ppK - studied with a Chiral SU(3)-based K bar N potential

ppK- studied with a Chiral SU(3)-based KbarN potential

Chiral ‘07　’ 07.11.14 @ Osaka univ. Conventional Center

1. Introduction

2. Model 　 - Simple Correlated Model (Revised) -

3. Local KbarN potential based on Chiral SU(3)

4. Result

5. Summary and Future plan

A. Dote (KEK), W. Weise (TU Munich)´

T. Hyodo (TU Munich)

１． Introduction

Kbar nuclei = Exotic system !?

I=0 KbarN potential … very attractive

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

To know in more detail …

ppKppK-- = Prototye of K = Prototye of Kbarbar nuclei nuclei

• Studied with various methods, because it is a three-body system. ATMS (Akaishi), Faddeev (Ikeda and Sato), Faddeev (Shevchenko and Gal), Variational approach (Noda, Sasaki, Hiyama and Hirenzaki), Skyrme model (Nishikawa) …

• Can be treated precisely …a bare NN potential can be used, without help of G-matrix method.

• Experiment done by FINUDA group B.E. = 116 MeV, Γ = 67 MeV

２．Model

Model wave function 　ー Simple Correlated Model (Revised) ー

1/ 22 0SCM mixN C

1 1 2 1 2 1 2 1 1 2 2 1 2 2 1

0 1 2 1 2 1 2 1 1 2 2 1 2 2 1

, , ' , ' , ' , ' , ' ,

, , ' , ' , ' , ' , ' ,

K K K K K K

K K K K K K

r r r G r G r G r F r r F r r F r r F r r F r r


��

��

1 1 1 2 1 1/ 2,1/ 2

0 0 1 2 0 1/ 2,1/ 2

, , 0

, , 0

N

N

K N T

K N T

r r r S NN K

r r r S NN K

��

��

NN correlation function

2

1 2 1 2, 1 expNN NNn n

n

F r r f r r ��

KN correlation function

2,' , 1 expKN KN

i K n na K in

aF r r f r r ��

2

2

exp

' exp

i i

K K

G r r

G r r

��

��

Nucleon

Kaon

Single-particle motion

… Nuclear part 　 Spin=0, Isospin=1


Thanks, Akaishi-san!

２．Model

1/ 22 0SCM mixN C

1 1 2 1 2 1 2 1 1 2 2 1 2 2 1

0 1 2 1 2 1 2 1 1 2 2 1 2 2 1

, , ' , ' , ' , ' , ' ,

, , ' , ' , ' , ' , ' ,

K K K K K K

K K K K K K



��

��

1 1 1 2 1 1/ 2,1/ 2

0 0 1 2 0 1/ 2,1/ 2

, , 0

, , 0

N

N

K N T

K N T

r r r S NN K

r r r S NN K

��

��


2


n

F r r f r r ��


2,' , 1 expKN KN

i K n na K in

aF r r f r r ��

2

2

exp

' exp

i i

K K

G r r

G r r

��

��

Nucleon

Kaon


Naïve “ppK-”



… Nuclear part 　 Spin=0, Isospin=0Isospin=0


２．Model

1/ 22 0SCM mixN C

1 1 2 1 2 1 2 1 1 2 2 1 2 2 1

0 1 2 1 2 1 2 1 1 2 2 1 2 2 1

, , ' , ' , ' , ' , ' ,

, , ' , ' , ' , ' , ' ,

K K K K K K

K K K K K K



��

��

1 1 1 2 1 1/ 2,1/ 2

0 0 1 2 0 1/ 2,1/ 2

, , 0

, , 0

N

N

K N T

K N T

r r r S NN K

r r r S NN K

��

��


2


n

F r r f r r ��


2,' , 1 expKN KN

i K n na K in

aF r r f r r ��

2

2

exp

' exp

i i

K K

G r r

G r r

��

��

Nucleon

Kaon






２．Model

1/ 22 0SCM mixN C

1 1 2 1 2 1 2 1 1 2 2 1 2 2 1

0 1 2 1 2 1 2 1 1 2 2 1 2 2 1

, , ' , ' , ' , ' , ' ,

, , ' , ' , ' , ' , ' ,

K K K K K K

K K K K K K



��

��

1 1 1 2 1 1/ 2,1/ 2

0 0 1 2 0 1/ 2,1/ 2

, , 0

, , 0

N

N

K N T

K N T

r r r S NN K

r r r S NN K

��

��


2


n

F r r f r r ��


2,' , 1 expKN KN

i K n na K in

aF r r f r r ��

2

2

exp

' exp

i i

K K

G r r

G r r

��

��

Nucleon

Kaon






２．Model

NN potential　…　 Av18 potential

-500

0

500

1000

1500

2000

2500

3000

3500

0.0 0.5 1.0 1.5 2.0

[fm]

[MeV

]

Av18

Av18- G3R

Av18- like

-200

0

200

400

600

800

1000

1200

1400

0.0 0.5 1.0 1.5 2.0

[fm]

[MeV

]

Av18

Av18- G3R

1E1E 1O1O

fitted with a few range Gaussians.

