Study of light kaonic nuclei with a Chiral SU(3)-based KN potential

A. Dote (KEK)W. Weise (TU Munich)

Introduction

ppK- studied with a simple model

• Simple Correlated Model• Test on two nucleons system• Result of ppK-

Summary and future planNuclear Physics at J-PARC2nd June ‘07 　 @ Ricotti in Tokai village

´

Introduction

0KN

IV Strongly attractive

Deeply bound and Dense

KN interactionKN interaction

Repulsive core at short distance

NN interactionNN interactionNon-mesonic decay mode

KNN → YN,

in addition to mesonic decay modeKN → Yπ

DecayDecayKaonic nuclei

Introduction

Repulsive core at short distance

Non-mesonic decay modeKNN → YN,

in addition to mesonic decay modeKN → Yπ

Kaonic nuclei

0KN

IV Strongly attractive

Deeply bound and Dense

KN interactionKN interaction

NN interactionNN interactionDecayDecay

ppK-

Chiral SU(3)-based KN potential

Av18-likeNN potential

H. Fujioka et al. @ FINUDA

B.E. = 116 MeV, Γ = 67 MeVB.E. = 116 MeV, Γ = 67 MeV

ppK- studied with a simple model

and Chiral SU(3)-based

KN potential

Prof. Akaishi gave advices on the few-body calculation.

1. Simple Correlated Model

1/ 21 2 1 1/ 2, 1/ 2, , 0

NSCM K T T Tz

N r r r S NN K

������������������������������������������

Model wave function of ppK-

NN spin: S=0NN isopin: TN=1Total isospin: T=1/2

NN spin: S=0NN isopin: TN=1Total isospin: T=1/2

Spatial part

1 2 1 2 1 2' , ' , ' ,K K KG r G r G r F r r F r r F r r������������������������������������������������������������������������������������������������������������������������������

Single-particle motion of nucleons and a kaon

Correlations

2

2

exp

' exp

i i

K K

G r r

G r r

����������������������������

����������������������������

nucleon

kaon

NN correlation functionNN correlation function

2

1 2 1 2, 1 expNN NNn n

n

F r r f r r ��������������������������������������������������������

KN correlation ' , 1i KF r r ����������������������������

Energy variation

Variational parameters

1 2

'

,

i

K

NNn

G r

G r

F r rf

��������������

��������������

����������������������������

1. Simple Correlated Model

1 2,NNn F r r

����������������������������

2

1

1

11

1NNn n

n

NN

n

r

rr r

r

Gaussians used for the NN correlation

… Kamimura Gauss

Included in the spatial part of the wave function

Real parameters

Determined by Simplex method to minimize the total energy

1. Simple Correlated ModelModel wave function of ppK-

Isospin state

Λ（１４０５）：　

ppK- ：　

Deuteron+K- ：　

1 1/ 2NT TNN K

0 1/ 2NT TNN K

0TN K

nucleon isospin=1

nucleon isospin=0

0Iv KN v

0 1

3 1

4 4I Iv KN v v

0 1

1 3

4 4I Iv KN v v

Very attractiv

e

2. Test on 2N system

Checked this model in case of pp system.

,

1/ 21 2 1, 1, 0

N N ZSCM T T

N r r S NN

����������������������������

1 2 1 2 1 2, ,r r G r G r F r r ������������������������������������������������������������������������������������

2expi iG r r

����������������������������

2

1 2 1 2, 1 expNN NNn n

n

F r r f r r ��������������������������������������������������������

Variational parameters are determined by the Simplex method.

are fixed to those of Kamimura Gauss.

NNnf

NNn

2. Test on 2N systemNN potential to test

- 500

0

500

1000

1500

2000

2500

3000

3500

0.0 0.5 1.0 1.5 2.0

Av18- like 1E

Dote_HC2

Enhanced the long-range attraction of the Av18-like potential slightly so as to make two protons bound.

2. Test on 2N system

Result

1

25

0.2 fm,

20 fm

r

r

Method SCM (Kamimura Gauss) GDM

Base number 5 6 7 8 9 25

Kinetic 26.84 23.93 23.51 24.15 24.00 24.21

Pot NN -31.25 -30.35 -29.01 -30.53 -30.42 -30.70

Total E. -4.41 -6.42 -5.50 -6.38 -6.42 -6.49

1 0.1 fm,

9.0 fmN

r

r

Converged

Result obtained by directly diagonalizing the relative Hamiltonian with a lot of Gaussian base.

