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 KbarN potential

3. ccCSM with an energy-dependent KbarN 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)’10.12.10 (7-11) @ Convention center, Osaka univ., Japan

1. Introduction

1. IntroductionKbar nuclei = Exotic system !?

I=0 KbarN potential … very attractive

Highly dense state formed in a nucleusInteresting structures that we have never seen in normal nuclei…

Recently, ones have focused on

K-pp= Prototye of Kbar nuclei

Recent results of calculation of K-pp and related experiments

50 60 70 80 90 100 110 120 130

-140

-120

-100

-80

-60

-40

-20

0

Width (KbarNN→πYN) [MeV]

- B.E

. [M

eV]

Dote, Hyodo, Weise (Variational, Chiral SU(3))

Akaishi, Yamazaki (Variational, Phenomenological) Shevchenko, Gal

(Faddeev, Phenomenological)

Ikeda, Sato(Faddeev, Chiral SU(3))

Exp. : FINUDAif K-pp bound state

Exp. : DISTOif K-pp bound state

Constrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc.

1. Introduction

Recent results of calculation of K-pp and related experiments

50 60 70 80 90 100 110 120 130

-140

-120

-100

-80

-60

-40

-20

0

Width (KbarNN→πYN) [MeV]

- B.E

. [M

eV]

Dote, Hyodo, Weise (Variational, Chiral SU(3))

Akaishi, Yamazaki (Variational, Phenomenological) Shevchenko, Gal

(Faddeev, Phenomenological)

Ikeda, Sato(Faddeev, Chiral SU(3))

Exp. : FINUDAif K-pp bound state

Exp. : DISTOif K-pp bound state

Constrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc.

1. Introduction

Recent results of calculation of K-pp and related experiments

50 60 70 80 90 100 110 120 130

-140

-120

-100

-80

-60

-40

-20

0

Width (KbarNN→πYN) [MeV]

- B.E

. [M

eV]

Dote, Hyodo, Weise (Variational, Chiral SU(3))

Akaishi, Yamazaki (Variational, Phenomenological) Shevchenko, Gal

(Faddeev, Phenomenological)

Ikeda, Sato(Faddeev, Chiral SU(3))

Exp. : FINUDAif K-pp bound state

Exp. : DISTOif K-pp bound state

Constrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc.

1. Introduction

Three-body system calculated with the effective KbarN potential

Σ Σ

K

N+ …

…=

πK

N

πK

N

K

N

N N N NN N

barK NNE

conserved

πΣN thee-body dynamics

1. IntroductionKbar nuclei = Exotic system !?

I=0 KbarN potential … very attractive

Highly dense state formed in a nucleusInteresting structures that we have never seen in normal nuclei…

Recently, ones have focused on

K-pp= Prototye of Kbar nuclei

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, we try to investigate KbarNN-πΣN resonant state with …

coupled-channel Complex Scaling Method

Kbar + N + N

“Kbar N N”

π + Σ + N

Λ(1405) : I=0 quasi-bound state of K-p … 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 KbarN potential

KbarN-πΣ coupled system with s-wave and isospin-0 state

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

Kbar + N

Λ(1405)

π + Σ

1435

1332 [MeV]

B. E. (KbarN) = 27 MeVΓ (πΣ) ～ 50 MeV

Jπ = 1/2-

I = 0

Kbar

(Jπ=0-, T=1/2)

N (Jπ=1/2+, T=1/2)

L=0

π(Jπ=0-, T=1)

Σ(Jπ=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 potentialY. 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

V

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),269E. 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 KbarN potential

Phenomenological potential (AY)Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005

Energy-independent potential

2, 0

436 412exp 0.66 fm MeV

412 0AY IV r

KbarN πΣ

1. free KbarN scattering data2. 1s level shift of kaonic hydrogen atom3. 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 !

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

EG /

2

[MeV]

q = 0 deg.

2q

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

EG /

2

[MeV]

q = 5 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

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

EG /

2

[MeV]

q =10 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

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

EG /

2

[MeV]

q =15 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

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

EG /

2

[MeV]

q =20 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

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

EG /

2

[MeV]

q =25 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

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

EG /

2

[MeV]

q =30 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

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

EG /

2

[MeV]

q =35 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

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

EG /

2

[MeV]

q =40 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

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

EG /

2

[MeV]

q trajectory

2q q =30 deg.

pS KbarN

pS continuum

KbarN continuum

Resonance!(E, Γ/2) = (75.8, 20.0)

Measured from KbarN thr.,

B. E. (KbarN) = 28.2 MeVΓ = 40.0 MeV… L(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

( 0)

( 0)28

ij

ij

Ii jI

i ji j

C M Mr rV

f s

Energy dependence is determined by Chiral low energy theorem. ← 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

( 0)

( 0)28

ij

ij

Ii jI

i ji j

C Mg

MV r r

f s

2

3/ 2 3

1 expg r ra

ap

a: range parameter [fm]

( 0)

( 0)28

ij

ij

Ii jI

i ji j

C M Mr rV

f s

Energy dependence is determined by Chiral low energy theorem. ← 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

( 0)

( 0)28

ij

ij

Ii jI

i ji j

C Mg

MV r r

f s

2

3/ 2 3

1 expg r ra

ap

a: range parameter [fm]Mi , mi : Baryon, Meson mass in channel iEi : Baryon energy, ωi : Meson energy

2 22 2 2 2

,2 2

i i i i

i i

s s M

sE

s

m M m

Reduced energy: i ii

i i

EE

( 0)332

4ij

IC

KbarN πΣ

Energy dependence of Vij is controlled by CM energy √s.

