Nov. 12, HAPHY.

A QCD sum rule analysis of the

PLB 594 (2004) 87, PLB 610 (2005) 50, and hep-ph/0506250

Hee-Jung LeeVicente Vento

(APCTP & U. Valencia)Nikolai Kochelev (JINR & SNU)

Contents

• Introduction• Interactions induced by Instantons• Diquark-Triquark model for Pentaquark with

in Instanton model for the QCD vac.• QCD sum rule : Triquark correlations and • Discussion

• QCD : The fundamental theory for the strong interaction - Degrees of freedom : quark and gluon • Hadrons = Baryons+Mesons Baryons = 3 quarks Mesons = quark+antiquark • Multiquark hadrons are possible? YES!? (S=+1 Exotic, etc) • Role of Instanton in Multiquark systems?

Quark Model

• T. Nakano et al : PRL 91(2003) 012002 Observed Pentaquark in n →K+K-n Mass=1540MeV, Width < 25MeV

MnK+

LEPS

1540 10MeV

25MeV

M

(LEPS)

nK

0SpK

F. Close and Q. Zhao, hep-ph/0404075

M

Status of

• ... 1997 : Prediction by Diakonov et al in the chiral soliton model, mass about 1530MeV and width < 15MeV.

• 2003 : Observation by LEPS. Following it, many (+) results were reported : Jefferson Lab, Russia, SAPHIR… (from reanalyzing data obtained in 1997-98)

• 2004-2005 : theoretical criticism and (-) results have been emerged.

- Null results : most from experiments with higher energy and large statistics using nuclei, hadrons rather than photons

Production depends on production mechanism strongly(?)

-Jeffeson Lab this year : hep-ex/0510061 : Upper limit for production cross section of < 0.2% (c.f. 40% in LEPS) Summary : hep-ph/0510309 by Burkert

F. Close, Nature,Vol435,19 May 2005

0p K K n

*/ (1520)

from soliton model

• At the top of Antidecouplet :

• Mass : 1540 MeV Width < 30MeV• Parity = +1(?)

ududs

• Simple constituent quark model based on independently moving quarks yields mass of pentaquark around 2 GeV

• Correlations of quarks inside of (?)

Quark model

Two models for pentaquark1. Jaffe and Wilczek, PRL91(2

003)232003• Potential by one gluon excha

nge• Two scalar diquarks (L=1) + a strange antiquark • Parity = +1 • Recent calculation gives

higher mass than observed. (PRL94 (2005) 062001)

2. Karliner and Lipkin, PLB575 (2003) 249

• Triquark + diquark + orbital motion (L=1)

• Interaction by one gluon exchange

• Triquark = diquark =

3 ,6 ,2fC S

,3 ,3 1C f S

• Mass=1385 (quark masses + OGE) + 207 (orbital motion L=1) = 1592 MeV• Parity = +1• Small width : Potential barrier by the orbital motion

• ?? :- Is there another important source for quark correl

ations in hadrons? instanton?- Only one state in 3q cluster in KL’s?

Instanton : Strong fluctuation of gluonic field in QCD v

acuum

Interactions induced by Instanton• Two body + Three body interactions

Some Properties

• Acts on only quarks with different flavors : flavor antisymmetric wave functions (Isospin=0)

• Minimal spin Maximal strength• Three body interaction by Instanton in color

less 3-quarks state vanishes• For interactions with anti-quark:

Our Model

• Interaction by one gluon exchange + Interaction by Instanton• Instanton supplies the much smaller stron

g interaction coupling (~0.5)• For pentaquark : with different flavors Diquark + Triquark : separated enough

Mass Formula for Hadrons

Confinement Masses, mu=md=m0

Diquark + Triquark System

• Diquark is in isospin = 0, spin = 0 : due to instanton • (ud) quarks in Triquark is in isospin = 0 : d

ue to instanton • Triquark is in spin = 1/2 • There are two possibilities in Triquark • Relative orbital motion : L=1

( )ud ( )uds

State of each cluster Diquark :• Flavor space :

• Color space :

• Spin space :

Triquark : - A state • Flavor space :

• Color space :

• Spin space :

3 3 6 3

3 3 6 3

- B state • Flavor space :

• Color space :

• Spin space :

3 3 6 3

6 3 15 3

After diagonalization

Masses of clusters from parameters

• Mass of Pentaquark : - With the energy from orbital motion

• Orbital motion gives Small width• Positive Parity

11 (1420) Me400 VL fM M M

1595 MeVM

3 2 280MeV / 4q qredM M

Short summary I• Instanton Induced Interaction plays a impor

tant role in pentaquark.• By the way, are there really two clusters of

quarks in pentaquark?

