1. K - pp studied with Coupled-channel Complex Scaling method Workshop on “Hadron and Nuclear Physics (HNP09)” ’09.11.19 @ Arata hall, Osaka univ., Ibaraki, Osaka, Japan 1.Introduction 2.Question --- Variational calculation and Faddeev --- 3.Complex Scaling Method 4.Result 5.Summary and Future plan A. Doté (KEK) and T. Inoue (Tsukuba univ.) Two-body K - p system … Λ(1405) studied with AY potential Not new calculation but confirmation of the past work

2. K bar nuclei (Nuclei with anti-koan) = Exotic system !? I=0 K bar N potential … very attractive Deeply bound (Total B.E. ～ 100MeV) Highly dense state formed in a nucleus Interesting structures that we have never seen in normal nuclei… To make the situation more clear … “K - pp” = Prototype of K bar nuclei (K bar NN, J p =1/2 -, T=1/2) K-K- 3 He 0.0 0.15 K - ppn 0.0 2.0 K - ppp 3 x 3 fm 2 0.0 2.0 [fm -3 ] Calculated with Antisymmetrized Molecular Dynamics method employing a phenomenological K bar N potential A. Doté, H. Horiuchi, Y. Akaishi and T. Yamazaki, PRC70, 044313 (2004) 1. Introduction

3. Variational calculation of K - pp with a chiral SU(3)-based K bar N potential Av18 NN potential Av18 NN potential … a realistic NN potential with strong repulsive core. Effective K bar N potential based on Chiral SU(3) theory Effective K bar N potential based on Chiral SU(3) theory Variational method Variational method … Investigate various properties with the obtained wave function. … reproduce the original K bar N scattering amplitude obtained with coupled channel chiral dynamics. with coupled channel chiral dynamics. Single channel, Energy dependent, Complex, Gaussian-shape potential Single channel, Energy dependent, Complex, Gaussian-shape potential Four variants of chiral unitary models × Total B. E. : 20 ± 3 MeV  （ K bar N→  Y ） : 40 ～ 70 MeV Total B. E. : 20 ± 3 MeV  （ K bar N→  Y ） : 40 ～ 70 MeV Shallow binding and large decay width NN distance = 2.2 fm K bar N distance = 2.0 fm NN distance = 2.2 fm K bar N distance = 2.0 fm ～ NN distance in normal nuclei A. Doté, T. Hyodo and W. Weise, Nucl. Phys. A804, 197 (2008) Phys. Rev. C79, 014003 (2009) A. Doté, T. Hyodo and W. Weise, Nucl. Phys. A804, 197 (2008) Phys. Rev. C79, 014003 (2009) I=0 K bar N resonance “Λ(1405)”appears at 1420 MeV, not 1405 MeV I=0 K bar N resonance “Λ(1405)”appears at 1420 MeV, not 1405 MeV I=0 K bar N resonance “Λ(1405)”appears at 1420 MeV, not 1405 MeV I=0 K bar N resonance “Λ(1405)”appears at 1420 MeV, not 1405 MeV

4. Width (K bar NN→πYN) [MeV] Doté, Hyodo, Weise [1] (Variational, Chiral SU(3)) Doté, Hyodo, Weise [1] (Variational, Chiral SU(3)) Akaishi, Yamazaki [2] (Variational, Phenomenological) Akaishi, Yamazaki [2] (Variational, Phenomenological) Exp. : FNUDA [5] Exp. : DISTO [6] (Preliminary) Exp. : DISTO [6] (Preliminary) Using S-wave K bar N potential constrained by experimental data. … K bar N scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc. Ikeda, Sato [4] (Faddeev, Chiral SU(3)) Ikeda, Sato [4] (Faddeev, Chiral SU(3)) Shevchenko, Gal [3] (Faddeev, Phenomenological) Shevchenko, Gal [3] (Faddeev, Phenomenological) [1] PRC79, 014003 (2009) [2] PRC76, 045201 (2007) [3] PRC76, 044004 (2007) [4] PRC76, 035203 (2007) [5] PRL94, 212303 (2005) [6] arXiv:0810.5182 [nucl-ex] Recent results of calculation of K - pp Recent results of calculation of K - pp and related experiments

