Table of contents Introduction Model Results

1. Toshitaka Uchino Tetsuo Hyodo, Makoto Oka Tokyo Institute of Technology 10 DEC 2010 Table of contents Introduction Model Results

2. I. Introduction The Λ* hypernuclei model Kbar nuclei bound states attract much attention. From several theoretical works, the Λ* N bound state is found to be a dominant component in the KbarNN bound state. Λ* and N bounds Bound state We take the viewpoint that the KbarNN bound state is regarded as the bound state of the Λ*N (Λ* hypernuclei). • Λ*-hypernuclei model has advantages ; • Including other two body, the ΛN and ΣN contributions. • Few body calculation.

3. Arai-Oka-Yasui model[1] and our model AOY model is constructed as follows : • s-wave : dominant for the lowest energy state. • Λ*is regarded as a elementary particle. • Potential : extending the YN OBEP. • For S=0, S=1. • Variational method. • Interaction : determined phenomenologically. • Purpose : fitting the results of FINUDA exp. Our model is following AOY model, but ; • Interaction : determined with the chiral unitary approach. • Purpose : finding possible bound states.

4. Λ* hypernuclei model with chiral dynamics By using the chiral unitary approach, the Λ* is described as meson-baryon multiple scattering. The coupling constant of the Λ* to MB channels are taken from[2,3]. The Λ* is dynamically generated as a superposition of two states. Each Λ* interact with nucleon, whereas the transition between each Λ*N state can take place. Then, we solve the coupled channel Schrödinger Eq.

5. II. Model The Λ*NOBEP We construct the Λ*N potential by extending the Juelich (Model A) potential[4]. It is the simplest one-boson-exchange potential which includes hyperon. Because isospin of the Λ* =0, isoscalar meson is exchanged, namely σ, ω. We further consider the Kbar exchange.

6. Kaon exchange 1 – spin dependence Considering that the parity of the Λ* is odd, the Λ*KbarN coupling is a scalar type. So the Kbar exchanged potential is essentially same as the scalar exchange in the NN potential, but it depends on the total spin S. Exchange factor Attractive for S=0 Repulsive for S=1

7. Kaon exchange 2 - effective Kaon mass In the Kaon propagator, since the energy transfer is not zero, we use the effective Kaon mass . It becomes smaller as the resonance energy is close to the KbarN threshold. Namely, in the upper energy state , Kaon exchange is stronger than the .

8. Coupling constants in our potential Coupling constants in our potential are classified into three types. Coupling constants determined by the chiral unitary approach are complex value, so we take its absolute value. : Chiral : Unknown : Juelich The unknown coupling constant is estimated by using the Λ*structure from chiral unitary approach analysis.

9. Estimation of the Λ*Λ*N(X=σ, ω) coupling By chiral dynamics, exchanged meson couples to the constituent baryonor meson in the . So the coupling constants can be estimated by summing up the microscopic contribution. We deal only dominant components, KbarN and πΣ. ππσis determined by σ decay : Chiral : KKbarσ is assumed to be 0 : Juelich Estimated coupling constants are complex. To obtain real value, we take their absolute value.

10. III. Results potentials

11. Bulk property of the potential To study the bulk property of the potential, we calculate the volume integral of the potential. • This results show that • The potential is attractive(repulsive) for S=0(S=1). • The potential is stronger than the , because of the stronger coupling constants and the lighter effective Kaon mass.

12. Bound states of the system S=1: No bound states S=0: Only bounds With no mixing With mixing S=0: More bounds

13. Wave function We obtain the wave function of the bound state for each . Each state is peaking at ～0.5 fm. The state is dominant, but the state is also important.

14. Decay width : B → πΣN We consider the case that the in the bound state decays with the nucleon being a spectator. The coupling constant is given by the chiral unitary approach.

15. Summary *As the strangeness S=-1, the baryon number B=2 Λ*-hypernuclei system, the Λ*Nbound state is studied. *The Λ*N one-boson-exchange potential is constructed by extending the Juelich potential. *The unknown coupling constant is estimated by using the information of the Λ* structure obtained from chiral unitary approach. *Solving the Schrödinger eq, we obtain the bound state solution for S=0 ;

16. Backup slides

17. Decay width If there exists the bound states, we can estimate the decay width with obtained wave function.

18. Cut-off mass The coupling strength depends on the exchanged meson momentum. This effect is taken into account as monopole type form factor. For vertices NNX(X=σ, ω), cut-off is given by Juelich potential. But, cut-off masses concerning the Λ* is unknown. We take into account the size of the Λ* and nucleon as parameter “c”. The unknown cut-off can be written with “c”. Considering the size of the Λ* [5], “c” is assumed to be 1.5.

19. c dependence Binding energies and decay width depend on the size of the Λ* ,parameter c. *Small “c “ leads to shallow bound. *πΣN decay is dominated by kinematics.

20. Future plans 1. Other decay modes 2. Extension of our model 3. Few body calculations

21. Other diagrams Using obtained wave function, other decay width, Non- mesonic decay, ΛN ,ΣN and mesonic decay πΛN, πΣN, can be estimated. Non-mesonic decay B → ΛN B → ΣN Mesonic decay B → πΛN B → πΣN

22. Extension of our model To include the information of the Λ* given by chiral dynamics more directly, we need model improvement. Energy dependence : Estimated : Juelich : Chiral The Λ* energy dependence in the Λ*Λ*X and Λ*KbarN vertices should be taken into account, when the Λ*N system bounds deeply. Including complexness Λ*N potential Complex Several parameters concerning the Λ*are complex value. So, our model needs an extension.

23. Other channel contribution Other two-body channels, the ΣN and ΛN contribution can be included within our model. Few body calculation Extension to few body studies, the Λ*NN and Λ*NNN can be calculated, using the Λ*N potential.