２．Model

NN potential　…　 Av18 potentialEffective KbarN potential … Akaishi’s

SCM ver2 SCM ver2 SCM ver1 Akaishi

(N=10, K=10) (N=6, K=6) (N=8, K=6)

Kinetic 164.26 165.38 145.93 167

NN pot -19.94 -20.22 -20.04 -19

KN pot -193.35 -193.76 -163.74 -196

B. E. 49.03 48.59 37.85 48

B(K) 79.11 79.43 64.42 68

Nucl. E 30.08 30.84 26.57 20

Width 60.52 60.76 59.15

Rel (NN) 1.84 1.81 1.70 1.90

Rel (KN) 1.24 1.23 1.54 1.57

Mixing ratio 5.9% 5.8%

Influence of the improvement

Remark:NN potential is Tamagaki potential in Akaishi-san’s calculation.

TN=1 onlyTN=1 onlyTN=1 + TN=0TN=1 + TN=0

３． Local KbarN potential based on Chiral SU(3)

Request for our KbarN potential

1. Reproduce the s-wave KbarN scattering amplitude calculated with Chiral unitary model

3. Local potential, r-space, Gaussian form

2. Single channel (KbarN channel) but energy-dependent and complex

To apply the structure study of ppK-,

, ,,S

EffKN S I K N KN S I a K Nv s V s g r r r r

2

23/ 2 3

1exp

S

K Na K N

SS

gaa

r rr r : Normalized Gaussian

: CM energy of KbarNs

Go to Hyodo-san’s poster!

３． Local KbarN potential based on Chiral SU(3)How to determine and the range parameter . ,KN S IV s

Chiral U.

Coupled Ch.

Vij

Tij

Step 1

• Exactly done in the framework of Chiral Unitary

Step 1

• Eliminate other channels than KbarN channel

K

N+ … + + …

…=

Σ

πK

N

K

N

K

N

π

Σ

K

N

K

N

VSingle, I V11 V1i Vj1

Single Ch.

T = T11

Single, IV s

Chiral U.

Sa

３． Local KbarN potential based on Chiral SU(3)How to determine and the range parameter . ,KN S IV s Sa

Chiral U.

Coupled Ch.

Vij

Tij

Step 1

Single Ch.

T = T11

Single, IV s

Chiral U.Step 2

Effective

Single Ch.

, ,EffKN S Iv s r

• Range parameter is determined so that the I=0 resonance appears at the same place as that in the Chiral unitary when we solve the Schroedinger equation with this potential.

Sa

Step 2

• Using , construct simply as SingleV s , ,EffKN S Iv s r

, Single,

1,

2 S

Eff NKN S I I a

Mv s V s g

s r r

３． Local KbarN potential based on Chiral SU(3)How to determine and the range parameter . ,KN S IV s Sa

Chiral U.

Coupled Ch.

Vij

Tij

Step 1

Single Ch.

T = T11

Single, IV s

Chiral U.Step 2

Effective

Single Ch.

, ,EffKN S Iv s r

Step 3

• Correct so as to reproduce the original scattering amplitude (T-matrix) better especially far below the threshold.

, ,EffKN S Iv s r

Corrected KbarN potential … “Corrected”

KbarN potential without correction … “Uncorrected”

,, ,Eff C

KN S Iv s r

Step 3

３． Local KbarN potential based on Chiral SU(3)I=0 KbarN scatteing amplitude

“Uncorrected” “Corrected”

Chiral Unitary

The scattering amplitude far below threshold is overestimated if “Uncorrected” effective potential is used. (about twice)

Chiral Unitary

1420

Resonance position in I=0 KbarN channel

In Chiral unitary model,

1420 MeV1420 MeV

not 1405 MeV !

Chiral unitary; T.Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003)

４． ResultNotes on actual calculation

We have tried “Corrected” and “Uncorrected” for four models:• “ORB” E. Oset, A. Ramos, and C. Bennhold, Phys. Lett. B527, 99 (2002)• “HNJH” T.Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003)• “BNW” B. Borasoy, R. Nissler, and W. Weise, Eur. Phys. J. A25, 79 (2005)• “BMN” B. Borasoy, U. G. Meissner, and R. Nissler, Phys. Rev. C74, 055201 (2006)

Imaginary part of the effective KbarN potential is treated perturbatively.

