1. Variational calculation of K - pp with Chiral SU(3)-based K bar N interaction The International Conference on Particles And Nuclei (PANIC08) ’08.11.13 @ Dan panorama hotel, Eilat, Israel 1.Introduction 2.Method -Model wave function -Effective K bar N potential derived from coupled channel Chiral dynamics 3.Result of systematic study with effective s-wave K bar N potential -Total binding energy and mesonic decay width -Structure of K - pp 4.Other effects -Dispersive effect, P-wave interaction, Two nucleons absorption 5.Summary A. Dote (KEK) T. Hyodo (TU Munich / YITP), W. Weise (TU Munich) ´ Nucl. Phys. A804, 197 (2008) arXiv:0806.4917v1 Nucl. Phys. A804, 197 (2008) arXiv:0806.4917v1

2. People have focused on K - pp, “a prototype of K bar nuclei” But, different theoretical studies give different results from each other… Width (K bar NN→  YN) [MeV] 1. Introduction Dote, Hyodo, Weise (Variational, Chiral SU(3)) Dote, Hyodo, Weise (Variational, Chiral SU(3)) Akaishi, Yamazaki (Variational, Phenomenological) Akaishi, Yamazaki (Variational, Phenomenological) Shevchenko, Gal (Faddeev, Phenomenological) Shevchenko, Gal (Faddeev, Phenomenological) Ikeda, Sato (Faddeev, Chiral SU(3)) Ikeda, Sato (Faddeev, Chiral SU(3)) Constrained by experimental data. … K bar N scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc. a beautiful night of Eilat !

3. Av18 NN potential Av18 NN potential … a realistic NN potential with strong repulsive core. with strong repulsive core. K bar N potential based on Chiral SU(3) theory K bar N potential based on Chiral SU(3) theory Variational method Variational method … Investigate various properties with the obtained wave function. with the obtained wave function. … Well describe S=-1 meson-baryon scattering and dynamical generation of  (1405) and dynamical generation of  (1405) 1. Introduction

4. 2 ． Model wave fucntion NN correlation function K bar N correlation function Nucleon Kaon Single-particle motion Thanks, Akaishi-san! Model wave function ー Simple Correlated Model (Revised) ー NN in : 1 E, T N =1 K - pp NN in : 1 O, T N =0 NN correlation is directly treated.

5. 2. NN potential A realistic NN potential … Av18 potential Use one-pion-exchange potential (Central and Spin-Spin) and L 2 potentials in addition to phenomenological central term (= repulsive core). 1E1E 1E1E 1O1O 1O1O Strong repulsive core (3 GeV) Strong repulsive core (3 GeV) Central potential

6. 2. Local K bar N potential based on Chiral SU(3) T. Hyodo and W. Weise, PRC77, 035204(2008) : Effective local potential Non-relativistic / Single Channel Single channel … only K bar N channel,  is eliminated. Energy dependent and Complex potential Local, Gaussian form : Chiral unitary Relativistic / Coupled Channel Effective K bar N potential Effective K bar N potential … reproduce the scattering K bar N amplitude obtained with Chiral unitary model = … T-matrix for K bar N channel

7. 2. Local K bar N potential based on Chiral SU(3) I=0 K bar N scatteing amplitude Chiral unitary; T. Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003) Chiral Unitary 1420 Resonance position in I=0 K bar N channel In Chiral unitary model, 1420 MeV not 1405 MeV ! Effective potential

8. 2. Local K bar N potential based on Chiral SU(3)  Four Chiral unitary models  Four Chiral unitary models : “ORB” E. Oset, A. Ramos, and C. Bennhold, Phys. Lett. B527, 99 (2002) “HNJH” T.Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003) “BNW” B. Borasoy, R. Nissler, and W. Weise, Eur. Phys. J. A25, 79 (2005) “BMN” B. Borasoy, U. G. Meissner, and R. Nissler, Phys. Rev. C74, 055201 (2006) I=0 channelI=1 channel 1420

9. 2 ． Self-consistency of kaon’s energy Self-consistency for anti-kaon’s energy  Hamiltonian Controlled by “ Anti-kaon’s binding energy ” : Hamiltonian of nuclear part treated perturbatively.  Try two ansatz for. K bar field surrounds two nucleons which are almost static. K bar NN NN K bar is bound by each nucleon with B(K)/2 binding energy.

