1. Application of coupled-channel Complex Scaling Method to the K bar N-πY system 1.Introduction 2.coupled-channel Complex Scaling Method For resonance states For scattering problem 3.Set up for the calculation of K bar N-πY system “KSW-type potential” Kinematics / Non-rela. Approximation 4.Results I=0 channel: Scattering amplitude / Resonance pole I=1 channel: Scattering amplitude 5.Summary and future plans A. Doté (KEK Theory center / IPNS/ J-PARC branch) T. Inoue (Nihon univ.) T. Myo (Osaka Inst. Tech. univ.) International workshop “Inelastic Reactions in Light Nuclei”, 09.Oct.’13 @ IIAS, The Hebrew univ., Jerusalem, Israel International workshop “Inelastic Reactions in Light Nuclei”, 09.Oct.’13 @ IIAS, The Hebrew univ., Jerusalem, Israel A. Dote, T. Inoue, T. Myo, Nucl. Phys. A912, 66-101 (2013) A. Dote, T. Inoue, T. Myo, Nucl. Phys. A912, 66-101 (2013)

2. 1. Introduction

3. “K bar ” Kaonic nuclei = Nuclear system with Anti-kaon “K bar ” Excited hyperon Λ(1405) = a quasi-bound state of K - and proton Attractive K bar N interaction! Difficult to explain by naive 3-quark model Consistent with “repulsive nature” indicated by K bar N scattering length and 1s level shift of kaonic hydron atom u bar s u u d s u d T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012) K bar N threshold = 1435MeV

4. “K bar ” Kaonic nuclei = Nuclear system with Anti-kaon “K bar ” Excited hyperon Λ(1405) = a quasi-bound state of K - and proton Attractive K bar N interaction! Difficult to explain by naive 3-quark model Consistent with “repulsive nature” indicated by K bar N scattering length and 1s level shift of kaonic hydron atom  Antisymmetrized Molecular Dynamics with a phenomenological K bar N potential Anti-kaon can be deeply bound and/or form dense state? Anti-kaon bound in light nuclei with ~ 100MeV binding energy Anti-kaon bound in light nuclei with ~ 100MeV binding energy Dense matter (average density = 2 ～ 4 ρ 0 ) Dense matter (average density = 2 ～ 4 ρ 0 )  Relativistic Mean Field applied to medium to heavy nuclei with anti-kaons AMD study A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313. RMF study D. Gazda, E. Friedman, A. Gal, J. Mares, PRC 76 (2007) 055204; PRC 77 (2008) 045206; A. Cieply, E. Friedman, A. Gal, D. Gazda, J. Mares, PRC 84 (2011) 045206. T. Muto, T. Maruyama, T. Tatsumi, PRC 79 (2009) 035207.

5. “K bar ” Kaonic nuclei = Nuclear system with Anti-kaon “K bar ” Excited hyperon Λ(1405) = a quasi-bound state of K - and proton Attractive K bar N interaction! Difficult to explain by naive 3-quark model Consistent with “repulsive nature” indicated by K bar N scattering length and 1s level shift of kaonic hydron atom  Antisymmetrized Molecular Dynamics with a phenomenological K bar N potential Anti-kaon can be deeply bound and/or form dense state? Anti-kaon bound in light nuclei with ~ 100MeV binding energy Anti-kaon bound in light nuclei with ~ 100MeV binding energy Dense matter (average density = 2 ～ 4 ρ 0 ) Dense matter (average density = 2 ～ 4 ρ 0 )  Relativistic Mean Field applied to medium to heavy nuclei with anti-kaons Prototype of kaonic nuclei = “K - pp”  Variational calculation with Gaussian base  Av18 NN potential  Chiral SU(3)-based K bar N potential B. E. ~ 20MeV, Γ = 40 ~70MeV pp distance ~ 2.1 fm A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009)

6. Theoretical studies of K - pp Doté, Hyodo, Weise PRC79, 014003(2009)Doté, Hyodo, Weise PRC79, 014003(2009) Variational with a chiral SU(3)-based K bar N potential Variational with a chiral SU(3)-based K bar N potential Akaishi, Yamazaki PRC76, 045201(2007)Akaishi, Yamazaki PRC76, 045201(2007) ATMS with a phenomenological K bar N potential ATMS with a phenomenological K bar N potential Ikeda, Sato PRC76, 035203(2007)Ikeda, Sato PRC76, 035203(2007) Faddeev with a chiral SU(3)-derived K bar N potential Faddeev with a chiral SU(3)-derived K bar N potential Shevchenko, Gal, Mares PRC76, 044004(2007)Shevchenko, Gal, Mares PRC76, 044004(2007) Faddeev with a phenomenological K bar N potential Faddeev with a phenomenological K bar N potential Barnea, Gal, Liverts PLB712, 132(2012)Barnea, Gal, Liverts PLB712, 132(2012) Hyperspherical harmonics with a chiral SU(3)-based K bar N potential Hyperspherical harmonics with a chiral SU(3)-based K bar N potential Wycech, Green PRC79, 014001(2009) Variational with a phenomenological K bar N potential (with p-wave) Arai, Yasui, Oka/ Uchino, Hyodo, Oka PTP119, 103(2008) Λ* nuclei model /PTPS 186, 240(2010) Nishikawa, Kondo PRC77, 055202(2008) Skyrme model “K - pp” = Prototype of K bar nuclei (K bar NN, J p =1/2 -, T=1/2) All studies predict that K - pp can be bound!

7. Theoretical studies of K - pp Doté, Hyodo, Weise PRC79, 014003(2009)Doté, Hyodo, Weise PRC79, 014003(2009) Variational with a chiral SU(3)-based K bar N potential Variational with a chiral SU(3)-based K bar N potential Akaishi, Yamazaki PRC76, 045201(2007)Akaishi, Yamazaki PRC76, 045201(2007) ATMS with a phenomenological K bar N potential ATMS with a phenomenological K bar N potential Ikeda, Sato PRC76, 035203(2007)Ikeda, Sato PRC76, 035203(2007) Faddeev with a chiral SU(3)-derived K bar N potential Faddeev with a chiral SU(3)-derived K bar N potential Shevchenko, Gal, Mares PRC76, 044004(2007)Shevchenko, Gal, Mares PRC76, 044004(2007) Faddeev with a phenomenological K bar N potential Faddeev with a phenomenological K bar N potential Barnea, Gal, Liverts PLB712, 132(2012)Barnea, Gal, Liverts PLB712, 132(2012) Hyperspherical harmonics with a chiral SU(3)-based K bar N potential Hyperspherical harmonics with a chiral SU(3)-based K bar N potential Wycech, Green PRC79, 014001(2009) Variational with a phenomenological K bar N potential (with p-wave) Arai, Yasui, Oka/ Uchino, Hyodo, Oka PTP119, 103(2008) Λ* nuclei model /PTPS 186, 240(2010) Nishikawa, Kondo PRC77, 055202(2008) Skyrme model “K - pp” = Prototype of K bar nuclei (K bar NN, J p =1/2 -, T=1/2) All studies predict that K - pp can be bound!

8. 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. : FINUDA [6] if K - pp bound state. Exp. : FINUDA [6] if K - pp bound state. Exp. : DISTO [7] if K - pp bound state. Exp. : DISTO [7] if K - pp bound state. 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, Mares [3] (Faddeev, Phenomenological) Shevchenko, Gal, Mares [3] (Faddeev, Phenomenological) [1] PRC79, 014003 (2009) [5] PLB94, 712 (2012) [2] PRC76, 045201 (2007) [3] PRC76, 044004 (2007) [4] PRC76, 035203 (2007) [6] PRL94, 212303 (2005) [7] PRL104, 132502 (2010) Typical results of theoretical studies of K - pp Barnea, Gal, Liverts [5] (HH, Chiral SU(3)) Barnea, Gal, Liverts [5] (HH, Chiral SU(3)) -100 MeV

9. 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. : FINUDA [6] if K - pp bound state. Exp. : FINUDA [6] if K - pp bound state. Exp. : DISTO [7] if K - pp bound state. Exp. : DISTO [7] if K - pp bound state. 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, Mares [3] (Faddeev, Phenomenological) Shevchenko, Gal, Mares [3] (Faddeev, Phenomenological) [1] PRC79, 014003 (2009) [5] PLB94, 712 (2012) [2] PRC76, 045201 (2007) [3] PRC76, 044004 (2007) [4] PRC76, 035203 (2007) [6] PRL94, 212303 (2005) [7] PRL104, 132502 (2010) Typical results of theoretical studies of K - pp Barnea, Gal, Liverts [5] (HH, Chiral SU(3)) Barnea, Gal, Liverts [5] (HH, Chiral SU(3)) K bar +N+N threshold π+Σ+N threshold From theoretical viewpoint, K - pp exists between K bar -N-N and π-Σ-N thresholds! From theoretical viewpoint, K - pp exists between K bar -N-N and π-Σ-N thresholds!

