1. Introduction. 2. Recent results of various studies of K pp. 3. Variational cal. vsfaddeev

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1 Theoretical studies of K - pp Akinobu Doté (KEK Theory Center) 1. Introduction 2. Recent results of various studies of K pp DHW (Variational with Chiral-based) vs AY (Variational with phenomenological) DHW (Variational with Chiral-based) vs IS (Faddeev with Chiral-based) 3. Variational cal. vsfaddeev πσn three-body dynamics 4. Application of Complex Scaling Method to K p (=Λ(1405)) 5. Summary and Future plan Collaboration with Dr. Takashi Inoue (Tsukuba univ.) New Frontiers of QCD (NFQCD10), Hadrons in Nuclei part, Seminar, Panasonic hall, YITP, Kyoto, Japan

2 1. Introduction

K bar nuclei = Exotic system!? 3 He K - ppn K - ppp Phenomenological K bar N potential very attractive in I=0 channel Y. Akaishi and T. Yamazaki, PRC 65, 044005 (2002) 3 x 3fm 2 AMD + G-matrix (eff.

it is a three-body system: A. Doté, H. Horiuchi, Y. Akaishi and T.

3 K bar nuclei = Exotic system!? 3 He K - ppn K - ppp Phenomenological K bar N potential very attractive in I=0 channel Y. Akaishi and T. Yamazaki, PRC 65, (2002) 3 x 3fm 2 AMD + G-matrix (eff. NN potential ) + AY K bar N potential [fm -3 ] Deeply bound (~100MeV), Highly dense state (ρ ave =2~4ρ 0 ), Interesting structures To make the situation more clear K - pp= Prototye of K bar nuclei Studied with various methods, because it is a three-body system: A. Doté, H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590, 51 (2004); PRC 70, (2004) Doté, Hyodo, Weise Variational with a chiral SU(3)-based K bar N potential PRC79, (2009) Akaishi, Yamazaki ATMS with a phenomenological K bar N potential PRC76, (2007) Ikeda, Sato Faddeev with a chiral SU(3)-derived K bar N potential PRC76, (2007) Shevchenko, Gal, Faddeev with a phenomenological K bar N potential PRC76, (2007) Mares Wycech, Green Variational with a phenomenological K bar N potential (with p-wave) PRC79, (2009) Arai, Yasui, Oka Λ* nuclei model PTP119, 103(2008) Nishikawa, Kondo Skyrme model PRC77, (2008) There are some experiments concerned to this subject (FINUDA, KEK, DISTO)

4 K bar nuclei = Exotic system!? 3 He K - ppn K - ppp Phenomenological K bar N potential very attractive in I=0 channel Y. Akaishi and T. Yamazaki, PRC 65, (2002) 3 x 3fm 2 AMD + G-matrix (eff. NN potential ) + AY K bar N potential [fm -3 ] Deeply bound (~100MeV), Highly dense state (ρ ave =2~4ρ 0 ), Interesting structures To make the situation more clear K - pp= Prototye of K bar nuclei Studied with various methods, because it is a three-body system: A. Doté, H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590, 51 (2004); PRC 70, (2004) Doté, Hyodo, Weise Variational with a chiral SU(3)-based K bar N potential PRC79, (2009) Akaishi, Yamazaki ATMS with a phenomenological K bar N potential PRC76, (2007) Ikeda, Sato Faddeev with a chiral SU(3)-derived K bar N potential PRC76, (2007) Shevchenko, Gal, Faddeev with a phenomenological K bar N potential PRC76, (2007) Mares Wycech, Green Variational with a phenomenological K bar N potential (with p-wave) PRC79, (2009) Arai, Yasui, Oka Λ* nuclei model PTP119, 103(2008) Nishikawa, Kondo Skyrme model PRC77, (2008) All calculations predict that K - pp can be bound. There are some experiments concerned to this subject (FINUDA, KEK, DISTO)

5 2. Recent status of various studies of K - pp

6 Recent results of calculation of K - pp and related experiments Width (K bar NN πyn) [MeV] Doté, Hyodo, Weise [1] (Variational, Chiral SU(3)) Akaishi, Yamazaki [2] (Variational, Phenomenological) Ikeda, Sato [4] (Faddeev, Chiral SU(3)) Shevchenko, Gal, Mares [3] (Faddeev, Phenomenological) Exp. : DISTO [6] (Finalized) 120 Exp. : FNUDA [5] 140 [1] PRC79, (2009) [2] PRC76, (2007) [3] PRC76, (2007) [4] PRC76, (2007) [5] PRL94, (2005) [6] arxiv: [nucl-ex] Using S-wave K bar N potential constrained by experimental data. K bar N scattering data, Kaonic hydrogen atom data, Λ(1405) etc.

