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
)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
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
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