10. Key points to study kaonic nuclei Similar to the bound-state calculation 1. Consider a coupled-channel problem K bar + N + N “K bar N N” π + Σ + N 2. Treat resonant states adequately.  Coupling of K bar N and πY  Bound below K bar N threshold, but a resonant state above πY threshold.  Their nature? 3. Obtain the wave function to help the analysis of the state. “coupled-channel Complex Scaling Method” K bar N (-πY) interaction?? 4. Confirmed that CSM works well on many-body systems. “Chiral SU(3)-based potential” … Anti-kaon = Nambu-Goldstone boson “Chiral SU(3)-based potential” … Anti-kaon = Nambu-Goldstone boson

11. 2. Complex Scaling Method Λ(1405) = a building block of kaonic nuclei K-K-K-K-Proton For Resonance states For Scattering problem

12. K bar N-πΣ coupled system with s-wave and isospin-0 state Λ(1405) with c.c. Complex Scaling Method K bar (J π =0 -, T=1/2) N (J π =1/2 +, T=1/2) L=0 π (J π =0 -, T=1) Σ (J π =1/2 +, T=1) L=0 K bar + N Λ(1405) π + Σ 1435 1332 [MeV]

13. Complex Scaling Method for Resonance By Complex scaling, … Complex rotation of coordinate (Complex scaling) Resonant state κ, γ : real, >0 Divergent Damping under some condition

14. Complex Scaling Method for Resonance Complex rotation of coordinate (Complex scaling) By Complex scaling,  Resonance wave function: divergent function ⇒ damping function  Resonance energy (pole position) doesn’t change. ABC theorem “The energy of bound and resonant states is independent of scaling angle θ.” † J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269. E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280 Diagonalize H θ with Gaussian base, we can obtain resonant states, in the same way as bound states!

15. Complex Scaling Method for Resonance πΣ K bar N   continuum K bar N continuum [MeV]   =30 deg. B. E. (K bar N) = 28.2 MeV Γ = 40.0 MeV … Nominal position of Λ(1405) Akaishi-Yamazaki potential † Local Gaussian form Energy independent K bar N πΣ † Y. Akaishi and T. Yamazaki, PRC65, 044005 (2002) Test calculation with a phenomenological potential

16. 2. Complex Scaling Method For Resonance states For Scattering problem Calculation of K bar N scattering amplitude

17. Calc. of scattering amplitude with CSM With help of the CSM, all problems for bound, resonant and scattering states can be treated with Gaussian base!

18. Calc. of scattering amplitude with CSM A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007) 1. Separate incoming wave Unknown Non square-integrable square-integrable for 0 < θ < π 2. Complex scaling r → re iθ 3. Calculate scattering amplitude with help of Cauchy theorem Born term is OK! Scattered part is unknown. Scattered wave function along re iθ is known! By Cauchy theorem, we can obtain as Expanding with square-integrable basis function Gaussian basis (ex: Gaussian basis) Expanding with square-integrable basis function Gaussian basis (ex: Gaussian basis) r 0 r e iθ Cauchy theorem

19. Calc. of scattering amplitude with CSM A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007) square-integrable for 0 < θ < π Gaussian basis can be expanded with Gaussian basis Scattering problem can be solved as bound-state problem by matrix calculation! Equation to be solved: Linear equation to be solved with matrix calculation!

20. 3. Set up for the calculation of K bar N-πY system “KSW-type potential” … Chiral SU(3)-based Kinematics Non-rela. approximation of KSW-type potential

21. “KSW-type potential” … chiral-SU(3) based KSW N. K aiser, P. B. S iegel and W. W eise, NPA 594, 325 (1995) Effective Chiral Lagrangian Pseudopotential Delta-function type (→ Yukawa / separable type) Up to order q 2 Local Gaussian form in r-space Weinberg-Tomozawa term Energy dependence “KSW-type potential” a (I) ij : range parameter [fm] → Easy to handle in many-body calculation with Gaussian base → Relative strength between channels determined by the SU(3) algebra ← Chiral SU(3) theory K bar NπΣ Constrained by K bar N scattering length (Martin’s value) a KN(I=0) = -1.70+i0.67fm, a KN(I=1) = 0.37+i0.60fm A.D.Martin, NPB179, 33(1979)

22. Kinematics 1. Non-relativistic 2. Semi-relativistic Reduced mass Reduced energy

23. Non-rela. approximation of KSW potential - Original Normalized Gaussian Range parameter for coupling potential → (NRv1) Non-rela. approx. version 1 Comparison of the flux factor for differential cross section between non-rela. and rela. → (NRv2) Non-rela. approx. version 2 @ non-rela. limit (small p 2 )

24. Kinematics - Original (NRv1) Non-rela. approx. ver. 1 (NRv2) Non-rela. approx. ver. 2 Non-rela. Semi-rela. Potential

25. 4. Result I=0 channel I=0 K bar N scattering length Range parameters of KSW-type potential Scattering amplitude Resonance pole … wave function and size Using Chiral SU(3) potential “KSW-type” … r-space, Gaussian form, Energy-dependent

26. I=0 K bar N scattering length Range parameters of KSW-type potential Range parameters: Pion decay constant “f π ” as a parameter f π = 90 ~ 120 MeV Found sets of range parameters (d KN,KN, d πΣ,πΣ ) to reproduce the Martin’s value. Two sets found in semi-rela. case. (SR-A, SR-B) d KN,KN d πΣ,πΣ Re a I=0 KN Im a I=0 KN f π =110 MeV

27. Scattering amplitude … Non-rela. / KSW-type NRv1 f π =110 MeV πΣK bar N πΣK bar N Resonance structure appears in K bar N and πΣ channels. Re Im K bar N → K bar N πΣ → πΣ Resonance structure at 1413 MeV Resonance structure at 1413 MeV Resonance structure at 1405 MeV Resonance structure at 1405 MeV

28. Scattering amplitude … Non-rela. / KSW-type NRv2 f π =110 MeV πΣK bar N πΣK bar N Essentially same as NRv1 case. Re Im K bar N → K bar N πΣ → πΣ Resonance structure at 1416 MeV Resonance structure at 1416 MeV Resonance structure at 1408 MeV Resonance structure at 1408 MeV

29. Scattering amplitude … Semi-rela. / KSW-type SR-A f π =110 MeV πΣK bar N πΣK bar N Qualitatively similar to non-rela. amplitudes. Re Im K bar N → K bar N πΣ → πΣ Resonance structure at 1410 MeV Resonance structure at 1410 MeV Resonance structure at 1398 MeV Resonance structure at 1398 MeV

30. Scattering amplitude … Semi-rela. / KSW-type SR-B f π =110 MeV πΣK bar N πΣK bar N Very different behavior of πΣ amplitude from other cases. Re Im K bar N → K bar N πΣ → πΣ (Another set of SR) ??? Resonance structure at 1419 MeV Resonance structure at 1419 MeV Resonance structure at 1421 MeV Resonance structure at 1421 MeV

31. Pole position of the resonance f π =90 - 120MeV - Γ / 2 [MeV] 90 120 90 120 90 120 M [MeV] 120 90 - Γ / 2 [MeV] Complex energy plane

32. Pole position of the resonance - Γ / 2 [MeV] f π =90 - 120MeV 90 120 90 120 90 120 90 M [MeV] Non-rela. (M, Γ/2) = (1418.2 ± 1.6, 21.4 ± 4.7) … NRv1 (1418.9 ± 1.1, 18.6 ± 4.6) … NRv2 (M, Γ/2) = (1418.2 ± 1.6, 21.4 ± 4.7) … NRv1 (1418.9 ± 1.1, 18.6 ± 4.6) … NRv2 - Γ / 2 [MeV]

33. Pole position of the resonance - Γ / 2 [MeV] f π =90 - 120MeV 90 120 90 120 90 120 M [MeV] 120 90 Semi-rela. (M, Γ/2) = (1420.5 ± 3, 24.5 ± 2) … SR-A (M, Γ/2) = (1420.5 ± 3, 24.5 ± 2) … SR-A - Γ / 2 [MeV]

34. Pole position of the resonance - Γ / 2 [MeV] f π =90 - 120MeV 90 120 90 120 90 120 90 M [MeV] Semi-rela. (Another set) (M, Γ/2) = (1419.5 ± 0.5, 13.0 ± 1.4) … SR-B (M, Γ/2) = (1419.5 ± 0.5, 13.0 ± 1.4) … SR-B Different behavior from other cases Different behavior from other cases - Γ / 2 [MeV]

