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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
)800(
)980( , )980( 00 fDf s
ss
scadron70 page 4
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
scadron70, page 6
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
scadron70 page 8
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
scadron70 page 19
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
scadron70 page 20
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
scadron70 page 23
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
scadron70 page 24
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
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 (?)
scadron70 page 30
Azimuthal anisotropy in Au + Au collisions with
= 200 GeV (STAR collaboration)NNS