10. 3. Result with effective s-wave K bar N potential ORB, HNJH BNW, BMN × 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 … Not so dependent on chiral unitary models 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 I=0,1 I=0 K bar N distance = 1.82 fm I=1 K bar N distance = 2.33 fm I=0 K bar N distance = 1.82 fm I=1 K bar N distance = 2.33 fm … I=0 K bar N potential is more attractive than I=1 one. K bar NN

11. Structure of K - pp K bar N potential based on “HNJH” “Corrected”, Density distribution K bar N pair in K - pp vs  (1405) For comparison, All densities are normalized to 1. I=0 in K - pp I=1 in K - pp Isolated I=0 K bar N …  (1405) Isolated I=0 K bar N …  (1405) Very different. The I=1 component distributes widely. Rather similar.  (1405) almost survives in K - pp!

12. 4. Other effects I. Dispersive correction (Effect of imaginary part) Estimate other possible effects, perturbatively… II. p-wave K bar N potential III. Two nucleon absorption Improve a phenomenological formula for a mean field model, taking into account the NN correlation, Using an updated version of the K bar N scattering volume, Guess from two-body K bar N case, Solving with the original complex potential vs only its real part of the potential Re V KN,P ～ +3 MeV …small and repulsive

13. 4. Other effects Dispersive correction (Effect of imaginary part) +6 ～ +18 MeV p-wave K bar N potential ～ -3 MeV10 ～ 35 MeV Two nucleon absorption 4 ～ 12 MeV s-wave K bar N potential B.E.Width 20 ± 3 MeV 40 ～ 70 MeV Rough estimation 20 ～ 40 MeV Total B.E. 55 ～ 120 MeV Total Width

14. 5 ． Summary Structure of K - pp  NN distance in K - pp is smaller than that of deuteron, rather comparable to that in normal nuclei. NN distance = ～ 2.2 fm K bar N distance = ～ 2.0 fm  The I=0 K bar N component in K - pp is found to be very similar to  (1405). Contribution of other effects Dispersive correction: +6 ～ +18MeV to B.E. p-wave K bar N potential: ～ -3MeV to B.E., 10 ～ 35MeV to width Two nucleon absorption: 4 ～ 12MeV to width  We studied K - pp with a variational method, using a realistic NN potential (Av18) and a Chiral SU(3)-based K bar N potential.  We tried four variants of Chiral unitary models and two ansatz of K bar N energy in the three-body system. However, the result doesn’t depend so much on them. Total binding energy and mesonic decay width are in the small window: Binding energy and mesonic decay width of K - pp  Total binding energy and mesonic decay width are in the small window: Total Binding energy = 20 ± 3 MeV  (K bar N → πY) = 40 ～ 70 MeV … Very shallow binding By rough estimation, Total Binding energy = 20 ～ 40 MeV  ( total ) = 55 ～ 120 MeV  (1405) almost survives in K - pp. But, such a state is very short-lived.

15. Thank you very much!!

16. Back-up slides

17. 4. Other effects I. Dispersive correction (Effect of imaginary part) (Effect of imaginary part) II. p-wave K bar N potential III. Two nucleon absorption Estimate other possible effects, perturbatively…

18. I. Dispersive correction ( Effect of imaginary potential ) Two-body system (I=0 K bar N) case Lippmann-Schwinger eq. with the complex potential Schroedinger eq. with only the real part B. E. ～ 10 MeV B. E. ～ 15 MeV 1420 Complex

19. I. Dispersive correction ( Effect of imaginary potential ) Two-body system (I=0 K bar N) case Lippmann-Schwinger eq. with the complex potential Schroedinger eq. with only the real part B. E. ～ 10 MeV B. E. ～ 15 MeV Dispersive correction for K bar N system ～ +5 MeV to B. E. Dispersive correction for K bar N system ～ +5 MeV to B. E. Dispersive correction for K bar NN +6 ～ +18 MeV ×2 Considering Four variants of models…

20. II. P-wave K bar N potential Estimate its contribution perturbatively. Derived from “Full” scattering volume. Re V KN,P ～ +3 MeV …small and repulsive For a p = 0.4 ～ 0.9 fm, Γ p-wave = -2 Im V KN,P = 10 ～ 35 MeV R. Brockmann, W. Weise and L. Tauscher, Nucl. Phys. A 308, 365 (1978)

21. III. Two nucleon absorption For independent particle model … no correlation between two nucleons Contact interaction N N Y (N) N (Y) K bar N N Y (N) N (Y) K bar Three-body correlation density Gaussian-type Interaction for NN In our model, particles are strongly correlated. The NN can’t touch each other, due to the strong NN repulsion!! Finite range interaction (between NN) J. Mares, E. Friedman and A. Gal, Nucl. Phys. A 770, 84 (2006)