• Triquark correlations in a QCD sum rule• Diquark correlation by Schaefer et al NPB 412(1994) 143

QCD sum rules• The Correlator of the interpolating current

with the quantum number of particle (state) under consideration

• Calculating it in deeply Euclidean region by the perturbative OPE

Condensates from the

nonperturbative vacuum

2 4( ) 0 | ( ) (0) | 0iq xHHq i d x e T x

OPE 2( ) :q

Nonperturbative Vacuum

( )H x

• is related to physical region by the dispersion relation

with a narrow resonance approx.

for spin ½ particle case

OPE 2( )q

2OPE 2 2

2 20

1 Im ( )( )

sq ds

s q

2 2 2 2

2 2 2 20 1 2

ˆIm ( ) ( ) ( )

ˆ( )( Im ( ) Im ( ))

H H Hs q M s M

s s q s s

0 | |H H HH u mass

threshold

2 2 2Im ( ) ( ) 0 | (0) | | (0) | 0n H Hn

p p M n n

• With the Borel transformation, the higher state contributions is exponentially suppressed.

• Two QCD sum rules :

• Quark-Hadron duality :

2 22 2

20

2 22 2

20

/OPE 2 2 / 2 211 1

/OPE 2 2 / 2 212 2

( ) Im ( )

( ) Im ( )

H

H

M Ms MHs

M Ms MH Hs

M ds e s e

M ds e s M e

2Im ( )i M

Borel Mass

21( )M

22 ( )M

• Mass of Particle can be determined in the three ways

• In general, if we include all contribution from OPE, the above ways must give same mass independent on .

• Actually, we cannot do it. Up to a certain energy dimension operator,

in some region of

2 1

3

/

( / 2 )

H

M i i

M

M

M

32 2 ( / 2 )H MM M M

• Parity of particle : - With the state of opposite

parity to the current is

-The interpolating current can create a state of opposite parity to the intrinsic parity of the current :

5i

0 | | H HH u

5 50 | | 0 | | H Hii uH H

Left hand side on the 2nd sum rule is + : the same parity as the current Left hand side on the 2nd sum rule is - : opposite parity to the current

2 2 2

22 2 2 2 2

Im ( ) ( ) 0 | (0) | | (0) | 0

( )( ) ( )( )

+higher

n H Hn

H H H H H H

p p M n n

q M q M q M q M

2 2/22 ( ) HM M

H HM M e

• The 2nd sum rule with massless u,d :

• OPE up to d=5 for nucleon & triquarks :

6 4 21 3 5

97 2

OO

OM O M O MM

• Quark Propagator :

By the Wick’s theorem : perturbative vacuum

[ ( )] 0 | ( ) (0) | 0ab

q a bS x i Tq x q

[ ( )] 0 | ( ) (0) | 0

0 |: ( ) (0) :| 0

ab

q a b

a b

S x i Tq x q

i q x q

Perturbative propagator

color spin

Nonperturbative effect : vacuum condensates

flavor

Nucleon and triquark states

• Interpolating current for nucleon of positive parity :

• Energy dimension of correlator for nucleon and color triquark states after the Borel transform : 7

• T-product of currents :

5

1 5

( ) [ ( ) ( )] ( )

( ) [ ( ) ( )] ( )

N abc Ta b c

N abc Ta b c

x u x C d x u x

x u x Cd x u x

1( ) ( ) ( )tot x f x t x

• QCD sum rule without direct instantons:

=+

• Instanton effect : zero mode of quark

*2 2 2Di : 2 / 3 /(2lute Instanton )