5. Width (K bar NN→πYN) [MeV] Doté, Hyodo, Weise [1] Chiral SU(3) (Variational, Chiral SU(3)) Doté, Hyodo, Weise [1] Chiral SU(3) (Variational, Chiral SU(3)) Akaishi, Yamazaki [2] (Variational, Phenomenological) Akaishi, Yamazaki [2] (Variational, Phenomenological) Exp. : FNUDA [5] Exp. : DISTO [6] (Preliminary) Exp. : DISTO [6] (Preliminary) Recent results of calculation of K - pp and related experiments Using S-wave K bar N potential constrained by experimental data. … K bar N scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc. Ikeda, Sato [4] Chiral SU(3) (Faddeev, Chiral SU(3)) Ikeda, Sato [4] Chiral SU(3) (Faddeev, Chiral SU(3)) Shevchenko, Gal [3] (Faddeev, Phenomenological) Shevchenko, Gal [3] (Faddeev, Phenomenological) [1] PRC79, 014003 (2009) [2] PRC76, 045201 (2007) [3] PRC76, 044004 (2007) [4] PRC76, 035203 (2007) [5] PRL94, 212303 (2005) [6] arXiv:0810.5182 [nucl-ex]

6. 2. Question --- Variational calc. and Faddeev --- The K bar N potentials used in both calculations are constrained with Chiral SU(3) theory, but … Total B. E. = 20±3 MeV, Decay width = 40 ～ 70 MeV A. Doté, T. Hyodo and W. Weise, Phys. Rev. C79, 014003 (2009) Variational calculation (DHW) Variational calculation (DHW) Faddeev calculation (IS) Faddeev calculation (IS) Total B. E. = 60 ～ 95 MeV, Decay width = 45 ～ 80 MeV Y. Ikeda, and T. Sato, Phys. Rev. C76, 035203 (2007) ? ? ? Discrepancy between Variational calc. and Faddeev calc. Separable potential used in Faddeev calculation? Non-relativistic (semi-relativistic) vs relativistic? Energy dependence of two-body system (K bar N) in the three-body system (K bar NN)? …??? Why ?

7. A possible reason is πΣN thee-body dynamics Y. Ikeda and T. Sato, Phys. Rev. C79, 035201(2009) This is correct for the two-body system. The effective K bar N potential has energy dependence due to the elimination of πΣ channel. It reproduces the original K bar N scattering amplitude calculated with K bar N-πΣ coupled channel model. Two-body system In the variational calculation (DHW), πΣ channel is eliminated and incorporated into the effective K bar N potential. K N + … ++ … … = Σ π K N K N K N π Σ K N K N V 11 V 1i V j1 2. Question --- Variational calc. and Faddeev ---

8. Three-body system calculated with the effective K bar N potential ΣΣ K N + … ++ … … = π K N K N K N π K N K N NNNNNN The energy of the intermediate πΣ state is fixed to the initial K bar N energy. Not true! Apply the effective K bar N potential to the three-body system… 2. Question --- Variational calc. and Faddeev ---

9. Three-body system calculated with the effective K bar N potential ΣΣ K N + … ++ … … = π K N K N K N π K N K N NNNNNN Due to the third nucleon, the intermediate πΣ energy can differ from the initial K bar N energy. NN Apply the effective K bar N potential to the three-body system… 2. Question --- Variational calc. and Faddeev ---

10. Three-body system calculated with the effective K bar N potential ΣΣ K N + … … = π K N π K N K N NNNN Due to the third nucleon, the intermediate πΣ energy can differ from the initial K bar N energy. NN Only the total energy of the three-body system should be conserved. The intermediate energy of the πΣ state can be variable, due to the third nucleon. conserved πΣN thee-body dynamics Apply the effective K bar N potential to the three-body system… 2. Question --- Variational calc. and Faddeev ---

11. πΣN three-body dynamics is taken into account in the Faddeev calculation, because the πΣ channel is directly treated in their calculation. Dr. Ikeda‘s talk in KEK theory center workshop “Nuclear and Hadron physics” (KEK, 11-13 Aug. ‘09) 2. Question --- Variational calc. and Faddeev ---

12. Coupled channel calculation Direct treatment of the “πΣ” degree of freedom The obtained state is possibly to appear below K bar NN threshold, but above πΣN threshold. The actual calculation is done in multi channels such as K bar N and πΣ. A bound state for K bar NN channel, A bound state for K bar NN channel, K bar + N + N “K bar N N” π + Σ + Nbut a resonant state for πΣN channel Resonant state can’t be treated with a variational method. Here, we employ “Complex Scaling method” to deal with resonant state. Resonant state can’t be treated with a variational method. Here, we employ “Complex Scaling method” to deal with resonant state. 3. Complex Scaling method

