Tokyo 2006, page 1

σ(600) and Pattern of Scalar Mesons from Lattice QCD

• a0 (1450) on the Lattice

• Tetraquark Mesonium – Sigma (600) on the Lattice

• Pattern of Scalar Mesons and Glueball

χQCD Collaboration: A. Alexandru, Y. Chen, S.J. Dong, T. Draper, I. Horvath, B.

Joo, F .X. Lee, K.F. Liu, N. Mathur, T. Streuer, S. Tamhankar,

H.Thacker, J.B. ZhangTokyo U., Nov. 17, 2006

Tokyo 2006, page 2

Introduction to Lattice Gauge TheoryIntroduction to Lattice Gauge Theory

• Path integral of the partition function of continuum QCD in Path integral of the partition function of continuum QCD in Minkowski space Minkowski space

• Imaginary time with Wick rotation Imaginary time with Wick rotation

andand thenthen

• Integrating Grassmann numbers and gives Euclidean Integrating Grassmann numbers and gives Euclidean partition functionpartition function

• Note the Grassmann number integrationNote the Grassmann number integration

• The Green’s function The Green’s function

Path Integral Formulation in Discrete Euclidean Space-TimePath Integral Formulation in Discrete Euclidean Space-Time

)det,det 11111 jklilkjiM

lkjijiM

ji MMMMMeDDMMeDD (

MiSeDDDAZ }])

2([

4

1{4 mAigiGGxdS a

aaa

M

ESeDDDAZE

}])2

([4

1{4 mAigGGxdS a

aaa

E

),( 40 ixxitt

}{},{,,, ,4

04

0 giiAA ii

mDMeMDAZ GSE ,det

...}...{det1

)...],,(),,([ 1121 AMMTrMeDA

ZAOAOT GS

Tokyo 2006, page 3

Lattice QCDLattice QCD

• RegularizationRegularization

– Lattice spacing aLattice spacing a– Hard cutoff, p ≤ Hard cutoff, p ≤ ππ/a/a– Scale introduced (dimensional transmutation)Scale introduced (dimensional transmutation)

• Renormalization Renormalization

– PerturbativePerturbative– Non-perturbativeNon-perturbative Regularization independent SchemeRegularization independent Scheme Schroedinger functionalSchroedinger functional Current algebra relationsCurrent algebra relations

• Numerical SimulationNumerical Simulation

– Quantum field theory classical statistical mechanicsQuantum field theory classical statistical mechanics– Monte Carlo simulation (importance sampling)Monte Carlo simulation (importance sampling)

Why Lattice?Why Lattice?

a

0det Me GS

Tokyo 2006, page 4

Lattice QCDLattice QCDCorrespondence between Euclidean field theory and classical statistical Correspondence between Euclidean field theory and classical statistical mechanicsmechanics

Euclidean field theoryEuclidean field theory Classical statistical mechanicsClassical statistical mechanics

