Chiral SU(3) Quack Model and

Multiquark State

Zhang Zong ye(Institute of High Energy Physics, Beijing)

Outline

• Introduction

• Chiral SU(3) Quark Model

and NN , YN and KN scatterings

• Dibaryon

• Pentaquark state

Introduction The NPQCD effect is very important for the light quark system. But up to now, there is no effective approach to solve the NPQCD problem seriously. We still need QCD inspired model to help.

In the framework of the constituent quark model, to understand the source of the constituent quark mass, the spontaneous vacuum breaking has to be considered, and as a consequence, the coupling between quark field and goldstone boson is introduced to restore the chiral symmetry. The chiral quark model can be regarded as a quite reasonable and useful model to describe the medium range NPQCD effect.

First we generalized the chiral SU(2) quark model to chiral SU(3) quark model for studying the system with s quark and try tosee if we can get a unified explanation of N-N and Y-N data on quark level. Then we extend the study to the multiquark system.. Since for the multiquark states, quarks should stay in a small area, the quark exchange effect must be important. We did an analysis to see the quark exchange effect in two baryon systems. We found that: the quark exchange effect of some cases is not important; some cases have very strong Pauli Block Effect, and for some cases it makes these two baryon cluster close together--- it is favorable to make two baryon to be bound. According to the analysis, using chiral SU(3) quark model with thesame parameters we used in the scattering calculations, we got : A new interesting dibaryon candidate - - - di-Omega .

We also studied the structure of the pentaquark state .

+0(ΩΩ)

+Θ

• The ModelIn the chiral SU(3) quark model, the coupling between chiral field

and quark is introduced to describe low momentum medium range

NPQCD effect. The interacting Lagrangian can be written as:

.

scalar nonet fields pseudo-scalar nonet fields

It is easy to prove that is invariant under the infinitesimal chiral

transformation. This can be regarded as an extension of the SU(2) - σ model for

studying the system with s quark.

8 8

I ch a a a a 5a=0 a=0

L = -g ψ( σ λ + i π λ γ )ψ

σ,σ', χ,ε π,K, η, η'

IL L RSU(3) SU(3)

IL

In chiral SU(3) quark model, we still employ an effective OGE

interaction to govern the short range behavior, and a confinement

potential to provide the NPQCD effect in the long distance.

Hamiltonian of the system:

( is taken as quadratic for

m.)

i G iji i<j

H = t - T + V ,

,conf ogeij ij i ij

chjV = V + V V+

ch s(a) ps(a)ij ij ij

a

V = (V + V ) .

confijV

The expressions of and :

2( )

, ( ), 2 ( ),( ) ( , )12

( ) ( ) ( ) + tensor term

ps ach ps a ps a ij

qi qj

i j a a

mC g m X m r

m m

i j

, ( ), 1 ( ),( ) ( , ) ( ) ( )

term,

ch ps a s a ij a aC g m X m r i j

l s

2

, 2 2( , ) .chC g m m

m

1( , , ) ( ) ( ),X m r Y mr Y rm

32( , , ) ( ) ( ) ( ),X m r Y mr Y r

m

1( ) ,xY x e

x

psijVs

ijV

Here we have only one coupling constant ,chg

s(a)ijV

2chg

4π

ps(a)ijV

2chg

4π

2 2 2ch u NNπ

2N

g m g9= .

4π 25 4πM

• Parameters:(1). Input part: taken to be the usual values.

(2). Chiral field part: and are adjustable.

( are taken to be experimental values, )

(3). OGE and confinement part: and are fixed by and .

are determined by the stability condition of

ub = 0.5fm, um = 313MeV, sm = 470MeV.

2 2 2ch u NNπ

2N

g m g9= ,

4π 25 M 4π

', , , Km m m m

ug

σm s

' 980 .m m m MeV

sg Δ NM - M Σ ΛM - M

cuu,a ... N, Λ,Ξ.

N-N,Y-N K-N

0.5

313

470

2.63

595 675

0 35.3 (-18)

0.886

0.755

48.1 52.4 (55.2)

63.7 75.3 (71.4)

ub (fm)

um (MeV)

sm (MeV)

chg

σm (MeV)

s (degree)

ug

sg2

cuua (MeV/fm )2

cusa (MeV/fm )

Baryon Mass

Expt. Theor. Expt. Theor.

N 939 939 Δ 1232 1237

Λ 1116 1116 1385 1375

Σ 1193 1194 1530 1515

Ξ 1319 1334 1672 1657

*Σ*Ξ

Ω

To study the two baryon system, we did a two-cluster dynamical RGM calculation

Phase shifts of N-N scattering

10S 3

1S

N-N P wave

N-N D wave

Y-N cross sections of ± ±Λp Λp, Σ p Σ p

Y-N cross sections of - 0 -Σ p Σ n, Σ p Λn

Phase shifts of K-N scattering

S wave

0 200 400 600 800 1000

-100

-80

-60

-40

-20

0

20

40 S11

(de

g)

Plab

(MeV)

0 200 400 600 800 1000-40

-30

-20

-10

0

10 S01

(de

g)

Plab

(MeV)

K-N P wave

K-N D wave

K-N F wave

•DibaryonSince our model can explain the scattering data quite well, using the same groups of parameters to study some two baryon states and multi-quark states is significant.

An analysis of quark exchange effectSince quarks are fermions, when the distance between two baryon clusters

is short enough, the quark exchange effect must be important.