22. III. Two nucleon absorption For independent particle model … no correlation between two nucleons Contact interaction N N Y (N) N (Y) K bar Suppressed by NN repulsive core Suppressed by NN repulsive core J. Mares, E. Friedman and A. Gal, Nucl. Phys. A 770, 84 (2006)

23. 1-I. Recent results with various calculations of K - pp B. E.Γ (mesonic) Method K bar N Int.Ref. DHW 20 ± 3 40 ～ 70 Variational Chiral SU(3) arXiv:0806.4917v1 AY 47 61 Variational Phenom. PRC76, 045201(2007) IS 79 74 Faddeev Chiral SU(3) PRC76, 035203(2007) (AGS) (Separable) SG 50 ～ 70 ～ 100 Faddeev Phenom. PRC76, 044004(2007) (AGS) (Separable) Exp. FINUDA 116 67 K - absorption, Λp inv. mass PRL94, 212303(2005) DISTO 105±2 118±5 p+p→K + +Λ+p, Λp inv. mass EXA’08 Yamazaki-san (Preliminary) All four calculations shown above are constrained by some of experimental data. … K bar N scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc.

24. 1-I. Recent results with various calculations of K - pp B. E.Γ (mesonic) Method K bar N Int.Ref. DHW 20 ± 3 40 ～ 70 Variational Chiral SU(3) arXiv:0806.4917v1 AY 47 61 Variational Phenom. PRC76, 045201(2007) IS 79 74 Faddeev Chiral SU(3) PRC76, 035203(2007) (AGS) (Separable) SG 50 ～ 70 ～ 100 Faddeev Phenom. PRC76, 044004(2007) (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.

25. 1-I. Recent results with various calculations of K - pp B. E.Γ (mesonic) Method K bar N Int.Ref. DHW 20 ± 3 40 ～ 70 Variational Chiral SU(3) arXiv:0806.4917v1 AY 47 61 Variational Phenom. PRC76, 045201(2007) IS 79 74 Faddeev Chiral SU(3) PRC76, 035203(2007) (AGS) (Separable) SG 50 ～ 70 ～ 100 Faddeev Phenom. PRC76, 044004(2007) (AGS) (Separable) DHW vs IS Separable approximation? Importance of πΣN three-body dynamics, which is not considered to included in DHW. (Y.Ikeda and T. Sato, arXiv:0809.1285v1) Although both calculations are based on Chiral SU(3) theory, results are very different from each other.

26. 1-I. Recent results with various calculations of K - pp B. E.Γ (mesonic) Method K bar N Int.Ref. DHW 20 ± 3 40 ～ 70 Variational Chiral SU(3) arXiv:0806.4917v1 AY 47 61 Variational Phenom. PRC76, 045201(2007) IS 79 74 Faddeev Chiral SU(3) PRC76, 035203(2007) (AGS) (Separable) SG 50 ～ 70 ～ 100 Faddeev Phenom. PRC76, 044004(2007) (AGS) (Separable) AY vs SG (IS) “Pole state” in Faddeev eq. (SG (IS)) ≠ “Decaying state” (experimentally observed state) … emitted particles are on-shell with real energies AY ⇒ SG (IS) arXiv: 0805.4382v1

27. 1-II. Problem in the theory side = Uncertainty in the sub-threshold region K bar nuclei are below the K bar N threshold. However, there are no experimental constrains for such region. K bar N K bar N threshold Σπ Σπ threshold

28. 1-II. Problem in the theory side = Uncertainty in the sub-threshold region K bar nuclei are below the K bar N threshold. However, there are no experimental constrains for such region. K bar N Scattering data K bar N threshold ??? No data Kaonic hydrogen atom KpX, DEAR Σπ Σπ threshold Σπ mass spectrum Σ + π -, Σ - π + and Σ 0 π 0 No data of Σπ (I=0)

29. 1-II. Problem in the theory side Because of no constraints for K bar N sub-threshold region, theory can extrapolate the K bar N scattering amplitude in various way into such region. Example1: I=0 scattering amplitude of DHW and AY Agree with each other, in on-shell region Agree with each other, in on-shell region But, very different from each other, in the sub-threshold region But, very different from each other, in the sub-threshold region

30. 1-II. Problem in the theory side Example2: T-matrix with a Separable potential Artificial enhancement by the form factor in the sub-threshold region Lippmann-Schwinger eq. Introducing a separable potential with a form factor: The form factor makes the Loop integral converged. If we adopt analytic continuation for sub-threshold region, namely, then, T-matrix is necessarily enhanced !