( 0)

( 0)28

ij

ij

Ii jI

i ji j

C M Mr rV

f s

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

Flavor SU(3) symmetry

Chiral SU(3) potential (KSW)Energy dependence

( 0) 0, @ 0.5 , 100ij

IV r s a fm f MeVp

√s [MeV]

KbarN-KbarN

πΣ-πΣ

KbarN-πΣ

N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)

KbarN thresholdπΣ threshold

Chiral SU(3) potential = Energy-dependent potential

Calculational procedure

Perform the Complex Scaling method.

Then, find a pole of resonance or bound state. Calculateds

Calculated Assumeds sCheck Finished !If Yes

Self consistency for the energy!

Assume the values of the CM energy √s.

MBH T V s M Assumeds

If No

Result

Range parameter (a) and pion-decay constant fπ are ambiguous in this model. Various combinations (a,fπ) are tried.

fπ = 95 ～ 105 MeV

Self consistency for real energy

-B (Assumed) [MeV]

-B (C

alcu

late

d) [M

eV]

a=0.60

a=0.56

a=0.54

a=0.52

a=0.51

a=0.50

a=0.49

a=0.48a=0.44a=0.45

fπ = 100 MeV

No resonance for a>0.60

√s [MeV]1435

KbarNResonant state

Self consistency for real energy

-B (Assumed) [MeV]

-B (C

alcu

late

d) [M

eV]

√s [MeV]1435

a=0.48

a=0.45

a=0.44

fπ = 100 MeV

πΣ bound state1331

a=0.43

No self-consistent solution for a<0.44

KbarNπΣ

Resonance

Self consistent solutions (KSW)

-B [MeV]

-Γ /

2 [M

eV]

fπ = 100 MeV

a=0.60

a=0.51a=0.50

a=0.49

a=0.48

a=0.47

a=0.46

a=0.45

a=0.48a=0.44 a=0.45

√s [MeV]14351331

KbarNπΣ

πΣ bound state

Resonant state

a=0.49 ~ 0.60 : Resonance onlya=0.45 ~ 0.48 : Resonance and Bound statea= 0.44 : Bound state only

Self consistent solutions (KSW)

-B [MeV]

-Γ /

2 [M

eV]

fπ = 100 MeV

a=0.60

a=0.51a=0.50

a=0.49

a=0.48

a=0.47

a=0.46

a=0.45

a=0.48a=0.44 a=0.45

√s [MeV]14351331

KbarNπΣ

πΣ bound state

Resonant state

a=0.49 ~ 0.60 : Resonance onlya=0.45 ~ 0.48 : Resonance and Bound statea= 0.44 : Bound state only

Resonance energy < 40 MeVBut , decay width increases, as “a” decreases.

Self-consistency for complex energy

ReCalculateds E

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

Calculated complex en ys E erg

More reasonable?

0T V V G ZTZ Z Z Z

Pole search of T-matrix is done on complex-energy plane.

Z: complex energy

Self consistency for complex energy KSWfπ = 100 MeV

a=0.47, θ=35°

-B [MeV]

1 step-Γ/2 [MeV]

obtained by the self-consistency for the real energy

KbarN

Self consistency for complex energy KSWfπ = 100 MeV

a=0.47, θ=35°

-B [MeV]

2 steps-Γ/2 [MeV]

KbarN

Self consistency for complex energy KSWfπ = 100 MeV

a=0.47, θ=35°

-B [MeV]

3 steps-Γ/2 [MeV]

KbarN

Self consistency for complex energy KSWfπ = 100 MeV

a=0.47, θ=35°

-B [MeV]

4 steps-Γ/2 [MeV]

KbarN

Self consistency for complex energy KSWfπ = 100 MeV

a=0.47, θ=35°

5 steps-Γ/2 [MeV]

-B [MeV]

Self consistent!

KbarN

Self consistency for complex energy KSWfπ = 100 MeV

Assumed

Calc.

Assumed

Calc.

Self consistency for complex energy KSWfπ = 100 MeV

-B [MeV]

-Γ /

2 [M

eV]

a=0.60

a=0.50

a=0.47

a=0.45

S.C. for real energy

KbarN

Self consistency for complex energy KSWfπ = 100 MeV

-B [MeV]

-Γ /

2 [M

eV]

a=0.60

a=0.50

a=0.47

a=0.45

S.C. for real energyS.C. for complex energy

KbarN

Repulsively shifted!

Mean distance between Kbar (π) and N (Σ)

Kbar (π)

N (Σ)

Distance

Chiral (HW-HNJH): B ～ 12 MeV, Distance = 1.86 fm

4. Summary and

Future plan

4. SummaryΛ(1405) studied with coupled-channel Complex Scaling Method using energy independent / dependent potentials

Coupled Channel problem = KbarN + 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 … KbarNN-πΣN system corresponding to “K-pp”

Effect of πΣN three-body dynamics…

1. Two-body system … KbarN-πΣ system corresponding to Λ(1405)

• Analyze the obtained wave function

• 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)

Thank you very much!