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

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 exp.@J-PARC.

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-

Skyrme model

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

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

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)

Bound kaon approach to the Skyrme model

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

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..

Description of the ppK- system

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

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

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

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

Results

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

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

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

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

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

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

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 !

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.

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.