Application of coupled-channel Complex Scaling Method to Λ(1405) 1.Introduction Recent status of theoretical study of K - pp 2.Application of ccCSM to Λ(1405) Coupled-channel complex scaling method (ccCSM) Energy-independent K bar N potential 3.ccCSM with an energy-dependent K bar N potential for Λ(1405) 4.Summary and Future plan A. Doté (KEK Theory center) T. Inoue (Nihon univ.) T. Myo (Osaka Tech. univ.) International conference on the structure of baryons (BARYONS ‘10) ’ Convention center, Osaka univ., Japan

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…  Recently, ones have focused on K - pp= Prototye of K bar nuclei K-K-

Recent results of calculation of K - pp and related experiments Width (K bar NN→πYN) [MeV] 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)) Exp. : FINUDA if K - pp bound state Exp. : FINUDA if K - pp bound state Exp. : DISTO if K - pp bound state Exp. : DISTO if K - pp bound state Constrained by experimental data. … K bar N scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc. 1. Introduction

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

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

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…  Recently, ones have focused on K - pp= Prototye of K bar nuclei K-K-  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, resonant we try to investigate K bar NN-πΣN resonant state with … coupled-channel Complex Scaling Method K bar + N + N “K bar N N” π + Σ + N

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

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

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 [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

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) = 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

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

Phenomenological potential (AY) Y. Akaishi and T. Yamazaki, PRC 52 (2002) 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 !

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

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

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

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

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

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

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

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

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

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

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

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 ← Kaon: Nambu-Goldstone boson

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] Energy dependence is determined by Chiral low energy theorem. ← Kaon: Nambu-Goldstone boson ← Kaon: Nambu-Goldstone boson

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

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

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

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

Self consistency for real energy -B (Assumed) [MeV] a=0.60 a=0.56 a=0.54 a=0.52 a=0.51 a=0.50 a=0.49 a=0.48 a=0.44 a=0.45 f π = 100 MeV No resonance for a>0.60 √s [MeV] 1435 K bar N Resonant state

Self consistency for real energy -B (Assumed) [MeV] √s [MeV] 1435 a=0.48 a=0.45 a=0.44 f π = 100 MeV πΣ bound state 1331 a=0.43 No self-consistent solution for a<0.44 K bar N πΣ Resonance

Self consistent solutions (KSW) -B [MeV] f π = 100 MeV a=0.60 a=0.51 a=0.50 a=0.49 a=0.48 a=0.47 a=0.46 a=0.45 a=0.48a=0.44a=0.45 √s [MeV] K bar N πΣ πΣ bound state Resonant state a=0.49 ~ 0.60 : Resonance only a=0.45 ~ 0.48 : Resonance and Bound state a= 0.44 : Bound state only a=0.49 ~ 0.60 : Resonance only a=0.45 ~ 0.48 : Resonance and Bound state a= 0.44 : Bound state only

Self consistent solutions (KSW) -B [MeV] f π = 100 MeV a=0.60 a=0.51 a=0.50 a=0.49 a=0.48 a=0.47 a=0.46 a=0.45 a=0.48a=0.44a=0.45 √s [MeV] K bar N πΣ πΣ bound state Resonant state a=0.49 ~ 0.60 : Resonance only a=0.45 ~ 0.48 : Resonance and Bound state a= 0.44 : Bound state only a=0.49 ~ 0.60 : Resonance only a=0.45 ~ 0.48 : Resonance and Bound state a= 0.44 : Bound state only Resonance energy < 40 MeV But, decay width increases, as “a” decreases. Resonance energy < 40 MeV But, decay width increases, as “a” decreases.

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

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

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

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

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

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

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

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 K bar N

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!

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

4. Summary and Future plan

4. Summary Λ(1405) studied with coupled-channel Complex Scaling Method using energy independent / dependent potentials Coupled Channel problem = K bar N + πΣ Solved with Gaussian base Energy-independent case A phenomenological potential (AY) is used. AY result is correctly reproduced: (B.E., Γ) = (28, 40) MeV A Chiral SU(3) potential (KSW) with Gaussian form is used. Take into account the self consistency for the real/complex energy 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.

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

Thank you very much!