Table of contentsI. Introduction

II. ModelIII. Results

Toshitaka UchinoTetsuo Hyodo, Makoto Oka

Tokyo Institute of Technology10 DEC 2010

I. Introduction The Λ* hypernuclei

model

We take the viewpoint that the KbarNN bound state is regarded as the bound state of the Λ*N (Λ* hypernuclei).

Kbar nuclei bound states attract much attention. From several

theoretical works, the Λ* N bound state is found to be a dominant component in the KbarNN bound state.

Bound state

Λ* and N bounds

Λ*-hypernuclei model has advantages ;• Including other two body, the ΛN and ΣN

contributions.• Few body calculation.

Arai-Oka-Yasui model[1] and our model

AOY model is constructed as follows :

• s-wave : dominant for the lowest energy state.• Λ*is regarded as a elementary particle.• Potential : extending the YN OBEP.• For S=0, S=1.• Variational method.

• Interaction : determined phenomenologically. • Purpose : fitting the results of FINUDA exp.

Our model is following AOY model, but ;

• Interaction : determined with the chiral unitary approach.

• Purpose : finding possible bound states.

Λ* hypernuclei model with chiral dynamics

Each Λ* interact with nucleon, whereas the transition between each Λ*N state can take place. Then, we solve

the coupled channel Schrödinger Eq.

The Λ* is dynamically generated as a superposition of two states.

The coupling constant of the Λ* to MB channels are taken from[2,3].

By using the chiral unitary approach, the Λ* is described as

meson-baryon multiple scattering.

II. ModelThe Λ*N OBEP

We construct the Λ*N potential by extending the Juelich(Model A) potential[4]. It is the simplest one-boson-

exchange potential which includes hyperon. Because isospin of the

Λ* =0, isoscalar meson is exchanged, namely σ, ω. We further

considerthe Kbar exchange.

Considering that the parity of the Λ* is odd, the Λ*KbarN

coupling is a scalar type. So the Kbar exchanged potential

is essentially same as the scalar exchange in the NN potential, but it depends on the total spin S.

Kaon exchange 1 – spin dependence

Exchange factor

Attractive for S=0Repulsive for S=1

Kaon exchange 2 - effective Kaon mass

In the Kaon propagator, since the energy transfer is not zero, we

use the effective Kaon mass . It becomes smaller as theresonance energy is close to the KbarN threshold. Namely,

in the upper energy state , Kaon exchange is stronger than the .

Coupling constants in our potential

Coupling constants in our potential are classified into three

types. Coupling constants determined by the chiral unitary

approach are complex value, so we take its absolute value.

: Chiral

: Juelich

: Unknown

The unknown coupling constant is estimated by using theΛ* structure from chiral unitary approach analysis.

Estimation of the Λ*Λ*N(X=σ, ω) coupling

By chiral dynamics, exchanged meson couples to the

constituent baryon or meson in the . So the coupling constants can be estimated by

summing up the microscopic contribution.

: Chiral: Juelich

ππσ is determined by σ decay:KKbarσ is assumed to be 0

Estimated coupling constants are complex. To obtain real

value, we take their absolute value.

We deal only dominant components, KbarN and πΣ.

potentia

ls

III. Results

Bulk property of the potential

To study the bulk property of the potential, we calculate the volume

integral of the potential.

This results show that• The potential is attractive(repulsive) for

S=0(S=1).• The potential is stronger than the

, because of the stronger coupling constants and the lighter effective Kaon mass.

With mixing

Bound states of the system

With no mixing

S=0: More bounds

S=1: No bound statesS=0: Only bounds

Wave function

We obtain the wave function of the bound state for

each .

Each state is peaking at ～ 0.5 fm. The state is

dominant, but the state is also important.

Decay width : B → πΣN

We consider the case that the in the bound state decays

with the nucleon being a spectator.

The coupling constant is given by the chiral unitary approach.

*As the strangeness S=-1, the baryon number B=2 Λ*-hypernuclei system, the Λ*N bound state is studied.

*The Λ*N one-boson-exchange potential is constructed by extending the Juelich potential.

*The unknown coupling constant is estimated by using the

information of the Λ* structure obtained from chiral unitary

approach.

*Solving the Schrödinger eq, we obtain the bound state solution

for S=0 ;

Summary

Backup slides

Decay width

If there exists the bound states, we can estimate the decay

width with obtained wave function.

Cut-off massThe coupling strength depends on the exchanged

meson momentum. This effect is taken into account as

monopole type form factor.

For vertices NNX(X=σ, ω), cut-off is given by Juelich potential.

But, cut-off masses concerning the Λ* is unknown. We take into

account the size of the Λ* and nucleon as parameter “c”.

The unknown cut-off can be written with “c”. Considering

the size of the Λ* [5], “c” is assumed to be 1.5.

c dependence

Binding energies and decay width depend on the size

of the Λ* , parameter c.

*Small “c “ leads to shallow bound.*πΣN decay is dominated by kinematics.

1. Other decay modes

2. Extension of our model

3. Few body calculations

Future plans

Other diagrams

B → ΛN B → ΣN

B → πΛN B → πΣN

Non-mesonic decay

Mesonic decay

Using obtained wave function, other decay width, Non-

mesonic decay, ΛN ,ΣN and mesonic decay πΛN, πΣN,

can be estimated.

Extension of our model

Including complexness

Energy dependence

: Chiral: Juelich: Estimated

The Λ* energy dependence in the Λ*Λ*X and Λ*KbarN vertices should be taken into account, when the Λ*N system bounds deeply.

Several parameters concerning the Λ* are complex value.

So, our model needs an extension.

To include the information of the Λ* given by chiral dynamics more directly, we need model

improvement.

Complex Λ*N potential

Few body calculation

Other channel contribution

Other two-body channels, the ΣN and ΛN contribution

can be included within our model.

Extension to few body studies, the Λ*NN and Λ*NNN can be calculated, using the Λ*N potential.