Hamiltonian

2

N KN

N K

M m B Ks M

M m B K

Re CMNN KN SH T V V s T

Binding energy of kaon NuclB K H H NuclH : Hamiltonian of nuclear part

We performed variational calculation. Since the KbarN potential is energy-dependent, we repeat the calculation until the self-consistency on the kaon energy is accomplished.

We have tried two prescriptions for . s

N N

Kbar

2B K

Kbar is bound by each nucleon with B(K)/2 binding energy.

2B K

４． Result

We have tried “Corrected” and “Uncorrected” for four models:• “ORB” E. Oset, A. Ramos, and C. Bennhold, Phys. Lett. B527, 99 (2002)• “HNJH” T.Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003)• “BNW” B. Borasoy, R. Nissler, and W. Weise, Eur. Phys. J. A25, 79 (2005)• “BMN” B. Borasoy, U. G. Meissner, and R. Nissler, Phys. Rev. C74, 055201 (2006)

Imaginary part of the effective KbarN potential is treated perturbatively.

Hamiltonian

2

N KN

N K

M m B Ks M

M m B K

Re CMNN KN SH T V V s T

Binding energy of kaon NuclB K H H NuclH : Hamiltonian of nuclear part

We performed variational calculation. Since the KbarN potential is energy-dependent, we repeat the calculation until the self-consistency on the kaon energy is accomplished.

We have tried two prescriptions for . s

I=0 channel I=1 channel

Notes on actual calculation

４． ResultTotal Binding Energy and Decay Width

“Corrected”, N Ks M m B K

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25 30

Total B. E. [MeV]

Wid

th [

MeV

] HNJ H-S-WfORB-S-WfBMN-S-WfBNW-S-Wf


“Corrected”, 2N Ks M m B K

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25 30

Total B. E. [MeV]

Wid

th [

MeV

] HNJ H-S- J fORB-S- J fBMN-S- J fBNW-S- J f


“Uncorrected”, N Ks M m B K

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25 30

Total B. E. [MeV]

Wid

th [

MeV

] HNJ H-U-WfORB-U-WfBMN-U-WfBNW-U-Wf


“Uncorrected”, 2N Ks M m B K

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25 30

Total B. E. [MeV]

Wid

th [

MeV

] HNJ H-U- J fORB-U- J fBMN-U- J fBNW-U- J f

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25 30

Total B. E. [MeV]

Wid

th [

MeV

]

HNJ H-S-WfORB-S-WfBMN-S-WfBNW-S-WfHNJ H-S- J fORB-S- J fBMN-S- J fBNW-S- J fHNJ H-U-WfORB-U-WfBMN-U-WfBNW-U-WfHNJ H-U- J fORB-U- J fBMN-U- J fBNW-U- J f


Total B. E. : 16 ～ 26 MeVWidth : 37 ～ 63 MeV

Total B. E. : 16 ～ 26 MeVWidth : 37 ～ 63 MeV

Structure of ppK-

N N

Kbar

KbarN potential based on “HNJH”


Structure of ppK-

Kbar

2.26 fm

2.00 fm

N N



Structure of ppK-

Kbar

NN distance = 2.26 fm KbarN distance = 2.00 fm

N N

T = 1/2T = 1/2TN=1 … 95.5 %TN=0 … 4.5 %



Structure of ppK-

Kbar

N N

Mixture of TN=0 component = 4.5 %

I=0 KbarN I=1 KbarN

1.83 fm 2.39 fm

2 0.3l 2 1.9l




Structure of ppK-

Kbar

N N

Mixture of TN=0 component = 4.5 %

I=0 KbarN I=1 KbarN

1.83 fm 2.39 fm

2 0.3l 2 1.9l




“Λ(1405)” as I=0 KbarN calculated with this potential

1.87 fm

2 0.0l

Almost “Λ(1405)”Almost “Λ(1405)”

Influence of P-wave KbarN potential• Estimate its contribution perturbatively.• Derived from “Full” scattering volume.

2

2, 3/ 2 3

1 4 1, exp

2K N

KN P wave K N KNPN P

sv C

aM a

r rr r

��

- 0.05

0.00

0.05

0.10

0.15

0.20

0.25

300 350 400 450 500

Kaon's energy w [MeV]

Cp,

cm [

fm3 ]

Re CpIm Cp

B(K) [MeV]46

The B(K) obtained with only the S-wave potential is very close to the position of Σ(1385) accidentally. (Slightly above it)

VKN,P ～ +3 MeV

…small and repulsive

For ap= 0.4 ～ 0.9 fm,

５． Summary and Future planSummary

• We have studied ppK- with a model that can treat the NN repulsive core directly. Here, we have used Av18 potential as a bare NN potential.

• Exactly speaking, the system we are now considering is [NNKbar]T=1/2, Tz=1/2. This system can contain TN (isospin of nuclear part) =0 component in addition to TN=1 component which corresponds to just ppK-. Although the mixture of the TN=0 component is small, typically 5 %, its contribution to the binding energy is rather large.