35. “Wave function” of the resonance pole f π =110 MeV θ=30° f π =110 MeV θ=30° Non-rela. (NRv2) πΣ component also localized due to Complex scaling. πΣ component also localized due to Complex scaling. Mean distance between meson and baryon B m NR: ～ 1.3 - 0.3 i [fm] SR: ～ 1.2 - 0.5 i [fm] ? Somehow small. cf) 1.9 fm @ M = 1423 MeV † (B = 12 MeV) - Difference of binding energy? - Different definition of pole? Gamow state or bound state † A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009) ? Somehow small. cf) 1.9 fm @ M = 1423 MeV † (B = 12 MeV) - Difference of binding energy? - Different definition of pole? Gamow state or bound state † A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009)

36. 4. Result I=1 channel Range parameters of the KSW-type potential Scattering amplitude Using Chiral SU(3) potential (KSW-type) … r-space, Gaussian form, Energy-dependent

37. I=1 channel … K bar N - πΣ - πΛ Data : a I=1 KbarN = 0.37 + i 0.60 fm (A. D. Martin) † Range parameters: † A. D. Martin, Nucl. Phys. B 179, 33 (1981) Potential … Weinberg-Tomozawa term only

38. I=1 channel … K bar N - πΣ - πΛ Range parameters: SU(3) C.G. coefficients for I=1 channel × × Data : a I=1 KbarN = 0.37 + i 0.60 fm (A. D. Martin) † Potential … Weinberg-Tomozawa term only

39. I=1 channel … K bar N - πΣ - πΛ Range parameters: SU(3) C.G. coefficients for I=1 channel × × Data : a I=1 KbarN = 0.37 + i 0.60 fm (A. D. Martin) † Potential … Weinberg-Tomozawa term only

40. I=1 channel … K bar N - πΣ - πΛ Range parameters: SU(3) C.G. coefficients for I=1 channel × × “Isospin symmetric choice” Ref.) chiral unitary model “Isospin symmetric choice” Ref.) chiral unitary model Data : a I=1 KbarN = 0.37 + i 0.60 fm (A. D. Martin) † Potential … Weinberg-Tomozawa term only

41. I=1 channel … K bar N - πΣ - πΛ A. D. Martin : a I=1 KbarN = 0.37 + i 0.60 fm “Iso-symmetric choice” : {d KN,KN, d πΣ,πΣ } fixed to I=0 channel ones Search d KN,πΣ to reproduce Re or Im a I=1 KbarN of Matin’s value. Difficult to reproduce Re a I=1 KbarN Martin’s value within our model. When Re a I=1 KbarN is reproduced, Im a I=1 KbarN deviates largely from Martin’s value. (fit) (I=0 one) (Search)

42. I=1 channel … K bar N - πΣ - πΛ NRv2 (c): a I=1 KbarN = 0.657 + i 0.599 fm f π =110 MeV Im a I=1 KbarN Search d KN,πΣ to reproduce Im a I=1 KbarN of Matin’s value. K bar NπΣπΛ K bar NπΣπΛ SR-A (c): a I=1 KbarN = 0.659 + i 0.600 fm

43. I=1 channel … K bar N - πΣ - πΛ Only d KN,KN fixed to I=0 channel ones In a study with separable potential, the cutoff parameter for K bar N is not so different between I=0 and 1 channels. † Search {d πΣ,πΣ, d KN,πΣ } to reproduce Re and Im a I=1 KbarN of Matin’s value. When the only d KN,KN is fixed iso-symmetrically, we can find a set of {d πΣ,πΣ, d KN,πΣ } to reproduce simultaneously Re and Im of Martin’s value (fit) † Y. Ikeda and T. Sato, PRC 76, 035203 (2007) (I=0 one) (Search)

44. I=1 channel … K bar N - πΣ - πΛ NRv2 (a): a I=1 KbarN = 0.376 + i 0.606 fm f π =110 MeV Re and Im a I=1 KbarN Search {d πΣ,πΣ, d KN,πΣ } to reproduce Re and Im a I=1 KbarN of Matin’s value. SR-A (a): a I=1 KbarN = 0.375 + i 0.605 fm K bar NπΣπΛ K bar NπΣπΛ A narrow resonance exists at a few MeV below πΣ threshold A narrow resonance exists at a few MeV below πΣ threshold πΣ repulsive ???

45. 5. Summary and Future plan

46. 5. Summary Scattering and resonant states of K bar N-πY system is studied with a coupled-channel Complex Scaling Method using a chiral SU(3) potential A Chiral SU(3) potential “KSW-type” … r-space, Gaussian form, energy dependence Calculated scattering amplitude with help of CSM Scattering states as well as resonant and bound states are treated with Gaussian base. Non-rela. / Semi-rela. kinematics and two types of non-rela. approximation of KSW-type potential are tried.  Determined by the K bar N scattering length obtained by Martin’s analysis of old data.  f π dependence (f π = 90 ～ 120MeV)  Found two sets of range parameters in SR case. K bar N-πY system is essential for the study of K bar nuclear system which is expected to be an exotic nuclear system with strangeness.

47. 5. Summary and future plans Future plans Three-body system (K bar NN-πYN); Updated data of K - p scattering length by SHIDDARTA Pole position “Size” of the I=0 pole state NR: ～ 1.3 – 0.3i fm SR: ～ 1.2 – 0.5i fm (M, Γ/2) NRv1 : (1418.2 ± 1.6, 21.4 ± 4.7) NRv2 : (1418.9 ± 1.1, 18.6 ± 4.6) SR-A : (1420.5 ± 0.5, 24.5 ± 2) SR-B : (1419 ± 1, 13.0 ± 2) (M, Γ/2) NRv1 : (1418.2 ± 1.6, 21.4 ± 4.7) NRv2 : (1418.9 ± 1.1, 18.6 ± 4.6) SR-A : (1420.5 ± 0.5, 24.5 ± 2) SR-B : (1419 ± 1, 13.0 ± 2)  I=0 channel (K bar N-πΣ)  I=1 channel (K bar N-πΣ-πΛ) Difficult to reproduce Re a I=1 KbarN of Martin’s value within our model, in case of “iso-symmetric choice” of range parameter. Another self-consistent solution NR: (~1360, 40~90), SR: (1350~1390, 30~100) … Lower pole of double pole?

48. Thank you very much! A. D. is thankful to Prof. Katō for his advice on the scattering-amplitude calculation in CSM, and to Dr. Hyodo for useful discussion.

49. Backup slides

50. Rough estimation of charge radius K-K- p Distance CM † Calculated from electric form factor with chiral unitary model T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)

51. Charge radius Chiral (HW-HNJH): B ～ 12 MeV, Distance = 1.86 fm → = -1.07 fm 2 Charge radius derived from electric form factor T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008) Rough estimation

52. 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… 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 ] Antisymmetrized Molecular Dynamics method with a phenomenological K bar N potential A. Dote, H. Horiuchi, Y. Akaishi and T. Yamazaki, PRC70, 044313 (2004) K-K- Relate to various interesting physics such as …  Restoration of chiral symmetry in dense matter  Interesting structure  Neutron star

53. Excited hyperon Λ （１４０ ５） I=0 Proton-K - bound state with 30MeV binding energy with 30MeV binding energy Not 3 quark state, Not 3 quark state, ← can’t be explained with a simple quark model… But rather a molecular state But rather a molecular state I=0 Proton-K - bound state with 30MeV binding energy with 30MeV binding energy Not 3 quark state, Not 3 quark state, ← can’t be explained with a simple quark model… But rather a molecular state But rather a molecular state u bar s u u d s u d T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012)

54. K - pp Λ(1405) Today, K - pp has been focused in theor. and exp. studies! As the first trial of our study with ccCSM, consider Λ(1405). K-K-K-K- PP Prototype of Kaonic nuclei 3 HeK -, pppK -, 4 HeK -, pppnK -, … 8 BeK -, …Complicated nuclear system with K - K-K-K-K- Proton Λ(1405) = Building block of Kaonic nuclei

55. Asymptotic behavior of the wave function before/after complex scaling Bound state Resonant state Continuum state κ B : real, >0 κ, γ : real, >0 k : real, >0 Damping Damping under some condition Oscillating Diverging Damping under some condition

56. Complex Scaling Method for Resonance Great success in nuclear physics … in particular, unstable nuclei. Well applied to usual calculation of bound states… diagonalize with Gaussian base (Gaussian Expansion Method, Correlated Gaussian) Tiny modification of matrix elements Possible to study many-body systems … up to 5-body system ( 4 He+n+n+n+n) “Cluster-Orbital Shell Model” T. Myo, R. Ando, K. Kato, PLB691, 150 (2010)

57. Calc. of scattering amplitude with CSM A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007) Complex scaling r → r e iΘ Unknown Non square-integrable Separate the incoming wave ! Point 1 Point 2 square-integrable for 0 < θ < π Expanding with square-integrable basis function Gaussian basis (ex: Gaussian basis)

58. Calculation of scattering amplitude We don’t have which is a solution along r, but we have which is a solution along re iθ. r 0 r e iθ Cauchy theorem Point 3 Calc. of scattering amplitude with CSM is independent of θ. expressed with Gaussian base A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007)

59. Test calculation with AY potential Resonance pole πΣ K bar N   continuum K bar N continuum [MeV]   =30 deg. B. E. (K bar N) = 28.2 MeV Γ = 40.0 MeV … Nominal position of Λ(1405) Akaishi-Yamazaki potential † Phenomenological potential Local Gaussian form Energy independent K bar N πΣ † Y. Akaishi and T. Yamazaki, PRC65, 044005 (2002)

60. Complex Scaling Method for Resonance Study of unstable nuclei … Resonance of 8 He “Cluster-Orbital Shell Model” T. Myo, R. Ando, K. Kato, PLB691, 150 (2010) Bound state Resonance Continuum on 2θ line Continuum on 2θ line

61. Test calculation with AY potential Unitarity violation of S-matrix Phase shift sum | |Det S| -1 | E KbarN [MeV] Checked by Continuum Level Density method (R. Suzuki, A. T. Kruppa, B. G. Giraud, and K. Katō, PTP119, 949(2008)) δ KbarN + δ πΣ [deg.]

