Faddeev 計算による K 中間子原子核

池田 陽一，佐藤 透（阪大理・原子核理論）池田 陽一，佐藤 透（阪大理・原子核理論）

FaddeevFaddeev 計算による計算による KK 中間子原子核中間子原子核

原子核・ハドロン物理：横断研究会＠ KEK 　 2007 年 11 月 19 日 -21日

based on PRC76, 035203 (2007).

ContentsContents

Our motivationOur motivation

KNN system with coupled channel KNN system with coupled channel

Numerical resultsNumerical results

Summary and future workSummary and future work

Our motivationOur motivation

Kaon absorption??(Magas et al.)

Kaon absorption??(Magas et al.)

Akaishi and Yamazaki, Akaishi and Yamazaki, PRC65(2002)PRC65(2002)

Agnello et al., PRL94(2005)Agnello et al., PRL94(2005)

Our Investigation We investigate the possible We investigate the possible

[KNN][KNN](I=1/2,J=0)(I=1/2,J=0) 3-body resonance 3-body resonance state.state.

PP

K-

K-pp system

S=-1, B=2, Q=+1Jπ=0-

(3-body s-wave state)

We consider s-wave state.We consider s-wave state. We expect We expect most strong attractive interactionmost strong attractive interaction in this in this configure.configure.

L=0 (s-wave interaction)

Λ(1405)

-> the resonance in the KN-πΣ coupled channel system

It will be very important It will be very important to to take into account the dynamics of KN-take into account the dynamics of KN-πΣπΣ system system in order to investigate whether KNN resonance may exist. in order to investigate whether KNN resonance may exist.

-> Coupled channel Faddeev equation-> Coupled channel Faddeev equation

Solve

-> KNN 3-body resonance-> KNN 3-body resonanceFind

KNN systemKNN system

with coupled channelwith coupled channel

Separable interaction :

Two-body t-matrix :

例えば、一部のダイアグラムは…例えば、一部のダイアグラムは…

＝

Kernel Kernel

ｔ

ｔ

g τ g

g τ g

Formal solution of AGS Formal solution of AGS equationequation

Fredholm type kernel

Eigenvalue equation for Fredholm kernelEigenvalue equation for Fredholm kernel

Formal solution for 3-boby amplitudeFormal solution for 3-boby amplitude

3-body resonance pole at W3-body resonance pole at Wpolepole

AGS AGS EquationEquation

The The ＫＮＮＫＮＮ -π-π ＹＮ ＹＮ resonanceresonance

: 1-particle exchange term: 1-particle exchange term

: 2-body scattering term: 2-body scattering term

KN-KN-ππY(Y(I=0, 1)I=0, 1)

ππN(I=1/2,3/2)N(I=1/2,3/2)

NN scatteringNN scattering

(Anti-symmetrized)(Anti-symmetrized)

ππ

Σ,ΛΣ,Λ

NN

NNKK NN

NN

KK

NN

ππΣ,ΛΣ,Λ

NN ππ

Σ,ΛΣ,ΛNN

本研究での枠組み：本研究での枠組み： KNKN 相互作用に注目相互作用に注目

Meson-Baryon Meson-Baryon scatteringscattering

Baryon-Baryon Baryon-Baryon scatteringscattering

φ : Meson field , B : Baryon field

2-body Meson-Baryon Potential2-body Meson-Baryon Potential

Chiral effective LagrangianChiral effective Lagrangian

Coupling Coupling const.const.

Form Form factorfactor

S-wave separable potentialS-wave separable potential

Parameter fit (KN interaction)Parameter fit (KN interaction)

Our parameters -> cut-off of dipole form factor

Fit ① : Scattering length given by MartinFit ① : Scattering length given by Martin

Fit ② : Fit ② : ΛΛ(1405) pole position given by Dalitz(1405) pole position given by Dalitz

Experimental data (total cross Experimental data (total cross section)section)

(I=1 channel)

Experimental data (total cross Experimental data (total cross section)section)

(I=0 channel)

πN scattering (S11 and S31)πN scattering (S11 and S31)

S11 phase shift

S31 phase shift

Exp. Exp.

Our scattering Our scattering lengthlength

Our scattering Our scattering lengthlength

NN potential -> 2-term Yamaguchi typeNN potential -> 2-term Yamaguchi type

AttractiveAttractive Repulsive Repulsive corecore

AGS equation for 3-body amplitude

K, N,π KN-π Ｙ , NN, πN

Eigenvalue equation for Fredholm kernel

Pole of 3-body Pole of 3-body amplitudeamplitude

WWpp = -B –iΓ/2 = -B –iΓ/2

Similar to πNN, ηNN, K-d analyses. 　　 (Matsuyama, Yazaki, ……)

Numerical resultsNumerical results

The pole trajectories of three-body systemThe pole trajectories of three-body system

a:KN onlya:KN onlyb:+NNb:+NNc:+c:+ππ ＹＹ in in ττdd:π :π exchangeexchange

KNN physicalKNN physicalπΣπΣN unphysicalN unphysical energy plane energy plane

Fit to Dalitz pole

The three-body resonance poleThe three-body resonance pole

Martin

-1.80-i0.68 fm

-1.70-i0.68 fm

-1.60-i0.68 fm

-1.70-i0.59 fm

-1.70-i0.78 fm

KNN physicalKNN physicalπΣπΣN unphysicalN unphysical energy plane energy plane

KK--pppp 研究の現状研究の現状

Shevchenko et al.Shevchenko et al.

Akaishi, YamazakiAkaishi, Yamazaki

MartinMartin-1.70-i0.68 -1.70-i0.68 fmfm DalitzDalitz

KNN physicalKNN physicalπΣπΣN unphysicalN unphysical energy plane energy planeNishikawa, KondoNishikawa, Kondo

Dote et Dote et al.al.

DAΦNEDAΦNE

The pole position strongly depends on KN interaction.　　 -> Structure dependence on Λ(1405) Arai, Oka, Yasui　　　　　　　　　　　　　 reaction

We solve 3-body equation directly.We solve 3-body equation directly. We can find the resonance pole We can find the resonance pole

in the in the KNN physicalKNN physical and and ππ ＹＹ N unphysicalN unphysical energy energy plane.plane.

SummarySummary

In the futureIn the future

Effects of KNN --> Ｙ N (p-wave interaction)