Coupled-channel complex scaling法
を用いたΛ(1405)の研究
A. Doté (KEK Theory center)
T. Inoue (Nihon univ.)
T. Myo (Osaka Tech. univ.)

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 potentialfor Λ(1405)
4. Summary and Future plan
KEK理論センター研究会『ストレンジネス核物理 2010』’10.12.03 @ KEK 小林ホール

1. Introduction

1. Introduction
Kbar nuclei = Exotic system !?
I=0 KbarN potential … very attractive
Highly dense state formed in a nucleus
Interesting structures that we have never seen in normal nuclei…
¾ Recently, ones have focused on

K-pp= Prototye of Kbar nuclei

1. Introduction
Recent results of calculation of K-pp and related experiments
Width (KbarNN→πYN) [MeV]
020406080

1001201400Dote, Hyodo, Weise(Variational, Chiral SU(3))-20-40

Akaishi, Yamazaki(Variational, Phenomenological)-60-80

Ikeda, Sato
(Faddeev, Chiral SU(3))
Exp. : DISTO
if K-pp bound state-100-120-140Shevchenko, Gal(Faddeev, Phenomenological)

Exp. : FINUDA
if 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
Width (KbarNN→πYN) [MeV]
0204060801001201400Dote, Hyodo, Weise(Variational, Chiral SU(3))SU(3)-20-40

Akaishi, Yamazaki(Variational, Phenomenological)-60-80

Ikeda, Sato
(Faddeev, Chiral SU(3))SU(3)
Exp. : DISTO
if K-pp bound state-100-120-140Shevchenko, Gal(Faddeev, Phenomenological)

Exp. : FINUDA
if 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
Width (KbarNN→πYN) [MeV]πΣN thee-bodyy dynamics
y020406080100120

1400

Three-body system calculated with the effective KbarN potential

Dote, Hyodo, Weise(Variational, Chiral SU(3))SU(3)-20-40KN

Akaishi,YamazakiK(Variational, =Phenomenological)
EK bar NNN-60

N -80

N

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

-140

NN

Σ

πK
…
Shevchenko,+GalN(Faddeev, Phenomenological)…

ΣN
Exp. : DISTO
if K-pp bound state-100-120KπExp. : FINUDA
if K-pp bound state

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

1. Introduction
Kbar nuclei = Exotic system !?
bar
I=0bar K N potential … very attractive

K+N+N

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

“Kbar N N”

¾ Recently, ones have focused on-π + Σ + NK pp= Prototye of Kbar nuclei¾ In the study of K-pp, it was pointed out that theπΣNΣN threethreeh -body
b d dynamics
dimight be important.

¾ 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
coupledKaonic nuclei sdtudied with Complex Scaling Method

Before K-pp, …

Λ(1405) : I=0 quasiquasi-bound state of K-p… twotwo--body system

2. Application of CSMt Λ(1405)
to• Coupled-channel Complex Scaling Method (ccCSM)• Energy-independent KbarN potentialΛ(1405) with c.c. Complex Scaling Method
Kbar + N

1435

Λ(1405)B. E. (KbarN) = 27 MeV
Γ (πΣ)～ 50 MeVJπ = 1/2I = 0

π +Σ

1332
[MeV]

L 0
L=0Kbar(Jπ=0-, T=1/2)

N(Jπ=1/2+, T=1/2)

L 0
L=0π
(Jπ=0-, T=1)

Σ(Jπ=1/2+, T=1)

KbarN-πΣ coupled system with ss-wave and isospin
isospin--0 stateΛ(1405) with c.c. Complex Scaling Method
Schrödinger equation to be solved

Phenomenological potential

Vl= Energy independent potential

Chiral SU(3) potential= Energy dependent potential

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

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

Wave function expanded with Gaussian base

: complex parameters to be determined

Complex--rotate
Complex, then diagonalize

with Gaussian base.

Λ(1405) with c.c. Complex Scaling Method
Schrödinger equation to be solved

Complexprotation of coordinate
Phenomenological potentialVl= Energy independent potential

Chiral SU(3) potential= Energy dependent potential

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

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

ABC theorem

Wave function expanded with Gaussian base

The energy of bound and resonant states is
independent of scaling
angle θ.
: complex parameters to be determined
Complex--rotate
ComplexJ. Aguilar and J. M. Combes
Combes,, Commun.
Commun. Math. Phys. 22 (1971),269
E. Balslev and J. M. Combes
Combes,, Commun.
Commun. Math. Phys. 22 (1971),280, then diagonalize

with Gaussian base.