31. 1-II. Problem in the theory side Example2: T-matrix with a Separable potential Artificial enhancement by the form factor in the sub-threshold region The enhancement amounts to 20% at 80 MeV below the threshold. The enhancement amounts to 20% at 80 MeV below the threshold. N. Kaiser, P.B. Siegel, W. Weise, NPA594, 325(1995) Munich Autumn School ‘08, “You can calculate the scattering amplitude for a separable potential!” Lecturers: T. Hyodo and W. Weise

32. 1-II. Problem in the theory side Example2: T-matrix with a Separable potential Artificial enhancement by the form factor in the sub-threshold region Therefore, if we construct an effective K bar N potential based on a separable potential in such a way of Hyodo and Weise †, we will get a potential which are artificially enhanced by a form factor in the sub-threshold region… † T. Hyodo and W. Weise, PRC77, 035204(2008) Remark Such an artificial enhancement is NOT expected to occur in Faddeev calculations (IS and SG), because the momentum is always real value appeared in their calculations. Effective K bar N potential (HW-type) is also artificially enhanced there.

33. 1-III. “What’s the object we are calculating?” A scenario in coupled-channel chiral dynamics Importance of πΣ channel Strong coupling between πΣ and K bar N channels Somewhat strong attraction in πΣ- πΣ channel Double pole nature of “Λ(1405)” A pole coupling strongly with K bar N and the other pole strongly coupling with πΣ “Λ(1405)” 1420MeV in K bar N 1405MeV in πΣ

34. 1-III. “What’s the object we are calculating?” A scenario in coupled-channel chiral dynamics The same story would happen in case of “K - pp” ? K bar N K bar N threshold -15 MeV πΣ K bar N threshold -30 MeV “Λ(1405)” K bar NN K bar NN threshold -20 MeV “K - pp” DHW ? πΣN K bar NN threshold -?? MeV Experimentally observed ??? Experimentally observed ??? Hyodo’s idea Calc. Exp.

35. 1-III. “What’s the object we are calculating?” Experimental side Λ p K-K- FINUDA, KEK E549/570 re-analysis Λ p K+K+ p p DISTO What we see is just “Λ+p” at the final step. “Something with B=2, S=-1” might have been formed in the reaction. But, should it be K - pp? How can we confirm?

36. 1-III. “What’s the object we are calculating?” Experimental side For example, DISTO K bar + N + N π + Σ + N DISTO-observed object … not K bar NN but πΣN bound object? (as for dominant component) The same discussion was held when pnnK - was reported… 0 -105 -104

37. 3 ． Summary Faddeev SG IS DHW Variational AY πΣ-πΣ interaction, which is respected in Chiral SU(3) theory πΣN three-body dynamics Separable? The state calculated is not that observed experimentally…

38. 6. Status of experiments searching for K bar nuclei 4 He (stopped K -, n) ppnK - M. Iwasaki et al. @ KEK Invariant mass of p and Λ H. Fujioka et al. @ FINUDA ppK - B.E. = 116 MeV Γ = 67 MeV ppK - B.E. = 116 MeV Γ = 67 MeV Heavy ion collision N. Herrmann et al. @ GSI ppnK - B.E. = 150 MeV Γ ～ 100MeV ppnK - B.E. = 150 MeV Γ ～ 100MeV 16 O (in-flight K -, n) 15 OK - T. Kishimoto et al. @ BNL 15 OK - B(K) = 90 MeV 15 OK - B(K) = 90 MeV ppnK - (T=0) B.E. = 169 MeV Γ < 25 MeV ppnK - (T=0) B.E. = 169 MeV Γ < 25 MeV