Hamiltonian

,( )N N CM NNN TestH T V T

-500

0

500

1000

1500

2000

2500

3000

3500

0.0 0.5 1.0 1.5 2.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Av18- like 1E

Dote_HC2

Kamimurabase N=25SCM KG N=5

SCM KG N=6

SCM KG N=7

SCM KG N=8

SCM KG N=9

2. Test on 2N systemRelative wave function

[fm]

[MeV

]

Test potential

SCMN=9

GDMN=25

Hamiltonian CMNN KN S KN PH T V V V T

CMNN KN S KN PH T V V V T

2

2, 3/ 2 3

2

2, 3/ 2 3

1 4 1, exp

2

1 4 1, exp

2

K iKN S wave K i KN

sN s

K iKN P wave K i KN

PN P

sv F

aM a

sv C

aM a

r rr r

r rr r

����������������������������

2 22

1

22

1

2 2

, 2 ,2

i K

i N K

CM N K i Ki

Tm m

T M m mM

p p

PP p p

2

,1

,KN S P K iKN S P wavei

V v

r r

21 11 2expNN NN n n

n

V V E V E b r r Coulomb force is neglected.

3. Result of ppK-

-500

0

500

1000

1500

2000

2500

3000

3500

0.0 0.5 1.0

• Short-range part; referring to Av18, fitted with a few range Gaussians. • Long-range part; Akaishi-san’s effective NN interaction for ppnK- (ρmax=9ρ0)

[fm]

[Me

V]

Av18-like

Av18

Akaishi

Respect the repulsive-core part

1E1EImportant in ppK-

NN potential

KN potentialNs M

1, Gaussian shape

2

2, 3/ 2 3

1 4 1, exp

2K N

KN S wave K N KNsN s

sv F

aM a

r rr r

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

����������������������������

S-wave potential

P-wave potential

2, Energy dependent

as=ap=a

KNF : KN scattering amplitude

KNC : KN scattering volume

3, P-wave potential including derivative operator.

Chiral SU(3) theory

S-wave scattering amplitude

KN potential

- 1.0

- 0.5

0.0

0.5

1.0

1.5

2.0

2.5

300 350 400 450 500

Kaon's energy w [MeV]

Fcm

[fm

]

Re F(Kp)sIm F(Kp)sRe F(Kn)sIm F(Kn)s

1405

Ns M

B. Borasoy, R. Niβler, and W. Weise, Euro. Phys. J. A 25, 79-96 (2005)

P-wave scattering volume

KN potential

- 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

2n pC C

1385

Ns M

R. Brockmann, W. Weise, and L. Taucher, Nucl. Phys. A 308, 365 (1978)※updated version

• Self-consistency of kaon’s energy is taken into account.

Procedure of the present calculation

Perform the energy variation by the Simplex method.

Then, calculate the binding energy of kaon with the obtained wave function. Obtained

B K

Obtained AssumedB K B KCheck Finished !If Yes

Assume the values of the binding energy of kaon itself “B(K)”.

Re Re CMNN KN S KN PH T V V V T KAssumedB K m

The Hamiltonian is determined.

If No

Procedure of the present calculation

Re Re

Im Im

CMNN KN S KN P

KN S KN P

H T V V V T

i V i V

• The imaginary parts are ignored in the current study.

Remarks

• The kaon’s binding energy “B(K)”

B(K) = -EK = -(Etotal – Enucl)

total SCM SCM

Nuclnucl SCM SCM

E H

E H

p+p+K

[pp] in ppK- + K

[ppK-]

0

Enucl

Etotal

B(K)

0

50

100

150

200

250

0 50 100 150 200 250Assumed B(K) [MeV]

Obt

aine

d B

(K) [M

eV]

a=0.67a=0.7a=0.8a=0.9a=1.0Ref

3. Result of ppK- Kamimura Gauss, N=9, r1=0.1 fm, rN=9.0 fmP-wave int. : non-perturbative

Self consistency

a; range parameter [fm]

a=1.00 fm

a=0.90 fm

a=0.80 fm

a=0.70 fm

a=0.67 fm

There doesn’t exist any self-consistent solution for the range parameter a < 0.67 fm.

