scadron70 page 1

Pattern of Light Scalar Mesons• a0 (1450) and K0*(1430) on the Lattice

• Tetraquark Mesonium – Sigma (600) on the Lattice

• Pattern of Scalar Mesons and Glueball

QCD 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. Zhang

SCADRON 70 Workshop on

"Scalar Mesons and Related Topics"

February 11-16, 2008, at IST in Lisbon, Portugal

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)

scadron70 page 3

Is aIs a00(980) is a state?(980) is a state?

• The corresponding KThe corresponding K00** would be ~ 1100 MeV which is would be ~ 1100 MeV which is

300 MeV away from both and .300 MeV away from both and .

• Hard to explain why aHard to explain why a00(980) and f(980) and f00(980) are narrow (980) are narrow while while σσ(600) and (600) 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 (980). How to explain the mass degeneracy then?explain the mass degeneracy then?

)1430(*0K

qq

)800(

)980( , )980( 00 fDf s

ss

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IsIs aa00(1450) the state?(1450) the state?qq

• Why is it higher than aWhy is it higher than a1 1 (1230) and (1230) and aa22(1320)? (1320)?

• Why is it degenerate with KWhy is it degenerate with K00*(1430)?*(1430)?

• Why is it higher than aWhy is it higher than a00 (980)(980) ?

σ(600)

f0(980)

f0(1370)

f0(1500)

f0(1710)?

Julian Alps, Slovenia 2007

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Masses of N, Masses of N, ρρ, and , and ππ in inQuenched Lattice Quenched Lattice CalculationCalculation

• 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

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

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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)

chiral07 page 13

E

|T|2 in continuum

E

W on lattice

E

L

E

L ?

Lnp ii

2

chiral07 page 14

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

chiral07 page 15

22

4

1mWp cmcm Lüscher

formula

2 ,mod)()(0

Lpqqp cm

cm

Tokyo 2005, page 16

2 I ,

, 0 I ,

)()(

)(2

1)(3)(

2

1)(2)0()(

tCtD

tGtAtCtDt

scadron70 page 17

)]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)?

Two Pion Energy Shift

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Scattering state and its volume dependenceScattering state and its volume dependence

Vspn

1,,|

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

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

Vx

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Scattering state and its volume dependenceScattering state and its volume dependence

Vspn

1 ,,|

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

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

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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

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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, PR D74, 094005 (2006) 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

scadron70, page 28

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(

arXiv:0706.1262 

scadron70, page 29

SummarySummary

• Plenty of tetraquark mesonium Plenty of tetraquark mesonium candidatescandidates

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

• ff00(1710) could be a fairly pure (1710) could be a fairly pure glueball.glueball.

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

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Azimuthal anisotropy in Au + Au collisions with

= 200 GeV (STAR collaboration)NNS