Light scalar nonets in pole-dominated QCD sum rules

T. Kojo (Kyoto Univ.)

D. Jido 　 (YITP)

Light scalar nonets ~ candidates of exotic hadrons

isospin: 0 1/2 1 0

mass (MeV): 600 ? 800 ? 980 980width: broad broad narrow narrow

2 - quark picture has the difficulty:

To obtain JP = 0+ state, P-wave excitation (~500MeV) is needed.

The masses exceed ~ 1 GeV.

The assignment assuming the ideal mixing:

wrong ordering

Natural explanation for mass ordering & decay mode & width

Possible strong diquark correlation (?) → mass < 1 GeV

4 -quark picture leads the favorable prescription (Jaffe, 1977)

The purpose of my talk

The purpose of my talk is:

to provide the information to consider

the relevant constituents possible mixing with 2q & 2qG

for light scalar nonets, using the QCD Sum Rules.

We already know the 2q operator analysis

fails to reproduce the lightness of light scalar nonets.

Therefore, we perform the QSR analysis

using the tetraquark operators.

2, Basics of QCD Sum Rules& typical artifacts in the application to the exotics

QCD Sum Rules (QSR)

？hard soft

OPE

small large

Borel window

OPE bad OPE good

Procedures for estimating the physical quantities

3, Select Sth to give the good stability against the variation of M.

2, Plot physical quantities as functions of M2:

effective mass:

E

Eth

1, Set the Borel window for each Sth :

Mmin < M < Mmax

constraint for OPE convergence constraint for continuum suppression

highest dim. / whole OPE < 10 % pole / whole spectral func. > 50 %

peak like structure stability against M variation

QSR for Exotics2. Good continuum suppression

3. small M2-dependence

1. Good OPE convergence

Difficulty to analyze Exotics

R.D.Matheus and S.Narison, hep-ph/0412063 M2

small

?

When quark number of operator is large, realizing the conditions 1, 2 becomes extremely difficult.

Indeed, in most of the previous works, OPE convergence is not good, and

In the M2 region where pole ratio is too small,

the artificial stability of the physical parameters emerges! ( Even in the meson and baryon cases )

To avoid the artificial stability, we must estimate the physical values in the Borel window.

pole / whole contribution in the spectral integral is less than ~20 %!

QSR artifact ~ pseudo peak artifact

: outside of the Borel window

dim 10 ~ 12 dim 0 ~ 8

E

Eth

dim 8 ~ 12

dim 0 ~ 6

E

Eth

pseudo peak ! dim 0 ~ 6

dim 8 ~ 12

Spectral function

Pseudo peak artifact ~ Impact on physical quantities

mass residue

spectral function pole dominance

These artifacts are easily rejected by Borel window. & inclusion of higher dimension (> 6) terms.

Impact of width on physical quantities

？

example: Breit-Wigner form (pole mass = 600 MeV)

effective mass:

We will estimate the physical quantities considering the error from width effects.

output: mass

width = 400 MeV

input: Breit-Wigner

3, Tetraquark operator analysis

Calculation

Linear combination:

θ will be chosen to achieve:

Set up of the operator:

up to dim12 within vacuum saturationOPE:Must be calculated to find the Borel window !

weak M – dep.

weak Eth – dep.

well-isolated peak-like structure(not strongly affected by background)

well-separated from threshold

Borel window most important for meaningful estimation

Annihilation diagrams ~ Flavor dependence

The number of annihilation diagrams strongly depends on the flavor.

2q - 4q, 2qG - 4q

& cyclicdiquark base:

pure singlet:

pure octet:

large

small

num. of annihilation

diagrams

mixing

Effective mass for pure singlet & octet( in the SU(3) chiral limit )

mass: 0.7 ~ 0.85 GeV Eth: 1.0 ~ 1.3 GeV

mass: 0.6 ~ 0.75 GeV Eth: 0.8 ~ 1.3 GeV

May be broad, or small pole to background ratio

Effective residue for pure singlet & octet

smaller than singlet residue small pole to background ratio?

σ(600)

mass: 0.6 ~ 0.75 GeV Eth: 0.9 ~ 1.3 GeV

f0(980) (preliminary)

mass: 0.75 ~ 0.90 GeV

Eth: 1.1 ~ 1.5 GeV

Effective mass plots for σ & f0

a0(980) (preliminary)

a0 - channel No stability in the Borel window　　　　 in the arbitrary θ

mass residue (×107 GeV8)

Results for pure octet ~ mass & residue for a0

κ- channel shows almost same behavior.

SummaryWe perform the tetraquark op. analysis within the Borel window.

To find the Borel window, the higher dim. calculation is inevitable to include the sufficient low energy correlation.

Our analysises imply ( within our operator combinations ):

σ(600) and f0 (980) are more likely 4q rather than 2q state. ( in 2q op. case, their masses are ~ 1.0 - 1.3 GeV)

singlet channel has well-developed enhancement around E~ 0.7GeV.

octet channel may be strongly affected by low energy contaminations.

( pole to background ratio may be large. )

( pole to background ratio may be small or no pole. )

The difference between singlet and octet originates from annihilation diagrams, 4q→2q or 4q→2qG.

