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