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Application of coupled-channel Complex Scaling Method to Λ(1405)

Application of coupled-channel Complex Scaling Method to Λ(1405). A. Doté ( KEK Theory center) T. Inoue (Nihon univ .) T. Myo (Osaka Tech. univ .). Introduction Recent status of theoretical study of K - pp Application of ccCSM to Λ(1405)

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Application of coupled-channel Complex Scaling Method to Λ(1405)

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  1. Application of coupled-channel Complex Scaling Methodto Λ(1405) A. Doté(KEK Theory center) T. Inoue (Nihon univ.) T. Myo (Osaka Tech. univ.) • Introduction • Recent status of theoretical study of K-pp • Application of ccCSM to Λ(1405) • Coupled-channel complex scaling method (ccCSM) • Energy-independent KbarN potential • ccCSM with an energy-dependent KbarN potential for Λ(1405) • Summary and Future plan International conference on the structure of baryons (BARYONS ‘10) ’10.12.10 (7-11) @ Convention center, Osaka univ., Japan

  2. 1. Introduction

  3. 1. Introduction I=0 KbarN potential … very attractive Highly dense state formed in a nucleus Interesting structures that we have never seen in normal nuclei… K- Kbarnuclei = Exotic system !? • Recently, ones have focused on K-pp= Prototye of Kbar nuclei

  4. 1. Introduction Recent results of calculation of K-pp and related experiments Width (KbarNN→πYN) [MeV] Dote, Hyodo, Weise (Variational, Chiral SU(3)) Akaishi, Yamazaki (Variational, Phenomenological) Shevchenko, Gal (Faddeev, Phenomenological) - B.E. [MeV] Ikeda, Sato (Faddeev, Chiral SU(3)) Exp. : DISTO if K-pp bound state Exp. : FINUDA if K-pp bound state Constrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc.

  5. 1. Introduction Recent results of calculation of K-pp and related experiments Width (KbarNN→πYN) [MeV] Dote, Hyodo, Weise (Variational, Chiral SU(3)) Akaishi, Yamazaki (Variational, Phenomenological) Shevchenko, Gal (Faddeev, Phenomenological) - B.E. [MeV] Ikeda, Sato (Faddeev, Chiral SU(3)) Exp. : DISTO if K-pp bound state Exp. : FINUDA if K-pp bound state Constrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc.

  6. 1. Introduction Recent results of calculation of K-pp and related experiments Three-body system calculated with the effective KbarN potential Width (KbarNN→πYN) [MeV] πΣN thee-body dynamics π π Dote, Hyodo, Weise (Variational, Chiral SU(3)) K K K … K = + … N N N N Σ Σ conserved Akaishi, Yamazaki (Variational, Phenomenological) N N N N N N Shevchenko, Gal (Faddeev, Phenomenological) - B.E. [MeV] Ikeda, Sato (Faddeev, Chiral SU(3)) Exp. : DISTO if K-pp bound state Exp. : FINUDA if K-pp bound state Constrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc.

  7. 1. Introduction I=0 KbarN potential … very attractive Highly dense state formed in a nucleus Interesting structures that we have never seen in normal nuclei… K- Kbarnuclei = Exotic system !? • Recently, ones have focused on K-pp= Prototye of Kbar nuclei Kbar + N + N π + Σ + N “Kbar N N” • In the study of K-pp, it was pointed out that the πΣN three-body dynamics might be important. • Based on the variational approach, and explicitly treating the πΣN channel, • we try to investigate KbarNN-πΣNresonant state with … coupled-channel Complex Scaling Method

  8. Kaonic nuclei sdtudied with Complex Scaling Method Before K-pp, … Λ(1405) : I=0 quasi-bound state of K-p … two-body system

  9. 2. Application of CSM to Λ(1405) • Coupled-channel Complex Scaling Method (ccCSM) • Energy-independent KbarN potential

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

  11. Λ(1405) with c.c. Complex Scaling Method Complex scaling of coordinate Schrödinger equation to be solved 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 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 Wave function expanded with Gaussian base : complex parameters to be determined Complex-rotate , then diagonalize with Gaussian base.

  12. 2. Application of CSM to Λ(1405) • Coupled-channel Complex Scaling Method (ccCSM) • Energy-independent KbarN potential

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

  14. Λ(1405) with c.c. Complex Scaling Method qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] 2q -G / 2 q= 0 deg.

  15. Λ(1405) with c.c. Complex Scaling Method qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] 2q -G / 2 q= 5 deg.

  16. Λ(1405) with c.c. Complex Scaling Method qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] 2q -G / 2 q=10 deg.

  17. Λ(1405) with c.c. Complex Scaling Method qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] -G / 2 2q q=15 deg.

  18. Λ(1405) with c.c. Complex Scaling Method qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] -G / 2 2q q=20 deg.

  19. Λ(1405) with c.c. Complex Scaling Method qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] -G / 2 q=25 deg.

  20. Λ(1405) with c.c. Complex Scaling Method qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] -G / 2 q=30 deg. 2q

  21. Λ(1405) with c.c. Complex Scaling Method qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] -G / 2 q=35 deg. 2q

  22. Λ(1405) with c.c. Complex Scaling Method qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] -G / 2 q=40 deg. 2q

  23. Λ(1405) with c.c. Complex Scaling Method qtrajectory Resonance! (E, Γ/2) = (75.8, 20.0) pS KbarN E [MeV] Measured from KbarN thr., B. E. (KbarN) = 28.2 MeV Γ = 40.0 MeV … L(1405) ! KbarN continuum -G / 2 pScontinuum 2q q=30 deg.

