Multiquark States in the Inherent Nodal Structure Analysis Approach

Yu-xin Liu

Department of Physics, Peking University, Beijing 100871

Outline

I. Introduction

II. The INS Analysis Approach

III. Application to Penta-quark System

IV. Application to Six-quark System

V. Remarks

References: P R L 82 (1999) 61; P L B 544 (2002) 280; P R C 67(2003) 055207 (nucl-th/0212069)

hep-ph/0401197

I. Introduction Multi-quark systems are Appropriate to investigate the quark behavior in short distance to explore exotic states of QCD

Many six-quark cluster states e.g., H, d’, d*, (ΩΩ), (ΩΞ), (ΩΞ*), … … have been predicted in many QCD approaches: Lattice QCD (e.g., Nucl. Phys. B-Proc. Sup. 73 (1999) 255) QCD Sum Rules (e.g., Nucl. Phys. A 580 (1994) 445) Bag Model (e.g., Phys. Rev. Lett. 38 (1978) 195, Sov. J. Nucl. Phys. 45 (1987) 445) Quark Delocalization and Color Screening Model (e.g., Phys. Rev. Lett. 69 (1992) 776) SU(3) Chiral Quark Model (e.g., Phys. Rev. C61 (2000) 065204)

No dibaryons have been observed in experiment after more than 25 years efforts.

An exciting point: It was claimed that Penta-quark state + was observed in LEPS, DIANA, CLAS, SAPHIR, HERMES, ZEUS, …

Many theoretical investigations have been accomplished in Chiral soliton model (ZPA 359(1997) 305)

Diquark-antiquark model (e.g., PRL 91(’03) 232003, etc. ), Skyrme model (PLB 575 (2003) 234) Diquark-triquark cluster model (PLB 575 (2003) 249) Chiral Q model (PLB 575(‘03)18, PLB 577(‘03) 242, hep-ph/0310040)

QCD Sum Rules (e.g., PRL 91 (2003) 2320020, etc.)

Large Nc QCD (e.g., hep-ph/0309150 )

Lattice QCD (e.g., hep-lat/0309090, hep-lat/0310014 ), …… ……

The parity has not been fixed commonly (model dependent). The narrow width has not been reproduced.

To neglect handling the complicated interactions in QCD, ……, we propose a model independent approach ---- INS Analysis.

II. The INS Analyzing Approach 1. General Point of View• Penta/Six - quark clusters involve flavors u, d and ( s )• Intrinsic space {color, flavor, spin} holds symmetry

• Coordinate space holds symmetry Geometric symmetry INS accessible• Penta/Six - quark clusters must have symmetry

all the quantum Numbers

)6()3( FSC SUSU

FS

FSFSC f

fff

][]2,2,2[

][]1,1,2[][][

6S Of ][

FSCO fff ][][][]1[

]1[6

4

Of ][

FSf ][

JSTsL ,,,,

s

2. Inherent Nodal Structure Analysis• Starting Point: The less nodal surfaces the wavefunction

contains, the lower energy the state has,

e.g., infinite square well

0n 1n 2n

2

22

1 2

4

maEn

2

22

0 2maEn

2

22

2 2

9

maEn

Dynamical Nodal SurfaceInherent Nodal Surface

• Nodal Surface

• Inherent Nodal Surface

Ψ eigenstate, a geometric configuration,

may be invariant to a specific operation , i.e., (1) The representation of the operation on is a matrix, Eq.(1) appears as a set of homogeneous linear equations.

In some cases, there exists solution () = 0

Inherent Nodal Surface (INS) which is imposed by the inherent geometric configuration and independent of dynamics exists. Then, the inherent nodeless states are accessible to the geometric configuration.

O() (ˆO () )

O

O

• An Example of Six-body System A 6-body system has several regular geometric shapes, for example: the regular octahedron (OCTA) the regular pentagon pyramid (PENTA) the triangular pyramid the regular hexagon

For OCTA , it is invariant to the operations:

(2) (3) (4) (5)

'

901ˆ)1432(ˆ kRPO

'

1202ˆ)146()253(ˆ OORPPO

'

1805623143ˆˆ iRPPPO

IPPPO ˆˆ5624134

Denoting the OCTA as and the basis of the representation of the rotation, space inversion and permutation as , for the , we have

(6)

The Solution depends on the , and . We obtain then the INS accessible and for each , and all the quantum numbers further.

• Since , S-wave nodeless state is the lowest

state in energy, then the P-wave nodeless one.