13. diagonalizing with Gaussian basis The resonant state can be obtained by diagonalizing with Gaussian basis, quite in the same way as calculating bound state. When, the wave function of a resonant state changes to a dumping function. Wave function of a resonant state (two-body system) Negative! Complex rotation of coordinate Advised by Dr. Myo (RCNP) Advised by Dr. Myo (RCNP) The energy of bound and resonant states is independent of scaling angle θ. (ABC theorem) The energy of bound and resonant states is independent of scaling angle θ. (ABC theorem)

14. Two-body system: K bar N-πΣ coupled system with s-wave and isospin zero state … Λ(1405) Here, we consider just two-channel problem; K bar N and πΣ channels Employ Akaishi-Yamazaki potential Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 Schrödinger equation to be solved : complex parameters to be determined Wave function expanded with Gaussian base Complex-rotate, then diagonalize with Gaussian base. 4. Result

15.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm] E [MeV]   = 0 deg. 

16. 4. Result E [MeV]   = 5 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm] 

17. 4. Result E [MeV]   =10 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm] 

18. 4. Result E [MeV]   =15 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm] 

19. 4. Result E [MeV]   =20 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm] 

20. 4. Result E [MeV]   =25 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm]

21. 4. Result E [MeV]   =30 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm] 

22. 4. Result E [MeV]   =35 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm] 

23. 4. Result E [MeV]   =40 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm] 

24. 4. Result E [MeV]   trajectory    =30 deg.  K bar N   continuum K bar N continuum Resonance! (E, Γ/2) = (75.8, 20.0) Measured from K bar N thr., B. E. (K bar N) = 28.2 MeV Γ = 40.0 MeV …  (1405) ! Measured from K bar N thr., B. E. (K bar N) = 28.2 MeV Γ = 40.0 MeV …  (1405) !

25. 4. Result b trajectory # Gauss base ( n ) = 30 Rotation angle    = 30 [deg] n trajectory Max range ( b ) = 10 [fm] Rotation angle    = 30 [deg] b = 50 [fm] Stability of the resonance for other parameters # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm] Analysis of θ trajectory with Resonance found at B. E. (K bar N) = 28.2 MeV Γ = 40.0 MeV Resonance found at B. E. (K bar N) = 28.2 MeV Γ = 40.0 MeV

26. 5. Summary and Future plan No other resonant state (no interaction between π and Σ) Solve the Coupled Channel problem = K bar N + πΣ Complex Scaling Method applying to Λ(1405) Resonance found at B. E. (K bar N) = 28.2 MeV Γ = 40.0 MeV Resonance found at B. E. (K bar N) = 28.2 MeV Γ = 40.0 MeV In case of AY potential, Comment In case of only V KN-KN, a bound state appears at B. E. = 3.2 MeV. πΣ continuum state Coupling with πΣ continuum state increases drastically the binding energy. Also, it gives the decay width. K bar N thr. V KN-πΣ V KN-KN

27. 5. Summary and Future plan Future plan Keep doing calculations! 2. Calculate three-body system … K bar NN-πΣN system corresponding to “ K - pp ” 1. Calculate two-body system … K bar N-πΣ system corresponding to Λ(1405) See what happen if we use a chiral SU(3)-based K bar N potential (HW potential, energy-dependent). T. Hyodo and W. Weise, PRC77, 035204(2008) For a test calculation, a phenomenological K bar N potential (AY potential, energy-independent) will be employed. Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 Done Done Investigate the property of the obtained wave function and thinking…

28. J-PARC will give us lots of interesting data! E15: A search for deeply bound kaonic nuclear states by 3 He(inflight K -, n) reaction --- Spokespersons: M. Iwasaki (RIKEN), T. Nagae (Kyoto) E17: Precision spectroscopy of kaonic 3 He atom 3d→2p X-rays --- Spokespersons: R. Hayano (Tokyo), H. Outa (Riken)

29. Thank you very much! Acknowledgement: Dr. Myo (RCNP) taught me CSM and supplied a subroutine for diagonalization of a complex matrix.