1. Bound kaon approach for the ppK- system in the Skyrme model
Tetsuo Nishikawa (Tokyo Ins. Tech.)
Yoshihiko Kondo (Kokugakuin U.)
2019/4/6
Nuclear Physics at J-PARC 01-02/06/07

2. ppK- bound state
Lightest K-nucleus proposed by Dote, Akaishi & Yamazaki.
Its existence is theoretically still unclear.
The interpretation of the experiment by FINUDA collab. is still controversial.
More clear evidence is expected to be seen in future

3. Our interests Does the Skyrme model suggest deeply bound ppK- states?
⇒ Can the energy of K- coupled with pp be considerably small ?
If yes, we want to know
mechanism responsible for the strong binding
possible structure of ppK-

4. Skyrme model

5. Skyrme model Skyrme, 1961
Nucleons = topological solitons of the pion field
Skyrme’s ansatz for chiral (pion) field:
The isospin is oriented to the radial direction, .
“Hedgehog” ansatz

6. Skyrme model Hedgehog ansatz: Soliton profile:
n : “winding number” classifies the mapping
Winding number (conserved) = Baryon number
minimizing the energy
, … S3 or
n=1
n=2

7. The action of the Skyrme model
Skyrme term
Wess-Zumino-Witten term
Skyrme term
stabilizes solitons
Wess-Zumino-Witten term
remove an extra symmetry of chiral Lagrangian
ensure the Skyrmion to behave like a fermion
total
Skyrme term
NL-sigma term
(E.Witten, 1983)

8. Bound kaon approach to the Skyrme model

9. Bound kaon approach to the Skyrme model
(Callan and Klevanov, 1985)
A kaon field fluctuates around the SU(2) Skyrmion.
The kaon field has bound states.
lowest-lying mode: T ( =L+I )=1/2, L=1
next mode: T=1/2, L=0
Wess-Zumino-Witten term split S=±1 states
S=-1 states: bound states hyperons
S=+1 states (e.g. pentaquark) continuum

10. Why bound kaon approach?
No parameter, once we adjust Fπ and e to fit N and Δ masses in SU(2)f sector.
KN interaction, which is a key ingredient for the study of K-nuclei, is unambiguously determined.
It reproduces the mass of Λ(1405) as well as Λ, Σ, Ξ etc..

11. Description of the ppK- system

12. Description of the ppK- system
(2-Skyrmion)
+ (kaon field fluctuating around the Skyrmion)
bound kaon

13. Description of the ppK- system
Derive the kaon’s EoM for Skyrmions at fixed positions
kaon’s energy
Solve the NN dynamics
the binding energy of the ppK- system
VNN R
ωK? R

14. Derivation of the kaon’s EoM
Substitute the ansatz
into the action
Expand up to O(K2) and neglect O(1/Nc) terms
Lagrangian of the kaon field
under the background of B=2 Skyrmion

15. Derivation of the kaon’s EoM
Collective coordinate quantization
projection onto (pp)S=0
Average the orientation of
Spherical partial wave analysis is allowed:
Equation of motion for the kaon

16. Results

17. Energy eigenvalue of K-
P-wave
S-wave
R=2.0 fm
BK=220 MeV
R=1.5 fm
BK=340 MeV

18. K- (s-wave) distribution
ωK=156MeV
Proton position
R=1.5fm

19. K- distribution and baryon# density
ωK=272MeV
Proton position
R=2.0fm

20. K- distribution and baryon# density
mK=495MeV
ωK=349MeV

21. K- distribution and baryon# density
Molecular
states ?
mK=495MeV
ωK=397MeV

22. Effects of the Wess-Zumino-Witten term
Switching off the WZW term,…

23. Roles of the Wess-Zumino-Witten term in the Skyrme model
Remove an extra symmetry the chiral Lagrangian possesses
Ensure the Skyrmion behaves as a fermion
(E.Witten, 1983)
Gives an attractive contribution to S=-1 state
correct values of hyperon masses
Λ(1405) is bound owing to the WZW term alone !

24. Roles of the WZW term Strong binding of K- to pp seems to be natural
Interaction from the WZW term,
attractive contribution VWZW to K-.
VWZW for B=2 is stronger than that for B=1,
since LWZW ∝(baryon#).
makes K- bound to pp very deeply
This scenario is very similar to that proposed by AY.
Strong binding of K- to pp seems to be natural
within the Skyrme model.

25. Summary
Within the product ansatz Skyrme approach, the energy of K- bound to pp can be considerably small.
K- is centered between pp for small R, while for larger separation it shows molecular nature.
Only the attractive interaction arising from the Wess-Zumino-Witten term binds Λ(1405).
This interaction plays an important role also for the strong binding of K- to pp.
Solving the p-p radial motion to obtain the total energy of ppK is now in progress.