7 Recent results of calculation of K - pp and related experiments Width (K bar NN πyn) [MeV] Doté, Hyodo, Weise [1] (Variational, Chiral SU(3)) 40 Akaishi, Yamazaki [2] (Variational, Phenomenological) Wycech, Green [7] 60 (Variational, phenomenological, P-wave) Ikeda, Sato [4] (Faddeev, Chiral SU(3)) Shevchenko, Gal, Mares [3] (Faddeev, Phenomenological) Exp. : DISTO [6] (Finalized) 120 Exp. : FNUDA [5] 140 [1] PRC79, (2009) [2] PRC76, (2007) [3] PRC76, (2007) [4] PRC76, (2007) [5] PRL94, (2005) [6] arxiv: [nucl-ex] Using S-wave K bar N potential constrained by experimental data. K bar N scattering data, Including P-wave K bar N potential, Kaonic hydrogen atom data, and other effects. Λ(1405) etc. [7] PRC79, (2009)

8 Recent results with various calculations of K - pp B. E. Γ (mesonic) Method K bar N Int. Channels at final step DHW 20 ± 3 40 ~ 70 Variational Chiral SU(3) K bar N AY Variational Phenom. K bar N IS 60 ~ ~ 80 Faddeev Chiral SU(3) K bar N, πy (AGS) (Separable) SGM 50~70 90 ~ 110 Faddeev Phenom. K bar N, πy (AGS) (Separable) Exp. FINUDA K - absorption, Λp inv. mass DISTO 105±2 118±5 p+p K + +Λ+p, Λp inv. mass (Finalized) All four calculations shown above are constrained by experimental data. K bar N scattering data, Kaonic hydrogen atom data, Λ(1405) etc. Only s-wave K bar N potential is used.

9 Recent results with various calculations of K - pp B. E. Γ (mesonic) Method K bar N Int. Channels at final step DHW 20 ± 3 40 ~ 70 Variational Chiral SU(3) K bar N AY Variational Phenom. K bar N IS 60 ~ ~ 80 Faddeev Chiral SU(3) K bar N, πy (AGS) (Separable) SGM 50~70 90 ~ 110 Faddeev Phenom. K bar N, πy (AGS) (Separable) DHW vs AY Difference of the used K bar N interactions.

10 Comparison of AY potential and Chiral-based potential AY potential Coupled channel Chiral dynamics Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian K bar N πσ ηλ KΞ Energy independent potential No πσ-πσ interaction Energy dependent potential Somewhat strongly attractive πσ-πσ interaction

11 Comparison of AY potential and Chiral-based potential AY potential Coupled channel Chiral dynamics Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian Λ(1405) = a quasi-bound state of I=0 K bar N at 1405MeV. Appears in I=0 K bar N channel. I=0 K bar N 1405MeV. Energy independent potential No πσ-πσ interaction Two poles (double pole); one couples strongly to K bar N, the other couples K bar N strongly πσ ηλto πσ. KΞ Λ(1405) (experimentally observed) appears in I=0 πσ-πσ channel. I=0 K bar N 1420MeV. Energy dependent potential Somewhat strongly attractive πσ-πσ interaction

12 Comparison of AY potential and Chiral-based potential I=0 K bar N full scattering amplitude Quite different in the sub-threhold region Almost same in the on-shell region

13 Recent results with various calculations of K - pp B. E. Γ (mesonic) Method K bar N Int. Channels at final step DHW 20 ± 3 40 ~ 70 Variational Chiral SU(3) K bar N AY Variational Phenom. K bar N IS 60 ~ ~ 80 Faddeev Chiral SU(3) K bar N, πy (AGS) (Separable) SGM 50~70 90 ~ 110 Faddeev Phenom. K bar N, πy (AGS) (Separable) DHW vs AY In Chiral SU(3) theory, the πσ-πσ interaction is so attractive to make a resonance, while AY potential doesn t have it. Λ(1405) is I=0 K bar N bound state at 1420 MeV or 1405 MeV? AY potential is twice more attractive than Chiral-based one.

14 Recent results with various calculations of K - pp B. E. Γ (mesonic) Method K bar N Int. Channels at final step DHW 20 ± 3 40 ~ 70 Variational Chiral SU(3) K bar N AY Variational Phenom. K bar N IS 60 ~ ~ 80 Faddeev Chiral SU(3) K bar N, πy (AGS) (Separable) SGM 50~70 90 ~ 110 Faddeev Phenom. K bar N, πy (AGS) (Separable) DHW vs IS Although both are based on Chiral SU(3) theory, results are very different from each other. Separable approximation? Different energy dependence of interaction kernel V ij? πσn three-body dynamics may not be included in DHW. (Y. Ikeda and T. Sato, PRC79, (2009))

15 3. πσn three-body dynamics in the study of K - pp

= 20±3 MeV, Decay width = 40~70 MeV A. Doté, T. Hyodo and W. Weise, Phys. Rev. C79, 014003 (2009) Total B. E. = 60~95 MeV, Decay width = 45~80 MeV Y.

