Yoichi Ikeda (Osaka Univ.)Yoichi Ikeda (Osaka Univ.)in collaboration within collaboration with

Hiroyuki Kamano (JLab) and Toru Sato (Osaka Univ.) Hiroyuki Kamano (JLab) and Toru Sato (Osaka Univ.)

IntroductionIntroduction

Our model of KN interactionOur model of KN interaction

Coupled-channel Faddeev equationsCoupled-channel Faddeev equations

Numerical ResultsNumerical Results

SummarySummary

IntroductionIntroduction

Our model of KN interactionOur model of KN interaction

Coupled-channel Faddeev equationsCoupled-channel Faddeev equations

Numerical ResultsNumerical Results

SummarySummary

The resonance pole of strange dibaryon The resonance pole of strange dibaryon in KNN – in KNN – YN systemYN system

Introduction Introduction

KN interaction in isospin I=0 channelKN interaction in isospin I=0 channelStrong Strong attractionattraction

The The (1405) (1405) resonanceresonance

Quasi-bound state of KN Quasi-bound state of KN statestate

CDD pole coupling with CDD pole coupling with mesonsmesons

Multi-quark stateMulti-quark state

Quasi-bound state of KN Quasi-bound state of KN statestate

CDD pole coupling with CDD pole coupling with mesonsmesons

Multi-quark stateMulti-quark state

KN - KN - coupled system coupled system

Strange dibaryon resonanceStrange dibaryon resonance

KNN – KNN – YN coupled systemYN coupled system

IntroductionIntroduction

Two poles on KN physical and Two poles on KN physical and unphysical sheet (chiral unitary model) unphysical sheet (chiral unitary model)

taken form Jido et al. NPAtaken form Jido et al. NPA725725 (2003). (2003). taken form Hyodo and Weise. PRCtaken form Hyodo and Weise. PRC7777 (200 (2008).8).

Structure of the Structure of the (1405)(1405)

Structure of strange dibaryonStructure of strange dibaryon

We investigate We investigate

possible strange dibaryon resonance poles.possible strange dibaryon resonance poles.

S=-1, B=2, Q=+1S=-1, B=2, Q=+1

JJππ=0=0--

(3-body s-wave state)(3-body s-wave state)

We consider s-wave state.We consider s-wave state. We can expect We can expect most strong attractive interactionmost strong attractive interaction

in this configure.in this configure.

L=0 (s-wave interaction)

NN

KK

NN

IntroductionIntroduction

Potential derived from Weinberg-Tomozawa termPotential derived from Weinberg-Tomozawa termPotential derived from Weinberg-Tomozawa termPotential derived from Weinberg-Tomozawa term

: : Meson field Meson field , , BB : : Baryon Baryon fieldfield

Chiral effective LagrangianChiral effective LagrangianChiral effective LagrangianChiral effective Lagrangian

…

on-shell factorizationon-shell factorization

Our model of KN interactionOur model of KN interaction

Unitarized by Lippmann-Schwinger equationUnitarized by Lippmann-Schwinger equation

Our model of KN interactionOur model of KN interaction

E-dep. potentialE-dep. potentialE-dep. potentialE-dep. potential

Cutoff parametersCutoff parameters

Our model of KN interactionOur model of KN interaction

Poles of the amplitudePoles of the amplitude

(KN bound state)(KN bound state)1428.8-i15.3(MeV)1428.8-i15.3(MeV)

(( resonance) resonance)1344.0-i49.0(MeV)1344.0-i49.0(MeV)

Hyodo, Weise PRCHyodo, Weise PRC7777(2008).(2008).Consistent with chiral unitary modelConsistent with chiral unitary model

(coupled-channel chiral dynamics)(coupled-channel chiral dynamics)

Faddeev EquationsFaddeev EquationsFaddeev EquationsFaddeev Equations

W : 3-body scattering energyW : 3-body scattering energy

i(j) = 1, 2, 3i(j) = 1, 2, 3 (Spectator particles) (Spectator particles)

T(W)=TT(W)=T11(W)+T(W)+T22(W)+T(W)+T33(W) (T : 3-body amplitude)(W) (T : 3-body amplitude)

ttii(W, E(p(W, E(pii)) : 2-body t-matrix )) : 2-body t-matrix with spectator particle iwith spectator particle i

GG00(W) : 3-body Green’s function ((W) : 3-body Green’s function (relativistic kinematicsrelativistic kinematics))

W : 3-body scattering energyW : 3-body scattering energy

i(j) = 1, 2, 3i(j) = 1, 2, 3 (Spectator particles) (Spectator particles)

