mass spectra of the low-lying nonet scalar mesons in the lattice qcd motoo sekiguchi kokushikan...
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![Page 1: Mass spectra of the low-lying nonet scalar mesons in the lattice QCD Motoo Sekiguchi Kokushikan University Scalar collaboration; T. Kunihiro, S. Muroya,](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d6a5503460f94a47fa6/html5/thumbnails/1.jpg)
Mass spectra of the low-lying nonet scalar mesons in the latt
ice QCD
Motoo SekiguchiKokushikan University
Scalar collaboration; T. Kunihiro, S. Muroya, A. Nakamura,
C. Nonaka, H. Wada, M. S.
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Plan of the talk
• Objective and Motivation
• The sigma meson in lattice simulation with dynamical quarks
• The kappa meson in quenched lattice simulation
• Current status of our new simulation for the scalar mesons
• Summary
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Objective of Scalar Collaboration
• Confidence level of the sigma meson and kappa meson has been increasing, and its physical significance in hadron physics and QCD is apparent.
• Using Lattice QCD, we have been addressing the following Question about the scalar mesons: the sigma and the low-lying scalar mesons are resonances in QCD or something else?
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• The nature of the
low-lying nonet
scalar mesons is
not understood yet.
• Experimentally well established scalar resonances below 1
GeV are a0 (980) and
f0(980).
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The significance of sigma meson (I=0, JPC=0++) in low energy hadorn physics
• There is a strong experimental evidence for a light sigma meson, whose pole was extracted with a small uncertainty from modern analyses in pi-pi scattering (Igi, Hikasa PR D (1999), I. Caprini, G. Colangero and H. Leutwyler PRL (2006)).
• The significant contributions of the sigma pole were identified in the D meson decay; D+ →π - π + π +
Fermilab E791; E.M. Aitala et al, Phys Rev. Lett. (86), 770 (2001).• Responsible for the intermediate range attraction in the nuclear forc
e.• Accounts for ΔI=1/2 enhancement in K0 →2π compared with K+→π
+π-.
T. Morozumi, C.S. Lim and I. Sanda, PRL (1990).• Particle Data Group, Physics Letters B667, 1 (2008)
Mass_sigma=(400-1200) MeV, Full width=(600-1000) MeV.
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Issues with the low-mass Sigma meson in QCD
• In the quark model; JPC=0++ mesons→3P0 the mass in t⇒he 1.2 -1.6 GeV region. Some mechanism need to down the mass with ~ 800 MeV
• Color magnetic interaction between the di-quarks (Jaffe1977) with the bag-model wave functions. ⇒ All the low-lying scalars are tetraquarks!
• The sigma is a superposition of qq-bar states. The sigma is indentified as the chiral partner of the π meson in Dynamical Chiral symmery Breaking in QCD.
• The π-π molecule as suggested in π-π scattering.
• A mixed state with scalar guleball state.
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The Kappa meson
• scalar meson with the Strangeness (I=1/2)
• Recent experimental candidates:– Fermilab E791: hep-ex/020
4018, (PRL89(2002)12801).
– BES:hep-ex/0304001.
• Both observed a candidate near 800MeV
• Even 660MeV! Eur. Phys. J C48(2006)543 (hep-ph/0607133)
0+
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The sigma in latticesimulations with dynamical quarks
• A first work on the sigma in lattice QCD with dynamical quarks. (Phys. Rev. D70, 034504(2004).)
• The full QCD simulation is necessary to properly describe the sigma with possible contents, ie ., the qq-bar, the glueball, tetra quarks and so on.
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Previous Lattice QCD simulations of the sigma mesons
• W. Lee and D. Weingarten, Phys.ReV.D61(1999)012015
Quenched simulation Mixing matrix between the Guleball and qq-bar Mass above 1 GeV• Alford and Jaffe Nucl. Phys.B578(2000)367. Quenched simulation Tetraquarks type interpolating operator Disconnected diagrams are omitted.
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I=0, scalar interpolating operator for sigma
• There is experimental evidence that the sigma consist of only uu-bar and dd-bar compnets.
3
1
4
1
3
1
2
)()()()(
)()()(
c
cccc
c
cc
xdxdxuxu
xxx
c = 1,2,3 ・・・ color
α=1,2,3,4 ・・・ Dirac spin
d
u
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• The bound state properties are obtained by calculating expectation values,
where Oi is the interpolating operator.
• The path integral is regulated by the introduction of a space-time lattice. The integral is computed in Euclidean space using Monte Carlo techniques on the computer.
ground state: m0 excited state: m1
• The results is a table of numbers. We average and fit exponentials to get masses.
