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.

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

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?

• 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).

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.

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.

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+

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.

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.

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

• 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

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

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)

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 !

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

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

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

We conclude the sigma shows a pole behavior and

m m m Here the disconnected diagram plays essential role. 　

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.

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.

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

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).

　　

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.

• 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.

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).

• Variational Method

It is to use several different interpolating operator Oi.

ntMj

niij e|n|O|n|OG(t) 00

Setting of Calculation

Old Interpolating operator Oi

New Interpolating operator Oi

(Additional)

plaqUqq ,

qqqq

qq

qq

ii

ii

55

4

Variational method with multiple interpolating operators an explicit inclusion of tetraquark operator

• We are preparing for the simulation of sigma and the other scalar mesons.

• We will start the simulation for the kappa meson at October.

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.

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

• 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)

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

Simulation parameter

• Quech approximation

• β=5.9

• Lattice size =20X20X20X24

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

π , ρ, σmesons (κ=0.1891 )

σmeson propagatorsConnected and Disconnected Parts ( κ=0.1891

)

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

Summary

• kappa (m*a = 0.8843)1.55 [GeV]

• axial victor(m*a =0.9448) 1.65 [GeV]

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

Setting of Calculation

• Moreover, we use also 2+1 flavor full QCD configurations by CP-PACS+JLQCD

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