Vacuum transition amplitudeVacuum transition amplitude

A = A = ∫∫DDφφ e e - S/h- S/h

Action SAction S

Vacuum EnergyVacuum Energy

Vacuum Expectation <0| Vacuum Expectation <0| OO |0>|0>

Time-ordered Product Time-ordered Product

Green’s Function <0|T(OGreen’s Function <0|T(O11OO22…O…Onn)|)|0>0>

Mass MMass M

Regularization with cutoff Regularization with cutoff ΛΛ

Renormalization: Renormalization: ΛΛ ∞ ∞

Vacuum ChangeVacuum Change

Partition functionPartition function

Z = ∑ ℮ Z = ∑ ℮ −−ββHH

Hamiltonian HHamiltonian H

Free EnergyFree Energy

Ensemble Average < O >Ensemble Average < O >

Ordinary ProductOrdinary Product

Correlation Function < OCorrelation Function < O11OO22…O…On n

>>

Correlation Length Correlation Length ξξ = 1/M = 1/M

Lattice Spacing aLattice Spacing a

Continuum Limit : a Continuum Limit : a 0 0

Phase TransitionPhase Transition

Tokyo 2006, page 5

Hadron Mass and Decay ConstantHadron Mass and Decay Constant

)(2

3

)(1

00

0

0

0 ||2

0||)()(||0

)()((),,(

ttE

p

p

p

ttEtt

x

xpiNN

p

p

eE

mE

VE

epNpN

xxTepttG

The two-point Green’s function decays exponentially at large separation of time

0t t

Mass M= Ep(p=0), decay constant ~ Φ

Tokyo 2005, page 6

QCD VacuumQCD Vacuum

ππ

ρρ

σσ NN

ΔΔφφ

ΣΣΛΛ

ΞΞ

ωω

KK

SS1111

NN**CreationCreationOperatoOperatorr

QCD QCD VacuumVacuum

CreationCreationOperatoOperatorr

QCD QCD VacuumVacuum

ΘΘ++

????PentaquarkPentaquark

TetraquarkTetraquark

????

Tokyo 2005, page 9

Le Taureau of Pablo Picasso (1945)

Dynamical chiral fermion Quenched approximation

with Chiral symmetry,

and light quark masses

5th stage 11th stage

Tokyo 2006, page 10

Masses of N, Masses of N, ρρ, and , and ππ

• 16163 3 x 28 quenched x 28 quenched lattice, Iwasaki action lattice, Iwasaki action with a = 0.200(3) fmwith a = 0.200(3) fm

• Overlap fermionOverlap fermion

• Critical slowing down Critical slowing down is gentleis gentle

• Smallest Smallest mmππ ~ 180 ~ 180 MeVMeV

• mmππ L > 3L > 3

Tokyo 2005, page 11

Quenched ArtifactsQuenched Artifacts

• Chiral log in mChiral log in mππ2 2

1)/ln(1 22

2 BmAmAmm

xη η

π π

Tokyo 2005, page 12

Evidence of Evidence of ηη’N GHOST State in S’N GHOST State in S11 11 (1535) (1535) ChannelChannel

-- --η η

W > 0

W<0

Tokyo 2006, page 13

Tetraquark MesoniumsTetraquark Mesoniums

QCD allows a state with more than three quarksFour quarks : Two quarks + two anti-quarks Like molecular state? Like di-quark anti-diquark state?

q1

2q

q21q

0¯ ¯(1)

1¯+(1)0++(0)0+ ¯(1)1+ ¯(1)

π(137)

0+ (1/2)

ρ(770)

σ(600)

f0(980)

f0(1370)

f0(1500)

a0(980)

a0(1450)

a1(1230)

K0*(1430)

JPG(I))

M (

MeV

)

a2(1320)

2+ ¯(1)

f0(1710)

K0*(800)

Tokyo 2006, page 15

Why aWhy a00(980) is not a (980) is not a state?state?

• The corresponding KThe corresponding K00 would be ~ 1100 MeV which is 300 MeV would be ~ 1100 MeV which is 300 MeV away from both and .away from both and .

• Cannot explain why aCannot explain why a00(980) and f(980) and f00(980) are narrow while (980) are narrow while σσ(600) (600) and and κκ(800) are broad.(800) are broad.

• Large indicates Large indicates

in fin f00(980), but cannot be in I=1 a(980), but cannot be in I=1 a00(980). How to explain the (980). How to explain the mass degeneracy then?mass degeneracy then?

)1430(*0K

qq

)800(

)980( , )980( 00 fDf s

ss

Scalar mesons

f (600)0

f (980)0

f (1710)0

(800)

(1430)0*

a (980)0

a (1450)0

(1950)0*

f (1500)0

f (1370)0

M. Pennington Charm 2006

Tokyo 2006, page 17

Is aIs a00 (1450) (0 (1450) (0++++) a two quark ) a two quark state?state?

Ground state : Ground state : ππ ηη ghost stateghost state..

First excited state : First excited state : aa00

CorrelationCorrelationfunctionfunctionfor for Scalar Scalar channelchannel

Our results shows scalar mass around 1400-1500 MeV, suggesting Our results shows scalar mass around 1400-1500 MeV, suggesting

aa00(1450)(1450) is a two quark state.is a two quark state.