2. Application of CSM
t Λ(1405)
to• 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
EnergyKbarNπΣ

⎛ −436 −412 ⎞2l⎡
V AY , I =0 = ⎜⎟ exp ⎣ − ( r 0.66 fm ) ⎤⎦0 ⎠⎝ −412[ MeV ]

1. free KbarN scattering data
2. 1s level shift of kaonic hydrogen atom
3. Binding energy and width of Λ(1405)= K- + proton※注意：このポテンシャルを作る際に実はccCSMが使われている。
論文には明記されていないが。
なので、このポテンシャルを使用した以下の計算は検証にすぎない。

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

θ trajectory• # Gauss base (n) = 30
• Max range (b) = 10 [fm]E2θ

θ = 0 deg.

[MV]
[MeV]

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

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

E[MV]
[MeV]

2θ

θ = 5 deg.

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

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

E
[MV]
[MeV]

2θ

θ =10 deg.

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

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

E
[MV]
[MeV]

2θ

θ =15 deg.

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

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

E[MV]
[MeV]

2θ

θ =20 deg.

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

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

E
[MV]
[MeV]

θ =25 deg.

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

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

E[MV]
[MeV]

2θ

θ =30 deg.

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

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

E
[MV]
[MeV]

2θ

θ =35 deg.

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

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

E[MV]
[MeV]

2θ

θ =40 deg.

Λ(1405) with c.c. Complex Scaling Methodθ trajectoryπΣKbarN

Resonance!
(E, Γ/2) = (75.8, 20.0)

E
[M V]
[MeV]

Measured from KbarN thr
thr.,.,
B. E. (K( barN)) = 28.2 MeV
Γ = 40.0 MeV… Λ(1405) ! KbarN continuum2θπΣ continuumθ =30 deg.

3.. ccCSM
CS with aanenergy-dependent potentialfor Λ(1405)

Chiral SU(3) potential (KSW)
N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)

Original: δδ-function type
( I = 0)Vij(r) = −C ij( I =0)
8f 2(ω + ω )ijEnergy dependence is determined byChiral low energy theorem.
← Kaon:Kaon: Nambu
Nambu--Goldstone boson

MiM jδ riiω
ij ( )sω

Present: Normalized Gaussian typeVij( I =0) ( r ) = −C ij( I =0)
8f 2

MiM j(ω + ω ) s ωi ωi g ( r )ijig (r ) =j( s) −m +M , ω = ( s) +m −M
E =i2ii

2 sReduced energy:

ii =ω22i

exp ⎡ − ( r a ) ⎤3/ 2 3⎣⎦π a2a: range parameter [fm]Mi , mi : Baryon, Meson mass in channel i
Ei : Baryon energy, ωi : Meson energy
212i

2 s

ωi Ei
ωi + Ei

Energy dependence of Vij iscontrolled by CM energy √s.

Flavor SU(3) symmetry
2i

KbarNπΣ⎛3 ⎞3 −⎟C ij( I =0) = ⎜⎜2 ⎟⎜4 ⎟⎠⎝Chiral SU(3) potential (KSW)
N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)

Energy dependence()

Vij( I = 0) r = 0, s @ a = 0.5 fm, fπ = 100 MeV

KbarN‐πΣ√s [MeV]KbarN‐KbarN

πΣ‐πΣπΣ threshold

KbarN thresholdCalculational procedure
Chiral SU(3) potential = Energy-dependent potential

Self consistency for the energy!
Assume the values of the CM energy √s.

s Assumedl = Tl + Vl MB
H( s ) + Mm

Perform the Complex Scaling method
method.
Then, find a pole of resonance or bound state.

Checks Calculated = s Assumed

s Calculated

If NoNCalculational procedure
Chiral SU(3) potential = Energy-dependent potential

Self consistency for the energy!
Assume the values of the CM energy √s.

s Assumedl = Tl + Vl MBH( s ) + Mm

Perform the Complex Scaling method
method.
Then, find a pole of resonance or bound state.

Checks Calculated = s Assumed

s Calculated

If YesFinished !

Calculational procedure
Chiral SU(3) potential = Energy-dependent potential

Self consistency for the energy!
Assume the values of the CM energy √s.

s Assumedl = Tl + Vl MB
H( s ) + Mm

Perform the Complex Scaling method
method.
Then, find a pole of resonance or bound state.