62. Test calculation with AY potential K bar N (I=0) scattering amplitude Scattering length (Scatt. amp. @ E KbarN =0) = -1.77 + i 0.47 fm Y. Akaishi and T. Yamazaki, PRC65, 044005 (2002) Scattering length = -1.76 + i 0.46 fm E KbarN [MeV]

63. Scattering amplitude … Non-rela. / KSW-type SR f π =90 MeV πΣK bar N πΣK bar N Kinematics-potential mismatched case Re Im K bar N → K bar N πΣ → πΣ … A virtual state exists. (a πΣ, r e )=(61,-6.3) fm satisfies -a πΣ /2 < r e †. † Y. Ikeda et al, PTP 125, 1205 (2011) Singular behavior at πΣ threshold

64. I=0 K bar N scattering length Range parameters of KSW-type potential f π =110 MeV Data : a I=0 KbarN = -1.70 + i 0.68 fm (A. D. Martin) Found sets of range parameters (d KN,KN, d πΣ,πΣ ) to reproduce the Martin’s value. Two sets found in semi-rela. case. (SR-A, SR-B) d KN,KN d πΣ,πΣ Re a I=0 KN Im a I=0 KN

65. Resonant structure in the scattering amplitude Energy @ Re f KbarN → KbarN = 0 Energy @ Re f πΣ → πΣ = 0 f π =90 - 120MeV

66. Resonant structure in the scattering amplitude Energy @ Re f KbarN → KbarN = 0 Energy @ Re f πΣ → πΣ = 0 f π =90 - 120MeV Strange behavior? SR-B SR-A

67. “Wave function” of the resonance pole f π =110 MeV θ=30° f π =110 MeV θ=30° πΣ component also localized due to Complex scaling. * The wave functions shown above are multiplied by a phase factor so that the K bar N wfn. becomes real at r=0. Non-rela. (NRv2) Semi rela. (SR-A) Semi rela. (SR-B)

68. “Size” of the resonance pole state Mean distance between meson and baryon Re Im f π =90 - 120MeV NR: ～ 1.3 - 0.3 i [fm] SR: ～ 1.2 - 0.5 i [fm] B M ? Somehow small. cf) 1.9 fm @ M = 1423 MeV † (B = 12 MeV) - Difference of binding energy? - Different definition of pole? Gamow state or bound state † A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009) ? Somehow small. cf) 1.9 fm @ M = 1423 MeV † (B = 12 MeV) - Difference of binding energy? - Different definition of pole? Gamow state or bound state † A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009)

69. Strange nuclear physicsNucleus Hypernuclei…ud s Hyperon baryon = qqq Strangeness is introduced through baryons.

70. Strange nuclear physicsNucleus Kaonic nuclei ! u bar s K - meson (anti-kaon, K bar ) Strangeness is introduced through mesons … meson = qq bar

71. Key points to study kaonic nuclei Similar to the bound-state calculation 1. Consider a coupled-channel problem K bar + N + N “K bar N N” π + Σ + N 2. Treat resonant states adequately.  K bar N couples with πY.  Bound state for K bar N channel but a resonant state for πY channel  Their nature? 3. Obtain the wave function to help the analysis of the state. “coupled-channel Complex Scaling Method” K bar N (-πY) interaction?? “Chiral SU(3)-based potential”

72. Excited hyperon Λ （１４０ ５） 939p,n 1115 Λ 1190 Σ p + K - 1435 Λ(1405) 1405 1250 Λ + π Σ + π 1325 Energy [MeV] s u d

73. Mysterious state; Λ(1405) Quark model prediction … calculated as 3-quark state N. Isgar and G. Karl, Phys. Rev. D18, 4187 (1978) q q q Λ(1405) can’t be well reproduced as a 3-quark state! Λ(1405) can’t be well reproduced as a 3-quark state! observed Λ(1405) calculated Λ(1405)

74. 赤字のファクターは、微分断面積で適切な 相対論的 flux factor を得るために導入されている。 KSW の（１４）式、 Born 近似での微分断面積の表式 （相対論的な場合） k i : meson momentum 、 ω i : reduced energy ３． KSW 非相対論版の考察

75. 非相対論的な場合の Born 近似での微分断面積 f ij : 散乱振幅 v i : 速度 k ij : 運動量 μ i : 換算質量 ３． KSW 非相対論版の考察

76. 相対論・非相対論での Born 近似微分断面積の表式の比較 相対論的 ： (KSW) 非相対論的 ： 微分断面積の表式は、相対論的なものと非相対論的なもので、 明らかに対応している。 非相対論的な表式の換算質量（ μ i ）が、 相対論的表式では換算エネルギー（ ω i ）に対応。 ３． KSW 非相対論版の考察

77. 推 測 推 測 KSW の pseudo-potential 中のファクター が微分断面積の表式とつじつまが合うように導入されているのならば、 このポテンシャルの非相対論版を考える際には、 前ページの対応関係からこのファクター中の換算エネルギーを 換算質量に置き換えればいいのでは？ つまり カイラル理論を尊重するので、メソンのエネルギー ω i は残しておく。 （続きは次ページ） 1 / √s はどうするか？そのまま、系の全エネルギー（ E Tot ）として残しておく？？ ３． KSW 非相対論版の考察

78. Comparison with other studies of I=0 channel

79. Comparison with other studies of I=0 K bar N-πΣ Kaiser-Siegel- Weise Hyodo-Weise Ikeda-Hyodo- Jido-Kamano- Sato-Yazaki Dote-Inoue- Myo BaseChargeIsospin Isopin TermsAll terms up to q 2 WT term Regularization Yukawa / Separable form Dimensional regularization / Separable form Gaussian form Constraint Low energy scattering data, Branching ratio at K bar N threshold Low energy scattering data, Branching ratio at K bar N threshold K bar N scattering length / Λ(1405) pole K bar N scattering length Ref. NPA 594, 325 (1995)PRC 77, 035204 (2008)PTP 125, 1205 (2011)NPA 912, 66 (2013)

80. Comparison with other studies of I=0 K bar N-πΣ IHJKSY, HW: DIM: Different energy dependence of interaction kernel Proportional to just meson energy Not proportional to meson energy … including relativistic q 2 correction and non-static effect of baryons and non-static effect of baryons

81. Comparison with other studies of I=0 K bar N-πΣ IHJKSY HW

82. Use the same interaction kernel as that of IHJKSY and HW in our ansatz (Flux factor and Gaussian form factor) Suppressed the enhancement near the πΣ threshold, but still different from the amplitudes of IHJKSY. Different interaction kernel Different interaction kernel  Use of flux factor  Different regularization scheme (Gaussian form factor vs Dimensional reg.) * Gaussian form factor enhances the amplitude far below the threshold as |E| exp [c|E|] with E → -∞. (c>0)

83. 1. Introduction Neutron star in universe Radius ～ 10km Mass ～ M sun High dense at core ～ > 2 ρ 0 Maybe … Involving strangeness K bar nuclear system on the Earth Self-bound K bar -nuclear system K-K-K-K- … light nucleus with K - mesons Involving strangeness via K - meson

84. 1. Introduction Neutron star in universe Radius ～ 10km Mass ～ M sun High dense at core ～ > 2 ρ 0 Maybe … Involving strangeness K bar nuclear system on the Earth Density [fm -3 ] 0.00 0.14 3 He Density [fm -3 ] 0.0 0.75 1.5 3 HeK - 4×4 fm 2 … light nucleus with K - mesons Involving strangeness via K - meson Strong attraction by K - meson ⇒ Dense nuclear system ⇒ Dense nuclear system † A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313. A phenomenological K bar N potential † ⇒ B(K) ～ 100MeV, 2 ～ 4ρ 0 ⇒ B(K) ～ 100MeV, 2 ～ 4ρ 0

