Transcript Multiquark states in the Inherent Nodal Structure Analysis

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 ( s )
•
Intrinsic space {color, flavor, spin} holds symmetry
SU C (3)  SU FS (6)
•
Coordinate space holds symmetry
Geometric symmetry
•
[2,1,1]  [ f ]FS
[ f ]C [ f ]FS  
[2,2,2]  [ f ]FS
[ f ]O
S6
INS
accessible
[ f ]O
Penta/Six - quark clusters must have symmetry
[14 ]
 [ f ]O [ f ]C [ f ]FS
6 
[1 ]
all the quantum Numbers
[ f ]FS

L , s, T , S , J 
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
n 1
n0

En  0 
 2 2
2ma
2
• Nodal Surface
En 1
4 2  2

2ma 2
n2

En  2

9 2  2

2ma 2
Dynamical Nodal Surface
Inherent Nodal Surface
• Inherent Nodal Surface
Ψ
eigenstate,
A
a geometric configuration,
A may be invariant to a specific operation Ô , i.e.,
Oˆ  ( A )   (Ô
)  (
A
)

Ô
(1)
The representation of the operation
on
A
is a matrix,
Eq.(1) appears as a set of homogeneous linear equations.
In some cases, there exists solution
(A) = 0
Inherent Nodal Surface (INS) which is imposed by the
inherent geometric configuration and independent
of dynamics exists. Then, the inherent nodeless states
•
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:
'
Oˆ1  P(1432) Rˆ k90

'
Oˆ 2  P(253) P(146) Rˆ OO
120
i'
Oˆ 3  P14 P23 P56 Rˆ180

Oˆ 4  P13P24 P56 Iˆ
(2)
(3)
(4)
(5)
Denoting the OCTA as A and the basis of the representation
of the rotation, space inversion and permutation as F
for the Oˆ (i  1,2,3,4) , we have
,
i
LSQ
i
(6)
i
i
Oˆ i F LSQ (A )  F LSQ (Oˆ iA
i
The Solution
F LSQ (
i
)  F LSQA(
)
A depends on the
We obtain then the INS accessible

and
,S
)
L
S
and
for each
.
L
,
and all the quantum numbers further.
Erot 
• Since
L( L  1)
r2
, S-wave nodeless state is the lowest
state in energy, then the P-wave nodeless one.
III. Application to Penta-quark System
1. Intrinsic States
Since [ f ]C  [2,1,1]  [ 1 ] , the orbital symmetry and the
flavor-spin symmetry has the following relation
The explicit quantum numbers and configurations
2. Accessibility of the spatial configurations
k'
ˆ
ˆ
O1  p12 p34 R180
i
ˆ
ˆ
O2  p12 R180 Pˆ
k' ˆ
Oˆ 3  p(1423) Rˆ90
P
n'
ˆ
ˆ
O4  p(234) R120
Oˆ1  p12 p34 Pˆ
k' ˆ
ˆ
ˆ
O2  R P
180
i'
Oˆ 3  p34 Rˆ180

k'
ˆ
ˆ
O4  p(1324) R90
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 [ f ]C  [2 2 2], the orbital symmetry and the
flavor-spin symmetry has the following relation
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 nodeless
accessible orbital symmetries as
for S-wave ( L  0  ) states,
[ f ]O  {} {{6}*,{4,2}*,{2,2,2}}
for P-wave ( L  1 ) states,
[ f ]O  {} {{5,1}*,{4,1,1},{3,3}*,{3,2,1},{2,2,1,1}}
The accessibilities of the states are listed in the following
tables.
The accessibility of the S-wave nodeless components
(continued)
The accessibility of P-wave nodeless components
(continued)
3. Possible low-lying S-wave dibaryon states
The 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)
Pauli principle, L+T+S=odd
4, 4, 4, 4
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 ) [ * ](1/ 2,0 )
?
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),
[]( 0, 2 )
(-2, 0, 4)
[ * *]( 0, 4 )
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
classified, the quantum numbers and the
configurations of the wave-functions are obtained.
•
The [](0,0 ) , [](1/ 2,1 ) , [ * ](1/ 2,0 ) , and [](0, 2 ) , [ * *](0, 4 ) 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.
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Thanks !!!