1. S. Aoki (Univ. of Tsukuba), T. Doi (Univ. of Tsukuba), T. Hatsuda (Univ. of Tokyo), T. Inoue (Univ. of Tsukuba), N. Ishii (Univ. of Tokyo), K. Murano (Univ. of Tokyo), H. Nemura (Tohoku Univ.), K. Sasaki (Univ. of Tsukuba) Kaon-Nucleon potential from lattice QCD Yoichi Ikeda (Univ. of Tokyo) for HAL QCD collaboration

2. Plan of this talk Plan of this talk Introduction and our motivation Introduction and our motivation Formalism Formalism Numerical results and discussions Numerical results and discussions Summary and future plans Summary and future plans

3. Experimental study of  + state by LEPS collaboration  nK + invariant mass distribution shows a narrow peak around 1.524 GeV  Statistical significance of the peak is 5.1   The obtained results support the evidence of the   state Introduction Introduction Many experiments at high energy show negative results. No narrow peak was observed by CLAS collaboration. Open problems Reaction-dependent production mechanism?

4. Introduction Introduction LQCD studies for signal of  + state Quenched LQCD for energy of penta-quark (J=1/2, 3/2) Quenched LQCD for energy of penta-quark (J=1/2, 3/2) (e.g., Csikor, Sasaki, Chiu, Mathur, Ishii, Takahashi, Alexandrou, Lasscock, Holland)  Penta-quark might be allowed, but most probably NK scattering state. but most probably NK scattering state.  Penta-quark might be allowed, but most probably NK scattering state. but most probably NK scattering state. Current status from LQCD One possible explanation of  + state One possible explanation of  + state  Hadronic molecule (NK) or bound state (N  K) Need precise information on NK interaction

5. Introduction Introduction NK scattering phase shift  repulsive NK (I=0, 1) scattering phase shifts Hashimoto, PRC 29. S01 S11 I=0 I=1 No attraction in any range were found. Potential from Quark model Barnes-Swanson, PRC49. Barnes-Swanson, PRC49. Consistent with meson-exchange model? e.g.,) Julich group, NPA506. LQCD simulations solve the ambiguities of the NK potential.

6. Our motivation Our motivation Previous LQCD studies of the  + are all quenched. Full LQCD simulation is necessary. PACS-CS config. (2+1 flavors) We are almost on the “physical point”. PRD79(2009).

7. Our motivation Our motivation Production mechanism of  + state might be reaction dependent. The nK + potential derived from QCD powerful tool to analyze the nature of the  +. Previous LQCD studies of the  + are all quenched. Full LQCD simulation is necessary. This study We derive the nK + potential from Full LQCD simulation.

8. Formalism (HAL procedure) Formalism (HAL procedure) Developed by Ishii, Aoki, and Hatsuda for NN system Developed by Ishii, Aoki, and Hatsuda for NN system PRL99, 022001(2007). 3) Derive the potential from Schrodinger Eq. 1) Define interpolating operators Wave func.  Potential  Observable 2) Calculate the equal-time BS amplitude

9. This work Formalism (This work) Formalism (This work) Schrodinger Eq. We investigate the s-wave nK + potential. S-wave projected nK + wave function and potential

10. Numerical set-up Numerical set-up 2+1 flavor full QCD configuration by CP-PACS/JLQCD 2+1 flavor full QCD configuration by CP-PACS/JLQCD RG improved gauge action & RG improved gauge action & O(a) improved Wilson-clover quark action O(a) improved Wilson-clover quark action Lattice spacing : a=0.1209 [fm] Lattice spacing : a=0.1209 [fm] Size of Lattice : 16 3 ×32  L=1.93 [fm] Size of Lattice : 16 3 ×32  L=1.93 [fm] Hopping parameters :, Hopping parameters :, # of conf. = 700 # of conf. = 700 Flat wall source to provide NK state. Flat wall source to provide NK state.

11. Numerical results (Hadron effective masses) Numerical results (Hadron effective masses) G 2 (t) : 2-point function Wall source Dirichlet B.C. Mass [MeV] Fit range Pion870.7(1.9)5-10 Kaon911.5(1.9)5-10 Nucleon1795.5(6.9)6-11 NK threshold 2707(8) MeV

12. nK + effective mass in J=1/2 - channel nK + effective mass in J=1/2 - channel Wall source Dirichlet B.C. Plateau NK threshold (s-wave) Single state saturation is achieved. Single state saturation is achieved. The best fit in the plateau gives M eff =2723(10) MeV. The best fit in the plateau gives M eff =2723(10) MeV. G 4 (t) : NK temporal correlator

13. S-wave nK + BS wave function and potential S-wave nK + BS wave function and potentialPotential Wave function  Repulsive core at short distance ( r<0.5[fm] )  Attractive pocket in middle range ( 0.5<r<1.2 [fm] ) I=0 I=1

14. S-wave nK + scattering phase shift S-wave nK + scattering phase shift S01 S11  Qualitatively consistent with experimental data experimental data  Pion mass is 871 MeV in our set-up  Note Gaussian core + (Yukawa x form factor) 2 Gaussian core + (Yukawa x form factor) 2  The range of Yukawa-type potential = 805 MeV Fitting function

15. Summary Summary  S-wave nK + potential derived from full LQCD is studied. - We found repulsive core at short distance and weak attractive pocket in the middle range. - We found repulsive core at short distance and weak attractive pocket in the middle range.  We calculate the scattering phase shift - Qualitatively consistent with experimental data - Qualitatively consistent with experimental data  We also study the effective mass of the nK + state in J=1/2 - channel. - The low-lying state is consistent with nK + threshold. - The low-lying state is consistent with nK + threshold. Future plans Future plans  We will study the quark mass dependence of the s-wave nK + potential.  P and D-wave nK + potentials  The small width of the  + might be explained  The small width of the  + might be explained due to the centrifugal barrier. due to the centrifugal barrier.

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18. Convergence of nK+ potential Convergence of nK+ potential

19. Potential fit Potential fit Gaussian core + (Yukawa x form factor) 2 Gaussian core + (Yukawa x form factor) 2  The range of Yukawa-type potential = 805 MeV

20. Scattering length Scattering length r  (r) fit Calculated from LS eq. with fitted potential Very weak interaction at threshold