Back up slide

0¯ ¯(1)1¯+(1)

0++(0)0+ ¯(1)1+ ¯(1)

π(137)

0+ (1/2)

ρ(770)

a1(1230)

JPG(I)

M (

MeV

)

a2(1320)

2+ ¯(1)

vector,axial vector, tensor

Scalar meson

(L=0, S=1)

(L=1, S=0)(L=1, S=1)

QQ

QQ

singlet – octet mixing

valence: (QQ)(QQ) (QQ)-(QQ)

GGor

σ(600)

f0(980)a0(980)

K0*(800)

f0(1370)

f0(1500)a0(1450)K0

*(1430)

f0(1710)

2q?(L=1, S=1)

Pole dominance ~ importance of higher dim. terms

Pole / Whole spectral function (σ- case )

Only after dim. 8 terms contributes, Mmax becomes large.

+dim.8

2-quark Annihilation diagrams

4-quark 3-loop, α suppression

4-loop, α suppression

no suppression factor, few loopsfew loops, but equal to zero

Annihilation diagrams have more importancein 4q op. case than in 2q op. case. split singlet & octet

Annihilation diagrams increase in higher dim. terms.important especially in low energy region.

4q-2q or 4q-2qG mixing

dim 10, 12 dim 8 dim 6 dim 0 ~ 4

E~ 1 GeV

2qG mixing 2q mixing

~ 2 GeV

singlet octet

qualitative behavior of

This 2qG mixing is turned out to be essential for the large correlation in low energy ~ 1 GeV.

4q-2q or 4q-2qG mixing

essential for low energy enhancement

can be interpreted asdiquark-diquark correlation ?

contributes mainly 1~ 2 GeV enhancement

can be interpreted as2q component above 1GeV ?

OPE diagrams ( for massless limit )

Theoretical suggestions 1:

mass (MeV): 600 ? 800 ? 980 980width(MeV): ~ 400 ? 50 ~ 300 50 ~ 300~500 ?

Natural explanation for mass ordering & decay mode & width

Possible strong diquark correlation (?) → mass < 1 GeV

4 -quark picture leads the favorable prescription.

(due to strong chromo-magnetic interaction)

1, Jaffe: ( MIT bag model ) PRD15, (1976) 267

Theoretical suggestions 2 & 3:2, Weinstein & Isgur: ( 4-particle Shrodinger eq. )

PRL48, (1982) 659qqqq bound states normally do not exist.

a0(980), f0(980) → loosely KK bound states .

T. Barnes (estimate a0, f0 → 2γ width) PLB165, 434 (1985)a0 (2q) → 2γ : width ~ 1.6 keV ~ 8 ×exp. widtha0 (KK) → 2γ: width ~ 0.6 keV ~ 3 ×exp. widthf0 (KK) → 2γ: width ~ exp. width

3, Narison: ( phenomelogy with QCD sum rules cal. ) PRD73, 114024(2006)

σ, f0 → strong 2q – glueball mixing.

a0 → 2q, not 4q

κ → 2q ( strong interference with nonresonant background)

( a0 → 2γwidth is 1/1000 small in 4q case)

( σ, f0 → 　 2π width is too small in 4q case)

( but all cal. of width in QSR is suspicious)

Experimental results:1, Exp. at Fermi lab. ( E791 Collaboration )

PRL12, 121801(2002)PRL86, 770(2001)PRL12, 765(2001)

σ

( mass = 478±17 MeV width = 324±21 MeV )

( no evidence for σ)

κ( mass = 797±62 MeV width = 410±130 MeV )

cs s

sW+

ud

cd d

dW+

ud

M (

MeV

)

Scalar meson

σ(600)

f0(980)a0(980)

K0*(800)

f0(1370)

f0(1500)a0(1450)K0

*(1430)

f0(1710)

Kentucky group

a0(1450), K0(1430) → 2q

σ(600) → 4q

overlap fermion (χ-symmetry)volume dep.

UK QCD groupNf=2 sea quark (partially quenched)

a0(980) → reproduced within 2q?

Scalar collaborationdynamical fermion(including glueball mixing)

disconnected diagram dominate(σ case)

light σ

Lattice:

No KK, using (ud) picture

Dispersion relation, OPE, quark-hadron duality

QCD side

, …

sum of local operators

Operator Product Expansion

information of QCD vacuum

（ OPE ）

hard

softq q

Hadronic sidespectral function

？

simple parametrization

Constraint for MBorel trans.

Information of low energy we want to know

OPE bad OPE good

small large

Borel window

Within the Borel window, we represent mass & residue as functions of the unphysical expansion parameter M ( & physical value Sth ).

,

physical parameter

should not depend on M

QSR artifact ~ pseudo peak artifact

E

Eth

dim 8 ~ 12

dim 0 ~ 6

E

Eth

pseudo peak ! dim 0 ~ 6

dim 8 ~ 12

: outside of the Borel window

Spectral function

Procedures for estimating the physical quantities

3, Select Sth to give the best stability against the variation of M.

2, Plot the physical quantities as functions of M2.