)(JI PCG ),0(1 )1(1

mmss

Tokyo 2006, page 19

What is the nature of What is the nature of σσ (600)?(600)?

r

σ (500): Johnson and Teller

Two-pion exchange potential:

Chembto, Durso, Riska; Stony Brook, Paris, …

σ enhancement of Δ I = ½ rule

The The σσ in in DD++→→ ππ¯̄ππ++ππ++

σσ

Without a Without a σσ pole pole

With a With a σσ pole pole

MMσσ= 478 = 478 ± ± 24242323 ± ± 17MeV 17MeV ΓΓσσ = 324 ± = 324 ± 4242

40 40 ± 21 MeV± 21 MeV

2423478M

E.M. Aitala et. al. Phys. Rev. Lett. 86, 770, (2001) E.M. Aitala et. al. Phys. Rev. Lett. 86, 770, (2001)

M. Ablikim et al. (BES), Phys. Lett. B598, 149 (2004)

Mσ = 541 ± 39 MeV, Γσ = 504 ± 84 MeV

J/ψ —> ωπ+π-

0

0.2

0.4

-0.4

-0.2

-0.2 0 0.2 0.4 0.6 0.8 1.0

Re s (GeV )2

Im s

(G

eV )2

: I = 0, J = 0

complex s-plane

E791

BES

CERN-Munich

ZQZXZWZQZXZW

Zhou, Qin, Zhang, Xiao, Zheng & WuZhou, Qin, Zhang, Xiao, Zheng & Wu

CCL

Caprini, Colangelo, & Leutwyler

M. Pennington Charm 2006

)(2

1

, 3

1

55

5

5

dduu

uddu

o

oo

ππππ four quark operator (I=0)four quark operator (I=0)

Tokyo 2005, page 24

2 I ,

, 0 I ,

)()(

)(2

1)(3)(

2

1)(2)0()(

tCtD

tGtAtCtDt

Tokyo 2005, page 25

E

|T|2 in continuum

E

W on lattice

E

L

E

L ?

Lnp ii

2

Tokyo 2005, page 26

K. Rummukainen andK. Rummukainen and S. Gottlieb, NP B450, 397 S. Gottlieb, NP B450, 397 (1995)(1995)

Tokyo 2005, page 27

22

4

1mWp cmcm Lüscher

formula

2 ,mod)()(0

Lpqqp cm

cm

Tokyo 2006, page 28

Scattering Length and energy Scattering Length and energy shiftshift

• ππππ energies : energies :

• Threshold energy shift on the finite latticeThreshold energy shift on the finite lattice ::

2

, 2)(

,2)0(

22

LappmppE

mpE

LLL

)(]12

2)0( 62

20

20

130 LO

L

aC

L

aC

L

ampE

Tokyo 2006, page 29

)]0(0 )(JI ,[ PCG55

Further study is needed to check the Further study is needed to check the volume dependencevolume dependence of the of the observed states.observed states.

Scattering statesScattering states(Negative scattering(Negative scattering length)length)

)0()0( pEpE

)1()1( pEpE

Scattering statesScattering states

Possible BOUND statePossible BOUND state

σσ(600)?(600)?

Tokyo 2006, page 30

Scattering state and its volume dependenceScattering state and its volume dependence ),,|

1,,| spn

Vspn

nn

n

tMn

x n

tM

n

x

M

nW

eW

eVM

n

txTtG

n

n

2

|)0(|0

2

|)0(|0

0|))0(),((|0)(

2

2

Normalization condition requires :

Two point function : Lattice Continuum

For one particle bound state spectral weight (W) will NOT be explicitly dependent on lattice volume

Vx

Tokyo 2006, page 31

Scattering state and its volume dependenceScattering state and its volume dependence ),,|

1,,| spn

Vspn

tEE

nn

nn

tEE

nn nnx

x

nn

nn

eV

WW

eVMVM

nn

txtxTtG

,

,

222

211

2121

11

21

21

11

21 21

2 2

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

0|))0()0(),(),((|0)(

Normalization condition requires :

Two point function : Lattice Continuum

For two particle scattering state spectral weight (W) WILL be explicitly dependent on lattice volume

Vx

Volume dependence of spectral weights

Volume independence suggests the observed state is an Volume independence suggests the observed state is an one particle stateone particle state

WW00

WW11

0¯ ¯(1)

1¯+(1)0++(0)0+ ¯(1)1+ ¯(1)

π(137)

0+ (1/2)

ρ(770)