39. 4 He (stopped K -, n) ppnK - M. Iwasaki et al. @ KEK Invariant mass of p and Λ H. Fujioka et al. @ FINUDA ppK - B.E. = 116 MeV Γ = 67 MeV ppK - B.E. = 116 MeV Γ = 67 MeV Heavy ion collision N. Herrmann et al. @ GSI ppnK - B.E. = 150 MeV Γ ～ 100MeV ppnK - B.E. = 150 MeV Γ ～ 100MeV 16 O (in-flight K -, n) 15 OK - T. Kishimoto et al. @ BNL 15 OK - B(K) = 90 MeV 15 OK - B(K) = 90 MeV ppnK - (T=0) B.E. = 169 MeV Γ < 25 MeV ppnK - (T=0) B.E. = 169 MeV Γ < 25 MeV Very preliminary Not recomfirmed Re-analyzing the data, strong YN correlation was found. What does it mean… ？ ppK - ??? Re-analyzing the data, strong YN correlation was found. What does it mean… ？ ppK - ??? 12 C (in-flight K -, N ) T. Kishimoto et al. @ KEK N = proton N = neutron Signals in the bound region. To reproduce the spectrum, K bar N-potential depths of 190MeV for N=n 160MeV for N=p are needed. T. Kishimoto et. al. PTP118, 181(2007) Signals in the bound region. To reproduce the spectrum, K bar N-potential depths of 190MeV for N=n 160MeV for N=p are needed. T. Kishimoto et. al. PTP118, 181(2007) Subtraction of background ?? Final State Interaction? Target dependence ?? Final State Interaction? Target dependence ?? Renewal experiment Statistics improved 10 times better than the previous one. Data taken on each target. Strong back-to-back correlation of Λp was confirmed on each target! But B.E. ??? Renewal experiment Statistics improved 10 times better than the previous one. Data taken on each target. Strong back-to-back correlation of Λp was confirmed on each target! But B.E. ??? 6. Status of experiments searching for K bar nuclei Invariant mass of Λp p+p → K + + Λ + p @ 2.85GeV (DISTO collabolation) ppK - ??? B. E.= 105 MeV Γ = 118 MeV Reported by Prof. Yamazaki in EXA ‘08 Invariant mass of Λp p+p → K + + Λ + p @ 2.85GeV (DISTO collabolation) ppK - ??? B. E.= 105 MeV Γ = 118 MeV Reported by Prof. Yamazaki in EXA ‘08

40. 7 ． Future plan Understanding of the difference between our result and those obtained by other groups. Especially, comparison with Faddeev (Dr. Ikeda and Prof. Sato) study Total B. E. = 79 MeV, Decay width = 74 MeV Separable potential? Non-relativistic (semi-relativistic) vs relativistic? Energy dependence of two-body system (K bar N) in the three-body system (K bar NN)? Important role of πΣN channel? …??? Their K bar N interaction is also constrained by Chiral SU(3) theory. However, it differs from ours very much. Y. Ikeda and T. Sato, arXiv:0809.1285

41. Tiny comment D Yes,yes! Deeply bound!! D Hyodo PANDA EXA ’08 Yes… they might be shallowly bound. EXA ’05

42. Tiny comment DD Hyodo PANDA EXA ’08 EXA ’05 ? ? ?

43. Remark : Difference of 1405 MeV and 1420 MeV 0 MeV 1420 MeV 1405 MeV Difference of 1405 MeV and 1420 MeV is very small. Not so important !

44. Remark : Difference of 1405 MeV and 1420 MeV 1420 MeV 1405 MeV 1435 MeV K bar N threshold 0 MeV Difference of 1405 MeV and 1420 MeV is very small. Not so important !

45. Remark : Difference of 1405 MeV and 1420 MeV 1420 MeV 1405 MeV 1435 MeV K bar N threshold 0 MeV 0 -15 MeV -30 MeV Since we measure the energy from the K bar N threshold, choice of “Λ(1405)” energy, 1405 or 1420 MeV, is very important. Depending on the choice, its binding energy changes to be doubled. Since we measure the energy from the K bar N threshold, choice of “Λ(1405)” energy, 1405 or 1420 MeV, is very important. Depending on the choice, its binding energy changes to be doubled.

46. Structure of ppK - K bar N potential based on “HNJH” “Corrected”, Density distribution NN pair in ppK - vs Deuteron Suppressed by NN repulsive core Suppressed by NN repulsive core Quite compacter than deuteron Quite compacter than deuteron

47. Two nucleon absorption For independent particle model … no correlation between two nucleons Contact interaction NN and K bar N correlations taken into account Finite range interaction (between NN) N N N N K bar N N N N Three-body correlation density Gaussian-type Interaction for NN

48. Two nucleon absorption For independent particle model … no correlation between two nucleons Contact interaction N N N N K bar Apply to our wave function… almost vanishes, because of NN correlation !! Two nucleons can’t be at the same position, due to the strongly repulsive core of NN potential. Suppressed by NN repulsive core Suppressed by NN repulsive core