Total Binding energy = 16 ～ 26 MeVWidth (S-wave) = 37 ～ 63 MeV … Very shallow binding

• I=0 KbarN component in the ppK- seems almost genuine Λ(1405), investigating its size and orbital angular momentum. This fact agrees with Akaishi-san’s picture.

• We have estimated the influence of the p-wave KbarN potential, derived from the “Full” scattering volume, perturbatively. Since the system is shallowly bound, its contribution is small and repulsive. (VKN,P ～ +3 MeV)

• We have constructed an effective s-wave KbarN potential which reproduces the scattering amplitude of KbarN calculated in the framework of Chiral unitary model. The present potential is energy-dependent, complex and local potential and has single Gaussian shape.

• We have investigated the four KbarN potentials which are based on different Chiral unitary model and tried two prescriptions for the relationship of √s and B(K). For all cases, results are not so different:

５． Summary and Future planFuture plan

Understanding of the difference between our result and those obtained by other groups.

• Comparison with Akaishi-san’s study

In the region of the sub-threshold, the absolute value of the KbarN scattering amplitude in addition to its behavior (energy-dependence) is definitely different from that derived from Chiral unitary model.

• Comparison with Faddeev (Ikeda-san and Sato-san) study

• Comparison with Faddeev (Shevchenko and Gal) study

Total B. E. = 79 MeV, Decay width = 74 MeV

Total B. E. = 50 – 70 MeV, Decay width = ～ 100 MeV

Why is their result so different from ours, although their KbarN interaction is based on Chiral SU(3) theory similarly to our study? Is there a problem in the treatment of the energy dependence of two-body system (KbarN) in the three-body system (KbarNN)???

Total B. E. = 48 MeV, Decay width = 61 MeV

In their study, ppK- can be bound by about 40 MeV even only in the KbarNN channel, namely without coupling to the πΣN channel.

ppK- calculated with various potentialsCorrected Uncorrected

√S = MN + mK

- B(K)

√S = MN + mK

- B(K) / 2

HNJH ORB BMN BNWa 0.47 0.52 0.41 0.51

Total B. E. 15.9 17.7 15.6 17.1Width 47.1 53.2 39.2 60.9

B (K) 38.9 40.1 41.0 39.3E (Nucl) 23.0 22.4 25.3 22.2Kin (Nucl) 37.3 37.5 39.3 36.6

Kin. 129.6 125.5 142.5 125.0V (NN) -14.3 -15.1 -14.0 -14.4V (KN-s) -131.2 -128.2 -144.1 -127.7

Rel (NN) 2.26 2.21 2.27 2.26Rel (KN) 2.00 1.96 2.02 2.00

HNJH ORB BMN BNWa 0.47 0.52 0.41 0.51

Total B. E. 19.8 21.6 20.8 19.6Width 58.6 64.5 53.7 71.7

B (K) 45.2 46.1 49.6 43.2E (Nucl) 25.4 24.5 28.8 23.6Kin (Nucl) 40.3 40.1 43.4 38.5

Kin. 141.5 136.0 160.0 132.1V (NN) -14.9 -15.6 -14.6 -14.9V (KN-s) -146.4 -142.0 -166.1 -136.8

Rel (NN) 2.19 2.15 2.20 2.21Rel (KN) 1.91 1.88 1.91 1.94

HNJH ORB BMN BNWa 0.47 0.52 0.41 0.51

Total B. E. 24.2 26.4 21.8 23.4Width 45.5 51.4 36.9 60.1

B (K) 51.9 53.3 51.2 49.0E (Nucl) 27.8 27.0 29.4 25.6Kin (Nucl) 43.1 43.1 44.0 40.9

Kin. 153.5 148.0 163.2 142.3V (NN) -15.3 -16.2 -14.6 -15.3V (KN-s) -162.4 -158.2 -170.4 -150.4

Rel (NN) 2.14 2.08 2.19 2.16Rel (KN) 1.84 1.80 1.89 1.86

HNJH ORB BMN BNWa 0.47 0.52 0.41 0.51

Total B. E. 21.2 21.7 22.5 16.0Width 56.6 59.8 53.4 63.0

B (K) 47.3 46.2 52.4 37.7E (Nucl) 26.2 24.6 29.9 21.7Kin (Nucl) 41.3 40.3 44.7 36.2

Kin. 145.4 136.2 165.4 122.1V (NN) -15.2 -15.8 -14.8 -14.5V (KN-s) -151.4 -142.1 -173.1 -123.6

Rel (NN) 2.16 2.14 2.17 2.25Rel (KN) 1.88 1.87 1.87 2.01

Documents

ppK - studied with a Chiral SU(3)-based K bar N potential