Y. Ikeda and T.Sato (Osaka Univ.)

Three-body resonanceThree-body resonancein the KNN-πin the KNN-π ＹＹ N coupled N coupled systemsystem

Table of contents

1. Motivation

2. Formalism (3-body equation)

3. Results (KNN resonance state)

4. Summary

Table of contents

1. Motivation

2. Formalism (3-body equation)

3. Results (KNN resonance state)

4. Summary

Kaon absorption??(Magas et al.)

Kaon absorption??(Magas et al.)

1. Our Motivation1. Our Motivation

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

φ : Meson field , B : Baryon field

2-body Meson-Baryon Potential2-body Meson-Baryon PotentialChiral effective LagrangianChiral effective Lagrangian

2. Formalism2. Formalism

The The ＫＮＮＫＮＮ -πΣ-πΣ Ｎ Ｎ resonanceresonance

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

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

ＫＮＮＫＮＮ -π-π ＹＮ 共鳴状態を理解するための全体の目ＹＮ 共鳴状態を理解するための全体の目標標本研究での枠組み本研究での枠組み ->KN interaction ->KN interaction に注目に注目

KN-KN-ππY(Y(I=0, 1)I=0, 1)

ππN(I=1/2,3/2)N(I=1/2,3/2)NNNN 散乱散乱

反対称化を考慮反対称化を考慮

π

Σ

Ｎ

Parameter fitParameter fit

-> cut-off of dipole form factor

fit : Martin’s analysis and Kaonic hydrogen datafit : Martin’s analysis and Kaonic hydrogen data

Our parameterOur parameter

Non relativistic

RelativisticI=0 (-1.70 + i0.68 fm)KN =946 MeVπΣ=988 MeV------------------------I=1 (0.72 + i0.59 fm)KN =920 MeVπΣ=960 MeVπΛ=640 MeV

I=0 (-1.70 + i0.68 fm)KN =1095 MeVπΣ=1450 MeV------------------------I=1 (0.68 + i0.60 fm)KN =1100 MeVπΣ=850 MeVπΛ=1250 MeV

ππN scattering (S11 and S31)N scattering (S11 and S31)

Rel.

Non-rel.

Rel.

Non-rel.

-> Fit to the scattering length and phase shift.

Phys. Lett. B 469 (1999) 25.

(S11) (S31)

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

Phys. Rev. 178 (1969) 1597.

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, ……)

spectator energyspectator energy

3. Numerical 3. Numerical ResultsResults

KN amplitude bellow threshold

WKN (MeV)

2-body T-matrix in the 3-body 2-body T-matrix in the 3-body systemsystem 　　 :spectator

energy

How this attractive KN interaction How this attractive KN interaction contributes to 3-body binding system contributes to 3-body binding system is determined by solving dynamical 3-body eq. directly!!is determined by solving dynamical 3-body eq. directly!!

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

The three-body resonance poles (other The three-body resonance poles (other parameters)parameters)

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

Rel. model

The pole position strongly depends on KN interaction. -> Form factor (dipole -> monopole)

-> Structure dependence on Λ(1405) Effects of kaon absorption (p-wave interaction)

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 πΣπΣN unphysicalN unphysical energy plane. energy plane. (W(Wpolepole = -79.7-i36.4 MeV using relativistic kinematics) = -79.7-i36.4 MeV using relativistic kinematics)

SummarySummary

In the futureIn the future