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

ストレンジ・ダイバリオンのストレンジ・ダイバリオンの質量と崩壊幅の研究質量と崩壊幅の研究

KNN resonance (Recent theoretical progress)KNN resonance (Recent theoretical progress)

Faddeev approach and variational approachFaddeev approach and variational approach

Numerical ResultsNumerical Results

SummarySummary

KNN resonance KNN resonance

-- Recent theoretical progress -- Recent theoretical progress ----

KNN resonance – KNN resonance – --

-> -> S-wave resonanceS-wave resonance in the KN-in the KN- coupled channel system coupled channel system

(Chiral unitary model)

Structure of Structure of Multi-quark state?Multi-quark state?

Bound state of KN?Bound state of KN?Large meson-baryon Large meson-baryon componentscomponents

Chiral SU(3) dynamicsChiral SU(3) dynamics

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

KNN resonance – Theoretical progress -KNN resonance – Theoretical progress -

Ikeda, SatoIkeda, Sato

(45-80, 45-75)MeV(45-80, 45-75)MeVShevchenko et al.Shevchenko et al.

(55-70, 90-110)MeV(55-70, 90-110)MeVFaddeev equationFaddeev equation

(B, (B, ))

Dote, Hyodo, WeiseDote, Hyodo, Weise(17-23, 40-70)MeV(17-23, 40-70)MeV

Akaishi, YamazakiAkaishi, Yamazaki

(48, 60)MeV(48, 60)MeVVariational MethodVariational Method

(B, (B, ))

Chiral SU(3)Chiral SU(3)PhenomenologicalPhenomenologicalKN KN interactioninteraction

3 body Method3 body Method

Faddeev equation -> Faddeev equation -> Full dynamicsFull dynamics of KNN- of KNN-N systemN system

Variational approach -> Variational approach -> N system is N system is effectivelyeffectively included. included.

Faddeev approachFaddeev approach

andand

Variational approachVariational approach

Faddeev EquationFaddeev EquationFaddeev EquationFaddeev Equation

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) : 2-body t-matrix (W) : 2-body t-matrix with spectator particlewith spectator particle ii

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

Faddeev approachFaddeev approach

AGS and Faddeev eqs. are equivalent within separable potential model.AGS and Faddeev eqs. are equivalent within separable potential model.

Separable potential :

Two-body t-matrix :

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

Faddeev approach v.s. Variational approachFaddeev approach v.s. Variational approach

Effective KN interactionEffective KN interaction Hyodo, Weise PRC77(2008).

NN

KK・・・

N N

K K

Effective KN interaction in KNN systemEffective KN interaction in KNN system

NN

KK

N -> Faddeev approach-> Faddeev approach-> Variational approach-> Variational approach

At least, the spectator momentum is neglected in At least, the spectator momentum is neglected in the the N Fock space.N Fock space.

KN potential modelKN potential model

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

Weinberg-Tomozawa interactionWeinberg-Tomozawa interactionWeinberg-Tomozawa interactionWeinberg-Tomozawa interaction

Coupling Coupling const.const.

Form Form factorfactor

S-wave Weinberg-Tomozawa potentialS-wave Weinberg-Tomozawa potentialS-wave Weinberg-Tomozawa potentialS-wave Weinberg-Tomozawa potential

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

Our parameters -> cut-off of dipole form factor

Fit 1 : Fit 1 : (1405) pole position given by Dalitz ((1405) pole position given by Dalitz (Model DalitzModel Dalitz))

Fit 2 : Hemingway’s experiment (Fit 2 : Hemingway’s experiment (Model Model HemingwayHemingway))

KK--

pp

NPB253, NPB253, 742(1985)742(1985)

Invariant massInvariant mass

Dalitz and Deloff JPG17, 289 (1991)., Nacher et al., PLB461, 299 (1999).

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

with with assumption assumption

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

(I=0 (I=0 channel)channel)

(I=1 (I=1 channel)channel)

i

j

i

j

=XXijij

i j

XXijij

nn

+

Summary of our frameworkSummary of our framework

K, NK, NK, NK, N KN-KN-Y, NNY, NNKN-KN-Y, NNY, NN

WWpolepole = -B –i = -B –i/2 /2

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

Eigenvalue equation for Fredholm kernelEigenvalue equation for Fredholm kernel

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

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

Numerical resultsNumerical results

The pole position of three-body KNN The pole position of three-body KNN systemsystem

DalitzDalitzWWpolepole=-67-i22=-67-i22

HemingwayHemingwayWWpolepole=-47-i25=-47-i25

KNN physicalKNN physicalYN unphysical sheetYN unphysical sheet

DalitzDalitz(Approx.)(Approx.)

WWpolepole=-42-i35=-42-i35

HemingwayHemingway(Approx.)(Approx.)

WWpolepole=-32-i26=-32-i26

The reason of less binding energiesThe reason of less binding energies

Model Dalitz (W=-67 MeV)Model Dalitz (W=-67 MeV) Model Hemingway (W=-47 MeV)Model Hemingway (W=-47 MeV)

KN interaction dependences of KNN polesKN interaction dependences of KNN poles

Model DalitzModel Dalitz Model Model HemingwayHemingway

　　　　　　　　　　　　　 reaction

We compare variational approach with Faddeev approachWe compare variational approach with Faddeev approach

by using the approximate 2-body KN amplitude.by using the approximate 2-body KN amplitude. We find the different pole energiesWe find the different pole energies

corresponding to KNN state for each approach.corresponding to KNN state for each approach. KNN state becomes the bound state KNN state becomes the bound state

as increasing KN interaction.as increasing KN interaction.

SummarySummary

In the futureIn the future

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