49. Two nucleon absorption For independent particle model … no correlation between two nucleons Contact interaction N N N N K bar Apply to our wave function… Finite range interaction (between NN) N N N N K bar Three-body correlation density Gaussian-type Interaction for NN

50. 要望 補足 K 原子核の束縛が計算によっては、深くなったり浅くなったりするのは、 結局 K bar N subthreshold での K bar N ポテンシャルの不定性のため。 I=0 K bar N full scattering amplitude Chiral unitary と現象論的ポテンシャル（赤石さん）の比較 On shell では ほとんど同じ。 Off shell では 全く違う。 K - pp の全束縛エネルギー 赤石さん： 48 MeV DHW : ～ 20 MeV

51. 最近の研究での K bar N 相互作用 Fit している物理量 Faddeev (AGS) Shevchenko, Gal Faddeev (AGS) Ikeda, Sato ATMS Akaishi Variational calc. Dote, Hyodo, Weise I=0 K bar N quasi-bound state I=0,1 K bar N scattering length K - p scattering length (kaonic hydrogen atom) Threshold branching ratio (K - p → π + Σ - / K - p → π - Σ + ) Total cross section (K - p → K - p, π + Σ -, …) πΣ mass spectrum ○ ○ Checked ○ ○○ Λ(1405) 1406.5 － i 25 ( 1420 – i 30 ) Λ(1405) 1406.5 － i 25 ○ ○ ○ ○ ( ～ 1420 – i 22 )

52. これは FULL Σπ scattering amplitude に関係してくる。 Chiral unitary での Λ(1405) の理解 I=0 K bar N quasi-bound state Λ （１４０５） 実験で “ １４０５ＭｅＶ ” と言っているのは Σπ mass spectrum を見て言っている。 1405 MeV

53. Chiral unitary での Λ(1405) の理解 しかし今、模型計算は K bar N のチャンネルで行う。 が必要。 Chiral unitary model による FULL K bar N scattering amplitude を再現するように作る。 πΣ では共鳴構造が 1405MeVK bar N では共鳴構造が 1420MeV

54. Chiral unitary での Λ(1405) の理解 K bar N を消去して、 πΣ チャンネルで見ると、 I=0 の共鳴は 1405 MeV 実験で見てるもの（ πΣ mass spectrum ）に対応 πΣ を消去して、 K bar N チャンネルで見ると、 I=0 の共鳴は 1420 MeV K - pp の計算は K bar N チャンネルで行うので、 その際に使用する K bar N ポテンシャルは、 I=0 共鳴を 1420MeV に出すようにすべき。 1405MeV ではない。 構造計算を行うチャンネル 見るチャンネルによって、共鳴構造が変わるのは “Double pole” のため。

55. Chiral unitary での Λ(1405) の理解 “Double pole structure” Chiral unitary では pole が二つ出てくる： z 1 と z 2 二つのポールでは、 K bar N 、 πΣ との結合の強さが異なる。 であるため、散乱振幅の形が異なる。 共鳴構造も異なる。

56. Σπ mass spectrum Peak position of Σπ spectrum is certainly 1405 MeV, which corresponds to Λ(1405). 1405

57. １． Introduction Y. Akaishi and T. Yamazaki, PRC76, 045201(2007) Super Strong Nuclear Force

58. ３． Local K bar N potential based on Chiral SU(3) How to determine and the range parameter. Chiral U. Coupled Ch. V ij T ij Step 1 Exactly done in the framework of Chiral Unitary Step 1 Eliminate other channels than K bar N channel K N + … ++ … … = Σ π K N K N K N π Σ K N K N V Single, I V 11 V 1i V j1 Single Ch. T = T 11 Chiral U.

59. ３． Local K bar N potential based on Chiral SU(3) How to determine and the range parameter. Chiral U. Coupled Ch. V ij T ij Step 1 Single Ch. T = T 11 Chiral U. Step 2 Effective Single Ch. Range parameter is determined so that the I=0 resonance appears at the same place as that in the Chiral unitary when we solve the Schroedinger equation with this potential. Step 2 Using, construct simply as

60. ３． Local K bar N potential based on Chiral SU(3) How to determine and the range parameter. Chiral U. Coupled Ch. V ij T ij Step 1 Single Ch. T = T 11 Chiral U. Step 2 Effective Single Ch. Step 3 Correct so as to reproduce the original scattering amplitude (T-matrix) better especially far below the threshold. Corrected K bar N potential … “Corrected” K bar N potential without correction … “Uncorrected” Step 3