Checks Calculated = s Assumed

s Calculated

If YesIf NoN
Finished !

R lt
ResultCase: self consistency for real energy
Rangeg pparameter ((a)) andpion--decay constant fπ are ambiguouspionin this model.
Various combinations (a,fπ) are tried.

fπ = 95 ～ 105 MeV

Self consistency for energy

fπ = 100 MeV
1435

Resonant state‐B (Assumed) [MeV
[MeV]]

a=0.60a=0.56
a=0.54
a=0.52a=0.51
a=0.50a=0.49a=0.48a=0.44
a=0.45

No resonance for a>0.60

KbarN√s [MeV[MeV]]Self consistency for energyπΣ bound state

fπ = 100 MeV13311435

πΣ

KbarN

‐B (Assumed) [MeV
[MeV]]√s [MeV[MeV]]

Resonance

a=0.48a=0.43a=0.45a=0.44

No self‐consistent solution for a<0.44

Self consistent solutions (KSW)fπ = 100 MeV

1331

1435

πΣ

KbarN

a=0.44 a=0.45‐B [MeV[MeV]]

a=0.60a=0.48πΣ bound state

a=0.51
a=0.50a=0.49

RResonantt statet ta=0.49 ~ 0.60 : Resonance only
a=0.45 ~ 0.48 : Resonance and Bound statea= 0.44: Bound state only

a=0.48a=0.47a=0.46
a=0.45√s [MeV[MeV]]

Self consistent solutions (KSW)fπ = 100 MeV

1331

1435

πΣKbarN

a=0.44 a=0.45‐B [MeV[MeV]]

a=0.60a=0.48πΣ bound state

a=0.51
a=0.50a=0.49

RResonantt statet t
a=0.49 ~ 0.60 : Resonance only
a=0.45 ~ 0.48 : Resonance and Bound statea= 0.44: Bound state only
Resonance energy < 40 MeV
But , decay width inceases, as a decrease.

a=0.48a=0.47a=0.46
a=0.45√s [MeV[MeV]]

Root--meanRoot
mean--square radius

Chiral SU(3) potential gives smaller Λ(1405),
in comparison with a phen.
phen. potential.

SelfSelf-consistency for complex energys Calculated = Re E

s Calculated = E( 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?
9Pole search of T-matrix is done on complex-energy plane.

T ( Z ) = V ( Z ) + V ( Z ) G0 ( Z ) T ( Z )Z: complex energy

Self consistency for complex energy

KSW
fπ = 100 MeV

AssumedCalc.

AssumedCalc.

Self consistency for complex energy‐B [MeV[MeV]]
a=0.60a=0.50a=0.47a=0.45
S.C. for real energy

KSW
fπ = 100 MeV

Self consistency for complex energy‐B [MeV[MeV]]
a=0.60a=0.50a=0.47a=0.45
S.C. for real energy

S.C. for complex energy

KSW
fπ = 100 MeV

4 Summary
4.
SandFuture plan4. SummaryΛ(1405) studied with coupledcoupled-channel Complex Scaling Method
using energy independent / dependent potentials
9 Coupled Channel problem = KbarN + πΣ
9 Solved with Gaussian baseEnergy--independent case
Energy9 A phenomenological potential (AY) is used.
9 AY result is correctly reproduced: (B.E., Γ) = (28, 40) MeV

Resonance found at
B. E. (KbarN) = 28.2 MeV
Γ = 40.0 MeV

4. Summary
Energy--dependent case
Energy9 A Chiral SU(3) potential (KSW) with Gaussian form is used.
9 Take into account the self consistencyy for the real/complex
p energygy¾ Self consistent solutions are found, also for the complex energy case.

¾ Only for the restricted range parameter a of Gaussian form,self consistent solutions can exist.
¾ As for resonant states in the self consistent solutions for real energy,
their binding energy don’t exceed 40 MeV for fπ = 95 ~ 105MeV.
¾ Self-consistency for the complex energy seems tocontribute repulsively to the binding energy.

4. Future plan
1. TwoTwo-body system … KbarN-πΣ system corresponding to Λ(1405)• For the case of energy dependent potential,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)• Analyze the obtained wave function
2. ThreeThree-body system … KbarNN
NN--πΣN system corresponding to “K-pppp””
Effect of πΣN three-body dynamics…

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