σ(600)

f0(980)

f0(1370)

f0(1500)

a0(980)

a0(1450)

a1(1230)

K0*(1430)

JPG(I))

M (

MeV

)

a2(1320)

2+ ¯(1)

f0(1710)

K0*(800)

MesoniumsKK Kπ Mesonium

ππ Mesonium

qq

Tokyo 2006, page 34

Mixing of Mixing of ssdduu and ,,

First order approximation: exact SU(3)

MeV 33 ,314703

)1370(

14706

2)1500(

14702

147000

014700

001470

0

0

0

xxssdduu

f

ssdduuf

dduua

xxx

xxx

xxxx is annihilation diagram

Tokyo 2006, page 35

Mixing of with GlueballMixing of with Glueballssdduu ,,

First order approximation: exact SU(3)

MeV 6.213701710

mixed;slightly are glueball and )1370(

mix)not (does 14706

2)1500(

14702

1710000

0147000

0014700

0001470

2

0

0

0

xm

f

ssdduuf

dduua

xxxx

xxxx

xxxx

xxxx

SU(3) Breaking and f0(1370), f0(1500), f0 (1710) mixing

p

p

f

fR

p

p

f

KKfR

ssdduuf

K

2

0

02

2

0

01

0

227

1

))1500((

))1500(( ,1

3

1

))1500((

))1500((

|)|(|)1500( e.g.

For SU(3) octet f0(1500), = -2 R1=0.21 vs. 0.2460.026 (expt)

R2=0 vs. 0.1450.027 (expt)

LQCD [Lee, Weingarten] y= 4331 MeV, y/ys=1.1980.072

y and x are of the same order of magnitude !

Need SU(3) breaking in mass matrix to lift degeneracy of a0(1450) and f0(1500)

Need SU(3) breaking in decay amplitudes to accommodate observed strong decays

SU(3) breaking effect is weak and can be treated perturbatively

H.Y. Cheng, C.K. Chua, and K.F. Liu, hep-ph/0607206

Consider two different cases of chiral suppression in G→PP:

(i)

(ii)

59.1:55.1:1:: ggg KK

74.4:15.3:1:: ggg KK

In absence of chiral suppression (i.e. g=gKK=g), the predicted f0(1710) width is too small (< 1 MeV) importance of chiral suppression in GPP decay

][Tr][Tr][TrX][Tr 3G21 PPXfPPfPPXfH GFSPP

SNGf

SNGf

SNGf

52.078.036.0)1370(

84.054.003.0)1500(

17.032.093.0)1710(

0

0

0

MS-MU 25 MeV is consistent with LQCD result

near degeneracy of a0(1450), K0*(1430), f0(1500)

(J/f0(1710)) = 4.1 ( J/ f0(1710)) versus 6.62.7(expt)

no large doubly OZI is needed

(J/ f0(1710)) >> (J/f0(1500))

: primarily a glueball

: tend to be an SU(3) octet

: SU(3) singlet + glueball content ( 13%)

MU=1474 MeV, MS=1498 MeV, MG=1666 MeV

Tokyo 2006, page 39

Scalar Mesons and Scalar Mesons and GlueballGlueball

)1500(0f)1470(0

0a

)1430(*0K)1430(*

0K

)1430(*0K)1430(*

0K

)1470(0a )1470(0

a

)1370(0f

)600(

)1710(0fglueball

qq

22qq )980(0f)980(0

a )980(0a

)980(00a

)800(

)800( )800(

)( KK

)( K

)800(

Constituent Quark Scaling

Anisotropy in Au + Au at = 200 GeV (STAR)Meson n=2 and Baryon n=3 groupingSome deviation due to internal hadron structure

NNS

Tokyo 2006, page 41

SummarySummary

• Plenty of tetraquark mesonium Plenty of tetraquark mesonium candidatescandidates

• σσ(600) is very likely to be a (600) is very likely to be a tetraquark mesonium.tetraquark mesonium.

• Pattern of light scalar mesons may Pattern of light scalar mesons may be repeated in the heavy-light be repeated in the heavy-light sectors (?)sectors (?)

Tokyo 2006, page 42

Azimuthal anisotropy in Au + Au collisions with

= 200 GeV (STAR collaboration)NNS