16 Variational cal. vs Faddeev???? Discrepancy between Variational calc. and Faddeev calc. The K bar N potentials used in both calculations are constrained with Chiral SU(3) theory, but Variational calculation (DHW) Faddeev calculation (IS) Total B. E. = 20±3 MeV, Decay width = 40~70 MeV A. Doté, T. Hyodo and W. Weise, Phys. Rev. C79, (2009) Total B. E. = 60~95 MeV, Decay width = 45~80 MeV Y. Ikeda, and T. Sato, Phys. Rev. C76, (2007) Why? 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)????

17 Variational cal. vs Faddeev A possible reason is πσn thee-body dynamics Y. Ikeda and T. Sato, Phys. Rev. C79, (2009) In the variational calculation (DHW), πσ channel is eliminated and incorporated into the effective K bar N potential. Two-body system K N K K K N = N N π K K N V 11 V 1i V j1 Σ Σ π N 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.

18 Variational cal. vs Faddeev Apply the effective K bar N potential to the three-body system Three-body system calculated with the effective K bar N potential K N K N = K N K N π K K N Σ Σ π N N N N N N N The energy of the intermediate πσ state is fixed to the initial K bar N energy. Not true!

19 Variational cal. vs Faddeev Apply the effective K bar N potential to the three-body system Three-body system calculated with the effective K bar N potential K N K N = K N K N π K K N Σ Σ π N N N N N N N Due to the third nucleon, the intermediate πσ energy can differ from the initial K bar N energy.

20 Variational cal. vs Faddeev Apply the effective K bar N potential to the three-body system Three-body system calculated with the effective K bar N potential K N K N = E K bar NN conserved N N N N K N π Σ Σ π K N + πσn thee-body dynamics Due to the third nucleon, the intermediate πσ energy can differ from the initial K bar N energy. 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.

21 Variational cal. vs Faddeev 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, Aug. 09)

22 4. Application of Complex Scaling Method to K - p(=λ(1405))

23 Coupled-Channel Complex Scaling Method Direct treatment of the πσ degree of freedom Coupled channel calculation K bar + N + N K bar N N π + Σ +N A bound state for K bar NN channel, but a resonant state for πσn channel The actual calculation is done in multi channels such as K bar N and πσ. The obtained state is possibly to appear below K bar NN threshold, but above πσn threshold.

24 Coupled-Channel Complex Scaling Method Direct treatment of the πσ degree of freedom Coupled channel calculation K bar + N + N K bar N N π + Σ +N The actual calculation is done in multi channels such as K bar N and πσ. A bound state for K bar NN channel, but 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. The obtained state is possibly to appear below K bar NN threshold, but above πσn threshold.

25 Coupled-Channel Complex Scaling Method Complex rotation of coordinate Advised by Dr. Myo (RCNP) Wave function of a resonant state (two-body system) The energy of bound and resonant states is independent of scaling angle θ. (ABC theorem J. Aguilar and J. M. Combes, Commun.. Math. Phys. 22 (1971),269 E. Balslev and J. M. Combes, Commun.. Math. Phys. 22 (1971),280 ) Negative! When, the wave function of a resonant state changes to a dumping function. Boundary condition is the same as that for a bound state. The resonant state can be obtained by diagonalizing quite in the same way as calculating bound state. with Gaussian basis,

of unstable nuclei. Resonant state of 6 He n 4 He n Resonance S.

26 Example of a study with Complex Scaling method Advised by Dr. Myo (RCNP) Succeeded in nuclear physics Especially used in the study of unstable nuclei. Resonant state of 6 He n 4 He n Resonance S. Aoyama, T. Myo, K. Kato and K. Ikeda, Prog. Theor. Phys. 116, 1 (2006) Continuum

27 Λ(1405) with C.C. Complex Scaling Method 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 Schrödinger equation to be solved Employ Akaishi-Yamazaki potential Y. Akaishi and T. Yamazaki, PRC 52 (2002) Wave function expanded with Gaussian base : complex parameters to be determined Complex-rotate, then diagonalize with Gaussian base.

28 Remark! The result that I show hereafter is not new, because the same calculation was done by Akaishi-san, when he made AY potential. Here, I want to show the result in order to show how the coupled-channel channel complex method works.

29 Λ(1405) with C.C. Complex Scaling Method θ trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm] E 2θ [MeV] θ = 0 deg.

30 Λ(1405) with C.C. Complex Scaling Method θ trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm] E [MeV] 2θ θ = 5 deg.