...10 10 tmij
tmijij eCeCtG
GF SSjiij exOyODDdU
ZxyG 1),(
)()( xxO
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Propagator
1 1
1 1
1 1
( , )
Tr ( , ) ( , )
2 Tr ( , ) Tr ( , )
2 Tr ( , ) Tr ( , )
G x y
D x y D y x
D y y D x x
D y y D x x
Connected diagramq
q
Disconnected diagram
- Vacuum contribution
Inverse of Fermion Matrix, i.e., Quark Propagators
1( , ) :D x y
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Details of our Calculation (1)
Full QCD Update by Hybrid Monte Carlo (SX5 at RCNP)
Wilson Fermions (2 flavors)
Plaquette Gauge ActionPhys.Rev. D70 (2004) 034504 (hep-ph/0310312)
Disconnected Part by Z2 Noise Method (SR8000 at KEK)
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Details of our Calculation (2)- Simulation parameters
Lattice size : 83 × 16
β = 4.8
κ = 0.1846, 0.1874, 0.1891 well established by CP-PACS, a = 0.197(2) fm , κc = 0.19286(14) , Lattice size:( CP - PACS, Phys. Rev. D60(1999)114508 )
Number of the Z2 noise = 1000
Wilson Fermions & Plaquette gauge action
Very small !
Very strong coupling !
Very large !
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Details of our Calculation (3)
Separation between configurations are 10 trajectories
[κ = 0.1846]
1110 configurations
[κ = 0.1874]
860 configurations
[κ = 0.1891]
730 configurations
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Details of our Calculation (4)
κ m_π/m_ρ(Our Results)
m_π/m_ρ
(CP-PACS)
0.1846 0.825±0.001 0.8291±0.0012
0.1874 0.760±0.002 0.7715±0.0017
0.1891 0.692±0.005 0.7026±0.0032
Using the same values of the hoping parameters except for the lattice size.
Our results and CP-PACS are nearly equal.The small errors indicate the high precision of our
simulation
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m_π^2, m_ρ and m_sigma as a function of the inverse hopping parameter
- Chiral Extrapolation -
5.1410±0.0747
κc = 0.1945±0.0029 ( CP-PACS κc = 0.19286(14) )
0.8093
a = 1.05×10 -3(MeV)-1
= 0.207fm (1=197MeV fm)
CP-PACSa = 0.197(2) fm
)123.0559.16(
1)0229.0221.3(2
m
)0819.04195.5(
1)0153.02116.1(
m
)1.5885.28(
1)958.0671.5(
m
0.270
334.08093.0
270.0
m
m
mσ=257MeV
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We conclude the sigma shows a pole behavior and
m m m Here the disconnected diagram plays essential role.
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Other Lattice QCD simulations of the sigma mesons
• The Kentucky lattice group (hep-ph/0607110) claimed to get a result for mass of Sigma from quenched lattice QCD with pion masses as low as 180 MeV. Using tetraquarks type operator.
However, the Kentucky lattice group reported the full QCD results obscured the tetraquark (arXiv/0810.5512, Lattice2008).
• UK-QCD (Phys. Rev. D74 (2006)114504.) claimed to get a result for Sigma using the glueb
all and qq-bar interpolating operators in the dynamical quark simulation. The mass of the glueball is below 1 GeV.
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The Kappa meson in quenched lattice simulation
• We perform quenched simulations on kappa meson so as to clarify the structure of the scalar meson rather than to reproduce the experimental value of the mass; a quenched-level simulation should give a rather clear perspective on whether the system can fit with the simple quark model picture or not.
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Kappa propagator
xyWyxWTr
WssuWuS
xuxsxuxssDuDsDuDUDDZ
xuxsTxyG
xuxsx
us
uG
aabb
ba
cc
cc
c
,,
exp
1
,
ˆ
1
4
1,
3
1,
4
1
3
1
+
du,
s
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Simulation parameter • Quech approximation• Lattice size =20X20X20X24• Wilson Fermions • Plaquette Gauge Action • Laticce spacing a=0.1038 fm, β=5.9• Hopping parameters; h_u,d=0.1589, 0.1583 and 0.1574 h_s=0.1557 and 0_1566
• We heve checked that the mass of the π, ρ, K and K* mesons obtained in our simulation are good agreement with those on a large lattice (CP-PACS, 32^3X56).
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The mass ratios m_K /m_K* and m_kappa/m_K* at chiral limit, and m_φ/m_K* for s quark hopping parameters h_s=0.1566 and 0.1557.
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• m_kappa ~ 1.7 GeV (K*0(1430)?)
• Large than twice the experimental mass.
• Our quenched lattice calculation suggests the kappa can not a normal qq-bar state.