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

  25. Chiral SU(3) potential (KSW) N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995) Original: δ-function type Energy dependence is determined by Chiral low energy theorem. ← Kaon: Nambu-Goldstone boson

  26. Chiral SU(3) potential (KSW) N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995) Original: δ-function type Energy dependence is determined by Chiral low energy theorem. ← Kaon: Nambu-Goldstone boson Present: Normalized Gaussian type a: range parameter [fm]

  27. Chiral SU(3) potential (KSW) N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995) Original: δ-function type Energy dependence is determined by Chiral low energy theorem. ← Kaon: Nambu-Goldstone boson Present: Normalized Gaussian type a: range parameter [fm] Mi , mi : Baryon, Meson mass in channel i Ei : Baryon energy, ωi : Meson energy Flavor SU(3) symmetry KbarN πΣ Reduced energy: Energy dependence of Vij is controlled by CM energy √s.

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

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

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

  31. Self consistency for real energy fπ = 100MeV 1435 √s [MeV] Resonant state KbarN -B (Assumed) [MeV] No resonance for a>0.60 a=0.60 a=0.56 a=0.54 -B (Calculated) [MeV] a=0.52 a=0.51 a=0.50 a=0.49 a=0.48 a=0.44 a=0.45

  32. Self consistency for real energy fπ = 100MeV 1331 1435 √s [MeV] πΣ bound state KbarN πΣ -B (Assumed) [MeV] Resonance -B (Calculated) [MeV] a=0.48 a=0.43 a=0.45 No self-consistent solution for a<0.44 a=0.44

  33. Self consistent solutions (KSW) fπ = 100MeV √s [MeV] 1331 1435 πΣ KbarN -B [MeV] a=0.60 a=0.44 a=0.45 a=0.48 πΣ bound state a=0.51 a=0.50 a=0.49 -Γ / 2 [MeV] a=0.48 Resonant state a=0.47 a=0.49 ~ 0.60 : Resonance only a=0.45 ~ 0.48 : Resonance and Bound state a= 0.44 : Bound state only a=0.46 a=0.45

  34. Self consistent solutions (KSW) fπ = 100MeV √s [MeV] 1331 1435 πΣ KbarN -B [MeV] a=0.60 a=0.44 a=0.45 a=0.48 πΣ bound state a=0.51 a=0.50 a=0.49 -Γ / 2 [MeV] a=0.48 Resonant state a=0.47 a=0.49 ~ 0.60 : Resonance only a=0.45 ~ 0.48 : Resonance and Bound state a= 0.44 : Bound state only a=0.46 a=0.45 Resonance energy < 40 MeV But , decay width increases, as “a” decreases.

  35. Self-consistency for complex 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

  36. Self consistency for complex energy KSW fπ = 100MeV a=0.47, θ=35° obtained by the self-consistency for the real energy KbarN -B [MeV] -Γ/2 [MeV] 1 step

  37. Self consistency for complex energy KSW fπ = 100MeV a=0.47, θ=35° KbarN -B [MeV] -Γ/2 [MeV] 2 steps

  38. Self consistency for complex energy KSW fπ = 100MeV a=0.47, θ=35° KbarN -B [MeV] -Γ/2 [MeV] 3 steps

  39. Self consistency for complex energy KSW fπ = 100MeV a=0.47, θ=35° KbarN -B [MeV] -Γ/2 [MeV] 4 steps

  40. Self consistency for complex energy KSW fπ = 100MeV a=0.47, θ=35° KbarN -B [MeV] Self consistent! -Γ/2 [MeV] 5 steps

  41. Self consistency for complex energy KSW fπ = 100MeV Assumed Calc. Assumed Calc.

  42. Self consistency for complex energy KSW fπ = 100MeV -B [MeV] KbarN a=0.60 a=0.50 -Γ / 2 [MeV] a=0.47 a=0.45 S.C. for real energy

  43. Self consistency for complex energy KSW fπ = 100MeV -B [MeV] KbarN a=0.60 a=0.50 -Γ / 2 [MeV] a=0.47 Repulsively shifted! a=0.45 S.C. for complex energy S.C. for real energy

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

  45. 4. Summary and Future plan

  46. 4. Summary Λ(1405) studied with coupled-channel Complex Scaling Method using energy independent / dependent potentials • A Chiral SU(3) potential (KSW) with Gaussian form is used. • Coupled Channel problem = KbarN + πΣ • Solved with Gaussian base • Take into account the self consistency for the real/complex energy Energy-independent case • A phenomenological potential (AY) is used. • AY result is correctly reproduced: (B.E., Γ)= (28, 40) MeV Energy-dependent case • Self consistent solutions are found, also for the complex energy case. • Self-consistency for the complex energy seems to • contribute repulsively to the binding energy.

  47. 4. Future plan 1. Two-body system … KbarN-πΣ system corresponding to Λ(1405) • For the case of energy dependent potential, • further investigation is needed. • Fix the combination of (a, fπ) • … experimental value such as I=0 KbarN scattering length. • Another pole ??? • … Double pole problem suggested by chiral unitary model D. Jido, J. A. Oller, E. Oset, A. Ramos and U. -G. Meissner, NPA725, 181 (2003) 2. Three-body system … KbarNN-πΣN system corresponding to “K-pp” • Analyze the obtained wave function Effect of πΣN three-body dynamics …

  48. Thank you very much!

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