61. ４． Result  We have tried “Corrected” and “Uncorrected” for four models: “ORB” E. Oset, A. Ramos, and C. Bennhold, Phys. Lett. B527, 99 (2002) “HNJH” T.Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003) “BNW” B. Borasoy, R. Nissler, and W. Weise, Eur. Phys. J. A25, 79 (2005) “BMN” B. Borasoy, U. G. Meissner, and R. Nissler, Phys. Rev. C74, 055201 (2006)  Hamiltonian  Binding energy of kaon : Hamiltonian of nuclear part  We performed variational calculation. Since the K bar N potential is energy-dependent, we repeat the calculation until the self-consistency on the kaon energy is accomplished.  We have tried two prescriptions for. Notes on actual calculation is treated perturbatively.

62. Effective K bar N potentials Central depth of the potential I=0 channelI=1 channel Real “Corrected” “Uncorrected” Imaginary Case: “HNJH” range parameter a s =0.47fm

63. ３． Local K bar N potential based on Chiral SU(3) T. Hyodo and W. Weise, PRC77, 035204(2008) 1. Chiral unitary / Relativistic / Coupled Channel 2. Chiral unitary / Relativistic / Single Channel 3. Effective local potential / Non-relativistic / Single Channel Energy dependent Complex Energy dependent Complex Local, Gaussian form “Corrected version” ; Energy-dependence improved in low-energy region

64. ３． Local K bar N potential based on Chiral SU(3) Request for our K bar N potential 1. Reproduce the s-wave K bar N scattering amplitude calculated with Chiral unitary model 3. Local potential, r-space, Gaussian form 2. Single channel (K bar N channel) but energy-dependent and complex To apply the structure study of ppK -, : Normalized Gaussian : CM energy of K bar N Hyodo’s talk

65. Deeply bound; Binding energy of K - > 100 MeV Discrete state; Below Σπ threshold Very attractive I=0 K bar N interaction makes …... Deeply bound kaonic nuclei 1.free K bar N scattering data 2.1s level shift of kaonic hydrogen atom 3.binding energy and width of Λ(1405) Phenomenological K bar N potential (AY K bar N potential) Strongly attractive. Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 = K - + proton Akaishi-san and Yamazaki-san’s study

66. According to the study with A ntisymmetrized M olecular D ynamics + G-matrix + Phenomenological K bar N interaction K bar nuclei have interesting properties… Collaboration with Akaishi-san and Yamazaki-san Collaboration with Akaishi-san and Yamazaki-san Total system is treated in a fully microscopic way. NN repulsive core is adequately smoothed out by following conventional nuclear physics. Strongly attractive, especially in I=0 channel

67. AMD + G-matrix + AY K bar N interaction studies revealed … A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313. 1.E(K) > 100 MeV for various light nuclei ( 3 He to 11 C) 2.Drastic change of the structure of 8 Be, isovector deformation in 8 BeK - 3.Highly dense state is formed in K nuclei. maximum density > 4ρ 0 averaged density 2 ～ 4ρ 0 4.Proton satellite in pppK - Rrms = 2.46 fm β = 0.63 Central density = 0.10 /fm^3 8 Be Density (/fm^3) 0.0 0.10 0.20 Rrms = 1.42 fm β = 0.55 Central density = 0.76 /fm^3 8 BeK - Density (/fm^3) 0.0 0.41 0.83 4.5 normal density Binding energy of K - = 104 MeV pppK - Proton satellite

68. A ntisymmetrized M olecular D ynamics + G-matrix + Phenomenological K bar N interaction Collaboration with Akaishi-san and Yamazaki-san Collaboration with Akaishi-san and Yamazaki-san Total system is treated in a fully microscopic way. NN repulsive core is adequately smoothed out by following conventional nuclear physics. Strongly attractive, especially in I=0 channel NN repulsive core is TOO smoothed out ??? Energy-dependence? π∑-π∑ potential?? Lots of questions (criticisms, claims) came…