85. FINUDA collaboration (DAΦNE, Frascati) K - absorption at rest on various nuclei ( 6 Li, 7 Li, 12 C, 27 Al, 51 V) Invariant-mass method Strong correlation between emitted p and Λ (back-to-back) Invariant mass of p and Λ K-K- p p p Λ If it is K - pp, … Total binding energy = 115 MeV Decay width = 67 MeV PRL 94, 212303 (2005) Experimental studies of K - pp

86. Re-analysis of KEK-PS E549 DISTO collaboration - p + p -> K + + Λ + p @ 2.85GeV - Λp invariant mass - Comparison with simulation data T. Yamazaki et al. (DISTIO collaboration), PRL104, 132502 (2010) - K - stopped on 4 He target - Λp invariant mass T. Suzuki et al (KEK-PS E549 collaboration), arXiv:0711.4943v1[nucl-ex] K - pp??? B. E.= 103 ±3 ±5 MeV Γ = 118 ±8 ±10 MeV K - pp??? B. E.= 103 ±3 ±5 MeV Γ = 118 ±8 ±10 MeV Strong Λp back-to-back correlation is confirmed. Unknown strength is there in the same energy region as FINUDA. Strong Λp back-to-back correlation is confirmed. Unknown strength is there in the same energy region as FINUDA. Experimental studies of K - pp

87. E15: A search for deeply bound kaonic nuclear states by 3 He(inflight K -, n) reaction (M. Iwasaki (RIKEN), T. Nagae (Kyoto)) E17: Precision spectroscopy of kaonic 3 He atom 3d→2p X-rays (R. Hayano (Tokyo), H. Outa (Riken)) E27: Search for a nuclear Kbar bound state K - pp in the d(π +, K + ) reaction (T. Nagae (Kyoto)) E31: Spectroscopic study of hyperon resonances below KN threshold via the (K -, n) reaction on deuteron (H. Noumi (Osaka)) as of 2011 as of 2011 Experiments of kaonic nuclei at J-PARC

88. E15: A search for deeply bound kaonic nuclear states by 3 He(inflight K -, n) reaction by 3 He(inflight K -, n) reaction (M. Iwasaki (RIKEN), T. Nagae (Kyoto)) E17: Precision spectroscopy of kaonic 3 He atom 3d→2p X-rays (R. Hayano (Tokyo), H. Outa (Riken)) E27: Search for a nuclear Kbar bound state K - pp in the d(π +, K + ) reaction (T. Nagae (Kyoto)) E31: Spectroscopic study of hyperon resonances below KN threshold via the (K -, n) reaction on deuteron (H. Noumi (Osaka)) as of 2011 as of 2011 Experiments of kaonic nuclei at J-PARC Dr. Fujioka’s talk (KEK workshop, 7-9. Aug. 08) at K1.8BR beam line 1.8GeV/c

89. E15: A search for deeply bound kaonic nuclear states by 3 He(inflight K -, n) reaction by 3 He(inflight K -, n) reaction (M. Iwasaki (RIKEN), T. Nagae (Kyoto)) E17: Precision spectroscopy of kaonic 3 He atom 3d→2p X-rays (R. Hayano (Tokyo), H. Outa (Riken)) E27: Search for a nuclear Kbar bound state K - pp in the d(π +, K + ) reaction (T. Nagae (Kyoto)) E31: Spectroscopic study of hyperon resonances below KN threshold via the (K -, n) reaction on deuteron (H. Noumi (Osaka)) as of 2011 as of 2011 Experiments of kaonic nuclei at J-PARC Missing mass spectroscopy at K1.8BR beam line 1.8GeV/c Invariant mass spectroscopy Dr. Fujioka’s talk (KEK workshop, 7-9. Aug. 08) All emitted particles will be measured. 「完全実験」 「完全実験」

90. E15: A search for deeply bound kaonic nuclear states by 3 He(inflight K -, n) reaction (M. Iwasaki (RIKEN), T. Nagae (Kyoto)) E17: Precision spectroscopy of kaonic 3 He atom 3d→2p X-rays (R. Hayano (Tokyo), H. Outa (Riken)) E27: Search for a nuclear Kbar bound state K - pp in the d(π +, K + ) reaction (T. Nagae (Kyoto)) E31: Spectroscopic study of hyperon resonances below KN threshold via the (K -, n) reaction on deuteron (H. Noumi (Osaka)) as of 2011 as of 2011 Experiments of kaonic nuclei at J-PARC d π+π+ K+K+ Λ(1405) n p p p K-K- Preceding E15, E27 experiment was performed in June. Analysis is now undergoing! Preceding E15, E27 experiment was performed in June. Analysis is now undergoing! Use lots of pion Missing mass

91. E15: A search for deeply bound kaonic nuclear states by 3 He(inflight K -, n) reaction (M. Iwasaki (RIKEN), T. Nagae (Kyoto)) E17: Precision spectroscopy of kaonic 3 He atom 3d→2p X-rays 3d→2p X-rays (R. Hayano (Tokyo), H. Outa (Riken)) E27: Search for a nuclear Kbar bound state K - pp in the d(π +, K + ) reaction (T. Nagae (Kyoto)) E31: Spectroscopic study of hyperon resonances below KN threshold via the (K -, n) reaction on deuteron via the (K -, n) reaction on deuteron (H. Noumi (Osaka)) as of 2011 as of 2011 Experiments of kaonic nuclei at J-PARC K bar N interaction Λ(1405) K - pp

92. K - pp These days, K - pp has been focused in both of theoretical and experimental studies! K-K-K-K-Proton Λ(1405) 3 HeK -, pppK -, 4 HeK -, pppnK -, … 8 BeK -, … Complicated nuclear system with K - = K-K-K-K- PP Most essential kaonic nucleus!

93. Coupled Channel Chiral Dynamics (Chiral Unitary model) We are thinking about K bar N interaction. S=-1 meson-baryon system is constrained by The leading couplings between Low mass pseudo-scalar meson octet (Nambu-Goldston bosons) and Baryon octet are determined by Spontaneous breaking of SU(3)×SU(3) Chiral symmetry Chiral SU(3) dynamics !

94. Chiral low-energy theorem tells us … K bar N πΣ ηΛ KΞ For I=0 channel, Remark There are no free parameters as for coupling. There is an attractive interaction in πΣ-πΣ channel, while AY potential doesn’t have it. Energy and mass of baryon in channel i and Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian : Psuedo-scalar meson decay constant Coupled Channel Chiral Dynamics (Chiral Unitary model)

95. 3. Solve coupled channel Bethe-Salpeter equation. T-matrix of coupled channel scattering + … ++ … … = 2. Using the WT-term as a building block, 1. Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian: K - p scattering length Threshold branching ratio Total cross section of K - p scattering Coupled Channel Chiral Dynamics (Chiral Unitary model)

96. Threshold branching ratio Total cross section of K - p scattering T. Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003) Coupled Channel Chiral Dynamics (Chiral Unitary model)

97. 3. Solve coupled channel Bethe-Salpeter equation. T-matrix of coupled channel scattering + … ++ … … = 2. Using the WT-term as a building block, Λ(1405) is dynamically generated as meson-baryon system. 1. Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian: K - p scattering length Threshold branching ratio Total cross section of K - p scattering Coupled Channel Chiral Dynamics (Chiral Unitary model)

98. I=0 πΣ mass distribution T. Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003) Dynamical generation of Λ(1405) ! Remark: Calculated with πΣ-πΣ scattering amplitude. Coupled Channel Chiral Dynamics (Chiral Unitary model)

99. Chiral SU(3) potential (KSW) N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995) a: range parameter [fm] M i, m i : Baryon, Meson mass in channel i E i : Baryon energy, ω i : Meson energy Reduced energy: K bar NπΣ Energy dependence of V ij is controlled by CM energy √s. Flavor SU(3) symmetry Original: δ-function type Present: Normalized Gaussian type a: range parameter [fm] Energy dependence is determined by Chiral low energy theorem. ← Kaon: Nambu-Goldstone boson ← Kaon: Nambu-Goldstone boson

100. Chiral SU(3) potential (KSW) Energy dependence √s [MeV] K bar N-K bar N πΣ-πΣ K bar N-πΣ N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995) K bar N threshold πΣ threshold

101. Chiral SU(3) potential = Energy-dependent potential Calculational procedure Perform the Complex Scaling method. Then, find a pole of resonance or bound state. Check Finished ! If Yes Self consistency for the energy! Assume the values of the CM energy √s. If No