If these quantities heavily depend on M2 in the Borel window, 1-pole + continuum approximation is bad..

We must consider another possibilities: 2 or 3 poles, smooth function for the scattering states and so on.

1, Set the Borel window for each Sth :

Mmin < M < Mmax

constraint for OPE convergence constraint for continuum suppression

highest dim. / whole OPE < 10 % pole / whole spectral func. > 50 %

750~790 770

2.3~2.5 2.36

Example when QSR is workable:

-meson case 　

0.8

0.6

0.4

1.0

1.2

0.4 0.6 0.8 1.21.0 1.4

1 2

3

Borel window

( up to dim. 6 )

Note for physical importance of higher dim. terms of OPE:

Only after including dim.6 terms of OPE (including low energy correlation) , stability emerges in the Borel window.

Dim.6 terms are responsible for the ρ － A1 mass splitting.(Without dim.6 terms, OPE forρand A1 give the same result.)

QSR for Exotics2. Good continuum suppression

3. small M2-dependence

1. Good OPE convergence

Difficulty to analyze Exotics

R.D.Matheus and S.Narison, hep-ph/0412063 M2

small

?

When quark number of operator is large, realizing the conditions 1, 2 becomes extremely difficult.

Indeed, in most of the previous works, OPE convergence is not good, and

In the M2 region where pole ratio is too small,

the artificial stability of the physical parameters emerges! ( Even in the meson and baryon cases )

To avoid the artificial stability, we must estimate the physical values in the Borel window.

pole / whole contribution in the spectral integral is less than ~20 %!

QSR artifact ~ pseudo peak artifact

E

Eth

E

Eth

pseudo peak ! dim 0 ~ 6

dim 8 ~ 12dim 8 ~ 12

dim 0 ~ 6

: outside of the Borel window

dim 10 ~ 12 dim 0 ~ 8

Pseudo peak artifact ~ examples

mass residue

spectral function pole dominance

Calculation

Linear combination:

θ will be chosen to give the best stability in the Borel window.

Set up of the operator:

up to dim12 within vacuum saturationOPE:

u, d-quark is treated in massless limit → x- rep. calculation

s-quark mass is kept finite → p- rep. calculation

regulate mass ×divergence termsresummation of the strange quark mass

treatment of current quark mass:

Must be calculated to find the Borel window !

diquark base:singlet octet

Classification of nonets

mass ordering:

ideal mixingassumption:

600 800 980 980

Annihilation diagrams ~ Flavor dependence

The number of annihilation diagrams strongly dependent on the flavor.

& cyclicdiquark base:

pure singlet:

pure octet:

large

small

num. of annihilation

diagrams

1, Sufficiently wide Borel window

2, Weak M dependence

3, Weak threshold dependence

4, The sufficient strength of the effective residue

Criterions on selection of operators

most important, well-satisfied for almost all θ

necessary to avoid the contaminations below Eth

necessary to avoid the contaminations from regions between “pole” and Eth

necessary to avoid the truncated OPE error

( for stong low energy correlation, pole isolation )

Singlet

Octet

Global analysis ~ θdependence

Singlet → better in Borel stability, larger residue

Except some θ region, behavior is similar.

θ

θ

residue mass

Effective mass plot ~ θ fixed to 7π/8

mass: 0.7 ~ 0.8 GeV Eth: 1.0 ~ 1.3 GeV

mass: 0.6 ~ 0.75 GeV Eth: 0.8 ~ 1.3 GeV

May be broad, or small pole to background ratio

Effective residue plot

smaller than singlet residue

σ(600)

mass: 0.6 ~ 0.75 GeV Eth: 0.9 ~ 1.3 GeV

f0(980) (preliminary)

mass: 0.75 ~ 0.90 GeV

Eth: 1.1 ~ 1.5 GeV

Effective mass plots for σ & f0

a0(980) (preliminary)

a0 - channel No stability in the Borel window　　　　 in the arbitrary θ

mass residue (×107 GeV8)

Results for pure octet ~ mass & residue for a0

κ- channel shows almost same behavior.

Experimental results:1, Exp. at Fermi lab. ( E791 Collaboration )

PRL12, 121801(2002)

Dalitz decay of D meson

PRL86, 770(2001)PRL12, 765(2001)

SummaryWe perform the tetraquark operator analysis within the Borel window.

The states including the SU(3) singlet component:

σ(600): f0 (980):0.60 ~ 0.75 GeV 0.75 ~ 0.90 GeV(preliminary)

The states including only the SU(3) octet component:

a0 (980): no stabilityκ(800): no stability

Real world

SU(3) chiral symmetric world

singlet: octet:0.70 ~ 0.85 GeV (stability not good)

The difference comes from self-annihilation processes (diagrams).

Much stronger low energy correlation than 2-quark case → Borel window is easily found .

Some important effects associated with strange quark mass & hadronic threshold seem to be underestimated.

0.6 ~ 0.75 GeV

Resummation of current quark mass

E E

effective mass shifts to high energy side.

cut

Mass × divergence

cut for spectral integral

regulation for integralof Feynman parameter

example:( dim.8 )

a) b)

c) d)

＝ ０

a) b)

c)