The antisymmetrization operator:

is the permutation operator of quark i and j, and of

baryon A and B. When two cluster is closed together and L=0,

Thus is very important to measure the quark exchange effect for various spin-flavor states.

r σfcij ij ijP = P P ABP

.rij< P > 1

σfcij< P >

)P)(1P(1A ABBjA,i

ij

When the quark exchange effect is not important,

the Pauli Block Effect is very serious,

the quark exchange effect makes two baryon cluster closer. It has so-called quantum coherent effect.

In all of two baryon systems, only 6 of them belong to this interesting case, they are:

Only has enough long lifetime, because it can’t decay through strong interactions.

ST=03(ΔΔ) , ST=30(ΔΔ) , ST=00(ΩΩ)*1

ST=0 ,2

(Ξ Ω)*5

ST=0 ,2

(Σ Δ) *1

ST=3 ,2

(Σ Δ)

σfcij

i A,j B

(1- < P >) 1,

σfcij

i A,j B

(1- < P >) 0,

σfcij

i A,j B

(1- < P >) 2,

ST=00(ΩΩ)

H partical ~ 2 (near 2Λ threshold)

State B(MeV)

1ST=0

2

1ST=1

2

3ST=0

2

(NΛ) 1 unbound

(NΛ) 1 unbound

10(NΣ

10Deuteron

)

9

2.139

1ST=1

2

ST=00

unbound

(NΣ) 1 unbound

(ΛΛ) 1 unbound

σfcij

ij

(1- <P >)

( )N The results are calculated by using chiral SU(3) quark model with the same parameters we used in the NN and YN scattering processes.

State B(MeV)

2 137 2 92 2 25 2 26 2 16 2 37 is the most interesting one, because it

can’t decay through strong interactions , and thus it has enough long lifetime.

ST=00(ΩΩ)*

1ST=0

2

(Ξ Ω)*

5ST=0

2

(Σ Δ)*

1ST=3

2

(Σ Δ)

ST=03(ΔΔ)

ST=30(ΔΔ)

σfcij

ij

(1- <P >)

ST=00(ΩΩ)

Main properties of :1). Binding energy several tens to hundred MeV,2). Distance between two , 0.8fm,3). It carries –2 charges,4). Mean lifetime sec. Decay modes:

All of them are weak decays.

+

- - 0

0(ΩΩ) Ω + Ξ + π ,

+

- 0 -

0(ΩΩ) Ω + Ξ + π ,

+

- -

0(ΩΩ) Ω + Ξ .

-1010

+0(ΩΩ)

Ω Ω-Ωd

ΩΩτ

+0

(ΩΩ)B

+

- - 0

0(ΩΩ) Ω + Ξ + π ,

How the result dependent on the parameters is.

Fit NN Fit KN

I II

137.4 61.2 134.4

Even the mixing of and is taken to be ideally mixing, i.e. there is no

meson exchange between two s quarks, the binding energy of is

still quite large , around 60 MeV. This result tells us again that the symmetry

property of is really very important to make it bound.

+0(ΩΩ)B (MeV)

0( 0 )s 0( 35.3 )s 0( 18.0 )s

0 8 00

( )

0( )

is a very interesting new dibaryon candidate,

but how to search it seems not so easy, because its

production rate is rare.

Some authors estimated its production rate in the relativistic heavy ion

collision processes at RHIC energy by using different models.

Their results show: the ratio of the production rate of

and single ,

Though the searching work is hard, RHIC-STAR group still plans to try to

search it in the heavy ion collision processes and has listed this work in

their research proposal.

+0(ΩΩ)

+0(ΩΩ) /Ω

+0(ΩΩ)

Ω

• Pentaquark state

4(0 ) (0 ), 0, S S L -

π 1J =

23(0 ) (0 )(0 ), 1, S P S L

+π 1

J =2

1[4] [31] 01, 0

21

10, 121

11, 121

21, 22

forb ts TS

ts TS

ts TS

ts TS

1[31] [4] 00, 0

21

11, 123

11, 123

22, 22

forb ts TS

ts TS

ts TS

ts TS

The structure of pentaquark state is studied by chiral SU(3) quark model.

4 configurations of and 4 of are considered. They are:

-π 1

J =2

+π 1

J =2

The trail wave function is taken as an expansion of the

5q states with different size b :

and solve to get

Results tell us that:

5q i 5q ii

Ψ = α Φ (b ),

5q 5q 5qHΨ = E Ψ 5qE .

+Θ

-π 1

J = and T = 0 state is always the lowest one 2

for 3 groups of parameter but its energy is about

250 - 350 MeV higher than the experimental

M

s,

.

Fit NN Fit KN

I II-

π 1J = T = 0 1799 1901 1897

2+

π

1J = T = 0 2153 2221 2

2

206

ΘM (MeV) (calculated)

It seems that in the framework of the chiral SU(3) quark model, when the parameters are taken in the reasonable region, it is difficult to get the mass of Θ to be closed to the experimental value (1540 MeV).

Summary:

Taking the same

Chiral SU(3) qu

parameters we

ark model is quite successful

used in the scattering calcula

in explaining

the N

tion,

a new interes

N, YN and K

ting dibary

N

o

s

n

cattering data

candida

,

te (ΩΩ) +0

-10

is predicted, it is

deeply bound with quite small size, its mean lifetime is about

10 sec and c

The calculated mass of the pentaquark state Θ in the chiral

SU(

arries - 2 char

3) quark model

ges

.

is about 250 - 350 MeV higher than the

experimental value.