102. 4. Experiments related to K bar nuclear physics

103. Ｋ原子核に関係する実 験 LEPS / SPring-8 πΣ invariant mass 測定  Ｋ原子 (Kaonic atom)  Λ(1405) CLAS / JLab Kaonic 4 He atom, 2p レベルシフト (3d→2p X 線測定 ) @ KEK (E570) S. Okada et. al., Phys. Lett. B653, 387 (2007) Kaonic hydrogen atom, 1s レベルシフト @ DEAR group, DAΦNE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories Kaonic hydrogen, deuterium @ SIDDHARTA group Kaonic 3 He atom, 2p レベルシフト (3d→2p X 線測定 ) @ J-PARC (E17, DAY-1) γ + p → K + + Λ(1405), Λ(1405) → π Σ K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010) J. K. Ahn, Nucl. Phys. A835, 329 (2010) π - Σ +, π 0 Σ 0, π + Σ - が全て押さえられた M. Bazzi et al., Phys. Lett. B704, 113 (2011)

104. Ｋ原子核に関係する実 験 LEPS / SPring-8 πΣ invariant mass 測定  Ｋ原子 (Kaonic atom)  Λ(1405) CLAS / JLab Kaonic 4 He atom, 2p レベルシフト (3d→2p X 線測定 ) @ KEK (E570) S. Okada et. al., Phys. Lett. B653, 387 (2007) Kaonic hydrogen atom, 1s レベルシフト @ DEAR group, DAΦNE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories Kaonic hydrogen, deuterium @ SIDDHARTA group Kaonic 3 He atom, 2p レベルシフト (3d→2p X 線測定 ) @ J-PARC (E17, DAY-1) γ + p → K + + Λ(1405), Λ(1405) → π Σ K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010) J. K. Ahn, Nucl. Phys. A835, 329 (2010) π - Σ +, π 0 Σ 0, π + Σ - が全て押さえられた M. Bazzi et al., Phys. Lett. B704, 113 (2011)

105. DEAR exp. for kaonic hydrogen atom シフトの符号は同じだが、ＫＥＫの前回の実験（ＫｐＸ）と重ならない！ Kaonic hydrogen atom, 1s のレベルシフト @ DEAR Collaboration, DAΦNE, Frascati National Laboratories G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) cf) KEK exp. M.Iwasaki et al., Phys. Rev. Lett. 78, 3067 (1997) B. Borasoy et al., Phys. Rev. Lett. 94, 213401 (2005) KEK exp. DEAR Coupled channel chiral dynamics (Chiral unitary model) で DEAR の結果を合わすのには苦労する。 かろうじてギリギリ合わせられる程度。。。

106. SHIDDARTA exp. for kaonic hydrogen atom KEK 実験（ＫｐＸ）とコンシステントな結果 Kaonic hydrogen atom, 1s のレベルシフト @ SHIDDARTA Collaboration, DAΦNE, Frascati National Laboratories M. Bazzi et al., Phys. Lett. B704, 113 (2011) K - p 散乱長が精密に決定 理論計算にとって重要な インプットに強い拘束条件 K bar N subthreshold での 散乱振幅の振る舞い、 Λ(1405) のポールの位置、 が制限される。 Y. Ikeda, T. Hyodo and W. Weise, Phys. Lett. B706, 63 (2011)

107. Ｋ原子核に関係する実 験 LEPS / SPring-8 πΣ invariant mass 測定  Ｋ原子 (Kaonic atom)  Λ(1405) CLAS / JLab Kaonic 4 He atom, 2p レベルシフト (3d→2p X 線測定 ) @ KEK (E570) S. Okada et. al., Phys. Lett. B653, 387 (2007) Kaonic hydrogen atom, 1s レベルシフト @ DEAR group, DAΦNE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories Kaonic hydrogen, deuterium @ SIDDHARTA group Kaonic 3 He atom, 2p レベルシフト (3d→2p X 線測定 ) @ J-PARC (E17, DAY-1) γ + p → K + + Λ(1405), Λ(1405) → π Σ K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010) J. K. Ahn, Nucl. Phys. A835, 329 (2010) π - Σ +, π 0 Σ 0, π + Σ - が全て押さえられた M. Bazzi et al., Phys. Lett. B704, 113 (2011)

108. KEK E570 for kaonic 4 He atom シフトは 0 eV と consisitent Kaonic 4 He atom, 2p のレベルシフト 3d→2p X 線測定 @ KEK, E570 理論の予言がほぼ 0eV に対して、 過去の実験ではシフトは平均 -43 eV “Kaonic helium puzzle” S. Okada et. al., Phys. Lett. B653, 387 (2007) パズルは解けた！ S. Hirenzaki et al., Phys. Rev. C61, 055205 (2000)

109. J-PARC for kaonic 3 He atom Kaonic 3 He atom, 2p のレベルシフト 3d→2p X 線測定 @ J-PARC, E17 DAY-1 S. Okada et. al., Phys. Lett. B653, 387 (2007) Kaonic 4 He atom, 2p のレベルシフト … ほぼ 0 eV と確定 赤石氏の計算 K bar N potential の強度が 二つの領域に絞られた。 ＋ さらに 3 He でシフトが測定されることで K bar N potential の強度に絞りを 掛けることが期待できる。 Y. Akaishi, Proceedings of EXA’05, Austrian Academy of Sciences press, Vienna, 2005, p.45 ※特定領域研究「ストレンジネスで探るクォーク多体系」研究会 2007 での岡田氏のスライドより引用

110. Ｋ原子核に関係する実 験 LEPS / SPring-8 πΣ invariant mass 測定  Ｋ原子 (Kaonic atom)  Λ(1405) CLAS / JLab Kaonic 4 He atom, 2p レベルシフト (3d→2p X 線測定 ) @ KEK (E570) S. Okada et. al., Phys. Lett. B653, 387 (2007) Kaonic hydrogen atom, 1s レベルシフト @ DEAR group, DAΦNE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories Kaonic hydrogen, deuterium @ SIDDHARTA group Kaonic 3 He atom, 2p レベルシフト (3d→2p X 線測定 ) @ J-PARC (E17, DAY-1) γ + p → K + + Λ(1405), Λ(1405) → π Σ K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010) J. K. Ahn, Nucl. Phys. A835, 329 (2010) π - Σ +, π 0 Σ 0, π + Σ - が全て押さえられた M. Bazzi et al., Phys. Lett. B704, 113 (2011)

111. Λ(1405) - πΣ invariant mass 測定 - 三つの異なる電荷状態が抑えられた LEPS / Spring-8 K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010) CLAS / Jefferson Laboratory γ + p → K + + Λ(1405), Λ(1405) → π Σ ピークの順番が理論 (chiral unitary) と違う？ Highest peak 実験： Σ + π - 理論： Σ - π + 理論 J. K. Ahn, Nucl. Phys. A835, 329 (2010) p (γ, K + π) Σ at E γ = 1.5-2.4GeV Charged πΣ を測定

112. What is the object observed experimentally? Only what we can say from only this spectrum is that “There is some object with B=2, S=-1, charge=+1”… DISTO collaboration

113. Self-consistent solutions for complex energy : complex energy Self-consistent solution … Ex) KSW ; f π = 100 MeV, a= 0.47 fm ( θ=35°) Self-consistency for complex energy should be considered! Chiral potential … Energy-dependent potential Self-consistency achieved after a few times iteration. A. D., T. Inoue and T. Myo, Proc. of BARYONS’ 10, AIP conf. proc. 1388, 549 (2010)

114. f π = 90 MeV Scattering amplitude

115. … KSW org. – Non. rela. Phase shift (πΣ) K bar N → K bar N Re Im πΣ → πΣ K bar N → πΣ δ （ πΣ ） πΣ→ K bar N f π =90 MeV πΣK bar N

116. Scattering amplitude … KSW NRv1 – Non. rela. f π =90 MeV Phase shift (πΣ) Re Im K bar N → K bar N K bar N → πΣ πΣ→ K bar N πΣ → πΣ δ （ πΣ ） πΣK bar N

117. Scattering amplitude … KSW NRv2 – Non. rela. f π =90 MeV Phase shift (πΣ) Re Im K bar N → K bar N K bar N → πΣ πΣ→ K bar N πΣ → πΣ δ （ πΣ ） πΣK bar N

118. Scattering amplitude … KSW org. – Semi rela. f π =90 MeV Phase shift (πΣ) Re Im K bar N → K bar N K bar N → πΣ πΣ→ K bar N πΣ → πΣ δ （ πΣ ） πΣK bar N

119. Succeeded in nuclear physics … Especially used in the study of unstable nuclei. Resonant state of 6 He 4 He n n S. Aoyama, T. Myo, K. Kato and K. Ikeda, Prog. Theor. Phys. 116, 1 (2006) Continuum Resonance Complex Scaling Method

