2)+d σ・(q×q’)

  34. 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.

  35. Faddeev three-body calculation for the proton-deuteron elastic scattering with the realistic NN potential and the three-nucleon force 3 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)

  36. Faddeev three-body calculation for the proton-deuteron elastic scattering with the realistic NN potential and the three-nucleon force 3 65 Tensor Polarization T20 Elab[MeV] 2NF only 3NF (original TM) 190 135 TM’ 3NF Urbana IX 3NF

  37. NNLO 3NF

  38. Low Energy Constant

  39. Chiral Perturbation Theory 2NF 3NF 4NF ν=0 & ν=2 π+N Δ+heavy meson expansion ν=3 ν=4

  40. 2006.11.17 35 Diagram 21 Diagram 8 Diagram

  41. 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

  42. 4NF

  43. Chiral Perturbation Theory 2NF 3NF 4NF ν=0 & ν=2 π+N Δ+heavy meson expansion ν=3 ν=4

  44. Possible Diagrams (NNNLO)

  45. 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)

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