A QCD sum rule study of f0(980)

Hee-Jung Lee (Chungbuk Nat’l Univ.)

Collaborators :

N. I. Kochelev (JINR)

Y. Oh (Kyungpook Nat’l Univ.)

Correlator of the interpolating current with the quantum number of the hadron under consideration

Calculating it in deeply Euclidean region by the perturbative OPE

Condensates from the

nonperturbative vacuum

QCD sum rule (SR)

2 4 ( ) 0 | ( ) (0) | 0iq x

S S Sq i d x e TJ x J

2( ) :OPE

S q

Nonperturbative QCD Vacuum

SJ

is related to physical region by the dispersion relation

Narrow resonance approx. in the phen. side

OPE 2( )q

22 2

2 20

Im ( )1( )OPE S

S

sq ds

s q

2 2 8 2 2 2 2 2

0Im ( ) 2 ( ) ( ) Im ( )OPE

S S S S Ss f M s M s s s

40 | | 2S S SJ S f M

threshold

2 2 2 Im ( ) ( ) 0 | (0) | | (0) | 0S n S S

n

q q m J n n J

Quark-hadron duality

Borel transform makes the contributions from the continuum suppressed exponentially.

QCD sum rules :

: Borel Mass

22 22 20 /2 / 2 2 81

0Im ( ) 2 S

sM Ms M OPE

S S Sds e s f M e

M

2( ) :S M Must be POSITIVE

Mass of Particle can be determined by

Generally, including all contributions from OPE, the mass must be independent on the Borel mass.

Actually, we cannot do it. Up to a certain energy dimension operators, mass plateau appears in some region of the Borel mass.

Borel window

Borel window must be opened in .

3( / 2 )S M S SM M

0M s

Light scalar meson nonet

Members :

Large decay widths :

Refs. : PDG, Phys. Rep. 389(2004) 61, 397(2004)257

0

0 0

0 0

0

1 : , ( )

1/ 2 : , , (8

980

00)

9800 : (600), ( )

I a a

I

I f

0 050 ~ 100MeV, 40 ~ 100MeV

600 ~ 1000MeV

a f

interpretation

With ideal mixing : for P=+1

(?1)Decays of : fraction of ?

0

0 0

0 0

0

0

1, ( ),

2

(800) , , ,

1(600) ( ),

(980)

(980) 2

a uu dd a du

us ds sd su

uu

a u

d

d

f ssd

1L

0a

0

0

[ (980) ]0.85 0.02

[ (980) ]KK

a

a

ss

qq

Amsler et al, Phys. Rep. 384(2004)61

(?2) Mass degeneracy in

1. From number of strange quarks

2. gives 400MeV more mass :

from the mass formula in a quark model

(Kochelev, H.-J. Lee, Vento, PLB 594 (2004) 87),

for example :

0 0, a f

0 0, f am mm m

1L

0 12

214 2 407 2 0 400 1425MeV

f conf s OGE I LM E m E E E

0 (980)f

interpretation

One gluon exchange & instanton :

strongest attraction in two quarks of : scalar (S) diquark

in two antiquarks of : S antidiquark

-Jaffe & Wilczek, Shuryak & Zahed

In flavor space :

Explicitly

[ ][ ]qq qq

3 3 63 3, 3 3 6

3 3 1 8

f f S f f S

A

A

A

A

[ ] , [ ] , [ ]

[ ] , [ ] , [ ]

A A A

A A A

ud s us d ds u

ud s us d ds u

| 3 , 3 ,1F C S

| 3 ,3 ,1F C S

In terms of S diquark & S antidiquark : L=0

Number of strange quark :

Strange quark component in :

0

0

0 0 0

0 0

(980) [ ][ ], , [ ][ ]

(800) [ ][ ], [ ][ ], [ ][ ], [ ][ ]

1([ ][ ] [ ][ ])

2

1 (980) ([ ][ ] [ ][ ](600) [ ][ ], )

2

a ds us a a us ds

ds ud us ud ud us u

ds ds us us

f ds ds u

d ds

ud u sd us

0 0f a m mm m

0 0, f a

0 0, f a KK

: Inverted mass spectrum

SRs for light scalar nonet

Interpolating currents : energy dim.=6

After Borel transform :

