バリオンのソリトン描像から見た K 中間子束縛核

1. バリオンのソリトン描像から見たK中間子束縛核バリオンのソリトン描像から見たK中間子束縛核 based on hep-ph/0703100; arXiv:0710.0948 西川哲夫 （東工大） 近藤良彦（國學院大学） 横断研究会@KEK 19/11/07

2. ppK-: the lightest kaonic nuclei Akaishi&Yamazaki, 2003 • Kaonic nuclei: • Kbar-nucleus bound states formed by strong attraction in (KbarN)I=0 • Experimental evidence (?) by FINUDA collaboration: • peak in the invariant mass spectrum of Λp from • B.E.=115MeV, Γ=67MeV • Still controversial • Future experiment planned at J-PARC:

3. Theoretical studies of ppK-

4. Anit-kaon nucleus bound states? Large Nc QCD • QCD at Nc→∞: weakly interacting meson theory • Fundamental degrees of freedom: mesons (Diagrams representing meson propagation are dominant.) • Interaction between mesons〜O(1/ Nc) (t’Hooft, 1974) • Baryons emerge as topological solitons (“Skyrmion”) of the meson field. (Skyrme, 1961; Witten, 1983) c.f. derived from the string theory via AdS/CFT correspondence (Sakai&Sugimoto, 2005; Nawa, Suganuma, Kojo, 2007)

5. Why soliton model? • The action, written in terms of NG boson fields U, respects chiral symmetry and reproduces anomaly (Wess-Zumino-Witten term). • KN interaction is unambiguously determined, once Fπ and e are fixed, e.g. fitted to MN and MΔ. • Hyperons can be well described as kaon-soliton bound states. Skyrme term NL-sigma model Wess-Zumino-Witten term

6. Skyrmion

7. Skyrmion (Skyrme, 1961) Skyrmion: topological soliton of the pion field • Ansatz for pion field: Isospin is oriented to the radial direction, “Hedgehog ansatz”

8. or S3 S3 S3 , … n=2 n=1 Skyrmion • Hedgehog represents a mapping: classified by n (“winding number”). • Winding number: conserved ∵ Mappings with different n cannot be smoothly connected with each other • Winding number = Baryon number

9. Zero mode (collective coordinate) quantization • Zero mode: displacement without changing the energy • invalidates the semi-classical approx. • full quantum mechanical treatment is necessary. • The Skyrme lagrangian is invariant under • Regarding A(t) as collective coordinate, quantize the rigid body rotation of the Skyrmion, Nucleon spectrum as rotational spectrum of a Skyrmion

10. Bound kaon approachto the strangeness in the Skyrme model

11. bound kaon Bound kaon approach in the Skyrme model (Callan and Klebanov, 1985) Hyperon • Kaon’s equation of motion under the b.g. of Skyrmion • Bound states of K-Skyrmion • Quantize the collective rotation of the bound system • Hyperon mass spectrum

12. Baryon masses Callan and Klebanov, 1985; Rho, Riska and Scoccola, 1992 (mπ=0,FK/Fπ=1.23) • Set I: Fπ and e are fitted to MN and MΔ(Adkins, Nappi&Witten). • Set II: fitted to MΛ(1405) and MΔ (present study)

13. Description of ppK- systemin the bound kaon approach

14. Description of the ppK- system Hedgehog skyrmion boundkaon

15. Description of the ppK- system • Kaon’s EoM for Skyrmions at fixed positions (adiabatic approximation) kaon’s energy • Solve the pp radial motion rough estimate of the binding energy of ppK- VNN+ωK R R

16. R Derivation of the kaon’s EoM • Ansatz for chiral field • U(1)and U(2): hedgehogs centered at r =±R/2 • UK: kaon field

17. Derivation of the kaon’s EoM • Substitute the ansatz into the action • Expand up to O(K2) and neglect O(1/Nc) terms • Lagrangian for K under the background B=2 Skyrmion (KN interaction is unambiguously determined, once the ansatz for the chiral field is given.)

18. Derivation of the kaon’s EoM • Collective coordinate quantization projection of the skyrmion rotation onto (pp)S=0 • Average the orientation of Spherical partial wave analysis is allowed: EoM for the kaon in S-, P-,..wave

19. Results

20. Energy of K- normal nuclei • Dependence on the choices, Set I or II, is weak. • S-wave K- is strongly bound even for relatively large R, e.g. • BK= 233MeV (R=1.5 fm) • BK= 139MeV (R=2.0 fm) R=2.0 fm BK= 139MeV

21. R Distribution of K- and baryon # density Molecular nature

22. K- K- ppK-: “molecular” state Molecular states ⇒ deep binding of K- approaching proton Suppose Λ(1405) is an “atomic” state

23. Origin of the strong binding: WZW term

24. Role of the Wess-Zumino-Witten term • Origin of the WZW term: anomaly in QCD • Effective theory should reproduce anomaly in QCD ⇒ WZW term • Effects： • An extra symmetry of the chiral Lagrangian is removed. • Skyrmion behaves like a fermion. • KN Interaction from the WZW term, • attractive potential VWZW to negative strangeness states • Correct mass spectrum of hyperons • Λ(1405) is bound owing to the VWZW alone. • gives a double-well potential for K- coupled with pp

25. p-p potential with and without K- pp potential without K- (VNN) pp potential in ppK- (VNN+VK) Energy of K- bound to pp (VK=ωK-mK)

26. p-p potential with and without K- Akaishi&Yamazaki,2007 pp potential without K- (VNN) pp potential in ppK- (VNN+VK) Energy of K- (VK=ωK-mK)

27. p-p radial motion • Assume p-p radial motion is governed by the Hamiltonian: • From VNN+ωK-mK R

28. Parameter set of Fπ and e • Set I: fitted to MN and MΔ. • Set II: fitted to MΛ(1405) and MΔ Binding energy of ppK-and its decomposition

29. Conclusion • π中間子場のソリトンとして表された陽子２個に結合したK-のエネルギーは著しく小さくなり得る。 （Wess-Zumino-Witten項が大きな役割を果たす。） • K-の空間分布は、ppK-が分子状態であることを示唆する。 • K-が生む非常に強い引力が、斥力的なppポテンシャルに勝って、ppK-を深く束縛させる。 〜我々のアプローチでのシナリオ〜

30. K- p π Σ K- p p π Σ p A comment • Λ(1405)は必ずしも KbarN 束縛状態ではない。K-soliton全体の回転を量子化して得られた状態。 • 我々のアプローチ • ２個のソリトンを独立に回して量子化し、ppに射影 • K-を束縛させる。 • より適切には、新たに集団座標を導入し、

31. total Skyrme term NL-sigma term The action of the Skyrme model • Skyrme term (included by hand) stabilizes solitons. • Wess-Zumino-Witten term • remove an extra symmetry of the chiral Lagrangian • makes a Skyrmion behave like a fermion Skyrme term NL-sigma model Wess-Zumino-Witten term