120. Result Scattering amplitude Scattering amplitude Position of resonant structure appeared in the scattering amplitude Position of resonant structure appeared in the scattering amplitude “Wave function” of the resonance pole “Wave function” of the resonance pole “Size” of the resonance pole state “Size” of the resonance pole state θ dependence of “Size” and “Wave function” θ dependence of “Size” and “Wave function” Comparison with Hyodo-Weise’s result Comparison with Hyodo-Weise’s result

121. Potential: KSW original Kinematics: Non. rela. Scattering amplitude

122. … KSW org. – Non. rela. f π =90 MeV Phase shift (πΣ) δ （ πΣ ） Re Im K bar N → K bar N πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ）

123. Scattering amplitude … KSW org. – Non. rela. f π =100 MeV Phase shift (πΣ) δ （ πΣ ） Re Im K bar N → K bar N πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ）

124. Scattering amplitude … KSW org. – Non. rela. f π =110 MeV Phase shift (πΣ) δ （ πΣ ） Re Im K bar N → K bar N πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ）

125. Scattering amplitude … KSW org. – Non. rela. f π =120 MeV Phase shift (πΣ) δ （ πΣ ） Re Im K bar N → K bar N πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ）

126. Potential: KSW NRv1 Kinematics: Non. rela. Scattering amplitude

127. … KSW NRv1 – Non. rela. f π =90 MeV Phase shift (πΣ) δ （ πΣ ） Re Im K bar N → K bar N πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ）

128. Scattering amplitude … KSW NRv1 – Non. rela. Phase shift (πΣ) f π =100 MeV δ （ πΣ ） πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ） Re Im K bar N → K bar N

129. Scattering amplitude … KSW NRv1 – Non. rela. Phase shift (πΣ) δ （ πΣ ） f π =110 MeV πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ） Re Im K bar N → K bar N

130. Scattering amplitude … KSW NRv1 – Non. rela. Phase shift (πΣ) πΣ → πΣ δ （ πΣ ） f π =120 MeV δ （ πΣ ） πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ） Re Im K bar N → K bar N

131. Potential: KSW NRv2 Kinematics: Non. rela. Scattering amplitude

132. … KSW NRv2 – Non. rela. Phase shift (πΣ) πΣ → πΣ δ （ πΣ ） f π =90 MeV δ （ πΣ ） Re Im K bar N → K bar N πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ）

133. Scattering amplitude … KSW NRv2 – Non. rela. Phase shift (πΣ) f π =100 MeV δ （ πΣ ） Re Im K bar N → K bar N πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ）

134. Scattering amplitude … KSW NRv2 – Non. rela. Phase shift (πΣ) πΣ → πΣ δ （ πΣ ） f π =110 MeV δ （ πΣ ） Re Im K bar N → K bar N πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ）

135. Scattering amplitude … KSW NRv2 – Non. rela. Phase shift (πΣ) πΣ → πΣ δ （ πΣ ） f π =120 MeV δ （ πΣ ） πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ） Re Im K bar N → K bar N

136. Potential: KSW original Kinematics: Semi rela. Scattering amplitude

137. … KSW org. – Semi rela. Phase shift (πΣ) πΣ → πΣ δ （ πΣ ） f π =90 MeV δ （ πΣ ） Re Im K bar N → K bar N πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ）

138. Scattering amplitude … KSW org. – Semi rela. Phase shift (πΣ) πΣ → πΣ δ （ πΣ ） f π =100 MeV δ （ πΣ ） Im K bar N → K bar N πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ） Re

139. Scattering amplitude … KSW org. – Semi rela. Phase shift (πΣ) πΣ → πΣ δ （ πΣ ） f π =110 MeV δ （ πΣ ） Im K bar N → K bar N πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ） Re

140. Scattering amplitude … KSW org. – Semi rela. Phase shift (πΣ) πΣ → πΣ δ （ πΣ ） f π =120 MeV δ （ πΣ ） Im K bar N → K bar N πΣ → πΣ K bar N → πΣ πΣ→ K bar N δ （ πΣ ） Re

141. Position of resonant structure appeared in the scattering amplitude

142. Position of resonant structure appeared in the scattering amplitude f π =90 MeV K bar N → K bar N πΣ → πΣ

143. Position of resonant structure appeared in the scattering amplitude f π =100 MeV K bar N → K bar N πΣ → πΣ

144. Position of resonant structure appeared in the scattering amplitude f π =110 MeV K bar N → K bar N πΣ → πΣ

145. Position of resonant structure appeared in the scattering amplitude f π =120 MeV K bar N → K bar N πΣ → πΣ

146. Position of resonant structure appeared in the scattering amplitude Resonance position in K bar N → K bar N Resonance position in K bar N → K bar N Resonance position in πΣ → πΣ Resonance position in πΣ → πΣ

147. “Wave function” of the resonance pole

148. “Wave function” of the resonance pole Org. – Non.rela. NRv1 – Non.rela. NRv2 – Non.rela. Org. – Semi rela. f π =90 MeV θ=30° f π =90 MeV θ=30°

149. “Wave function” of the resonance pole Org. – Non.rela. Org. – Semi rela. NRv1 – Non.rela. NRv2 – Non.rela. f π =100 MeV θ=30° f π =100 MeV θ=30°

150. “Wave function” of the resonance pole Org. – Non.rela. NRv1 – Non.rela. NRv2 – Non.rela. Org. – Semi rela. f π =110 MeV θ=30° f π =110 MeV θ=30°

151. “Wave function” of the resonance pole Org. – Non.rela. NRv1 – Non.rela. NRv2 – Non.rela. Org. – Semi rela. f π =120 MeV θ=30° f π =120 MeV θ=30°

152. “Size” of the resonance pole state

153. “Size” of the resonance pole state f π =90 MeV θ=30° f π =90 MeV θ=30°

154. “Size” of the resonance pole state f π =100 MeV θ=30° f π =100 MeV θ=30°

155. “Size” of the resonance pole state f π =110 MeV θ=30° f π =110 MeV θ=30°

156. “Size” of the resonance pole state f π =120 MeV θ=30° f π =120 MeV θ=30°

157. “Size” of the resonance pole state θ=30° Mean distance between meson and baryon Re Im

158. Pole position of the resonance - Γ / 2 [MeV] M [MeV] 90 100 110 120 90 100 110 120 90 100 110 120 90 100 110 120

159. θ dependence of “Size” and “Wave function”

160. θ dependence of “Size” NRv2-NR f π =90 MeV NRv2-NR f π =90 MeV θ [deg] [fm]

161. θ dependence of “Size” Org-SR f π =90 MeV Org-SR f π =90 MeV θ [deg] [fm]

162. θ dependence of “Wave function” NRv2-NR f π =90 MeV NRv2-NR f π =90 MeV K bar N bound state πΣ resonant state K bar N πΣ r [fm] θ =10 deg.

163. θ dependence of “Wave function” NRv2-NR f π =90 MeV NRv2-NR f π =90 MeV K bar N bound state πΣ resonant state K bar N πΣ r [fm] θ =15 deg.

164. θ dependence of “Wave function” NRv2-NR f π =90 MeV NRv2-NR f π =90 MeV K bar N bound state πΣ resonant state K bar N πΣ r [fm] θ =20 deg.

165. θ dependence of “Wave function” NRv2-NR f π =90 MeV NRv2-NR f π =90 MeV K bar N bound state πΣ resonant state K bar N πΣ r [fm] θ =25 deg.

166. θ dependence of “Wave function” NRv2-NR f π =90 MeV NRv2-NR f π =90 MeV K bar N bound state πΣ resonant state K bar N πΣ r [fm] θ =30 deg.

167. θ dependence of “Wave function” NRv2-NR f π =90 MeV NRv2-NR f π =90 MeV K bar N bound state πΣ resonant state K bar N πΣ r [fm] θ =35 deg.