69. １． Introduction K bar nuclei = Exotic system !? I=0 K bar N potential … very attractive Highly dense state formed in a nucleus Interesting structures that we have never seen in normal nuclei… To make the situation more clear … ppK - = Prototye of K bar nuclei K-K- Studied with various methods, because it is a three-body system: Akaishi, Yamazaki ATMS B.E. = 47MeV, Γ= 61MeV PRC76, 045201(2007) Ikeda, Sato Faddeev B.E. = 79MeV, Γ= 74MeV PRC76, 035203(2007) Shevchenko, Gal Faddeev B.E. = 50 ～ 70MeV, Γ= ～ 100MeV PRC76, 044004(2007) Arai, Yasui, Oka (Λ* nuclei model), Nishikawa, Kondo (Skyrme model), Noda, Yamagata, Sasaki, Hiyama, Hirenzaki (Variational method) … FINUDA experiment B.E. = 116MeV, Γ= 67MeV PRL94, 212303(2005)

70. ２． Model NN correlation function K bar N correlation function Nucleon Kaon Single-particle motion Thanks, Akaishi-san! Model wave function ー Simple Correlated Model (Revised) ー

71. ２． Model NN correlation function K bar N correlation function Nucleon Kaon Single-particle motion Model wave function ー Simple Correlated Model (Revised) ー ALL parameters are determined by energy-variation, namely to minimize the expectation value of Hamiltonian. Variational calculation … Variational calculation ALL parameters are determined by energy-variation, namely to minimize the expectation value of Hamiltonian. Variational calculation … Variational calculation

72. ２． Model NN potential … Tamagaki potential Effective K bar N potential … Akaishi-Yamazaki potential Influence of the improvement T N =1 only T N =1 + T N =0 Additional K bar N attraction

73. ４． Result Total Binding Energy and Decay Width ORB HNJH BNW BMN × Total B. E. : 20 ± 3 MeV Γ （ K bar N→πY ） : 40 ～ 70 MeV Total B. E. : 20 ± 3 MeV Γ （ K bar N→πY ） : 40 ～ 70 MeV Small model and ansatz dependence Small model and ansatz dependence

74. Structure of K - pp NN K bar K bar N potential based on “HNJH” “Corrected”,

75. Structure of K - pp K bar 2.21 fm 1.97 fm NN K bar N potential based on “HNJH” “Corrected”, Cf) NN distance in normal nuclei ～ 2 fm Size of deuteron ～ 4 fm

76. Structure of K - pp K bar NN distance = 2.21 fm K bar N distance = 1.97 fm NN T = 1/2 T N =1 … 96.2 % T N =0 … 3.8 % K bar N potential based on “HNJH” “Corrected”,

77. Structure of K - pp K bar NN Mixture of T N =0 component = 3.8 % K bar N potential based on “HNJH” “Corrected”, NN distance = 2.21 fm K bar N distance = 1.97 fm 1.97 fm

78. Structure of K - pp K bar NN Mixture of T N =0 component = 3.8 % I=0 K bar N 1.82 fm K bar N potential based on “HNJH” “Corrected”, NN distance = 2.21 fm K bar N distance = 1.97 fm

79. Structure of K - pp K bar NN Mixture of T N =0 component = 3.8 % I=0 K bar NI=1 K bar N 1.82 fm2.33 fm K bar N potential based on “HNJH” “Corrected”, NN distance = 2.21 fm K bar N distance = 1.97 fm

80. Structure of K - pp N Mixture of T N =0 component = 3.8 % I=0 K bar NI=1 K bar N 1.82 fm2.33 fm K bar N potential based on “HNJH” “Corrected”, NN distance = 2.21 fm K bar N distance = 1.97 fm “Λ(1405)” as I=0 K bar N calculated with this potential 1.86 fm K bar N Almost “Λ(1405)”

81. ４． Result Total Binding Energy and Decay Width “Corrected”,

82. ４． Result Total Binding Energy and Decay Width “Corrected”,

83. ４． Result Total Binding Energy and Decay Width “Uncorrected”,

84. ４． Result Total Binding Energy and Decay Width “Uncorrected”,

85. ppK - calculated with various potentials Corrected Uncorrected √S = M N + m K - B(K) √S = M N + m K - B(K) / 2

86. ５． Summary Structure of ppK -  NN distance = ～ 2.2 fm K bar N distance = ～ 2.0 fm  Mixture of T N =0 component = ～ 4 % Genuine ppK - → T N =1 component  P-wave K bar N potential : small and repulsive ( ～ +3 MeV)  K bar N in [ppK - ] T=1/2 I=0 : Distance = 1.82 fm, =0.4 “Λ(1405)” Size = 1.86 fm, =0 I=0 K bar N in ppK - is almost “Λ(1405)” ! … This picture doesn’t contradict with Akaishi-san’s one. Smaller than deuteron, comparable to NN in normal nuclei. I=1 : Distance = 2.33 fm, =1.9