chiral dynamics を尊重した場合の ppk -

Chiral dynamics を尊重した場合の ppK-

“Physics at J-PARC: New aspects of hadron and nuclear studies”　’ 08.08.07 @ KEK 3rd building Seminar hall

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 / YITP)

arXiv:0806.4917v1arXiv:0806.4917v1

( How is ppK- if we respect Chiral SU(3) dynamics? )

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 nuclei…

To make the situation more clear …

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

Studied with various methods, because it is a three-body system:

Akaishi, Yamazaki ATMS B.E. = 47MeV, Γ= 61MeV PRC76, 045201(2007)

Ikeda, Sato Faddeev B.E. = 79MeV, Γ= 74MeV PRC76, 035203(2007)

Shevchenko, Gal Faddeev B.E. = 50 ～ 70MeV, Γ= ～ 100MeV PRC76, 044004(2007)

Arai, Yasui, Oka (Λ* nuclei model), Nishikawa, Kondo (Skyrme model), Noda, Yamagata, Sasaki, Hiyama, Hirenzaki (Variational method) …

FINUDA experiment B.E. = 116MeV, Γ= 67MeV PRL94, 212303(2005)

１． Introduction

Av18 NN potentialAv18 NN potential

… … a realistic NN potential a realistic NN potential with strong repulsive core.with strong repulsive core.

KKbarbarN potential based on Chiral SU(3) theoryN potential based on Chiral SU(3) theory

Variational methodVariational method

… … Investigate various properties Investigate various properties with the obtained wave function.with the obtained wave function.

We also study ppK- with …

…… Well describe S=-1 meson-baryon scattering Well describe S=-1 meson-baryon scattering and dynamical generation of Λ(1405)and dynamical generation of Λ(1405)

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

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

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

Thanks, Akaishi-san!

NN correlation is directly treated.

Model wave function 　ー Simple Correlated Model (Revised) ー

10 ,

2J T

NN in : 1E, TN=1 ppK-

NN in : 1O, TN=0

1

0

２． Model

NN potential 　 …　 Tamagaki potentialEffective KbarN potential … Akaishi-Yamazaki potential

Influence of the improvement

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

SCM ver2 SCM ver1 ATMSCmix Finite Zero ---

B. E. 51.4 39.0 48

Γ (K bar N → π Y ) 61.0 60.0 61

B(K) 80.0 65.8 68Nucl. E 28.7 26.8 20

Kinetic 162.4 147.0 167NN pot -19.2 -19.8 -19KbarN pot -194.6 -166.2 -196

Rel (NN) 1.83 1.75 1.90Rel (KN) 1.55 1.54 1.57

Mixing ratio 5.9% 0.0%

1 1 0 1

0 0 0 1

1 0 1

3 1

4 41 3

4 4

3

4

KN KNKN I I

KN KNKN I I

KN KNKN I I

V V V

V V V

V V V

1 1 0 1

0 0 0 1

1 0 1

3 1

4 41 3

4 4

3

4

KN KNKN I I

KN KNKN I I

KN KNKN I I

V V V

V V V

V V V

Additional KbarN attractionAdditional KbarN attraction

２． Model

NN potential 　 …　 Av18 potential• Fitted with a few range Gaussians.• Use one-pion-exchange potential (Central and Spin-Spin) and L2 potentials in addition to phenomenological central term (= repulsive core).

1E1E 1O1O

Strong repulsive core(3 GeV)

Strong repulsive core(3 GeV)

Central potential

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

ij ij ik k kjT V V G T

T. Hyodo and W. Weise, PRC77, 035204(2008)

11 11 11 1 11Eff EffT V V G T

ijV

11EffV s

11 1 11T U U G T

11

1,

2EffN

a

MU r s V s g r

s

3/ 2

2

2

1expag r r a

a

,CorrU r s

1. Chiral unitary / Relativistic / Coupled Channel

2. Chiral unitary / Relativistic / Single Channel

3. Effective local potential / Non-relativistic / Single Channel

• Energy dependent• Complex

• Energy dependent• Complex• Local, Gaussian form

“Corrected version” ; Energy-dependence improved in low-energy region

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

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

“Uncorrected”

Chiral Unitary

“Corrected”

1420

Resonance position in I=0 KbarN channel

In Chiral unitary model,

1420 MeV1420 MeV

not 1405 MeV !

４． ResultNotes on actual calculation

Hamiltonian Re CMNN KN SH T V V s T

Controlled by “Anti-kaon’s binding energy”

NuclB K H H NuclH : Hamiltonian of nuclear part

Self-consistency on anti-kaon’s energy

treated perturbatively. Im KN SV s

2

1 2 2NN C ss LV V r V r V r L

KN SV s

2

N KN

N K

M m B Ks M

M m B K

Try two prescriptions for . s

N N

Kbar

2B K

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

2B K

Kbar field surrounds two nucleons which are almost static.