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Current Status of our new full QCD simulation for Scalar mesons
• We use gauge configurations from International Lattice Data Grid (ILDG).
• We employ the all-to-all propagator method with the dilution techniques (Trinlat Collab. Compt. Phys. Commun. 172 (2005) 145.).
• Smearing (Jacobi, Gaussian, Derivative quark) source and sink for errors reduction (Graz group, PR D78 (2008) 034501).
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• Variational Method
It is to use several different interpolating operator Oi.
ntMj
niij e|n|O|n|OG(t) 00
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Setting of Calculation
Old Interpolating operator Oi
New Interpolating operator Oi
(Additional)
plaqUqq ,
qqqq
ii
ii
55
4
Variational method with multiple interpolating operators an explicit inclusion of tetraquark operator
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• We are preparing for the simulation of sigma and the other scalar mesons.
• We will start the simulation for the kappa meson at October.
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Summary
• The sigma meson and other low-lying scalar mesons are still a source of debates.
• A full QCD lattice simulation suggests the existence of a low-lying sigma as a pole in QCD; the physical content is obscure: the disconnected diagram gives the dominate contribution.
• A quenched lattice calculation suggests that the kappa can not be a normal qq-bar state.
• We will present the new results of sigma and kappa meson in the near future.
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Propagator
3 4
, 1 , 1
3 4
, 1 , 1
( , ) ( ) ( )
( ) ( ) ( ) ( )1
2
( ) ( ) ( ) ( )
2
1 1( ) ( ) ( ) ( )
2
( ) ( ) ( ) ( )
g
b b b b
a b
a a a aS uDu dDd
b b a a
a b
b b a a
G y x y x
u y u y d y d yDUduDuDdDd
Z
u x u x d x d x
DUDuDuDdDd u y u y u x u xZ
d y d y d x d x u
e
†
†
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) g
b b a a
S uDu dDdb b a a
y u y d x d x
d y d y u x u x e
![Page 36: Mass spectra of the low-lying nonet scalar mesons in the lattice QCD Motoo Sekiguchi Kokushikan University Scalar collaboration; T. Kunihiro, S. Muroya,](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d6a5503460f94a47fa6/html5/thumbnails/36.jpg)
• The flavored scalar mesons are not light as obseved
• m_kappa ~ 1.8 GeV (Exp. 0.8 GeV)
• m_a0 ~ 1.9 GeV > (Exp. 0.98GeV)
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Details of our Calculation
Quench Full QCDColdStart
κ=0.1846
1500trajectory
500trajectories
On every 10 trajectry, we calculate propagators.
κ=0.1891
κ=0.1874
ca. 10,000 trajectories
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Simulation parameter
• Quech approximation
• β=5.9
• Lattice size =20X20X20X24
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Alford-Jaffe, Nucl. Phys. B578, 367 (2000), (hep-lat/0001023,hep-lat/0306037)
Quench Calculation
They consider these Diagrams
Deviation from Lueshcer
Scattering formula.Bound state ?
h 1 h2
1 2 1 2
1 2 03
1 2
2 60 01 2
( )
2 ( )
1 ( ) ( )
h h h h
h h
h h
E m m
m m a
m m L
a ac c O L
L L
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π , ρ, σmesons (κ=0.1891 )
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σmeson propagatorsConnected and Disconnected Parts ( κ=0.1891
)
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6.2578 ( c =0.1598 )
CP-PACSの値0.1598 (外挿の仕方による)
M_rho a = 0.43955 (縦軸読み取り値)
M_rho = 770 MeVより1/a = 770 / 0.43955 [MeV] = 1.75 [GeV]
a = 0.112 [fm]
CP-PACS : 1/a = 1.934(16) GeVa = 0.1020(8) fm
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Summary
• kappa (m*a = 0.8843)1.55 [GeV]
• axial victor(m*a =0.9448) 1.65 [GeV]
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Setting of Calculation
• Configurations:
We use gauge configurations from CP-PACS (Lattice QCD Archive).
βLattice Size
kappa configurations
1.80 123X
24
0.1409 5951.80 123X
24
0.1430 4721.80 123X
24
0.1445 3221.80 123X
24
0.1464 1481.95 163X
32
0.1375 5991.95 163X
32
0.1390 6821.95 163X
32
0.1400 2941.95 163X
32
0.1410 223
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Setting of Calculation
• Moreover, we use also 2+1 flavor full QCD configurations by CP-PACS+JLQCD
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Project Start
• We will start new project for the scalar nonet mesons at October .
• Now we write simulation program for SX-9 at RCNP Osaka University.
• To do list
• Program improvred.• Speed up (time over)• MPI