T(W)=TT(W)=T11(W)+T(W)+T22(W)+T(W)+T33(W) (T : 3-body amplitude)(W) (T : 3-body amplitude)

ttii(W, E(p(W, E(pii)) : 2-body t-matrix )) : 2-body t-matrix with spectator particle iwith spectator particle i

GG00(W) : 3-body Green’s function ((W) : 3-body Green’s function (relativistic kinematicsrelativistic kinematics))

W : 3-body scattering energyW : 3-body scattering energy

i(j) = 1, 2, 3i(j) = 1, 2, 3 (Spectator particles) (Spectator particles)

Z(pZ(pii,p,pjj;W) : Particle exchange potentials;W) : Particle exchange potentials

(p(pnn;W) : Isobar propagators;W) : Isobar propagators

Faddeev equation with separable potentialsFaddeev equation with separable potentials

i

j

i

j

=XXijij

i j

XXijij

nn

+n

Alt-Grassberger-Sandhas(AGS) EquationAlt-Grassberger-Sandhas(AGS) EquationAlt-Grassberger-Sandhas(AGS) EquationAlt-Grassberger-Sandhas(AGS) Equation

KNN-pYN coupled-channel systemKNN-pYN coupled-channel system

Alt-Grassberger-Sandhas(AGS) EquationAlt-Grassberger-Sandhas(AGS) EquationAlt-Grassberger-Sandhas(AGS) EquationAlt-Grassberger-Sandhas(AGS) Equation

i

j

i

j

=XXijij

i j

XXijij

nn

+n

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

ππ

Σ,ΛΣ,Λ

NN

NNKK NN

NN

KK

NN

ππΣ,ΛΣ,Λ

NN ππ

Σ,ΛΣ,ΛNN

NN potential -> Two-term separable potentialNN potential -> Two-term separable potential

AttractionAttraction Repulsive coreRepulsive core

XXijij

Two-body potentials –NN interaction-Two-body potentials –NN interaction-

NN

NN

KK

NNNN

Two-body potentials –Two-body potentials – interaction- interaction-

XXijij

Σ,ΛΣ,Λ

ππ

NN

E-dep. potentialE-dep. potentialE-dep. potentialE-dep. potential

I=1/2I=1/2 I=3/2I=3/2=500 (MeV)=500 (MeV) =500 (MeV)=500 (MeV)

Scattering lengthScattering length Scattering lengthScattering length

YN potential -> One-term separable potentialYN potential -> One-term separable potential

XXijijΣ,ΛΣ,Λ

ππ

NN YNYN

Two-body potentials –YN interaction-Two-body potentials –YN interaction-

I=1/2I=1/2

I=3/2I=3/2

Scattering lengthScattering length Scattering lengthScattering length

Torres, Dalitz, Deloff, PLBTorres, Dalitz, Deloff, PLB174174 (1986). (1986).

Pole of the AGS amplitudesPole of the AGS amplitudes

WWpolepole = -B –i = -B –i/2 /2

Eigenvalue equation for Fredholm kernelEigenvalue equation for Fredholm kernel

three-body resonance pole at Wthree-body resonance pole at Wpolepole

Formal solution for three-boby amplitudesFormal solution for three-boby amplitudes

Fredholm kernelFredholm kernel

Possible singularities of the amplitudesPossible singularities of the amplitudes

Z(pZ(pii,p,pjj;W) : Particle exchange potentials;W) : Particle exchange potentials

(p(pnn;W) : Isobar propagators;W) : Isobar propagators

We search for three-body resonance poles We search for three-body resonance poles

on KNN physical, on KNN physical, YN unphysical, and YN unphysical, and “………”“………” sheet. sheet.

Numerical resultsNumerical results

We construct the model of We construct the model of energy-dependent KN interactionenergy-dependent KN interaction. .

(chiral unitary approach)(chiral unitary approach)

We solve the Faddeev equations.We solve the Faddeev equations.

　　　：　　　： We found We found two poles of strange dibaryontwo poles of strange dibaryon..

　　　：　　　： -B-i -B-i /2 = (-13.7-i29.0, -37.2-i93.3) MeV/2 = (-13.7-i29.0, -37.2-i93.3) MeV

Pole I -> KNN physical, Pole I -> KNN physical, YN unphysical, YN unphysical, *N physical sheet*N physical sheet

Pole II -> KNN physical, Pole II -> KNN physical, YN unphysical, YN unphysical, *N unphysical sheet*N unphysical sheet

SummarySummary

FutureFuture　　　　　　　　　　　　　reaction

This production mechanism will be investigated by LEThis production mechanism will be investigated by LEPS and CLAS collaborations. @SPring8, JlabPS and CLAS collaborations. @SPring8, Jlab