Kbar

N N

Km B K

４． Result

Four Chiral unitary 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)

Notes on actual calculation

I=0 channel I=1 channel

1420

４． ResultTotal Binding Energy and Decay Width

ORBHNJHBNWBMN

Corrected 2

N K

N K

M m B Ks

M m B K

× ×

Total B. E. : 20 ± 3 MeVΓ （ KbarN→πY ） : 40 ～　 70 MeV

Total B. E. : 20 ± 3 MeVΓ （ KbarN→πY ） : 40 ～　 70 MeV

Small model and ansatz dependence

Small model and ansatz dependence

Structure of ppK-

N N

Kbar

KbarN potential based on “HNJH”

“Corrected”, N Ks M m B K

Structure of ppK-

Kbar

2.21 fm

1.97 fm

N N



Cf)　

　　　　　　NN　distance　in　normal　nuclei　　　～ 2　fm　　　　　　Size　of　deuteron　　　　　　　　　　　　　　　　　　　　　～ 4　fm

Structure of ppK-

Kbar

NN distance = 2.21 fm KbarN distance = 1.97 fm

N N

T = 1/2T = 1/2TN=1 … 96.2 %TN=0 … 3.8 %



Structure of ppK-

Kbar

N N

Mixture of TN=0 component = 3.8 %




1.97 fm

Structure of ppK-

Kbar

N N


I=0 KbarN

1.82 fm

2 0.4l




Structure of ppK-

Kbar

N N


I=0 KbarN I=1 KbarN

1.82 fm 2.33 fm

2 0.4l 2 1.9l




Structure of ppK-

N


I=0 KbarN I=1 KbarN

1.82 fm 2.33 fm

2 0.4l 2 1.9l




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

1.86 fm

2 0.0l Kbar

NAlmost “Λ(1405)”Almost “Λ(1405)”

Structure of ppK- KbarN potential based on “HNJH”


Density distribution NN pair in ppK- vs Deuteron

0.0E+00

5.0E- 03

1.0E- 02

1.5E- 02

2.0E- 02

2.5E- 02

3.0E- 02

3.5E- 02

4.0E- 02

0.0 1.0 2.0 3.0 4.0 5.0

r　[fm]

[fm

-3]

HNJ HORBBMNBNWDeuteron

2NNr r NN r

0.0E+00

5.0E- 03

1.0E- 02

1.5E- 02

2.0E- 02

2.5E- 02

3.0E- 02

3.5E- 02

4.0E- 02

4.5E- 02

0.0 2.0 4.0 6.0 8.0 10.0

r　[fm]

[fm

-1]

HNJ HORBBMNBNWDeuteron

Suppressed by NN repulsive core

Suppressed by NN repulsive core

Quite compacter than deuteron

Quite compacter than deuteron

Structure of ppK- KbarN potential based on “HNJH”


Density distribution KbarN pair in ppK- vs Λ(1405)

2 Normalized

KNr r For comparison,

All densities are normalized to 1.

0.00

0.01

0.02

0.03

0.04

0.05

0.0 2.0 4.0 6.0 8.0

r　[fm]

[fm

-1]

I=0 in ppK-I=0 in ppK-

I=1 in ppK-I=1 in ppK-

Λ(1405)Λ(1405)

Very different.The I=1 component distributes widely.

Rather similar.

Other effects

Dispersive correction (Effect of imaginary part)

～ +10 MeV

p-wave KbarN potential ～ -3 MeV 10 ～ 35 MeV

two nucleon absorption 4 ～ 12 MeV

s-wave KbarN potentialB .E. Width

20 ± 3 MeV 40 ～ 70 MeV

～ 25 MeV 55 ～ 120 MeV

B .E. Width

Rough estimation

５． Summary

Total Binding energy = 20 ± 3 MeVΓ(KbarN → πY) = 40 ～ 70 MeV … Very shallow binding

ppK- is studied with

NN potential = Av18 potential … Strongly repulsive core Spin-spin, L2 terms KbarN potential = Local potential based on Chiral unitary model … Scattering amplitude is reproduced. Single Gaussian form Strong energy dependence

Simple correlated model (Revised) … Respect Short-range correlation. Contain TN=0 component as well as TN=1 component.

+

Four Chiral unitary models and

Two prescriptions on B(K)

５． Summary

Structure of ppK-

NN distance in ppK- is smaller than that of deuteron, rather comparable to that in normal nuclei.

NN distance = ～ 2.2 fmKbarN distance = ～ 2.0 fm

TN=0 component is slightly ( ～ 4%) mixed, while TN=1 is dominating in ppK-.

Calculating the size and orbital angular momentum of I=0 KbarN component, it is found to be very similar to the isolated I=0-KbarN bound system, “Λ(1405)”.

I=0 KbarN in ppK- is almost “Λ(1405)” !

… This picture doesn’t contradict with Akaishi-san’s one.

Contribution of other effects

PRC76, 045201(2007)

• Dispersive correction: ～ +10MeV to B.E.

• p-wave KbarN potential: ～ -3MeV to B.E. , 10 ～ 35MeV to width

• Two nucleon absorption: 4 ～ 12MeV to width

5 ． Future plan

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

Especially, comparison with Faddeev (Ikeda-san and Sato-san) study

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

Separable potential?

Non-relativistic (semi-relativistic) vs relativistic?

Energy dependence of two-body system (KbarN) in the three-body system (KbarNN)?

…???

Their KTheir KbarbarN interaction is also constrained by Chiral SU(3) theory. N interaction is also constrained by Chiral SU(3) theory. However, it differs from ours very much. However, it differs from ours very much.

chiral dynamics を尊重した場合の ppk -

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