Shur

liquid mo

yak, NPB 203(1982) 930

d

el eff q cn m Size of Instan 1/ mton 3 f

• Mass and Residue : With threshold 1.75 GeV and f=1, t=-1

Without Instanton

A state in Triquark

• Interpolating currents of (-) parity :

• QCD sum rule without direct instantons:

1( ) ( ) ( )tot x f x t x

• Interpolating current of (–) parity :

• QCD sum rule without direct instantons

B state in Triquark

Instanton effect

• Two body effects :

• Three body effect : no in B

Masses and residues

• For A state : threshold 1.8 GeV, f~-t

800 , MeV Parity 1AM

• For B state : threshold 1.72 GeV

MeV, Parit930 1yBM

Short summary II• Schaefer et al showed (ud)-diquark correlat

ion with mass around 420MeV.• Are there really three quark correlations in

pentaquark? Two quasi-bound color quark correlations are there.

• These estimation gives the similar value of mass to in our MODEL.

• With a current of 5 quarks ?

uds

QCD sum rule in 5-quarks

• Two color clusters in the model : - Diquark and Triquark clusters• Diquark cluster : Spin(S)=0, Isospin(I)=0, Color(C)= • Triquark cluster : S=1/2, I=0, C=3• Two states in triquark cluster (ud) diquark : in A, S=1/2, I=0, C= in B, S=1, I=0, C=6

3

3

Interpolating currents• Current for 5-quarks without derivative=(current for triquark)*(current for diquark)• With A state : Two currents

1 5 5

5 5

1 1[ ] [ ]

4 21

[ ][ ]4 2

T T TA abc bde d e c afg f g

T T Tabc bde afg d e f g c

u C d Cs u C d

u C d u C d Cs

5 5

5 5

1 1[ ] [ ]

4 21

[ ][ ]4 2

T T TA abc bde d e c afg f g

T T Tabc bde afg d e f g c

u Cd Cs u C d

u Cd u C d Cs

0!

5 5

1 1[ ] [ ]

4 3 2T T T T

B a b b a b afg f gu C d u C d Cs u C d

• With B state :

• Energy dimension of correlator for the 5-quark after the Borel transform : 13

• The 2nd sum rule with massless u,d :

12 1 1501 3 13 2

OOM O M O

M

Sum rules by other authors

1. Shi-Lin Zhu, PRL91 (2003) 232002 : - Current : - Sum rule : 1st sum rule up to d=12 - Mass ~1.56 GeV, Parity = ? 2. R. D. Matheus et al, PLB578(2004) 323 - Two currents : not proper for - Sum rule : 1st sum rule up to d=12 - Mass~1.54, Parity = ?

5 5

1[ ]{ [ ] ( )}

2T

abc a b e e cu C d u s i d u d

3. M. Eidelmuller, PLB 597 (2004) 314 - Current : P-wave current - Sum rule : 1st sum rule up to d=6 - Mass~1.64, Parity = ?4. J. Sugiyama et al, PLB 581 (2004) 167 - Current : - Sum rule : Both sum rules up to d=6 - Mass~1.63, Parity = -1 - Current is very similar to our A state

5[ ][ ] T T Tabc def cfg a b d e gu Cd u C d Cs

2nd sum rule for A up to d=13 :diagrammatic representation

Parameters in Matheus et al, hep-ph/0412063

LHS

+ or - ?

Left hand side (LHS) of 2nd sum rule :

1512 10 8 61 3 5 7 1 23OM O M O M O M

OO

M

0 2.0GeVs

Sign of LHS is

changed with

going from d=5 to d=7 : Parity of state is

changed.

Contribution from Instanton

• Differently from model calculation, there is instanton effect in two quark pairs of

only.• No instanton effect in thre

e quarks by the color-spin structure of the current!!!

• Instanton effect is small in

, us ds

1.0 2.0M

Mass fitting with Instanton

32 2 ( / 2 )H MM M

Final summary

• For parity : higher operators are important since correlator of Pentaquark has higher energy dimension.

• Instanton effect is unexpectively small• Mass is around 1.75MeV• + parity : one quark is in excited state? Tw

o clusters in L=1?• B state in the triquark cluster?