31 Λ(1405) with C.C. Complex Scaling Method θ trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm] E [MeV] 2θ θ =10 deg.

32 Λ(1405) with C.C. Complex Scaling Method θ trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm] E [MeV] 2θ θ =15 deg.

33 Λ(1405) with C.C. Complex Scaling Method θ trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm] E [MeV] θ =20 deg. 2θ

34 Λ(1405) with C.C. Complex Scaling Method θ trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm] E [MeV] θ =25 deg.

35 Λ(1405) with C.C. Complex Scaling Method θ trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm] E [MeV] 2θ θ =30 deg.

36 Λ(1405) with C.C. Complex Scaling Method θ trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm] E [MeV] 2θ θ =35 deg.

37 Λ(1405) with C.C. Complex Scaling Method θ trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm] E [MeV] 2θ θ =40 deg.

38 Λ(1405) with C.C. Complex Scaling Method θ trajectory πσ K bar N Resonance! (E, Γ/2) = (75.8, 20.0) E [MeV] K bar N continuum 2θ πσ continuum θ =30 deg.

39 Λ(1405) with C.C. Complex Scaling Method θ trajectory πσ K bar N Resonance! (E, Γ/2) = (75.8, 20.0) E [MeV] Measured from K bar N thr., B. E. (K bar N) = 28.2 MeV Γ = 40.0 MeV Λ(1405)! K bar N continuum 2θ πσ continuum θ =30 deg.

trajectory with # Gauss base (n) = 30 Max range (b) = 10 [fm] Resonance found at B. E.

40 Λ(1405) with C.C. Complex Scaling Method Stability of the resonance for other parameters Analysis of θ trajectory with # Gauss base (n) = 30 Max range (b) = 10 [fm] Resonance found at B. E. (K bar N) = 28.2 MeV Γ = 40.0 MeV b trajectory # Gauss base (n) = 30 Rotation angle (θ) = 30 [deg] n trajectory Max range (b) = 10 [fm] Rotation angle (θ) = 30 [deg] b = 50 [fm]

41 5. Summary and Future plan

42 Current status of studies of K - pp 5. Summary The most essential K bar nuclei K pp (K bar NN, J p =1/2, T=0) has been investigated in various ways. But the situation is still controversial Theory Variational + Phenom. K bar N B.E. = 47MeV, Γ= 61MeV PRC76, (2007) Variational + Chiral-based K bar N B.E. = 20±3MeV, Γ= 40-70MeV PRC79, (2009) Faddeev + Phenom. K bar N B.E. = 50~70MeV, Γ=~100MeV PRC76, (2007) Faddeev + Chiral-based K bar N B.E. = 79MeV, Γ= 74MeV PRC76, (2007) Experiment (Unknown object which seems related to K - pp) FINUDA B.E. = 116MeV, Γ= 67MeV PRL94, (2005) DISTO B.E. = 105MeV, Γ= 118MeV arxiv: [nucl-ex] Discrepancy between theoretical ti studies of K - pp DHW (Variational with Chiral based) vs AY (Variational with phenomenological) Difference of K bar N attraction Λ(1420) scheme and Λ(1405) scheme DHW (Variational with Chiral based) vs IS (Faddeev with Chiral based) πσnthree body dynamics (might be also differenct energy dependence of interaction kernel? )

5. Summary Complex Scaling Method applying to Λ(1405) Solve the Coupled Channel problem = K bar N + πσ In case of AY potential, Comment Resonance found at B. E. (K bar N) = 28.2 MeV Γ = 40.

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

phenomenological K bar N potential (AY potential, energy-independent) will be employed. Y. Akaishi and T.

potential, energy-dependent). T. Hyodo and W. Weise, PRC77, 035204(2008) system corresponding to K - pp 2.

44 5. Future plan 1. Calculate two-body system K bar N-πΣ system corresponding to Λ(1405) For a test calculation, a phenomenological K bar N potential (AY potential, energy-independent) will be employed. Y. Akaishi and T. Yamazaki, PRC65, (2002) 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, (2008) system corresponding to K - pp 2. Calculate three-body system K bar NN-πΣN system corresponding to Effect of πσn three-body dynamics Comparison with experimental results Interpretation of DISTO result 3. Calculate four-body system K bar NNN-πΣNN system

45 g{tç~ çéâ äxüç Åâv{4 Acknowledgement: Dr. Myo (RCNP) taught me CSM and supplied a subroutine for diagonalization of a complex matrix.

Application of the complex scaling method to hadronic resonances Λ(1405) and K - pp resonances

Application of the complex scaling method to hadronic resonances Λ(1405) and K - pp resonances Akinobu Doté (KEK Theory Center, IPS / J-PARC branch) Takashi Inoue (ihon university) Takayuki Myo (Osaka