168. Comparison with Hyodo-Weise’s result

169. K bar N → K bar N πΣ → πΣ Scattering amplitudes DIM HW HW: PRC 77, 035204 (2008) Fig. 4 I=0 K bar N/πΣ scattering amplitude in two channel model

170. Comparison with Hyodo-Weise’s result Pole position HW: PRC 77, 035204 (2008) DIM HW Fig. 5 Pole positions of I=0 scattering amplitude (Upper pole in two channel model) z = 1419 – 14.4i MeV

171. Schrödinger equation to be solved : complex parameters to be determined Wave function expanded with Gaussian base c.c. CSM for resonant states Phenomenological potential Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 = Energy independent potential Chiral SU(3) potential N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995) = Energy dependent potential

172. Schrödinger equation to be solved : complex parameters to be determined Wave function expanded with Gaussian base Complex-rotate, then diagonalize with Gaussian base. c.c. CSM for resonant states Phenomenological potential Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 = Energy independent potential Chiral SU(3) potential N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995) = Energy dependent potential Complex scaling of coordinate ABC theorem The energy of bound and resonant states is independent of scaling angle θ. J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269 E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280 ABC theorem The energy of bound and resonant states is independent of scaling angle θ. J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269 E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280

173. Chiral SU(3) potential (KSW) N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995) Original: δ-function type Present: Normalized Gaussian type a: range parameter [fm] M i, m i : Baryon, Meson mass in channel i E i : Baryon energy, ω i : Meson energy Reduced energy: K bar NπΣ Energy dependence of V ij is controlled by CM energy √s. Energy dependence is determined by Chiral low energy theorem. ← Kaon: Nambu-Goldstone boson ← Kaon: Nambu-Goldstone boson Flavor SU(3) symmetry

174. Chiral SU(3) potential (KSW) Energy dependence √s [MeV] K bar N-K bar N πΣ-πΣ K bar N-πΣ N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995) K bar N threshold πΣ threshold

175. Chiral SU(3) potential = Energy-dependent potential Calculational procedure Perform the Complex Scaling method. Then, find a pole of resonance or bound state. Check Finished ! If Yes Self consistency for the energy! Assume the values of the CM energy √s. If No

176. やったこと 1. Chiral SU(3) potential で、 r-space ガウス型を仮定 レンジパラメータ a 、相互作用の強さ f π の決定が必要 2. A.D. Martin の解析による I=0 K bar N 散乱帳 a KbarN (I=0) を再現するような (a, f π ) を探す。 散乱振幅の計算は、 Kruppa 流に複素スケーリングを使って計算 3. 見つけた (a, f π ) に対して、これまでの結合チャネル複素スケーリングで、 Λ(1405) に相当する K bar N-πΣ 系の共鳴エネルギーを求める。 Chiral SU(3) potential はエネルギー依存性をもつが、 複素エネルギーに対して self consistency を課して計算。 N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995) A. T. Kruppa, R. Suzuki and K. Kato, PRC 75, 044602 (2007)

177. 結果のまとめ 1. この形のポテンシャルでは Martin の散乱長の実部・虚部を同時に 合わせることはできない。 f π =90 ～ 120MeV の範囲では。 f π =75.5MeV というカイラルリミットより小さい値を使えば、一応再現できる。 2. 散乱長の実部・虚部、一方だけなら再現できるポテンシャルが存在。 各々の場合で、複素エネルギーに対して self-consistent な K bar N-πΣ 系の 共鳴エネルギーが求まる。 散乱長実部を尊重した場合： (E KbarN, Γ/2) = (-19, 27) ～ (-5,8) MeV 虚部を尊重した場合 : (E KbarN, Γ/2) = (-32 ～ -24, 28 ～ 66) MeV 3. 散乱長の再現に関わらず、このポテンシャルでは PDG に載っている Λ （ 1405 ）のエネルギー及び幅 (E KbarN, Γ/2) = (-29±4, 25±1) MeV は 再現できない。 ※ f π =75.5 で無理やり散乱長を再現した場合 : (E KbarN, Γ/2) = (-23, 34) MeV

178. Mysterious state; Λ(1405) Quark model prediction … calculated as 3-quark state N. Isgar and G. Karl, Phys. Rev. D18, 4187 (1978) q q q Λ(1405) can’t be well reproduced as a 3-quark state! Λ(1405) can’t be well reproduced as a 3-quark state! observed Λ(1405) calculated Λ(1405)

179. K bar N-πΣ coupled system with s-wave and isospin-0 state Λ(1405) with c.c. Complex Scaling Method K bar + N Λ(1405) π + Σ 1435 1332 [MeV] B. E. (K bar N) = 27 MeV Γ (πΣ) ～ 50 MeV J π = 1/2 - I = 0 K bar (J π =0 -, T=1/2) N (J π =1/2 +, T=1/2) L=0 π (J π =0 -, T=1) Σ (J π =1/2 +, T=1) L=0

180. Schrödinger equation to be solved : complex parameters to be determined Wave function expanded with Gaussian base Complex-rotate, then diagonalize with Gaussian base. Λ(1405) with c.c. Complex Scaling Method Phenomenological potential Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 = Energy independent potential Chiral SU(3) potential N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995) = Energy dependent potential Complex scaling of coordinate ABC theorem The energy of bound and resonant states is independent of scaling angle θ. J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269 E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280 ABC theorem The energy of bound and resonant states is independent of scaling angle θ. J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269 E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280

181. Calc. of scattering amplitude with CSM “Kruppa method” A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007) Radial Schroedinger eq. Asymptotic behavior of wave func. Equation to be solved V(r) : short-range potential Asymptotic behavior of scattering part of wave func. : Ricatti-Bessel, Hankel func. is not square-integrable Search the solution in the form Separate the incoming wave ! Point 1

182. Calc. of scattering amplitude with CSM “Kruppa method” A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007) Complex scaling Equation to be solved Expanding with square-integrable basis function (ex: Gaussian basis) square-integrable for 0 < θ < π Point 2 : square integrable function Linear equation:

183. Calc. of scattering amplitude with CSM “Kruppa method” A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007) Calculation of scattering amplitude We don’t have which is a solution along r, but we have which is a solution along re iθ. r 0 r e iθ Cauchy theorem Point 3

184. 2. Application of CSM to Λ(1405) Coupled-channel Complex Scaling Method (ccCSM) Energy-independent K bar N potential

185. Phenomenological potential (AY) Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 Energy-independent potential K bar N πΣ 1.free K bar N scattering data 2.1s level shift of kaonic hydrogen atom 3.Binding energy and width of Λ(1405) = K - + proton The result that I show hereafter is not new, because the same calculation was done by Akaishi-san, when he made AY potential. Remark !

186.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm] E [MeV]   = 0 deg.  Λ(1405) with c.c. Complex Scaling Method

187. E [MeV]   = 5 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm]  Λ(1405) with c.c. Complex Scaling Method

188. E [MeV]   =10 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm]  Λ(1405) with c.c. Complex Scaling Method

189. E [MeV]   =15 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm]  Λ(1405) with c.c. Complex Scaling Method

190. E [MeV]   =20 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm]  Λ(1405) with c.c. Complex Scaling Method

191. E [MeV]   =25 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm] Λ(1405) with c.c. Complex Scaling Method

192. E [MeV]   =30 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm]  Λ(1405) with c.c. Complex Scaling Method

193. E [MeV]   =35 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm]  Λ(1405) with c.c. Complex Scaling Method

194. E [MeV]   =40 deg.   trajectory # Gauss base ( n ) = 30 Max range ( b ) = 10 [fm]  Λ(1405) with c.c. Complex Scaling Method

195. 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) ! Λ(1405) with c.c. Complex Scaling Method

196. 3. ccCSM with an energy-dependent potential for Λ(1405)

197. Result Range parameter (a) and pion-decay constant f π are ambiguous in this model. Various combinations (a,f π ) are tried. f π = 95 ～ 105 MeV

198. Self-consistency for complex energy both of real and imaginary parts of energy Search for such a solution that both of real and imaginary parts of energy are identical to assumed ones. (B.E., Γ) Calculated = (B.E., Γ) Assumed More reasonable? Pole search of T-matrix is done on complex-energy plane. Z: complex energy

199. Self consistency for complex energy KSW f π = 100 MeV a=0.47, θ=35° -B [MeV] 1 step -Γ/2[MeV] obtained by the self-consistency for the real energy K bar N

200. Self consistency for complex energy KSW f π = 100 MeV a=0.47, θ=35° -B [MeV] 2 steps -Γ/2[MeV] K bar N

201. Self consistency for complex energy KSW f π = 100 MeV a=0.47, θ=35° -B [MeV] 3 steps -Γ/2[MeV] K bar N

202. Self consistency for complex energy KSW f π = 100 MeV a=0.47, θ=35° -B [MeV] 4 steps -Γ/2[MeV] K bar N

203. Self consistency for complex energy KSW f π = 100 MeV a=0.47, θ=35° 5 steps -Γ/2[MeV] -B [MeV] Self consistent! K bar N

204. Self consistency for complex energy KSW f π = 100 MeV Assumed Calc. Assumed Calc.

205. Self consistency for complex energy KSW f π = 100 MeV -B [MeV] a=0.60 a=0.50 a=0.47 a=0.45 S.C. for real energy S.C. for complex energy K bar N Repulsively shifted!

206. Mean distance between K bar (π) and N (Σ) K bar (π) N (Σ) Distance Chiral (HW-HNJH): B ～ 12 MeV, Distance = 1.86 fm

207. Λ （１４０５）、 他の計算との比較

208. Rough estimation of charge radius K-K- p Distance Chiral (HW-HNJH): B ～ 12 MeV, Distance = 1.86 fm → = -1.07 fm 2 CM

209. Charge radius Chiral (HW-HNJH): B ～ 12 MeV, Distance = 1.86 fm → = -1.07 fm 2 Charge radius derived from electric form factor T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008) Rough estimation