This result is the same as that obtained in the previous

AMD study reported in YKIS’06 and so on.

There doesn’t exist any self-consistent solution for the range parameter a < 0.67 fm.

This result is the same as that obtained in the previous

AMD study reported in YKIS’06 and so on.

3. Result of ppK- Property

a 0.67 0.70 0.80 0.90 1.00

Assumed B(K) 200 147 106 87 73

Kinetic 517.24 371.31 264.89 211.47 173.63

Pot (NN) -11.64 -22.78 -27.81 -27.86 -26.75

Pot (KNs) -410.15 -321.22 -242.88 -198.04 -167.15

Pot (KNp) -171.68 -87.57 -46.66 -31.32 -22.16

Total E. -76.24 -60.27 -52.46 -45.75 -42.44

B(K) 200.13 148.49 108.15 87.73 75.28

Rel (NN) 1.06 1.15 1.29 1.40 1.49

Rel (KN) 0.70 0.79 0.91 1.00 1.09

[fm]

[MeV]

[MeV]

[MeV]

[fm]

The total binding energy of ppK- is 42 – 76 MeV.

cf) It doesn’t exceed 53 MeV in the previous AMD study.

The total binding energy of ppK- is 42 – 76 MeV.

cf) It doesn’t exceed 53 MeV in the previous AMD study.

3. Result of ppK- Property

a 0.67 0.70 0.80 0.90 1.00

Assumed B(K) 200 147 106 87 73

Kinetic 517.24 371.31 264.89 211.47 173.63

Pot (NN) -11.64 -22.78 -27.81 -27.86 -26.75

Pot (KNs) -410.15 -321.22 -242.88 -198.04 -167.15

Pot (KNp) -171.68 -87.57 -46.66 -31.32 -22.16

Total E. -76.24 -60.27 -52.46 -45.75 -42.44

B(K) 200.13 148.49 108.15 87.73 75.28

Rel (NN) 1.06 1.15 1.29 1.40 1.49

Rel (KN) 0.70 0.79 0.91 1.00 1.09

[fm]

[MeV]

[MeV]

[MeV]

[fm]

The relative distance between two nucleons is larger than 1.0 fm. If the size of a nucleon core is 0.5 fm, they don’t touch.

This result is the same as that of the previous AMD study.　

The relative distance between two nucleons is larger than 1.0 fm. If the size of a nucleon core is 0.5 fm, they don’t touch.

This result is the same as that of the previous AMD study.　

The total binding energy is 42 ～ 76 MeV, when the range parameter changes from 1.00 fm to 0.67 fm. There exists a lower limit in the range parameter due to the self consistency. The mean distance between the two nucleons is larger than 1fm. Essentially, the present result is very similar to the previous one by the AMD study.

Summary• We are now investigating “prototype of a K cluster” ppK- with a simple model respecting the NN short-range correlation.

• In the present study, we adopt a NN potential which has a strongly repulsive core. (Av18-like) • The present KN potential is based on the Chiral SU(3) theory. It includes the p-wave interaction in addition to the s-wave interaction.

• The model wave function is very simple. The nuclear part is assumed to be purely L=S=0 and T=1 state. But in this model we introduce a correlation function between the two nucleons so as to avoid the repulsive core adequately.

• Difference from the previous AMD study The present calculation performs “Variation After Projection” with respect to the total angular momentum and the total isospin. The p-wave interaction is treated non-perturbatively.

•Result

Future plansDouble counting problem 　 (claimed by Prof. Akaishi and Prof. Morimatsu)

+ … + + … …= K

N

K

N

K K

N N

π π π

Σ Σ Σ

K

N

K

N

t matrix

K

N

N

effv

If we solve the three body system, ppK-, with this …eff

v

Although it has already been considered that a KN pair interacts infinite times, such a process is incorporated again and again in the three-body calculation…

In the , the KN pair interacts again and again, coupling to the Σπ pair.

effv

Future plans

• We should directly treat the imaginary part of the KN potential. This is important to estimate the decay width. In addition, this will give an influence to the total energy because the imaginary potential is expected to give a repulsive contribution.

• We should determine the range parameter of the KN interaction.

• We would like to introduce a correlation between two nucleons into the AMD calculation so as to investigate larger system.