Fi

LSQ

)4,3,2,1(ˆ iOi

(ˆ Fi

LSQiO

i

i

LSQOF ˆ()

() Fi

LSQ

)

(Fi

LSQ

) L S

S L

2

)1(

r

LLErot

III. Application to Penta-quark System

1. Intrinsic States

Since , the orbital symmetry and the flavor-spin symmetry has the following relation

]1[]1,1,2[][ Cf

The explicit quantum numbers and configurations

2. Accessibility of the spatial configurations

'

18034121ˆˆ kRppO PRpO i ˆˆˆ

180122

PRpO k ˆˆ)1423(ˆ '

903 '

1204ˆ)234(ˆ nRpO

PppO ˆˆ34121 PRO k ˆˆˆ '

1802 '

180343ˆˆ iRpO

'

904ˆ)1324(ˆ kRpO

The accessibility of the ETH and square configurations to the (L) wave-functions

3. Possible low-lying penta-quark states

Consistent with the results in chiral soliton model,

general framework of QCD, Chiral quark model,

diquark-triquark cluster model, …… ……

IV. Application to Six-quark System

1. Intrinsic States

Since [2 2 2], the orbital symmetry and the

flavor-spin symmetry has the following relation

Cf ][

The strangeness, isospin and spin of the states listed above and the baryon-baryon and hidden color channel correspondence

2.Accessible Orbital Symmetries

Solving the sets of linear equations in Eq.(6) at geometric configurations OCTA and C-PENTA, we obtain the nodele

ss accessible orbital symmetries as

for S-wave ( ) states,

for P-wave ( ) states,

The accessibilities of the states are listed in the following tables.

0L

1L

}}2,2,2{}*,2,4{}*,6{{}{][ Of

}}1,1,2,2{},1,2,3{}*,3,3{},1,1,4{}*,1,5{{}{][ Of

The accessibility of the S-wave nodeless components

(continued)

The accessibility of P-wave nodeless components

(continued)

3. Possible low-lying S-wave dibaryon statesThe configuration with large nodeless accessibility:s=-6, (T, S)=(0, 0) 3

s=-5, (T, S)=(1/2, 1), (1/2, 0) 4, 3

s=-4, (T, S)=(0, 1), (1, 0), (1, 1), (1, 2) 8, 7, 5, 6

s=-1, -2, -3, many configurations

s=0, (T, S)=(0, 1), (1, 0), (1, 2), (2, 1) 4, 4, 4, 4Pauli principle, L+T+S=odd decay to two free baryons

low-lying stable S-wave dibaryons: (s, T, S)=(-6, 0, 0) , (-5, ½, 1), (-5, ½, 0), (-4, 1, 1)

?)0,0(][

)1,2/1(][

)0,2/1(]*[

4. Possible low-lying P-wave dibaryon states

P-wave resonance may have narrow width, but higher energy

P-wave accessible, but S-wave inaccessible configurations

being taken as P-wave dibaryon states

(s, T, S)=(-6, 0, 1), (-4, 0, 0), (-2, 0, 3), (0, 0, 0),

(0, 0, 2),(0,2,0), (0,1,3), (0,3,1), (0,3,3)

Pauli Principle being taken into account, Possible ones are

(s, T, S)=(-6, 0, 1), (-2, 0, 3) Spin-orbital interaction : high J states may have low energy

Possible low-lying stable P-wave dibaryons are

(s, T, J) = (-6, 0, 2), (-2, 0, 4))2,0(

][ )4,0(

*]*[

5. Comparison with other theoretical studies and experimental results

The candidates are consistent with the results in

Quark-Delocalization and Color-Screening Model (QDCSM) and

Chiral SU(3) quark model d* is possible, since the accessibility for (s, T, S)=(0, 0, 3) is 3, if its energy is very low.

Consistent with QDCSM result. d’ is impossible, since the accessibility for L=1, (s, T, S)=(0, 0, 1) is only 1. Inconsistent with Bag model and Chiral quark model, but consistent with p-p collision results (PLB 550 (2002) 147, EPJA 18 (2003) 171, 297)

V. Remarks• The inherent nodal structure analysis approach for few-

body system is proposed• The wave-functions of penta/six-quark systems are class

ified, the quantum numbers and the configurations of the wave-functions are obtained.

• The , , , and , and the hidden-color channel states are proposed to be dibaryon states, which may be observed in exp.

• The d* is also a possible dibaryon, but the d’ is not. • The parity of the + is proposed to be positive. • The INS analysis approach is independent of dynamics. To obtain numerical result both the INS analysis and the

dynamical calculation are required.

)0,0(][

)1,2/1(][

)0,2/1(]*[

)2,0(][ )4,0(

*]*[

Thanks !!!