1. Y. Ikeda and T. Sato (Osaka Univ.) ストレンジ・ダイバリオンの 質量と崩壊幅の研究 KNN resonance (Recent theoretical progress) KNN resonance (Recent theoretical progress) Faddeev approach and variational approach Faddeev approach and variational approach Numerical Results Numerical Results Summary Summary

2. KNN resonance -- Recent theoretical progress --

3. KNN resonance – L(1405) - KNN resonance – L(1405) - -> S-wave resonance in the KN- pS coupled channel system in the KN- pS coupled channel system (Chiral unitary model) Structure of L(1405)  Multi-quark state?  Bound state of KN? Large meson-baryon components Chiral SU(3) dynamics Chiral SU(3) dynamics It will be very important It will be very important to take into account the full dynamics of KN- pS system to take into account the full dynamics of KN- pS system in order to investigate whether KNN resonance may exist. in order to investigate whether KNN resonance may exist.

4. KNN resonance – Theoretical progress - KNN resonance – Theoretical progress - Ikeda, Sato (45-80, 45-75)MeV Shevchenko et al. (55-70, 90-110)MeV Faddeev equation (B, G ) Dote, Hyodo, Weise (17-23, 40-70)MeV Akaishi, Yamazaki (48, 60)MeV Variational Method (B, G ) Chiral SU(3) Phenomenological KN interaction 3 body Method  Faddeev equation -> Full dynamics of KNN- pS N system  Variational approach -> pS N system is effectively included.

5. Faddeev approach and Variational approach

6. Faddeev Equation  W : 3-body scattering energy  i(j) = 1, 2, 3 (Spectator particles)  T(W)=T 1 (W)+T 2 (W)+T 3 (W) (T : 3-body amplitude) t i (W) : 2-body t-matrix with spectator particle i  t i (W) : 2-body t-matrix with spectator particle i  G 0 : 3-body Green’s function (relativistic kinematics) Faddeev approach Faddeev approach

7. AGS and Faddeev eqs. are equivalent within separable potential model. Separable potential : Two-body t-matrix : Alt-Grassberger-Sandhas(AGS) Equation i j i j = X ij i j tntntntn + n

8. Faddeev approach v.s. Variational approach Faddeev approach v.s. Variational approach  Effective KN interaction Hyodo, Weise PRC77(2008). NN K K ・・・・・・ S p NN KK  Effective KN interaction in KNN system NN K K N -> Faddeev approach -> Variational approach At least, the spectator momentum is neglected in the pS N Fock space.

9. KN potential model KN potential model F : Meson field, B : Baryon field Weinberg-Tomozawa interaction Coupling const. Form factor S-wave Weinberg-Tomozawa potential

10. Parameter fit (KN interaction) Parameter fit (KN interaction) Our parameters -> cut-off of dipole form factor Fit 1 : L (1405) pole position given by Dalitz (Model Dalitz) Fit 2 : Hemingway’s experiment (Model Hemingway) K-K-K-K-p p-p-p-p- p+p+p+p+ p S S + ( 1660 ) L(1 405) NPB253, 742(1985) Invariant mass

11. Dalitz and Deloff JPG17, 289 (1991)., Nacher et al., PLB461, 299 (1999). Parameter fit (KN interaction) Parameter fit (KN interaction) with assumption

12. Experimental data (total cross section) Experimental data (total cross section) (I=0 channel) (I=1 channel)

13. i j i j = X ij i j tntntntn + Summary of our framework Summary of our framework K, N KN- p Y, NN W pole = -B –i G /2 Similar to πNN, ηNN, K - d analyses. (Matsuyama, Yazaki, ……) Eigenvalue equation for Fredholm kernel 3-body resonance pole at W pole Alt-Grassberger-Sandhas(AGS) Equation

14. Numerical results

15. The pole position of three-body KNN system The pole position of three-body KNN system Dalitz W pole =-67-i22 Hemingway W pole =-47-i25 KNN physical p YN unphysical sheet Dalitz(Approx.) W pole =-42-i35 Hemingway(Approx.) W pole =-32-i26

16. The reason of less binding energies The reason of less binding energies Model Dalitz (W=-67 MeV) Model Hemingway (W=-47 MeV)

17. KN interaction dependences of KNN poles KN interaction dependences of KNN poles Model Dalitz Model Hemingway

18. reaction We compare variational approach with Faddeev approach by using the approximate 2-body KN amplitude. by using the approximate 2-body KN amplitude. We find the different pole energies We find the different pole energies corresponding to KNN state for each approach. KNN state becomes the bound state KNN state becomes the bound state as increasing KN interaction. as increasing KN interaction. Summary In the future This production mechanism will be investigated by LEPS and CLAS collaborations. @SPring8, Jlab