Energy dimension of the correlator =10

Some details for sigma :

Vacuum expectation value of currents :

Quark propagator :

1st :

2nd :

3rd :

4th :

5th :

QCD SR for sigma :

QCD SR for the nonet : ,

Continuum contributions :

Physical Borel region is around 1GeV :

operators up to dim=8 would be important

2 2

10 6 4

0 4,0 4 4, 2 6, 1

/2 2 8

8, 0 10, 10,

6,

2

8, 12,

( )

/ 2 S

S S S

S i i

M MS S S

i i i i S S

i i

i

M C E C E C E

C E C O C M

O M O M O M

O M f M e

0u dm m ( )sO m

220 /

2 2 0

1( )

( 1)

sx M n

n nE M dx e x

n M

LHS of SRs with scalar diquark

Values of condensates and mass :

0 0, f a

What we have seen…

SRs up to d=6 ops. :

- Multiquark system has large energy dim. :

SRs up to d=6 ops. could not be enough.

Large negative contribution from d=8 ops. :

destroys physical meaning of SR.

Effect from Instanton?

Any possibility to kill large contributions from higher ops. ?

Generally, five types of relativistic currents :

General interpolating currents :

1, 2, 3, 4,

1, 2, 3, 4,

3 3 : [ ] [ ]

6 6 : {[ ] ( )}{[ ] ( )}

i T A T A

C C S abc b i c ade d i e

i T S T S

C C S a i b a i b

J q q q q

J q q a b q q a b

? ,

0 0with , and , A T A S T S

i i i i

5 5( ), ( ), ( )

( ), ( )

A

i

S

i

C S C PS C V

C AV C T

'S PS V AV T

S S S S S SJ J J vJ v J tJ

Chen et al., Phys.Lett.B650:369-372,2007

SR for sigma again

‘t Hooft instanton induced interaction for u,d :

Fierz trans.

1 1 0 ' 0, , , , 1/ 4 for 3Cv v t N

From PDG:

Interpolating current of the tetraquark can couple to the two pion state : Fierz transf.

We need to modify the phenomenological side.

2 2 2 Im ( ) ( ) 0 | (0) | | (0) | 0S n S S

n

q q m J n n J

Narrow resonance + two pion state

in the phen. side :

PCAC gives :

222

2 2 2 2 2 2 2

2 2 2

22 2

2

1 6( ) ( ) ( ) ( 2 )

16 4 4

4 1 ( 4 )

fqqq q m

f

mq m

q

Instanton effects :

QCD sum rules :

2 2

Including the form factor, for a=-b=1, there can be stable result!

Other members with diquarks

For f0(980) and a0(980)

Upper sign : f0(980)

Mass degeneracy in f0(980) and a0(980)

Mass fitting

~ 700MeV

For kappa(800)

With , mass fitting :

~ 730MeV

Bound state of two psedoscalar mesons?

For f0(980) : bound state of two etas? Y.U. Surovtsev et al., Int. J. Mod. Phys. A 26, 610 (2011)

-From analysis of resonances appearing in

Interpolating current :

, ,

/ ,

KK

J KK

Left Hand side of SR

Another possibility :

For f0(980) : bound state of two Kaons? - Weinstein and Isgur, PRL 48, 659 (1982), PRD 27, 588 (1983) : Using the color hyperfine and harmonic oscillator potentials. - T. Branz, et. Al. , Eur. Phys. J. A 37, 303 (2008) : Using a phenomenological Lagrangian.

Interpolating current :

Left hand side of SR :

Discussion For sigma(600) : it could be a diquark-antidiquark bound state. Effect from width? Are other members diquark-antidiquark bound states?

- Mass splitting from sigma is too small even though they have strange quark.

Can f0(980) be a bound state of two mesons? - we did not see a signal which f0(980) is a bound

state of the two etas or the two kaons. Mixing tetraquarks and two quark state, or glueballs…

Instaton induced interaction in three flavors could give

a hint for understanding the scalar mesons. Thank you! Спасибо!